a time-frequency calculus for time-varying systems … · and nonstationary processes with...

Dissertation

A TIME-FREQUENCY CALCULUS

FOR TIME-VARYING SYSTEMS

AND NONSTATIONARY PROCESSES

WITH APPLICATIONS

Gerald Matz([email protected])

Institute of Communicationsand Radio-Frequency Engineering

Vienna University of Technology

This dissertation is available online at

http://www.nt.tuwien.ac.at/dspgroup/tfgroup/doc/psfiles/GM-phd.ps.gz

NACHRICHTENTECHNIKINSTITUT FÜR

UND HOCHFREQUENZTECHNIK

DISSERTATION

A TIME-FREQUENCY CALCULUS

FOR TIME-VARYING SYSTEMS

AND NONSTATIONARY PROCESSES

WITH APPLICATIONS

ausgefuhrt zum Zwecke der Erlangung des akademischen Grades eines

Doktors der technischen Wissenschaften

unter der Leitung von

Ao. Univ.-Prof. Dipl.-Ing. Dr. Franz Hlawatsch

Institut fur Nachrichtentechnik und Hochfrequenztechnik

eingereicht an der Technischen Universitat Wien

Fakultat fur Elektrotechnik

von

Gerald Matz

Servitengasse 13/9

1090 Wien

Wien, im November 2000

Die Begutachtung dieser Arbeit erfolgte durch:

1. Ao. Univ.-Prof. Dipl.-Ing. Dr. F. Hlawatsch


Technische Universitat Wien

2. O. Univ.-Prof. Dipl.-Ing. Dr. W. Mecklenbrauker


Technische Universitat Wien

—— To Petra ——

Abstract

This thesis introduces an approximate time-frequency calculus for underspread linear time-varying

systems (i.e., time-varying systems that effect only small time-frequency shifts of the input signal)

and underspread nonstationary random processes (i.e., nonstationary processes that feature only small

time-frequency correlations).

After briefly describing the major difficulties encountered with time-varying systems and non-

stationary processes, we introduce an extended definition of underspread systems. Our extended

underspread concept is based on weighted integrals and moments of the system’s generalized spread-

ing function. Subsequently, numerous approximations are presented which show that in the case of

underspread systems the generalized Weyl symbol constitutes an approximate time-frequency trans-

fer function. As a mathematical underpinning of our transfer function approximations, we provide

bounds on the associated approximation errors that involve the previously defined weighted integrals

and moments of the generalized spreading function.

We then consider nonstationary random processes and provide an extended definition of under-

spread processes. This extended underspread concept is based on weighted integrals and moments of

the generalized expected ambiguity function of the process. Subsequently, two fundamental classes of

time-varying power spectra are introduced and analyzed: “type I” spectra that extend the generalized

Wigner-Ville spectrum and “type II” spectra that extend the generalized evolutionary spectrum. We

show that in the case of underspread processes, the various members of these two classes of spectra

are approximately equivalent to each other and (at least) approximately satisfy several desirable prop-

erties. Our approximations are again supported by bounds on the associated approximation errors.

These bounds are formulated in terms of the previously defined weighted integrals and moments of

the generalized expected ambiguity function. The definition and analysis of time-frequency coherence

functions concludes our discussion of time-varying power spectra.

Finally, we illustrate the practical relevance of our theoretical findings by considering several ap-

plications in the areas of statistical signal processing and wireless communications. These applications

include nonstationary signal estimation and detection, the sounding of mobile radio channels, multi-

carrier communications over time-varying channels, and the analysis of car engine signals.

ix

Kurzfassung

Diese Dissertation beschreibt einen Zeit-Frequenz-Kalkul fur lineare zeitvariante Systeme mit

geringen Zeit-Frequenz-Verschiebungen (“underspread”-Systeme) und fur instationare stochastische

Prozesse mit schwachen Zeit-Frequenz-Korrelationen (“underspread”-Prozesse).

Nach einer kurzen Darstellung der Probleme, welche bei zeitvarianten Systemen und instationaren

Prozessen auftreten, stellen wir ein erweitertes Konzept von “underspread”-Systemen vor. Dieses

beruht auf gewichteten Integralen und Momenten der verallgemeinerten Spreading-Funktion des Sys-

tems. Im Weiteren werden zahlreiche Approximationen formuliert, welche zeigen, dass fur die Klasse

der “underspread”-Systeme das verallgemeinerte Weyl-Symbol naherungsweise als Zeit-Frequenz-

Ubertragungsfunktion interpretiert und verwendet werden kann. Die angesprochenen Approxima-

tionen werden durch obere Schranken fur die zugehorigen Approximationsfehler mathematisch un-

termauert, wobei diese Schranken mit den zuvor definierten gewichteten Integralen und Momenten

formuliert werden.

Danach betrachten wir instationare stochastische Prozesse und geben eine erweiterte Definition

von “underspread”-Prozessen. Diese beruht auf einer globalen Charakterisierung der Zeit-Frequenz-

Korrelation des Prozesses mittels gewichteter Integrale und Momente des Erwartungswertes der ver-

allgemeinerten Ambiguitatsfunktion. Schließlich werden zwei Klassen von zeitvarianten Spektren

vorgestellt und analysiert: Spektren vom Typ I (eine Erweiterung des verallgemeinerten Wigner-

Ville-Spektrums) und Spektren vom Typ II (eine Erweiterung des verallgemeinerten evolutionaren

Spektrums). Wir zeigen, dass im Fall von “underspread”-Prozessen die verschiedenen Spektren bei-

der Klassen naherungsweise aquivalent sind und gewisse wunschenswerte Eigenschaften (zumindest)

naherungsweise erfullen. Wieder werden alle Naherungen durch obere Schranken fur die entsprechen-

den Approximationsfehler untermauert. Die Diskussion zeitvarianter Spektren wird mit der Definition

und Analyse von Zeit-Frequenz-Koharenzfunktionen beendet.

Abschließend illustrieren wir die Anwendung unserer theoretischen Ergebnisse auf gewisse Be-

reiche der statistischen Signalverarbeitung und der Mobilkommunikation. Diese Bereiche umfassen

die instationare Signalschatzung und -detektion, die Messung von Mobilfunkkanalen, Mehrtrager-

Ubertragungsverfahren fur zeitvariante Kanale und die Analyse instationarer Motorsignale.

x

Acknowledgements

It is my pleasure to express my sincere thanks to several people who have contributed to this thesis

in various ways.

⊲ I am particularly indebted to F. Hlawatsch who continually furthered my personal and pro-

fessional development. His constant advice and thorough proofreading resulted in countless useful

suggestions that helped a lot to improve this thesis with regard to both technical content and presen-

tation.

⊲ W. Mecklenbrauker kindly agreed to act as a referee and pointed me to generalized Chebyshev

inequalities. His support and continual interest in the progress of this work are gratefully acknowl-

edged.

⊲ W. Kozek pioneered the theory of underspread systems and processes. His influence on this

thesis and my research in general is sincerely appreciated.

⊲ I am grateful to J. F. Bohme, S. Carstens-Behrens, and M. Wagner for introducing me to the

problems of car engine diagnosis and providing me with the car engine data used in Chapter 4.

⊲ I am indebted to A. Molisch for several enlightening discussions in which he generously shared

with me his expertise on mobile radio.

⊲ Finally and most importantly, I have been permanently backed up by my wife Petra. Her

sympathy, encouragement, love, and support have been vital for the completion of this thesis. I owe

her more than anyone else.

xi

Contents

1 Introduction 11.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Review of Time-Invariant/Stationary Theory . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Transfer Functions of Time-Invariant and Frequeny-Invariant Linear Sytems . . . . 3

1.2.2 Power Densities of Stationary and White Processes . . . . . . . . . . . . . . . . . 4

1.3 Time-Varying Systems and Nonstationary Random Processes . . . . . . . . . . . . . . . . 5

1.3.1 Time-Varying Systems and the Generalized Weyl Symbol . . . . . . . . . . . . . . 6

1.3.2 Nonstationary Processes and Time-Varying Power Spectra . . . . . . . . . . . . . 6

1.4 The Importance of Being Underspread . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Underspread Linear Time-Varying Systems . . . . . . . . . . . . . . . . . . . . . . 8

1.4.2 Underspread Nonstationary Processes . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Signal Processing Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7 Overview of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Underspread Systems 152.1 Operators with Compactly Supported Spreading Function . . . . . . . . . . . . . . . . . . 16

2.1.1 General Support Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.2 Definition of Displacement-limited Underspread Operators . . . . . . . . . . . . . 17

2.1.3 Unitary Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.4 Operator Sums, Adjoints, Products, and Inverses . . . . . . . . . . . . . . . . . . 20

2.2 Operators with Rapidly Decaying Spreading Function . . . . . . . . . . . . . . . . . . . . 21

2.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.2 Weighted Integrals and Moments of the Generalized Spreading Function . . . . . . 22

2.2.3 Unitary Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.4 Underspread Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.5 Operator Sums, Adjoints, Products, and Inverses . . . . . . . . . . . . . . . . . . 32

2.2.6 Non-Band-Limited Parts of Operators with Rapidly Decaying Spreading Function . 35

2.3 Underspread Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3.1 Approximate Uniqueness of the Generalized Weyl Symbol . . . . . . . . . . . . . . 37

xiii

xiv

2.3.2 The Generalized Weyl Symbol of Operator Adjoints . . . . . . . . . . . . . . . . . 40

2.3.3 Approximate Real-Valuedness of the Generalized Weyl Symbol . . . . . . . . . . . 41

2.3.4 Composition of Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3.5 Composition of H with H+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.3.6 Operator Inversion Based on the Generalized Weyl Symbol—Part I . . . . . . . . . 50

2.3.7 Operator Inversion Based on the Generalized Weyl Symbol—Part II . . . . . . . . 57

2.3.8 Approximate Eigenvalues and Eigenfunctions . . . . . . . . . . . . . . . . . . . . 61

2.3.9 Approximate Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.3.10 Input-Output Relation for Deterministic Signals Based on the Generalized Weyl

Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.3.11 (Multi-Window) STFT Filter Approximation of Time-Varying Systems . . . . . . . 73

2.3.12 Infimum and Supremum of the Weyl Symbol . . . . . . . . . . . . . . . . . . . . 77

2.3.13 Approximate Non-Negativity of the Generalized Weyl Symbol of Positive Semi-

Definite Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

2.3.14 Boundedness of the Generalized Weyl Symbol of Self-Adjoint Operators . . . . . . 84

2.3.15 Maximum System Gain (Operator Norm) . . . . . . . . . . . . . . . . . . . . . . 87

2.3.16 Approximate Commutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

2.3.17 Approximate Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

2.3.18 Sampling of the Generalized Weyl Symbol of Underspread Operators . . . . . . . . 92

3 Underspread Processes 973.1 Time-Frequency Correlation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.1.2 Time-Frequency Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . 98

3.1.3 The Expected Ambiguity Function . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.1.4 Extended Concept of Underspread Processes . . . . . . . . . . . . . . . . . . . . . 103

3.1.5 Innovations System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.1.6 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.2 Elementary Time-Varying Power Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.2.1 Generalized Wigner-Ville Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.2.2 Generalized Evolutionary Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 120

3.3 Type I Time-Varying Power Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

3.3.1 Definition and Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

3.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

3.3.3 Approximate Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

3.3.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

3.4 Type II Time-Varying Power Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

3.4.1 Definition and Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

3.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

3.4.3 Approximate Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

3.4.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

3.5 Equivalence of Time-Varying Power Spectra . . . . . . . . . . . . . . . . . . . . . . . . . 146

3.5.1 Equivalence of Generalized Wigner-Ville Spectrum and Generalized Evolutionary

Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

xv

3.5.2 Equivalence of Type I and Type II Spectra . . . . . . . . . . . . . . . . . . . . . . 149

3.6 Input-Output Relations for Nonstationary Random Processes . . . . . . . . . . . . . . . . 151

3.6.1 Input-Output Relation Based on the Generalized Wigner-Ville Spectrum . . . . . . 151

3.6.2 Input-Output Relation Based on the Generalized Evolutionary Spectrum . . . . . . 154

3.7 Approximate Karhunen-Loeve Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

3.8 Time-Frequency Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

3.8.1 Spectral Coherence and Coherence Operator . . . . . . . . . . . . . . . . . . . . . 159

3.8.2 Time-Frequency Formulation of the Coherence Operator . . . . . . . . . . . . . . 160

3.8.3 The Generalized Time-Frequency Coherence Function . . . . . . . . . . . . . . . . 164

4 Applications 1694.1 Nonstationary Signal Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

4.1.1 Time-Varying Wiener Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

4.1.2 Time-Frequency Formulation of the Time-Varying Wiener Filter . . . . . . . . . . 170

4.1.3 Time-Frequency Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

4.1.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

4.2 Nonstationary Signal Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4.2.1 Optimal Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

4.2.2 Time-Frequency Formulation of Optimal Detectors . . . . . . . . . . . . . . . . . 176

4.2.3 Time-Frequency Detector Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 178


4.3 Sounding of Mobile Radio Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

4.3.1 Channel Sounder Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

4.3.2 Analysis of Measurement Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

4.3.3 Optimization of PN Sequence Length . . . . . . . . . . . . . . . . . . . . . . . . 185


4.4 Multicarrier Communication Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

4.4.1 Pulse-Shaping OFDM and BFDM Systems . . . . . . . . . . . . . . . . . . . . . 188

4.4.2 Approximate Input-Output Relation for OFDM/BFDM Systems . . . . . . . . . . 189


4.5 Analysis of Car Engine Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

4.5.1 Time-Varying Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

4.5.2 TF Coherence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

4.5.3 Subspace Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

5 Conclusions 1975.1 Summary of Novel Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

5.2 Open Problems for Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

A Linear Operator Theory 207A.1 Basic Facts about Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

A.2 Kernel Representation of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

A.3 Eigenvalue Decomposition and Singular Value Decomposition . . . . . . . . . . . . . . . 210

xvi

A.4 Special Types of Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

B Time-Frequency Analysis Tools 213B.1 Time-Frequency Representations of Linear, Time-Varying Systems . . . . . . . . . . . . . 214

B.1.1 Generalized Spreading Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

B.1.2 Generalized Weyl Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

B.1.3 Generalized Transfer Wigner Distribution and Generalized Input and Output Wigner

Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

B.2 Time-Frequency Signal Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

B.2.1 Short-Time Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

B.2.2 Generalized Wigner Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

B.2.3 Spectrogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

B.2.4 Generalized Ambiguity Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

B.3 Time-Frequency Representations of Random Processes . . . . . . . . . . . . . . . . . . . 224

B.3.1 Generalized Wigner-Ville Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 225

B.3.2 Generalized Evolutionary Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 226

B.3.3 Physical Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

B.3.4 Generalized Expected Ambiguity Function . . . . . . . . . . . . . . . . . . . . . . 227

C The Symplectic Group and Metaplectic Operators 229C.1 The Symplectic Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

C.2 Metaplectic Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

C.3 Effects on Time-Frequency Representations . . . . . . . . . . . . . . . . . . . . . . . . . 233

Bibliography 237

List of Abbreviations 249

1

Introduction

“[. . .] there is no doubt that linear systems will continue to be an objectof study for as long as one can foresee.” Thomas Kailath

LINEAR systems and random processes are the foundations of numerous signal modelling, analysis,

and processing schemes. In a formal sense, the characterization of linear systems and the second-

order description of random processes can be based on the same mathematics: linear operator theory.

This viewpoint will be emphasized throughout this thesis.

A fundamental distinction must be made between time-invariant systems and stationary processes

on the one hand and time-varying systems and nonstationary processes on the other. The majority of

books and research papers restrict to time-invariant systems and stationary processes, both of which

can be treated in an efficient and intuitively appealing manner using Fourier analysis. In contrast,

time-varying systems and nonstationary processes have received much less attention.

In this introductory chapter, we review some basic facts of the theories of time-invariant linear

systems and stationary processes (Section 1.2). Further, in Section 1.3 we discuss the fundamental

problems encountered when dealing with time-varying/nonstationary scenarios. Section 1.4 outlines

the central results of this thesis, which consist in the (approximate) solution of these basic problems

by means of an approximate time-frequency calculus of time-varying transfer functions and time-

varying power spectra. This time-frequency calculus is valid for the practically important classes of

underspread systems and underspread processes and it has the advantage of being conceptually simple,

computationally attractive, and physically intuitive. The chapter continues with an outline of signal

processing applications of our results in Section 1.5, a brief account of related work in Section 1.6, and

a summary of the major contributions of this thesis in Section 1.7.

1

2 Chapter 1. Introduction

1.1 General Remarks

Linear systems and random processes are of fundamental importance in many engineering applica-

tions. In particular, linear systems provide useful models for communication channels and speech

production, are vital in transmitter/receiver design, and are used as filters in processing schemes

for signal separation, enhancement, and detection. Similarly, random processes are used to model

phenomena as diverse as speech and audio, communication signals, visual data, biological signals, or

signals arising in machine monitoring. Furthermore, undesired disturbances (noise and interference)

are usually modelled as being random, too. Hence, linear systems and random processes lie at the

heart of numerous (statistical) signal processing methods. The above short list of their applications

is far from being complete and, indeed, falls short of illustrating their ubiquity.

In a formal sense, linear systems and (second-order statistics of) random processes allow a unified

mathematical treatment via linear operator theory1 (a brief review of certain elements of linear oper-

ator theory is given in Appendix A; far more comprehensive treatments can be found in [69,158]). In

particular, a linear system can be associated to a linear operator H whose kernel equals the system’s

impulse response h(t, t′) relating the input signal x(t) and the output signal y(t) as2

y(t) = (Hx)(t) =

∫

t′h(t, t′)x(t′) dt′. (1.1)

Throughout this thesis, when talking about linear systems we will restrict our attention to Hilbert-

Schmidt (HS) operators (see Appendix A). The only exceptions to this general rule are i) linear

time-invariant (LTI) systems and linear frequency-invariant (LFI) systems (which are never HS); and

ii) unitary systems (see Subsection 2.1.3). In a similar way, a correlation operator Rx can be used

as second-order description of a random process x(t) in the sense that the kernel of Rx equals the

correlation function rx(t, t′) = E{x(t)x∗(t′)} of x(t) (here, E denotes expectation). We note that

the set of all correlation operators equals the subclass of positive semi-definite linear operators (see

Appendix A). Except in the case of stationary or white processes, we will implicitly assume the

processes involved to have finite mean energy, Ex , E{‖x‖22} <∞. This implies that the corresponding

correlation operator is trace-class (or nuclear, see Appendix A).

LTI systems and stationary processes can be efficiently dealt with using convolution and Fourier

transform techniques. The corresponding theories are well developed and allow to gain useful insights

(as discussed in Section 1.2). Unfortunately, Fourier transform techniques lose much of their appeal

and usefulness in the case of linear time-varying (LTV) systems and nonstationary processes. This

explains why LTV systems and nonstationary processes, while providing a very general framework

(much more general than LTI systems and stationary processes), are considerably more difficult to

treat (see Section 1.3). This problem motivated much of the work in this thesis, which is concerned

1We note that while the theoretical treatment of linear systems and random processes can be cast in the same

mathematical framework, the interpretation of the corresponding objects is quite different. In particular, in the case of

a linear system we are mainly interested in characterizing its effects on various input signals. In contrast, in the case of

a random process we are concerned with a description of its correlative properties and its power distribution.2Throughout this thesis, integrals are from −∞ to ∞ unless stated otherwise.

1.2 Review of Time-Invariant/Stationary Theory 3

with the development of a time-frequency (TF) calculus for LTV systems and nonstationary processes.

This TF calculus is valid for underspread systems and underspread processes which will be discussed

in Section 1.4.

1.2 Review of Time-Invariant/Stationary Theory

LTI systems and their dual, LFI systems, as well as stationary processes and their dual, white processes,

are quite restrictive models (see Section A.4). However, their treatment using convolution and Fourier-

type methods is comparatively simple. Thus, we subsequently outline several well-known results

for LTI (LFI) systems and stationary (white) random processes (details can be found in standard

textbooks like [160, 163]). Our emphasis will be placed on those properties whose extension to time-

varying systems and nonstationary processes will be developed in Chapters 2 and 3, respectively.

1.2.1 Transfer Functions of Time-Invariant and Frequeny-Invariant Linear Sytems

LTI systems are characterized by an impulse response of the form h(t, t′) = g(t − t′), so that the

general input-output relation (1.1) specializes to a convolution (denoted by an asterisk),

y(t) = (g ∗ x)(t) =

∫

t′g(t− t′)x(t′) dt′.

LFI systems are characterized by an impulse response of the form h(t, t′) = m(t) δ(t − t′); the input-

output relation (1.1) here reduces to a time-domain multiplication,

y(t) =

∫

t′m(t) δ(t − t′)x(t′) dt′ = m(t)x(t).

LTI (LFI) systems are always normal and any two LTI systems (LFI systems) commute with each

other. The spectral transfer function (frequency response) of LTI systems is given by the Fourier

transform of g(τ),

G(f) ,

∫

τg(τ) e−j2πfτdτ , (1.2)

and the “temporal transfer function” of LFI systems is given by the multiplier function m(t). These

spectral and temporal transfer functions are extremely simple and efficient system descriptions. This

is due to the following properties:

• The complex sinusoids {ej2πft} (parametrized in a physically meaningful manner by frequency f)

are the generalized eigenfunctions [68] of any LTI system, with G(f) the associated generalized

eigenvalues, i.e. (Hx)(t) = G(f) ej2πft for x(t) = ej2πft. Similarly, the Dirac impulses δ(t − t′)

(parametrized in a physically meaningful manner by time t′) are the generalized eigenfunctions

of LFI systems, with m(t′) the associated generalized eigenvalues: (Hx)(t) = m(t′) δ(t − t′) for

x(t) = δ(t− t′).


• As a consequence of the previous property, for LTI systems the Fourier transform of (Hx)(t)

equals the Fourier transform of x(t) multiplied by G(f), and for LFI systems the output signal

(Hx)(t) equals the input signal x(t) multiplied by m(t). Hence, for LTI and LFI systems, the

input-output relation simplifies to the multiplication of two functions in the frequency domain

and in the time domain, respectively.

• The spectral (temporal) transfer function of the series connection (composition) of two LTI

(LFI) systems with transfer functions G1(f) (m1(t)) and G2(f) (m2(t)) equals G1(f)G2(f)

(m1(t)m2(t)).

• The adjoint H+ of an LTI (LFI) system has impulse response g∗(−τ) (m∗(t) δ(t′− t)), and hence

its spectral (temporal) transfer function is simply the complex conjugate of G(f) (m(t)).

• The inverse H−1 of LTI (LFI) system—if it exists—corresponds to the reciprocal of the spectral

(temporal) transfer function.

• For LTI (LFI) systems the maximum system gain (i.e., operator norm, see Section A.1) is equal

to the supremum of |G(f)| (|m(t)|).

1.2.2 Power Densities of Stationary and White Processes

The correlation function rx(t1, t2) = E{x(t1)x∗(t2)} of (wide-sense) stationary processes depends only

on the difference t1 − t2, i.e., rx(t1, t2) = rx(t1 − t2). Hence, the 1-D correlation function rx(τ) is a

complete second-order description. The power spectral density (PSD) [163] of the stationary process

x(t) is defined as the Fourier transform of the correlation function (Wiener-Khintchine relation), i.e.,

Px(f) ,

∫

τrx(τ) e−j2πfτ dτ . (1.3)

Similarly, the correlation function of a white process is of the form rx(t1, t2) = qx(t1) δ(t1 − t2) and

the temporal “power density” is given by the mean instantaneous intensity [163] qx(t). The PSD and

mean instantaneous intensity are very simple, physically intuitive, and useful second-order statisti-

cal descriptions of the process. The following properties and interpretations of the PSD and mean

instantaneous intensity are of fundamental importance:

• The complex exponentials ej2πft are the generalized eigenfunctions of the (convolution type)

correlation operator Rx of stationary processes. This implies that the Fourier transform diago-

nalizes the correlation operator and thus provides a decorrelation of stationary processes,

E{X(f)X∗(f ′)

}= Px(f) δ(f − f ′) .

Here, X(f) =∫t x(t) e

−j2πft dt is the process’ Fourier transform. For white processes, the corre-

lation operator is diagonalized by the Dirac impulses δ(t − t′) and decorrelation is obtained in

the time domain, E {x(t)x∗(t′)} = qx(t) δ(t− t′).

1.3 Time-Varying Systems and Nonstationary Random Processes 5

• Let h(τ) denote the impulse response of a (time-invariant) innovations system of x(t) [38, 163],

i.e., x(t) =∫τ h(τ)n(t − τ) dτ , with n(t) normalized stationary white noise. Then it is known

that

Px(f) = |H(f)|2, (1.4)

where H(f) is the transfer function of the innovations system as defined in (1.2). Similarly, if

h(t, t′) = m(t) δ(t − t′) is the (frequency-invariant) innovations system of a white process, i.e.,

x(t) = m(t)n(t), then qx(t) = |m(t)|2.

• The PSD (mean instantaneous intensity) is a complete second-order statistics of stationary

(white) processes since the correlation function rx(t1, t2) can be completely recovered from Px(f)

(qx(t)).

• The PSD and the mean instantaneous intensity are nonnegative quantities that integrate to the

mean temporal and spectral power, respectively:

Px(f) ≥ 0 ,

∫

fPx(f) df = E

{|x(t)|2

},

qx(t) ≥ 0 ,

∫

tqx(t) dt = E

{|X(f)|2

}.

These properties are essential for an interpretation as average “power densities.”

• If a stationary process x(t) is passed through an LTI system with impulse response k(τ), the

output process y(t) = (k ∗ x)(t) is again stationary with PSD

Py(f) = |K(f)|2 Px(f) . (1.5)

Furthermore, the cross PSD of y(t) and x(t) is given by

Py,x(f) ,

∫

τry,x(τ) e

−j2πfτ dτ = K(f)Px(f) , (1.6)

where ry,x(τ) = E{y(t)x∗(t − τ)} is the cross correlation function of y(t) and x(t). Similarly,

passing a white process x(t) through an LFI system with temporal transfer function m(t) again

yields a white process y(t) = m(t)x(t) with mean instantaneous intensity qy(t) = |m(t)|2 qx(t)and mean instantaneous cross intensity qy,x(t) = m(t) qx(t).

1.3 Time-Varying Systems and Nonstationary Random Processes

LTV systems and nonstationary random processes provide very general and thus powerful models

for a large variety of engineering applications. Unfortunately, as discussed below, this generality is

paid for with an increased difficulty in describing LTV systems and nonstationary processes. The

situation gets more comforting if one considers the classes of underspread systems and underspread

processes [117–120, 126, 127, 129, 144, 145] which can be viewed as natural extensions of LTI (LFI)


systems and stationary (white) processes, respectively. The basic concepts related to underspread

systems and underspread processes as well as their importance for an approximate TF calculus will

be outlined in Section 1.4, while a detailed discussion of the approximate TF calculus will be provided

in Chapters 2 and 3.

1.3.1 Time-Varying Systems and the Generalized Weyl Symbol

For LTV systems, the generalized Weyl symbol (GWS) is defined as [114,118,144]

L(α)H

(t, f) ,

∫

τh(α)(t, τ) e−j2πfτ dτ , (1.7)

with h(α)(t, τ) = h(t+ (1/2−α)τ, t− (1/2+α)τ). Important properties and relations of the GWS are

summarized in Section B.1.2. The GWS has been recognized as a potential candidate for a TF (or,

time-varying) transfer function, i.e., as a generalization of the spectral (temporal) transfer function.

Unfortunately, in contrast to LTI (LFI) systems, LTV systems and the GWS are generally much more

cumbersome to work with. This is due to the following reasons :

• Contrary to LTI and LFI systems, LTV systems are not always normal. In the non-normal

case, one has to deal with (numerically more expensive) singular value decompositions instead

of eigenvalue decompositions [69,158] (see also Appendix A).

• In general, the eigenfunctions or singular functions of distinct LTV system are different and

possess no simple specific structure.

• Since the singular value decomposition (eigenvalue decomposition) of LTV systems has no specific

structure, it can only be interpreted using signal space concepts, thus lacking a specific physical

interpretation. In particular, the parameter k in (A.4) and (A.5) has in general no physical

meaning. Furthermore, there exists no fast implementation (like the FFT for LTI systems) of

the transform associated to these unstructured singular functions (eigenfunctions).

• None of the practically convenient properties of the spectral (temporal) transfer function of LTI

(LFI) systems (see the list in Subsection 1.2.1) is valid any longer for the GWS of LTV systems.

1.3.2 Nonstationary Processes and Time-Varying Power Spectra

In the context of random processes, the eigenvalue decomposition of the correlation operator, referred

to as Karhunen-Loeve (KL) decomposition [136], has a specific interpretation that is important for

statistical signal processing schemes. We shall thus discuss very briefly the KL expansion of non-

stationary processes. Let us consider a finite-energy, zero-mean, nonstationary process x(t). The

associated correlation operator Rx is trace-class (cf. Appendix A) and has (orthonormal) eigenfunc-

tions uk(t) and absolutely summable nonnegative eigenvalues λk. The KL theorem [136] states that

x(t) can be expanded into the eigenfunctions uk(t) (in the mean-square sense) and that the expansion

1.3 Time-Varying Systems and Nonstationary Random Processes 7

coefficients are uncorrelated with mean power equal to the eigenvalues λk, i.e.,3

x(t) =∞∑

k=1

〈x, uk〉uk(t) , E {〈x, uk〉〈x, ul〉∗} = λk δkl . (1.8)

The double orthogonality of the KL expansion (orthogonality of the basis functions uk(t) and orthog-

onality of the random coefficients 〈x, uk〉) is the reason for the (theoretical) optimality and usefulness

of the KL transform (i.e., the transform mapping the random process x(t) to the coefficients 〈x, uk〉)in various applications like transform coding or signal detection.

In the case of stationary processes, the KL transform reduces to the Fourier transform. With X(f)

denoting the Fourier transform of x(t), (1.8) reads

x(t) =

∫

fX(f) ej2πft df , E

{X(f)X∗(f ′)

}= Px(f) δ(f − f ′) ,

i.e., one formally has uk(t) −→ ej2πft and λk −→ Px(f), with the continuous parameter f (frequency)

replacing the discrete parameter k. For white processes there is a similar integral representation where

formally uk(t) −→ δ(t − t′) and λk −→ qx(t′), with the continuous parameter t′ (time) replacing the

discrete parameter k.

Hence, in a certain sense, the KL eigenvalues provide a generalization of the PSD and the mean

instantaneous intensity to general nonstationary processes. Unfortunately, the KL transform in general

suffers from several drawbacks:

• The KL basis functions uk(t) are not known a priori and in general are different for different

processes.

• In general, the basis {uk(t)} lacks a simple structure; this is the reason why typically no efficient

implementation of the KL transform exists.

• In general, the parameter k of the KL transform has no physical meaning like frequency f or

time t.

These drawbacks are well recognized and have been a major driving force for research into approxi-

mations of the KL transform [50, 60, 61, 118, 120, 137, 138]: depending on the specific application, the

goal has been either to develop “approximate” KL transforms using highly structured and efficiently

implementable bases or to provide physically more relevant definitions of spectra for nonstationary

processes. Potential candidates for the latter are the generalized Wigner-Ville spectrum (GWVS),

defined as [60,61,63,140,145]

W(α)

x (t, f) ,

∫

τr(α)x (t, τ) e−j2πfτ dτ ,

with r(α)x (t, τ) = rx(t+ (1/2−α)τ, t− (1/2 +α)τ), and the generalized evolutionary spectrum (GES),

defined as [145,148] (cf. also [49,118,170,171])

G(α)x (t, f) ,

∣∣L(α)H

(t, f)∣∣2,

3The inner product is defined as usual, 〈x, y〉 =R

tx(t) y∗(t) dt.


with H an innovations system of x(t), i.e., a system satisfying HH+ = Rx. Further details about the

GWVS and the GES can be found in Sections B.3.1 and B.3.2, respectively. Unfortunately, none of

the practically important and convenient properties of the PSD or the mean instantaneous intensity

as detailed in Subsection 1.2.2 are satisfied by the GWVS or GES of general nonstationary processes.

1.4 The Importance of Being Underspread

In the foregoing subsections, we saw that in general the GWS of an LTV system and the GWVS or

GES of a nonstationary process fail to provide tools that are as efficient and meaningful as the transfer

function of LTI (LFI) systems and the power densities of stationary (white) processes, respectively.

However, for underspread systems (see Chapter 2) the GWS features similar properties as the transfer

function (at least in an approximate sense) and for underspread processes (see Chapter 3) the GWVS

and GES (approximately) satisfy similar properties as the PSD. Making this statement precise and

proving this claim is the main theme of this thesis. In the next two subsections, we outline the main

results of the theory developed in Chapters 2 and 3.

1.4.1 Underspread Linear Time-Varying Systems

For our subsequent discussion, it is necessary to consider the TF shifts introduced by an LTV system

H. These can be characterized by the generalized spreading function (GSF) [114,118]

S(α)H

(τ, ν) ,

∫

th(α)(t, τ) e−j2πνt dt ,

with h(α)(t, τ) as before. Important properties and relations of the GSF are summarized in Section

B.1.1. In particular,

y(t) = (Hx)(t) =

∫

τ

∫

νS

(α)H

(τ, ν) (S(α)τ,νx)(t) dτ dν , (1.9)

with S(α)τ,ν denoting the generalized TF shift operator,

(S(α)τ,νx)(t) = x(t− τ) ej2πνt ej2πτν(α−1/2) .

It is seen from (1.9) that the spread, or extension, of the GSF about the origin of the (τ, ν)-plane

provides a global characterization of the TF shifts introduced by the system. The GSF extension in

the τ direction characterizes the system’s “length of memory” whereas the extension in the ν direction

determines the fastness of the system’s time-variations or fluctuations. Conceptually, an LTV system

is called underspread if its GSF is concentrated in a small region about the origin of the (τ, ν) plane,4

which indicates that the system introduces only small TF shifts τ , ν or, in other words, that the

system’s memory is short and/or its time-variations are slow (see Subsections 2.1.2 and 2.2.4). In

contrast, systems introducing large TF shifts τ , ν are referred to as overspread .

4In fact, in most cases it suffices that the GSF is concentrated around some point (τ0, ν0) since the corresponding

offset TF shift can be split off from the system.

1.4 The Importance of Being Underspread 9

The underspread concept was first used for random LTV systems in the context of fading multipath

channels [11,109,172,203] and for “doubly spread” radar targets [72,74,203]. Adopting the terminology

of the random case, Kozek [118–120] introduced underspread deterministic LTV systems by requiring

that their GSF be exactly zero outside a small rectangular support region about the origin of the (τ, ν)

plane. Systems being underspread in Kozek’s sense, i.e., having small compact GSF support, will be

treated in Section 2.1 and we will refer to them as displacement-limited (DL) operators.

In practice, the condition of small compact GSF support is often not satisfied exactly but only

effectively . This raises the question of how to choose the effective support region and how the modeling

error resulting from a specific choice of this effective support region affects the validity of the results

obtained using the compact support model. Furthermore, the small compact support requirement is

often unnecessarily restrictive since several important results hold for much wider classes of systems,

including systems with a GSF that satisfies certain support constraints but still has infinite support.

Thus, in this thesis we introduce and use an extended underspread concept based on operators with

rapidly decaying GSF (see Section 2.2). As a foundation for this extension, we use weighted integrals

and moments of the GSF as measures of the global TF shifts of a system, without requiring the GSF

to have compact support.

For LTV systems of the underspread type (be they DL operators or operators with rapidly decaying

GSF), all of the difficulties connected with LTV systems and the GWS as listed in Subsection 1.3.1 are

(at least approximately) removed. In fact, one can establish a GWS-based approximate TF transfer

function calculus that yields the following useful results (the details and bounds on the associated

approximation errors will be presented in Section 2.3):

• Underspread operators are approximately normal (see Subsection 2.3.17) and have approximate

eigenfunctions which, being time and frequency translates of a prototype signal, are highly

structured (see Subsection 2.3.8). Hence, these approximate eigenfunctions also allow an intuitive

physical interpretation.

• The GWS approximately reflects the system’s maximum gain (see Subsection 2.3.15).

• The GWS of the adjoint operator H+ is approximately equal to the complex conjugate of the

GWS of H (see Subsection 2.3.2).

• The GWS of the product (composition) of two (jointly) underspread operators (systems) is

approximately given by the product of the respective individual GWS (see Subsection 2.3.4).

• Jointly underspread LTV systems are approximately commuting (see Subsection 2.3.16).

• In a certain sense to be discussed in Subsection 2.3.6, the GWS of the inverse of H approximately

corresponds to 1/L(α)H

(t, f).

All these approximations can be summarized by stating that the GWS of (jointly) underspread systems

can be interpreted as a “TF transfer function” which can be used in exactly the same manner as the

spectral (temporal) transfer function of LTI (LFI) systems.


1.4.2 Underspread Nonstationary Processes

TF correlations play a similar role for random processes as TF shifts do for linear systems. They can

be compactly characterized by the generalized expected ambiguity function (GEAF) [117,118,126]

A(α)x (τ, ν) , S

(α)Rx

(τ, ν) =

∫

tr(α)x (t, τ) e−j2πνt dt ,

with r(α)x (t, τ) as before (further details can be found in Sections 3.1 and B.3.4). In particular, the

GEAF can be interpreted as a global measure of the TF correlation of process components separated

by τ in time and by ν in frequency. Hence, the extension of the GEAF about the origin of the (τ, ν)

plane provides a global characterization of the TF correlations of the process. The GEAF extension

in the τ direction describes the temporal correlation width whereas the extension in the ν direction

characterizes the spectral correlation width of the process. Conceptually, we will refer to a random

process as underspread if its GEAF is sufficiently concentrated about the origin of the (τ, ν) plane.

This is equivalent to the requirement that the process features only small TF correlations. Since

the GEAF is the GSF of the correlation operator Rx, it follows that the correlation operator of an

underspread process is underspread in the sense of Subsection 1.4.1. In contrast, processes with large

TF correlations are referred to as overspread .

Underspread processes were first introduced by Kozek [117, 118, 126] in analogy to underspread

LTV systems. Kozek’s definition of underspread processes was similarly based on the requirement

that their GEAF is exactly zero outside a small rectangular support region about the origin of the

(τ, ν) plane. We will refer to processes that are underspread according to this original definition as

correlation–limited (CL) processes.

Similar to the case of LTV systems, the condition of small compact GEAF support will often be

satisfied only effectively . Hence, for the same reasons as mentioned in the foregoing subsection, we

shall introduce and use in this thesis an extended underspread concept that is based on rapid decay

of the GEAF. The GEAF decay (and thus the global TF correlations) will be measured via weighted

integrals and moments. For this extended version of the underspread property, no compact GEAF

support is required.

The important fact about underspread processes is that the difficulties connected with nonstation-

ary random processes and the GWVS/GES as listed in Subsection 1.3.2 are alleviated. In particular,

one can show that the GWVS and GES (as well as other time-varying spectra—see Chapter 3) of

underspread processes (approximately) satisfy the following useful properties (the details and bounds

on the associated approximation errors will be presented in Chapter 3):

• Time and frequency translates of a reasonable prototype signal are approximate KL eigen-

functions of underspread processes. These approximate KL eigenfunctions allow a physically

meaningful interpretation and, due to their underlying structure, the associated transform can

be efficiently implemented.

• The GWVS and GES are (approximately) positive and describe the mean TF energy distribution

of the process in a meaningful way.

1.5 Signal Processing Applications 11

• For both the GWVS and GES, simple and intuitive approximate input-output relations similar

to (1.5) and (1.6) can be derived.

• Almost all definitions of time-varying power spectra proposed up to now in the literature (in

particular, the GWVS and GES) are approximately equivalent.

• In an approximate sense, most of the time-varying power spectra are complete second-order

characterizations of the process (for the GWVS, this holds exactly, see Subsection B.3.1).

In essence, the above approximations imply that the GWVS and GES of underspread processes are

reasonable definitions of time-varying power spectra which extend the PSD (mean instantaneous in-

tensity) of stationary (white) processes in a meaningful way to the nonstationary case.

1.5 Signal Processing Applications

As explained above, Chapters 2 and 3 establish that the GWS is a meaningful TF transfer function for

underspread LTV systems and that the GWVS and GES are meaningful mean TF energy distributions

of underspread nonstationary random processes. These findings have important implications for the

following applications that will be discussed in more detail in Chapter 4.

Nonstationary Signal Estimation and Enhancement. The Wiener filter is known to be

the optimal linear signal estimator with respect to a mean square error criterion [106, 187, 197, 202].

Unfortunately, in the case of nonstationary processes, the design of the Wiener filter is based on

the solution of an operator equation and requires a computationally costly and potentially unstable

operator inversion. Applying the results of Subsection 2.3.7 allows to use a computationally efficient,

numerically stable, and physically intuitive approximate TF design of time-varying Wiener filters

[92,111] (see Section 4.1).

Nonstationary Signal Detection. In the case of nonstationary Gaussian processes, the design

of the likelihood ratio detector [108,168,187,202] and the deflection-optimal detector [7,168] involves the

solution of an operator equation that requires expensive and potentially unstable operator inversions.

Similar to the signal estimation problem, we show how the results of Subsection 2.3.6 can be used

to obtain a computationally less costly, stable, and intuitive approximate TF design of time-varying

signal detectors [141–143,146] (see Section 4.2).

Sounding of Mobile Radio Channels. Accurate wideband measurements of mobile radio

channels by means of correlative channel sounders are the cornerstone of any design or simulation

of mobile radio systems with high data rate [40, 52, 164]. While most channel sounders assume the

channel to be time-invariant, practical mobile radio channels are time-varying. For this reason, the

measurements are typically affected by systematic errors [149,150,156]. Using the results of Subsections

2.3.1, 2.3.4, and 2.3.18, these errors can be quantified and bounded (see Section 4.3).

Multicarrier Communication Systems. Multicarrier communication systems like orthogonal

frequency division multiplex (OFDM) and discrete multi-tone (DMT) [28–30, 133, 182, 209, 215] are


closely related to TF signal expansions. Recent work showed that pulses other than the usual rectan-

gular one as well as an extension to biorthogonal transmit and receive filters can be advantageous in

the case of fast time-varying channels [18,19,128]. In Section 4.4, we will point out a close relation of

these recent results with approximate eigenfunctions of LTV systems as discussed in Subsection 2.3.8.

Analysis of Car Engine Signals. Pressure and vibration signals measured in combustion

engines can be modelled successfully as nonstationary random processes. They are important for

knock detection and other car engine diagnosis tasks [17,26,27,113,143,146,181,207]. Application of

the time-varying spectra and TF coherence function considered in Chapter 3 allows to extract useful

time-varying and nonstationary features of this type of real data (see Section 4.5).

1.6 Related Work

As mentioned above, underspread LTV systems with a GSF satisfying a rectangular support constraint

were first considered in the pioneering work of W. Kozek [118–120, 127]. Several transfer function

approximations have been derived within this framework [118–120, 127]. Furthermore, results in a

similar spirit have been obtained for the symbol calculi in the context of quantum-mechanical quan-

tization [64, 167, 206] and pseudo-differential operators [64, 73, 94, 112], with the difference that these

theories define specific symbol classes directly in phase space whereas in our approach we formulate

growth conditions in the (dual) spreading function domain. Also, whereas in quantization theory

and pseudo-differential operator theory one studies the operators corresponding to a given symbol,

we consider a given LTV system (operator) and investigate how close its GWS comes to the intu-

itive engineering notion of a TF transfer function. We will further comment on these differences and

similarities in Sections 2.2 and 2.3.

Several results for underspread random processes with rectangularly supported GEAF that are

related to the approximations in Chapter 3 were also derived by W. Kozek in [115, 118, 120, 126].

Furthermore, results close in spirit to our considerations have been presented in [191–193]. There, the

analysis and processing of random processes with slowly time-varying statistics based on innovation

systems and the evolutionary spectrum is discussed. In [60,61,63], observations regarding the Wigner-

Ville spectrum and the evolutionary spectrum were made which are closely related to our discussion of

time-varying power spectra. Finally, [32] considers LTV systems and nonstationary random processes

that are “regionally underspread,” i.e., require the underspread condition to hold only for a portion

of the system (process) that is localized in a specific TF region.

1.7 Overview of Contributions

We conclude this introductory chapter with an overview of the major contributions of this thesis (in

order of appearance).

LTV systems with compactly supported GSF (Section 2.1): We introduce novel parameters

1.7 Overview of Contributions 13

characterizing the TF shifts of LTV systems with compact GSF support, and we extend the

previous definitions [118–120,127] of (jointly) underspread systems to accommodate oblique ori-

entations of GSF support regions. Furthermore, the spread parameters of unitarily transformed

systems as well as of sums, products, and adjoints of underspread LTV systems are analyzed.

LTV systems with rapidly decaying GSF (Section 2.2): We present an extension of the un-

derspread concept to LTV systems whose GSF does not necessarily have compact support but

features rapid decay. This extension uses weigthed integrals and moments of the GSF that

provide novel measures of the TF shifts introduced by a system. We derive various relations

for weighted GSF integrals and moments of unitarily transformed LTV systems and of sums,

products, and adjoints of LTV systems. Furthermore, we present Chebyshev-like inequalities

inequalities that are useful for assessing the errors made by approximating an arbitrary LTV

system by a system with compactly supported GSF.

TF transfer function approximations (Section 2.3): This large section contains numerous re-

sults that establish a GWS–based approximate TF transfer function calculus for underspread

LTV systems. All TF transfer function approximations are underpinned by bounds on the as-

sociated approximation errors which are formulated in terms of weighted GSF integrals and

moments. While numerous TF transfer function approximations are completely new, others are

extensions of existing result for the special case of LTV systems with compactly supported GSF

in [118–120,127].

TF correlation analysis (Section 3.1): We present methods for the analysis of the TF correlations

of a random process. Furthermore, we provide a novel concept of underspread processes that is

based on weighted integrals and moments of the GEAF and that extends previous definitions

of underspread processes that were based on the assumption of a compactly supported GEAF

[117,118,126].

Elementary time-varying spectra (Section 3.2): For underspread processes, the GWVS and

GES are shown to be smooth, effectively real-valued, and positive quantities. We furthermore

discuss the occurrence of “statistical cross-terms in the case of overspread processes and we

present uncertainty relations for the GWVS and GES.

Type I and type II spectra (Sections 3.3–3.5): “Type I time-varying power spectra (previously

considered in [3,60,61,63]) are introduced in an axiomatic fashion and shown to be extensions of

the GWVS. Similarly, we extend the GES by introducing the novel class of “type II time-varying

power spectra. We show that these spectra satisfy (at least approximately) several desirable

properties. Furthermore, we prove that for underspread processes the members of these two

classes of spectra are approximately equivalent (for the special case of GWVS and GES and

processes with compactly supported GEAF, part of these results has been shown previously

in [118,126]).


Input-output relations (Section 3.6): We present novel approximations that relate the GWVS

(GES) of the nonstationary output process of an LTV system to the GWVS (GES) of the

nonstationary input process.

Approximate KL expansion (Section 3.7): We present approximate KL expansions for under-

spread nonstationary random processes. This extends existing results derived for processes with

compactly supported GEAF [115, 116, 118, 120] to more general scenarios and is furthermore

related to results obtained in [137,138] for locally stationary processes.

TF coherence function (Section 3.8): We discuss the concept of coherence for nonstationary ran-

dom processes and introduce a novel class of TF coherence functions. While a TF coherence

has been defined in [210] in an ad hoc fashion, we prove that our TF coherence functions can be

viewed as approximate TF formulations of a coherence operator and we provide several approx-

imations that justify their interpretation as a coherence function.

Applications (Chapter 4): Chapter 4 applies the theoretical results developed in Chapters 2 and

3 to the problems of signal estimation, signal detection, channel sounding, multicarrier commu-

nications, and car engine diagnosis (see Section 1.5 for an outline of these applications).

2

Underspread Systems

“My work always tried to unite the truth with the beautiful, but when I had tochoose one or the other, I usually chose the beautiful.” Hermann Weyl

ALTHOUGH there exist several useful time-frequency tools for characterizing linear time-varying

systems (linear operators), such as the generalized Weyl symbol and the generalized spreading

function, these time-frequency representations are in general difficult to work with. For example,

series connections and inverses of linear time-varying systems in general cannot be expressed via the

generalized Weyl symbol in as simple a manner as it is possible with the transfer function of linear

time-invariant systems. Hence, there remain several problems as to how these representations are to

be interpreted and how they can be used in specific signal processing applications.

A key concept that allows to answer these questions is that of underspread systems. Such systems

introduce only limited time-frequency shifts and are characterized by a spreading function concentrated

around the origin of the (τ, ν)-plane. Underspread systems essentially come in two flavors: The first

type, proposed in this thesis and discussed in detail in Section 2.2, builds on the requirement of rapid

decay of the spreading function such that specific weighted integrals and moments are sufficiently

small. The second type (which can be viewed as a special case of the first), is based on a strict

support constraint of the spreading function, similar to strictly bandlimited signals. This latter type

of underspread systems, reviewed in Section 2.1, has been introduced and extensively analyzed in the

pioneering work of W. Kozek.

In Section 2.3 of this chapter, numerous approximations based on specific underspread assumptions

are proved that establish an approximate time-frequency transfer function calculus. We note that some

parts of our analysis are parallel to Kozek’s work, although our (more general) definition of underspread

systems using weighted integrals and moments and the approximations based on it are original.

15

16 Chapter 2. Underspread Systems

2.1 Operators with Compactly Supported Spreading Function

In this section we consider operators with compactly supported GSF and review Kozek’s original def-

inition of underspread operators. We furthermore discuss the effect of unitary metaplectic transforms

on the operator’s GSF support and study the GSF supports of the sum, product, and inverses of

operators.

2.1.1 General Support Constraints

The definition of underspread (deterministic) LTV systems by requiring the support of their GSF to

be confined to a rectangular region about the origin has first been proposed and extensively studied

in the pioneering work of W. Kozek [117–120, 123, 126]. This support constraint on the GSF implies

that the system introduces only limited time displacements and frequency displacements and hence

we refer to such systems as displacement-limited (DL) systems (operators). Since GSF and GWS are

2-D Fourier transform pairs, it further follows that the GWS of a DL system is a 2-D band-limited

function. The existing extensive theory of band-limited signals [162] thus serves as an additional

motivation for this definition of underspread systems. In particular, a strict band-limitation of the

GWS allows the GWS to be sampled on a 2-D sampling grid without information loss.

The following discussion is intended as a review of some of the concepts introduced in [117–119,

123,126], with some slight modifications and extensions comprising more general support constraints

than rectangular ones (e.g., oblique regions in the (τ, ν)-plane). This will also yield bounds on the

errors incurred by the transfer function approximations in Section 2.3 that are slightly more tight

and/or valid under more general conditions than previous bounds in [117–119,123,126].

Consider an LTV system (operator) H with GSF S(α)H

(τ, ν). We require that the support of the

GSF is confined to a compact region GH, i.e., |SH(τ, ν)| = 0 for (τ, ν) 6∈ GH. For such systems, we can

write1

GH ={(τ, ν) : |SH(τ, ν)| > 0

}.

Thus, with the indicator function IGH(τ, ν) of GH, defined as

IGH(τ, ν) =

1 , (τ, ν) ∈ GH ,

0 , (τ, ν) 6∈ GH ,

the GSF satisfies the condition

S(α)H

(τ, ν) IGH(τ, ν) = S

(α)H

(τ, ν). (2.1)

While in general there exist infinitely many indicator functions which satisfy (2.1) for a given S(α)H

(τ, ν),

our definition corresponds to the indicator function with minimal support, i.e., where the associated

region GH has minimal area. The multiplicative relation (2.1) in the (τ, ν)-domain corresponds to a

1 Note that the GSF magnitude is independent of α, |S(α)H

(τ, ν)| = |SH(τ, ν)| (see Subsection B.1.1).

2.1 Operators with Compactly Supported Spreading Function 17

convolution relation in the TF domain, i.e.,∫

t′

∫

f ′

L(α)H

(t′, f ′)L(α)T

(t− t′, f − f ′) dt′ df ′ = L(α)H

(t, f). (2.2)

Here, L(α)T

(t, f) is the GWS of an operator T defined via its GSF as

S(α)T

(τ, ν) = IGH(τ, ν). (2.3)

Note (2.2) holds for any operator T with GSF satisfying S(α)eT

(τ, ν) = 1 for (τ, ν) ∈ GH (while being

arbitrary for (τ, ν) 6∈ GH). However, due to (B.9), the operator T defined via (2.3) has minimum norm

within the class of all operators satisfying (2.2).

To each (τ, ν)-region G or indicator function IG(τ, ν), there corresponds a linear subspace SG that

consists of all linear operators H satisfying the corresponding support constraint (2.1),

SG ={H : S

(α)H

(τ, ν)IG(τ, ν) = S(α)H

(τ, ν)}.

The orthogonal complement of SG corresponds to the complement G = R2\G of G whose indicator

function is IG(τ, ν) = 1 − IG(τ, ν), such that for any operator Hc in the complement space ScG = SG

there is S(α)Hc

(τ, ν) IG(τ, ν) = 0 or equivalently S(α)Hc

(τ, ν) IG(τ, ν) = S(α)Hc

(τ, ν) [1− IG(τ, ν)] = S(α)Hc

(τ, ν).

Obviously, we also have

H ∈ SG1 and G1 ⊆ G2 =⇒ H ∈ SG2 .

Furthermore, given a region G or indicator function IG(τ, ν), any operator H can be split into two

orthogonal parts lying respectively inside and outside the associated operator space SG ,

H = HG + HG . (2.4)

The orthogonal compononents HG and HG are found by projecting H onto the respective subspace.

This projection can easily be accomplished in the GSF domain according to

S(α)

HG (τ, ν) = S(α)H

(τ, ν)IG(τ, ν), S(α)

HG(τ, ν) = S

(α)H

(τ, ν) [1 − IG(τ, ν)] . (2.5)

We call HG the DL part of H corresponding to the region G. It is easily checked using (B.8) that

these two systems are indeed orthogonal, i.e.⟨HG ,HG

⟩= Tr

{HG(HG)+

}= 0. Note that this does

not in general imply HGHG = 0 or HGHG = 0. These concepts will be important in Sections 2.3.6

and 2.3.18 when discussing operator inverses and TF-sampling of the GWS, respectively.

2.1.2 Definition of Displacement-limited Underspread Operators

In the following, we recall from [118] the definition of underspread operators in the constrained support

sense:

Definition 2.1. Let H be a DL system with GSF support GH and let τ(max)H

and ν(max)H

be the maximum

time and frequency shift, respectively, introduced by the system,

τ(max)H

, max(τ,ν)∈GH

|τ | , ν(max)H

, max(τ,ν)∈GH

|ν| . (2.6)


ν ν

τ τ

ν(max)H

−ν(max)H

τ(max)H

−τ(max)H

GH

GH

(a) (b)

Figure 2.1: Illustration of (a) a rectangular and (b) a “hyperbolic” compact support constraint.

Then,

σH = 4τ(max)H

ν(max)H

is called the (strict-sense) displacement spread of the DL system H, and H is called (strict-sense) DL

underspread if

σH = 4τ(max)H

ν(max)H

≪ 1 . (2.7)

Condition (2.7) was first introduced in [118–120]. We note that 4τ(max)H

ν(max)H

is the area of a

rectangle with sides of length 2τ(max)H

and 2ν(max)H

, respectively, which contains the GSF support

region GH. Hence, σH measures the support of the GSF of H via a circumscribed rectangle (cf. Fig.

2.1(a)). Sometimes, it will suffice to require the less restrictive condition

µH , max(τ,ν)∈GH

|τν| ≪ 1 . (2.8)

Since µH measures only the maximum product |τν| for points within GH, we have

µH ≤ σH/4 , (2.9)

with equality if and only if one of the four “corner” points (±τmax,±νmax) ∈ GH. Condition (2.8) is

somewhat different in spirit from (2.7) since it does not require the area of the support of the GSF to

be small or even finite; however, it still implies a support constraint since the GSF has to lie within

the hyperbolae defined by |τν| = µH (see also Fig. 2.1(b)).

It is important to note that DL operators [118–120] are a special case of operators with rapidly

decaying GSF to be defined in Section 2.2. This will further be discussed in Subsection 2.2.2.

2.1.3 Unitary Transformations

Let us now consider the influence of some specific unitary transformations of operators on the GSF

support, i.e., we consider the transformed operator H = UHU+ with some unitary operator U.

2.1 Operators with Compactly Supported Spreading Function 19

We first study the effects of TF shifts, i.e., H = St,fHS+t,f where S

(α)t,f denotes the joint TF shift

operator (see Appendix B). Since TF shifts leave the GSF magnitude unchanged, the quantities σH

and µH remain unchanged too, i.e.,

σ eH= σH, µ eH

= µH.

Hence, any class of operators defined by some type of GSF support constraint also comprises all TF

shifted versions of each member of this class,

H ∈ SG =⇒ St,fHS+t,f ∈ SG .

In particular, all TF shifted versions of an underspread DL system are also DL underspread.

Next let us consider the class M of metaplectic transformations U = µ(A) that correspond to the

area-preserving linear (symplectic) TF coordinate transforms (see Appendix C and [46,64,162,208]).

For any metaplectic operator U = µ(A), the GSF with α = 0 of H = UHU+ is given by (cf. (C.4))

S(0)eH

(τ, ν) = S(0)H

(aτ + bν, cτ + dν).

Since |S(α)H

(τ, ν)| is independent of α (cf. Section B.1.1), there is also2

∣∣S(α)eH

(τ, ν)∣∣ =

∣∣S(α)H

(aτ + bν, cτ + dν)∣∣ for all α. (2.10)

Specific metaplectic operators which depend only on a single parameter are the TF scaling operator

(a = 1/d, b = c = 0), the Fourier transform operator (a = d = 0, b = −1/c = −T 2), the chirp

multiplication operator (a = d = 1, b = 0), the chirp convolution operator (a = d = 1, c = 0), and the

fractional Fourier transform operator (a = d = cos θ, b = −T 2 sin θ, c = (sin θ)/T 2).

It is straightforward to show that only for TF scalings and Fourier transforms there is σ eH= σH

and µ eH= µH. In all other cases, σH and µH must be expected to change. In some cases, it is

undesirable that systems whose GSFs are equal up to area-preserving linear TF coordinate transforms

are assigned different displacement spreads. Hence, we extend the definition of the DL underspread

property in (2.7) such that the displacement spread of all operators whose GSFs are obtained from

each other by symplectic group TF coordinate transforms is equal:

Definition 2.2. An operator H is called wide-sense DL underspread if there exists a metaplectic

operator U ∈ M such that the displacement spread of H = UHU+ satisfies σ eH≪ 1. We call

σminH , inf

U∈Mσ eH

the wide-sense displacement spread of H.

Note that all systems which are unitarily equivalent via some U ∈ M are thus assigned the

same wide-sense displacement spread. This is of specific importance when discussing transfer function

approximations for the case α = 0 where shearings and rotations of the GSF can be of particular

interest (see, e.g., Subsections 2.3.4 and 2.3.15).2 Note that for α 6= 0 this equality is valid only for the magnitude of the GSF and not for the GSF itself.


2.1.4 Operator Sums, Adjoints, Products, and Inverses

We shall next consider the displacement spreads of sums, adjoints, products, and inverses of operators.

Sum. With regard to the sum of two operators H1 and H2, we obviously have

|SH1+H2(τ, ν)| =∣∣∣S(α)

H1(τ, ν) + S

(α)H2

(τ, ν)∣∣∣ ≤ |SH1(τ, ν)| + |SH2(τ, ν)| .

Without making any additional assumptions, we have to deal with the worst case, i.e., GSFs SH1(τ, ν),

SH2(τ, ν) which do not cancel anywhere in the sum S(α)H1

(τ, ν) + S(α)H2

(τ, ν). Here, the region of sup-

port of SH1+H2(τ, ν) is given by GH1,H2 , GH1 ∪ GH2 and the corresponding indicator function is

IGH1,H2(τ, ν) = IGH1

(τ, ν) + IGH2(τ, ν) − IGH1

(τ, ν)IGH2(τ, ν). For the displacement spreads we then

obtain

σH1+H2 ≤ σH1,H2 , 4max{τ

(max)H1

, τ(max)H2

}max

{ν

(max)H1

, ν(max)H2

}, (2.11)

µH1+H2 ≤ µH1,H2 , max {µH1 , µH2} .

It follows that σH1,H2 ≥ max {σH1 , σH2} and hence, in general, σH1+H2 must be expected to be larger

than both σH1 and σH2 (except if the support of one GSF is totally contained within that of the other

GSF, or the GSFs add to zero in specific peripheral regions of the (τ, ν)-plane). This shows that the

sum of two DL underspread operators is DL but not necessarily DL underspread. On the other hand,

µH1,H2 is never larger than the maximum of µH1 and µH2 . Hence, the class of operators defined by

µH ≤ A is closed under addition.

In [118] two systems are called jointly underspread if the area of the union of their GSF supports

can be circumscribed by an axis-parallel rectangle of area less than one. However, based on the above

observations and using the definition

σminH1,H2

, infU∈M

σUH1U+,UH2U

+

we here introduce the following definition of jointly DL operators:

Definition 2.3. Two systems H1 and H2 are said to be jointly strict-sense (wide-sense) DL under-

spread if σH1,H2 ≪ 1 (σminH1,H1

≪ 1).

Hence, jointly DL underspread systems are individually DL underspread and also their sum H1+H2

is DL underspread. We will call σH1,H2 (σminH1,H1

) the strict-sense (wide-sense) joint displacement spread

of H1 and H2. In essence, jointly DL underspread operators have GSFs satisfying similar support

constraints.

Adjoint. Since∣∣SH+(τ, ν)

∣∣ =∣∣S∗

H(−τ,−ν)

∣∣ according to (B.6), the displacement spreads of a

system and its adjoint are equal, σH+ = σH, and furthermore µH+ = µH.

Product. The following inequalities can be easily deduced from the bound (B.13):

τ(max)H2H1

≤ τ(max)H1

+ τ(max)H2

, ν(max)H2H1

≤ ν(max)H1

+ ν(max)H2

.

2.2 Operators with Rapidly Decaying Spreading Function 21

In particular, the above inequalities imply that σH2 can be as large as 4σH but not larger,

σH2 = 4τ(max)H2 ν

(max)H2 ≤ 4(2τ

(max)H

)(2ν(max)H

) = 4σH.

This bound can be shown to hold also for the composition of a system and its adjoint, i.e., σHH+ ≤ 4σH

and σH+H ≤ 4σH.

Inverse. The GSF of the inverse of an operator H is difficult to analyze. From the Neumann

series [158]

H−1 =

∞∑

k=0

(I − H)k ,

it is seen that due to the higher powers of (I − H) the support of the GSF of H−1 may grow ad

infinitum. Thus, in general the inverse of a DL underspread operator need not be even DL. Yet, this

does not imply that the GSF of H−1 may not be concentrated around the origin. This will further be

discussed in Subsection 2.3.6.

2.2 Operators with Rapidly Decaying Spreading Function

After this discussion of DL and DL underspread operators, we now turn to a novel extended concept

of underspread operators that replaces the compact support constraint on the GSF by generalized

decay constraints.

2.2.1 Motivation

In many cases, the assumption that the support of the GSF of an operator is exactly confined to some

small area around the origin (as used in [118, 119, 127] for the definition of underspread operators)

appears to be too restrictive. Often, the GSF is merely concentrated around the origin and has rapid

decay away from the origin. A useful measure of the decay of the GSF is in terms of weighted GSF

integrals and moments which describe the effective, rather than exact, support of the GSF. Hence,

a reasonable and practically relevant extension of the underspread concept can be based on such

measures of effective GSF support. This is the point of view we adopt in this section.

Based on our extended concept of underspread operators with rapidly decaying GSF, we will

prove the validity of several underspread approximations in Section 2.3. This approach has several

advantages as compared to the results obtained for DL operators:

• In many practical situations, a theory based on weighted GSF integrals and moments is closer

to physical reality than a theory based on exact support constraints.

• The results can be used to judge how well the results obtained for DL operators apply to operators

with rapidly decaying GSF (see Subsection 2.2.6).

• In general, an operator and its inverse cannot be simultaneously underspread in the DL sense,

whereas it is possible that they both have fast decaying GSF.


(a) (b) (c) (d) (e)

ν ν ν ν ν

τ τ τ τ τ

Figure 2.2: Gray-scale plots (darker shades correspond to larger values) of several specific weighting

functions: (a) φ(τ, ν) = |τ |k, (b) φ(τ, ν) = |ν|k, (c) φ(τ, ν) = |τν|k, (d) φ(τ, ν) = |1 − A(0)s (τ, ν)|, (e)

φ(τ, ν) =∣∣∣1 − 1

A(0)s (τ, ν)

∣∣∣ with A(0)s (τ, ν) the ambiguity function (see Section B.2.4) of a normalized

Gaussian function.

On the other hand, a drawback of our extended theory of underspread operators is that it does not

allow for an exact TF sampling of the GWS as discussed in [118] for the case of DL underspread

operators. We will further comment on the sampling problem in Subsection 2.3.18.

2.2.2 Weighted Integrals and Moments of the Generalized Spreading Function

To circumvent the problems and limitations associated the DL underspread concept as mentioned

above, we here propose to globally characterize the TF shifts of a system H by means of the weighted

GSF integrals

m(φ)H

,

∫

τ

∫

νφ(τ, ν) |SH(τ, ν)| dτdν

∫

τ

∫

ν|SH(τ, ν)| dτdν

=1

‖SH‖1

∫

τ

∫

νφ(τ, ν) |SH(τ, ν)| dτ dν , (2.12-a)

M(φ)H

,

∫

τ

∫

νφ2(τ, ν) |SH(τ, ν)|2 dτdν∫

τ

∫

ν|SH(τ, ν)|2 dτdν

1/2

=1

‖H‖2

[∫

τ

∫

νφ2(τ, ν) |SH(τ, ν)|2 dτ dν

]1/2

, (2.12-b)

which are normalized by the L1 or L2 norm of S(α)H

(τ, ν) (recall that ‖SH‖2 = ‖H‖2). We note that

due to (1.9), the implicit assumption that the GSF has finite L1 norm, ‖SH‖1 < ∞, is a sufficient

condition for the bounded input bounded output stability of the system H. Typically, φ(τ, ν) in (2.12-a)

and (2.12-b) is a nonnegative weighting function which satisfies φ(τ, ν) ≥ φ(0, 0) = 0 and penalizes

GSF contributions lying away from the origin. We note that m(φ)H

and M(φ)H

do not depend on the

GWS parameter α. Since the GSF magnitude of the adjoint system H+ is given by |SH+(τ, ν)| =

|SH(−τ,−ν)|, we have m(φ)H+ = m

(φ)H

and M(φ)H+ = M

(φ)H

whenever the weighting function is even-

symmetric, i.e., φ(τ, ν) = φ(−τ,−ν). Fig. 2.2 shows some specific weighting functions which will be

used in later subsections.


Absolute Moments of the GSF. As a special case of the above weighted integrals using the

weighting functions φ(τ, ν) = |τ |k|ν|l, we also introduce the normalized moments

m(k,l)H

,1

‖SH‖1

∫

τ

∫

ν|τ |k |ν|l |SH(τ, ν)| dτ dν , (2.13-a)

M(k,l)H

,1

‖H‖2

[∫

τ

∫

ντ2k ν2l |SH(τ, ν)|2 dτ dν

]1/2

, (2.13-b)

with integers3 k, l ∈ N0. If one views |SH(τ, ν)|/‖SH(τ, ν)‖1 and |SH(τ, ν)|2/‖H‖22 as probability

density functions (they are positive and integrate to one), the above moments are analogous to the

(absolute) moments of random variables [163]; thus, we call m(k,l)H

and M(k,l)H

the (absolute) moments

of order (k, l) of H. Note that m(0,0)H

= M(0,0)H

= 1 for any system H. In almost all cases we will use

those (absolute) moments where either k = 0 or l = 0 or k = l.

• Moments with k = 0 or l = 0 penalize mainly GSF contributions located away from the τ axis or

away from the ν axis, respectively. Thus, systems with GSF concentrated along the τ axis (i.e.,

quasi-LTI systems) have small m(0,l)H

and small M(0,l)H

, whereas systems with GSF concentrated

along the ν axis (i.e., quasi-LFI systems) have small m(k,0)H

and small M(k,0)H

(cf. Figs. B.1 and

2.2).

• Moments with k = l penalize mainly GSF contributions located away from the τ and ν axes,

i.e., lying in oblique directions in the (τ, ν) plane. This is due to the fact that the corresponding

weighting function is constant along the hyperbolae |τν| = c (cf. Fig. 2.2). In particular, a

superposition (i.e., parallel connection) of a (quasi-) LTI system and a (quasi-) LFI system has

a GSF concentrated along the τ and ν axes and will thus have small m(k,k)H

and M(k,k)H

.

Note thatM(k,l)H

in (2.13-b) is well-defined only for Hilbert-Schmidt (HS) operators, i.e., for systems

with finite HS norm (cf. Appendix A). As such, this definition is not directly applicable to LTI and

LFI systems. However, for LTI and LFI systems appropriately modified moment definitions can be

given in terms of the impulse response g(τ) or the Fourier transform of the modulation function,

M(ν) = (Fm)(ν), respectively:

M(k,0)HLTI

,1

‖g‖2

[∫

ττ2k |g(τ)|2 dτ

]1/2

, M(0,l)HLFI

,1

‖M‖2

[∫

νν2l |M(ν)|2 dν

]1/2

. (2.14)

Note that these specific moments characterize the time displacements and frequency displacements of

LTI and LFI systems, respectively; since LTI systems do not cause frequency displacements and LFI

systems do not cause time displacements, it does not make sense to ask about moments characterizing

the frequency displacements of LTI systems or the time displacements of LFI systems. Furthermore,

for systems with GSF perfectly concentrated along the τ and ν axes, i.e., for any superposition of LTI

and LFI systems with impulse response h(t, t′) = g(t − t′) +m(t) δ(t − t′), one has m(k,l)H

= 0 for any

k, l both > 0.

3Note that theoretically k, l could be any positive real-valued numbers. However, in the subsequent sections only

integer values are required.


Due to Schwarz’ inequality, the moments satisfy the following inequalities:

m(k,l)H

≤√m

(2k,0)H

m(0,2l)H

, M(k,l)H

≤√M

(2k,0)H

M(0,2l)H

. (2.15)

Since m(2k,0)H

and m(0,2l)H

and similarly M(2k,0)H

and M(2k,0)H

measure the average extension of the GSF

in the τ and ν direction, respectively, their product is a measure of the effective “off origin” support

of the GSF defined via an “equivalent rectangle.” The above inequalities hence are mathematical

formulations of the fact that the effective “off axes” support is always less than the effective “off

origin” support.

Weighted GSF Integrals/Moments of DL Operators. For the case of a DL operator H

with GSF exactly zero outside a compact region GH, the following result relates the GSF integrals

and moments to the quantities τ(max)H

, ν(max)H

, and µH defined in Section 2.1.

Proposition 2.4. The GSF integrals of a DL operator H satisfy the bounds

m(φ)H

≤ φ(max)H

, M(φ)H

≤ φ(max)H

, (2.16)

with φ(max)H

, max(τ,ν)∈GHφ(τ, ν). For the GSF moments of a DL operator H, (2.16) further implies

the bounds

m(k,l)H

≤[τ

(max)H

]k [ν

(max)H

]l, M

(k,l)H

≤[τ

(max)H

]k [ν

(max)H

]l, (2.17)

and for k = l one has the tighter bounds

m(k,k)H

≤ µkH ≤

(σH

4

)k, M

(k,k)H

≤ µkH ≤

(σH

4

)k. (2.18)

Proof. The bound for m(φ)H

is obtained by noting that

m(φ)H

=1

‖SH‖1

∫

τ

∫

νφ(τ, ν) |SH(τ, ν)| dτ dν =

1

‖SH‖1

∫∫

GH

φ(τ, ν) |SH(τ, ν)| dτ dν

≤ 1

‖SH‖1

∫∫

GH

[max

(τ,ν)∈GH

φ(τ, ν)]|SH(τ, ν)| dτ dν

= φ(max)H

1

‖SH‖1

∫∫

GH

|SH(τ, ν)| dτ dν = φ(max)H

.

From this, the bound for m(k,l)H

follows by noting that for φ(τ, ν) = |τ |k |ν|l

φ(max)H

= maxGH

{|τ |k |ν|l} ≤ maxGH

{|τ |k} maxGH

{|ν|l} =[τ

(max)H

]k [ν

(max)H

]l,

and the bound for m(k,k)H

follows by noting that for φ(τ, ν) = |τ |k|ν|k = |τν|k

φ(max)H

= maxGH

{|τν|k} = µkH .

The bounds on M(φ)H

, M(k,l)H

and M(k,k)H

can be derived in a similar manner.


This result implies that for systems which are DL underspread, the weighted GSF integrals and

moments are small. Hence, DL underspread operators are a special case of the extended underspread

framework based on weighted GSF integrals and moments. The above bounds will also allow us

in Section 2.3 to compare the results obtained for operators with rapidly decaying GSF with those

obtained in [118] for DL operators.

Derivatives of the GWS. Due to the 2-D Fourier relation (B.20) of the GSF and GWS, the

following bounds and expressions in terms of the GSF moments can be derived for for the (partial)

derivatives of the GWS in (1.7):

Proposition 2.5. The L∞ and L2 norms of the partial derivatives of the GWS are related to the

(absolute) GSF moments via

∣∣∣∣∂k+lL

(α)H

(t, f)

∂tl ∂fk

∣∣∣∣ ≤ (2π)k+l‖SH‖1m(k,l)H

,

∥∥∥∥∂k+lL

(α)H

∂tl ∂fk

∥∥∥∥2

= (2π)(k+l)‖H‖2M(k,l)H

.

Proof. Since differentiating in the TF domain corresponds to multiplications in the spreading domain,

there is

∂k+lL(α)H

(t, f)

∂tl ∂fk=

∫

τ

∫

ν(−1)l(j2π)(k+l) τk νl S

(α)H

(τ, ν) ej2π(τf−νt) dτ dν.

Therefore, the magnitude of the partial derivatives of the GWS is bounded as

∣∣∣∣∂k+lL

(α)H

(t, f)

∂tl ∂fk

∣∣∣∣ =

∣∣∣∣∫

τ

∫

ν(−1)l(j2π)(k+l) τk νl S

(α)H

(τ, ν) ej2π(τf−νt) dτ dν

∣∣∣∣

≤∫

τ

∫

ν(2π)(k+l)

∣∣τk νl S(α)H

(τ, ν)∣∣dτ dν = (2π)(k+l)‖SH‖1m

(k,l)H

,

from which the L∞ bound follows. Similarly, by Parseval’s theorem,

∥∥∥∥∂k+lL

(α)H

∂tl ∂fk

∥∥∥∥2

2

=

∫

τ

∫

ν(2π)2(k+l)


(τ, ν)∣∣2dτ dν

= (2π)2(k+l)

∫

τ

∫

ντ2k ν2l |S(α)

H(τ, ν)|2dτ dν = (2π)2(k+l)‖H‖2

2

[M

(k,l)H

]2,

thus yielding an exact expression for the L2 norm of the partial derivatives of the GWS in terms of

M(k,l)H

.

From Proposition 2.5 it is seen that the GSF moments m(k,l)H

and M(k,l)H

essentially determine

the smoothness of the GWS. In particular, the GWS’s first derivative with respect to time will be

small—implying smoothness of the GWS in the time direction—if the moments m(0,1)H

and M(0,1)H

are

small. Similarly, the GWS will be smooth in the frequency direction if m(1,0)H

and M(1,0)H

are small.

We note that the first relation (bound) bears some resemblance to the definition of specific symbol

classes of pseudo-differential operators [64, 73, 94, 95, 112, 206], with the difference that our bound is

uniform over the entire TF plane whereas in pseudo-differential operator theory there is an additional

term controlling the growth of the partial derivatives of the symbol.


Displacement Spread Parameters. A specifically interesting characterization of the TF shifts

of an LTV system is in terms of the moments M(1,0)H

and M(0,1)H

for which we will use special symbols.

We define the temporal displacement spread τH and the spectral displacement spread νH as

τH , M(1,0)H

=1

‖H‖2

[∫

τ

∫

ντ2 |SH(τ, ν)|2 dτ dν

]1/2

=1

‖H‖2

[∫

t

∫

ττ2 |h(α)(t, τ)|2 dt dτ

]1/2

, (2.19)

νH , M(0,1)H

=1

‖H‖2

[∫

τ

∫

νν2 |SH(τ, ν)|2 dτ dν

]1/2

=1

2π‖H‖2

[∫

t

∫

τ

∣∣∣ ∂∂th(α)(t, τ)

∣∣∣2dt dτ

]1/2

. (2.20)

From the right-most expressions, it is seen that the temporal and spectral displacement spreads char-

acterize, respectively, the memory and the time-variations of H.

Sometimes we will also use the displacement radius ρH(T ) [86] where T > 0 is an arbitrary time

constant introduced for reasons of compatibility of physical dimensions. The displacement radius is

defined as weighted GSF integral M(φT )H

with weighting function equal to the normalized Euclidian

distance of (τ, ν) from the origin, φT (τ, ν) =√(

τT

)2+ (νT )2 =

∥∥(τ/TνT

)∥∥2,

ρH(T ) , M(φT )H

=1

‖H‖2

[∫

τ

∫

ν

[( τT

)2+ (νT )2

]|SH(τ, ν)|2 dτ dν

]1/2

.

Hence, all GSF contributions are weighted by their Euclidian distance from the origin. Thus, ρH(T )

is a measure of the “off origin” support of the GSF. It may be large even for systems whose “off axes”

support as measured by m(k,k)H

is small. The square of the displacement radius is easily shown to equal

the normalized sum of the two squared displacement spreads, i.e.,

ρ2H(T ) =

τ2H

T 2+ T 2 ν2

H .

By using this relation and completing the square one obtains

ρ2H(T ) =

τ2H

T 2+ T 2 ν2

H =

(τH

T− νHT

)2

+ 2 τHνH .

It is thus seen that the constant T minimizing ρ2H

(T ) is T0 =√τH/νH for which τH

T − νHT = 0, and

the minimum is given by

ρ2H , min

Tρ2H(T ) = ρ2

H

(√τH

νH

)= 2τHνH ,

from which it follows that ρ2H

(T ) ≥ 2τHνH .

We will furthermore occasionally use the weighted GSF integral

κH ,1

‖H‖22

∫

τ

∫

ντν |SH(τ, ν)|2 dτ dν .

It is not a special case of m(φ)H

or M(φ)H

and in general does not describe the effective support of

S(α)H

(τ, ν). Rather, it is a measure of the orientation (or skewness) of the GSF, i.e., it indicates whether

the GSF is concentrated along an axis or rather along oblique directions. Since a large magnitude of


κH implies that the time and frequency shifts caused by H are strongly coupled, i.e., large (small) time

shifts imply large (small) frequency shifts and vice versa, we refer to κH as displacement correlation

of H. An inequality which we will need later is

κ2H ≤ τ2

Hν2H. (2.21)

In some cases it is convenient to arrange the (normalized) temporal and spectral displacement

spreads and the displacement correlation into a matrix which will be called the displacement spread

matrix ,

D(T )H

,

τ2H

T 2 κH

κH ν2HT 2

.

The determinant of D(T )H

(which is independent of T ) will be denoted by η2H

,

η2H , detD

(T )H

= τ2Hν

2H − κ2

H.

Note that (2.21) implies that detD(T )H

≥ 0 and thus D(T )H

is seen to be positive semi-definite. Further-

more, the squared displacement radius is the trace of the displacement spread matrix, i.e.,

ρ2H(T ) = Tr

{D

(T )H

}.

All moment quantities introduced in this subsection and their interrelations are summarized in Table

2.1.

2.2.3 Unitary Transformations

Similarly to Subsection 2.1.3, we now consider the influence of some specific unitary transformations

of operators on the weighted integrals and moments of the GSF.

TF Shifts. Since TF shifts of an operator H leave the GSF magnitude unchanged, the weighted

GSF integrals and moments as defined by (2.12-a), (2.12-b), (2.13-a), and (2.13-b) remain unchanged

too, i.e., the weighted GSF integrals and moments of the TF shifted operator H = St,fHS+t,f are given

by

m(φ)eH

= m(φ)H, M

(φ)eH

= M(φ)H, m

(k,l)eH

= m(k,l)H

, and M(k,l)eH

= M(k,l)H

.

Metaplectic Transformations. Next, let us consider unitary transformations U ∈ M corre-

sponding to area-preserving linear TF coordinate transforms [64, 162] (see Subsection 2.1.3 and Ap-

pendix C). Using the metaplectic covariance (2.10) of the GSF, the following expression is obtained

for the weighted GSF integrals of H = UHU+,

m(φ)eH

=1

‖S eH‖1

∫

τ

∫

νφ(τ, ν) |S eH

(τ, ν)| dτ dν

=1

‖SH‖1

∫

τ

∫

νφ(τ, ν) |SH(aτ + bν, cτ + dν)| dτ dν


Quantity Definition Interrelations

m(k,l)H

1‖SH‖1

∫τ

∫ν |τ |k |ν|l |SH(τ, ν)| dτ dν 0 ≤ m

(k,l)H

≤√m

(2k,0)H

m(0,2l)H

M(k,l)H

1‖H‖2

[∫τ

∫ντ2k ν2l |SH(τ, ν)|2 dτ dν

]1/2

0 ≤M(k,l)H

≤√M

(2k,0)H

M(0,2l)H

τH1

‖H‖2

[∫τ

∫ντ2 |SH(τ, ν)|2 dτ dν

]1/2

τH = M(1,0)H

νH1

‖H‖2

[∫τ

∫ν ν

2 |SH(τ, ν)|2 dτ dν]1/2

νH = M(0,1)H

κH1

‖H‖22

∫τ

∫ντ ν |SH(τ, ν)|2 dτ dν

∣∣κH

∣∣ ≤ τH νH

ρ2H

(T ) 1‖H‖2

2

∫τ

∫ν

[ (τT

)2+ (νT )

2]|SH(τ, ν)|2 dτ dν ρ2

H(T ) =

τ2H

T 2 + T 2ν2H

= Tr{D

(T )H

}≥ 2 τH νH

D(T )H

τ2H

T 2 κH

κH ν2HT 2

D

(T )H

≥ 0, Tr{D

(T )H

}= ρ2

H(T ) ≥ 2 τH νH

η2H

detD(T )H

η2H

= τ2Hν2H− κ2

H

Table 2.1: Moments and displacement spread parameters and their interrelations.

=1

‖SH‖1

∫

τ

∫

νφ(dτ − bν, aν − cτ) |SH(τ, ν)| dτ dν = m

(φ)H,

with φ(τ, ν) = φ(dτ − bν, aν − cτ). Similarly, M(φ)eH

= M(φ)H

.

In Section 2.3, weighting functions of the type φs(τ, ν) = |1−A(0)s (τ, ν)| and φ′s(τ, ν) =

∣∣∣∣1− 1

A(0)s (τ,ν)

∣∣∣∣(with A

(0)s (τ, ν) the ambiguity function, see Section B.2.4) will be of specific importance. Here, using

the fact that the ambiguity function is covariant to metaplectic transforms, it can be shown that

φs(τ, ν) = |1 −A(0)s (dτ − bν, aν − cτ)| = |1 −A

(0)s (τ, ν)| = φs(τ, ν) , (2.22)

φ′s(τ, ν) =

∣∣∣∣1 − 1

A(0)s (dτ − bν, aν − cτ)

∣∣∣∣ =∣∣∣∣1 − 1

A(0)s (τ, ν)

∣∣∣∣ = φ′s(τ, ν) , (2.23)

with s(t) = (U+s)(t). Thus, we here obtain m(φs)UHU+ = m

(φU+s)

Hand m

(φ′s)

UHU+ = m(φ′

U+s)

H.

Further special cases of the above result M(φ)eH

= M(φ)H

which are of particular interest are the

following expressions for the temporal and spectral displacement spread,

τ 2eH

= d2 τ2H + b2 ν2

H − 2bd κH

ν2eH

= c2 τ2H + a2 ν2

H − 2ac κH . (2.24)

In a similar manner, we obtain for the displacement correlation

κeH= −cd τ 2

H − ab ν2H + (ad+ bc)κH. (2.25)


TF scalingFourier

transform

Chirpmultiplication

Chirpconvolution

A(

1/d 0

0 d

) (0 −T 2

1/T 2 0

) (1 0

c 1

) (1 b

0 1

)

(Ux)(t) 1√|d|x(

td

)1T X( t

T 2 ) e−jπct2x(t) x(t) ∗ 1√|b|e−jπt2/b

τ2eH

τ2H/a2 ν2

HT 4 τ2

Hτ2H− 2bκH + b2ν2

H

ν2eH

a2ν2H

τ2H/T 4 ν2

H− 2cκH + c2τ2

Hν2H

κeHκH −κH κH − cτ2

HκH − bν2

H

Table 2.2: The effect of specific unitary transformations U ∈ M (corresponding to area-preserving

linear TF coordinate transforms A) on the displacement spread parameters τ2H

, ν2H

, and κH.

Simplifications of these expressions for the case of specific unitary transformations are shown in Table

2.2. The above three relations can be rewritten more compactly in terms of the displacement spread

matrix,

D(T )eH

= A−TT D

(T )H

A−1T ,

where AT =(

1/T 00 T

)(a bc d

)(T 00 1/T

)and A−T denotes the transpose of A−1

T . With detAT = 1, it follows

that

η2eH

= detD(T )eH

= det{A−TD

(T )H

A−1}

=detD

(T )H

det2 A = detD(T )H

= η2H , (2.26)

i.e., the determinant of the displacement spread matrix is invariant with respect to metaplectic unitary

operator transformations.

It is difficult to obtain general expressions for the moments of unitarily transformed systems.

However, the following results for specific metaplectic transformations (cf. Table 2.2) can be found:

TF scaling: m(k,l)eH

= |a|l−k m(k,l)H

, M(k,l)eH

= |a|l−k M(k,l)H

,

Fourier transform: m(k,l)eH

= |b|k−lm(l,k)H

, M(k,l)eH

= |b|k−lM(l,k)H

,

Chirp multiplication: m(k,0)eH

= m(k,0)H

, M(k,0)eH

= M(k,0)H

,

Chirp convolution: m(0,l)eH

= m(0,l)H

, M(0,l)eH

= M(0,l)H

.

Minimization of τ2Hν2H. The following interesting result yields a closed-form solution for the min-

imum of the product τ eHν eH

over all metaplectic operators. Its derivation is based on the determinant

η2H

of the displacement spread matrix D(T )H

.

Theorem 2.6. The minimum of τ eHν eH

, where H = UHU+, over all metaplectic operators U ∈ Mis equal to

infU∈M

{τ eHν eH

} =

√detD

(T )H

= ηH


and is achieved by the following systems (among others, in particular all their TF scaled and Fourier

transformed versions):

• H1 = F (T )θ0

HF (T )+θ0

, where F (T )θ0

= µ(R(T )θ0

) is the fractional Fourier transform operator corre-

sponding to the rotation matrix R(T )θ (cf. Appendix C) with the angle θ0 given by

θ0 =1

2arctan

2κH

T 2ν2H− τ2

H

T 2

.

• H2 = CcHC+c , where Cc is the chirp multiplication operator (cf. Appendix C) corresponding to

Cc =(

1 0c 1

)with chirp rate c = κH

τ2H

.

• H3 = BbHB+b , where Bb is the chirp convolution operator (cf. Appendix C) corresponding to

Bb =(

1 b0 1

)with chirp rate 1/b =

ν2H

κH.

Proof. The proof of this theorem is based on the fact that, as expressed by (2.26), the determinant

of the displacement spread matrix is invariant to area-preserving linear coordinate transforms. Eq.

(2.26) can be rewritten as η2eH

= τ2eHν2

eH− κ2

eH= η2

Hfrom which τ eH

ν eH=√η2H

+ κ2eH. For fixed H, this

expression is obviously minimized by minimizing |κeH|. In fact, since there are three free parameters

in (2.25) (one of the four parameters a, b, c, and d is related to the other three by the condition

detA = 1), it is possible to choose two parameters arbitrarily and solve for the third free parameter

such that κ2eH

= 0. Instead of giving the general solution (which actually consists of all matrices

diagonalizing D(T )H

), we discuss three important special solutions that achieve κ2eH

= 0.

First, we restrict the minimization to rotational transforms of the type R(T )θ =

( cos θ −T 2 sin θ(sin θ)/T 2 cos θ

)

with θ, the rotation angle, now being the only parameter (apart from the normalization constant T ).

The corresponding metaplectic representation U is given by the fractional Fourier transform operator.

With the assumed form of A, κeHcan be simplified as follows (cf. (2.25)):

κeH= sin θ cos θ

(τ2H

T 2− T 2ν2

H

)+ (cos2 θ − sin2 θ)κH

=1

2

(τ2H

T 2− T 2ν2

H

)sin 2θ + κH cos 2θ

=

[1

4

(τ2H

T 2− T 2ν2

H

)2

+ κ2H

]1/2

sin

(2θ + arctan

2κH

τ2H

T 2 − T 2ν2H

).

