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TITLE PAGE

MIMO TRANSMISSION FOR 4G WIRELESS COMMUNICATIONS

Pedro M. M. Marques

Dissertation for the Degree of Doctor of Philosophy inElectrical and Computer Engineering

Faculdade de Engenharia da Universidade do Porto

Porto, October 1, 2008

Dissertation submitted toFaculdade de Engenharia da Universidade do Porto

in partial fulfillment of the requirements forthe Degree of Doctor of Philosophy inElectrical and Computer Engineering

Author: Pedro Manuel Martins Marquesdepartment, faculty, code: DEEC, FEUP, 030535009

Supervisor: Prof. Sílvio Almeida Abrantes Moreiradepartment, faculty: DEEC, FEUP

DEDICATION v

DEDICATION

To my family, and Raquel

vi DEDICATION

ACKNOWLEDGMENTS vii

ACKNOWLEDGMENTS

I would like to express my honest gratitude towards my thesis supervisor, Prof.Sílvio Abrantes, for his initial encouragement, his timely orientation and contin-ued support. I thank him for his humane character and accessibility, for alleviat-ing the pressure and for being patient when the pace of investigation becameslow. I am indebted with him.

I also thank my family for being there at all times. Without their incentiveand comfort this thesis would be hardly possible. I also cannot forget Raquel,whose dedication and appreciation brought new inspiration to my work.

Finally, I have to thank two institutions: 1) the Faculdade de Engenhariada Universidade do Porto (FEUP), Porto, Portugal, for approving my Ph.D pro-gram and providing electronic access to research literature; and 2) the Ministérioda Ciência e Ensino Superior, Fundação para a Ciência e Tecnologia (FCT), Lis-bon, Portugal, for the financial support of Grant SFRH/BD/17131/2004.

viii ACKNOWLEDGMENTS

ABSTRACT ix

ABSTRACT

The pervasive challenge of increasingly higher spectral efficiencies in wirelesscommunications has fomented the onset of various exceptional technologies.One of the most notorious is the concept of multiple-input multiple-output(MIMO) systems, a remarkable invention that brought the promise of extraordi-nary data rates of transmission. As it is gradually understood and shaped by theresearch community, that promise is progressively becoming reality.

The thesis herein investigates some open issues in present-day MIMO de-siderata, and has the aim of filling important gaps. After a preliminary under-standing of the mobile channel characteristics and the implications of wirelessdiversity, its primary focus is on an information-theoretic perspective of MIMOcommunications. Even though the capacity of ideal MIMO channels has beenknown for some time, that is not the case for more realistic scenarios such as thepresence of cross-correlation between the antennas of MIMO systems. An in-depth study of the impact of receiver-sided correlated fading on the channel ca-pacity is conducted, and novel, exact analytic (closed-form) formulas for the er-godic capacity are found in a unified derivation. The derivation involves the studyof non-trivial multivariate statistical distributions, matrix integrals and the theo-ries of groups and group representations. In the process, some results of therealm of mathematical statistics are derived, such as expressions for several com-plex Jacobians and the universal Wishart density function.

The thesis then departs from Information Theory and focuses on morepractical aspects of MIMO transmission. No longer assuming that the channel isflat-fading, the goal is to study transmission overspreading in a unified frame-work, such that a more robust MIMO receiver can be implemented. A basebanddescription of the linear time-varying MIMO channel sets the ground to orthog-onalization and full discretization, where a novel approach using continuous, dis-crete, and hybrid operators becomes indispensable foundation. A novel, optimalmatched filter, the ORTHO-TS-MMF, is derived. To round out the study, a fullfrequency description using spectral operators is also obtained, and novel formu-las for the spectral factorization of time-varying MIMO filters are recovered. Fi-nally, after deriving three instances of MMSE based linear detectors, the optimalone is used to test the error performance of the devised discrete model usingMonte Carlo simulation. The thesis finalizes with the numerical results.

x ABSTRACT

RESUMO xi

RESUMO

O desafio permanente do aumento da eficiência espectral das comunicações semfios tem fomentado o aparecimento de várias tecnologias excepcionais. Uma dasmais notórias é o conceito de sistemas de múltipla-entrada múltipla-saída (co-nhecidos na literatura inglesa pela sigla MIMO), uma invenção notável quetrouxe consigo a promessa de taxas de transmissão extraordinárias. À medidaque o conceito é gradualmente compreendido e moldado pela comunidade de in-vestigadores, a promessa tem-se tornado efectivamente realidade.

A presente tese investiga algumas questões em aberto no âmbito do desi-derato MIMO actual, e almeja preencher lacunas importantes. Depois de umacompreensão preliminar das características do canal de comunicações móveis eas implicações da diversidade sem fios, o seu foco primeiro é numa perspectivateórica da informação em comunicações MIMO. Apesar da capacidade de canaisMIMO ideais ser conhecida já há algum tempo, o mesmo não se pode dizer daque concerne a cenários mais realistas tal como a presença de correlação cruzadaentre as antenas dos sistemas MIMO. É realizado um estudo aprofundado do im-pacto de desvanecimentos correlacionados do lado do receptor, sendo obtidasnovas fórmulas analíticas e exactas para a capacidade ergódica do canal, deduzi-das numa derivação unificada. A derivação envolve o estudo de distribuições es-tatísticas multivariadas não triviais, integrais de matrizes e teorias de grupos e derepresentações de grupos. No processo, são derivados alguns resultados dodomínio da estatística matemática tal como expressões para vários Jacobianoscomplexos e a função densidade de Wishart universal.

A tese afasta-se então da Teoria da Informação e centra-se em aspectosmais práticos da transmissão MIMO. Abandonando a condição de um canal nãoselectivo em frequência, o objecto passa a ser o estudo da sobre-dispersão datransmissão numa conjuntura unificada, viabilizando deste modo a implemen-tação de receptores MIMO mais robustos. Uma descrição em banda base do ca-nal MIMO linear e variável no tempo alicerça os processos de ortogonalização ediscretização completa, onde uma nova abordagem baseada em operadorescontínuos, discretos e híbridos, se traduz num fundamento indispensável. Adi-cionalmente, é derivado um novo filtro adaptado óptimo, designado porORTHO-TS-MMF. Para alargar a abrangência do estudo, são ainda definidosoperadores espectrais que conduzem a uma descrição completa no domínio das

xii RESUMO

frequências, em sequência recuperando novas fórmulas para a factorização es-pectral de filtros MIMO variáveis no tempo. Finalmente, depois de derivar trêsinstâncias de detectores lineares baseados em estimação MMSE, o detector linearóptimo é usado para testar o desempenho do modelo discreto projectado, sendo,para tal, implementadas eficientes simulações Monte Carlo. A tese termina comos resultados numéricos.

RÉSUMÉ xiii

RÉSUMÉ

Le défi permanent de láugmentation de l'efficacité spectrale des communica-tions sans fils (wireless) a fomenté lápparition de plusieurs technologies excep-tionnelles. Lúne des plus notoires cést le concept de multiple-entrée etmultiple-sortie (connu dans la littérature anglaise par la sigle MIMO), une inven-tion notable qui a porté sur soi une promesse de taux de transmission extraordi-naires. À mesure que le concept est graduellement compris et modelé par lacommunauté de chercheurs, la promesse devient effectivement une réalité.

Cette thèse recherche quelques questions en ouvert, dans ce qui concerneláctuel désidératum MIMO, et prétend remplir dímportantes lacunes. Aussitôtaprès la compréhension préliminaire des caractéristiques du canal des communi-cations mobiles et ses implications dans la diversité sans fils, son premier ciblecést une perspective théorique de línformation dans les communications MI-MO. Bien que la connaissance de la capacité des canaux MIMO idéaux soitconnue il y a longtemps, on ne peut pas affirmer le même concernant des scènesplus réalistes tel que la présence de la corrélation croisée entre les antennes dessystèmes MIMO. Un étude approfondi sur límpact d´évanouissements corréla-tionnés du côté du récepteur a été réalisé, et on a obtenu de nouvelles formulesanalytiques e exactes pour la capacité ergodique du canal, déduites dans une for-mule unifiée. La dérivation comprend l'étude des distributions statistiques multi-variées non triviaux, intégraux des matrices et des théories des groupes et des re-présentations de groupes. Dans ce processus, sont dérivés quelques résultats dudomaine de la statistique mathématique tel que des expressions pour divers Ja-cobiens complexes et pour la fonction densité d'Wishart universelle.

Alors, cette thèse s'éloigne de la Théorie de lÍnformation et se centre dansdes aspects plus pratiques de la transmission MIMO. Renonçant la conditiondún canal non sélectif en fréquence, lóbjet cést l´étude de la sur-dispersion dela transmission dans une conjunture unifiée, viabilisant ainsi límplémentation derécepteurs MIMO plus robustes. Une description en bande-base du canal MIMOlinéaire et variable dans le temps consolide les processus dórthogonalisation etde discrétisation complète, où un nouvel abordage fondé dans des opérateurscontinus, discrètes et hybrides, constitue un fondement indispensable. Addition-nellement, un nouveau filtre adapté optime, désigné par ORTHO-TS-MMF, estdérivé. Pour élargir cet étude, sont encore définis des opérateurs spectraux qui

xiv RÉSUMÉ

conduisent à une description complète dans le domaine des fréquences, en sé-quence récupérant de nouvelles formules pour la factorisation spectralle des fil-tres MIMO variables dans le temps. Finalement, après dériver trois instances dedétecteurs linéaires fondés en estimation MMSE, le détecteur linéaire optime estutilisé pour tester láccomplissement du modèle discret projeté, cést pourquoidéfficaces simulations Monte Carlo ont été réalisées. Cette thèse termine avecdes résultats numériques.

PUBLICATIONS xv

PUBLICATIONS

Paper 1 - IEEE Transactions on Information Theory: CLN: 5-170Title: “On the Derivation of the Closed-Form Capacity of MIMO Channels

under Correlated Fading”Authors: P. M. Marques, S. A. AbrantesStatus: submitted on March 10, 2005, accepted for revision.

Paper 2 - IEEE Transactions on Information Theory: CLN: 5-351Title: “On the Exact, Closed-Form Capacity Formulas for MIMO Channels

with an Arbitrary number of Inputs and Outputs, under CorrelatedFading”

Authors: P. M. Marques, S. A. AbrantesStatus: submitted on April 14, 2005, accepted for revision.

Paper 3 - Unified development submitted on July 23, 2007, and published in theMarch 2008 issue of the IEEE Transactions on Information Theory:

P. M. Marques and S. A. Abrantes, “On the Derivation of the Exact,Closed-Form Capacity Formulas for Receiver-Sided Correlated MIMOChannels,” IEEE Trans. Inf. Theory, vol. 54, no. 3, pp. 1139–1161, Mar.2008.

Paper 4 - Provisional Title: “Overspread Transmission over Wireless LTV MIMO

Systems”Authors: P. M. Marques, S. A. AbrantesStatus: in preparation; to be submitted to an international journal of reference.

xvi PUBLICATIONS

CONTENTS xvii

CONTENTS

Dedication v Acknowledgments vii Abstract ix Resumo xi Résumé xiii Publications xv Contents xvii List of Figures xxi

CHAPTER 1 INTRODUCTION 11.1 Motivation 21.2 Thesis Contributions 21.3 Thesis Outline 3

CHAPTER 2 MOBILE CHANNEL CHARACTERIZATION 12.1 Introduction 12.2 Fundamental Propagation Aspects 2

2.2.1 Free-Space Power Transmission 32.2.2 Received Signal for Static Channels 42.2.3 Received Signal for Dynamic Channels 5

2.3 Large-Scale Path Loss 62.3.1 Path Loss Power Law 62.3.2 Wave Scattering 82.3.3 Log-Normal Shadowing 9

2.4 Small-Scale Fading 102.4.1 Statistical Channel Characterization 122.4.2 Delay Profile Measurements 132.4.3 Delay Profile Modelling 142.4.4 Time and Frequency Autocorrelation 15

xviii CONTENTS

2.4.5 Frequency Selectivity Simulation 182.4.6 Frequency Selectivity Modelling 192.4.7 Statistical Fading Characterization 212.4.8 Time Selectivity Simulation 262.4.9 Time Selectivity and Doppler Spectrum 272.4.10 Bello Functions and Relations 29

CHAPTER 3 DIVERSITY RECEPTION 313.1 Introduction 313.2 Envelope Autocorrelation 323.3 The Maximal-Ratio Combiner 343.4 Statistical Characterization of the MRC 353.5 Error Performance of the MRC 38

3.5.1 Independent Fading 393.5.2 Correlated Fading 41

3.6 Information-theoretic Capacity of the MRC 44

CHAPTER 4 INFORMATION-THEORETIC ASPECTS OF MIMO CHANNELS 474.1 Introduction 474.2 MIMO Channel Modelling 484.3 Maximum Entropy 494.4 Information-theoretic Capacity 514.5 Capacity under Rayleigh Fading 55

4.5.1 Channel Matrix Distribution 564.5.2 Jacobians and Exterior Products 594.5.3 The Stiefel Manifold and the Unitary Group 664.5.4 Nonsingular Wishart Distribution 684.5.5 Eigenvalue Distribution 704.5.6 Independent Fading 714.5.7 Correlated Fading 814.5.8 Spatial Correlation Modelling 964.5.9 Numerical Results and Discussion 984.5.10 Conclusion 102

4.6 Capacity when the Wishart Matrix is Singular 1024.6.1 Statistical Characterization 1034.6.2 Volume Elements of Transformations 1044.6.3 The Wishart and Eigenvalue Densities 1074.6.4 Capacity under Receiver-sided Correlated Fading 1104.6.5 Capacity under Transmitter-sided Correlated Fading 1124.6.6 Numerical Results and Discussion 1134.6.7 Conclusion 116

CHAPTER 5 OVERSPREAD TRANSMISSION OVER WIRELESS LTV MIMO SYSTEMS 1195.1 Introduction 119

CONTENTS xix

5.2 Input/Output Model 1215.2.1 Baseband MIMO Channel Model 1215.2.2 Baseband MIMO Input Design 1265.2.3 Baseband MIMO Output Design 129

5.3 Discretizing the MIMO Input-Output Model 1305.3.1 Defining a Basis for MIMO Response Expansion 1315.3.2 Time-Varying Maximum Likelihood Sequence Estimation (TV-

MLSE) 1385.3.3 Time-Shear Matrix Matched Filter (TS-MMF) 1405.3.4 Drawbacks of the TS-MMF 1425.3.5 Optimality of the Discrete MIMO Model in 1435.3.6 Decomposing the TS-MMF into Discrete and Semi-Orthonormal

Factors 1455.4 Frequency-Domain Description 148

5.4.1 Frequency-Domain Operators 1505.4.2 The Discrete MIMO Model in the Space 1535.4.3 The TS-MMF in the Frequency Domain 155

5.5 Noise Whitening and Full Discretization 1585.5.1 Whitening the Noise at the TS-MMF Output 1585.5.2 Fully Discretizing the TS-MMF and ORTHO-TS-MMF 160

5.6 Linear Detection for the Overspread MIMO Channel 1645.6.1 Constrained Linear Detection 1645.6.2 Unconstrained Linear Detection 1665.6.3 Zero-forcing Linear Detection 168

5.7 Error Performance of the ORTHO-TS-MMF with Linear Detection 1695.7.1 Channel Model for Simulation 1695.7.2 Discrete Channel Model for Simulation 1715.7.3 Building the ORTHO-TS-MMF with Block-wise Householder

Reflections 1715.7.4 Monte Carlo Simulation, Numerical Results and Discussion 173

5.8 Conclusion 179

CHAPTER 6 CONCLUSIONS AND FUTURE RESEARCH 1816.1 Conclusions 1816.2 Future Research 182

APPENDIX 185Appendix A 185Appendix B 186Appendix C 187

REFERENCES 191

( )R Φ

( )ΦF

R

xx CONTENTS

LIST OF FIGURES xxi

LIST OF FIGURES

Figure 2.1 Received power as a function of T-R separation for the ground reflection model, and asymptotic behaviour for large . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Figure 2.2 Surface roughness model for Rayleigh criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Figure 2.3 Typical measured excess delay power density for an urban environment. . . . . . . . . 14

Figure 2.4 Spaced-Time and Spaced-Frequency correlation functions. . . . . . . . . . . . . . . . . . . . 17

Figure 2.5 Simulated frequency response of multipath propagation channel for consecutive time instants, within a 5 MHz bandwidth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Figure 2.6 Tapped delay line model for a noiseless frequency-selective multipath channel. . . . 21

Figure 2.7 Possible probability density functions for local area signal fading characterization. 23

Figure 2.8 Nakagami-m pdf for several parameters, shown with . . . . . . . . . . . . . . . . . 26

Figure 2.9 Rayleigh fading behaviour of the time varying stochastic tap weights for a frequency-se-lective mobile propagation channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Figure 2.10 Typical Doppler power spectrum for mobile radio channels in cluttered outdoor envi-ronments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Figure 2.11 Average Doppler spectrum for a fixed link and dynamic environment (e.g. traffic road) at 40 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Figure 2.12 Relations among the correlation and power spectra functions for the WSSUS channel model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Figure 3.1 Illustration of the angular spread of multipath components at a base station.. . . . . 33

Figure 3.2 The effect of MRC diversity combining on the error performance of coherent BPSK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Figure 3.3 Error performance of MRC combiner for a linear array of antennas, correlated fading with fading figure , and uniform distribution of wave arrival. . . . . . . . . . . . . . 43

Figure 3.4 Error performance of MRC combiner for a linear array of antennas, correlated fading with fading figure , and uniform distribution of wave arrival. . . . . . . . . . . . . 43

Figure 3.5 Information-theoretic capacity of MRC combiner for several effective diversity orders, in the case of independent fading, and uniform distribution of wave arrival. . . . . . 45

Figure 3.6 Information-theoretic capacity of MRC combiner for different antenna separations, in

d

m 1Ω =

M1m =

M2m =

xxii LIST OF FIGURES

the case of correlated fading, and uniform distribution of wave arrival. . . . . . . . . . 46

Figure 4.1 Illustrating the phase difference between different antennas of a moving receiver.. 57

Figure 4.2 Average information-theoretic capacity of the MIMO channel as a function of the branch SNR at the receiver side, in the special cases of (a) one receiver branch, and (b) two receiver branches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Figure 4.3 Average information-theoretic capacity of the MIMO channel as a function of the branch SNR at the receiver side, in the special cases of (a) four receiver branches, and (b) eight receiver branches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Figure 4.4 Average information-theoretic capacity of the MIMO channel as a function of the branch SNR at the receiver side, in the special cases of (a) one transmitter branch, and (b) two transmitter branches.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Figure 4.5 Average information-theoretic capacity of the MIMO channel as a function of the branch SNR at the receiver side, in the special cases of (a) four transmitter branches, and (b) eight transmitter branches.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Figure 4.6 Average MIMO capacity as a function of the number of transmitter and receiver anten-nas, for an average SNR of 5 dB at each receiver branch. . . . . . . . . . . . . . . . . . . . . . 80

Figure 4.7 Capacity as a function of the number of transmitter and receiver antennas, for an average SNR of 5 dB at each receiver branch. The left figure retrieves some isolines from Figure 4.6.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Figure 4.8 Receiver spatial correlation for the isotropic scattering model.. . . . . . . . . . . . . . . . . 98

Figure 4.9 Average capacity of the MIMO channel as a function of the branch SNR at the receiver side, in the special cases of two and four receiver antennas. . . . . . . . . . . . . . . . . . . . 99

Figure 4.10 Average capacity of the MIMO channel as a function of the branch SNR at the receiver side, in the special case of eight receiver antennas. . . . . . . . . . . . . . . . . . . . . . . . . . 100

Figure 4.11 Optimizing the capacity of the MIMO channel by selecting four receiver antennas from an equispaced linear array with eight and sixteen antennas. . . . . . . . . . . . . . . . . . . 101

Figure 4.12 Optimizing the capacity of the MIMO channel by selecting eight receiver antennas from an equispaced linear array with sixteen and thirty two antennas. . . . . . . . . . . . . . . 101

Figure 4.13 Average MIMO channel capacity as a function of and , for . The black line is for receiver-sided correlated fading with isotropic scattering, and the gray line is for two-sided independent fading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Figure 4.14 Average MIMO channel capacity as a function of , for several values of , and . For receiver-sided correlated fading, isotropic scattering is assumed. . . 114

Figure 4.15 Uplink and downlink average MIMO channel capacities as a function of , for several values of , and . Correlated fading and isotropic scattering are assumed for the mobile unit only. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Figure 4.16 Average MIMO channel capacity as a function of , for , receiver-sided correlated fading, and nonisotropic scattering modelled by (4.297) with and

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Figure 5.1 Illustration of up-conversion and down-conversion operations in the complex domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Figure 5.2 Equivalent passband and baseband models for the MIMO wireless channel. . . . . 126

Rn Tn 5 dBγ =

Tn Rn5 dBγ =

MUn

BSn 5 dBγ =

T Rn n= 5 dBγ = 0, /2cφ π=

2, ..., 8κ =

LIST OF FIGURES xxiii

Figure 5.3 Transmission model with discrete-time vector-valued input sequence. . . . . . . . . . 127

Figure 5.4 Transmission domains of a time-varying linear channel as seen by the receiver. . . 130

Figure 5.5 Decomposition of the MIMO transfer matrix into discrete and orthonormal operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Figure 5.6 Discretizing the MIMO output with the semi-orthonormal channel operator . . 137

Figure 5.7 Discrete MIMO model after projecting the output onto . . . . . . . . . . . . . . . 138

Figure 5.8 Mapping of the MIMO channel response onto its associated matched filter. . . . . 142

Figure 5.9 Matched filtering of the MIMO time-varying channel model. . . . . . . . . . . . . . . . . 142

Figure 5.10 Delayed matched filtering of the MIMO time-varying channel model. . . . . . . . . . 143

Figure 5.11 Decomposition of the TS-MMF and global function into discrete and semi-orthonor-mal filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Figure 5.12 Skewed Doppler convolution induced by LTV MIMO channels.. . . . . . . . . . . . . . 151

Figure 5.13 Frequency perspective of the discrete MIMO model after channel decomposition and output projection onto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Figure 5.14 Discrete MIMO model after matched filtering and spectral decomposition.. . . . . 158

Figure 5.15 Orthonormalizing the TS-MMF using the Moore-Penrose pseudoinverse of the discrete channel filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Figure 5.16 Effect of sampling the received signal at twice the signalling rate. . . . . . . . . . . . . . 162

Figure 5.17 Optimal detection based, fully discretized receiver model for the MIMO LTV channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Figure 5.18 Illustrating the different multipath phases of a moving array with colinear axis. . . 170

Figure 5.19 Average error rates for several symbol constellation sizes and two signalling intervals, for and with equispaced fit within . . . . . . . . . . . . . . . . . . . 174

Figure 5.20 Average error rates for two receiver sizes with equispaced fit within and in-creasing number of transmitter antennas, for . . . . . . . . . . . . . . . . . . . . 175

Figure 5.21 Average error rates for one transmitter antenna and several receiver sizes (equispaced fit within and free scaling considered), for . . . . . . . . . . . . . . . 176

Figure 5.22 Average error rates for two transmitter antennas and several receiver sizes (equispaced fit within and free scaling considered), for . . . . . . . . . . . . . . 176

Figure 5.23 Average error rates for four transmitter antennas and several receiver sizes (equispaced fit within and free scaling considered), for .. . . . . . . . . . . . 177

Figure 5.24 Average error rates for several signalling rates, and a MIMO configuration of and with equispaced receiver antenna fit within . . . . . . . . . . . 177

Figure 5.25 Average error rates for several maximum Doppler shifts, and a MIMO configuration of and several with equispaced receiver antenna fit within , for .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Figure 5.26 Average error rates for several maximum Doppler shifts, and a MIMO configuration of and several with equispaced receiver antenna fit within , for

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Figure 5.27 Average error rates for several maximum receiver antenna separations, and a MIMO

Φ

( )R Φ

( )ΦF

R

2Tn = 4Rn = max 0.5d λ=

max 0.5d λ=100sT sμ=

max 0.5d λ= 100sT sμ=


max 0.5d λ= 0.1sT sμ=

2Tn =4Rn = max 0.5d λ=

2Tn = max 0.5d λ= 1sT s=

2Tn = Rn max 0.5d λ=1sT sμ=

xxiv LIST OF FIGURES

configuration of and with equispaced receiver antenna fit within , for . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

2Tn = 4Rn =


CHAPTER 1

INTRODUCTION

Wireless mobile communications at the physical level is an exceedingly challenging topicin both theoretical and practical terms. The mobile propagation channel inherently suffersfrom severe impairments such as random small-scale fading and frequency selectivity that,without powerful mitigation techniques, deem reliable communication virtually impossi-ble. Besides these impairments, the wireless medium is effectively shared by a lot of serv-ices to different users that are increasingly eager of high data rates, which means that eitherhigh spectral efficiencies are attained by proficient technologies or it will not be possibleto accommodate all of them. Relevant sources of information regarding the modelling andcharacterization of the mobile channel are [1], [2], [3] and [4], and an extensive review isprovided in [5].

One ineluctable tool to combat the wireless impairments is to take advantage of therandom nature of the channel fading to explore spatial diversity. Spatial diversity is consid-ered one of the most important instruments for wireless communications’ reliability, as itintroduces impressive channel amelioration by the employment of diversity combiningtechniques. Spatial diversity also allows for spatial multiplexing which, in essence, permitssubstantial data rate improvements and user separation in multiuser systems. A broad re-view of these facts is given in [6], [7], [8] and [9].

Multiple-input multiple-output (MIMO) systems have originated a full discipline ofstudy which sole purpose is to formulate methods to fully exploit the enhancements ofspatial diversity that are steering wireless mobile communications. Ranging from transmitbeamforming to space-time processing and space-time coding, the goal is to obtain eitherdiversity gain, spatial multiplexing gain, or interference reduction. In what concerns spatialmultiplexing, Information Theory has provided results that foresee an almost linear scal-ing of the information-theoretic capacity with MIMO array sizes, confirming that the si-multaneous use of transmit and receive diversity offers spectral efficiency improvements

1

2 CHAPTER 1 INTRODUCTION

[10], [11] and [12]. MIMO literature is vast, but for an introductory study see [13], [14],[15], [16] and [17], and for a comprehensive instalment [18], [19].

1.1 Motivation

The fundamental motivation for this thesis is the global urge for data rates in the order ofhundreds or even thousands of megabits per second in a 4th generation (4G) mobile wire-less communications setting. Inherent complexity makes the subject difficult, yet challeng-ing, and the ability to grasp the mathematics that tackles the problem and moulds asolution to one’s advantage is no less than stimulating. MIMO systems are an enthusiasticnew technology that is becoming a landmark in our time, and will definitely drive the fu-ture of wireless communications.

All the beauty of MIMO systems can be directly drawn from theoretical principles ofa multidimensional extension of classical Information Theory, and this is where the firstfar-reaching incursion is made. The goal is to deprive the formulas for attainable informa-tion rates in frequency non-selective MIMO systems from an open-form, computationallydemanding situation, within a propagation context as general as possible. Due to the ma-tricial nature of the MIMO model, this task is anything but easy when the degrees of free-dom that MIMO is supposed to introduce are not so free after all. These are the correlatedfading scenarios that the thesis investigates in pursuit of a possible solution.

Having concluded the information-theoretic study, the motivation becomes morepractical. Now, in a frequency-selective and/or Doppler-spread set-up, the aim is to un-derstand how to optimally design the MIMO input/output model when no space-timechannel coding is used. MIMO channels also suffer from overspreading in multipath delayand in Doppler shift, so a basic challenge is to devise a receiver that copes with the prob-lem in a time-variability aware manner. Instead of designing a receiver and then adaptingit to the time-varying nature of the wireless MIMO channel, the approach is to assumetime-variability from the beginning so as to explore Doppler diversity.

In the process of solving these questions, the thesis finds additional motivation in ex-citing new mathematical tools that it builds to reach its conclusions.

1.2 Thesis Contributions

So that the reader gets acquainted with the innovative scope of this thesis, the purpose ofthis section is to provide a brief summary of its most relevant contributions to the currentresearch work. They are:

1. Derivation of the probability density function of the Nakagami-m distribution of enve-lope fading, by resorting to hyperspherical coordinates;

2. Derivation of the probability density function of the output signal-to-noise ratio of themaximal-ratio combiner under Nakagami-m distributed correlated envelope fading;

3. Derivation of closed-form formulas for the error probability and information-theoret-ic capacity of the maximal-ratio combiner under Nakagami-m distributed correlated en-velope fading;

1.3 Thesis Outline 3

4. Derivation of the complex Jacobians of four matrix decompositions, namely: 1) theCholesky factorization; 2) the QR factorization; 3) the congruence transformation; and 4)the eigenvalue decomposition; the Jacobians were derived for both nonsingular and sin-gular complex matrices, making them universal;

5. Derivation of the density function of both nonsingular and singular, complex Wishartmatrices, based on the transformations associated with the given Jacobians;

6. Derivation of the unordered eigenvalue distributions of both nonsingular and singular,complex Wishart matrices;

7. Extension of the Itzykson-Zuber integral to integration over a quotient space, usingthe theory of group representations;

8. Derivation of exact, closed-form formulas, for the information-theoretic capacity ofwireless MIMO channels under receiver-sided correlated fading, in the configurations: 1)no more receiver antennas than transmitter antennas; and 2) more receiver antennas thantransmitter antennas;

9. Full description of the input/output model of the linear time-varying MIMO channelusing continuous, discrete and hybrid operators;

10. Orthogonalization of the MIMO channel response and determination of a fully dis-crete input/output model of confirmed optimality in the maximum likelihood sense; thelatter model is obtained from the first by filtering with a new matrix matched filter, theORTHO-TS-MMF; it is understood that the matched filter is always the hermitian adjointoperator;

11. Derivation of the time-varying noise whitening filter that applied at the output of theTS-MMF produces the same output as the ORTHO-TS-MMF;

12. Full discretization of the matched filter operators, which allows for a fully discrete re-ceiver after sampling;

13. Derivation of constrained, unconstrained and zero-forcing linear detectors operatorsby the MMSE method of error energy minimization;

14. Implementation of a novel method of MIMO channel operator orthogonalization,based on backward block-wise Householder reflections; the method is numerically stableand extremely time-efficient (in certain cases, dozens or even hundreds of times fasterthan the conventional, unstable, backward block-wise Gram-Schmidt procedure); thespeed-up made the ensuing Monte Carlo simulations possible to complete in due time.

1.3 Thesis Outline

The thesis is organized in logical progression. Chapter 2 is devoted to a preliminary studyof the wireless mobile channel. It begins with the fundamentals of electromagnetic prop-agation and retrieves a model for the received signal of dynamic channels. After presentingsome notions from large-scale path loss, it concentrates on the small-scale fading phe-nomenon. The channel is statistically characterized in terms of time and frequency auto-correlation functions and models are provided for the power delay profile and angulardistribution of wave arrival. The spaced-time spaced-frequency autocorrelation function

4 CHAPTER 1 INTRODUCTION

of the wide-sense stationary uncorrelated (isotropic) scattering (WSSUS) channel modelis derived, which leads to formulas for the coherence time and coherence bandwidth of the wire-less channel. Simulations of the behaviour of time-selective and frequency-selective chan-nels are performed. Also, the fading is statistically characterized in terms of the Rayleigh,Rice, and Nakagami-m distributions.

Chapter 3 evolves into the study of spatial diversity at the receiver end. A brief incur-sion into autocorrelation of the fading envelope sets the ground to the study of the max-imal-ratio combiner as a predetection combining technique. Information-theoretic anderror performance results expose first evidence of the benefit of spatial diversity even un-der correlation scenarios.

In Chapter 4 the information-theoretic scaffolding pertaining to MIMO systems ispresented. Beginning with the model for a frequency non-selective, slowly fading channel,and the principle of maximum entropy, an all-round derivation of the information-theo-retic capacity of MIMO channels is performed until exact, analytic formulas, are obtained.The complexity of the study lies in the presence of fading correlation at the receiver endof the MIMO system, which is a realistic situation in space-constrained (reduced antennaseparations) mobile transceivers. The initial study is for the case of no more receiver an-tennas than transmitter antennas, and the more complex case of more receiver antennasis treated thereafter. Abundant plots are presented to demonstrate and validate the results.

Chapter 5 departs from the simple flat-fading assumption and aims to study over-spreading in MIMO transmission. A baseband input/output model is conceived and, ori-ented by a maximum likelihood estimation approach, an optimal input/output discretemodel is obtained. Arising from a general theory of matrix operators, time-variabilityaware matched filters are proposed. Then, a frequency domain description of the input/output model is made, trying to purvey new insight into the model construction and op-eration. To complete, a set of linear detectors for the mentioned model is obtained, andMonte Carlo simulations are used for testing the error performance of the global system.

Finally, Chapter 6 concludes the thesis and anticipates new vectors of research.

CHAPTER 2

MOBILE CHANNEL CHARACTERIZATION

2.1 Introduction

Due to its distinctive features, the mobile radio propagation channel places fundamentallimits in the performance of wireless communication systems. Various types of phenom-ena affect the transmission of radio signals, the most severe of which are responsible formultipath propagation between the transmitter and the receiver. These are:• Reflection - resulting from specular reflections at objects with dimensions larger than

the wavelength of the propagating waves (e.g. ground, buildings, and walls).• Diffraction - resulting from surface irregularities (edges) between the transmitter and

receiver, which “bend” the propagating waves.• Scattering - resulting from heaped up objects with dimensions shorter than the wave-

length (e.g. rough surfaces, trees, and lamp posts).

Furthermore, due to the mobility of the transceivers (i.e. the mobile units), all thesephenomena are time-variant, meaning that they become unpredictable and need to betreated stochastically. The first issue that arises when designing a communication systemis what model to choose for the mobile radio environment, so that the performance canbe maximized for a particular application. A thorough knowledge of the channel behav-iour is needed to pursuit this goal, beginning with the study of fundamental aspects ofelectromagnetic theory, and the stochastic estimation of the channel output for a giventransmitted signal. With this information at hand, signal processing can be employed tomaximize the performance of the whole communication system. Also, the transmitted sig-nal can be “intelligently formatted” to even improve the results. Some fundamental ques-tions stand up: How can one exploit the knowledge of the channel characteristics to obtainan optimum design solution? And, may it be possible to put the channel impairments toone’s favour?

In general, the mobile radio propagation environment is characterized by two partially

1

2 CHAPTER 2 MOBILE CHANNEL CHARACTERIZATION

separable effects: large-scale path loss, which is a function of the distance between transmitterand receiver, and the presence of large obstructions along the path; and small-scale fading,which results from constructive and destructive interference of multipath propagation,and varies over distances proportional to the signal wavelength.

2.2 Fundamental Propagation Aspects

This discussion firstly introduces some basic antenna theory, and then proceeds to esti-mate the channel dependent, received signal characteristics. A good direction to take with-in the analysis is to begin with ideal conditions of propagation (i.e. no multipathpropagation phenomena and a free-space medium) and after that, generalize the results tonon-ideal conditions.

Using spherical coordinates, the far zone - ( is the phase constant and isthe distance from the radiating element) - electric and magnetic fields produced by a radi-ating element (antenna) operating in free-space can be written generically in phasor (base-band) notation as:

where

Also, for , where is the antenna dimension and is the wavelength, theelectromagnetic field is within the Fraunhofer region, which means that may be con-sidered independent of [20]. The average power density in a given direction may then beexpressed as

The directivity of an antenna is a power measure of its directional pattern, and is de-fined as the ratio of the radiated power density to the power density averaged over all di-rections (equivalent isotropic radiation), that is

(2.1)

(2.2)

(2.3)

(2.4)

1dβ β d

0( )

( , )−

≈j d

meE S

d

ϕ β

θ φ θE a

0( )

( , )−

≈j d

mE e Sd

ϕ β

θ φη φH a

0

amplitude function proportional to excitation current

( , ) E-plane or

characteristic impedance of the medium

/ elevation/azimuth angle

fixed phase offset

mE

S radiation pattern space factorθ φ

η

θ φ

ϕ

=

=

=

=

=

22 /d D λ≥ D λ

( , )S θ φ

d

*

2

22

2

1Re

2

2

( , )2

av

m

S E H

E

SEd

η

θ φη

= ×

=

=

2( , )/ 4av

rad

SD

P dθ φ

π=

2.2 Fundamental Propagation Aspects 3

where is the total radiated power. To expand the conduction, dielectric, and mismatchlosses in the antenna, (2.4) can be reformulated to

where is the total input power at the transmitter side and is the total radiation effi-ciency of the antenna. Neglecting polarization losses, the effective directional power gainto be used in link calculations may be obtained from (2.5) as

2.2.1 FREE-SPACE POWER TRANSMISSION

Looking at (2.6), it becomes apparent that the power delivered to a matched load connect-ed to a receiver antenna (with effective area ) in line-of-sight with the transmitter is

A general expression for the effective area of a receiving antenna can be derived by ex-changing the transmission/reception roles of two communicating antennas. For a linear,passive, and isotropic medium, and when both antennas are matched to the connectedloads, the following relation must be true

Equation (2.8) interprets the fact that a change in the input-output direction of a two-an-tenna system does not affect the system response and, in this particular case, the powerresponse. It is a direct consequence of the reciprocity and reaction theorems of electromag-netic theory. Thus, simplifying (2.8), the relation

states that the effective area of a receiving antenna is related to its directional power gainby the constant . More specifically, since the effective area of reception should beindependent of the transmitter antenna, it asserts that the quotient may be a con-stant independent of the type of antenna. It can be proved generically that it is so. In par-ticular, the well known effective area and power gain of an infinitesimal dipole are givenby [21]

and their ratio is

(2.5)

(2.6)

radP

2( , )/ 4av

T T

SD

P dθ φ

ε π=

TP Tε

2( , ) ( , )/ 4

avT T

T

SG D

P dθ φ ε θ φ

π= =

(2.7)

(2.8)

(2.9)

(2.10)

(2.11)

RA

2( , )4

RL T T

AP P G

dθ φ

π=

2 2( , ) ( , )4 4

R TL T T T R

A AP P G P G

d dθ φ θ φ

π π= =

( , )( , )T

R RT

AA G

Gθ φ

θ φ=

/T TA G

/T TA G

22

2

3sin

83

sin2

T

T

A

G

λθ

π

θ

=

=

2

4T

T

AG

λπ

=


Substituting (2.11) into (2.9), the desired expression for the effective area of an antenna isobtained as

which, after direct substitution into (2.7), relates the power (delivered to the receiverload) to the input power of the transmitting antenna - the Friis Transmission Formula:

The term is called the free-space loss factor, and it takes into consideration the lossesdue to the spherical spreading of the energy by the antenna.

2.2.2 RECEIVED SIGNAL FOR STATIC CHANNELS

Although (2.13) assumes line-of-sight and free-space conditions, it may be used as a basisto find a general expression for the voltage induced at the receiver load .For a matched load, and neglecting the antenna input reactance , the delivered poweris also given by

which equated to (2.13) gives

Moreover, the voltage phase should be equal to the phase of the impinging wave plus aphase shift due to the receiver circuits. The complete voltage expression is

To inspect how the load voltage relates to the impinging wave, (2.6) can be substitutedinto (2.16), giving

(2.12)

(2.13)

2

( , )4R RA Gλ

θ φπ

=

LP

TP

( )2

( , ) ( , )4L T T RP P G G

dλ

θ φ θ φπ

=

( )2/ 4 dλ π

(2.14)

(2.15)

(2.16)

(2.17)

LV r r rZ R jX= +

rX

*

*

2

1Re

21

Re2

2

L L L

L L

r

L

r

P V I

V VR

VR

=

⎧ ⎫⎪ ⎪⎪ ⎪= ⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

=

( )2

2 ( , ) ( , )4

L r L

r T T R

V R P

R P G Gd

λθ φ θ φ

π

=

=

δ

( ) 0( )2 ( , ) ( , )4

j dL r T T RV R P G G e

dϕ β δλ

θ φ θ φπ

− −=

( )( )

0

0

0

2 ( )

22 ( )

( )

2 (4 ) ( , )4

2 (4 ) ( , )2 4

( , )4

( , )

j dL r av R

j dr R

r R j j d

rad

V R d S G ed

ER d G e

d

R Ge E e

E L

ϕ β δ

ϕ β δ

δ ϕ β

λπ θ φ

π

λπ θ φ

η π

θ φλ

πηθ φ

− −

− −

− −

=

=

=

=

2.2 Fundamental Propagation Aspects 5

where and . Apart from a phase shift,(2.17) shows a direct proportion relationship between incident field and load voltage. Itcan be used to induct the general load voltage when the propagation conditions of the mo-bile radio channel are not the ideal ones.

In a realistic situation, there may exist “objects” nearby the transmitting and receivingantennas. These “objects” include natural and man-made structures, such as the ground,trees, hills, buildings, walls, and towers. For frequencies above 100 MHz, these structuresreflect/scatter much of the incoming waves, possibly leading to several replicas of thesame signal arriving at the receiving antenna. Using the superposition principle, the totalimpinging field may be written as

where is the equivalent reflection coefficient associated with the -th path,which includes all the reflections undergone by the path. It is assumed that the polariza-tion mismatches between the incident waves and the receiving antenna are negligible,which is a good approximation for vertical and horizontal reflectors. Also, for high fre-quencies (>100 MHz) and grazing reflection angles, irrespective of the polariza-tion. The total voltage delivered to the load will be given by

Each signal component experiences a different multipath environment which determinesthe amplitude , and the associated phase shift

. Furthermore, the transmitter may linearly modulate the propagat-ing wave with a baseband signal, say

where the transmission rate is , represents the sequence of complex symbolsmapped from k-bit blocks, and is a pulse shaping function. If this is the case, the loadvoltage must be rewritten as

where is the time delay of the -th propagation path.

2.2.3 RECEIVED SIGNAL FOR DYNAMIC CHANNELS

Equation (2.21) is only valid for static radio channel conditions, that is, when neither theenvironment “objects” nor the transceivers are in relative motion. When the multipath en-vironment is “dynamic”, there is a continuous change in the length of each propagation

(2.18)

(2.19)

(2.20)

(2.21)

( , ) ( , )/ 4 jr RL R G e δθ φ θ φ πη λ −= 0( )j d

radE E e ϕ β−=

rad i i

i

E E= Γ∑ij

i ieαρΓ = i

1iΓ ≈ −

0( )

( , )

2 ( , ) ( , )4

i i

i

L i i i i

i

j dr T T i i R i i i

ii

ji

i

V E L

R P G G ed

e

ϕ α β δ

ϕ

θ φ

λθ φ θ φ ρ

π

υ

+ − −

= Γ

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠

=

∑

∑∑

( )2 ( , ) ( , ) / 4i r T T i i R i i i iR P G G dυ θ φ θ φ λρ π=

0i i idϕ ϕ α β δ= + − −

( ) ( )b n

n

s t I g t nT∞

=−∞

= −∑

1/T nI

( )g t

( ) ( )ijL i b i

i

V t e s tϕυ τ= −∑

iτ i


path, the reflection coefficients, and angles of wave departure and incidence. Thus, the re-ceived voltage amplitudes and phase shifts are a function of the transceivers’ spatial loca-tion and, therefore, a function of time.