It is thus seen that κeH= 0 can be achieved via a rotation of the GSF by an angle θ0 =

−12 arctan 2κH

τ2H

/T 2−T 2ν2H

, which solves the minimization of τ eHν eH

. Note that this rotation, up to the

normalization constant T , is an orthogonal transform that diagonalizes D(T )H

, thus corresponding to

the eigenvector matrix of D(T )H

.

Alternatively, the minimization of κeHcan be performed using chirp multiplications or chirp con-

volutions. It is seen from (2.25) that κeH= 0 can be achieved by a chirp multiplication by choosing

a = d = 1, b = 0, c = κH/τ2H

or by a chirp convolution by setting a = d = 1, c = 0, b = κH/ν2H

, thus


yielding further systems with minimal τ eHν eH

. Finally, since TF scalings and Fourier transforms do

not affect τ eHν eH

, any such modification of a system with minimal τ eHν eH

yields again a system with

minimal τ eHν eH

.

Hence, any system H can be unitarily transformed such that the resulting system H = UHU+

has κeH= 0 and τ eH

ν eHachieves the lower bound ηH.

2.2.4 Underspread Systems

The GSF integrals m(φ)H

and M(φ)H

and, specifically, the GSF moments m(k,l)H

and M(k,l)H

measure

the spread of |SH(τ, ν)| about the origin of the (τ, ν) plane. Hence, without being forced to assume

that the GSF has finite support, we can consider a system H to be underspread if suitable GSF

integrals/moments are “small.” LTV systems that are not underspread are called overspread. We

note that this is not a clear-cut definition of underspread systems. Indeed, we will see in Section

2.3 that bounds on various error quantities associated with specific transfer function approximations

can be formulated using different GSF integrals/moments. Hence, in order that these approximation

errors be small, different GSF integrals/moments are required to be small. This is somewhat similar

to the many different quantities that can be used to measure the effective bandwidth of a signal.

Our concept of underspread systems is thus more complicated but also more flexible than Kozek’s

definition that was based on the area of the (assumedly) compact support of the GSF. It is precisely

this flexibility which, in Section 2.3, will allow us to establish error bounds for the transfer function

approximation without unnecessarily restrictive assumptions. To be specific, several of the following

conditions satisfied by underspread systems will be of importance in Section 2.3:

m(1,1)H

≪ 1, M(1,1)H

≪ 1,

m(1,0)H

m(0,1)H

≪ 1, τHνH = M(1,0)H

M(0,1)H

≪ 1, (2.27)

m(φs)H

≪ 1, M(φs)H

≪ 1,

where in the last line either φs(τ, ν) = |1 − A(0)s (τ, ν)| or φs(τ, ν) =

∣∣∣1 − 1

A(0)s (τ,ν)

∣∣∣. Examples for

underspread systems satisfying the above constraints are illustrated in Fig. 2.3. It should be noted

that the concept of underspread systems is not equivalent to that of slowly time-varying (quasi-LTI)

systems which requires |SH(τ, ν)| to be narrow with respect to ν (i.e., small moments m(0,l)H

and

M(0,l)H

, see part (d) of Fig. 2.3). In particular, a slowly time-varying system may be overspread

(i.e., not underspread) if its memory is very long, while a system with fast time-variations may be

underspread if its memory is short enough.

The conditions in the first two lines of (2.27) do not allow the GSF to be oriented in oblique

directions (for the weighted GSF integrals in the last line, this depends on the choice of s(t)). Yet,

in some cases (especially when deriving transfer function approximations for the case α = 0), oblique

orientations of the GSF will be allowed. Often specific moments of a system H are rather large whereas

the same moments of a unitarily equivalent system UHU+ with U ∈ M are small. Examples of such

systems include systems having GSF oriented along oblique directions (see Fig. 2.3(e)). To cover these


(a) (b) (c) (d) (e)

τ τ τ τ τ

ν ν ν ν ν

Figure 2.3: Schematic representation of the GSF magnitude of (a) an underspread system with

small m(1,1)H

and M(1,1)H

; (b) an underspread system with small m(1,0)H

m(0,1)H

and M(1,0)H

M(0,1)H

; (c) an

underspread system with small m(1,0)UHU+m

(0,1)UHU+ for U corresponding to a rotation; (d) a quasi-LTI

system (small m(0,l)H

and M(0,l)H

); (e) a quasi-LFI system (small m(k,0)H

and M(k,0)H

).

cases in a satisfactory way, only the minima of certain moment quantities among all systems related

by metaplectic transforms may be required to be small, e.g.,

infU∈M

{m

(1,0)UHU+m

(0,1)UHU+

}≪ 1, inf

U∈M

{τUHU+νUHU+

}≪ 1. (2.28)

Here, an appropriate U ∈ M produces a coordinate transform such that the oblique orientation of

|SH(τ, ν)| is converted into an orientation of |SUHU+(τ, ν)| along the τ axis and/or ν axis (cf. Theorem

2.6).

2.2.5 Operator Sums, Adjoints, Products, and Inverses

We conclude our discussion of operators with rapidly decaying GSF by considering the weighted GSF

integrals and moments of sums, adjoints, products, and inverses of such operators.

Sum. Using the inequality

|SH1+H2(τ, ν)| = |SH1(τ, ν) + SH2(τ, ν)| ≤ |SH1(τ, ν)| + |SH2(τ, ν)| ,

one straightforwardly obtains

m(φ)H1+H2

≤ ‖SH1‖1

‖SH1 + SH2‖1m

(φ)H1

+‖SH2‖1

‖SH1 + SH2‖1m

(φ)H2

and, as a special case,

m(k,l)H1+H2

≤ ‖SH1‖1

‖SH1 + SH2‖1m

(k,l)H1

+‖SH2‖1

‖SH1 + SH2‖1m

(k,l)H2

. (2.29)

Using the fact that |SH1+H2(τ, ν)|2 ≤ 2|SH1(τ, ν)|2 + 2|SH2(τ, ν)|2 (due to the Schwarz inequality

| 〈x, y〉 |2 ≤ ‖x‖2‖y‖2 with x = (SH1(τ, ν) SH2(τ, ν))T and y = (1 1)T ) and the inequality

√a2 + b2 ≤

a+ b (valid for a ≥ 0, b ≥ 0), one can furthermore show that

M(φ)H1+H2

≤√

2‖H1‖2

‖H1 + H2‖2M

(φ)H1

+

√2‖H2‖2

‖H1 + H2‖2M

(φ)H2


and

M(k,l)H1+H2

≤√

2‖H1‖2

‖H1 + H2‖2M

(k,l)H1

+

√2‖H2‖2

‖H1 + H2‖2M

(k,l)H2

.

These bounds imply that the sum of two underspread operators will again be underspread. Further-

more, if the norms ‖SH1‖1 and ‖SH2‖1 or the norms ‖H1‖2 and ‖H2‖2 are not similar, then the

moments of the operator sum tend to be determined by the moments of the operator with the larger

norm. Therefore, also the sum of two operators where only one is underspread may be underspread if

the underspread operator dominates the other one in an appropriate sense. This property is in striking

constrast to DL operators (cf. Subsection 2.1.4).

We also generalize the concept of jointly underspread systems, introduced for DL operators in

Definition 2.3, using a moment-based approach. In particular, we call two systems H1 and H2 jointly

underspread if both systems are individually underspread and if in addition also their sum H1 +H2 is

underspread. With regard to the underspread condition m(1,0)H1+H2

m(0,1)H1+H2

≪ 1, it follows from (2.29)

that

m(1,0)H1+H2

m(0,1)H1+H2

≤( ‖SH1‖1

‖SH1 + SH2‖1m

(1,0)H1

+‖SH2‖1

‖SH1 + SH2‖1m

(1,0)H2

)

·( ‖SH1‖1

‖SH1 + SH2‖1m

(0,1)H1

+‖SH2‖1

‖SH1 + SH2‖1m

(0,1)H2

)

=‖SH1‖2

1

‖SH1 + SH2‖21

m(1,0)H1

m(0,1)H1

+‖SH2‖2

1

‖SH1 + SH2‖21

m(1,0)H2

m(0,1)H2

+‖SH1‖1 ‖SH2‖1

‖SH1 + SH2‖21

(m

(1,0)H1

m(0,1)H2

+m(1,0)H2

m(0,1)H1

).

Thus it is seen that in addition to m(1,0)H1

m(0,1)H1

≪ 1 and m(1,0)H2

m(0,1)H2

≪ 1, jointly underspread systems

have to satisfy the condition

m(1,0)H1

m(0,1)H2

+m(1,0)H2

m(0,1)H1

≪ 1 . (2.30)

This latter requirement will become particularly important in Subsection 2.3.4. In essence, the concept

of jointly underspread implies that, in addition to the systems being individually underspread, their

memory lengths and amounts of temporal variation are respectively comparable. For example, an LTI

system with long memory (i.e., large m(1,0)HLTI

) and an LFI system with rapidly varying multiplication

function (i.e., large m(0,1)HLFI

), although being individually underspread, are not jointly underspread since

m(1,0)HLTI

m(0,1)HLFI

will be large in that case.

Adjoint. With∣∣S(0)

H+(τ, ν)∣∣ =

∣∣S(0)H

(−τ,−ν)∣∣, we obtain

m(φ)H+ = m

(φ)H, M

(φ)H+ = M

(φ)H

,

with φ(τ, ν) = φ(−τ,−ν), which further specializes to

m(k,l)H+ = m

(k,l)H

, M(k,l)H+ = M

(k,l)H

.

Product. The GSF of the product of two operators is given by the so-called twisted convolution

(cf. Eq. (B.12) in Section B.1.1) which is difficult to analyze in general. However, a simple consequence


of (B.12) is

|SH2H1(τ, ν)| ≤ |SH1(τ, ν)| ∗∗ |SH2(τ, ν)| , (2.31)

with ∗∗ denoting 2-D convolution. Unfortunately, this bound is often very loose. Simply think of a

unitary system U and its adjoint U+ which can have arbitrarily large TF shifts. In this case, UU+ = I

whose GSF is perfectly concentrated at the origin of the (τ, ν) plane although the above inequality

suggests that the the support of S(α)H2H1

(τ, ν) may be larger than the support of S(α)U

(τ, ν) or that of

S(α)U+(τ, ν).

Using (2.31) we can derive bounds on the moments of the system H2H1. With the normalization

m(k,l)H

= m(k,l)H

/(k! l!), we arrive at the following

Proposition 2.7. The normalized moments of the product H2H1 are bounded by the 2-D convolution

of the normalized moments of H1 and H2 up to order k and l,

m(k,l)H2H1

≤ ‖SH2‖1 ‖SH1‖1

‖SH2H1‖1

k∑

i=0

l∑

j=0

m(i,j)H1

m(k−i,l−j)H2

. (2.32)

Proof. Inserting (2.31) into the moment definition (2.13-a), one has

‖SH2H1‖1m(k,l)H2H1

≤∫

τ

∫

ν

[∫

τ1

∫

ν1

|SH1(τ1, ν1)| |SH2(τ − τ1, ν − ν1)| dτ1 dν1

]|τ |k |ν|l dτ dν

=

∫

τ1

∫

ν1

∫

τ2

∫

ν2

|SH1(τ1, ν1)| |SH2(τ2, ν2)| |τ1 + τ2|k |ν1 + ν2|l dτ1 dν1 dτ2 dν2

≤∫

τ1

∫

ν1

∫

τ2

∫

ν2

|SH1(τ1, ν1)| |SH2(τ2, ν2)|[

k∑

i=0

(k

i

)|τ1|i |τ2|k−i

]

·

l∑

j=0

(l

j

)|ν1|j |ν2|l−j

dτ1 dν1 dτ2 dν2

=

k∑

i=0

l∑

j=0

k! l!

i! j! (k − i)! (l − j)!

∫

τ1

∫

ν1

|SH1(τ1, ν1)| |τ1|i |ν1|j dτ1 dν1

·∫

τ2

∫

ν2

|SH2(τ2, ν2)| |τ2|k−i |ν2|l−j dτ2 dν2

=

k∑

i=0

l∑

j=0

k! l!

i! j! (k − i)! (l − j)!‖SH1‖1m

(i,j)H1

‖SH2‖1m(k−i,l−j)H2

= ‖SH1‖1‖SH2‖1 k! l!k∑

i=0

l∑

j=0

m(i,j)H1

i! j!

m(k−i,l−j)H2

(k − i)! (l − j)!,

which finally gives (2.32).

Important special cases of this result are obtained for (k, l) = (1, 0), (k, l) = (0, 1), and (k, l) =

(1, 1):

m(1,0)H2H1

≤ ‖SH2‖1 ‖SH1‖1

‖SH2H1‖1

[m

(1,0)H1

+m(1,0)H2

], (2.33)


m(0,1)H2H1

≤ ‖SH2‖1 ‖SH1‖1

‖SH2H1‖1

[m

(0,1)H1

+m(0,1)H2

], (2.34)

m(1,1)H2H1

≤ ‖SH2‖1 ‖SH1‖1

‖SH2H1‖1

[m

(1,1)H1

+m(0,1)H1

m(1,0)H2

+m(1,0)H1

m(0,1)H2

+m(1,1)H2

]. (2.35)

Inverse. In general, it is difficult to characterize the moments of the inverse of an operator. From

the Neumann series [158]

H−1 =

∞∑

k=0

(I − H)k , (2.36)

it is seen that, due to the higher powers of (I − H), it is possible that the support of the GSF of

H−1 will grow ad infinitum. Yet, under sufficient regularity conditions the inverse of an underspread

operator can also be underspread in the sense of having small moments. Inverses of underspread

operators are further discussed in Subsection 2.3.6.

2.2.6 Non-Band-Limited Parts of Operators with Rapidly Decaying SpreadingFunction

In this section, we will relate DL operators and operators with rapidly decaying GSF by studying

the norms of the non-DL part of the latter. To this end, recall the splitup (2.4) of any operator H

into its DL part HG and its non-DL part HG . The next few results derive bounds on the norms of

SHG (τ, ν) for regions G of different shape using a Chebyshev inequality-like approach4. The bounds

on ‖SHG‖1 =

∫∫G |SH(τ, ν)| dτ dν and ‖HG‖2

2 = ‖SHG‖2

2 =∫∫

G |SH(τ, ν)|2 dτ dν will be important in

Section 2.3. Furthermore, they serve as a measure of the errors made when replacing H by its DL

part HG . Indeed, the difference L(α)H

(t, f) − L(α)

HG (t, f) satisfies

∣∣L(α)H

(t, f) − L(α)

HG (t, f)∣∣

‖SH‖1=

∣∣L(α)

HG(t, f)

∣∣‖SH‖1

≤ ‖SHG‖1

‖SH‖1,

∥∥L(α)H

− L(α)

HG

∥∥2

‖SH‖2=

∥∥HG∥∥

2

‖H‖2. (2.37)

Furthermore, using the definition (A.1) of the operator norm ‖H‖O and the fact that ‖H‖O ≤ ‖H‖2,

the L2 norm of the difference (Hx)(t) − (HGx)(t) can be bounded as

‖Hx− HGx‖2

‖H‖2 ‖x‖2=

‖HGx‖2

‖H‖2 ‖x‖2≤ ‖HG‖O‖x‖2

‖H‖2 ‖x‖2≤∥∥HG

∥∥2

‖H‖2.

The first result considers rectangular support constraints (see Fig. 2.1(a)).

Proposition 2.8. For any rectangular region G , [−τG , τG ] × [−νG , νG ], the non-DL part HG of any

operator H is bounded as

‖SHG‖1

‖SH‖1≤ m

(1,0)H

τG+m

(0,1)H

νG,

‖HG‖2

‖H‖2≤[(

M(1,0)H

τG

)2+

(M

(0,1)H

νG

)2 ]1/2

≤ M(1,0)H

τG+M

(0,1)H

νG. (2.38)

4We are grateful to Prof. W. Mecklenbrauker for drawing our attention to this method.


-1

1

I|τ |≥τG(τ, ν)

ττG

τ 2

τ 2G

|τ |τG

0 1

Figure 2.4: Illustration of Chebyshev-like inequalities.

Proof. The basis for the proof of the above bounds is given by the inequalities

I|τ |≥τG (τ, ν) ≤ |τ |τG

, I|ν|≥νG(τ, ν) ≤ |ν|νG

, (2.39)

I|τ |≥τG (τ, ν) ≤ τ2

τ2G

, I|ν|≥νG(τ, ν) ≤ ν2

ν2G

, (2.40)

which are illustrated in Fig. 2.4. Using (2.39), the first bound in (2.38) is shown as follows:

‖SHG‖1

=

∫∫

G

|SH(τ, ν)| dτ dν =

∫∫

|τ |≥τG|ν|≥νG

|SH(τ, ν)| dτ dν

≤∫∫

|τ |≥τG

|SH(τ, ν)| dτ dν +

∫∫

|ν|≥νG


≤∫

τ

∫

ν

|τ |τG

|SH(τ, ν)| dτ dν +

∫

τ

∫

ν

|ν|νG


= ‖SH‖1

[m

(1,0)H

τG+m

(0,1)H

νG

]. (2.41)

The HS norm bound can be shown similarly using (2.40),

‖HG‖2

2 = ‖SHG‖2

2=

∫∫

G

|SH(τ, ν)|2 dτ dν

≤∫∫

|τ |≥τG

|SH(τ, ν)|2 dτ dν +

∫∫

|ν|≥νG


≤∫

τ

∫

ν

τ2

τ2G

|SH(τ, ν)|2 dτ dν +

∫

ν

∫

ν

ν2

ν2G


= ‖H‖22

[(M

(1,0)H

τG

)2+

(M

(0,1)H

νG

)2 ].

The looser bound in (2.38) follows from the inequality x2 + y2 ≤ (x+ y)2 for x, y ≥ 0.

2.3 Underspread Approximations 37

Next, we consider regions G with hyperbolic boundaries (see Fig. 2.1(b)).

Proposition 2.9. For any region G , {(τ, ν) : |τν| ≤ µG}, the non-DL part HG of any operator H is

bounded as‖S

HG‖1

‖SH‖1≤ m

(1,1)H

µG,

‖HG‖2

‖H‖2≤ M

(1,1)H

µG. (2.42)

Proof. The integral of |SH(τ, ν)| over G can be written as∫τ

∫ν |SH(τ, ν)| IG(τ, ν) dτ dν with IG(τ, ν)

the indicator function of G. Using a Chebyshev inequality-like approach, we can bound this indicator

function by the weighting function |τν|/µG , i.e.,5 IG(τ, ν) ≤ |τν|/µG , and obtain

‖SHG‖1

=

∫∫

G

|SH(τ, ν)| dτ dν =

∫

τ

∫

ν|SH(τ, ν)| IG(τ, ν) dτ dν

≤ 1

µG

∫

τ

∫

ν|SH(τ, ν)| |τν| dτ dν = ‖SH‖1

m(1,1)H

µG.

The HS norm bound follows similarly by noting that furthermore IG(τ, ν) ≤ τ2ν2/µ2G .

2.3 Underspread Approximations

In this section, we show that for LTV systems that are underspread in the extended sense of Section 2.2,

the GWS L(α)H

(t, f) is an approximate “TF transfer function” that generalizes the spectral (temporal)

transfer function of LTI (LFI) systems. As a mathematical underpinning of this approximate TF

transfer function calculus, we establish explicit upper bounds6 on the approximation errors associated

with it. These bounds are formulated in terms of the GSF integrals and/or moments introduced in

Subsection 2.2.2 and do not require the GSF to have finite support. Hence, our subsequent results will

show that a GWS-based transfer function calculus is valid for a significantly wider class of systems

than that considered previously [118–120].

2.3.1 Approximate Uniqueness of the Generalized Weyl Symbol

In general, two GWSs with different α values will yield different results. For example, the Weyl symbol

(α = 0) will be different from Zadeh’s function (α = 1/2). The interrelation of the individual members

of the GWS family is given by (B.21) with the dual interrelation of the corresponding members of

the GSF family given by (B.5). This situation is different from the LTI and LFI cases (where the

transfer function of a given system is uniquely defined) and may be considered an inconvenience since

in practice it may often not be clear which α value should be selected. Fortunately, the subsequently

derived bounds on the GWS’s α-dependence show that in the underspread case the choice of α is not

critical. These bounds extend existing bounds of Kozek [118]. Furthermore, results in a similar spirit5That IG(τ, ν) ≤ |τν|/µG is seen as follows: for (τ, ν) ∈ G one has |τν|/µG ≥ 0 = IG(τ, ν), and for (τ, ν) 6∈ G, i.e.,

|τν| > µG , one has |τν|/µG ≥ 1 = IG(τ, ν).6We note that in some situations our error bounds may be rather coarse. However, in practical situations they are

still useful as there are often no other ways to assess the accuracy of specific transfer function approximations.


have been derived by Kohn and Nirenberg [112] for the special cases α = ±1/2 and by Folland [64]

for the difference between Weyl symbol and Zadeh’s time-varying transfer function.

Theorem 2.10. For any LTV system H, the difference

∆(α1,α2)1 (t, f) , L

(α1)H

(t, f) − L(α2)H

(t, f)

between two GWSs with parameters α1 and α2 is bounded as

∣∣∆(α1,α2)1 (t, f)

∣∣‖SH‖1

≤ 2π|α1 − α2|m(1,1)H

,

∥∥∆(α1,α2)1

∥∥2

‖H‖2

≤ 2π|α1 − α2|M (1,1)H

. (2.43)

Proof. With (B.20) and (B.5), the 2-D Fourier transform of ∆(α1,α2)1 (t, f) is given by

∆(α1,α2)1 (τ, ν) =

∫

t

∫

f∆

(α1,α2)1 (t, f) e−j2π(νt−τf) dt df = S

(α1)H

(τ, ν)[1 − ej2π(α1−α2)τν

].

It is thus seen that this difference is essentially determined by the deviation of e−j2π(α1−α2)τν from 1

which is obviously small for (τ, ν) values around the origin. The first bound (L∞ bound) in (2.43) is

then shown as

∣∣∆(α1,α2)1 (t, f)

∣∣ =

∣∣∣∣∫

τ

∫

ν∆

(α1,α2)1 (τ, ν) ej2π(tν−fτ) dτ dν

∣∣∣∣ ≤∫

τ

∫

ν

∣∣∆(α1,α2)1 (τ, ν)

∣∣ dτ dν

=

∫

τ

∫

ν|SH(τ, ν)|

∣∣∣1 − ej2π(α1−α2)τν∣∣∣ dτ dν

= 2

∫

τ

∫

ν|SH(τ, ν)| |sin(π(α1 − α2)τν)| dτ dν (2.44)

≤ 2π|α1 − α2|∫

τ

∫

ν|SH(τ, ν)| |τ | |ν| dτ dν = 2π|α1 − α2| ‖SH‖1m

(1,1)H

,

where we used | sinx| ≤ |x|. The second bound (L2 bound) is shown similarly,

∥∥∆(α1,α2)1

∥∥2

2=∥∥∆(α1,α2)

1

∥∥2

2=

∫

τ

∫

ν|SH(τ, ν)|2

∣∣∣1 − ej2π(α1−α2)τν∣∣∣2dτ dν

= 4

∫

τ

∫

ν|SH(τ, ν)|2 sin2(π(α1 − α2)τν) dτ dν

≤ 4π2(α1 − α2)2

∫

τ

∫

ν|SH(τ, ν)|2τ2ν2dτ dν = 4π2(α1 − α2)

2‖H‖22

[M

(1,1)H

]2.

Discussion. The bounds (2.43) depend only on the difference of α1 and α2 and approach zero

with decreasing α1 − α2, thus correctly reflecting the obvious fact that for α1 = α2 the difference

∆(α1,α2)1 (t, f) becomes zero. From the proof of the above theorem, it is also seen that ∆

(α1,α2)1 (t, f)

vanishes for systems whose GSF is perfectly localized along the hyperbolae |τν| = k|α1−α2|

with integer

k, since here sin(π(α1 −α2)τν) = 0. A specific class of such systems (corresponding to k = 0) is given

by any superposition of LTI and LFI systems with impulse response h(t, t′) = g(t− t′) +m(t) δ(t− t′)

and GSF located on the τ and ν axis. The GWS of such systems is given by L(α)H

(t, f) = G(f) +m(t)

and is therefore independent of α.


In the case that the bounds (2.43) are small, which calls for small moments m(1,1)H

and M(1,1)H

, it

is seen that two GWSs obtained with different parameters α1 and α2 are approximately equal,

L(α1)H

(t, f) ≈ L(α2)H

(t, f) .

The GWS is thus approximately independent of α and can therefore be considered as an (approx-

imately) unique TF transfer function. Small m(1,1)H

and M(1,1)H

in particular require H to be an

underspread system with GSF concentrated along the τ and/or ν axis. Examples of such systems

include quasi-LTI systems (i.e., systems with slow time-variations), quasi-LFI systems (i.e., systems

with short memory), and any parallel connection (superposition) thereof. On the other hand, the

moments m(1,1)H

and M(1,1)H

may be large for GSF oriented in oblique directions. This is consistent

with the known fact that the Weyl symbol, i.e. the GWS with α = 0, has unique properties regarding

systems having obliquely oriented GSF.

DL Operators. For the special case of DL operators as defined in Section 2.1, using (2.18) in (2.43)

directly yields the bound 2π|α1−α2|µH for both the normalized L∞ and L2 norm of ∆(α1,α2)1 . However,

a slight refinement of the proof of Theorem 2.10 that uses the fact that max(τ,ν)∈GH| sin(π(α1 −

α2)τν)| ≤ sin(π|α1 − α2|µH) (valid for 2|α1 − α2|µH ≤ 1) yields the tighter bounds

∣∣∆(α1,α2)1 (t, f)

∣∣‖SH‖1

≤ 2 sin(π|α1 − α2|µH),

∥∥∆(α1,α2)1

∥∥2

‖H‖2

≤ 2 sin(π|α1 − α2|µH), (2.45)

which are valid for DL operators satisfying7 2|α1 − α2|µH ≤ 1. These bounds may be compared to

the following bounds obtained in [118] for DL operators with rectangular GSF support,

∣∣∆(α1,α2)1 (t, f)

∣∣‖SH‖1

≤ 2 sin(π|α1 − α2|

σH

4

),

∥∥∆(α1,α2)1

∥∥2

‖H‖2

≤ 2 sin(π|α1 − α2|

σH

4

),

which are valid for |α1 − α2|σH ≤ 2. From this comparison, it is seen that our bounds are tighter

(since µH ≤ σH/4 according to Subsection 2.1.2) and valid under more general conditions (e.g., for

superpositions of quasi-LTI and quasi-LFI systems for which σH may be infinite).

Non-DL Operators. We will now consider the case that one erroneously assumes that the system

under analysis is DL with presumed GSF support region G, while actually the operator is not DL but

rather has a rapidly decaying GSF. Specifically, let us consider the case where the system’s GSF is

effectively (but not exactly) contained within G = {(τ, ν) : |τν| ≤ µG}8. In that case, neglecting the

GSF contributions outside of G, (2.45) would suggest that the L∞ and L2 norms of the difference

∆(α1,α2)1 (t, f) are (approximately) less than 2π sin(π|α1 − α2|µG). Of course, this approximate bound

might be wrong, i.e., too small. The following result presents correct bounds that also show by how

much one might be wrong when erroneously using the bounds (2.45).

7This condition is not very restrictive since the error bounds in (2.45) will typically be used only for small µH.8This effective support region G could be determined by thresholding the GSF, i.e., G = {(τ, ν) : |SH(τ, ν)| ≥ ǫ}, so

that µG = max(τ,ν):|SH(τ,ν)|≥ǫ |τν|.


Proposition 2.11. For any LTV system H and any µG such that 2|α1 − α2|µG ≤ 1, the difference

∆(α1,α2)1 (t, f) = L

(α1)H

(t, f) − L(α2)H

(t, f) is bounded as

∣∣∆(α1,α2)1 (t, f)

∣∣‖SH‖1

≤ 2 sin(π|α1 − α2|µG) + 2m

(1,1)H

µG, (2.46)

∥∥∆(α1,α2)1

∥∥2

‖H‖2

≤ 2

[sin2(π|α1 − α2|µG) +

(M

(1,1)H

µG

)2]1/2

≤ 2 sin(π|α1 − α2|µG) + 2M

(1,1)H

µG. (2.47)

Proof. Consider the region G = {(τ, ν) : |τν| ≤ µG}. Starting from (2.44) and splitting the integral

yields

∣∣∆(α1,α2)1 (t, f)

∣∣ ≤ 2

∫∫

G

|SH(τ, ν)| |sin(π(α1 − α2)τν)| dτ dν + 2

∫∫

G

|SH(τ, ν)| |sin(π(α1 − α2)τν)| dτ dν

≤ 2 |sin(π(α1 − α2)µG)|∫∫

G

|SH(τ, ν)| dτ dν + 2

∫∫

G


≤ 2‖SH‖1 sin(π|α1 − α2|µG) + 2‖SHG‖1

,

where the first term in the bound holds for 2|α1 − α2|µG ≤ 1. The second term is bounded according

to (2.42), which finally yields (2.46). The proof of (2.47) is analogous.

Thus, for non-DL operators, the bounds (2.45) with an arbitrarily chosen but in any case incorrect

µG might deviate from the correct bounds (2.46) and (2.47) by as much as 2m(1,1)H

/µG and 2M(1,1)H

/µG ,

respectively.

2.3.2 The Generalized Weyl Symbol of Operator Adjoints

The GWS of the adjoint operator H+ is given by (B.23), i.e., L(α)H+(t, f) = L

(−α)∗H

(t, f), and thus is

generally not equal to the complex conjugate of the GWS of H (this is only true for α = 0, i.e.,

L(0)H+(t, f) = L

(0)∗H

(t, f)). This is a difference from the LTI and LFI cases where the transfer functions

of the adjoint of a system H can be obtained by complex conjugation of the transfer function of

H. Yet, the subsequently presented results will show that for an underspread system H the complex

conjugate of its GWS is a good approximation to the GWS of H+. Our bounds on the resulting

approximation error are straightforward consequences of the approximate α-invariance of the GWS as

discussed in the preceding subsection. For the special case α = −1/2, a result in a similar spirit has

been derived by Kohn and Nirenberg [112].

Corollary 2.12. For any LTV system H, the difference

∆(α)2 (t, f) , L

(α)H+(t, f) − L

(α)∗H

(t, f)

is bounded as ∣∣∆(α)2 (t, f)

∣∣‖SH‖1

≤ 4π|α|m(1,1)H

,

∥∥∆(α)2

∥∥2

‖H‖2

≤ 4π|α|M (1,1)H

. (2.48)


Proof. With L(α)H+(t, f) = L

(−α)∗H

(t, f), we have ∆(α)2 (t, f) = L

(−α)∗H

(t, f) − L(α)∗H

(t, f) = ∆(−α,α)∗1 (t, f).

Hence, the above bounds follow straightforwardly by applying the bounds (2.43) with α1 = −α and

α2 = α.

Discussion. The bounds in Corollary 2.12 correctly reflect that ∆(α)2 (t, f) = 0 for α = 0. Further-

more, for an arbitrary parallel connection of LTI and LFI systems, with GWS L(α)H

(t, f) = G(f)+m(t),

there is ∆(α)2 (t, f) = 0 since here m

(1,1)H

= 0. More generally, for an underspread system whose GSF

is concentrated along the τ and ν axes so that m(1,1)H

and M(1,1)H

are small, the bounds show that

the GWS of the adjoint of a system can approximately be obtained by complex conjugation of the

system’s GWS,

L(α)H+(t, f) ≈ L

(α)∗H

(t, f) . (2.49)

Due to the similar bounds in Corollary 2.12 and Theorem 2.10, the above approximation is valid

in the same situations in which the GWS is approximately α-independent, e.g. for (superpositions

of) quasi-LTI and quasi-LFI systems. On the other hand, (2.49) is not valid for systems with GSF

oriented in oblique directions. For such systems one has thus to resort to the case α = 0 where

L(0)H+(t, f) = L

(0)∗H

(t, f).

DL Operators. For the special case of DL operators, setting α1 = −α and α2 = α in (2.45), one

obtains

∣∣∆(α)2 (t, f)

∣∣‖SH‖1

≤ 2 sin(2π|α|µH) ≤ 4π|α|µH,

∥∥∆(α)2

∥∥2

‖H‖2

≤ 2 sin(2π|α|µH) ≤ 4π|α|µH, (2.50)

where the tighter bounds involving the sine term are valid only for 4|α|µH ≤ 1.

Non-DL Operators. Similarly to the results (2.46) and (2.47) derived in connection with the

uniqueness of the GWS, it can be shown that for any LTV system H and any µG such that 4|α|µG ≤ 1,

one has

∣∣∆(α)2 (t, f)

∣∣‖SH‖1

≤ 2 sin(2π|α|µG) + 2m

(1,1)H

µG, (2.51)

∥∥∆(α)2

∥∥2

‖H‖2

≤ 2

[sin2(2π|α|µG) +

(M

(1,1)H

µG

)2]1/2

≤ 2 sin(2π|α|µG) + 2M

(1,1)H

µG. (2.52)

Thus, in the case of non-DL operators, the deviations of the bounds (2.50) from the correct bounds

(2.51) and (2.52) can be as large as 2m(1,1)H

/µG and 2M(1,1)H

/µG , respectively.

2.3.3 Approximate Real-Valuedness of the Generalized Weyl Symbol

A simple but important consequence of Corollary 2.12 concerns self-adjoint operators, i.e., operators

satisfying H = H+. Unlike the transfer functions of LTI and LFI systems, the GWS of such operators

is generally not real-valued unless α = 0. Fortunately, Corollary 2.12 implies that for underspread

self-adjoint operators, the imaginary part of the GWS is negligible.


Corollary 2.13. For a self-adjoint operator H, the imaginary part of its GWS, ℑ{L

(α)H

(t, f)}

=12j

[L

(α)H

(t, f) − L(α)∗H

(t, f)], is bounded as

∣∣ℑ{L

(α)H

(t, f)}∣∣

‖SH‖1

≤ 2π|α|m(1,1)H

,

∥∥ℑ{L

(α)H

}∥∥2

‖H‖2

≤ 2π|α|M (1,1)H

. (2.53)

Proof. The above bounds are a straightforward consequence of Corollary 2.12 since for a self-adjoint

operator ℑ{L

(α)H

(t, f)}

= 12j

[L

(α)H

(t, f) − L(α)∗H

(t, f)]

= 12j

[L

(α)H+(t, f) − L

(α)∗H

(t, f)]= 1

2j ∆(α)2 (t, f).

Discussion. As a consequence of the foregoing corollary, self-adjoint operators having their GSF

localized along the τ and/or ν axis, i.e., underspread operators with small m(1,1)H

and M(1,1)H

, have

approximately real-valued GWSs,

ℑ{L

(α)H

(t, f)}≈ 0, or L

(α)H

(t, f) ≈ L(α)∗H

(t, f).

An example illustrating this approximation will be provided in Subsection 2.3.14. Furthermore, this

result is important in the context of time-varying spectral analysis, since it shows that in the un-

derspread case the GWVS in (B.45) is approximately real-valued even for α 6= 0 (see Subsection

3.2.1).

DL Operators. For the special case of DL operators, the following bounds can be obtained from

(2.50),

∣∣ℑ{L

(α)H

(t, f)}∣∣

‖SH‖1

≤ sin(2π|α|µH) ≤ 2π|α|µH,

∥∥ℑ{L

(α)H

}∥∥2

‖H‖2

≤ sin(2π|α|µH) ≤ 2π|α|µH , (2.54)

where the tighter bounds in terms of the sine function are valid for 4|α|µH ≤ 1. Similar to the

previous subsections, the deviations involved in incorrectly using these bounds for non-DL systems

are upper-bounded by m(1,1)H

/µG and M(1,1)H

/µG .

2.3.4 Composition of Systems

One of the most important properties of LTI systems is the fact that the transfer function of a

composition (series connection) of two LTI systems equals the product G1(f)G2(f) of the transfer

functions of the individual systems. Similarly, the temporal transfer function of the composition

of two LFI systems is given by m1(t)m2(t). This composition property of conventional transfer

functions is the cornerstone of many signal processing techniques used in applications like filtering,

estimation, detection, and channel equalization. Unfortunately, contrary to the LTI/LFI case, a

similar composition property no longer holds true for general LTV systems: The GWS of the operator

composition H2H1 can not be obtained by multiplying the individual GWSs of H1 and H2. This

prohibits an exact transfer function based formulation of several signal processing techniques for the

time-varying/nonstationary case. (In the context of quantum mechanics, such a multiplicative symbol

calculus corresponds to ideal quantization [64].) Besides the cases where H1 and H2 are either both

LTI or both LFI (or, for α = 0, metaplectic transformations thereof), for Hilbert-Schmidt (HS)


LTI LTV

LTV LFI

α = 1/2

LTV LTI

LFI LTV

α = −1/2

LTI LTI

LFI LFI

|α| 6= 1/2

Figure 2.5: Situations where L(α)H2H1

(t, f) = L(α)H2

(t, f)L(α)H1

(t, f) with H1 corresponding to the left-hand

system and H2 corresponding to the righ-hand system.

operators the relation L(α)H2H1

(t, f) = L(α)H1

(t, f)L(α)H2

(t, f) is correct only in the following situations (see

Figure 2.5):

• For α = 1/2, if H1 is an LTI system with spectral transfer function G1(f) or H2 is an LFI

system with temporal transfer function m2(t), it can be shown that

L(1/2)H2H1

(t, f) = L(1/2)H1

(t, f)L(1/2)H2

(t, f) =

G1(f)L

(1/2)H2

(t, f) for H1 LTI

L(1/2)H1

(t, f)m2(t) for H2 LFI.(2.55)

• In the dual case α = −1/2, if H1 is LFI with temporal transfer function m1(t) or H2 is LTI with

spectral transfer function G2(f), one obtains

L(−1/2)H2H1

(t, f) = L(−1/2)H1

(t, f)L(−1/2)H2

(t, f) =

m1(t)L

(−1/2)H2

(t, f) for H1 LFI

L(−1/2)H1

(t, f)G2(f) for H2 LTI.(2.56)

Hence, we see that the situations where a multiplicative relation of the GWS holds exactly are

rare. However, the situation is more comforting if one restricts attention to underspread operators. In

the following, it is shown that for underspread systems the desired composition property for the GWS

is approximately valid. The bounds on the approximation error given below generalize and extend

the bounds derived for DL operators in [118–120]. Results in a similar spirit have been obtained in

the context of quantization [54,64,206] and pseudo-differential operators [64,73,94,95,112,206]. Note

that an approximate “product formula” for the GWS of a composition of LTV systems also implies

the approximate commutation of the systems involved (cf. Subsection 2.3.16).

Using the twisted product in the form given by (B.31), it is seen that the error in replacing

L(α)H2H1

(t, f) =(L

(α)H2

#L(α)H1

)(t, f) by L

(α)H2

(t, f)L(α)H1

(t, f) is equal to

∆(α)3 (t, f) , L

(α)H2H1

(t, f) − L(α)H2

(t, f)L(α)H1

(t, f) =∑

k+l>0

ckl

(j2π)k+l

∂k+lL(α)H2

(t, f)

∂tl∂fk

∂k+lL(α)H1

(t, f)

∂tk∂f l,


with ck,l = (α+1/2)k(α−1/2)l/(k! l!). This already shows that this error will be small for sufficiently

smooth GWSs. The following result gives bounds on the magnitude of this error.

Theorem 2.14. For any two LTV systems H1 and H2, the difference

∆(α)3 (t, f) , L

(α)H2H1

(t, f) − L(α)H1

(t, f)L(α)H2

(t, f)

is bounded as

∣∣∆(α)3 (t, f)

∣∣‖SH1‖1‖SH2‖1

≤ 2π B(α)H1,H2

with B(α)H1,H2

,

∣∣∣α+1

2

∣∣∣m(0,1)H1

m(1,0)H2

+∣∣∣α− 1

2

∣∣∣m(1,0)H1

m(0,1)H2

. (2.57)

Proof. The GSF of the operator product H2H1 is given by the twisted convolution (B.12). Hence,

the Fourier transform of ∆(α)3 (t, f) is obtained as

∆(α)3 (τ, ν) =

(S

(α)H2♮ S

(α)H1

)(τ, ν) −

(S

(α)H2

∗∗S(α)H1

)(τ, ν)

=

∫

τ ′

∫

ν′

S(α)H2

(τ ′, ν ′)S(α)H1

(τ − τ ′, ν − ν ′)[e−j2πφα(τ,ν,τ ′,ν′) − 1

]dτ ′ dν ′ ,

with φα(τ, ν, τ ′, ν ′) = (α+ 1/2)τ ′(ν − ν ′) + (α− 1/2)(τ − τ ′)ν ′. Thus, we obtain

∣∣∆(α)3 (t, f)

∣∣ =

∣∣∣∣∫

τ

∫

ν∆

(α)3 (τ, ν) ej2π(tν−fτ) dτ dν

∣∣∣∣ ≤∫

τ

∫

ν

∣∣∆(α)3 (τ, ν)

∣∣ dτ dν

=

∫

τ

∫

ν

∣∣∣∣∫

τ ′

∫

ν′

S(α)H2

(τ ′, ν ′)S(α)H1

(τ − τ ′, ν − ν ′)[e−j2πφα(τ,ν,τ ′,ν′) − 1

]dτ ′dν ′

∣∣∣∣ dτ dν

≤ 2

∫

τ

∫

ν

∫

τ ′

∫

ν′

∣∣SH2(τ′, ν ′)

∣∣ ∣∣SH1(τ − τ ′, ν − ν ′)∣∣ ∣∣ sin

(πφα(τ, ν, τ ′, ν ′)

)∣∣ dτ dν dτ ′dν ′ .

(2.58)

Substituting τ1 = τ − τ ′, ν1 = ν − ν ′ and using | sinx| ≤ |x| yields further

∣∣∆(α)3 (t, f)

∣∣ ≤ 2

∫

τ ′

∫

ν′

∫

τ1

∫

ν1

|SH2(τ′, ν ′)| |SH1(τ1, ν1)| (2.59)

·∣∣∣sin

(π[(α+

1

2

)τ ′ν1 +

(α− 1

2

)τ1ν

′])∣∣∣dτ ′dν ′dτ1 dν1 (2.60)

≤ 2π

∫

τ ′

∫

ν′

∫

τ1

∫

ν1

|SH2(τ′, ν ′)| |SH1(τ1, ν1)|

∣∣∣(α+

1

2

)τ ′ν1 +

(α− 1

2

)τ1ν

′∣∣∣dτ ′dν ′dτ1 dν1

≤ 2π∣∣∣α+

1

2

∣∣∣∫

τ ′

∫

ν′

|τ ′| |SH2(τ′, ν ′)| dτ ′dν ′

∫

τ1

∫

ν1

|ν1| |SH1(τ1, ν1)| dτ1 dν1

+ 2π∣∣∣α− 1

2

∣∣∣∫

τ ′

∫

ν′

|ν ′| |SH2(τ′, ν ′)| dτ ′dν ′

∫

τ1

∫

ν1

|τ1| |SH1(τ1, ν1)| dτ1 dν1 , (2.61)

from which the result (2.57) follows.

Discussion. By virtue of its symmetry with respect to H1 and H2, the bound in (2.57) also

applies to the differences L(α)H1H2

(t, f) − L(α)H1

(t, f)L(α)H2

(t, f) and L(α)H1⋆H2

(t, f) − L(α)H1

(t, f)L(α)H2

(t, f),

where H1 ⋆H2 = [H1H2 + H2H1]/2 is the (commutative) Jordan product [64] of H1 and H2.


(a) (b) (c) (d)t- t- t- t-

f

6

f

6

f

6

f

6

Figure 2.6: Transfer function approximation (with α = 0) for composition of systems: (a) Weyl

symbol of H1, (b) Weyl symbol of H2, (c) Weyl symbol of H2H1, and (d) product of L(0)H1

(t, f) and

L(0)H2

(t, f). The number of samples is 128 and (normalized) frequency ranges from −1/4 to 1/4.

The bound shows that if H1 and H2 are “jointly underspread” such that m(0,1)H1

m(1,0)H2

and

m(1,0)H1

m(0,1)H2

are both small, then we have an approximate multiplicative composition property,

(L

(α)H2

#L(α)H1

)(t, f) = L

(α)H2H1

(t, f) ≈ L(α)H1

(t, f)L(α)H2

(t, f) . (2.62)

In other words, for underspread operators the twisted GWS product is approximately equal to the

pointwise GWS product. An example illustrating the validity of this approximation (with α = 0) is

shown in Fig. 2.6. In this example, the maximum normalized error ismaxt,f |∆

(0)3 (t,f)|

‖SH1‖1‖SH2

‖1

= 0.065 while

the bound in (2.57) is 2πB(0)H1,H2

= 0.084.

In the following, we briefly discuss this approximation for the cases α = ±1/2. The case α = 0

deserves special attention and will be considered in more detail afterwards.

For α = 1/2, B(α)H1,H2

simplifies to B(1/2)H1,H2

= m(0,1)H1

m(1,0)H2

, which will be small if |SH1(τ, ν)| is located

along the τ axis and/or |SH2(τ, ν)| is located along the ν axis. Thus, for α = 1/2 the approximation

(2.62) is good when H1 is a quasi-LTI system and/or H2 is a quasi-LFI system (note the consistency

with the exact multiplicative calculus in (2.55)).

For α = −1/2, we have B(−1/2)H1,H2

= m(1,0)H1

m(0,1)H2

. This will be small if |SH1(τ, ν)| is located along the

ν axis and/or |SH2(τ, ν)| is located along the τ axis, i.e., if H1 is quasi-LFI and/or H2 is quasi-LTI

(note the consistency with (2.56)). For both α = 1/2 and α = −1/2, the bound is large, and thus the

approximation (2.62) is poor, if the systems’ GSFs are oriented in oblique directions.

For |α| ≤ 1/2 one has∣∣α+ 1

2

∣∣+∣∣α− 1

2

∣∣ = 1; hence B(α)H1,H2

in (2.57) is here a convex combination9

of m(0,1)H1

m(1,0)H2

and m(1,0)H1

m(0,1)H2

and thus assumes values between the two “extreme” cases α = ±1/2.

Case α = 0. The case α = 0 is of particular importance. Here, B(α)H1,H2

simplifies to

B(0)H1,H2

=1

2

[m

(0,1)H1

m(1,0)H2

+m(1,0)H1

m(0,1)H2

],

which is symmetric with respect to H1 and H2. Thus, the approximation (2.62) will be good if both

m(0,1)H1

m(1,0)H2

and m(1,0)H1

m(0,1)H2

are small, which amounts to the condition that H1 and H2 are jointly9The convex combination of two quantities x and y is defined by (1 − ǫ)x + ǫy with 0 ≤ ǫ ≤ 1.


underspread in the sense of Subsection 2.2.5. This requires that the GSFs of H1 and H2 are both

concentrated about the origin of the (τ, ν) plane, with similar orientation parallel to the τ or ν axis.

However, this requirement can be relaxed by a refinement of the bound 2πB(0)H1,H2

that is stated in

the next theorem and that allows for systems H1 and H2 with GSFs oriented along similar oblique

directions. This improved bound is due to the covariance of the Weyl symbol to unitary metaplectic

operators U ∈ M that correspond to area-preserving, linear TF coordinate transforms (cf. Subsection

2.1.3).

Theorem 2.15. For any two LTV systems H1 and H2, the difference ∆(0)3 (t, f) = L

(0)H2H1

(t, f) −L

(0)H1

(t, f)L(0)H2

(t, f) obeys the (generally tighter) bound

∣∣∆(0)3 (t, f)

∣∣‖SH1‖1‖SH2‖1

≤ 2π infU∈M

B(0)UH1U

+,UH2U+ . (2.63)

Proof. Specializing the proof of Theorem 2.14 to α = 0, it is seen that

∣∣∆(0)3 (t, f)

∣∣ ≤ π

∫

τ1

∫

ν1

∫

τ2

∫

ν2

|SH2(τ1, ν1)| |SH1(τ2, ν2)| |τ1ν2 − τ2ν1| dτ1 dν1 dτ2 dν2 , (2.64)

where τ1ν2 − τ2ν1 is the symplectic form on R2. Performing a symplectic coordinate transform

(τiνi

)=

A(τ ′

iν′

i

), where A =

(a bc d

)with detA = ad − bc = 1, and using the invariance of the symplectic

form [64,154], τ1ν2 − τ2ν1 = τ ′1ν′2 − τ ′2ν

′1, the right hand side of (2.64) becomes

π

∫

τ ′1

∫

ν′1

∫

τ ′2

∫

ν′2

∣∣SH2(aτ′1 + bν ′1, cτ

′1 + dν ′1)

∣∣ ∣∣SH1(aτ′2 + bν ′2, cτ

′2 + dν ′2)

∣∣ ∣∣τ ′1ν ′2 − τ ′2ν′1

∣∣ dτ ′1 dν ′1 dτ ′2 dν ′2 .

Let Hi = UHiU+ where U ∈ M is the unitary operator corresponding to A in the sense of Subsection

2.1.3. Using the covariance property S(0)eH

(τ, ν) = S(0)H

(aτ + bν, cτ + dν) and noting that |SH(τ, ν)| =

|S(0)H

(τ, ν)|, the above expression becomes

π

∫

τ ′1

∫

ν′1

∫

τ ′2

∫

ν′2

|S eH2(τ ′1, ν

′1)| |S eH1

(τ ′2, ν′2)|∣∣τ ′1ν ′2 − τ ′2ν

′1

∣∣ dτ ′1 dν ′1 dτ ′2 dν ′2.

Inserting |τ ′1ν ′2 − τ ′2ν′1| ≤ |τ ′1ν ′2| + |τ ′2ν ′1| and using ‖S eHi

‖1

= ‖SHi‖1 (which holds since the GSF

magnitude of Hi is obtained from that of Hi via an area-preserving coordinate transform), we obtain

∣∣∆(0)3 (t, f)

∣∣ ≤ π

∫

τ ′1

∫

ν′1

|S eH2(τ ′1, ν

′1)| |τ ′1| dτ ′1 dν ′1

∫

τ ′2

∫

ν′2

|S eH1(τ ′2, ν

′2)| |ν ′2| dτ ′2 dν ′2

+ π

∫

τ ′1

∫

ν′1

|S eH2(τ ′1, ν

′1)| |ν ′1| dτ ′1 dν ′1

∫

τ ′2

∫

ν′2

|S eH1(τ ′2, ν

′2)| |τ ′2| dτ ′2 dν ′2

= π ‖SH1‖1 ‖SH2‖1

[m

(0,1)eH1

m(1,0)eH2

+m(1,0)eH1

m(0,1)eH2

].

As this bound is valid for all U ∈ M, we finally obtain the bound (2.63).

Since the symplectic group contains TF rotations and TF shearings, this theorem shows that

∆(0)3 (t, f) may be small even if the GSFs of H1 and H2 are oriented in (similar) oblique directions.

This is not true for α 6= 0, which once again shows the exceptional position of the Weyl symbol.


DL Operators. For DL operators, using (2.17), the bound (2.57) can be further bounded as

∣∣∆(α)3 (t, f)

∣∣‖SH1‖1‖SH2‖1

≤ 2π[∣∣∣α+

1

2

∣∣∣ τ (max)H2

ν(max)H1

+∣∣∣α− 1

2

∣∣∣ τ (max)H1

ν(max)H2

]. (2.65)

A tighter bound is obtained by adapting the proof of Theorem 2.14 for DL operators, starting from

(2.60). For∣∣α+ 1

2

∣∣ τ (max)H2

ν(max)H1

+∣∣α− 1

2

∣∣ τ (max)H1

ν(max)H2

≤ 1/2, this yields

∣∣∆(α)3 (t, f)

∣∣‖SH1‖1‖SH2‖1

≤ 2 sin(π[∣∣∣α+

1

2

∣∣∣ τ (max)H2

ν(max)H1

+∣∣∣α− 1

2

∣∣∣ τ (max)H1

ν(max)H2

]). (2.66)

This latter bound may be compared to the corresponding bound in [118] (valid for (1+2|α|)σH1 ,H2 ≤ 2

where σH1,H2 has been defined in (2.11))

∣∣∆(α)3 (t, f)

∣∣‖SH1‖1‖SH2‖1

≤ 2 sin(π(1

2+ |α|

)σH1,H2

2

).

It is seen that our bound shows more explicitly which quantities have to be small for various values

of α in order that the approximation (2.62) is valid; furthermore, our bound is also tighter since

∣∣∣α+1

2

∣∣∣ τ (max)H2

ν(max)H1

+∣∣∣α− 1

2

∣∣∣ τ (max)H1

ν(max)H2

≤(1

2+ |α|

)[τ

(max)H2

ν(max)H1

+ τ(max)H1

ν(max)H2

]

≤(1

2+ |α|

)max

{τ

(max)H1

, τ(max)H2

}(ν

(max)H1

+ ν(max)H2

)

≤ 2(1

2+ |α|

)max

{τ

(max)H1

, τ(max)H2

}max

{ν

(max)H1

, ν(max)H2

}=(1

2+ |α|

)σH1,H2

2.

Non-DL Operators. We now investigate the potential error resulting from using the bound

(2.66) derived for DL operators in the case of non-DL operators.

Proposition 2.16. For any two LTV systems H1 and H2 and any τG1, νG1, τG2 , νG2 such that∣∣α+ 12

∣∣ τG2νG1 +∣∣α− 1

2

∣∣ τG1νG2 ≤ 1/2, the difference ∆(α)3 (t, f) is bounded as

∣∣∆(α)3 (t, f)

∣∣‖SH1‖1 ‖SH2‖1

≤ 2 sin(π[∣∣∣α+

1

2

∣∣∣ τG2νG1 +∣∣∣α− 1

2

∣∣∣ τG1νG2

])+ 2

(m

(1,0)H1

τG1

+m

(0,1)H1

νG1

)(m

(1,0)H2

τG2

+m

(0,1)H2

νG2

).

(2.67)

Proof. Let G1 = [−τG1, τG1 ] × [−νG1 , νG1 ] and G2 = [−τG2 , τG2 ] × [−νG2 , νG2 ]. Starting with (2.60) and

splitting the integral yields

∣∣∆(α)3 (t, f)

∣∣ ≤ 2

∫∫∫∫

G1 G2

|SH2(τ′, ν ′)| |SH1(τ1, ν1)|

∣∣∣sin(π[(α+

1

2

)τ ′ν1 +

(α− 1

2

)τ1ν

′])∣∣∣dτ ′dν ′dτ1 dν1

+ 2

∫∫∫∫

G1 G2

|SH2(τ′, ν ′)| |SH1(τ1, ν1)| dτ ′dν ′dτ1 dν1 , (2.68)

where we also used | sinx| ≤ 1 to bound the second integral. The first term in (2.68) can be bounded

using the inequality

max(τ1,ν1)∈G1

(τ ′,ν′)∈G2

∣∣∣sin(π[(α+

1

2

)τ ′ν1 +

(α− 1

2

)τ1ν

′])∣∣∣ ≤ sin

(π[∣∣∣α+

1

2

∣∣∣τG2νG1 +∣∣∣α− 1

2

∣∣∣τG1νG2

]),


which is valid for∣∣α + 1

2

∣∣ τG2νG1 +∣∣α − 1

2

∣∣ τG1νG2 ≤ 1/2. Furthermore, the second term in (2.68) can

be rewritten as∫∫∫∫

G1 G2

|SH2(τ′, ν ′)| |SH1(τ1, ν1)| dτ ′dν ′dτ1 dν1 =

∥∥∥SH

G11

∥∥∥1

∥∥∥SH

G22

∥∥∥1,

and the L1-bound in (2.38) can be applied to∥∥∥S

HG11

∥∥∥1

and∥∥∥S

HG22

∥∥∥1. This finally establishes the

proposition.

It is thus seen that for non-DL operators, the bound (2.66) (incorrectly used) deviates from the

correct bound (2.67) by as much as 2(

m(1,0)H1τG1

+m

(0,1)H1νG1

)(m

(1,0)H2τG2

+m

(0,1)H2νG2

).

2.3.5 Composition of H with H+

In some applications, the composition of H with H+ is of importance. Examples are the innovations

system representation of random processes [148] (see also Chapter 3) and the proofs of Theorems 2.22

and 2.32. For LTI or LFI systems, the transfer function of H+H is |G(f)|2 or |m(t)|2, respectively.

For general LTV systems, the TF transfer function (GWS) of H+H is no longer given by the squared

magnitude of the GWS of H. However, the following result can be obtained from Theorem 2.14 and

Corollary 2.12. We note that for the case α = 0 a similar result for DL operators is given in [118].

Corollary 2.17. For any LTV system H, the difference

∆(α)4 (t, f) , L

(α)H+H

(t, f) −∣∣L(α)

H(t, f)

∣∣2 (2.69)

is bounded as

∣∣∆(α)4 (t, f)

∣∣‖SH‖2

1

≤ 2π C(α)H

with C(α)H

, cαm(0,1)H

m(1,0)H

+ 2 |α|m(1,1)H

, (2.70)

where cα = |α+ 1/2| + |α− 1/2|.

Proof. Subtracting and adding L(α)H

(t, f)L(α)H+(t, f) to ∆

(α)4 (t, f), we obtain

∣∣∆(α)4 (t, f)

∣∣ =∣∣L(α)

H+H(t, f) − L

(α)H

(t, f)L(α)H+(t, f) + L

(α)H

(t, f)L(α)H+(t, f) − |L(α)

H(t, f)|2

∣∣

≤∣∣L(α)

H+H(t, f) − L

(α)H

(t, f)L(α)H+(t, f)

∣∣+∣∣L(α)

H(t, f)L

(α)H+(t, f) − |L(α)

H(t, f)|2

∣∣

=∣∣L(α)

H+H(t, f) − L

(α)H

(t, f)L(α)H+(t, f)

∣∣+∣∣L(α)

H+(t, f) − L(α)∗H

(t, f)∣∣∣∣L(α)

H(t, f)

∣∣ (2.71)

≤ ‖SH‖21 2πcαm

(1,0)H

m(0,1)H

+ ‖SH‖1 4π|α|m(1,1)H

∣∣L(α)H

(t, f)∣∣ ,

where we used (2.57) with m(k,l)H+ = m

(k,l)H

and ‖SH+‖1 = ‖SH‖1 as well as (2.48). From this, (2.70)

follows with∣∣L(α)

H(t, f)

∣∣ ≤ ‖SH‖1.

Discussion. Hence, for an underspread system with small m(0,1)H

m(1,0)H

and small m(1,1)H

, we have

L(α)H+H

(t, f) ≈∣∣L(α)

H(t, f)

∣∣2 ,


such that “squaring” a system (i.e., composing H and H+) is approximately equivalent to squaring

its GWS. We note that the bound (2.70), and thus the above approximation, remain valid if H+H is

replaced by HH+ or H ⋆H+.

For |α| ≤ 1/2, we have cα = 1 and thus

C(α)H

= m(1,0)H

m(0,1)H

+ 2 |α|m(1,1)H

≤ m(1,0)H

m(0,1)H

+m(1,1)H

, |α| ≤ 1/2 .

The bound C(α)H

is tightest for α = 0, in which case C(0)H

= m(0,1)H

m(1,0)H

. This is due to the fact that

the Weyl symbol of the adjoint is exactly obtained by complex conjugation. Furthermore, for α = 0

the above result can be refined similarly to Theorem 2.15, yielding the tighter bound

|∆(0)4 (t, f)|‖SH‖2

1

≤ 2π infU∈M

C(0)UHU+ = 2π inf

U∈M

{m

(0,1)UHU+m

(1,0)UHU+

}. (2.72)

These results for the GWS of the composition of H and H+ are interesting also since they show that

the generalized input Wigner distribution IW(α)H

(t, f) and the generalized output Wigner distribution

OW(α)H

(t, f) (see (B.36) and [90]) of an underspread system are approximately equal to the squared

magnitude of the GWS, thus reducing essentially to the GWS. Furthermore, since both L(α)HH+(t, f) and

L(α)H+H

(t, f) are approximately equal to |L(α)H

(t, f)|2, generalized input and output Wigner distribution

are also approximately equal to each other. More precisely, by applying the triangle inequality to

IW(α)H

(t, f) − OW(α)H

(t, f) = L(α)H+H

(t, f) − L(α)HH+(t, f) = L

(α)H+H

(t, f) − |L(α)H

(t, f)|2 + |L(α)H

(t, f)|2 −L

(α)HH+(t, f) and twice using (2.70), we obtain

∣∣IW(α)H

(t, f) − OW(α)H

(t, f)∣∣

‖SH‖21

≤ 4π C(α)H

,

which, in the case α = 0 can be refined by using (2.72), i.e.,

∣∣IW(α)H

(t, f) − OW(α)H

(t, f)∣∣

‖SH‖21

≤ 4π infU∈M

{m

(0,1)UHU+m

(1,0)UHU+

}.

Thus, for underspread systems the generalized input Wigner distribution and the generalized output

Wigner distribution are approximately equal. The approximation L(α)H+H

(t, f) ≈ L(α)HH+(t, f) also

suggests that HH+ ≈ H+H, i.e., that an underspread operator H is approximately normal. This

subject will be discussed in more detail in Subsection 2.3.17.

DL Operators. According to (2.71), we can write

∣∣∆(α)4 (t, f)

∣∣ ≤∣∣∆(α)

3 (t, f)∣∣+∣∣∆(α)

2 (t, f)∣∣ ∣∣L(α)

H(t, f)

∣∣ , (2.73)

with H1 and H2 in ∆(α)3 (t, f) replaced by H and H+, respectively. For a DL operator, the bounds

(2.66) on∣∣∆(α)

3 (t, f)∣∣ and (2.50) on

∣∣∆(α)2 (t, f)

∣∣ apply, and we obtain (recall that τ(max)H+ = τ

(max)H

,

ν(max)H+ = ν

(max)H

, and µH ≤ τ(max)H

ν(max)H

= σH/4)

∣∣∆(α)4 (t, f)

∣∣ ≤ 2 ‖SH‖1 ‖SH+‖1 sin(π[∣∣∣α+

1

2

∣∣∣ τ (max)H

ν(max)H+ +

∣∣∣α− 1

2

∣∣∣ τ (max)H+ ν

(max)H

])


+ 2 ‖SH‖1 sin(2π|α|µH)∣∣L(α)

H(t, f)

∣∣

≤ 2 ‖SH‖21 sin

(π4cασH

)+ 2 ‖SH‖2

1 sin(π

2|α|σH

),

where we furthermore used∣∣L(α)

H(t, f)

∣∣ ≤ ‖SH‖1. This yields the bound

∣∣∆(α)4 (t, f)

∣∣‖SH‖2

1

≤ 2 sin(π

4cασH

)+ 2 sin

(π2|α|σH

), (2.74)

which is valid for DL operators with cασH ≤ 2 (recall that cα = |α+ 1/2| + |α− 1/2|).

Non-DL Operators. For any (i.e., not necessarly DL) operator H and any τG , νG such that

2cατGνG ≤ 1, (2.67) and (2.51) (in combination with µG ≤ τGνG) can be applied to (2.73), which yields

∣∣∆(α)4 (t, f)

∣∣‖SH‖2

1

≤ 2 sin(πcα τGνG

)+2

(m

(1,0)H

τG+m

(0,1)H

νG

)2

+2 sin(2π|α|τGνG)+2

(m

(1,0)H

τG+m

(0,1)H

νG

). (2.75)

Hence, the error incurred by applying (2.74) to a non-DL operator that is erroneously assumed to be

DL with σH = 4τGνG is bounded by 2(

m(1,0)H

τG+

m(0,1)H

νG

)2+ 2(

m(1,0)H

τG+

m(0,1)H

νG

).

2.3.6 Operator Inversion Based on the Generalized Weyl Symbol—Part I

In this subsection, we consider the approximate solution of an important operator equation via ap-

proximate operator inversions in the TF domain. We will consider equations of the type

H1GH2 = H3, (2.76)

where H1, H2, and H3 are HS operators that are jointly DL10, i.e., their GSFs are contained within

the same centered rectangle G, and G is some operator that is to be calculated.

Eq. (2.76) can be solved for G by computing the (pseudo-)inverses of H1 and H2. However, our

aim is to find approximate expressions for the GWS of G since it is desirable to replace potentially nu-

merically unstable and computationally intensive operator inversions by simple TF domain inversions

(i.e., scalar divisions).

Equations of the type (2.76) are important for the following reasons:

• In Section 3.8, we consider a coherence operator Γx,y for nonstationary random processes x(t)

and y(t) that is defined by R1/2x Γx,yR

1/2y = Rx,y (with Rx, Ry, and Rx,y the auto- and cross-

correlation operators of x(t) and y(t)), i.e., by an equation of the type (2.76) with H1 = R1/2x ,

G = Γx,y, H2 = R1/2y , and H3 = Rx,y. Our subsequent results will enable us to formulate an

approximate TF coherence function that applies to jointly underspread nonstationary processes

(see Section 3.8).

10The restriction to DL operators is made for reasons of tractability. The extension of our results to operators with

rapidly decaying spreading function would be desirable but appears to be difficult.


• In the Gaussian signal detection problem discussed in Section 4.2, the likelihood ratio test

statistic is a quadratic form 〈HLRx, x〉 with HLR defined by R0HLRR1 = R1 − R0 (here, R0

and R1 are the correlation operators to be discriminated). This equation is of the type (2.76)

with H1 = R0, G = HLR, H2 = R1, and H3 = R1 − R0. Similarly, the deflection-optimal test

is given by 〈HDx, x〉 with HD defined by R1HLRR1 = R1 − R0. This equation is again of the

type (2.76) with H1 = H2 = R1, G = HD, and H3 = R1 − R0. The results of this subsection

will be the basis for an approximate TF formulation of optimal detectors that is valid for jointly

underspread processes (see Section 4.2).

In order to obtain an approximation for the GWS of G in (2.76), it is tempting to twice apply the

product formula (2.62) directly to (2.76). Unfortunately, in most cases G will not be DL underspread,

thereby prohibiting direct application of (2.62). Alternatively, one could think of applying (2.62) to

the equation G = H−11 H3H

−12 , obtained by multiplying (2.76) by the (pseudo-)inverses H−1

1 and

H−12 . Unfortunately, H−1

1 and H−12 will typically not be DL either (see Subsection 2.1.4), and hence

(2.62) again cannot be applied directly. In the following, we present two theorems which show that

a smoothed version of the GWS of G is approximately equal to the following ratio formed with the

GWSs of H3, H1, and H2,

(L

(α)G

∗∗ ψ)(t, f) ≈

L(α)H3

(t, f)

L(α)H1

(t, f)L(α)H2

(t, f). (2.77)

Here, ψ(t, f) denotes a 2-D lowpass function to be discussed in more detail later.

DL Approximation of G. Our approximate TF solution of (2.76) is based on splitting the

operator G into a DL part and a non-DL part according to (2.4). Thus we have G = GG + GG with

S(α)

GG (τ, ν) = S(α)G

(τ, ν) IG(t, f) where G is the joint support of the GSFs of H1, H2, and H3. We next

show that the non-DL part GG = G − GG is “negligible” in the sense that removing it from G does

not greatly influence the validity of (2.76).

Theorem 2.18. Consider an operator G satisfying H1GH2 = H3, where H1, H2, and H3 are jointly

DL HS operators with GSF support contained in the rectangle G =[−τG , τG

]×[−νG , νG

]so that

the joint displacement spread is given by σG = 4τGνG. Let GG denote the DL part of G defined

by S(α)


(τ, ν) IG(τ, ν) and let GG = G − GG denote the non-DL part of G. Then, the

difference H1GGH2 − H3 is bounded as

∥∥H1GGH2 − H3

∥∥2

‖H1‖2 ‖GG‖2 ‖H2‖2

≤ 3√σG . (2.78)

Proof. In the (τ, ν)-domain, (2.76) corresponds to S(α)H1GH2

(τ, ν) = S(α)H3

(τ, ν). Since S(α)H1GH2

(τ, ν)

equals the twisted convolution of S(α)H1

(τ, ν), S(α)G

(τ, ν), and S(α)H2

(τ, ν) (see (B.12)), we obtain

(S

(α)H1

♮ S(α)G

♮ S(α)H2

)(τ, ν) = S

(α)H3

(τ, ν) .


Splitting G into a DL part and a non-DL part corresponding to the rectangular region G, i.e., G =

GG + GG , further yields

(S

(α)H1

♮ S(α)

GG ♮ S(α)H2

)(τ, ν) +

(S

(α)H1

♮ S(α)

GG♮ S

(α)H2

)(τ, ν) = S

(α)H3

(τ, ν). (2.79)

Since the supports of S(α)H1

(τ, ν), S(α)

GG (τ, ν), and S(α)H2

(τ, ν) are confined to G, (B.13) implies that

the support of(S

(α)H1

♮ SGG ♮ SH2

)(τ, ν) is confined to G′ =

[−3τG , 3τG

]×[−3νG , 3νG

]. The crucial

observation now is that since S(α)H3

(τ, ν) is confined to G and(S

(α)H1

♮ S(α)

GG ♮ S(α)H2

)(τ, ν) is confined to G′, it

follows from (2.79) that(S

(α)H1

♮ S(α)

GG♮ S

(α)H2

)(τ, ν) must also be confined to G′, irrespective of the fact that

the support of SGG(τ, ν) lies totally outside G. Indeed, any contributions of

(S

(α)H1

♮ S(α)

GG♮ S

(α)H2

)(τ, ν)

outside of G′ cannot be canceled by(S

(α)H1

♮ S(α)

GG ♮ S(α)H2

)(τ, ν) and thus would contradict (2.79).

Proceeding with our proof of the bound (2.78), we note that

H1GGH2 − H3 = H1G

GH2 − H1GH2 = H1

(GG − G

)H2 = −H1G

GH2, (2.80)

and hence, using (B.12) there is

∥∥H1GGH2 −H3

∥∥2

2=∥∥H1G

GH2

∥∥2

2=

∫∫

G′

∣∣∣(S

(α)H1

♮ S(α)

GG♮ S

(α)H2

)(τ, ν)

∣∣∣2dτ dν

=

∫∫

G′

∣∣∣∣∫

τ ′

∫

ν′

(S

(α)H1

♮ S(α)

GG

)(τ ′, ν ′)S

(α)H2

(τ − τ ′, ν− ν ′) e−j2πφα(τ,ν,τ ′,ν′) dτ ′ dν ′∣∣∣∣2

dτ dν,

where we used the support constraint of(S

(α)H1

♮ S(α)

GG♮ S

(α)H2

)(τ, ν) discussed above. By applying the

Schwarz inequality and the inequality (B.15), and by noticing that the area of G′ is 9σG , we finally

obtain

∥∥H1GGH2 − H3

∥∥2

2≤∫∫

G′

[∫

τ1

∫

ν1

∣∣∣(S

(α)H1

♮ S(α)

GG

)(τ1, ν1)

∣∣∣2dτ1 dν1

]

·[∫

τ2

∫

ν2

|SH2(τ − τ2, ν − ν2)|2 dτ2 dν2

]dτ dν

≤∫∫

G′

‖SH1‖22 ‖SGG‖2

2 ‖SH2‖22 dτ dν

= ‖H1‖22

∥∥GG∥∥2

2‖H2‖2

2

∫∫

G′

dτ dν = ‖H1‖22

∥∥GG∥∥2

2‖H2‖2

2 9σG .

Discussion. From the foregoing theorem, it is seen that if σG is small, i.e., if H1, H2, and H3

are jointly DL underspread, then removing the non-DL part GG from G does not greatly affect the

validity of H1GH2 = H3:

H1GH2 = H3 =⇒ H1GGH2 ≈ H3 . (2.81)

We note that small σG requires the GSFs of H1, H2, and H3 to be essentially located along the τ or ν

axis, respectively. Yet, Theorem 2.18 can easily be extended to allow for arbitrary GSF orientations.

Let us assume that the GSFs of H1, H2, and H3 are supported within a region G that is obtained


by a rotation of the rectangular region G using the rotation matrix A =(

cos θ0−T 2 sin θ0

(sin θ0)/T 2 cos θ0

). Then,

with the metaplectic operator U ∈ M that corresponds to A (see Appendix C) we obtain

∥∥H1GeGH2 −H3

∥∥2

=∥∥U(H1G

eGH2 − H3)U+∥∥

2=∥∥UH1G

eGH2U+ − UH3U

+∥∥

2

=∥∥UH1U

+UGeGU+UH2U

+ − UH3U+∥∥

2

=∥∥H1G

GH2 − H3

∥∥2

(2.82)

where the systems Hi = UHiU+, i = 1, 2, 3, have GSFs supported within G and G = UGU+. Hence,

Theorem 2.18 directly applies to (2.82) and it is thus seen that the orientation of G is not relevant to

(2.78); all that counts is the area σeG = σG of the GSF support.

Since the operator norm is upper-bounded by the HS norm (see Appendix A), the approximation

(2.81) is also valid in the sense that the operator norm ‖H1GGH2 − H3‖O is small. Furthermore, we

note that the above theorem can be viewed as an improvement on inequality (B.16). Indeed, (B.16) is

valid for arbitrary operators (i.e., also for non-DL operators) and, together with (2.80), implies that

∥∥H1GGH2 − H3

∥∥2

=∥∥H1G

GH2

∥∥2≤ ‖H1‖2

∥∥GG∥∥

2‖H2‖2 or

∥∥H1GGH2 − H3

∥∥2

‖H1‖2

∥∥GG∥∥

2‖H2‖2

≤ 1 .

Approximation for the GWS of G. The product (2.5) defining the GSF of GG corresponds

to a 2-D convolution of GWSs,

L(α)

GG (t, f) =(L

(α)G

∗∗ L(α)T

)(t, f) , (2.83)

where T is defined by S(α)T

(τ, ν) = IG(τ, ν) (see (2.3)). Since the GWS of T is a 2-D lowpass function

(that equals the function ψ(t, f) in (2.77)), (2.83) means that the GWS of GG is a smoothed version

of the GWS of G. The previous theorem, which shows that for jointly DL underspread operators H1,

H2, and H3 the non-DL part of G is “negligible” with respect to the operator equation (2.76), is the

basis for the next result which formulates an approximation for the GWS of GG (i.e., for the smoothed

GWS of G according to (2.83)) by the ratio L(α)H3

(t, f)/[L(α)H1

(t, f)L(α)H2

(t, f)].

Theorem 2.19. Let the operators H1, H2, H3, G, GG and the rectangular region G be defined as in

Theorem 2.18. Then, the difference

∆(α)5 (t, f) , L

(α)H1

(t, f)L(α)

GG (t, f)L(α)H2

(t, f) − L(α)H3

(t, f)

is bounded as∣∣∆(α)

5 (t, f)∣∣

‖SH1‖1 ‖SG‖∞ ‖SH2‖1

≤ 3π

2cα σ

2G + 9σG ,

∥∥∆(α)5

∥∥2

‖H1‖2‖G‖2 ‖H2‖2≤ πcασ

2G + 8πcα

√σ3G + 3

√σG ,

(2.84)

with cα = |α+ 1/2| + |α− 1/2|.

Proof. To prove the first bound (L∞ bound), we subtract/add both L(α)H1

(t, f)L(α)

GGH2(t, f) and

L(α)

H1GGH2

(t, f) from/to ∆(α)5 (t, f) and apply the triangle inequality twice. This gives

∣∣∆(α)5 (t, f)

∣∣ ≤∣∣L(α)

H1(t, f)L

(α)

GG (t, f)L(α)H2

(t, f) − L(α)H1

(t, f)L(α)

GGH2(t, f)

∣∣

+∣∣L(α)

H1(t, f)L

(α)

GGH2(t, f) − L

(α)

H1GGH2

(t, f)∣∣+∣∣L(α)

H1GGH2

(t, f) − L(α)H3

(t, f)∣∣ .

(2.85)


By noting that the maximum time shift and maximum frequency shift of both GG and H2 are τG and

νG, respectively, the first term on the right hand side of (2.85) can be bounded by using (2.65) (with

H1 in (2.65) replaced by GG) and by subsequently applying the inequalities |L(α)H1

(t, f)| ≤ ‖SH1‖1 and

‖SGG‖1 ≤ ‖SGG‖∞σG ≤ ‖SG‖∞σG ,

∣∣L(α)H1

(t, f)L(α)

GG (t, f)L(α)H2

(t, f) − L(α)H1

(t, f)L(α)

GGH2(t, f)

∣∣ ≤

≤∣∣L(α)

H1(t, f)

∣∣ ∣∣L(α)

GG (t, f)L(α)H2

(t, f) − L(α)

GGH2(t, f)

∣∣

≤∣∣L(α)

H1(t, f)

∣∣ ‖SGG‖1‖SH2‖1 2π[∣∣∣α+

1

2

∣∣∣ τGνG +∣∣∣α− 1

2

∣∣∣ τGνG]

≤ ‖SH1‖1‖SGG‖1‖SH2‖1 2π cα τGνG

≤ π

2cα σ

2G ‖SH1‖1‖SG‖∞‖SH2‖1 . (2.86)

In a similar way, the second term on the right hand side of (2.85) can be bounded by noting that the

GSFs of H1 and GGH2 are confined to G and [−2τG , 2τG ]× [−2νG , 2νG ], respectively. Hence, applying

(2.65) (with H2 in (2.65) replaced by GGH2) and subsequently using the inequalities ‖SGGH2‖1≤

‖SGG‖1‖SH2‖1 and ‖SGG‖1 ≤ ‖SG‖∞σG yields

∣∣L(α)H1

(t, f)L(α)

GGH2(t, f) − L

(α)

H1GGH2

(t, f)∣∣ ≤

≤ ‖SH1‖1‖SGGH2‖12π[∣∣∣α+

1

2

∣∣∣ (2τG)νG +∣∣∣α− 1

2

∣∣∣ τG(2νG)]

≤ ‖SH1‖1‖SGG‖1‖SH2‖14π cα τGνG

≤ π cα σ2G ‖SH1‖1‖SG‖∞‖SH2‖1 . (2.87)

Finally, the third term on the right hand side of (2.85) can be developed and bounded as

∣∣L(α)

H1GGH2

(t, f) − L(α)H3

(t, f)∣∣ =

∣∣L(α)

H1GGH2

(t, f)∣∣ =

∣∣∣∣∫

τ

∫

ν

(S

(α)H1

♮ S(α)

GG♮ S

(α)H2

)(τ, ν) dτ dν

∣∣∣∣

≤∫∫

G′

∣∣∣(S

(α)H1

♮ S(α)

GG♮ S

(α)H2

)(τ, ν)

∣∣∣ dτ dν

≤∥∥∥S(α)

H1♮ S

(α)

GG♮ S

(α)H2

∥∥∥∞

∫∫

G′

dτ dν

≤ 9σG ‖S(α)H1

♮ SGG‖∞ ‖SH2‖1

≤ 9σG ‖SH1‖1 ‖SG‖∞ ‖SH2‖1 , (2.88)

where we twice used Young’s inequality (B.14) (with p = ∞, q = 1) and the fact that the support of(S

(α)H1

♮ S(α)

GG♮ S

(α)H2

)(τ, ν) is confined to G′, whose area is given by 9σG (see the proof of Theorem 2.18).

The L∞ bound in (2.84) finally follows by combining (2.86), (2.87), and (2.88).

The second bound (L2 bound) in (2.84) is derived by again subtracting/adding both

L(α)H1

(t, f)L(α)

GGH2(t, f) and L

(α)

H1GGH2

(t, f) from/to ∆(α)5 (t, f) and by twice applying the triangle in-

equality,

∥∥∆(α)5

∥∥2≤∥∥L(α)

H1L

(α)

GGL(α)H2

−L(α)H1L

(α)

GGH2

∥∥2+∥∥L(α)

H1L

(α)

GGH2−L

(α)

H1GGH2

∥∥2+∥∥L(α)

H1GGH2

−L(α)H3

∥∥2. (2.89)


The first term on the right hand side of (2.89) can be bounded as follows. First, it is easily seen that

∥∥L(α)H1L

(α)

GGL(α)H2

− L(α)H1L

(α)

GGH2

∥∥2≤∥∥L(α)

H1

∥∥∞

∥∥L(α)

GGL(α)H2

− L(α)

GGH2

∥∥2.

Noting that the maximum time shift and maximum frequency shift of both GG and H2 are τG and

νG , respectively, and by applying the (slightly refined) bound [118,119]

∥∥L(α)

GGL(α)H2

− L(α)

GGH2

∥∥2≤ π cα

√64τ3

Gν3G ‖GG‖2‖H2‖2 = π cα

√σ3G ‖GG‖2‖H2‖2

we obtain

∥∥L(α)H1L

(α)

GGL(α)H2

− L(α)H1L

(α)

GGH2

∥∥2≤∥∥L(α)

H1

∥∥∞π cα

√σ3G ‖GG‖2‖H2‖2

≤ π cα σ2G ‖H1‖2‖G‖2‖H2‖2 , (2.90)

where we further used the inequalities ‖GG‖2 ≤ ‖G‖2 and ‖L(α)H

‖∞ ≤ ‖SH‖1 ≤√σG ‖H‖2 (the last

inequality holds for DL operators and can be shown using the Schwarz inequality). The second term

on the right hand side of (2.89) can be similarly bounded by noting that the maximum time shift and

maximum frequency shift of GG H2 are 2τG and 2νG , respectively. Then applying the (slightly refined)

bound [118,119]

∥∥L(α)H1L

(α)

GGH2− L

(α)

H1GGH2

∥∥2≤ π cα

√64(2τG)3 (2νG)3 ‖H1‖2‖GGH2‖2 ,

as well as σG = 4τGνG, ‖SGGH2‖2≤ ‖GG‖2‖H2‖2, and ‖GG‖2 ≤ ‖G‖2, we obtain

∥∥L(α)H1L

(α)

GGH2− L

(α)

H1GGH2

∥∥2≤ π cα

√(4σG

)3 ‖H1‖2‖GGH2‖2

≤ 8π cα

√σ3G ‖H1‖2‖G‖2‖H2‖2 . (2.91)

By noting that∥∥L(α)

H1GGH2

− L(α)H3

∥∥2

=∥∥H1G

GH2 − H3

∥∥2, the third term on the right hand side of

(2.89) is bounded according to (2.78). The L2 bound for ∆(α)5 (t, f) in (2.84) is finally obtained by

combining (2.90), (2.91), and (2.78).

Discussion. The above result shows that for jointly DL underspread HS operators H1, H2, and

H3, i.e., operators with GSFs supported within a rectangular region G of small area σG , it follows

from H1GH2 = H3 that

L(α)H1

(t, f)L(α)

GG (t, f)L(α)H2

(t, f) ≈ L(α)H3

(t, f) . (2.92)

We note that small σG requires the GSFs of H1, H2, and H3 to be essentially located along the τ or

ν axis, respectively. Yet, in the case of α = 0, Theorem 2.19 can be extended using a refinement of

the involved bounds via metaplectic transformations like that performed in Subsection 2.3.4.

Regularized Inversion in the TF Domain. The approximation (2.92) can be used to obtain

an approximate solution of (2.76) via a regularized inversion in the TF domain. To this end, let us

define a new operator G via its GWS as

L(α)eG

(t, f) ,

L(α)H3

(t, f)

L(α)H1

(t, f)L(α)H2

(t, f), for (t, f) ∈ R

0 , for (t, f) 6∈ R ,

(2.93)


where

R ,

{(t, f) :

∣∣L(α)H1

(t, f)L(α)H2

(t, f)∣∣

‖SH1‖1 ‖SH2‖1

≥ κ ǫ}, with ǫ ,

3π

2cα σ

2G + 9σG , (2.94)

and κ ≥ 0. Since∣∣L(α)

H1(t, f)L

(α)H2

(t, f)∣∣ =

∣∣L(α)H1

(t, f)∣∣ ∣∣L(α)

H2(t, f)

∣∣ ≤ ‖SH1‖1 ‖SH2‖1, the quan-

tity ‖SH1‖1 ‖SH2‖1 is an upper bound on the maximum value of∣∣L(α)

H1(t, f)L

(α)H2

(t, f)∣∣ and hence

the region R corresponds to the TF locations where the denominator in (2.93) is larger than

κǫ = κ[

3π2 cα σ

2G +9σG

]times the upper bound ‖SH1‖1 ‖SH2‖1 on its maximum value. Now, Theorem

2.19 implies that, for (t, f) ∈ R,

1

‖SG‖∞∣∣L(α)

GG (t, f) − L(α)eG

(t, f)∣∣ =

1

‖SG‖∞

∣∣∣∣∣L(α)

GG (t, f) −L

(α)H3

(t, f)

L(α)H1

(t, f)L(α)H2

(t, f)

∣∣∣∣∣

=1

‖SG‖∞

∣∣∣∣∣L

(α)H1

(t, f)L(α)

GG (t, f)L(α)H2

(t, f) − L(α)H3

(t, f)

L(α)H1

(t, f)L(α)H2

(t, f)

∣∣∣∣∣

≤ 1

‖SG‖∞

‖SH1‖1 ‖SG‖∞ ‖SH2‖1

(3π2 cασ

2G + 9σG

)

∣∣L(α)H1

(t, f)L(α)H2

(t, f)∣∣

=‖SH1‖1 ‖SH2‖1 ǫ∣∣L(α)H1

(t, f)L(α)H2

(t, f)∣∣

≤ ǫ

κ ǫ=

1

κ.

Hence, for κ large enough, it follows that within R

L(α)

GG (t, f) ≈ L(α)eG

(t, f) . (2.95)

For the purpose of illustration, let us consider an example where σG = 10−5. In order that the

regularized TF inverse L(α)eG

(t, f) in (2.93) approximates L(α)

GG (t, f) with an accuracy of 1% in the sense

that 1‖SG‖∞

∣∣L(α)


(t, f)∣∣ ≤ 0.01, κ = 100 is required. It follows that κǫ = κ [3π cα σ

2G/2 +

9σG ] ≈ 0.01. Hence, the region R where the approximation (2.95) holds with the desired accuracy of

1% is obtained from (2.94) as

R ={

(t, f) :

∣∣L(α)H1

(t, f)L(α)H2

(t, f)∣∣

‖SH1‖1 ‖SH2‖1

≥ 0.01},

i.e., R consists of those TF points where∣∣L(α)

H1(t, f)L

(α)H2

(t, f)∣∣ exceeds 1% of the upper bound

‖SH1‖1 ‖SH2‖1 on its maximum value.

Since GG approximately satisfies (2.76), i.e., H1GGH2 ≈ H3, the approximation (2.95) implies

that for κ large enough, the operator G obtained by the regularized TF inversion (2.93) approximately

satisfies (2.76) as well. It is thus possible to approximately solve operator equations of the type (2.76)

using algebraic operations in the TF domain, thereby avoiding computationally costly and potentially

unstable operator inversions:

H1GH2 = H3 =⇒ H1GGH2 ≈ H3


=⇒(L

(α)G

∗∗ ψ)(t, f) ≈ L

(α)eG

(t, f) =

L(α)H3

(t, f)

L(α)H1

(t, f)L(α)H2

(t, f), for (t, f) ∈ R,

0 , for (t, f) 6∈ R ,

with the smoothing function ψ(t, f) = L(α)T

(t, f). The operator G is finally obtained from L(α)eG

(t, f)

by an inverse Weyl transform (B.18). Applications of this result to the definition of a TF coherence

function and to nonstationary signal detection will be presented in Subsections 3.8 and 4.2, respectively.

We note that in a certain sense, the regularized TF domain inversion resembles the computation

of pseudo-inverses of (numerically) rank deficient matrices (or operators) via a thresholding of the

singular values [70]. However, in our case the thresholding is performed in the TF domain, which is

considerably more intuitive.

2.3.7 Operator Inversion Based on the Generalized Weyl Symbol—Part II

We will next consider the operator equation

GH2 = H3, (2.96)

where H2 and H3 are jointly DL HS operators with GSF support region G and G is the operator to

be calculated. Due to our assumptions of HS operators H1, H2, H3 in the context of (2.76), (2.96)

cannot be viewed as a special case of (2.76) since the identitiy operator I has infinite HS norm. Still,

most of the preceding arguments apply to (2.96) as well, leading to similar theorems with similar

proofs. We further note that all results presented below for GH2 = H3 are valid for equations of the

type H1G = H3 as well.

Eq. (2.96) can be solved for G by computing the (pseudo-)inverse of H2. However, our aim

again is to find approximate expressions for the GWS of G in order to replace potentially numerically

unstable and computationally intensive operator inversions by simple TF domain inversions (i.e., scalar

divisions).

Eq. (2.96) will be important for studying nonstationary linear signal estimation in Section 4.1.

There, we will see that the defining equation for the Wiener filter HW is given by HWRy = Rx,y,

with Ry being the auto-correlation operator of the observation y(t) and Rx,y being the cross-correlation

operator of the desired signal x(t) and the observation y(t). This equation is of the type (2.96) with

G = HW , H2 = Ry, and H3 = Rx,y. The results of this subsection will be useful for an approximate

TF formulation of the Wiener filter that is valid for jointly underspread processes (see Section 4.1).