The received signal variations can be related to the motion of the propagation envi-ronment using the Doppler shift effect. When a receiver moves with a velocity relative tothe transmitter, and with an angle towards the incoming wave, the apparent change inangular frequency is

Hence, to take channel related time variations (i.e. spatial motion) into account, (2.21)should be modified to

where is the Doppler shift and is the carrier frequency.

2.3 Large-Scale Path Loss

The Friis Transmission Formula - (2.13), states that the path loss of radio waves propagatingthrough free-space is proportional to the inverse of , where is the distance between thetransmitter and receiver. However, propagation in a mobile radio channel is neither free-space nor line-of-sight. Determining the received signal power in a multipath environmentis a difficult problem, nevertheless a basic understanding may be achieved by separatingthe essential effects of reflections.

2.3.1 PATH LOSS POWER LAW

Reflections can be roughly divided into two types: horizontal reflections and vertical re-flections. Horizontal reflections occur from vertical objects, such as building walls, and arecharacterized by an approximate power decay as (i.e. the free-space decay). Vertical re-flections occur from horizontal objects, such as the ground, and combine at the receiverwith the line-of-sight paths. To study the vertical reflection of waves, (2.19) can be simpli-fied by assuming constant gain patterns for the direct and reflected rays, which is a goodassumption for grazing reflection angles. Moreover, for large distances between transmit-ter and receiver (as compared to their heights), the load voltage can be modified to

where a simplified two-ray model is assumed, with single direct and reflectedrays. Also, and . We shall consider that the transmit antenna is lo-

(2.22)

(2.23)

ν

φ

2 cosi iν

ω π φλ

=

( )

( )( ) ( ) ( ( ))

( )

i

i i

j tL i b i

i

ij ti b i

ci

V t t e s t t

t e s t t

ϕ

ϕ ω

υ τ

ωυ τ

ω+

= −

⎛ ⎞⎟⎜= − + ⎟⎜ ⎟⎜⎝ ⎠

∑

∑

iω cω

2d d

(2.24)

2d−

( ) 0

2( )

1

24

i ij dL r T T R i

i

V R P G G ed

ϕ α β δλρ

π+ − −

=

= ∑( 1)i =

( 2)i = 1 1Γ = 2 1Γ ≈ −

2.3 Large-Scale Path Loss 7

cated at height above a flat ground, and the receive antenna is located at height . Thetotal power delivered to the load can now be expressed as

where and . For large , (2.25) can be simpli-fied to (see e.g. [3])

Equation (2.26) shows that, in the limiting case of very large distances, for a two-rayground reflection model the received power falls off as , which is much more severecompared to that of free-space.

Figure 2.1 shows a plot of (2.25) as a function of the distance between transmitter and

receiver for a typical WCDMA reverse link. The plot parameters are: ;; ; ; ; . Up to a critical dis-

tance , the wave experiences constructive and destructive interference of thetwo rays, and the average power fall off with distance corresponds to free space.For the average power fall off with distance is approximated by the fourth powerlaw in (2.26).

Empirical results have shown that this is the worst case propagation path loss decay,and fundament the decay law as being inversely proportional to , . However,

(2.25)

(2.26)

FIGURE 2.1 Received power as a function of transmitter-receiver (T-R) separation for the ground reflection

model, and asymptotic behaviour for large .

th rh

( ) 2 1

2 2( )14

j d dL T T RP P G G e

dβλ

π− −= −

2 21 ( )t rd d h h= + − 2 2

2 ( )t rd d h h= + + d

( )( )

2 22 /

2

2

14

t rj h h dL T T R

t rT T R

P P G G ed

h hP G G

d

βλπ

−≈ −

≈

4d−

d

24 dBmTP =

18 dBiTG = 50 mth = 2 dBiRG = 2 mrh = 2 GHzf =

5 kmcd ≈

cd d>

pd 2 4p≤ ≤


neither (2.13) nor (2.26) hold for . For this reason, large-scale propagation modelsuse a reference distance , known as the close-in distance. For instance, the free-space trans-mission equation is changed as follows

In general, the received power will be written as

where the value may be predicted from (2.13), or may be measured by averaging thereceived power at many points near . When multipath fading is present, the measure-ments should span distances between and , so that the short-term fading compo-nents are removed and the details of the local means are maintained [22]. From (2.28), theaverage power path loss in dB is given by

which represents a straight line with a slope equal to dB per decade.In practice, actual propagation paths will be more complicated than the idealized cas-

es, due to the possibility of multiple reflections, diffraction, wave scattering, and shadow-ing. The following sections will study the last two.

2.3.2 WAVE SCATTERING

When the propagating waves are incident on smooth surfaces, specular reflections occurand Snell’s law applies. However, if the surfaces contain protuberances with dimensionssimilar to the wavelength of the incident waves, a diffuse reflection occurs which may notbe characterized by a simple reflection coefficient. A fraction of the incident energy maybe scattered in the direction of the receiving antenna, making the received signal oftenstronger than what is predicted by reflection models.

The model shown in Figure 2.2 may be used to characterize surface roughness by an-alysing what happens to a wavefront when it impinges on a protuberance with height .

(2.27)

(2.28)

(2.29)

0d =

0d

( )( )

2 20

02

00

( , ) ( , )4

( )

L T T R

L

dP P G G

d d

dP d

d

λθ φ θ φ

π⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠

=

( )00( )

p

L Ld

P P dd

=

0( )LP d

0d

40λ 200λ

0 100

( ) 10 logdB dBd

L L d pd⎛ ⎞⎟⎜= + ⎟⎜ ⎟⎜⎝ ⎠

10p

FIGURE 2.2 Surface roughness model for Rayleigh criterion.

η

θ θ

rayA

rayB

1δ

2δ2θ

η

2.3 Large-Scale Path Loss 9

If one ray of the wave front hits the protuberance (ray B) and other ray misses it (ray A),the difference in path length of the two rays when they meet again is

and the phase difference is

When the protuberances are very small compared to the wavelength, then , and thetwo reflected rays will add almost in phase. When , the reflected rays will be outof phase and cancel each other. A valid boundary for roughness is then , whichafter direct substitution into (2.31) yields

where is the critical protuberance height. The Rayleigh criterion for roughness states thata surface is considered rough if (see e.g. [23]).

In practice, the value used as a measure of surface undulation height is , the stand-ard deviation of the surface irregularities relative to the mean height. Additionally, for typ-ical mobile radio scenarios, the incidence angles are grazing, and justify theapproximation . Replacing by in (2.31), we obtain the following roughnessmeasure

The usual criterion for roughness is , whereas the criterion for smoothnessis . For example, at 2 GHz and for , the value of necessary to make a sur-face rough is 7 m. This means that typical objects such as vehicles, trees, and houses willcause wave scattering.

Wave scattering may be accounted for in the propagation models by adjusting the re-flection coefficient of surfaces

where is the scattering loss factor, and can be modelled as [26]

where is the zeroth order modified Bessel function of the first kind.

2.3.3 LOG-NORMAL SHADOWING

When large obstructions are present between the transmitter and receiver, the path loss asgiven by (2.29) may be very different at two different locations with the same T-R separa-

(2.30)

(2.31)

(2.32)

(2.33)

(2.34)

(2.35)

1 2 1 1 cos2

cos2 (1 cos2 )sin sin sin2 sin

δ δ δ δ δ θη η η

θ θθ θ θ

η θ

= − = −

= − = −

=

1 22

( )

4 sin

πφ δ δ

λπη θλ

Δ = −

=

0φΔ ≈

φ πΔ ≈

/2φ πΔ =

8 sincλ

ηθ

=

cη

cη η>

ησ

sin θ θ≈ η ησ

4 dπσ θλ

Δ =

10Δ >

0.1Δ < 1ºθ = ησ

rough sρΓ = Γ

sρ

2 2

0sin sin

exp 8 8s Iη ηπσ θ πσ θρ

λ λ

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎟ ⎟⎢ ⎥ ⎢ ⎥⎜ ⎜= − ⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

0( )I ⋅


tion. Different levels of clutter on the propagation path lead to random shadowing effectsat the receiver, and are usually modelled by a log-normally distributed, slowly varying ran-dom process [24], [25]. The random process is usually considered to be a multiplicativeprocess

where is the expected signal level obtained from (2.28). As the T-R separation changes,the presence or not of an obstruction on the path will cause a change in the expected sig-nal level given by

where is the attenuation/gain in nepers. Moreover, in a multiple obstructions scenariothe change in signal level will be

where the are assumed to be zero-mean, independent random variables. Thus, thepower loss variation may be expressed stochastically as

For a large number of obstructions, the central limit theorem suggests that will be ap-proximately a zero-mean, Gaussian distributed random variable with standard devia-tion (dB). In addition, may be regarded as a sample function of a log-normallydistributed random process. represents the path loss in dB and can be included in(2.29) to give

In practice, is computed from measured data. The probability distribution of maythen be used to determine the probability that the received signal level will exceed or fallbelow a particular level.

2.4 Small-Scale Fading

The small-scale fading of the signal in a mobile radio environment is a phenomenon thatoccurs when either the transmitter or the receiver are surrounded by nearby scatterers,such as houses, buildings, walls, and trees. These scatterers will create multiple propagationpaths between the transmitter and the receiver, each arriving with a particular phase delay.

Furthermore, (2.32) reveals that as the operating frequency rises the more intense thewave scattering becomes, because more and more objects of smaller dimensions are thenprone to scatter the incoming waves. For instance, the light frequencies are so high (in theorder of Hz), that even the irregularities of a small sheet of paper produce wave scat-tering, and that is why the paper does not perform as a specular reflective surface, appear-ing white.

(2.36)

(2.37)

(2.38)

(2.39)

(2.40)

( ) ( ) ( )r t a t s t=

( )s t

( )( ) ta t eα=

( )tα

( )( )

ii

ta t e

α∑=

( )i tα

10

10

20 log ( )

20 ( )logi

i

X a t

t e

σ

α

=

= ∑

Xσ

σ ( )a t

Xσ

0 100

( ) 10 logdB dBd

L L d p Xd σ

⎛ ⎞⎟⎜= + +⎟⎜ ⎟⎜⎝ ⎠

σ Xσ

1510

2.4 Small-Scale Fading 11

Therefore, and because there is an increasing tendency for wideband wireless com-munications and higher operating frequencies, the scattering effects become of more sig-nificance and must be considered.

The equivalent baseband received signal for a multipath environment is given by(2.23) and reproduced here for convenience

where

Some considerations may be used to simplify (2.41) while keeping the small-scale fad-ing information. First, since the Doppler shift is very small compared to the carrier frequen-cy, the delay variation given by is negligible during a symbol period. For example,at 2 GHz and for a mobile unit velocity of 200 km/h, the maximum Doppler shift is about400 Hz, while the data/chip rates are around 5 Mbps (e.g. WCDMA), which means thatapproximately 50000 symbols are transmitted before a delay variation of 1% the symbolperiod is reached. These 50000 symbols correspond approximately to a displacementof the mobile unit, which is enough to preserve the fading behaviour of the signal. Second,since is small compared to the multipath distances between transmitter and receiver,the voltage amplitude of each propagation path may be considered time-invariantduring the transmission. Assuming large distances between transmitter and receiver, andnear-horizontal wave propagation, may be reformulated as

where is still the close-in distance, the received power reference point. Introducingthese simplifications, (2.41) reduces to

where .Apart from the deterministic digital baseband signal , (2.43) clearly shows that the

received signal is the sum of complex valued stochastic time-variant phasors, which, de-pending on the relative phases, will add in or cancel out, causing the multipath fading ef-

(2.41)

(2.42)

(2.43)

( )

1

0

( ) ( ) i i

Nij t

b i b ici

r t t e s t tϕ ω ωυ τ

ω

−+

=

⎛ ⎞⎟⎜= − + ⎟⎜ ⎟⎜⎝ ⎠∑

th

th

th

( ) baseband digital transmitted signal

( ) amplitude of the i propagation path

N = number of multipath replicas

phase shift of the i propagation path

Doppler shift of the i propagatio

b

i

i

i

s t

tυ

ϕ

ω

=

=

=

=th

n path

time delay of the i propagation path

carrier frequency

i

c

τ

ω

=

=

( / )i c tω ω

4λ

4λ id

( )i tυ

( )i tυ

0

0

00

( ) 2 ( , ) ( , )4

24

i r T T i i R i i ii

r T T R ii

ii

t R P G Gd

dR P G G

d ddd

λυ θ φ θ φ ρ

πλ

ρπ

υ ρ

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠

=

0d

( )

1

0

0

( ) ( )i i

Nj t

b i b i

i

r t c e s tϕ ωυ τ−

+

=

= −∑

0( / )i i ic d dρ=

( )bs t


fect. It should be noticed that relatively slow motions of the medium can cause significantphase changes. For instance, a displacement of the mobile unit causes a phase changeof and radians in the multipath components arriving at 0º and 180º relative tothe velocity vector, respectively. If the components were initially in-phase, after the move-ment they will be out-of-phase and will cancel each other. This example agrees with ex-perimental data, which shows that the deep fades of the signal occur in intervals of displacements of the mobile unit.

2.4.1 STATISTICAL CHANNEL CHARACTERIZATION

The equivalent baseband time-variant response of the mobile radio channel to an inputequivalent baseband signal, may be written as the convolution

where represents the channel response at time due to an impulse applied attime . Comparing (2.44) with (2.43), the equivalent baseband channel impulse re-sponse is obtained as a sequence of delayed impulses, each scaled by a multipath propaga-tion stochastic phasor, that is

As can be seen, the phase is the only time-variant parameter, meaning that the amplitude(and power) associated with each propagation path remains constant.

To analyse how the power distributes among the multiple propagation paths, the au-tocorrelation function of the channel impulse response may be computed as

Assuming that the propagation paths are uncorrelated with one another (independentscatterers), the autocorrelation function is only non-zero for , that is

Also, if the channel is wide-sense stationary, the autocorrelation function only depends onthe difference of observation times

and is called the delay cross-power density. This type of channel is called a wide-sensestationary - uncorrelated scattering (WSSUS) channel, and it has shown to be a good model forthe mobile radio channel over small-scale distance or time intervals [27]. The WSSUSchannel impulse response is approximated by (2.45), where the stochastic parame-ters , , , and are considered statistically independent and identically distributed(i.i.d.) from path to path (uncorrelated scattering) and time-invariant for small-scale dis-

/4λ

/2π /2π−

/2λ

(2.44)

(2.45)

(2.46)

(2.47)

(2.48)

( ) ( , ) ( )b b br t h t s t dτ τ τ∞

−∞= −∫

( , )bh tτ t

t τ−

( )

1

0

0

( , ) ( )i i

Nj t

b i i

i

h t c e ϕ ωτ υ δ τ τ−

+

=

= −∑

*1 2 1 2 1 1 2 2

1( , ; , ) ( , ) ( , )

2hh b bR t t E h t h tτ τ τ τ⎡ ⎤= ⎣ ⎦

1 2τ τ=

*1 2 1 2 1 1 1 2 1 2

1( , ; , ) ( , ) ( , ) ( )

2hh b bR t t E h t h tτ τ τ τ δ τ τ⎡ ⎤= −⎣ ⎦

1 2t t tΔ = −

*1 2 1 2 1 1 1 1 1 2

1 1 2

1( , ; , ) ( , ) ( , ) ( )

2( , ) ( )

hh b b

hh

R t t E h t h t t

R t

τ τ τ τ δ τ τ

τ δ τ τ

⎡ ⎤= + Δ −⎣ ⎦

= Δ −

( , )hhR tτ Δ

ic iϕ iω iτ


tances (wide-sense stationarity).

2.4.2 DELAY PROFILE MEASUREMENTS

Setting in the impulse response autocorrelation function, results. It iscalled the delay power density or power delay profile of the channel, and is measurable bytransmitting very narrow probe pulses and squaring the received signal amplitude. If theprobe pulse is defined as

where is the pulse width, the received signal will be

Since any practical measuring system is bandwidth-limited, it is often difficult to resolveall multipath components arriving at the receiver. Some multipath components will thenvectorially combine to yield a multipath amplitude fading process, which means that themeasured will be time-variant over small-scale distances. If the bandwidth of themeasuring system is high enough to resolve every multipath component, avoiding the fad-ing process, an approximate replica of may be obtained as

where the bandwidth of the measuring system is and should be sufficient to resolveall multipaths. If the bandwidth is not high enough, then a small-scale distance aver-age must be obtained to conceal the fading processes. A novel approach based onpseudo-random sequences for high resolution delay profile measurements was presentedin [28]. A typical measure of the delay power density (relative to the time and power of thefirst arriving multipath component) is shown in Figure 2.3.

The delay dispersion parameters that characterize the delay power density are the first

(2.49)

(2.50)

(2.51)

0tΔ = ( )hhR τ

2/ 0( )

0

p p

p

T t Tp t

T t

⎧ ≤ ≤⎪⎪⎪= ⎨⎪ <⎪⎪⎩

pT

( )

1'

0

0

( ) ( )i i

Nj t

b i i

i

h t c e p tϕ ωυ τ−

+

=

= −∑

( )hhR τ

( )hhR τ

( )

' ' ' *

1 120

0 0

12 2 20

0

1( ) ( ) ( )

2

1( ) ( )

2

1( )

2

1( )

i j i j

hh b b

N Nj t t

i j i j

i j

N

i i

i

hhp p

R h t h t

c c e p t p t

c p t

tR rect

T T

ϕ ϕ ω ω

τ

υ τ τ

υ τ

τ

− −− + −

= =

−

=

=

= − −

= −

⎛ ⎞⎟⎜= ∗ ⎟⎜ ⎟⎟⎜⎝ ⎠

∑∑

∑

1/ pT

' ( )hhR τ


moment or mean excess delay

and the square root of the second central moment or RMS delay spread

2.4.3 DELAY PROFILE MODELLING

The exponentially decaying attribute of the delay power density is easily understood. In anurban propagation environment, the receiver is far from the transmitter and cluttered withnearby reflectors and scatterers, so that the distance spread of thearising multipath components is much less than the propagation distance and, therefore,that the difference in attenuation between the multipath components will be mainly dueto the effective reflection coefficient of each component. If the -th component expe-riences reflections, each of which with reflection coefficient , the power it carries canbe expressed as

where is a multiplicative constant and . Moreover, the number of reflections canbe estimated to be proportional to the propagation delay of the multipath compo-nent, , and (2.54) may be rewritten as

where is the channel dependent RMS delay spread. If the multipathspreading is assumed a continuous process, the power delay profile is approximated by

FIGURE 2.3 Typical measured excess delay power density for an urban environment.

(2.52)

(2.53)

-20

-10

0

0 2 1 3 4 -1 5 6

Excess delay (μs)

No

rmal

ized

Rec

eive

d P

ow

er (d

B)

RMS delay spread

Mean Excess delay

Noise Threshold

0 0( ) ( )hh hhR d R dτ τ τ τ τ τ

∞ ∞

= ∫ ∫

22

0 0( ) ( )hh hhR d R dτσ τ τ τ τ τ τ

∞ ∞⎛ ⎞⎟⎜= −⎟⎜ ⎟⎜⎝ ⎠∫ ∫

(2.54)

(2.55)

(2.56)

( max min )i id d dΔ = −

d

iρ i

in ρ

2 20 0

ini iP P Pρ ρ= =

0P iniρ ρ=

i in kτ=

20

20

2 ln(1/ )0

/0

i

i

i

i

ni

k

k

P P

P

P e

P e τ

τ

τ ρ

τ σ

ρ

ρ

−

−

=

=

=

=

[ ] 12 ln(1/ )kτσ ρ −=

/0( )hhR P e ττ στ −=


Equation (2.56) justifies the exponential decay shown in Figure 2.3, but naturally does nottake into account the dominant reflections of the waves, which may appear at the receiveras strong signals with large excess delays. Furthermore, the delay probability distributioncan be expressed as

In fact, if the propagation delay is partitioned into small delay bins, and since many of themultipath components arrive closely in time and few arrive spaced far apart, the powerreceived within each delay bin should be approximately proportional to the number ofmultipath components within the bin. Therefore, from (2.57), the multipath arrival de-lay may be treated as an exponentially distributed random variable [24], [25].

The RMS delay spread is used as a measure of the range of values of overwhich is essentially nonzero or, in other words, as a measure of the delay dispersionof the different multipath components relative to the mean delay. The typical mean RMSdelay spreads in suburban and urban areas are 0.5 μs and 1.3 μs, respectively, and the max-imum RMS delay spread generally used as a rule of thumb for calculations is 3 μs. Thismeans that it is highly probable that any signalling rate in the order of megabits per secondwill cause intersymbol interference.

2.4.4 TIME AND FREQUENCY AUTOCORRELATION

The delay spread has a close relationship with the frequency response of the mobile radiochannel, which will now be statistically characterized. Given that the mobile radio channelimpulse response must be modelled as a time-variant stochastic process, the sameapplies to the Fourier transform . Its autocorrelation function is

showing a Fourier transform relationship with the delay cross-power density ,and implying a frequency difference dependence only, which is the result of the uncorre-lated scattering assumption. The spaced-time, spaced-frequency correlation func-tion outlines two important channel measures: coherence time andcoherence bandwidth . These characterize the time and frequency distortion (selec-tivity) of the channel, respectively.

For the channel model in (2.45), can be derived as follows. The Fourier

(2.57)0

/0 /

/0

0

( )( )

( )

1

hh

hh

Rp

R d

P ee

P e d

ττ

τ

τ στ σ

ττ σ

ττ

τ τ

στ

∞

−−

∞−

=

= =

∫

∫

τ

τσ τ

( )hhR τ

(2.58)

( , )bh tτ

( , )bH f t

1 1 2 2

1

*1 2 1 2

2 ( )1 1 2 1 2

21 1

1( , ; ) ( ; ) ( ; )

2

( ; ) ( )

( ; ) ( ; )

HH b b

j f fhh

j fhh HH

R f f t E H f t H f t t

R t e d d

R t e d R f t

π τ τ

π τ

τ δ τ τ τ τ

τ τ

∞−

−∞∞

− Δ

−∞

⎡ ⎤Δ = + Δ⎣ ⎦

= Δ −

= Δ = Δ Δ

∫∫

( , )hhR tτ Δ

( ; )HHR f tΔ Δ ( )ctΔ

( )cfΔ

( ; )HHR f tΔ Δ


transform of the channel impulse response is

where is the maximum Doppler shift. Then, the time-frequency correlationof is given by

The random variable is uniformly distributed in the interval , exceptwhen , where it equals with probability . Considering the statistical independenceof all the random variables , , , and , this means that taking the expecta-tion in (2.60) over gives a non-zero value only for . This greatly simplifies theresult in (2.60) to

where, with no loss of generality, it is assumed that . Furthermore, it is eas-ily checked that

and that the following relations are valid (they are the consequence of time and frequencywide-sense stationarity)

As expected from the WSSUS channel assumptions, the channel autocorrelationfunction only depends on time and frequency separations. To com-plete the expectation process, the probability density function of and must be speci-fied. The delay was already inferred in (2.57) to be exponentially distributed. Also, if themultipath scatterers are numerous and distributed 360º around the receiver, the waves willarrive from all angles in the azimuth plane with equal probability [29], i.e.

(2.59)

(2.60)

(2.61)

(2.62)

(2.63)

(2.64)

2

1( 2 cos 2 )

0

0

( , ) ( , )

( ; ) ( ; )i D i i

j fb b

Nj f t f

i

i

H f t h t e d

c e I f t jQ f t

π τ

ϕ π φ π τ

τ τ

υ

∞−

−∞−

+ −

=

=

= = +

∫

∑

/Df ν λ=

( , )bH f t

[ ] ( )

*

12( ) 2 (cos ( ) cos ) 2 20

0

1( , ; ) ( ; ) ( ; )

2

2k i D k i b k a i

HH a b b a b b

Nj f t t t f f

i k

i

R f f t E H f t H f t t

E c c E eϕ ϕ π φ φ π τ π τυ

−− + +Δ − − +∗

=

⎡ ⎤Δ = + Δ⎣ ⎦

⎡ ⎤= ⎢ ⎥⎣ ⎦∑

i jϕ ϕ− (0,2 )π

i j= 0 1

ic iϕ iφ iτ

i jϕ ϕ− i j=

[ ] ( )

[ ] ( )

( )

122 cos 2 ( )0 2

02

2 cos 2 ( )0 2

2 cos 2

( , ; )2

2

( ; )

D i a b i

D a b

D

Nj f t f f

HH a b i

i

j f t f fi

j f t fHH

R f f t E c E e

NE c E e

E e R f t

π φ π τ

π φ π τ

π φ π τ

υ

υ

−Δ + −

=

Δ + −

Δ + Δ

⎡ ⎤Δ = ⎢ ⎥⎣ ⎦

⎡ ⎤= ⎢ ⎥⎣ ⎦

⎡ ⎤= = Δ Δ⎢ ⎥⎣ ⎦

∑

[ ]2 202/iE c Nv=

[ ] [ ]( , ; ) ( ; ) ( ; ) ( ; ) ( ; )

( , ; ) ( , ; )

HH a b a b a b

II a b IQ a b

R f f t E I f t I f t t jE I f t Q f t t

R f f t jR f f t

Δ = + Δ + + Δ

= Δ + Δ

[ ] [ ]

[ ] [ ]

( ; ) ( ; ) ( ; ) ( ; )

( ; ) ( ; ) ( ; ) ( ; )

a b a b

a b a b

E I f t I f t t E Q f t Q f t t

E Q f t I f t t E I f t Q f t t

+ Δ = + Δ

+ Δ = − + Δ

( , ; , )HH a bR f f t t t+ Δ

φ τ

1( ) 0 2

2p φ φ π

π= ≤ ≤


Thus, the in-phase correlation function becomes

Performing a double integration by parts in the inner integral, (2.65) may be simplified to

where is the zeroth order Bessel function of the first kind. In much the same way, itcan be shown that

Hence, the spaced-time, spaced-frequency autocorrelation function may be written as

One should notice from (2.68) that for , , and also that thechannel will present a phase deviation for , as expected. Plots of (2.68) for the twocases and are shown in Figures 2.4(a) and 2.4(b), respectively. These clear-

ly illustrate that, the higher are the Doppler shift and the RMS delay spread, the more se-lective is the channel in the time and frequency domains, respectively.

To find the coherence bandwidth of the channel, one may put in (2.68), and use

(2.65)

(2.66)

(2.67)

(2.68)

( )[ ]

( )2

/

0 0

( ; ) cos 2 cos 2

1 1cos 2 cos 2

2

II D

D

R f t E f t f

f t f e d dτ

πτ σ

τ

π φ π τ

π φ π τ τ φπ σ

∞−

Δ Δ = Δ + Δ

= Δ + Δ∫ ∫

( )2

20

02

1 1( ; ) cos 2 cos

21 (2 )

(2 )1 (2 )

II D

D

R f t f t df

J f tf

π

τ

τ

π φ φππ σ

ππ σ

⎡ ⎤⎢ ⎥Δ Δ = Δ⎢ ⎥+ Δ ⎣ ⎦

Δ=

+ Δ

∫

0( )J ⋅

( )2

20

02

1 2( ; ) cos 2 cos

21 (2 )

(2 )2

1 (2 )

IQ D

D

fR f t f t d

f

J f tf

f

πτ

τ

ττ

π σπ φ φ

ππ σπ

π σπ σ

⎡ ⎤Δ⎢ ⎥Δ Δ = Δ⎢ ⎥+ Δ ⎣ ⎦Δ

= Δ+ Δ

∫

1

0 2

0 tan (2 )2

( ; ) ( ; ) ( ; )

1 2(2 )

1 (2 )

(2 )

1 (2 )

HH II IQ

D

D j f

R f t R f t jR f t

j fJ f t

f

J f te

fτ

τ

τ

π σ

τ

π σπ

π σπ

π σ

− Δ

Δ Δ = Δ Δ + Δ Δ

+ Δ= Δ

+ ΔΔ

=+ Δ

0t fΔ = Δ = ( ; ) 1HHR f tΔ Δ =

0fΔ >

0fΔ = 0tΔ =

(a) (b)

FIGURE 2.4 Spaced-Time (a) and Spaced-Frequency (b) correlation functions.

-0.01 -0.005 0 0.005 0.01-0.5

0

0.5

1

Time separation (s)

Spa

ced-

Tim

e C

orre

latio

n

fD = 100 HzfD = 200 Hz

-0.5 -0.25 0 0.25 0.50.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Spa

ced-

Freq

uenc

y C

orre

latio

n

Frequency separation (MHz)

στ = 1 μsστ = 2 μs

0tΔ =


the -3 dB cutoff criterion

which gives the coherence bandwidth as

The inverse proportionality between the coherence bandwidth and the RMS delayspread was already expected because of the Fourier transform relationship between thedelay power density and the spaced-frequency autocorrelation function, as given by (2.58)and rewritten here with :

For the rule of thumb RMS delay spread , the coherence bandwidth is approxi-mately , which means that, besides intersymbol interference, widebandmobile communications will experience frequency distortion of the modulated signal.

Similarly to the coherence bandwidth, the coherence time can be found by putting in (2.68) and writing

which numerically gives

2.4.5 FREQUENCY SELECTIVITY SIMULATION

To confirm the frequency selectivity of the multipath propagation channel, a frequencyresponse numerical simulation based on (2.59) is plotted in Figure 2.5. The simulation pa-rameters are: number of propagation paths , RMS delay spread , maxi-mum Doppler shift , LOS distance , frequency band ,and angle of arrival , . For simplicity, the multipath compo-nents were assumed equally spaced in their delay and, in addition, their amplitudes wereassumed exponentially decaying. One observes that the response is periodic and also thatthe rate of frequency response variation is necessarily proportional to the Doppler shift ofthe channel. The time step in the simulation was , which means that for sig-nalling rates of 5 MHz the channel frequency response remains virtually unchanged for asymbol/chip period ( ). We also gather that the spacing of two consecutive dipsin the frequency response is approximately 0.5 MHz, which is almost equal to the recip-rocal of the RMS delay spread.

(2.69)

(2.70)

(2.71)

(2.72)

(2.73)

2

1 1(( ) ; 0)

21 (2 ( ) )HH c

c

R f tf τπ σ

Δ Δ = = =+ Δ

1( )

2cfτπσ

Δ =

( )cfΔ

τσ

0tΔ =

2( ) ( ) j fHH hhR f R e dπ ττ τ

∞− Δ

−∞Δ = ∫

3 sτσ = μ

( ) 50 kHzcfΔ =

0fΔ =

01

( 0;( ) ) (2 ( ) )2HH c D cR f t J f tπΔ = Δ = Δ =

1.126 9 1( )

2 16 2cD D D

tf f fπ π π

Δ ≈ ≈ ≈

10N = 3 sτσ = μ

200 HzDf = 1 kmd = 5 MHzB =

( )2 /i N iφ π= 0,.., 1i N= −

0.167 mstΔ =

cT 0.2 s= μ


2.4.6 FREQUENCY SELECTIVITY MODELLING

The noiseless received signal from a time-variant mobile propagation channel can be ex-pressed as

where is the baseband Fourier transform of the input signal. If the channel is fre-quency-selective, i.e. its coherence bandwidth is less than the bandwidth of the basebandinput signal, cannot be considered constant within the input signal bandwidth, andso it cannot be removed from the integral expression. However, due to the limited band-width condition of , the output signal should also be bandwidth-limited, meaningthat both the input and output channel signals can be represented by their respective sam-ples at the Nyquist rate.

If the symbol rate is , for less than excess bandwidth pulses (e.g. raised cosine puls-es with less than unity roll-off factors), the maximum bandwidth of the input and outputsignals is and the minimum sample period is . The frequency

FIGURE 2.5 Simulated frequency response of multipath propagation channel for consecutive time instants, within a 5 MHz bandwidth.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 106

-20

-15

-10

-5

0

5

10

15

20

Frequency (Hz)

Am

plitu

de (d

B)

t = 0t = 1/(10πfD)t = 2/(10πfD)

(2.74)

(2.75)

2

( ) ( ; ) ( )

( ; ) ( )

b b b

j ftb b

r t h t s t d

H f t S f e dfπ

τ τ τ∞

−∞∞

−∞

= −

=

∫∫

( )bS f

( ; )bH f t

( )bS f ( )br t

( ) ( ) ( )

( ) ( ) ( )

bs b s

n

bs b s

n

s t s t t nT

r t r t t nT

δ

δ

∞

=−∞∞

=−∞

= −

= −

∑

∑

R 100%

W R= 1/ 1/2s sT f R= =


spectrum of is given by

and the frequency spectrum of is

Moreover, the samples of the channel response obtained at the same samplingrate are needed to perform the convolution with the input signal, giving the frequency do-main response as

Now, noticing (2.76) and (2.77), it is clear that the channel output can be related to thefrequency response of the discrete impulse response as

The discrete channel impulse response is derived from by partitioning thedelay axis into equal time delay segments called excess delay bins , where

is the partition width or time resolution of the multipath delay profile.In effect, the finite bandwidth of the input signal and the receiver leads to an upperbound for the resolution of structures in the impulse response due to the smearing of thepulses over a width . Thus, all the multipath components received within theith partition are assumed to have the same time delay , and are considered unresolvable,which means that they approximately combine to produce a complex stochastic process.Ultimately, the discrete impulse response is obtained by separating resolvable paths fromunresolvable paths, yielding

where is a complex-valued wide-sense stationary random process. The Fourier trans-

(2.76)

(2.77)

(2.78)

(2.79)

(2.80)

( )bss t

( ) ( ) ( )

( ) ( )

( )

bs b s

n

s b s

n

s b s

n

S f S f t nT

f S f f nf

f S f nf

δ

δ

∞

=−∞∞

=−∞∞

=−∞

⎧ ⎫⎪ ⎪⎪ ⎪= ∗ ℑ −⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

= ∗ −

= −

∑

∑

∑( )bsr t

( ) ( )bs s b s

n

R f f R f nf∞

=−∞

= −∑

( ; )bh tτ

1( ) ( ; ) ( )bs bs bs

sR f H f t S f

f=

( ; )bsh tτ

( )/ /2( )

0 /2

1( ; ) ( )

bs s s

bs

bs bs

R f f f fR f

f f

H f t S ff

⎧ ≤⎪⎪= ⎨⎪ >⎪⎩

=

( ; )bsh tτ ( ; )bh tτ

τ i iτ τ= Δ

1/2 /2R TτΔ = =

W

1(2 )Wτ −Δ =

iτ

( )

1 1

0 0

0 0

1

0

0

( , ) ( )

( ) ( )

i

ik ik

N Lj t

bs ik s

i k

N

i s

i

h t c e iT

v t iT

ϕ ωτ υ δ τ τ

δ τ τ

− −+

= =

−

=

⎡ ⎤⎢ ⎥= − −⎢ ⎥⎢ ⎥⎣ ⎦

= − −

∑ ∑

∑

( )iv t


form of (2.80) is

which, after direct substitution into (2.79), and taking the inverse Fourier transform, yieldsthe baseband response

where

From (2.82) it is clear that the baseband output signal of the frequency selective mobilepropagation channel is a finite convolution sum of the input signal with the discrete chan-nel impulse response. A tapped delay line model of (2.83) is shown in Figure 2.6, which isparticularly useful for channel numerical simulations, provided that the stochastic tapweights are correctly modelled.

Short-term channel variations may differ for each delay bin , and may be ideally charac-terized by the WSSUS model

which is a modified version of (2.48) to include the tap fading processes. Also, the timeautocorrelation function of each tap fading process is the same of (2.68) with foruniformly distributed angles-of-arrival (AOA).

2.4.7 STATISTICAL FADING CHARACTERIZATION

The statistical characterization of the ’s can be done by assuming is very large, and

(2.81)

(2.82)

(2.83)

FIGURE 2.6 Tapped delay line model for a noiseless frequency-selective multipath channel.

(2.84)

0

12 ( )

0

( , ) ( ) s

Nj f iT

bs i

i

H f t v t e π τ−

− +

=

= ∑

0

2

2

12 ( )

01

0

0

( ) ( )

1( ; ) ( )

1( ) ( )

( ) ( )

s

j ftb b

j ftbs b

s

Nj f t iT

i bsi

N

i b s

i

r t R f e df

H f t S f e dff

v t S f e dff

h t s t iT

π

π

π τ

τ

∞

−∞∞

−∞− ∞

− −

−∞=−

=

=

=

=

= − −

∫∫

∑ ∫

∑

( )

10

0

( )i

ik ik

Lj t

i iks k

h t c ef

ϕ ωυ−

+

=

= ∑

( )ih t

Delay Delay Delay

h1(t) h2(t) hN(t)h3(t)

Input sb(t)

Frequency-selective output rb(t)

Delayτ0

iτ

*1( , ; ) ( ) ( ) ( )

2hh i j i j i jR t E h t h t tτ τ δ τ τ⎡ ⎤Δ = + Δ −⎣ ⎦

0fΔ =

( )ih t iL


using the central limit theorem to estimate the probability distribution. Equation (2.83)can be rewritten as

Now, since , and are considered statistically independent random variables andthe phases are uniformly distributed in the interval , , andby the central limit theorem both and are zero mean Gaussian random variables,then

Moreover, since , they have equal variance , and be-cause , they are also uncorrelated. The Gaussian assumption and the un-correlatedness are sufficient conditions for the statistical independence of and ,and thus their joint density is

If the amplitude and phase are

then a solution pair is

and the joint distribution of and is

The envelope distribution is obtained by integrating with respect to , giving

In conclusion, the envelope of the stochastic tap values is Rayleigh distributed for anyfixed value of and the phase is uniformly distributed in the interval . The envelopestatistic characterizes the small-scale fading behaviour of the mobile channel signal when

(2.85)

(2.86)

(2.87)

(2.88)

(2.89)

(2.90)

(2.91)

(2.92)

( ) ( )

1 1

0 0

2 2 arctan( ( )/ ( ))

( ) cos sin

( ) ( )

( ) ( )

i iL L

i ik ik ik ik ik ik

k k

j Y t X t

h t a t j a t

X t jY t

X t Y t e

ϕ ω ϕ ω− −

= =

= + + +

= +

= +

∑ ∑

ika ikϕ ikω

ikϕ (0,2 )π [ ] [ ]( ) ( ) 0E X t E Y t= =

( )X t ( )Y t

2 2/21( )

2xx

Xx

f x e σ

πσ−=

2 2/21( )

2yy

Yy

f y e σ

πσ−=

[ ] [ ]2 2( ) ( )E X t E Y t= 2 2 2x yσ σ σ= =

[ ]( ) ( ) 0E X t Y t =

( )X t ( )Y t

2 2 2( )/22

( , ) ( ) ( )

12

XY X Y

x y

f x y f x f y

e σ

πσ− +

=

=

2 2( ) ( ) ( ) ( ) arctan( ( )/ ( ))R t X t Y t t Y t X tθ= + =

( ) ( )cos ( ) ( ) ( )sin ( )x t r t t y t r t tθ θ= =

( )R t ( )tθ

2 2/22

( , ) ( , ) ( , )

( cos , sin )

2

R XY

XY

r

f r f x y J r

x xrf r ry yr

re

θ

σ

θ θ

θθ θ

θ

πσ−

=

∂ ∂∂ ∂=∂ ∂∂ ∂

=

θ

2 2/22( ) r

Rr

f r e σ

σ−=

( )ih t

t (0,2 )π


a receiver moves over small distances as compared to the wavelength.The Rayleigh distribution obtained is valid only when the multipath phenomenon oc-

curs mainly because of local scatterers around the receiver (e.g. urban areas), meaning thatthe zero mean Gaussian distribution of and is an acceptable assumption, asshown in Figure 2.7 a).Whenever there is a LOS multipath component reaching the re-

ceiver, or some dominant specular reflections of the signal are present, the zero mean as-sumption of and is no longer valid because the signal strength of the scatteredmultipath components will be much lower than the former’s. In this case, the complexmultipath process can be assumed a nonzero mean Gaussian process, as depicted inFigure 2.7 b), and with joint pdf given by

where and are the in-phase and quadrature specular component amplitudes. Fol-lowing the same procedure from (2.88) to (2.91), it follows that

where and . Thus

where is the zeroth order modified Bessel function of the first kind. The pdf of theenvelope given by (2.95) is called the Rice distribution [30]. When plotted, the Rician pdfconfirms a less severe fading effect than the Rayleigh pdf, making it suitable for the char-

FIGURE 2.7 Possible probability density functions for local area signal fading characterization.

(2.93)

(2.94)

(2.95)

( )X t ( )Y t

( )X t ( )Y t

2 2

2

( ) ( )

22

1( , )

2

x yx m y m

XYf x y e σπσ

− + −−

=

xm ym

( )

2 2

2

2 2 2 20

( cos ) ( sin )

22

/2 cos( )/2

( , )2

2

x yr m r m

R

r A rA

rf r e

re e

θ θσθ

σ θ θ σ

θπσ

πσ

− + −−

− + −

=

=

2 2x yA m m= + 0 arctan( / )y xm mθ =

( )

( )

( )

2 2 2 20

2 2 2 2

2 2 2

2/2 cos( )/

20

2/2 cos /

20

/202 2

( )2

12

r A rAR

r A rA

r A

rf r e e d

re e d

r rAe I

πσ θ θ σ

πσ θ σ

σ

θπσ

θπσ

σ σ

− + −

− +

− +

=

⎡ ⎤⎢ ⎥= ⎢ ⎥⎣ ⎦⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠

∫∫

0( )I ⋅


acterization of rural or sub-urban areas.While the Rayleigh and Rice distributions can indeed be used to model the envelope

of fading channels in many cases of interest, it has been found experimentally that the Na-kagami-m distribution offers a better fit for a wider range of fading conditions (see [25],[31], [32], [33], and [34]. The Rice distribution can only describe better-than-Rayleigh fad-ing conditions, whereas the Nakagami-m distribution can describe Rayleigh, Rice andworst-than-Rayleigh fading conditions.

A roughly generic form for the expression of the fading tap weights is obtainedwhen the scattering process at time delay is assumed to be the sum of the contributionsof several clusters of scatterers, each cluster approximately producing an independentcomplex zero-mean Gaussian process. If this is the case, and in (2.85) can be writtenas

where for simplicity of presentation the time dependency of the random variables hasbeen omitted. The joint probability density function of the ’s and the ’s is given by

The squared envelope of the random fading process can be written as

Due to the independence of the scatterers and, as a result, the uncorrelatedness of the ’sand the ’s, the second term in (2.98) is negligible as compared with the first term, andso a reasonable approximation is to equate it to zero, and thus

Furthermore, expressing the joint probability density function in 2m-dimensional hyper-

(2.96)

(2.97)

(2.98)

(2.99)

( )ih t

iτ

X Y

1 1

0 01 1

0 0

cos

sin

m m

k k k

k km m

k k k

k k

X X

Y Y

α θ

α θ

− −

= =− −

= =

= =

= =

∑ ∑

∑ ∑

kX kY

( )

( )

2 2 2

12 2 2

0

1/2

20

/2

2

1(.., ,.., ,..)

2

1(2 )

k kk k

m

k kk

mx y

X Y k k

k

x y

m m

f x y e

e

σ

σ

πσ

π σ

−

=

−− +

⋅⋅ ⋅⋅ ⋅⋅=

− +

=

∑=

∏

( )

2 2 2

2 21 1

0 0

1 1 12 2

0 0 0

( )

m m

k k

k k

m m m

k k k l k l

k k ll k

R X Y

X Y

X Y X X Y Y

− −

= =

− − −

= = =≠

= +

⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜= +⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠

= + + +

∑ ∑

∑ ∑∑

kX

kY

( )1 1

2 2 2 2

0 0

m m

k k k

k k

R X Y α− −

= =

= + =∑ ∑


spherical coordinates , , where

for , yields

Now, since the vector differential length in 2m-dimensional hyperspherical coordinates isgiven by

the Jacobian in (2.101) may be written as the product of all differential lengths

which gives the envelope probability density function as

The integral in (2.104) is over all solid angles subtended by an hypersphere of unit radiusin 2m-dimensional space, giving its total hyper-surface area. To evaluate it, write

and make the change of variables , yielding

Defining the gamma function as

the area of the hypersphere in 2m-dimensional space is

(2.100)

(2.101)

(2.102)

(2.103)

(2.104)

(2.105)

(2.106)

(2.107)

(2.108)

R 0 2 2.. mθ θ −

2 1

2

02

2 1

02 1

1

0

cos sin 0,.., 1

cos sin 0,.., 2

sin

k

k k l

lk

k k l

lm

m l

l

X R k m

Y R k m

Y R

θ θ

θ θ

θ

−

=

+=

−

−=

= = −

= = −

=

∏

∏

∏0 lθ π≤ ≤

2 2

0 2 2

/20 2 0 2 22

1( , ,.., ) ( , ,.., )

(2 )m

rR m mm mf r e J rσθ θ θ θ θ θ

π σ−−

⋅⋅ − −=

0 1 2 2

2 3

0 0 1 2 2

0

sin ... sinm

m

k m

k

d dr rd r d r dθ θ θθ θ θ θ θ−

−

−=

⎡ ⎤⎢ ⎥= + + + + ⎢ ⎥⎢ ⎥⎣ ⎦∏ra a a al

( )2 3

2 22 10 2 2

0

( , ,.., ) sinm

m kmm k

k

J r rθ θ θ−

− −−−

=

= ∏

( )2 2

0 2 2

2 32 12 2/2

0 2 22.. 0

( ) sin ..(2 )

m

mmm kr

R k mm mk

rf r e d dσ

θ θθ θ θ

π σ−

−−− −−

−=

= ∏∫

2 22 1

/22 2

0 0( ) 1

(2 )

mr

R m m mr

f r dr S e drσ

π σ

∞ ∞ −−= =∫ ∫

2 2/2t r σ=

1

20

12

mt

m mt

S e dtπ

∞ −− =∫

1

0( ) m tm t e dt

∞− −Γ = ∫

22( )

m

mSmπ

=Γ


Substitution into (2.104) gives

and defining , (2.109) may be expressed as

The envelope probability density function in (2.110) is called the Nakagami-m distribution,and can be used to statistically characterize the fading processes that appear in the fre-quency-flat or frequency-selective mobile radio propagation channel. An alternative ap-proach to the derivation of the above expression based on the physical properties of theradio channel and on the Hankel transformation of radially symmetric pdfs appears in[35]. Figure 2.8 illustrates the Nakagami-m pdf for different values of . Varying the fading

figure , various channel fading conditions can be statistically emulated. For instance, when (a single scattering cluster) the Nakagami-m distribution is identical to the Rayleigh

distribution, and when it approaches the Rice distribution (i.e. when strong specularcomponents appear at the receiver). For the Nakagami-m distribution canmodel worst-than-Rayleigh scenarios that may arise when the ’s and the ’s in (2.96)cannot be considered Gaussian uncorrelated random variables.

2.4.8 TIME SELECTIVITY SIMULATION

In the absence of specular signal components, and when the delay spread of the channelis small compared to the symbol period ( or, i.e., in (2.82)), the channel willproduce a frequency-flat signal fading output. Otherwise, if , the channel outputwill be the sum of several time shifted independent stochastic processes, producing fre-quency distortion and small-scale fading at the same time. A simulation of (2.83) is shownin Figure 2.9 assuming that the multipath components’ amplitudes are equal, the phas-es are uniformly distributed in the interval , , and , where

and , . The carrier frequency is .

(2.109)

(2.110)

FIGURE 2.8 Nakagami-m pdf for several parameters, shown with .

2 22 1

/22

2( )

(2 ) ( )

mr

R mr

f r em

σ

σ

−−=

Γ

[ ]2 22E R mσΩ = =

( ) 22 1

/2( )

( )

m mmr

Rr m

f r em

−− Ω=

Γ Ω

m

0 0.5 1 1.5 2 2.50

0.5

1

1.5

Envelope r

m=1/2

m=3/4

m=1

m=3/2

m=2 m=3

Nak

agam

i fR(r)

m 1Ω =

m

1m =

1m >

1/2 1m< <

kX kY

Tτσ 1N =

Tτσ

ikc

ikϕ (0,2 )π 10iL = 2 cosik D ikfω π φ=

200 HzDf = (2 / )ik ik Lφ π= 0,.., 1ik L= − 2 GHzcf =


Since the multipath phenomenon is slowly fading for wideband communica-tions , the deep fades in Figure 2.9 occur at bursts, that is, many symbolsapart, reducing the instantaneous signal-to-noise ratio at the receiver. Consequently, theerror performance of a communications system is severely degraded in multipath fadingenvironments. The frequency domain manifestation of signal fading is the time variationof the frequency response, as already depicted in Figure 2.5.

2.4.9 TIME SELECTIVITY AND DOPPLER SPECTRUM

A spectral “picture” of the time selectivity of the channel, or in other words, the time var-iation of channel due to the receiver motion relative to the transmitter, can be obtained bytaking the Fourier transform of (2.68) with respect to the variable, when , that is

Now, since is injective in the interval , the change of variable isapplicable, and (2.111) can thus be written

Equation (2.112) is called the wavenumber spectrum or Doppler power spectrum of the mo-bile channel, and has an U-shaped appearance as shown in Figure 2.10. It characterizes thefrequency shift (spectral broadening) of the waves as perceived at the receiver due to itsmotion relative to the transmitter and, as expected, shows that the multipath components

FIGURE 2.9 Rayleigh fading behaviour of the time varying stochastic tap weights for a frequency-selective mobile propagation channel.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05-20

-15

-10

-5

0

5

10

15

20

Time (s)

h i(t) fa

ding

env

elop

e (d

B)

( )ih t

( 1 MHz)B >

(2.111)

(2.112)

tΔ 0fΔ =

20

2 cos 2

0

2 ( cos )

0

0

( ) (2 )

1

1

1( cos )

D

D

j tHH D

j f t j t

j f t

D

S J f t e d t

e e d d t

e d t d

f d

πκ

ππ φ πκ

ππ φ κ

π

κ π

φπ

φπ

δ κ φ φπ

∞− Δ

−∞∞

Δ − Δ

−∞∞

− Δ

−∞

= Δ Δ

= Δ

= Δ

= −

∫∫ ∫∫ ∫∫

cosφ 0 φ π≤ ≤ cosDx f φ=

2 2

2 2

1( ) ( )

1

D

D

f

HHf D

D

S x dxf x

f

κ δ κπ

π κ

−= −

−

=−

∫


that contribute the most to the frequency shift are the ones arriving at 0º and 180º relativeto the receiver velocity vector.

Although the Doppler spectrum in Figure 2.10 is an experimentally confirmed, wellaccepted model for the mobile radio propagation channel, it is not the better model forall channel conditions. For instance, if both the transmitter and receiver are standing still,but the propagation environment is densely occupied with randomly moving scatterers,namely wind-blown leaves or random street traffic, measurements confirm that the Dop-pler power spectrum is concentrated heavily at , and quickly diminishes for largervalues of , approximating a Laplacian-shaped function as illustrated in Figure 2.11 [36](cf. [37]). Thus, in practice a receiver will perceive a Doppler spectrum which is a “mix-ture” of Figure 2.10 with Figure 2.11.

Another time-varying channel example is the tropospheric scatter channel, which hasthe Doppler spectrum approximated by a Gaussian-shaped function

where is some arbitrary constant and is the RMS Doppler spread.

FIGURE 2.10 Typical Doppler power spectrum for mobile radio channels in cluttered outdoor environments.

(2.113)

FIGURE 2.11 Average Doppler spectrum for a fixed link and dynamic environment (e.g. traffic road) at 40 GHz.

-fd -fd/2 0 fd/2 fdDoppler frequency κ

Dop

pler

spe

ctru

m S

HH( κ

)

0κ =

κ

2 2/20( )HHS S e κκ σκ −=

0S κσ


2.4.10 BELLO FUNCTIONS AND RELATIONS

Taking the Fourier transform of in the variable

gives the average power output of the channel as a function of the time delay and theDoppler frequency . is called the scattering function of the channel, and is particularlyuseful in obtaining the power delay profile and Doppler spectrum functions by a directintegration

The relations among the correlation and power spectra functions described so far consti-tute the Bello relations [38] for time-varying channels, and are shown in Figure 2.12. Thefunctions are called the Bello system functions of the WSSUS channel, and are related by aloop of Fourier transform pairs in the , , and variables.

To complete the picture, one can easily deduce from the wide-sense stationarity prop-erty of the channel in the time and frequency domains that 1)

(2.114)

(2.115)

FIGURE 2.12 Relations among the correlation and power spectra functions for the WSSUS channel model.

(2.116)

( , )hhR tτ Δ tΔ

2( , ) ( , ) j tHH hhS R t e d tπκτ κ τ

∞− Δ

−∞= Δ Δ∫

τ

κ ( , )HHS τ κ

( ) = ( , )

( ) = ( , )

hh HH

HH HH

R S d

S S d

τ τ κ κ

κ τ κ τ

∞

−∞∞

−∞

∫∫

tΔ τ fΔ κ

Rhh(τ t)

RHH( f t)SHH(τ κ)

SHH( f, )

SHH( )

RHH( f)

RHH( t)

Rhh(τ)

ℑτ

ℑ t

ℑ t

ℑτ

t = 0

f = 0

τ

f = 0

t = 0

Doppler Spectrum

Power Delay Profile

ScatteringFunction

Spaced-timeAutocorrelation

Spaced-frequencyAutocorrelation

2 ( ' ) 2 ( ' )

2 ( ' ' ' ' )

' '

( , ) ( , ', , ') ( ' , ' )

( , )

( , ) ( ' ) ( ' ) ' '

HH HH HH

j f f j t tHH

j f t f tHH

R f t R f f t t R f f t t

S e e d d

S e d d d d

π τ πκ

τ κ

π τ κ τ κ

τ τ κ κ

τ κ ν τ

τ κ δ τ τ δ κ κ κ κ τ τ

− − −

− + + −

Δ Δ = = − −

=

= − −

∫ ∫∫ ∫ ∫ ∫


and also 2)

which must imply that

showing that the uncorrelated scattering in a WSSUS channel is present in both delay andDoppler domains.

(2.117)

(2.118)

2 ( ' ' ' ' )

' '

2 ( ' ' ' ' )

' '

1( , ', , ') [ ( ', ') ( , )]

21

[ ( ', ') ( , )] ' '2

( , ', , ') ' '

HH

j f t f t

j f t f tHH

R f f t t E H f t H f t

E H H e d d d d

R e d d d d

π τ κ τ κ

τ τ κ κ

π τ κ τ κ

τ τ κ κ

τ κ τ κ κ κ τ τ

τ τ κ κ κ κ τ τ

∗

∗ − + + −

− + + −

=

=

=

∫ ∫ ∫ ∫∫ ∫ ∫ ∫

( , ', , ') ( , ) ( ' ) ( ' )HH HHR Rτ τ κ κ τ κ δ τ τ δ κ κ= − −

CHAPTER 3

DIVERSITY RECEPTION

3.1 Introduction

It was shown in Section 2.4 that, in a typical urban or sub-urban propagation environ-ment, the reflection and scattering of electromagnetic waves is responsible for signal fad-ing at the receiver. The severity of the fading process will depend on the operatingfrequency, the data signalling rate, and the statistical properties of the radio channel. Forbroadband communications (high signalling rates) the fading is typically slow and frequen-cy selective, that is, it results from several delay-spaced fading processes (Figure 2.6). De-pending on the channel conditions, each fading process can be statistically characterizedby a Rayleigh, Rice or, more generically, by a Nakagami-m probability distribution.

The error performance of wireless communication systems is harshly degraded by sig-nal fading conditions, encouraging the employment of efficient techniques to overcomethe problem. Diversity reception techniques are based on the fact that the transmitted sig-nal has several degrees of freedom, namely time, frequency and space. In fact, the way inwhich the multipath components add at the receiver should be nearly independent fromtime to time, frequency to frequency and spatial position to spatial position, as long asenough separation is provided. Indeed, there is an intuitive fundamental premise in spatialstatistics that states that nearby things are in average more alike than remote things, and itfinds applications in geostatistics, ecology (e.g. biodiversity), archeology and even agricul-ture. As a result, it may seem reasonable to foresee that the fading statistics associated witha multipath propagation channel will also be nearly independent. For instance, in the caseof spatial diversity, if and are the sampled envelopes of the re-ceived signal at two different receiver positions separated by , then the probability

1 ( )R R t= 2 ( )R R t t= + Δ

d v t= Δ

31

32 CHAPTER 3 DIVERSITY RECEPTION

that both signal envelopes will be below a certain threshold is

and the independence condition leads to . For independent fading sig-nals the joint probability is further reduced to . A simple method to ob-tain several replicas of the transmitted signal is to place several antennas at the receiver,and if they are spaced sufficiently far apart so that their received signals fade independent-ly, then they can be used for diversity reception. Furthermore, since the uncorrelatednessof the fading signals is a manifestation of their independence, the autocorrelation functionmay be used to find the necessary spatial separation of the antennas.

3.2 Envelope Autocorrelation

Because time separation can be easily converted to space separation and it is easier to ma-nipulate the square of the envelope, it is appropriate to start with the spaced-time auto-correlation function of the squared envelope, that is

where and are given in (2.85) and

for uniform distribution of wave arrival. Now, assuming for simplicity that both and are zero-mean, Gaussian distributed random variables with variance , and using the

equality [39]

Equation (3.2) simplifies to

The second term in (3.5) expands to

where is the power of the received signal’s envelope. The spaced-time autocorre-

(3.1)

A

1 2

2 1 1

( ; )

( | ) ( )

AP P R A R A

P R A R A P R A

= < <

= < < <

21( )AP P R A= < L

1( )LAP P R A= <

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

[ ]

[ ] [ ]

22 2

2 2 2 2

( ) ( ) ( )

2 ( ) ( ) 2 ( )

R t E R t R t t

E X t X t t E X t

φ Δ = + Δ

= + Δ +

( )X t ( )Y t

[ ] [ ]( ) ( ) ( ) ( ) 0E X t Y t t E Y t X t t+ Δ = + Δ =

( )X t

( )Y t 2σ

[ ] [ ] [ ] [ ]2 2 2 2 21 2 1 2

21

222E X X E X E X E X X= +

[ ] [ ][ ] [ ]

[ ] [ ]

22 2

2 2 2

2 2 2

( ) 2 ( ) ( )

4 ( ) ( ) 2 ( )

4 ( ) 4 ( ) ( )

R t E X t E X t t

E X t X t t E X t

E X t E X t X t t

φ Δ = + Δ

+ + Δ +

= + + Δ

[ ]

[ ] ( ) ( )

[ ] [ ]

[ ]

1 1

0 0

2

20

( ) ( )

cos cos ( )

cos(2 cos / )2

( ) (2 / )

L L

i k k k i i

i k

i i

E X t X t t

E a a E t t t

LE a E v t

E X t J v t

ϕ ω ϕ ω

π φ λ

π λ

− −

= =

+ Δ =

⎡ ⎤= + + + Δ⎢ ⎥⎣ ⎦

= Δ

= Δ

∑∑

[ ]2iLE a

3.2 Envelope Autocorrelation 33

lation function of the squared envelope is finally

The correlation coefficient is given by

Substitution of (3.7) into (3.8) gives

where is the phase constant and is the antenna separation distance. This simple equa-tion gives the spatial autocovariance between the squared envelopes of the signals receivedby the antennas. It may also be derived that the correlation coefficient of the envelope isapproximated by [22]

The signal envelopes will be uncorrelated when the correlation coefficient becomes zero,and it can be found numerically that the first zero of the Bessel function occurs for

. However, due to the lack of uniform distribution of wave arrival (e.g. sub-urbanareas), and the possibility of specular components arriving at the receiver, measurementsshow that the first null is at about . For mobile units located in urban/sub-urbanareas a good rule of thumb for the antenna separation distance is usually .

The values just estimated often do not apply to base station receivers due to the re-duced angular spread of the incoming waves, which introduces extra correlation betweenthe antennas. As illustrated in Figure 3.1, on account of base stations being typically locat-ed in isolated places (above the clutter), the angles of wave arrival span only a small frac-tion of 360º, and hence higher antenna separations are required so that the propagationdifferences between the multipath components becomes significant. Moreover, the corre-

lation also depends on the angle and the height of the base station antennas. To achievelow correlation between base station antennas, separations as high as are usuallyrequired [40].

(3.7)

(3.8)

(3.9)

(3.10)

FIGURE 3.1 Illustration of the angular spread of multipath components at a base station.

[ ][ ]22 2 2

0( ) 4 ( ) 1 (2 / )R t E X t J v tφ π λΔ = + Δ

[ ][ ]

2

2

2

2 2

2 2

( ) ( )

(0) ( )R

RR

t E R t

E R t

φρ

φ

Δ −=

−

220

20

(2 / )

( )

R J v t

J d

ρ π λ

β

= Δ

=

β d

20 ( )R J dρ β≈

0.38d λ=

0.8d λ=

0.5d λ=

α

10-20λ


3.3 The Maximal-Ratio Combiner

When multiple antennas are employed at the receiver side, the original single-input single-output (SISO) channel is converted to a single-input multiple-output (SIMO) channel.This type of channel makes several signal replicas available to a receiver signal processorwhich may combine them in order to strengthen the output signal. The optimum combin-ing technique depends on the fading characteristics of the input signal replicas but, as weshall see, whenever it is possible by some procedure to convert the type of fading to flat-fading, then the maximal-ratio combiner (MRC) is the optimum combining scheme for max-imum output SNR.

Let represent a baseband-equivalent flat-fading process associated with thechannel response between the transmitter antenna and the -th receiver antenna, and letit be expressed as

Then, the output signal from the -th antenna may be written as

where is a complex-valued zero-mean Gaussian noise process. Thediversity combiner weighs and sums the signals from all the antennas, producing the out-put signal

where is the number of receiver antennas (or diversity order), and the ’s are the com-biner weights. This technique is also a type of predetection combining, meaning that the detec-tion process for will occur after the branch combining. The performance ofpredetection combining and postdetection combining is identical when coherent detection is used,but for nonlinear detectors (e.g. square-law detectors) predetection is the better combiningscheme since it provides the maximum output SNR [41]. Now, since the noise processesfrom different antennas are uncorrelated, the average noise power at the output of thecombiner is given by

where is the total noise power in each branch before detection and is the

(3.11)

(3.12)

(3.13)

(3.14)

( )kh t

k

( )

1

0

( )

( )

( )

k

ik ik

k

Lj t

k ik

i

j tk

h t a e

t e

ϕ ω

θα

−+

=

=

=

∑

k

0

( )0

( ) ( ) ( ) ( )

( ) ( ) ( )k

bk k b k

j tk b k

r t h t s t n t

t e s t n tθ

τ

α τ

= − +

= − +

( ) ( ) ( )k kI kQn t n t jn t= +

1

01 1

( )0

0 0

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )k

M

b k bk

kM M

j tk k b k k

k k

r t w t r t

w t t e s t w t n tθα τ

−

=− −

= =

=

= − +

∑

∑ ∑

M kw

0( )bs t τ−

12 2

01

2

0

1( ) ( )

2

( )

M

k k

kM

k k

k

w t E n t

w t

η

η

−

=−

=

⎡ ⎤= ⎣ ⎦

=

∑

∑

0k Bη = N B

3.4 Statistical Characterization of the MRC 35

bandwidth occupied by the signal. Furthermore, the instantaneous power of the outputsignal may be written as

where the last inequality follows from Cauchy-Schwarz. Consequently, the instantaneousoutput SNR is

The maximum output SNR will occur when , that is, when theweights are proportional to the conjugate of each fading signal and inversely proportionalto the average noise power. In fact, since the factor is identical for all the branch-es, all that the combiner weights have to do is to compensate for the phase shift of thedifferent channels and assign a larger weight to stronger signals, that is .Rewriting (3.16) in the case of maximal-ratio combining, gives

Thus, the SNR at the combiner’s output is the sum of the branch SNRs. The former willbe responsible for the error performance of the receiver system and should be as high aspossible. A more important SNR measure is obtained by averaging (3.17) over a symbolperiod , yielding

where is the symbol energy of the bandpass signal , and it is assumed that isinvariant during the symbol period. To evaluate the error performance of MRC, the prob-ability density function of the output SNR must be obtained and, if possible, should ac-count for the correlation between the antennas, since in most situations it is not feasibleto provide enough separations among them.

3.4 Statistical Characterization of the MRC

In Section 2.4.7 it was shown that the envelope of a multipath fading signal may be statis-tically described by the Nakagami-m distribution. This conclusion was withdrawn fromthe fact that the squared envelope of the channel response between the transmitterantenna and the -th receiver branch can be approximated by the sum of the squares of

zero-mean independent Gaussian random variables, as given by (2.98). This means

(3.15)

(3.16)

(3.17)

(3.18)

21

0

0

1 122

0

0 0

1( ) ( ) ( )

2

1( ) ( ) ( ) /

2

M

t k k b

k

M M

k k k b k

k k

S w t h t s t

w t t s t

τ

η α τ η

−

=

− −

= =

= −

⎛ ⎞⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜≤ −⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎝ ⎠⎝ ⎠

∑

∑ ∑

12

0

0

1( ) ( ) /

2

Mt

k b k

k

St s tγ α τ η

η

−

=

= ≤ −∑* *

0( ) ( ) ( )/k k b kw t K h t s t τ η= −

*0( )bs t τ−

*( ) ( )/k k kw t K h t η=

12

max 0

01 1

2

0 0

1( ) ( ) /

2

1/

2

M

k b k

kM M

k k k

k k

t s t

r

γ α τ η

η γ

−

=− −

= =

= −

= =

∑

∑ ∑

1/T B=

0

0

1 1 122 2

000 0 0

1 1 1( ) ( ) ( )

2

M M MTs

s k b k skk k k k

t s t dt tT

τ

τγ α τ α γ

η

− − −+

= = =

⎡ ⎤⎢ ⎥= − = =⎢ ⎥⎣ ⎦

∑ ∑ ∑∫ EN

sE ( )s t 2( )k tα

2( )k tα

k

2m


that (3.18) can alternatively be written as

where is a zero-mean Gaussian random variable associated with the -th branch and is the fading figure of the multipath channel (it is assumed that the fading figure is the

same for all antenna branches). The joint characteristic function of the ’s is given by

where and . Expanding the vectors and noting thatthe ’s are statistically independent from the ’s for , (3.20) gives

where and . Now, since the ’s are jointlyGaussian random variables, the probability density function of is given in vector nota-tion as

where is a positive-definite symmetrical covariance matrix. Taking the ex-pectation in (3.21) it follows that

This result easily generalizes for the case of different fading figures in each branch. Rear-ranging the branches such that , (3.21) becomes

where , and the difference of the fading param-

(3.19)

(3.20)

(3.21)

(3.22)

(3.23)

(3.24)

1 1 2 1 1 2 12 2 2

0 00 0 0 0 0

1 2 1( ) ( )

2 2

M M m M ms s

s k ki ki

k k i k i

t t xγ α α− − − − −

= = = = =

= = =∑ ∑∑ ∑∑E EN N

kix k

m

skγ

( )T

ss

jE e⎡ ⎤Φ = ⎢ ⎥⎣ ⎦Ω Γ

Γ Ω

0 1( ,.., )TMω ω −=Ω 0 ( 1)( ,.., )Ts s s Mγ γ −=Γ

ix∗ jx∗ i j≠

1

0

2 1 12

0 0

21 212 2

0

( ) exp

1exp

2

1exp

2

s

T

M

k sk

k

m M

k ki

i k

mM mj

k ki

k

E j

E j x

E j x E e

ω γ

ω

ω

−

=

− −

= =

−

=

⎡ ⎛ ⎞⎤⎟⎜⎢ ⎥⎟⎜Φ = ⎟⎜⎢ ⎥⎟⎜ ⎟⎟⎜⎢ ⎥⎝ ⎠⎣ ⎦⎡ ⎛ ⎞⎤⎟⎜⎢ ⎥⎟⎜= ⎟⎜⎢ ⎥⎟⎜ ⎟⎟⎜⎢ ⎥⎝ ⎠⎣ ⎦⎡ ⎛ ⎞⎤ ⎡ ⎤⎟⎜⎢ ⎥⎟⎜ ⎢ ⎥= =⎟⎜⎢ ⎥⎟ ⎢ ⎥⎜ ⎟⎟⎜⎢ ⎥ ⎣ ⎦⎝ ⎠⎣ ⎦

∑

∑∑

∑ Ω

Γ Ω

X D X

( )0 ( 1),..,T

i M ix x −=X ( )0 1,.., Mdiag ω ω −=DΩ kix

X

( ) ( ) ( )1/2/2 112 exp

2M Tf π −− −= −X XX C X C X

[ ]TE=XC XX

( )( )1

21

1/2/2 2

21/2 1/21

( ) 2 ..T

s

m

jM

m

m

e d

j

π−

∞ ∞− −−−

−∞ −∞

− −−

−

⎡ ⎤⎢ ⎥Φ = ⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤= −⎢ ⎥⎣ ⎦

= −

∫ ∫ ΩΓ

Ω

Ω

Ω XX C D XX

X X

X

C X

C jD C

I D C

1k km m +≤

1 2 12

0 0

211 1 12 2

0 0

1

0

1( ) exp

2

1exp

2

k

s

l lTll l

l

l l

M m

k ki

k i

nMM M nj

k ki

l lk l

Mn

l

l

E j x

E j x E e

j

ω

ω

− −

= =

−− −

= ==

−−

=

⎡ ⎛ ⎞⎤⎟⎜⎢ ⎥⎟⎜Φ = ⎟⎜⎢ ⎥⎟⎜ ⎟⎟⎜⎢ ⎥⎝ ⎠⎣ ⎦⎡ ⎛ ⎞⎤ ⎡ ⎤⎟⎜⎢ ⎥⎟⎜ ⎢ ⎥= =⎟⎜⎢ ⎥⎟ ⎢ ⎥⎜ ⎟⎟⎜⎢ ⎥ ⎣ ⎦⎝ ⎠⎣ ⎦

= −

∑∑

∑∏ ∏

∏

Ω

Γ

Ω

Ω

X D X

XI D C

( )( 1),..,T

l li M ix x −=X ( )( 1),..,T

l li M iω ω −=Ω

3.4 Statistical Characterization of the MRC 37

eters is

Since the antennas are usually close to one another, it is often reasonable to assume equalfading figures for all the branches, and therefore (3.23) will serve as the starting point forthe ensuing analysis.

From (3.23), the characteristic function (CHF) of the maximum achievable outputSNR in (3.17) is obtained by replacing by , giving

Furthermore, since is a real-symmetric matrix, it is orthogonally similar to a real-diag-onal matrix , that is

where is a vector of real and positive eigenvalues of (assumed distinctwithout loss of generality), and is a unitary matrix with a complete set of orthogonaleigenvectors forming its columns. Equation (3.26) simplifies to

An alternative interpretation of (3.28) stems from the fact that is independent ofthe coordinate system chosen to express , and therefore (3.22) may be expressed as theproduct of independent Gaussian pdfs

in which is an isometry in that diagonalizes the quadratic form .Consequently, from (3.19), can be regarded as the sum of independent chi-squaredrandom variables with two degrees of freedom, and hence (3.28) results. Furthermore, itis a rational function in , and therefore can be expanded into partial fractions

where the coefficients are computed from the equation

Taking the inverse Fourier transform of (3.30), the probability density function of the

(3.25)

(3.26)

(3.27)

(3.28)

(3.29)

(3.30)

(3.31)

0

1

, 0

, 1 1ll l

m ln

m m l M−

=⎧⎪⎪= ⎨⎪ − ≤ ≤ −⎪⎩

DΩ ωI

( )s

mjγ ω ω −Φ = − XI C

XC

λD

T= λXC QD Q

( )0 1,.., Mλ λ −=λ XC

Q M

( )

( )1

0

( )

1

s

mT

Mm

k

k

j

j

γ ω ω

ωλ

−

−−

=

Φ = −

= −∏

Q I D Qλ

( )sγ ωΦ

XC

M

( ) ( ) ( )( )

1/2/2 1

1 21/2

0

12 exp

2

2 exp2

M T

Mk

kkk

f

y

π

πλλ

−− −

−−

=

= −

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠∏

Y D Y D Yλ λ

T=Y Q X Mℜ 1T −XX C X

sγ mM

jω

( )

1

0 1

( ) 1s

M ml

kl k

k l

A jγ ω ωλ−

−

= =

Φ = −∑∑

klA

( )( )[ ]

1/

11 ( )

!( ) ( ) s

k

m lm

kl km l m lk j

dA j

m l d jγ

ω λ

ωλ ωλ ω

−

− −=

⎡ ⎤⎢ ⎥= − Φ⎢ ⎥− − ⎣ ⎦


output SNR is

In words, the pdf of is the linear combination of independent gamma variableswith pdfs . Of particular interest in (3.32) are two special cases:

1. Rayleigh fading - for the SNR pdf in (3.32) reduces to

2. Independent Fading - the branches are uncorrelated and is a diagonal matrix witheigenvalues

These two cases are noteworthy mainly because they are simple, easier to manipulatemathematically, and often lead to reasonable approximations to the performance of a par-ticular diversity system.

If the average SNR in each branch is admitted to be equal for all branches, then(3.32) simplifies to

where for . It is clear from (3.35) that is afunction of , which is the product of the fading figure of each branch and the numberof branches. This is equivalent to say that the effect of having several uncorrelated branch-es is an increase in the overall fading figure of the single-input multiple-output (SIMO)channel, resulting in a less severe signal fading condition.

Traditional communication systems were optimized for Additive White Gaussian Noise(AWGN) and Intersymbol Interference (ISI) as restraints to error probability and maximumachievable data rates of communication channels. Using the output SNR pdf , it ispossible to extend the error performance and capacity estimates to fading channels anddiversity combining systems. This is done in the mean sense by statistical averaging anySNR dependent function over , that is

3.5 Error Performance of the MRC

This section will aim to illustrate the effects of fading, diversity combining and branch cor-relation in the error performance of a wireless communication system. It will be assumedthat the channel is frequency non-selective, or otherwise can be converted to frequency

(3.32)

(3.33)

(3.34)

(3.35)

(3.36)

sγ

( )

( )

1

0 11 11

/

0 1 0 1

1( ) 1

2

( , , )1 !

s

s k

M ml j

s kl k

k lM m M ml

skl kl sl

kk l k l

f A j e d

A e A f k ll

ωγ

γ λ

γ ωλ ωπ

γγ

λ

− ∞− −

−∞= =− −−

−

= = = =

= −

= =−

∑∑ ∫

∑∑ ∑∑

sγ mM

( , , )sf k lγ

1m =

1/

0

0

1( ) s k

M

s kkk

f A e γ λγλ

−−

=

= ∑

XC

[ ] [ ]2 1 1k ki sk skE x E

m mλ γ= = = Γ

skΓ

( ) ( )1

/( )1 !

s s

mMmMs mM

smM

f emM

γγγ

−− Γ=

− Γ

[ ] [ ]s s skE MEγ γΓ = = 0,1,.., 1k M= − ( )sf γ

mM

( )sf γ

( )E γ ( )f γ

0( ) ( )avgE E f dγ γ γ

∞

= ∫

3.5 Error Performance of the MRC 39

non-selective using a special type of receiver (e.g. a receiver capable of combining the mul-tiple flat-fading processes of a frequency selective channel). Maximal-ratio combining andbinary PSK modulation will be used throughout the analysis.

When all the receiver branches have the same noise power, the combiner weights maybe written as , and from (3.13) the combiner output reduces to

Also, for BPSK modulation a single matched filter is sufficient, and since is virtuallyconstant during a symbol period (slow fading), the decision variable is given by

where is still the symbol energy of the bandpass signal and is given by

For fixed ’s, the variable is Gaussian distributed with mean

and, using the equality , with variance

When a “positive” is sent, an error will occur when , for which the receiver in-correctly decides in favour of , and the error probability is

where is the SNR at the combiner’s output, averaged over a symbol period, as given by(3.18). Hence, the average error probability may be written as

3.5.1 INDEPENDENT FADING

Inserting (3.35) into (3.43) and expanding the function, the average error probability

(3.37)

(3.38)

(3.39)

(3.40)

(3.41)

(3.42)

(3.43)

( )( ) ( ) kj tk kw t t e θα −=

1 12 ( )

0

0 0

( ) ( ) ( ) ( ) ( ) k

M Mj t

b b k k k

k k

r t s t t t n t e θτ α α− −

−

= =

= − +∑ ∑

( )k tα

0 0

0 0

0 0

0 0

0 0

1 12 2 ( )

0 0

0 01

2

0 0

1( ) ( ) e ( ) ( )

2

( )1( ) ( ) e ( ) ( )

2 2

( ) ( )w

k

T T

b b

M MT Tk j t

b k k b

k kM M

s k k kr

k k

d r t s t dt r t s t dt

ts t dt t e n t s t dt

t t

τ τ

τ τ

τ τθ

τ τ

τ τ

ατ α τ

α α

+ +∗

− −+ +− ∗

= =−

= =

⎧ ⎫⎪ ⎪⎪ ⎪= − = ℜ −⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

⎛ ⎞ ⎧ ⎫⎪ ⎪⎟ ⎪ ⎪⎜ ⎟= − + ℜ −⎜ ⎨ ⎬⎟⎜ ⎟⎜ ⎪ ⎪⎝ ⎠ ⎪ ⎪⎩ ⎭

= +

∫ ∫

∑ ∑∫ ∫

∑E1−

∑

sE ( )s t wkr

0

0

( )0

1w e ( ) ( )

2k

Tj t

kr k be n t s t dtτ

θ

ττ

+− ∗⎧ ⎫⎪ ⎪⎪ ⎪= ℜ −⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭∫

kα d

12

0

( )M

d s k

k

tμ α−

=

= ∑E

[ ] 0( ) ( ) 2 ( )k kE n t n u t uδ∗ = −N

[ ]1 1

02 2 2 2

0 0

w ( ) ( )2

M Ms

d kr k k

k k

E t tσ α α− −

= =

= =∑ ∑N E

( )bs t 0d <

( )bs t−

( )

( )

2

de s

d

s

P Q

Q

μγ

σγ

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠

=

sγ

( )0

2 ( )e s s sP Q f dγ γ γ∞

= ∫

()Q ⋅


is written in integral form as

where is the MRC effective diversity order and is the average SNR at theoutput of the combiner. Using the equality

and integrating (3.44) by parts yields

A plot of Equation (3.46) as a function of the average output SNR , for several effective

diversity orders , is shown in Figure 3.2. It is noticeable the reduction in error probabilityfor the same output SNR when diversity is utilized. Figure 3.2 also emphasizes how theuse of diversity makes the error probability curves bend towards the nonfaded signal curvewhen the number of receiver branches increases. For instance, achieving a 10-4 error prob-ability target with no diversity and a Rayleigh fading scenario would require approximately34 dB of SNR. Using eight antennas at the receiver (or four antennas in the case of in each branch) the SNR requirement drops to more or less 11 dB, which is a remarkableSNR gain of 23 dB.

(3.44)

(3.45)

(3.46)

FIGURE 3.2 The effect of MRC diversity combining on the error performance of coherent BPSK.

( )( )

2 /2 1 /

0 2

/2 1 !

s s

s

s xe s sP e dx e dγ

γγ γ

π

Λ ∞ ∞− Λ− −Λ Γ⎡ ⎤Λ Γ ⎡ ⎤⎢ ⎥= ⎣ ⎦⎢ ⎥Λ − ⎣ ⎦∫ ∫

mMΛ = [ ]s sE γΓ =

10

!( )!

n n kn ax ax

kk

n xx e dx e

n k a

−− −

+=

= −−∑∫

( ) ( )

( )( )

2 /2

2 01 1

1 / 3/2

001 11/2

00

12

/12 ( 1)!

1 /11/2

2 ( 1)!

s

s s

xe

ks k

s s

k

kkss

lk

P e dx

e dk

lk

γ

γ

π

γ γπ

∞∞−

Λ− + −Λ ∞− +Λ Γ Λ− −

=Λ− Λ− −+ −Λ

==

⎡ ⎤⎢ ⎥= −⎢ ⎥⎣ ⎦

Γ Λ−

Λ − −

+ Γ ΛΓ= + −

Λ Λ − −

∫

∑ ∫

∑ ∏

sΓ

−10 0 10 20 30 4010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Average SNR − s (dB)

Err

or P

roba

bilit

y −

Pe

= 2

= 1 (Rayleigh)

= 4

= 8

Nonfaded

Λ

2m =


3.5.2 CORRELATED FADING

Figure 3.2 reveals a clear error performance advantage of MRC diversity combining forgeneric fading channels. Nevertheless, in practical situations there is always some level ofcorrelation between the signals of different receiver branches that will diminish the diver-sity gain of any system. Obtaining the correlation between the ’s as a function of thecorrelation between the ’s defined in (3.19) is a straightforward process. Putting

in (3.23), the joint moment-generating function (MGF) of the’s is given by

Letting , differentiation of (3.47) gives

and also

where is the matrix with the -th row differentiated with respect to . To normalize(3.49) one may use the correlation coefficient

where . As expected, (3.50) is independent of the fading pa-rameter . Now, invoking the result of (3.6), rewritten as

the correlation coefficient between the SNR of two different branches can finally beexpressed as

where is the phase constant and is the separation distance between the antennas ofbranches and . This equation generalizes (3.9), which was only applicable in Rayleighfading conditions. It has, however, two significant drawbacks: 1) it is not a good approxi-mation for irregular distributions of wave arrival (e.g. within sub-urban areas); 2) for lowangular spreads, it depends upon the direction of wave arrival (e.g. near base stations). Thestatistical consequence of such scenarios will be the emergence of some cross-correlation

(3.47)

(3.48)

(3.49)

(3.50)

(3.51)

(3.52)

skγ

kix

1( ,.., )Mj diag s s= =Ω SD D

skγ

( )s

m−= −Γ S XM S I D C

− S XA = I D C

[ ]

[ ]

1

2

ssk

k

mk

ki

Es

m

mE x

γ

− −

∂⎡ ⎤⎢ ⎥=∂⎢ ⎥⎣ ⎦

⎡ ⎤= −⎣ ⎦=

Γ

S=0

S=0

M

A A

[ ]

[ ] [ ]

2

2 1

2 2 2 2

( 1)

ssk sl

k l

km mk l

l

ki ki li

Es s

m m ms

m E x mE x x

γ γ

− − − −

⎡ ⎤∂⎢ ⎥= ⎢ ⎥∂ ∂⎣ ⎦⎡ ∂ ⎤

= + −⎢ ⎥∂⎢ ⎥⎣ ⎦

= +

Γ

S=0

S=0

M

AA A A A

kA A k ks

[ ] [ ]

[ ] [ ]

[ ]

[ ]

2

2 2

22

2 2

sk sl

ki li

sk sl sk

sk sk

ki lix x

ki

E EE E

E x xE x

γ γγ γ γ

ργ γ

ρ

−=

−

= =

[ ] [ ]2 / /ki s sE x E mMγ= = Γ Λ

m

[ ] [ ]20 0( ) ( )s

ki li ki kl klE x x E x J d J dβ βΓ

= =Λ

sk slγ γρ

20 ( )

sk sl klJ dγ γρ β=

β kld

k l


between the in-phase and quadrature components of different ’s, that is

which, for the Rayleigh case, is equivalent to say that (3.3) is not valid. Also, the in-phaseand quadrature components are given generically by

for a receiver (branch) that moves with a velocity relative to the transmitter, and with anangle towards the incoming wave of the -th scatterer. Carrying out the same stepsin (3.19)-(3.23) and (3.47)-(3.50), but this time abiding by (3.53), it easily follows that theSNR correlation coefficient is given by

for any .For the sake of simplicity, the following analysis will be performed assuming a clut-

tered urban environment, so that uniform distribution of wave arrival is a good approxi-mation, and hence (3.52) can be used. Suppose that an evenly spaced linear antenna arrayis used to receive signals that suffer from Nakagami fading. The antennas are connectedto a MRC combiner and the SNR of the detector’s output is modelled by the probabilitydensity function in (3.32). The covariance matrix of the ’s may be written as

where is the normalized covariance matrix. Denoting an eigenvalue of by , andgiven that (3.32) is simply a linear combination of (3.35), the average error probability atthe detector’s output is similarly a linear combination of (3.46) with the substitutions

and , that is

The ’s can be found from (3.31) using the ’s and the help of a recursive algorithm thatautomates the calculation for any diversity order and fading figure . Figures 3.3 and3.4 show plots of (3.57) for several situations: two antenna spacings, and , andfour / conditions. These figures illustrate the following:• the presence of correlation between the signals of different antennas reduces the

SNR gain obtained by employing diversity at the receiver;

(3.53)

(3.54)

(3.55)

(3.56)

(3.57)

kγ

(2 ) (2 )

(2 ) (2 1)

0

0

k i l i

k i l i

E x x

E x x +

⎡ ⎤ ≠⎢ ⎥⎣ ⎦⎡ ⎤ ≠⎢ ⎥⎣ ⎦

( ) ( )( )

12 cos

2 2 1

0

kilki lki

Lj t

k i k i lki

l

x jx a eν

ϕ π φλ

−+

+=

+ = ∑

ν

lkiφ lki

( ) ( )[ ] ( ) ( )[ ]( )[ ]

2 22 2 2 2 1

2 22

sk sl

k i l i k i l i

k i

E x x E x x

E xγ γρ ++=

0,.., 1i m= −

kix

( ) ( )( )

( ) ( )( )

( )( ) ( )( )

0 01 0 0 1

0 10 0 1 1

0 1 0 0 1 1

1

1

1

M

Ms

M M

s

J d J d

J d J d

J d J d

β β

β β

β β

−

−

− −

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟Γ ⎜ ⎟= ⎟⎜ ⎟⎜Λ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠Γ

=Λ

…

…

…

X

X

C

C

XC XC kλ

/ /ks k sλ λΓ Λ → = Γ Λ lΛ →

( )( )

1/21 1 1

00 1 0

1 /11/2

2 ( 1)!

i lM m l l ik ss

kl ke

jk l i

P A jl i

λλ

+ −− − − −

== = =

⎡ ⎡ ⎤ ⎤+ Γ ΛΓ⎢ ⎢ ⎥ ⎥= + −⎢ ⎢ ⎥ ⎥Λ − −⎢ ⎢ ⎥ ⎥⎣ ⎣ ⎦ ⎦

∑∑ ∑ ∏

klA kλ

M m

/4λ /8λ

m M


• the reduction in SNR gain is more significant for higher diversity orders and forsmaller antenna separation;

• as the separation between the antennas increases the error performance approachesthat of independent fading;

Thus, improving the error performance in a diversity system with correlated fading isequivalent to increasing the diversity order and/or increasing the antenna separation.One may also deduce that sometimes it is preferable to increase the order of the array thanto increase the aperture (i.e. the sum the antenna separations in a linear array) of a lowerorder array. Either way, the cost of diversity is always higher system complexity and size,which is often an expensive price to pay in mobile units that require low power consump-

FIGURE 3.3 Error performance of MRC combiner for a linear array of antennas, correlated fading with fading figure , and uniform distribution of wave arrival.