In order to obtain an approximation for the GWS of G, it is again tempting to apply (2.62) directly

to (2.96) or to the equation G = H3H−12 obtained by multiplying (2.96) by H−1

2 . Unfortunately, in

most cases neither G nor H−12 will be DL underspread, and hence (2.62) cannot be applied directly. In

analogy to the discussion of equations of the type (2.76), we subsequently present two theorems which

show that a smoothed version of the GWS of G is approximately equal to the ratio of the GWSs of

H3 and H2,

(L

(α)G

∗∗ ψ)(t, f) ≈

L(α)H3

(t, f)

L(α)H2

(t, f).


Here, ψ(t, f) = L(α)T

(t, f) is a 2-D lowpass function as in (2.83).

DL Approximation of G. We again split the operator G into a DL part and a non-DL part

such that G = GG + GG (with S(α)


(τ, ν) IG(τ, ν) where G denotes the joint GSF support

of H2 and H3). We will then show that the non-DL part GG is “negligible” for the validity of (2.96).

Theorem 2.20. Consider an operator G satisfying GH2 = H3, where H2 and H3 are jointly DL

with GSF support contained in the rectangular region G =[−τG , τG

]×[−νG , νG

]so that the joint

displacement spread is given by σG = 4τGνG. Let GG denote the DL part of G defined by S(α)

GG (τ, ν) =

S(α)G

(τ, ν) IG(τ, ν) and let GG = G−GG denote the non-DL part of G. Then, the difference GGH2−H3

is bounded as ∥∥GGH2 − H3

∥∥2

‖GG‖2 ‖H2‖2

≤ 2√σG . (2.97)

Proof. The proof of this theorem is essentially parallel to that of Theorem 2.18. In particular, the

equation(S

(α)

GG ♮ S(α)H2

)(τ, ν) +

(S

(α)

GG♮ S

(α)H2

)(τ, ν) = S

(α)H3

(τ, ν) , (2.98)

obtained by rewriting GH2 = GGH2 + GGH2 = H3 in the (τ, ν)-domain, implies that the support of(SGG ♮ SH2

)(τ, ν) is confined to G′ =

[−2τG , 2τG

]×[−2νG , 2νG

], and hence (since S

(α)H3

(τ, ν) is confined

to G)(S

(α)

GG♮ S

(α)H2

)(τ, ν) is confined to G′. Indeed, any contributions of

(S

(α)

GG♮ S

(α)H2

)(τ, ν) outside of

G′ cannot be canceled by(S

(α)

GG ♮ S(α)H2

)(τ, ν) (which is supported within G′) and thus would contradict

(2.98). With

GGH2 − H3 = GGH2 − GH2 =(GG − G

)H2 = −GGH2, (2.99)

there is

∥∥GGH2 − H3

∥∥2

2=∥∥GGH2

∥∥2

2=

∫∫

G′

∣∣∣(S

(α)

GG♮ S

(α)H2

)(τ, ν)

∣∣∣2dτ dν

=

∫∫

G′

∣∣∣∣∫

τ ′

∫

ν′

S(α)

GG(τ ′, ν ′)S

(α)H2

(τ − τ ′, ν − ν ′) e−j2πφα(τ,ν,τ ′,ν′) dτ ′ dν ′∣∣∣∣2

dτ dν

≤∫∫

G′

[∫

τ1

∫

ν1

∣∣SGG (τ1, ν1)

∣∣2 dτ1 dν1

] [∫

τ2

∫

ν2

|SH2(τ − τ2, ν − ν2)|2 dτ2 dν2

]dτ dν

=∥∥GG

∥∥2

2‖H2‖2

2

∫∫

G′

dτ dν =∥∥GG

∥∥2

2‖H2‖2

2 4σG ,

where we used the support constraint of(S

(α)

GG♮ S

(α)H2

)(τ, ν), applied the Schwarz inequality, and noted

that the area of G′ is 4σG .

Discussion. The foregoing theorem shows that if σG is small, i.e., if H2 and H3 are jointly

DL underspread, then removing the non-DL part GG from G does not greatly affect the validity of

GH2 = H3:

GH2 = H3 =⇒ GGH2 ≈ H3 . (2.100)


Whereas small σG requires the GSFs of H2 and H3 to be essentially located along the τ or ν axis,

respectively, Theorem 2.20 can easily be extended to allow for arbitrary GSF orientations. Let us

assume that the GSFs of H2 and H3 are supported within a rectangular region G obtained by a

rotation of the rectangular region G using a rotation matrix A that corresponds to a metaplectic

operator U. We then have (cf. (2.82))

∥∥GeGH2 − H3

∥∥2

=∥∥GGH2 − H3

∥∥2, (2.101)

where the systems Hi = UHiU+, i = 2, 3, have GSFs supported within G and G = UGU+. Hence,

Theorem 2.20 directly applies to (2.101) and it is seen that the orientation of G is not relevant to

(2.97).

Since the operator norm is upper-bounded by the HS norm (see Appendix A), the approximation

(2.100) is also valid in the sense that the operator norm ‖GGH2 − H3‖O is small. Again, the bound

(2.97) can be viewed as an improvement on inequality (B.16). Note that (B.16)) holds for arbitrary

operators (i.e., also for non-DL operators) and, together with (2.99), implies that

∥∥GGH2 − H3

∥∥2

=∥∥GGH2

∥∥2≤∥∥GG

∥∥2‖H2‖2 or

∥∥GGH2

∥∥2∥∥GG

∥∥2‖H2‖2

≤ 1 .

Approximation for the GWS of G. The previous theorem, which shows that for H2, H3 jointly

DL underspread the non-DL part of G is “negligible” with respect to the operator equation (2.96), is

the basis for the next theorem which allows to approximate the GWS of GG (i.e., the smoothed GWS

of G according to (2.83)) by the ratio L(α)H3

(t, f)/L(α)H2

(t, f).

Theorem 2.21. Let the operators H2, H3, G, and GG be defined as in Theorem 2.20. Then, the

difference

∆(α)6 (t, f) , L

(α)

GG (t, f)L(α)H2

(t, f) − L(α)H3

(t, f)

is bounded as

∣∣∆(α)6 (t, f)

∣∣‖SG‖∞ ‖SH2‖1

≤ π

2cα σ

2G + 4σG ,

∥∥∆(α)6

∥∥2

‖G‖2 ‖H2‖2≤ π cα

√σ3G + 2

√σG ,

where cα = |α+ 1/2| + |α− 1/2|.

Proof. To prove the first bound (L∞ bound), we subtract and add L(α)

GGH2(t, f) from/to ∆

(α)6 (t, f) and

apply the triangle inequality,

∣∣∆(α)6 (t, f)

∣∣ ≤∣∣L(α)

GG (t, f)L(α)H2

(t, f) − L(α)

GGH2(t, f)

∣∣+∣∣L(α)

GGH2(t, f) − L

(α)H3

(t, f)∣∣ . (2.102)

The first term on the right-hand side of (2.102) can be bounded by using (2.65) and ‖SGG‖1 ≤‖SG‖∞ σG ,

∣∣L(α)

GG (t, f)L(α)H2

(t, f) − L(α)

GGH2(t, f)

∣∣ ≤ ‖SH2‖1 ‖SGG‖1

π

2cα σG ≤ ‖SH2‖1 ‖SG‖∞

π

2cα σ

2G .


The second term on the right hand side of (2.102) can be developed and bounded as

∣∣L(α)

GGH2(t, f) − L

(α)H3

(t, f)∣∣ =

∣∣L(α)

GGH2(t, f)

∣∣ ≤∫∫

G′

∣∣∣(S

(α)

GG♮ S

(α)H2

)(τ, ν)

∣∣∣ dτ dν

≤∥∥∥S(α)

GG♮ S

(α)H2

∥∥∥∞

∫∫

G′

dτ dν =∥∥∥S(α)

GG♮ S

(α)H2

∥∥∥∞

4σG

≤ ‖SH2‖1 ‖SGG‖∞ 4σG ≤ 4 ‖SH2‖1 ‖SG‖∞ σG ,

where we used Young’s inequality (B.14) with p = ∞, q = 1, ‖SGG‖∞ ≤ ‖SG‖∞, and the fact that the

support of(S

(α)

GG♮ S

(α)H2

)(τ, ν) is confined to G′, whose area is given by 4σG (see the proof of Theorem

2.20).

The second bound (L2 bound) is derived by again subtracting and adding L(α)

GGH2(t, f) from/to

∆(α)6 (t, f) and applying the triangle inequality,

∥∥∆(α)6

∥∥2≤∥∥L(α)

GGL(α)H2

− L(α)

GGH2

∥∥2+∥∥L(α)

GGH2− L

(α)H3

∥∥2

=∥∥L(α)

GGL(α)H2

− L(α)

GGH2

∥∥2+∥∥GGH2 − H3

∥∥2. (2.103)

The first term in (2.103) obeys the bound

∥∥L(α)

GGL(α)H2

− L(α)

GGH2

∥∥2≤ π cα

√σ3G

∥∥GG∥∥

2‖H2‖2 ≤ π cα

√σ3G

∥∥G∥∥

2‖H2‖2 (2.104)

that has been derived in [118,119]. The second term in (2.103) can be bounded using (2.97). The L2

bound for ∆(α)6 (t, f) is finally obtained by noting that ‖GG‖2 ≤ ‖G‖2.

Discussion. The above result shows that for jointly DL underspread operators H2, H3, i.e.,

operators having small joint displacement spread σG , it follows from GH2 = H3 that there is approx-

imately

L(α)

GG (t, f)L(α)H2

(t, f) ≈ L(α)H3

(t, f) . (2.105)

We note that small σG requires the GSFs of H2 and H3 to be essentially located along the τ or ν

axis, respectively. Yet, in the case of α = 0, Theorem 2.21 can be extended using a refinement of the

involved bounds via metaplectic transformations like that performed in Subsection 2.3.4.

Regularized Inversion in the TF Domain. The approximation (2.105) can again be used as

a basis for an approximate solution of (2.96) via a regularized inversion in the TF domain. Let us

define the operator G via its GWS as

L(α)eG

(t, f) ,

L(α)H3

(t, f)

L(α)H2

(t, f), for (t, f) ∈ R

0 , for (t, f) 6∈ R ,

(2.106)

where

R ,

{(t, f) :

∣∣L(α)H2

(t, f)∣∣

‖SH2‖1

≥ κ ǫ}, with ǫ ,

(π2cα σ

2G + 4σG

).


Since∣∣L(α)

H2(t, f)

∣∣ ≤ ‖SH2‖1, R can again be interpreted as the TF region where the magnitude of the

denominator L(α)H2

(t, f) in (2.106) is larger than κǫ = κ[

π2 cα σ

2G + 4σG

]times the upper bound ‖SH2‖1

on its maximum value. Theorem 2.21 implies that, for (t, f) ∈ R,

1

‖SG‖∞∣∣L(α)


(t, f)∣∣ =

1

‖SG‖∞

∣∣∣∣∣L(α)

GG (t, f) −L

(α)H3

(t, f)

L(α)H2

(t, f)

∣∣∣∣∣

=1

‖SG‖∞

∣∣∣∣∣L

(α)

GG (t, f)L(α)H2

(t, f) − L(α)H3

(t, f)

L(α)H2

(t, f)

∣∣∣∣∣

≤ 1

‖SG‖∞

‖SG‖∞ ‖SH2‖1

(π2 cα σ

2G + 4σG

)

∣∣L(α)H2

(t, f)∣∣

=‖SH‖1 ǫ∣∣L(α)H2

(t, f)∣∣

≤ ǫ

κ ǫ=

1

κ.

Hence, for κ large enough, it follows that within R

L(α)

GG (t, f) ≈ L(α)eG

(t, f) .

Since GG approximately satisfies (2.96), i.e., GGH2 ≈ H3, the preceding approximation implies that

the operator G obtained by the regularized TF inversion (2.106) approximately satisfies (2.96) as well.

This allows to approximately solve operator equations of the type (2.96) using algebraic operations in

the TF domain, thereby avoiding a computationally costly operator inversion:

GH2 = H3 =⇒ GGH2 ≈ H3

=⇒(L

(α)G

∗∗ ψ)(t, f) ≈ L

(α)eG

(t, f) =

L(α)H3

(t, f)

L(α)H2

(t, f), for (t, f) ∈ R,

0 , for (t, f) 6∈ R ,

with the smoothing function ψ(t, f) = L(α)T

(t, f). The operator G can finally be obtained from its

GWS via an inverse Weyl transformation (B.18). The application of this result to nonstationary signal

estimation will be discussed in Section 4.1.

2.3.8 Approximate Eigenvalues and Eigenfunctions

As mentioned in Subection 1.2.1, the complex sinusoids ef0(t) = ej2πf0t—i.e., signals with perfect

frequency concentration—are the generalized eigenfunctions [68] of any LTI system, with the transfer

function at frequency f0, G(f0), being the associated generalized eigenvalue. Thus, the response of

an LTI system to ef0(t) = ej2πf0t is G(f0) ej2πf0t. Similarly, the Dirac impulses δt0(t) = δ(t − t0)—

i.e., signals perfectly localized in time—are the generalized eigenfunctions of any LFI system, with

the temporal transfer function at time t0, m(t0), being the associated generalized eigenvalue. Thus,


the response of an LFI system to a Dirac impulse δ(t − t0) is given by m(t0) δ(t − t0). Note that

in the LTI and LFI cases the eigenfunctions are highly structured: the complex sinusoids ef (t) are

frequency-shifted versions of each other, and the Dirac impulses δt′(t) are time-shifted versions of each

other,

ef2(t) = (Ff2−f1ef1)(t) , δt2(t) = (Tt2−t1δt1)(t) ,

with Tτ and Fν denoting the time shift operator and the frequency shift operator, respectively (see

Subsection A.4). Moreover, the eigenvalues are given by the values of the transfer function. Thus, for

LTI and LFI systems the mathematical notion of an eigenvalue spectrum coincides with the engineering

notion of a transfer function.

The situation is different in the case of general LTV systems. The eigenfunctions (singular func-

tions) of different LTV systems are different (unless the systems commute [64, 158]). Furthermore,

the eigenfunctions (singular functions) of general LTV systems are not localized and structured in any

sense and the eigenvalues (singular values) are not equal to the TF transfer function (i.e. GWS) values.

However, we will now show that underspread systems have a well-structured set of TF-localized “ap-

proximate eigenfunctions,” with the associated “approximate eigenvalues” given by the GWS values.

(Note that it is not necessary to consider approximate singular functions since according to Subsection

2.3.17 underspread operators are approximately normal.)

Let s(t) be a normalized function that is well concentrated about the origin of the TF plane (e.g.,

a Gaussian function). We consider the family of functions11

st0,f0(t) =(S

(1/2)t0,f0

s)(t) = s(t− t0) e

j2πf0t

obtained by TF-shifting s(t) to the TF point (t0, f0). By construction, st0,f0(t) is then well TF-

concentrated about (t0, f0) and two functions out of this set are related (up to a phase factor) by a

TF shift,

st2,f2(t) = ej2πf1(t1−t2)(S

(1/2)t2−t1,f2−f1

st1,f1

)(t) .

In the following, we shall state conditions under which the response of an LTV system to st0,f0(t) is

approximately L(α)H

(t0, f0) st0,f0(t), which implies that st0,f0(t) is an “approximate eigenfunction” of

H with L(α)H

(t0, f0) the associated “approximate eigenvalue.” The families of TF shifted functions

st0,f0(t) are sometimes called “Weyl-Heisenberg” function sets. They have previously been considered

as approximate eigenfunctions of DL operators [118, 127], and the next theorem is essentially an

adaptation and extension of these results.

Theorem 2.22. For any LTV system H, TF point (t0, f0), and normalized function s(t) (i.e., ‖s‖2 =

1), the difference

∆(α)7 (t) ,

(Hst0,f0

)(t) − L

(α)H

(t0, f0) st0,f0(t)

11Note that the specific choice of α = 1/2 for the joint TF shift S(α)t0,f0

is made for notational simplicity; choosing a

different α leaves the subsequent arguments unchanged. Furthermore, this α value is not related to that in L(α)H

(t, f)

used in our subsequent development.


is bounded as

∥∥∆(α)7

∥∥2

‖SH‖1

≤ D(α)H,s with D

(α)H,s ,

√2π C

(α)H

+m(φs)H+H

+ 2m(φs)H

, (2.107)

with the weighting function φs(τ, ν) =∣∣1 − A

(α)s (τ, ν)

∣∣ where A(α)s (τ, ν) is the generalized ambiguity

function (see B.2.4) and C(α)H

as defined in (2.70).

Proof. First, we develop the square of∥∥∆(α)

7

∥∥2

as

∥∥∆(α)7

∥∥2

2= ‖Hst0,f0‖2

2 − 2ℜ{L

(α)∗H

(t0, f0) 〈Hst0,f0, st0,f0〉}

+∣∣L(α)

H(t0, f0)

∣∣2 ‖st0,f0‖22.

Using ‖st0,f0‖2 = ‖s‖2 = 1, subtracting and adding 2L(α)∗H

(t0, f0)L(α)H

(t0, f0), and applying the triangle

inequality yields

∥∥∆(α)7

∥∥2

2= ‖Hst0,f0‖2

2 − 2ℜ{L

(α)∗H

(t0, f0)[〈Hst0,f0 , st0,f0〉 − L

(α)H

(t0, f0)]}

−∣∣L(α)

H(t0, f0)

∣∣2

≤∣∣∣‖Hst0,f0‖2

2 −∣∣L(α)

H(t0, f0)

∣∣2∣∣∣+ 2

∣∣L(α)H

(t0, f0)∣∣ ∣∣〈Hst0,f0, st0,f0〉 − L

(α)H

(t0, f0)∣∣. (2.108)

By subtracting and adding L(α)H+H

(t0, f0), applying (B.10) to the quadratic form 〈H+H st0,f0 , st0,f0〉,using A

(α)st0,f0

(τ, ν) = A(α)s (τ, ν) ej2π(τf0−νt0), and recalling the definition of ∆

(α)4 (t, f) in Corollary 2.17,

the first term in (2.108) becomes

∣∣∣‖Hst0,f0‖22 −

∣∣L(α)H

(t0, f0)∣∣2∣∣∣ =

∣∣∣⟨H+H st0,f0, st0,f0

⟩− L

(α)H+H

(t0, f0) + L(α)H+H

(t0, f0) − |L(α)H

(t0, f0)|2∣∣∣

=

∣∣∣∣∫

τ

∫

νS

(α)H+H

(τ, ν)A(α)∗st0,f0

(τ, ν) dτ dν−∫

τ

∫

νS

(α)H+H

(τ, ν) ej2π(νt0−τf0) dτ dν + ∆(α)4 (t0, f0)

∣∣∣∣

≤∣∣∣∣∫

τ

∫

νS

(α)H+H

(τ, ν)[A(α)∗

s (τ, ν) − 1]ej2π(νt0−τf0) dτ dν

∣∣∣∣+∣∣∆(α)

4 (t0, f0)∣∣

≤∫

τ

∫

ν|SH+H(τ, ν)|

∣∣1 −A(α)s (τ, ν)

∣∣dτ dν +∣∣∆(α)

4 (t0, f0)∣∣ (2.109)

≤ ‖SH+H‖1m(φs)H+H

+ ‖SH‖21 2π C

(α)H

≤ ‖SH‖21

[m

(φs)H+H

+ 2π C(α)H

],

where we furthermore used the triangle inequality, the bound (2.70), and Young’s inequality (B.14)

with p = q = r = 1, i.e., ‖SH+H‖1 ≤ ‖SH‖21. In a similar manner, the factor of the second term in

(2.108) can be developed and bounded as

∣∣ 〈Hst0,f0, st0,f0〉 − L(α)H

(t0, f0)∣∣ =

∣∣∣∣∫

τ

∫

νS

(α)H

(τ, ν)[A(α)∗

s (τ, ν) − 1]ej2π(νt0−τf0) dτ dν

∣∣∣∣

≤∫

τ

∫

ν|SH(τ, ν)|

∣∣1 −A(α)s (τ, ν)

∣∣dτ dν = ‖SH‖1m(φs)H

. (2.110)

Inserting these two bounds into (2.108) and using |L(α)H

(t0, f0)| ≤ ‖SH‖1 yields the bound (2.107).

Discussion. For an underspread system H where C(α)H

, m(φs)H+H

, and m(φs)H

can be made small such

that D(α)H,s will be small, we thus have the “approximate eigenvalue relation”

(H st0,f0

)(t) ≈ L

(α)H

(t0, f0) st0,f0(t) , (2.111)


(a) (b) (c) (d)

t- t- t-

t- t- t- t-

f

6

f

6

f

6

f

6

Figure 2.7: Approximate eigenfunctions and eigenvalue interpretation of GWS: (a) Wigner distri-

bution (top) and real and imaginary part (bottom) of input signal st0,f0(t), (b) Weyl symbol of H, (c)

output signal (Hst0,f0)(t), and (d) input signal multiplied by corresponding TF transfer function value

LH(t0, f0). The signal length is 128 samples and the (normalized) frequency ranges from −1/4 to 1/4.

which implies that properly TF-localized functions are approximate eigenfunctions and the GWS

L(α)H

(t, f) is an approximate eigenvalue distribution over the TF plane. In particular, since the effective

duration and bandwidth of the normalized function s(t) correspond to the second-order derivatives of

A(α)s (τ, ν) at the origin [162,212],

T 2s ,

∫

tt2|s(t)|2dt = − 1

4π2

∂2A(α)s (τ, ν)

∂ν2

∣∣∣(0,0)

, F 2s ,

∫

ff2|S(f)|2df = − 1

4π2

∂2A(α)s (τ, ν)

∂τ2

∣∣∣(0,0)

,

the GAF of a well TF-localized function s(t) (i.e., having small TsFs) satisfies A(α)s (τ, ν) ≈ A

(α)s (0, 0) ≡

1 around the origin which further implies φs(τ, ν) ≈ 0 for small τ , ν. Thus, it is seen that small m(φs)H

and m(φs)H+H

requires that the effective supports of |SH(τ, ν)| and of |SH+H(τ, ν)| are small, i.e., that

H is underspread12.

The boundD(α)H,s is tightest for α = 0 since here C

(α)H

is smallest. Furthermore, C(0)H

can be replaced

by infU∈M C(0)UHU+ which allows for oblique orientation of the GSF of H. We note that also m

(φs)H

and m(φs)H+H

can be adapted to oblique GSF orientation; this can be achieved by proper choice of the

function s(t) according to (2.22).

An example illustrating the approximation (2.111) for the case α = 0 is shown in Fig. 2.7. In

this example, the normalized error is

∥∥∆(0)7

∥∥2

‖SH‖1= 0.38 while the corresponding bound in (2.107) is

D(0)H,s = 0.73.

DL Operators. From the proof of Theorem 2.22 (see (2.108), (2.109), and (2.110)), it follows

12Note that S(α)

H+H(τ, ν) =

“S

(α)

H+♮S(α)H

”(τ, ν) and that twisted convolution enlarges the effective support maximally

by a factor of four [118].


that

∥∥∆(α)7

∥∥2

2≤∫

τ

∫

ν|SH+H(τ, ν)|

∣∣1 −A(α)s (τ, ν)

∣∣dτ dν +∣∣∆(α)

4 (t0, f0)∣∣

+ 2‖SH‖1

∫

τ

∫

ν|SH(τ, ν)|

∣∣1 −A(α)s (τ, ν)

∣∣dτ dν .(2.112)

In the case of a DL operator H with GSF support contained in GH = [−τ (max)H

, τ(max)H

] ×[−ν(max)

H, ν

(max)H

], we can further develop (2.112) by applying (2.74) (valid for cασH ≤ 2 which also

ensures |α|σH ≤ 1 since 2|α| ≤ cα) and by noting that the support of |SH+H(τ, ν)| is contained in

G′ = [−2τ(max)H

, 2τ(max)H

] × [−2ν(max)H

, 2ν(max)H

],

∥∥∆(α)7

∥∥2

2≤∫∫

G′

|SH+H(τ, ν)|∣∣1 −A(α)

s (τ, ν)∣∣ dτ dν + 2‖SH‖1

∫∫

GH

|SH(τ, ν)|∣∣1 −A(α)

s (τ, ν)∣∣ dτ dν

+ ‖SH‖21

[2 sin

(π4cασH

)+ 2 sin

(π2|α|σH

)]

≤ φ(max)2

∫∫

G′

|SH+H(τ, ν)| dτ dν + 2‖SH‖1φ(max)1

∫∫

GH


+ 2‖SH‖21

[sin(π

4cασH

)+ sin

(π2|α|σH

)]

= φ(max)2 ‖SH+H‖1 + 2‖SH‖2

1

[φ

(max)1 + sin

(π4cασH

)+ sin

(π2|α|σH

)],

where φ(max)1 , max(τ,ν)∈GH

φs(τ, ν) and φ(max)2 , max(τ,ν)∈G′ φs(τ, ν). Thus, using ‖SH+H‖1 ≤ ‖SH‖2

1,

we finally obtain the bounds∥∥∆(α)

7

∥∥2

2

‖SH‖21

≤ φ(max)2 + 2φ

(max)1 + 2 sin

(π4cασH

)+ 2 sin

(π2|α|σH

)≤ 3φ

(max)2 + 4 sin

(π4σH

), (2.113)

where the second (coarser but simpler) bound requires |α| ≤ 1/2 (in which case cα = 1) in addition

to cασH ≤ 2 and exploits the fact that φ(max)1 ≤ φ

(max)2 .

Non-DL Operators. Next , let us consider the case where H is not DL but erroneously assumed

to be DL with GSF support region G = [−τG, τG ] × [−νG , νG ]. Then, the first term on the right hand

side of (2.112) can be developed as∫

τ

∫

ν|SH+H(τ, ν)|

∣∣1 −A(α)s (τ, ν)

∣∣dτ dν

=

∫∫

G′

|SH+H(τ, ν)|∣∣1 −A(α)

s (τ, ν)∣∣dτ dν +

∫∫

G′

|SH+H(τ, ν)|∣∣1 −A(α)

s (τ, ν)∣∣dτ dν

≤ φ(max)2 ‖SH+H‖1 + 2

∫∫

G′

|SH+H(τ, ν)| dτ dν

≤ φ(max)2 ‖SH‖2

1 + 2‖SH+H‖1

(m

(1,0)H+H

2τG+m

(0,1)H+H

2νG

)

≤ φ(max)2 ‖SH‖2

1 + 2‖SH‖21

(m

(1,0)H

τG+m

(0,1)H

νG

), (2.114)


where we used G′ = [−2τG , 2τG ] × [−2νG , 2νG ], φ(max)2 = max(τ,ν)∈G′ φs(τ, ν), the fact that

∣∣1 −A

(α)s (τ, ν)

∣∣ ≤ 2, the first Chebyshev-like inequality in (2.38), as well as (2.33) and (2.34) (with H1 = H

and H2 = H+). Similarly, the third term on the right hand side of (2.112) can be developed as

2‖SH‖1

∫

τ

∫

ν|SH(τ, ν)|

∣∣1 −A(α)s (τ, ν)

∣∣dτ dν

= 2‖SH‖1

∫∫

G

|SH(τ, ν)|∣∣1 −A(α)

s (τ, ν)∣∣dτ dν + 2‖SH‖1

∫∫

G

|SH(τ, ν)|∣∣1 −A(α)

s (τ, ν)∣∣dτ dν

≤ 2‖SH‖21 φ

(max)1 + 4‖SH‖1

∫∫

G


≤ 2‖SH‖21 φ

(max)1 + 4‖SH‖2

1

(m

(1,0)H

τG+m

(0,1)H

νG

), (2.115)

where we used φ(max)1 = max(τ,ν)∈G φs(τ, ν),

∣∣1−A(α)s (τ, ν)

∣∣ ≤ 2, and the first Chebyshev-like inequality

in (2.38). Furthermore, (2.75) can be applied to the second term∣∣∆(α)

4 (t0, f0)∣∣ on the right hand side

of (2.112). Upon combining (2.75) with (2.114) and (2.115) and using σG = 4τGνG , we finally obtain∥∥∆(α)

7

∥∥2

2

‖SH‖21

≤ φ(max)2 + 2φ

(max)1 + 2 sin

(π4cασG

)+ 2 sin

(π2|α|σG

)(2.116)

+ 8

(m

(1,0)H

τG+m

(0,1)H

νG

)+ 2

(m

(1,0)H

τG+m

(0,1)H

νG

)2

. (2.117)

By comparing this expression (which is valid for any system H) with the first bound in (2.113), it

is seen that the bound (2.113), erroneously applied to a non-DL operator (erroneously assumed to

have displacement spread σG) might deviate from the correct bound (2.117) by as much as 8(

m(1,0)H

τG+

m(0,1)H

νG

)+ 2(

m(1,0)H

τG+

m(0,1)H

νG

)2.

2.3.9 Approximate Diagonalization

A normal operator is diagonalized by its eigenfunctions {uk(t)}, i.e.,

〈Huk, uk′〉 = 〈λk uk, uk′〉 = λk〈uk, uk′〉 = λk δkk′ .

This diagonalization property is of major importance in many applications. In the subsequent, we shall

establish an approximate diagonalization result for underspread operators. To this end, we consider

biorthogonal13 Weyl-Heisenberg function sets [56,100] {uk,l(t)}, {vk,l(t)} (note that these function sets

are now doubly indexed). These function sets are obtained by TF shifting two normalized functions

u(t), v(t),

uk,l(t) = u(t− kT ) ej2πlF t , vk,l(t) = v(t− kT ) ej2πlF t ,

with14 TF ≥ 1, and they are assumed to satisfy the biorthogonality condition

〈uk,l, vk′,l′〉 = δkk′ δll′ .13Orthogonal Weyl-Heisenberg function sets suffer from poor TF localization [56] and are thus not considered here.14Note that the condition TF ≥ 1 implies that the function sets {uk,l(t)}, {vk,l(t)} are incomplete. This is no serious

restriction since we here are not concerned with complete operator expansions (cf. [118]).


An important consequence of this biorthogonality condition is the fact that the generalized cross

ambiguity function (see Subsection B.2.4) of u(t) and v(t) vanishes on the lattice (kT, lF ) for k 6= 0

and l 6= 0, i.e.,∣∣A(α)

u,v(kT, lF )∣∣ = δk0 δl0 . (2.118)

This property will be important for the interpretation of the following theorem. We note that ap-

proximate operator diagonalization has previously been considered (partly at a mathematically much

more sophisticated level) in [36,42,54,64,118,120, 128,137,138,153,179].

Theorem 2.23. For any LTV system H and any biorthogonal Weyl-Heisenberg sets {uk,l(t)},{vk,l(t)}, the difference

∆(α)8 [k, l; k′, l′] ,

⟨Huk,l, vk′,l′

⟩− L

(α)H

(kT, lF ) δkk′ δll′

is bounded as ∣∣∆(α)8 [k, l; k′, l′]

∣∣‖SH‖1

≤ m

(φ

(k−k′,l−l′)u,v

)H

, (2.119)

with φ(k,l)u,v (τ, ν) =

∣∣δk0 δl0 −A(α)v,u

(τ + kT, ν + lF

)∣∣.

Proof. First note that ∆(α)8 [k, l; k′, l′] can be developed as

∆(α)8 [k, l; k′, l′] =


⟩− L

(α)H

(kT, lF )δkk′ δll′

=⟨S

(α)H, A(α)

vk′ ,l′ ,uk,l

⟩− δkk′ δll′

∫

τ

∫

νS

(α)H

(τ, ν) ej2π(νkT−τlF ) dτ dν

=

∫

τ

∫

νS

(α)H

(τ, ν)[A(α)∗

vk′,l′ ,uk,l(τ, ν) − δkk′ δll′ e

j2π(νkT−τlF )]dτ dν .

Specializing this expression to k = k′, l = l′ and using A(α)vk,l,uk,l(τ, ν) = A

(α)v,u(τ, ν) e−j2π(νkT−τlF ), we

obtain

∣∣∆(α)8 [k, l; k, l]

∣∣ =

∣∣∣∣∫

τ

∫

νS

(α)H

(τ, ν)[A(α)∗

v,u (τ, ν) − 1]ej2π(νkT−τlF ) dτ dν

∣∣∣∣

≤∫

τ

∫

ν

∣∣SH(τ, ν)∣∣∣∣∣1 −A(α)

v,u(τ, ν)∣∣∣ dτ dν = ‖SH‖1m

(φ(0,0)u,v )

H,

which proves (2.119) for the case k = k′, l = l′. For k 6= k′, l 6= l′, the bound in (2.119) is shown

similarly,

∣∣∆(α)8 [k, l; k′, l′]

∣∣ =

∣∣∣∣∫

τ

∫

νS

(α)H

(τ, ν)A(α)∗vk′ ,l′ ,uk,l

(τ, ν) dτ dν

∣∣∣∣

≤∫

τ

∫

ν

∣∣SH(τ, ν)∣∣ ∣∣A(α)

v,u

(τ + (k − k′)T, ν + (l − l′)F

)∣∣ dτ dν = ‖SH‖1m

(φ


)H

,

where we used∣∣A(α)

vk′,l′ ,uk,l(τ, ν)∣∣ =

∣∣A(α)v,u

(τ + (k − k′)T, ν + (l − l′)F

)∣∣ .


k′@

@@l′

-3 -2 -1 0 1 2 3

-3 -172 -160 -147 -153 -155 -160 -166

-2 -130 -117 -104 -110 -113 -117 -123

-1 -101 -86 -69 -54 -74 -91 -112

0 -84 -66 -49 0 -35 -53 -70

1 -106 -86 -68 -54 -70 -86 -106

2 -123 -116 -98 -91 -107 -116 -121

3 -165 -159 -141 -134 -150 -158 -164

Table 2.3: Ratio 〈Hu0,0, vk′,l′〉/〈Hu0,0, v0,0〉 of off-diagonal elements to diagonal element in dB for

k′ = −3, . . . , 3 and l′ = −3, . . . , 3.

Discussion. The preceding theorem shows that for underspread systems where m

(φ

(k,l)u,v

)H

can be

made small by suitable choice of u(t) and v(t), one has


⟩≈ L

(α)H

(kT, lF ) δkk′ δll′ =

L

(α)H

(kT, lF ) , for k = k′, l = l′ ,

0 , for k 6= k′, l 6= l′ .(2.120)

For well TF-localized u(t), v(t), it can be shown that A(α)v,u(τ, ν) ≈ A

(α)v,u(0, 0) = 1 or equivalently

φ(0,0)u,v (τ, ν)

∣∣1−A(α)v,u(τ, ν)

∣∣ ≈ 0 about the origin of the (τ, ν)-plane. Thus, it is seen that small m(φ

(0,0)u,v )

H

requires that |SH(τ, ν)| is concentrated about the origin, i.e., that H is underspread. Furthermore,

(2.118) implies that for k 6= 0, l 6= 0,

φ(k,l)u,v (0, 0) =

∣∣A(α)u,v(kT, lF )

∣∣ = 0 , k 6= 0, l 6= 0 .

Thus, for H underspread (with |SH(τ, ν)| concentrated about the origin) also m

(φ


)H

for k 6= k′,

l 6= l′ will be small. We can thus conclude that well TF-localized biorthogonal Weyl-Heisenberg func-

tion sets approximately diagonalize underspread operators in the sense that the off-diagonal elements⟨Huk,l, vk′,l′

⟩, k 6= k′, l 6= l′, are approximately zero and the GWS values at the grid points (kT, lF )

approximately equal the “diagonal” elements⟨Huk,l, vk,l

⟩.

The approximation (2.120) is illustrated in Fig. 2.8 and Table 2.3 for α = 0 and the same LTV

system as in Fig. 2.7. It is seen from Table 2.3 that the “off-diagonal” values 〈Hu0,0, vk′,l′〉 (with

k′ 6= 0, l′ 6= 0) are substantially smaller than the “diagonal element” 〈Hu0,0, v0,0〉. In particular,

in this example, the normalized difference between 〈Hu0,0, v0,0〉 and L(0)H

(0, 0) is|∆

(0)8 [0,0;0,0]|‖SH‖1

= 0.03

and the corresponding bound is m(φu,v)H

= 0.14. Furthermore, the largest normalized “off-diagonal”

element and associated bound are|∆

(0)8 [0,0;1,0]|‖SH‖1

|〈Hu0,0,v1,0〉|‖SH‖1

= 0.021 and m

(φ

(−1,0)u,v

)H

= 0.098, respectively.


−0.5 −0.25 0 0.25 0.5−100

−80

−60

−40

−20

0

20

−0.5 −0.25 0 0.25 0.5−100

−80

−60

−40

−20

0

20

(a) (b) (c)f

t

f

t

-

-

-

-

ν

τ

6

-

Figure 2.8: Prototypes u(t) and v(t) of a biorthogonal Weyl-Heisenberg function set with T = 16

and F = 1/8. (a) Prototype u(t) (top) and magnitude of its Fourier transform (bottom, in dB); (b)

prototype v(t) (top) and magnitude of its Fourier transform (bottom, in dB); (c) magnitude of cross

ambiguity function of u(t) and v(t); note the zeros on the grid (kT, lF ). The signals have length 128

and the (normalized) frequency ranges from −1/2 to 1/2.

2.3.10 Input-Output Relation for Deterministic Signals Based on the GeneralizedWeyl Symbol

In this subsection, we discuss a “TF input-output relation” for deterministic signals based on the

short-time Fourier transform [61,84,157,169] (STFT) and the GWS. We note that TF input-output

relations for random processes will be presented in Section 3.6.

For LTI systems, due to the fact that the complex sinusoids are eigenfunctions, the Fourier trans-

form of the output signal (Hx)(t) is given by G(f)X(f). Similarly, for LFI systems (Hx)(t) equals

m(t)x(t). Note that these relations involve the signals and the transfer functions in a linear manner.

In the case of LTV systems, a similar multiplicative “input-output relation” in the TF domain may

be desirable; this input-output relation should involve the signals and the transfer function (GWS) in

a linear manner, too. We shall select the STFT [61, 84, 157, 169] as TF signal representation since it

is the only linear TF signal representation that is covariant, up to an unavoidable phase factor, to TF

shifts [93]. According to Subsection B.2.1, the STFT is given by

STFT(s)x (t, f) ,

∫

t′x(t′) s∗(t′ − t) e−j2πft′ dt′, (2.121)

where s(t) is a normalized window. Assuming s(t) to have good TF concentration and H to be

underspread, we know from Subsection 2.3.8 that the TF shifted functions st0,f0(t) = s(t− t0)ej2πf0t

are approximate eigenfunctions of H, i.e.,(Hst0,f0

)(t) ≈ L

(α)H

(t0, f0) st0,f0(t). Hence, applying H to

both sides of the STFT inversion formula (see (B.38))

x(t) =

∫

t0

∫

f0

STFT(s)x (t0, f0) st0,f0(t) dt0 df0 , (2.122)

we obtain

(Hx)(t) =

(H

{∫

t0

∫

f0

STFT(s)x (t0, f0)st0,f0 dt0 df0

})(t)


=

∫

t0

∫

f0

STFT(s)x (t0, f0)(Hst0,f0)(t) dt0 df0

≈∫

t0

∫

f0

STFT(s)x (t0, f0)L

(α)H

(t0, f0) st0,f0(t) dt0 df0 .

Comparing this with

(Hx)(t) =

∫

t0

∫

f0

STFT(s)Hx(t0, f0) st0,f0(t) dt0 df0

(obtained from (2.122) by replacing x(t) with (Hx)(t)) suggests that there might be STFT(s)Hx(t, f) ≈

STFT(s)x (t, f)L

(α)H

(t, f) (although of course equal integrals do not imply that the integrands are equal).

The next theorem gives a bound on the quality of this approximation.

Theorem 2.24. For any LTV system H, any signal x(t), and any normalized function s(t), the

difference

∆(α)9 (t, f) , STFT

(s)Hx(t, f) − L

(α)H

(t, f) STFT(s)x (t, f)


9 (t, f)∣∣

‖SH‖1 ‖x‖2

≤ D(α)H+,s

+ 4π|α|m(1,1)H

,

∥∥∆(α)9

∥∥2

‖H‖2 ‖x‖2

≤√

2M(φ′

s)H

+ 4π|α|M (1,1)H

, (2.123)

with the weighting functions φs(τ, ν) =∣∣1 − A

(α)s (τ, ν)

∣∣ (used in D(α)H+,s

, see (2.107)) and φ′s(τ, ν) =√1 − Re{A(α)

s (τ, ν)} (used in M(φ′

s)H

).

Proof. Using the STFT definition (B.37), we have

∆(α)9 (t, f) =

⟨Hx, st,f

⟩− L

(α)H

(t, f) 〈x, st,f 〉 =⟨x,[H+ − L

(α)∗H

(t, f)I]st,f

⟩

=⟨x,[H+ − L

(α)H+(t, f) I + L

(α)H+(t, f) I − L

(α)∗H

(t, f) I]st,f

⟩

= ∆(α)A (t, f) + ∆

(α)B (t, f) (2.124)

with

∆(α)A (t, f) ,

⟨x,[H+ − L

(α)H+(t, f)I

]st,f

⟩, ∆

(α)B (t, f) ,

[L

(α)H+(t, f) − L

(α)∗H

(t, f)]∗⟨

x, st,f

⟩.

Using Schwarz’ inequality, (2.107), and ‖SH+‖1 = ‖SH‖1, we obtain

∣∣∆(α)A (t, f)

∣∣ ≤ ‖x‖2

∥∥H+st,f − L(α)H+(t, f) st,f

∥∥2≤ ‖x‖2 ‖SH+‖1D

(α)H+,s

= ‖x‖2 ‖SH‖1D(α)H+,s

, (2.125)

while applying Schwarz’ inequality to∣∣∆(α)

B (t, f)∣∣ and using (2.48) as well as ‖st,f‖2 = 1 yields

∣∣∆(α)B (t, f)

∣∣ ≤∣∣L(α)

H+(t, f) − L(α)∗H

(t, f)∣∣ ‖x‖2 ‖st,f‖2 ≤ ‖x‖2 ‖SH‖1 4π|α|m(1,1)

H. (2.126)

The first (L∞) bound in (2.123) then follows upon inserting (2.125) and (2.126) into∣∣∆(α)

9 (t, f)∣∣ ≤∣∣∆(α)

A (t, f)∣∣+∣∣∆(α)

B (t, f)∣∣ (cf. (2.124)).

To derive the second (L2) bound in (2.123), we note that

∥∥∆(α)A

∥∥2

2=

∫

t

∫

f

∣∣⟨x,H+st,f − L(α)H+(t, f) st,f

⟩∣∣2 dt df ≤∫

t

∫

f‖x‖2

2 ‖H+st,f − L(α)H+(t, f)st,f‖2

2 dt df


= ‖x‖22

∫

t

∫

f

[‖H+st,f‖2

2 − 2ℜ{L

(α)∗H+ (t, f) 〈H+st,f , st,f 〉

}+ |L(α)

H+(t, f)|2‖st,f‖22

]dt df .

(2.127)

Next, we develop the three terms in this expression. Using (B.10), it can be shown that

∫

t

∫

f‖H+st,f‖2

2 dt df =

∫

t

∫

f〈HH+st,f , st,f 〉 dt df

=

∫

t

∫

f

[ ∫

τ

∫

νS

(α)HH+(τ, ν)A(α)∗

s (τ, ν) ej2π(νt−τf) dτ dν

]dt df

=

∫

τ

∫

νS

(α)HH+(τ, ν)A(α)∗

s (τ, ν) δ(τ) δ(ν) dτ dν

= S(α)HH+(0, 0) = Tr

{HH+

}= ‖SH‖2

2 , (2.128)

where we used A(α)s (0, 0) = ‖s‖2

2 = 1 and (B.7). Similarly,

∫

t

∫

fL

(α)∗H+ (t, f) 〈H+st,f , st,f 〉 dt df =

∫

t

∫

fL

(α)∗H+ (t, f)

[ ∫

τ

∫

νS

(α)H+(τ, ν)A(α)∗

s (τ, ν) ej2π(νt−τf) dτ dν

]dt df

=

∫

τ

∫

ν|SH+(τ, ν)|2A(α)∗

s (τ, ν) dτ dν . (2.129)

Finally, with ‖st,f‖22 = 1 we have

∫

t

∫

f

∣∣L(α)H+(t, f)

∣∣2 ‖st,f‖22 dt df =

∥∥L(α)H

∥∥2

2= ‖SH‖2

2. (2.130)

Inserting (2.128), (2.129), and (2.130) in (2.127) yields the following L2 bound for ∆(α)A (t, f),

∥∥∆(α)A

∥∥2

2≤ ‖x‖2

2

[2

∫

τ

∫

ν|SH(τ, ν)|2 dτ dν − 2ℜ

{∫

τ

∫

ν|SH(τ, ν)|2A(α)∗

s (τ, ν) dτ dν}]

= 2 ‖x‖22

∫

τ

∫

ν|SH(τ, ν)|2

[1 −ℜ

{A(α)

s (τ, ν)}]dτ dν = 2 ‖x‖2

2 ‖H‖22

[M

(φ′s)

H

]2. (2.131)

The L2 norm of ∆(α)B (t, f) can be bounded using Schwarz’ inequality and (2.48),

∥∥∆(α)B

∥∥2

2≤∫

t

∫

f

∣∣L(α)H+(t, f) − L

(α)∗H

(t, f)∣∣2 ‖x‖2

2 ‖st,f‖22 dt df ≤ ‖x‖2

2 ‖H‖22

[4π|α|M (1,1)

H

]2. (2.132)

The second bound (L2 bound) in (2.123) then follows by inserting (2.131) and (2.132) into∥∥∆(α)

9

∥∥2≤∥∥∆(α)

A

∥∥2+∥∥∆(α)

B

∥∥2

(cf. (2.124)).

Discussion. Theorem 2.24 shows that if D(α)H+,s

(see Subsection 2.3.8) and M(φ′

s)H

can be made

small by suitable choice of s(t), and if m(1,1)H

and M(1,1)H

are small, we obtain the approximate input-

output relation

STFT(s)Hx(t, f) ≈ L

(α)H

(t, f) STFT(s)x (t, f) . (2.133)

We note that small M(φ′

s)H

requires that ℜ{A

(α)s (τ, ν)

}≈ A

(α)s (0, 0) ≡ 1 on the effective support of

|SH(τ, ν)|, which in turn requires H to be underspread with small effective GSF support. Small m(1,1)H


(a) (b) (c) (d)t- t- t- t-

f

6

f

6

f

6

f

6

Figure 2.9: Approximate multiplicative input-output relation: (a) STFT magnitude of chirp-like input

signal x(t), (b) Weyl symbol of H, (c) STFT magnitude of output signal, and (d) magnitude of the

product of STFT of input signal and TF transfer function, L(0)H

(t, f) STFT(s)x (t, f). The signal length

is 128 samples and the (normalized) frequency ranges from −1/4 to 1/4.

and M(1,1)H

require that S(α)H

(τ, ν) be localized along the τ and/or ν axis respectively. The latter

condition can be relaxed in the case α = 0. Here, the second terms in the bounds (2.123) are zero and

the bounds are tightest in this case. Furthermore, these bounds can further be refined by exploiting

the covariance of the Weyl symbol to metaplectic transforms. These refined bounds also allow for

systems with GSF oriented in oblique directions (see the discussion at the end of Subsections 2.3.4,

2.3.6, and 2.3.8).

An example of the approximation (2.133) is shown in Fig. 2.9 for α = 0. In this example, the

normalized errors were maxt,f|∆

(0)9 (t,f)|

‖SH‖1 ‖x‖2= 0.16 and

‖∆(0)9 ‖

2‖H‖2 ‖x‖2

= 0.075 while the corresponding bounds

in (2.123) were D(0)H+,s

= 0.62 and√

2M(φ′

s)H

= 0.116, respectively.

DL Operators. Since a comparison of the L∞ bounds for non-DL and DL operators is cumber-

some, we restrict ourselves to the L2 bounds. Here, one can show that for DL H with GSF support

contained within GH = [−τ (max)H

, τ(max)H

] × [−ν(max)H

, ν(max)H

] and for |α|σH ≤ 1, one has

∥∥∆(α)9

∥∥2≤

√2φ(max) + 2 sin

(π2|α|σH

)(2.134)

with φ(max) = max(τ,ν)∈GHφ′s(τ, ν).

Non-DL Operators. On the other hand, for non-DL operators and any region G = [−τG , τG ] ×[−νG, νG ] of area σG = 4τGνG , one can show that for |α|σG ≤ 1,

∥∥∆(α)9

∥∥2≤

√2φ(max) + 2 sin

(π2|α|σG

)+ 2

√2

(M

(1,0)H

τG+M

(0,1)H

νG

), (2.135)

with φ(max) = max(τ,ν)∈G φ′s(τ, ν). It is thus seen that the bound (2.134), when erroneously applied to

non-DL operators, might deviate from the bound (2.135) by as much as 2√

2(

M(1,0)H

τG+

M(0,1)H

νG

)from

the correct bound (2.135).


2.3.11 (Multi-Window) STFT Filter Approximation of Time-Varying Systems

The input-output relation STFT(s)Hx(t, f) ≈ L

(α)H

(t, f) STFT(s)x (t, f) valid in the underspread case sug-

gests that the output signal (Hx)(t) can approximately be obtained by applying an inverse STFT

to L(α)H

(t, f) STFT(s)x (t, f) (note, however, that in general this product is not a valid STFT). This

corresponds to an approximate TF implementation of H that consists of calculating the STFT of

the input signal, STFT(s)x (t, f), multiplying by the TF transfer function L

(α)H

(t, f), and applying the

inverse STFT (2.122) to L(α)H

(t, f) STFT(s)x (t, f). The resulting LTV system is called an STFT filter

and will be denoted by H(α). Using (2.122) and (2.121), the STFT filter’s input-output relation is

(the subscript in y1 and H(α)1 will become clear shortly)

y1(t) ,(H

(α)1 x

)(t) =

∫

t′

∫

f ′

STFT(s)x (t′, f ′)L

(α)H

(t′, f ′) st′,f ′(t) dt′ df ′

=

∫

t′

∫

f ′

〈x, st′,f ′〉L(α)H

(t′, f ′) st′,f ′(t) dt′ df ′ .

Such STFT filters have been considered previously [24, 39, 42, 47, 118, 123, 125, 157, 169], mostly with

weighting functions other than L(α)H

(t, f). They are an intuitively appealing, practical way of imple-

menting an LTV filter. In a TF-discretized form, they permit time-varying subband signal processing

using the Gabor expansion [53,55,56,66] or, equivalently, DFT filter banks [22,23,39,135,166,196,204].

Furthermore, in a mathematical context STFT filters are related to Toeplitz operators [118,159,179].

The question now is whether the STFT filter H(α)1 with weighting function L

(α)H

(t, f) is a reasonable

approximation to H. We shall postpone the answer to this question and first discuss multi-window

STFT filters [118, 123, 125], which are an extension of the conventional STFT filter using several

orthonormal window functions. Here, N STFTs are calculated using N orthonormal window func-

tions {sk(t)}k=1...N . Each of these STFTs is multiplied by the same weighting function (in our case

L(α)H

(t, f)), and to each product L(α)H

(t, f) STFT(sk)x (t, f) an inverse STFT (2.122) using the respective

window sk(t) is applied. The output is finally obtained as a weighted superposition of the individual

STFT filter outputs,

yN (t) =(H

(α)N x

)(t) ,

N∑

k=1

λk

[ ∫

t′

∫

f ′

STFT(sk)x (t′, f ′)L

(α)H

(t′, f ′)(sk)t′,f ′(t) dt′ df ′], (2.136)

where H(α)N denotes the overall multi-window STFT filter and the λk satisfy

∑Nk=1 λk = 1. Note that

for N = 1 this reduces to the single-window case as discussed above. In general, H will be different

from its multi-window STFT filter approximation H(α)N . Yet, in the following we will establish a

relation and bound for the difference between the GWS of the original system H and that of the

multi-window STFT filter H(α)N ; subsequently, we will bound the difference between the exact output

signal y(t) = (Hx)(t) and the multi-window STFT filter’s output signal yN (t) = (H(α)N x)(t).

Theorem 2.25. Let H(α)N be the multi-window STFT filter operator defined in ( 2.136), with orthonor-

mal window functions {sk(t)}k=1...N and∑N

k=1 λk = 1. Then, the difference

∆(α)10 (t, f) = L

(α)H

(t, f) − L(α)

bH(α)N

(t, f)


satisfies ∣∣∆(α)10 (t, f)

∣∣‖SH‖1

≤ m(φN )H

,

∥∥∆(α)10

∥∥2

‖H‖2

= M(φN )H

, (2.137)

with the weighting function φN (τ, ν) =∣∣1 − ∑N

k=1 λkA(α)sk (τ, ν)

∣∣ =∣∣1 − S

(α)TN

(τ, ν)∣∣ where TN =

∑Nk=1 λk sk⊗s∗k. In the single-window case there is φ1(τ, ν) =

∣∣1 −A(α)s1 (τ, ν)

∣∣.

Proof. The GSF of H(α) can be written as [64,118,125,190]

S(α)

bH(α)N

(τ, ν) =

N∑

k=1

λk S(α)H

(τ, ν)A(α)∗sk

(τ, ν) = S(α)H

(τ, ν)

N∑

k=1

λk A(α)∗sk

(τ, ν) = S(α)H

(τ, ν)S(α)∗TN

(τ, ν).

The first (L∞) bound is then shown as

∣∣∆(α)10 (t, f)

∣∣ =

∣∣∣∣∫

τ

∫

ν

[S

(α)H

(τ, ν) − S(α)

bH(α)N

(τ, ν)]ej2π(νt−τf)dτ dν

∣∣∣∣

≤∫

τ

∫

ν|SH(τ, ν)|

∣∣1 − S(α)∗TN

(τ, ν)∣∣dτ dν

=

∫

τ

∫

ν|SH(τ, ν)|

∣∣1 − S(α)TN

(τ, ν)∣∣ dτ dν (2.138)

= ‖SH‖1m(φN )H

.

The expression for the L2 norm of ∆(α)10 (t, f) is shown by noting that

∥∥∆(α)10

∥∥2

2=∥∥∥S(α)

H− S

(α)

bH(α)N

∥∥∥2

2=

∫

τ

∫

ν

∣∣∣S(α)H

(τ, ν)[1 − S

(α)∗TN

(τ, ν)]∣∣∣

2dτ dν

=

∫

τ

∫

ν|SH(τ, ν)|2

∣∣1 − S(α)TN

(τ, ν)∣∣2dτ dν (2.139)

= ‖SH‖22

[M

(φN )H

]2.

Discussion. If m(φN )H

and M(φN )H

are small, the foregoing theorem implies that the GWS of the

multi-window STFT filter is approximately equal to the GWS of H (i.e., to the weighting function in

(2.136)),

L(α)

bH(α)N

(t, f) ≈ L(α)H

(t, f).

Small m(φN )H

and M(φN )H

require that TN (i.e., the orthonormal set {sk(t)}k=1,...,N and the coefficients

{λk}k=1,...,N ) can be chosen such that |STN(τ, ν)| ≈ 1 on the effective support of |SH(τ, ν)|, which in

turn requires that the effective support of |SH(τ, ν)| is small, i.e., that H is underspread.

Furthermore, since∥∥∆(α)

10

∥∥2

equals the HS norm∥∥H − H

(α)N

∥∥2

and since the operator norm is

bounded from above by the HS norm (see Section A.1), there is

∥∥H − H(α)N

∥∥O≤∥∥H − H

(α)N

∥∥2

= ‖H‖2M(φN )H

. (2.140)

Thus, it follows from (2.137) that, if M(φN )H

is small, the multi-window STFT filter H(α)N will be close

to H in operator and HS norm, H(α)N ≈ H. Note that the weighting function φN provides increasing

flexibility with increasing N . In fact, for N = 1 one has S(α)T1

(τ, ν) = A(α)s1 (τ, ν) and the situation


reduces to the single-window or rank-one case. By exploiting the inherent flexibility of the rank-N

operator TN , it is possible to achieve smallm(φN )H

andM(φN )H

and thus a good approximation H(α)N ≈ H

even in those cases where the conventional (single-window) STFT filter (which is obtained for N = 1)

fails to yield acceptable results.

As a consequence of the foregoing theorem, the performance of the multi-window STFT filter can

also be assessed in terms of the output signals y(t) = (Hx)(t) and yN (t) =(H

(α)N x

)(t).

Corollary 2.26. For any LTV system H and any signal x(t), the difference

∆(α)11 (t) , (Hx)(t) −

(H

(α)N x

)(t)

is bounded as ∣∣∆(α)11 (t)

∣∣‖SH‖1 ‖x‖∞

≤ m(φN )H

,

∥∥∆(α)11

∥∥2

‖H‖2 ‖x‖2

≤M(φN )H

, (2.141)

with the weighting function φN (τ, ν) as in Theorem 2.25.

Proof. Using (B.3), the expression S(α)

bH(α)N

(τ, ν) = S(α)H

(τ, ν)S(α)∗T

(τ, ν), and15∣∣x(α)

τ,ν (t)∣∣ ≤

∥∥x(α)τ,ν

∥∥∞

=

‖x‖∞, the first (L∞) bound is shown as

∣∣∆(α)11 (t)

∣∣ =

∣∣∣∣∫

τ

∫

ν

[S

(α)H

(τ, ν) − S(α)

bH(α)N

(τ, ν)]x(α)

τ,ν (t) dτ dν

∣∣∣∣

=

∣∣∣∣∫

τ

∫

νS

(α)H

(τ, ν)[1 − S

(α)∗TN

(τ, ν)]x(α)

τ,ν (t) dτ dν

∣∣∣∣

≤∫

τ

∫

ν|SH(τ, ν)|

∣∣1 − S(α)∗TN

(τ, ν)∣∣ ∣∣x(α)

τ,ν (t)∣∣ dτ dν ≤ ‖x‖∞ ‖SH‖1m

(φN )H

.

The second (L2) bound is shown by noting that

∥∥∆(α)11

∥∥2

2=∥∥(H − H

(α)N

)x∥∥2

2≤∥∥H − H

(α)N

∥∥2

O‖x‖2

2

and invoking (2.140).

Discussion. It is seen that for an underspread system H for which m(φN )H

and M(φN )H

can be

made small by suitable choice of the STFT windows sk(t) and the weighting factors λk, we have(H

(α)N x

)(t) ≈ (Hx)(t), i.e., the output of the multi-window STFT filter is a good approximation

to the output of the given LTV system H. In [118] it was shown that for DL operators, an exact

multi-window STFT filter implementation is possible by choosing T such that S(α)T

(τ, ν) = IH(τ, ν)

as in (2.3) (which in general requires infinitely many windows, i.e., N → ∞). However, also in our

framework the approximation error can be made arbitrary small if N is sufficiently large.

DL Operators. Let us assume that H is a DL operator with GSF support GH. Suitably intro-

ducing (2.16) in the previous proofs yields the bounds

∣∣∆(α)10 (t, f)

∣∣‖SH‖1

≤ φ(max)N ,

∥∥∆(α)10

∥∥2

‖H‖2

≤ φ(max)N (2.142)

15Here, x(α)τ,ν (t) ,

`S

(α)τ,νx)(t) with the generalized joint TF shift operator S

(α)τ,ν (see (B.4)).


and

∣∣∆(α)11 (t)

∣∣‖SH‖1 ‖x‖∞

≤ φ(max)N ,

∥∥∆(α)11

∥∥2

‖H‖2 ‖x‖2

≤ φ(max)N , (2.143)

with φ(max)N = max(τ,ν)∈GH

{φN (τ, ν)

}. In general, these bounds are less tight than the corresponding

bounds (2.137) and (2.141) which also hold for DL operators.

Non-DL Operators. For non-DL operators and arbitrary rectangular region G = [−τG, τG ] ×[−νG, νG ], the following bounds for ∆

(α)10 (t, f) and ∆

(α)11 (t) can be derived in terms of φ

(max)N and

appropriate moments.

Proposition 2.27. For any LTV system H, any multi-window STFT filter H(α)N as defined in ( 2.136),

and any rectangular support region G = [−τG , τG ] × [−νG , νG ], the difference ∆(α)10 (t, f) is bounded as

∣∣∆(α)10 (t, f)

∣∣‖SH‖1

≤ φ(max)N + φ

(max)N

[m

(1,0)H

τG+m

(0,1)H

νG

],

∥∥∆(α)10

∥∥2

‖H‖2

≤[(φ

(max)N

)2+

(φ

(max)N

M(1,0)H

τG

)2

+

(φ

(max)N

M(0,1)H

νG

)2]1/2

≤ φ(max)N + φ

(max)N

[M

(1,0)H

τG+M

(0,1)H

νG

],

and the difference ∆(α)11 (t) is bounded as

∣∣∆(α)11 (t)

∣∣‖SH‖1 ‖x‖∞

≤ φ(max)N + φ

(max)N

[m

(1,0)H

τG+m

(0,1)H

νG

],

∥∥∆(α)11

∥∥2

‖H‖2 ‖x‖2

≤[(φ

(max)N

)2+

(φ

(max)N

M(1,0)H

τG

)2

+

(φ

(max)N

M(0,1)H

νG

)2]1/2

≤ φ(max)N + φ

(max)N

[M

(1,0)H

τG+M

(0,1)H

νG

].

Here, φ(max)N = max(τ,ν)∈G{φN (τ, ν)} and φ

(max)N = max(τ,ν)∈G{φN (τ, ν)} with φN (τ, ν) defined as in

Theorem 2.25.

Proof. Starting directly with the expression (2.138) and splitting this integral yields

∣∣∆(α)10 (t, f)

∣∣ ≤∫∫

G

|SH(τ, ν)|∣∣1 − S

(α)T

(τ, ν)∣∣ dτ dν +

∫∫

G

|SH(τ, ν)|∣∣1 − S

(α)T

(τ, ν)∣∣ dτ dν

=

∫∫

G

|SH(τ, ν)|φN (τ, ν) dτ dν +

∫∫

G

|SH(τ, ν)|φN (τ, ν) dτ dν

≤ φ(max)N

∫∫

G

|SH(τ, ν)| dτ dν + φ(max)N

∫∫

G


= φ(max)N ‖SHG‖1 + φ

(max)N ‖S

HG‖1,


where we used φN (τ, ν) ≤ φ(max)N for (τ, ν) ∈ G and φN (τ, ν) ≤ φ

(max)N for (τ, ν) ∈ G. The first

term in this expression is bounded by ‖SH‖1φ(max)N , and a bound for the second term is given by the

first inequality in (2.38). Combining these bounds yields the result to be proved. The L2 bound for

∆(α)10 (t, f) and the bounds for ∆

(α)11 (t) can be derived in a completely analogous way. The coarser

bounds for the L2 norms of ∆(α)10 (t) and ∆

(α)11 (t) follow from the inequality

√a2 + b2 ≤ a+ b, valid for

a ≥ 0, b ≥ 0.

The foregoing result shows that the bounds (2.142) and (2.143), when erroneously applied to

non-DL operators, might deviate from the correct bounds on the L∞ and L2 norm, respectively, of

the differences ∆(α)10 (t, f) and ∆

(α)11 (t) in Proposition 2.27 by as much as φ

(max)N

[m(1,0)H

τG+

m(0,1)H

νG

]and

φ(max)N

[M(1,0)H

τG+

M(0,1)H

νG

].

2.3.12 Infimum and Supremum of the Weyl Symbol

We have seen in Subsection 2.3.8 that the GWS of an underspread system can be interpreted as an

approximate eigenvalue distribution over the TF plane. We now assume H to be self-adjoint, such that

the eigenvalues λk of H are real-valued. Furthermore, we consider the Weyl symbol L(0)H

(t, f) of H (i.e.,

α = 0) which is real-valued as well. We shall investigate how close inft,f L(0)H

(t, f) and supt,f L(0)H

(t, f)

are to λinfH

, infk λk and λsupH

, supk λk, respectively. Note that in the case of self-adjoint LTI/LFI

systems the infimum (supremum) of the eigenvalues is always equal to the infimum (supremum) of

the respective transfer function. However, a similar relation unfortunately is not true for the GWS of

general LTV systems. This topic will be seen to be of importance in Subsections 2.3.15 and 2.3.17.

Related issues in the theory of quantization and pseudo-differential operators are the boundedness

or positivity of operators corresponding to bounded or positive symbols, the results being theorems

of the Calderon-Vaillancourt type [25, 64, 94, 95] and Garding inequalities [36, 54, 64, 94], respectively.

The next theorem is an extension and adaptation of previous results [118].

Theorem 2.28. For any self-adjoint LTV system H, there is

∣∣ inft,f L(0)H

(t, f) − λinfH

∣∣‖SH‖1

≤ m(φs)H

,

∣∣ supt,f L(0)H

(t, f) − λsupH

∣∣‖SH‖1

≤ m(φs)H

, (2.144)

with the weighting function φs(τ, ν) =∣∣∣1 − 1

A(0)s (τ,ν)

∣∣∣ where s(t) is an arbitrary normalized function.

Proof. Let us consider the “lower symbol” [64,118]

LLH(t, f) , 〈Hst,f , st,f 〉 =

∫

τ

∫

νS

(0)H

(τ, ν)A(0)∗s (τ, ν) ej2π(tν−fτ) dτ dν (2.145)

and the “upper symbol” [64,118]

LUH(t, f) ,

∫

τ

∫

ν

S(0)H

(τ, ν)

A(0)s (τ, ν)

ej2π(tν−fτ) dτ dν, (2.146)


λsupH

inft,fLU

H(t, f) inf

t,fLL

H(t, f) sup

t,fLU

H(t, f)sup

t,fLL

H(t, f)

λinfH

Figure 2.10: Illustration of inequalities (2.147) and (2.148).

where s(t) is a normalized function. (For Gaussian s(t), the lower and upper symbols can be related

to the Wick and anti-Wick symbols of quantum mechanics [14,64].) It is known [64,118] that (see Fig.

2.10)

inft,fLU

H(t, f) ≤ λinfH ≤ inf

t,fLL

H(t, f) (2.147)

supt,f

LLH(t, f) ≤ λsup

H≤ sup

t,fLU

H(t, f) . (2.148)

We now have

∣∣∣L(0)H

(t, f) − LLH(t, f)

∣∣∣ =∣∣∣∣∫

τ

∫

νS

(0)H

(τ, ν)[1 −A(0)∗

s (τ, ν)]ej2π(τf−νt) dτ dν

∣∣∣∣

≤∫

τ

∫

ν|SH(τ, ν)|

∣∣∣1 −A(0)s (τ, ν)

∣∣∣ dτ dν = ‖SH‖1m(φs)H

, (2.149)

with φs(τ, ν) =∣∣1 −A

(0)s (τ, ν)

∣∣. Similarly, we have

∣∣∣L(0)H

(t, f) − LUH(t, f)

∣∣∣ =

∣∣∣∣∣

∫

τ

∫

νS

(0)H

(τ, ν)

[1 − 1

A(0)s (τ, ν)

]ej2π(τf−νt) dτ dν

∣∣∣∣∣

≤∫

τ

∫

ν|SH(τ, ν)|

∣∣∣∣∣1 − 1

A(0)s (τ, ν)

∣∣∣∣∣ dτ dν = ‖SH‖1m(φs)H

, (2.150)

with φs(τ, ν) =∣∣∣1 − 1

A(0)s (τ,ν)

∣∣∣. Furthermore, due to |A(0)s (τ, ν)| ≤ ‖s‖2

2 = 1 there is

m(φs)H

=1

‖SH‖1

∫

τ

∫

ν|SH(τ, ν)| |1 −A

(0)s (τ, ν)|

|A(0)s (τ, ν)|

dτ dν

≥ 1

‖SH‖1

∫

τ

∫

ν|SH(τ, ν)|

∣∣∣1 −A(0)s (τ, ν)

∣∣∣ dτ dν = m(φs)H

. (2.151)

The inequality (2.149) is equivalent to the two inequalities

L(0)H

(t, f) − ‖SH‖1m(φs)H

≤ LLH(t, f) , (2.152)

LLH(t, f) ≤ L

(0)H

(t, f) + ‖SH‖1m(φs)H

, (2.153)

and similarly, (2.150) is equivalent to

L(0)H


≤ LUH(t, f) , (2.154)

LUH(t, f) ≤ L

(0)H


. (2.155)


Since inft′,f ′ LLH

(t′, f ′) ≤ LLH

(t, f), it follows from (2.153) that inft′,f ′ LLH

(t′, f ′) ≤ L(0)H

(t, f) +

‖SH‖1m(φs)H

for all (t, f) and hence that

inft′,f ′

LLH(t′, f ′) ≤ inf

t,fL

(0)H


≤ inft,fL

(0)H


, (2.156)

where the second inequality is due to (2.151). In a similar manner, (2.154) can be shown to imply

inft,fL

(0)H


≤ inft′,f ′

LUH(t′, f ′) . (2.157)

Inserting (2.156) and (2.157) into (2.147), it follows that

inft,fL

(0)H


≤ λinfH ≤ inf

t,fL

(0)H


or, equivalently, | inft,f L(0)H

(t, f) − λinfH| ≤ ‖SH‖1m

(φs)H

.

By similar reasoning, it can be shown that (2.152) (together with (2.151)) implies that

supt,f

L(0)H


≤ supt,f

L(0)H


≤ supt′,f ′

LLH(t′, f ′) , (2.158)

and that (2.155) implies

supt′,f ′

LUH(t′, f ′) ≤ sup

t,fL

(0)H


. (2.159)

Inserting (2.158) and (2.159) into (2.148) shows that

supt,f

L(0)H


≤ λsupH

≤ supt,f

L(0)H


or, equivalently, | supt,f L(0)H

(t, f) − λsupH

| ≤ ‖SH‖1m(φs)H

.

Discussion. Whenever m(φs)H

can be made small by suitable choice of s(t), it follows from Theorem

2.28 that

inft,fL

(0)H

(t, f) ≈ λinfH , sup

t,fL

(0)H

(t, f) ≈ λsupH

.

Small m(φs)H

requires As(τ, ν) ≈ As(0, 0) ≡ 1 on the effective support of |SH(τ, ν)|, thus implying that

this effective support is small, i.e., that H is underspread. Since φs(τ, ν) =∣∣∣1 − 1

A(0)s (τ,ν)

∣∣∣ typically

grows very rapidly, |SH(τ, ν)| must decrease extremely fast in order to allow for small m(φs)H

. Small

m(φs)H

thus seems to be the strongest constraint on H imposed in this chapter. Note, however, that

m(φs)H

can be adapted to oblique orientation of |SH(τ, ν)| by proper choice of the function s(t).

2.3.13 Approximate Non-Negativity of the Generalized Weyl Symbol of PositiveSemi-Definite Operators

In contrast to positive semi-definite16 LTI (LFI) systems, which have nonnegative spectral (temporal)

transfer functions, a similar property does not hold for the GWS of a general positive semi-definite16We note that a completely analogous discussion applies to negative semi-definite HS operators and their GWS.


HS operator H. However, since for positive semi-definite HS operators λinfH

= 0, the discussion in the

preceding subsection suggests that in the underspread case inft,f L(0)H

(t, f) ≈ λinfH

= 0, i.e., that the

Weyl symbol of an underspread positive semi-definite operator is approximately nonnegative. For the

general case α 6= 0 (where L(α)H

(t, f) is complex-valued), the nonnegativity property can be formulated

as L(α)H

(t, f) = P{L

(α)H

(t, f)}

with the positive real part of the GWS

P{L

(α)H

(t, f)}

,1

2

[∣∣ℜ{L

(α)H

(t, f)}∣∣+ ℜ

{L

(α)H

(t, f)}]

≥ 0 .

Subsequently, we will show that the GWS of underspread positive semi-definite operators is approx-

imately nonnegative, i.e., L(α)H

(t, f) ≈ P{L

(α)H

(t, f)}. Specifically, we will show that i) the imaginary

part ℑ{L

(α)H

(t, f)}

of the GWS is small (this was already demonstrated in Subsection 2.3.3); and ii)

the negative real part of the GWS (see Fig. 2.11)

N{L

(α)H

(t, f)}

, P{L

(α)H

(t, f)}−ℜ

{L

(α)H

(t, f)}

=1

2

[∣∣ℜ{L

(α)H

(t, f)}∣∣−ℜ

{L

(α)H

(t, f)}]

(2.160)

is small (note that N{L

(α)H

(t, f)}

≥ 0). To this end, we recall and adapt several of the preceding

results:

• Theorem 2.28 shows that the infimum of the Weyl symbol of an underspread positive HS operator

approximately equals 0. In particular, for α = 0, the GWS is real-valued and N{L

(0)H

(t, f)}

≤∣∣ inft,f L(0)H

(t, f)∣∣ (this is trivial in the case inft,f L

(0)H

(t, f) ≥ 0 where N{L

(0)H

(t, f)}

= 0; in the

case inft,f L(0)H

(t, f) < 0, there is supt,f N{L

(0)H

(t, f)}

= − inft,f L(0)H

(t, f) =∣∣ inft,f L

(0)H

(t, f)∣∣).

Further using the left-hand inequality of (2.144) with λinfH

= 0 yields

N{L

(0)H

(t, f)}

≤∣∣ inf

t,fL

(0)H

(t, f)∣∣ =

∣∣ inft,fL

(0)H

(t, f) − λinfH

∣∣ ≤ ‖SH‖1m(φ1)H

(2.161)

where φ1(τ, ν) =∣∣∣1 − 1

A(0)s (τ,ν)

∣∣∣ with s(t) any normalized function. However, this result will not

be used in the following since the bound in (2.161) is always less tight than the bound (2.166)

below (see (2.151) and the proof of Theorem 2.29).

• The positive semi-definite square-root Hr =√

H of a positive semi-definite operator H (see

Appendix A) has minimal TF displacements among all systems G factoring H according to

GG+ = H [79, 90]. That is, for H underspread, Hr will be underspread too. Corollary 2.17

shows that for Hr underspread, the GWS of H approximately equals the squared magnitude of

the GWS of Hr, i.e., L(α)H

(t, f) = L(α)

HrH+r(t, f) ≈

∣∣L(α)Hr

(t, f)∣∣2. This suggests that the negative

part of L(α)H

(t, f) is small.

• Similarly, for an underspread positive semi-definite operator H, (2.149) shows that the Weyl

symbol is approximately equivalent to the lower symbol LLH

(t, f) (defined in (2.145)). But

LLH

(t, f) ≥ λinfH

= 0 according to (2.147). Below, we will show that for an underspread systen H

the approximation L(α)H

(t, f) ≈ LLH

(t, f) holds for arbitrary α. This in turn indicates that the

negative part of the GWS of an underspread positive semi-definite operator H is small.


ℜ{L

(α)H

(t, f0)}

0

LLH

(t, f0)

t

P{L

(0)H

(t, f0)}λsup

H

}N{L

(α)H

(t, f0)}

L<H

(t, f0)

L>H

(t, f0)

{

Figure 2.11: Illustration of N{L

(α)H

(t, f)}, P{L

(α)H

(t, f)}, L>

H(t, f), L<

H(t, f), and LL

H(t, f) for a

positive semi-definite system. The solid curve shows the real part of the GWS, whereas the thick

dashed curve shows the lower symbol LLH

(t, f), both as a function of t for fixed f = f0.

By making the above arguments precise, the following theorem is obtained.

Theorem 2.29. For any positive semi-definite system H, the negative real part N{L

(α)H

(t, f)}

of the

GWS is bounded as

N{L

(α)H

(t, f)}

‖SHr‖21

≤ min{β1, β2} , (2.162)

∥∥N{L

(α)H

}∥∥2

‖H‖2

≤ inf‖s‖2=1

{M

(φs)H

}, (2.163)

where β1 = inf‖s‖2=1

{m

(φs)H

}, β2 = 2π C

(α)Hr

(with C(α)H

defined in (2.70) and Hr the positive semi-

definite square-root of H), and φs(τ, ν) =∣∣1 −A

(α)s (τ, ν)

∣∣ with ‖s‖2 = 1.

Proof. With P (t, f) denoting an arbitrary real-valued and positive TF function, the Lp norm of

N{L

(α)H

(t, f)}

can be bounded as

∥∥N{L

(α)H

}∥∥p

=1

2

∥∥∥∣∣ℜ{L

(α)H

}∣∣−ℜ{L

(α)H

}∥∥∥p

=1

2

∥∥∥∣∣ℜ{L

(α)H

}∣∣− P + P −ℜ{L

(α)H

}∥∥∥p

≤ 1

2

∥∥∥∣∣ℜ{L

(α)H

}∣∣− P∥∥∥

p+

1

2

∥∥∥P −ℜ{L

(α)H

}∥∥∥p

≤∥∥P −ℜ

{L

(α)H

}∥∥p

≤∥∥L(α)

H− P

∥∥p. (2.164)

Here, we used the triangle inequality and the fact that the difference between P (t, f) and the∣∣ℜ{L

(α)H

(t, f)}∣∣ is ≤ the difference between P (t, f) and ℜ

{L

(α)H

(t, f)}

(since P (t, f) is positive), which

in turn is ≤ the difference between P (t, f) and L(α)H

(t, f) (since P (t, f) is real-valued).


To prove (2.162) we note that with p = ∞ and17 P (t, f) = LLH

(t, f), (2.164) can be bounded as

∣∣L(α)H

(t, f) − LLH(t, f)

∣∣ ≤∫

τ

∫

ν

∣∣S(α)H

(τ, ν) − S(α)H

(τ, ν)A(α)∗s (τ, ν)

∣∣ dτ dν

=

∫

τ

∫

ν

∣∣S(α)H

(τ, ν)∣∣ ∣∣1 −A(α)

s (τ, ν)∣∣ dτ dν

= ‖SH‖1m(φs)H

(2.165)

≤ ‖SHr‖21m

(φs)H

.

With (2.164), we thus obtain

N{L

(α)H

(t, f)}≤∣∣L(α)

H(t, f) − LL

H(t, f)∣∣ ≤ ‖SH‖1m

(φs)H

(2.166)

≤ ‖SHr‖21m

(φs)H

,

which holds for all s(t) (assumed to be normalized in order that φs(0, 0) = 0). Taking the infimum

with regard to s(t) finally gives

N{L

(α)H

(t, f)}≤ ‖SHr‖2

1 inf‖s‖2=1

{m

(φs)H

}= β1 ‖SHr‖2

1 . (2.167)

Similarly, with p = ∞ and P (t, f) =∣∣L(α)

Hr(t, f)

∣∣2, (2.164) reduces to (2.70) (with H replaced by Hr),

and thus

N{L

(α)H

(t, f)}≤ ‖SHr‖2

1 2π C(α)H

= β2 ‖SHr‖21 . (2.168)

The final bound (2.162) follows upon combining (2.167) and (2.168). Note that φs(τ, ν) ≤ φ1(τ, ν)

(where φ1(τ, ν) is the weighting function used in (2.161)) and hence m(φs)H

≤ m(φ1)H

for all normalized

s(t). Therefore, as mentioned previously, the bound (2.161) is always less tight than that in (2.166).

To prove the bound (2.163), we note that

∥∥L(α)H

− LLH

∥∥2

2=

∫

τ

∫

ν

∣∣S(α)H

(τ, ν) − S(α)H

(τ, ν)A(α)∗s (τ, ν)

∣∣2 dτ dν

=

∫

τ

∫

ν|S(α)

H(τ, ν)|2

∣∣1 −A(α)s (τ, ν)

∣∣2 dτ dν = ‖H‖22

[M

(φs)H

]2. (2.169)

Since this expression holds for all s(t), we obtain with (2.164), P (t, f) = LLH

(t, f), and p = 2

∥∥N{L

(α)H

}∥∥2≤ inf

‖s‖2=1

∥∥L(α)H

− LLH

∥∥2≤ ‖H‖2 inf

‖s‖2=1

{M

(φs)H

},

which finally establishes (2.163).

In combination with Corollary 2.13, the foregoing theorem immediately implies the following result.

Theorem 2.30. For any positive semi-definite system H, the difference

∆(α)12 (t, f) , L

(α)H

(t, f) −P{L

(α)H

(t, f)}

17That LLH(t, f) is indeed non-negative follows from LL

H(t, f) = 〈Hst,f , st,f 〉 ≥ 0. In the following derivation, note also

that (B.10) implies LLH(t, f) =

Rτ

Rν

S(α)H

(τ, ν) A(α)∗s (τ, ν) ej2π(tν−fτ) dτ dν , which generalizes the expression in (2.145).


between the GWS and its positive real part is bounded as

∣∣∆(α)12 (t, f)

∣∣‖SHr‖2

1

≤ 2π|α|m(1,1)H

+ min{β1, β2} , (2.170)

∥∥∆(α)12

∥∥2

‖H‖2

≤ 2π|α|M (1,1)H

+ inf‖s‖2=1

{M

(φs)H

}, (2.171)

where β1, β1, and φs(τ, ν) are as defined in Theorem 2.29.

Proof. The GWS can be written as

L(α)H

(t, f) = ℜ{L

(α)H

(t, f)}

+ j ℑ{L

(α)H

(t, f)}

= P{L

(α)H

(t, f)}−N

{L

(α)H

(t, f)}

+ jℑ{L

(α)H

(t, f)}.

It follows that

∆(α)12 (t, f) = jℑ

{L

(α)H

(t, f)}−N

{L

(α)H

(t, f)}, (2.172)

and hence by the triangle inequality that

∥∥∆(α)12

∥∥p≤∥∥ℑ{L

(α)H

}∥∥p+∥∥N{L

(α)H

}∥∥p

(2.173)

where for our purposes p = 2 or p = ∞. Bounds on the first term on the right hand side of (2.173)

(involving the imaginary part of the GWS) have been provided in Corollary 2.13, i.e.,

∥∥ℑ{L

(α)H

}∥∥∞

≤ ‖SH‖1 2π|α|m(1,1)H

≤ ‖SHr‖21 2π|α|m(1,1)

H, (2.174)

∥∥ℑ{L

(α)H

}∥∥2≤ ‖H‖2 2π|α|M (1,1)

H, (2.175)

where in the L∞ bound we further used ‖SH‖1 ≤ ‖SHr‖21. The bound (2.170) then follows via (2.173)

(with p = ∞) upon combining the bounds (2.174) and (2.162). Similarly, the bound (2.171) follows

via (2.173) (with p = 2) upon combination of the bounds (2.175) and (2.163).

Discussion. The foregoing theorems show that for an underspread positive semi-definite operator

having small m(1,1)H

, M(1,1)H

, m(φs)H

or C(α)Hr

, and M(φs)H

, the GWS approximately equals its positive real

part,

L(α)H

(t, f) ≈ P{L

(α)H

(t, f)},

i.e., both the imaginary part and the negative real part of the GWS are small,

ℑ{L

(α)H

(t, f)}≈ 0 , N

{L

(α)H

(t, f)}≈ 0 .

The bounds in the theorems are tightest for α = 0 (since here the imaginary part ℑ{L

(α)H

(t, f)}

vanishes completely). Furthermore, for α = 0 the bounds allow to account for oblique orientation of

the GSF of H and Hr by choosing a function s(t) with obliquely oriented ambiguity function A(0)s (τ, ν),

and furthermore, β2 can be replaced by the refined bound 2π infU∈M

{m

(0,1)UHrU

+m(1,0)UHrU

+

}according

to (2.72).


DL Operators. For a positive semi-definite DL operator H with GSF support GH, applying

(2.16)–(2.18) to the bounds (2.162), (2.163), (2.170), and (2.171), yields the bounds18

N{L

(α)H

(t, f)}

‖SH‖1

≤ β1 ,

∥∥N{L

(α)H

}∥∥2

‖H‖2

≤ β1 , (2.176)

∣∣∆(α)12 (t, f)

∣∣‖SH‖1

≤ π

2|α|σH + β1 ,

∥∥∆(α)12

∥∥2

‖H‖2

≤ π

2|α|σH + β1 , (2.177)

where β1 = inf‖s‖2=1 φ(max)s with φ

(max)s = max(τ,ν)∈GH

∣∣1 − A(α)s (τ, ν)

∣∣, and σH = 4 τ(max)H

ν(max)H

with

τ(max)H

, ν(max)H

as defined in (2.6).

Non-DL Operators. For any (i.e., not necessarily DL) system H, and any rectangular region

G = [−τG , τG ] × [−νG , νG ], the results of Section 2.2.6 can be used to show that

N{L

(α)H

(t, f)}

‖SH‖1

≤ β1 + 2

(m

(1,0)H

τG+m

(0,1)H

νG

),

∥∥N{L

(α)H

}∥∥2

‖H‖2

≤ β1 + 2

(M

(1,0)H

τG+M

(0,1)H

νG

),

∣∣∆(α)12 (t, f)

∣∣‖SH‖1

≤ 2π|α| τG νG + β1 + 3

(M

(1,0)H

τG+M

(0,1)H

νG

),

∥∥∆(α)12

∥∥2

‖H‖2

≤ 2π|α| τG νG + β1 + 3

(M

(1,0)H

τG+M

(0,1)H

νG

),

where β1 = inf‖s‖2=1 φ(max)s with φ

(max)s = max(τ,ν)∈G

∣∣1 − A(α)s (τ, ν)

∣∣. Hence, it is seen that the error

resulting from erroneously applying (2.176) and (2.177) (with GH = G) to a non-DL system is upper

bounded in terms ofM

(1,0)H

τG+

M(0,1)H

νG.

2.3.14 Boundedness of the Generalized Weyl Symbol of Self-Adjoint Operators

In the preceding subsection we showed that the GWS of underspread positive semi-definite HS oper-

ators cannot be too negative (i.e., not much smaller than λinfH

= 0). Subsequently, we present a dual

discussion which shows that the GWS cannot exceed the norm ‖H‖O of a self-adjoint HS operator

by too much. This differs from the results of Subsection 2.3.15 in the sense that i) only the part of

|L(α)H

(t, f)| exceeding ‖H‖O is considered (i.e., the case |L(α)H

(t, f)| < ‖H‖O is not relevant here); and

ii) bounds on the L∞ and L2 norms of the GWS part above ‖H‖O are developed.

Let H be a self-adjoint operator so that λk ∈ R. While ‖H‖O = supk |λk| = max{λsupH,−λinf

H}

[158], we shall subsequently consider only the case ‖H‖O = λsupH

and note that a completely parallel

discussion applies to the case ‖H‖O = −λinfH

.

We define the part of ℜ{L

(α)H

(t, f)}

which is less than λsupH

by the thresholding (see Fig. 2.11)

L<H

(t, f) ,

ℜ{L

(α)H

(t, f)}, for ℜ

{L

(α)H

(t, f)}≤ λsup

H,

λsupH

, for ℜ{L

(α)H

(t, f)}> λsup

H.

18Note that the term involving β2 is ignored here for simplicity.


The part of ℜ{L

(α)H

(t, f)}

exceeding λsupH

is defined similarly (see Fig. 2.11),

L>H

(t, f) ,

ℜ{L

(α)H

(t, f)}− λsup

H, for ℜ

{L

(α)H

(t, f)}> λsup

H,

0 , for ℜ{L

(α)H

(t, f)}≤ λsup

H.

(2.178)

Note that L<H

(t, f) + L>H

(t, f) = ℜ{L

(α)H

(t, f)}

and hence

L(α)H

(t, f) = L<H

(t, f) + L>H

(t, f) + jℑ{L

(α)H

(t, f)}. (2.179)

The thresholding of ℜ{L

(α)H

(t, f)}

by λsupH

in the definition (2.178) is equivalent to a thresholding of

ℜ{L

(α)H

(t, f)}− λsup

H= ℜ

{L

(α)

H−λsupH

I(t, f)

}with threshold 0. Hence, L>

H(t, f) can be rewritten as

L>H

(t, f) , N{L

(α)

λsupH

I−H(t, f)

}= N

{λsupH

− L(α)H

(t, f)}, (2.180)

i.e., as the negative real part of the GWS of the operator λsupH

I − H. Similarly,

L<H

(t, f) = λsupH

− P{L

(α)

λsupH

I−H(t, f)

}.

Since we assumed ‖H‖O = λsupH

, the inequality 〈Hx, x〉 ≤ ‖H‖O‖x‖22 implies

⟨(λsup

HI − H)x, x

⟩=⟨λsupH

Ix, x⟩− 〈Hx, x〉 = ‖H‖O‖x‖2

2 − 〈Hx, x〉 ≥ 0 .

Thus, the operator λsupH

I− H is positive semi-definite. Together with (2.180), this allows to take

advantage of the results of the previous subsection in the derivation of the next theorem.

Theorem 2.31. For any self-adjoint, bounded HS system H with ‖H‖O = λsupH

, the difference

∆(α)13 (t, f) , L

(α)H

(t, f) − L<H

(t, f)


13 (t, f)∣∣

‖SH‖1≤ 2π|α|m(1,1)

H+ inf

‖s‖2=1

{m

(φs)H

},

∥∥∆(α)13

∥∥2

‖H‖2≤ 2π|α|M (1,1)

H+ inf

‖s‖2=1

{M

(φs)H

}, (2.181)

with φs(τ, ν) as in Theorem 2.30.