FIGURE 3.4 Error performance of MRC combiner for a linear array of antennas, correlated fading with

fading figure , and uniform distribution of wave arrival.

−10 0 10 20 30 4010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100


Err

or P

roba

bilit

y -

Pe

Nonfaded (Gaussian)Independent FadingCorrelated Fading − d

min = /4

Correlated Fading − dmin

= /8

m = 1M = 2

m = 1M = 8

M1m =

−10 0 10 20 30 4010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100


Nonfaded (Gaussian)Independent FadingCorrelated Fading − d

min = /4

Correlated Fading − dmin

= /8

m = 2M = 2

m = 2M = 8

Err

or P

roba

bilit

y -

Pe

M2m =

M


tion and reduced sizes.

3.6 Information-theoretic Capacity of the MRC

In Section 3.5 it was possible to analyse from a general viewpoint the error performanceof a slowly-fading multipath propagation channel with spatial diversity reception and max-imal-ratio combining. It is well known that the error performance of a communicationsystem is directly related to the data rate of transmission, in the sense that increasing thelatter for a given SNR has the consequence of degrading the former. Here, the com-mon denominator is . From Shannon’s information capacity theorem, a communicationchannel perturbed by AWGN noise has a capacity given by

where a “cycle” (or ) is the time of two channel transmissions (i.e. two sam-ples) if the sampling rate of the input signal equals the Nyquist rate. It means that it is the-oretically possible to employ some channel coding technique so that the maximumachievable signalling rate with zero error probability is , and also that it is theoreticallyimpossible to have error-free communication when the signalling rate is above . It istherefore important to know to what extent is the channel capacity reduced in the case ofa fading channel, and how much can diversity reception compete against this reduction.

In a generic fading channel must be interpreted as a random variable, which meansthat (3.58) has to be obtained in the average sense, that is

When the fading severity is characterized by an integral fading parameter , is givenby (3.32). Thus, integrating (3.59) by parts,

performing the change of variables ,

and using the binomial theorem,

the average channel capacity is rewritten as

(3.58)

(3.59)

(3.60)

(3.61)

(3.62)

(3.63)

sγ

sγ

( )2log 1 bits/cyclesC γ= +

seconds Hz×

C

C

sγ

( )20

( ) log 1 bits/cycleavg s s sC f dγ γ γ∞

= +∫m ( )sf γ

( )( )

1 1/

200 1

1 1/

200 1 0

log 11 !

1 1log

1!

s k

s k

M m ls

avg kl s slkk l

M m li

kl s siskk l i

C A e dl

e A e di

γ λ

γ λ

γγ γ

λ

γ γγλ

− ∞ −−

= =− − ∞

−

= = =

= +−

=+

∑∑ ∫

∑∑ ∑ ∫1sγ η= −

( )1 1

1/ /2

10 1 0

1 1log 1

!k k

M m li

avg kl ikk l i

C e A e e di

λ η λη ηηλ

− − ∞−

= = =

= −∑∑ ∑ ∫

( ) ( )

0

1 1i

i i jj

j

i

jη η −

=

⎛ ⎞⎟⎜ ⎟− = −⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠∑

( )1 1

1/2

0 1 0 0

1 1log ,1/

!k

M m l i i j

avg kl kkk l i j

iC e A e j

jiλ λ

λ

− − −

= = = =

⎛ ⎞⎛ ⎞⎟⎜ ⎟⎟⎜= − Γ⎜ ⎟⎟⎜ ⎟⎜ ⎜⎟⎝ ⎠⎟⎜⎝ ⎠∑∑ ∑ ∑

3.6 Information-theoretic Capacity of the MRC 45

where

is the incomplete gamma function. If the branches fade independently, is given by(3.35) and the expression for the average channel capacity is

Equation (3.65) is plotted in Figure 3.5 as a function of the average SNR , for effectivediversity orders . The Gaussian channel capacity is also represented for com-

parison (solid curve), and it is equivalent to the limiting case . Therefore, the ca-pacity of a multipath fading channel with diversity at the receiver converges to Shannon’sresult for an AWGN channel. However, Figure 3.5 surprisingly shows that in the range of

between 0 and 20 dB, the difference in capacity is always less than 1 bit/s/Hz, meaningthat in the case of independent fading the loss of information-theoretic capacity is negli-gible. This conclusion applies without modification to the case of correlated fading. Com-puting the covariance matrix in (3.56) for two antenna separations, and , in alinear array of four antennas, finding the eigenvalues and the coefficients , and plot-ting (3.63), Figure 3.6 results. For comparison, the capacity curves for the Rayleigh andGaussian channels are also depicted. As long as the channel is a fading channel, the worstscenario is always Rayleigh fading (no diversity) and the best scenario is always independ-ent fading, since the capacity difference from Shannon’s result is maximum and minimum,respectively. When the fading is correlated, the higher is the antenna separation, the ishigher the achievable capacity for a given diversity order , and for a given antenna sep-aration, the higher is the diversity order , the higher is the achievable capacity. The par-

(3.64)

(3.65)

FIGURE 3.5 Information-theoretic capacity of MRC combiner for several effective diversity orders, in the case of independent fading, and uniform distribution of wave arrival.

( ) 1

1/,1/

k

jkj e dη

λλ η η

∞− −Γ = ∫

( )sf γ

( )1/

0 0

1, /

ln2 !

si i j

avg ssi j

ieC j

ji

Λ− −Λ Γ

= =

⎛ ⎞⎛ Λ ⎞⎟⎜ ⎟⎟⎜= − Γ Λ Γ⎜ ⎟⎟⎜ ⎟⎜ ⎜⎟⎝ ⎠⎟ Γ⎜⎝ ⎠∑ ∑

sΓ

1,2, 4, 8Λ =

0 5 10 15 200

1

2

3

4

5

6

7


Cha

nnel

Cap

acity

− C

(bi

ts/s

/Hz)

Nonfaded (Gaussian) = 8

= 4

= 2

= 1 (Rayleigh)

Λ = ∞

dΓ

/4λ /8λ

kλ klA

M

M


ticular results shown in Figure 3.6 are fully generalizable for any diversity order , branchfading figure , and antenna separation. Nevertheless, this generalization assumes inte-gral fading figures . One should remember that worst-than-Rayleigh scenarios are pos-sible for , but since a mathematical treatment of nonintegral fading figures isrequired, these scenarios will be omitted here

FIGURE 3.6 Information-theoretic capacity of MRC combiner for different antenna separations, in the case of correlated fading, and uniform distribution of wave arrival. .

0 5 10 15 200

1

2

3

4

5

6

7


Cha

nnel

Cap

acity

− C

(bi

ts/s

/Hz)

Nonfaded (Gaussian)Indep. Fading; M = 4d

min = /4; M = 4

dmin

= /8; M = 4

Rayleigh; M = 1

1m =

M

m

m

1/2 1m≤ <

CHAPTER 4

INFORMATION-THEORETIC ASPECTS OF MIMO CHANNELS

4.1 Introduction

The ever-growing need for high data-rates of communication triggered an “explosion” ofresearch activity with the aim of finding revolutionary techniques to combat fading, andto increase the spectral efficiency of wireless communication systems. Based on the fun-damental premise of geostatistics and biodiversity that nearby things are on average morealike than distant things, the invention of spatial diversity techniques naturally occurred.

Clearly, the way in which the received phasors add up at different locations should benearly independent provided that the respective scatterers are also independent. To vali-date this argument, one could build an array of spaced antennas at the receiver and, hence,obtain a set of nearly independent linear equations with a single unknown: the transmittedsignal. At this point we should be able to compute a better estimate of the transmitted sig-nal. To enhance the system, we could add several unknowns to it or, equivalently, set upan array of spaced antennas at the transmitter side and send out independent signals fromeach antenna. If we could quantify the channel responses between each pair of transmit-ting/receiving antennas, we could also estimate a solution to the system of equations. Still,because of channel noise, the solution had to be necessarily an approximation. This is thebasic concept behind single-user multiple-input multiple-output (MIMO) wireless sys-tems, that is, to take advantage of multipath and diversity to improve communications.

There is an ongoing study within the re-search community on the actual possibilitiesof such concept. A review of the principles and recent advances of MIMO wireless sys-tems can be found in [16], [17]. The primary aim of this chapter is to unveil a simple, com-pact, and generic analytic solution to the problem of finding the limit of the spectralefficiency (i.e. the capacity) that can be offered by single-user MIMO systems. The initialassumptions are: 1) the receiver has perfect channel state information (CSI); 2) the signalsat different receiver antennas are correlated; 3) the signals from different transmitter an-

47

48 CHAPTER 4 INFORMATION-THEORETIC ASPECTS OF MIMO CHANNELS

tennas are independent; and 4) there is no CSI at the transmitter.

4.2 MIMO Channel Modelling

A multiple-input multiple-output wireless channel, as its name indicates, is a channel com-posed of several antennas at the transmitter (input) and receiver (output) sides. To under-stand the advantages of such a channel, let the channel response between the j-thtransmitter antenna and the i-th receiver antenna be modelled by a baseband-equiva-lent flat-fading process like the one in (3.11), rewritten as

This expression can be simplified by assuming that:

1. each wave that impinges on one antenna, impinges on all the others by a parallel path;essentially this means that the Doppler shift is identical for all the receiver antennas, i.e.

where is the phase constant, is the magnitude of the receiver’s velocity relative to thetransmitter, and is the angle between the k-th impinging wave (from the j-th transmitterantenna) and the receiver’s velocity vector relative to the transmitter; we also have that thenumber of multipaths is simply ;

2. each wave that impinges on one antenna, impinges on all the others with the same am-plitude, i.e. ;

Introducing these simplifications, it follows that

The output signal from the i-th receiver antenna is now obtained as the sum of the con-tributions of the input signals of the transmitter antennas, weighted by the individualchannel responses, that is

where is the delayed, baseband version of the input signal, is thecomplex information symbol, is a delayed unit energy shaping function, and

is a complex valued baseband AWGN noise process. If the channelis slowly fading, it may be considered time-invariant during a symbol period, and conse-quently the time dependence in (4.4) can be disregarded. To recover the information sym-bol, the receiver must correlate the output signals with the pulse shaping function, yielding

(4.1)

(4.2)

(4.3)

(4.4)

(4.5)

( )1

0

( )ij

ijk ijk

Lj t

ij ijk

k

h t a e ϕ ω−

+

=

= ∑

cosijk jk jkω ω βν φ=∼

β ν

jkφ

ij jL L∼

ijk jka a∼

( )1

( )

0

( ) ( ) ( ) ( )j

ijk jk ij

Lj t j t

ij jk ij ij ij

k

h t a e t e u t jv tϕ ω θα−

+

=

= = = +∑

Tn

1

0

0

( ) ( ) ( ) ( )Tn

bi ij bj bi

j

r t h t s t z tτ−

=

= − +∑

0 0( ) ( )bj js t x g tτ τ− = − jx

0( )g t τ−

( ) ( ) ( )QIbi bi biz t z t jz t= +

0

0

1

0

0

( ) ( )TnT

i bi ij j i

j

y r t g t dt h x nτ

ττ

−+

=

= − = +∑∫

4.3 Maximum Entropy 49

where is a complex zero-mean gaussian random variable given by

The global channel response is now naturally written in vector/matrix form

where is the transmit vector, is the receive vector, is the noisevector, and is the channel matrix. In expanded form, the MIMO channel re-sponse is described by the equation

4.3 Maximum Entropy

A measure of the uncertainty about a continuous random vector is its differential entropy, given by

where is the probability density function of , and denotes the expectation over. Let be the complex random vector , and its covariance matrix be ex-

pressed as

The following relations apply

If is the joint Gaussian pdf, then it equals

where is the length of and is the covariance matrix

The corresponding characteristic function is

Now, notice that if is a circularly symmetric vector such that

(4.6)

(4.7)

(4.8)

in

0

0

0( ) ( )T

i bin z t g t dtτ

ττ

+

= −∫

= +y Hx n

x 1Tn × y 1Rn × n 1Rn ×

H R Tn n×

1 1 111 12 1

2 2 221 22 2

1 2

T

T

R T RR R R T

n

n

n n nn n n n

y x nh h hy x nh h h

y x nh h h

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞⎟ ⎟ ⎟⎜ ⎜ ⎜⎟⎜⎟ ⎟ ⎟⎟⎜ ⎜ ⎜⎜⎟ ⎟ ⎟⎟⎜ ⎜ ⎜⎜⎟ ⎟ ⎟⎟⎜ ⎜ ⎜⎜⎟ ⎟ ⎟⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎟= +⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎟⎟ ⎟ ⎟⎜ ⎜ ⎜⎟⎜⎟ ⎟ ⎟⎜ ⎜ ⎜⎟⎜⎟ ⎟ ⎟⎜ ⎜ ⎜⎟⎜⎟ ⎟ ⎟⎟⎜ ⎜ ⎜⎜⎟ ⎟ ⎟⎟⎜⎟ ⎟ ⎟⎜ ⎜ ⎜⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠

……

…

(4.9)

(4.10)

(4.11)

(4.12)

(4.13)

(4.14)

(4.15)

x

dH

[ ]2( ) log ( ) (bits)d ff E f= − xH

( )f x x ()fE ⋅

f x I Qj= +x x x

[ ]

( )

H

I I Q Q I Q Q I

fE

j

=

= + − −

xx

x x x x x x x x

C xx

C C C C

T

I I I I=x x x xC C T

I Q Q I=x x x xC C

( )f x

( )[ ] ( )1/2 11( , ) det 2 exp

2T T

I

I Q I QQ

f π − −⎛ ⎛ ⎞⎞⎟⎟⎜ ⎜= − ⎟⎟⎜ ⎜ ⎟⎟⎟⎟⎜ ⎜⎝ ⎝ ⎠⎠

xx x D x x D x

n x D

I I I Q

Q I Q Q

⎛ ⎞⎟⎜ ⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠

x x x x

x x x x

C CD

C C

( )1( ) exp

2T T

I

I QQ

⎛ ⎛ ⎞⎞⎟⎟⎜ ⎜Φ = − ⎟⎟⎜ ⎜ ⎟⎟⎟⎟⎜ ⎜⎝ ⎝ ⎠⎠

ww w w D w

x

I I Q Q=x x x xC C

I Q Q I= −x x x xC C


then the following equality holds

Therefore, taking the inverse Fourier transform of (4.16), it is found that, under the statedassumptions, the joint Gaussian pdf has the form

which is a well known result from probability theory [39]. This form will be important inthe following explanation. To find the pdf that maximizes the entropy , let be an-other pdf such that , and compute the difference

But if follows (4.17), then the expectations over and are equal

where denotes the trace of a matrix. As a result, the entropy difference is

To transform (4.20) into an inequality one uses a property of the logarithm:

Therefore

with equality only if . This equation proves that the pdf of a circularly symmetriccomplex gaussian random vector , given in (4.17), achieves the maximum possible entro-py. Finding this maximum is as simple as taking the expectation in (4.9), yielding

Entropy maximization is an important result that can be used in the determination of theinformation-theoretic capacity of a MIMO channel.

(4.16)

(4.17)

(4.18)

(4.19)

(4.20)

(4.21)

(4.22)

(4.23)

( )

( )

1( ) exp

2

1exp

4

T T

H

I

I QQ

⎛ ⎛ ⎞⎞⎟⎟⎜ ⎜Φ = − ⎟⎟⎜ ⎜ ⎟⎟⎟⎟⎜ ⎜⎝ ⎝ ⎠⎠

= − xx

ww w w D w

w C w

( )1 1( ) det( ) exp Hf π − −= −xx xxx C x C x

dH ( )g x

[ ] [ ]H Hg fE E=xx xx

[ ] [ ]2 2( ) ( ) log ( ) log ( )d d f gg f E f E g− = − +x xH H

( )f x f g

[ ] [ ][ ][ ][ ]

[ ]

12 2

12

2

log ( ) log ln det( )

log ln det( ) ( )

log ( )

H

H

f f

f

g

E f e E

e tr E

E f

π

π

−

−

= − −

= − −

=

xx xx

xx xx

x C x C x

C xx C

x

()tr ⋅

[ ] [ ]2 2

2

( ) ( ) log ( ) log ( )

( )log

( )

d d g g

g

g f E g E f

fE

g

− = − +

⎡ ⎤= ⎢ ⎥

⎢ ⎥⎣ ⎦

x x

xx

H H

ln 1x x≤ −

2( )

( ) ( ) log 1 0( )d d g

fg f e E

g⎡ ⎤

− ≤ − =⎢ ⎥⎢ ⎥⎣ ⎦

xx

H H

f g=

x

[ ]

[ ]

2

12 2

2

( ) log ( )

log det( ) (log ) ( )

log det( )

H

d f

f

f E f

e tr E

e

π

π

−

= −

= +

=

xx xx

xx

x

C xx C

C

H

4.4 Information-theoretic Capacity 51

4.4 Information-theoretic Capacity

The maximum theoretically achievable signalling rate with error-free detection is called thechannel capacity, and it has been recently determined for the case of a MIMO channel (see[10], [11]). Here, and for the sake of completeness, the process of finding the capacity isoutlined. If denotes the uncertainty about the input random vector of the MIMOchannel, and denotes the uncertainty about after observing the output vector ,then the reduction in uncertainty is given by

and it is called the mutual information between the input and the output. For a power limitedinput, the information-theoretic capacity of a channel is defined as the maximum reduc-tion in uncertainty, or maximum mutual information, over all statistical distributions of theinput

Denoting the joint pdf of the input and output vectors by , substitution of (4.9) into(4.23) leads to the symmetry property of

Now, since the input and the output of the MIMO channel are related by (4.7), thedistribution of conditioned on is

provided that is independent of . Assuming that the noise vector is a normal randomvariable with covariance matrix , where is the noise power in thesystem’s bandwidth, the conditional entropy in (4.26) evaluates to

Therefore, the mutual information in (4.26) will be maximum when the differential entro-py of the output is maximized. Hence, from Section 4.3, must be a circularly sym-metric complex Gaussian vector, and its entropy is given by (4.23), that is

Furthermore, the output covariance matrix is

(4.24)

(4.25)

(4.26)

(4.27)

(4.28)

(4.29)

(4.30)

( )d fxH x| ( )d fx yH x y

|( ; ) ( ) ( )d df f= − x yxx yI H H

( )max ( ; )f

C =x

x yI

( , )h x y

( ; )x yI

[ ] [ ]

|

|2 2

| |2 2

|

( ; ) ( ) ( )

log ( ) log ( )

( ) ( )log log

( ) ( )

( ; ) ( ) ( )

d d

h h

h h

d d

h h

E h E h

h hE E

h h

h h

= −

= − +

⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

= = −

x yx

y xy

x y

x x y

x y y xx y

y x

I H H

I H H

x y

y x

| |( ) ( ; ) ( )h f f= − =n ny x y Hx H x n

n x n2[ ] 2H

Rn nE σ= =nnC nn I 2nσ

|

22 2

( ) ( )

log det( ) log det(2 )R

dd

n n

h f

e eπ π σ

=

= =

y x nn

nnC I

H H

( )d hyH y

2( ) log det( )d h eπ=yyyCH

[ ]

( )( )[ ]

H

H

H

E

E

=

= + +

= +

yy

xx nn

C yy

Hx n Hx n

HC H C


Finally, substituting (4.28) and (4.29) into (4.26), the information-theoretic capacity of theMIMO channel is expressed as

At first glance, the channel capacity equation does not say much about how large the ca-pacity can really be. However, a closer inspection can unveil some facts. First, if the trans-mitter is unable to estimate the channel matrix , the best it can do is to evenly distributethe available power among all the antennas, i.e. .Assuming Nyquist sampling and pulse shaping the capacity is then

Moreover if is normalized so that

then the available power incorporates the average path loss and hence

where and are the average power and average SNR at the input of each receiverbranch, respectively. In the limit of very large it follows that

where .Some notes about the capacity:

• it is directly proportional to the SNR of the input signal in each transmitter antenna,and it depends on the stochastic properties of the channel transfer function;

• interesting enough, it looks very similar to Shannon’s capacity formula for the one-dimensional case, which is

where is the normalized channel power transfer characteristic ( in Gaus-sian channels);

• it is the true channel capacity if and only if the receiver has perfect knowledge of thechannel matrix ;

• if the coherence time of the channel (given in (2.73)) is greater than the duration ofeach symbol transmitted through the channel, it is sufficient to have a single exactestimate of for the entire coherence time;

• if the coherence time of the channel is smaller than the duration of each symboltransmitted through the channel, the capacity expression (4.34) is valid provided that

(4.31)

(4.32)

(4.33)

(4.34)

(4.35)

(4.36)

( )[ ]

2 2

2 2

2 2

log det( ) log det( )

log det log ( )

1log det bits/vector sample

2

H

H

Rnn

C e e

e e

π π

π π

σ

= −

= + −

⎛ ⎞⎟⎜= + ⎟⎜ ⎟⎜⎝ ⎠

yy nn

xx nn nn

xx

C C

HC H C C

I HC H

H2 /2T T jP n E x⎡ ⎤= ⎢ ⎥⎣ ⎦ ( )2 /

TT T nP n=xxC I

2 2log det (bits/s/Hz)H

R

Tn

T n

PC

n σ⎛ ⎞⎟⎜= + ⎟⎜ ⎟⎜⎝ ⎠I HH

H

2 1ijE h⎡ ⎤ =⎢ ⎥⎣ ⎦

2 22log det log detH H

R Rn nTT n

PC

nnγ

σ⎛ ⎞ ⎛ ⎞⎟⎜ ⎟⎜= + = +⎟ ⎟⎜ ⎜ ⎟⎜⎟⎜ ⎝ ⎠⎝ ⎠I HH I HH

P γ

Tn

( )2log detT Rn nC γ↑ ≈ +I Σ

[ ]Hj jE= h hΣ

22 2log 1

n

PC H

σ⎛ ⎞⎟⎜= + ⎟⎜ ⎟⎜⎝ ⎠

2H 2 1H =

H

H

4.4 Information-theoretic Capacity 53

the stochastic process that generates is locally ergodic, i.e. provided that the time-averaged statistics (moments) that can be obtained for the duration of the symbol areunbiased estimates of the statistics of the stochastic process; in this case the effectivecapacity is the time-average of (4.34), or symbolically ;

• the capacity in (4.34) is formally called the fast-fading capacity of the MIMO channel,and its expectation the ergodic capacity;

Another close look to (4.34), and one distinguishes the factor

Recalling that the coefficients of the channel matrix are stochastic and given by (4.1), itis possible to determine the coefficients of the matrix

Also, invoking the strong law of large numbers, when the number of transmitter antennas islarge and the are independent random variables, the coefficients are

where . Therefore, in the limit of large transmitter antenna arrays, the capacityis upper-bounded by

which is times larger than that of the Gaussian SISO channel. When is considerablylarge, say , the capacity in (4.40) may be approximated as

showing that the capacity grows linearly with the branch SNR in decibels, and that for each increase in the SNR the capacity scales approximately as more bits/cycle. Even

though it is obtained under ideal conditions, this is incredibly higher than the Shannonlimit for a Gaussian SISO channel.

The result in (4.40) comes as no surprise if one examines the channel response in(4.8). Supposing that the channel matrix entries are independent, the rank of the channelmatrix for will be , meaning that if the receiver knows , it will be able to solvethe system of equations and find out of input variables, i.e. flows of information.The only setback is the presence of the noise term, which could produce an error whensolving the system. More formally, writing the MIMO channel equation as

(4.37)

(4.38)

(4.39)

(4.40)

(4.41)

(4.42)

H

TC

1 H

Tn=T HH

H

T

( )

1

1T

ik jk

nj

ij ik jkT k

t en

θ θα α −

=

= ∑

ijh

[ ]

( )

12

( )

1lim

0

T

ik jk

T

ik jk

nj

ij ik jkn T k

ikjik jk

t en

E i jE e

i j

θ θ

θ θ

α α

αα α

−

↑=

−

⎛ ⎞⎟⎜ ⎟⎜= ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎧ =⎪⎪⎡ ⎤≈ = ⎨⎢ ⎥⎣ ⎦ ⎪ ≠⎪⎩

∑

[ ]2 1ikE α =

( )lim 2log 1R TC n nγ= + → ∞

Rn γ

5 dBγ

( )2 dBlim log 1 0.33 5 dB ; R R TC n n nγ γ γ γ↑ = + ≈ ⋅ → ∞

3 dB Rn

T Rn n≥ Rn H

Rn Tn Rn

( )1

1 T

T

x

x

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜= +⎟⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜⎝ ⎠

y h h n…


and applying the standard Gauss-Jordan method, one arrives at the equivalent system

where is a linear function of . This is a consistent system of inde-pendent equations

where the middle term can be forced to zero if desired, yielding

As a result, the receiver is capable of separating noise-limited, identical and independ-ent flows of information, and hence (4.40) follows intuitively.

Further analysis of the MIMO channel capacity is possible by diagonalizing the matrixproduct , which is self-adjoint

Consequently, it has a complete set of orthonormal eigenvectors that can perform the ei-genvalue decomposition

where , with real and non-negative eigenvalues. The capacity expres-sion in (4.34) simplifies to

When the factorization is not full-rank, which may bring some difficulties,and therefore the capacity expression in (4.34) should be reexpressed as

The eigenvalue decomposition is in this case given by

leaving the capacity expression in (4.48) in the form

(4.43)

(4.44)

(4.45)

(4.46)

(4.47)

(4.48)

(4.49)

(4.50)

(4.51)

11 1 1 1 1

1

01( ) ( ) ( ) ( )

0 1( ) ( ) ( ) ( )

R T

R R R R T R

n n

Tn n n n n n

x

x

ϕ ϕ ϕ ϕ

ϕ ϕ ϕ ϕ

+

+

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞⎟ ⎟ ⎟⎜ ⎜ ⎜⎟⎜⎟ ⎟ ⎟⎟⎜ ⎜ ⎜⎜⎟ ⎟ ⎟⎟⎜ ⎜ ⎜⎜⎟ ⎟ ⎟= +⎟⎜ ⎜ ⎜⎜⎟ ⎟ ⎟⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎟⎟⎜⎟ ⎟ ⎟⎜ ⎜ ⎜⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠

y h h n

y h h n

… …

… …

1( ( ,.., ))Ri na aϕ =a a Rn

1

( ) ( ) ( ) 1,...,T

R

n

i i k i k i R

k n

x x i nϕ ϕ ϕ= +

= + + =∑y h n

( ) ( ) 1,...,i i i Rx i nϕ ϕ= + =y n

Rn

HHH

( )HH H=HH HH

H H=HH Q QΛ

1 2( , ,.., )Rndiag λ λ λ=Λ

2

2 2

1

2

1

log det

log det log 1

log 1

H

R

R

R

R

nT

n

n kT Tk

n

kTk

Cn

n n

n

γ

γ γλ

γλ

=

=

⎛ ⎞⎟⎜= + ⎟⎜ ⎟⎜⎝ ⎠⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥⎟ ⎟⎜ ⎜= + = +⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎢ ⎥⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

⎛ ⎞⎟⎜= + ⎟⎜ ⎟⎜⎝ ⎠

∏

∑

I Q Q

I

Λ

Λ

T Rn n< HHH

2 2log det log detH H

R Tn nT T

Cn nγ γ⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜= + = +⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠

I HH I H H

H H=H H Q QΛ

2

1

log 1Tn

kTk

Cnγ

λ=

⎛ ⎞⎟⎜= + ⎟⎜ ⎟⎜⎝ ⎠∑

4.5 Capacity under Rayleigh Fading 55

A capacity expression that covers both cases, and , is

From this equation one acknowledges that a MIMO system may be described in terms of SISO sub-channels, or eigenmodes, and thus the total capacity is the sum of the

capacities of these sub-channels. Applying the law of large numbers once more, but thistime to (4.49), it easily follows that when is very large and the entries of areindependent, the capacity can be approximated as

4.5 Capacity under Rayleigh Fading

The initial assumption in the derivation of the information-theoretic capacity of theMIMO channel was that it was modelled by the channel response in (4.7), but nothing wassaid about the stochastic properties of the channel matrix . In a mobile wireless environ-ment, matrix will be time-variant and its entries will suffer from the fading phenome-non. It is therefore imperative that the channel capacity be given in the average sense

where the expectation is over the pdf of each eigenvalue .In [45], has been computed at the expense of restricting the quadratic form

to a positive definite matrix, and hence . It then appeals to some results from math-ematical statistics to find the required pdfs, but at the same time circumvents the eigenval-ue approach and bases its results on hypergeometric functions of matrix arguments,characteristic functions and infinite sums. Moreover, [45] assumes that the transmit cov-ariance matrix is a multiple of the identity, which unfolds in a mutual information that isnot the true capacity when there is transmitter-sided correlation. More recently, in [46],doubly-correlated channels have been studied, but not without the same limitations. Othertechniques that circumvent the eigenvalue approach are found in [47], [48], and resort tomoment generating functions and direct integration. [49] does not circumvent the eigen-value approach but fails to give a closed-form expression and only treats the case of nomore receiver antennas than transmitter antennas. No such limitations and “worka-rounds” exist within the author’s results.

When the joint pdf of the eigenvalues is a symmetric function (i.e. a function that isunchanged by any permutation of its variables), then (4.54) simplifies to

The order of magnitude of (4.55) can be assessed as follows:

(4.52)

(4.53)

T Rn n≥ T Rn n<

min( , )

2

1

log 1R Tn n

kTk

Cnγ

λ=

⎛ ⎞⎟⎜= + ⎟⎜ ⎟⎜⎝ ⎠∑

min( , )R Tn n

max( , )R Tn n H

2 2max( , )

min( , ) log 1 min( , ) log 1min( , )

R T RR T R T

T R T

n n nC n n n n

n n nγ γ

⎛ ⎞⎛ ⎞ ⎟⎟ ⎜⎜≈ + = + ⎟⎟ ⎜⎜ ⎟⎜ ⎟⎜⎝ ⎠ ⎝ ⎠

(4.54)

(4.55)

H

H

min( , )

2

1

log 1R Tn n

avg kTk

C Enγ

λ=

⎡ ⎛ ⎞⎤⎟⎜= +⎢ ⎥⎟⎜ ⎟⎜⎝ ⎠⎢ ⎥⎣ ⎦∑

kλ

avgC HHH

0kλ >

2min( , ) log 1avg R TT

C n n Enγ

λ⎡ ⎛ ⎞⎤⎟⎜= +⎢ ⎥⎟⎜ ⎟⎜⎝ ⎠⎢ ⎥⎣ ⎦


• since the logarithm is a concave function, (4.55) can be upper-bounded as

• and using the properties of the trace

• yields

• therefore the upper bound on the average MIMO channel capacity is

which is identical to (4.53). Besides defining a bound to the capacity, it also tells us thatwhen the average capacity is always less than times the Shannon capacity.However, it does not yet unveil the essence of the true capacities in Rayleigh fading envi-ronments. The calculation of the exact capacities will be presented in the following sec-tions.

4.5.1 CHANNEL MATRIX DISTRIBUTION

For a better understanding of the subsequent analysis the author recommends the read-ings [50], [51], [52] and [53]. Finding the distribution of the eigenvalues of is a somewhatintricate process which starts by outlining some particular assumptions about the channelmatrix:

1. the number of multipath components between the -th transmitter antenna and the-th receiver antenna is very large and there is a single cluster of scatterers between the

transmitter and receiver; therefore, by the central limit theorem, the entries of matrix can be statistically characterized as zero-mean complex Gaussian random variables,

with jointly Gaussian in-phase and quadrature components;

2. the ’s corresponding to different transmitter antennas are statistically independent,thus uncorrelated, and the ’s corresponding to different receiver antennas are correlatedrandom variables;

In view of the two conditions above, the columns of the channel matrix can be regarded as independent samples from a complex multivariate zero-mean Gaus-

sian distribution, that is, for every , , where is thecovariance matrix of the column entries.

Determining the pdf of is equivalent to finding the joint pdf of its columns, and so

(4.56)

(4.57)

(4.58)

(4.59)

[ ]2min( , )log 1avg R TT

C n n Enγ

λ⎛ ⎞⎟⎜≤ + ⎟⎜ ⎟⎜⎝ ⎠

( ) ( ) ( )min( , )

1

R T

H H H

n n

k

k

tr tr tr λ=

= = = ∑H H HH Q QΛ

[ ]( )[ ]

/min( , )min( , )

H

R T R TR T

E trE n n n n

n nλ = =

H H

2min( , )log 1min( , )

Ravg R T

R T

nC n n

n nγ

⎛ ⎞⎟⎜ ⎟⎜≤ + ⎟⎜ ⎟⎟⎜⎝ ⎠

R Tn n n= = n

H

j

i

ijh

H

ijh

ijh

( )1 2| | .. |Tnh h h

H1 Tj n≤ ≤ (0, )Rnjh ∼ N Σ [ ]H

j jE= h hΣ

H


it is useful to introduce the notation

The covariance matrix of the entries of is then

where denotes the Kronecker product of matrices. The entries of are therefore theentries of . They are given by

It is not difficult to evaluate these since the ’s can be obtained from the ’s. Re-member that the indexes i and r are identifiers for the different receiver antennas (branch-es). From (4.3) we have

and therefore

Assuming an environment with uncorrelated scattering, (4.64) is clearly non-zero only for, which reduces it to

where the normalization of (4.33) is used as a simplification. Now refer to the situationdepicted in Figure 4.1, where the wavefront of a single propagation path impinges on twoof the receiver antennas, say, antenna i and antenna r. Both antennas travel in the samedirection, but within parallel paths. is the angle between the direction of motion andthe k-th incoming wave, is the angle between the direction of motion and the line join-ing the two antennas, is the distance between the antennas, and is the k-th

(4.60)

(4.61)

(4.62)

(4.63)

(4.64)

(4.65)

( )

1

Tn

vec

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟= = ⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

h

h H

h

H

[ ]

0

0

H

TnE

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟= = ⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

hhC hh I

…

…

Σ

⊗ Σ =

Σ

⊗ hhC

Σ

[ ] [ ]( ) ( )ij rjir E h t h t∗Σ =

( )rjh t ( )ijh t

( )1

0

( )j

ijk jk

Lj t

ij jk

k

h t a e ϕ ω−

+

=

= ∑

[ ] [ ] ( )1

( )

, 0

( ) ( )j

ijk rjp jk jp

Lj t

ij rj jk jp

k p

E h t h t E a a E e ϕ ϕ ω ω−

− + −∗

=

⎡ ⎤= ⎢ ⎥⎣ ⎦∑

k p=

[ ] [ ] ( ) [ ] ( )

( ) ( )

1

2 2

01

2

( ) ( )

( )

j

ijk rjk ijk rjk

ijk rjk ijk rjk

Lj j

ij rj jk j jk

k

j jij

E h t h t E a E e L E a E e

E h t E e E e

ϕ ϕ ϕ ϕ

ϕ ϕ ϕ ϕ

−− −∗

==

− −

⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤⎡ ⎤= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

∑

jkφ

irθ

ird ijk rjkϕ ϕ−


path phase difference between the antennas. The phase difference can be written as

and a direct substitution into (4.65) results in

From this expression, it is easy to see that the following relations are true

where . The elements at the left are merely the entries of the covariancematrices of the real and imaginary parts of . It is now a simple assertion to writethe identities between the covariance matrices

Then again, these relations constitute the definition (4.15) of a circularly symmetric vector,and therefore the pdf of is identical to (4.17), rewritten as

Equation (4.70) can be put to another form by noticing that

where is the trace function. Hence, the pdf of is

FIGURE 4.1 Illustrating the phase difference between different antennas of a moving receiver.

(4.66)

(4.67)

(4.68)

(4.69)

(4.70)

(4.71)

(4.72)

dir

i

r

ϕijk

ϕrjk

Wavefront of thek -th path from the

j -th transmitter antenna

φjk

θir

cos( )ijk rjk ir jk irdϕ ϕ β φ θ− = − +

[ ] ( )[ ] ( )[ ]( ) ( ) cos cos( ) sin cos( )ij rj ir jk ir ir jk irE h t h t E d jE dβ φ θ β φ θ∗ = + − +

[ ] [ ] ( )[ ]

[ ] [ ] ( )[ ]

1( ) ( ) ( ) ( ) cos cos( )

21

( ) ( ) ( ) ( ) sin cos( )2

ijI rjI ijQ rjQ ir jk ir

ijI rjQ ijQ rjI ir jk ir

E h t h t E h t h t E d

E h t h t E h t h t E d

β φ θ

β φ θ

= = +

= − = − +

ij ijI ijQh h jh= +

jIh jQh jh

TjI jI jQ jQ jI jI

TjI jQ jQ jI jI jQ

= =

= − = −

h h h h h h

h h h h h h

C C C

C C C

h

( )

( )

1 1

1

1 1

( ) det( ) exp

det( ) exp

T

H

H

n

j j

j

f π

π

− −

=

− −

= −

= −

∏h

hh hh

h h h

C h C h

Σ Σ

[ ]

( ) ( )

1 1 1

1 1

1 1

T T

H H H

H H

n n

j j jjj j

tr tr

− − −

= =

− −

= =

= =

∑ ∑hhh C h h h H H

H H HH

Σ Σ

Σ Σ

()tr ⋅ H

( )[ ]1( ) det( ) exp HTnf trπ − −= −H H HHΣ Σ


4.5.2 JACOBIANS AND EXTERIOR PRODUCTS

Let and be two -dimensional real vectors such that the transformation (or equiva-lently, change of coordinates) is valid. Then, there exists an associated lineartransformation of differentials

associated with it that conveys the relation between the differential lengths in the two co-ordinate systems. is the Jacobian matrix

of the transformation, and it equals in the case of linear transformations . Thevolume element (ignoring sign) in the coordinate system of vector equals the exteriorproduct of all the distinct elements in

where it is understood that the differential lengths are implicitly associated with a certaindirection in the coordinate system, and thus can be regarded as differential vectors. Theexterior product has the following basic rules

which means that products of repeated differentials disappear. Denoting the elements ofthe Jacobian matrix by , the volume elements in the two coordinate systems are relatedby

where the summation is taken over the permutations of . Using theexterior product rules in (4.76) it is possible to write

where

The first factor of (4.78) is simply the determinant of the Jacobian matrix , and

(4.73)

(4.74)

(4.75)

(4.76)

(4.77)

(4.78)

(4.79)

x y n

( )=y T x

( )d d= →y J y x x

( )→J y x

1 1

1

1

( )n

n n

n

y yx x

y yx x

⎛ ⎞∂ ∂ ⎟⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜→ = ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∂ ∂

J y x

…

T =y Tx

x

dx

( ) 1 21

..n

n ii

d dx dx dx dx∧

== ∧ ∧ ∧ = ∧x

; 0dx dy dy dx dx dx dx dx∧ = − ∧ ∧ = − ∧ =

ijt

( )

1 2 1 21 2

11 1

1 2( , ,.., )

.. ..n n

n

nm m

i ij jji i

p p np p p pp p p p

d dy t dx

t t t dx dx dx

∧

== =

=

= =

= ∧ ∧ ∧

Σ

Σ

∧ ∧y

!n 1 2( , ,.., )np p p (1,2,.., )n

( )1 2

1 2

1 21 2

1 2 1 2( , ,.., )

1 2( , ,.., ) 1

( ) .. ..

( ) ..

nn

nn

p p np np p p p

n

p p np jp p p p j

d p t t t dx dx dx

p t t t dx

σ

σ

∧

=

= =

= ∧ ∧ ∧

⎡ ⎤⎢ ⎥= ⎢ ⎥⎣ ⎦

Σ

Σ ∧

y

1 (1,2,.., )( )

1 (1,2,.., )

np

n

p even

p odd σ

⎧+⎪⎪= ⎨⎪−⎪⎩

if is an permutation of if is an permutation of

( )→J y x


so it follows that the equation

establishes the relation between the volume elements of the two coordinate systems. TheJacobian determinant of the transformation is denoted here by .

The Jacobian determinant in the complex case (complex vectors and transformationmatrices) is slightly different from (4.80). For simplicity, only linear transformations willbe discussed. Writing

and separating the real and imaginary parts, gives

Therefore, using (4.80) and observing that , the relation between theright and left volume elements is

Finally, invoking the invariance of the determinant under elementary row and column op-erations, we may multiply the first row by and add it to the second row,

and then multiply the second column by and add it to the first column, yielding

This result for complex vector transformations easily generalizes to complex linear trans-formations of complex matrices . Letting denote the number of columns of thematrices and , it follows that

and accordingly .Now, let and be generic random matrices (real or complex) and let and be the respective joint pdfs of their random entries. Define a one-to-one generic trans-

(4.80)

(4.81)

(4.82)

(4.83)

(4.84)

(4.85)

(4.86)

( ) [ ]( )

( )

det ( )

( )

d d

J d

∧ ∧

∧

= →

= →

y J y x x

y x x

( )=y T x ( )J →y x

( )( )

( ) ( )

r i r i r i

r r i i i r r i

d d

d jd j d jd

d d j d d

=

+ = + +

= − + +

y T x

y y T T x x

T x T x T x T x

r r i r

i ri i

d d

d d

−⎛ ⎞ ⎛ ⎞⎛ ⎞⎟ ⎟⎟⎜ ⎜⎜⎟ ⎟⎟⎜ ⎜⎜=⎟ ⎟⎟⎜ ⎜⎜⎟ ⎟⎟⎟⎟ ⎟⎜⎜ ⎜⎝ ⎠⎝ ⎠ ⎝ ⎠

y T T x

T Ty x

( ) ( ) ( )r id d d∧ ∧= ∧y y y

( ) ( ) ( )( ) detr i

i rd J d d∧ ∧ ∧

−⎛ ⎞⎟⎜ ⎟⎜= → = ⎟⎜ ⎟⎟⎜⎝ ⎠

T Ty y x x x

T T

j

( ) ( )detr i

r i r id d

j j∧ ∧

−⎛ ⎞⎟⎜ ⎟⎜= ⎟⎜ ⎟+ − ⎟⎜⎝ ⎠

T Ty x

T T T T

j−

( ) ( )

( )

( )22

det

det( )det( )

det( ) det

r i i

r i

r i r i

r i

jd d

j

j j d

j d d

∧ ∧

∧

∧

+ −⎛ ⎞⎟⎜ ⎟⎜= ⎟⎜ ⎟− ⎟⎜⎝ ⎠

= + −

= + =

T T Ty x

0 T T

T T T T x

T T x T x

=Y TX n

X Y

( ) ( ) ( )

( ) ( )

2

1 1

2 2

1

det

det det

n n

iii in

n ni

i

d d d

d d

∧ ∧ ∧

∧∧

= =

=

= =

= =

∧ ∧

∧

Y y T x

T x T X

( ) 2det

nJ → =Y X T

X Y ( )fX X

( )fY Y


formation between the random matrices, and write the relationship between thevolume elements, i.e.

where and are the positive volume elements of and , respectively. Thenit must be the case that

and in the same manner

This means that the statistics of can be determined in terms of the statistics of by theexpression (4.89). Together with (4.85), this important result leads to a fundamental prop-erty of complex Gaussian distributions such as (4.72): they are invariant under right-sidedunitary transformations

The Jacobians of a few particular cases of linear matrix transformations will be useful inthe subsequent sections, and so the details of their derivation will be given here.

1. Jacobian of the Cholesky factorization - let be the Cholesky factorizationof the complex positive definite matrix , where is an upper-triangular ma-trix with positive and real diagonal entries. In matrix form, is given by

Differentiating each element of

it is easy to assess that, since only the differentials elements that do not appear in aprevious column or row prevail in the exterior product, solely the last term in each matrixentry must be accounted for

(4.87)

(4.88)

(4.89)

(4.90)

(4.91)

( )=Y T X

( ) ( )( )d J d∧ ∧= →Y Y X X

( )d ∧X ( )d ∧Y X Y

( ) ( )( ) ( )f d f d∧ ∧=Y XY Y X X

( )

( )1( ) ( ) ( ( )) ( )

df f f J

d

∧

∧−= = →Y X X

XY X T Y X Y

Y

Y X

=Y XQ

( )

( )2( ) ( ) ( ) det ( )H nd

f f f fd

∧

∧= = =Y X X XX

Y YQ Y Q YY

H=S R R

n n× S R n n×

S

211 11 12 11 13

212 12 22 12 13 22 23

213 13 23 23 33

r r r r r

r r r r r r r

r r r r r

∗ ∗

∗ ∗ ∗

∗ ∗

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟× + +⎜ ⎟⎜ ⎟⎟⎜= ⎟⎜ ⎟⎜× × + + ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠

S

…

…

…

S

(4.92)

11 11 11 12 11 12 11 13 11 13

12 12 12 12 22 22 12 13 12 13 22 23 22 23

13 13 13 13 23 23 23 23 33 33

2

2

2

r dr dr r r dr dr r r dr

dr r r dr r dr dr r r dr dr r r drd

dr r r dr dr r r dr r dr

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

+ +⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜× + + + + + ⎟⎜ ⎟⎜ ⎟= ⎟⎜ ⎟⎜× × + + + + ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠

S

…

…

…

(4.93)

ijdr

11 11 11 12 11 13

22 22 22 23

33 33

2

2

2

r dr r dr r dr

r dr r drd

r dr

+ ∗ + ∗⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜× + ∗ + ∗ ⎟⎜ ⎟⎜ ⎟= ⎟⎜ ⎟⎜× × + ∗ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠

S

…

…

…


Finally, taking exterior products and noticing that is a complex dif-ferential for , and thus , the volume elementof matrix is

The Jacobian determinant is

2. Jacobian of the QR factorization - let be the QR factorization of the com-plex ( ) matrix of rank , where is an unitary matrix and is an

upper-triangular matrix. The identity defines independent equations.There are two options to make the factorization unique:• the entries of matrix are all complex, which means that has independ-

ent parameters; the diagonal elements of matrix are real, and hence has inde-pendent parameters, making a total of independent parameters with ;

• the diagonal elements of are real, which means that has independ-ent parameters; the entries of matrix are all complex, and hence has independent parameters, making a total of independent parameters with ;

Only the Jacobian of the first case will be derived, since the second case follows by thesame lines. Differentiating and introducing the volume element notation, yields

In the present form, this expression makes the desired calculation exceedingly complicat-ed. Nevertheless, a worthy simplification is introduced by the following fact:• matrix possesses orthogonal columns and can be extended with columns

(i.e. the orthogonal complement) to form an unitary matrix ,such that using (4.86) it is possible to write

and therefore

(4.94)

(4.95)

(4.96)

(4.97)

(4.98)

[ ] [ ]ij ij ijr cdr dr j dr= +

i j< ( ) [ ] [ ] ( )2 2ii ij ii ij ij ii ijr cr dr r dr dr r dr

∧ ∧= ∧ =

S

( ) ( )

( )

( )

2 1 2 311 22

1

(2 2 1)

1

2

2 ...

2

R R

n nn

ij ii iji j i j

n nn n n

n n ii iji i j

nn kn

kkk

d ds r dr

r r r dr dr

r d

∧∧

∧

∧

≤ ≤

− −= <

− +

=

= =

⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟= ∧⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠

= ∏

∧ ∧

∧ ∧

S

R

( )( )

( )(2 2 1)

1

2n

n knkk

k

dJ r

d

∧

∧− +

=→ = = ∏SS R

R

=S QR

m n× m n≥ S n Q m n× R

n n× H =Q Q I 2n

Q Q 22mn n−

R R 2n

2mn Q

Q Q 22mn n n− −

R R 2n n+

2mn Q

S

( ) ( )d d d∧ ∧= +S QR Q R

Q n m n−

m m× ⊥⎡ ⎤= ⎢ ⎥⎣ ⎦Q Q Q

( ) ( )

( )

2det

nd d

d

∧∧

∧

=

=

Q S Q S

S

( ) ( )

( ) ( )

H

H

H

H H

dd d

d

d d d

∧∧

∧

⊥

∧ ∧⊥

⎛ ⎞⎟⎜ ⎟⎜= = ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

= + ∧

Q SS Q S

Q S

Q QR R Q QR


Moreover, because is skew-hermitian ( ), matrix is given by

It is obvious looking at (4.99) that only the last term of the diagonal and subdiagonal en-tries is not redundant, in the sense that the factor does not appear in a previous col-umn and products of repeated differentials are zero

In addition, the (not redundant) diagonal terms are purely imaginary, and thus independ-ent of the real diagonal elements of . In other words, and make a perfect fitin (4.98), thereby reducing it to

Hence, denoting the columns of by , the first factor is given by

The second factor in (4.101) follows from (4.86)

Substituting (4.102) and (4.103) into (4.101), and evaluating the product, gives

In words, the Jacobian determinant of the one-to-one QR factorization with strictly com-plex is

Going by the same steps but now with a strictly complex , and verifying that the factors are redundant and so may be ignored in the exterior products, it follows that the

HdQ Q H Hd d= −Q Q Q Q HdQ QR

(4.99)

11 1 1 12 1 1 22 2 1 13 1 1 23 2 1 33 3 1

11 2 1 12 2 1 22 2 2 13 2 1 23 2 2 33 3 2

11 3 1 12 3 1 22 3 2 13 3 1 23 3 2 33 3 3

( ) ( ) ( )

( )

H H H H H H

H H H H H H

H

H H H H H H

r d r d r d r d r d r d

r d r d r d r d r d r dd

r d r d r d r d r d r d

∗ ∗ ∗

∗

⎛ − − −⎜⎜+ + −

=+ + +

⎝

q q q q q q q q q q q q

q q q q q q q q q q q qQ QR

q q q q q q q q q q q q

…

…

…

⎞⎟⎟⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜ ⎠

(4.100)

(4.101)

(4.102)

(4.103)

(4.104)

(4.105)

Hi jdq q

( ) ( ) ( ) ( )( ) 0H H H Hi j i j i j i jd d d d

∧ ∧ ∧ ∧∗∧ = ∧ =q q q q q q q q

dR HdQ QR dR

( ) ( ) ( ) ( )H Hd d d d∧ ∧∧ ∧⊥= ∧ ∧S Q QR Q QR R

Q iq

( ) ( ) ( )

( ) ( )

( )

2

(2 2 1)

1

HH H

H H

H

n n

i j jj j jjii j i jn n

j jj j jji ii j i jnn

n kjikk i jk

d d r d r

d r d r

r d

∧ ∧∧

∧ ∧

∧

≥ ≥

> =

− +

≥=

= =

=

= ∏

∧ ∧

∧ ∧

∧

Q QR q q q q

q q q q

q q

( ) ( ) ( )

( )

2 2

(2 2 )

1 11

detH H

H

m n

n mnm n

jikk j i nk

d d

r d

∧ ∧⊥ ⊥

∧

−

−

= = +=

=

= ∏ ∧ ∧

Q QR R Q Q

q q

( ) ( ) ( )

( ) ( ) ( ) ( )

(2 2 1)

11

(2 2 1) (2 2 1)

1 1

H

H

n mnn k

jikk j i jkn n

m k m kkk kk

k k

d r d d

r d d r d d

∧∧ ∧

∧∧ ∧ ∧

− +

= ==

− + − +

= =

= ∧

≡ ∧ = ∧

∏

∏ ∏

∧∧S q q R

Q Q R Q R

Q

( )( )

( ) ( )(2 2 1)

1

,n

m kkk

k

dJ r

d d

∧

∧ ∧− +

=→ = =

∧ ∏SS Q R

Q R

RHi idq q


Jacobian is

3. Jacobian of the transformation - let be a fixed complex nonsingular matrix and and be two hermitian positive definite matrices, such that

. Since and are positive definite they have a Cholesky factorization, namely and , and for that reason the transformation can be rewritten as

Introduce two unitary matrices and and write (4.107) as

Matrices and can be properly chosen to make the relation between the left and writemembers an obvious one, that is . Thus, one may write the following rela-tions

The expression for the Jacobian determinant follows immediately

Clearly , so from (4.86) it follows that

Also, using the Jacobians of the Cholesky and QR factorizations, (4.95) and (4.105) re-spectively with , leads to the following equalities

Therefore, the Jacobian determinant in (4.110) is

However, there is a unitary matrix (regarded as a member of the space of cosets span-ning the unitary group) such that , yielding

and the final result for the Jacobian determinant is then1(see next page)

(4.106)

(4.107)

(4.108)

(4.109)

(4.110)

(4.111)

(4.112)

(4.113)

(4.114)

(4.115)

( )( )

( ) ( )2( )

1

,n

m kkk

k

dJ r

d d

∧

∧ ∧−

=→ = =

∧ ∏SS Q R

Q R

H=S B AB B

n n× A S n n×H=S B AB A S

1 1H=S R R 2 2

H=A R R

1 1 2 2

H

H H H

=

=

S B AB

R R B R R B

n n× 1Q 2Q

( ) ( )

1 1 1 1 2 2 2 2

1 1 1 1 2 2 2 2

H H H H H

H H

= =

= =

S R Q Q R B R Q Q R B

Q R Q R Q R B Q R B

1Q 2Q

1 1 2 2=Q R Q R B

1 1 1 1 1 1 2 2 2 2 ; ; ; H H= = = =S R R X Q R Y Q R A R R

( )( )

( )( )

( )( )( )

( )( )

( )( )

( )( )

1 1 1 2

1 1 1 2

d d d d d dJ

d d d d d d

∧ ∧ ∧ ∧∧ ∧

∧ ∧∧ ∧ ∧ ∧→ = =S S R X Y R

S AA R X Y R A

1 1=X Y B

( )( )

1 2

1det nd

d

∧

∧ =X

BY

n m=

( )

( )( )( ) ( )

( )( )

( )( )

( )1 1 2 2

1 1 1 2

2 ;

2

n

nd d d d dd d d d d

∧ ∧ ∧ ∧∧

∧∧ ∧ ∧ ∧= =S R Y R QR X Q R A

( )( )( )

22

1det n d

Jd

∧

∧→ =Q

S A BQ

Q

2 1=Q QQ

( )( )

2 2

1det 1nd

d

∧

∧ = =Q

QQ

( )( )

( )2det nd

Jd

∧

∧→ = =S

S A BA


4. Jacobian of the Eigenvalue decomposition - let be an hermitian positive definite matrix with the eigenvalue decomposition given by

where is a diagonal matrix with real and distinct elements. It is important to make a briefpause here and analyse the above decomposition. Matrix has independent parame-ters, the same number as . Therefore the decomposition is not one-to-one because theright side of (4.150) has a total of independent parameters. Let the columns of the

unitary matrix be given by . Clearly, introducing a random phase in each of itscolumns will always determine the same matrix

which confirms that the transformation in (4.150) is not one-to-one or, in other words,does not have a unique solution. Since a one-to-one mapping is required, at least one entryin each column must have its phase fixed, for example, by requiring that the diagonal ele-ments of matrix be real.

Differentiating (4.116), the volume element of is written as

To find a closed expression for (4.118), it is helpful to use the same reasoning already em-ployed in the case of the QR factorization (4.96)-(4.105). Multiplying the left and rightsides of the differentials by and , respectively, (4.118) reassembles to

But using (4.115), the left member of (4.119) becomes

which is the required volume element in (4.118). In addition, using the fact that isskew-hermitian, it follows that only the diagonal (purely imaginary) and subdiagonal ele-ments of are relevant. These are given by

Furthermore, since the sum in (4.119) is hermitian, its diagonal ele-ments are all zero, which makes this sum a perfect fit with

1. Compare this result to the transformation with regular matrices and : in this case.

(4.116)

(4.117)

(4.118)

(4.119)

(4.120)

(4.121)

(4.122)

H=S B AB A S

( ) 4det nJ → =S A B

S

n n×

HS = Q QΛ

Λ

Q 2n

S2n n+

n n× Q iq

S

1 1

1 1

1 1 1

1 1 1

( ,.., )diag( ,..., )( ,.., )

( ,.., )diag( ,..., )( ,.., )

H Hn nR RR R R

Hn nR RR R R

H

j jj jn n n

j jj jn n n

e e e e

e e e e

φ φφ φφ φ

φ φφ φ

λ λ

λ λ −−

=

=

= =

Q Q q q q q

q q q q

Q Q S

Λ

Λ

Q

S

( ) ( )H H Hd d d ∧∧ = + +dS Q Q Q Q Q QΛ Λ Λ

HQ Q

( ) ( )

( )( )

H H H

HH H

d d d

d d d

∧ ∧

∧

= + +

= + +

Q dSQ Q Q Q Q

Q Q Q Q

Λ Λ Λ

Λ Λ Λ

( ) ( ) ( )2detH n∧ ∧ ∧= =Q dSQ Q dS dS

HdQ Q

HdQ QΛ

[ ]H Hi j jijd d i jλ= ≥Q Q q qΛ

( )HH Hd d+Q Q Q QΛ ΛdΛ

( ) ( )

( ) ( )( )

H H

HH H

d d d

d d d

∧∧

∧∧

= + +

= ∧ +

dS Q Q Q Q

Q Q Q Q

Λ Λ Λ

Λ Λ Λ


The matrix elements of the hermitian sum are

and therefore the volume element for this factor is

Inserting this into (4.122) gives the final expression for the volume element of matrix

The Jacobian determinant follows directly

4.5.3 THE STIEFEL MANIFOLD AND THE UNITARY GROUP

Consider the set of all ( ) unitary matrices , that is, the set of all complexmatrices with orthonormal columns (frames) in . All such matrices satisfy the equa-tion (i.e. independent conditions), and thus define a -dimensionalalgebraic variety in usually called the complex Stiefel manifold . The set of all unitary matrices with the operation of matrix multiplication is called the unitary group

.Choosing a matrix which spans and its orthogonal complement, the

volume element on the Stiefel manifold may be written as

which defines a differential form of maximum degree, namely . Thus, it repre-sents a measure, and hence the total volume of the Stiefel manifold can be obtained bydirect integration

To evaluate this integral, set in (4.72) and define as an ( ) complexmatrix with independent zero-mean Gaussian entries, then integrate it and equate the re-sult to unity, that is

Now, assume has a unique QR factorization with strictly complex , and with the help

(4.123)

(4.124)

(4.125)

(4.126)

( )[ ] ( )HH H Hi j j iij

d d d i jλ λ+ = − ≥Q Q Q Q q qΛ Λ

( )( ) ( )

( )2

( )

( )

HH H H

n

i j j ii jn

j ii j

d d d

d

λ λ

λ λ

∧ ∧

∧

≥

>

+ = −

= −∏

Q Q Q Q q q

Q

Λ Λ Λ

S

( ) ( ) ( )( )

( ) ( )2( )

HH H

n

j ii j

d d d

d dλ λ

∧∧∧

∧∧

>

= ∧ +

= − ∧∏

dS Q Q Q Q

Q

Λ Λ Λ

Λ

( )( )

( ) ( )2, ( )

n

j ii j

dJ

d dλ λ

∧

∧∧

>→ = = −

∧ ∏SS Q

QΛ

Λ

(4.127)

(4.128)

(4.129)

m n× m n≥ Q

n mCH =Q Q I 2n ( )22mn n−

2mn,n mV n n×

Q

( )U n

⊥⎡ ⎤= ⎢ ⎥⎣ ⎦Q Q Q Q

( ) ( )Hd d

∧∧ =Q Q Q

( )22mn n−

( ) ( ),

,Volumen m

n mV

V d ∧= ∫ Q

=Σ Ι HH m n× m n≥

( )[ ]( )exp H mn

Vtr d π∧− =∫

H

HH H

HH Q


of (4.105) write

The first integral can be further simplified by separating the real and imaginary parts of, and grouping factors

The first integral in (4.131) is the ordinary Gaussian integral, and the second integral canbe transformed to the Gamma function by the transformation . Therefore

where the complex multivariate gamma function is defined as

It now follows that the total volume of the complex Stiefel manifold is

Recall that this volume was computed assuming a QR factorization with strictly complexunitary matrix and an upper-triangular matrix with real diagonal. Nevertheless, it issometimes desirable to put an additional restriction in matrix . As mentioned before, thisis the case of the Eigenvalue decomposition, where it is necessary to force the imaginaryparts of the diagonal elements of to zero, in order to make the decomposition unique.But this restricts the range of , and thus introduces an additional constraint to the Stiefelmanifold that changes the total volume. The constrained Stiefel manifold will be denotedby . To find this volume one must follow the same steps as above, but this time usethe alternative QR factorization for which the Jacobian determinant is the one given in(4.106). Equation (4.105) is now

(4.130)

( )[ ] ( ) ( )

( ) ( )

,

,

(2 2 1)

1

2 (2 2 1)

1

exp

exp

H

n m

n m

n

i j

nm kmn

kkV Vk

n nm k

ij ijkk i jV Vk

tr r d d

r r dr d

π ∧ ∧

∧ ∧

≤

− +

=

− +

≤=

= −

⎛ ⎞⎟⎜ ⎟= ⎜− ⎟⎜ ⎟⎟⎜⎝ ⎠

∏

∏∑

∫ ∫

∫ ∫R

R

R R R Q

QΛ

ijr

(4.131)

[ ] [ ] [ ] [ ] [ ] ( )

[ ]( )[ ] [ ]( ) [ ] ( )

,

,

2 2 (2 2 1)

11

22 (2 2 1)2

1

exp

exp exp

n m

n m

n

i j

n n nm kmn

ij ij ij ij ii IkkI Q I Qi j iV Vk

n nm i

ij ij ii iiiiI II IVi j i

r r r dr dr dr d

r dr r r dr d

π ∧

∧

≤

− +

< ==

∞ ∞− +

−∞ −∞< =

⎛ ⎞⎟⎜ ⎟= ⎜− + ∧⎟⎜ ⎟⎟⎜⎝ ⎠

⎡ ⎤ ⎡ ⎤= − −⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∏∑∫ ∫

∏ ∏∫ ∫ ∫

R

Q

Q

Λ Λ

(4.132)

(4.133)

(4.134)

(4.135)

[ ]2ii rx r=

( )

( ) ( )

,

, ,

( 1)/2 ( )

1

( 1)/2

1

1exp( )

2

1( 1) ( )

2 2

n m

n m n m

nmn n n m i

Vinn n

nn nV Vi

x x dx d

m i d m d

π π

π

∧

∧ ∧

∞− −

−∞=

−

=

⎡ ⎤= −⎢ ⎥

⎢ ⎥⎣ ⎦

= Γ − + = Γ

∏ ∫ ∫

∏ ∫ ∫

Q

Q Q

( 1)/2

1

( ) ( 1)n

n nn

i

m m iπ −

=

Γ = Γ − +∏

[ ] ( ),

1

,

1

2 2Volume

( 1) ( )n m

n m i n mn

n mnV i

V dm i mπ π∧

− +

=

= = =Γ − + Γ∏∫ Q

Q R

Q

Q

Q

,n mV⊂

( )[ ] ( ) ( )

( ) ( )

,

,

2( )

1

2 2( )

1

exp

exp

H

n m

n m

n

i j

nm kmn

kkV Vk

n nm k

ij kk iji jV Vk

tr r d d

r r dr d

π ∧ ∧

⊂

∧ ∧

⊂≤

−

=

−

≤=

= −

⎛ ⎞⎟⎜ ⎟= ⎜− ⎟⎜ ⎟⎟⎜⎝ ⎠

∏

∏∑

∫ ∫

∫ ∫R

R

R R R Q

QΛ


Once again, separating the real and imaginary parts of , and grouping factors leads to

Introducing polar coordinates

(4.136) simplifies to

Therefore, from (4.138) we find that the volume subtended by the set of all unitary ( ) matrices with real diagonal elements is

This is simply (4.134) divided by , which accounts for the arbitrary phases of the di-agonal elements of . An immediate corollary of this result is that whenever a function

that does not depend on the particular phase of the columns of is integrated over, the following equality holds

The volumes for the unitary group and constrained unitary group are obtained bysimply putting in the general expressions. Finally, it should be pointed out that thecomputed volumes are invariant under right and left transformations with unitary squarematrices.

4.5.4 NONSINGULAR WISHART DISTRIBUTION

Returning to the problem at hand, and letting , the challenge is to find the pdfof from the pdf of channel matrix , rewritten here for convenience

Regarding the columns of , as samples from a multivariate normal distribution, thenwe may write

(4.136)

(4.137)

(4.138)

(4.139)

(4.140)

ijr

[ ] [ ]( ) [ ] [ ] ( ),

2 2 2( )( 1)/2

1

expn m

nm imn n n

ii ii ii ii iiI Q I QVi

r r r dr dr dπ π ∧

⊂

∞−−

−∞=

⎡ ⎤= − − ∧⎢ ⎥

⎢ ⎥⎣ ⎦∏ ∫ ∫ Q

[ ] [ ]

[ ] [ ]

; cos ; sin ;i ii ii i ii iI Q

ii ii i iI Q

r r r r r r

dr dr r dr d

θ θ

θ

= = =

∧ = ∧

( ) ( )

( ) ( )

,

, ,

2(2 2 1)( 1)/2 2

01

( 1)/2

1

exp

(2 )( 1) ( )

2

n m

n m n m

nm imn n n

i iiVi

nn n nn

nnV Vi

r r dr d d

m i d m d

π

π π θ

π ππ

∧

⊂

∧ ∧

⊂ ⊂

∞− +−

−∞=

−

=

⎡ ⎤⎢ ⎥= −⎢ ⎥⎣ ⎦

= Γ − + = Γ

∏ ∫ ∫ ∫

∏ ∫ ∫

Q

Q Q

m n×

m n≥ Q

( ),

( 1)

,Volume( )n m

n m

n mnV

V dm

π∧

⊂

−⊂⎡ ⎤ = =⎢ ⎥⎣ ⎦ Γ∫ Q

(2 )nπ

Q

( )h Q Q

,n mV⊂

( ) ( ), ,

1( ) ( )

(2 )n m n m

nV V

h d h dπ

∧ ∧

⊂=∫ ∫Q Q Q Q

n m=

(4.141)

(4.142)

H=A HH

A H

( )[ ]1( ) det( ) exp HTnf trπ − −= −H H HHΣ Σ

jh H

1

T

H H

n

j j

j=

= = ∑A HH h h


which clearly shows that is proportional to the sample covariance matrix

The joint density of the elements of in the real case was first computed by J. Wishart[42], and naturally inherited his name: it is called the Wishart matrix. However, in orderto derive the pdf of we require the complex case, which is somewhat different. To thatend it is useful to characterize the channel matrix :• since the rank of the matrix equals the rank of , and since is posi-

tive semidefinite, matrix is nonsingular (and thus positive definite) if and only if;

• recalling that the columns of are statistically independent, it follows that, in statis-tical terms, contains linearly independent columns; therefore, the rank of thematrix is , which means that it has a QR factorization

where is an unitary matrix and is an upper-triangular matrix withpositive and real diagonal entries;

This transforms into the product of lower-triangular and upper-triangular matrices: theCholesky factorization

and consequently the pdf of can be expressed as

Moreover, using the Jacobian of the QR factorization in (4.105) it is possible to write

Hence, inserting (4.147) into (4.146) and expanding the pdf of , yields

Finally, inserting the volume of the Stiefel manifold (4.134) and the Jacobian of theCholesky factorization (4.95) into (4.148), the pdf of matrix is

This is indeed the pdf of the matrix that was defined in the MIMO channel capacityexpression (4.32), and it is known in the “realm” of multivariate statistics as the complexWishart distribution .

(4.143)

(4.144)

(4.145)

(4.146)

(4.147)

(4.148)

(4.149)

A

1

11

T

H

n

j jT j

n=

=− ∑W h h

W

W

A

R Tn n× H

R Rn n× A H HHH

A

T Rn n≥

H

H RnHH Rn

H =H QR

Q T Rn n× R R Rn n×

A

H H H= =A R Q QR R R

A

( )

( )( )

( )

( ),

( ) ( ) ( , )R Tn nV

d df f f d

d d

∧ ∧∧

∧ ∧

⎡ ⎤⎢ ⎥= = ⎢ ⎥⎣ ⎦∫A R QR

R RA R Q R Q

A A

( )( )

( )( )(2 2 1)

1

( , ) ( ) ( )R

H H H H T

nn k

kkk

df d f f r d

d

∧∧ ∧

∧− +

== = ∏QR H H

HQ R Q R Q R Q Q

R

H

( )[ ] ( )( )

( ),

(2 2 1)1

1

( ) det( ) expR T

RTT

n n

nn kn

kkVk

df tr r d

dπ

∧∧

∧− +− −

=

⎡ ⎤⎢ ⎥= −⎢ ⎥⎣ ⎦

∏ ∫AR

A A QA

Σ Σ

A

( )[ ]

( )[ ]( )

2( )1

1

1

( ) expdet( ) ( )

1exp det

det ( )

RT RT R

TR

T R

TR

nn nn n

kknn T k

n nn

n T

f tr rn

trn

ππ

−−

=

−−

= −Γ

= −Γ

∏A A A

A A

ΣΣ

ΣΣ

HHH

( , )Rn TnA ∼ W Σ


4.5.5 EIGENVALUE DISTRIBUTION

To take the expectation in the capacity expression (4.54), the distribution of the eigenval-ues of must be known. Even though the joint pdf of the eigenvalues of the Wishart ma-trix is already known (e.g. [43], and more recently [44]), its derivation has been done in adifferent context, disregarding the statistical peculiarities of the 's (e.g. the correlationbetween its elements), and is in fact somewhat intricate.

To round out the section, the author understood it was important to give a brief, yetlucid derivation. From (4.47) the eigenvalue decomposition is given by

where . To make the factorization unique, we fix such that the ei-genvalues are ordered from greatest to least , and write explicitly

We also impose no restriction to apart from a diagonal with real elements. The jointpdf of and satisfies

Expanding the Wishart pdf with (4.149) and inserting the Jacobian of the Eigenvaluetransformation in (4.126), we get

Once again, an integration is required to eliminate the dependence from (4.153), that is

where, since is a unitary matrix with real diagonal elements, the integration is over theconstrained Stiefel manifold or, in light of (4.140), over the unitary group .

Equation (4.155) gives the joint density function of the strictly positive and ordered ei-genvalues of .

To find the unordered distribution of the eigenvalues one may notice that the orderedeigenvalues define a -th part of positive -dimensional space. Thus, imposing noorder to the eigenvalues is equivalent to extending the ordered to the whole extentof positive -dimensional space, which can be done by finding the pdfs of all permuta-

(4.150)

(4.151)

(4.152)

(4.153)

(4.154)

(4.155)

A

jh

H H= =A HH Q QΛ

1 2( , ,..., )Rndiag λ λ λ=Λ Λ

1 2 ... 0Rnλ λ λ> > > >

o H=A Q QΛ

QoΛ Q

( )( )

( )o o

o( ) ( )H df d f

d

∧∧

∧=Q AA

Q Q Q QΛ Λ , ΛΛ

( ) ( ) ( )

2

o 1 o

( ) (det

( ) expdet ( )

R

T R

TR

nn n

j ii j H

nn T

f d tr dn

λ λ∧ ∧

−

< −

−⎡ ⎤= −⎣ ⎦Γ

∏Q Q Q Q Q QΛ

Λ)Λ , Σ Λ

Σ

Q

( )[ ]( ),

( ) 2

1o 1 o

( )

( ) expdet ( )

R RT R

H

TR n nR R

n nn n

j iii i j

nn VT

f tr dn

λ λ λ∧

⊂

−

= > −

−= −

Γ

∏ ∏∫ Q Q QΛ Λ Σ Λ

Σ

Q

,R Rn nV⊂

( )RU n

( )( )[ ]( )

( ) 2

1o 1 o

( )

( )

( ) exp2 det ( )

R RT R

H

TRR R

n nn n

j iii i j

nnn U nT

f tr dn

λ λ λ

π∧

−

= > −

−= −

Γ


Σ

1 2 ... 0Rnλ λ λ> > > > H=A HH

(1/ !)Rn Rno( )fΛ Λ

Rn


tions of the eigenvalues. Hence, we must have that

where the ’s are the non-overlapping pdfs of the ordered eigenvalues, and the sum-mation is over all permutations of the eigenvalues. It then follows that , and byconsequence the pdf of the unordered eigenvalues is given by

which is simply an average of (4.155). This expression is not yet written in closed-formbecause the integral must still be evaluated. This integral depends on the covariance matrix

of the channel matrix columns or, in other words, it depends on the corre-lation between the fading processes at different receiver antennas.

4.5.6 INDEPENDENT FADING

It turns out that a special case of the matrix makes the integration in (4.157) straight-forward. When the fading processes at different receiver antennas are uncorrelated withone another (i.e. independent fading) and therefore , substitutioninto the integral of (4.157) and using (4.134) leads to

which inserted into (4.157) leads to the joint pdf of the unordered eigenvalues in this par-ticular fading scenario

Recall that this distribution was obtained by always assuming that the number of receiverantennas was smaller than or equal to the number of transmitter antennas, i.e. ,otherwise the Wishart matrix would be singular and consequently would renderthe problem much difficult. However, in the case of independent fading at the receiverside this restriction can be relaxed. This is because the pdf of the channel matrix as givenby (4.141) will now have the form

and as a result the Wishart matrix for may be written as , which is clearly

(4.156)

(4.157)

11

1,.., ( ,.., )

( ) .. 1R

nR nR

p n

p

K f p d dλ λ λ λ

λ λ=

⎡ ⎤⎢ ⎥ =⎢ ⎥⎢ ⎥⎣ ⎦∑∫

( )pf p

1/ !RK n=

( )( )[ ]( )

1

o

( ,.., )

( ) 2

1 1 o

( )

1 1( ) ( ) ( )

! !

( )

exp! 2 det ( )

nR

R RT R

H

TRR R

pR Rp

n nn n

j iii i j

nnn U nR T

f f p fn n

tr dn n

λ λ

λ λ λ

π∧

=

−

= < −

= =

−= −

Γ

∏ ∏

∑

∫ Q Q Q

Λ ΛΛ Λ

Σ ΛΣ

[ ]Hj jE= h hΣ

(4.158)

(4.159)

(4.160)

Σ

2R Rij n nE h⎡ ⎤= =⎢ ⎥⎣ ⎦ I IΣ

( )[ ]( ) ( )[ ] ( )o o

( ) ( )

1

exp exp

2exp

( )

H

R R

RR R R

R

U n U n

nn n n

in iR

tr d tr d

nπ

λ

∧ ∧

=

− = −

⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎜⎝ ⎠Γ∑

∫ ∫Q Q Q QΛ Λ

2( 1)

1

1

( )

( ) exp! ( ) ( )

R RT R

RR R

R R

n nn n

j ii nn ni i j

in nR iT R

fn n n

λ λ λπ

λ=

−−

= <

=

−⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎜⎝ ⎠Γ Γ

∏ ∏∑IΣΛ Λ

T Rn n≥H=A HH

( )[ ] ( )[ ]( ) exp expH HR T R Tn n n nf tr trπ π− −= − = −H H HH H H

T Rn n< H=A H H


nonsingular. To cover both cases one may define the variables

and write the joint density of the unordered eigenvalues as

Now, noting that (4.162) is a symmetric function of the eigenvalues, the average MIMOchannel capacity can be computed by means of (4.55), i.e.

The only thing left to compute is the pdf of some eigenvalue, e.g. , something thatmay be accomplished by integrating (4.162), for instance, with respect to . Thus

A possible method to perform the integration in (4.164) is as follows. First, notice that thefirst factor can be put in determinant form using the following sequence of steps

which equals the well known Vandermonde determinant. Then, carry out a sequence of rowoperations that may be useful to simplify the integration

(4.161)

(4.162)

(4.163)

(4.164)

(4.165)

(4.166)

min( , )

max( , )

R T

R T

m n n

n n n

=

=

2( 1)

1

1

( )

( ) exp! ( ) ( )

m mn mi j i mm m

i i ji

m m if

m n m

λ λ λπ

λ=

−−

= <

=

−⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎜⎝ ⎠Γ Γ

∏ ∏∑IΣΛ Λ

2

20

min( , ) log 1

log 1 ( )

avg R TT

T

C n n En

m f dn λ

γλ

γλ λ λ

∞

⎡ ⎛ ⎞⎤⎟⎜= +⎢ ⎥⎟⎜ ⎟⎜⎝ ⎠⎢ ⎥⎣ ⎦⎛ ⎞⎟⎜= + ⎟⎜ ⎟⎜⎝ ⎠∫

1λ λ=

2,..., mλ λ

2

1 22

1,...,( ) ( ) ...

!i

m

m mn m

j i i mi j i

Kf e d d

mλ

λλ λ

λ λ λ λ λ λ=− −

< == −∏ ∏∫IΣ

[ ][ ] [ ]

[ ] [ ]

[ ]

1 2 1 2 1 2 1

2 1 2 1 2 11

12 1 1 2 1

2 2 22 1 1 1

1

( ) ( ) ( )( ) ... ( )...( )

1 1( )( ) ... ( )...( )

1 11 1 1

... ( )...( )

m

j i m m m m m m mi j

m m m m mm m

mm m m m

m m m m mm

λ λ λ λ λ λ λ λ λ λ λ λ

λ λ λ λ λ λ λ λλ λ

λ λλ λ λ λ λ λ λ

λ λ λλ λ

− − − −<

− − −−

− −

− − − −

− = − − − − −

= − − − −

= − − =

∏

……

…

1 1 1

1 2 1 22

1 111

21

1 1 ( ) ( )

( ) ( )( )

( ) ( )

( ) ( )i

mm

m mj i

i j

m mm m mm

m

i pip

V K

K p

ϕ λ ϕ λλ λ ϕ λ ϕ λ

λ λ

ϕ λ ϕ λλ λ

σ ϕ λ

<− −

=

= − = =

⎡ ⎤⎢ ⎥= ⎢ ⎥⎣ ⎦

∏

∏∑

… …… …

… …


The eigenvalue density function then becomes

The integral inside (4.162) is recognized as the inner product of polynomial functions with a weighting function of . Thus, if the functions are orthonor-

mal, we get the simple result

where is the Kronecker delta. The eigenvalue pdf now reduces to

where the constant may be found by interpreting (4.169) as a density function

Therefore, the pdf of a single eigenvalue of the Wishart matrix in the case of independentfading is expressed as

for a set of orthonormal polynomial functions. Recall from (4.162) that thesefunctions were obtained by performing row operations on the Vandermonde determi-nant, which means that they are linear combinations of . The latter areclearly independent functions because the Wronskian matrix is nonsingular

and thus they may be orthonormalized as required. The orthonormalization can be ac-complished using the well established Gram-Schmidt procedure. Writing

the orthonormalized functions are given by

(4.167)

(4.168)

(4.169)

(4.170)

(4.171)

(4.172)

(4.173)

(4.174)

2

11 1

21 2

21 1,..., ,

21 2

1 1 1, 2

( ) ( ) ( ) ( ) ( ) ...!

( ) ( ) ( ) ( ) ( ) ( )!

ii i

m

ii i

i

m mn m

p i s i i mi ip s

mn m n m

p s p i s i i ip s i

K Kf p s e d d

m

K Kp s e e d

m

λλ

λ λ

λ λ

λ

λ σ σ ϕ λ ϕ λ λ λ λ

σ σ ϕ λ ϕ λ λ ϕ λ ϕ λ λ λ

=− −

= =

− − − −

=

⎡ ⎤⎢ ⎥= ⎢ ⎥⎣ ⎦

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦

∏ ∏∑

∑

∫

∏ ∫

IΣ

( )ip iϕ λ in m

i e λλ − − ( )ip iϕ λ

, ( ) ( ) ii i i i i i

i

n mp s p i s i i i p se dλ

λϕ ϕ ϕ λ ϕ λ λ λ δ− −= =∫

i ip sδ

21 2 2

1

( 1)!( ) ( )

!

mn m

k

k

K K mf e

mλ

λ λ ϕ λ λ=− −

=

−= ∑IΣ

21 2 2 2

1 2

1

( 1)!( ) ( ) 1

!

mn m

k

k

K K mf d e d K K

mλ

λλ λ

λ λ ϕ λ λ λ=− −

=

−= = =∑∫ ∫IΣ

2

1

1( ) ( )

mn m

k

k

f em

λλ λ ϕ λ λ=

− −

=

= ∑IΣ

( )kϕ λ

2 11, , ,..., mλ λ λ −

2 1

2

3

1

0 1 2 ( 1)0 0 2det 0( 1)( 2)

0 0 0 ( 1)!

m

m

m

m

m m

m

λ λ λλ λ

λ

−

−

−

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜ ≠− − ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ − ⎠

……

11 2( ) 1 ; ( ) ; ... ; ( ) m

mf f fλ λ λ λ λ −= = =

1

1 2 2 1 1 11 2 1

1 2 2 1 1

1

( ) ,( ) ( ) ,

( ) ; ( ) ; ... ; ( )( ) ( ) ,

( ) ,

m

k m i ii

m m

m m i ii

f ff f ff f f

f f

λ ϕ ϕλ λ ϕ ϕ

ϕ λ ϕ λ ϕ λλ λ ϕ ϕ

λ ϕ ϕ

−

=−

=

−−

= = =−

−

∑

∑


Using the notation and observing that , these func-tions can be found recursively as follows:• the first orthonormal function is computed as

• the second orthonormal function is computed as

• the third orthonormal function is computed as

• the same procedure leads to the fourth function

• and henceforth, by induction

which concludes the procedure. Having found the orthonormal polynomials of the pdfexpression in (4.171), the next step is to insert the latter into (4.163), yielding

Now, substituting (4.179) into (4.180) one obtains

(4.175)

(4.176)

(4.177)

(4.178)

(4.179)

(4.180)

(4.181)

( ) ( )!t i n m i= − + , ( )i j t i jλ λ = +

1

1/22

1 1 1 10

( ) 1

( ) , ( ) (0)!n m

f

f f f f e d tλ

λ

λ λ λ λ∞

− −

=

⎛ ⎞⎟⎜= = =⎟⎜ ⎟⎜⎝ ⎠∫1/2

11

1

( ) 1( )

( ) (0)!ff tλ

ϕ λλ

⎛ ⎞⎟⎜= = ⎟⎜ ⎟⎜⎝ ⎠

2 2 10

1/22

2 2 1 10

(1)( ) ; ,

(0) (0)

(1)( ) , (1)

(0)

n m

n m

tf f e d

t t

tf f e d t

t

λ

λ

λλ λ ϕ λ λ

λ ϕ ϕ λ λ λ

∞− −

∞− −

= = =

⎡ ⎤⎛ ⎞⎟⎢ ⎥⎜− = − =⎟⎜ ⎟⎜⎢ ⎥⎝ ⎠⎣ ⎦

∫

∫1/2

2(1)!1

( )(1)! (0)!

tt t

ϕ λ λ⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜= −⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠

22

3 3 10

1/23 2

3 20

(2)( ) ; ,

(0) (0)

(1) 2 (2)1,

(1) (0) (1)

n m

n m

tf f e d

t t

t tf e d

t t t

λ

λ

λλ λ ϕ λ λ

ϕ λ λ λ λ

∞− −

∞− −

= = =

⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜= − =⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠

∫

∫1/22 2

23 3

01

1/2 2

3

(2) (2)( ) , 2 2 (2)

(1) (0)

(2) (2)2( )

(2) 2 (1) 2 (0)

n mi i

i

t tf f e d t

t t

t tt t t

λλ ϕ ϕ λ λ λ λ

λϕ λ λ

∞− −

=

⎡ ⎤⎛ ⎞⎟⎢ ⎥⎜− = − + =⎟⎜ ⎟⎜⎢ ⎥⎝ ⎠⎣ ⎦⎛ ⎞⎛ ⎞ ⎟⎟ ⎜⎜= − + ⎟⎟ ⎜⎜ ⎟⎟⎜ ⎟⎜⎝ ⎠ ⎝ ⎠

∑ ∫

1/2 32

4(3) (3) (3)6

( )(3) 6 2 (2) 2 (1) 6 (0)

t t tt t t t

λϕ λ λ λ

⎛ ⎞⎛ ⎞ ⎟⎟ ⎜⎜= − + − ⎟⎟ ⎜⎜ ⎟⎟⎜ ⎟⎜⎝ ⎠ ⎝ ⎠

1/2

1

0

( )!( ) ( 1) 0,..., 1

( ) ( ) !( )!

k ik i

k

i

t kkk m

t k t i i k iλ

ϕ λ −+

=

⎛ ⎞⎟⎜= − = −⎟⎜ ⎟⎜⎝ ⎠ −∑

12

1 200

( ) log 1m

n mavg k

Tk

C e dn

λ γϕ λ λ λ λ

− ∞− −

+=

⎛ ⎞⎟⎜= + ⎟⎜ ⎟⎜⎝ ⎠∑∫

[ ]1

200 , 0

( 1) 1/ !( )!! ( ) log 1

( ) ( ) !( )!

m k i jn m i j

avgTk i j

j k jC k t k e d

t i t j i k i nλ γ

λ λ λ− ∞+

− + + −

= =

− − ⎛ ⎞⎟⎜= + ⎟⎜ ⎟⎜⎝ ⎠−∑ ∑ ∫


To complete the derivation, the integral

must be evaluated. Integrating by parts

and using the equality in (3.45), leads to

Performing the change of variables

and using the binomial theorem

it follows that

where in the above expression

is the incomplete gamma function. The final expression for the average capacity of theMIMO channel in independent Rayleigh fading conditions is thus given by

where

As a matter of curiosity, consider the case of a MISO system, i.e. let the transmitter haveseveral antennas and the receiver have a single antenna. Then , and the av-

(4.182)

(4.183)

(4.184)

(4.185)

(4.186)

(4.187)

(4.188)

2 20 0

log 1 log ln 1r r

T Te d e e d

n nλ λγ γ

λ λ λ λ λ λ∞ ∞

− −⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜+ = +⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠∫ ∫

0

00

ln 1

/ln 1

1 ( / )

r

T

Tr r

T T

e dn

ne d e d d

n n

λ

λ λ

γλ λ λ

γγλ λ λ λ λ λ

γ λ

∞−

∞ ∞− −

⎛ ⎞⎟⎜ + =⎟⎜ ⎟⎜⎝ ⎠

⎡ ⎛ ⎞ ⎤ ⎡ ⎤⎟⎜ + −⎢ ⎥ ⎢ ⎥⎟⎜ ⎟⎜⎝ ⎠ + ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

∫∫ ∫ ∫

0 00

/!ln 1

! 1 ( / )

rTr p

T Tp

nre d e d

n p nλ λγγ

λ λ λ λ λγ λ

∞ ∞− −

=

⎛ ⎞⎟⎜ + =⎟⎜ ⎟⎜⎝ ⎠ +∑∫ ∫( 1)/Tnλ η γ= −

//

0 10

! 1ln 1 ( 1)

! ( / )

TT

r nr p n

pT Tp

r ee d e d

n p n

γλ η γγ

λ λ λ η ηγ η

∞ ∞− −

=

⎛ ⎞⎟⎜ + = −⎟⎜ ⎟⎜⎝ ⎠ ∑∫ ∫

( ) ( )

0

1 1p

p p ll

l

p

lη η −

=

⎛ ⎞⎟⎜ ⎟− = −⎜ ⎟⎜ ⎟⎜⎝ ⎠∑

( )

( )

/1 /

0 10 0

/

0 0

!ln 1 1

! ( / )

!, /

!