Proof. To prove this theorem, we first note that according to (2.179),

∆(α)13 (t, f) = L

(α)H

(t, f) − L<H

(t, f) = jℑ{L

(α)H

(t, f)}

+ L>H

(t, f) ,

and hence, by the triangle inequality,

∥∥∆(α)13

∥∥p≤∥∥ℑ{L

(α)H

}∥∥p+∥∥L>

H

∥∥p

(2.182)

where for our purposes p = 2 or p = ∞. The first term in this bound (involving ℑ{L

(α)H

(t, f)}) can

again be bounded using Corollary 2.13 (see (2.174) and (2.175)). The second term can be further devel-

oped by inserting the definition of the negative real part (2.160) into (2.180) and by adding/subtracting

LLH

(t, f) followed by application of the triangle inequality,

∥∥L>H

∥∥p

=1

2

∥∥∥∣∣ℜ{L

(α)

λsupH

I−H

}∣∣−ℜ{L

(α)

λsupH

I−H

}∥∥∥p


=1

2

∥∥∥∣∣λsup

H−ℜ

{L

(α)H

}∣∣− λsupH

+ ℜ{L

(α)H

}∥∥∥p

=1

2


H−ℜ

{L

(α)H

}∣∣− λsupH

+ LLH − LL

H + L(α)H

∥∥∥p

≤ 1

2


H−ℜ

{L

(α)H

}∣∣− (λsupH

− LLH)∥∥∥

p+

1

2

∥∥L(α)H

− LLH

∥∥p

≤ 1

2

∥∥λsupH

−ℜ{L

(α)H

}− (λsup

H− LL

H)∥∥

p+

1

2

∥∥L(α)H

− LLH

∥∥p

≤ 1

2

∥∥LLH −ℜ

{L

(α)H

}∥∥p+

1

2

∥∥L(α)H

− LLH

∥∥p

≤ 1

2

∥∥LLH − L

(α)H

∥∥p+

1

2

∥∥L(α)H

− LLH

∥∥p

=∥∥L(α)

H− LL

H

∥∥p. (2.183)

Here, we exploited the fact that LLH

(t, f) is real-valued and that λsupH

− LLH

(t, f) ≥ 0 according to

(2.148). Combining (2.183) with (2.165) (for p = ∞) and (2.169) (for p = 2) an taking the infimum

with regard to s(t) further yields

∥∥L>H

∥∥∞

≤ ‖SH‖1 inf‖s‖2=1

{m

(φs)H

},

∥∥L>H

∥∥2≤ ‖H‖2 inf

‖s‖2=1

{M

(φs)H

}. (2.184)

The L∞ bound in (2.181) then follows by combining the left hand inequality in (2.184) and (2.174)

in (2.182) (with p = ∞). Similarly, the L2 bound in (2.181) follows by combining the right hand

inequality in (2.184) and (2.175) in (2.182) (with p = 2).

Discussion. The foregoing theorem shows that for underspread operators H where m(1,1)H

, M(1,1)H

are small and m(φs)H

, M(φs)H

can be made small by suitable choice of s(t), one has

L(α)H

(t, f) ≈ L<H

(t, f) ,

or quivalently

ℑ{L

(α)H

(t, f)}≈ 0 , L>

H(t, f) ≈ 0 .

The bounds in the theorem are tightest for α = 0 (since here the imaginary part ℑ{L

(α)H

(t, f)}

vanishes

completely). Furthermore, for α = 0 the bounds (2.181) allow to account for oblique orientation of

the GSF of H by choosing a function s(t) with obliquely oriented ambiguity function.

Theorems 2.30 and 2.31 impose severe restrictions on the possible values of the GWS of positive

semi-definite HS operators (note, however, that Theorem 2.31 holds for the broader class of self-adjoint

operators). This restricted range of GWS values is illustrated in Fig. 2.12 for the case α = 1/2 (we here

do not consider α = 0 since we want to illustrate the approximation ℑ{L

(α)H

(t, f)}≈ 0 but there is

ℑ{L

(0)H

(t, f)}

= 0). In the example in part (b) of this figure, the maximum normalized magnitude and

L2 norm of ℑ{L

(1/2)H

(t, f)}

are maxt,f

∣∣ℑ{

L(1/2)H

(t,f)}∣∣

‖SH‖1= 0.054 and

∥∥ℑ{

L(1/2)H

}∥∥2

‖H‖2= 0.032, respectively,

while the corresponding bounds are πm(1,1)H

= 0.315 and πM(1,1)H

= 0.087, respectively. Similarly,

the maximum magnitude and L2 norm of L>H

(t, f) aremaxt,f L>

H(t,f)

‖SH‖1= 0.007 and

‖L>H‖2

‖H‖2= 0.026,


(a)

‖H‖O

ℑ{L

(α)H

(t, f)}

ℜ{L

(α)H

(t, f)}

2πm(1,1)H

‖SH‖1

−2πm(1,1)H

‖SH‖1

‖H‖O + m(φs)H

‖SH‖1

−m(φs)H

‖SH‖1

(b)

0 1 2 3 4−0.5

0

0.5

ℑ{L

(1/2)H

(t, f)}

ℜ{L

(1/2)H

(t, f)}

Figure 2.12: (a) Schematic illustration of the possible values of the GWS of an underspread positive

semi-definite HS operator H. The thick line represents the desired range of GWS values and the dashed

line represents the bounds of Theorems 2.30 and 2.31. (b) Actual values of the GWS (with α = 1/2)

of an underspread positive semi-definite operator with ‖H‖O = 4.

respectively, while the corresponding bounds are m(φs)H

= 0.145 and M(φs)H

= 0.046.19 The bounds

on L>H

(t, f) also apply to the negative part N{L

(1/2)H

(t, f)}

(see Theorem 2.29) which has maximum

normalized magnitude maxt,fN{

L(1/2)H

(t,f)}

‖SH‖1= 0.006 and normalized L2 norm

‖N{L(1/2)H

}‖2

‖H‖2= 0.026,

respectively.

Combining the results for ℑ{L

(1/2)H

(t, f)}

and L>H

(t, f) and N{L

(1/2)H

(t, f)}

yields

maxt,f|∆

(1/2)12 (t,f)|‖SH‖1

≈ maxt,f|∆

(1/2)13 (t,f)|‖SH‖1

= 0.06 and‖∆

(1/2)12 ‖2

‖H‖2≈ ‖∆

(1/2)13 ‖2

‖H‖2= 0.058 as compared to

the bounds πm(1,1)H

+m(φs)H

= 0.46 and πM(1,1)H

+M(φs)H

= 0.133.

An important special case of a positive semi-definite operator H is an orthogonal projection oper-

ator (see Section A.4 in Appendix A). If the associated signal space is non-sophisticated [81,87], the

orthogonal projection operator H will be underspread. Here, it follows that the GWS of H (termed

the generalized Wigner distribution of the signal space corresponding to H in [81, 87]) essentially is

limited to values between 0 and 1 (see Fig. 2.11 with λsupH

= 1).

2.3.15 Maximum System Gain (Operator Norm)

We now return to a general (i.e., not necessarily self-adjoint) LTV system H. For LTI and LFI

systems, the maximum system gain (operator norm) ‖H‖O as defined by (A.1) equals the supremum

19Here, instead of looking for the infimum of m(φs)H

and M(φs)H

within all s(t) with ‖s‖ = 1, we restricted to Gaussian

signals of varying length.


of the magnitude of the transfer function. We now ask whether for a general LTV system ‖H‖O is

similarly related to the magnitude of L(α)H

(t, f). The following result extends an existing bound for

DL operators [118].

Theorem 2.32. For any LTV system H, the difference between the supremum of |L(α)H

(t, f)|2 and the

squared maximum system gain is bounded as∣∣∣ supt,f

∣∣L(α)H

(t, f)∣∣2 − ‖H‖2

O

∣∣∣‖SH‖2

1

≤ 2πm(0,1)H

m(1,0)H

+ 4π|α|m(1,1)H

+m(φs)H+H

, (2.185)

with the weighting function φs(τ, ν) as in Theorem 2.28.

Proof. By subtracting and adding supt,f

∣∣L(0)H

(t, f)∣∣2 and supt,f L

(0)H+H

(t, f) and by using ‖H‖2O =

λsupH+H

as well as the triangle inequality, supt,f

∣∣L(α)H

(t, f)∣∣2 − ‖H‖2

O can be bounded as∣∣∣ sup

t,f

∣∣L(α)H

(t, f)∣∣2 − ‖H‖2

O

∣∣∣ ≤∣∣∣ sup

t,f

∣∣L(α)H

(t, f)∣∣2 − sup

t,f

∣∣L(0)H

(t, f)∣∣2∣∣∣

+∣∣∣ sup

t,f

∣∣L(0)H

(t, f)∣∣2 − sup

t,fL

(0)H+H

(t, f)∣∣∣+∣∣∣ sup

t,fL

(0)H+H

(t, f) − λsupH+H

∣∣∣. (2.186)

We next derive a bound for the difference ∆(α)14 (t, f) ,

∣∣L(α)H

(t, f)∣∣2 −

∣∣L(0)H

(t, f)∣∣2 by noting that the

2-D Fourier transform of∣∣L(α)

H(t, f)

∣∣2 is given by S(α)H

(τ, ν)∗∗S(α)∗H

(−τ,−ν) and by using S(α)H

(τ, ν) =

S(0)H

(τ, ν) e−j2πατν ,

∣∣∣∆(α)14 (t, f)

∣∣∣ ≤∫

τ

∫

ν

∣∣∣∆(α)14 (t, f)

∣∣∣ dτ dν

=

∫

τ

∫

ν

∣∣∣∣∫

τ ′

∫

ν′

S(0)H

(τ ′, ν ′)S(0)∗H

(τ ′ − τ, ν ′ − ν)[e−j2πα[τ ′ν′−(τ ′−τ)(ν′−ν)] − 1

]dτ ′ dν ′


≤ 2

∫

τ1

∫

ν1

∫

τ ′

∫

ν′

|SH(τ ′, ν ′)| |SH(τ1, ν1)|∣∣sinπα(τ ′ν ′ − τ1ν1)

∣∣ dτ ′ dν ′ dτ1 dν1

≤ 2π|α|∫

τ1

∫

ν1

∫

τ ′

∫

ν′

|SH(τ ′, ν ′)| |SH(τ1, ν1)|[|τ ′ν ′| + |τ1ν1|

]dτ ′ dν ′ dτ1 dν1

= 4π|α| ‖SH‖21m

(1,1)H

.

Using arguments similar to those in the proof of Theorem 2.28, it can be shown that this bound also

implies that the first term on the right hand side of (2.186) is bounded as∣∣∣ sup

t,f

∣∣L(α)H

(t, f)∣∣2 − sup

t,f

∣∣L(0)H

(t, f)∣∣2∣∣∣ ≤ 4π|α| ‖SH‖2

1m(1,1)H

. (2.187)

Furthermore, it similarly follows from (2.70) with α = 0 that the second term on the right hand side

of (2.186) is bounded as∣∣∣ sup

t,f

∣∣L(0)H

(t, f)∣∣2 − sup

t,fL

(0)H+H

(t, f)∣∣∣ ≤ 2π ‖SH‖2

1m(0,1)H

m(1,0)H

. (2.188)

Finally, since H+H is self-adjoint, the bounds in (2.144) hold for H+H and we obtain that the third

term on the right hand side of (2.186) is bounded as∣∣∣ sup

t,fL

(0)H+H

(t, f) − λsupH+H

∣∣∣ ≤ ‖SH+H‖1m(φs)H+H

≤ ‖SH‖21m

(φs)H+H

, (2.189)


where we furthermore used ‖SH+H‖1 ≤ ‖SH+‖1‖SH‖1 = ‖SH‖21. Inserting the bounds (2.187), (2.188),

and (2.189) into (2.186) finally yields (2.185).

Discussion. Theorem 2.32 shows that if m(φs)H+H

can be made small by suitable choice of the

function s(t), and if m(1,1)H

and m(0,1)H

m(1,0)H

are small too, then

supt,f

|L(α)H

(t, f)| ≈ ‖H‖O .

In particular, small m(φs)H+H

requires that As(τ, ν) ≈ As(0, 0) ≡ 1 on the effective support of

|SH+H(τ, ν)|, thus implying that H+H is underspread with very rapidly decreasing GSF. Further-

more, small m(1,1)H

and m(0,1)H

m(1,0)H

amount to the requirement that the GSF of H be localized along

the τ or ν axis. The latter requirement can be relaxed in the case α = 0. Here, the bound in (2.185)

will be tightest since its second term vanishes. Moreover, using (2.72) we obtain the tighter bound

∣∣∣ supt,f

∣∣L(0)H

(t, f)∣∣2 − ‖H‖2

O

∣∣∣‖SH‖2

1

≤ 2π infU∈M

{m

(0,1)UHU+m

(1,0)UHU+

}+m

(φs)H+H

. (2.190)

Since the metaplectic operators U ∈ M allow shearings and rotations of the GSF, the first term in

this refined bound may be small even if the GSF is oriented in an oblique direction in the (τ, ν) plane.

The second term, m(φs)H+H

, can be adapted to oblique directions by proper choice of s(t).

2.3.16 Approximate Commutation

In contrast to the LTI or LFI case, two LTV systems G and H typically do not commute, i.e.,

GH 6= HG or equivalently [G,H] 6= 0, where [G,H] , GH−HG is the commutator of G and H. A

necessary and sufficient condition for two operators to commute is the existence of a common system

of eigenfunctions [64]. However, even if two operators do not have a common system of eigenfunctions,

the following theorem shows that these operators will approximately commutate if they are jointly

underspread. In the following we consider the operator norm∥∥[G,H]

∥∥O. We note that bounds on the

HS norm∥∥[G,H]

∥∥2

have been formulated previously for DL operators [118].

Theorem 2.33. The operator norm of the commutator of two LTV systems G and H is bounded as

∥∥[G,H]∥∥2

O

‖SG‖21‖SH‖2

1

≤ 16π2[

infU∈M

B(0)

UGU+,UHU

+

]2

+ 8π infU∈M

{[m

(0,1)

UGU+ +m

(0,1)

UHU+

][m

(1,0)

UGU+ +m

(1,0)

UHU+

]}+ 4m

(φs)[G,H]+[G,H]

, (2.191)

with B(0)G,H = 1

2

[m

(0,1)G

m(1,0)H

+m(1,0)G

m(0,1)H

]and with φs(τ, ν) as in Theorem 2.28.

Proof. Applying (2.190) to [G,H] yields

∥∥[G,H]∥∥2

O≤ sup

t,f

∣∣L(0)[G,H](t, f)

∣∣2 +∥∥S[G,H]

∥∥2

1

[2π inf

U∈M

{m

(0,1)U[G,H]U+m

(1,0)U[G,H]U+

}+m

(φs)[G,H]+[G,H]

].

(2.192)


In order to bound supt,f

∣∣L(0)[G,H](t, f)

∣∣2, we consider L(0)[G,H](t, f). Subtracting and adding

L(0)G

(t, f)L(0)H

(t, f) and applying (2.63) twice, we obtain

∣∣∣L(0)[G,H](t, f)

∣∣∣ =∣∣∣L(0)

GH(t, f) − L

(0)HG

(t, f)∣∣∣

≤∣∣∣L(0)

GH(t, f) − L

(0)G

(t, f)L(0)H

(t, f)∣∣∣+∣∣∣L(0)

G(t, f)L

(0)H

(t, f) − L(0)HG

(t, f)∣∣∣

≤ ‖SG‖1‖SH‖1 4π infU∈M

B(0)

UGU+,UHU

+ ,

which gives the bound

supt,f

∣∣L[G,H](t, f)∣∣2 ≤ ‖SG‖2

1‖SH‖21 16π2

[inf

U∈MB

(0)

UGU+,UHU

+

]2. (2.193)

With regard to the second term on the right hand side of (2.192), we have

‖S[G,H]‖1m

(0,1)U[G,H]U+ =

∫

τ

∫

ν|ν|∣∣SU(GH−HG)U+(τ, ν)

∣∣ dτ dν

=

∫

τ

∫

ν|ν|∣∣SUGU+UHU+(τ, ν) − SUHU+UGU+(τ, ν)

∣∣ dτ dν

≤∫

τ

∫

ν|ν|[|SUGU+UHU+(τ, ν)| + |SUHU+UGU+(τ, ν)|

]dτ dν

= ‖SUGU+UHU+‖1m(0,1)UGU+UHU+ + ‖SUHU+UGU+‖1m

(0,1)UHU+UGU+

≤ ‖SUGU

+‖1‖SUHU+‖1

(m

(0,1)UGU+ +m

(0,1)UHU+

)

+ ‖SUHU

+‖1‖SUGU+‖1

(m

(0,1)UHU+ +m

(0,1)UGU+

)

= ‖SG‖1‖SH‖1 2(m

(0,1)UGU+ +m

(0,1)UHU+

),

where we twice applied (2.34) (once for H1 = UGU+, H2 = UHU+ and once for H1 = UHU+,

H2 = UGU+) and in the last step used the fact that ‖SUGU+‖1 = ‖SG‖1, ‖SUHU+‖1 = ‖SH‖1

for U ∈ M. In a similar manner, one can show using (2.33) that ‖S[G,H]‖1m

(1,0)U[G,H]U+ ≤

‖SG‖1‖SH‖1 2(m

(1,0)UGU+ + m

(1,0)UHU+

). Inserting these bounds into (2.192) and using ‖S[G,H]‖1

≤‖SGH‖1 + ‖SHG‖1 ≤ 2‖SG‖1‖SH‖1, the bound (2.191) follows.

Discussion. In the case of two jointly underspread systems where the weighted GSF integrals and

moments in the bound (2.191) are small, the operator norm of the commutator [G,H] is also small,

which shows that two jointly underspread systems approximately commute, i.e., GH ≈ HG. Since

two systems commute if and only if they have a common set of eigenfunctions [64,158], this result is

consistent with Theorem 2.22.

2.3.17 Approximate Normality

Normal systems have the advantage of allowing an eigenvalue decomposition instead of a (numerically

more expensive) singular value decomposition. In contrast to LTI or LFI systems, general LTV systems

may be non-normal, i.e., HH+ 6= H+H or equivalently [H,H+] = HH+ − H+H 6= 0. A bound on


∥∥[H,H+]∥∥

Ocould be obtained as a special case of (2.191). However, the following theorem exploits

the self-adjointness of [H,H+] to yield a tighter and simpler bound. We note that bounds on the HS

norm∥∥[H,H+]

∥∥2

have been formulated previously for DL operators [118].

Theorem 2.34. The operator norm of the commutator of an LTV system H and its adjoint H+ is

bounded as ∥∥[H,H+]∥∥

O

‖SH‖21

≤ 4π infU∈M

{m

(0,1)

UHU+m

(1,0)

UHU+

}+m

(φs)HH+ +m

(φs)H+H

, (2.194)

with the weighting function φs(τ, ν) as in Theorem 2.28.

Proof. It is known [158] that ‖[H,H+]‖O = sup |λ[H,H+]| = max{−λinf[H,H+], λ

sup[H,H+]

} =

max{λsup−[H,H+]

, λsup[H,H+]

}. First assume that ‖[H,H+]‖O = λsup[H,H+]

. In this case, the second bound in

(2.144) yields

∥∥[H,H+]∥∥

O= λsup

[H,H+]≤ sup

t,fL

(0)[H,H+]

(t, f) +∥∥S[H,H+]

∥∥1m

(φs)[H,H+]

≤ supt,f

∣∣∣L(0)[H,H+]

(t, f)∣∣∣+∥∥S[H,H+]

∥∥1m

(φs)[H,H+]

. (2.195)

Specializing (2.193), the first term in (2.195) can be bounded as

supt,f

∣∣∣L(0)[H,H+](t, f)

∣∣∣ ≤ ‖SH‖21 4π inf

U∈Mm

(0,1)

UHU+m

(1,0)

UHU+ , (2.196)

where we used that B(0)H,H+ = m

(0,1)H

m(1,0)H

due to m(k,l)H+ = m

(k,l)H

. The second term in (2.195) can be

bounded as

∥∥S[H,H+]

∥∥1m

(φs)[H,H+]

=

∫

τ

∫

νφs(τ, ν)|SHH+(τ, ν) − SH+H(τ, ν)| dτ dν (2.197)

≤∫

τ

∫

νφs(τ, ν)

[|SHH+(τ, ν)| + |SH+H(τ, ν)|

]dτ dν

= ‖SHH+‖1m(φs)HH+ + ‖SH+H‖1m

(φs)H+H

≤ ‖SH‖21

[m

(φs)HH+ +m

(φs)H+H

], (2.198)

where once again we used Young’s inequality. Inserting (2.196) and (2.198) into (2.195), the bound

(2.194) follows. If ‖[H,H+]‖O = −λinf[H,H+] = λsup

−[H,H+], the bound is shown similarly by applying the

second bound in (2.144) to −[H,H+].

Discussion. From Theorem 2.34, it follows that an underspread system H for which

inf U∈M

{m

(0,1)

UHU+m

(1,0)

UHU+

}as well as m

(φs)HH+ and m

(φs)H+H

are small satisfies HH+ ≈ H+H and is

thus approximately normal. In most cases, the latter two requirements will be the stronger con-

straints, requiring very rapid GSF decay. Note however that it is not necessary that the GSFs of H,

H+H, or HH+ are located along the τ or ν axis since all moments and weighted integrals involved in

the bound allow for oblique GSF orientation.


2.3.18 Sampling of the Generalized Weyl Symbol of Underspread Operators

In this section we shall investigate a sampling expansion of the GWS. GWS sampling is of practical

interest since it yields sparse representations of the underlying system [118]. This topic has previously

been considered for DL operators in [118] and in the context of random LTV channels in [11,102,103].

Here, we will consider two methods for the sampling of the GWS of non-DL operators and investigate

the associated reconstruction errors. In what follows, we will restrict to rectangular sampling grids,

although other grids might also be useful [118, 120, 121]. The following short-hand notation will be

used, where T and F denote the temporal and spectral sampling periods, respectively,

L(α)H

[k, l] , L(α)H

(kT, lF ).

In order to reconstruct the continuous GWS from its discrete samples, a weighted superposition of TF

shifted versions of an interpolation function ψ(t, f), with the GWS samples as weights, can be used,

L(α)H

(t, f) = TF∑

k

∑

l

L(α)H

[k, l]ψ(t − kT, f − lF ) .

If consistency is desired in the sense that the TF samples of L(α)H

(t, f) equal those of L(α)H

(t, f), i.e.,

L(α)H

(kT, lF ) = L(α)H

[k, l], one has to require that

ψ(kT, lF ) =

1

TF, k = l = 0

0, otherwise.

Using standard techniques of 2-D sampling theory, the 2-D Fourier transform of L(α)H

(t, f) can be

shown to be given by

S(α)H

(τ, ν) ,

∫

t

∫

fL

(α)H

(t, f) e−j2π(νt−τf) dt df = Ψ(τ, ν)∑

k

∑

l

S(α)H

(τ +

k

F, ν +

l

T

), (2.199)

where Ψ(τ, ν) =∫t

∫f ψ(t, f) e−j2π(νt−τf) dt df is the 2-D Fourier transform of ψ(t, f) and sum-

mations are from −∞ to ∞. A particularly interesting choice for the reconstruction kernel is

ψ(t, f) = 1TF sinc

(πtT

)sinc

(πfF

), with sinc(x) = sin(x)/x, or, equivalently, Ψ(τ, ν) = IG(τ, ν) with

G =[− 1

2F ,1

2F

]×[− 1

2T ,1

2T

]. In that case, the reconstruction formulas for the GWS and GSF read

L(α)H

(t, f) =∑

k

∑

l

L(α)H

[k, l] sinc( πT

(t− kT ))

sinc( πF

(f − lF ))

(2.200)

and

S(α)H

(τ, ν) = IG(τ, ν)∑

k

∑

l

S(α)H

(τ +

k

F, ν +

l

T

), (2.201)

The latter relation, which describes GWS sampling and reconstruction in the GSF domain, is illus-

trated in Fig. 2.13.

In the case of DL operators whose GSF support is completely contained in G, the continuous GWS

is perfectly re-obtained from its samples if we use Ψ(τ, ν) = IG(τ, ν). In fact, it follows from (2.201)

that in this case S(α)H

(τ, ν) = S(α)H

(τ, ν) and hence L(α)H

(t, f) = L(α)H

(t, f).


ν ν

τ τ

1/T

1/F

G

(a) (b)

Figure 2.13: Illustration of the effect of GWS sampling in the GSF domain: (a) GSF of original

system, (b) GSF of sampled GWS, reconstruction function Ψ(τ, ν) = IG(τ, ν) (dotted rectangle), and

GSF of reconstructed GWS (dark gray). A properly DL operator has been assumed resulting in error-

free reconstruction. In the case of an operator that is not properly DL, the various GSF components

in (b) would overlap and error-free reconstruction would be impossible.

In the case of non-DL operators or DL operators whose GSF support is not contained in G,

GWS sampling and interpolation using ψ(t, f) = 1TF sinc

(πtT

)sinc

(πfF

)yields a reconstructed GWS

L(α)H

(t, f) which equals L(α)H

(t, f) only on the sampling grid,

L(α)H

(kT, lF ) = L(α)H

(kT, lF ) ,

but is generally different from L(α)H

(t, f) otherwise. Let us consider the operator H defined by

L(α)bH

(t, f) = L(α)H

(t, f),

with L(α)H

(t, f) given by (2.200). With S(α)bH

(τ, ν) = S(α)H

(τ, ν) and (2.201), it follows that H is DL.

Furthermore, the GWS of H equals that of H on the sampling grid but is generally different otherwise.

A general bound on the error introduced by sampling the GWS of a system H that is not properly

DL, i.e., by using the “reconstructed operator” H instead of H, is stated in the following theorem.

Theorem 2.35. For any operator H and any sampling periods T and F , the difference

∆(α)15 (t, f) , L

(α)bH

(t, f) − L(α)H

(t, f) = L(α)H

(t, f) − L(α)H

(t, f)


15 (t, f)∣∣

‖SH‖1

≤ 4(m

(1,0)H

F +m(0,1)H

T),

∥∥∆(α)15

∥∥2

‖H‖2

≤ 4(M

(1,0)H

F +M(0,1)H

T). (2.202)

Proof. As before, let G =[− 1

2F ,1

2F

]×[− 1

2T ,1

2T

]. Splitting H into its DL part20 HG and its non-DL

part HG according to (2.5), we obtain

∆(α)15 (t, f) = L

(α)

HG+HG(t, f) − L

(α)

HG+HG(t, f)

20Note that in general HG 6= bH due to the aliasing components in bH.


= L(α)

HG (t, f) + L(α)

HG(t, f) − L

(α)

HG (t, f) − L(α)

HG(t, f)

= L(α)

HG(t, f) − L

(α)

HG(t, f) , (2.203)

since the DL part HG can be perfectly reconstructed, i.e., L(α)

HG (t, f) = L(α)

HG (t, f). Using the triangle

inequality, we have ∣∣∆(α)15 (t, f)

∣∣ ≤∣∣L(α)

HG(t, f)

∣∣+∣∣L(α)

HG(t, f)

∣∣. (2.204)

Using (2.201), the first term on the right hand side can be bounded as

∣∣L(α)

HG(t, f)

∣∣ ≤∫

τ

∫

ν

∣∣SHG (τ, ν)

∣∣ dτ dν

≤∫

τ

∫

νIG(τ, ν)

[∑

k

∑

l

∣∣∣SHG

(τ +

k

F, ν +

l

T

)∣∣∣]dτ dν

=

∫

τ ′

∫

ν′

[∑

k

∑

l

IG

(τ ′ − k

F, ν ′ − l

T

)

︸︷︷︸1

]∣∣SHG

(τ ′, ν ′

)∣∣ dτ ′ dν ′

=

∫

τ ′

∫

ν′

∣∣SHG (τ ′, ν ′)

∣∣ dτ ′ dν ′ =

∫∫

G

|SH(τ ′, ν ′)| dτ ′ dν ′ . (2.205)

The second term on the right-hand side of (2.204) can be bounded as

∣∣L(α)

HG(t, f)

∣∣ ≤∫

τ

∫

ν|S

HG (τ, ν)| dτ dν =

∫∫

G

|SH(τ, ν)| dτ dν . (2.206)

By inserting (2.205) and (2.206) in (2.204), it follows that∣∣∆(α)

15 (t, f)∣∣ ≤ 2

∫∫

G

|SH(τ, ν)| dτ dν. From

this, the first (L∞) bound in (2.202) finally follows upon applying the first bound in (2.38) with

τG = 1/(2F ) and νG = 1/(2T ). With regard to the second (L2) bound in (2.202), one can show

that∥∥∆(α)

15

∥∥2≤ 2

[ ∫∫

G

|SH(τ, ν)|2 dτ dν]1/2

, from which the final bound is obtained upon applying the

second bound in (2.38).

Note that (2.203) implies that the 2-D Fourier transform of ∆(α)15 (t, f) equals

∆(α)15 (τ, ν) ,

∫

t

∫

f∆

(α)15 (t, f) e−j2π(νt−τf) dt df = S

(α)

HG(t, f) − S

(α)

HG(τ, ν) ,

with S(α)H

(t, f) given by (2.201). According to (2.201) and (2.5), these two components are supported

within the disjoint regions G and G, respectively. The first component (corresponding to L(α)

HG(t, f))

is the aliasing error resulting from sampling L(α)H

(t, f) which is not properly bandlimited. The second

component (corresponding to L(α)

HG(t, f)) is the error which results since the non-DL part of H cannot

be reconstructed by bandlimited interpolation. We note that the application of Theorem 2.35 to the

sounding of mobile radio channels will be discussed in Section 4.3.

The reconstruction method described above preserves the GWS values on the sampling grid (t, f) =

(kT, lF ), but suffers from a reconstruction error due to aliasing if we sample the GWS of an operator


H that is not properly DL. In order to avoid this aliasing error, one could alternatively try to first

approximate H by a properly DL operator whose GWS can then be perfectly reconstructed from its

samples. Let us approximate H in a least-squares sense by a DL operator with GSF support contained

in G. Straightforward application of the orthogonality principle yields as best approximation the

operator HG (cf. also Subsection 2.1.1),

arg minG:GG=G

‖H − G‖2 = HG.

The next theorem quantifies the errors induced by this alternative sampling method.

Theorem 2.36. For any operator H and any sampling periods T , F with corresponding region G =[− 1

2F ,1

2F

]×[− 1

2T ,1

2T

], the difference

∆(α)16 (t, f) , L

(α)H

(t, f) − L(α)

HG (t, f)

is bounded as

∣∣∆(α)16 (t, f)

∣∣‖SH‖1

≤ 2(m

(1,0)H

F +m(0,1)H

T),

∥∥∆(α)16

∥∥2

‖H‖2

≤ 2(M

(1,0)H

F +M(0,1)H

T). (2.207)

Proof. Since HG is properly DL, it can be perfectly reconstructed, i.e., HG = HG and L(α)

HG (t, f) =

L(α)

HG (t, f). Hence, the reconstruction error is given by

∆(α)16 (t, f) = L

(α)H

(t, f) − L(α)

HG (t, f) = L(α)H

(t, f) − L(α)

HG (t, f) = L(α)

HG(t, f) .

The bounds in (2.207) then follow straightforwardly upon combining (2.37) and in (2.38).

By comparing (2.207) with (2.202), it is seen that the bounds on the reconstruction error ∆(α)16 (t, f)

are smaller than the respective bounds on the reconstruction error ∆(α)15 (t, f) (resulting from sampling

L(α)H

(t, f) directly). This corresponds to the fact that no aliasing errors are introduced when sampling

L(α)

HG (t, f). On the other hand, the GWS samples of H and its DL approximation HG are not equal

in general, L(α)H

[k, l] 6= L(α)

HG [k, l], and thus this second method does not give perfect reconstruction on

the sampling grid.

An example illustrating the sampling and reconstruction of the GWS of a non-DL operator is shown

in Fig. 2.14. It is seen that irrespective of the fact that H is non-DL, both sampling/reconstruction

methods yield satisfactory results, the reason being that H is sufficiently underspread. In this exam-

ple, the normalized errors introduced by sampling the GWS directly are|∆

(α)15 (t,f)|‖SH‖1

= 1.7 · 10−4 and

‖∆(α)15 ‖2

‖H‖2= 0.002, while the corresponding L∞ and L2 bounds in (2.202) are 0.01 and 0.1, respectively.

The normalized errors obtained by approximating H by its DL part HG (which can be perfectly recon-

structed from its GWS samples) are|∆

(α)16 (t,f)|‖SH‖1

= 10−4 and‖∆

(α)16 ‖2

‖H‖2= 0.001, while the corresponding

L∞ and L2 bounds in (2.207) are 0.005 and 0.05, respectively. These errors as well as the correspond-

ing bounds confirm that the second approach, based on the least-squares approximation of H by HG,

is superior to sampling the GWS directly. Finally, we note that in this example the bounds are larger


(a) (b) (c) (d)

t- t- t- t-

f

6

f

6

f

6

f

6

Figure 2.14: Example illustrating the sampling/reconstruction of the GWS (with α = 0) of a non-DL

system: (a) true Weyl symbol L(0)H

(t, f), (b) reconstructed Weyl symbol L(0)H

(t, f), (c) Weyl symbol

of DL part HG, (d) reconstruction kernel ψ(t, f). The number of samples is 128, the (normalized)

frequency range covered is [−1/4, 1/4], and the temporal and spectral sampling periods were T = 16

and F = 1/16, respectively.

by a factor of about 50 than the true errors. This can be attributed to the fact that our error bounds

are based on the Chebyshev-like inequalities in (2.38), which often tend to be rather loose.

3

Underspread Processes

“All stable processes we shall predict. All unstableprocesses we shall control.” John von Neumann

THIS chapter discusses the analysis and characterization of nonstationary random processes via

time-frequency methods. In Section 3.1, we consider the description of the statistical time-

frequency correlations of random processes by means of time-frequency correlation functions and the

expected ambiguity function, and we discuss the concept of underspread and overspread processes. In

Section 3.2, we study two different families of time-varying power spectra—the generalized Wigner-

Ville spectrum and the generalized evolutionary spectrum—and we show that these two families of

spectra become approximately equivalent in the case of underspread processes. We also discuss the

relation of time-frequency correlations with statistical cross or interference terms appearing in time-

varying power spectra. The suppression of such interference terms via smoothing provides a motivation

for the definition of two fundamental classes of time-varying power spectra which we refer to as type

I spectra (Section 3.3) and type II spectra (Section 3.4). These two classes of spectra extend the

generalized Wigner-Ville spectrum and the generalized evolutionary spectrum, respectively. It turns

out that for underspread processes, type I and type II spectra satisfy desirable mathematical properties

at least in an approximate way. Furthermore, we show in Section 3.5 that in the underspread case

type I and type II spectra are almost identical.

Further topics discussed in this chapter include time-frequency input-output relations for nonsta-

tionary random processes (Section 3.6), approximate Karhunen-Loeve expansions (Section 3.7), and

the concept of time-frequency coherence functions (Section 3.8).

97

98 Chapter 3. Underspread Processes

3.1 Time-Frequency Correlation Analysis

In Chapter 2, the characterization of time-frequency (TF) displacements of linear systems was seen

to be of fundamental importance. For random processes, a similar role is played by TF correlations.

In what follows, this will be explained in more detail.

3.1.1 Motivation

The basic second-order statistic of a nonstationary random process1 x(t) is the (temporal) correlation

function rx(t1, t2) = E {x(t1)x∗(t2)} (or, equivalently, the correlation operator Rx). Another useful

second-order characterization is the spectral correlation function, i.e., the correlation of the process’

Fourier transformX(f), RX(f1, f2) = E {X(f1)X∗(f2)} [60,63,136]. The spectral correlation function

is also referred to as Loeve’s spectrum and has recently gained new interest (see e.g. [76,199]). In the

stationary case, the spectral correlation function is nonzero only for f1 = f2, which shows that different

spectral components are uncorrelated and thus only temporal correlations are present. Similarly, the

temporal correlation function of a white process is nonzero only for t1 = t2, which reflects that

different temporal components are uncorrelated (while spectral correlations may well be present). In

the extreme case of a stationary and white process, neither temporal nor spectral correlations are

present. General nonstationary processes, however, feature both temporal and spectral correlations.

Joint descriptions of these TF correlations are considered next.

3.1.2 Time-Frequency Correlation Functions

To obtain a function that jointly describes the temporal and spectral correlations of a random process,

we present a modification and generalization of the line of arguments given in [118, 126]. There, the

correlation function of the short-time Fourier transform (STFT, see Subsection B.2.1) of x(t) using

analysis window2 g(t)

R(g)x (t1, f1; t2, f2) , E

{STFT(g)

x (t1, f1) STFT(g)∗x (t2, f2)

}

was introduced as a measure of the correlation between two process components localized around the

TF points (t1, f1) and (t2, f2), respectively. Since STFT(g)x (t, f) = 〈x, gt,f 〉 with gt,f (t′) = (St,fg)(t

′),

R(g)x (t1, f1; t2, f2) can be rewritten as

R(g)x (t1, f1; t2, f2) = E {〈x, gt1,f1〉〈x, gt2,f2〉∗} = E

{⟨PgS

+t1,f1

x,S+t2,f2

x⟩}

= Tr{St2,f2PgS

+t1,f1

Rx

},

(3.1)

where Pg = g⊗g∗ denotes the rank-one projection operator on span{g(t)}. Note that the temporal

and spectral correlation functions are re-obtained from R(g)x (t1, f1; t2, f2) with g(t) = δ(t) and g(t) = 1,

respectively:

g(t) = δ(t) , =⇒ R(g)x (t1, f1; t2, f2) ≡ rx(t1, t2) ,

1Throughout this chapter, we assume that all random processes have zero mean.2The window g(t) is assumed to be normalized and well-localized about the origin of the TF plane.

3.1 Time-Frequency Correlation Analysis 99

g(t) = 1 , =⇒ R(g)x (t1, f1; t2, f2) ≡ RX(f1, f2) .

Furthermore, for t1 = t2 = t and f1 = f2 = f , there is

R(g)x (t, f ; t, f) = E

{SPEC(g)

x (t, f)}

= PS(g)x (t, f) ,

i.e., the “diagonal” of the TF correlation function equals the physical spectrum (see Subsections 3.3.2

and B.3.3).

By replacing Pg in (3.1) with a trace-normalized TF localization operator T [42–44,80,81,174,175],

a generalization of R(g)x (t1, f1; t2, f2) can be defined as

R(T)x (t1, f1; t2, f2) , E

{⟨TS+

t1,f1x,S+

t2,f2x⟩}

= Tr{St2,f2TS+

t1,f1Rx

}. (3.2)

This is a more flexible measure of the TF correlations of x(t) than R(g)x (t1, f1; t2, f2). For compact and

normal T with eigendecomposition T =∑

k γk gk⊗g∗k =∑

k γk Pgk(see Subsection A.3), one obtains

R(T)x (t1, f1; t2, f2) = Tr

{St2,f2

(∑

k

γk Pgk

)S+

t1,f1Rx

}=∑

k

γkTr{St2,f2Pgk

S+t1,f1

Rx

}

=∑

k

λkR(gk)x (t1, f1; t2, f2) .

This shows that R(g)x (t1, f1; t2, f2) in (3.1) is a special case of R

(T)x (t1, f1; t2, f2) obtained using a rank-

one localization operator T = Pg. A further special case is T = 1N P where P is the projection operator

on a given N -dimensional subspace X . To each non-sophisticated subspace X , there corresponds a TF

region R such that L(α)P

(t, f) ≈ IR(t, f) with IR(t, f) the indicator function of R [80, 81]. Assuming

that P is such that the corresponding TF region R is localized around the origin of the TF plane, the

TF correlation R(T)x (t1, f1; t2, f2) can be interpreted as correlation of the process components localized

within IR(t− t1, f − f1) and the process components localized within IR(t− t2, f − f2) (see Fig. 3.1).

In the subsequent, the following coordinate-transformed version of the TF correlation will be

convenient,

R(T)x (t, f ;∆t,∆f) , R(T)

x

(t+

∆t

2, f +

∆f

2; t− ∆t

2, f − ∆f

2

).

In Section 3.3, the “diagonal” R(T)x (t, f ; t, f) = R

(T)x (t, f ; 0, 0) will be seen to correspond to a specific

class of time-varying spectra.

3.1.3 The Expected Ambiguity Function

The correlation function R(T)x (t1, f1; t2, f2) is physically intuitive but has two drawbacks: first, it

depends on the specific choice of the TF localization operator T and thus is non-unique; second, as a

function of four variables it is cumbersome to work with. We will see presently that the generalized

expected ambiguity function [118,120,126] (GEAF) resolves the above two problems; however, it does

not directly lend itself to an intuitive physical interpretation.


��

��

��

��

��

��

��

��

��

��

��

��

x(t)

t1

f1

f2

f IR(t − t2, f − f2)

t2t

IR(t − t1, f − f1)

Figure 3.1: Illustration of the TF correlation R(T)x (t1, f1; t2, f2) between components of the random

process x(t) localized in regions around the TF analysis points (t1, f1) and (t2, f2). In this example,

T = P/3 where P is the projection operator on the space spanned by the first three Hermite functions

[64, 81, 99] (each represented by a hatched annular region). The corresponding TF region R has

approximately area 3. The shaded region illustrates the TF support of the nonstationary process x(t).

The GEAF is defined as (cf. (B.52))

A(α)x (τ, ν) =

∫

tr(α)x (t, τ) e−j2πνt dt , (3.3)

with r(α)x (t, τ) = rx

(t+

(12 − α

)τ, t−

(12 + α

)τ). Under certain weak conditions, it equals the expec-

tation of the process’ generalized ambiguity function, A(α)x (τ, ν) = E

{A

(α)x (τ, ν)

}= E

{⟨x,S

(α)τ,νx

⟩}.

Furthermore, by comparing (3.3) with (B.1), it is seen that the GEAF is the GSF of the correlation

operator Rx,

A(α)x (τ, ν) = S

(α)Rx

(τ, ν) .

Further properties of the GEAF can be found in Subsection B.3.4 of Appendix B and, in [118,126].

What is important in the context of the TF correlation function R(T)x (t, f ;∆t,∆f) is the fact that

R(T)x (t, f ;∆t,∆f) = e−j2πt∆f

∫

τ

∫

νS

(0)T

(∆t− τ,∆f − ν) A(0)x (τ, ν) ej2π(νt−τ f) dτ dν . (3.4)

If∣∣ST(τ, ν)

∣∣ is well localized around the origin, (3.4) shows that R(T)x (t, f ;∆t,∆f), and hence also

R(T)x (t1, f1; t2, f2), is essentially determined by the values of A

(0)x (τ, ν) around τ = ∆t and ν = ∆f .

Furthermore, (3.4) implies that the magnitude of the TF correlation function is bounded by the

convolution of |ST(τ, ν)| and Ax(τ, ν)|, i.e.,

∣∣R(T)x (t, f ;∆t,∆f)

∣∣ ≤∫

τ

∫

ν|ST(τ − ∆t, ν − ∆f)| |Ax(τ, ν)| dτ dν . (3.5)


This upper bound does not depend on the absolute location (center point) (t, f) of the TF analysis

points (t1, f1) and (t2, f2) but only on their time separation ∆t = t1 − t2 and frequency separation

∆f = f1 − f2. Thus, for well localized |ST(τ, ν)|, (3.5) shows that the magnitude |Ax(τ, ν)| of the

GEAF at a point (τ, ν) characterizes the correlation of all process components separated by ∆t = τ

in time and by ∆f = ν in frequency.

It is insightful to formally consider two extreme cases3 for the choice of the operator T underlying

R(T)x (t, f ;∆t,∆f):

• T = I: Choosing T as the identity operator implies that no TF localization is performed at all.

Consequently, all components of x(t) separated by ∆t in time and by ∆f in frequency contribute

to the TF correlation function. This is reflected by

R(I)x (t, f ;∆t,∆f) = A(0)

x (∆t,∆f) e−j2πf∆t ,∣∣R(I)

x (t, f ;∆t,∆f)∣∣ =

∣∣Ax(∆t,∆f)∣∣ ,

which shows that here TF correlation function and GEAF become perfectly equivalent. In

particular, |R(T)x (t, f ;∆t,∆f)| does not depend on the absolute location (center point) (t, f) of

the TF analysis points (t1, f1) and (t2, f2).

• T = L(0): Choosing T to equal the TF reflection operator L(0) defined by (B.22) yields “perfect”

TF localization since L(0)

L(0)(t, f) = δ(t)δ(f). Here, R(T)x (t, f ;∆t,∆f) is obtained as

R(L(0))x (t, f ;∆t,∆f) = e−j2πt∆f W

(0)x (t, f) ,

∣∣R(L(0))x (t, f ;∆t,∆f)

∣∣ =∣∣W (0)

x (t, f)∣∣ .

It is seen that in this case,∣∣R(L(0))

x (t, f ;∆t,∆f)∣∣ is independent of the time separation ∆t and

the frequency separation ∆f .

We close this section on the GEAF by considering the special case of a Gaussian random process.

This sheds additional light on the interpretation of the GEAF as a TF correlation. We consider

the covariance function of the generalized Wigner distribution (GWD) W(α)x (t, f) of x(t) (see Section

B.2.2), defined as

C(α)x (t1, f1; t2, f2) , cov

{W (α)

x (t1, f1),W(α)x (t2, f2)

}

= E{[W (α)

x (t1, f1) −W(α)x (t1, f1)

][W (α)

x (t2, f2) −W(α)x (t2, f2)

]∗}

= E{W (α)

x (t1, f1)W(α)∗x (t2, f2)

}−W

(α)x (t1, f1)W

(α)∗x (t2, f2) . (3.6)

In the following, we use a coordinate-transformed version of C(α)x (t1, f1; t2, f2) defined as

C(α)x (t, f ; τ, ν) = C(α)

x (t+ α−τ, f + α+ν; t− α+τ, f − α−ν) , (3.7)

with α+ = 1/2 + α and α− = 1/2 − α.3Note that these two cases do not satisfy our assumption of T having normalized trace.


Theorem 3.1. For a circular complex Gaussian random process x(t),

∫

t

∫

fC(α)

x (t, f ; τ, ν) dt df =∣∣Ax(τ, ν)

∣∣2. (3.8)

Proof. We start by developing the correlation function of the generalized Wigner distribution using

Isserli’s formula [177] for circular complex Gaussian random processes,

E {x(t1)x∗(t2)x(t3)x∗(t4)} = rx(t1, t2) r∗x(t4, t3) + rx(t1, t4) r

∗x(t2, t3) .

Using this in (B.39) yields

E{W (α)

x (t1, f1)W(α)∗x (t2, f2)

}

= E

{∫

τ1

x(t1 + α−τ1)x∗(t1 − α+τ1) e

−j2πf1τ1 dτ1

∫

τ2

x∗(t2 + α−τ2)x(t2 − α+τ2) ej2πf2τ2 dτ2

}

=

∫

τ1

∫

τ2

E{x(t1 + α−τ1)x

∗(t1 − α+τ1)x(t2 − α+τ2)x∗(t2 + α−τ2)

}e−j2π(f1τ1−f2τ2) dτ1 dτ2

=

∫

τ1

∫

τ2

[rx(t1 + α−τ1, t1 − α+τ1) r

∗x(t2 + α−τ2, t2 − α+τ2) (3.9)

+ rx(t1 + α−τ1, t2 + α−τ2) r∗x(t1 − α+τ1, t2 − α+τ2)

]e−j2π(f1τ1−f2τ2) dτ1 dτ2

= W(α)x (t1, f1)W

(α)∗x (t2, f2)

+

∫

τ1

∫

τ2

rx(t1 + α−τ1, t2 + α−τ2) r∗x(t1 − α+τ1, t2 − α+τ2) e

−j2π(f1τ1−f2τ2) dτ1 dτ2

= W(α)x (t1, f1)W

(α)∗x (t2, f2) + W(α)

Rx(t1, f1; t2, f2) , (3.10)

where W(α)Rx

(t1, f1; t2, f2) is the generalized transfer Wigner distribution of Rx as defined in (B.32).

Inserting (3.10) into (3.6) shows that Cx(t1, f1; t2, f2) = W(α)Rx

(t1, f1; t2, f2) and hence

C(α)x (t, f ; τ, ν) = C(α)

x (t+ α−τ, f + α+ν; t− α+τ, f − α−ν)

= W(α)Rx

(t+ α−τ, f + α+ν; t− α+τ, f − α−ν) = W(α)Rx

(t, f ; τ, ν) ,

with W(α)H

(t, f ; τ, ν) as defined in (B.33). The final result (3.8) then follows from (B.35) (with H

replaced by Rx), i.e.,

∫

t

∫

fC(α)

x (t, f ; τ, ν) dt df =

∫

t

∫

fW(α)

Rx(t, f ; τ, ν) dt df =

∣∣SRx(τ, ν)∣∣2 ,

by noting that S(α)Rx

(τ, ν) = A(α)x (τ, ν).

The foregoing theorem confirms that for Gaussian processes the squared magnitude of the GEAF

at (τ, ν) can be interpreted as the integrated covariance of all pairs of GWD values separated by τ in

time and by ν in frequency.


3.1.4 Extended Concept of Underspread Processes

In many situations, a simple global characterization of TF correlations by just a few parameters is

desired. Such a characterization will be provided next. The basis of our discussion is the fact that

the GEAF of a random process x(t) equals the GSF of the correlation operator Rx of x(t), i.e.,

A(α)x (τ, ν) = S

(α)Rx

(τ, ν). Hence, all of the considerations in Sections 2.1 and 2.2 regarding the global

characterization of TF displacements of linear systems (as described in detail by the GSF) carry over

to the global characterization of TF correlations of random processes (as described in detail by the

GEAF).

In the following, we shall thus rephrase (with minor modifications) the main ideas of Sections 2.1

and 2.2 in the context of TF correlation analysis of random processes. For a better understanding,

we note that the unitary transformation URxU+ of a correlation operator Rx corresponds to the

transformation (Ux)(t) of the underlying random process, i.e.,

x(t) = (Ux)(t) =⇒ Rx = URxU+.

For metaplectic operators U ∈ M this corresponds to a symplectic coordinate transformation of the

GEAF (see Appendix C).

Correlation-Limited Processes. In [118,120,126], the support of the GEAF was used to define

a global measure of TF correlations that is similar to the global characterization of TF shifts discussed

in Section 2.1.

Let us consider a random process x(t) with GEAF supported within a region Gx, i.e.,

A(α)x (τ, ν) = A(α)

x (τ, ν) IGx(τ, ν) , (3.11)

where IGx(τ, ν) is the indicator function of Gx and Gx is the minimal region such that (3.11) holds.

Any such process will be termed correlation limited (CL). The temporal correlation width τ(max)x and

the spectral correlation width ν(max)x of a CL process are defined as

τ (max)x , max

(τ,ν)∈Gx

|τ | , ν(max)x , max

(τ,ν)∈Gx

|ν| .

Assuming in addition that the area of the region Gx is sufficiently small leads us to the definition of

CL underspread processes:

Definition 3.2. Let x(t) be a CL process with temporal correlation width τ(max)x and spectral correlation

width ν(max)x . Then,

σx , 4τ (max)x ν(max)

x

is called the (strict-sense) correlation spread of the CL process x(t), and x(t) is called strict-sense CL

underspread if

σx ≪ 1 . (3.12)

If x(t) is not strict-sense CL underspread but there exists a metaplectic operator U ∈ M such that the

correlation spread of (Ux)(t) satisfies σUx ≪ 1, then x(t) is called wide-sense CL underspread. The


quantitity

σ(min)x , min

U∈MσUx

is referred to as wide-sense correlation spread of x(t).

The CL underspread definition (3.12) was introduced in [117, 118, 120, 126] and further used in

[92, 115, 116, 122–124, 129, 141]. We note that the wide-sense correlation spread σ(min)x measures the

area of a rectangle circumscribed about the region Gx. An alternative measure of the global TF

correlations of a process is given by

µx , max(τ,ν)∈Gx

|τν| .

It can be used as basis for the alternative CL underspread condition

µx ≪ 1 , (3.13)

which corresponds to a hyperbolic support constraint (cf. Fig. 2.1 for an illustration of rectangular

and hyperbolic support constraints). Since

µx = max(τ,ν)∈Gx

|τν| ≤ max(τ,ν)∈Gx

|τ | max(τ,ν)∈Gx

|ν| = τ (max)x ν(max)

x = σx/4 ,

the condition (3.13) is less restrictive than (3.12). Note that A(α)x (τ, ν) = S

(α)Rx

(τ, ν) implies that

σx = σRx and µx = µRx , where σH and µH were defined in Subsection 2.1.2.

Processes with Rapidly Decaying Expected Ambiguity Function. Processes with GEAF

having exactly limited support seem unrealistic in practical applications. Using an effective GEAF

support instead is problematic since there is no unique choice for the effective support and furthermore

the modelling errors incurred by a specific choice are difficult to judge. Hence, similar to Section 2.2,

we propose to use weighted integrals of the GEAF as alternative global measures of TF correlation,

i.e. (cf. (2.12-a), (2.12-b)),

m(φ)x ,

∫

τ

∫

νφ(τ, ν)

∣∣Ax(τ, ν)∣∣ dτdν

∫

τ

∫

ν

∣∣Ax(τ, ν)∣∣ dτdν

=1∥∥Ax

∥∥1

∫

τ

∫

νφ(τ, ν)

∣∣Ax(τ, ν)∣∣ dτ dν , (3.14a)

M (φ)x ,

∫

τ

∫

νφ2(τ, ν)

∣∣Ax(τ, ν)∣∣2 dτdν

∫

τ

∫

ν

∣∣Ax(τ, ν)∣∣2 dτdν

1/2

=1∥∥Ax

∥∥2

[∫

τ

∫

νφ2(τ, ν)

∣∣Ax(τ, ν)∣∣2 dτ dν

]1/2

, (3.14b)

where φ(τ, ν) is a nonnegative weighting function as described in Subsection 2.2.2. A special case of

these weighted integrals are the GEAF moments obtained with weighting function φk,l(τ, ν) = |τ |k |ν|l(cf. (2.13-a), (2.13-b) and Fig. 2.2),

m(k,l)x , m

(φk,l)x , M (k,l)

x , M(φk,l)x . (3.15)


Due to A(α)x (τ, ν) = S

(α)Rx

(τ, ν), the weighted integrals and moments of the GEAF of x(t) equal the

corresponding weighted integrals and moments of the GSF of Rx, i.e.,

m(φ)x = m

(φ)Rx, M (φ)

x = M(φ)Rx, (3.16)

m(k,l)x = m

(k,l)Rx

, M (k,l)x = M

(k,l)Rx

. (3.17)

Since the weighted GEAF integrals m(φ)x and M

(φ)x and the GEAF moments m

(k,l)x and M

(k,l)x

measure the spread of∣∣Ax(τ, ν)

∣∣ about the origin of the (τ, ν) plane, we can consider a nonstationary

random process x(t) to be underspread if suitable GEAF integrals/moments are “small,” without

being forced to assume that the GEAF has compact support. We note that while this is not a clear-

cut definition of underspread processes, this underspread concept is less restrictive and provides us

with more flexibility than an underspread definition based on compact GEAF support. Some of the

conditions (satisfied by underspread systems) that will be important in subsequent sections are the

following:

m(1,1)x ≪ 1, M (1,1)

x ≪ 1,

m(1,0)x m(0,1)

x ≪ 1, M (1,0)x M (0,1)

x ≪ 1, (3.18)

m(φs)x ≪ 1, M (φs)

x ≪ 1,

where in the last line either φs(τ, ν) = |1−A(α)s (τ, ν)| or φs(τ, ν) =

∣∣∣1− 1

A(α)s (τ,ν)

∣∣∣ with some normalized

function s(t). Examples for underspread systems satisfying these constraints are illustrated in Fig. 3.2.

Note that this extended concept of underspread processes is not equivalent to that of quasistationary

processes (which require the GEAF to be concentrated along the τ axis, see part (d) of Fig. 3.2).

In particular, a quasistationary process may be overspread (i.e., not underspread) if its effective

temporal correlation width is large while a highly nonstationary process (i.e., a process with large

effective spectral correlation width) may be underspread if its effective temporal correlation is short

enough.

Since oblique orientations of the GEAF are not allowed by the conditions in the first two lines of

(3.18), in such situations the less restrictive conditions

minU∈M

{m

(1,0)Ux m

(0,1)Ux

}≪ 1, min

U∈M

{M

(1,0)Ux M

(0,1)Ux

}≪ 1

can be used, with M denoting the set of metaplectic transforms. Here, an appropriate U ∈ Mproduces a coordinate-transformed GEAF such that the oblique orientation of |Ax(τ, ν)| is converted

into an orientation of |AUx(τ, ν)| along the τ axis and/or ν axis (see Appendix C).

While the interpretation of the weighted GEAF integrals and GEAF moments in terms of corre-

lations is different from that of the weighted GSF integrals and GSF moments defined in Subsection

2.2.2, the discussion in Subsections 2.2.2 through 2.2.6 concerning the mathematical properties of

the weighted GSF integrals and GSF moments straightforwardly carries over to the weighted GEAF

integrals and GEAF moments and thus need not be repeated here. Only the case of sums of random

processes deserves some special attention.


(a) (b) (c) (d) (e)

τ τ τ τ τ

ν ν ν ν ν

Figure 3.2: Schematic representation of the GEAF magnitude of (a) an underspread process with

small m(1,1)x and M

(1,1)x ; (b) an underspread process with small m

(1,0)x m

(0,1)x and M

(1,0)x M

(0,1)x ; (c) an

underspread process with small m(1,0)Ux m

(0,1)Ux with U corresponding to a rotation; (c) a quasistationary

process (small m(0,l)x and M

(0,l)x ); (d) a quasiwhite process (small m

(k,0)x and M

(k,0)x ).

Sums of Random Processes. The difficulty with the sum x(t) = x1(t) + x2(t) of two random

processes x1(t) and x2(t) stems from the fact that the correlation Rx does not equal the sum of the

individual correlation operators Rx1 and Rx2, unless x1(t) and x2(t) are uncorrelated. In general, one

has

Rx = Rx1 + Rx2 + Rx1,x2 + Rx2,x1 = Rx1 + Rx2 + 2RHx1,x2

,

where Rx1,x2 denotes the cross-correlation operator associated to the cross-correlation function

E {x1(t)x∗2(t

′)} and RHx1,x2

, (Rx1,x2 + R+x1,x2

)/2 is the hermitian (self-adjoint) part of Rx1,x2. Since

A(α)x (τ, ν) = S

(α)Rx

(τ, ν) = S(α)Rx1

(τ, ν) + S(α)Rx2

(τ, ν) + 2S(α)

RHx1,x2

(τ, ν)

= A(α)x1

(τ, ν) + A(α)x2

(τ, ν) + 2H{A(α)

x1,x2(τ, ν)

},

with H{A

(α)x1,x2(τ, ν)

}=[A

(α)x1,x2(τ, ν) + A

(α)∗x1,x2(−τ,−ν)

]/2 denoting the (hermitian) even part of

A(α)x1,x2(τ, ν), it follows that the magnitude of A

(α)x (τ, ν) is bounded as

∣∣A(α)x (τ, ν)

∣∣ ≤∣∣A(α)

x1(τ, ν)

∣∣+∣∣A(α)

x2(τ, ν)

∣∣+ 2∣∣H{A(α)

x1,x2(τ, ν)

}∣∣ .

Thus, it is seen that in order that the GEAF of x(t) is concentrated about the origin, not only the

individual GEAFs of x1(t) and x2(t) but also the cross GEAF A(α)x1,x2(τ, ν) has to be concentrated

about the origin. This observation is the basis for our concept of jointly underspread processes:

Definition 3.3. Two processes x1(t) and x2(t) are referred to as being jointly underspread if they both

are individually underspread and furthermore the cross-correlation operator Rx1,x2 is underspread.

Note that two uncorrelated underspread processes are thus jointly underspread.

3.1.5 Innovations System

An alternative characterization of the TF correlations of a process is in terms of innovations systems.

The innovations system representation of a random process x(t) is given by [38,148,152]

x(t) = (Hn)(t) =

∫

t′h(t, t′)n(t′) dt′ , (3.19)


where n(t) denotes normalized stationary white noise with correlation operator I. An innovations

system H is obtained as a solution of the equation HH+ = Rx. This solution is not unique since,

given a valid innovations system H and unitary system U such that UU+ = I, also the system

H = HU satisfies the defining equation, HH+ = HUU+H+ = HH+ = Rx, and thus also H is a

valid innovations system. Subsequently, the set of all innovations systems for a given random process

x(t) will be denoted by Ix, i.e., Ix , {H : HH+ = Rx}.The relation Rx = HH+ implies that A

(α)x (τ, ν) = S

(α)Rx

(τ, ν) = S(α)HH+(τ, ν) and thus by virtue of

(B.13)

|Ax(τ, ν)| ≤ |SH(τ, ν)| ∗∗ |SH(−τ,−ν)| . (3.20)

Hence, if the innovations system H is underspread (cf. Chapter 2) and thus |SH(τ, ν)| is concentrated

about the origin, also |Ax(τ, ν)| will be concentrated about the origin and thus the process x(t) will

be underspread too. Conversely, if x(t) is an underspread process, one can always find an underspread

innovations system H. In particular, the innovations system H =√

Rx is maximally underspread

since the positive square root has minimal displacement effects among all innovations systems H ∈ Ix

[86,90,148]. We furthermore note that (3.20) also implies that the innovations system of an overspread

process is necessarily overspread too.

From these relations, it follows that for characterizing the TF correlations of a random process x(t),

the weighted integrals m(φ)H

, M(φ)H

and moments m(k,l)H

, M(k,l)H

of the GSF of the innovations system

H =√

Rx provide an alternative to the weighted integrals and moments m(φ)x , M

(φ)x and moments

m(k,l)x , M

(k,l)x of the GEAF of x(t). Proposition 2.7 (with H2 = H and H1 = H+) establishes a relation

between the normalized moments m(k,l)x = m

(k,l)x /(k! l!) and m

(k,l)H

= m(k,l)H

/(k! l!),

m(k,l)x ≤ ‖SH‖2

1

‖Ax‖1

k∑

i=0

l∑

j=0

m(i,j)H

m(k−i,l−j)H

.

Important special cases of this relation are (cf. (2.33)–(2.35))

m(1,0)x ≤ 2

‖SH‖21

‖Ax‖1m

(1,0)H

, m(0,1)x ≤ 2

‖SH‖21

‖Ax‖1m

(0,1)H

, m(1,1)x ≤ 2

‖SH‖21

‖Ax‖1

[m

(1,1)H

+m(0,1)H

m(1,0)H

].

Finally, (3.20) implies that a displacement-limited (DL) innovations system H with GSF support

region GH = [−τ (max)H

, τ(max)H

] × [−ν(max)H

, ν(max)H

] generates a CL process x(t) with GEAF support

region Gx = [−τ (max)x , τ

(max)x ] × [−ν(max)

x , ν(max)x ] where τ

(max)x = 2τ

(max)H

and ν(max)x = 2ν

(max)H

.

3.1.6 An Example

We next present two example processes that illustrate the difference between underspread and over-

spread processes. The processes will be specified by their Karhunen-Loeve (KL) expansions.

Underspread Process. Let {v(1)k (t)}k=1...N and {v(2)

k (t)}k=1...N denote two orthonormal sets of

N signals that are well TF localized in two TF regions R1 and R2, respectively. The regions R1 and

R2 are separated in time by ∆t and in frequency by ∆f (see Fig. 3.3(a)). We assume that the signals


t

τ

(a)

R1

R2

∆f

∆t

f νν

−∆t

∆t

−∆f

∆f

(b) (c)

τ

Figure 3.3: Illustration of underspread and overspread processes: (a) sketch of TF regions R1 and R1

underlying the processes’ KL expansion, (b) GEAF magnitude of underspread process, (c) GEAF mag-

nitude of overspread process. In (b) and (c), the number of signal samples is 256 and the (normalized)

frequency lag ranges from −1/2 to 1/2.

in these sets are mutually orthogonal and TF disjoint,

〈v(1)k , v

(2)l 〉 = 0 , W

v(1)k

(t, f)Wv(2)l

(t, f) ≡ 0 .

Let us first consider a random processes x(t) given by

x(t) =

N∑

k=1

ξ(1)k v

(1)k (t) +

N∑

k=1

ξ(2)k v

(2)k (t) , (3.21)

where the ξ(i)k (k = 1 . . . N , i = 1, 2) are mutually orthogonal, zero-mean random coefficients with mean

power E{∣∣ξ(i)k

∣∣2}

= β(i)k . Since this expansion involves orthogonal basis functions and orthogonal

coefficients, it constitutes the KL expansion of x(t). The corresponding correlation operator and

expected ambiguity function are given by

Rx =N∑

k=1

β(1)k v

(1)k ⊗ v

(1)∗k +

N∑

k=1

β(2)k v

(2)k ⊗ v

(2)∗k , (3.22)

A(α)x (τ, ν) =

N∑

k=1

β(1)k A

(α)

v(1)k

(τ, ν) +

N∑

k=1

β(2)k A

(α)

v(2)k

(τ, ν) .

Since the KL expansion of x(t) consists of uncorrelated, well TF localized basis functions, only small

TF correlations are present and the GEAF of x(t) is concentrated about the origin (at least for N not

too small). Thus, the process x(t) is underspread.

The GEAF of an example process with N = 30 is shown in Fig. 3.3(b). It is indeed highly

concentrated about the origin which is further corroborated by the correspondingly small moments

m(1,0)x m

(0,1)x = 0.035, m

(1,1)x = 0.249, M

(1,0)x M

(0,1)x = 0.008, M

(1,1)x = 0.012.

Overspread Process. In contrast to the process x(t) discussed previously, consider the process

x(t) =

N∑

k=1

ξ(1)k v

(1)k (t) +

N∑

k=1

ξ(2)k v

(2)k (t) , (3.23)


where the power of each component is the same as for the process x(t), i.e., E{∣∣ξ(i)k

∣∣2}

= β(i)k , but

now the coefficents are correlated in the sense that E{ξ(1)k ξ

(2)∗l

}= γk δk,l with γk > 0 (note that for

the underspread process above, γk = 0). This process features significant TF correlations since the

coefficients of the components v(1)k (t) and v

(2)k (t), which are located in the TF regions R1 and R2,

respectively, are correlated. Since the ξ(i)k are not mutually orthogonal, (3.23) is not the KL expansion

of x(t) which actually is given by

x(t) =

N∑

k=1

ξ(1)k v

(1)k (t) +

N∑

k=1

ξ(2)k v

(2)k (t) , (3.24)

where

v(1)k (t) = cos(φk) v

(1)k (t) + sin(φk) v

(2)k (t) , ξ

(1)k = cos(φk) ξ

(1)k + sin(φk)ξ

(2)k , (3.25a)

v(2)k (t) = − sin(φk) v

(1)k (t) + cos(φk) v

(2)k (t) , ξ

(2)k = − sin(φk) ξ

(1)k + cos(φk)ξ

(2)k , (3.25b)

with φk = 12arctan

(2γk

β(1)k −β

(2)k

). The coefficients ξ

(i)k are mutually orthogonal with respective powers

E{∣∣ξ(1)k

∣∣2}

= β(1)k ,

β(1)k + β

(2)k

2+

√[β

(1)k − β

(2)k

]2+ 4|γk|2

2,

E{∣∣ξ(2)k

∣∣2}

= β(2)k ,

β(1)k + β

(2)k

2−

√[β

(1)k − β

(2)k

]2+ 4|γk|2

2.

While the KL expansion (3.24) is doubly orthogonal, the underlying basis functions v(i)k (t) are no

longer well TF localized since they are supported in both R1 and R2. Hence, in the KL representation

(3.24) the TF correlations of the process x(t) are hidden in the TF structure of the basis functions.

In subsequent sections, it will be seen that these TF correlations lead to the occurrence of “statistical

cross terms in time-varying spectra. The correlation operator and GEAF of x(t) can be written as

Rx =N∑

k=1

β(1)k v

(1)k ⊗ v

(1)∗k +

N∑

k=1

β(2)k v

(2)k ⊗ v

(2)∗k (3.26)

=

N∑

k=1

β(1)k v

(1)k ⊗ v

(1)∗k +

N∑

k=1

β(2)k v

(2)k ⊗ v

(2)∗k +

N∑

k=1

γk v(1)k ⊗ v

(2)∗k +

N∑

k=1

γ∗k v(2)k ⊗ v

(1)∗k ,

(3.27)

A(α)x (τ, ν) =

N∑

k=1

β(1)k A

(α)

v(1)k

(τ, ν) +N∑

k=1

β(2)k A

(α)

v(2)k

(τ, ν)

=

N∑

k=1

β(1)k A

(α)

v(1)k

(τ, ν) +

N∑

k=1

β(2)k A

(α)

v(2)k

(τ, ν) +

N∑

k=1

γk A(α)

v(1)k ,v

(2)k

(τ, ν) +

N∑

k=1

γ∗k A(α)

v(2)k ,v

(1)k

(τ, ν) .

It is seen that the existence of TF correlations is also reflected by the GEAF of x(t) which has

significant components Av(i)k ,v

(j)k

(τ, ν) (i 6= j) off the origin about the points (τ, ν) = (∆t,∆f) and

(τ, ν) = (−∆t,−∆f). Hence, x(t) is an overspread process.


The GEAF of an example process with N = 30 is shown Fig. 3.3(c)). The GEAF components

off the origin about (τ, ν) = (∆t,∆f) and (τ, ν) = (−∆t,−∆f) lead to substantially larger moments

m(1,0)x m

(0,1)x = 14.31, m

(1,1)x = 15.25, M

(1,0)x M

(0,1)x = 7.57, and M

(1,1)x = 11.35.

3.2 Elementary Time-Varying Power Spectra

We shall next discuss two fundamental classes of time-varying power spectra: the generalized Wigner-

Ville spectrum and the generalized evolutionary spectrum.

3.2.1 Generalized Wigner-Ville Spectrum

Extending the Wiener-Khintchine relation (1.3) between the temporal correlation function rx(τ) and

the power spectral density Px(f) of a stationary random process, a time-varying spectrum of a nonsta-

tionary process can be defined as the 2-D (symplectic) Fourier transform of the GEAF (TF correlation

function) A(α)x (τ, ν),

W(α)

x (t, f) ,

∫

τ

∫

νA(α)

x (τ, ν) e−j2π(fτ−tν) dτ dν (3.28)

=

∫

τr(α)x (t, τ) e−j2πfτ dτ

= L(α)Rx

(t, f) . (3.29)

This time-varying power spectrum is known as generalized Wigner-Ville spectrum (GWVS, see also

Section B.3.1) [60, 61, 63, 126, 140, 145]. For α = 0, the GWVS reduces to the ordinary Wigner-Ville

spectrum [12,60,63,67, 140] whereas for α = 1/2 the Rihaczek spectrum [60,63,178] is obtained. The

GWVS satisfies a large number of desirable mathematical properties and proved to be useful for certain

signal processing applications [92,111,141,143]. A potential drawback of the GWVS from the point of

view of interpretation, namely the occurrence of negative values, will be discussed in Subsection 3.3.4.

From the 2-D Fourier relationship connecting the GWVS and the GEAF, it follows immediately

that the GWVS of an underspread process is a smooth 2-D lowpass function. For convenience,

we reformulate Proposition 2.5 for nonstationary random processes using the relations W(α)

x (t, f) =

L(α)Rx

(t, f), A(α)x (τ, ν) = S

(α)Rx

(τ, ν), m(k,l)x = m

(k,l)Rx

, and M(k,l)x = M

(k,l)Rx

:

Corollary 3.4. For any process x(t), the partial derivatives of the GWVS satisfy

∣∣∣∣∂k+lW

(α)x (t, f)

∂tl ∂fk

∣∣∣∣ ≤ (2π)k+l‖Ax‖1m(k,l)x ,

∥∥∥∥∂k+lW

(α)x

∂tl ∂fk

∥∥∥∥2

= (2π)(k+l)‖Ax‖2M(k,l)x .

It is thus seen that for underspread processes with small GEAF moments, the GWVS is a smooth

function. On the other hand, if the process is overspread, (3.28) implies that the GWVS contains

severe oscillations that correspond to GEAF contributions off the origin and hence can be viewed

as indicators of TF correlation. These oscillatory GWVS components also comprise negative GWVS

3.2 Elementary Time-Varying Power Spectra 111

α = 1/2

α = 0

t2t1

f2

f1

f

α = 1

t

(c)

t0

f0

ν/2

α = −1/2

ν/2τ/2

τ/2

t

f α = 0

(a) (b)

ν

τ

t0

f0

t

α = 1/2f

(t2, f2)

(t2, f2)

(t1, f1) (t1, f1)

Figure 3.4: Illustration of TF points (t1, f1), (t2, f2) contributing to W(α)x (t0, f0) for (a) α = 0

(Wigner-Ville spectrum), (b) α = 1/2 (Rihaczek spectrum); (c) TF locus of statistical cross terms of

the GWVS for several values of α.

values and can be interpreted as “cross terms” or “interference terms.” Thus, TF correlations cause

“statistical cross or interference terms in the GWVS [145] (similar observations have been reported

for a special case in [198]).

For Gaussian random processes, the cross terms of the GWVS and their relation to TF correlations

can further be analyzed using the covariance function C(α)x (t, f ; τ, ν) of the GWD of x(t) as defined in

(3.7). We have the following corollary to Theorem 3.1.

Corollary 3.5. For a circular complex Gaussian random process x(t),∫

τ

∫

νC(α)

x (t, f ; τ, ν) dτ dν =∣∣W (α)

x (t, f)∣∣2. (3.30)

Proof. In the proof of Theorem 3.1 it was shown that C(α)x (t, f ; τ, ν) = W(α)

Rx(t, f ; τ, ν). From this,

(3.30) follows by using (B.34) (with H replaced by Rx) and L(α)Rx

(t, f) = W(α)

x (t, f).

Let us give an interpretation of Corollary 3.5. We consider (3.30) for fixed TF analysis point (t, f) =

(t0, f0). According to (3.7), C(α)x (t0, f0; τ, ν) = cov

{W

(α)x (t+α−τ, f +α+ν),W

(α)x (t−α+τ, f −α−ν)

}.

Hence, (3.30) says that the value of the GWVS at the TF point (t0, f0) subsumes the covariance of

the GWD at all TF points (t1, f1) = (t0 + α−τ, f0 + α+ν) and (t2, f2) = (t0 − α+τ, f0 − α−ν). Note

that while these TF points are always separated by τ in time and by ν in frequency, their particular

TF location is determined by α (see Fig. 3.4(a),(b) for the cases α = 0 and α = 1/2). The other way

around, if two TF points (t1, f1) and (t2, f2) of the generalized Wigner distribution are correlated,

i.e., C(α)x (t1, f1; t2, f2) 6= 0, then this correlation will contribute to the GWVS value at the TF point

(t, f) =(

t1+t22 − α(t1 − t2),

f1+f2

2 + α(f1 − f2))

(see Fig. 3.4(c)). This explains why∣∣W (α)

x (t, f)| > 0

even if x(t) features no energy at this TF point.

The basic mechanisms of statistical cross terms are further illustrated by the following example.

Example (continued). We reconsider the random processes x(t) and x(t) defined by (3.21) and

(3.23), respectively, with N = 30. In Subsection 3.1.6, it was seen that x(t) is an underspread process


(a)

f

(b) (c) (d)t t t t

fff

Figure 3.5: Illustration of the GWVS of the underspread example process x(t) and the overspread

example process x(t): (a) W(0)x (t, f), (b) real part of W

(1/2)x (t, f), (c) W

(0)x (t, f), (d) real part of

W(1/2)x (t, f). In (c) and (d), statistical cross terms are clearly visible. The number of signal samples

is 256, (normalized) frequency ranges from −1/4 to 1/4.

since its GEAF is highly concentrated about the origin and features small moments. From (3.22), the

GWVS of x(t) is obtained as

W(α)

x (t, f) =N∑

k=1

β(1)k W

(α)

v(1)k

(t, f) +N∑

k=1

β(2)k W

(α)

v(2)k

(t, f) .

Since the basis functions v(1)k and v

(2)k are well TF localized in R1 and R2, respectively, W

(α)x (t, f)

essentially features energetic contributions in these two TF regions (see Fig. 3.5(a)–(b)).

In contrast, it follows from (3.26) that the GWVS of x(t) is given by

W(α)

x (t, f) =N∑

k=1

β(1)k W

(α)

v(1)k

(t, f) +N∑

k=1

β(2)k W

(α)

v(2)k

(t, f) +N∑

k=1

γk W(α)

v(1)k ,v

(2)k

(t, f) +N∑

k=1

γ∗k W(α)

v(2)k ,v

(1)k

(t, f) .

Compared to W(α)

x (t, f), W(α)

x (t, f) is seen to feature additional cross GWD terms W(α)

v(1)k ,v

(2)k

(t, f),

W(α)

v(2)k ,v

(1)k

(t, f) (see Fig. 3.5(c) and (d)). These additional contributions correspond to oscillating

statistical cross terms that can be attributed to nonzero the TF correlations γk present in the process

x(t). While the strength of the statistical cross terms is determined by the correlation parameters

γk, their location and geometry is determined by the (α-dependent) interference geometry of the

GWD [85]. The suppression or reduction of these cross terms by smoothing the GWVS will be

considered in Section 3.3.

Approximate Uniqueness of the GWVS

In contrast to the PSD which is unique, the GWVS depends on the parameter α. This non-uniqueness

might be considered an inconvenience. An important result regarding the choice of the GWVS pa-

rameter α results from specializing Theorem 2.10.

Corollary 3.6. For any random process x(t), the difference W(α1)x (t, f) −W

(α2)x (t, f) between two


GWVS with parameters α1 and α2 is bounded as

∣∣W (α1)x (t, f) −W

(α2)x (t, f)

∣∣‖Ax‖1

≤ 2π|α1 − α2|m(1,1)x ,

∥∥W (α1)x −W

(α2)x

∥∥2

‖Ax‖2

≤ 2π|α1 − α2|M (1,1)x .

(3.31)

Proof. The corollary follows from Theorem 2.10 with H = Rx and the moment equality (3.17).

The bound (3.31) shows that for small m(1,1)x and M

(1,1)x , i.e., for underspread processes, GWVS

with different α are approximately equal,

W(α1)x (t, f) ≈W

(α2)x (t, f) .

An example for this approximation with α1 = 0 and α2 = 1/2 is shown in parts (a) and (b) of Fig. 3.5.

In this example, the normalized errors were maxt,f

∣∣W (0)x (t,f)−W

(1/2)x (t,f)

∣∣‖Ax‖1

= 10−4 and

∥∥W(0)x −W

(1/2)x

∥∥2

‖Rx‖2=

2.7 · 10−3 while the corresponding bounds in (3.31) were πm(1,1)x = 3 · 10−3 and πM

(1,1)x = 3.8 · 10−2.

Small m(1,1)x , M

(1,1)x essentially requires that the process’ GEAF is concentrated along the τ axis

and/or ν axis. Processes having a GEAF oriented in oblique directions will not have small m(1,1)x ,

M(1,1)x . For such processes, the Wigner-Ville spectrum (i.e., the GWVS with α = 0) plays an out-

standing role [60,61,63] due to its metaplectic covariance properties (which are analogous to those of

the GWS with α = 0), see Appendix C.

CL Processes. For CL processes, application of 2.18 (with m(1,1)x = m

(1,1)Rx

, M(1,1)x = M

(1,1)Rx

, and

µx = µRx) to (3.31) yields the bounds

∣∣W (α1)x (t, f) −W

(α2)x (t, f)

∣∣‖Ax‖1

≤ 2π|α1 − α2|µx,

∥∥W (α1)x −W

(α2)x

∥∥2

‖Ax‖2

≤ 2π|α1 − α2|µx.

Moreover, we can formulate a result analogous to Proposition 2.11 which illuminates the interrelation

of the bounds for the CL case with those for the non-CL case.

Real-Valuedness and Positivity

The PSD is real-valued and positive. In contrast, the GWVS is real-valued only for α = 0 and its real

part is only rarely everywhere positive, i.e., there may be

ℜ{W

(α)x (t, f)

}6= P

{W

(α)x (t, f)

},

where

P{W

(α)x (t, f)

},

1

2

[∣∣ℜ{W

(α)x (t, f)

}∣∣+ ℜ{W

(α)x (t, f)

}]

denotes the positive real part of the GWVS W(α)

x (t, f). However, the approximate α-invariance of the

GWVS stated in the foregoing corollary suggests that for underspread processes the imaginary part of

the GWVS with α 6= 0 is negligible. Furthermore, our qualitative discussion of statistical cross terms

suggested that these cross terms are the main source for negative GWVS values. Since the GWVS

of underspread processes contain only small statistical cross terms, they might be suspected to be

“almost positive.” Indeed, reformulation of Corollary 2.13 and Theorem 2.29 yields


Corollary 3.7. For any random process x(t), the imaginary part of the GWVS, ℑ{W

(α)x (t, f)

},

12j

[W

(α)x (t, f) −W

(α)∗x (t, f)

], is bounded as

∣∣ℑ{W

(α)x (t, f)

}∣∣‖Ax‖1

≤ 2π|α|m(1,1)x ,

∥∥ℑ{W

(α)x

}∥∥2

‖Ax‖2

≤ 2π|α|M (1,1)x . (3.32)

Furthermore, the negative real part of the GWVS, N{W

(α)x (t, f)

}, P

{W

(α)x (t, f)

}−ℜ

{W

(α)x (t, f)

},

is bounded as

N{W

(α)x (t, f)

}

‖SHp‖21

≤ min{β1, β2} ,∥∥N{W

(α)x

}∥∥2

‖Ax‖2

≤ inf‖s‖2=1

{M (φs)

x

}, (3.33)

where β1 = inf‖s‖2=1

{m

(φs)x

}with φs(τ, ν) =

∣∣1 − A(α)s (τ, ν)

∣∣ and β2 = 2π C(α)Hp

with Hp the positive

semi-definite innovations system of x(t).

Proof. Using the moment equalities (3.16), (3.17), and the inequality∥∥Ax

∥∥1≤ ‖SHp‖2

1, the above

bounds follow from Corollary 2.13 and Theorem 2.29 with H = Rx.

Combination of (3.32) and (3.33) gives the following reformulation of Corollary 2.30.

Corollary 3.8. For any random process x(t), the difference W(α)

x (t, f) − P{W

(α)x (t, f)

}is bounded

as∣∣W (α)

x (t, f) − P{W

(α)x (t, f)

}∣∣‖SHp‖2

1

≤ 2π|α|m(1,1)x + min{β1, β2} , (3.34a)

∥∥W (α)x − P

{W

(α)x

}∥∥2

‖Ax‖2

≤ 2π|α|M (1,1)x + inf

‖s‖2=1

{M (φs)

x

}, (3.34b)

Proof. We have

W(α)

x (t, f) − P{W

(α)x (t, f)

}= jℑ

{W

(α)x (t, f)

}+ N

{W

(α)x (t, f)

},

and thus, by the triangle inequality,

∥∥W (α)x − P

{W

(α)x

}∥∥p≤∥∥ℑ{W

(α)x

}∥∥p+∥∥N{W

(α)x

}∥∥p, (3.35)

where for our purposes either p = ∞ or p = 2. Applying the bounds on the left-hand sides of (3.32)

and (3.33) to (3.35) (with p = ∞) then yields (3.34a). Similarly, (3.34b) is obtained by applying the

bounds on the right-hand sides of (3.32) and (3.33) to (3.35) (with p = 2).

For underspread processes, i.e., processes where m(1,1)x , M

(1,1)x , inf‖s‖2=1

{m

(φs)x

},

inf‖s‖2=1

{M

(φs)x

}, and C

(α)Hp

are small, the above bounds show that

ℑ{W

(α)x (t, f)

}≈ 0 , N

{W

(α)x (t, f)

}≈ 0 ,

and thus

W(α)

x (t, f) ≈ P{W

(α)x (t, f)

}.


Hence, for underspread processes the imaginary and negative real part of the GWVS will be negligible.

Parts (a) and (b) of Fig. 3.5 are examples for this approximation with α = 0 and α = 1/2,

respectively. For α = 0, the normalized errors were maxt,fN{

W(0)x (t,f)

}

‖SHp‖2

1

= 4 · 10−6 and

∥∥N{

W(0)x

}∥∥2

‖Ax‖2=

3.3 · 10−4, while the corresponding bounds in (3.33) (obtained with a Gaussian signal s(t) of optimal

duration) were m(φs)x = 5.2 · 10−4 and M

(φs)x = 2.9 · 10−3, respectively. For α = 1/2, the normalized

errors were maxt,f

∣∣W (1/2)x (t,f)−P

{W

(1/2)x (t,f)

}∣∣‖SHp‖

2

1

= 8.3 · 10−5 and

∥∥W(1/2)x −P

{W

(1/2)x

}∥∥2

‖Ax‖2= 0.033, while the

corresponding bounds in (3.34) (using the same Gaussian signal s(t) as before) were πm(1,1)x +m

(φs)x =

3.6 · 10−3 and πM(1,1)x +M

(φs)x = 0.041, respectively.

If the GEAF of x(t) and the GSF of Hp are oriented along the τ axis or ν axis, all of the relevant

weighted integrals and moments will be small. For processes having an obliquely oriented GEAF, only

the GWVS with α = 0 (which is always real-valued as correctly reflected by (3.32)) will approximately

equal its positive part. In such cases, the weighted integrals m(φs)x and M

(φs)x can be made small by

suitable choice of s(t), and β2 can be refined using the metaplectic covariance of the Weyl symbol by

replacing C(0)Hp

with minU∈MC(0)UHpU

+ (cf. (2.72)).

CL Processes. If x(t) is a CL process with GEAF support region Gx, applying (2.18) and (2.16)

(with m(1,1)x = m

(1,1)Rx

, M(1,1)x = M

(1,1)Rx

, and M(φs)x = M

(φs)Rx

) to (3.34) yields the bounds4

∣∣W (α)x (t, f) − P

{W

(α)x (t, f)

}∣∣‖SHp‖2

1

≤ 2π|α|µx + inf‖s‖2=1

{φ(max)s } ,

∥∥W (α)x − P

{W

(α)x

}∥∥2

‖Ax‖2

≤ 2π|α|µx + inf‖s‖2=1

{φ(max)s } ,

where φ(max)s = max(τ,ν)∈Gx

φs(τ, ν).

Uncertainty Relations

The effective duration Tx and the effective bandwidth Fx of a finite-energy, deterministic signal x(t)

are defined by

T 2x =

∫t t

2 |x(t)|2 dt∫t |x(t)|2 dt

=1

‖x‖22

∫

tt2 |x(t)|2 dt , F 2

x =

∫f f

2 |X(f)|2 df∫f |X(f)|2 df =

1

‖x‖22

∫

ff2 |X(f)|2 df .

Furthermore, a joint measure of the duration and bandwidth of x(t) is given by

ρ2x ,

(Tx

T

)2+ (TFx)2,

where T is an arbitrary normalization constant. According to the uncertainty principle [45,64–66], ρx

is bounded from below as

ρ2x =

(Tx

T

)2+ (TFx)2 ≥ 1

2π. (3.36)

4For simplicity, we here disregard the term in (3.34a) involving β2.


A special case of this inequality, obtained with T 2 = Tx/Fx, reads

TxFx ≥ 1

4π, (3.37)

which shows that Tx and Fx cannot be simultaneously small. Equality in (3.36) and (3.36) is attained

if and only if x(t) is a Gaussian signal [64–66]. The uncertainty inequality can also be viewed as a

lower bound on the spread of the GWD since the GWD’s marginal properties imply

T 2x =

1

‖x‖22

∫

t

∫

ft2W (α)

x (t, f) dt df , F 2x =

1

‖x‖22

∫

t

∫

ff2W (α)

x (t, f) dt df ,

and thus (3.36) can be rewritten as

ρ2x =

1

‖x‖22

∫

t

∫

f

[( tT

)2+ (Tf)2

]W (α)

x (t, f) dt df ≥ 1

2π. (3.38)

This shows that ρx can be interpreted as a “TF radius” measuring the TF concentration of W(α)x (t, f).

Next, let x(t) be a nonstationary random process. Since (3.36) holds for any (finite-energy) re-

alization of a random process, multiplying by ‖x‖22, taking the expectation, and renormalizing by

E2x = E

{‖x‖2

2

}yields

ρ2x ≥ 1

2π, Tx Fx ≥ 1

4π, (3.39)

with

T 2x ,

∫t t

2 rx(t, t) dt∫t rx(t, t) dt

=1

E2x

∫

tt2 rx(t, t) dt , (3.40a)

F 2x ,

∫f f

2 rX(f, f) df∫f rX(f, f) df

=1

E2x

∫

ff2 rX(f, f) df , (3.40b)

and

ρ2x ,

( Tx

T

)2+ (T Fx)2.