TT

T

r pnp lr l n

pT Tp l

r p p lTn

T

p l

pr ee d e d

ln p n

pr ne l n

lp

γλ η γ

γ

γλ λ λ η η

γ

γγ

∞ ∞−− − −

= =

−

= =

⎛ ⎞⎛ ⎞ ⎟⎜⎟⎜ ⎟+ = −⎜⎟⎜ ⎟⎟⎜ ⎜ ⎟⎝ ⎠ ⎜⎝ ⎠

⎛ ⎞⎛ ⎞⎟⎜ ⎟⎜⎟= − Γ⎜ ⎟⎜⎟ ⎟⎜⎜ ⎟⎝ ⎠⎜⎝ ⎠

∑ ∑∫ ∫

∑ ∑

( ) 1

/, /

T

lT

nl n e dη

γγ η η

∞− −Γ = ∫

(4.189)[ ]

( )1/

0 , 0 0 0

( ) 1/ !( )! 1! ( ) ( 1) , /

ln2 ( ) ( ) !( )! !

Tm k n m i j p p ln

Ti javg T

k i j p l

pt i j j k je nC k t k l n

lt i t j i k i p

γγ

γ

− − + + −+

= = = =

⎛ ⎞+ − ⎛ ⎞⎟⎜ ⎟⎜⎟= − − Γ⎜ ⎟⎜⎟ ⎟⎜⎜ ⎟⎝ ⎠⎜− ⎝ ⎠∑ ∑ ∑ ∑

( , )

( , )

( ) ( )!

R T

R T

m n n

n n n

t i n m i

γ →

=

=

= − +

average SNR at each receiver branchminmax

1m = Tn n=


erage channel capacity reduces to

Equation (4.190) is exactly the capacity expression in (3.65) for the maximal-ratio combin-er in a SIMO system, and here the effective diversity order is replaced by . This means,for instance, that in Rayleigh fading conditions a MISO system with transmitter anten-nas has the same average information-theoretic capacity than the SIMO system with receiver antennas and an MRC combiner at the output. This is clearly an extension of theresult by Alamouti [54], which showed that in flat-fading channels and under a specificcombining technique, a two-branch transmit diversity system is equivalent to a two-branchreceive diversity MRC system, and hence provides the same diversity order. Moreover, ina SIMO system, i.e. and , the average channel capacity is

Still, since , where and is the average SNR at the MRC output,(4.191) is also exactly equal to (3.65), showing that the maximal-ratio combiner is optimalfrom an information-theoretic perspective.

Now, returning to the general capacity expression in (4.189), an effective way to fullyunderstand its behaviour is to produce several plots in diverse array dimensions at boththe transmitter and receiver sides, each one as a function of the average SNR at eachreceiver branch. In this manner, the scaling of the capacity with the key variables at stakein a MIMO system will become more evident, and makes it possible to include relevantcomparison curves to help in the analysis. More specifically, the plots in Figures 4.2 and4.3 show how the capacity scales by fixing the number of receiver antennas and varyingthe number of transmitter antennas and the average SNR. They also incorporate the

Gaussian SISO (Shannon) curve (Equation (4.36) with ), and the limiting capacity

(4.190)

(4.191)

( )1/

0 0

1, /

ln2 !

TT

n p p lnT

avg T

p l

pe nC l n

lp

γγ

γ

− −

= =

⎛ ⎞⎛ ⎞⎟⎜ ⎟⎜⎟= − Γ⎜ ⎟⎜⎟ ⎟⎜⎜ ⎟⎝ ⎠⎜⎝ ⎠∑ ∑

Λ Tn

Tn

Tn

1m = Rn n=

( )11/

0 0

1 1,1/

ln2 !

Tn p p l

avg

p l

peC l

lp

γγ

γ

− −

= =

⎛ ⎞⎛ ⎞⎟⎜ ⎟⎜⎟= − Γ⎜ ⎟⎜⎟ ⎟⎜⎜ ⎟⎝ ⎠⎜⎝ ⎠∑ ∑

/s Rnγ = Γ Rn = Λ sΓ

γ

(a) (b)

FIGURE 4.2 Average information-theoretic capacity of the MIMO channel as a function of the branch SNR at the receiver side, in the special cases of (a) one receiver branch, and (b) two receiver branches.

0 5 10 15 200

1

2

3

4

5

6

7

Average SNR at each receiver branch − (dB)

Cha

nnel

Cap

acity

− C

(bi

ts/s

/Hz)

Gaussian SISO / Rayleigh nR

= 1; nT

Rayleigh nR

= 1; nT = 1

Rayleigh nR

= 1; nT = 2

Rayleigh nR

= 1; nT = 4

Rayleigh nR

= 1; nT = 8

0 5 10 15 200

2

4

6

8

10

12

14


Cha

nnel

Cap

acity

− C

(bi

ts/s

/Hz)

Gaussian SISORayleigh n

R = 2; n

T = 1

Rayleigh nR

= 2; nT = 2

Rayleigh nR

= 2; nT = 4

Rayleigh nR

= 2; nT = 8

Rayleigh nR

= 2; nT

2 1H =


for large (Equation (4.40)). From these plots one may infer that:• when the receiver side has a single antenna, the achievable capacity with a MISO sys-

tem is always lower than the Gaussian SISO capacity, but is higher than the RayleighSISO when ;

• in a Rayleigh fading environment, the average MISO capacity is equal to the capacityof the SIMO system with a maximal-ratio combiner, i.e. Figure 4.2(a) is equivalent toFigure 3.5;

• for a fixed , the capacity is upper-bounded by the limit (4.40) as ;• the higher is the number of receiver antennas, the better the capacity scales with

increasing relative to ; for example, at a branch SNR of and with, the capacity can scale up to approximately bits/cycle, for it can

scale up to bits/cycle, for it can scale up to bits/cycle, and for itcan scale a maximum of bits/cycle; however

• the higher is , the higher must be to get within some percentage of the limitingcapacity; this fact is shown in Table 4.1, which gives the minimum to reach 90%of the limiting capacity as ;

A different situation is depicted in the plots of Figures 4.4 and 4.5, which show how thecapacity scales by fixing the number of transmitter antennas and varying the number ofreceiver antennas. From these cases one may conclude that:

(a) (b)

FIGURE 4.3 Average information-theoretic capacity of the MIMO channel as a function of the branch SNR at the receiver side, in the special cases of (a) four receiver branches, and (b) eight receiver branches.

0 5 10 15 200

5

10

15

20

25

30


Cha

nnel

Cap

acity

− C

(bi

ts/s

/Hz)


R = 4; n

T = 1

Rayleigh nR

= 4; nT = 2

Rayleigh nR

= 4; nT = 4

Rayleigh nR

= 4; nT = 8

Rayleigh nR

= 4; nT

0 5 10 15 200

10

20

30

40

50

60


Cha

nnel

Cap

acity

− C

(bi

ts/s

/Hz)


R = 8; n

T = 1

Rayleigh nR

= 8; nT = 2

Rayleigh nR

= 8; nT = 4

Rayleigh nR

= 8; nT = 8

Rayleigh nR

= 8; nT

TABLE 4.1 The number of transmitter antennas required to attain (4.40).

Tn

1Tn >

Rn limC Tn → ∞

Tn 1Tn = 20 dB

1Rn = 0.8 2Rn =

6 4Rn = 18 8Rn =

44

Rn Tn

Tn

Tn → ∞

lim0.9avgC C≥

1 2 4 8

4 2 4 9 17

8 2 4 9 17

12 2 4 8 15

16 2 4 7 13

20 2 3 6 12

SNR (dB)nR


• the capacity of the SIMO channel (Figure 4.4(a)) is higher than the Gaussian SISOcapacity, provided that , contrary to the MISO channel (Figure 4.2(a)) where itis always lower; moreover

• it may seem that Figure 4.4(a) asserts that the maximal-ratio combining technique isdeficient; in fact, the capacity curves of the SIMO system with MRC (Figure 3.5) arealways lower that the curve of the Gaussian SISO capacity, whereas in Figure 4.4(a)they are higher when ; this is explained by the different abscissas in the plots,

in Figure 3.5 and in Figure 4.4(a); as explained earlier, the MRC isoptimal;

• the higher is the number of receiver antennas, the better the capacity scales withincreasing average SNR;

• at least in theory, the capacity is not upper-bounded with increasing ; among otherthings, this means that it is possible to attain the limiting capacity (4.40) as

(a) (b)

FIGURE 4.4 Average information-theoretic capacity of the MIMO channel as a function of the branch SNR at the receiver side, in the special cases of (a) one transmitter branch, and (b) two transmitter branches.

(a) (b)

FIGURE 4.5 Average information-theoretic capacity of the MIMO channel as a function of the branch SNR at the receiver side, in the special cases of (a) four transmitter branches, and (b) eight transmitter branches.

0 5 10 15 200

1

2

3

4

5

6

7

8

9

10


Cha

nnel

Cap

acity

− C

(bi

ts/s

/Hz)


T = 1; n

R = 1

Rayleigh nT = 1; n

R = 2

Rayleigh nT = 1; n

R = 4

Rayleigh nT = 1; n

R = 8

0 5 10 15 200

2

4

6

8

10

12

14

16

18


Cha

nnel

Cap

acity

− C

(bi

ts/s

/Hz)


T = 2; n

R = 1

Rayleigh nT = 2; n

R = 2

Rayleigh nT = 2; n

R = 4

Rayleigh nT = 2; n

R = 8

0 5 10 15 200

5

10

15

20

25

30


Cha

nnel

Cap

acity

− C

(bi

ts/s

/Hz)


T = 4; n

R = 1

Rayleigh nT = 4; n

R = 2

Rayleigh nT = 4; n

R = 4

Rayleigh nT = 4; n

R = 8

0 5 10 15 200

5

10

15

20

25

30

35

40

45


Cha

nnel

Cap

acity

− C

(bi

ts/s

/Hz)


T = 8; n

R = 1

Rayleigh nT = 8; n

R = 2

Rayleigh nT = 8; n

R = 4

Rayleigh nT = 8; n

R = 8

1Rn >

1Rn >

sΓ /s Rnγ = Γ

Rn

limC


, and hence that in certain situations it may be more convenient to increase instead of to achieve a greater capacity; as shown in Table 4.2, increasing the

number of receiver antennas may convey theoretical capacities that cannot beobtained by increasing solely;

Practical example. Consider the particular case of the downlink channel in theWCDMA-FDD air interface. The multiuser separation is achieved by spreading the userdata with orthogonal channelization codes of length 4-512 chips, and a chip rate of

. Each chip is shaped by a root-raised cosine filter with roll-off factor, and hence the total occupied bandwidth is . QPSK

modulation is used for the downlink. With a single code, the maximum possible bit rate(including control symbols) is , corresponding to a spectral efficiencyper user of . At the receiver, the multiuser signal is despreadusing the original codes, and the usual requirement for the signal to noise-plus-interference ratioafter despreading is in the interval , which is satisfactory for achiev-ing a given error probability. The reason why the symbol rate may not be increased whilemaintaining the same bandwidth, is because the ratio will also decrease if the trans-mitter power is not increased (e.g. duplicating the symbol rate reduces by 3 dB), andthis will have the effect of increasing the average error probability. Increasing the trans-mission power is not a option in a multiuser system because the interference power mustbe kept to a minimum. Even though it seems we have reached a deadlock, theoretical re-sults show, however, that higher spectral efficiencies are certainly possible. For instance,at an average SNR of 5 dB, the capacities1 that can be obtained with asymptotic MIMOarrangements ( ) is summarized in Table 4.3. With a simple SISO system in aRayleigh environment it is possible to obtain, on average, a spectral efficiency of

without receiver errors, which is more than the WCDMA spectral efficien-cy per user. Increasing the number of antennas dramatically increases the achievable ca-

TABLE 4.2 MIMO capacity comparison for a few particular array sizes, at .

1. Computational note. Numerical evaluation of the capacity expression in (4.189) may lead to incorrectresults when the number of transmitter and/or receiver antennas is large. This happens because when asubstantial number of additions and products is performed using finite-precision arithmetic, the finalresult may be inaccurate. Usually, software oriented to efficient numerical computation (such as Mat-lab), represents numbers using double-precision (64 bits) floating point binary values, meaning thatlarge factorials and numerical integrals (for instance) will have limited precision. It is clearly the case ofEquation (4.189). To achieve a good capacity precision, when the sum of with is larger than 18,Computer Algebra Software (CAS) should be used to perform the capacity calculation. For example,Mathematica or Maple are powerful CAS’s that can be forced to use arbitrary precision arithmetic whenperforming calculations, producing results to the exact precision requested.

Tn → ∞

Rn Tn

Tn

20 dBγ =

Array size C lim bits/s/Hz( ) Array size Cavg bits/s/Hz( )

n nR T= → ∞1 ; 6.7 n nR T= =2 1 ; 7.3

n nR T= → ∞2 ; 13.3 n nR T= =4 2 ; 14.5

n nR T= → ∞4 ; 26.6 n nR T= =7 4 ; 27.8

n nR T= → ∞8 ; 53.3 n nR T= =13 8 ; 54.3

3.84 Mchips/scR =

0.22α = (1 ) 5 MHzcB R α= + ≈

2. / 4 1920 kb/scR =

1.92/5 0.38 bits/s/Hzρ = =

00 / 5 (dB)bE≤ ≤I

0/bE I

0/bE I

Rn Tn

R Tn n=

1.716 bits/s/Hz


pacity. Notice that duplicating the number of transmitter and receiver antennas almostduplicates the average capacity and, as the number of antennas increases, the dupli-cation becomes more accurate. This fact is a manifestation of the linearity of the capacitywith increasing array sizes (i.e. when is large) and it is justified by (4.53). Thisis in fact what happens with (in the fourth column of the table), the limit of the ca-pacity for very large . Table 4.3 also shows the capacity per sub-channel (or eigenmode)of the MIMO system, which is essentially , revealing that it decreases mo-notonically and is always less than the Gaussian and Rayleigh SISO capacities. For further

reference, the achievable capacities as a function of the array sizes at the transmitter andreceiver sides are depicted in Figures 4.6 and 4.7. The plots in Figure 4.7 are bidimensionalreplicas of Figure 4.6, and can be used to discover the array sizes that achieve a given in-formation-theoretic capacity. All these plots assume independent fading, and thus repre-

TABLE 4.3 Average capacities of asymptotic MIMO systems, for an average SNR at each receiver branch of 5 dB.

MIMO type avg bits/s/Hz( ) Cavg/sub-channel bits/s/Hz( ) C lim/sub-channel

Gaussian SISO 2.057 2.057 - -

n nR T= =1 1 ; 1.716 1.716 2.057 2.057

n nR T= =2 2 ; 3.306 1.653 4.115 2.057

n nR T= =4 4 ; 6.553 1.638 8.229 2.057

n nR T= =8 8 ; 13.080 1.635 16.459 2.057

n nR T= =16 16 ; 26.146 1.6341 32.918 2.057

n nR T= =32 32 ; 52.286 1.6339 65.836 2.057

C C lim

FIGURE 4.6 Average MIMO capacity as a function of the number of transmitter and receiver antennas, for an average SNR of 5 dB at each receiver branch.

avgC

max( , )R Tn n

limC

Tn

/min( , )avg R TC n n

24

68

1012

1416

24

68

1012

1416

0

5

10

15

20

25

30

Number of transmitter antennas − nT

Number of receiver antennas − nR

Cha

nnel

Cap

acity

− C

avg (

bits

/s/H

z)


sent a best-case scenario.The bottom line of this discussion is that using appropriate signal processing tech-

niques together with multi-antenna systems that circumvent, in the better possible way, thespatial correlation between the antennas (among other issues), it may be possible to equipmultiuser wireless mobile networks with remarkable data transfer capabilities still un-known to the present time.

4.5.7 CORRELATED FADING

Estimation of the spectral efficiency limit (or capacity) of multiple-input multiple-outputwireless channels under the most general conditions is a current issue within the researchcommunity. One of these conditions is the existence of signal correlation between the an-tennas of a moving receiver. The concept of MIMO channels has emerged from the prin-ciples of diversity and independence between the antennas of transmitters and receivers.However, practical evidence has shown that strict independence is difficult to attain, es-pecially when either or both the transmitters and receivers are located in empty, isolatedareas [55], [56]. It is expected that the antennas be spaced several wavelengths apart so thatthe correlation becomes negligible, but often these spacings are not feasible because ofsize restrictions in communication units. These restrictions will inevitably lead to an ad-justment of the enormous capacities promised by [10], [11]. This adjustment must be the-oretically quantified, and should be as generic as possible (i.e. irrespective of thecorrelation scenario).

The author has found several approaches to this problem in the literature, roughly di-vided into two groups: asymptotic (see, e.g., [57]) and exact (see, e.g., [58]). The secondgroup shall be tackled. Some papers have appeared that successfully present some inter-esting results, but deal with the problem in a limited manner. For instance, [59] addressesthe correlated case for large matrix sizes, solely. Another paper worth mentioning is [60].It discusses power allocation when the correlation is transmitter-sided and partial channelstate information (CSI) is known at the transmitter. One shall not dwell on this issue. Oth-

FIGURE 4.7 Capacity as a function of the number of transmitter and receiver antennas, for an average SNR at each receiver branch of 5 dB. The left figure retrieves some isolines from Figure 4.6.


Num

ber

of r

ecei

ver

ante

nnas

− n

R

2 4 6 8 10 12 14 16

2

4

6

8

10

12

14

16

Cavg = 20

Cavg = 4

Cavg = 6

Cavg = 8

Cavg = 10

Cavg = 12

Cavg = 14

Cavg = 16

Cavg = 18

Cavg = 2

Cavg = 22

Cavg = 24

2 4 6 8 10 12 14 160

2

4

6

8

10

12

14

16

18

20

22

24

26


Cha

nnel

Cap

acity

− C

avg (

bits

/s/H

z)

nT = (1,2,..,16)


er approaches will be referred accordingly throughout the text.Evaluating the MIMO capacity when the signals at the receiver antennas do not fade

independently is a more complicated task. In this situation, the correlation matrix is notthe identity matrix as in the case of independent fading, and therefore the integral in thejoint eigenvalue density is not readily obtained. The generic pdf is given by (4.157) as

The integral to be evaluated is the famous Itzykson-Zuber integral [61]. In what followsit will be shown that it is solvable by means of a powerful method that, at the same time,is beautiful and clear. It is based on the theory of group representations and character ex-pansions.

Little was said so far about , except that it was given by

where the ’s are the columns of the channel matrix . From (4.193) one deduces thefollowing about :• it is self-adjoint or, in other words, it is an hermitian matrix;• it has an eigenvalue decomposition , where is an unitary matrix

and with real eigenvalues;• since it is a covariance matrix, one may expect that it can also be factorized as

, where is a scaled matrix of multivariate samples (recall (4.143)); hence,all the eigenvalues of are nonnegative because

• since it must be assumed from (4.192) that is nonsingular, it must also follow that

which implies that

and therefore, together with (4.194), one concludes that the eigenvalues of must bereal and positive;

Using the above conclusions, it is possible to rewrite the eigenvalue pdf in (4.192) in sim-plified form as

where the introduced constant is given by

(4.192)

(4.193)

(4.194)

(4.195)

(4.196)

(4.197)

(4.198)

Σ

( )( )[ ]( )

( ) 2

1 1 o

( )

( )

( ) exp2 ! det ( )

R RT R

H

TRR R

n nn n

j iii i j

nnn U nR T

f tr dn n

λ λ λ

π∧

−

= < −

−= −

Γ


Σ

Σ

[ ]Hj jE= h hΣ

jh H

Σ

H= E EΣ ϒ E R Rn n×

1( ,..., )Rndiag υ υ=ϒ

H= B BΣ B

Σ

2

2 0H H

H

i i ii

i i iυ = = ≥

x B Bx Bxx x x

Σ

= ⇒ =Ax 0 x 0

for υ= ≠ ≠Ax x 0 x 0

Σ

( )[ ]( )1 o,

( )( ) ( ) exp H H

R T

R

n nU n

f G tr d ∧−= −∫ E Q Q E QΛ Λ Λ,ϒ ϒ Λ

( )

2

1,

1

( )

( )2 ! ( )

R RT R

R T RTR

R

n nn n

j iii i j

n n nnn

nR T ii

Gn n

λ λ λ

π υ

−

= <

=

−=

Γ

∏ ∏

∏Λ,ϒ


Moreover, putting , where is an orthogonal symmetric permutation matrix,writing , and noting that , leads to

which shows that the eigenvalue pdf is a symmetric function of the eigenvalues of the co-variance matrix and the Wishart matrix .

To make the integration in (4.192) as easy as possible, one may substitute the inte-grand function by its series expansion, yielding

where the integrand is now a polynomial function in the elements of matrices and .Now, since the trace of a matrix equals the sum of its eigenvalues, we may notice that

where are the eigenvalues of the matrix . Expanding (4.201)results in a finite linear combination of monomials of degree , i.e.

where the sum is over all -tuples that sum to . As varies, the monomi-als in (4.202) generate the space of all polynomials of degree , which are homogeneousand symmetric in the eigenvalues of . Now, in the same way as any vector space can bedecomposed into a set of irreducible invariant subspaces under a given linear operator, sodoes has a decomposition into a set of irreducible subspaces , invariant under the uni-tary group. More formally, this means that

where the symbol denotes the direct sum of the subspaces over all distinct partitions, , of the integer . Only the distinct partitions of

need to be considered, given that the monomials in (4.202) can be grouped into equiva-lence classes in relation to the space they generate. Each subspace can be generated bya homogeneous symmetric polynomial (commonly referred in the literature as thezonal polynomial) in the eigenvalues of the matricial argument . Hence, (4.202) can be al-ternatively expressed as

The reason for expressing (4.202) as (4.204) is that a fundamental property of the zonalpolynomials is given by the integral, proved in [43]:

(4.199)

(4.200)

(4.201)

(4.202)

(4.203)

(4.204)

(4.205)

o H= P PΛ Λ PH=U E QP ( ) ( ) ( )2 2det (det )R Rn nd d d∧ ∧ ∧= =U E P Q Q

( )[ ]( )1,

( )( ) ( ) exp H

R T

R

n nU n

f G tr d ∧−= −∫ U U UΛ Λ Λ,ϒ ϒ Λ

Σ A

( )[ ] ( )1,

( )0

1( ) ( )

!H

R T

R

kn n

U nk

f G tr dk

∧

∞−

=

= −∑ ∫ U U UΛ Λ Λ,ϒ ϒ Λ

Λ ϒ

( )[ ] ( )[ ] ( )11 2 ...H

R

k kkntr tr x x x−− = = + + +U U Xϒ Λ

1 2, ,...,Rnx x x 1 H−= −X U Uϒ Λ

k

( )[ ] 1 21 21

R

R

n

n R

kk k kk k ntr d x x x⋅⋅⋅= ⋅ ⋅ ⋅∑X

Rn ( )1 2, ,...,Rnk k k k X

kV k

X

kV Vκ

kk

V Vκκ

= ⊕

kκ⊕

1 2( , ,..., )Rnk k kκ = 1 20 ...

Rnk k k≤ ≤ ≤ ≤ k k

Vκ

( )Cκ XX

( )[ ] ( )k

k

tr Cκκ

= ∑X X

( )( ) [ ] ( ) ( )

( )( )Volume ( )H

nU n

C CC d U n

Cκ κ

κκ

∧ =∫ A BAUBU U

I


and thus the substitution of (4.204) into (4.200) gives

where is the simplest case of the complex hypergeometric function of two matrix argu-ments, defined generically as

Even though Equation (4.206) is an analytical expression for the joint pdf of the eigenval-ues, it is expressed as a function of an infinite series which has poor convergence proper-ties. In fact, the hypergeometric functions of matrix argument are hard to approximate,and require the evaluation of a very high number of its terms ([62], [63] and [64]. Discov-ering an alternative expression for is therefore of practical importance. Aftersome study, one realizes that the answer may be found in the context of the group theory ofrepresentations and characters. For a detailed discussion of these topics and much more be-sides, the author vividly recommends [65], [66], [67], and [68]. Some important resultsfrom this theory are now outlined:

1. denotes the group of invertible linear operators (e.g. invertible matrices) thatact on the vector space ;

2. if is a basis for , , and , , then we may alsowrite , where and denote the coordinate vectors with respect to ,and is the coordinate matrix of with respectto ; furthermore we have the composition of linear operators

3. a subspace of is an invariant subspace under whenever ; in this case can be considered as a linear operator on , and is denoted ; the trivial subspaces

of are the subspace and itself; a subspace with no non-trivial invariant subspacesis called irreducible;

4. let be an irreducible subspace of , and let ; if is a basis for, then is a basis for ; in this case, the coordinate matrix of a lin-

ear operator with respect to can be expressed equivalently as a block diagonal ma-trix

(4.206)

(4.207)

(4.208)

(4.209)

[ ]( ) ( )

( )

[ ] 0 0

1

,

0

1,

( ) Volume ( ) ( )!

Volume ( ) ( ) ( , )

R T

R

R T

n nnk

n n

C Cf U n G

C k

U n G F

κ κ

κκ

∞ −

=−

−=

= −

∑∑ IΛϒ Λ

Λ Λ,ϒ

Λ,ϒ ϒ Λ

0 0( , )F A B

( ) ( )

( )1

1 110

1

( ) ( )( ,..., ; ,..., ; , )

( ) ( ) !

( ) ( 1 )/ ( 1)

pp q p q

q nk kn

i

i

a a C CF a a b b

b b C k

a a i k a i

κ κ κ κ

κ κ κκ

κ

∞

=

=

⋅ ⋅ ⋅=

⋅ ⋅ ⋅

= Γ − + + Γ − +

∑∑

∏

X YX Y

I

0 0( , )F A B

GL( )V T

V

1 2 , ,..., n= v v vB V ∈x V ( )=y T x GL( )∈T V

[ ] [ ] [ ]=y T xB B B [ ]x B [ ]y B B

[ ] [ ] [ ]( )1 2[ ] ( ) | ( ) | | ( )n= ⋅ ⋅ ⋅T T v T v T vB B B B T

B

2 1 2 1[ ] [ ] [ ]=T T T TB B B

U V T ( ) ⊆T U U

T U /T UV 0 V V

iV V 1 2 r= ⊕ ⊕ ⋅ ⋅ ⋅ ⊕V V V V iB

iV '1 2 r= ∪ ∪ ⋅ ⋅ ⋅ ∪B B B B V

T 'B

[ ]

[ ]

1 1

'

/

1

/

[ ] [ ]

r r

−

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟= = ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠

T 0

T E T E

0 T

…V B

BB

V B


where is the change of basis matrix; in turn, this means that

i.e. the linear operator can be decomposed into a set of irreducible linear operators ;

5. the homomorphism is a linear action of a group on the vector space; if , then is a linear operator on the vector space ; is called a represen-

tation of ;

6. from (4.) it follows that can be decomposed into a set of linear operators ;these linear operators are also representations because

and thus the homomorphisms are the irreducible subrepre-sentations of associated with the irreducible invariant subspaces of ; in other words,there is an equivalent representation such that

7. now, for simplicity, write and call it the i-th irreducible matrix repre-sentation of the group element ; then is equivalent to an irreducible unitarymatrix representation, because writing

and choosing where is a unitary matrix and is a diagonal matrix with realand positive entries, we have

- therefore, one simply has to define the eigenvalue decomposition

which substituted into (4.214) yields

proving that is unitary; from this point on we shall assume that the irreducible matrixrepresentations are all unitary;

8. let and be two inequivalent, unitary, and irreducible matrix representations

(4.210)

(4.211)

(4.212)

(4.213)

(4.214)

(4.215)

(4.216)

E

[ ]' ' ' /[ ] [ ] [ ] [ ] [ ]i i ii

= → =y T x y T xB B B B V BB

T / iT V

: GL( )Gρ → V G

V g G∈ ( ) ggρ = T V ρ

G

( )gρ /( )i

gρ V

[ ] [ ] [ ]/ / //( ) ( ) ( ) ( ) ( )i i ii

gh g h g hρ ρ ρ ρ ρ= =V V VV

/ /: GL( ) : ( )i iiG g gρ ρ→ →V VV

ρ V

[ ]

[ ]

1 1

'

/

1

/

( )

[ ( )] [ ( )]

( )r r

g

g g

g

ρ

ρ ρ

ρ

−

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟= = ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠

0

E E

0

…V B

BB

V B

[ ]/( ) ( )i i

i g gρ= V Bπ

g G∈ ( )i gπ

[ ] [ ] [ ]1 1 1( ) ( ) ( ) ( ) ( ) ( )HH HH

i i i i i ig g g g g g− − −′ ′ ′= → =E E E EE Eπ π π π π π

1/2=E UD U D

[ ] [ ]( )1/2 1/ 2( ) ( ) ( ) ( )H HH H

i i i ig g g g− −′ ′ = D U UDU UDπ π π π

[ ]( ) ( ) HHi i

Gx x dx= ∫UDU π π

[ ] [ ]

[ ]

[ ]

1/2 1/2

1/2 1/ 2

1/2 1/2 1/2 1/2

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

( ) ( )

H HH

HH

HH

i i i i i iG

i iG

i iG

g g g x g x dx

gx gx dx

x x dx

− −

− −

− − − −

⎛ ⎞⎟⎜′ ′ = ⎟⎜ ⎟⎜⎝ ⎠⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠⎛ ⎞⎟⎜= = =⎟⎜ ⎟⎜⎝ ⎠

∫∫∫

D U UD

D U UD

D U UD D DD I

π π π π π π

π π

π π

( )i g′π

( )i gπ ( )j gπ


of a group ; also, let be a compact topological group (so that it is measurable),and and be generic matrices such that

- multiply (4.217) on the left by and on the right by , yielding

- thus one must have , and choosing

- to find we put and , and sum over , which gives

where is the size of , i.e. the dimension of the irreducible subspace ; now use thesimple identity

where is the volume of ; comparison with (4.220) leads to , and final in-sertion into (4.219) reveals the so called Schur orthogonality relations

- these relations show that the entries of irreducible representations of a compacttopological group are not only independent functions in the space of all functions

, but are also orthogonal functions;

9. the character of a representation is defined as the trace of thecoordinate matrix associated with , i.e.

where and is a basis for the vector space ; is the (irreducible) character as-sociated with the irreducible representation ; the character of the identity element givesthe dimension of :

(4.217)

(4.218)

(4.219)

(4.220)

(4.221)

(4.222)

(4.223)

(4.224)

G U G⊆

A B

[ ]( ) ( )H

i jU

u u du= ∫B Aπ π

( )i xπ [ ]( ) H

i xπ

[ ] [ ] [ ]

[ ] [ ]

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

H H H

H H

i j i i j jU

i j i jU U

x x x u u du x

xu xu du u u du

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠

= = =

∫∫ ∫

B A

A A B

π π π π π π

π π π π

ijλδ=B I [ ]lnδ=A

[ ] [ ] [ ]( ) ( )ij km i jkm kl mnU

u u duλδ δ ∗= = ∫B π π

λ i j= k m= k

[ ] [ ]1

( ) ( )id

i i ikl knUk

u u du dλ∗

=

=∑∫ π π

id ( )i gπ iV

[ ] [ ] [ ]1

( ) ( ) ( ) ( )i

H

d

i i U i i U lnkl knU Uk

u u du V u u du V δ∗

=

= → =∑∫ ∫Iπ π π π

VU U /U ln iV dλ δ=

[ ] [ ] [ ] [ ]*V

1 1( ) , ( ) ( ) ( )i j i j ij km lnkl klmn mn

U iUu u u u du

dδ δ δ= =∫π π π π

[ ]( )i kluπ

:f U →

: Gχ → : GL( )Gρ → V

( )gρ

( ) /

1 1

( ) ([ ( )] ) ( ( )) ( ) ( )i

r r

i

i i

g tr g tr g tr g gχ ρ χ= =

= = = =∑ ∑B Vπ π

g G∈ B V / iχ V

/ iρ V

iV

[ ]( ) ( )/ /(1) (1)i i ii

i dd tr trχ ρ= = = IV V B


10. since the irreducible characters are given by

they are clearly orthonormal functions, because from (4.218) we have

11. another fundamental identity relating characters can be obtained as follows: - let be a group and ; also, let be a compact topological group and

let ; now suppose we want to compute the integral

- substitution of (4.225) yields

- if the integrand is expanded as

then the integral can be rewritten as

- now, the integration can be performed directly using the orthogonality relations in(4.218), yielding a nonzero result only for and , i.e.

where is the volume of the group ; this is a very important result that will be usefullater;

12. a class function on is a function such that whenever ,for ; class functions are constant on the conjugacy classes ;clearly, the irreducible characters of a representation are (linearly independent) class

(4.225)

(4.226)

(4.227)

(4.228)

(4.229)

(4.230)

(4.231)

[ ]/1

( ) ( )i

di

i kkk

g gχ=

= ∑V π

[ ] [ ]

[ ] [ ]

/ / / /1 1

1 1 1 1

V V

V

1 1( ), ( ) ( ) ( ) ( ) ( )

1 1( ) ( )

ji

i j i j

j ji i

dd

i jkk ppG GG G k p

d dd d

i j ij kp ijkk ppG iGk p k p

g g g g dg g g dg

g g dgd

χ χ χ χ

δ δ δ

∗ ∗

= =

∗

= = = =

= =

= = =

∑ ∑

∑∑ ∑∑

∫ ∫

∫

V V V V π π

π π

G ,a bg g G∈ U G⊆

u U∈

1/ ( )

i a bU

g ug u duχ −∫ V

[ ]

[ ]

1 1/

1

1

( ) ( )

( ) ( ) ( ) ( )

i

H

di

a b i a b kkU Uk

di

i a i i b i kkUk

g ug u du g ug u du

g u g u du

χ − −

=

=

=

=

∑

∑

∫ ∫

∫

V π

π π π π

[ ] [ ] [ ] [ ] [ ]1 1 1

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )H

d d di i i

i a i i b i i a i i b ikk kp pq qs ksp q s

g u g u g u g u∗

= = == ∑∑∑π π π π π π π π

[ ] [ ] [ ] [ ]1/

1 1 1 1

( ) ( ) ( ) ( ) ( )i

d d d di i i i

a b i a i b i ikp qs pq ksU Uk p q s

g ug u du g g u u duχ − ∗

= = = == ∑∑∑∑∫ ∫V π π π π

p k= q s=

[ ] [ ]1/

1 1

/ /

/

V

V V

1( ) ( ) ( )

( ) ( )1( ( )) ( ( ))

(1)

i

i i

i

d di i

a b U i a i bkk qqiU k q

a bU i a i b U

i

g ug u du g gd

g gtr g tr g

d

χ

χ χχ

−

= ==

= =

∑ ∑∫ V

V V

V

π π

π π

VU U

G : Gϕ → ( ) ( )g hϕ ϕ= 1h ugu−=

, ,g h u G∈ 1 : Gg ugu g G−= ∈

ρ


functions since

13. furthermore, it is not difficult to prove that the number of irreducible characters isequal to the number of conjugacy classes, and hence they form an orthonormal basisfor the space of all class functions on , i.e.

where putting , and using (4.226), gives the completeness relation

Having described these fundamental aspects of the theory of representations andcharacters, it is now easy to find their application in the problem at hand. Recall that themain objective is to find a mathematically tractable expression for the function

, . Letting denote the function space generated by thefunction , we may say that a transformation

where , acts on by means of the function . Hence it is reasonable to spec-ify a mapping

which defines a representation of in function space, but now by means of the linearoperator which will act on . As a result, we may:

• decompose into a set of irreducible invariant subspaces under ;• for each , discover the irreducible representation associated with ;

• find a function basis for each such that , andconsequently

• the matrix representation is called functional since its entries are

coordinates with respect to of functions in the elements of ;

Considering, for instance, the case of a similarity transformation ,

(4.232)

(4.233)

(4.234)

(4.235)

(4.236)

1/ / / / /

1/ / /

/ /

( ) ([ ( )] ) ([ ( ) ( ) ( )] )

([ ( )] [ ( )] [ ( )] )

([ ( )] ) ( )

i i i i i i i

i i i i i i

i i i

h tr h tr u g u

tr u g u

tr g g

χ ρ ρ ρ ρ

ρ ρ ρ

ρ χ

−

−

= =

=

= =

V V B V V V B

V B V B V B

V B V

r

G

/ /

1

( ) ( ), ( ), ( )i i

r

i i

i

g c g c g gϕ χ ϕ χ=

= =∑ V V

( ) ( , )g g hϕ δ=

/ /

1

( ) ( ) V ( , )i i

r

G

i

g h g hχ χ δ∗

=

=∑ V V

( )[ ]( ) expf tr=X X GL( , )Rn∈X V

f

( )1

( )

( ) ( )f f −

→

→

X P X

X P X

GL( , )Rn∈P V f

: GL( , ) GL( ) : ( )Rnρ ρ→ P PV

P

( )ρ P V

V κV ( )ρ P

κ / ( )κ

ρ PV κV

1 2 , ,..., df f fκκ =B κV 1 2= ∪ ∪ ⋅ ⋅ ⋅B B B

(4.237)( )[ ] [ ] [ ] [ ]

[ ]

[ ][ ]

[ ]

1 1 1

2 22

/

1/

( ) ( )

( ) ( )( ( )) ( ) ( ) ( ) ( )

f

f f f f

ρ

ρ ρ ρ−

⎛ ⎞⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟= = = ⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎝ ⎠⎝ ⎠

P 0 X

P X P X P X 0 P X

…

…

V B B

VB B B BB B

[ ]/( ) ( )κ κ

κ ρ=P PV Bπ

B P

1( ) −=P X PXP


, it induces a transformation in function space given by

showing that is a class function on , and hence from (4.233) it follows that

The summation is, as before, over all irreducible representations indexed by . The func-tions are the characters of the irreducible representations of the linear group of invertible matrices of complex entries. Thesecan be found from [69] to be given by the Schur polynomials

where is the vector of eigenvalues of the matrix , isthe Vandermonde determinant1, and is the generalized Vandermonde deter-minant. In fact, it is well known (see e.g. [68], [70]) that the Schur polynomials constitute abasis for the vector space of homogeneous symmetric polynomials, which is exactly whatis needed for each term in the power series expansion

At this point, we need to deduce the expansion of into charactersas given by (4.239), this way proving the decomposition of the function space into irre-ducible invariants. First, since the trace of the matrix equals the sum of its eigenvalues

, the function can be rewritten as

Multiplying on both sides by the Vandermonde determinant in the eigenvalues , yields

Moreover, expanding the exponential inside the determinant into series of powers gives

To simplify (4.239) the determinant must be expanded using the standard sum over allpermutations p of

(4.238)

(4.239)

(4.240)

1. The notation refers to the determinant of the square matrix whose entry in the i-th row andj-th column is .

(4.241)

(4.242)

(4.243)

(4.244)

(4.245)

GL( , )Rn∈P

( ) ( )[ ] ( )[ ]1 1( ) exp exp ( )f tr tr f− −= = =P X P XP X X

f GL( , )Rn

( )[ ] /( ) exp ( )f tr cκκ

κ

χ= = ∑X X XV

κ

/ /( ) ([ ( )] )trκ κ κ

χ ρ=X XV V B / ( )κ

ρ XV

GL( , )Rn R Rn n× X

/

det( )( ) ( )

( ) det

R

R

jk n ji

n ji

xs

xκ

κκχ

+ −

−

⎡ ⎤Δ ⎢ ⎥⎣ ⎦= = =⎡ ⎤Δ ⎣ ⎦

xX x

xV

( )1 2, ,...,Rnx x x=x X ( ) det Rn j

ix −⎡ ⎤Δ = ⎣ ⎦x

( ) det Rjk n jixκ

+ −⎡ ⎤Δ = ⎢ ⎥⎣ ⎦x

[ ]det ijx X

ijx

( )[ ] ( )[ ]0

1exp

!k

k

tr trk

∞

=

= ∑X X

( )[ ]( ) expf tr=X X

V

X

( )1 2, ,...,Rnx x x

( )[ ] 1 2( ) exp Rnxx xf tr e e e= =X X

ix

( )[ ] 1 2det exp det detR R RRn ixn j n j n jx x xi i ix tr x e e e x e− − −⎡ ⎤ ⎡ ⎤ ⎡ ⎤= =⎣ ⎦ ⎣ ⎦ ⎣ ⎦X

( )[ ]0

1det exp det

!R Rn j k n j

i ik

x tr xk

∞− + −

=

⎡ ⎤⎡ ⎤ ⎢ ⎥=⎣ ⎦ ⎢ ⎥⎣ ⎦

∑X

( )1,2,..., Rn

[ ]1 1

det ( ) ( )R R

i j

n n

ij ip p jp pi j

a p a p a= =

= =∑ ∑∏ ∏ε ε


where for briefness of notation . This yields

Interchanging the order of summation in (4.246) we obtain

Now it is helpful to make the temporary change of variables

which produces

We notice from (4.246) that whenever two columns of the matrix are equal, the de-terminant will be zero, and hence the outer sum may be restricted to distinct ’s.Moreover, instead of considering all sequences of distinct ’s, one may consider solely theordered sequences, and then sum over all permutations of each sequence, i.e.