Note, however, that in general T 2x 6= E

{T 2

x

}, F 2

x 6= E{F 2

x

}, ρ2

x 6= E{ρ2

x

}. Similarly, we straightfor-

wardly obtain from (3.38) that

ρ2x =

1

E2x

∫

t

∫

f

[( tT

)2+ (Tf)2

]W

(α)x (t, f) dt df . (3.41)

and hence, the TF radius ρx is a measure of the TF concentration of W(α)

x (t, f). Together with the

left-hand inequality in (3.39), this indicates that the GWVS of a finite-energy process cannot be too

much concentrated.

In the follwoing we shall present bounds that are tighter than those in (3.39) and that relate the

spread of the GWVS to an “effective rank” parameter of the correlation operator. We note that

uncertainty relations for positive (semi-)definite operators (like correlation operators) are well known

in quantum mechanics (cf. [158]) and have previously been considered in [81, 99, 101]. The following

theorem is essentially an adapted/rephrased result from [99]. We recall that the KL expansion of a

correlation operator is Rx =∑∞

k=1 λk uk ⊗ u∗k, with the KL eigenvalues λk ≥ 0 (sorted in decreasing

order) and the (orthogonal) KL eigenfunctions uk(t).


Theorem 3.9. The TF radius ρx and the duration-bandwidth product TxFx of any finite-energy ran-

dom process x(t) are bounded from below as

ρ2x ≥ 1

π

(Λx − 1

2

), Tx Fx ≥ 1

2π

(Λx − 1

2

), (3.42)

where

Λx =

∑∞k=1 k λk∑∞k=1 λk

.

Proof. We need the Hermite operator HT defined as

(HTx

)(t) =

( tT

)2x(t) −

( T2π

)2 d2

dt2x(t) =

((M2 + D2)x

)(t) ,

where M is an LFI operator performing a multiplication by t/T and D = Tj2π

ddt . The operator

HT is positive definite and unbounded [69, 158]. Its eigenvalues are given by ηk = 12π (2k − 1) and

its eigenfunctions are the Hermite functions [45, 64, 81, 99], which we denote by hk(t). Thus, we

have HT =∑∞

k=1 ηk hk ⊗ h∗k. Furthermore, it can straightforwardly be shown that L(α)HT

(t, f) =(tT

)2+ (Tf)2 [45, 81, 99]. Starting with (3.41), the unitarity of the GWS (cf. (B.25)) together with

W(α)

x (t, f) = L(α)Rx

(t, f) then implies

ρ2x =

1

E2x

⟨L

(α)HT,W

(α)x

⟩=

1

E2x

⟨HT ,Rx

⟩=

1

E2x

∞∑

k=1

λk 〈HTuk, uk〉 .

According to Lemma 3.1 in [99], there is

∞∑

k=1

ck 〈HT fk, fk〉 ≥∞∑

k=1

ck ηk , (3.43)

for any orthonormal basis {fk(t)} and any ordered nonnegative sequence ck ≥ 0, c1 ≥ c2 ≥ · · · . With

ck = λk and fk(t) = uk(t), we hence obtain

ρ2x ≥ 1

E2x

∞∑

k=1

λk ηk =1

E2x

∞∑

k=1

λk1

2π(2k − 1) =

1

πE2x

[ ∞∑

k=1

k λk −∞∑

k=1

λk

2

]

=1

π

[∑∞k=1 k λk

E2x

− 1

2

∑∞k=1 λk

E2x

]=

1

π

(Λx − 1

2

),

where we further used∑∞

k=1 λk = Tr{Rx} = E2x. This concludes the proof of the left-hand inequality

in (3.42). The right-hand inequality in (3.42) is a special case of the left-hand inequality obtained by

setting T =√Tx/Fx in the expression ρ2

x =(

TxT

)2+ (T Fx)2.

Discussion. The parameter Λx can be viewed as a measures the effective rank of the correlation

operator Rx =∑∞

k=1 λk uk ⊗ u∗k. Hence, Theorem 3.9 shows that lower bound for the TF radius ρx

and the duration-bandwidth product TxFx is effectively determined by the eigenvalue spread of Rx. It

can furthermore be shown that Λx ≥ 1, and hence the lower bounds in (3.42) are tighter (larger) than

the lower bounds in (3.39). In particular, the lower bounds in (3.42) and (3.39) become equivalent


in the case Λx = 1, i.e., for a rank-one correlation operator Rx = E2x u1 ⊗ u∗1. Finally, we note that

equality in (3.43) is attained for uk(t) = hk(t) and hence the lower bounds in (3.42) are attained with

equality for Rx =∑∞

k=1 λk hk ⊗ h∗k, i.e., in the case of random processes whose KL eigenfunctions are

the Hermite functions.

Since the GWVS in general is neither positive nor real-valued (except for α = 0), we next consider

an alternative measure of GWVS concentration,

ρ2x(α) ,

∫t

∫f

[(tT

)2+ (Tf)2

] ∣∣W (α)x (t, f)

∣∣2 dt df∫t

∫f

∣∣W (α)x (t, f)

∣∣2 dt df=

1∥∥W (α)

x

∥∥2

2

∫

t

∫

f

[( tT

)2+ (Tf)2

] ∣∣W (α)x (t, f)

∣∣2 dt df .

A lower bound on this modified TF radius is slightly more difficult to establish. The following theorem

builds upon results from [99] and [101] and shows that the Wigner-Ville spectrum, i.e., the GWVS

with α = 0, has maximum TF concentration, i.e., minimum TF radius.

Theorem 3.10. The TF radius ρ2x(α) of any finite-energy random process x(t) satisfies

ρ2x(α) = ρ2

x(0) + α2[(M (1,0)

x

T

)2+ (TM (0,1)

x )2]. (3.44)

It is bounded from below as

ρ2x(α) ≥ 1

2π

(Λx − 1

2

)+ α2

[(M (1,0)x

T

)2+ (TM (0,1)

x )2], (3.45)

where

Λx =

∑∞k=1 k λ

2k∑∞

k=1 λ2k

.

Proof. We first consider the case α = 0. Using W(0)x (t, f) = L

(0)Rx

(t, f) =∑λkW

(0)uk (t, f) (see (B.29)),

we have∫

t

∫

f

( tT

)2[W

(0)x (t, f)

]2dt df

=

∫

t

∫

f

( tT

)2[ ∞∑

k=1

λk

∫

τ1

uk

(t+

τ12

)u∗k

(t− τ1

2

)e−j2πfτ1 dτ1

]

·[ ∞∑

l=1

λl

∫

τ2

u∗l

(t+

τ22

)ul

(t− τ2

2

)ej2πfτ2 dτ2

]dt df

=

∫

t

∫

τ1

∫

τ2

( tT

)2∞∑

k=1

∞∑

l=1

λk λl uk

(t+

τ12

)u∗k

(t− τ1

2

)u∗l

(t+

τ22

)ul

(t− τ2

2

)δ(τ1 − τ2) dτ1 dτ2 dt

=

∫

t

∫

τ1

( tT

)2∞∑

k=1

∞∑

l=1

λk λl uk

(t+

τ12

)u∗k

(t− τ1

2

)u∗l

(t+

τ12

)ul

(t− τ1

2

)dτ1 dt

=

∫

t1

∫

t2

( t1 + t22T

)2∞∑

k=1

∞∑

l=1

λk λl uk(t1)u∗k(t2)u

∗l (t1)ul(t2) dt1 dt2

=1

4T 2

∫

t1

∫

t2

(t21 + 2t1t2 + t22)

∞∑

k=1

∞∑

l=1

λk λl uk(t1)u∗k(t2)u

∗l (t1)ul(t2) dt1 dt2


=1

4

∞∑

k=1

∞∑

l=1

λk λl

[〈M2uk, ul〉〈ul, uk〉 + 2 〈Muk, ul〉〈Mul, uk〉 + 〈M2ul, uk〉〈uk, ul〉

]

=1

2

∞∑

k=1

∞∑

l=1

λk λl

[〈M2uk, uk〉 δk,l +

∣∣〈Muk, ul〉∣∣2]

≥ 1

2

∞∑

k=1

λ2k 〈M2uk, uk〉 .

In a similar way, one can show that

∫

t

∫

f(Tf)2

[W

(0)x (t, f)

]2dt df ≥ 1

2

∞∑

k=1

λ2k 〈D2uk, uk〉 .

Hence, recalling that HT = M2 + D2, we obtain

∫

t

∫

f

[( tT

)2+ (Tf)2

] [W

(0)x (t, f)

]2dt df ≥ 1

2

∞∑

k=1

λ2k 〈HTuk, uk〉 ≥ 1

2

∞∑

k=1

λ2k

1

2π(2k − 1) = ,

where we have used (3.43) with ck = λ2k and fk(t) = uk(t). Hence, with

∥∥W (α)x

∥∥2

2=∑

k λ2k, we have

ρ2x =

1∥∥W (α)

x

∥∥2

2

∫

t

∫

f

[( tT

)2+ (Tf)2

] ∣∣W (α)x (t, f)

∣∣2 dt df ≥ 1∥∥W (α)

x

∥∥2

2

1

2π

∞∑

k=1

λ2k

(k − 1

2

)

=1

2π

[∑∞k=1 kλ

2k∥∥W (α)

x

∥∥2

2

−∑∞

k=1 λ2k

2∥∥W (α)

x

∥∥2

2

]=

1

2π

(Λx − 1

2

)(3.46)

Next, we consider the case α 6= 0. We note that the 2-D Fourier transform of tW(α)

x (t, f) equals

1

j2π

∂

∂νA(α)

x (τ, ν) =1

j2π

∂

∂ν

[e−j2πατν A(0)

x (τ, ν)]

=1

j2πe−j2πατν

[ ∂∂νA(0)

x (τ, ν) − j2πατ A(0)x (τ, ν)

],

where (B.54) has been used. Using Parseval’s relation, we then obtain

∫

t

∫

f

( tT

)2∣∣W (α)x (t, f)

∣∣2 dt df =1

T 2

∫

τ

∫

ν

∣∣∣∣1

j2πe−j2πατν

[ ∂∂νA(0)

x (τ, ν) − j2πατ A(0)x (τ, ν)

]∣∣∣∣2

dτ dν

=1

(2πT )2

[ ∫

τ

∫

ν

∣∣∣ ∂∂νA(0)

x (τ, ν)∣∣∣2dτ dν + 2ℜ

{∫

τ

∫

νI(τ, ν) dτ dν

}

+ 4π2 α2

∫

τ

∫

ντ2∣∣Ax(τ, ν)

∣∣2 dτ dν]

where I(τ, ν) = j2πατ A(0)∗x (τ, ν) ∂

∂ν A(0)x (τ, ν). Since I∗(τ, ν) = −I(−τ,−ν), the second term vanishes

and we have

∫

t

∫

f

( tT

)2∣∣W (α)x (t, f)

∣∣2 dt df =1

(2πT )2

∫

τ

∫

ν

∣∣∣ ∂∂νA(0)

x (τ, ν)∣∣∣2dτ dν +

α2

T 2

∫

τ

∫

ντ2∣∣Ax(τ, ν)

∣∣2 dτ dν

=

∫

t

∫

f

( tT

)2[W

(0)x (t, f)

]2dt df +

α2

T 2‖Ax‖2

2

[M (1,0)

x

]2. (3.47)


In a similar way it can be shown that

∫

t

∫

f(Tf)2

∣∣W (α)x (t, f)

∣∣2 dt df =

∫

t

∫

f(Tf)2

[W

(0)x (t, f)

]2dt df + (αT )2‖Ax‖2

2

[M (0,1)

x

]2. (3.48)

The expression (3.44) then follows upon combination of (3.47) and (3.48). Finally, (3.45) follows by

applying (3.46) to (3.44).

The foregoing theorem shows two things. First, from (3.44) it is seen that the Wigner-Ville

spectrum, i.e., the GWVS with α = 0, has maximum TF concentration within the entire GWVS

family. Only in the case of processes with small moments M(1,0)x and M

(0,1)x , i.e., for underspread

processes, will the TF concentration of the other GWVS members be nearly as good as that of the

Wigner-Ville spectrum, i.e., ρ2x(α) ≈ ρ2

x(0). Second, according to (3.45), the TF concentration is

bounded from below, with the lower bound being determined by Λx and the moments M(1,0)x and

M(1,0)x . The parameter Λx again is a measure the spread of the eigenvalues λk of the correlation

operator Rx and hence can be interpreted as the effective rank of Rx. We note that α = 0 yields the

smallest lower bound in (3.45), i.e.,

ρ2x(0) ≥ 1

2π

(Λx − 1

2

).

Since Λx ≥ 1, it finally is seen that ρ2x(0) ≥ 1

4π for any finite-energy process.

From the uncertainty relations derived above, we can conclude an important general rule: for

greater effective rank (broader KL eigenvalue spectrum) of the correlation operator Rx, the GWVS

will have a larger TF support region.

3.2.2 Generalized Evolutionary Spectrum

Besides the GWVS, a second approach to defining a time-varying spectrum is based on the innovations

system representation (3.19) of the random process x(t). The generalized evolutionary spectrum (GES)

can be defined in analogy to (1.4) as squared magnitude of the TF transfer function (generalized Weyl

symbol) of an innovations system H ∈ Ix of x(t) [147,148,170],

G(α)x (t, f) ,

∣∣L(α)H

(t, f)∣∣2.

Note that G(α)x (t, f) depends on the choice of H (we recall from Subsection 3.1.5 that H is not uniquely

defined).

From the above definition, it is seen that the GES, in contrast to the GWVS, is always nonnegative.

For α = 1/2, the GES reduces to the (ordinary) evolutionary spectrum as originally defined by Priestley

[170]. For α = −1/2, the transitory evolutionary spectrum [49] is obtained. (We note that in the case of

a self-adjoint innovations system H = H+, the evolutionary spectrum and the transitory evolutionary

spectrum are equal since L(α)H+(t, f) = L

(−α)∗H

(t, f).) A further special case of the GES is the Weyl

spectrum [147] obtained with α = 0 and positive semi-definite innovations system H =√

Rx. Desirable

properties of the GES are discussed in detail in [148].


The usefulness of the GES definition relies on the interpretation of the GWS L(α)H

(t, f) as a TF

transfer function (i.e., TF domain weighting). In Chapter 2 it was seen that such an interpretation

requires the system H to be underspread. According to Subsection 3.1.5, underspread innovations

systems H produce underspread processes x(t), and thus it follows that the GES has a useful in-

terpretation only for underspread processes. For overspread processes, the innovations system H is

necessarily overspread as well. Here, the GWS of H will feature oscillating cross terms whose squared

magnitude will appear in the GES. Thus, for overspread processes, the GES features “statistical cross”

terms just as the GWVS. However, the appearance of the GES cross terms is different from that of

the GWVS cross terms (see further below).

The following result establishes smoothness properties of the GES in terms of moments of the

innovations system (see also Section 3.1.5).

Theorem 3.11. For any random process x(t) and any innovations system H ∈ Ix, the partial deriva-

tives of the GES (defined using H) are bounded as

∣∣∣∣∂k+lG

(α)x (t, f)

∂tl ∂fk

∣∣∣∣ ≤ (2π)k+l‖SH‖21

k∑

i=0

l∑

j=0

(k

i

)(l

j

)m

(i,j)H

m(k−i,l−j)H

.

Proof. The 2-D Fourier transform of G(α)x (t, f) = L

(α)H

(t, f)L(α)∗H

(t, f) is given by the convolution of

S(α)H

(τ, ν) and S(α)∗H

(−τ,−ν). Since S(α)∗H

(−τ,−ν) = S(−α)H+ (τ, ν), there is

G(α)x (t, f) =

∫

τ

∫

ν

(S

(α)H

∗∗ S(−α)H+

)(τ, ν) ej2π(νt−τf) dτ dν

and thus

∂k+lG(α)x (t, f)

∂tl ∂fk=

∫

τ

∫

ν(−1)l(j2π)k+l τk νl

(S

(α)H

∗∗ S(−α)H+

)(τ, ν) ej2π(νt−τf) dτ dν.

Therefore, the magnitude of the partial derivatives of the GES is bounded as

∣∣∣∣∂k+lG

(α)H

(t, f)

∂tl ∂fk

∣∣∣∣ ≤∫

τ

∫

ν(2π)k+l

∣∣∣∣τk νl

∫

τ ′

∫

ν′

S(α)H

(τ ′, ν ′)S(−α)H+ (τ − τ ′, ν − ν ′) dτ ′ dν ′


≤ (2π)k+l

∫

τ

∫

ν

∫

τ ′

∫

ν′


(τ ′, ν ′)S(−α)H+ (τ − τ ′, ν − ν ′)

∣∣ dτ ′ dν ′ dτ dν

= (2π)k+l

∫

τ1

∫

ν1

∫

τ ′

∫

ν′

|τ1 + τ ′|k |ν1 + ν ′|l |SH(τ ′, ν ′)| |SH+(τ1, ν1)| dτ1 dν1 dτ′ dν ′

≤ (2π)k+l

∫

τ1

∫

ν1

∫

τ ′

∫

ν′

[ k∑

i=0

(k

i

)|τ1|i |τ ′|k−i

][ l∑

j=0

(l

j

)|ν1|j |ν ′|l−j

]

· |SH(τ ′, ν ′)| |SH+(τ1, ν1)| dτ1 dν1 dτ′ dν ′

= (2π)k+lk∑

i=0

l∑

j=0

(k

i

)(l

j

)[∫

τ1

∫

ν1

|τ1|i |ν1|j |SH+(τ1, ν1)| dτ1 dν1

]

·[ ∫

τ ′

∫

ν′

|τ ′|k−i |ν ′|l−j |SH(τ ′, ν ′)| dτ ′ dν ′]


= (2π)k+l ‖SH‖21

k∑

i=0

l∑

j=0

(k

i

)(l

j

)m

(i,j)H+ m

(k−i,l−j)H

,

from which the final result follows since m(k,l)H+ = m

(k,l)H

.

It is thus seen that for underspread processes with underspread innovations systems having small

GSF moments, the GES is a 2-D lowpass function, i.e., smooth. In contrast, for overspread processes

the GES features statistical cross terms. These statistical GES cross terms can be related to the TF

displacements of the innovations system H as follows.

Corollary 3.12. For any innovations system H ∈ Ix of a nonstationary random process x(t), there

is ∫

τ

∫

νW(α)

H(t, f ; τ, ν) dτ dν = G(α)

x (t, f) , (3.49)

with G(α)x (t, f) defined using H.

Proof. Equation (3.49) is a restatement of (B.34) with∣∣L(α)

H(t, f)

∣∣2 = G(α)x (t, f).

Since according to [90] the generalized transfer Wigner distribution W(α)H

(t, f ; τ, ν) characterizes

the energy transfer between the TF points (t1, f1) = (t+α−τ, f+α+ν) and (t2, f2) = (t−α+τ, f−α−ν)

which are separated by τ in time and by ν in frequency, Corollary 3.12 states that the value of the

GES at any particular TF analysis point (t0, f0) subsumes all TF displacements of the innovations

system H. The other way around, if H causes TF displacements between the TF points (t1, f1) and

(t2, f2), i.e., W(α)H

(t1, f1; t2, f2) 6= 0, then these displacements will contribute to the GES value at the

TF point (t, f) =(

t1+t22 − α(t1 − t2),

f1+f2

2 + α(f1 − f2)).

The cross terms of the GES will be further illustrated by the following example.

Example (continued). We again reconsider the example processes x(t) and x(t) from Subsections

3.1.6 and 3.2.1.

From the eigenexpansion (3.22) of the correlation operator Rx, the positive semi-definite innova-

tions system Hp =√

Rx of the process x(t) is obtained as

Hp =N∑

k=1

√β

(1)k v

(1)k ⊗ v

(1)∗k +

N∑

k=1

√β

(2)k v

(2)k ⊗ v

(2)∗k .

Since the basis functions v(i)k (t) are well TF localized, the above eigenexpansion shows that the inno-

vations systen Hp introduces only small TF displacements, i.e., it is underspread. The GES of x(t) is

given by

G(α)x (t, f) =

∣∣∣∣N∑

k=1

√β

(1)k W

(α)

v(1)k

(t, f) +N∑

k=1

√β

(2)k W

(α)

v(2)k

(t, f)

∣∣∣∣2

=

∣∣∣∣N∑

k=1

√β

(1)k W

(α)

v(1)k

(t, f)

∣∣∣∣2

+

∣∣∣∣N∑

k=1

√β

(2)k W

(α)

v(2)k

(t, f)

∣∣∣∣2


+ 2ℜ{ N∑

k=1

N∑

l=1

√β

(1)k β

(2)l W

(α)

v(1)k

(t, f)W(α)∗

v(2)l

(t, f)

}.

Since the basis functions v(1)k and v

(2)k are assumed to be TF disjoint, i.e., W

(α)

v(1)k

(t, f) W(α)∗

v(2)l

(t, f) = 0,

the last term in the above expression vanishes. Evaluating the square in the remaining two terms and

neglecting the cross terms W(α)

v(i)k

(t, f)W(α)∗

v(i)l

(t, f), k 6= l, on the basis of the orthogonality5 of the sets

{v(i)k }i=1...N , the following approximation for the GES of x(t) is obtained,

G(α)x (t, f) ≈

N∑

k=1

β(1)k

∣∣∣W (α)

v(1)k

(t, f)∣∣∣2+

N∑

k=1

β(2)k

∣∣∣W (α)

v(2)k

(t, f)∣∣∣2.

Thus, it is seen that the GES of x(t) essentially features contributions only in the TF regions R1 and

R2 where the basis functions v(1)k and v

(2)k are respectively localized. This is illustred in Figs. 3.6(a)

and (b) for an example process with N = 30.

Next, we consider the overspread process x(t). Using (3.25), (3.26), and (3.27), it can be verified

that the positive semidefinite innovations system Hp =√

Rx of x(t) is given by

Hp =N∑

k=1

√β

(1)k v

(1)k ⊗ v

(1)∗k +

N∑

k=1

√β

(2)k v

(2)k ⊗ v

(2)∗k

=

N∑

k=1

η(1)k v

(1)k ⊗ v

(1)∗k +

N∑

k=1

η(2)k v

(2)k ⊗ v

(2)∗k +

N∑

k=1

ζk

[v(1)k ⊗ v

(2)∗k + v

(2)k ⊗ v

(1)∗k

],

with

η(1)k =

[√β

(1)k cos2(φk) +

√β

(2)k sin2(φk)

],

η(2)k =

[√β

(1)k sin2(φk) +

√β

(2)k cos2(φk)

],

ζk =[√

β(1)k −

√β

(2)k

]cos(φk) sin(φk) .

From this expression, it is seen that the innovations system Hp is able to transfer energy between

the TF regions R1 and R2, and thus represents an overspread system. The GES obtained with the

innovations system Hp is (approximately) given by

G(α)x (t, f) =

∣∣L(α)Hp

(t, f)∣∣2

=

∣∣∣∣∣

N∑

k=1

η(1)k W

(α)

v(1)k

(t, f) +N∑

k=1

η(2)k W

(α)

v(2)k

(t, f) +N∑

k=1

ζk

[W

(α)

v(1)k ,v

(2)k

(t, f) +W(α)

v(2)k ,v

(1)k

(t, f)]∣∣∣∣∣

2

≈∣∣∣∣

N∑

k=1

η(1)k W

(α)

v(1)k

(t, f)

∣∣∣∣2

+

∣∣∣∣N∑

k=1

η(2)k W

(α)

v(2)k

(t, f)

∣∣∣∣2

+

∣∣∣∣N∑

k=1

ζk

[W

(α)

v(1)k ,v

(2)k

(t, f)+W(α)

v(2)k ,v

(1)k

(t, f)]∣∣∣∣

2

,

5Via Moyal’s relation [151], the orthogonality conditionDv(i)k , v

(i)l

E= δkl actually implies

fiW

(α)

v(i)k

, W(α)

v(i)l

fl= δkl. This

is equivalent to W(α)

v(i)k

(t, f) W(α)∗

v(i)l

(t, f) ≈ 0, k 6= l, if the GWDs involved are effectively positive.


(a)

f

(b) (c) (d)t t t t

fff

Figure 3.6: Illustration of the GES of the underspread example process x(t) and the overspread

example process x(t) (the same processes as in Fig. 3.5): (a) G(0)x (t, f), (b) G

(1/2)x (t, f), (c) G

(0)x (t, f),

(d) G(1/2)x (t, f). In (c) and (d), statistical cross terms are clearly visible. The number of signal samples

is 256, (normalized) frequency ranges from −1/4 to 1/4.

where we used the TF disjointness of {v(1)k (t)} and {v(2)

k (t)} and neglected all terms involving

W(α)

v(i)k

(t, f)[W

(α)

v(i)k ,v

(j)k

(t, f) + W(α)

v(j)k ,v

(i)k

(t, f)]

(which is justified if the TF regions R1 and R2 are well

separated). It is seen that G(α)x (t, f) features cross GWDs W

(α)

v(i)k ,v

(j)k

(τ, ν); these correspond to oscil-

lating statistical cross terms that are due to TF correlations (see Figs. 3.6(c)–(d)). In the GES, these

cross terms appear squared and thus cannot be reduced by smoothing the GES. However, they can be

suppressed or reduced by smoothing the GWS of the innovations system before squaring. This will be

considered in Section 3.4.

Approximate Uniqueness of the GES

Like the GWVS, the GES depends on the parameter α. However, we next show that for underspread

processes the GES is approximately independent of α. (A related result for DL underspread innovations

systems can be found in [118].) We recall from Subsection 3.1.5 that for an underspread processes

x(t), we can always find an underspread innovations system H ∈ Ix.

Theorem 3.13. For any random process x(t) and any innovations system H ∈ Ix, the difference

between two GES with parameters α1 and α2, both defined using H, is bounded as

∣∣G(α1)x (t, f) −G

(α2)x (t, f)

∣∣‖SH‖2

1

≤ 4π|α1 − α2|m(1,1)H

. (3.50)

Proof. There is

∣∣G(α1)x (t, f) −G(α2)

x (t, f)∣∣ ≤

∫

τ

∫

ν

∣∣G(α1)x (τ, ν) − G(α2)

x (τ, ν)∣∣ dτ dν , (3.51)

with G(α)x (t, f) denoting the 2-D Fourier transform of G

(α)x (t, f). Since G

(α)x (t, f) equals the convolution

of S(α)H

(τ, ν) and S(α)∗H

(−τ,−ν), using (B.5) the 2-D Fourier transform of G(α1)x (t, f) −G

(α2)x (t, f) can

be written as

G(α1)x (τ, ν) − G(α2)

x (τ, ν) = S(α1)H

(τ, ν) ∗∗S(α1)∗H

(−τ,−ν) − S(α2)H

(τ, ν) ∗∗S(α2)∗H

(−τ,−ν)


=

∫

τ ′

∫

ν′

[S

(α1)H

(τ ′, ν ′)S(α1)∗H

(τ ′ − τ, ν ′ − ν) − S(α2)H

(τ ′, ν ′)S(α2)∗H

(τ ′ − τ, ν ′ − ν)]dτ ′ dν ′

=

∫

τ ′

∫

ν′

S(α1)H

(τ ′, ν ′)S(α1)∗H

(τ ′ − τ, ν ′ − ν)[1 − ej2π(α1−α2)[τ ′ν′−(τ ′−τ)(ν′−ν)]

]dτ ′ dν ′ .

Inserting the last expression in (3.51) and substituting τ = τ ′ − τ1 and ν = ν ′ − ν1, we further obtain

∣∣G(α1)x (t, f) −G(α2)

x (t, f)∣∣ ≤

∫

τ ′

∫

ν′

∫

τ1

∫

ν1

|SH(τ ′, ν ′)| |SH(τ1, ν1)|

·∣∣∣1 − ej2π(α1−α2)[τ ′ν′−τ1ν1]

∣∣∣ dτ ′ dν ′ dτ1 dν1

= 2

∫

τ ′

∫

ν′

∫

τ1

∫

ν1

|SH(τ ′, ν ′)| |SH(τ1, ν1)|∣∣∣sin

(π(α1 − α2)[τ

′ν ′ − τ1ν1])∣∣∣ dτ ′ dν ′ dτ1 dν1

≤ 2π|α1 − α2|∫

τ ′

∫

ν′

∫

τ1

∫

ν1

|SH(τ ′, ν ′)| |SH(τ2, ν2)|[|τ ′ν ′| + |τ1ν1|

]dτ ′ dν ′ dτ1 dν1

= 4π|α1 − α2| ‖SH‖21m

(1,1)H

.

The bound (3.50) shows that for small m(1,1)H

, GES with different α (but using the same innovations

system) are approximately equal,

G(α1)x (t, f) ≈ G(α2)

x (t, f) .

An example for this approximation with α1 = 0, α2 = 1/2, and H =√

Rx is shown in parts (a) and

(b) of Fig. 3.6. The maximum normalized error in this example was maxt,f

∣∣G(0)x (t,f)−G

(1/2)x (t,f)

∣∣‖SH‖2

1

= 0.005

while the corresponding bound in (3.50) was 2πm(1,1)H

= 0.054. Note that small m(1,1)H

implies that

the innovations system H (and hence the process x(t)) is underspread with GSF concentrated along

the τ axis and/or ν axis. Innovations systems whose GSF is oriented in oblique directions will not

have small m(1,1)H

. Indeed, the GES with α = 0 is known to be particularly suited to processes with

components that are obliquely oriented in the TF plane [147,148].

CL Processes. If the innovations system H is DL, then the resulting process will be CL and

furthermore µx ≤ 4µH. In that case, straightforward application of (2.18) to (3.50) yields the bound

∣∣G(α1)x (t, f) −G

(α2)x (t, f)

∣∣‖SH‖2

1

≤ 4π|α1 − α2|µH .

Uncertainty Relations for the Generalized Evolutionary Spectrum

The GES, like the GWVS, can be shown to obey a TF uncertainty principle if the underlying innova-

tions system H is positive (semi)definite. Hence, in the following we assume H = Hp =√

Rx. Let us

measure the GES extension by the TF radius of the GES defined as

ρ2x(α) ,

∫t

∫f

[(tT

)2+ (Tf)2

]G

(α)x (t, f) dt df

∫t

∫f G

(α)x (t, f) dt df

=1

E2x

∫

t

∫

f

[( tT

)2+ (Tf)2

]G(α)

x (t, f) dt df .

By adapting and combining results from [99] and [101], we then obtain the following theorem.


Theorem 3.14. For any finite-energy process x(t), the TF radius ρ2x of the GES (defined with the

positive (semi)definite innovations system Hp) satisfies

ρ2x(α) = ρ2

x(0) + α2[(M (1,0)

Hp

T

)2+ (TM

(0,1)Hp

)2].

It is bounded from below as

ρ2x(α) ≥ 1

2π

(Λx − 1

2

)+ α2

[(M (1,0)Hp

T

)2+ (TM

(0,1)Hp

)2], (3.52)

where

Λx =

∑∞k=1 k λk∑∞k=1 λk

with λk denoting the KL eigenvalues of x(t).

Proof. The proof of this theorem is completely analogous to that of Theorem 3.10 with Rx replaced

by Hp.

The foregoing theorem shows that the TF concentration of the GES (using the positive semi-

definite innovations system) is maximal for α = 0. For α 6= 0, the TF radius will be larger by an

amount determined by the moments M(1,0)Hp

and M(0,1)Hp

. Hence, only in the case of underspread systems

where M(1,0)Hp

and M(0,1)Hp

are small will the TF concentration of the GES with α 6= 0 be comparable

to that of the GES with α = 0. Furthermore, the theorem shows that ρ2x(α) is bounded from below

with the lower bound determined by Λx which measures the effective rank of the correlation operator

Rx =∑∞

k=1 λk uk ⊗u∗k (and of the positive semi-definite innovations system Hp =∑∞

k=1

√λk uk ⊗u∗k).

For the important special case α = 0, the lower bound in (3.52) is minimal, i.e.,

ρ2x(0) ≥ 1

2π

(Λx − 1

2

).

Furthermore, since Λx ≥ 1 (with Λx = 1 for a rank-one correlation operator), there is the bound

ρ2x(0) ≥ 1

4π which is independent of the KL eigenvalues λk.

We conclude that the TF concentration of the GES depends on the eigenvalue spread of Rx and

Hp, i.e., the TF support region of the GES is larger for processes whose correlation operator has a

large eigenvalue spread Λx.

3.3 Type I Time-Varying Power Spectra

According to Subsection 3.2.1, the GWVS of an overspread process contains cross terms that are due

to the TF correlations of the process. While these “statistical cross terms” are potentially useful as

they indicate TF correlations inherent in an overspread process, they may also be inconvenient as

they tend to mask the “auto terms” that characterize the mean TF energy distribution of the process

and thus indicate the TF location of the process’ components. In order to suppress or reduce cross

terms, a TF smoothing can be applied, similar to the TF smoothing used in deterministic quadratic

3.3 Type I Time-Varying Power Spectra 127

TF analysis [61,84,85]. This motivates a generalization of the GWVS which will be termed the class

of type I time-varying power spectra. In this section, we provide an axiomatic definition of this class

of time-varying power spectra. Furthermore, we discuss some specific members of this class and we

show that for underspread processes all type I spectra are approximately equivalent and approximately

satisfy desirable mathematical properties.

3.3.1 Definition and Formulations

The PSD Px(f) of a stationary process x(t) is a linear transformation of the correlation function rx(τ)

that is frequency-shift covariant in the sense that

x(t) = x(t) ej2πf0t =⇒ Px(f) = Px(f − f0) . (3.53)

Similarly, for nonstationary processes, we will use linearity and the TF shift covariance property to

define a class of time-varying spectra which we term type I time-varying power spectra. This class of

spectra has previously been defined and studied in [3, 60,61,63].

The cross-correlation operator Rx,y depends linearly on x(t) and sesquilinearly on y(t), i.e.,

x(t) = a1 x1(t) + a2 x2(t) =⇒ Rx,y = a1 Rx1,y + a2 Rx2,y ,

y(t) = b1 y1(t) + b2 y2(t) =⇒ Rx,y = b∗1 Rx,y1 + b∗2 Rx,y2

Let us require that the time-varying (cross) power spectrum to be defined, denoted Cx,y(t, f) hereafter,

x(t) = a1 x1(t) + a2 x2(t) =⇒ Cx,y(t, f) = a1 Cx1,y(t, f) + a2 Cx2,y(t, f) , (3.54a)

y(t) = b1 y1(t) + b2 y2(t) =⇒ Cx,y(t, f) = b∗1 Cx,y1(t, f) + b∗2 Cx,y2(t, f) . (3.54b)

This condition is satisfied if and only if Cx,y(t, f) depends linearly on the cross-correlation operator

Rx,y, i.e.,

Cx,y(t, f) =

∫

t1

∫

t2

c(t, f ; t1, t2) rx,y(t1, t2) dt1 dt2 ,

where c(t, f ; t1, t2) is the kernel of the linear transformation rx,y(t1, t2) → Cx(t, f). This expression

of Cx,y(t, f) can alternatively be viewed as the inner product of the correlation operator Rx,y and a

TF-parameterized operator Ct,f ,

Cx,y(t, f) = 〈Ct,f ,Ry,x〉 = Tr{Ct,fRx,y} ,

where the kernel of Ct,f is given by ct,f (t1, t2) = c(t, f ; t2, t1). If Ct,f is a TF localization operator

about the TF point (t, f) [42–44, 81, 88, 174, 175], then by the above formulation, Cx(t, f) can be

interpreted as the mean amount of signal energy the TF localization operator Ct,f picks up about the

TF point (t, f).

In the above general formulation, the localization properties of the operator Ct,f may still vary

with the TF analysis point (t, f). This is no longer the case if, in addition to (3.54), one requires that

the time-varying power spectrum is TF shift covariant, i.e.,

x(t) =(S

(α)t0,f0

x)(t) , y(t) =

(S

(α)t0,f0

y)(t) =⇒ Cx,y(t, f) = Cx,y(t− t0, f − f0) (3.55)


(note that Rx,y = S(α)t0,f0

Rx,yS(α)+t0,f0

). This covariance property is the TF analog of the frequency-shift

covariance in (3.53). It can be shown [93] that (3.55) is satisfied if and only if Ct,f has the form6 [60,63]

Ct,f = S(α)t,f CS

(α)+t,f , ct,f (t1, t2) = c(t1 − t, t2 − t) ej2πf(t1−t2),

where c(t1, t2) is the kernel of a fixed TF localization operator C. This means that the TF localization

operator Ct,f with parameters t and f equals a fixed TF localization operator C shifted by t in time

and by f in frequency. We thus arrive at the following general definition of type I time-varying (cross)

power spectra:

Cx,y(t, f) , Tr{Ct,fRx,y} = 〈Ct,f ,Ry,x〉 with Ct,f = S(α)t,f CS

(α)+t,f . (3.56)

This class of spectra is axiomatically defined by the two properties (3.54) and (3.55). The class is

parameterized by the operator C or, equivalently, its kernel c(t1, t2).

Subsequently, we shall mostly consider auto spectra for which we simply write Cx(t, f) = Cx,x(t, f).

Furthermore, we assume that the spectra integrate to the total mean energy,

∫

t

∫

fCx(t, f) dt df = Ex ,

which amounts to a trace normalization of C, i.e.,

Tr{C} =

∫

t

∫

fL

(α)C

(t, f) dt df = S(α)C

(0, 0) = 1 . (3.57)

By comparing (3.56) with (3.2), it is seen that Cx(t, f) can be viewed as the “diagonal” of the TF

correlation function R(T)x (t1, f1; t2, f2) (with T replaced by C), i.e.,

Cx(t, f) = R(C)x (t, f ; t, f) or Cx(t, f) = R(C)

x (t, f ; 0, 0) .

Furthermore, under weak conditions

Cx(t, f) = E {Cx(t, f)} , with Cx(t, f) , 〈Ct,fx, x〉 ,

where Cx(t, f) is recognized as a member of Cohen’s class of quadratic TF signal representations [35,

61,84]. Hence, type I spectra can be viewed as the stochastic analogue of Cohen’s class of deterministic

TF representations.

There exist four “canonical” formulations of type I spectra that are based on representations of

the operators C and Rx in four different domains (related by Fourier transforms):7

Cx(t, f) = 〈Ct,f ,Rx〉 =

∫

t1

∫

t2

c(t1 − t, t2 − t) r∗x(t1, t2) ej2πf(t1−t2) dt1 dt2 (3.58a)

6Note that TF shifts S(α)t,f HS

(α)+t,f of an operator H are independent of α.

7The expression in (3.58b) uses the bi-frequency function BC(f1, f2) that is defined in Section A.2. Furthermore, in

the formulations (3.58c) adn (3.58d) that involve the GWS and GSF with α = 0, other choices for α are possible as well.

However, the choice α = 0 leads to more convenient expressions.


=

∫

f1

∫

f2

BC(f1 − f, f2 − f) r∗X(f1, f2) e−j2πt(f1−f2) df1 df2 (3.58b)

=

∫

t

∫

fL

(0)C

(t′ − t, f ′ − f)W(0)x (t′, f ′) dt′ df ′ (3.58c)

=

∫

τ

∫

νS

(0)C

(τ, ν) A(0)∗x (τ, ν) e−j2π(νt−τf) dτ dν . (3.58d)

From (3.58c), it is seen that the time-varying power spectrum Cx(t, f) results from convolving the

Wigner-Ville spectrum with the Weyl symbol of C. Hence, the 2-D Fourier transform of Cx(t, f)

equals the product of the expected ambiguity function of x(t) and the spreading function of C, as

expressed by (3.58d). If C is underspread, i.e., |SC(τ, ν)| is concentrated about the origin of the (τ, ν)

plane, L(0)C

(t, f) will be a 2-D lowpass function and Cx(t, f) will hence be a smoothed version of the

Wigner-Ville spectrum. This subclass of type I time-varying power spectra, generated by underspread

TF localization operators C, will be termed smoothed type I spectra. Obviously, smoothing can be

used to suppress or at least reduce the oscillatory cross terms occurring in the Wigner-Ville spectrum

of overspread processes (see also Fig. 3.8).

3.3.2 Examples

We now consider a few examples of type I spectra, all of which correspond to prominent members of

Cohen’s class.

Generalized Wigner-Ville Spectrum [60, 63, 140]: The GWVS W(α)

x (t, f) discussed in Sub-

section 3.2.1 represents a specific subclass of type I spectra obtained with C = L(α) (see Subsection

B.1.2), i.e., S(0)C

(τ, ν) = ej2πατν . Note that L(α) is not underspread, i.e., the GWS of L(α) is not a

lowpass function and hence the GWVS members are not smoothed type I spectra.

Page’s (Instantaneous Power) Spectrum [71,77,161]: Page was concerned about the causal-

ity of time-varying power spectra, which means that the spectrum should only depend on past values

of the process. This condition led him to the definition of the instantaneous power spectrum,

Cx(t, f) = E

{d

dt|X−

t (f)|2}

with X−t (f) =

∫ t

−∞x(τ) e−j2πfτ dτ . (3.59)

The instantaneous power spectrum can be rewritten as

Cx(t, f) = 2ℜ{∫ ∞

0rx(t, t− τ) e−j2πfτ dτ

}

and can be shown to be a type I spectrum with the corresponding operator C defined by S(0)C

(τ, ν) =

ejπν|τ |.

Levin’s Spectrum [71, 77, 161]: Levin augmented Page’s definition by adding an anti-causal

analogue of (3.59), i.e.,

Cx(t, f) = E

{d

dt|X−

t (f)|2}

+ E

{d

dt|X+

t (f)|2},


with X+t (f) =

∫∞t x(τ) e−j2πfτ dτ and X−

t (f) as in (3.59). Levin’s spectrum can alternatively be

expressed as

Cx(t, f) = ℜ{∫

τrx(t, t− τ) e−j2πfτ dτ

}.

The spreading function of the corresponding operator C is given by S(0)C

(τ, ν) = cos(πτν). We note

that Levin’s spectrum can also be viewed as real part of the Rihaczek spectrum (i.e., the GWVS with

α = 1/2).

Physical Spectrum [67, 139]: Guided by the idea of measuring the mean energy around a TF

analysis point, Mark introduced the physical spectrum (see also Subsection B.3.3), which can be shown

to equal the expectation of the spectrogram (see Subsection B.2.3), i.e.,

PS(g)x (t, f) = E

{SPEC(g)

x (t, f)}

= E{∣∣STFT(g)

x (t, f)∣∣2}

= E{∣∣〈x, gt,f 〉

∣∣2}

= 〈Rxgt,f , gt,f 〉 =⟨St,f

(g ⊗ g∗

)S+

t,f ,Rx

⟩,

with g(t) a normalized analysis window. The localization operator underlying the physical spectrum

has rank one and is given by C = g ⊗ g∗, so that L(0)C

(t, f) = W(0)

g (t, f) and S(0)C

(τ, ν) = A(0)g (τ, ν).

Multi-Window Physical Spectrum: If the localization operator is compact and normal, it has

an eigenexpansion

C =∑

k

γk gk⊗ g∗k .

Plugging this decomposition into (3.56), it is seen that

Cx(t, f) =⟨St,f

[∑kγk gk⊗ g∗k

]S+

t,f ,Rx

⟩=∑

kγk

⟨St,f

(gk⊗ g∗k

)S+

t,f ,Rx

⟩

=∑

kγk PS(gk)

x (t, f) . (3.60)

Hence, the time-varying spectrum Cx(t, f) corresponds to a multi-window physical spectrum with

orthonormal analysis windows gk(t). We note that the number of physical spectra involved equals

the rank of the localization operator C. Furthermore, L(0)C

(t, f) =∑

k γkW(0)gk (t, f) and S

(0)C

(τ, ν) =∑

k γkA(0)gk (τ, ν). For typical localization operators with fast decaying eigenvalues γk, a few terms in the

multi-window expansion (3.60) are sufficient for a reasonable approximation of Cx(t, f). Choosing the

TF localization operator proportional to the orthogonal projection operator P =∑N

k=1 gk ⊗ g∗k on the

N -dimensional space span{g1(t), . . . , gN (t)}, i.e., C = P/N , we obtain Cx(t, f) = 1N

∑Nk=1 PS

(gk)x (t, f).

In that case, the type I spectrum is simply the arithmetic mean of N physical spectra computed with

N orthogonal windows gk(t).

We note that multi-window expansions of TF representations have previously been considered, e.g.

in [4, 41,118].

3.3.3 Approximate Equivalence

Many definitions for a type I time-varying power spectrum can be given via different choices of the

TF localization operator C. Of course, the specific properties of the resulting spectrum depend on


this choice. While in general the spectra resulting from different TF localization operators are quite

different, the following shows that for underspread processes, most type I time-varying power spectra

yield effectively similar results.

Theorem 3.15. For any random process x(t), the difference of two type I time-varying power spectra

C(1)x (t, f) =

⟨C

(1)t,f ,Rx

⟩and C

(2)x (t, f) =

⟨C

(2)t,f ,Rx

⟩generated by operators C(1) and C(2), respectively,

satisfies ∣∣C(1)x (t, f) − C

(2)x (t, f)

∣∣‖Ax‖1

≤ m(φ)x ,

∥∥C(1)x − C

(2)x

∥∥2

‖Ax‖2

= M (φ)x , (3.61)

where φ(τ, ν) =∣∣S(0)

C(1)(τ, ν) − S(0)

C(2)(τ, ν)∣∣.

Proof. Using (3.58d), the 2-D Fourier transform of ∆(t, f) , C(1)x (t, f) − C

(2)x (t, f) is obtained as

∆(τ, ν) = A(0)∗x (τ, ν)

[S

(0)

C(1)(τ, ν) − S(0)

C(2)(τ, ν)]. The bound and expression for the L∞ and L2 norms

of ∆(t, f) then follow from |∆(t, f)| ≤∫τ

∫ν |∆(τ, ν)| dτ dν and ‖∆‖2 = ‖∆‖2, respectively.

Discussion. The above theorem shows that for processes with small m(φ)x and M

(φ)x , i.e., for

underspread processes, two different type I time-varying spectra are approximately equal,

C(1)x (t, f) ≈ C(2)

x (t, f) .

Note that φ(0, 0) = 0 due to our assumption of trace normalized TF localization operators (cf. (3.57)).

The requirements for small m(φ)x and M

(φ)x shall be illustrated next by considering some special cases.

The approximate equivalence of two GWVS with different α has already been discussed in Corollary

3.6. It can be viewed as specialization of (3.61) with C(1) = L(α1), C(2) = L(α2), and the additional

inequality φ(τ, ν) = |ej2πα1τν − ej2πα2τν | ≤ 2π|α1 − α2||τ ||ν| which yields m(φ)x ≤ 2π|α1 − α2|m(1,1)

x

and M(φ)x ≤ 2π|α1 − α2|M (1,1)

x .

A further interesting special case is C(1) = L(α) and C(2) = g⊗g∗ where φ(τ, ν) = |ej2πα1τν −A

(0)g (τ, ν)| = |1−A(α)

g (τ, ν)|. Note that in this case m(φ)x , M

(φ)x will be small if x(t) is underspread and

if the effective support region of∣∣A(α)

g (τ, ν)∣∣ covers the effective support region of |Ax(τ, ν)|. Here,

(3.61) implies the approximate equivalence W(0)x (t, f) ≈ P

(g)x (t, f). Together with the positivity of the

physical spectrum, this implies that the Wigner-Ville spectrum is approximately positive. This issue

was explored in Subsection 3.2.1.

CL Processes. In the case of CL processes, i.e., processes with compact GEAF support region

Gx, straightforward application of (2.16) yields the bounds

∣∣C(1)x (t, f) − C

(2)x (t, f)

∣∣‖Ax‖1

≤ φ(max)x ,

∥∥C(1)x − C

(2)x

∥∥2

‖Ax‖2

≤ φ(max)x ,

with φ(max)x = max(τ,ν)∈Gx

∣∣S(0)

C(1)(τ, ν) − S(0)

C(2)(τ, ν)∣∣. In particular, this implies that in cases where

the DL parts of C(1) and C(2) are equal, i.e., S(0)

C(1)(τ, ν) = S(0)

C(2)(τ, ν) for (τ, ν) ∈ Gx or equivalently

[C(1)]Gx = [C(2)]Gx , one has φ(max)x = 0 and thus C

(1)x (t, f) = C

(2)x (t, f) for a CL process with GEAF

support region Gx.


3.3.4 Properties

A large part of the literature on deterministic TF representations is dedicated to the investigation

of their mathematical properties [35, 61, 84]. Among the more important desirable properties are the

time and frequency marginal property, unitarity (Moyal’s relation), TF shift and scale covariance, real-

valuedness, and positivity. The properties of a type I spectrum Cx(t, f) are completely characterized

by the operator C or, equivalently, by the Weyl symbol L(0)C

(t, f) or the spreading function S(0)C

(τ, ν).

A given property is satisfied by Cx(t, f) if the associated operator C (or, equivalently, L(0)C

(t, f)

or S(0)C

(τ, ν)) satisfies a corresponding condition. In the following, we will consider some specfifc

properties and show that in a stochastic setting these properties are satisfied by many type I time-

varying spectra at least in an approximate manner if the process under consideration is underspread.

Real-valuedness

Whereas the PSD is a real-valued function, this ist not necessarily true for a type I spectrum. Hence,

the first property we consider is real-valuedness, i.e., Cx(t, f) = C∗x(t, f). According to (3.56), this

property is satisfied if and only if the operator C is self-adjoint (i.e., its eigenvalues have to be real,

ℑ{γk} = 0). From (B.6) it is seen that C is self-adjoint if and only if the GSF of C features Hermitian

symmetry, S(0)C

(τ, ν) = S(0)∗C

(−τ,−ν). However, the next result shows that even if C is not self-adjoint,

Cx(t, f) is approximately real-valued in the case of underspread processes.

Theorem 3.16. For any random process x(t), the imaginary part ℑ{Cx(t, f)} = 12j

[Cx(t, f) −

C∗x(t, f)

]of any type I spectrum Cx(t, f) = 〈Ct,f ,Rx〉 satisfies

|ℑ{Cx(t, f)}|‖Ax‖1

≤ 1

2m(φ)

x ,‖ℑ{Cx}‖2

‖Ax‖2=

1

2M (φ)

x , (3.62)

with the weight function φ(τ, ν) =∣∣S(0)

C(τ, ν) − S

(0)∗C

(−τ,−ν)∣∣.

Proof. Using (3.58d), it follows that

ℑ{Cx(t, f)} =1

2j

∫

τ

∫

ν

[S

(0)C

(τ, ν) − S(0)∗C

(−τ,−ν)]A(0)∗

x (τ, ν) e−j2π(νt−τf) dτ dν .

The L∞ bound in (3.62) is then shown as

|ℑ{Cx(t, f)}| =1

2

∣∣∣∣∫

τ

∫

ν

[S

(0)C

(τ, ν) − S(0)∗C

(−τ,−ν)]A(0)∗

x (τ, ν) e−j2π(νt−τf) dτ dν

∣∣∣∣

≤ 1

2

∫

τ

∫

ν

∣∣S(0)C

(τ, ν) − S(0)∗C

(−τ,−ν)∣∣ ∣∣Ax(τ, ν)

∣∣ dτ dν

=1

2m(φ)

x ‖Ax‖1 .

In a similar way one obtains

‖ℑ{Cx}‖22 =

1

4

∫

τ

∫

ν

∣∣S(0)C

(τ, ν) − S(0)∗C

(−τ,−ν)∣∣2 |Ax(τ, ν)|2dτ dν

=[12M (φ)

x ‖Ax‖2

]2,

which proves the expression for the L2 norm in (3.62).


Discussion. The above result shows that for underspread processes with small m(φ)x and M

(φ)x ,

there is

Cx(t, f) ≈ C∗x(t, f) , or ℑ{Cx(t, f)} ≈ 0 .

Due to φ(τ, ν) =∣∣S(0)

C(τ, ν) − S

(0)∗C

(−τ,−ν)∣∣, small m

(φ)x and M

(φ)x requires that S

(0)C

(τ, ν) approxi-

mately features Hermitian symmetry on the effective support of∣∣Ax(τ, ν)

∣∣. For typical TF localization

operators C, this in turn implies that∣∣Ax(τ, ν)

∣∣ has to be concentrated about the origin, i.e., the pro-

cess x(t) has to be underspread.

Of the example spectra in Subsection 3.3.2, all but the GWVS with α 6= 0 and the multiwindow

spectra with ℑ{γk} 6= 0 are real-valued. For the GWVS case, i.e., Cx(t, f) = W(α)

x (t, f) or equivalently

C = L(α), simpler but coarser bounds in terms of the GEAF moments m(1,1)x and M

(1,1)x are provided

by Corollary 3.7. These bounds can also be obtained from Theorem 3.16 by noting that here φ(τ, ν) =

|ej2πατν−e−j2πατν | = 2| sin(2πατν)| ≤ 4π|α||τ ||ν| so that m(φ)x ≤ 4π|α|m(1,1)

x and M(φ)x ≤ 4π|α|M (1,1)

x .

CL Processes. In the case of CL processes with compact GEAF support region Gx, straightfor-

ward application of (2.16) yields the bounds

|ℑ{Cx(t, f)}|‖Ax‖1

≤ 1

2φ(max)

x ,‖ℑ{Cx}‖2

‖Ax‖2≤ 1

2φ(max)

x

with φ(max)x = max(τ,ν)∈Gx

∣∣S(0)C

(τ, ν) − S(0)∗C

(−τ,−ν)∣∣. In particular, this implies that in cases where

the DL part of C is self-adjoint, i.e., S(0)C

(τ, ν) = S(0)∗C

(−τ,−ν) for (τ, ν) ∈ Gx or equivalently CGx =

[CGx ]+, we have φ(max)x = 0 and thus Cx(t, f) will be real-valued.

Marginal Properties

In a stochastic context, the marginal property in time requires that

∫

fCx(t, f) df = E

{|x(t)|2

}= rx(t, t) .

Similarly, the marginal property in frequency requires that

∫

tCx(t, f) dt = E

{|X(f)|2

}= rX(f, f) .

For a type I spectrum Cx(t, f), (3.58a) and (3.58b) respectively imply that

∫

fCx(t, f) df =

∫

t′c(t′ − t, t′ − t) rx(t′, t′) dt′ ,

∫

tCx(t, f) dt =

∫

f ′

BC(f ′ − f, f ′ − f) rX(f ′, f ′) df ′ .

Hence, the marginal property in time is satisfied if and only if c(t, t) = δ(t), or equivalently

S(0)C

(0, ν) = 1. Similarly, the marginal property in frequency is satisfied if and only if BC(f, f) = δ(f),

or equivalently S(0)C

(τ, 0) = 1. Even if these conditions are not met exactly, the following result shows

that in the case of underspread processes the marginal properties will approximately be satisfied.


Theorem 3.17. For any random process x(t) and any type I spectrum Cx(t, f) = 〈Ct,f ,Rx〉, the

differences ∆1(t) ,∫f Cx(t, f) df − rx(t, t) and ∆2(f) ,

∫t Cx(t, f) dt − rX(f, f) satisfy

|∆1(t)|‖Ax‖1,ν

≤ m(1)x ,

‖∆1‖2

‖Ax‖2,ν

= M (1)x ,

|∆2(f)|‖Ax‖1,τ

≤ m(2)x ,

‖∆2‖2

‖Ax‖2,τ

= M (2)x , (3.63)

where

m(1)x ,

1

‖Ax‖1,ν

∫

ν

∣∣1 − S(0)C

(0, ν)∣∣ |Ax(0, ν)| dν ,

M (1)x ,

1

‖Ax‖2,ν

[ ∫

ν

∣∣1 − S(0)C

(0, ν)∣∣2 |Ax(0, ν)|2 dν

]1/2

,

m(2)x ,

1

‖Ax‖1,τ

∫

τ

∣∣1 − S(0)C

(τ, 0)∣∣ |Ax(τ, 0)| dτ ,

M (2)x ,

1

‖Ax‖1,τ

[ ∫

τ

∣∣1 − S(0)C

(τ, 0)∣∣2 |Ax(τ, 0)|2 dτ

]1/2

,

with ‖Ax‖p,ν ,[∫

ν|Ax(0, ν)|p dν

]1/pand ‖Ax‖p,τ ,

[∫

τ|Ax(τ, 0)|p dτ

]1/p.

Proof. First note that (B.53) and (B.55) imply that rx(t, t) =∫ν A

(0)∗x (0, ν) e−j2πνt dν. Furthermore,

due to (3.58d) there is∫f Cx(t, f) df =

∫ν S

(0)C

(0, ν) A(0)∗x (0, ν) e−j2πνt dν. We thus obtain

∣∣∆1(t)∣∣ =

∣∣∣∣∫

ν

[1 − S

(0)C

(0, ν)]A(0)∗

x (0, ν) e−j2πνt dν

∣∣∣∣

≤∫

ν

∣∣1 − S(0)C

(0, ν)∣∣ ∣∣Ax(0, ν)

∣∣ dν = ‖Ax‖1,νm(1)x .

Similarly,

∥∥∆1

∥∥2

2=

∫

ν

∣∣1 − S(0)C

(0, ν)∣∣2 ∣∣Ax(0, ν)

∣∣2 dν = ‖Ax‖22,ν

[M (1)

x

]2.

The expressions for ∆2(f) can be shown in an analogous way by noting that rX(f, f) =∫τ A

(0)∗x (τ, 0) ej2πτf dτ and

∫t Cx(t, f) dt =

∫τ S

(0)C

(τ, 0) A(0)∗x (τ, 0) ej2πτf dτ .

Discussion. The above theorem shows that for underspread processes where m(1)x , m

(2)x , M

(1)x ,

and M(2)x are small, type I spectra Cx(t, f) that might not satisfy the marginal properties exactly will

satisfy them at least approximately, i.e.,∫

fCx(t, f) df ≈ E

{|x(t)|2

},

∫

tCx(t, f) dt ≈ E

{|X(f)|2

}.

Note that m(1)x , M

(1)x , m

(2)x , and M

(2)x can be viewed as degenerate weighted GEAF integrals that

measure the extension of∣∣Ax(τ, ν)

∣∣ along the ν axis and τ axis, respectively. Small m(1)x and M

(1)x

requires that E{|x(t)|2

}= rx(t, t) = r

(α)x (t, 0) varies slowly over time and small m

(2)x and M

(2)x requires

that E{|X(f)|2

}= rX(f, f) = r

(α)X (f, 0) varies slowly over frequency. Note however, that r

(α)x (t, τ)

and r(α)X (f, ν) may vary arbitrarily fast for τ > 0 and ν > 0, respectively.


Of the example spectra in Subsection 3.3.2, all but the physical spectrum and the multi-window

physical spectrum satisfy the marginal property in time and frequency, respectively. For the physical

spectrum, i.e., the case C = g⊗g∗ (with g(t) a normalized window function), there is S(0)C

(τ, ν) =

A(0)g (τ, ν) and thus S

(0)C

(0, ν) =∫f G(f+ν/2)G∗(f−ν/2) df and S

(0)C

(τ, 0) =∫t g(t+τ/2) g

∗(t−τ/2) dt.With the truncated Taylor series expansion of A

(0)g (τ, ν) about the origin (valid for small |τ |, |ν|),

A(0)g (τ, ν) ≈ 1 − 4π2 F 2

g τ2 − 4π2 T 2

g ν2 , (3.64)

where T 2g =

∫t t

2 |g(t)|2 dt and F 2g =

∫f f

2 |G(f)|2 df measure the effective duration and bandwidth of

g(t), we obtain the approximations

m(1)x ≈ 4π2 T 2

g

∫

νν2 |Ax(0, ν)| dν

‖Ax‖1,ν

, M (1)x ≈ 4π2 T 2

g

[ ∫

νν4 |Ax(0, ν)|2 dν

]1/2

‖Ax‖2,ν

(3.65)

m(2)x ≈ 4π2 F 2

g

∫

ττ2 |Ax(τ, 0)| dτ

‖Ax‖1,τ

, M (1)x ≈ 4π2 F 2

g

[ ∫

ττ4 |Ax(τ, 0)|2 dτ

]1/2

‖Ax‖2,τ

, (3.66)

which are valid for |Ax(τ, ν)| being concentrated about the origin. The fractions in (3.65) and (3.66) can

be interpreted as measures of the time-variation of rx(t, t) and of the frequency variation of rX(f, f),

respectively. Thus it is seen that in order to have small m(1)x , M

(1)x in the case of the physical spectrum,

the instantaneous power rx(t, t) = E{|x(t)|2

}must vary slowly compared to the window length Tg.

Similarly, small m(2)x , M

(2)x requires that the expected energy density rx(f, f) = E

{|X(f)|2

}varies

slowly compared to the window bandwidth Fg. We conclude that in order that the physical spectrum

approximately satisfies the marginal properties, the duration and bandwidth of the window g(t) must

be matched to the GEAF extension of the process x(t) in the ν and τ direction, respectively. In

particular, quasistationary processes require a long, narrowband analysis window, whereas quasiwhite

processes require a short, wideband window.

CL Processes. In the case of CL processes with compact GEAF support region Gx, it can be

shown using similar arguments as in Proposition 2.4 that

m(1)x ≤ φ

(max)1 , M (1)

x ≤ φ(max)1 , m(2)

x ≤ φ(max)2 , M (2)

x ≤ φ(max)2 ,

where φ(max)1 = max(τ,ν)∈Gx

∣∣1 − S(0)C

(0, ν)∣∣ and φ

(max)2 = max(τ,ν)∈Gx

∣∣1 − S(0)C

(τ, 0)∣∣. Applying these

inequalities to (3.63) yields the bounds

|∆1(t)|‖Ax‖1,ν

≤ φ(max)1 ,

‖∆1‖2

‖Ax‖2,ν

≤ φ(max)1 ,

|∆2(f)|‖Ax‖1,τ

≤ φ(max)2 ,

‖∆2‖2

‖Ax‖2,ν

≤ φ(max)2 .

Hence, if the spreading function of the operator C satisfies S(0)C

(0, ν) = 1 and/or S(0)C

(τ, 0) = 1 for

(τ, ν) ∈ Gx, we have, respectively, φ(max)1 = 0 and/or φ

(max)2 = 0, so that the marginal property in time

and/or frequency is satisfied.


Moyal-Type Relation

The Moyal-type relation ⟨Cx, Cy

⟩= Tr{RxRy} = 〈Rx,Ry〉 (3.67)

with ⟨Cx, Cy

⟩=

∫

t

∫

fCx(t, f) C ∗

y (t, f) dt df

can be shown to be satisfied if and only if |SC(τ, ν)| ≡ 1 (note that this condition cannot be met by

HS operators C). Even if this condition is not satisfied exactly, the following result shows that in the

case of underspread processes, (3.67) holds at least approximately.

Theorem 3.18. For any two random processes x(t) and y(t) and any type I spectrum Cx(t, f) =

〈Ct,f ,Rx〉, the difference between⟨Cx, Cy

⟩and 〈Rx,Ry〉 is bounded as

∣∣ ⟨Cx, Cy

⟩− 〈Rx,Ry〉

∣∣‖Rx‖2 ‖Ry‖2

≤ min{M (φ′)

x M (φ′)y ,M (φ)

x ,M (φ)y

}, (3.68)

with φ′(τ, ν) =√∣∣1 − |SC(τ, ν)|2

∣∣ and φ(τ, ν) =∣∣1 − |SC(τ, ν)|2

∣∣.

Proof. Using 〈Rx,Ry〉 =⟨A

(0)x , A

(0)x

⟩and (3.58d) in combination with Parseval’s relation, we have

∆ ,⟨Cx, Cy

⟩− 〈Rx,Ry〉

=

∫

τ

∫

νA(0)∗

x (τ, ν)S(0)C

(τ, ν)A(0)y (τ, ν)S

(0)∗C

(τ, ν) dτ dν −⟨A(0)

x , A(0)y

⟩

=

∫

τ

∫

νA(0)∗

x (τ, ν) A(0)y (τ, ν)

[|SC(τ, ν)|2 − 1

]dτ dν.

From this, we obtain the intermediate bound

∣∣∆∣∣ ≤

∫

τ

∫

ν

∣∣Ax(τ, ν)∣∣ ∣∣Ay(τ, ν)

∣∣ ∣∣1 − |SC(τ, ν)|2∣∣ dτ dν . (3.69)

Applying the Schwarz inequality to (3.69) in three different ways yields, respectively,

∣∣∆∣∣ ≤

[ ∫

τ

∫

ν

∣∣Ax(τ, ν)∣∣2 ∣∣1 − |SC(τ, ν)|2

∣∣ dτ dν]1/2[ ∫

τ

∫

ν

∣∣Ay(τ, ν)∣∣2 ∣∣1 − |SC(τ, ν)|2

∣∣ dτ dν]1/2

= ‖Ax‖2M(φ′)x ‖Ay‖2M

(φ′)y ,

∣∣∆∣∣ ≤

[ ∫

τ

∫

ν

∣∣Ax(τ, ν)∣∣2 dτ dν

]1/2[ ∫

τ

∫

ν

∣∣Ay(τ, ν)∣∣2 ∣∣1 − |SC(τ, ν)|2

∣∣2 dτ dν]1/2

= ‖Ax‖2 ‖Ay‖2M(φ)y ,

∣∣∆∣∣ ≤

[ ∫

τ

∫

ν

∣∣Ax(τ, ν)∣∣2 ∣∣1 − |SC(τ, ν)|2

∣∣2 dτ dν]1/2[ ∫

τ

∫

ν

∣∣Ay(τ, ν)∣∣2 dτ dν

]1/2

= ‖Ax‖2M(φ)x ‖Ay‖2 .

The bound (3.68) finally follows by simultaneously invoking the above three inequalities and by noting

that ‖Ax‖2 = ‖Rx‖2 and ‖Ay‖2 = ‖Ry‖2.


Discussion. The foregoing theorem shows that for underspread processes x(t) and y(t) where

M(φ′)x M

(φ′)y or M

(φ)x or M

(φ)y is small, type I spectra that might not satisfy the Moyal-type relation

(3.67) exactly will satisfy it at least approximately, i.e.,

⟨Cx, Cy

⟩≈ 〈Rx,Ry〉 .

We further note that the intermediate result (3.69) gives a tighter bound that shows that only the

product of the individual GEAFs has to be concentrated about the origin. However, the bound (3.68),

while being coarser, has the advantage of being expressed directly in terms of weighted integrals of

the individual processes. For the special case y(t) = x(t), (3.69) yields the bound

∣∣‖Cx‖22 − ‖Rx‖2

2

∣∣‖Rx‖2

2

≤[M (φ′)

x

]2.

Hence, Theorem 3.18 also shows that for an underspread process x(t) the L2 norm of Cx(t, f) approx-

imately equals the HS norm of the correlation operator Rx,

‖Cx‖2 ≈ ‖Rx‖2 ,

where the relative error of this approximation is bounded in terms of M(φ′)x .

Of the example spectra in Subsection 3.3.2, only the GWVS and Page’s spectrum satisfy the

Moyal-type relation (3.67) exactly. For Levin’s spectrum where S(0)C

(τ, ν) = cos(πτν), we obtain

φ′(τ, ν) =√∣∣1 − |SC(τ, ν)|2

∣∣ = | sin(πτν)| ≤ π|τ ||ν| and φ(τ, ν) =∣∣1 − |SC(τ, ν)|2

∣∣ = sin2(πτν) ≤π2τ2ν2, so that M

(φ′)x ≤ πM

(1,1)x , M

(φ)x ≤ π2M

(2,2)x . For the physical spectrum, approximations for

M(φ′)x and M

(φ)x can be obtained using a truncated Taylor series expansion in a similar way as it was

demonstrated for the marginal properties.

CL Processes. In the case of CL processes with GEAF support intersection region Gx,y = {(τ, ν) :

|Ax(τ, ν) Ay(τ, ν)| > 0}, starting from (3.69) we obtain

∣∣ ⟨Cx, Cy

⟩− 〈Rx,Ry〉

∣∣ ≤∫∫

Gx,y

∣∣Ax(τ, ν)∣∣ ∣∣Ay(τ, ν)

∣∣ ∣∣1 − |SC(τ, ν)|2∣∣ dτ dν

≤ φ(max)x

∫∫

Gx,y

∣∣Ax(τ, ν)∣∣ ∣∣Ay(τ, ν)

∣∣ dτ dν ≤ φ(max)x ‖Ax‖2 ‖Ay‖2 ,

where φ(max)x = max(τ,ν)∈Gx,y

φ(τ, ν) = max(τ,ν)∈Gx,y

∣∣1 − |SC(τ, ν)|2∣∣. Hence,

∣∣ ⟨Cx, Cy

⟩− 〈Rx,Ry〉

∣∣‖Rx‖2 ‖Ry‖2

≤ φ(max)x .

This furthermore implies that if C satisfies |SC(τ, ν)| = 1 for (τ, ν) ∈ Gx,y (or equivalently

|SC

Gx,y (τ, ν)| = IGx,y(τ, ν)), then φ(max) = 0 and thus⟨Cx, Cy

⟩= 〈Rx,Ry〉. Similar remarks ap-

ply to the special case y(t) = x(t).


Positivity

The PSD Px(f) of a stationary process and the mean instantaneous intensity qx(t) of a white process

are positive quantities. Positivity is necessary for a proper interpretation as mean power or energy

distribution. Hence, positivity of time-varying power spectra has been a topic of longstanding interest.

In [57,60,63], positivity of the Wigner-Ville spectrum was related to a “sufficient amount of random-

ness” of the process to be analyzed. Reinterpreting “sufficient amount of randomness” as “limited TF

correlations,” we may expect that type I spectra of underspread processes are at least approximately

(real-valued and) positive, i.e., Cx(t, f) ≈ P{Cx(t, f)

}, with the positive real-valued part of Cx(t, f)

defined as in Subsection 2.3.13

P{Cx(t, f)

},

1

2

[∣∣ℜ{Cx(t, f)

}∣∣+ ℜ{Cx(t, f)

}].

A necessary and sufficient condition for a time-varying power spectrum Cx(t, f) to be real-valued

and positive (strictly speaking, non-negative) for all nonstationary random processes is that the cor-

responding operator C be positive (semi-)definite, i.e., C ≥ 0 or equivalently γk ≥ 0. However,

the subsequent results show that if one restricts to underspread random processes, (approximate)

positivity can be obtained.

We note that by virtue of (3.29), Theorem 2.30 (with H replaced by Rx) directly applies to the

GWVS. Hence for underspread processes the GWVS is approximately real-valued and positive. This

was also stated in Corollary 3.8. In the following, we will generalize this result to arbitrary type I

spectra.

Theorem 3.19. For any random process x(t) and any type I spectrum Cx(t, f) = 〈Ct,f ,Rx〉, the

difference between Cx(t, f) and P{Cx(t, f)

}is bounded as

∣∣Cx(t, f) − P{Cx(t, f)

}∣∣‖Ax‖1

≤ m(φ)x + inf

C′≥0

{m

(φC′ )

x

}, (3.70a)

∥∥Cx − P{Cx

}∥∥2

‖Ax‖2

≤ M (φ)x + inf

C′≥0

{M

(φC′ )

x

}, (3.70b)

with φ(τ, ν) =∣∣S(0)

C(τ, ν) − S

(0)∗C

(−τ,−ν)∣∣ and φC′(τ, ν) =

∣∣S(0)C

(τ, ν) − S(0)C′ (τ, ν)

∣∣.

Proof. The proof is largely analogous to that of Theorem 2.30. With the negative real part of Cx(t, f)

N{Cx(t, f)

}, P

{Cx(t, f)

}−ℜ

{Cx(t, f)

}=

1

2

[∣∣ℜ{Cx(t, f)

}∣∣−ℜ{Cx(t, f)

}]≥ 0 ,

the type I spectrum Cx(t, f) can be split up as

Cx(t, f) = P{Cx(t, f)

}−N

{Cx(t, f)

}+ j ℑ

{Cx(t, f)

},

so that Cx(t, f) −P{Cx(t, f)

}= j ℑ

{Cx(t, f)

}−N

{Cx(t, f)

}. Hence, the triangle inequality yields

∥∥Cx − P{Cx

}∥∥p≤∥∥ℑ{Cx

}∥∥p+∥∥N{Cx

}∥∥p

(3.71)


where for our purposes p = 2 or p = ∞. Bounds on the first term on the right-hand side of (3.71)

(involving the imaginary part of Cx(t, f)) have been provided in Theorem 3.16. With C ′x(t, f) denoting

an arbitrary positive type I spectrum generated by a positive semi-definite operator C′ ≥ O, the second

term (involving the negative real part of Cx(t, f)) can be further bounded as

∥∥N{Cx

}∥∥p

=1

2

∥∥∥∣∣ℜ{Cx

}∣∣−ℜ{Cx

}∥∥∥p

=1

2

∥∥∥∣∣ℜ{Cx

}∣∣− C ′x + C ′

x −ℜ{Cx

}∥∥∥p

≤ 1

2

∥∥∥∣∣ℜ{Cx

}∣∣− C ′x

∥∥∥p+

1

2

∥∥∥C ′x −ℜ

{Cx

}∥∥∥p

≤∥∥C ′

x −ℜ{Cx

}∥∥p

≤∥∥Cx − C ′

x

∥∥p. (3.72)

Here, we used the triangle inequality and the fact that the difference between C ′x(t, f) and the magni-

tude of the real part of Cx(t, f) is ≤ the difference between C ′x(t, f) and the real part of Cx(t, f) (since

C ′x(t, f) is positive) which in turn is ≤ the difference between C ′

x(t, f) and Cx(t, f) (since C ′x(t, f)

is real-valued). By noting that (3.72) holds for arbitrary positive C ′x(t, f) and by applying (3.61) to

(3.72) (with C(1) = C, C(2) = C′ and p = ∞ or p = 2), we obtain further

∥∥N{Cx

}∥∥∞

≤ ‖Ax‖1 infC′≥0

{m

(φC′ )

x

},

∥∥N{Cx

}∥∥2≤ ‖Ax‖2 inf

C′≥0

{M

(φC′ )

x

}. (3.73)

The bounds in (3.70) finally follow by inserting theses bounds in (3.71).

Discussion. The foregoing theorem shows that type I spectra of processes with small m(φ)x , M

(φ)x ,

infC′≥0m(φ

C′ )x , and infC′≥0M

(φC′ )

x are approximately nonnegative,

Cx(t, f) ≈ P{Cx(t, f)

}, or ℑ

{Cx(t, f)

}≈ 0 , N

{Cx(t, f)

}≈ 0 .

The bounds (3.70) are tightest for real-valued type I spectra since here m(φ)x = 0 and M

(φ)x = 0.

Furthermore, if Cx(t, f) is indeed positive (i.e., C ≥ 0), this is correctly indicated by the bounds

(3.70) since with C′ = C we have m(φ

C′ )x = 0 and M

(φC′ )

x = 0 in addition to m(φ)x = 0, M

(φ)x = 0.

Note that oblique orientation of |Ax(τ, ν)| and/or |SC(τ, ν)| can be accommodated via metaplectic

transformations of C′, since C′ ≥ 0 and U ∈ M imply C′ = UC′U+ ≥ 0 (and C′ has spreading

function |SC′(aτ + bν, cτ + dν)|).As noted above, type I spectra induced by positive semi-definite operators C are real-valued and

positive. In particular, this is true for the physical spectrum (C = g ⊗ g∗ ≥ 0) and multi-window

physical spectra (C =∑N

k=1 γk gk⊗g∗k) with real-valued γk ≥ 0. For the GWVS, the bounds (3.70)

can further be specialized since φ(τ, ν) = |ej2πατν − e−j2πατν | = 2| sin(2πατν)| ≤ 4π|α||τ ||ν| and

φC′(τ, ν) =∣∣ej2πατν − S

(0)C′ (τ, ν)

∣∣ =∣∣1 − S

(α)C′ (τ, ν)

∣∣. In the case of rank-one TF localization operators

C′ = g ⊗ g∗, i.e., C ′x(t, f) = PS

(g)x (t, f), we have φC′(τ, ν) = |1 − A

(α)g (τ, ν)| so that M

(φC′ )

x can be

approximated using a truncated Taylor series of A(α)g (τ, ν) similarly as it was done for the marginal

properties.


CL Processes. In the case of CL processes with compact GEAF support region Gx, straightfor-

ward application of (2.16) yields the bounds

∣∣Cx(t, f) − P{Cx(t, f)

}∣∣‖Ax‖1

≤ φ(max) + infC′≥0

{φ

(max)C′

},

∥∥Cx − P{Cx

}∥∥2

‖Ax‖2

≤ φ(max) + infC′≥0

{φ

(max)C′

},

with φ(max) = max(τ,ν)∈Gxφ(τ, ν) and φ

(max)C′ = max(τ,ν)∈Gx

φC′(τ, ν). This further implies that in cases

where the DL part of C is self-adjoint, i.e., S(0)C

(τ, ν) = S(0)∗C

(−τ,−ν) for (τ, ν) ∈ Gx (or equivalently

CGx = [CGx ]+), and if there is a positive semi-definite operator C′ whose DL part equals that of C,

i.e., S(0)C

(τ, ν) = S(0)C′ (τ, ν) for (τ, ν) ∈ Gx (or equivalently CGx = (C′)Gx), we have φ(max) = 0 and

φ(max)C′ = 0, respectively, and thus Cx(t, f) will be real-valued and positive in this case.

3.4 Type II Time-Varying Power Spectra

The GES of an overspread process contains cross terms that are due to the TF correlations of the

process (see Subsection 3.2.2). Sometimes these “statistical cross terms” are undesirable since they

may mask the “auto terms” characterizing the process’ mean TF energy distribution or they may

erroneously be taken for auto terms. The GES cross terms can be suppressed or reduced by an

appropriate TF smoothing. This motivates a generalization of the GES which we call type II time-

varying power spectra.

3.4.1 Definition and Formulations

The PSD Px(f) can be interpreted as squared magnitude of the frequency response H(f) of an LTI

innovations system H ∈ Ix (see (1.4)),

Px(f) = |H(f)|2.

The frequency response H(f) is obtained from the innovations system’s impulse response h(τ) via a

Fourier transform, which is a linear, frequency shift-covariant transform.

In a similar spirit, we use the innovations system representation (3.19) of nonstationary processes

as a basis for defining type II time-varying power spectra as

Gx(t, f) ,∣∣DH(t, f)

∣∣2 , (3.74)

where H ∈ Ix is an innovations system of x(t) and DH(t, f) is a “time-varying transfer function”

of the innovations system H. The definition of this time-varying transfer function needs further

specification. In particular, if one requires that DH(t, f) is a linear transform of H that is moreover

TF shift covariant in the sense that

H = St0,f0HS+t0,f0

=⇒ DeH(t, f) = DH(t− t0, f − f0) ,

3.4 Type II Time-Varying Power Spectra 141

then it can be shown that DH(t, f) must be of the form

DH(t, f) , Tr{HDt,f} =⟨H,D+

t,f

⟩=⟨H,S

(α)t,f D+S

(α)+t,f

⟩(3.75)

=

∫

t′

∫

f ′

L(0)D

(t′ − t, f ′ − f)L(0)H

(t′, f ′) dt′ df ′ , (3.76)

where the operator D completely specifies the TF transfer function DH(t, f). Usually, D will be

chosen as a TF localization operator [42–44, 81, 88, 174, 175]. For convenience, we henceforth assume

that |SD(τ, ν)| ≤ 1 (this merely amounts to a proper normalization of D).

The class of TF transfer functions given by (3.75) has previously been considered in a quantum-

mechanical context in [33,34] and in a signal analysis context in [62,96]. The “lower” symbol LLH

(t, f)

in (2.145) and the “upper” symbol LUH

(t, f) in (2.146) are specific members of this class obtained with

S(0)D

(τ, ν) = A(0)s (τ, ν) and S

(0)D

(τ, ν) = 1/A(0)∗s (τ, ν), respectively.

Like type I spectra, type II spectra admit four different canonical formulations in terms of repre-

sentations of the operator D in four different domains:

Gx(t, f) =∣∣⟨H,D+

t,f

⟩∣∣2 =

∣∣∣∣∫

t1

∫

t2

d(t2 − t, t1 − t)h(t1, t2) e−j2πf(t1−t2) dt1 dt2

∣∣∣∣2

(3.77a)

=

∣∣∣∣∫

f1

∫

f2

BD(f2 − f, f1 − f)BH(f1, f2) ej2πt(f1−f2) df1 df2

∣∣∣∣2

(3.77b)

=

∣∣∣∣∫

t

∫

fL

(0)D

(t′ − t, f ′ − f)L(0)H

(t′, f ′) dt′ df ′∣∣∣∣2

(3.77c)

=

∣∣∣∣∫

τ

∫

νS

(0)∗D+ (τ, ν)S

(0)H

(τ, ν) ej2π(νt−τf) dτ dν

∣∣∣∣2

. (3.77d)

It is seen from (3.77c) that for underspread operators D whose Weyl symbol L(0)D

(t, f) is a 2-D lowpass

function, Gx(t, f) incorporates a smoothing of L(0)H

(t, f). This smoothing can be used to suppress or

at least reduce oscillating components that are contained in L(0)H

(t, f) when the innovations system H

(and hence x(t)) is overspread (see Fig. (3.8)). Type II spectra with underspread D will henceforth

be referred to as smoothed type II spectra. Since the smoothing is applied before taking the squared

magnitude, Gx(t, f) cannot be interpreted as smoothed version of the GES. Indeed, the cross terms

of the GES are positive and could not be suppressed by a post-smoothing. This is different from type

I spectra which can be interpreted as smoothed versions of the Wigner-Ville spectrum.

3.4.2 Examples

We briefly consider two classes of type II spectra which are of particular interest.

Generalized Evolutionary Spectrum: The GES G(α)x (t, f) discussed in Subsection 3.2.2 repre-

sents a specific family of type II spectra obtained with operator D = L(α) (see Subsection B.1.2), i.e.,

S(0)D

(τ, ν) = ej2πατν . Note, however, that L(α) is not underspread, i.e., the GWS of L(α) is not a lowpass

function and hence the GES members are not smoothed type II spectra. The Weyl spectrum [147,148],

the (ordinary) evolutionary spectrum [170,171], and the transitory evolutionary spectrum [49,148] are

important special cases of the GES obtained with α = 0, α = 1/2, and α = −1/2, respectively.


Rank-One TF Localization Operator: The lower symbol LLH

(t, f) = 〈Hgt,f , gt,f 〉 in (2.145)

[64, 118] is a TF transfer function that measures the gain of H around the TF analysis point (t, f).

This corresponds to using a rank-one TF localization operator D = g⊗g∗, i.e., S(0)D

(τ, ν) = A(0)g (τ, ν).

Since this approach parallels that taken in the definition of the physical spectrum [139] (see Subsection

3.3.2), we refer to

Gx(t, f) =∣∣LL

H(t, f)∣∣2 (3.78)

as type II physical spectrum.

3.4.3 Approximate Equivalence

Depending on the choice of the TF localization operator D, different TF transfer functions DH(t, f)

and hence different type II time-varying power spectra are obtained. In some situations this non-

uniqueness might be considered an inconvenience. However, the next result shows that for underspread

processes most type II spectra yield effectively equivalent results.


of two type II time-varying power spectra G(1)x (t, f) =

∣∣⟨D(1)t,f ,H

⟩∣∣2 and G(2)x (t, f) =

∣∣⟨D(2)t,f ,H

⟩∣∣2

generated by the operators D(1) and D(2), respectively, is bounded as

∣∣G(1)x (t, f) − G

(2)x (t, f)

∣∣‖SH‖2

1

≤ 2m(φ)H, (3.79)

with φ(τ, ν) =∣∣S(0)

D(1)+(τ, ν) − S(0)

D(2)+(τ, ν)∣∣.

Proof. With (3.77d), the Fourier transform G(i)x (τ, ν) of G

(i)x (t, f) is obtained as

G(i)x (τ, ν) =

(S

(0)∗

D(i)+(τ, ν)S(0)H

(τ, ν))∗∗(S

(0)

D(i)+(−τ,−ν)S(0)∗H

(−τ,−ν))

=

∫

τ ′

∫

ν′

S(0)∗

D(i)+(τ ′, ν ′)S(0)H

(τ ′, ν ′)S(0)

D(i)+(τ ′ − τ, ν ′ − ν)S(0)∗H

(τ ′ − τ, ν ′ − ν) dτ ′ dν ′ .

Hence, it follows that

∣∣G(1)x (t, f) − G(2)

x (t, f)∣∣ ≤

∫

τ

∫

ν

∣∣G(1)x (τ, ν) − G(2)

x (τ, ν)∣∣ dτ dν

≤∫

τ1

∫

ν1

∫

τ2

∫

ν2

∣∣S(0)

D(1)+(τ1, ν1)S(0)∗

D(1)+(τ2, ν2) − S(0)

D(2)+(τ1, ν1)S(0)∗

D(2)+(τ2, ν2)∣∣

· |SH(τ1, ν1)| |SH(τ2, ν2)| dτ1 dν1 dτ2 dν2

≤∫

τ1

∫

ν1

∫

τ2

∫

ν2

[∣∣S(0)

D(1)+(τ1, ν1) − S(0)

D(2)+(τ1, ν1)∣∣+∣∣S(0)∗

D(1)+(τ2, ν2) − S(0)∗

D(2)+(τ2, ν2)∣∣]

· |SH(τ1, ν1)| |SH(τ2, ν2)| dτ1 dν1 dτ2 dν2

≤∫

τ1

∫

ν1

∣∣S(0)

D(1)+(τ1, ν1) − S(0)

D(2)+(τ1, ν1)∣∣ |SH(τ1, ν1)| dτ1 dν1

∫

τ2

∫

ν2

|SH(τ2, ν2)| dτ2 dν2

+

∫

τ1

∫

ν1

|SH(τ1, ν1)| dτ1 dν1

∫

τ2

∫

ν2

∣∣S(0)

D(1)+(τ2, ν2) − S(0)

D(2)+(τ2, ν2)∣∣ |SH(τ2, ν2)| dτ2 dν2


= 2‖SH‖21m

(φ)H,

where we used the fact that |ab− cd| ≤ |a− c|+ |b− d| for |a|, |b|, |c|, |d| ≤ 1 together with our general

assumption that∣∣S

D(i)(τ, ν)∣∣ ≤ 1.

Discussion. The foregoing theorem implies that for an underspread innovations system H with

small m(φ)H

(which is possible only if the corresponding process x(t) is underspread), two different type

II spectra using H are approximately equal,

G(1)x (t, f) ≈ G(2)

x (t, f) .

The following special cases illustrate the requirements for small m(φ)H

.

The approximate equivalence of GES with different α has previously been considered in Theorem

3.13. It can alternatively be obtained as a special case of Theorem 3.20 with D(1) = L(α1), D(2) = L(α2),

and the inequality φ(τ, ν) = |ej2πα1τν − ej2πα2τν | ≤ 2| sin(π(α1 − α2)τν)| ≤ 2π|α1 − α2||τ ||ν| whence

m(φ)H

≤ 2π|α1 − α2|m(1,1)H

.

A further interesting special case is D(1) = L(α) and D(2) = s⊗s∗ (i.e., the difference between

a GES and a type II physical spectrum). Here, φ(τ, ν) = |ej2πατν − A(0)s (τ, ν)| = |1 − A

(α)s (τ, ν)| so

that m(φ)H

will be small if H is underspread and if the effective support region of |As(τ, ν)∣∣ covers the

effective support region of |SH(τ, ν)|.CL Processes. A DL innovations system H with compact GSF support region GH produces a CL

process whose GEAF support region is twice as large. In this case, applying (2.16) to (3.79) results

in the bounds ∣∣G(1)x (t, f) − G

(2)x (t, f)

∣∣‖SH‖2

1

≤ 2φ(max)H

,

with φ(max)H

= max(τ,ν)∈GH

∣∣S(0)

D(1)(τ, ν)−S(0)

D(2)(τ, ν)∣∣. In particular, this implies that in cases where the

DL parts of D(1) and D(2) are equal, i.e., S(0)

D(1)+(τ, ν) = S

(0)

D(2)+(τ, ν) for (τ, ν) ∈ GH (or equivalently

[D(1)+]GH = [D(2)+]GH), we have φ(max)H

= 0 and thus G(1)x (t, f) = G

(2)x (t, f).

3.4.4 Properties

Type II spectra have the advantage of being always real-valued and nonnegative. Thus, in the following

we only consider the total mean energy property and the marginal properties.

Mean Energy

It can be shown that in order for a type II spectrum to integrate to the total mean energy of the

process, i.e., ∫

t

∫

fGx(t, f) dt df = Ex = Tr{Rx} , (3.80)

the operator D underlying Gx(t, f) has to satisfy |SD(τ, ν)| = 1. This condition is met by all GES

members but not for most other type II spectra. However, the following result shows that for an

underspread innovations system H, (3.80) is satisfied at least in an approximate way.



∆ , Ex −∫t

∫f Gx(t, f) dt df with Gx(t, f) = |〈H,D+

t,f 〉|2 equals

∆

‖H‖22

=[M

(φ)H

]2, (3.81)

with the weighting function φ(τ, ν) =√

1 −∣∣SD+(τ, ν)

∣∣2.

Proof. Since the 2-D Fourier transform of DH(t, f) =⟨H,D+

t,f

⟩H equals DH(τ, ν) =

S(0)∗D+ (τ, ν)S

(0)H

(τ, ν), Parseval’s relation implies that∫

t

∫

fGx(t, f) dt df =

∫

t

∫

f|〈H,D+

t,f 〉|2 dt df =

∫

τ

∫

ν

∣∣DH(τ, ν)∣∣2 dτ dν

=

∫

τ

∫

ν|SH(τ, ν)|2|SD+(τ, ν)|2 dτ dν .

On the other hand, the mean energy Ex can be rewritten as

Ex = Tr{Rx} = Tr{HH+

}= ‖H‖2

2 =

∫

τ

∫

ν|SH(τ, ν)|2dτ dν .

Hence, we obtain

∆ =

∫

τ

∫

ν|SH(τ, ν)|2

[1 − |SD+(τ, ν)|2

]dτ dν ,

which implies the final result. Note that we have assumed |SD(τ, ν)| ≤ 1 so that 1 − |SD(τ, ν)|2 ≥ 0

and hence ∆ ≥ 0.

Discussion. The foregoing theorem implies that for an innovations system with small M(φ)H

, there

holds the approximation ∫

t

∫

fGx(t, f) dt df ≈ Ex .

The weighted GSF integrals M(φ)H

will be small as long as |SD+(τ, ν)| ≈ 1 on the effective support

of |SH(τ, ν)|, which is favored by a small effective support of |SH(τ, ν)|, i.e., by an underspread

innovations system (and hence, an underspread process).

As a special case, let us consider the type II physical spectrum where |SD+(τ, ν)| = |As(τ, ν)| ≈1 − 4π2 T 2

s ν2 − 4π2 F 2

s τ2 (see (3.64)) and hence

[M

(φ)H

]2 ≈ 4π2

∫

τ

∫

ν|SH(τ, ν)|2

[T 2

s ν2 + F 2

s τ2]dτ dν = 4π2 T 2

s

[M

(0,1)H

]2+ 4π2 F 2

s

[M

(1,0)H

]2.

Achieving small M(φ)H

in this case requires a well-balanced choice of the duration/bandwidth of the

analysis window s(t) as compared to the innovations system’s moments M(1,0)H

and M(0,1)H

.

DL Innovations System. For a DL innovations system H with compact GSF support region

GH, it follows upon combining (3.81) and (2.16) that

∆

‖H‖22

≤[φ

(max)H

]2,

with φ(max)H

= max(τ,ν)∈GH

√1 − |SD(τ, ν)|2. Furthermore, if |SD(τ, ν)| = 1 for (τ, ν) ∈ GH (or

equivalently |S[D+]GH(τ, ν)| = IGH

(τ, ν)), then φ(max)H

= 0 and thus∫t

∫f Gx(t, f) = Ex.


Marginal Properties

A type II spectrum is said to satisfy the marginal properties in time and frequency if, respectively,∫

fGx(t, f) df = rx(t, t) ,

∫

tGx(t, f) dt = rX(f, f) .

These properties will be satisfied if the operator D respectively satisfies the following two conditions:

SD(τ ′, ν ′)S∗D(τ ′, ν ′ − ν) e−jπντ ′

= 1 , SD(τ ′, ν ′)S∗D(τ ′ − τ, ν ′) e−jπν′τ = 1 .

In particular, the condition on the left-hand side, and thus the marginal property in time, is satisfied

for D = L(1/2), i.e., in the case of the evolutionary spectrum (the GES with α = 1/2). Similarly,

the condition on the right-hand side, and thus the marginal property in frequency, is satisfied for

D = L(−1/2), i.e., in the case of the transitory evolutionary spectrum (the GES with α = −1/2).

While explicit but slightly involved expressions for the differences∫f Gx(t, f) df − rx(t, t) and∫

t Gx(t, f) dt−rX(f, f) can be derived, we here consider bounds on these differences only for the GES,

i.e., the special case D = L(α) for which Gx(t, f) = G(α)x (t, f). These bounds extend previous bounds

in [148].

Theorem 3.22. For any random process x(t) and any innovations system H ∈ Ix, the differences

∆1(t) ,∫f G

(α)x (t, f) df − rx(t, t) and ∆2(f) ,

∫tG

(α)x (t, f) dt− rX(f, f) (with G

(α)x (t, f) using H) are

bounded as

|∆1(t)|‖SH‖τ ‖SH‖1

≤ 4π∣∣∣α− 1

2

∣∣∣m(1,1)H

,|∆2(f)|

‖SH‖ν ‖SH‖1

≤ 4π∣∣∣α+

1

2

∣∣∣m(1,1)H

, (3.82)

with ‖SH‖τ , supτ

{∫ν |SH(τ, ν)| dν

}and ‖SH‖ν , supν

{∫τ |SH(τ, ν)| dτ

}.

Proof. The twisted convolution (B.12) (with α = 0) implies that

A(0)x (τ, ν) = S

(0)HH+(τ, ν) =

∫

τ ′

∫

ν′

S(0)H

(τ ′, ν ′)S(0)H+(τ − τ ′, ν − ν ′) e−jπ(τ ′ν−ν′τ) dτ ′ dν ′ .

Since furthermore S(0)H+(τ, ν) = S

(0)∗H

(−τ,−ν), the mean power rx(t, t) of x(t) can be rewritten as

rx(t, t) =

∫

νA(0)

x (0, ν) ej2πνt dν =

∫

ν

[ ∫

τ ′

∫

ν′

S(0)H

(τ ′, ν ′)S(0)∗H

(τ ′, ν ′ − ν) e−jπντ ′dτ ′ dν ′

]ej2πνt dν.

In contrast, using (3.77d) and the fact that for the GES S(0)D

(τ, ν) = ej2πατν , the time marginal of the

GES can be shown to equal

∫

fG(α)

x (t, f) df =

∫

ν

[ ∫

τ ′

∫

ν′

S(0)H

(τ ′, ν ′)S(0)∗H

(τ ′, ν ′ − ν) e−j2πατ ′ν dτ ′ dν ′]ej2πνt dν.

Hence, we obtain

|∆1(t)| ≤∫

ν

∫

τ ′

∫

ν′

|SH(τ ′, ν ′)| |SH(τ ′, ν ′ − ν)|∣∣e−j2πατ ′ν − e−jπντ ′∣∣ dτ ′ dν ′ dν

=

∫

ν1

∫

τ ′

∫

ν′

|SH(τ ′, ν ′)| |SH(τ ′, ν1)|∣∣1 − ej2π(α− 1

2)τ ′(ν′−ν1)

∣∣ dτ ′ dν ′ dν1


= 2

∫

ν1

∫

τ ′

∫

ν′

|SH(τ ′, ν ′)| |SH(τ ′, ν1)|∣∣∣ sin

[π(α− 1

2

)τ ′(ν ′ − ν1)

]∣∣∣ dτ ′ dν ′ dν1

≤ 2π∣∣∣α− 1

2

∣∣∣∫

ν1

∫

τ ′

∫

ν′

|SH(τ ′, ν ′)| |SH(τ ′, ν1)| |τ ′|(|ν ′| + |ν1|) dτ ′ dν ′ dν1

= 2π∣∣∣α− 1

2

∣∣∣∫

ν1

∫

τ ′

∫

ν′

|SH(τ ′, ν ′)| |SH(τ ′, ν1)| |τ ′| |ν ′| dτ ′ dν ′ dν1

+ 2π∣∣∣α− 1

2

∣∣∣∫

ν1

∫

τ ′

∫

ν′

|SH(τ ′, ν ′)| |SH(τ ′, ν1)| |τ ′| |ν1| dτ ′ dν ′ dν1

≤ 2π∣∣∣α− 1

2

∣∣∣ supτ ′

{∫

ν1

|SH(τ ′, ν1)| dν1

}∫

τ ′

∫

ν′

|τ ′| |ν ′| |SH(τ ′, ν ′)| dτ ′ dν ′

+ 2π∣∣∣α− 1

2

∣∣∣ supτ ′

{∫

ν′

|SH(τ ′, ν ′)| dν ′}∫

τ ′

∫

ν1

|τ ′| |ν1| |SH(τ ′, ν1)| dτ ′ dν1

= 4π∣∣∣α− 1

2

∣∣∣ ‖SH‖τ‖SH‖1m(1,1)H

,

which proves the bound for the time marginal on the left-hand side of (3.82). The bound for the

frequency marginal on the right-hand side of (3.82) can be shown in an analogous manner.

Discussion. The foregoing theorem shows that for an underspread innovations system with small

m(1,1)H

(which is possible only if x(t) is underspread), the marginal properties will be approximately

satisfied, ∫

fG(α)

x (t, f) df ≈ rx(t, t) ,

∫

tG(α)

x (t, f) dt ≈ rX(f, f) .

We recall that small m(1,1)H

requires |SH(τ, ν)| to be concentrated along the τ axis and/or ν axis.

Furthermore, the bounds in (3.82) correctly reflect the fact that the marginal property in time is

exactly satisfied by the evolutionary spectrum (i.e., the GES with α = 1/2) and the marginal property

in frequency is exactly satisfied by the transitory evolutionary spectrum (i.e., the GES with α = −1/2).

CL Processes. A DL innovations system H with compact GSF support region GH generates a

CL process. In this case, it follows upon combining (3.82) and (2.18) that

|∆1(t)|‖SH‖τ ‖SH‖1

≤ 4π∣∣∣α− 1

2

∣∣∣µH ,|∆2(f)|

‖SH‖τ ‖SH‖1

≤ 4π∣∣∣α+

1

2

∣∣∣µH ,

with µH as defined in (2.8).

3.5 Equivalence of Time-Varying Power Spectra

In Sections 3.3 and 3.4, we have introduced the classes of type I and type II time-varying power spectra

with the GWVS and GES, respectively, as central members. These classes present two alternative

broad frameworks for the definition of time-varying spectra. However, it was seen in Subsections 3.2.1

and 3.2.2 that in the case of an underspread process, the GWVS and GES are essentially independent

of α. Moreover, Subsections 3.3.3 and 3.4.3 generalized this approximate equivalence to any two

members within either the class of type I spectra or the class of type II spectra. In the following, we

3.5 Equivalence of Time-Varying Power Spectra 147

further generalize these equivalence results. First, we prove the approximate equivalence of GWVS

and GES for underspread processes. Then, we combine the available equivalence results to show that

for underspread processes any type I spectrum is approximately equal to any type II spectrum.

3.5.1 Equivalence of Generalized Wigner-Ville Spectrum and Generalized Evolu-tionary Spectrum

The common element of the GWVS and GES is their formulation in terms of the GWS. Keeping in

mind the relation Rx = HH+ with H ∈ Ix, it is seen that the GWVS is the GWS of the “squared”

innovations system HH+,

W(α)

x (t, f) = L(α)HH+(t, f) , (3.83)

whereas the GES is the squared magnitude of the GWS of the innovations system H,

G(α)x (t, f) =

∣∣L(α)H

(t, f)∣∣2. (3.84)

This observation and Corollary 2.17 are the basis for the following result.

Corollary 3.23. For any random process x(t) and any innovations system H ∈ Ix, the difference

between the GWVS and the GES (using H) with the same parameter α is bounded as

∣∣W (α)x (t, f) −G

(α)x (t, f)

∣∣‖SH‖2

1

≤ 2π C(α)H

with C(α)H

, cαm(0,1)H

m(1,0)H

+ 2 |α|m(1,1)H

, (3.85)

where cα = |α+ 1/2| + |α− 1/2|.

Proof. With (3.83) and (3.84), we have W(α)

x (t, f) − G(α)x (t, f) = L

(α)HH+(t, f) −

∣∣L(α)H

(t, f)∣∣2, which

equals ∆(α)4 (t, f) in (2.69) except for the fact that H+H in (2.69) is replaced by HH+. As mentioned

in Subsection 2.3.5, the bound (2.70) remains valid even if H+H is replaced by HH+. This gives

(3.85).

Discussion. If an innovations system H exists such that m(0,1)H

m(1,0)H

and m(1,1)H

are small (which

essentially requires the GSF of H—and hence the GEAF of x(t)—to be concentrated along the τ axis

and/or ν axis), then the foregoing corollary shows that

W(α)

x (t, f) ≈ G(α)x (t, f) . (3.86)

For α = 0, this approximation is illustrated by the similarity of Fig. (3.5)(a) and Fig. (3.6)(a).

Similarly, for α = 1/2 the approximation (3.86) is illustrated by the similarity of Fig. (3.5)(b)

and Fig. (3.6)(b). In these examples, the normalized errors were

∣∣W (0)x (t,f)−G

(0)x (t,f)

∣∣‖SH‖2

1

= 0.003 and∣∣W (1/2)

x (t,f)−G(1/2)x (t,f)

∣∣‖SH‖2

1

= 0.003 while the corresponding bounds were 2π C(0)H

= 0.018 and 2π C(1/2)H

=

0.073, respectively.

The fact that the positive semi-definite root Hp =√

Rx causes minimal TF displacements within

the class of normal innovations systems [90, 148] suggests that the GWVS and GES agree best when


the positive (semi)definite innovations systems is used in the GES. Furthermore, the bound (3.85) is

tightest for α = 0 where the second term of C(α)H

vanishes. The case α = 0 deserves further attention

due to the metaplectic covariance of the GWS with α = 0. Indeed, it follows from (2.72) that for

α = 0 one has the refined bound

∣∣W (0)x (t, f) −G

(0)x (t, f)

∣∣‖SH‖2

1

≤ 2π infU∈M

m(0,1)UHU+ m

(1,0)UHU+ . (3.87)

By combining Corollary 3.6, Theorem 3.13, and Corollary 3.23, we finally obtain the following

equivalence result for GWVS and GES with arbitrary α.

Corollary 3.24. For any random process x(t) and any innovations system H ∈ Ix, the difference

W(α1)x (t, f) −G

(α2)x (t, f) (with G

(α2)x (t, f) using H) is bounded as

∣∣W (α1)x (t, f) −G

(α2)x (t, f)

∣∣‖SH‖2

1

≤ 2π min{B1, B2} , (3.88)

where

B1 ,[cα1 m

(0,1)H

m(1,0)H

+ 2 ( |α1| + |α1 − α2|)m(1,1)H

], (3.89)

B2 ,[cα2 m

(0,1)H

m(1,0)H

+ 2 |α2|m(1,1)H

+ |α1 − α2|m(1,1)x

]

≤[(cα2 + 2|α1 − α2|

)m

(0,1)H

m(1,0)H

+ 2(|α2| + |α1 − α2|

)m

(1,1)H

], (3.90)

with cα = |α+ 1/2| + |α− 1/2|.

Proof. Subtracting/adding G(α1)x (t, f) from/to W

(α1)x (t, f) −G

(α2)x (t, f), we have

∣∣∣W (α1)x (t, f) −G(α2)

x (t, f)∣∣∣ =

∣∣∣W (α1)x (t, f) −G(α1)

x (t, f) +G(α1)x (t, f) −G(α2)

x (t, f)∣∣∣

≤∣∣∣W (α1)

x (t, f) −G(α1)x (t, f)

∣∣∣+∣∣∣G(α1)

x (t, f) −G(α2)x (t, f)

∣∣∣ .

Applying (3.85) (with α = α1) and (3.50), we obtain the bound 2πB1. Alternatively, by subtract-

ing/adding W(α2)x (t, f) from/to W

(α1)x (t, f) −G

(α2)x (t, f), we have

∣∣∣W (α1)x (t, f) −G(α2)

x (t, f)∣∣∣ =

∣∣∣W (α1)x (t, f) −W

(α2)x (t, f) +W

(α2)x (t, f) −G(α2)

x (t, f)∣∣∣

≤∣∣∣W (α1)

x (t, f) −W(α2)x (t, f)

∣∣∣+∣∣∣W (α2)

x (t, f) −G(α2)x (t, f)

∣∣∣.

Applying (3.31) (in combination with ‖Ax‖1 = ‖SH ♮ SH+‖1 ≤ ‖SH‖21 according to (B.14)) and (3.85)

(with α = α2), we obtain the bound 2πB2. The bound (3.90) on B2 follows from the fact that

according to (2.35), m(1,1)x ≤ 2[m

(1,1)H

+ m(0,1)H

m(1,0)H

]. Combining the bounds 2πB1 and 2πB2 gives

(3.88).

Discussion. Due to (3.89) and (3.90), the moments m(0,1)H

, m(1,0)H

and m(1,1)H

of the innovations

system suffice for discussing the foregoing corollary. For innovations system H such that m(0,1)H

m(1,0)H

3.5 Equivalence of Time-Varying Power Spectra 149

and m(1,1)H

are small, i.e., for underspread innovations systems and thus for underspread processes, the

corollary implies that

W(α1)x (t, f) ≈ G(α2)

x (t, f) .

Small m(0,1)H

m(1,0)H

and m(1,1)H

essentially requires that the GSF of H is oriented along the τ axis and/or

ν axis.

3.5.2 Equivalence of Type I and Type II Spectra

The following result shows that for underspread processes all type I and type II spectra yield effectively

equivalent results. This general result on the equivalence of time-varying power spectra is obtained

by combining the bounds in (3.61), (3.79), and (3.87) via the triangle inequality.


between a type I spectrum Cx(t, f) =⟨Ct,f ,Rx

⟩and a type II spectrum Gx(t, f) =

∣∣⟨H,D+t,f

⟩∣∣ induced

by operators C and D, respectively, is bounded as∣∣Cx(t, f) − Gx(t, f)

∣∣‖SH‖2

1

≤ m(φ1)x + 2m

(φ2)H

+ 2π infU∈M

{m

(0,1)UHU+m

(1,0)UHU+

}, (3.91)

with φ1(τ, ν) = |1 − S(0)C

(τ, ν)| and φ2(τ, ν) = |1 − S(0)D+(τ, ν)|.

Proof. Subtracting/adding W(0)

x (t, f) and G(0)x (t, f) to Cx(t, f) − Gx(t, f) and applying the triangle

inequality yields

|Cx(t, f) − Gx(t, f)| =∣∣Cx(t, f) −W

(0)x (t, f) +W

(0)x (t, f) −G(0)

x (t, f) +G(0)x (t, f) − Gx(t, f)

∣∣

≤∣∣Cx(t, f) −W

(0)x (t, f)

∣∣+∣∣W (0)

x (t, f) −G(0)x (t, f)

∣∣+∣∣G(0)

x (t, f) − Gx(t, f)∣∣ . (3.92)

According to Theorem 3.15, with C(1) replaced by C and C(2) replaced by L(0), the first term in (3.92)

is bounded as ∣∣Cx(t, f) −W(0)

x (t, f)∣∣ ≤ ‖Ax‖1m

(φ1)x . (3.93)

A bound on the second term in (3.92) is given by (3.87). Finally, Theorem 3.20, with D(1) replaced

by L(0) and D(2) replaced by D, implies that the third term in (3.92) is bounded as

∣∣G(0)x (t, f) − Gx(t, f)

∣∣ ≤ ‖SH‖21m

(φ2)H

. (3.94)

The bound (3.91) then follows by combining (3.93), (3.87), and (3.94), and by noting that according

to (B.14) ‖Ax‖1 = ‖SHH+‖1 ≤ ‖SH‖21.

Discussion. The foregoing result shows that if the process x(t) and the innovations system H are

underspread such that the respective weighted integrals and moments are small, we have

Cx(t, f) ≈ Gx(t, f) .

Hence, in the underspread case, type I and type II power spectra will yield effectively equivalent results.

On the other hand, for overspread processes, different spectra can yield dramatically different results.


(d) (e) (f)

(a) (b) (c)

t t t

t t t

f f f

f f f

Figure 3.7: (a)–(c) Type I power spectra and (d)–(f) type II power spectra (with positive semi-definite

innovations system) of an underspread process: (a) Wigner-Ville spectrum, W(0)x (t, f), (b) real part

of Rihaczek spectrum, ℜ{W

(1/2)x (t, f)

}, (c) (type I) physical spectrum, PS

(g)x (t, f), with Gaussian

window g(t), (d) Weyl spectrum, G(0)x (t, f), (e) evolutionary spectrum, G

(1/2)x (t, f) (simultaneously

the transitory evolutionary spectrum, G(−1/2)x (t, f)), (f) type II physical spectrum, Gx(t, f) in (3.78)

with Gaussian window g(t). The signal length is 256 samples, normalized frequency ranges from −1/4

to 1/4.

An example illustrating the approximate equivalence of time-varying power spectra in the case of an

underspread process (which has been synthesized following [89]) is shown in Fig. 3.7. Here, all spectra

yield practically identical results and correctly characterize the process’ mean TF energy distribution.

As a counterexample, Fig. 3.8 shows the same time-varying spectra for the case of an overspread

process that has been obtained from the underspread process by introducing correlations between the

‘T’ component and the ‘F’ component. These TF correlations are indicated by oscillating cross terms

in parts (a), (b), (d), and (e) of Fig. 3.8. On the other hand, in the type I and type II physical spectra

in parts (c) and (f), respectively, these cross terms are effectively suppressed and the process’ mean TF

energy distribution is better visible. However, these smoothed spectra no longer indicate the existing

strong TF correlations and hence do not completely characterize the process’ second-order statistics.

3.6 Input-Output Relations for Nonstationary Random Processes 151

(d) (e) (f)

(a) (b) (c)

t t t

t t t

f f f

f f f

Figure 3.8: The same time-varying spectra as in Fig. 3.7, but now for an overspread process.

3.6 Input-Output Relations for Nonstationary Random Processes

One reason for the importance of the PSD is its usefulness for describing the action of LTI systems.

When a wide-sense stationary random process x(t) with power spectral density Px(f) is passed through

an LTI system K with frequency response K(f), the output y(t) = (Kx)(t) is again wide-sense sta-

tionary with power spectral density Py(f) = |K(f)|2 Px(f). Similarly, the response of an LFI system

with temporal transfer function m(t) to a (nonstationary) white process x(t) with mean instanta-

neous intensity [163] qx(t) is again white with mean instantaneous intensity qy(t) = |m(t)|2 qx(t). In

this section, we investigate whether these simple multiplicative “input-output relations” relating the

second-order statistics of x(t) with those of y(t) = (Kx)(t) can be extended to the general case where

nonstationary random processes are passed through LTV systems. Both GWVS-based and GES-based

input-output relations will be considered.

3.6.1 Input-Output Relation Based on the Generalized Wigner-Ville Spectrum

Let us consider an LTV system K whose input x(t) is a zero-mean, nonstationary random process.

The system output y(t) = (Kx)(t) is a zero-mean, nonstationary random process whose correlation


operator is given by

Ry = E {(Kx) ⊗ (Kx)∗} = KE {x⊗ x∗}K+ = KRxK+.

Note that in this relation, like in the LTI/LFI case, the system K enters in a quadratic manner. We

now look for a TF reformulation of the above input-output relation in terms of the GWS. Specifically,

if K and Rx are jointly underspread in the sense of Subsection 2.3.4, then (2.62) and (2.49) imply

L(α)Ry

(t, f) = L(α)KRxK

+(t, f) ≈ L(α)K

(t, f)L(α)Rx

(t, f)L(α)K+(t, f) ≈

∣∣L(α)K

(t, f)|2 L(α)Rx

(t, f) ,

or, equivalently, using the GWVS,

W(α)

y (t, f) ≈ |L(α)K

(t, f)|2W (α)x (t, f) . (3.95)

The next theorem bounds the error of this approximation.

Theorem 3.26. For any random process x(t) and any LTV system K, the difference

∆(α)(t, f) , W(α)Kx (t, f) − |L(α)

K(t, f)|2W (α)

x (t, f)

is bounded as

|∆(α)(t, f)|‖SK‖2

1 ‖Ax‖1

≤ 2π cα

[m

(0,1)K

m(1,0)x +m

(1,0)K

m(0,1)x +m

(0,1)K

m(1,0)K

]+ 4π|α|m(1,1)

K(3.96)

with cα = |α+ 1/2| + |α− 1/2|.

Proof. Applying the triangle inequality after subtracting and adding L(α)K

(t, f)W(α)

x (t, f)L(α)K+(t, f)

yields

∣∣∆(α)(t, f)∣∣ =

∣∣∣W (α)KRxK

+(t, f) − L(α)K

(t, f)W(α)


+ L(α)K

(t, f)W(α)

x (t, f)L(α)K+(t, f) − |L(α)

K(t, f)|2W (α)

x (t, f)∣∣∣

=∣∣∆(α)

A (t, f) + ∆(α)B (t, f)

∣∣ ≤∣∣∆(α)

A (t, f)∣∣+∣∣∆(α)

B (t, f)∣∣ (3.97)

with

∆(α)A (t, f) , L

(α)KRxK

+(t, f) − L(α)K

(t, f)W(α)


∆(α)B (t, f) , L

(α)K

(t, f)W(α)

x (t, f)[L

(α)K+(t, f) − L

(α)∗K

(t, f)].

Since the 2-D Fourier transform of L(α)KRxK

+(t, f) is given by S(α)KRxK

+(τ, ν) =(S

(α)K♮ A

(α)x ♮ S

(α)K+

)(τ, ν)

(see (B.12)) and since the 2-D Fourier transform of L(α)K

(t, f)W(α)

x (t, f)L(α)K+(t, f) is given by(

S(α)K

∗∗ A(α)x ∗∗S(α)

K+

)(τ, ν), the 2-D Fourier transform of ∆

(α)A (t, f) is obtained as

∆(α)A (τ, ν) =

(S

(α)K♮ A(α)

x ♮ S(α)K+

)(τ, ν) −

(S

(α)K

∗∗ A(α)x ∗∗S(α)

K+

)(τ, ν)

=

∫

τ1

∫

ν1

∫

τ2

∫

ν2

S(α)K

(τ1, ν1) A(α)x (τ2, ν2)S

(α)K+(τ − τ1 − τ2, ν − ν1 − ν2)


·[e−j2π[φα(τ−τ1,ν−ν1,τ2,ν2)+φα(τ,ν,τ1,ν1)] − 1

]dτ1 dν1 dτ2 dν2 ,

where (B.12) has been used. This leads to the following bound on ∆(α)A (t, f),

∣∣∆(α)A (t, f)

∣∣ ≤∫

τ

∫

ν

∣∣∆(α)A (τ, ν)

∣∣ dτ dν

≤ 2

∫

τ

∫

ν

∫

τ1

∫

ν1

∫

τ2

∫

ν2

|SK(τ1, ν1)| |Ax(τ2, ν2)| |SK+(τ − τ1 − τ2, ν − ν1 − ν2)|

·∣∣ sin

(π[φα(τ − τ1, ν − ν1, τ2, ν2) + φα(τ, ν, τ1, ν1)]

)∣∣ dτ1 dν1 dτ2 dν2 dτ dν .

Using | sinx| ≤ |x| and substituting τ3 = τ − τ1 − τ2 and ν3 = ν − ν1 − ν2, this expression can further

be developed as

∣∣∆(α)A (t, f)

∣∣ ≤ 2π

∫

τ1

∫

ν1

∫

τ2

∫

ν2

∫

τ3

∫

ν3

|SK(τ1, ν1)| |Ax(τ2, ν2)| |SK+(τ3, ν3)|

·[|φα(τ2 + τ3, ν2 + ν3, τ2, ν2)| + |φα(τ1 + τ2 + τ3, ν1 + ν2 + ν3, τ1, ν1)|

]

· dτ1 dν1 dτ2 dν2 dτ3 dν3

≤ 2π

∫

τ1

∫

ν1

∫

τ2

∫

ν2

∫

τ3

∫

ν3

|SK(τ1, ν1)| |Ax(τ2, ν2)| |SK+(τ3, ν3)|

·[∣∣∣α+

1

2

∣∣∣[|τ1ν2| + |τ1ν3| + |τ2ν3|

]+∣∣∣α− 1

2

∣∣∣[|τ2ν1| + |τ3ν1| + |τ3ν2|

]]


= 2π ‖SK‖21 ‖Ax‖1 cα

[m

(0,1)K

m(1,0)x +m

(1,0)K

m(0,1)x +m

(0,1)K

m(1,0)K

], (3.98)

where the final expression is obtained by collecting corresponding terms in the integral and using

m(k,l)K+ = m

(k,l)K

. A bound on ∆(α)B (t, f) is obtained by using

∣∣L(α)K

(t, f)∣∣ ≤ ‖SK‖1,

∣∣W (α)x (t, f)

∣∣ ≤ ‖Ax‖1,

and by applying (2.48),

∣∣∆(α)B (t, f)

∣∣ ≤ ‖SK‖1 ‖Ax‖1

∣∣∣L(α)K+(t, f) − L

(α)∗K

(t, f)∣∣∣ ≤ ‖SK‖2

1 ‖Ax‖1 4π|α|m(1,1)K

. (3.99)

Inserting (3.98) and (3.99) in (3.97) finally yields the bound (3.96).

Discussion. Due to Theorem 3.26, the approximate input-output relation (3.95) will be valid if

the LTV system K and the correlation operator Rx are jointly underspread such that the moments

appearing in the bound (3.96) are small. This requires the GSF of K and the GEAF of x(t) to be

similarly localized along the τ axis and/or ν axis. The bound (3.96) is tightest for α = 0, in which case

also a refined bound similar to (2.63) and (2.72) can be obtained by using metaplectic transformations

of K and x(t); this allows the GSF of K and the GEAF of x(t) to be oriented in (similar) oblique

directions. An example illustrating the approximation (3.95) for the case α = 0 is shown in Fig. 3.9.

In this example, the normalized error was maxt,f|∆(α)(t,f)|

‖SK‖21 ‖Ax‖1

= 1.3 · 10−3 while the corresponding

bound in (3.96) was 3.3 · 10−3.


(a) (b) (c) (d)t t t t

f f f f

Figure 3.9: Illustration of approximate input-output relation for the GWVS: (a) Wigner-Ville spec-

trum W(0)x (t, f) of input process x(t), (b) Weyl symbol L

(0)K

(t, f) of LTV system K, (c) Wigner-Ville

spectrum W(0)y (t, f) of filtered process y(t) = (Kx)(t), (d) approximation |L(0)

K(t, f)|2W (0)

x (t, f) for

Wigner-Ville spectrum of y(t). The signal length is 256 samples, the normalized frequency range is

[−1/4, 1/4].

CL Processes. In the case of a CL process x(t) with GEAF support contained in the rectangular

region Gx = [−τx, τx] × [−νx, νx] and a DL system K with GSF support contained in the rectangular

region GK = [−τK, τK] × [−νK, νK], applying (2.17) and (2.18) to (3.96) and using (2.9) yields

|∆(α)(t, f)|‖SK‖2

1 ‖Ax‖1

≤ 2π cα

[νKτx + τKνx + τKνK

]+ 4π|α|µK ≤

(3π

2cα + π|α|

)σK,Rx , (3.100)

where σK,Rx is the joint displacement spread of K and Rx as defined in (2.11). For |α| ≤ 1/2, we

have cα = 1 so that the bound in (3.100) is further bounded by the simple expression 2π σK,Rx.

3.6.2 Input-Output Relation Based on the Generalized Evolutionary Spectrum

An input-output relation for the GES was presented in [148] without a bound on the associated

approximation error and based on the assumption that a CL process x(t) is passed through a DL system

K. The following result is less restrictive in that the GEAF of x(t) and the GSF of K are only required

to have rapid decay. Furthermore, it exploits the fact that according to y(t) = (Kx)(t) = (KHn)(t),

the operator KH is an innovations system of y(t) if H is an innovations system of x(t), i.e.,

H ∈ Ix =⇒ KH ∈ Iy .

In the following, we assume that the GES of x(t) is computed using the innovations system H and the

GES of y(t) is computed using the innovations system KH. We note that if H is the positive semi-

definite root of Rx, KH will not be the positive semi-definite root of Ry unless K and H commute.

In contrast, if H is a causal innovations systems of x(t) and if K is causal, too, then KH is also a

causal innovations system of y(t).

Theorem 3.27. For any random process x(t) and any LTV system K, the difference

∆(α)(t, f) = G(α)Kx(t, f) − |L(α)

K(t, f)|2G(α)

x (t, f)


(with G(α)x (t, f) based on the innovations system H and G

(α)Kx(t, f) based on the innovations system

KH) is bounded as

|∆(α)(t, f)|‖SK‖2

1 ‖SH‖21

≤ 4πcα[m

(1,0)H

m(0,1)K

+m(1,0)K

m(0,1)H

](3.101)

with cα =∣∣α+ 1

2

∣∣+∣∣α− 1

2

∣∣.

Proof. With G(α)x (t, f) =

∣∣L(α)H

(t, f)∣∣2 and G

(α)Kx(t, f) =

∣∣L(α)KH

(t, f)∣∣2, we have

∆(α)(t, f) = L(α)KH

(t, f)L(α)∗KH

(t, f) − L(α)K

(t, f)L(α)∗K

(t, f)L(α)H

(t, f)L(α)∗H

(t, f) .

Hence, the 2-D Fourier transform of ∆(α)(t, f) is given by

∆(α)(τ, ν) =[S

(α)KH

∗∗S(−α)(KH)+

−(S

(α)K

∗∗S(α)H

∗∗S(−α)K+ ∗∗S(−α)

H+

)(τ, ν)

=[(S

(α)K

♮ S(α)H

)∗∗(S

(−α)H+ ♮ S

(−α)K+

)](τ, ν) −

(S

(α)K

∗∗S(α)H

∗∗S(−α)K+ ∗∗S(−α)

H+

)(τ, ν)

where we used S(α)∗H

(−τ,−ν) = S(−α)H+ (τ, ν) (see (B.6)). Using (B.12), we further obtain

|∆(α)(t, f)| ≤∫

τ

∫

ν|∆(α)(τ, ν)| dτ dν

=

∫

τ

∫

ν

∣∣∣∣∫

τ1

∫

ν1

(S

(α)K

♮ S(α)H

)(τ1, ν1)

(S

(−α)H+ ♮ S

(−α)K+

)(τ − τ1, ν − ν1) dτ1 dν1

−∫

τ1

∫

ν1

(S

(α)K

∗∗S(α)H

)(τ1, ν1)

(S

(−α)H+ ∗∗S(−α)

K+

)(τ − τ1, ν − ν1) dτ1 dν1


=

∫

τ

∫

ν

∣∣∣∣∫

τ1

∫

ν1

[ ∫

τ2

∫

ν2

S(α)K

(τ2, ν2)S(α)H

(τ1 − τ2, τ1 − ν2) e−j2πφα(τ1,ν1,τ2,ν2) dτ2 dν2

]

·[ ∫

τ3

∫

ν3

S(−α)K+ (τ3, ν3)S

(−α)H+ (τ − τ1 − τ3, ν − ν1 − ν3)

· e−j2πφ−α(τ−τ1,ν−ν1,τ3,ν3)] dτ3 dν3

]dτ1 dν1

−∫

τ1

∫

ν1

[ ∫

τ2

∫

ν2

S(α)K

(τ2, ν2)S(α)H

(τ1 − τ2, τ1 − ν2) dτ2 dν2

]

·[ ∫

τ3

∫

ν3

S(−α)K+ (τ3, ν3)S

(−α)H+ (τ − τ1 − τ3, ν − ν1 − ν3) dτ3 dν3

]dτ1 dν1


=

∫

τ

∫

ν

∣∣∣∣∫

τ1

∫

ν1

∫

τ2

∫

ν2

∫

τ3

∫

ν3

S(α)K

(τ2, ν2)S(α)H

(τ1 − τ2, τ1 − ν2)S(−α)K+ (τ3, ν3)

· S(−α)H+ (τ − τ1 − τ3, ν − ν1 − ν3)

[e−j2π[φα(τ1,ν1,τ2,ν2)+φ−α(τ−τ1,ν−ν1,τ3,ν3)] − 1

]



≤∫

τ4

∫

ν4

∫

τ ′

∫

ν′

∫

τ2

∫

ν2

∫

τ3

∫

ν3

∣∣SK(τ2, ν2)∣∣ ∣∣SH(τ ′, ν ′)

∣∣ ∣∣SK+(τ3, ν3)∣∣ ∣∣SH+(τ4, ν4)

∣∣

·∣∣∣e−j2π[φα(τ ′+τ2,ν′+ν2,τ2,ν2)+φ−α(τ3+τ4,ν3+ν4,τ3,ν3)] − 1

∣∣∣ dτ4 dν4 dτ′ dν ′ dτ2 dν2 dτ3 dν3

≤ 2π

∫

τ4

∫

ν4

∫

τ ′

∫

ν′

∫

τ2

∫

ν2

∫

τ3

∫

ν3

∣∣SK(τ2, ν2)∣∣ ∣∣SH(τ ′, ν ′)

∣∣ ∣∣SK+(τ3, ν3)∣∣ ∣∣SH+(τ4, ν4)

∣∣


(a) (b) (c) (d)t t t t

f f f f

Figure 3.10: Illustration of approximate input-output relation for the GES: (a) GES (Weyl spec-

trum) G(0)x (t, f) of input process x(t), (b) Weyl symbol L

(0)K

(t, f) of LTV system K, (c) GES

G(0)y (t, f) of filtered process y(t) = (Kx)(t) (using innovations system K

√Rx), (d) approximation

|L(0)K

(t, f)|2G(0)x (t, f) for GES of y(t). The signal length is 256 samples, the normalized frequency

range is [−1/4, 1/4].

·[∣∣∣1

2+ α

∣∣∣(|τ2ν ′| + |τ4ν3|

)+∣∣∣12− α

∣∣∣(|τ ′ν2| + |τ3ν4|

)]dτ4 dν4 dτ

′ dν ′ dτ2 dν2 dτ3 dν3

= 4πcα‖SH‖21‖SK‖2

1

[m

(1,0)H

m(0,1)K

+m(1,0)K

m(0,1)H

],

where in the last step we split the integral, collected corresponding terms, and used m(k,l)H+ = m

(k,l)H

.

Discussion. The foregoing theorem shows that if m(1,0)H

m(0,1)K

+m(1,0)K

m(0,1)H

is small, the GES of

y(t) = (Kx)(t) with innovations system KH can be approximated as

G(α)y (t, f) ≈ |L(α)

K(t, f)|2G(α)

x (t, f) . (3.102)

Small m(1,0)H

m(0,1)K

+m(1,0)K

m(0,1)H

requires that H and K (and hence x(t) and y(t)) are jointly under-

spread (see (2.30)), i.e., that the GSFs of H and K are both effectively concentrated in a similar

way along the τ axis and/or ν axis. In the case α = 0, a refined bound similar to (2.63) can be

obtained by using metaplectic transformations of H and K; this allows the GSFs of H and K to be

oriented in (similar) oblique directions. An example illustrating the approximation (3.102) for α = 0

is shown in Fig. 3.10. In this example, the normalized error was maxt,f|∆(0)(t,f)|

‖SK‖21 ‖SH‖2

1

= 6 · 10−4 while

the corresponding bound in (3.101) was 2.3 · 10−3.

CL Processes. In the case of a DL innovations system H with GSF support contained in the

rectangular region GH = [−τH, τH] × [−νH, νH], the process x(t) is CL with GEAF support region

Gx = [−2τH, 2τH]× [−2νH, 2νH]. If in addition K is a DL system with GSF support contained in the

rectangular region GK = [−τK, τK]× [−νK, νK], then y(t) = (Kx)(t) is also a CL process with GEAF

support region Gy = [−2(τH + τK), 2(τH + τK)] × [−2(νH + νK), 2(νH + νK)]. In this case, applying

(2.17) to (3.101) yields the bound

|∆(α)(t, f)|‖SK‖2

1 ‖SH‖21

≤ 4π cα

[τHνK + τKνH

]≤ 2π cα σK,H ,

3.7 Approximate Karhunen-Loeve Expansion 157

where σK,H is the joint displacement spread of H and K as defined in (2.11). For |α| ≤ 1/2 we have

cα = 1 so that the last bound becomes 2π σK,H.

3.7 Approximate Karhunen-Loeve Expansion

As mentioned in Subection 1.3.2, the complex sinusoids ef0(t) = ej2πf0t are the Karhunen-Loeve (KL)

eigenfunctions of any stationary random process x(t), with the PSD at frequency f0, Px(f0), the

associated KL eigenvalue. Similarly, the Dirac impulses δt0(t) = δ(t− t0) are the KL eigenfunctions of

any white random process, with the mean instantaneous intensity at time t0, qx(t0), the associated KL

eigenvalue. Note that in the stationary and white cases the KL eigenfunctions are highly structured,

i.e., they are related by frequency shifts and time shifts, respectively.

The situation is different in the case of general nonstationary processes. The KL eigenfunctions

of different processes are different (unless the associated correlation operators commute [64, 158]).

Furthermore, the KL eigenfunctions of a general nonstationary process are not localized and structured

in any sense and the KL eigenvalues are not equal to the values of any conventional time-varying power

spectrum. However, we will now show that underspread processes have a well-structured set of TF-

localized “approximate KL eigenfunctions,” with the associated “approximate KL eigenvalues” given

by the GWVS values. We note that our discussion of approximate KL eigenvalues and eigenfunctions

is essentially an adaptation of previous results of Kozek [115, 116, 118, 120] and is also conceptually

similar to the approach presented in [137, 138]. Furthermore, the subsequent discussion is closely

related to the results in Subsections 2.3.8 and 2.3.9.

Let s(t) be a normalized function that is well concentrated about the origin of the TF plane

(e.g., a Gaussian function). We consider the family of functions st0,f0(t) = s(t − t0) ej2πf0t obtained

by TF-shifting s(t) to the TF point (t0, f0). By construction, st0,f0(t) is then well TF-concentrated

about (t0, f0). If x(t) is an underspread process, then Rx is an underspread operator and Theorem

2.23, with H replaced by Rx, states that the st0,f0(t) are approximate eigenfunctions of Rx with

W(α)x (t0, f0) = L

(α)Rx

(t0, f0) the associated approximate eigenfunctions. This suggests that for an un-

derspread process x(t), the TF shifted functions st0,f0(t) are approximate KL eigenfunctions with

W(α)x (t0, f0) the associated approximate KL eigenvalues. However, an essential feature of the KL

expansion is its double orthogonality, i.e., (see (1.8))

x(t) =∞∑

k=1

〈x, uk〉 uk(t) , E {〈x, uk〉〈x, ul〉∗} = λk δkl , 〈uk, ul〉 = δkl .

A similar property cannot be achieved by the function set {st0,f0(t)} which is continously parameterized

by time t0 and frequency f0. Hence, we alternatively consider the “approximate KL expansion”

x(t) =

∞∑

k=1

〈x, uk,l〉 vk,l(t) (3.103)

with discrete parameters k and l. Here, {uk,l(t)}, {vk,l(t)} are biorthogonal Weyl-Heisenberg sets


[56,100] obtained by TF shifting two functions u(t), v(t) (for convenience we assume ‖u‖ = 1),

uk,l(t) = u(t− kT ) ej2πlF t , vk,l(t) = v(t− kT ) ej2πlF t ,

with TF ≥ 1. The biorthogonality condition reads

〈uk,l, vk′,l′〉 = δkk′ δll′ .

In order for (3.103) to constitute an approximate KL expansion, we furthermore require (statistical)

orthohonality of the expansion coefficients,

E{〈x, uk,l〉

⟨x, uk′,l′

⟩∗}=⟨Rxuk,l, uk′,l′

⟩≈W

(α)x (kT, lF ) δkk′ δll′ .

The error incurred by this approximation is bounded in the next corollary that is a modified version

of Theorem 2.23 with H = Rx.

Corollary 3.28. For any random process x(t) and any biorthogonal Weyl-Heisenberg set {uk,l(t)},the difference

∆(α)9 [k, l; k′, l′] ,

⟨Rxuk,l, uk′,l′

⟩−W

(α)x (kT, lF ) δkk′δll′

is bounded as ∣∣∆(α)[k, l; k′, l′]∣∣

‖Ax‖1

≤ m

(φ

(k−k′,l−l′)u

)x , (3.104)

with φ(k,l)u (τ, ν) =

∣∣δk0 δl0 −A(α)u

(τ + kT, ν + lF

)∣∣.

Discussion. The preceding corollary shows that for underspread processes where m

(φ

(k,l)u

)x can be

made small by suitable choice of u(t) one has

E{〈x, uk,l〉

⟨x, uk′,l′

⟩∗}=⟨Rxuk,l, uk′,l′

⟩≈ W

(α)x (kT, lF ) δkk′ δll′ , (3.105)

and hence approximate double (bi-)orthogonality of the expansion (3.103). For well TF-localized u(t)

it can be shown that A(α)u (τ, ν) ≈ A

(α)u (0, 0) = 1 or equivalently φ

(0,0)u (τ, ν) = |1 − A

(α)u (τ, ν)| ≈ 0

about the origin of the (τ, ν)-plane. Thus, it is seen that small m(φ

(0,0)u )

x requires that |Ax(τ, ν)| is

concentrated about the origin, i.e., that x(t) is an underspread process. Furthermore, for k 6= 0, l 6= 0

there is φ(k,l)u (0, 0) =

∣∣Au(kT, lF )∣∣. Thus, small m

(φ(k,l)u )

x requires that |Ax(τ, ν)| is concentrated about

the origin and∣∣Au(kT, lF )

∣∣ ≈ 0 for k 6= 0 or l 6= 0, i.e., the ambiguity function of u(t) should decay

quickly outside the effective support of |Ax(τ, ν)|. This means that the effective support of |Au(τ, ν)|should be matched to the effective support of |Ax(τ, ν)|.

3.8 Time-Frequency Coherence

The last section in this chapter is dedicated to the introduction and discussion of a TF coherence

function. We will first briefly review the definition and properties of the ordinary spectral coherence

function for stationary processes. For nonstationary processes, a coherence operator will be intro-

duced. Using results from Subsection 2.3.6, we show that the GWS of the coherence operator can

be approximated by a ratio involving the (cross) GWVS of the processes under consideration. Some

efforts regarding the definition of a TF coherence function have previously been presented in [210].

3.8 Time-Frequency Coherence 159

3.8.1 Spectral Coherence and Coherence Operator

For jointly stationary processes x(t) and y(t) with PSDs Px(f), Py(f) and cross-PSD Px,y(f), the

(spectral) coherence function is defined as [13,67,171]

γx,y(f) ,Px,y(f)√Px(f)Py(f)

.

This is similar to a correlation coefficient of the Fourier transforms of x(t) and y(t) at frequency f . In

particular, the spectral coherence function satisfies

0 ≤ |γx,y(f)|2 ≤ 1. (3.106)

The coherence function is important since it has unit magnitude in the case of processes related by

linear transforms,8 i.e., for y(t) = (k ∗ x)(t) one has

|γx,y(f)|2 =|Px,y(f)|2Px(f)Py(f)

=|Px(f)K∗(f)|2

Px(f) |K(f)|2 Px(f)≡ 1 . (3.107)

This property has been exploited in numerous signal processing applications [13].

Let x(t), y(t) be two zero-mean, generally nonstationary random processes. As a nonstationary

counterpart of the coherence function, we now introduce a coherence operator as the cross-correlation

operator of the whitened processes (R−1/2x x)(t) and (R

−1/2y y)(t), i.e.,9

Γx,y , E{

(R−1/2x x) ⊗ (R−1/2

y y)∗}

= R−1/2x Rx,yR

−1/2y , (3.108)

where Rx,y = E {x⊗ y∗}, Rx and Ry are assumed invertible with positive definite roots R1/2x =

√Rx,

R1/2y =

√Ry. Note that there is Γy,x = Γ+

x,y. Furthermore, in the case of jointly stationary processes

x(t) and y(t), Rx, Ry, and Rx,y are convolution operators. Hence, in this case Γx,y is a convolution

operator as well and its kernel (Γx,y)(t1, t2) = γx,y(t1−t2) corresponds to the inverse Fourier transform

of γx,y(f), i.e., γx,y(τ) =∫f γx,y(f) ej2πfτ df . In this sense, the definition of Γx,y is consistent with the

coherence function of the stationary case.

Apart from the consistency with the stationary case, a meaningful interpretation of Γx,y as coher-

ence requires that it satisfies properties similar to (3.106) and (3.107). The property (3.106) can be

extended to the nonstationary case as follows:

Theorem 3.29. The operator norm of the coherence operator satisfies

∥∥Γx,y

∥∥O≤ 1 . (3.109)

Furthermore, the quadratic form induced by the “squared” coherence operators Γx,yΓ+x,y and Γ+

x,yΓx,y

satisfies

0 ≤ 〈Γx,yΓ+x,yg, g〉 ≤ 1 , 0 ≤ 〈Γ+

x,yΓx,yg, g〉 ≤ 1 , (3.110)

where g(t) is any normalized function (i.e., ‖g‖2 = 1).8Note that due to the assumption of joint stationarity of x(t) and y(t), admissible linear transforms necessarily

correspond to LTI systems.9A similar approach in a discrete-time framework has recently been taken in [189].


Proof. We start from the singular value decomposition of the coherence operator,

Γx,y =∑

k

σk uk ⊗ v∗k .

Here, the orthonormal bases {uk(t)} and {vk(t)} are the left and right singular functions, respectively

and σk > 0 are the singular values given by σk = 〈Γx,yvk, uk〉. According to (3.108), Γx,y = E {x⊗ y∗}where x(t) = (R

−1/2x x)(t) and y(t) = (R

−1/2y y)(t) are stationary white processes with correlation

Rx = Ry = I. Hence, we further obtain

σk = 〈Γx,yvk, uk〉 = 〈(E {x⊗ y∗})vk, uk〉 = E {〈x, uk〉〈y, vk〉∗}

≤√

E {|〈x, uk〉|2}E {|〈y, vk〉|2} =√

〈Rxuk, uk〉〈Ryvk, vk〉 =√

‖uk‖22 ‖vk‖2

2 = 1 ,

where we used the Schwarz inequality for random variables. Since the supremum of the singular values

determines the operator norm [69], i.e., ‖Γx,y‖O = supk{σk}, the inequality (3.109) follows. The left-

hand inequality in (3.110) is obtained by noting that Γx,yΓ+x,y is a positive semi-definite operator

and

〈Γx,yΓ+x,yg, g〉 = ‖Γ+

x,yg‖22 ≤ ‖Γx,y‖2

O ‖g‖22 ≤ 1 ,

where in the last step we used the previously derived bound∥∥Γx,y

∥∥O

≤ 1. The inequality involving

Γ+x,yΓx,y is shown in a similar way.

With regard to (3.107), let us consider two processes related by an invertible linear system K, i.e.,

y(t) = (Kx)(t). Here, we have Rx,y = RxK+ and Ry = KRxK

+, and hence

Γx,yΓ+x,y = R−1/2

x Rx,yR−1y R+

x,yR−1/2x = R−1/2

x RxK+(K+)−1R−1

x K−1KRxR−1/2x

= R1/2x R−1

x R1/2x = I .

That is, the “squared” coherence operator Γx,yΓ+x,y equals the identity operator. Similarly, it can be

shown that Γ+x,yΓx,y = I. Hence, we conclude that the coherence operator of linearly related processes

is unitary. This extends the central property (3.107) of the “stationary” coherence function γx,y(f) to

the nonstationary case.

3.8.2 Time-Frequency Formulation of the Coherence Operator

While the coherence operator Γx,y satisfies similar mathematical properties as the coherence function,

it involves computationally costly and unstable operator inversions and lacks physical intuition. The

GWS of Γx,y,

L(α)Γx,y

(t, f) = L(α)

R−1/2x Rx,yR

−1/2y

(t, f) ,

yields a TF formulation of the coherence operator; however, it still involves operator inverses. There-

fore, we now introduce a more convenient TF reformulation of Γx,y and show that it is an approximation

to L(α)Γx,y

(t, f).


We first note that the coherence operator can be defined alternatively by

HxΓx,yHy = Rx,y , (3.111)

where Hx =√

Rx and Hy =√

Ry are the positive (semi-)definite innovations systems of x(t) and

y(t), respectively. Next, we assume that Hx, Hy, and Rx,y are jointly DL operators with joint GSF

support region G. Note that this implies that Rx and Ry are jointly DL underspread as well; hence,

x(t) and y(t) are jointly CL underspread.

In order to obtain an approximation to the GWS of Γx,y, one could think of directly applying

(2.62) to (3.111), i.e.,

L(α)HxΓx,yHy

(t, f) ≈ L(α)Hx

(t, f)L(α)Γx,y

(t, f)L(α)Hy

(t, f) = L(α)Rx,y

(t, f) .

Unfortunately, even though Hx, Hy, and Rx,y are DL underspread operators, it is not obvious whether

Γx,y is underspread as well, i.e., whether the GSF of Γx,y is sufficiently concentrated about the origin.

However, by noting that (3.111) is of the form (2.76) with H1 = Hx, H2 = Hy, H3 = Rx,y, and

G = Γx,y, we can use the results of Subsection 2.3.6.

DL approximation of Γx,y. We split Γx,y into a DL part and a non-DL part according to

(2.4), i.e., Γx,y = ΓGx,y + ΓG

x,y. Then, Theorem 2.18 states that the non-DL part ΓGx,y = Γx,y − ΓG

x,y is

“negligible” in the sense that removing it from Γx,y does not greatly influence the validity of (3.111).

For convenience, we reformulate this theorem as a corollary incorporating the necessary notational

modifications.

Corollary 3.30. Consider a coherence operator Γx,y defined by HxΓx,yHy = Rx,y, where Hx =√

Rx,

Hy =√

Ry, and Rx,y are jointly DL with GSF support contained in the rectangle G =[− τG , τG

]×[

− νG , νG]

so that the joint displacement spread is given by σG = 4τGνG. Let ΓGx,y denote the DL part

of Γx,y defined by S(α)

ΓGx,y

(τ, ν) = S(α)Γx,y

(τ, ν) IG(τ, ν) and let ΓGx,y = Γx,y −ΓG

x,y denote the non-DL part of

Γx,y. Then, the difference HxΓGx,yHy − Rx,y is bounded as

∥∥HxΓGx,yHy − Rx,y

∥∥2

‖Hx‖2 ‖Γx,y‖2 ‖Hy‖2≤ 3

√σG . (3.112)

Hence, it is seen that if σG is small, i.e., if x(t) and y(t) are jointly CL underspread, then removing

the non-DL part ΓGx,y from Γx,y does not greatly affect the validity of HxΓx,yHy = Rx,y:

HxΓx,yHy = Rx,y =⇒ HxΓGx,yHy ≈ Rx,y .

Note that since Rx = H2x and Ry = H2

y, x(t) and y(t) are CL processes with GEAF support region

G2 =[− 2τG , 2τG

]×[− 2νG , 2νG

]. Since by assumption the cross-GEAF of x(t) and y(t) (i.e., the

GSF of Rx,y) is contained in G ⊂ G2, it follows that x(t) and y(t) are jointly CL processes with

joint correlation spread σx,y = 4σG . Hence, small σG essentially requires small σx,y, i.e., jointly CL

underspread processes.


Approximation of the GWS of Γx,y. The foregoing corollary is the basis for the next result

which states an approximation of the GWS of the DL part ΓGx,y of the coherence operator. We note

that L(α)

ΓGx,y

(t, f) is a smoothed version of the GWS of Γx,y,

L(α)

ΓGx,y

(t, f) =(L

(α)Γx,y

∗∗L(α)T

)(t, f) ,

where T is defined via S(α)T

(τ, ν) = IG(τ, ν) (see Subsection 2.1.1). The next corollary is a reformulation

of Theorem 2.19 with some notational adaptations.

Corollary 3.31. Let Hx, Hy, Rx,y, Γx,y, ΓGx,y, and the rectangular region G be defined as in Corollary

3.30. Then, the difference

∆(α)(t, f) , L(α)Hx

(t, f)L(α)

ΓGx,y

(t, f)L(α)Hy

(t, f) −W(α)x,y (t, f) ,

where W(α)x,y (t, f) is the cross-GWVS as defined in (B.45), is bounded as

∣∣∆(α)(t, f)∣∣

‖SHx‖1 ‖SΓx,y‖∞ ‖SHy‖1

≤ 3π

2cα σ

2G + 9σG ,

∥∥∆(α)∥∥

2

‖Hx‖2‖Γx,y‖2 ‖Hy‖2≤ π cασ

2G + 8π cα

√σ3G + 3

√σG ,

with cα = |α+ 1/2| + |α− 1/2|.

The above result shows that for small σG, it follows from H1GH2 = H3 that

L(α)Hx

(t, f)L(α)

ΓGx,y

(t, f)L(α)Hy

(t, f) ≈ L(α)Rx,y

(t, f) = W(α)

x,y (t, f) .

As explained above, small σG essentially requires that x(t) and y(t) are jointly CL underspread pro-

cesses. Using the regularized inversion techniques of Subsection 2.3.6, we can now obtain a TF

approximation for the GWS of (the DL part of) Γx,y. In particular, let us define an operator Γx,y via

its GWS as (cf. (2.93), (2.94))

L(α)eΓx,y

(t, f) ,

W(α)

x,y (t, f)

L(α)Hx

(t, f)L(α)Hy

(t, f), for (t, f) ∈ R ,

0 , for (t, f) 6∈ R ,

(3.113)

where

R ,

{(t, f) :

∣∣L(α)Hx

(t, f)L(α)Hy

(t, f)∣∣

‖SHx‖1 ‖SHy‖1

≥ κ ǫ

}, with ǫ ,

(3π

2cα σ

2G + 9σG

),

is the TF region where∣∣L(α)

Hx(t, f)L

(α)Hy

(t, f)∣∣ is essentially nonzero. According to the relevant derivation

in Subsection 2.3.6, Corollary 3.31 now implies that within R,

1

‖SΓx,y‖∞∣∣L(α)

ΓGx,y

(t, f) − L(α)eΓx,y

(t, f)∣∣ ≤ 1

κ.

Hence, for large enough κ it follows that within R

L(α)

ΓGx,y

(t, f) ≈ L(α)eΓx,y

(t, f) . (3.114)


Since ΓGx,y approximately satisfies (3.111), i.e., HxΓ

Gx,yHy ≈ Rx,y, the approximation (3.114) shows

that for κ large enough, the operator Γx,y defined via (3.113) approximately satisfies (3.111) as well:

HxΓx,yHy = Rx,y =⇒ HxΓGx,yHy ≈ Rx,y

=⇒(L

(α)Γx,y

∗∗ L(α)T

)(t, f) ≈ L

(α)eΓx,y

(t, f) =

W(α)

x,y (t, f)

L(α)Hx

(t, f)L(α)Hy

(t, f), for (t, f) ∈ R,

0 , for (t, f) 6∈ R .

Finally, we can rephrase the TF approximation of L(α)

ΓGx,y

(t, f) in a still more suggestive form. Since

Hx and Hy are positive semi-definite DL underspread innovations systems, combining Corollaries 3.23

and 3.8 yields the approximations

|L(α)Hx

(t, f)|2 ≈W(α)

x (t, f) ≈ P{W

(α)x (t, f)

}, |L(α)

Hy(t, f)|2 ≈W

(α)y (t, f) ≈ P

{W

(α)y (t, f)

}.

By virtue of Theorem 2.30, there is |L(α)Hx

(t, f)| ≈ L(α)Hx

(t, f) and |L(α)Hy

(t, f)| ≈ L(α)Hy

(t, f). Hence, we ob-

tain L(α)Hx

(t, f) ≈√

P{W

(α)x (t, f)

}and L

(α)Hy

(t, f) ≈√

P{W

(α)y (t, f)

}. Plugging these approximations

into (3.113), we can finally rewrite (3.114) as

L(α)

ΓGx,y

(t, f) ≈ Γ(α)x,y (t, f) ,

W(α)

x,y (t, f)√P{W

(α)x (t, f)

}P{W

(α)y (t, f)

} , for (t, f) ∈ R ,

0 , for (t, f) 6∈ R .

(3.115)

Here, Γ(α)x,y(t, f) defines a TF coherence function that is formulated in terms of GWVS. For α = 0, the

TF coherence function Γ(0)x,y(t, f) is similar to the TF coherence function introduced in [210]. There,

however, R was chosen as the TF region where W(0)x (t, f) and W

(0)y (t, f) are positive and the sodefined

TF coherence was proposed in an ad hoc fashion without establishing its relation to the coherence

operator.

Finally, with Corollary 2.17 it follows from (3.115) that the GWS of the “square” of the DL part

of the coherence operator is approximately given by

L(α)

ΓGx,yΓ

G+x,y

(t, f) ≈∣∣L(α)

ΓGx,y

(t, f)∣∣2 ≈

∣∣Γ(α)x,y (t, f)

∣∣2 =

∣∣W (α)x,y (t, f)

∣∣2

P{W

(α)x (t, f)

}P{W

(α)y (t, f)

} , for (t, f) ∈ R ,

0 , for (t, f) 6∈ R .

Discussion. The foregoing results have shown that for processes where√

Rx,√

Ry, and Rx,y

are jointly DL with GSF support region σG, then the non-DL part ΓGx,y is negligible in the sense that

√RxΓ

Gx,y

√Ry ≈ Rx,y. Furthermore, the GWS of ΓG

x,y (which is a smoothed version of the GWS of

Γx,y) is approximately equal to the TF coherence function Γ(α)x,y(t, f). We note that if

√Rx,

√Ry, and

Rx,y are jointly DL with GSF support area σG, the processes x(t) and y(t) are jointly CL with joint

correlation spread σx,y = 4σG (see Subsection 3.1.5).

To illustrate the approximation (3.115), we analyze the coherence of the input process x(t) and

the noise-contaminated output process y(t) = (Kx)(t) + n(t) of an underspread, self-adjoint LTV


0

0.5

1

1.5

2

0

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8(a) (b) (c)

6 6 6f f f

- - -t t t

Figure 3.11: Illustration of approximation of L(0)Γx,y

(t, f) by TF coherence function Γ(0)x,y (t, f) for

y(t) = (Kx)(t) + n(t): (a) Weyl symbol of the LTV system K, L(0)K

(t, f); (b) magnitude of Weyl

symbol of coherence operator Γx,y, |L(0)Γx,y

(t, f)|; (c) magnitude of TF coherence function, |Γx,y(t, f)|.The signal length is 128 samples, normalized frequency ranges from −1/4 to 1/4.

system K (the Weyl symbol of K is shown in Fig. 3.11(a)). Input signal x(t) and noise n(t) are

assumed to be zero-mean, uncorrelated, with respective correlations Rx = I and Rn = η I. It follows

that Ry = KK+ + η I and Rx,y = K+. In the noise-free case, x(t) and y(t) would be completely

coherent, i.e., Γx,yΓ+x,y = I. However, the noise n(t) causes an SNR-dependent reduction of coherence.

In particular, with K =∑

k κk wk⊗w∗k one can show that Γx,yΓ

+x,y =

∑k

|κk|2

|κk|2+ηwk⊗w∗

k. The SNR

dependence of coherence is clearly visible in the Weyl symbol of Γx,y that is shown in Fig. 3.11(b).

In particular, in those regions of the TF plane where L(0)K

(t, f) is large, the output SNR is large (i.e.,

(Kx)(t) is the dominant part of the output signal) and the coherence of x(t) and y(t) in these regions

is large. Similarly, in TF regions where L(0)K

(t, f) is small, the output SNR is small (i.e., the output

signal is dominated by n(t)) and the coherence in these TF regions will be small. The TF coherence

function Γ(0)x,y(t, f) is shown for comparison in Fig. 3.11(c). It is seen to be practically identical to the

Weyl symbol of Γx,y, thereby confirming the approximation (3.115).

3.8.3 The Generalized Time-Frequency Coherence Function

In the previous subsection, we considered an approximate TF formulation of the coherence operator

in terms of the (cross-) GWVS. For (jointly) underspread processes, the GWVS could be replaced

by other type I time-varying power spectra to which it is approximately equal (cf. Theorem 3.15).

This motivates the following generalized definition of a generalized TF coherence function using type

I (cross-)spectra of the processes x(t) and y(t) as defined in Subsection 3.3,

Γx,y(t, f) ,Cx,y(t, f)√

P{Cx(t, f)

}P{Cy(t, f)

} . (3.116)

Here, the positive parts of Cx(t, f) and Cy(t, f) are used in order to make the square-root in the

denominator well-defined. Note that according to Theorem 3.19, neglecting the negative parts of

Cx(t, f) and Cy(t, f) does not result in a loss of information if x(t) and y(t) are underspread processes.

This does not restrict the applicability of Γx,y(t, f) since, as will be seen below, for a meaningful


interpretation of a TF coherence function the underspread condition is required anyway. Taking the

positive parts is not necessary if only the squared magnitude of the TF coherence function is of interest.

Here, we simply write∣∣Γx,y(t, f)

∣∣2 ,

∣∣Cx,y(t, f)∣∣2

∣∣Cx(t, f) Cy(t, f)∣∣ . (3.117)

A particularly intuitive interpretation of the above TF coherence function is obtained when the

type I spectrum C·(t, f) is chosen as the physical spectrum (which is always positive, P{PS

(g)x (t, f)

}=

PS(g)x (t, f)), i.e.,

Γx,y(t, f) =PS

(g)x,y(t, f)√

PS(g)x (t, f) PS

(g)y (t, f)

=E {〈x, gt,f 〉〈y, gt,f 〉∗}√

E {| 〈x, gt,f 〉 |2} E {| 〈y, gt,f 〉 |2}.

Here, the correlation and mean power of the nonstationary processes x(t) and y(t) at the TF analysis

point (t, f) are measured via a TF shifted analysis window gt,f (t′) = g(t′− t) ej2πft′ , thereby endowing

Γx,y(t, f) with an immediate physical interpretation as a TF correlation coefficient. Due to the following

result, positive type I spectra (satisfying P{Cx(t, f)

}= Cx(t, f) for all x(t)) are of particular interest.

Theorem 3.32. The squared magnitude of the TF coherence function in (3.117) is bounded as

0 ≤∣∣Γx,y(t, f)

∣∣2 ≤ 1

if and only if the underlying type I spectrum C·(t, f) = Tr{Ct,fR·} is induced by a semi-definite

operator, i.e., C ≥ 0 or C ≤ 0.

Proof. The lower bound 0 ≤ |Γx,y(t, f)| is of course trivial. Since

Cx,y(t, f) = E{〈Ct,fx, y〉} = E{〈CS+t,fx,S

+t,fy〉} = E{〈Cx, y〉} = Cx,y(0, 0) ,

with x = (S+t,fx)(t), y = (S+

t,fy)(t), we can restrict attention to∣∣Γx,y(0, 0)

∣∣. Furthermore, since only the

magnitudes of Cx,y(t, f), Cx(t, f), and Cy(t, f) are of interest, the case C ≤ 0 can be reduced to the

case C ≥ 0 by the substitution C → −C. Hence, let us next assume C ≥ 0 so that C =∑K

k=1 γk gk⊗g∗kwith γk > 0 (where possibly K = ∞). We obtain,

Cx,y(0, 0) = E{〈Cx, y〉} = E

{ K∑

k=1

γk 〈x, gk〉〈y, gk〉∗}

= 〈x, y〉E

with the vectors x = (〈x, g1〉, . . . 〈x, gK〉)T , y = (〈y, g1〉, . . . 〈y, gK〉)T , and the inner product 〈x, y〉E

,

E{yHΓx} where Γ = diag{γ1, . . . , γK} (it is easily checked that this defines a valid inner product for

the vectors x and y). With ‖x‖E = 〈x, x〉E and the Schwarz inequality, we further obtain

∣∣Γx,y(0, 0)∣∣2 =

|Cx,y(0, 0)|2∣∣Cx(0, 0)Cy(0, 0)| =|〈x, y〉

E|2

‖x‖2E ‖y‖2

E

≤‖x‖2

E ‖y‖2E

‖x‖2E ‖y‖2

E

= 1 ,

which proves the “if” part of the theorem. To prove the “only if” part, we show that for any indef-

inite operator C one can find processes x(t) and y(t) such that |Γx,y(0, 0)| > 1, i.e.,∣∣Cx,y(0, 0)

∣∣2 >∣∣Cx(0, 0)∣∣ ∣∣Cy(0, 0)

∣∣ or ∣∣〈C,Ry,x〉∣∣2 >

∣∣〈C,Rx〉∣∣ ∣∣〈C,Ry〉

∣∣ . (3.118)


To this end, we use the eigenfunctions gk(t) of C =∑

k γk gk⊗g∗k to construct two correlated ran-

dom processes x(t) =∑

k〈x, gk〉 gk(t) and y(t) =∑

k〈y, gk〉 gk(t) with E {〈x, gk〉〈x, gl〉∗} = ξk δkl,

E {〈y, gk〉〈y, gl〉∗} = ηk δkl, and E {〈x, gk〉〈y, gl〉∗} = ρk δkl so that Rx =∑

k ξk gk ⊗ g∗k, Ry =∑

k ηk gk ⊗ g∗k, and Rx,y =∑

k ρk gk ⊗ g∗k. The parameters ξk, ηk, ρk have to satisfy the conditions

ξk ≥ 0, ηk ≥ 0, and |ρk| ≤√ξkηk (the latter being due to the Schwarz inequality) and will be further

specified below. Since C is indefinite, at least two of its eigenvalues have different sign. Without loss

of generality, we assume that the eigenvalues are ordered such that the first eigenvalue is positive,

γ1 > 0, and the second eigenvalue is negative, γ2 < 0. By choosing ξ2 = −γ1

γ2ξ1 > 0 and ξk = 0 for

k ≥ 3, we achieve 〈C,Rx〉 =∑

k γk ξk = 0, so that the right-hand side of (3.118) equals zero. In con-

trast, the left-hand side of (3.118) equals∣∣〈C,Ry,x〉

∣∣2 =∣∣γ1 ρ

∗1 + γ2 ρ

∗2

∣∣2 since enforcing the condition

|ρk| ≤√ξkηk in our case implies ρk = 0 for k ≥ 3. However, ρ1 and ρ2 can be chosen arbitrarily since

ξ1, ξ2, η1, η2 can be chosen as arbitrarily large positive numbers (while still obeying ξ2 = −γ1

γ2ξ1). We

conclude that∣∣〈C,Ry,x〉

∣∣2 =∣∣γ1 ρ

∗1 + γ2 ρ

∗2

∣∣2 can be made arbitrarily large by suitable choice of ρ1, ρ2,

thereby verifying (3.118). This proves the “only if” part of the theorem.

The above theorem establishes a property of the TF coherence function∣∣Γx,y(t, f)

∣∣2 that is analo-

gous to (3.106). We next present an analogue of property (3.107) for the case of nonstationary random

processes related by an LTV system. To this end, we restrict to the case where the type I spectrum

chosen is the GWVS.

Theorem 3.33. For any two linearly related random processes y(t) = (Kx)(t), the difference

∆(α)(t, f) ,∣∣W (α)

x,y (t, f)∣∣2 −W

(α)x (t, f)W

(α)y (t, f)

is bounded as ∣∣∆(α)(t, f)∣∣

‖Ax‖21 ‖SK‖2

1

≤ 2πB(α)x,K + 4π |α|

(m(1,1)

x +m(1,1)K

)(3.119)

with

B(α)x,K , cα

[5m

(0,1)K

m(1,0)x + 5m

(1,0)K

m(0,1)x +m

(0,1)K

m(1,0)K

]

where cα = |1/2 + α| + |1/2 − α|.

Proof. Noting that W(α)

x,y (t, f) = L(α)RxK+(t, f) and W

(α)y (t, f) = L

(α)KRxK

+(t, f), subtracting/adding∣∣W (α)

x (t, f)∣∣2 ∣∣L(α)

K+(t, f)∣∣2, and applying the triangle inequality, we obtain

∣∣∆(α)(t, f)∣∣ ≤

∣∣∆A(t, f)∣∣+∣∣∆B(t, f)

∣∣

where

∆A(t, f) =∣∣L(α)

RxK+(t, f)

∣∣2 −∣∣W (α)

x (t, f)∣∣2 ∣∣L(α)

K+(t, f)∣∣2 ,

∆B(t, f) =∣∣W (α)

x (t, f)∣∣2 ∣∣L(α)

K+(t, f)∣∣2 − W

(α)x (t, f)W

(α)y (t, f) .

Let us first consider ∆A(t, f). Applying the inequality∣∣|a|2 − |b|2

∣∣ ≤ 2|a − b|(|a| + |b|) with a =

L(α)RxK

+(t, f) and b = W(α)

x (t, f)L(α)K+(t, f) to

∣∣∆A(t, f)∣∣ yields

∣∣∆A(t, f)∣∣ =

∣∣∣∣∣L(α)

RxK+(t, f)

∣∣2 −∣∣W (α)

x (t, f)∣∣2 ∣∣L(α)

K+(t, f)∣∣2∣∣∣


≤ 2∣∣L(α)

RxK+(t, f) −W(α)


∣∣ ·[∣∣L(α)

RxK+(t, f)

∣∣+∣∣W (α)


∣∣].

Applying (2.57) with H1 = Rx and H2 = K+ and subsequently using the relations∣∣W (α)

x (t, f)∣∣ ≤

‖Ax‖1,∣∣L(α)

K(t, f)

∣∣ ≤ ‖SK‖1, ‖SRxK+‖1 ≤ ‖SRx‖1‖SK+‖1 (see (B.14)), ‖SK+‖1 = ‖SK‖1, m(k,l)K+ =

m(k,l)K

, we further have

∣∣∆A(t, f)∣∣ ≤ 2

[2π ‖Ax‖1 ‖SK+‖1B

(α)Rx,K

] [∣∣L(α)RxK+(t, f)

∣∣+∣∣W (α)


∣∣]

≤ 2[2π ‖Ax‖1 ‖SK+‖1B

(α)Rx,K

] [‖SRxK+‖1 + ‖Ax‖1 ‖SK+‖1

]

≤ 8π‖Ax‖21 ‖SK‖2

1

[∣∣∣α+1

2

∣∣∣m(0,1)x m

(1,0)K

+∣∣∣α− 1

2

∣∣∣m(1,0)x m

(0,1)K

]

≤ 8π cα ‖Ax‖21 ‖SK‖2

1

[m(0,1)

x m(1,0)K

+m(1,0)x m

(0,1)K

], (3.120)

where in the last step we used |α± 1/2| ≤ cα. For ∆B(t, f) we obtain

∣∣∆B(t, f)∣∣ =

∣∣∣∣∣W (α)

x (t, f)∣∣2 ∣∣L(α)

K+(t, f)∣∣2 − W

(α)x (t, f)W

(α)y (t, f)

∣∣∣

=∣∣W (α)

x (t, f)∣∣∣∣∣∣∣L(α)

K(t, f)

∣∣2W (α)∗x (t, f) − W

(α)Kx (t, f)

∣∣∣ .

Subtracting/adding∣∣L(α)

K(t, f)

∣∣2W (α)x (t, f) inside the second term and applying the triangle inequality,

we obtain further

∣∣∆B(t, f)∣∣ =

∣∣W (α)x (t, f)

∣∣∣∣∣∣∣∣L(α)

K(t, f)

∣∣2(W (α)∗

x (t, f) −W(α)x (t, f)

)

+[∣∣L(α)

K(t, f)

∣∣2W (α)x (t, f) −W

(α)Kx (t, f)

]∣∣∣∣

≤∣∣W (α)

x (t, f)∣∣[∣∣L(α)

K(t, f)

∣∣2 2∣∣ℑ{W (α)

x (t, f)}∣∣ +

∣∣∣∣L(α)K

(t, f)∣∣2W (α)

x (t, f) −W(α)Kx (t, f)

∣∣].

Successively applying the inequalities∣∣W (α)

x (t, f)∣∣ ≤ ‖Ax‖1,

∣∣L(α)K

(t, f)∣∣ ≤ ‖SK‖1, the bound (3.32) on∣∣ℑ{W (α)

x (t, f)}∣∣, and the bound (3.96) on

∣∣∣∣L(α)K

(t, f)∣∣2W (α)

x (t, f) −W(α)Kx (t, f)

∣∣, we obtain

∣∣∆B(t, f)∣∣ ≤ ‖Ax‖1

[∣∣L(α)K

(t, f)∣∣2 2∣∣ℑ{W (α)

x (t, f)}∣∣ +

∣∣∣∣L(α)K

(t, f)∣∣2W (α)

x (t, f) −W(α)Kx (t, f)

∣∣]

≤ ‖Ax‖1

[‖SK‖2

1 2∣∣ℑ{W (α)

x (t, f)}∣∣ +

∣∣∣∣L(α)K

(t, f)∣∣2W (α)

x (t, f) −W(α)Kx (t, f)

∣∣]

≤ ‖Ax‖1

[‖SK‖2

1 ‖Ax‖1 4π |α|m(1,1)x +

∣∣∣∣L(α)K

(t, f)∣∣2W (α)

x (t, f) −W(α)Kx (t, f)

∣∣]

≤ ‖Ax‖1

[‖SK‖2

1 ‖Ax‖1 4π |α|m(1,1)x +

2π‖Ax‖1 ‖SK‖21

[2 |α|m(1,1)

K+ cα

(m

(0,1)K

m(1,0)x +m

(1,0)K

m(0,1)x +m

(0,1)K

m(1,0)K

)]]

= ‖Ax‖21 ‖SK‖2

1

[4π|α|

(m(1,1)

x +m(1,1)K

)+ 2π cα

(m

(0,1)K

m(1,0)x +m

(1,0)K

m(0,1)x +m

(0,1)K

m(1,0)K

)].

(3.121)

The final bound (3.119) follows upon combining (3.120) and (3.121).

Discussion. The foregoing theorem shows that for small m(0,1)K

m(1,0)x , m

(1,0)K

m(0,1)x , m

(1,1)x , and

m(1,1)K

, there is ∣∣W (α)x,y (t, f)

∣∣2 ≈W(α)

x (t, f)W(α)

y (t, f)


and thus ∣∣Γx,y(t, f)∣∣2 ≈ 1 . (3.122)

In order that the above moments be small, Rx and K have to be jointly strictly underspread operators,

i.e., x(t) has to a an underspread process and K has to be an underspread system and their GEAF/GSF

have to be similarly oriented along the τ axis or ν axis. Note that this implies that y(t) = (Kx)(t)

will be an underspread process as well. In the case α = 0, the bound (3.119) is tightest and can be

tightened even further by using metaplectic transformations of Rx and K in a similar manner as in

Theorem 2.15.

As an illustration of the approximation (3.122), we reconsider the example given at the end of

Subsection 3.8.2 without noise, i.e., y(t) = (Kx)(t). Here, x(t) is stationary white noise and K

is an LTV system with GWS as shown in Fig. 3.11(a). Since x(t) and y(t) are linearly related,

there is Γx,yΓ+x,y = I. The TF coherence function

∣∣Γx,y(t, f)∣∣ computed using the Wigner-Ville spec-

trum (i.e., α = 0), also correctly indicates the linear relationship since there is∣∣Γx,y(t, f)

∣∣ ≈ 1 with

max(t,f)

∣∣∣∣∣Γx,y(t, f)

∣∣− 1∣∣∣ = 0.0164, thereby confirming the validity of (3.122).

We can finally conclude from the foregoing two theorems that within the underspread framework,

the TF coherence function is a meaningful concept since it features similar properties as the spectral

coherence function of the stationary case. However, for overspread (i.e., not jointly underspread)

processes x(t) and y(t), the TF coherence function in general is only of limited usefulness, i.e., its

magnitude may get larger than one and/or it may not correctly indicate existing linear relationships.

As an example, consider two correlated random processes x(t) = β u(t+ t0) e−j2πf0t and y(t) = β u(t−

t0) ej2πf0t where u(t) = e−πt2/T 2

/√

2T , t0 and f0 are fixed, and β is random with E{|β|2} = γ > 0.

One obtains

W(0)

x,y (t, f) = γ e−2π[t2/T 2+f2T 2] ej4π(t0f−f0t)

W(0)

x (t, f) = γ e−2π[(t+t0)2/T 2+(f+f0)2T 2]

W(0)

y (t, f) = γ e−2π[(t−t0)2/T 2+(f−f0)2T 2] .

It is seen that W(0)

x (t, f) and W(0)

y (t, f) are localized about (−t0,−f0) and (t0, f0), respectively. How-

ever, W(α)

x,y (t, f) is localized (and oscillatory) about (0, 0), corresponding to a “statistical cross term”

(see Section 3.2). It follows that

|Γx,y(t, f)| = e2π[t20/T 2+f20T 2] ≥ 1 ,

which for increasing t0, f0 can become arbitrarily large. Furthermore, apart of not being upper

bounded by 1, the TF coherence function |Γx,y(t, f)| in this example also fails to indicate that x(t)

and y(t) are linearly related. In fact, y(t) = ej4πf0t0 (S(1/2)−2t0,2f0

x)(t) but |Γx,y(t, f)| 6= 1 for t0 6= 0,

f0 6= 0. The large values of |Γx,y(t, f)| are seen to be due to TF correlations, i.e., correlations between

components of x(t) and y(t) located in different parts of the TF plane. These TF correlations are

indicated by statistical cross terms in W(α)

x,y (t, f) and by the fact that |Ax,y(τ, ν)| is localized about the

point (−2t0,−2f0) in the (τ, ν) plane (i.e., not about the origin). Hence, it is seen that the processes

x(t) and y(t) indeed are not jointly underspread.

4

Applications

“Before creation God did just pure mathematics. Then He thought it would bea pleasant change to do some applied.” John E. Littlewood

THIS chapter discusses several practical applications of the theory developed in Chapters 2 and

3. First, in Sections 4.1 and 4.2 we show that the results of Subsections 2.3.6 and 2.3.7 allow a

time-frequency formulation and design of mean-square optimal time-varying Wiener filters and time-

varying likelihood ratio detectors. In Section 4.3, we demonstrate the relevance of Subsections 2.3.1,

2.3.4, and 2.3.18 to the analysis of systematic measurement errors of mobile radio channel sounders.

Then, in Section Section 4.4 we apply results of Subsection 2.3.8 to (bi-)orthogonal frequency division

multiplexing communication systems operating over linear time-varying channels. Finally, in Section

4.5, we illustrate the usefulness of time-varying spectra and time-frequency coherence functions (see

Chapter 3) for the analysis of signals measured in car engines.

169

170 Chapter 4. Applications

4.1 Nonstationary Signal Estimation

Signal estimation is important in many practical applications such as in speech enhancement and

interference excision. Subsequently, we will briefly outline the relevance of the results of Chapters

2 and 3 to this problem. Further details are discussed in [92, 111]. Other work on TF aspects of

time-varying Wiener filters can be found in [9, 110,131,132,186, 191,192].

4.1.1 Time-Varying Wiener Filter

Let us assume that we observe a zero-mean nonstationary random process y(t) and we would like

to form an estimate x(t) of a desired nonstationary zero-mean (“signal”) process x(t) by passing

y(t) through a linear, generally time-varying system H, i.e., x(t) = (H y)(t). The cross-correlation

operator Rx,y, describing the statistical relation of x(t) and y(t), and the correlation operator Ry

are assumed to be known. Adopting a minimum mean-square error (MSE) criterion, the optimal

estimator minimizes the expected energy Ee = E{‖e‖22} of the estimation error e(t) = x(t)− x(t), i.e.,

HW , arg minH Ee. Using the orthogonality principle [106,187,197,202], this optimization problem

can be reduced to the solution of the Wiener-Hopf equation

HWRy = Rx,y. (4.1)

With R−1y denoting the (pseudo-)inverse of Ry, the time-varying Wiener filter is given by [92,106,111,

187,197,202]

HW = Rx,y R−1y . (4.2)

Note that this involves a computationally intensive and potentially unstable operator inversion. The

minimum MSE achieved with HW is given by EeW= Es − 〈HW ,Rx,y〉 = Es − Tr

{HW R+

x,y

}.

In the special case of jointly stationary processes, the design of the Wiener filter can be performed

in the physically intuitive frequency domain. The transfer function of HW is then given by

HW (f) =

Px,y(f)Py(f)

, for Py(f) > 0 ,

0 , for Py(f) = 0 ,(4.3)

where Px,y(f), Py(f) denote the (cross) power spectral densities (PSDs) of x(t) and y(t).

4.1.2 Time-Frequency Formulation of the Time-Varying Wiener Filter

We next assume that x(t) and y(t) are jointly CL underspread processes (cf. Section 3.1) with GEAF

support region G and joint correlation spread σx,y = σG . Then, (4.1) is seen to be of the type (2.96)

with G = HW , H2 = Ry, and H3 = Rx,y. Hence, Theorem 2.20 implies that the DL part HGW of the

Wiener filter HW , defined by

S(α)

HGW

(τ, ν) = S(α)HW

(τ, ν) IG(τ, ν) ,

constitutes an approximate solution of (4.1), i.e. (see also (2.100)),

HGWRy ≈ Rx,y , (4.4)

4.1 Nonstationary Signal Estimation 171

with the associated approximation error being bounded as

∥∥HGW Ry − Rx,y

∥∥2

‖HGW ‖2 ‖Ry‖2

≤ 2√σx,y . (4.5)

This further implies the following result on the (suboptimal) MSE EeGWachieved with HG

W .

Corollary 4.1. The excess MSE EeGW− EeW

is bounded as

0 ≤ EeGW− EeW

≤ 2‖HW ‖22 ‖Ry‖2

√σx,y. (4.6)

Proof. The left-hand inequality in (4.6) is trivial since EeWis by definition the minimum achievable

MSE. Furthermore, it can be shown that the MSE achieved with an arbitrary system H can be written

as

Ee = Tr{Rx − HR+

x,y − Rx,yH+ + HRyH

+}.

With H = HW as defined in (4.2), this simplifies to EeW= Tr

{Rx − HWR+

x,y

}, so that we obtain

EeGW− EeW

= Tr{Rx − HG

WR+x,y − Rx,y

(HG

W

)++ HG

W Ry

(HG

W

)+}− Tr{Rx − HWR+

x,y

}

= Tr{Rx} + Tr{[

HGW Ry − Rx,y

](HG

W

)+ − HGW R+

x,y

}− Tr{Rx} + Tr

{HW R+

x,y

}

= Tr{[

HGW Ry − Rx,y

](HG

W

)+}+ Tr

{HG

W R+x,y

}.

Due to the unitarity of the GSF (see (B.8)) and the fact that the supports of S(α)

HGW

(τ, ν) and A(α)x,y(τ, ν)

do not overlap, there is Tr{HG

W R+x,y

}=⟨S

(α)

HGW

, A(α)x,y

⟩= 0. By applying the Schwarz inequality, we

further arrive at

EeGW− EeW

= Tr{[

HGW Ry − Rx,y

](HG

W

)+}=⟨HG

W Ry − Rx,y,HGW

⟩≤ ‖HG

WRy − Rx,y‖2 ‖HGW‖2 .

The final bound (4.6) now follows by applying (4.5) and by noting that ‖HGW ‖2 ≤ ‖HW ‖2 and

‖HGW ‖2 ≤ ‖HW ‖2.

The preceding corollary shows that in the case of jointly CL underspread processes with small joint

correlation spread σx,y, the MSE achieved with HGW is close to the minimum achievable MSE EeW

.

We next turn to a TF formulation of the time-varying Wiener filter. Since the Weyl symbols of

Rx,y and Ry equal the (cross) Wigner-Ville spectra of x(t) and y(t), Theorem 2.21 (with G = HW ,

H2 = Ry, H3 = Rx,y and σG = σx,y) implies that the Weyl symbol of HGW satisfies (cf. (2.105)

and [92,111])

L(0)

HGW

(t, f)W(0)

y (t, f) ≈W(0)

x,y (t, f) ,

where the approximation error is bounded as

∣∣L(0)

HGW

(t, f)W(0)

y (t, f) −W(0)

x,y (t, f)∣∣

‖SHW‖∞ ‖Ay‖1

≤ π

2σ2

x,y +4σx,y ,

∥∥L(0)

HGW

W(0)y −W

(0)x,y

∥∥2

‖HW ‖2 ‖Ry‖2≤ π

√σ3

x,y +2√σx,y .


Hence, using regularized inversion as described in Subsection 2.3.7 (cf. (2.106)), we obtain

L(0)

HGW

(t, f) ≈

W(0)

x,y (t, f)

W(0)

y (t, f), for (t, f) ∈ R ,

0 , for (t, f) 6∈ R ,

(4.7)

where R =

{(t, f) :

∣∣W (0)y (t,f)

∣∣‖SH2

‖1

≥ κ ǫ}

with ǫ =(

π2 cα σ

2x,y +4σx,y

)is the TF region where the Wigner-

Ville spectrum of y(t) is essentially positive and κ is chosen to meet prescribed accuracy requirements

(see Subsection 2.3.7). The approximation (4.7) constitutes a TF formulation of the time-varying

Wiener filter that extends the frequency-domain formulation (4.3) of the time-invariant Wiener filter

to the nonstationary case. We recall that HGW is “nearly optimal” in the sense that it achieves an MSE

only slightly larger than that achieved by HW . Furthermore, we recall that L(0)

HGW

(t, f) is a smoothed

version of the Weyl symbol of HW .