The factor must be introduced because the sign of the determinant dependsupon the order of the ’s. The inner sum is now recognized as a determinant, and thus wewrite (4.250) as

Finally, the change of variables in (4.250) can be reverted, which clearly leads to weaklydecreasing sequences indexing the sum, i.e.

Division by the Vandermonde determinant and substitution of (4.240) givesthe final result for the character expansion

(4.246)

(4.247)

(4.248)

(4.249)

(4.250)

(4.251)

(4.252)

(4.253)

( ) sign( )p p=ε

( )[ ]

1

01

0 0 1

1det exp ( )

!

1( )

!R

RR

R

Rj

j

j

Rj

j

n

nk n jn jpi

jp kj

nk n jp

jp k k j

x tr p xk

p xk

∞+ −−

==

∞ ∞+ −

= = =

⎡ ⎤ =⎣ ⎦

=

∑ ∑∏

∑ ∑ ∑ ∏

X ε

ε

( )[ ]1

1

0 0 1 1

0 0 1

1det exp ( )

!

1det

!

R

R

RR

R

R Rj

j

n

Rj

n

n nk n jn jpi

jk k pj j

nk n ji

jk k j

x tr p xk

xk

∞ ∞+ −−

= = = =

∞ ∞+ −

= = =

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎡ ⎤ =⎣ ⎦ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎡ ⎤

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥ ⎣ ⎦⎢ ⎥⎣ ⎦

∑ ∑ ∑∏ ∏

∑ ∑ ∏

X ε

j j Rt k n j= + −

( )[ ]1 21 2 0 1

1det exp det

( )!R

R

R R

Rj

n

ntn j

i ij Rt n t n t j

x tr xt n j

∞ ∞ ∞−

= − = − = =

⎡ ⎤⎢ ⎥ ⎡ ⎤⎡ ⎤ = ⎢ ⎥⎣ ⎦ ⎢ ⎥ ⎣ ⎦− +⎢ ⎥⎣ ⎦

∑ ∑ ∑ ∏X

jtix⎡ ⎤⎢ ⎥⎣ ⎦

det jtix⎡ ⎤⎢ ⎥⎣ ⎦ jt

jt

( )[ ]1 2 0 1

1det exp ( ) det

( )!R

R

Rj

jn

ntn j

i ip Rt t t p j

x tr p xt n j

−

> > > ≥ =

⎡ ⎤⎢ ⎥ ⎡ ⎤⎡ ⎤ = ⎢ ⎥⎣ ⎦ ⎢ ⎥ ⎣ ⎦− +⎢ ⎥⎣ ⎦

∑ ∑ ∏X ε

( )pε det jtix⎡ ⎤⎢ ⎥⎣ ⎦

jt

( )[ ]1 2 0

1det exp det det

( )!R

R j

n

tn ji i

j Rt t t

x tr xt n i

−

> > > ≥

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎢ ⎥= ⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥− +⎣ ⎦∑X

1 2 0Rnk k k≥ ≥ ≥ ≥

( )[ ]1 2 0

1det exp det det

( )!R j R

nR

k n jn ji i

jk k k

x tr xk i j

+ −−

≥ ≥ ≥ ≥

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎢ ⎥= ⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥+ −⎣ ⎦∑X

det Rn jix −⎡ ⎤⎣ ⎦

( )[ ]

1 2 0

/

det1exp det

( )! det

1det ( )

( )!

R

j R

nR

k n ji

n jj ik k k

j

xtr

k i j x

k i j κ

κ

χ

+ −

−≥ ≥ ≥ ≥

⎡ ⎤⎡ ⎤ ⎢ ⎥⎣ ⎦⎢ ⎥=⎡ ⎤⎢ ⎥+ −⎣ ⎦ ⎣ ⎦

⎡ ⎤⎢ ⎥=⎢ ⎥+ −⎣ ⎦

∑

∑

X

XV


where the summation is over all weakly decreasing partitions ,, into no more than parts. We now reach the conclusion that the

irreducible representations in the space generated by the function are labelled by the partitions of all integers into no more than parts. Naturally, thecoefficients in the expansion are given by

Using the character expansion in (4.253) it is now easy to compute the integral withinthe eigenvalue pdf of (4.199). Write

and the integration of the exponential reduces to the integration of the irreducible char-acter. Since the integration is over the unitary group, and the latter is a compact topologicalgroup, the general result of (4.231) applies directly, i.e.

and since the volume of the unitary group is already known from (4.134), we have

where the ’s are the eigenvalues of (and ) and the ’s are the eigenvalues of . No-tice that character of the identity matrix could not be obtained directly because of its in-determinate form

It is however possible to compute the limit with the help of L’Hôpital’s rule

and the rule of determinant differentiation

1 2( , ,..., )Rnk k kκ =

1 2 0Rnk k k≥ ≥ ≥ ≥ Rn

V ( )[ ]( ) expf tr=X X

κ Rn

(4.254)[ ]1

1

( )1 1

det det det ( )( )! ( )! ( )!

R

R

R

n

j np i n i

j Rj j R p Rp

k p j

c k n jk i j k n j k n pκ

= + −

=

⎡ ⎤⎢ ⎥+ −⎢ ⎥ ⎡ ⎤⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥= = = + −⎢ ⎥ ⎢ ⎥⎢ ⎥+ − + − + −⎣ ⎦ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

∏∏

(4.255)

(4.256)

( )[ ]( )

( )

1,

( )

1, /

( )

( ) ( ) exp

( ) ( )

H

R T

R

H

R T

R

n nU n

n nU n

f G tr d

G c dκκ

κ

χ

∧

∧

−

−

= −

= −

∫

∑ ∫

U U U

U U UV

Λ Λ Λ,ϒ ϒ Λ

Λ,ϒ ϒ Λ

( )1

/ /1/

/( )V

( ) ( )( ) olume ( )

( )H

RR

RnU n

d U n κ κκ

κ

χ χχ

χ∧

−− −

⎡ ⎤− = ⎣ ⎦∫ U U UI

V VV

V

ϒ Λϒ Λ

(4.257)

[ ]

2

2

1/ /

,/

1,

1/

( ) ( )2( ) ( )

( )( )

det ( ) det( )2( )( ) det ( ) det

R R

R TR R

j Rj RR R

R T

RRR R

n n

n nn nR

k n jk n jn n i in nn jn j

n nR i i

f G cn

Gc

n

κ κ

κ

κ

κκ

κκ

χ χπχ

υ λπχυ λ

−

+ −+ −−

−− −

−=

Γ

⎡ ⎤⎡ ⎤− ⎢ ⎥⎣ ⎦ ⎣ ⎦=⎡ ⎤Γ − ⎣ ⎦

∑

∑

I

I

V V

V

V

Λϒ Λ

Λ Λ,ϒ

Λ,ϒ

(4.258)

(4.259)

(4.260)

iυ ϒ Σ iλ Λ

/ /1

det 0( ) lim ( ) lim

0detRR

j R

Rn i

k n ji

n n jx i

x

xκ κχ χ

+ −

−→ →

⎡ ⎤⎢ ⎥⎣ ⎦= = =⎡ ⎤⎣ ⎦X I

I XV V

( )

( )

1

1

1/

1

1

detlim ( ) lim

det

R

RR R

nj R

R

nn i

R

rrk n ji

nrrx n j

in

xx x

xx x

κχ

+ −

→ → −

∂ ∂ ⎡ ⎤⎢ ⎥⎣ ⎦∂ ∂

=∂ ∂ ⎡ ⎤⎣ ⎦∂ ∂

X IXV

1 2det

det det det nx∂

= + + +∂A

D D D


where is identical to except that the entries in the i-th row are replaced by derivativeswith respect to x. Since the Vandermonde matrix has a different variable in each row, dif-ferentiating the Vandermonde determinant is equivalent to differentiating the respectiverow of the Vandermonde matrix, i.e.

Now, since the ’s are all unity in the limit of (4.259), the only possibility for having a non-zero determinant in both numerator and denominator is to differentiate a differentnumber of times with respect to each , say, differentiate times with respect to ,i.e. . Thus

We are now in conditions to compute the two limits in (4.259). We have

and

Substitution of (4.263) and (4.264) into (4.259) gives the character of the identity matrixand dimension of as

Comparing this expression with the coefficients expression in (4.254), the latter can be re-written as follows

and therefore, replacing into (4.257) yields

(4.261)

(4.262)

(4.263)

(4.264)

(4.265)

(4.266)

iD A

( )1

0

det det ( )kk

j j k ikik

rrt t r

ji ik p

x t p xx

δδ−

−

=

⎡ ⎤∂ ⎢ ⎥⎡ ⎤ = −⎢ ⎥ ⎢ ⎥⎣ ⎦∂ ⎢ ⎥⎣ ⎦∏

ix

ix Rn i− ix

k Rr n k= −

( )11 0

( )

1 0

det det ( )RR

j j R

R

n int t n i

ji in p

x t p xx x

− −−− −

=

⎡ ⎤∂ ∂ ⎢ ⎥⎡ ⎤ = −⎢ ⎥ ⎢ ⎥⎣ ⎦∂ ∂ ⎢ ⎥⎣ ⎦∏

( )

[ ]

11 0

1 1 0

1

lim det det ( )

det ( ) ( )!

R

RR

i R

R

R

n inn j

Rix n pn

n iR R

p

x n j px x

n j n p

− −−−

→=

−

=

⎡ ⎤∂ ∂ ⎢ ⎥⎡ ⎤ = − −⎣ ⎦ ⎢ ⎥∂ ∂ ⎢ ⎥⎣ ⎦

= − = −

∏

∏

( )

[ ]

11 0

1 1 0

lim det det ( )

det ( )

R

RRj

i R

R

n ink n j

j Rix n p

n ij R

x k n j px x

k n j

− −−+ −

→=

−

⎡ ⎤∂ ∂ ⎡ ⎤ ⎢ ⎥= + − −⎢ ⎥ ⎢ ⎥⎣ ⎦∂ ∂ ⎢ ⎥⎣ ⎦= + −

∏

κV

[ ][ ]/ /

det ( )( ) lim ( )

det ( )R

R

R Rn

n ij R

n n iR

k n jd

n jκ κκ χ χ−

−→

+ −= = =

−X II XV V

[ ] /1

1

/1 1

1det ( ) ( )

( )!

1! ( )

( )!

R

RR

R R

R

nn i

R np Rp

n n

np Rp p

c n jk n p

pk n p

κ

κ

κ χ

χ

−

=

−

= =

⎡ ⎤⎢ ⎥= −⎢ ⎥+ −⎢ ⎥⎣ ⎦⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= ⎢ ⎥ ⎢ ⎥+ −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∏

∏ ∏

I

I

V

V

(4.267)[ ]

2

1,1

11 1

2( )

1( )( ) ! det ( ) det

( )!det ( ) det

R R

R RR TR j Rj R

RR

n n

n nn nn k n jk n jR

i in jn jp Ri i p p

Gn

f pk n p

κ

π

υ λυ λ

−+ −+ −−

−− −= =

⎡ ⎤ ⎡ ⎤Γ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎡ ⎤= − ⎢ ⎥⎣ ⎦⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎡ ⎤ + −− ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦∏ ∏∑Λ

Λ,ϒΛ


Recall that the inner summation is given by

and that by making the change of variables of (4.248) one obtains a strictly ordered index-ation of the sum, or more formally

As such, the pdf expression in (4.247) can be written more simply as

In order to further simplify the sum we need to expand the determinants inside the sum-mation using (4.245). Expanding the one on the left yields the sum

Here, we see that for each sequence , the summand is a complete setof permutations, and hence it may be included in the outer sum by disregarding the orderof the ’s. Since for each permutation of the ’s, the sign of changes, it obviouslycancels the sign of the inner permutation , and thus the sum simplifies to

The final step involves expanding the remaining determinant and rearranging terms,which gives

Surprisingly, one realizes that the expression for the joint pdf of the eigenvalues of theWishart matrix is as simple as

Notice that this expression only depends upon three matrix determinants, which are easilycomputed in contrast with the infinite series of (4.206). The interrogation that now emerg-es is that of if the equation is sufficiently amenable to mathematical manipulation, since itmust be integrated with respect to the eigenvalues in . Expanding in (4.198)

(4.268)

(4.269)

(4.270)

(4.271)

(4.272)

(4.273)

(4.274)

1 2 0nRk k kκ ≥ ≥ ≥ ≥∑ ∑∼

1 2 1 20 0

R Rn n

j R j

k k k t t t

k n j t

≥ ≥ ≥ ≥ > > > ≥

+ − →

∑ ∑

[ ]

2

1 2

1,1

11 10

2( )

1( )( ) ! det ( ) det

!det ( ) det

R R

R RR TR jj

RR

nR

n n

n nn nn ttR

i in jn jpi i p pt t t

Gn

f pt

π

υ λυ λ

−−

−− −= => > > ≥

⎡ ⎤ ⎡ ⎤Γ ⎢ ⎥ ⎢ ⎥ ⎡ ⎤⎡ ⎤= − ⎢ ⎥⎣ ⎦⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎡ ⎤− ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦∏ ∏∑Λ

Λ,ϒΛ

1 2

1

1 10

1( ) ( ) det

!i

R Rjq

nR

n ntt

i ipq i pt t t

qt

υ λ−

= => > > ≥

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥ ⎡ ⎤− ⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∑ ∏ ∏∑ ε

1 2 0Rnt t t> > > ≥

it it det jtiλ⎡ ⎤⎢ ⎥⎣ ⎦

( )qε

1

1

10 0

1( ) det

!R

Rjp

n

ntt

p ippt t

tυ λ

∞ ∞−

== =

⎡ ⎤⎢ ⎥ ⎡ ⎤⋅ ⋅ ⋅ − ⎢ ⎥⎢ ⎥ ⎣ ⎦⎢ ⎥⎣ ⎦∏∑ ∑

( )

1

1

1 10 0

/ /

01 1

1( ) ( )

!

1( ) / ( ) det

!

R

R R

i ii

n

R Ri q i i ji

i

i

n nt t

i qi qi it t

n nt

q iiq t qi i

qt

q q e et

λ υ λ υ

υ λ

λ υ

∞ ∞−

= == =

∞− −

== =

⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥⋅ ⋅ ⋅ − =⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ ⎤⎡ ⎤⎢ ⎥= − = = ⎢ ⎥⎢ ⎥ ⎣ ⎦⎢ ⎥⎣ ⎦

∑∏ ∏

∑ ∑ ∑∏ ∏

∑ ∑ ε

ε ε

[ ]

2 /1

, 11

det2( ) ( ) !

( ) det ( ) det

i jRR R

R T RRR

nn n

n n n jn jn R i ip

ef G p

n

λ υπ

υ λ

−−

−− −=

⎡ ⎤⎡ ⎤ ⎢ ⎥⎣ ⎦⎢ ⎥= ⎢ ⎥ ⎡ ⎤Γ −⎢ ⎥ ⎣ ⎦⎣ ⎦∏Λ Λ Λ,ϒ

Λ , ( )R Tn nG Λ,ϒ


and observing that

the joint density can be reexpressed as

where

is a function that only depends on the MIMO channel dimensions and on the eigenvalues of the covariance matrix . It is clear from (4.276) that, in the correlated MIMO sce-

nario, the joint pdf of the eigenvalues is not a symmetric function of the eigenvalues.Hence, one requires (4.54) to compute the average capacity, i.e.

Expanding the pdf yields

Expanding the determinants yet again leads to

where is the kronecker delta. Remembering Equation (4.245), one acknowledges thatthe inner sum is in fact an expanded determinant where may index the matrix rows, andaccordingly, a more compact version of Equation (4.280) is

Moreover, each permutation p induces a column interchange in the matrix of the inner de-terminant, which brings in a factor of +1 or -1 and cancels . Therefore, since there are

permutations, we may write the average capacity as

For efficient evaluation of this expression, a closed form solution of the integral is clearly

(4.275)

(4.276)

(4.277)

(4.278)

det ( )R

R

nn j

i jii j

λ λ λ−

<

⎡ ⎤ = −⎣ ⎦ ∏

/,

1

1( ) ( )det det

!

Ri j R T R

R T

nn j n n

n n i iR i

f H en

λ υ λ λ− − −

=

⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎣ ⎦⎣ ⎦ ∏Λ Λ ϒ

[ ]

1( 1)

,11

1

1( ) !

( ) ( ) det ( )

RR R

R T RR R TR

nn n

n n nn n nn jR T p

i ii

H pn nπ

υ υ

−−

− −=

=

⎡ ⎤⎢ ⎥= ⎢ ⎥Γ Γ ⎢ ⎥⎣ ⎦ − ∏∏ϒ

iυ Σ

2 2

1 1

log 1 log 1 ( )( )R Rn n

avg k kT Tk k

C E f dn nγ γ

λ λ ∧

= =

⎡ ⎛ ⎞⎤ ⎛ ⎞⎟ ⎟⎜ ⎜= + = +⎢ ⎥⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦∑ ∑∫ ΛΛ

Λ Λ

(4.279)[ ]1

, 2 11,...,1

1( ) det det exp( / ) log 1 ...

!R

R RR T R

R T R

n

n nn j n n

avg n n i j k ni iR T ik

C H d dn nλ λ

γλ λ υ λ λ λ λ− −

==

⎛ ⎞⎟⎡ ⎤ ⎜= − + ⎟⎜⎣ ⎦ ⎟⎜⎝ ⎠∏∑∫ϒ

(4.280)1

/, 2 1

1,...,1 1

/, 2

0

1( ) ( ) ( ) log 1 ...

!

1( ) ( ) ( ) log 1

!

R

RR Ri qR i T Ri

R T R

n

kqiqT i i

R T

nn nn p n n

avg n n k ni iR T ik p q i

n pn n

R Tp q i

C H p q e d dn n

H p q e dn n

λ υ

λ λ

δλ υ

γλ λ λ λ λ

γλ λ λ

−− −

== =

∞−−

=

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎟⎜= + ⎟⎢ ⎥ ⎜ ⎟⎜⎝ ⎠⎢ ⎥⎣ ⎦

⎡ ⎛ ⎞⎤⎟⎜= +⎢ ⎥⎟⎜ ⎟⎜⎝ ⎠⎢ ⎥⎣ ⎦

∏∑ ∑ ∑ ∏∫

∑ ∑ ∫

ε ε

ε ε

ϒ

ϒ1 1

RR nn

k=∑ ∏

(4.281)

(4.282)

ikqδ

iq

/, 2

01

1( ) ( )det log 1

!

R kiT i i

R T

nn p

avg n nR Tk p

C H p e dn n

δλ υ γ

λ λ λ∞

− −

=

⎡ ⎤⎡ ⎛ ⎞⎤⎟⎢ ⎥⎜= +⎢ ⎥⎟⎜ ⎟⎜⎢ ⎥⎝ ⎠⎢ ⎥⎣ ⎦⎣ ⎦∑∑ ∫εϒ

( )pε!Rn

/, 2

01

( ) det log 1R ki

T iR T

nn j

avg n nTk

C H e dn

δλ υ γ

λ λ λ∞

− −

=

⎡ ⎤⎡ ⎛ ⎞⎤⎟⎢ ⎥⎜= +⎢ ⎥⎟⎜ ⎟⎜⎢ ⎥⎝ ⎠⎢ ⎥⎣ ⎦⎣ ⎦∑ ∫ϒ


required. First we make the change of variable and arrive at

Now, one knows from (4.187) that

and for the case , the basic identity

may be used. Hence, we write with no further delay that the average capacity of the MIMOchannel in a Rayleigh fading environment with branch correlation solely at the receiver isgiven by

where is the incomplete gamma function defined in (4.188), is given in(4.277), and the remaining variables are

- Two-sided Independent Fading scenarioThe average capacity under independent fading, i.e. when , has been comput-

ed in [71] in terms of the integral of Laguerre polynomials, and a closed form expressionhas been derived in (4.189). With the help of the mathematical concepts introduced so farit is possible to devise an alternative closed-form expression for the capacity in this par-ticular scenario. Notice that this result cannot be obtained directly from (4.286). Instead,one has to replace by in (4.257) and then proceed by the same linesas (4.258)-(4.276). We find that

and after some manipulations that

(4.283)

(4.284)

(4.285)

(4.286)

(4.287)

(4.288)

iλ υ λ→

1, 2

01

( ) det log 1R ki

T TR T

nn j in j

avg n n iTk

C H e dn

δλ υ γ

υ λ λ λ∞

− + − −

=

⎡ ⎤⎡ ⎛ ⎞⎤⎟⎢ ⎥⎜= +⎢ ⎥⎟⎜ ⎟⎜⎢ ⎥⎝ ⎠⎢ ⎥⎣ ⎦⎣ ⎦∑ ∫ϒ

( )/

20 0 0

1log 1 ! , /

ln2 !

Tr p p ln

TrT

T p l

pe ne d r l n

ln p

γλ γ

λ λ λ γγ

−∞−

= =

⎛ ⎞⎛ ⎞⎛ ⎞ ⎟⎜ ⎟⎟ ⎜⎜ ⎟+ = − Γ⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎟⎜ ⎜⎜ ⎟⎝ ⎠ ⎝ ⎠⎜⎝ ⎠∑ ∑∫0kiδ =

( )0

1 !re d r rλλ λ∞

− = Γ + =∫

/( )

, ( 1)1 0 0

( )! 1( ) det ,

ln 2 !

kiR TT i

R T T

n n j p p lnT T T

avg n n n ji iik p l

pn j e n nC H l

lp

δυ γ

υ γ υ γυ

− −

− − += = =

⎡ ⎤⎡ ⎤⎛ ⎞⎛ ⎞ ⎡ ⎤−⎢ ⎥⎢ ⎥⎟⎜ ⎟⎜⎟= − Γ ⎢ ⎥⎢ ⎥⎜ ⎟⎜⎢ ⎥⎟ ⎟⎜⎜ ⎟⎜ ⎝ ⎠ ⎢ ⎥⎢ ⎥⎝ ⎠ ⎣ ⎦⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦∑ ∑ ∑ϒ

( ),a bΓ , ( )R Tn nH ϒ

T

R

i R R

ki

n n

n n

n n

γ

υ

δ

→

→

→

→ ×

→

average SNR at each receiver branchumber of transmitter antennasumber of receiver antennas

eigenvalues of the covariance matrix kronecker delta

Σ

Rn= IΣ

( )1/ κ

χ −−ΣV / ( )Rnκ

χ −IV

[ ]/

det ( ) ( 1)( )

det ( ) ( 1)

jR

R R

k i jn ij R

n n i i jR

k n j

n jκχ

+ −−

− −

⎡ ⎤+ − −⎣ ⎦− =− −

IV

,

0

( 1)1( ) ( )det ( 1) det

! !

RTR

R T

t n jn jn j t

n n i iR t

tf H

n tλ λ=

∞ −−−

=

⎡ ⎤−⎢ ⎥ ⎡ ⎤= − ⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

∑IΣΛ Λ 1


where

The eigenvalue pdf can be simplified by noting the equality

where

which leads to

Now, completing the same steps as (4.278)-(4.286), the average MIMO capacity under in-dependent fading is found to be

This equation is only valid for . The coefficients can be computed using a sim-ple algorithm presented in Appendix A.

4.5.8 SPATIAL CORRELATION MODELLING

Before materializing the capacity expression into a readable plot, it is necessary to adopta model for the covariance matrix . Several correlation models have been proposed inthe literature. One of them is the exponential correlation model, which was introduced in[74] and has been extensively used by researchers in the performance analysis of space di-versity techniques. In this section we shall follow an alternative approach.

Recall that if is a column from the MIMO channel response matrix, then is givenby

which means that we only need to determine the general entry . To this end,Equation (4.67) can be used directly, i.e.

(4.289)

(4.290)

(4.291)

(4.292)

(4.293)

[ ]

1( 1)

,1

1( ) !

det ( ) ( 1)( ) ( )

RR R

R T RR R

nn n

n n n i i jn n RR T p

H pn jn n

π −−

− −=

⎡ ⎤⎢ ⎥= ⎢ ⎥ − −Γ Γ ⎢ ⎥⎣ ⎦∏1

,

0 1

( 1)( 1)

!

nk nk x k k

k n

k k

kx e a x

k

∞−

= =

−= −∑ ∑

1

, 1 2

0

...n k

k

k n n k

r r

a r r r−

−< ≤⋅⋅⋅≤

= ∑

, ,

1

1( ) ( )det ( 1) det

!

R

Ti RR T R

n jn jt n j t

n n t n j i iR t

f H e an

λ λ λ=

−−− + −

−=

⎡ ⎤⎢ ⎥ ⎡ ⎤= −⎢ ⎥ ⎣ ⎦⎢ ⎥⎣ ⎦

∑IΣΛ Λ 1

, ,

1 1

/

0 0

( ) det ( 1) ( )!

1 ,

ln2 !

R R

RR T R

kiTT

n n it n i

avg n n t n i T

k t

t n j p p lnT T

p l

C H a t n j

pe n nl

lp

δγ

γ γ

=

−+ −

−= =

+ − −

= =

⎡⎢= − + − ⋅⎢⎢⎣

⎤⎡ ⎤⎛ ⎞⎛ ⎞ ⎡ ⎤ ⎥⎢ ⎥⎟⎜ ⎟⎜⎟⋅ − Γ ⎢ ⎥⎜ ⎥⎟⎢ ⎥⎜⎟ ⎟⎜⎜ ⎟⎝ ⎠⎜ ⎢ ⎥ ⎥⎝ ⎠ ⎣ ⎦⎢ ⎥⎣ ⎦ ⎥⎦

∑ ∑

∑ ∑

IΣ 1

T Rn n≥ ,k na

(4.294)

(4.295)

Σ

jh Σ

[ ] [ ]

[ ] [ ]

1 1 1

1

R

R R R

j j j n j

Hj j

n j j n j n j

E h h E h h

E

E h h E h h

∗ ∗

∗ ∗

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤= = ⎜ ⎟⎣ ⎦ ⎜ ⎟⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

h h

…

Σ

[ ]ij rjE h h∗

[ ] ( )[ ] ( )[ ]( ) ( ) cos cos( ) sin cos( )ij rj ir jk ir ir jk irE h t h t E d jE dβ φ θ β φ θ∗ = + − +


where is the phase constant, is the angle between the direction of motion and thek-th incoming wave, is the angle between the direction of motion and the line joiningthe i-th and r-th receiver antennas, and is the distance between the antennas. To com-plete the expectation process, one may reasonably assume a rich scattering environmentwhere the multipath waves arrive from every direction around the receiver. This is a goodapproximation for urban environments cluttered with objects of different dimensions,such as buildings, cars, and trees, which scatter the waves in all directions (isotropic scat-tering). Moreover, there is no line-of-sight in these scenarios, justifying the Rayleigh mod-el. The angle spread is exactly radians and the density function of the ’s is uniformfrom 0 to , i.e.

When the scattering is non-isotropic (e.g. because of the “street-canyon” effect), the widthof the angle-of-arrival (AOA) is not and (4.296) may become a rough approximation.In [75], a generalization of (4.296) has been proposed, and is based on the von Mises densityfunction

where is the mean AOA, is the zeroth-order modified Bessel function of the firstkind, and controls the width of the AOA.

We shall consider solely the case , which corresponds to the isotropic scatteringmodel (ISM). It follows that the in-phase correlation is

and that the quadrature correlation is

and thus the entries of the covariance matrix are given by

where is the zeroth order Bessel function of the first kind. Although Equation (4.296)also depends on , the angle between the direction of motion and the lines joining eachpair of antennas, taking the expectation in (4.295) makes it clear that the correlation be-tween the channel responses at two different antennas is independent of . This is a dis-tinctive feature of the uniform distribution of wave arrival.

From the plot of (4.300) in Figure 4.8, it is seen that it is possible to achieve low cor-relation by placing the antennas near the zeros of the Bessel function. The zeros are ap-proximately equispaced by , and the first zero occurs at , which means that for1-D or 2-D arrays with more than 2 or 3 receiver antennas, respectively, it is not possibleto attain zero correlation between all the antennas. One should, however, expect an im-

(4.296)

(4.297)

(4.298)

(4.299)

(4.300)

β jkφ

irθ

ird

2π jkφ

2π

1( ) 0 2

2ij ijp φ φ ππ

= ≤ ≤

2π

cos( )

0

1( ) e

2 ( )jk c

jk jkpI

κ φ φφ π φ ππ κ

−= − ≤ ≤

cφ 0( )I ⋅[0, [κ ∈ ∞

0κ =

( )[ ] ( )

( )

2

02

0

1cos cos( ) cos cos( )

2

1cos cos( ) ( )

2

ir

ir

ir jk ir ir jk ir jk

ir jk jk ir

E d d d

d d J d

π

π θ

θ

β φ θ β φ θ φπ

β φ φ βπ

+

+ = +

= =

∫∫

( )[ ] ( )2

0

1sin cos( ) sin cos( ) 0

2ir jk ir ir jk ir jkE d d dπ

β φ θ β φ θ φπ

+ = + =∫

[ ] [ ] 0( ) ( ) ( )ij rj irir E h t h t J dβ∗= =Σ

0( )J ⋅

irθ

irθ

0.5λ 0.38λ


provement in capacity by arranging the antennas so that the average correlation betweenthem is minimized. In practice, and depending on the particular mobile scenario, the spa-tial correlation function between any pair of antennas will diverge from the ISM. Never-theless, if the receiver is able to accurately track the channel matrix, it may also be able toapproximate (4.300) by time-averaging the product during some multiple of thechannel coherence time. Now, and for instance, if the receiver is equipped with an arraywith many closely-spaced antennas, it may select (for signal detection) the ones thatachieve the lowest average correlation be-tween all pairs of antennas, and thus, in princi-ple, improve the capacity.

4.5.9 NUMERICAL RESULTS AND DISCUSSION

Due to receiver size constraints, the antenna arrays of typical mobile units must be restrict-ed in their end-to-end dimensions. Therefore, it seems reasonable to confine the analysisto receiver arrays of fixed maximum length, say , assume a minimum separa-tion between antennas of , and a maximum of eight antennas. In the case oflinear equispaced arrays, there are two options to increase the number of antennas fromone to eight: 1) comply with the maximum size, and evenly distribute the antennas within;and 2) comply with the minimum separation, thereby reducing the maximum length. Inwhat follows, we will use the first option as a case study, from which more general conclu-sions may be drawn. The entries of the covariance matrix are thus given by

with which can be built and its eigenvalues be found. The capacity in the special casesof and is plotted in Figure 4.9, where the curves for the Gaussian andRayleigh SISO channels are also included for comparison. It is clear that the capacity inthe correlated case is always lower than in the independent case, and this reduction be-comes more significant when the number of receiver antennas increases. For instance, at

, and for and , the capacity decreases by approximately 0.20 bits/s/Hz relative to the independent case, while for the loss is more than 1 bits/s/Hz.It is noticeable, however, that the utmost increase in capacity is obtained by increasing ,and increasing only provides a slight capacity gain. It is also seen that the higher is ,the better the capacity scales with increasing SNR, which means that the MIMO systemtakes superior advantage of receive diversity. Moreover, and despite the presence of cor-

FIGURE 4.8 Receiver spatial correlation for the isotropic scattering model.

0.5 1 1.5 2dirΛ

-0.4

-0.2

0.2

0.4

0.6

0.8

1

J02Π dirΛ

( ) ( )ij rjh t h t∗

(4.301)

max 1.2d λ=

min 0.15d λ=

[ ] 0 max( / )Rir J i r d nβ= −Σ

Σ

2Rn = 4Rn =

10 dBγ = 2Rn = 8Tn =4Rn =

Rn

Tn Rn


relation, it is evident that the use of multiple antenna receivers/transmitters offers remark-ably improved capacities as compared to SISO configurations. Even with a scenario assimple as , and , and a Rayleigh fading environment, we are grant-ed a capacity gain of 2.5 bits/s/Hz which almost doubles the capacity of the RayleighSISO channel, and for we watch an increase of 251%. Nonetheless, recall that thelength of the receiver array is being restricted to , and that larger capacities arecertainly possible by increasing this length.

Figure 4.10 illustrates the case and in addition incorporates the limiting situ-ation of , as given by (4.40). One concludes that it is not possible to achieve thecapacity under independent fading even with an infinite number of transmitter antennas.In fact, doubling offers a negligible increase in capacity. Even so, we find that at

and , the capacity in the correlated case is 521% superior to theRayleigh SISO capacity. These remarkable capacities are obtained without ever increasingthe length of the array, but by fitting all the antennas within , which is a veryimportant consideration if the receiver unit is to be portable and, hence, small-sized.

A question that now arises is: What is the best fit of the antennas within so thatthe capacity is maximized? So far we have been using equispaced linear arrays, and havealso been assuming that the receiver has the same number of detectors as the number ofantennas. As previously mentioned, if the receiver has several equispaced antennas, it mayselect the ones that achieve the highest capacity. For example, imagine a receiver with alinear array of eight equispaced antennas, but that only has four signal detectors available.Then it must choose the four antennas that will maximize capacity. Maximizing (4.286) di-rectly over does not seem an easy problem. We know, however, that (4.35) is maximizedwhen (see [76]), which is consistent with the fact that the independent fading sce-narios yield the highest capacities. Intuitively, one may now expect to achieve higher ca-

FIGURE 4.9 Average capacity of the MIMO channel as a function of the branch SNR at the receiver side, in the special cases of two and four receiver antennas.

0 5 10 15 20 0

5

10

15

20

25

Average SNR at each receiver branch (dB)

Average

Channel

Capacity

(bits/s/Hz )

Gaussian SISO; Rayleigh SISO; Rayleigh; nT = 2; Rayleigh; nT = 4; Rayleigh; nT = 8;

Indep. Rayleigh; nT = 8;

nR = 4

nR = 2

10 dBγ = 2Rn = 2Tn =

4Rn =

max 1.2d λ=

8Rn =

Tn → ∞

Tn

10 dBγ = 8R Tn n= =

max 1.2d λ=

maxd

Σ

Rn= IΣ


pacities by reducing the average correlation between all pairs of receiver antennas or,similarly, by minimizing the sum of the absolute values of the entries of , i.e.

Notice that (4.302) is equivalent to minimizing the Frobenius norm of . Since thereceiver has to choose four of eight available antennas, it must find all principal sub-matrices of , that is, all submatrices that lie on the same set of rows and columns, andthen apply (4.302) to each of these. It then chooses the principal submatrix with smallerFrobenius norm, and finally selects the antennas corresponding to the original rows (andcolumns) of . A simple algorithm for finding the antenna positions within thatmaximize capacity is given in Appendix B. For the example in consideration, the antennapositions are the subset 1, 3, 6, 8. When the receiver has 16 equispaced antennas and 4detectors, the positions are 1, 6, 11, 16 and, in the case of 32 equispaced antennas, 1,11, 22, 32. Figure 4.11 shows the capacity improvement in the ISM model by followingthis procedure. Since an increase in the number of receiver antennas is accompanied by anenhanced spatial resolution, we perceive a substantial increase in available capacity. By em-ploying 32 receiver antennas and effectively using four of them, it is almost possible to at-tain the capacity under independent fading. This a very interesting result, yet it is notgeneralizable to situations where the receiver has a greater number of detectors. For in-stance, if the receiver has eight detectors, then the optimum antenna positions are 1, 2,6, 7, 11, 12, 13, 16 and 1, 2, 11, 12, 21, 22, 31, 32 for the cases of 16 and 32 receiverantennas, respectively. From the plot in Figure 4.12 it is seen that the capacity improve-ment is subtle, and approximating the capacity under independent fading is not feasible.This happens because with eight detectors, the spatial resolution is already high, and anyantennas added to the configuration will be severely correlated with the ones already

FIGURE 4.10 Average capacity of the MIMO channel as a function of the branch SNR at the receiver side, in the special case of eight receiver antennas.

(4.302)

0 5 10 15 20 0

5

10

15

20

25

30

35

40

45


Average

Channel

Capacity

(bits/s/Hz)

Gaussian SISO; Rayleigh SISO; Rayleigh; nT = 8; Rayleigh; nT = 16;

Rayleigh; nT → ∞;

Indep. Rayleigh; nT = 8;

nR = 8

Σ

[ ],

max minavg iri r

C ↔ ∑Σ ΣΣ

Σ

4 4×

Σ

Σ maxd


present. Nevertheless, this exposition shows that equispaced linear arrays do not offer themaximum available capacity, and that by using some simple signal processing and calcula-tion techniques, any MIMO system can optimize the data rates delivered to users, and thespectral efficiency of the whole communication system.

FIGURE 4.11 Optimizing the capacity of the MIMO channel by selecting four receiver antennas from an equispaced linear array with eight and sixteen antennas.

FIGURE 4.12 Optimizing the capacity of the MIMO channel by selecting eight receiver antennas from an equispaced linear array with sixteen and thirty two antennas.

0 5 10 15 20 0

5

10

15

20

25


Average

Channel

Capacity

(bits/s/Hz )

Gaussian SISO; Rayleigh SISO; Rayleigh; nR = 4 out of 4; Rayleigh; nR = 4 out of 8; Rayleigh; nR = 4 out of 16; Indep. Rayleigh; nR = 4;

nT = 4

0 5 10 15 20 0

5

10

15

20

25

30

35

40

45


Average

Channel

Capacity

(bits/s/Hz)

Gaussian SISO; Rayleigh SISO; Rayleigh; nR = 8 out of 8; Rayleigh; nR = 8 out of 16; Rayleigh; nR = 8 out of 32; Indep. Rayleigh; nR = 8;

nT = 8


4.5.10 CONCLUSION

Section 4.5 has presented a clear and complete derivation of the closed-form expressionfor the average capacity of MIMO channels with , in the realistic scenario of signalcorrelation between different receiver antennas. It did so by introducing some simple con-cepts from the theory of linear transformations and the theory of group representations,making the exposition virtually self-contained. The main result was extended to independ-ent fading scenarios, yielding a new closed-form expression. Additionally, plots of the ca-pacity for the isotropic scattering model in distinct MIMO arrangements were obtained,showing a substantial capacity reduction in the presence of correlation. Finally, it wasproved that equispaced linear arrays do not attain the maximum capacity, and that byequipping the receiver with more antennas than signal detectors it becomes possible to re-alize capacity gains.

4.6 Capacity when the Wishart Matrix is Singular

In Section 4.5 an exact, closed-form, analytic solution was found for the particular case ofno more receiver antennas than transmitter antennas ( ). Although important anduseful, that solution did not contemplate the case , rendering it somewhat non-universal. Correlation is being assumed solely at one side of the communication link, andconsequently one must assume that the antennas at the other side are sufficiently spacedso as to provide independent transmission/reception diversity and avoid mutual couplingeffects. This is often a reasonable assumption if the radio links consist of a base station(BS) at one end and a mobile unit (MU) at the other end, provided that the antennas atthe base station have separations of several wavelengths. In such a scenario, the totalnumber of antennas at the base station must be limited, possibly being less than thenumber of antennas at the mobile unit. Recall that, even under correlation conditions andfixed array lengths, the utmost increase in capacity is obtained by increasing the numberof receiver antennas.

In this section we fill the aforementioned gap by following an approach identical tothe previous, and try to extend it to transmitter-sided correlation and two-sided independ-ent fading. It is advisable for the reader to be acquainted with Section 4.5, since some ofthe results and techniques that will be used here (without proof) rely on material that hasbeen properly introduced there. Namely:1. if is a complex generic ( ) random matrix with statistically independent

entries, it clearly defines a -dimensional Euclidean space; if, in addition, matrix satisfies the condition , it is implicit that a set of independent equationsmust be satisfied, which restricts the original space to a topological space of dimen-sion ; this topological space, denoted by , is closed and bounded, hencecompact, and is called the complex Stiefel manifold; if , then the Stiefel manifold iscalled the unitary group, and is denoted by ;

2. since is compact, it is measurable and has a finite volume; this volume is given by

T Rn n≥

(4.303)

T Rn n≥

T Rn n<

Q m n× m n≥

2mn QH =Q Q I 2n

22mn n− ,n mV

m n=

, ( )n nV U n∼

,n mV

[ ] ( ),

,2

Volume( )n m

n mn

n mV n

V dm

π∧= =Γ∫ Q

4.6 Capacity when the Wishart Matrix is Singular 103

where is the complex multivariate gamma function defined in (4.133);3. adding the extra restriction to of real diagonal elements (i.e. fixing the phase of the

columns so that the diagonal is real), one obtains the restricted Stiefel manifold ,which has the volume

4. the following equality was also obtained:

which is valid provided that the function does not depend on the particularphase of the columns of ;

5. in addition, the following integral over the unitary group has been evaluated

where and are generic, full-rank, complex matrices, and and aretheir respective eigenvalues.