4.1.3 Time-Frequency Filter Design

While (4.7) provides an approximate expression for the Weyl symbol of the Wiener filter, let us now

define another LTV system HW by setting its Weyl symbol equal to the right-hand side of (4.7) [92,111]:

L(0)eHW

(t, f) ,

W(0)

x,y (t, f)

W(0)

y (t, f), for (t, f) ∈ R ,

0 , for (t, f) 6∈ R .

(4.8)

We refer to HW as TF pseudo-Wiener filter [92,111]. For jointly CL underspread processes x(t) and

y(t), where (4.7) is a good approximation, combination of (4.8) and (4.7) yields L(0)eHW

(t, f) ≈ L(0)

HGW

(t, f)

and thus HW ≈ HGW , i.e., the TF pseudo-Wiener filter HW is a close approximation to the DL

part HGW of the Wiener filter HW and therefore nearly optimal. For processes that are not jointly

underspread, however, the performance of HW must be expected to be far from optimal (see the

overspread simulation example below). Compared to the Wiener filter HW , the TF pseudo-Wiener

filter HW has two advantages:

• Modified a priori knowledge. The design of HW is based on the (cross) Wigner-Ville spectra

W(0)

x,y (t, f) and W(0)

y (t, f), and thus it is physically more intuitive than the design of HW which

is based on correlation operators.

• Reduced and stable computation. The design of HW according to (4.2) requires a computationally

intensive and potentially unstable operator inversion. By using (4.8), this operator inversion is

replaced by a simple and stable scalar division.

We note that for jointly underspread processes, the (cross) Wigner-Ville spectra occurring in the

right-hand side of (4.8) can be replaced by the GWVS or any other type I spectrum since in the

underspread case these spectra are approximately equivalent (cf. Section 3.5.1 and [118,126,148]).

4.1 Nonstationary Signal Estimation 173

t

f

t

f

τ

ν

τ

ν

t

f

t

f

t

f

(a) (b) (c) (d)

(e) (f) (g)

Figure 4.1: Illustration of the TF formulation of the Wiener filter: (a) Wigner-Ville spectrum of

x(t), (b) Wigner-Ville spectrum of n(t), (c) expected ambiguity function of x(t), (d) expected ambiguity

function of n(t), (e) real part of Weyl symbol of Wiener filter HW , (f) real part of Weyl symbol of

DL part HGW of Wiener filter, (g) Weyl symbol of TF pseudo-Wiener filter HW . The rectangles in (c)

and (d) have area 1 and thus allow an assessment of the underspread property of x(t) and n(t). The

signal length is 128 samples.

4.1.4 Simulation Results

Underspread Example. Using the TF synthesis technique introduced in [89], we synthesized a

signal process x(t) and a noise process n(t) (uncorrelated with x(t)), whose sum constitutes the

observation, i.e., y(t) = x(t) + n(t). Hence, Ry = Rx + Rn and Rx,y = Rx. The expected energies

of these processes were Ex = 9.09 and En = 11.89, respectively, corresponding to a mean input

SNR of Ex/En = −1.17 dB. The Wigner-Ville spectra and expected ambiguity functions of x(t) and

n(t) are shown in Fig. 4.1(a)–(d). The expected ambiguity functions in parts (c) and (d) show that

the processes are jointly CL underspread. The Weyl symbols of the Wiener filter HW , its DL part

HGW , and the TF pseudo-Wiener filter HW are shown in Fig. 4.1(e)–(g). It is verified that HW

closely approximates HGW and also HW . This is further corroborated by similarity of the mean SNR

improvements1 of 6.14 dB and 6.11 dB achieved by HW and HW , respectively.

Overspread Example. We next consider an example where the underspread assumption is

violated. Again, the observation y(t) = x(t) + n(t) is the sum of a signal process x(t) and a noise

process n(t). Parts (a) and (c) of Fig. 4.2 show the Wigner-Ville spectrum und expected ambiguity

function, respectively, of the signal process x(t) that is seen to be a CL underspread process. In

1The SNR improvement is defined as the difference SNRout − SNRin of the output SNR SNRout = Ex/Ee and the

input SNR SNRin = Ex/En.


t

f

t

f

τ

ν

τ

ν

t

f

t

f

t

f

(a) (b) (c) (d)

(e) (f) (g)

Figure 4.2: A filtering experiment involving an overspread noise process: (a) Wigner-Ville spectrum

of x(t), (b) Wigner-Ville spectrum of n(t), (c) expected ambiguity function of x(t), (d) expected ambi-

guity function of n(t), (e) real part of Weyl symbol of Wiener filter HW , (f) real part of Weyl symbol

of DL part HGW of Wiener filter, (g) Weyl symbol of TF pseudo-Wiener filter HW . The rectangles in

parts (c) and (d) have area 1 and thus allow to assess the under-/overspread property of s(t) and n(t).

(In particular, part (d) shows that n(t) is overspread.) The signal length is 128 samples.

contrast, the noise process n(t), shown in parts (b) and (d) of Fig. 4.2, is fairly quasi-stationary but

nonetheless overspread since its temporal correlation width is too large (this is also indicated by the

statistical cross terms in the Wigner-Ville spectrum of n(t)). Consequently, the TF pseudo-Wiener

filter HW (shown in Fig. 4.2(g)) is significantly different from the Wiener filter HW (shown in Fig.

4.2(e)) and the DL part HGW of the Wiener filter (see Fig. 4.2(f)) does not approximate HW . The

processes x(t) and n(t) were constructed such that they lie in linearly independent (disjoint) signal

spaces. Here, HW can be shown to be an oblique projection operator [10] that perfectly reconstructs

x(t), thereby achieving zero MSE and infinite SNR improvement. On the other hand, the TF pseudo-

Wiener filter HW and the DL part HGW of the Wiener filter merely achieve SNR improvements of

4.79 dB and 4.24 dB, respectively. We conclude that in an overspread scenario, the TF designed

pseudo-Wiener filter achieves an SNR improvement that is far from optimal.

4.2 Nonstationary Signal Detection

In this section, we briefly discuss the relevance of the results of Chapters 2 and 3 to the practically

important problem of signal detection [108, 168, 187, 202]. Further details on TF detectors can be

found in [58,59,141–143,146,183,185].

4.2 Nonstationary Signal Detection 175

4.2.1 Optimal Detectors

We consider the binary hypothesis test

H0 : y(t) = x0(t) vs. H1 : y(t) = x1(t) , (4.9)

where x0(t) and x1(t) are zero-mean, nonstationary, mutually uncorrelated random processes with

known correlation operators R0 and R1, respectively. Typically, a decision between the hypotheses is

obtained by comparing a suitable test statistic Λ(y) derived from the observation y(t) to a prescribed

threshold ξ. Here, we will consider test statistics that can be written as a quadratic form

Λ(y) = 〈H y, y〉 , (4.10)

induced by a linear operator H. Note that the test statistic is completely specified by H.

Likelihood Ratio Detector

It can be shown that the optimal test statistic (optimal both in the Neyman-Pearson and in the

Bayesian sense) is given by the (log-)likelihood ratio [108, 168, 187, 202]. If both x0(t) and x1(t) are

Gaussian processes, the likelihood ratio leads to the equivalent test statistic

ΛLR(y) = 〈HLR y, y〉,

where the operator HLR is defined by [108,168,187,202]

R0HLRR1 = R1 − R0 . (4.11)

With the (pseudo-)inverses R−10 , R−1

1 of R0 and R1, the solution of (4.11) can be written as

HLR = R−10 (R1 − R0)R

−11 .

The computation of HLR requires two operator inversions which are computationally costly and poten-

tially unstable. The design of the likelihood ratio detector simplifies if x0(t) and x1(t) are stationary

processes. In this case, one has ΛLR(y) =

∫

fHLR(f) |Y (f)|2 df with

HLR(f) =

Px1(f) − Px0(f)Px0(f)Px1(f)

, for Px0(f)Px1(f) > 0 ,

0 , for Px0(f)Px1(f) = 0 ,(4.12)

where Y (f) is the Fourier transform of y(t) and Px0(f), Px1(f) are the PSDs of x0(t) and x1(t),

respectively.


Deflection-Optimal Detector

There are situations where the distribution of the observation under hypothesis H1 is not known or

the corresponding likelihood ratio is difficult to compute. In such situations, the deflection-optimal

detector can be used as an alternative to the optimal likelihood ratio detector. This is the quadratic

test statistic (4.10) with the operator H chosen to maximize deflection, defined as [7, 108]

d2 ,

(E {Λ|H1} − E {Λ|H0}

)2

var{Λ|H0},

with E {Λ|Hi} and var{Λ|Hi} denoting the expectation and variance of Λ under hypothesis Hi. The

deflection measures how well the conditional PDFs of Λ under H0 and H1 are separated. If the

distribution of y(t) under H0 is Gaussian, the deflection can be shown [7] to equal

d2 =Tr2{H(R1 − R0)}

Tr{(HR1)2}.

It can furthermore be shown [7] that the maximal deflection equals d2max = ‖R−1/2

1 (R1 −R0)R−1/21 ‖2

2

and is achieved with the operator HD that satisfies

R1HDR1 = R1 − R0 . (4.13)

The operator equation (4.13) can be solved using the (pseudo-)inverse R−11 of R1:

HD = R−11 (R1 − R0)R

−11 .

However, the computation of R−11 requires a large computational expense and may be numerically

unstable. A simplification is possible if x0(t) and x1(t) are stationary processes with PSDs Px0(f) and

Px1(f); here, ΛD(y) =∫f HD(f) |Y (f)|2 df with

HD(f) =

Px1(f) − Px0(f)

P 2x1

(f), for Px1(f) > 0 ,

0 , for Px1(f) = 0 .

(4.14)

4.2.2 Time-Frequency Formulation of Optimal Detectors

We next consider the TF reformulation of the above optimal detectors. To this end, we first note that

according to (B.27) the quadratic test statistic (4.10) can be rewritten as

Λ(y) =⟨L

(0)H,W (0)

y

⟩=

∫

t

∫

fL

(0)H

(t, f)W (0)y (t, f) dt df ,

where in the last expression we used the fact that the Wigner distribution W(0)x (t, f) is always real-

valued. In the following, we will present approximations for the Weyl symbol of the operators HLR

and HD that are valid for jointly CL processes and allow for a simple and intuitive TF formulation of

the likelihood-ratio detector and the deflection-optimal detector.


Time-Frequency Formulation of the Likelihood Ratio Detector

Let us assume that x0(t) and x1(t) are jointly CL underspread processes (cf. Section 3.1) with GEAF

support region G and correlation spread σx0,x1 = σG . It is then recognized that (4.11) is of the type

(2.76) with G = HLR, H1 = R0, H2 = R1, and H3 = R1 − R0. Hence, Theorem 2.18 implies that

the DL part HGLR of HLR approximately satisfies (4.11), i.e. (cf. (2.81))

R0HGLRR1 ≈ R1 − R0 ,

and the resulting approximation error is bounded as

∥∥R0HGLRR1 − (R1 − R0)

∥∥2

‖R0‖2 ‖HGLR‖2 ‖R1‖2

≤ 3√σx0,x1 .

With regard to an approximate TF formulation of the likelihood ratio detector, we next note that it

follows from Theorem 2.19 (with G = HLR, H1 = R0, H2 = R1, H3 = R1 − R0, and σG = σx0,x1)

that the Weyl symbol of HGLR approximately satisfies (see (2.92))

W(0)

x0(t, f)L

(0)

HGLR

(t, f)W(0)

x1(t, f) ≈W

(0)x1

(t, f) −W(0)

x0(t, f) ,

where we used the fact that the Weyl symbol of Ri equals the Wigner-Ville spectrum of xi(t), i.e.,

L(0)Ri

(t, f) = W(0)

xi(t, f). The associated approximation error is bounded as (cf. (2.84))

∣∣∣W (0)x0

(t, f)L(0)

HGLR

(t, f)W(0)

x1(t, f) −

[W

(0)x0

(t, f) −W(0)

x1(t, f)

]∣∣∣‖Ax0‖1 ‖SHLR

‖∞ ‖Ax1‖1

≤ 3π

2σ2

x0,x1+ 9σx0,x1 ,

∥∥∥W (0)x0L

(0)

HGLR

W(0)x1

−[W

(0)x0

−W(0)x1

]∥∥∥2

‖R0‖2‖HLR‖2 ‖R1‖2≤ πσ2

x0,x1+ 8π

√σ3

x0,x1+ 3√σx0,x1 .

Using the regularized TF inversion technique described at the end of Subsection 2.3.6, it finally follows

that

L(0)

HGLR

(t, f) ≈

W(0)

x1(t, f) −W

(0)x0

(t, f)

W(0)

x0(t, f)W

(0)x1

(t, f), (t, f) ∈ R

0 , (t, f) 6∈ R .

(4.15)

Here, R is the TF region where W(0)

x0(t, f)W

(0)x1

(t, f) ≥ ǫ with ǫ appropriately chosen to meet specific

accuracy requirements (see Subsection 2.3.6). The approximate TF formulation (4.15) extends the

frequency domain formulation (4.12) valid in the stationary case to the case of jointly CL nonstationary

processes.

Time-Frequency Formulation of the Deflection-Optimal Detector

The TF formulation of the deflection-optimal detector is completely analogous. It is again based on

the assumption that x0(t) and x1(t) are jointly CL underspread processes whose GEAFs are supported

within a region G such that the joint correlation spread of x0(t) and x1(t) is σx0,x1 = σG . Then, (4.13)


is again recognized to be of the type (2.76), now with G = HD, H1 = H2 = R1, and H3 = R1 − R0.

Hence, according to Theorem 2.18, the DL part HGD of HD approximately satisfies (4.13),

R1HGDR1 ≈ R1 − R0 .

The resulting approximation error is bounded as∥∥R1H

GDR1 − (R1 − R0)

∥∥2

‖R1‖22 ‖HG

LR‖2

≤ 3√σx0,x1 .

To obtain an approximate TF formulation of the deflection-optimal detector, we next apply Theorem

2.19 (with G = HD, H1 = H2 = R1, H3 = R1 − R0, and σG = σx0,x1). With L(0)Ri

(t, f) = W(0)

xi(t, f),

this yields the approximation

L(0)

HGLR

(t, f)[W

(0)x1

(t, f)]2

≈W(0)

x1(t, f) −W

(0)x0

(t, f) ,

for the Weyl symbol of HGD. Furthermore, the corresponding approximation error is bounded as

∣∣∣L(0)

HGD

(t, f)[W

(0)x1

(t, f)]2

−[W

(0)x0

(t, f) −W(0)

x1(t, f)

]∣∣∣

‖Ax1‖21 ‖SHD

‖∞≤ 3π

2σ2

x0,x1+ 9σx0,x1 ,

∥∥∥L(0)

HGD

[W

(0)x1

]2−[W

(0)x0

−W(0)x1

]∥∥∥2

‖R1‖22‖HD‖2

≤ πσ2x0,x1

+ 8π√σ3

x0,x1+ 3√σx0,x1 .

Hence, with the regularized TF inversion technique described in Subsection 2.3.6, we finally obtain

the approximation

L(0)

HGD

(t, f) ≈

W(0)

x1(t, f) −W

(0)x0

(t, f)[W

(0)x1

(t, f)]2 , (t, f) ∈ R

0 , (t, f) 6∈ R .

(4.16)

Here, R is the TF region whereW(0)

x1(t, f) ≥ ǫ with ǫ chosen according to certain accuracy requirements

(see Subsection 2.3.6). This approximate TF formulation of the deflection-optimal detector extends

the frequency domain formulation (4.14) valid for jointly stationary processes to the case of jointly

CL underspread nonstationary processes.

4.2.3 Time-Frequency Detector Design

The expressions (4.15) and (4.16) provide approximate TF formulations of the DL parts of the oper-

ators HLR and HD, respectively. This motivates the definition of alternative test statistics (termed

TF pseudo-likelihood ratio detector and TF pseudo-deflection-optimal detector) that are based on a

TF design of the underlying operators. The TF designed detectors proposed below offer the same

advantages as the TF pseudo-Wiener filter discussed in Subsection 4.1.3:

• Modified a priori knowledge. The a priori information required for the design of the TF detectors

is specified in the physically intuitive TF domain.


• Reduced and stable computation. The design of HLR and HD requires computation of the op-

erator inverses R−10 and R−1

1 . In contrast, the TF designed detectors are based on simple and

stable scalar inversions.

Time-Frequency Pseudo-Likelihood Ratio Detector

Motivated by the approximation (4.15), we define a TF test statistic

ΛLR(y) , 〈HLRy, y〉 =⟨L

(0)eHLR

,W (0)y

⟩

where the operator HLR is defined by setting its Weyl symbol equal to the right-hand side of (4.15),

i.e.,

L(0)eHLR

(t, f) ,

W(0)

x1(t, f) −W

(0)x0

(t, f)

W(0)

x0(t, f)W

(0)x1

(t, f), (t, f) ∈ R

0 , (t, f) 6∈ R .

We refer to ΛLR(y) as TF pseudo-likelihood ratio detector. For jointly CL underspread processes,

(4.15) implies that L(0)eHLR

(t, f) ≈ L(0)

HGLR

(t, f) and thus HLR ≈ HGLR ≈ HLR as well as ΛLR(y) ≈ ΛLR(y).

Hence, in the underspread case the TF pseudo-likelihood ratio detector will perform nearly as well as

the likelihood ratio detector.

Time-Frequency Pseudo-Deflection-Optimal Detector

Motivated by the approximation (4.16), we consider the TF test statistic

ΛD(y) , 〈HDy, y〉 =⟨L

(0)eHD,W (0)

y

⟩

defined with the operator HD whose Weyl symbol is chosen to equal the right-hand side of (4.16), i.e.,

L(0)eHD

(t, f) ,

W(0)

x1(t, f) −W

(0)x0

(t, f)[W

(0)x1

(t, f)]2 , (t, f) ∈ R

0 , (t, f) 6∈ R .

We refer to ΛD(y) as TF pseudo-deflection-optimal detector. For jointly CL underspread processes,

(4.16) implies that L(0)eHD

(t, f) ≈ L(0)

HGD

(t, f) and thus HD ≈ HGD ≈ HD as well as ΛD(y) ≈ ΛD(y).

Thus, in the underspread case the TF pseudo-deflection-optimal detector performs nearly as well as

the deflection-optimal detector.


Underspread Example. We next present Monte Carlo simulations for the detection of a nonsta-

tionary Gaussian signal s(t) in uncorrelated nonstationary Gaussian noise n(t) (both processes have

been synthesized using the technique introduced in [89]). This corresponds to the hypothesis test (4.9)

with x0(t) = n(t) and x1(t) = s(t) +n(t), i.e., R0 = Rn and R1 = Rs +Rn. Figs. 4.3(a),(c) show the


0.01 0.1 10.6

0.7

0.8

0.9

1

0.01 0.1 10.6

0.7

0.8

0.9

1

0.01 0.1 10.6

0.7

0.8

0.9

1

0.01 0.1 10.6

0.7

0.8

0.9

1

t

f

t

f

τ

ν

τ

ν

t

f

t

f

- -

6 6

PF PF

PD

PD

t

f

t

f

- -

6 6

PF PF

PD

PD

(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

Figure 4.3: Illustration of the TF formulation of likelihood ratio and deflection-optimal detector: (a)

Wigner-Ville spectrum of s(t), (b) Wigner-Ville spectrum of n(t), (c) magnitude of expected ambiguity

function of s(t), (d) magnitude of expected ambiguity function of n(t), (e) Weyl symbol of HLR, (f)

Weyl symbol of HLR, (g) ROC of ΛLR(y), (h) ROC of ΛLR(y), (i) Weyl symbol of HD, (j) Weyl

symbol of HD, (k) ROC of ΛD(y), (l) ROC of ΛD(y). The rectangles in (c) and (d) have area 1 and

thus allow to assess the underspread property of s(t) and n(t). The signal length is 128 samples.

Wigner-Ville spectrum and expected ambiguity function of s(t) and Figs. 4.3(b),(d) show the Wigner-

Ville spectrum and expected ambiguity function of n(t). It is seen that x(t) and n(t) are jointly CL

underspread processes. The Weyl symbols of the operators HLR and HLR are shown in Figs. 4.3(e)

and (f), from which the approximation L(0)HLR

(t, f) ≈ L(0)eHLR

(t, f) can be verified. Furthermore, the

receiver operating characteristics (ROCs) are shown in Figs. 4.3 (g) and (h) (the ROCs, empirically

obtained by averaging over 3 · 105 experiments, show the detection probability PD versus the false

alarm probability PF [108,168,187,202]). Since they are practically indistinguishable, likelihood ratio

detector and the TF pseudo-likelihood ratio detector perform equally well. Similar remarks apply to

the Weyl symbols of HD and HD (Fig. 4.3(i),(j)) and the ROCs of the deflection-optimal detector and

the TF pseudo-deflection-optimal detector (Fig. 4.3(k),(l)).

4.3 Sounding of Mobile Radio Channels 181

0.01 0.1 10.01

0.02

0.040.060.080.1

0.2

0.40.60.8

1

0.01 0.1 10.01

0.02

0.040.060.080.1

0.2

0.40.60.8

1

t

f

t

f

τ

ν

τ

ν

t

f

t

f

- -

6 6

PF PF

PD

PD

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 4.4: Discrimination between an overspread process x0(t) and an underspread process x1(t):

(a) Wigner-Ville spectrum of x0(t), (b) Wigner-Ville spectrum of x1(t), (c) magnitude of expected

ambiguity function of x0(t), (d) magnitude of expected ambiguity function of x1(t), (e) Weyl symbol

of HLR, (f) Weyl symbol of HLR, (g) ROC of ΛLR(y), (h) ROC of ΛLR(y). The rectangles in (c) and

(d) have area 1 and thus allow to assess the underspread/overspread property of x0(t) and x1(t). The

signal length is 128 samples.

Overspread Example. We next consider the discrimination between an overspread process

x0(t) and an underspread process x1(t). The Wigner-Ville spectra and expected ambiguity functions

of x0(t) and x1(t) are shown in Figs. 4.4(a)–(d). It is seen that both processes consist of two energetic

components that are concentrated in disjoint TF regions. However, while these components are un-

correlated in the case of x1(t), they are strongly correlated in the case of the overspread process x0(t).

The hypothesis test is thus intended to find out whether the two TF disjoint process components are

correlated (the overspread case x0(t)) or uncorrelated (the underspread case x1(t)). Figs. 4.4(e)–(h)

compare the likelihood ratio detector HLR and the TF pseudo-likelihood ratio detector HLR. Since

x0(t) is an overspread process, the Weyl symbol of the TF pseudo-likelihood ratio detection operator

HLR (shown in Fig. 4.4(f)) significantly differs from the Weyl symbol of HLR (shown in Fig. 4.4(e)).

This difference is furthermore reflected by the fact that the ROC of ΛLR(y) (see Fig. 4.4(h)) is sig-

nificantly worse than the (optimal) ROC of ΛLR(y) (see Fig. 4.4(g)). We conclude that in overspread

scenarios, the performance of TF designed detectors is far from optimum.

4.3 Sounding of Mobile Radio Channels

Accurate wideband measurements of mobile radio channels are important for the design or simula-

tion of mobile radio systems with high data rate. The typical channel sounders used to obtain such


∆(t) - G

transmitfilter

-x(t)H

time-varyingchannel

-y(t)R

receivefilter

- u(t)

Figure 4.5: Generic channel sounder model.

measurements are based on correlation/pulse compression techniques [40, 52, 164]. While these tech-

niques are theoretically exact in the case of time-invariant channels, in time-varying environments the

measurements obtained are affected by systematic measurement errors. In this section, these errors

will briefly be discussed for the case of uncalibrated sounders. For further details and a discussion of

calibrated channel sounders, we refer to [149,150,156].

4.3.1 Channel Sounder Model

We first describe a generic model for correlative channel sounders (see Fig. 4.5). The sounding signal

(i.e., the input to the mobile radio channel H) is obtained by periodic excitation ∆(t) =∑

m δ(t−mT )

of an LTI transmit filter G with impulse response g(t), i.e.,

x(t) = (G∆)(t) = (g ∗ ∆)(t) =∑

m

g(t−mT ) .

In the following, we assume that the duration of g(t) is less than T . The output signal of the equivalent

complex baseband channel H (assumed to be bandlimited to [−B,B]) is given by y(t) = (Hx)(t). At

the receiver front end, y(t) is passed through an LTI receive filter R with impulse response r(t). This

results in the sounder output signal

u(t) = (r ∗ y)(t) = (Ry)(t) = (RHx)(t) = (RHG∆)(t).

We note that for proper operation, g(t) and r(t) should be designed such that (r ∗ f)(t) ≈ δ(t) (pulse

compression), equivalently

RG ≈ I . (4.17)

Assuming that channel and receive filter commute, i.e.,

RH = HR , (4.18)

the pulse compression property (4.17) implies

u(t) = (RHG∆)(t) = (HRG∆)(t) ≈ (H∆)(t) =∑

m

h(t,mT ).

Therefore, under these assumptions channel sounders essentially achieve a direct “impulse sounding”

of the channel (i.e., the “efective sounding signal” is the impulse train ∆(t)) and u(t) approximately

consists of subsequent channel impulse response snapshots. An estimate of h(−1/2)(t, τ) = h(t + τ, t)


(cf. (B.2)) at t = mTrep = mKT (with the repetition factor K; note that Trep = KT ) can be calculated

from u(t) as

h(−1/2)(mTrep, τ) = u(τ +mTrep)w(τ) , (4.19)

where w(τ) equals 1 for 0 ≤ τ < T and 0 else. Furthermore, an estimate of h(−1/2)(t, τ) can then be

obtained by interpolation according to

h(−1/2)(t, τ) =∑

m

h(−1/2)(mTrep, τ) sinc

(π

Trep(t−mTrep)

)

=∑

m

u(τ +mTrep)w(τ) sinc

(π

Trep(t−mTrep)

)= (Υu)(t, τ) = (ΥRHG∆)(t, τ) .

(4.20)

Here, Υ denotes the mapping of the output signal u(t) to the measured impulse response. We note

that the above generic model covers the three most important practical channel sounders, namely the

PN-sounder [40,52,164], swept time-delay cross-correlator [37], and chirp sounder [180,194].

4.3.2 Analysis of Measurement Errors

As was shown in [149,150], the difference (error) between the measured impulse response h(−1/2)(t, τ) =

(ΥRHG∆)(t, τ) and the impulse response h(1/2)(t, τ) = h(t, t − τ) usually desired in mobile radio

applications can be split into four components, i.e.,

h(−1/2)(t, τ) − h(1/2)(t, τ) =

4∑

i=1

ei(t, τ) . (4.21)

We next discuss the definitions and interpretations of the error components ei(t, τ). Furthermore,

upper bounds on specific error norms are provided. These bounds are formulated in terms of important

channel and sounder parameters. We note that for an arbitrary norm ‖ · ‖, application of the triangle

inequality to (4.21) gives

‖h(−1/2) − h(1/2)‖ ≤4∑

i=1

‖ei‖ . (4.22)

Thus, our upper bounds on the error components ei(t, τ) also yield upper bounds on the total error

hOH

(t, τ) − hIH

(t, τ).

Commutation Error

For time-varying channels, the commutation property RH = HR in (4.18) is not satisfied exactly

(recall that correlative sounding assumes the operator ordering HRG but in fact we have RHG).

This causes a commutation error

e1(t, τ) , h(−1/2)(t, τ) − (ΥHRG∆)(t, τ) = (ΥRHG∆)(t, τ) − (ΥHRG∆)(t, τ) = (Υ[R,H]x)(t, τ) ,

where [R,H] = RH − HR denotes the commutator of R and H. In Subsection 2.3.16, we saw that

(jointly) underspread systems approximately commute. Indeed, it can be shown (by adapting the


proof of Theorem 2.14) [149], that the commutation error is bounded as

|e1(mTrep, τ)|‖g‖∞‖SH‖1‖r‖1

≤ ε1 , 2πm(1,0)R

m(0,1)H

,

∫τ |e1(mTrep, τ)| dτ‖g‖∞‖SH‖1‖r‖1

≤ T ε1 (4.23)

where m(1,0)R

measures the effective duration of the receive filter impulse response r(t) and m(0,1)H

measures the channel’s effective Doppler spread. From (4.23), it is seen that the commutation error

will be small if the receive filter is not too long and the channel does not vary too fast.

We note that the error bound (4.23) also proved important in the context of scattering function

estimation for random time-varying channels [5, 6].

Pulse Compression Error

According to (4.17), correlative channel sounders theoretically require that the transmit filter and the

receive filter have perfect pulse compression properties. Unfortunately, practical transmit and receive

filters do not yield perfect pulse compression, i.e., (r ∗ g)(t) 6= δ(t) or RG 6= I. This leads to a pulse

compression error

e2(t, τ) , (ΥHRG∆)(t, τ) − (ΥH∆)(t, τ) .

This error is determined by the transmit and receive filters and is not related to the channel’s time

variation. In practical channel sounders, calibration is used to reduce the effects of imperfect pulse

compression. However, it is shown in [150,156] that for time-varying channels conventional calibration

procedures are affected by systematic errors as well.

The pulse compression error is bounded as [149]

|e2(mTrep, τ)|‖SH‖1

≤ ε2 ,1

T

∑

|k|≤BT

∣∣∣R( kT

)G( kT

)− 1∣∣∣ ,

∫τ |e2(mTrep, τ)| dτ

‖SH‖1

≤ Tε2 , (4.24)

where B is the channel bandwidth. The bound (4.24) is intuitive since the error resulting from im-

perfect correlation/pulse compression properties of transmit and receive filter is obviously determined

by the deviation of the composite transfer function R(f)G(f) from the ideal value 1.

Aliasing Error

In order that subsequent snapshots do not overlap, the channel’s maximum delay τ(max)H

must satisfy

τ(max)H

≤ T . Similarly, in order that the channel variation is properly tracked using the repetition rate

Trep = KT , the channel’s maximum Doppler shift ν(max)H

must satisfy ν(max)H

≤ 1/(2KT ). Combining

these two requirements yields the “perfect identification condition” [103,104]

τ(max)H

≤ T ≤ 1

2Kν(max)H

. (4.25)

Note that it is necessary for this condition to hold that τ(max)H

ν(max)H

≤ 1/(2K), i.e., that the channel is

underspread. If the condition in (4.25) is not met, there will occur aliasing errors (du to overlapping

snapshots and/or insufficient channel tracking). The associated error component is given by

e3(t, τ) , (ΥH∆)(t, τ) − h(−1/2)(t, τ) .


Since correlative channel sounding can be shown to correspond to a sampling of the channel’s time-

varying transfer function (i.e., of the GWS L(α)H

(t, f)), Theorem 2.35 directly applies (with F =

1/(2KT )). It follows that the aliasing error e3(t, τ) is bounded as [149]

∫τ |e3(t, τ)| dτ

‖SH‖1

≤ 2

[m

(1,0)H

T+ 2KT m

(0,1)H

],

‖e3‖2

‖SH‖2

≤ 2

[M

(1,0)H

T+ 2KT M

(0,1)H

].

Misinterpretation Error

Usually, in the context of practical channel sounders the measured function h(−1/2)(t, τ) = (Υu)(t, τ) is

erroneously interpreted as an estimate of the impulse response h(1/2)(t, τ) rather than of h(−1/2)(t, τ).

This corresponds to a misinterpretation error

e4(t, τ) , h(−1/2)(t, τ) − h(1/2)(t, τ) .

Using the α-invariance results of Subsection 2.3.1, e4(t, τ) can be bounded as

∫τ |e4(t, τ)| dτ

‖SH‖1

≤ 2πm(1,1)H

,‖e4‖2

‖SH‖2

≤ 2πM(1,1)H

. (4.26)

Note that the effect of misinterpretation can easily be avoided by converting h(−1/2)(t, τ) into

h(1/2)(t, τ) via the relation h(1/2)(t, τ) = h(−1/2)(t− τ, τ).

4.3.3 Optimization of PN Sequence Length

We next analyze the dependence of the measurement errors of a PN sounder (i.e., a sounder using

pseudo-noise sequences as transmit signal) on the PN sequence length N for constant measurement

bandwidth. Disregarding aliasing and misinterpretation error and further developing and bounding

(4.23) and (4.24), the following bound can be shown:

|h(−1/2)H

(mTrep, τ) − h(−1/2)H

(mTrep, τ)|‖SH‖1

≤ |e1(mTrep, τ)|‖SH‖1

+|e2(mTrep, τ)|

‖SH‖1

≤ Nπm

(0,1)H

2B+

2

N. (4.27)

It is seen that the bound for the commutation error, Nπm

(0,1)H

2B , increases with increasing N while the

bound for the pulse compression error, 2/N , decreases with increasing N . The total bound on the

right-hand side of (4.27) is minimized by the following value of the PN sequence length N :

N ′ =

√4B

πν(1)H

.

However, for practical implementations N + 1 has to be a power of two so that proper PN sequences

can be used. Thus, our final design rule is to choose a PN sequence length Nopt = 2lopt − 1 where

lopt = round{ld(N ′ + 1)} . (4.28)


0.1 1 10 100 1000−250

−200

−150

−100

−50

0

0.1 1 10 100 1000−250

−200

−150

−100

−50

0

0.1 1 10 100 1000−250

−200

−150

−100

−50

0

(a) (b) (c)

Figure 4.6: Systematic measurement errors (dashed line) and corresponding upper bounds (solid line)

for a synthetic two-path channel (in dB): (a) Maximum integrated commutation error, (b) maximum

integrated pulse-compression error, (c) maximum integrated misinterpretation error. (In this example,

there was no aliasing error.) The horizontal axis shows the channel’s Doppler shift ν1 in Hz.


Sounding of a Twopath Channel. To illustrate the systematic sounding errors described above, we

simulated the sounding of a synthetic two-path channel with carrier frequency 1.8GHz. The baseband

channel’s impulse response is

h(1/2)(t, τ) = a0 δ(τ) + a1 cos(2πν1t) δ(τ − τ1) .

This channel consists of a direct path with constant amplitude a0 = 1 and a second path with delay

τ1 = 2µs and sinusoidally varying amplitude (peak amplitude a1 = 0.4). The Doppler shift ν1 was

varied between 0.1Hz and 1000Hz, corresponding to a velocity ranging from 0.06 km/h to 600 km/h.

We assumed a PN sounder using a PN sequence of length N = 127 and repetition factor K = 1 (i.e.,

Trep = T . With the sampling frequency (double measurement bandwidth B) assumed as 10MHz, the

duration of the PN sequence (= sounding period) is T = 12.7µs.

In this example, the aliasing error is zero since with τ1 = 2µs, ν1 ≤ 1000Hz, T = 12.7µs, and

K = 1 condition (4.25) is satisfied. The other three error components and their upper bounds are

compared for various values of ν1 in Fig. 4.6. The short PN sequence caused the pulse-compression

error to dominate. The maximum integrated pulse-compression error maxm

{1T

∫τ |e2(mT, τ)| dτ

}and

the corresponding upper bound ‖SH‖1ε2 according to (4.24) were calculated as 5.9 · 10−3 and 7.8 ·10−3, respectively, independently of ν1 (see Fig 4.6(b)). The maximum integrated commutation error

maxm

{1T


}and the corresponding upper bound ‖g‖∞ ‖SH‖1‖r‖1ε1 according to (4.23)

are shown in Fig. 4.6(a) as a function of ν1. Similarly, the maximum integrated misinterpretation

error maxm

{1T


}and the corresponding upper bound 2π ‖SH‖1m

(1,1)H

/T according to

(4.26) are shown in Fig. 4.6(c). It is seen that the commutation and misinterpretation errors and the

corresponding upper bounds grow with increasing ν1; however, in this example the misinterpretation

error always stays well below the commutation and pulse-compression errors.

4.4 Multicarrier Communication Systems 187

0.1 1 10 100 1000−60

−50

−40

−30

−20

−10

0.1 1 10 100 1000−60

−50

−40

−30

−20

−10

(a) (b)

Figure 4.7: Total systematic measurement error (dashed line) and corresponding upper bound (solid

line) in dB for a synthetic two-path channel: (a) for PN sequence length N = 127, (b) for PN sequence

length N = 1023. The horizontal axis shows the channel’s Doppler shift ν1 in Hz.

Finally, the total error maxm

{1T

∫τ |h

(−1/2)H

(mT, τ) − h(1/2)H

(mT, τ)| dτ}

and the associated bound

according to (4.22) are shown in Fig. 4.7(a). Comparing with Fig. 4.6, we see that whereas up to about

ν1 = 200Hz the (constant) pulse-compression error dominates, for ν1 > 200Hz the commutation error

dominates. For comparison, Fig. 4.7(b) shows the total error and corresponding bound when the same

two-path channel is sounded with a PN sequence of length N = 1023. It is seen that the value of ν1

where the commutation error starts to dominate has dropped to about 50Hz. This shows that for

different maximum Doppler shifts ν1 different values of N are preferable. The choice of N for a given

νmax is considered next.

Optimization of PN Sequence Length. To illustrate our result regarding optimal PN sequence

length, we sounded the same two-path channel as in the previous example, with ν1 = 60Hz, using

PN sequences of length N = 2l − 1 with l = 5, . . . , 11. Fig. 4.8 shows the maximum magnitude of

the commutation error, maxm,τ

|e1(mT, τ)|, of the pulse-compression error, maxm,τ

|e2(mT, τ)|, and of their

sum, maxm,τ

|e1(mT, τ) + e2(mT, τ)|, as a function of N . It is seen that these errors are best balanced

for N = Nmin = 511 since maxm,τ

|e1(mT, τ) + e2(mT, τ)| is minimal at this point. This agrees with our

theoretical guideline (4.28) which yields lopt = 9 and thus Nopt = 2lopt − 1 = 511 as well.

4.4 Multicarrier Communication Systems

Orthogonal frequency division multiplexing (OFDM) [28, 30, 133, 182, 209, 215] is a multicarrier com-

munication scheme used or proposed for digital audio broadcasting (DAB), digital video broadcasting

(DVB), wireless local area networks (WLANs), and data transmission over digital subscriber lines

(xDSL services, often also referred to as discrete multi-tone (DMT)). Recently, biorthogonal frequency

division multiplexing (BFDM) has been proposed as a generalization of OFDM that is particularly

attractive in time-varying environments [20,21,128]. Subsequently, we will briefly discuss the relevance

of Subsection 2.3.8 (which deals with approximate eigenfunctions and approximate diagonalization)


31 63 127 255 511 1023 2047−70

−60

−50

−40

−30

−20

- N

Figure 4.8: Maximum magnitude (in dB) of the commutation error (dotted line), of the pulse-

compression error (dashed line), and of their sum (solid line) as a function of the PN sequence length

N for a synthetic two-path channel with ν1 = 60Hz.

to pulse-shaping OFDM systems. We note that similar results have been presented for random time-

varying channels in [127].

4.4.1 Pulse-Shaping OFDM and BFDM Systems

Recently, pulse-shaping OFDM and BFDM systems [19–21, 128, 201] have been recognized to offer

improved robustness to time-frequency dispersion caused by time-varying (mobile radio) channels.

Subsequently, we consider an OFDM/BFDM system as shown in Fig. 4.9, with L subcarriers, symbol

duration T , and subcarrier spacing F (chosen such that TF ≥ 1). The baseband transmit signal for

such a system can be written as

x(t) =

∞∑

k=−∞

L−1∑

l=0

ak,l gk,l(t) ,

where ak,l is the transmit symbol associated to the kth symbol period and the lth subcarrier and

gk,l(t) = gk,l(t) = g(t − kT ) ej2πlF (t−kT ) is the corresponding transmit pulse that is obtained by TF

shifting a prototype filter g(t). The receiver determines the symbol estimates according to ak,l =⟨y, γk,l

⟩where γk,l(t) = γ(t − kT ) ej2πlF (t−T ) is the corresponding receive pulse (obtained by TF

shifting the prototype receive filter2 γ(t)) and y(t) = (Hx)(t) is the received signal when transmitting

x(t) over an LTV channel H.3 Transmit and receive pulses are assumed to satisfy the (bi)orthogonality

condition

〈gk,l, γk′,l′〉 = δkk′ δll′ . (4.29)

For H = I, (4.29) implies ak,l = ak,l, i.e., perfect recovery of the transmit symbols.

2Note that γ(t) = g(t) in the case of OFDM.3For simplicity, we here consider a noisefree scenario.

4.4 Multicarrier Communication Systems 189

kT

kT

kT

ak,0

x(t) y(t) γ∗(−t) ej2πF t

γ∗(−t)

H

ak,0

g(t) ej2π(L−1)F t

g(t) ej2πF t

g(t)

γ∗(−t) ej2π(L−1)F t

ak,1

ak,L−1 ak,L−1

ak,1

Figure 4.9: OFDM/BFDM communication over a time-varying channel H.

4.4.2 Approximate Input-Output Relation for OFDM/BFDM Systems

We first note that the received symbols can be written as (cf. also [128])

ak,l =⟨Hx, γk,l

⟩=

∞∑

k′=−∞

L−1∑

l′=0

ak′,l′⟨Hgk′,l′ , γk,l

⟩.

Hence, due to the time and frequency dispersion of the time-varying channel, a given the received

symbol ak,l depends not only on ak,l but also on ak′,l′ with k 6= k′ and l 6= l′. This parasitic dependence

is known as intersymbol interference (ISI) and interchannel interference (ICI) [128].

However, the results of Subsection 2.3.8 imply that for underspread LTV channels and properly

chosen prototype pulses g(t) and γ(t), this ISI/ICI is small. Specifically, Theorem 2.23 (with u(t)

and v(t) replaced by g(t) and γ(t), respectively) directly applies to the BFDM transmission system

considered here. It implies that for a reasonably underspread channel H and well TF-localized transmit

and receive filter prototypes4 g(t) and γ(t), one has

⟨Hgk′,l′ , γk,l

⟩≈ L

(α)H

(kT, lF ) δkk′δll′ , (4.30)

where the approximation error is bounded as (cf. (2.119))∣∣⟨Hgk′,l′ , γk,l

⟩− L

(α)H

(kT, lF ) δkk′δll′∣∣

‖SH‖1

≤ m

(φ

(k−k′,l−l′)g,γ

)H

,

with φ(k,l)g,γ (τ, ν) =

∣∣δk0 δl0 −A(α)γ,g

(τ + kT, ν + lF

)∣∣. The approximation (4.30) further implies

ak,l ≈ L(α)H

(kT, lF ) ak,l . (4.31)

This approximate multiplicative input-output relation for pulse-shaping OFDM and BFDM systems

operating over underspread LTV channels is practically important since it allows the use of simple

methods for channel estimation and equalization [29, 48, 51, 75, 134, 182, 209]. Consequently, a sim-

ilar input-output relation is almost always used in the relevant literature. Note that the results in

Subsection 2.3.9 provide a theoretical justification of this approximate input-output relation.


We next illustrate the approximate input-output relation (4.31) using a practical OFDM transmission

scheme with rectangular transmit and receive pulses g(t) = γ(t) of duration T = 1ms and 64 subcar-4We note that a procedure for optimally matching the pulses g(t) and γ(t) to the channel has been proposed in [128].


8 16 24 32 40 48 56

l

64-50

-40

-30

-20

-10

0

Figure 4.10: Intersymbol and interchannel interference occurring in a conventional OFDM system

with 64 subcarriers. The single symbol ak0,32 was transmitted. ‘—◦—’ shows |ak0,l|2 and ‘- -×- -’

shows |ak0+1,l|2 (in dB relative to |ak0,32|2) as a function of the subcarrier index l. (All other ak,l were

zero.)

riers with subcarrier spacing F = 1kHz. No cyclic prefix [165] was used since it cannot help avoid

interchannel interference in the case of time-varying channels. A single symbol was transmitted at sym-

bol period k = k0 and subcarrier l = 32. The channel was obtained as a single realization of a standard

random WSSUS channel [11] with Jakes Doppler profile (maximum Doppler frequency νmax = 156Hz)

and exponential delay profile (τdecay ≈ 120µs) [176] (note that due to νmaxτdecay = 0.018 this channel

is reasonably underspread). Fig. 4.10 shows the power of the received symbols ak,l for the symbol

periods k0 and k0 + 1 as a function of the subcarrier index l. Since the channel was causal and its

memory was shorter than the symbol period T , all received symbols ak,l for k < k0 and k > k0+1 were

exactly zero. On the other hand, it is seen that the received symbols ak0,l, l 6= 32 as well as ak0+1,l

are not exactly zero which means that there is some ISI/ICI. However, the power of these symbols is

more than 20 dB weaker than that of ak0,32, thereby confirming the approximation (4.31).

4.5 Analysis of Car Engine Signals

In this section, we consider time-varying spectral analysis of pressure and vibration data measured in

a car engine.5 A TF analysis of this kind of data has previously been considered in [17, 26, 27, 113,

143,146,148,155,181].

Our data consisted of vibration signals obtained from acceleration sensors mounted on the engine

housing and corresponding pressure signals obtained from pressure sensors mounted inside the cylinder

of a BMW engine. Several data sets were recorded at different engine speeds. Each data set contains

the vibration and pressure signals corresponding to several knocking combustion cycles (for more

details on knock and its detection see [17, 26, 27, 113, 143, 146]). The load and the engine speed were

5These data have kindly been made available to us by J. F. Bohme, D. Konig, S. Carstens-Behrens, and M. Wagner

(courtesy of Aral-Forschung, Bochum).

4.5 Analysis of Car Engine Signals 191

i speed Li Ni

2 2000 360 126

3 3000 248 343

4 4000 186 199

Table 4.1: Number Ni and length Li of available combustion signals at various engine speeds.

kept constant during each measurement. The measured signal segments corresponding to different

combustion cycles can be viewed as independent realizations of a nonstationary discrete-time and

finite-length random process. Thus, the data constitutes an ensemble (i.e., multiple realizations) on

which the subsequent spectral analysis can be based.

Let us denote the kth measured pressure signal at engine speed i · 1000 rpm, i = 2, 3, 4, by x(i)k [n]

and similarly for the vibration signals y(i)k [n]. Here, n = 1, . . . , Li and k = 1, . . . ,Ni with Li the

signal length and Ni the number of combustions measured at speed i · 1000 rpm (see Table 4.1). Note

that the signal length depends on engine speed since the sampling rate was kept constant while the

duration of a combustion decreases with engine speed.

4.5.1 Time-Varying Spectral Analysis

We first discuss time-varying spectral analysis of the pressure and vibration signals using the GWVS

and GES. For each engine speed, sample (i.e., estimated) correlation operators were computed from

the available data according to6

R(i)x =

1

Ni

Ni∑

k=1

x(i)k ⊗ x

(i)∗k , R(i)

y =1

Ni

Ni∑

k=1

y(i)k ⊗ y

(i)∗k .

Note that N2 = 126 ≤ L2 = 360 so that R(2)x and R

(2)y were singular. Furthermore, we also determined

the positive semi-definite square roots H(i)x and H

(i)y of R

(i)x and R

(i)y , respectively. Finally, estimates

of the GWVS and GES with α = 0 and α = 1/2 were computed according to (B.45) and (B.49),

W(α)

x(i) (t, f) = L(α)

bR(i)x

(t, f) , G(α)

x(i)(t, f) =∣∣L(α)

bH(i)x

(t, f)∣∣2 ,

W(α)

y(i) (t, f) = L(α)

bR(i)y

(t, f) , G(α)

y(i)(t, f) =∣∣L(α)

bH(i)y

(t, f)∣∣2 .

The resulting spectrum estimates are shown in Figs. 4.11 and 4.12.

It can be seen from Fig. 4.11 that all spectra succeed in displaying the time-varying and nonsta-

tionary features of the pressure signals. In particular, it can be recognized that at all engine speeds the

pressure signals consist of several resonances with decreasing resonance frequencies. These results are

consistent with physical considerations involving cylinder geometry, coupling of resonance frequencies,

and decreasing cylinder temperature [17, 113]. Fig. 4.12 shows that the vibration signals feature a

6Note that all signals and operators involved in this example are discrete-time, and thus the correaltion operators bR(i)x

and bR(i)x become matrices of size Li ×Li. Nevertheless, we continue using the continuous-time notation for convenience.


0 30 60 90

25

20

15

10

5

00 30 60 90

25

20

15

10

5

00 30 60 90

25

20

15

10

5

00 30 60 90

25

20

15

10

5

0

0 30 60 90

25

20

15

10

5

00 30 60 90

25

20

15

10

5

00 30 60 90

25

20

15

10

5

00 30 60 90

25

20

15

10

5

0

0 30 60 90

25

20

15

10

5

00 30 60 90

25

20

15

10

5

00 30 60 90

25

20

15

10

5

00 30 60 90

25

20

15

10

5

0

Wigner-Ville spectrum Rihaczek spectrum Weyl spectrum evolutionary spectrum

4000

rpm

3000

rpm

2000

rpm

Figure 4.11: Time-varying spectra of cylinder pressure signals x(i)[n]. First row: 2000 rpm, second

row: 3000 rpm, third row: 4000 rpm; first column: Wigner-Ville spectrum, second column: real part

of Rihaczek spectrum, third column: Weyl spectrum, fourth column: evolutionary spectrum (equal to

transitory evolutionary spectrum). Horizontal axis: crank angle (in degrees, proportional to time),

vertical axis: frequency (in kHz).

similar behavior although the spectra are much more affected by engine noise (especially at higher

engine speeds) and by dispersion effects caused by the the engine housing.

With regard to the different spectra used, it is seen that the Wigner-Ville spectrum and Weyl

spectrum (i.e., GWVS and GES with α = 0) are preferable to the Rihaczek spectrum and evolutionary

spectrum (i.e., GWVS and GES with α = 1/2) since they better display the decreasing resonance

frequencies. Specifically, they feature a better concentration of the resonance components along the

decreasing resonance frequencies. This can be attributed to the fact that due to its metaplectic

covariance, the Weyl symbol (i.e., the GWS with α = 0, which underlies the definition of Wigner-Ville

spectrum and Weyl spectrum), is particularly suited for structures with oblique locations in the TF

plane. We finally note using (slightly) smoothed type I and type II spectra (see Sections 3.3 and 3.4)

tends to yield further improvements regarding the readability of the resulting spectra (in particular

for the noisier vibration signals).


0 30 60 90

25

20

15

10

5

00 30 60 90

25

20

15

10

5

00 30 60 90

25

20

15

10

5

00 30 60 90

25

20

15

10

5

0

0 30 60 90

25

20

15

10

5

00 30 60 90

25

20

15

10

5

00 30 60 90

25

20

15

10

5

00 30 60 90

25

20

15

10

5

0

0 30 60 90

25

20

15

10

5

00 30 60 90

25

20

15

10

5

00 30 60 90

25

20

15

10

5

00 30 60 90

25

20

15

10

5

0

Wigner-Ville spectrum Rihaczek spectrum Weyl spectrum evolutionary spectrum4000

rpm

3000

rpm

2000

rpm

Figure 4.12: Time-varying spectra of engine vibration signals y(i)[n]. First row: 2000 rpm, second

row: 3000 rpm; third row: 4000 rpm; first column: Wigner-Ville spectrum, second column: real part

of Rihaczek spectrum, third column: Weyl spectrum, fourth column: evolutionary spectrum (equal to

transitory evolutionary spectrum). Horizontal axis: crank angle (in degrees, proportional to time),

vertical axis: frequency (in kHz).

4.5.2 TF Coherence Analysis

We next analyze the TF coherence of pressure and vibration signals. The goal is to see whether

corresponding pressure and vibration signals are linearly related (as assumed e.g. in [27]), i.e., whether

the relation between pressure signal and vibration signal can be modelled by an LTV system (such an

approach was taken in [17,113]).

The magnitudes of the estimated TF coherence functions (cf. (3.116))

Γx(i),y(i)(t, f) =ˆCx(i),y(i)(t, f)

√ˆCx(i)(t, f) ˆCy(i)(t, f)

,

are shown in Fig. 4.13 for engine speeds of 2000, 3000, and 4000 rpm. Here, the spectrum estimates


0 30 60 90

25

20

15

10

5

0 0

0.2

0.4

0.6

0.8

1

0 30 60 90

25

20

15

10

5

0 0

0.2

0.4

0.6

0.8

1

0 30 60 90

25

20

15

10

5

0 0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

(a) (b) (c)

Figure 4.13: Magnitude of estimated TF coherence function of corresponding pressure and vibration

signals for (a) 2000 rpm, (b) 3000 rpm, (c) 4000 rpm. The upper plots show gray scale representations

where the horizontal axis is crank angle (in degrees, proportional to time) and the vertical axis is

frequency (in kHz). The lower plots show cuts in the frequency direction taken at crank angle 30

degree (indicated by the dashed line in the gray scale plots).

ˆC·(t, f) were obtained by averaging rank two multiwindow spectrograms (see Subsection B.2.3), i.e.,

ˆCx(i),y(i)(t, f) =1

Ni

Ni∑

k=1

(1

2SPEC

(g1)

x(i)k ,y

(i)k

(t, f) +1

2SPEC

(g2)

x(i)k ,y

(i)k

(t, f))

and similarly for ˆCx(i)(t, f) and ˆCy(i)(t, f). The windows g1(t) and g2(t) were chosen as chirped

versions of the first and second Hermite function with window length and chirp rate chosen to match

the duration and frequency decay of the resonances in the combustion signals. It is seen that in those

TF regions where the resonances are localized,∣∣Γx(i),y(i)(t, f)

∣∣ is significantly larger than zero. In

particular, at all engine speeds the estimated TF coherence function in the region corresponding to

the first resonance frequency is ≈ 0.9, which clearly indicates a linear relationship between pressure

and vibration signals. The TF coherence of the higher resonances is smaller but still suggests a linear

relationship of pressure and vibration, though apparently contaminated by measurement noise and

interference from extraneous sources (note that coherence drops with decreasing SNR).

4.5.3 Subspace Identification

Recently, matched subspace detectors [187, 188] have been proposed for the detection of knock in car

engines [146]. The prior knowledge required to design such detectors consists of the subspace X in

which the signals to be detected are supposed to live. A conventional method for estimating the


subspace X (whose dimension p is assumed known) from N observations yi(t) is maximum-likelihood

(ML) subspace identification [187]. This method consists of the following steps:

i) Compute the sample correlation

Ry =1

N

N∑

j=1

yi⊗y∗i .

ii) Solve the eigenproblem for the sample correlation Ry, i.e., compute the real numbers λk and

the orthonormal functions uk(t) satisfying

(Ry uk

)(t) = λk uk(t) . (4.32)

It is assumed that the eigenvalues (and corresponding eigenfunctions) are sorted in order of de-

creasing magnitude.

iii) Finally, the estimated subspace X is obtained as

X = span{u1(t), . . . , up(t)} .

and the corresponding estimated orthogonal projection is given by PX =∑p

k=1 uk ⊗ u∗k.

Unfortunately, the solution of the eigenproblem (4.32) is computationally expensive and sensitive

to noise. However, if y(t) is an underspread process (so that Ry and, for N large enough, also Ry are

underspread operators), the results of Subsection 2.3.8 can be applied. There, it was shown that TF

shifted versions skT,lF (t) of a well-localized prototype function s(t) are approximate eigenfunctions of

underspread operators and the corresponding GWS values L(α)H

(kT, lF ) are the associated approximate

eigenvalues (cf. (2.111)),

(HskT,lF )(t) ≈ L(α)H

(kT, lF ) skT,lF (t) .

For TF = 1, each approximate eigenfunction skT,lF (t) approximately covers a separate TF region of

area 1. Hence, the space spanned by p approximate eigenfunctions skT,lF (t) approximately corresponds

to a TF region of area p. Picking the TF region with the largest GWS values thus approximately

corresponds to picking the subspace spanned by the eigenfunctions associated with the largest eigen-

values. (A more comprehensive treatment of the correspondence between subspaces and TF regions

can be found in [81]).

Recalling that the GWS of Ry is the GWVS of W(α)

y (t, f), and using α = 0, the above considera-

tions suggest the following TF version of ML subspace identification [146]:

i) Compute the Wigner-Ville spectrum estimate

W(0)

y (t, f) = L(α)

Ry(t, f) =

1

N

N∑

j=1

W (0)yi

(t, f) .

For underspread processes or low SNR, an additional TF smoothing is advantageous [61,129,140].

ii) Determine an estimate of the TF region R by thresholding W(0)

y (t, f),

R ={

(t, f) : W(0)

y (t, f) ≥ ε},

with ε chosen such that the area of R equals p, i.e.,∫t

∫f IR(t, f) dt df = p


0 30 60 90

25

20

15

10

5

00 30 60 90

25

20

15

10

5

00 30 60 90

25

20

15

10

5

0

0 30 60 90

25

20

15

10

5

00 30 60 90

25

20

15

10

5

00 30 60 90

25

20

15

10

5

0

(a) (b) (c)

Figure 4.14: Illustration of ML subspace identification and its TF counterpart for engine vibration

signals at (a) 2000 rpm, (b) 3000 rpm, and (c) 4000 rpm. Upper row: Weyl symbols L(0)bPX

(t, f)

of orthogonal projection operators PX estimated using ML subspace identification. Lower row: TF

regions R estimated using the TF technique for subspace identification. Horizontal axis: crank angle

(in degrees, proportional to time); vertical axis: frequency (in kHz).

iii) Compute the kernel of the “approximate orthogonal projection operator” P via an inverse Weyl

transform (cf. (B.18)) of the indicator function of R,

p(t, t′) =

∫

fIR

(t+ t′

2, f)ej2πf(t−t′) df .

(Thus, the Weyl symbol of P equals the indicator function IR(t, f).) Note that P is self-adjoint but

not idempotent. If idempotency is indispensable, a least squares approach can be used to determine

the orthogonal projection operator that best approximates P [81].

Numerical Simulations. Examples illustrating ML subspace identification and its TF analogue

for the vibration signals y(i)k [n] discussed further above are shown in Fig. 4.14. Here, all Ni available

signals corresponding to a given engine speed were used to determine the ML estimate of the orthog-

onal projection operator PX as well as the estimated TF region R according to the two procedures

outlined above. It is seen that the TF version of ML subspace identification yields clearer pictures

of the dominant components of the vibrations signals. The corresponding TF designed approximate

orthogonal projection operators P where also observed to yield better performance than the estimated

orthogonal projection operators PX when used for the design of TF subspace detectors [146].

5

Conclusions

“Whoever in the pursuit of science seeks after immediate practical utility mayrest assured that he seeks in vain.” Hermann von Helmholtz

IN this concluding chapter, we first give a concise summary of the novel contributions of this thesis.

In particular, we review underspread linear systems, time-frequency transfer function approxima-

tions, underspread random processes, time-varying power spectra, and their applications. In addition,

we outline several open problems that extend the results of this thesis into various directions and may

serve as suggestions for future research.

197

198 Chapter 5. Conclusions

5.1 Summary of Novel Contributions

The following summary of novel contributions reviews the main theoretical results and practical appli-

cations discussed in the previous chapters. For each subject, a reference to the corresponding section(s)

is provided.

Generalized Concept of Underspread LTV Systems −→ Sections 2.1 and 2.2

The concept of underspread systems is of central importance for most of the results derived in this

thesis. Our contributions here are threefold:

• First, a previous definition of (jointly) underspread linear time-varying (LTV) systems [118–120,

127] that is based on the assumption of compactly supported generalized spreading function

(GSF) was generalized in order to accommodate oblique orientations of compact GSF support

regions. We also introduced novel parameters for characterizing the amount of time-frequency

(TF) shifts introduced by such systems and investigated their behavior when the system is

subjected to various transformations.

• Second, we extended the underspread concept to LTV systems whose GSF does not have compact

support but features rapid decay. This extension was based on the introduction of weighted

integrals and moments of the GSF as measures of the amount of TF shifts introduced by a

system. Again, the effect of various system transformations on these novel parameters was

studied.

• Finally, generalized Chebyshev inequalities were used to derive bounds on the error made by ap-

proximating an arbitrary LTV system by a system with compactly supported GSF. In particular,

this allows to relate the two different concepts of underspread systems (compact support/rapid

decay).

Time-Frequency Transfer Function Approximations −→ Section 2.3

A major part of this thesis was concerned with the development of a “TF transfer function cal-

culus” for LTV systems that is based on the generalized Weyl symbol (GWS). We provided various

approximations that are valid in the case of (jointly) underspread systems and allow an interpretation

of the GWS similar to the transfer function (frequency response) of linear time-invariant systems.

In particular, we proved that for underspread systems the following GWS properties are valid in an

approximate sense:

• The composition of jointly underspread systems approximately corresponds to a multiplication

of the corresponding GWSs.

• TF translates of a well TF localized function are approximate eigenfunctions of underspread

LTV systems, with the associated GWS values being the approximate eigenvalues.

5.1 Summary of Novel Contributions 199

• The input-output relation of underspread LTV systems can approximately be formulated in

terms of a multiplication of the input signal’s short-time Fourier transform by the GWS.

• The maximum gain of an underspread LTV system approximately equals the supremum of the

GWS.

• Underspread LTV systems are approximately normal and they approximately commute.

• The GWS of positive semi-definite systems is approximately nonnegative.

• In a certain sense, operator inversion for underspread LTV systems can approximately be re-

placed by pointwise inversion of the GWS.

As a mathematical underpinning, each approximation was accompanied by an upper bound on the

associated approximation error that is formulated in terms of in terms of weighted GSF integrals and

moments.

Time-Frequency Correlation Analysis and Underspread

Nonstationary Random Processes −→ Section 3.1

TF correlations play a similar role for nonstationary random processes as TF displacements do for

LTV systems. Our contributions regarding TF correlations are as follows:

• We presented intuitive methods for the TF correlation analysis of nonstationary random process

that involves TF correlation functions and the generalized expected ambiguity function (GEAF);

they show some similarity to the analysis of TF displacements of LTV systems.

• We reviewed and generalized an existing definition of underspread random processes that is

based on the assumption of compactly supported GEAF.

• Furthermore, we provided a novel, extended concept of underspread processes that is based on

weighted integrals and moments of the GEAF and does not require the GEAF to have compact

support.

• We also considered innovations system representations of random processes and related the TF

correlations of a process to the TF displacements of its innovations systems.

Elementary Time-Varying Power Spectra −→ Sections 3.2, 3.6, and 3.7

The generalized Wigner-Ville spectrum (GWVS) and the generalized evolutionary spectrum (GES)

are well-established elementary classes of time-varying power spectra. We provided the following novel

results regarding the GWSV and GES:

• We showed that for an underspread process the spectra within both classes are smooth, (ap-

proximately) real-valued and (approximately) nonnegative. In contrast, for overspread (i.e., not


underspread) processes we demonstrated that the GWVS and GES contain “statistical cross-

terms” that are indicative of TF correlations. We furthermore derived uncertainty relations for

both the GWVS and GES that link the maximum TF concentration of these spectra to the

effective rank of the correlation operator of the underlying process.

• We provided approximations that relate the GWVS (GES) of the nonstationary output process

of an underspread LTV system to the GWVS (GES) of an underspread nonstationary input

process. Finally, we provided an approximate Karhunen-Loeve (KL) expansion in which the

GWVS acts as an approximate KL eigenvalue distribution and two biorthogonal bases obtained

by TF shifting two well TF localized prototype functions act as approximate KL eigenfunctions.

Type I and Type II Spectra −→ Sections 3.3–3.5

Another major part of this thesis is dedicated to two classes of generalized time-varying power

spectra: type I spectra and type II spectra, which are obtained by incorporating a TF domain con-

volution in the GWVS and GES, respectively. Whereas type I spectra have already been considered

in the literature, our definition of type II spectra is new. We showed that in the case of underspread

processes, any type I and type II spectrum

• satisfies desirable mathematical properties (at least approximately);

• is approximately equivalent to any other (type I or type II) spectrum and approximately con-

stitutes a complete second-order statistic;

• describes the mean TF energy distribution of the process in a satisfactory way.

On the other hand, in the case of an overspread process different (type I and/or type II) spectra may

differ dramatically. Non-smoothed spectra are complete second-order statistics but in the overspread

case they contain statistical cross-terms. While indicating TF correlations inherent in overspread

processes, they tend to obscure the process’ mean TF energy distribution. Smoothed spectra, on the

oher hand, are not complete second-order statistics (they fail to indicate TF correlations) but feature

attenuated cross-terms and thereby better represent the process’ mean TF energy distribution.

Time-Frequency Coherence Functions −→ Section 3.8

We introduced a coherence operator and TF coherence functions for the coherence analysis of

two nonstationary random processes. We proved that the TF coherence functions constitute an ap-

proximate TF formulation of the coherence operator. Furthermore, both the coherence operator an

the coherence function possess similar or analogous properties as the ordinary coherence function

of stationary processes. Compared to the coherence operator, the TF coherence functions have the

advantage of being more intuitive and computationally less intensive.

5.2 Open Problems for Future Research 201

Applications −→ Sections 4.1–4.5

An important part of this thesis was concerned with the application of our theoretical results to

practically important problems in the areas of statistical signal processing and communications.

• We showed that our results yield an approximate TF formulation and design of time-varying

Wiener filters for (jointly) underspread processes. The resulting “TF Wiener filters” are com-

putationally efficient and feature nearly optimal performance.

• Similarly, we derived an efficient approximate TF formulation and design of optimal detectors

for (jointly) underspread random processes.

• We used several of the bounds derived previously to develop upper bounds on the systematic

measurement errors of correlative mobile radio channel sounders.

• The eigenfunction approximation for underspread LTV systems was used to quantify the in-

tersymbol and interchannel interference occuring in orthogonal frequency division multiplexing

(OFDM) communications systems.

• Finally, we considered the application of time-varying spectral analysis and TF coherence anal-

ysis to car engine signals.

5.2 Open Problems for Future Research

The following discussion suggests some topics for future research. Most of these topics concern the

extension of the approach taken in this thesis in various directions.

• Our discussion of approximate GWS-based inversion of LTV systems for the solution of operator

equations pf the types H1GH2 = H3 and GH2 = H3 was restricted to displacement-limited

(DL) systems (see Subsections 2.3.6 and 2.3.7). Up to now, we were not able to derive similar

results for the non-DL case. While simulation results indicate that approximate GWS based

inversion of non-DL underspread operators is possible, a theoretical analysis and verification of

this empirical observation is still an open problem.

• The main results of this thesis concern the development of various approximations that establish

a TF calculus for underspread LTV systems and underspread nonstationary random processes.

In general, these approximations are not valid for overspread systems and processes. However, we

feel that at least part of our results can be extended to specific subclasses of overspread scenarios.

To be specific, we note that an arbitrary LTV system H can be written as a superposition of

TF shifted underspread systems Hk,l, i.e.,

H =∑

k

∑

l

S(α)kτ0,lν0

Hk,lS(α)+kτ0,lν0

.


Here, the individual subsystems Hk,l are defined by S(α)Hk,l

(τ, ν) = S(α)H

(τ − kτ0, ν − lν0)Ψ(τ, ν)

where Ψ(τ, ν) is a function that is concentrated in a region [−τ0/2, τ0/2]×[−ν0/2, ν0/2] about the

origin (with τ0ν0 ≪ 1 in order that Hk,l is underspread) and satisfies∑

k

∑l Ψ(τ−kτ0, ν− lν0) =

1. If H is an overspread system with GSF effectively contained in a region [−τH, τH]× [−νH, νH]

with τHνH ≫ 1, there will effectively be K = (τHνH)/(τ0ν0) subsystems Hk,l. While our

approximate TF transfer function then applies to the individual subsystems Hk,l with errors in

the order of τ0ν0, the total error for H will generally be in the order of Kτ0ν0 = τHνH so that

nothing is gained. However, if only K ′ ≪ K subsystems Hk,l are effectively nonzero, tolerable

approximation errors in the order of K ′τ0ν0 = K ′

K τHνH can be expected. A detailed analysis of

the usefulness of such an approach is left for future work.

• This thesis was restricted to deterministic LTV systems. In some applications (e.g., mobile com-

munications), the underlying systems (channels) are often modelled as being random in addition

to being time-varying. Some TF transfer function approximations for underspread random LTV

systems satisfying the so-called wide-sense stationary uncorrelated scattering (WSSUS) assump-

tion have been provided in [91, 127]. However, a systematic study of such approximations for

underspread random LTV systems is still lacking. Furthermore, since the WSSUS assumption

is satisfied only approximately by real channels, non-WSSUS random LTV systems also are

practically relevant and thus should be studied in more detail.

• Another possible extension of our TF calculus for underspread systems and processes concerns

multi-input/multi-output (MIMO) LTV systems and multivariate nonstationary random pro-

cesses (i.e., vector processes). Recently, these systems and processes have gained considerable

interest in connection with communications systems using antenna arrays. MIMO LTV systems

can be viewed as matrices of operators, i.e.,

y1(t)...

yN (t)

=

H11 · · · H1M

.... . .

...

HN1 · · · HNM

·

x1(t)...

xM (t)

,

with M andN the dimensionality at the input and output side, respectively. In a similar manner,

the correlations of multivariate processes can be described by matrices of correlation operators,

i.e.,

E

x1(t)...

xN (t)

·(y∗1(t

′) · · · y∗M (t′))

corresponds to

R11 · · · R1M

.... . .

...

RN1 · · · RNM

,

where Rkl = Rxk,yl. The formulation of an underspread concept and the development of an

associated approximate TF calculus poses an interesting problem for future research. In fact, it

is known from the case of LTI systems that MIMO systems can feature properties very different

from single-input/single-ouput systems [105].

5.2 Open Problems for Future Research 203

• Recently, generalized TF symbols for LTV systems (operators) and nonstationary random pro-

cesses have been introduced which are covariant to TF displacements other than TF shifts [96,97].

The basic theory of these generalized symbols and associated generalized spreading functions is

reasonably developed and some applications have also been investigated. However, a modifica-

tion of the underspread concept that is adapted to these new symbol classes and the formulation

of an associated approximate symbolic calculus have not been considered until now. (We note

that a symbolic calculus for affine symbols in a mathematical/quantum-mechanical context has

been presented in [15].)

Instead of developing an underspread theory for various generalized TF symbols separately, it

may be possible to take advantage of the generalized covariance theory for linear and quadratic

TF signal representations [82,83,93,184]. The steps to be taken here are are: i) the development

of an operator symbol that is covariant to an arbitrary (but fixed) TF displacement operator

(this can be based on the eigenvalue or singular value decomposition of operators as in (B.28));

ii) the derivation of the associated generalization of the GSF; iii) the formulation of an associated

underspread property for systems and processes with regard to the underlying TF displacement

operator; iv) the establishment of an approximate TF symbol calculus for these generalized

underspread systems and processes.

• While an LTV system or a nonstationary random process might not satisfy the underspread

assumption for all times and/or frequencies, its restriction to certain TF regions might well

be reasonably underspread. This observation motivates the concept of “regionally underspread”

systems and processes. Such an approach has recently been introduced and studied in [32], where

some approximate regional TF transfer function approximations (essentially based on [144]) are

also provided. However, there are still some open questions regarding regionally underspread

systems and processes that might be studied in future work.

• Finally, we suggest to consider further applications of our TF calculus of underspread systems

and processes. We feel that our results may be particularly useful in the areas of communications

over LTV channels (with potential applications such as channel estimation, channel equalization,

interference cancellation, and diversity combining) and nonstationary statistical signal processing

(with potential applications such as nonstationary prediction and transform coding using TF

signal expansions). Furthermore, nonstationary statistical modelling of time-varying channels

involves aspects of both linear time-varying systems and nonstationary random processes.

Appendices

205

A

Linear Operator Theory

“The theory of [. . .] operators [. . .] has attracted the ever increasing attention of mathematicians andphysicists, and sometimes of engineers also.” Israel C. Gohberg and Mark G. Krein

LINEAR operators (linear systems) play a fundamental role in this thesis and generally in many

areas of engineering. This is due to the fact that they are a sufficiently general concept to yield

satisfactory and tractable models for a wide variety of physical phenomena. Hence, in this appendix

we present a self-contained discussion of linear operator theory as far as is necessary for this thesis. We

consider norms of linear operators, representations of linear operators via kernel functions, eigenvalue

and singular value decompositions, and important special types of operators. Excellent and far more

comprehensive treatments of linear operator theory can be found in [69,158].

207

208 Appendix A. Linear Operator Theory

A.1 Basic Facts about Linear Operators

Operators are abstract mathematical objects which describe the interrelation between an input quan-

tity x and an output quantity y which are elements of two linear spaces X and Y, respectively. This

interrelation or transformation can symbolically be written as y = Hx where H denotes the opera-

tor and Hx denotes the result of H acting on x (the “output” associated to the “input” x). Linear

operators (systems) are a special type of operators which satisfy the following two requirements:

• Homogeneity: H(cx) = cHx for all complex numbers c and for all x ∈ X

• Additivity: H(x1 + x2) = Hx1 + Hx2 for all x1 ∈ X and x2 ∈ X .

The above two conditions are satisfied if and only if

H

(N∑

k=1

ckxk

)=

N∑

k=1

ckHxk ,

which is well-known as superposition principle.

In almost all cases we consider, the linear spaces X and Y are Hilbert spaces, i.e., complete

linear spaces equipped with an inner product denoted by1 〈x1, x2〉. In particular, in most cases

X and Y are equal to the space of square-integrable functions L2(R) with the usual inner product

〈x1, x2〉 =∫t x1(t)x

∗2(t) dt. Using these inner products, the adjoint operator H+ is defined by

〈Hx, y〉 =⟨x,H+y

⟩, for all x ∈ X , y ∈ Y .

The adjoint can be shown to satisfy (H+)+ = H. Operators which are equal to their adjoint, i.e.

H = H+, are called self-adjoint or Hermitian.

Note that by the relation ‖x‖ =√〈x, x〉 an inner product always induces a norm. Based on

this norm, we can define the important subclass of bounded linear operators, i.e. those satisfying

‖Hx‖ ≤M ‖x‖ with finite M . Bounded linear operators constitute themselves a normed linear space,

with the operator norm defined as

‖H‖O , sup‖x‖=1

‖Hx‖ = sup‖x‖6=0

‖Hx‖‖x‖ . (A.1)

Note that ‖H+‖O = ‖H‖O. Further subclasses of linear operators can be obtained by imposing

additional constraints. Of specific interest to us are Hilbert-Schmidt (HS) operators which satisfy

‖H‖2 ,

√∑k‖Hxk‖2 <∞ ,

where {xk} is an arbitrary orthonormal basis of X and ‖H‖2 denotes the Hilbert-Schmidt (HS) norm.

Observe that ‖H+‖2 = ‖H‖2. We note that ‖H‖O ≤ ‖H‖2 which in particular implies that any HS

1 Note that the inner products in X and Y are possibly different. Nevertheless, we do not use different symbols since

in most cases it will be clear from the context to which space we refer.

A.2 Kernel Representation of Operators 209

operator is bounded. HS operators on a particular space X can themselves be viewed as elements of

a Hilbert space with an inner product defined as

〈H,G〉 ,∑

k

〈Hxk,Gxk〉 .

The trace of an operator is defined by

Tr{H} ,∑

k

〈Hxk, xk〉 ,

which by comparison shows that 〈H,G〉 = Tr{G+H} = Tr{HG+} and furthermore ‖H‖22 =

Tr{H+H} = Tr{HH+}. Operators satisfying

∑

k

| 〈Hxk, xk〉 | <∞ (A.2)

are referred to as trace class (or nuclear) operators [69] and form a subclass of HS operators.

A further fundamental concept is the bilinear form of an operator given by the inner product

QH(x, y) , 〈Hx, y〉 ,

which in the case of y = x is referred to as quadratic form. Note that QH(x, y) = Q∗H+(y, x) which

implies that the quadratic form of a self-adjoint operator is always real-valued. If furthermoreQH(x, x)

is positive (non-negative) for all x, the operator is referred to as being positive definite (positive semi-

definite).

A.2 Kernel Representation of Operators

It is often useful to view the action of a linear operator as an integral of x(t) with kernel function

(impulse response) h(t, t′), i.e.

(Hx)(t) =

∫

t′h(t, t′)x(t′) dt′ (A.3)

(note that for HS operators such a representation is guaranteed to exist with h(t, t′) ∈ L2(R2)). The

kernel of the adjoint operator H+ is given by h∗(t′, t) and the kernel of the operator Hc = H2H1

composed of H1 and H2 reads

hc(t, t′) =

∫

t′′h2(t, t

′′)h1(t′′, t′) dt′′ .

The trace, HS norm, inner product, and bilinear form can be reformulated in terms of the kernel as

Tr{H} =

∫

th(t, t) dt ,

‖H‖22 =

∫

t

∫

t′|h(t, t′)|2 dt dt′ ,

〈H,G〉 =

∫

t

∫

t′h(t, t′) g∗(t, t′) dt dt′ ,


QH(x, y) =

∫

t

∫

t′h(t, t′)x(t′) y∗(t) dt dt′ .

We note that there exists a similar kernel representation in the frequency domain, which is referred

to as bi-frequency function [214] and is defined as

BH(f, f ′) =

∫

t

∫

t′h(t, t′) e−j2π(ft−f ′t′) dt dt′ .

The bi-frequency function maps the Fourier transformX(f) of the input signal to the Fourier transform

Y (f) of the output signal y(t) = (Hx)(t),

Y (f) =

∫

f ′

BH(f, f ′)X(f ′) df ′ .

A.3 Eigenvalue Decomposition and Singular Value Decomposition

Operators satisfying HH+ = H+H (which necessarily requires X = Y) are called normal . Normal HS

operators possess an eigenexpansion or eigenvalue decomposition (EVD) in the sense that they can be

written as2

H =∑

k

λk uk⊗u∗k , h(t, t′) =∑

k

λk uk(t)u∗k(t

′) , (A.4)

with generally complex-valued eigenvalues λk and orthonormal eigenfunctions uk(t). This in particular

implies that H leaves its eigenfunctions unchanged besides a multiplication by a scalar, i.e.,

(Huk)(t) = λk uk(t) .

Inserting (A.4) into (A.3) yields

(Hx)(t) =

∞∑

k=0

λk〈x, uk〉uk(t),

which is a representation of the system output as superposition of the eigenfunctions uk(t) weighted

by the eigenvalues λk and the inner product of the input signal x(t) with the eigenfunctions uk(t). In

the case of self-adjoint operators it can be shown that the eigenvalues are real-valued. If in addition

the operator is positive definite there is furthermore λk > 0. We further note that a fundamental

result of operator theory states that two operators commute (i.e., their order can be interchanged,

HG = GH) if and only if they have identical sets of eigenfunctions. In that case the eigenvalues of

the operator product are given by λ(HG)k = λ

(GH)k = λ

(G)k λ

(H)k .

In the case of non-normal HS operators, the eigenvalue decomposition is replaced by the singular

value decomposition (SVD)

H =∑

k

σk uk⊗ v∗k , h(t, t′) =∑

k

σk uk(t) v∗k(t′), (A.5)

2 Here, x ⊗ y∗ denotes a rank one operator with kernel x(t)y∗(t′).

A.4 Special Types of Linear Operators 211

with real, nonnegative singular values σk ≥ 0 and {uk(t)} and {vk(t)} being two sets of orthonormal

functions referred to as left and right singular functions, respectively. The singular values and left/right

singular functions can be found by solving two eigenproblems for the operators HH+ and H+H,

respectively:

(HH+uk)(t) = σ2k uk(t) , (H+Hvk)(t) = σ2

k vk(t) .

Inserting (A.5) into (A.3) yields

(Hx)(t) =

∞∑

k=0

σk〈x, vk〉uk(t),

which is a representation of the system output as superposition of the left singular functions uk(t)

weighted by the singular values σk and the inner product of the input signal x(t) with the right singular

functions vk(t). The operator thus projects the input x(t) onto the signal space spanned by {vk(t)}and uses the resulting coefficients to compose the output as superposition of the signals {uk(t)}.

Sometimes, the following expressions for the trace, operator norm, HS norm, and quadratic form

of H in terms of the eigenvalues (singular values) and eigenfunctions (singular functions) are useful:

Tr{H} =∑

k

λk , Tr{H} =∑

k

σk ,

‖H‖O = supk

{|λk|} , ‖H‖O = supk

{|σk|} ,

‖H‖22 =

∑

k

|λk|2 , ‖H‖22 =

∑

k

|σk|2 ,

QH(x, y) =∑

k

λk 〈x, uk〉〈y, uk〉∗ , QH(x, y) =∑

k

σk 〈x, vk〉〈y, uk〉∗ .

A.4 Special Types of Linear Operators

We conclude this brief dicussion of linear operator theory with a survey of some important special

types of linear operators.

Normal Operators. Normal operators are defined by the equation H+H = HH+ and possess

an eigenexpansion (see (A.4)).

Self-Adjoint Operators. A particularly important special case of normal operators is given by

self-adjoint or Hermitian operators, defined by H = H+. It is straightforward to show that their

eigenvalues λk as well as their quadratic form QH(x, x) are real-valued.

Positive Definite Operators. Positive definite (positive semi-definite) operators are special

self-adjoint operators characterized by having a positive (nonnegative) quadratic form QH(x, x) for

all x 6= 0. We write H > 0 (H ≥ 0). All eigenvalues of a positive definite (semi-definite) operator

are positive (nonnegative). Note that operators of the form HH+ and H+H are always positive semi-

definite. The most important positive semi-definite operators in this thesis are correlation operators

R of random processes whose kernel equals the correlation function r(t, t′).


The positive (semi-)definite square root Hr =√

H of a positive (semi-)definite operator H can

be defined by HrHr = H and Hr ≥ 0. Then, it is possible to assign to each normal operator a

corresponding magnitude operator via |H| =√

HH+. We note that the trace class condition (A.2)

can equivalently be written as Tr{|H|} < ∞ [69]. The positive part of a self-adjoint operator H is

defined as H+ = (|H| + H)/2.

Unitary Operators. Unitary operators U satisfy UU+ = U+U = I, from which it follows that

U−1 = U+. The magnitude of the eigenvalues of a unitary operator is equal to one, i.e., |λk| = 1. An

important example of a unitary operator is the TF shift operator (see Section B.1.1).

Projection Operators. Projection operators are characterized by being idempotent, P2 = P.

If furthermore P is self-adjoint, it is referred to as orthogonal projection, otherwise the projector is

called oblique. Any eigenvalue of an orthogonal projector equals either zero or one.

Linear Time-Invariant Systems. Linear time-invariant (LTI) systems are systems which com-

mute with a time shift, i.e.,

HTτ = TτH .

Here, Tτ is the time shift operator defined by (Tτx)(t) = x(t− τ). The impulse response (kernel) of

such systems is of the type h(t, t′) = g(t − t′), which shows that the I/O relation (A.3) reduces to a

convolution, y(t) = (Hx)(t) =∫t′ g(t − t′)x(t′) dt′ , which, via the Fourier transform, corresponds to

a multiplication in the frequency domain, Y (f) = G(f)X(f) . LTI systems are the most prominent

operators that are not HS.

Linear Frequency-Invariant Systems. Linear frequency-invariant (LFI) systems are systems

which commute with the frequency shift operator Fν , i.e.,

HFν = FνH ,

where Fν is defined by (Fνx)(t) = x(t) ej2πνt. The impulse response (kernel) of such systems is of the

type h(t, t′) = m(t)δ(t − t′). Thus, LFI systems are dual to LTI systems in the sense that the I/O

relation (A.3) here reduces to a multiplication in the time domain, i.e., y(t) = (Hx)(t) = m(t)x(t) .

LFI systems also do not belong to the HS class.

B

Time-Frequency Analysis Tools

“The simplicities of natural laws arise through the complexities of thelanguages we use for their expression.” Eugene P. Wigner

IN this appendix we present a self-contained discussion of time-frequency representations of linear,

time-varying systems, of deterministic signals, and of random processes as far as is necessary for this

thesis. In the case of time-varying systems, we describe mainly two concepts: the generalized spreading

function and the generalized Weyl symbol . The former is related to a decomposition of a system into

elementary time-frequency shifts and the latter is a time-frequency parametrized representation of

the system with the interpretation of a time-frequency weighting or a time-varying transfer function.

These linear time-frequency system representations are complemented by the generalized transfer

Wigner distribution, the generalized input Wigner distribution, and the generalized output Wigner

distribution, which are quadratic time-frequency representations of linear systems.

For deterministic signals, there exist many time-frequency representations. However, for our pur-

poses, it suffices to discuss the short-time Fourier transform, the generalized Wigner distribution, the

spectrogram, and the generalized ambiguity function.

Finally, the time-frequency representations of random processes we discuss are the generalized

Wigner-Ville spectrum, the generalized evolutionary spectrum, the physical spectrum, and the gener-

alized expected ambiguity function.

For these time-frequency representations, the properties relevant to this thesis are discussed, and

several interrelations and parallels are pointed out. We note that further details on time-frequency

analysis can be found in several textbooks and tutorial papers [16,35,56,61,84,151,173,200].

213

214 Appendix B. Time-Frequency Analysis Tools

B.1 Time-Frequency Representations of Linear, Time-Varying Systems

In Appendix A we gave a brief discussion of linear operators which are mathematical models for

linear time-varying (LTV) systems. In addition to the system descriptions considered there, there also

exist time-frequency (TF) representations that characterize LTV systems in different ways. These TF

descriptions will be reviewed in the following two subsections.

B.1.1 Generalized Spreading Function

In contrast to linear time-invariant and linear frequency-invariant systems, which produce only time

shifts and frequency shifts, respectively, of various components of the input signal, general LTV systems

introduce both time shifts and frequency shifts. A transparent description of the TF shifts of a linear

system H with kernel h(t, t′) is given by the generalized spreading function (GSF) [114,118]

S(α)H

(τ, ν) ,

∫

th(α)(t, τ) e−j2πνt dt , (B.1)

with

h(α)(t, τ) , h

(t+

(1

2− α

)τ, t−

(1

2+ α

)τ

), (B.2)

where α is a real-valued parameter. The kernel of H can be reobtained from the GSF via

h(t, t′) =

∫

νS

(α)H

(t− t′, ν) ej2πν(( 12+α)t+( 1

2−α)t′) dν ,

which shows that the GSF contains all information about H. For a specific LTV system H, the GSF

describes the weights in a representation of the output signal y(t) as a weighted superposition of TF

shifted versions of the input signal x(t),

y(t) = (Hx)(t) =

∫

τ

∫

νS

(α)H

(τ, ν) (S(α)τ,νx)(t) dτ dν , (B.3)

with S(α)τ,ν denoting the generalized TF shift operator,1

(S(α)τ,νx)(t) = x(t− τ) ej2πνt ej2πτν(α−1/2) . (B.4)

With α = 1/2 the ordinary spreading function (delay-Doppler spread function) as introduced by

Bello [11, 109, 195] is reobtained from the GSF definition, and for α = −1/2 Bello’s Doppler-delay

spread function is reobtained,

S(1/2)H

(τ, ν) =

∫

th(t, t− τ) e−j2πνt dt ,

1 The parameter α is due to the infinitely many possibilities to define a joint TF shift by combining the time shift

operator Tτ with the frequency shift operator Fν . The case α = 1/2 corresponds to first shifting in time and then

shifting in frequency, S(1/2)τ,ν = Fν Tτ , while α = −1/2 corresponds to first shifting in frequency and then in time,

S(1/2)τ,ν = Tτ Fν . Note that the generalized TF shift operator corresponds to the Schrodinger representation of the

Heisenberg group. The map S(α)H

(τ, ν) → H is then referred to as integrated representation of the convolution algebra

on the Heisenberg group [64].

B.1 Time-Frequency Representations of Linear, Time-Varying Systems 215

τ

ννν

(d) (e) (f)

τ ττ

(a) (b) (c)

τ

ννν

τ

Figure B.1: Schematic representation of the GSF magnitude of some generic types of linear systems:

(a) identity operator, (b) TF shift operator, (c) LTI system, (d) LFI system, (e) quasi-LTI system,

(f) quasi-LFI system.

S(−1/2)H

(τ, ν) =

∫

th(t+ τ, t) e−j2πνt dt .

For different α values the various GSFs differ from each other merely by a phase factor:

S(α2)H

(τ, ν) = S(α1)H

(τ, ν) ej2π(α1−α2)τν . (B.5)

Due to (B.5), the magnitude of the GSF is α–invariant,

∣∣S(α1)H

(τ, ν)∣∣ =

∣∣S(α2)H

(τ, ν)∣∣.

We will therefore usually neglect the superscript α and simply write |SH(τ, ν)|.We briefly consider some special cases which help clarify the interpretation of the GSF (see Fig.

B.1):

• The GSF of the identity operator I with impulse response h(t, t′) = δ(t − t′) is given by

S(α)I

(τ, ν) = δ(τ)δ(ν). This is consistent with the fact that the identity operator shifts a signal

neither in time nor in frequency.

• For a TF shift operator S(α0)τ0,ν0 with impulse response h(t, t′) = δ(t−t′−τ0) ej2πν0t ej2πτ0ν0(α0−1/2),

the GSF is obtained as S(α)

S(α0)τ0,ν0

(τ, ν) = δ(τ − τ0)δ(ν − ν0) ej2π(α−α0)τ0ν0, i.e., a 2-D Dirac impulse

at the point (τ0, ν0). This agrees with the fact that S(α0)τ0,ν0 shifts a signal by τ0 in time and by ν0

in frequency.

• For an LTI system H with impulse response h(t, t′) = g(t− t′), we obtain S(α)H

(τ, ν) = g(τ)δ(ν),

which is consistent with the fact that LTI systems do not introduce any frequency shifts.

• For an LFI system H with h(t, t′) = m(t)δ(t − t′), the GSF is given by S(α)H

(τ, ν) = M(ν)δ(τ),

where M(ν) is the Fourier transform of m(t). This correctly reflects that LFI systems do not

introduce any time shifts.

Sometimes we will need the GSF of the adjoint system H+ which is related to the GSF of H as

S(α)H+(τ, ν) = S

(−α)∗H

(−τ,−ν) =[S

(α)H

(−τ,−ν) ej4πατν]∗. (B.6)


It is seen directly from the definition of the GSF that

S(α)H

(0, 0) = Tr{H} . (B.7)

Furthermore, the GSF is also a unitary operator representation in the sense that it preserves inner

products, i.e.,⟨S

(α)H, S

(α)G

⟩=

∫

τ

∫

νS

(α)H

(τ, ν)S(α)∗G

(τ, ν) dτ dν = 〈H,G〉 , (B.8)

which implies that the L2 norm of S(α)H

(τ, ν) equals the HS norm of H,

∥∥S(α)H

∥∥2

2=

∫

τ

∫

ν

∣∣S(α)H

(τ, ν)∣∣2 dτ dν = ‖H‖2

2 , (B.9)

and, furthermore, that the bilinear formQH(x, y) can be reformulated as a (τ, ν)-domain inner product,

QH(x, y) = 〈Hx, y〉 =⟨S

(α)H, A(α)

y,x

⟩=

∫

τ

∫

νS

(α)H

(τ, ν)A(α)∗y,x (τ, ν) dτ dν , (B.10)

where A(α)y,x(τ, ν) denotes the generalized cross ambiguity function of y(t) and x(t) (cf. Subsection

B.2.4). Using the SVD (A.5) of H, the GSF can also be written as a weighted superposition of

generalized cross ambiguity functions of the (left and right) singular functions, i.e.

S(α)H

(τ, ν) =∑

k

σk A(α)uk,vk

(τ, ν) . (B.11)

For normal operators, this simplifies to the following weighted superposition of generalized auto am-

biguity functions of the eigenfunctions (cf. (A.4)),

S(α)H

(τ, ν) =∑

k

λk A(α)uk,uk

(τ, ν) .

In Subsection 2.3.4 we consider operator products of the type H2H1. The GSF of such operator

products is given by the so-called twisted convolution [64,118],

S(α)H2H1

(τ, ν) =(S

(α)H2♮ S

(α)H1

)(τ, ν) ,

∫

τ ′

∫

ν′

S(α)H2

(τ ′, ν ′)S(α)H1

(τ − τ ′, ν − ν ′) e−j2πφα(τ,ν,τ ′,ν′) dτ ′ dν ′ (B.12)

with φα(τ, ν, τ ′, ν ′) = (α+1/2)τ ′(ν−ν ′)+(α−1/2)(τ−τ ′)ν ′. In the case α = 0, the phase φα(τ, ν, τ ′, ν ′)

simplifies (up to the factor of 1/2) to the symplectic form on R2, i.e., φ0(τ, ν, τ

′, ν ′) = 12 (τ ′ν − τν ′).

The name “twisted convolution” stems from the fact that apart from the phase factor φα(τ, ν, τ ′, ν ′),

(B.12) looks like an ordinary convolution. Using (B.12), the magnitude of S(α)H2H1

(τ, ν) can be shown

to be bounded as

∣∣S(α)H2H1

(τ, ν)∣∣ =

∣∣(S(α)H2♮ S

(α)H1

)(τ, ν)

∣∣ ≤∣∣S(α)

H1(τ, ν)

∣∣ ∗∗∣∣S(α)

H2(τ, ν)

∣∣ , (B.13)

with ∗∗ denoting ordinary 2-D convolution. The twisted convolution also satisfies Young’s inequality

[64],∥∥S(α)

H2♮ S

(α)H1

∥∥r≤∥∥S(α)

H2

∥∥p

∥∥S(α)H1

∥∥q,

1

p+

1

q=

1

r+ 1 , (B.14)


and even the stronger bound ∥∥S(α)H2♮ S

(α)H1

∥∥2≤∥∥S(α)

H2

∥∥2

∥∥S(α)H1

∥∥2. (B.15)

We note that with (B.9) and (B.12), this is equivalent to

∥∥H2H1

∥∥2≤∥∥H2

∥∥2

∥∥H1

∥∥2. (B.16)

However, the twisted convolution is not commutative. In particular, interchanging H2 and H1 reverses

the sign of φα(τ, ν, τ ′, ν ′). By expanding the phase factor into its Taylor series,

e−j2πφα(τ,ν,τ ′,ν′) = e−j2π(α+1/2)τ ′(ν−ν′)e−j2π(α−1/2)(τ−τ ′)ν′

=

∞∑

k=0

[−j2π(α + 1/2)τ ′(ν − ν ′)]k

k!

∞∑

l=0

[−j2π(α − 1/2)(τ − τ ′)ν ′]l

l!

=∞∑

k,l=0

(−j2π)k+l ckl τ′kν ′

l(τ − τ ′)l(ν − ν ′)k ,

with ckl = (α+ 1/2)k(α − 1/2)l/(k! l!), and suitably substituting, the twisted convolution (B.12) can

be expressed as an infinite sum of ordinary convolutions (for α = 0, this was already done in [64]),

S(α)H2H1

(τ, ν) =∞∑

k,l=0

(−j2π)k+l ckl

∫

τ ′

∫

ν′

τ ′kν ′

lS

(α)H2

(τ ′, ν ′) (τ − τ ′)l(ν − ν ′)kS(α)H1

(τ − τ ′, ν − ν ′) dτ ′ dν ′

=(S

(α)H1

∗∗S(α)H2

)(τ, ν) +

∑

k+l>0

(−j2π)k+l ckl

[τk νl S

(α)H2

(τ, ν)]∗∗[τ l νk S

(α)H1

(τ, ν)],

where in the last step we have split off the k = l = 0 term since it gives the ordinary convolution of

S(α)H1

(τ, ν) and S(α)H2

(τ, ν).

B.1.2 Generalized Weyl Symbol

An alternative TF description of an LTV system that is related to TF weightings instead of TF shifts

(see Section 2.3) is the generalized Weyl symbol (GWS), defined by [114,118]

L(α)H

(t, f) ,

∫

τh(α)(t, τ) e−j2πfτ dτ , (B.17)

with h(α)(t, τ) given by (B.2). For α = 0, the GWS reduces to the ordinary Weyl symbol [64,99,114,

118,190],

L(0)H

(t, f) =

∫

τh(t+

τ

2, t− τ

2

)e−j2πfτ dτ .

With α = 1/2, Zadeh’s time-varying transfer function [214] is re-obtained,

L(1/2)H

(t, f) =

∫

τh(t, t− τ) e−j2πfτ dτ ,

and for α = −1/2 the GWS reduces to the Kohn-Nirenberg symbol [64, 112] which is equivalent to

Bello’s frequency-dependent modulation function [11],

L(−1/2)H

(t, f) =

∫

τh(t+ τ, t) e−j2πfτ dτ .


The kernel of H can be reobtained from the GWS via

h(t, t′) =

∫

fL

(α)H

( t+ t′

2+ α(t− t′), f

)ej2πf(t−t′) df , (B.18)

which shows that the GWS contains all information about H. Based on (B.18), the I/O relation (A.3)

can be reformulated as

y(t) = (Hx)(t) =

∫

t′

∫

fL

(α)H

(t+ t′

2+ α(t− t′), f

)x(t′) ej2πf(t−t′) df dt′.

For α = ±1/2 this reduces to the particularly simple expressions

y(t) =

∫

fL

(1/2)H

(t, f)X(f) ej2πft df , Y (f) =

∫

tL

(−1/2)H

(t, f)x(t) e−j2πft dt . (B.19)

It is important to note that the GWS is in 2-D Fourier correspondence with the GSF, i.e.,

S(α)H

(τ, ν) =

∫

t

∫

fL

(α)H

(t, f) e−j2π(νt−τf) dt df. (B.20)

Hence, as a consequence of (B.5), for two different parameters α1 and α2 the corresponding GWSs are

connected via a two-dimensional convolution,

L(α2)H

(t, f) = L(α1)H

(t, f) ∗∗ ej2πft 1α1−α2 . (B.21)

Furthermore, taking the 2-D Fourier transform of (B.3), one obtains the alternative I/O relation

(Hx)(t′) =

∫

t

∫

fL

(α)H

(t, f)(L

(α)t,f x

)(t′) dt df , with L

(α)t,f = St,fL

(α)S+t,f ,

where L(α) is a linear operator with kernel l(α)(t, t′) given by

l(α)(t, t′) = δ

(t+ t′

2+ α(t− t′)

). (B.22)

For α = ±1/2 and α = 0 the action of the operator L(α)t,f is given by

(L

(1/2)t,f x

)(t′) = X(f) δ(t′ − t) ej2πft

(L

(−1/2)t,f x

)(t′) = x(t) ej2πf(t′−t)

(L

(0)t,fx

)(t′) = 2x(2t− t′) ej4πf(t′−t) .

Note that L(0)t,f performs a “TF reflection” about (t, f).

Some examples for the GWS of specific systems are as follows:

• The GWS of the identity operator I with impulse response h(t, t′) = δ(t − t′) is given by

L(α)I

(t, f) = 1, which correctly reflects that the identity operator leaves all signals unchanged.

• For a TF shift operator S(α0)τ0,ν0 with impulse response h(t, t′) = δ(t−t′−τ0) ej2πν0t ej2πτ0ν0(α0−1/2),

the GWS is given by L(α)

S(α0)τ0,ν0

(t, f) = ej2π(ν0t−τ0f) ej2π(α−α0)τ0ν0.


• The GWS of an LTI system with impulse response h(t, t′) = g(t − t′) reduces to the ordinary

transfer function for all t, L(α)H

(t, f) = G(f) where G(f) is the Fourier transform of g(τ).

• The GWS of an LFI system with impulse response h(t, t′) = m(t)δ(t−t′) reduces to the temporal

transfer function for all f , i.e. L(α)H

(t, f) = m(t).

Due to (B.20) and (B.6), the GWS of the adjoint system H+ can be expressed (for α 6= 0) in terms of

the GWS of H as

L(α)H+(t, f) = L

(−α)∗H

(t, f) =[L

(α)H

(t, f) ∗∗ ejπft 1α

]∗, (B.23)

and for α = 0 there is simply

L(0)H+(t, f) = L

(0)∗H

(t, f), (B.24)

from which it follows that the Weyl symbol (GWS with α = 0) of a self-adjoint system is real-valued.

By integrating (B.17) over t and f , it is seen that

∫

t

∫

fL

(α)H

(t, f) dt df = Tr{H} .

Furthermore, like the GSF, the GWS preserves inner products and norms of HS operators,

⟨L

(α)H, L

(α)G

⟩=

∫

t

∫

fL

(α)H

(t, f)L(α)∗G

(t, f) dt df = 〈H,G〉 , (B.25)

∥∥L(α)H

∥∥2

2=

∫

t

∫

f

∣∣L(α)H

(t, f)∣∣2 dt df = ‖H‖2

2 . (B.26)

Furthermore, the bilinear form can be expressed in terms of the GWS as

QH(x, y) = 〈Hx, y〉 =⟨L

(α)H,W (α)

y,x

⟩=

∫

t

∫

fL

(α)H

(t, f)W (α)∗y,x (t, f) dt df , (B.27)

where W(α)y,x (t, f) is the generalized cross Wigner distribution (see Subsection B.2.2) of y(t) and x(t).

Combining the SVD (A.5) of H with the GWS definition (B.17), it is seen that the GWS can be written

as a weighted superposition of generalized cross Wigner distributions of the singular functions,

L(α)H

(t, f) =∑

k

σkW(α)uk,vk

(t, f) . (B.28)

In the case of normal operators, this simplifies to the weighted superposition of generalized auto

Wigner distributions of the eigenfunctions,

L(α)H

(t, f) =∑

k

λkW(α)uk

(t, f) . (B.29)

The GWS of operator products H2H1 can be obtained by the taking the 2-D Fourier transform of

(B.12), which yields

L(α)H2H1

(t, f) =(L

(α)H2

#L(α)H1

)(t, f) ,

∞∑

k,l=0

ckl

(j2π)k+l

∂k+lL(α)H2

(t, f)

∂tl∂fk

∂k+lL(α)H1

(t, f)

∂tk∂f l(B.30)


= L(α)H2

(t, f)L(α)H1

(t, f) +∑

k+l>0

ckl

(j2π)k+l

∂k+lL(α)H2

(t, f)

∂tl∂fk

∂k+lL(α)H1

(t, f)

∂tk∂f l, (B.31)

with ckl = (α+ 1/2)k(α− 1/2)l/(k! l!). Note that if the phase factor in the twisted convolution (B.12)

was not present, then due to the 2-D Fourier relation (B.20), the GWS of H2H1 would be exactly

equal to the product of the individual GWSs. According to the naming convention used for α = 0 in

the mathematics literature [64], we refer to(L

(α)H2

#L(α)H1

)(t, f) as twisted product (or star product) of

L(α)H2

(t, f) and L(α)H1

(t, f).

B.1.3 Generalized Transfer Wigner Distribution and Generalized Input and OutputWigner Distribution

An interesting alternative approach to describe the TF weightings and TF displacements of an LTV

system H is in terms of the generalized transfer Wigner distribution, defined as [8, 90,130]

W(α)H

(t1, f1; t2, f2) =

∫

τ1

∫

τ2

h

(t1 +

(1

2− α

)τ1, t2 +

(1

2− α

)τ2

)

· h∗(t1 −

(1

2+ α

)τ1, t2 −

(1

2+ α

)τ2

)e−j2π(f1τ1−f2τ2) dτ1 dτ2 . (B.32)

The generalized transfer Wigner distribution describes the mapping of the generalized Wigner distri-

bution (see B.2.2) of the input signal x(t) to the generalized Wigner distribution of the output signal

(Hx)(t) according to

W(α)Hx (t1, f1) =

∫

t2

∫

f2

W(α)H

(t1, f1; t2, f2)W(α)x (t2, f2) dt2 df2 .

We next present a Theorem that will be important in Chapter 3 and that generalizes a result

presented for the special case α = 0 in [90]. To this end, we define a coordinate transformed version

of the generalized transfer Wigner distribution as

W(α)H

(t, f ; τ, ν) = W(α)H

(t+ α−τ, f + α+ν; t− α+τ, f − α−ν) (B.33)

where α+ = 1/2 + α and α− = 1/2 − α.

Theorem B.1. The generalized transfer Wigner distribution of any linear system H satisfies∫

τ

∫

νW(α)

H(t, f ; τ, ν) dτ dν =

∣∣L(α)H

(t, f)∣∣2, (B.34)

∫

t

∫

fW(α)

H(t, f ; τ, ν) dt df =

∣∣SH(τ, ν)∣∣2 . (B.35)

Proof. To prove (B.34), we insert (B.32) into (B.33) and integrate with respect to τ and ν,

∫

τ

∫

νW(α)

H(t, f ; τ, ν) dτ dν

=

∫

τ

∫

ν

∫

τ1

∫

τ2

h(t+ α−τ + α−τ1, t− α+τ + α−τ2)h∗(t+ α−τ − α+τ1, t− α+τ − α+τ2)


· e−j2π[(f+α+ν)τ1−(f−α−ν)τ2] dτ1 dτ2 dτ dν

=

∫

τ

∫

τ1

∫

τ2


·[ ∫

νe−j2πν(α+τ1+α−τ2) dν

]

︸︷︷︸δ(α+τ1 + α−τ2)

e−j2πf(τ1−τ2) dτ1 dτ2 dτ

=

∫

τ

∫

τ ′1

∫

τ ′2

h(t+ α−τ ′1, t− α+τ + α−(τ ′2 − τ))h∗(t+ α−τ − α+(τ ′1 − τ), t− α+τ ′2)

· δ(α+τ ′1 + α−τ ′2 − τ) e−j2πf(τ ′1−τ ′

2) dτ ′1 dτ′2 dτ

=

∫

τ ′1

∫

τ ′2

h(t+ α−τ ′1, t− α+τ ′1)h∗(t+ α−τ ′2, t− α+τ ′2) e

−j2πf(τ ′1−τ ′

2) dτ ′1 dτ′2

= L(α)H

(t, f)L(α)∗H

(t, f) ,

which completes the proof of (B.34). In a similar way, integrating W(α)H

(t, f ; τ, ν) with respect to t

and f yields∫

f

∫

fW(α)

H(t, f ; τ, ν) dt df

=

∫

t

∫

f

∫

τ1

∫

τ2


· e−j2π[(f+α+ν)τ1−(f−α−ν)τ2] dτ1 dτ2 dt df

=

∫

t

∫

τ1

∫

τ2


·[ ∫

fe−j2πf(τ1−τ2) df

]e−j2π(α+ντ1+α−ντ2) dτ1 dτ2 dt

=

∫

t

∫

τ1

∫

τ2


· δ(τ1 − τ2) e−j2π(α+ντ1+α−ντ2) dτ1 dτ2 dt

=

∫

t

∫

τ1

h(t+ α−τ + α−τ1, t− α+τ + α−τ1)h∗(t+ α−τ − α+τ1, t− α+τ − α+τ1) e

−j2πντ1 dτ1 dt

=

∫

t1

∫

t2

h(t1 + α−τ, t1 − α+τ)h∗(t2 + α−τ, t2 − α+τ) e−j2πν(t1−t2) dt1 dt2

= S(α)H

(τ, ν)S(α)∗H

(τ, ν) ,

and hence finally (3.8) since |S(α)H

(τ, ν)| is independent of α.

Integrating the generalized transfer Wigner distribution W(α)H

(t1, f1; t2, f2) with respect to (t1, f1)

and (t2, f2), respectively, yields the generalized input Wigner distribution and the generalized output

Wigner distribution of H [90], i.e.,

IW(α)H

(t, f) ,

∫

t1

∫

f1

W(α)H

(t1, f1; t, f) dt1 df1 ,

OW(α)H

(t, f) ,

∫

t2

∫

f2

W(α)H

(t, f ; t2, f2) dt2 df2 .


The generalized input Wigner distribution describes the TF regions where the system H can pick up

energy, and the generalized output Wigner distribution describes the TF regions where the output

signals are potentially located. We note that the generalized input and output Wigner distribution

can be related to the GWS via

IW(α)H

(t, f) = L(α)H+H

(t, f) and OW(α)H

(t, f) = L(α)HH+(t, f). (B.36)

Hence, it is seen that in the case of normal systems, the generalized input Wigner distribution and

the generalized output Wigner distribution coincide and are then simply referred to as “generalized

Wigner distribution of the system H.”

B.2 Time-Frequency Signal Representations

In this section, we give a brief account of linear and quadratic TF signal representations [16, 35, 56,

61, 84, 151, 173, 200]. Quadratic TF representations of a signal x(t) can be viewed as special cases of

TF operator representations (see Section B.1) obtained for the rank one operator H = x⊗ x∗.

B.2.1 Short-Time Fourier Transform

An obvious way of analyzing a signal as to which frequencies occur at a given time is to apply a

sliding window to the original signal and to compute the Fourier transform of the windowed signal.

The resulting linear TF representation is referred to as short-time Fourier transform (STFT) [84,157]

and is given by

STFT(g)x (t, f) ,

⟨x,S

(1/2)t,f g

⟩=

∫

t′x(t′)g∗(t′ − t)e−j2πft′ dt′ , (B.37)

where g(t) is an analysis window. Thus, the STFT describes the local frequency content of the signal

in a neighbourhood of the respective analysis time t. The STFT is unique in the sense that it is the

only TF shift covariant linear TFR (up to a phase factor). It is possible to recover the signal x(t)

from its STFT using the “synthesis formula”

x(t) =

∫

t′

∫

f ′

STFT(g)x (t′, f ′)

(S

(1/2)t′,f ′ w

)(t) dt′ df ′ =

∫

t′

∫

f ′

STFT(g)x (t′, f ′)w(t − t′)ej2πf ′t dt′ df ′ , (B.38)

provided that the analysis window g(t) and the synthesis window w(t) satisfy the (not very restrictive)

condition 〈g,w〉 = 1 [84]. Clearly, the STFT is significantly influenced by the choice of the window

function g(t). In order that the STFT correctly localizes the signal in the TF plane, the window has

to be localized around the origin of the TF plane.

B.2.2 Generalized Wigner Distribution

The generalized (cross) Wigner distribution (GWD) is a bilinear TF representation defined as [31,46,

98,151]

W (α)x,y (t, f) ,

∫

τq(α)x,y (t, τ)e−j2πfτ dτ , (B.39)

B.2 Time-Frequency Signal Representations 223

where

q(α)x,y (t, τ) , qx,y

(t+

(1

2− α

)τ, t−

(1

2+ α

)τ), with qx,y(t, t

′) = x(t) y∗(t′) . (B.40)

It is an extension of the ordinary (cross) Wigner distribution which was originally introduced in

quantum mechanics by Wigner [211], rediscovered for signal analysis by Ville [205] and more recently

made popular by Claasen and Mecklenbrauker [31]. The Wigner distribution is re-obtained from the

GWD with α = 0,

W (0)x,y (t, f) =

∫

τx(t+

τ

2

)y∗(t− τ

2

)e−j2πfτ dτ. (B.41)

A further special GWD member, obtained with α = 1/2, is the (cross) Rihaczek distribution [178],

W (1/2)x,y (t, f) =

∫

τx(t)y∗(t− τ)e−j2πfτ dτ = x(t)Y ∗(f) e−j2πft.

Furthermore, two GWD members with different α values are related by a 2-D convolution as

W (α2)x,y (t, f) = W (α1)

x,y (t, f) ∗∗ ej2πft 1α1−α2 .

The GWD is a bilinear TF representation well-known for the exceptionally large number of de-

sirable properties it satisfies. It allows a loose interpretation as an energy distribution in the TF

plane (we note that a pointwise interpretation as “TF energy density” is a priori prohibited by the

uncertainty principle. We note that the GWD is the deterministic counterpart of the generalized

Wigner-Ville spectrum (see Section B.3.1).

The cross GWD of x(t) and y(t) can also be written in terms of the GWS of the rank one operator

x⊗ y∗ whose kernel is the outer signal product qx,y(t, t′) in (B.40),

W (α)x,y (t, f) = L

(α)x⊗y∗(t, f).

Thus, any GWS property (see Subsection B.1.2) carries over to a corresponding GWD property.

B.2.3 Spectrogram

An important quadratic TF representation is the spectrogram that is defined as the squared magnitude

of the STFT (B.37) [84],

SPEC(g)x (t, f) ,

∣∣STFT(g)x (t, f)

∣∣2. (B.42)

Correspondingly, the cross spectrogram is defined as

SPEC(g)x,y(t, f) = STFT(g)

x (t, f) STFT(g)∗y (t, f) .

The spectrogram is connected with the GWD via a double convolution,

SPEC(g)x (t, f) =

∫

t′

∫

f ′

W (α)x (t′, f ′)W (α)∗

g (t′ − t, f ′ − f) dt′ df ′.

Thus, for usual windows g(t), the spectrogram can be interpreted as a smoothed GWD.


B.2.4 Generalized Ambiguity Function

Unlike the WD and the spectrogram, the generalized (cross) ambiguity function (GAF) [78]

A(α)x,y(τ, ν) ,

∫

tq(α)x,y (t, τ)e−j2πνt dt = 〈x,S(α)

τ,ν y〉 (B.43)

describes signals in a correlative domain. The GAF is an extension of the ordinary ambiguity function,

originally introduced by Woodward in 1953 [213], and re-obtained from the GAF with α = 0,

A(0)x,y(τ, ν) =

∫

tx(t+

τ

2

)y∗(t− τ

2

)e−j2πνt dt .

It is furthermore in (two-dimensional) Fourier correspondence with the GWD,

A(α)x,y(τ, ν) =

∫

t

∫

fW (α)

x,y (t, f) e−j2π(νt−τf) dt df.

The GAFs for various α values differ from each other merely by a phase factor, i.e.,

A(α2)x,y (τ, ν) = A(α1)

x,y (τ, ν) ej2π(α1−α2)τν . (B.44)

In many situations we are interested only in the magnitude of the GAF, which is α-invariant,

∣∣A(α1)x,y (τ, ν)

∣∣ =∣∣A(α2)

x,y (τ, ν)∣∣ .

The auto GAF A(α)x (τ, ν) , A

(α)x,x(τ, ν) features hermitian even symmetry,

A(α)x (τ, ν) = A(α)∗

x (−τ,−ν),

and satisfies the inequality2

∣∣A(α)x (τ, ν)

∣∣ ≤ A(α)x (0, 0) = Ex ,

which indicates that the GAF behaves like a correlation function. The shape of the auto GAF is

constrained by the so-called “radar uncertainty principle,”∫

τ

∫

ν

∣∣A(α)x (τ, ν)

∣∣2dτ dν =[A(α)

x (0, 0)]2

= E2x ,

which imposes a strong limitation on the relation of volume and height of the auto GAF.

The GAF is furthermore equal to the GSF of the outer signal product operator x⊗ y∗,

A(α)x,y(τ, ν) = S

(α)x⊗y∗(τ, ν).

Thus, the discussion of the GSF properties in Subsection B.1.1 applies to the GAF as well.

B.3 Time-Frequency Representations of Random Processes

In this section, we discuss TF representations of nonstationary random processes. These TF repre-

sentations are statistical characterizations of the process considered. Some of them can be viewed as

special cases of the TF operator representations discussed in Section B.1, obtained with the (positive

semi-definite) correlation operator Rx of the random process x(t), or, equivalently, as expectations of

some of the TF signal representations discussed in Section B.2.2 Ex denotes the energy of the signal x(t), i.e., Ex = ‖x‖2

2 =R

t|x(t)|2 dt.

B.3 Time-Frequency Representations of Random Processes 225

B.3.1 Generalized Wigner-Ville Spectrum

The generalized (cross) Wigner–Ville spectrum (GWVS) [60,139,140] of two (generally nonstationary)

random processes x(t), y(t) is defined by

W(α)

x,y (t, f) ,

∫

τr(α)x,y (t, τ) e−j2πfτ dτ , (B.45)

with r(α)x,y (t, τ) , rx,y

(t+(

12 − α

)τ, t−

(12 + α

)τ), where rx,y(t, t

′) = E {x(t) y∗(t′)} is the cross cor-

relation function of x(t) and y(t) (E denotes expectation). With α = 0 the ordinary Wigner-Ville

spectrum [12,60,139,140] is obtained,

W(0)

x,y (t, f) =

∫

τrx,y

(t+

τ

2, t− τ

2

)e−j2πfτ dτ ,

while α = 1/2 yields the Rihaczek spectrum [60]

W(1/2)x,y (t, f) =

∫

τrx,y(t, t− τ) e−j2πfτ dτ .

The GWVS satisfies a large number of desirable properties and thus is an interesting approach to

defining a TF function that describes the spectral properties of nonstationary random processes. Note

that (B.45) is a one-to-one mapping that can be inverted as

rx,y(t, t′) =

∫

fW

(α)x,y

(t+ t′

2+ α(t− t′), f

)ej2πf(t−t′) df .

Thus, for y(t) = x(t) the GWVS is a complete second-order statistic of the process x(t). Under certain

mild conditions, the GWVS equals the expectation of the GWD in (B.39),

W(α)

x,y (t, f) = E{W (α)

x,y (t, f)}.

Comparing the definition of the GWVS with (B.17) shows that the WVS can equivalently be written

as the GWS (see (B.17)) of the correlation operator Rx,y = E {x⊗ y∗} with kernel rx,y(t, t′),

W(α)

x,y (t, f) = L(α)Rx,y

(t, f) .

It can furthermore be shown that the GWVS constitutes a 2-D Fourier transform pair with the GEAF

(to be discussed in Subsection B.3.4),

W(α)

x,y (t, f) =

∫

τ

∫

νA(α)

x,y(τ, ν) ej2π(νt−τf) dτ dν . (B.46)

This is of particular importance since it makes explicit the connection between the smoothness of the

GWVS and the extension of the GEAF about the origin of the (τ, ν) plane (see Subsection 3.2.1).

To further illustrate the interpretation of the GWVS, we finally consider specific examples of

random processes:

• The GWVS of a stationary and white process with correlation operator Rx = η I is given by

W(α)

x (t, f) ≡ η. This shows that stationary white processes cover the entire TF plane in an

ideally homogeneous manner.


• For a stationary process with correlation function rx(t, t′) = rx(t − t′), the GWVS equals the

power spectral density for all t, W(α)

x (t, f) ≡ Sx(f).

• For a (generally nonstationary) white process with correlation function rx(t, t′) = qx(t)δ(t− t′),

the GWVS equals the mean instantaneous intensity qx(t) for all f , W(α)

x (t, f) ≡ qx(t).

B.3.2 Generalized Evolutionary Spectrum

In 1965, Priestley [170] defined the evolutionary spectrum which is based on considerations related to

the Karhunen Loeve expansion [1,107,136]. Another interpretation of the evolutionary spectrum is in

terms of an innovations system representation of the underlying process. Based on this innovations

system interpretation, the generalized evolutionary spectrum (GES) has been introduced recently [145,

147,148].

Any nonstationary process can be modelled as output of an LTV innovations system H which is

excited by stationary white noise n(t),

x(t) = (Hn)(t) =

∫

t′h(t, t′)n(t′) dt′ , with E

{n(t)n∗(t′)

}= δ(t − t′) . (B.47)

Computing the correlation operator of x(t) from (B.47) results in

Rx = HH+. (B.48)

There exists no unique innovations system since for any valid innovations system H and any system

U satisfying UU+ = I, it can easily be checked that HU is a valid innovations system as well. In

analogy to the power spectral density of stationary processes, the GES can then be defined as squared

magnitude of the innovations system’s transfer function (i.e., GWS),

G(α)x (t, f) , |L(α)

H(t, f)|2 . (B.49)

To compute the GES, the so-called factorization problem has to be solved, i.e., one has to find a valid

innovations system H satisfying (B.48) for given Rx. Note that the GES is nonunique since different

choices of the innovations system H yield different results for G(α)x (t, f). The evolutionary spectrum

is a special case of the GES re-obtained with α = 1/2. Another special case is given by the transitory

evolutionary spectrum [49,147,148], which is dual to the evolutionary spectrum and re-obtained from

the GES with α = −1/2. A further special case, the Weyl spectrum [147,148], is obtained by choosing

α = 0 and using the the positive semidefinite root of Rx (see Appendix A) as innovations system.

B.3.3 Physical Spectrum

Mark, in his 1970 paper [139], defined the physical spectrum as the expectation of the spectrogram

(see (B.42)),

PS(g)x (t, f) , E

{SPEC(g)

x (t, f)}

=⟨RxS

(1/2)t,f g,S

(1/2)t,f g

⟩. (B.50)

B.3 Time-Frequency Representations of Random Processes 227

It can readily be verified that (B.50) can be written as 2-D convolution of the GWVS and the GWD

of the analysis window [139],

PS(g)x (t, f) =

∫

t′

∫

f ′

W(α)x (t′, f ′)W (α)∗

g (t′ − t, f ′ − f) dt′ df ′ . (B.51)

B.3.4 Generalized Expected Ambiguity Function

The generalized expected (cross) ambiguity function (GEAF) is defined as [118,126]

A(α)x,y(τ, ν) ,

∫

tr(α)x,y (t, τ) e−j2πνt dt. (B.52)

The cross correlation function rx,y(t, t′) can be recovered from the GEAF by

rx,y(t, t′) =

∫

νA(α)

x,y(t− t′, ν) ej2πν(( 12+α)t+( 1

2−α)t′) dν . (B.53)

Thus, for y(t) = x(t) the GEAF is a complete second-order statistics of the random process x(t).

Under mild conditions, the GEAF can be shown to equal the expectation of the GAF in (B.43),

A(α)x,y(τ, ν) = E

{A(α)

x,y(τ, ν)}.

Furthermore it equals the GSF (see (B.1)) of the correlation operator Rx,y,

A(α)x,y(τ, ν) = S

(α)Rx,y

(τ, ν).

Thus, it follows from (B.5) or from (B.44) that the members of the GEAF family obtained with

different α values differ from each other merely by a phase factor,

A(α2)x,y (τ, ν) = A(α1)

x,y (τ, ν) ej2π(α1−α2)τν . (B.54)

Clearly, the properties of the GAF/GSF carry over to the GEAF. In particular, the (auto) GEAF

A(α)x (τ, ν) , A

(α)x,x(τ, ν) features Hermitian symmetry,

A(α)x (τ, ν) = A(α)∗

x (−τ,−ν) , (B.55)

and attains its maximum at the origin,

|A(α)x (τ, ν)| ≤ A(α)

x (0, 0) = Ex ,

with Ex , E{‖x‖2

2

}=∫t rx(t, t) dt the expected energy of x(t). Furthermore, the GEAF is in 2-D

Fourier correspondence with the GWVS (cf. (B.46)),

A(α)x,y(τ, ν) =

∫

t

∫

fW

(α)x,y (t, f) e−j2π(νt−τf) dt df .

An important interpretation of the GEAF as TF correlation function is presented in Subsection 3.1.3.

We finally consider some specific examples to clarify the interpretation of the GEAF (see Fig. B.2):


ττ τ

ν

τ τ

(a) (c) (d) (e)(b)

ν ν νν

Figure B.2: Schematic representation of the GEAF magnitude of some generic types of random

processes: (a) stationary white process, (b) stationary (non-white) process, (c) white (nonstationary)

process, (d) quasi-stationary process, (e) quasi-white process.

• The GEAF of a stationary and white process with correlation operator Rx = η I is given by

A(α)x (τ, ν)(τ, ν) = η δ(τ)δ(ν). This is consistent with the fact that the stationary white processes

feature neither temporal correlations nor spectral correlations.

• For a stationary process with correlation function rx(t, t′) = rx(t − t′), we obtain A(α)x (τ, ν) =

rx(τ)δ(ν), which is consistent with the fact that stationary processes feature no spectral corre-

lations.

• For a (generally nonstationary) white process with correlation function rx(t, t′) = qx(t)δ(t− t′),

the GEAF is given by A(α)x (τ, ν) = Qx(ν)δ(τ), where Qx(ν) is the Fourier transform of qx(t).

This result is intuitive since white processes feature no temporal correlations.

C

The Symplectic Groupand Metaplectic Operators

“God exists since mathematics is consistent, and the Devilexists since we cannot prove it.” Andre Weil

LINEAR coordinate transforms in the time-frequency or time lag/frequency lag domain are im-

portant in time-frequency analysis. They can be associated to what is known in mathematics as

the symplectic group and its metaplectic representation. The latter in essence yields a class of uni-

tary operators (termed metaplectic operators) that includes the time-frequency scaling operator, the

Fourier transform operator, and the chirp convolution and multiplication operators as special cases.

In this appendix, we outline the definition and essential properties of the symplectic group and meta-

plectic operators. Furthermore, we study the relevance of these concepts to certain time-frequency

representations of linear systems and random processes.

229

230 Appendix C. The Symplectic Group and Metaplectic Operators

C.1 The Symplectic Group

We first discuss the two-dimensional1 symplectic group in a notation adapted to our purposes. More

detailed discussions can be found in [46,64,154,208].

A symplectic form on a vector space V over a field F is a function f(x, y) (defined for all x, y ∈ V

and taking values in F ) that satisfies

f(ax1 + b x2, y) = a f(x1, y) + b f(x2, y) and f(y, x) = −f(x, y)

for all x1, x2, x, y ∈ V and a, b ∈ F . Hence, f(x, y) can be viewed as an antisymmetric bilinear form.

For V = R2 and F = R, it can be shown that any symplectic form f(ξ

1, ξ

2) of two vectors

ξ1

= (τ1 ν1)T , ξ

2= (τ2 ν2)

T , can be written as

f(ξ1, ξ

2) = β ξT

1J ξ

2, with J =

(0 1

−1 0

), β ∈ R.

(Note that J T = J−1 = −J .) The standard symplectic form is obtained with β = 1 and denoted

by [64][ξ1, ξ

2

], ξT

1J ξ

2= τ1ν2 − ν1τ2.

The set of real-valued 2 × 2 matrices A =(

a bc d

)preserving the symplectic form in the sense that

[Aξ

1,Aξ

2

]=[ξ1, ξ

2

]

is referred to as symplectic group2 [64,154] and denoted by Sp. Any A ∈ Sp will be called symplectic

matrix . The following equivalences can be shown:

A ∈ Sp ⇐⇒ ATJA = J ,

A ∈ Sp ⇐⇒ A−1 = JATJ−1 =

(d −b−c a

),

A ∈ Sp ⇐⇒ detA = 1 .

The last property shows that the symplectic group consists of all matrices that correspond to area-

preserving linear coordinate transforms ξ = Aξ.We next consider some important subsets of the symplectic group Sp. In particular, it can be

shown that the matrices

Bb =

(1 b

0 1

), Cc =

(1 0

c 1

), Dd =

(1/d 0

0 d

), and Rθ = e−θJ =

(cos θ − sin θ

sin θ cos θ

)(C.1)

1The symplectic group is generally considered for R2n [64]. For our purposes, the case n = 1 is sufficient.

2The corresponding group operation is matrix multiplication, the identity element equals the identity matrix, and the

inverse element is given by the matrix inverse.

C.2 Metaplectic Operators 231

(with b, c, d, θ ∈ R, |d| > 0) generate subgroups of Sp. For reasons of proper physical dimension, we

will sometimes use a modified version of Rθ that incorporates a normalization, i.e.,

R(T )θ =

(cos θ −T 2 sin θ

(sin θ)/T 2 cos θ

)= D1/TRθDT .

Any A =(

a bc d

)∈ Sp can be written as a decomposition of the three3 matrices Bb, Cc, and Dd in (C.1)

in at least one of the following forms [64,162]:

a 6= 0 : A = Cc1D1/aBb1 , with b1 =b

a, c1 =

c

a(C.2a)

b 6= 0 : A = Cc1Bb Cc2 , with c1 =d− 1

b, c2 =

a− 1

b(C.2b)

c 6= 0 : A = Bb1Cc Bb2 , with b1 =a− 1

c, b2 =

d− 1

c(C.2c)

d 6= 0 : A = Bb1Dd Cc1 , with b1 =b

d, c1 =

c

d. (C.2d)

These decompositions will be important in the following. Note that there exists no unique decompo-

sition of an arbitrary symplectic matrix A [64,162].

C.2 Metaplectic Operators

To each symplectic matrix A ∈ Sp, one can associate in a unique manner a unitary operator U = µ(A)

on L2(R) (for details see [64]). The mapping A → µ(A) is termed the metaplectic representation of

the symplectic group Sp. We will refer to any unitary operator U = µ(A) as metaplectic operator

and denote the class of all µ(A) by M. The group structure of Sp implies the important composition

property

A1 ∈ Sp, A2 ∈ Sp ⇐⇒ µ(A2A1) = µ(A1)µ(A2) . (C.3)

Unfortunately, it is difficult to calculate µ(A) explicitely for arbitrary A. However, it is comparatively

simple to determine µ(A) for the following special cases [64,162] (see also Table C.1):

Scaling (A = Dd): The metaplectic operator Dd = µ(Dd) corresponding to Dd (with |d| > 0) per-

forms a scaling according to

(Ddx)(t) =1√|d|

x( td

).

Fourier Transform (A = R(T )π/2): The metaplectic operator F (T ) = µ(R(T )

π/2) corresponding to

R(T )π/2 = D1/TRπ/2DT = −D1/TJDT can be shown to act like a Fourier transform, i.e.,

(F (T )x)(t) =1

TX( t

T 2

).

3Note that due to detA = 1, any symplectic matrix A has only three free parameters.


TF scalingFourier

transform

Chirpmultiplication

Chirpconvolution

A Dd =

(1/d 0

0 d

)R(T )

π/2 =

(0 −T 2

1/T 2 0

)Cc =

(1 0

c 1

)Bb =

(1 b

0 1

)

(µ(A)x

)(t) 1√

|d|x(

td

)1T X

(t

T 2

)e−jπct2x(t) x(t) ∗ 1√

|b|e−jπt2/b

Table C.1: Basic symplectic matrices and associated metaplectic operators.

Chirp Convolution (A = Bb): The metaplectic operator Bb = µ(Bb) corresponding to Bb acts as

an LTI system with impulse response 1√|b|e−jπt2/b, i.e.,

(Bbx)(t) =1√|b|

∫

t′x(t′) e−jπ(t−t′)2/b dt′ .

Chirp Multiplication (A = Cc): The metaplectic operator Cc = µ(Cc) corresponding to Cc per-

forms a multiplication by e−jπct2,

(Ccx)(t) = x(t) e−jπct2 .

Note that this corresponds to an LFI system that is dual to the LTI system Bb.

Since any symplectic matrix A ∈ Sp can be decomposed according to at least one of the equations

in (C.2), the foregoing simple unitary operators can be combined according to (C.3) to describe the

action of any µ(A). According to (C.2), the following cases have to be distinguished.

Case 1 (a 6= 0): For a 6= 0, we can apply (C.3) twice to (C.2a) in order to obtain

µ(A) = µ(Cc1D1/aBb1) = Bb1D1/aCc1 with b1 =b

a, c1 =

c

a.

The action of this operator can be compactly written as

(µ(A)x

)(t) =

√|a||b1|

∫

t′e−jπ(t−t′)2/b1 x(at′) e−jπc1(at′)2 dt′ .

Case 2 (b 6= 0): In the case b 6= 0, application of (C.3) to (C.2b) yields

µ(A) = µ(Cc1Bb Cc2) = Cc2BbCc1 with c1 =d− 1

b, c2 =

a− 1

b.

The action of this operator can be compactly written as

(µ(A)x

)(t) = e−jπc2t2 1√

|b|

∫

t′x(t′) e−jπc1t′2 e−jπ(t−t′)2/b dt′ .

Case 3 (c 6= 0): The case c 6= 0 is dual to the case b 6= 0. Here, applying (C.3) to (C.2c) gives

µ(A) = µ(Bb1Cc Bb2) = Bb2CcBb1 with b1 =a− 1

c, b2 =

d− 1

c.

C.3 Effects on Time-Frequency Representations 233

This operator acts as

(µ(A)x

)(t) =

1√|b1b2|

∫

t1

∫

t2

x(t2)e−jπ(t1−t2)2/b1 e−jπ(t−t1)2/b2 e−jπct21 dt1 dt2 .

Case 4 (d 6= 0): Finally, for d 6= 0, application of (C.3) to (C.2d) yields the operator

µ(A) = µ(Bb1Dd Cc1) = Cc1DdBb1 with b1 =b

d, c1 =

c

d,

which acts according to

(µ(A)x

)(t) = e−jπc1t2 1√

|db1|

∫

t′x(t′) e−jπ(t/d−t′)2/b1 dt′ .

We finally note that the fractional Fourier transform (see [2] and the references therein) is a

specialization of the foregoing expressions for µ(A) to the case A = R(T )θ , i.e., for F (T )

θ = µ(R(T )θ ) we

have

(F (T )

θ x)(t) =

1

T√| sin θ|

∫

t′x(t′) ejπ

cos θsin θ

(t2+t′2

T2

)e−j2π tt′

T2 sin θ dt′ , sin θ 6= 0 ,

x(t) , θ = 0,±2π, . . . ,

x(−t) , θ = ±π,±3π, . . . .

C.3 Effects on Time-Frequency Representations

We next consider the behavior of TF representations of LTV systems and nonstationary random

processes when the system/process is subjected to unitary transformations by a metaplectic operator.

LTV Systems. We first consider unitarily transformed LTV systems H = UHU+. It can be

shown that for U = µ(A) the spreading function (i.e., the GSF with α = 0, see Subsection B.1.1) of

H can be written as

S(0)eH

(τ, ν) = S(0)H

(A(τ, ν)

)= S

(0)H

(aτ + bν, cτ + dν). (C.4)

We emphasize that this property holds only for α = 0. However, since |S(α)H

(τ, ν)| is independent of α

(cf. Section B.1.1), there is also

∣∣S(α)eH

(τ, ν)∣∣ =

∣∣S(α)H

(aτ + bν, cτ + dν)∣∣ for all α.

Similarly, the Weyl symbol (i.e., the GWS with α = 0, see Subsection B.1.2) of H can be shown to be

given by

L(0)eH

(t, f) = L(0)H

(A(t, f)

)= L

(0)H

(at+ bf, ct+ df). (C.5)

Again, this property holds only for the case α = 0. From (C.4) and (C.5), we conclude that metaplectic

transformations of LTV systems indeed correspond to area-preserving linear TF coordinate transforms.

The following special cases are of particular practical importance (see also Fig. C.1):


τ τ τ

ν ν ν

(a) (b) (c)

τ τ τ

ν ν ν

(d) (e) (f)

Figure C.1: GSF magnitude of unitarily transformed systems: (a) original system, (b)–(f) systems

transformed by (b) TF scaling, (c) Fourier transform, (d) chirp multiplication, (e) chirp convolution,

(f) fractional Fourier transform.

• Scaling: H = DdHD+d ⇐⇒

S(0)eH

(τ, ν) = S(0)H

(τd, dν)

L(0)eH

(t, f) = L(0)H

( td, df)

• Fourier transform: H = F (T )HF (T )+ ⇐⇒

S(0)eH

(τ, ν) = S(0)H

(− ν T 2,

τ

T 2

)

L(0)eH

(t, f) = L(0)H

(− f T 2,

t

T 2

)

• Chirp convolution: H = BbHB+b ⇐⇒

S

(0)eH

(τ, ν) = S(0)H

(τ + bν, ν)

L(0)eH

(t, f) = L(0)H

(t+ bf, f)

• Chirp multiplication: H = CcHC+c ⇐⇒

S

(0)eH

(τ, ν) = S(0)H

(τ, ν + cτ)

L(0)eH

(t, f) = L(0)H

(t, f + ct)

Nonstationary Random Processes. The discussion of unitary metaplectic transformations

of random processes is parallel to that of LTV systems provided previously. Indeed, the correlation

operators of the random processes x(t) and x(t) = (Ux)(t) are related by

Rx = URxU+.

By noting that the expected ambiguity function (i.e., the GEAF with α = 0, see Subsection B.3.4)

and the Wigner-Ville spectrum (i.e., the GWVS with α = 0, see Subsection B.3.1) can be written as

C.3 Effects on Time-Frequency Representations 235

A(0)x (τ, ν) = S

(0)Rx

(τ, ν) and W(0)x (t, f) = L

(0)Rx

(t, f), respectively, the foregoing results in (C.4), (C.5)

can be applied immediately:

x(t) =(µ(A)x

)(t) ⇐⇒

A

(0)x (τ, ν) = A

(0)x (A(τ, ν)) = A

(0)x (aτ + bν, cτ + dν) ,

W(0)x (t, f) = W

(0)x (A(τ, ν)) = W

(0)x (at+ bf, ct+ df) .

Next, we consider the Weyl spectrum (i.e., the GES with α = 0, see Subsection B.3.2) G(0)x (t, f) =∣∣L(0)

H(t, f)

∣∣2, where Hx is an innovations system of x(t). If Hx is an innovations system of x(t), it can

easily be checked that Hx = µ(A)Hxµ(A)+ is an innovations systems of x(t) =(µ(A)x

)(t). Thus,

the Weyl spectrum of x(t) using Hx = µ(A)Hxµ(A)+ can be related to the Weyl spectrum of x(t)

using Hx according to

G(0)x (t, f) = G(0)

x (A(τ, ν)) = G(0)x (at+ bf, ct+ df) .

We emphasize that all of the above relations for the GWVS, GEAF, and GES only hold for α = 0.

For convenience, we explicitely state the the following special cases:

• Scaling: x(t) = (Ddx)(t) ⇐⇒

A(0)x (τ, ν) = A

(0)x

(τd, dν)

W(0)x (t, f) = W

(0)x

( td, df)

G(0)x (t, f) = G

(0)x

( td, df)

• Fourier transform: x(t) = (FTx)(t) ⇐⇒

A(0)x (τ, ν) = A

(0)x

(− ν T 2,

τ

T 2

)

W(0)x (t, f) = W

(0)x

(− f T 2,− t

T 2

)

G(0)x (t, f) = G

(0)x

(− f T 2,− t

T 2

)

• Chirp convolution: x(t) = (Bbx)(t) ⇐⇒

A(0)x (τ, ν) = A

(0)x (τ + bν, ν)

W(0)x (t, f) = W

(0)x (t+ bf, f)

G(0)x (t, f) = G

(0)x (t+ bf, f)

• Chirp multiplication: x(t) = (Ccx)(t) ⇐⇒

A(0)x (τ, ν) = A

(0)x (τ, ν + cτ)

W(0)x (t, f) = W

(0)x (t, f + ct)

G(0)x (t, f) = G

(0)x (t, f + ct) .

Bibliography

[1] V. R. Algazi and D. J. Sakrison, On the optimality of Karhunen-Loeve expansion, IEEE Trans. Inf.

Theory, (1969), pp. 319–321.

[2] L. B. Almeida, The fractional Fourier transform and time-frequency representations, IEEE Trans. Signal

Processing, 42 (1994), pp. 3084–3091.

[3] M. Amin, Time-frequency spectrum analysis and estimation for non-stationary random processes, in

Advances in Spectrum Estimation, B. Boashash, ed., Melbourne: Longman Cheshire, 1992, pp. 208–232.

[4] M. G. Amin, Spectral decomposition of time–frequency distribution kernels, IEEE Trans. Signal Process-

ing, 42 (1994), pp. 1156–1165.

[5] H. Artes, G. Matz, and F. Hlawatsch, An unbiased scattering function estimator for fast time-

varying channels, in Proc. 2nd IEEE Workshop on Signal Processing Advances in Wireless Communica-

tions, Annapolis, MD, May 1999, pp. 411–414.

[6] , A novel unbiased scattering function estimator allowing data-driven operation, , (to be submitted).

[7] C. R. Baker, Optimum quadratic detection of a random vector in Gaussian noise, IEEE Trans. Comm.

Technol., 14 (1966), pp. 802–805.

[8] M. J. Bastiaans, Application of the Wigner distribution function in optics, in The Wigner Distribution

— Theory and Applications in Signal Processing, W. Mecklenbrauker and F. Hlawatsch, eds., Elsevier,

Amsterdam (The Netherlands), 1997, pp. 375–424.

[9] A. A. Beex and M. Xie, Time-varying filtering via multiresolution parametric spectral estimation, in

Proc. IEEE ICASSP-95, Detroit, MI, May 1995, pp. 1565–1568.

[10] R. T. Behrens and L. L. Scharf, Signal processing applications of oblique projection operators, IEEE

Trans. Signal Processing, 42 (1994), pp. 1413–1424.

[11] P. A. Bello, Characterization of randomly time-variant linear channels, IEEE Trans. Comm. Syst., 11

(1963), pp. 360–393.

[12] J. S. Bendat and A. G. Piersol, Measurement and Analysis of Random Data, Wiley, New York, 1966.

[13] , Engineering Applications of Correlation and Spectral Analysis, Wiley, New York, 2nd ed., 1993.

[14] F. A. Berezin, Wick and anti-Wick operator symbols, Math. USSR Sb., 15 (1971), pp. 577–606.

[15] J. Bertrand and P. Bertrand, Symbolic calculus on the time-frequency half-plane, J. Math. Phys.,

39 (1998), pp. 4071–4090.

[16] B. Boashash, ed., Time-Frequency Signal Analysis, Melbourne: Longman Cheshire and New York:

Wiley, 1992.

237

238 Bibliography

[17] J. F. Bohme and D. Konig, Statistical processing of car engine signals for combustion diagnosis, in

Proc. IEEE-SP Workshop on Statistical Signal and Array Proc., Quebec, CA, June 1994, pp. 369–374.

[18] H. Bolcskei, Blind estimation of symbol timing and carrier frequency offset in pulse shaping OFDM

systems, in Proc. IEEE ICASSP-99, Phoenix, Arizona, March 1999. to appear.

[19] , Design of pulse shaping OFDM systems for mobile radio applications, in Proc. Second IEEE Work-

shop on Signal Processing Applications in Wireless Communication, Annapolis, MD, May 1999. submitted.

[20] H. Bolcskei, P. Duhamel, and R. Hleiss, Design of pulse shaping OFDM/OQAM systems for wireless

communications with high spectral efficiency, submitted to IEEE Trans. Signal Processing, (1998).

[21] H. Bolcskei, P. Duhamel, and R. Hleiss, Design of pulse shaping OFDM/OQAM systems for high

data-rate transmission over wireless channels, in IEEE-ICC99, Vancouver, Canada, June 1999, pp. 559–

564.

[22] H. Bolcskei and F. Hlawatsch, Oversampled modulated filter banks, in Gabor Analysis and Algo-

rithms: Theory and Applications, H. G. Feichtinger and T. Strohmer, eds., Birkhauser, Boston, MA,

1998, ch. 9, pp. 295–322.

[23] H. Bolcskei, F. Hlawatsch, and H. G. Feichtinger, Oversampled FIR and IIR DFT filter banks

and Weyl-Heisenberg frames, in Proc. IEEE ICASSP-96, vol. 3, Atlanta, GA, May 1996, pp. 1391–1394.

[24] R. Bourdier, J. F. Allard, and K. Trumpf, Effective frequency response and signal replica genera-

tion for filtering algorithms using multiplicative modifications of the STFT, Signal Processing, 15 (1988),

pp. 193–201.

[25] A. Calderon and R. Vaillancourt, On the boundedness of pseudodifferential operators, J. Math. Soc.

Japan, 23 (1971), pp. 374–378.

[26] S. Carstens-Behrens, M. Wagner, and J. F. Bohme, Detection of multiple resonances in noise,

Int. J. Electron. Commun. (AEU), 52 (1998), pp. 285–292.

[27] S. Carstens-Behrens, M. Wagner, and J. F. Bohme, Improved knock detection by time-variant

filtered structure-borne sound, in Proc. IEEE ICASSP-99, Phoenix, AZ, March 1999, pp. 2255–2258.

[28] R. W. Chang, Synthesis of band-limited orthogonal signals for multi-channel data transmission, Bell

Syst. Tech. J., 45 (1966), pp. 1775–1796.

[29] J. S. Chow, J. C. Tu, and J. M. Cioffi, A discrete multitone transceiver system for HDSL applications,

IEEE J. Sel. Areas Comm., 9 (1991), pp. 895–908.

[30] L. J. Cimini, Analysis and simulation of a digital mobile channel using orthogonal frequency division

multiplexing, IEEE Trans. Comm., 33 (1985), pp. 665–675.

[31] T. A. C. M. Claasen and W. F. G. Mecklenbrauker, The Wigner distribution—A tool for time-

frequency signal analysis; Part I: Continuous-time signals, Part II: Discrete-time signals, Part III: Re-

lations with other time-frequency signal transformations, Philips J. Research, 35 (1980), pp. 217–250,

276–300, and 372–389.

[32] M. Coates, Time-Frequency Modelling, PhD thesis, Department of Engineering, University of Cam-

bridge, 1998.

[33] L. Cohen, Generalized phase-space distribution functions, J. Math. Phys., 7 (1966), pp. 781–786.

[34] , Quantization problem and variational principle in the phase-space formulation of quantum mechan-

ics, J. Math. Phys., 17 (1976), pp. 1863–1866.

[35] , Time-Frequency Analysis, Prentice Hall, Englewood Cliffs (NJ), 1995.

Bibliography 239

[36] A. Cordoba and C. Fefferman, Wave packets and Fourier integral operators, Comm. Partial Diff. Eq.,

3 (1978), pp. 979–1005.

[37] D. Cox, Delay Doppler characteristics of multipath propagation in a suburban mobile radio environment,

IEEE Trans. Antennas and Propagation, 20 (1972), pp. 625–635.

[38] H. Cramer, On some classes of nonstationary stochastic processes, in Proc. 4th Berkeley Symp. on Math.

Stat. and Prob., Univ. Calif. Press, 1961, pp. 57–78.

[39] R. E. Crochiere and L. R. Rabiner, Multirate Digital Signal Processing, Prentice Hall, Englewood

Cliffs (NJ), 1983.

[40] P. J. Cullen, P. C. Fannin, and A. Molina, Wide-band measurement and analysis techniques for

the mobile radio channel, IEEE Trans. Vehicular Technology, 42 (1993), pp. 589–603.

[41] G. S. Cunningham and W. J. Williams, Kernel decomposition of time-frequency distributions, IEEE

Trans. Signal Processing, 42 (1994), pp. 1425–1442.

[42] I. Daubechies, Time-frequency localization operators: A geometric phase space approach, IEEE Trans.

Inf. Theory, 34 (1988), pp. 605–612.

[43] , The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inf. Theory,

36 (1990), pp. 961–1005.

[44] I. Daubechies and T. Paul, Time-frequency localization operators: A geometric phase space approach—

II. The use of dilations, Inverse Problems, 4 (1988), pp. 661–680.

[45] N. G. de Bruijn, Uncertainty principles in Fourier analysis, in Inequalities, O. Shisha, ed., Academic

Press, New York, 1967, pp. 57–71.

[46] , A theory of generalized functions, with applications to Wigner distribution and Weyl correspondence,

Nieuw Archief voor Wiskunde (3), XXI (1973), pp. 205–280.

[47] A. Dembo and D. Malah, Signal synthesis from modified discrete short-time transform, IEEE Trans.

Acoust., Speech, Signal Processing, 36 (1988), pp. 168–181.

[48] V. Demoulin and M. Pecot, Vector equalization: An alternative approach for OFDM systems, Annales

des Telecommunications, 52 (1997), pp. 4–11.

[49] C. S. Detka and A. El-Jaroudi, The transitory evolutionary spectrum, in Proc. IEEE ICASSP-94,

Adelaide, Australia, April 1994, pp. 289–292.

[50] D. L. Donoho, M. Vetterli, R. A. DeVore, and I. Daubechies, Data compression and harmonic

analysis, IEEE Trans. Inf. Theory, 44 (1998), pp. 2435–2477.

[51] O. Edfors, M. Sandell, J.-J. van de Beek, S. Wilson, and P. Borjesson, OFDM channel

estimation by singular value decomposition, IEEE Trans. Comm., 46 (1998), pp. 931–939.

[52] P. C. Fannin, A. Molina, S. S. Swords, and P. J. Cullen, Digital signal processing techniques

applied to mobile radio channel sounding, Proc. IEE-F, 138 (1991), pp. 502–508.

[53] S. Farkash and S. Raz, Linear systems in Gabor time-frequency space, IEEE Trans. Signal Processing,

42 (1994), pp. 611–617.

[54] C. L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc., 9 (1983), pp. 129–206.

[55] H. G. Feichtinger and K. Grochenig, Gabor wavelets and the Heisenberg group: Gabor expansions

and short time Fourier transform from the group theoretical point of view, in Wavelets — A Tutorial in

Theory and Applications, C. Chui, ed., Academic Press, Boston, 1992, pp. 359–397.

[56] H. G. Feichtinger and T. Strohmer, eds., Gabor Analysis and Algorithms: Theory and Applications,

Birkhauser, Boston (MA), 1998.

240 Bibliography

[57] P. Flandrin, On the positivity of the Wigner-Ville spectrum, Signal Processing, 11 (1986), pp. 187–189.

[58] P. Flandrin, A time-frequency formulation of optimum detection, IEEE Trans. Acoust., Speech, Signal

Processing, 36 (1988), pp. 1377–1384.

[59] P. Flandrin, Time-frequency receivers for locally optimum detection, in Proc. IEEE ICASSP-83, New

York, 1988, pp. 2725–2728.

[60] , Time-dependent spectra for nonstationary stochastic processes, in Time and Frequency Represen-

tation of Signals and Systems, G. Longo and B. Picinbono, eds., Springer, Wien, 1989, pp. 69–124.

[61] , Time-Frequency/Time-Scale Analysis, Academic Press, San Diego (CA), 1999.

[62] P. Flandrin, B. Escudie, and J. Grea, Regles des correspondance et ⋆-produits dans le plan temps-

frequence, C. R. Acad. Sc. Paris, serie I, 294 (1982), pp. 281–284.

[63] P. Flandrin and W. Martin, The Wigner-Ville spectrum of nonstationary random signals, in

The Wigner Distribution — Theory and Applications in Signal Processing, W. Mecklenbrauker and

F. Hlawatsch, eds., Elsevier, Amsterdam (The Netherlands), 1997, pp. 211–267.

[64] G. B. Folland, Harmonic Analysis in Phase Space, vol. 122 of Annals of Mathematics Studies, Princeton

University Press, Princeton (NJ), 1989.

[65] G. B. Folland and A. Sitaram, The uncertainty principle: A mathematical survey, J. Fourier Anal.

Appl., 3 (1997), pp. 207–238.

[66] D. Gabor, Theory of communication, J. IEE, 93 (1946), pp. 429–457.

[67] W. A. Gardner, Statistical Spectral Analysis, Prentice Hall, New Jersey, 1988.

[68] I. M. Gelfand and G. E. Schilow, Verallgemeinerte Funktionen (Distributionen), VEB, Berlin, 1960.

[69] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Amer.

Math. Soc., Providence (RI), 1969.

[70] G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore,

2 ed., 1989.

[71] O. D. Grace, Instantaneous power spectra, J. Acoust. Soc. Amer., 69 (1981), pp. 191–198.

[72] P. E. Green, Jr., Radar measurements of target scattering properties, in Radar Astronomy, J. V. Evans

and T. Hagfors, eds., McGraw-Hill, New York, 1968, ch. 1.

[73] A. Grossmann, G. Loupias, and E. M. Stein, An algebra of pseudodifferential operators and quantum

mechanics in phase space, Ann. Inst. Fourier, 18 (1968), pp. 343–368.

[74] T. Hagfors and J. M. Moran, Detection and estimation practices in radio and radar astronomy, Proc.

IEEE, 58 (1970), pp. 743–759.

[75] L. Hanzo, W. Webb, and T. Keller, Single- and multi-carrier quadrature amplitude modulation,

Wiley, Chichester, UK, 2000.

[76] S. Haykin and D. J. Thomson, Signal detection in a nonstationary environment reformulated as an

adaptive pattern classification problem, Proc. IEEE, 86 (1998), pp. 2325–2345.

[77] R. D. Hippenstiel and P. M. De Oliveira, Time-varying spectral estimation using the instantaneous

power spectrum (IPS), IEEE Trans. Acoust., Speech, Signal Processing, 38 (1990), pp. 1752–1759.

[78] F. Hlawatsch, Duality and classification of bilinear time-frequency signal representations, IEEE Trans.

Signal Processing, 39 (1991), pp. 1564–1574.

[79] , Wigner distribution analysis of linear, time-varying systems, in Proc. IEEE ISCAS-92, San Diego,

CA, May 1992, pp. 1459–1462.

Bibliography 241

[80] , Time-frequency analysis and synthesis of linear signal spaces, Tech. Rep. #95-03, Institute of

Communications and Radio-Frequency Engineering, Vienna University of Technology, Vienna, 1996.

[81] , Time-Frequency Analysis and Synthesis of Linear Signal Spaces: Time-Frequency Filters, Signal

Detection and Estimation, and Range-Doppler Estimation, Kluwer, Boston, 1998.

[82] , The covariance principle in time-frequency analysis, in Encyclopedia of Signal Processing,

A. Poularikas, ed., CRC Press, Boca Raton (FL), 2001.

[83] F. Hlawatsch and H. Bolcskei, Covariant time-frequency distributions based on conjugate operators,

IEEE Signal Processing Letters, 3 (1996), pp. 44–46.

[84] F. Hlawatsch and G. F. Boudreaux-Bartels, Linear and quadratic time-frequency signal represen-

tations, IEEE Signal Processing Magazine, 9 (1992), pp. 21–67.

[85] F. Hlawatsch and P. Flandrin, The interference structure of the Wigner distribution and related

time-frequency signal representations, in The Wigner Distribution — Theory and Applications in Signal

Processing, W. Mecklenbrauker and F. Hlawatsch, eds., Elsevier, Amsterdam, The Netherlands, 1997,

pp. 59–133.

[86] F. Hlawatsch and W. Kozek, Time-frequency weighting and displacement effects in linear, time-

varying systems, in Proc. IEEE ISCAS-92, San Diego, CA, May 1992, pp. 1455–1458.

[87] , The Wigner distribution of a linear signal space, IEEE Trans. Signal Processing, 41 (1993),

pp. 1248–1258.

[88] , Time-frequency projection filters and time-frequency signal expansions, IEEE Trans. Signal Pro-

cessing, 42 (1994), pp. 3321–3334.

[89] , Second-order time-frequency synthesis of nonstationary random processes, IEEE Trans. Inf. Theory,

41 (1995), pp. 255–267.

[90] F. Hlawatsch and G. Matz, Quadratic time-frequency analysis of linear time-varying systems, in

Wavelet Transforms and Time-Frequency Signal Analysis, L. Debnath, ed., Birkhauser, Boston (MA),

2000, ch. 9.

[91] , Time-frequency characterization of random time-varying channels, in Encyclopedia of Signal Pro-

cessing, A. Poularikas, ed., CRC Press, Boca Raton (FL), 2001.

[92] F. Hlawatsch, G. Matz, H. Kirchauer, and W. Kozek, Time-frequency formulation, design, and

implementation of time-varying optimal filters for signal estimation, IEEE Trans. Signal Processing, 48

(2000), pp. 1417–1432.

[93] F. Hlawatsch, G. Taubock, and T. Twaroch, Covariant time-frequency analysis, in Wavelets and

Signal Processing, L. Debnath, ed., Birkhauser, Boston (MA), 2001, ch. 7.

[94] L. Hormander, The Weyl calculus of pseudo-differential operators, Comm. Pure Appl. Math., 32 (1979),

pp. 359–443.

[95] R. Howe, Quantum mechanics and partial differential equations, J. Funct. Anal., 38 (1980), pp. 188–254.

[96] B. Iem, Generalization of the Weyl symbol and the spreading function via time-frequency warpings: Theory

and applications, PhD thesis, Univ. Rhode Island, 1998.

[97] B. Iem, A. Papandreou-Suppappola, and G. F. Boudreaux-Bartels, Time-frequency symbols for

statistical signal processing, in Encyclopedia of Signal Processing, A. Poularikas, ed., CRC Press, Boca

Raton (FL), 2001.

[98] A. J. E. M. Janssen, On the locus and spread of pseudo-density functions in the time-frequency plane,

Philips J. Research, 37 (1982), pp. 79–110.

242 Bibliography

[99] , Wigner weight functions and Weyl symbols of non-negative definite linear operators, Philips J.

Research, 44 (1989), pp. 7–42.

[100] , Duality and biorthogonality for Weyl-Heisenberg frames, J. Fourier Analysis and Applications, 1

(1995), pp. 403–436.

[101] , Positivity and spread of bilinear time-frequency distributions, in The Wigner Distribution — Theory

and Applications in Signal Processing, W. Mecklenbrauker and F. Hlawatsch, eds., Elsevier, Amsterdam

(The Netherlands), 1997, pp. 1–58.

[102] T. Kailath, Channel characterization: time-variant dispersive channels, in Lectures on Communication

Theory, E. J. Baghdady, ed., McGraw–Hill, 1961, pp. 95–123.

[103] , Measurements on time-variant communication channels, IEEE Trans. Inf. Theory, 8 (1962),

pp. 229–236.

[104] , Report on progress in information theory in the USA, 1960–1963, Part IV: Time-variant commu-

nication channels, IEEE Trans. Inf. Theory, 9 (1963), pp. 233–237.

[105] , Linear Systems, Prentice Hall, Englewood Cliffs (NJ), 1980.

[106] , Lectures on Wiener and Kalman Filtering, vol. 140 of CISM Courses and Lectures, Springer, Wien,

1981.

[107] K. Karhunen, Uber lineare Methoden in der Wahrscheinlichkeitsrechnung, Ann. Acad. Sci. Fennicae,

ser. A (I), 18 (1947).

[108] S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory, Prentice Hall, Upper Saddle

River (NJ), 1998.

[109] R. S. Kennedy, Fading Dispersive Communication Channels, Wiley, New York, 1969.

[110] H. A. Khan and L. F. Chaparro, Nonstationary Wiener filtering based on evolutionary spectral theory,

in Proc. IEEE ICASSP-97, Munich, Germany, May 1997, pp. 3677–3680.

[111] H. Kirchauer, F. Hlawatsch, and W. Kozek, Time-frequency formulation and design of nonsta-

tionary Wiener filters, in Proc. IEEE ICASSP-95, Detroit, MI, May 1995, pp. 1549–1552.

[112] J. J. Kohn and L. Nirenberg, An algebra of pseudo-differential operators, Comm. Pure Appl. Math.,

18 (1965), pp. 269–305.

[113] D. Konig and J. F. Bohme, Application of cyclostationary and time-frequency signal analysis to car

engine diagnosis, in Proc. IEEE ICASSP-94, Adelaide, Australia, April 1994, pp. 149–152.

[114] W. Kozek, On the generalized Weyl correspondence and its application to time-frequency analysis of linear

time-varying systems, in Proc. IEEE-SP Int. Sympos. Time-Frequency Time-Scale Analysis, Victoria,

Canada, Oct. 1992, pp. 167–170.

[115] , Matched generalized Gabor expansion of nonstationary processes, in Proc. 27th Asilomar Conf.

Signals, Systems, Computers, Pacific Grove, CA, Nov. 1993, pp. 499–503.

[116] , Optimally Karhunen-Loeve-like STFT expansion of nonstationary random processes, in Proc. IEEE

ICASSP-93, Minneapolis, MN, April 1993, pp. 428–431.

[117] , On the underspread/overspread classification of nonstationary random processes, in Proc. Int.

Conf. Industrial and Applied Mathematics (Hamburg, July 1995), K. Kirchgassner, O. Mahrenholtz,

and R. Mennicken, eds., vol. 3 of Mathematical Research, Berlin, 1996, Akademieverlag, pp. 63–66.

[118] , Matched Weyl-Heisenberg Expansions of Nonstationary Environments, PhD thesis, Vienna Univer-

sity of Technology, March 1997.

Bibliography 243

[119] , On the transfer function calculus for underspread LTV channels, IEEE Trans. Signal Processing,

45 (1997), pp. 219–223.

[120] , Adaptation of Weyl-Heisenberg frames to underspread environments, in Gabor Analysis and Algo-

rithms: Theory and Applications, H. G. Feichtinger and T. Strohmer, eds., Birkhauser, Boston (MA),

1998, ch. 10, pp. 323–352.

[121] W. Kozek and H. Feichtinger, Time-frequency structured decorrelation of speech signals via nonsep-

arable Gabor frames, in Proc. IEEE ICASSP-97, Munich, Germany, 1997.

[122] W. Kozek and H. G. Feichtinger, Time-frequency structured decorrelation of speech signals via non-

separable Gabor frames, in Proc. IEEE ICASSP-97, Munich, Germany, April 1997, pp. 1439–1442.

[123] W. Kozek, H. G. Feichtinger, and J. Scharinger, Matched multiwindow methods for the estimation

and filtering of nonstationary processes, in Proc. IEEE ISCAS-96, Atlanta, GA, May 1996, pp. 509–512.

[124] W. Kozek, H. G. Feichtinger, and T. Strohmer, Time-frequency synthesis of statistically matched

Weyl-Heisenberg prototype signals, in Proc. IEEE Int. Symp. Time-Frequency/Time-Scale Analysis,

Philadelphia (PA), Oct. 1994, pp. 417–420.

[125] W. Kozek and F. Hlawatsch, A comparative study of linear and nonlinear time-frequency filters, in

Proc. IEEE-SP Int. Sympos. Time-Frequency Time-Scale Analysis, Victoria, Canada, Oct. 1992, pp. 163–

166.

[126] W. Kozek, F. Hlawatsch, H. Kirchauer, and U. Trautwein, Correlative time-frequency analysis

and classification of nonstationary random processes, in Proc. IEEE-SP Int. Sympos. Time-Frequency

Time-Scale Analysis, Philadelphia, PA, Oct. 1994, pp. 417–420.

[127] W. Kozek and A. F. Molisch, On the eigenstructure of underspread WSSUS channels, in Proc.

IEEE Workshop on Signal Processing Applications in Wireless Communication, Paris, France, April

1997, pp. 325–328.

[128] , Nonorthogonal pulseshapes for multicarrier communications in doubly dispersive channels, IEEE J.

Sel. Areas Comm., 16 (1998), pp. 1579–1589.

[129] W. Kozek and K. Riedel, Quadratic time-varying spectral estimation for underspread processes, in

Proc. IEEE-SP Int. Sympos. Time-Frequency Time-Scale Analysis, Philadelphia, PA, Oct. 1994, pp. 460–

463.

[130] B. V. K. Kumar and K. J. deVos, Linear system description using Wigner distribution functions, in

Proc. SPIE, “Advanced Algorithms and Architectures for Signal Processing II,” vol. 826, 1987, pp. 115–

124.

[131] P. Lander and E. J. Berbari, Time-frequency plane Wiener filtering of the high-resolution ECG:

Background and time-frequency representations, IEEE Trans. Biomedical Engineering, 44 (1997), pp. 247–

255.

[132] , Time-frequency plane Wiener filtering of the high-resolution ECG: Development and application,

IEEE Trans. Biomedical Engineering, 44 (1997), pp. 256–265.

[133] B. LeFloch, M. Alard, and C. Berrou, Coded orthogonal frequency division multiplex, Proc. of

IEEE, 83 (1995), pp. 982–996.

[134] Y. Li, L. Cimini, and N. Sollenberger, Robust channel estimation for OFDM systems with rapid

dispersive fading channels, IEEE Trans. Comm., 46 (1998), pp. 902–915.

[135] Y. Lin and P. P. Vaidyanathan, Application of DFT filter banks and cosine modulated filter banks in

filtering, in Proc. APCCAS-94, Taipei, Taiwan, 1994, pp. 254–259.

[136] M. Loeve, Probability Theory, Van Nostrand, Princeton (NJ), 3rd ed., 1962.

244 Bibliography

[137] S. G. Mallat, A Wavelet Tour of Signal Processing, Academic Press, San Diego, 1998.

[138] S. G. Mallat, G. Papanicolaou, and Z. Zhang, Adaptive covariance estimation of locally stationary

processes, Annals of Stat., 26 (1998).

[139] W. D. Mark, Spectral analysis of the convolution and filtering of non-stationary stochastic processes, J.

Sound Vib., 11 (1970), pp. 19–63.

[140] W. Martin and P. Flandrin, Wigner-Ville spectral analysis of nonstationary processes, IEEE Trans.

Acoust., Speech, Signal Processing, 33 (1985), pp. 1461–1470.

[141] G. Matz and F. Hlawatsch, Time-frequency formulation and design of optimal detectors, in Proc.

IEEE-SP Int. Sympos. Time-Frequency Time-Scale Analysis, Paris, France, June 1996, pp. 213–216.

[142] , Time-frequency formulation and design of optimal detectors in nonstationary environments, Tech.

Rep. #96-05, Dept. of Communications and Radio-Frequency Engineering, Vienna University of Technol-

ogy, Sept. 1996.

[143] , Time-frequency methods for signal detection with application to the detection of knock in car engines,

in Proc. IEEE-SP Workshop on Statistical Signal and Array Proc., Portland, OR, Sept. 1998, pp. 196–199.

[144] , Time-frequency transfer function calculus (symbolic calculus) of linear time-varying systems (linear

operators) based on a generalized underspread theory, J. Math. Phys., Special Issue on Wavelet and Time-

Frequency Analysis, 39 (1998), pp. 4041–4071.

[145] , Time-varying spectra for underspread and overspread nonstationary processes, in Proc. 32nd Asilo-

mar Conf. Signals, Systems, Computers, Pacific Grove, CA, Nov. 1998, pp. 282–286.

[146] , Time-frequency subspace detectors and application to knock detection, Int. J. Electron. Commun.

(AEU), 53 (1999), pp. 379–385.

[147] G. Matz, F. Hlawatsch, and W. Kozek, Weyl spectral analysis of nonstationary random processes, in

IEEE UK Sympos. Applications of Time-Frequency and Time-Scale Methods, Univ. of Warwick, Coventry,

UK, Aug. 1995, pp. pp. 120–127.

[148] , Generalized evolutionary spectral analysis and the Weyl spectrum of nonstationary random pro-

cesses, IEEE Trans. Signal Processing, 45 (1997), pp. 1520–1534.

[149] G. Matz, A. F. Molisch, F. Hlawatsch, M. Steinbauer, and I. Gaspard, On the systematic

measurement errors of correlative mobile radio channel sounders. Submitted to IEEE Trans. Comm.

[150] G. Matz, A. F. Molisch, M. Steinbauer, F. Hlawatsch, I. Gaspard, and H. Artes, Bounds on

the systematic measurement errors of channel sounders for time-varying mobile radio channels, in Proc.

IEEE VTC-99 Fall, Amsterdam, The Netherlands, Sept. 1999, pp. 1465–1470.

[151] W. Mecklenbrauker and F. Hlawatsch, eds., The Wigner Distribution — Theory and Applications

in Signal Processing, Elsevier, Amsterdam (The Netherlands), 1997.

[152] G. Melard, Proprietes du spectre evolutif d’un processus non-stationnaire, Ann. Inst. H. Poincare B,

XIV (1978), pp. 411–424.

[153] Y. Meyer, Wavelets and operators, in AMS Proc. of Symposia in Appl. Math.: Different Perspectives

on Wavelets, I. Daubechies, ed., AMS, Providence (RI), 1993, pp. 35–58.

[154] W. Miller, Jr., Topics in harmonic analysis with applications to radar and sonar, in Radar and Sonar,

Part I, R. E. Blahut, W. Miller, Jr., and C. H. Wilcox, eds., Springer, New York, 1991, pp. 66–168.

[155] F. Molinaro and F. Castanie, A comparison of time-frequency methods, in Signal Processing V:

Theories and Applications, L. Torres, E. Masgrau, and M. A. Lagunas, eds., Amsterdam, 1990, Elsevier,

pp. 145–148.

Bibliography 245

[156] A. F. Molisch, G. Matz, and F. Hlawatsch, Systematic errors of calibrated correlative channel

sounders and improved calibration. To be submitted.

[157] S. H. Nawab and T. F. Quatieri, Short-time Fourier transform, in Advanced Topics in Signal Process-

ing, J. S. Lim and A. V. Oppenheim, eds., Prentice Hall, Englewood Cliffs, NJ, 1988, ch. 6, pp. 289–337.

[158] A. W. Naylor and G. R. Sell, Linear Operator Theory in Engineering and Science, Springer, New

York, 2nd ed., 1982.

[159] K. Nowak, Some eigenvalue estimates for wavelet related Toeplitz operators, Colloquium Mathematicum,

LXV, Fasc. 1 (1993), pp. 149–156.

[160] A. V. Oppenheim, A. S. Willsky, and I. T. Young, Signals and Systems, Prentice Hall, Englewood

Cliffs (NJ), 1983.

[161] C. H. Page, Instantaneous power spectra, J. Appl. Phys., 23 (1952), pp. 103–106.

[162] A. Papoulis, Signal Analysis, McGraw-Hill, Singapore, 1984.

[163] , Probability, Random Variables, and Stochastic Processes, McGraw-Hill, New York, 3rd ed., 1991.

[164] J. D. Parsons, D. A. Demery, and A. M. D. Turkmani, Sounding techniques for wideband mobile

radio channels: A review, Proc. IEE-I, 138 (1991), pp. 437–446.

[165] A. Peled and A. Ruiz, Frequency domain data transmission using reduced computational complexity

algorithms, in Proc. IEEE ICASSP-80, Denver, CO, 1980, pp. 964–967.

[166] M. Poize, M. Renaudin, and P. Venier, The Gabor transform as a modulated filter bank system, in

Quatorzieme Colloque GRETSI, Juan-Les-Pins, France, Sept. 1993, pp. 351–354.

[167] J. C. T. Pool, Mathematical aspects of the Weyl correspondence, J. Math. Phys., 7 (1966), pp. 66–76.

[168] H. V. Poor, An Introduction to Signal Detection and Estimation, Springer, New York, 1988.

[169] M. R. Portnoff, Time-frequency representation of digital signals and systems based on short-time

Fourier analysis, IEEE Trans. Acoust., Speech, Signal Processing, 28 (1980), pp. 55–69.

[170] M. B. Priestley, Evolutionary spectra and non-stationary processes, J. Roy. Stat. Soc. Ser. B, 27 (1965),

pp. 204–237.

[171] , Spectral Analysis and Time Series – Part II, Academic Press, London, 1981.

[172] J. G. Proakis, Digital Communications, McGraw-Hill, New York, 3rd ed., 1995.

[173] S. Qian and D. Chen, Joint Time-Frequency Analysis, Prentice Hall, Englewood Cliffs (NJ), 1996.

[174] J. Ramanathan and P. Topiwala, Time-frequency localization via the Weyl correspondence, SIAM J.

Matrix Anal. Appl., 24 (1993), pp. 1378–1393.

[175] , Time-frequency localization and the spectrogram, Applied and Computational Harmonic Analysis,

1 (1994), pp. 209–215.

[176] T. S. Rappaport, Wireless Communications: Principles & Practice, Prentice Hall, Upper Saddle River,

New Jersey, 1996.

[177] K. Riedel, Optimal data-based kernel estimation of evolutionary spectra, IEEE Trans. Signal Processing,

41 (1993), pp. 2439–2447.

[178] A. W. Rihaczek, Signal energy distribution in time and frequency, IEEE Trans. Inf. Theory, 14 (1968),

pp. 369–374.

[179] R. Rochberg, Toeplitz and Hankel operators, wavelets, NWO sequences, and almost diagonalization of

operators, Lecture Notes in Pure and Appl. Math., 51 (1990), pp. 425–444.

246 Bibliography

[180] S. Salous, N. Nikandrou, and N. Bajj, Digital techniques for mobile radio chirp sounders, IEE Proc.

Commun., 145 (1998), pp. 191–196.

[181] B. Samimy and G. Rizzoni, Mechanical signature analysis using time-frequency signal processing: Ap-

plication to internal combustion engine knock detection, Proc. IEEE, 84 (1996), pp. 1330–1343.

[182] M. Sandell, Design and analysis of estimators for multicarrier modulation and ultrasonic imaging, PhD

thesis, Lulea University of Technology, Lulea, Sweden, 1996.

[183] A. M. Sayeed and D. L. Jones, Optimal detection using bilinear time-frequency and time-scale repre-

sentations, IEEE Trans. Signal Processing, 43 (1995), pp. 2872–2883.

[184] A. M. Sayeed and D. L. Jones, A canonical covariance-based method for generalized joint signal

representations, IEEE Signal Processing Letters, 3 (1996), pp. 121–123.

[185] A. M. Sayeed and D. L. Jones, Optimal reduced-rank time-frequency/time-scale detectors, in Proc.

IEEE-SP Int. Sympos. Time-Frequency Time-Scale Analysis, Paris, June 1996, pp. 209–212.

[186] A. M. Sayeed, P. Lander, and D. L. Jones, Improved time-frequency filtering of signal-averaged

electrocardiograms, J. of Electrocardiology, 28 (1995), pp. 53–58.

[187] L. L. Scharf, Statistical Signal Processing, Addison Wesley, Reading (MA), 1991.

[188] L. L. Scharf and B. Friedlander, Matched subspace detectors, IEEE Trans. Signal Processing, 42

(1994), pp. 2146–2157.

[189] L. L. Scharf and J. B. Thomas, Wiener filters in canonical coordinates for transform coding, filtering,

and quantizing, IEEE Trans. Signal Processing, 46 (1998), pp. 647–654.

[190] R. G. Shenoy and T. W. Parks, The Weyl correspondence and time-frequency analysis, IEEE Trans.

Signal Processing, 42 (1994), pp. 318–331.

[191] J. A. Sills, Nonstationary Signal Modeling, Filtering, and Parameterization, PhD thesis, Georgia Insti-

tute of Technology, Atlanta, March 1995.

[192] J. A. Sills and E. W. Kamen, Wiener filtering of nonstationary signals based on spectral density

functions, in Proc. 34th IEEE Conf. Decision and Control, Kobe, Japan, Dec. 1995, pp. 2521–2526.

[193] , Time-varying matched filters, Circuits, Systems, Signal Processing, 15 (1996), pp. 609–630.

[194] M. Skolnik, Radar Handbook, McGraw-Hill, New York, 1984.

[195] K. A. Sostrand, Mathematics of the time-varying channel, Proc. NATO Advanced Study Inst. on Signal

Processing with Emphasis on Underwater Acoustics, 2 (1968), pp. 25.1–25.20.

[196] K. Swaminathan and P. P. Vaidyanathan, Theory and design of uniform DFT, parallel, quadrature

mirror filter banks, IEEE Trans. Circuits and Systems, 33 (1986), pp. 1170–1191.

[197] C. W. Therrien, Discrete Random Signals and Statistical Signal Processing, Prentice Hall, Englewood

Cliffs (NJ), 1992.

[198] R. Thoma, J. Steffens, and U. Trautwein, Statistical cross-terms in quadratic time-frequency dis-

tributions, in Proc. Int. Conf. on DSP, Nicosia, Cyprus, July 1993.

[199] D. J. Thomson, Multi-window spectrum estimation for non-stationary data, in Proc. IEEE-SP Workshop

on Statistical Signal and Array Proc., Portland, OR, Sept. 1998, pp. 344–347.

[200] R. Tolimieri and M. An, Time-Frequency Representations, Birkhauser, Boston, 1998.

[201] A. Vahlin and N. Holte, Optimal finite duration pulses for OFDM, IEEE Trans. Comm., 4 (1996),

pp. 10–14.

Bibliography 247

[202] H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I: Detection, Estimation, and

Linear Modulation Theory, Wiley, New York, 1968.

[203] , Detection, Estimation, and Modulation Theory, Part III: Radar-Sonar Signal Processing and Gaus-

sian Signals in Noise, Krieger, Malabar (FL), 1992.

[204] M. Vetterli, A theory of multirate filter banks, IEEE Trans. Acoust., Speech, Signal Processing, 35

(1987), pp. 356–372.

[205] J. Ville, Theorie et applications de la notion de signal analytique, Cables & Transm., 2eme A. (1948),

pp. 61–74.

[206] A. Voros, An algebra of pseudodifferential operators and the asymptotics of quantum mechanics, J. Funct.

Anal., 29 (1978), pp. 104–132.

[207] M. Wagner, E. Karlsson, D. Konig, and C. Tork, Time variant system identification for car

engine signal analysis, in Proc. EUSIPCO-94, Edinburgh, Sept. 1994, pp. 1409–1412.

[208] A. Weil, Sur certaines groupes d’operateurs unitaires, Acta Math., 111 (1964), pp. 143–211.

[209] S. B. Weinstein and P. M. Ebert, Data transmission by frequency division multiplexing using the

discrete Fourier transform, IEEE Trans. Comm. Tech., 19 (1971), pp. 628–634.

[210] L. B. White and B. Boashash, Cross spectral analysis of nonstationary processes, IEEE Trans. Inf.

Theory, 36 (1990), pp. 830–835.

[211] E. P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), pp. 749–

759.

[212] C. H. Wilcox, The synthesis problem for radar ambiguity functions, in Radar and Sonar, Part I, R. E.

Blahut, W. Miller, Jr., and C. H. Wilcox, eds., Springer, New York, 1991, pp. 229–260.

[213] P. M. Woodward, Probability and Information Theory with Application to Radar, Pergamon Press,

London, 1953.

[214] L. A. Zadeh, Frequency analysis of variable networks, Proc. of IRE, 76 (1950), pp. 291–299.

[215] W. Y. Zou and Y. Wu, COFDM: An overview, IEEE Trans. Broadc., 41 (1995), pp. 1–8.

List of Abbreviations

BFDM biorthogonal frequency division multiplexing

CL correlation-limited

DL displacement-limited

EVD eigenvalue decomposition

GAF generalized ambiguity function

GEAF generalized expected ambiguity function

GES generalized evolutionary spectrum

GSF generalized spreading function

GWD generalized Wigner distribution

GWS generalized Weyl symbol

GWVS generalized Wigner-Ville spectrum

HS Hilbert-Schmidt

ICI interchannel interference

ISI intersymbol interference

KL Karhunen-Loeve

LFI linear frequency-invariant

LTI linear time-invariant

LTV linear time-varying

OFDM orthogonal frequency division multiplexing

PSD power spectral density

ROC receiver operating characteristics

STFT short-time Fourier transform

SVD singular value decomposition

TF time-frequency

249

a time-frequency calculus for time-varying systems … · and nonstationary processes with...

Documents