4.6.1 STATISTICAL CHARACTERIZATION

We are solely interested in studying frequency-flat multipath wireless channels, that is,channels that have a small delay spread as compared to the period of the transmitted sym-bols. If the channel is frequency-selective, it is assumed that it can be converted to fre-quency-flat. Also, for typical data rates, it is a good approximation to assume that thechannel is slowly-fading, i.e. virtually time-invariant during a symbol period. Under theseconditions, it is possible to model each MIMO channel transmission with the equation

where is the input vector, is the output vector, is the AWGNnoise vector and is the random channel matrix. The average (ergodic) capacityof the MIMO channel was determined to be given by

where is the average SNR at each receiver branch, and are the number of trans-mitter and receiver antennas, respectively, and denotes the expectation over . Wecall matrix the Wishart matrix and let denote the joint density function

(4.304)

(4.305)

(4.306)

( )n mΓ

Q

,n mV⊂

( ),

( 1)

,Volume( )n m

n m

n mnV

V dm

π∧

⊂

−⊂⎡ ⎤ = =⎢ ⎥⎣ ⎦ Γ∫ Q

( ) ( ), ,

1( ) ( )

(2 )n m n m

nV V

h d h dπ

∧ ∧

⊂=∫ ∫Q Q Q Q

( )h Q

Q

( ) ( )

[ ]

( )

Vdet ( 1)!

olume ( ) ,det det

H

i i

tr

U n

a b

n j n ji i

e d

i eU n

a b

∧

− −

=

⎡ ⎤−⎣ ⎦=⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

∫ AUBU U

A B n n× ia ib

(4.307)

(4.308)

= +y Hx n

x 1Tn × y 1Rn × n 1Rn ×

H R Tn n×

2log det H

Ravg nT

C Enγ⎡ ⎛ ⎞⎤⎟⎜= +⎢ ⎥⎟⎜ ⎟⎜⎝ ⎠⎢ ⎥⎣ ⎦

H I HH

γ Tn Rn

[ ]E ⋅H HH= HHA ( )fH H


of 's entries. Using the eigenvalue decomposition , where is a unitary ma-trix, we may also write (4.308) as

where is the number of non-zero eigenvalues of , is some eigenvalue,, is the joint pdf of the eigenvalues, which must be derived from

, and is the volume element in the eigenvalue coordinates. Assuming aRayleigh fading environment, and denoting by and the transmit and receive corre-lation matrices, respectively, it was found that if correlation exists only at the receiver side,then

If correlation exists only at the transmitter side, then it is easy to verify that

It was previously said that the capacity for the case was computed inSection 4.5. The problem with the case is that the Wishart matrix be-comes rank-deficient (i.e. not full-rank, hence singular), which leads to serious complica-tions in the derivation of the eigenvalue pdf . In the following subsections we showthat it is possible to overcome these difficulties.

4.6.2 VOLUME ELEMENTS OF TRANSFORMATIONS

Before proceeding with the derivation of the density functions and , one mustfirst obtain the relations between the volume elements associated with the following trans-formations:

1. The Cholesky factorization of the hermitian, positive definite, matrix, where is an upper-triangular matrix with positive and real diagonal entries.

From (4.95) we have

2. The QR factorization of the complex ( ) matrix of rank , where is a unitary matrix, is an upper-triangular matrix,and is an matrix. Each matrix has the following number of functionallyindependent entries: matrix - ; matrix - (due to ); matrix -

; and matrix - . To make the factorization a one-to-one mapping weforce the diagonal elements of to be real, so that the right side of the factorization hasa total of functionally independent entries. Using (4.86), we write

(4.309)

(4.310)

(4.311)

H H= Q QΛA Q

2

1

log 1 ( )( )m

avg kTk

C f dnγ

λ ∧

=

⎛ ⎞⎟⎜= + ⎟⎜ ⎟⎜⎝ ⎠∑∫ ΛΛ

Λ Λ

m A kλ

1( ,.., )mdiag λ λ=Λ ( )fΛ Λ

( )fH H ( )d ∧Λ

TΣ RΣ

[ ] ( )[ ]T

R R1( ) det( ) expTn HTnf trπ= − −= −I

H H HHΣ Σ Σ

[ ] ( )[ ]R

T T1( ) det( ) expRn HRnf trπ= − −= −I

H H H HΣ Σ Σ

T Rn n≥

T Rn n< H= HHA

( )fΛ Λ

(4.312)

(4.313)

( )fA A ( )fΛ Λ

H=S R R n n×

S R n n×

( ) ( )2 2 1

1

2n

n n kkk

k

d r d∧ ∧− +

== ∏S R

( )1 2= =S QR Q R R m n× n m> S

m Q m m× 1R m m×

2R ( )m n m× −

S 2mn Q 2m H =Q Q I 1R2m m+ 2R 22 2mn m−

1R

2mn

( ) ( ) ( )

( ) ( )( )

2

1 2 1 2

det

.

H H

H

nd d d

d d d

∧∧ ∧

∧

= =

= +

S Q S Q S

Q Q R R R R


Also, we have

Since is skew-hermitian, and noting that products of repeated differentials are zero,i.e. , the entries in (4.314) for donot contribute to the exterior product, and only the last term of the diagonal and sub-di-agonal entries of is not redundant. It follows that

and because

we hence have

3. The Congruence transformation between two hermitian matrices and of rank , where is a fixed nonsingular matrix. It is assumed that

there is an matrix such that factorizes as . The Congruence transfor-mation can be written in block form as

where and are hermitian, positive definite, matrices of rank . Matrix con-tains functionally independent elements, which correspond, for in-stance, to the elements of plus the diagonal and above diagonal elements of . Thesame applies to matrix . The volume elements of and are thus given by

We need to find

(4.314)

(4.315)

(4.316)

(4.317)

(4.318)

(4.319)

(4.320)

( )[ ]

[ ]

11 2

1

H

H

H

j

i k kjkmij

i k kjk

d r j m

d

d r j m

=

=

⎧⎪⎪ ≤⎪⎪⎪⎪⎡ ⎤ = ⎨⎢ ⎥⎣ ⎦ ⎪⎪⎪ >⎪⎪⎪⎩

∑

∑

q qQ Q R R

q q

HdQ Q

( ) ( ) ( ) ( ) 0H H H Hi k i k i k k id d d d∧ ∧ ∧ ∧∧ = ∧ =q q q q q q q q j m>

1HdQ QR

( )( ) ( )2 2 11 2

1

H H

mmm k

i jkk i jk

d r d∧ ∧− +

≥== ∏ ∧Q Q R R q q

( ) ( ) ( )H H

m

i ji j

d d d∧ ∧ ∧

≥= =∧ q q Q Q Q

( ) ( ) ( )2 2 1

1

mm k

kkk

d r d d∧ ∧ ∧− +

== ∧∏S Q R

H=S B AB n n×

A S q n< B n n×

n q× X A H=A X X

11 12 11 1211 12 11 12

21 22 21 2212 22 12 22

H

H H

⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎟⎟ ⎟ ⎟⎜⎜ ⎜ ⎜⎟⎟ ⎟ ⎟⎜⎜ ⎜ ⎜= ⎟⎟ ⎟ ⎟⎜⎜ ⎜ ⎜⎟⎟ ⎟ ⎟⎟ ⎟⎟ ⎜⎜ ⎜⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

S S A AB B B B

B B B BS S A A

11S 11A q q× q S

( 1) 2 ( )q q q n q+ + −

12S 11S

A S A

( ) ( ) ( )

( ) ( ) ( )

11 12

11 12

d d d

d d d

∧ ∧∧

∧ ∧ ∧

= ∧

= ∧

S S S

A A A

( )

( )( ) ( )( ) ( )

( )( )

( )( )

12 1111 12 11

11 12 11 12

|dd d d dd d d d d

∧∧ ∧ ∧∧

∧ ∧ ∧ ∧ ∧

∧= =

∧S SS S S S

A A A A A


where denotes the differential of matrix given . From (4.318) it is foundthat

Differentiating these equations, and using (4.86) and (4.115), yields

and

where the following property of the determinant has been used:

Consequently, we find

4. The Eigenvalue decomposition of the hermitian matrix of rank, where is a diagonal matrix with the non-zero eigenvalues of , and is

an semi-unitary matrix (such that ). Alternatively, the decomposition canbe rewritten as

where and is the orthogonal complement of . Defining, where is , by (4.325) we know that

It is easily checked that, for and , we have

Moreover, since is skew hermitian, it follows that , and thus onlythe diagonal and subdiagonal elements given by (4.328) are relevant when taking exterior

(4.321)

(4.322)

(4.323)

(4.324)

(4.325)

(4.326)

(4.327)

(4.328)

( )12 11|d ∧S S 12S 11S

( )

11 11 11 11 21 12 11 11 12 21 21 22 21

12 11 11 12 21 12 12 11 12 22 21 22 22

1 111 12 22 21 11 12 11 11 12

121 22 21 11 12 21 22 22

H H H H H

H H H H H

H

H H

− −

−

= + + +

= + + +

= − + −

− +

S B A B B A B B A B B A B

S B A B B A B B A B B A B

B A B B B B S B B

B A B B B B A B

( ) ( )211 11 11det qd d ∧∧ =S B A

( ) ( )2( ) 2 212 11 11 11| det detn q q qd d∧ ∧− −=S S B B A

( )11 12

111 22 21 11 12

21 22det det det det −

⎛ ⎞⎟⎜ ⎟⎜= = −⎟⎜ ⎟⎟⎜⎝ ⎠

B BB B B B B B

B B

( ) ( )2( ) 211det detn q qd d ∧∧ −=S B B A

1 1HS = Q QΛ n n× S

q n< Λ q q× S 1Q

n q× 1 1H =Q Q I

H⎛ ⎞⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠

0S Q Q

0 0Λ

( )1 1⊥=Q Q Q 1

⊥Q 1Q

( )1 11 12H H H=Q Q Q 11Q q q×

( ) ( )2( )11det

.

H

HH H

n q

dd d

∧∧

∧

−=

⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎛ ⎞⎞ ⎟⎜ ⎟ ⎟ ⎟⎟⎜ ⎜ ⎜ ⎜ ⎟= + +⎟ ⎟ ⎟⎟⎜⎜ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎟⎟⎜ ⎟ ⎟ ⎟⎟⎜ ⎜ ⎜ ⎜ ⎟⎜⎝ ⎠ ⎝ ⎠ ⎝ ⎝ ⎠⎠⎝ ⎠

Q dSQ Q dS

Q Q Q Q0 0 0Λ Λ Λ

i n≤ j q≤

( )H

H H Hj ii j

ij

d d d λ λ⎡ ⎤⎛ ⎞ ⎛ ⎛ ⎞⎞⎟ ⎟⎟⎜ ⎜ ⎜⎢ ⎥+ = −⎟ ⎟⎟⎜ ⎜ ⎜⎟ ⎟⎟⎟ ⎟⎟⎢ ⎥⎜ ⎜ ⎜⎝ ⎠ ⎝ ⎝ ⎠⎠⎣ ⎦Q Q Q Q q q

0 0Λ Λ

HdQ Q H H

i j j id d= −q q q q


products (i.e. ). This leads to

which, after plugging exterior product factors together, and using the identities

yields

Substituting this result back into (4.327), one reaches the final relation between the volumeelements

4.6.3 THE WISHART AND EIGENVALUE DENSITIES

We now determine the pdf of the Wishart matrix . Since we are considering thecase we note that, in statistical terms, matrix contains linearly independentcolumns. Consequently, we can use the factorization , where is a unitary

matrix and is an matrix with upper-triangular.Expressing the joint pdf of and as

and inserting (4.317) gives

Now, since is given by

where is , we notice the dependence , which means that

(4.329)

(4.330)

(4.331)

(4.332)

i j≥

( ) ( )

( ) ( )

2( )

1 11

2

1( ) ,

H H

H H

j q i nq nq

n qi jj j i qj

j i q i jq qq

j i i j i i ii j ii j

d

d d d

λ

λ λ λ

∧ ∧

∧ ∧

< < ≤

−

= = +=

< ≤ =

> =>

= ∧

∧ − ∧ ∧

∏

∏

∧ ∧

∧ ∧

Q dSQ q q

q q q q

( ) ( ) ( )1 11

det ( ),

H H

q n

i jj i j

qq j

i jii j

d d d

λ λ λ

∧ ∧∧

= =

−

<

= =

⎡ ⎤ = −⎣ ⎦ ∏

∧ ∧Q Q Q q q

( ) ( ) ( ) ( ) ( )22( )

1det detH n q q j

i d dλ∧ ∧∧− −⎡ ⎤= ∧⎣ ⎦Q dSQ QΛ Λ

( )( )( )

( ) ( )

22( )

11 12( )

detdet

det

q jiq n

q nd d

λ∧∧∧

−−

−

⎡ ⎤⎣ ⎦= ∧dS Q Q ΛΛ

(4.333)

(4.334)

(4.335)

(4.336)

H= HHA

T Rn n< H TnH =H QR Q

T Tn n× ( )1 2=R R R T Rn n× 1R T Tn n×

Q R

( )( )

( )T( , ) ( )Tn H H d

f d fd

∧=∧

∧= IQR H

HQ R Q R Q

RΣ

( ) ( )T 2 2 1

1

( , ) ( )T

Tn H H T

nn k

kkk

f d f r d=∧ ∧− +

== ∏I

QR HQ R Q R Q QΣ

A

11 121 1 1 2

12 222 1 2 2

H H

H

HH H

⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜= = =⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜ ⎟⎟⎜ ⎝ ⎠⎝ ⎠

R R R RR R

R R R R

A AA

A A

11A T Tn n× 122 12 11 12

H −=A A A A

( ) ( ) ( )11 12 11 12 11 12( ) ( , )f d f d d∧ ∧ ∧= ∧A A AA A A A A A


Hence, we can also write

From result (4.312) for the Cholesky factorization

and from (4.86)

Expanding (4.310) in (4.334), inserting the result into (4.337), and then using (4.338) and(4.339), we find that the pdf of the complex, singular Wishart matrix is given by

It is now possible to derive the joint pdf of the non-zero eigenvalues of .Since , it has non-zero eigenvalues. Thus, we define the Eigenvalue de-composition , where is the diagonal matrix of the ordered eigen-values of , and is a semi-unitary ( ) matrix with real diagonal.Furthermore, we write the joint pdf of and as

Substitution of (4.332) and (4.340) yields

where and is the orthogonal complement of . Equation (4.342) canbe integrated over the restricted Stiefel manifold to give

where

(4.337)

(4.338)

(4.339)

(4.340)

(4.341)

(4.342)

(4.343)

(4.344)

( ) ( )

( ) ( )

( )

( )( )( )

( )

1 2

,

1 21 2

11 12

2 11

11 12

( ) ( , )

|( , ) .

T Tn nV

d df f

d d

ddf d

d d

∧ ∧

∧ ∧

∧∧∧

∧ ∧

∧=

∧

= ∫

R R

QR

R RR R

R RRQ R Q

A A A A

A A

( )

( )

12 2 11

11 1

2T

TT

nn kn

kkk

dr

d

∧

∧

−− +

=

⎡ ⎤⎢ ⎥=⎢ ⎥⎣ ⎦

∏RA

( )( )

( )2 1 2( )1 11

12

|det det T RT R n nn nd

d

∧

∧−−= =

R RR A

A

( )( ) ( )R

1

R

( )

11( ) detdet ( )

T T RT R

TT

n n ntr n n

nn T

f en

π −−− −=

ΓΣ

ΣA

A A A

( )fΛ Λ A

( ) Trank n=A Tno

1 1H= Q QΛA oΛ T Tn n×

Λ 1Q R Tn n× T Rn n<oΛ 1Q

( )( )( )1

o o1 1 1 1 o( ) ( )H d

f d fd

∧∧

∧=Q Q Q Q QΛ Λ , ΛΛAA

( )( )

( ) ( )R

1R

o1

( )o

1 1

020 0

11

( )det ( )

det ,

T T R

TT

HT

R T T

n n n

nn T

ntrn n n jj i

j

f dn

e d

π

λ λ

∧

−∧

−

⎛ ⎛ ⎞ ⎞⎟ ⎟⎜ ⎜ ⎟ ⎟− ⎜ ⎜ ⎟ ⎟⎜ ⎜ ⎟ ⎟ − −⎝ ⎝ ⎠ ⎠

=

= ⋅Γ

⎡ ⎤⋅ ⎣ ⎦∏

Q

Q Q

Q Q

Q

Λ

ΛΣ

Λ ,Σ

( )1 1⊥=Q Q Q 1

⊥Q 1Q

,T Rn nV⊂

( ) ( )( )

R

R

o1

1 1 1 1

,

o o,

1

( ) ( , )

,T R

R T

H

n n

n n

tr

V

f G

e d− ⊥ ⊥

∧

⊂

⎛ ⎛ ⎞ ⎞⎟ ⎟⎜ ⎜ ⎟ ⎟− ⎜ ⎜ ⎟ ⎟⎜ ⎜ ⎟ ⎟⎝ ⎝ ⎠ ⎠

′= ⋅

⋅ ∫0

Q Q Q Q0 0 Q

Λ

ΛΣ

Λ Λ Σ

( )( )

2( )

,det (det

( )det ( )

n jn n m m ni

m n nn

Gn

π λ −− −⎡ ⎤⎣ ⎦′ =Γ

Λ)Λ,Σ

Σ


Averaging over all permutations, applying (4.305), and making the transformation, where is orthogonal and , we are led to

At this point, we require a closed-form expression for the integral in (4.345). This problemcan be solved by extending (4.306) to the case of a singular matrix, and noting that in(4.345) we are in fact integrating over a confined domain, more specifically the quotientspace . Ignoring the singularity of matrix , we could thuswrite

where and are the eigenvalues of and , respectively. Since eigen-values of are zero, say , (4.346) is the indetermination . The indetermi-nation can be dealt with using the fact that the integrand function is uniformly convergentas the eigenvalues of approach zero, which allows us to write

It is now possible to use L'Hôpital's rule by differentiating both numerator and denomi-nator inside the limit, times with respect to , for . We get

where has value 1 if , and 0 otherwise. If we substitute (4.348) into (4.345) andrearrange the result, we finally find that the joint eigenvalue density is

where is the vector of eigenvalues of ,

(4.345)

(4.346)

(4.347)

(4.348)

(4.349)

(4.350)

1 1=U Q P P T Tn n× o H= P PΛ Λ

( ) ( )( )

R

R

11 1 1 1

,

,

1

( )( )

!(2 )

T R

R T

T

H

n n

n nn

T

tr

V

Gf

n

e d

π− ⊥ ⊥

∧

⎛ ⎛ ⎞ ⎞⎟ ⎟⎜ ⎜ ⎟ ⎟− ⎜ ⎜ ⎟ ⎟⎜ ⎜ ⎟ ⎟⎝ ⎝ ⎠ ⎠

′= ⋅

⋅ ∫0

U U U U0 0 U

Λ

ΛΣ

Λ,ΣΛ

B

( )/ ( )R R TU n U n n− ( , )diag=D 0Λ

( ) ( )( ) ( )

[ ][ ]

R1

1 1 1 1

,

1

/

, 1 V

det ( 1)!olume

det ( ) det

T R

R

H

n n

i j

T R R

tr

V

d

n n n jn ji i

e d

i eV

d

υ

υ

− ⊥ ⊥∧−

−

−− −=

=

⎡ ⎤−⎢ ⎥⎣ ⎦⎡ ⎤− ⎣ ⎦

∫U U D U U

UΣ

iυ id RΣ D R Tn n−

D 1,..,T Rn nd d+ 0/0

D

( ) ( )( ) ( )

[ ][ ]

R1

1 1 1 1

,

1

1

/,

1 ,.., 0

V

det ( 1)!olumelim .

det ( ) det

T R

RT R

H

n n

i jT R

Rn n

tr

V

dn n

n jn j d di i

e d

i eV

d

υ

υ

− ⊥ ⊥∧

+

−

−

−− − →=

=

⎡ ⎤−⎢ ⎥⎣ ⎦⎡ ⎤− ⎣ ⎦

∫U U D U U

UΣ

Rn i− id Ti n>

( ) ( )( )

[ ][ ]

( )

R1

1 1 1 1

,

1

( ) /1,

1

V detolume! ,

det ( ) (det det

T R

T

R

H

n n

R in i jRT R

TR TR T

tr

V

i nnjn n

n jn j n ni p n n i

e d

eVp

σ λ υυ

υ λ

− ⊥ ⊥∧

⎛ ⎛ ⎞ ⎞⎟ ⎟⎜ ⎜ ⎟ ⎟− ⎜ ⎜ ⎟ ⎟⎜ ⎜ ⎟ ⎟⎝ ⎝ ⎠ ⎠

− −−

−− − −= −

=

=

⎡ ⎤−⎡ ⎤ ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥ ⎡ ⎤− ⎢ ⎥⎣ ⎦ ⎣ ⎦∏

∫0

U U U U0 0 UΛ

Σ

Λ)

abσ a b>

( )( )R ( ), /

1

( )( ) det

detTR inR T i j

T

i nn njn j

i

Hf eσ λ υυ

λ

− −−−

′ ⎡ ⎤= −⎢ ⎥⎣ ⎦⎡ ⎤⎣ ⎦Λ Λ

υ

Rυ RΣ

[ ]

( )

,

1,V

( )!(2 ) ( )

olume ! ,

det ( 1)

n n m

m n nn

mn m

n m jm jp m ni

Hn n

Vp

ππ

υ

−

−

− +−= −

′ = ⋅Γ

⎡ ⎤⎢ ⎥⋅ ⎢ ⎥⎡ ⎤− ⎢ ⎥⎣ ⎦⎣ ⎦∏

υ


and is obtained from (4.303).

4.6.4 CAPACITY UNDER RECEIVER-SIDED CORRELATED FADING

Having found the joint pdf of the eigenvalues of the complex singular Wishart matrix, wenow derive the average capacity of the MIMO wireless channel for the scenario and . To that end we evaluate (4.309) with (4.349), yielding

Expanding the determinants, the multiple integral becomes

In (4.352), we switch integration and summation operations, and incorporate the loga-rithm within the product, so that (4.351) becomes

where is the Kronecker function. Keep in mind that the summation is over all -tu-ples (bounded by ), or equivalently, over all permutations of thenatural sequence .

Following the same steps as in (4.182)-(4.189) (essentially, integrating by parts andthen using the binomial theorem) the integral in (4.353) can be expressed in closed-form.The average capacity formula becomes

where is the incomplete gamma function. This completes the derivation for. The expression for the case is given in (4.286).

Evidence has shown that these capacity expressions are prone to numerical overflow/underflow when the eigenvalues of the covariance matrix are very small and/or is large.This leads to a large , and the consequence is that overflows and

[ ],Volume n mV

(4.351)

(4.352)

(4.353)

(4.354)

T Tn= IΣ

T Rn n<

( )

R

1

, 2

1

( ) /1

( ) log 1

det det ... .

T

T

T

R T

n

R in i jTTT T

n

avg n n kTk

i nn jj ni n n

C Hn

e d d

λ λ

σ λ υ

γλ

λ υ λ λ

=− −−

×

⎛ ⎞⎟⎜′= ⋅ ⋅ ⋅ + ⋅⎟⎜ ⎟⎜⎝ ⎠

⎡ ⎤⎡ ⎤⋅ −⎢ ⎥⎣ ⎦ ⎣ ⎦

∑∫ ∫υ

( )

1

2

/11

1 1

log 1 ( ) ( )

... .

T

R T

n

R T

R i qT i il T

T

n n

kT q p

n nn l n p

q nil n i

q pn

e d d

λ λ

λ υ

γλ

υ λ λ λ− −−−

= + =

⎛ ⎞⎟⎜⋅ ⋅ ⋅ + ⋅⎟⎜ ⎟⎜⎝ ⎠

⎡ ⎤⎢ ⎥⋅ −⎢ ⎥⎢ ⎥⎣ ⎦

∑ ∑∫ ∫

∏ ∏

ε ε

( )R1

,

11

/2

0

( ) ( )

det log 1

T R R

R

R T l

T

kjqT i

T T

n n nn l

avg n n q

l nk q

n j

T n n

C H q

e dn

δλ υ

υ

γλ λ λ

−−

= +=

∞−−

×

⎡ ⎤⎢ ⎥′= − ⋅⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤⎡ ⎛ ⎞⎤⎟⎢ ⎥⎜⋅ +⎢ ⎥⎟⎜ ⎟⎜⎢ ⎥⎝ ⎠⎢ ⎥⎣ ⎦⎣ ⎦

∑∑ ∏

∫

υ ε

abδ Rn

1( ,.., )Rnq q q= 0 i Rq n< ≤

(1,.., )Rn

( )RR

/ 1, ( 1)

11

/( )

0 0

( )!( ) ( ) det

1,

ln2 !

T R R

RT RR T l T

iT

kjTT qi

i i

T

n n nn ln n T

avg n n q n jql nk q

n j p p lnT T

q qp ln

n jC H q

pe n nl

lp

δυ γ

υυ

υ γ υ γ

−< −− − +

= +=

− −

= = ×

⎡ ⎤ ⎡ −⎢ ⎥ ⎢′= ⋅ − ⋅⎢ ⎥ ⎢⎢ ⎥ ⎢⎣⎣ ⎦⎤⎡ ⎤⎛ ⎞⎛ ⎞ ⎡ ⎤ ⎥⎢ ⎥⎟⎜ ⎟⎜ ⎢ ⎥⎟⋅ − Γ⎟⎜ ⎥⎜⎢ ⎥⎟ ⎟⎜ ⎟⎜⎟ ⎢ ⎥⎜ ⎝ ⎠ ⎥⎝ ⎠⎢ ⎥⎣ ⎦⎣ ⎦ ⎥⎦

∑∑ ∏

∑ ∑

Σ υ ε

Tn

( ),1/l aΓ

T Rn n< T Rn n≥

Tn

/( )i T ix n υ γ= ixe [ ], il xΓ


underflows. For we deal with this problem by noting that

and hence we compute . For and say, , it may be suitable to usethe asymptotic expansion

where should be chosen sufficiently large (e.g. is usually adequate).

- Two-sided Independent Fading scenarioEquation (4.293) provides a closed-form expression for the average capacity of a

MIMO channel in a two-sided independent fading scenario when . Now, we tryto find a similar expression for . In (4.348), we draw attention to the limit

Making the change of variables , and using the identity

where is the falling factorial

we apply L'Hôpital's rule and differentiate the numerator and denominator of (4.357) times with respect to , thus getting

Setting and using the identity

(4.360) is equivalent to

(4.355)

(4.356)

(4.357)

(4.358)

(4.359)

(4.360)

(4.361)

(4.362)

0l >

[ ]

1

0

, ( 1)! 0 (integer)!

l kx

k

xl x e l l

k

−−

=

Γ = − >∑

[ ],ixie l xΓ 0l = 310ix >

[ ]1

0

!0, ( 1) ( )

Nx k

kk

kx e x

x−

+=

Γ − → ∞∑∼

N 10N >

T Rn n≥

T Rn n<

( )

[ ]1

/( )

1,.., 1

det 0lim

0det ( )

T

RR

j iR jn

n

j ni

n ji

e λ υσ

υ υ

υ

υ

−−

− −→

⎡ ⎤−⎢ ⎥⎣ ⎦ =−

1i ix υ−= −

( )

0

( ) ( ) ( )n

a bx n bx n k a kk

k

nf x x e f x e a b x

k− −

=

⎛ ⎞⎟⎜ ⎟= → = ⎜ ⎟⎜ ⎟⎜⎝ ⎠∑

( )ka

1

0

( ) ( )k

k

p

a a p−

=

= −∏

Rn i− ix

1 1

( ) ( )0

,.., 1 ,.., 1

(( ) )det

detlim lim

det ( )det

T T

R R

R

R Tj iR jn i j R jnR

Rn nR

n in i R jn sx

n j x s s n js i ni j is

n j i jx x x x R n i ii

n je

x e x

n j xx

λσ λ σ

σ

λ

−⎛ ⎞− ⎟⎜ ⎟⎜ ⎟⎜− ⎟ − −⎜ ⎟ + −⎝ ⎠

=− −→− →− −

⎡ ⎤−⎢ ⎥⎢ ⎥⎡ ⎤ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦=

⎡ ⎤ ⎡ ⎤−⎣ ⎦⎣ ⎦

∑

1ix = −

[ ] [ ]det ( ) ( 1) det ( ) RR

i j n iR n i Rn j n j− −

−− − = −

[ ]

( )0

(( ) )det

( 1)

det ( )

T

RR Ti

R jnR

R

n jn j R in s

s n js j nkis

n iR

n ie

n j

λσ

σ

λ

−⎛ ⎞− ⎟⎜− ⎟⎜ ⎟⎜ ⎟⎜ − −⎟ + −⎟⎜⎝ ⎠

=−

⎡ ⎤−⎢ ⎥⎢ ⎥−⎢ ⎥⎣ ⎦

−

∑


Replacing (4.357) by (4.362) in (4.348) we find that the joint pdf of the eigenvalues is givenby

where

By the same reasoning as in (4.351)-(4.354), the capacity under independent fading for is found to be

Evaluating (4.293) and (4.365) for several parameter choices and comparing the capacityvalues with the results from [71], an exact match was observed, which clearly validates theprocedure employed to derive the capacity formulas.

A final remark is required. Despite the fact that the configurations and have been treated independently in this thesis, the author has proved (see [72])

that it is indeed possible to perform a unified derivation that contemplates both situations.The universal complex Wishart density function obtained therein provides the necessarymomentum for the derivation of a set of all-embracing analytic formulas for the averagecapacity of the MIMO wireless channel under receiver-sided correlated fading.

4.6.5 CAPACITY UNDER TRANSMITTER-SIDED CORRELATED FADING

When correlation exists only between signals departing from different transmitter anten-nas (i.e. ), which may happen if the transmitter antennas are close to one another,then the capacity derivation must follow from the pdf in (4.311). The situation is a littlebit different because of the dependence on the quadratic form , which is differentfrom (4.308). However, using the properties of block determinants, we may write

where is . Hence, assuming that no channel state information is available at thetransmitter, (4.308) is equivalent to

where we have to note that, for simplicity, the optimization with respect to the input co-variance matrix is being avoided. For , the quadratic form is full-rank, andtherefore if we write , then has linearly independent columns and the average

(4.363)

(4.364)

(4.365)

, ( )0

(( ) )( ) det det

( 1) T

RR TT i

R T R jnR

n jn j R in sn j

n n i s n js j nkis

n if H e λ

σ

σλ

λ=

−⎛ ⎞− ⎟⎜− − ⎟⎜ ⎟⎜ ⎟⎜ ⎟ − −+ −⎟⎜⎝ ⎠

=

⎡ ⎤−⎢ ⎥⎡ ⎤′= ⎢ ⎥⎣ ⎦ −⎢ ⎥⎣ ⎦∑IΣ

Λ Λ

[ ][ ]

1( ) ,,

Volume!

det ( )!(2 ) ( )

mn n m n mm n m in

n p m n

VH p

m jn nππ

−−

−= −

⎡ ⎤⎢ ⎥′ = ⎢ ⎥−Γ ⎢ ⎥⎣ ⎦∏

T Rn n<

[/,

11

/

0 0

( ) ( 1) ( ) det ( )!

1,

ln 2 !

T R R

T R lR T R l

T

kjR T iT

T T

n n nn n q l

avg n n R n q R T i

l nk q

n n j q p p lnT T

p ln n

C H q n l n n j q

pe n nl

lp

δγ

γ γ

= < −−

= +=

+ − − −

= = ×

⎡ ⎤⎢ ⎥′= − − + − − ⋅⎢ ⎥⎢ ⎥⎣ ⎦

⎤⎡ ⎤⎛ ⎞⎛ ⎞ ⎡ ⎤ ⎥⎢ ⎥⎟⎜ ⎟⎜⎟⋅ − Γ ⎢ ⎥⎜ ⎥⎟⎢ ⎥⎜⎟ ⎟⎜⎜ ⎟⎝ ⎠⎜ ⎢ ⎥ ⎥⎝ ⎠ ⎣ ⎦⎢ ⎥⎣ ⎦ ⎥⎦

∑∑ ∏

∑ ∑

IΣ ε

T Rn n≥

T Rn n<

(4.366)

(4.367)

R Rn= IΣ

HH H

( ) ( )det det detH

H HH H

m mmm n

n n n

+⎡⎛ ⎞⎛ ⎞⎛ ⎞⎤⎟ ⎟ ⎟⎜ ⎜ ⎜⎢ ⎥⎟ ⎟ ⎟⎜ ⎜ ⎜ = + = +⎟ ⎟ ⎟⎜ ⎜ ⎜⎢ ⎥⎟ ⎟ ⎟−⎟ ⎟ ⎟⎜ ⎜ ⎜⎝ ⎠⎝ ⎠⎝ ⎠⎢ ⎥⎣ ⎦

I 0 I 0I AA AI AA I A A

A I 0 I A I

A m n×

2log det H

Tavg nT

C Enγ⎡ ⎛ ⎞⎤⎟⎜= +⎢ ⎥⎟⎜ ⎟⎜⎝ ⎠⎢ ⎥⎣ ⎦

H I H H

T Rn n≤ HH HH=G H G Tn


capacity is evaluated by eigendecomposing . As a result, we simply have to exchange by , by , and finally make in (4.286) to obtain the desired ca-

pacity expression

where is the vector of eigenvalues of . Finally, when and there is no fadingcorrelation at the receiver, matrix is rank-deficient. This was exactly the conditionaimed to be solved by this section, and for which the solution was found to be (4.354).Therefore, we again exchange by , by , and finally make in(4.354) to obtain

Equations (4.368) and (4.369) are the consequence of capacity duality in single-sided cor-related MIMO systems, that is, the fact that the capacity in one direction for equals the capacity in the other direction for , where and are the respectivenumbers of transmitting antennas.

4.6.6 NUMERICAL RESULTS AND DISCUSSION

We shall consider several distinct situations in the subsequent capacity analysis. Neverthe-less, at the correlated side of the communication link, the analysis will be always confinedto linear antenna arrays of fixed maximum length , and a maximum of eightantennas. Moreover, the antennas will be considered evenly distributed within . Whenisotropic scattering is assumed, the entries of the covariance matrix are given by

where the zeroth-order Bessel function of the first kind, is the phase con-stant, and is the number of antennas at the correlated side of the link.

First, we consider the downlink capacity of a typical cellular mobile communicationsystem, and assume that correlation is present only at the mobile unit. The scattering isisotropic around the MU, mutual coupling effects are ignored, and the received SNR isfixed at . Figure 4.13 shows a 3-dimensional capacity plot as a function of thenumber of transmitter and receiver antennas, which includes the scenario of two-sided in-dependent fading for comparison. The capacity values were obtained using expressions(4.286), (4.354), (4.293) and (4.365). We conclude that the capacity under receiver-sidedcorrelated fading always lags behind the capacity under two-sided independent fading, andthis condition becomes more pronounced as the number of antennas increases. Given that

(4.368)

(4.369)

HGG

RΣ TΣ Rn Tn ( )/R Tn nγ γ→

TT

/, ( 1)

1

/( )

0 0

( )!( ) det

1 , ,

ln 2 !

T

T RT R R

kiRT i

nn n R

avg n n n jik

n j p p lnT T

i ip l

n jC H

pe n nl

lp

δυ γ

υ

υ γ υ γ

≤− − +

=

− −

= =

⎡ −⎢= ⋅⎢⎣

⎤⎡ ⎤⎛ ⎞⎛ ⎞ ⎡ ⎤ ⎥⎢ ⎥⎟⎜ ⎟⎜⎟⋅ − Γ ⎢ ⎥⎜ ⎥⎟⎢ ⎥⎜⎟ ⎟⎜⎜ ⎟⎝ ⎠⎜ ⎢ ⎥ ⎥⎝ ⎠ ⎣ ⎦⎢ ⎥⎣ ⎦ ⎥⎦

∑

∑ ∑

Σ υ

Tυ TΣ T Rn n>HGG

RΣ TΣ Rn Tn ( )/R Tn nγ γ→

( )TT

/ 1, ( 1)

11

/( )

0 0

( )!( ) ( ) det

1,

ln2 !

R T T

TT RT R l R

iR

kjRT qi

i i

R

n n nn ln n R

avg n n q n jql nk q

n j p p lnT T

q qp ln n

n jC H q

pe n nl

lp

δυ γ

υυ

υ γ υ γ

−> −− − +

= +=

− −

= = ×

⎡ ⎤ ⎡ −⎢ ⎥ ⎢′= − ⋅⎢ ⎥ ⎢⎢ ⎥ ⎢⎣⎣ ⎦⎤⎡ ⎤⎛ ⎞⎛ ⎞ ⎡ ⎤ ⎥⎢ ⎥⎟⎜ ⎟⎜ ⎢ ⎥⎟⋅ − Γ⎟⎜ ⎥⎜⎢ ⎥⎟ ⎟⎜ ⎟⎜⎟ ⎢ ⎥⎜ ⎝ ⎠ ⎥⎝ ⎠⎢ ⎥⎣ ⎦⎣ ⎦ ⎥⎦

∑∑ ∏

∑ ∑

Σ υ ε

R

0Nγ γ=

0Mγ γ= N M

(4.370)

max 1.2d λ=

maxd

[ ]R/T 0 max /( ( ) / )R Tir J i r d nβ= −Σ

0()J ⋅ 2 /β π λ=

/R Tn

5 dBγ =


the capacity is limited as (4.35), and due to the imposed at the receiver, add-ing more antennas will eventually become negligible, since the capacity increase will tendto stagnate. A different plot is shown in Figure 4.14 which confirms this behaviour. We

observe that in the correlated scenarios the most significant capacity increase occurs while, and that substantial stagnation occurs otherwise. Moreover, we observe that to

attain an equivalent capacity increase as by switching from to , we need to

FIGURE 4.13 Average MIMO channel capacity as a function of and , for . The black line is for receiver-sided correlated fading with isotropic scattering, and the gray line is for two-sided independent fading.

FIGURE 4.14 Average MIMO channel capacity as a function of , for several values of , and . For receiver-sided correlated fading, isotropic scattering is assumed.

2 4

6 8

2

4

6

8 0

5

10

15

Number of

transmitter a

ntennas − nT


Average

Capacity

− bits/s/Hz

Rn Tn 5 dBγ =

Tn → ∞ maxd

1 2 3 4 5 6 7 8 0

2

4

6

8

10

12

14


Average

Capacity

− bits/s/H

z nR = 8

nR = 4

nR = 2

nR = 1

Receiver-sided correlated fading; Two-sided independent fading;

Tn Rn 5 dBγ =

T Rn n<

2Rn = 4Rn =


switch from to . This means that under receiver-sided correlated fadingwith fixed , adding more receiver antennas has a diminished impact on the capacity ascompared to two-sided independent fading. In fact, the higher is , the lesser is the im-pact of increasing the former, and one should expect this impact to eventually become ir-relevant.

To analyse the effects of transmitter-sided correlation, we now consider the uplink ofthe cellular communication system, and plot the capacity as a function of the number ofmobile unit antennas , and the number of base station antennas . This is shown inFigure 4.15, where the downlink capacity is also included for comparison. As expected,

we notice that the uplink capacity is higher than the downlink capacity if , andthe opposite if , while an exact match occurs if . Naturally, this onlyhappens because is assumed equal on both sides. We also note a peculiar behaviour ofthe uplink capacity, as it is virtually non-increasing for . This fact makes it moredifficult to increase the uplink capacity by increasing than to increase the downlinkcapacity by increasing . If one wishes to obtain uplink capacity enlargements, eitherone improves , or one has to increase . Moreover, since the downlink capacity is mostoften the more demanding, one may usually require that , where approxi-mately fixes the uplink capacity, and varies so as to achieve the desired downlink ca-pacity.

To conclude, we study the implications of receiver-sided correlation with non-iso-tropic scattering on the average capacity of the MIMO channel. Non-isotropy is worthconsidering, since it may frequently apply to the propagation conditions of some sub-ur-ban/urban environments. To that end, we use the correlation model proposed in [73], for

FIGURE 4.15 Uplink and downlink average MIMO channel capacities as a function of , for several values of , and . Correlated fading and isotropic scattering are assumed for the mobile unit

only.

4Rn = 8Rn =

maxd

Rn

MUn BSn

1 2 3 4 5 6 7 8 1

2

3

4

5

6

7

8

9

10

Number of antennas at the mobile unit − nMU

Average

Capacity

− bits/s/H

z

nBS = 2

nBS = 1

nBS = 4 nBS = 8

nBS = 2

nBS = 1

nBS = 4

nBS = 8 Uplink; Downlink;

Up = Down;

MUnBSn 5 dBγ =

BS MUn n>

BS MUn n< BS MUn n=γ

3MUn >

MUn

BSn

γ BSn

MU BSn n≥ BSn

MUn


which

where is the mean angle-of-arrival (AOA) relative to the direction of motion, is the zeroth-order modified Bessel function of the first kind, controls the

width of the AOA, and for the array under consideration we have .Equation (4.365) is easily obtained from (4.295) and (4.297), by setting (i.e. themobile unit moves in parallel with the axis of the linear array). Average capacity curves inthe configuration are shown in Figure 4.16. We observe that non-isotropic scat-

tering has the effect of decreasing the average capacity, and this feature is more pro-nounced for (or equivalently ), that is, when the receiver moves towardor away from the mean direction of multipath arrival. This condition might have been ex-pected since the multipath components that arrive nearly parallel to the axis of the lineararray (which, in this case, coincides with the direction of motion), are the ones that mostcontribute to the correlation between the antennas, and hence most likely contribute toreduce the capacity. From Figure 4.16 we also infer a substantial capacity reduction as thewidth of the AOA narrows, i.e. as increases, which is consistent with the fact that theantennas become more and more correlated.

4.6.7 CONCLUSION

Section 4.6 has presented the derivation of the closed-form expression for the average ca-pacity of correlated MIMO channels when the number of receiver antennas exceeds thenumber of transmitter antennas. In addition, closed-form formulas for the capacity undertransmitter-sided correlation and two-sided independent fading were obtained. Numerical

(4.371)

FIGURE 4.16 Average MIMO channel capacity as a function of , for , receiver-sided correlated fading, and nonisotropic scattering modelled by (4.297) with and

.

[ ] ( )R2 2 2

0 0( 2 cos / ( )ir ir cir I d j d Iκ β βκ φ κ= − +Σ

[ , )cφ π π∈ −

0( )I ⋅ [0, [κ ∈ ∞

max( ) /ir Rd i r d n= −

irθ π=

T Rn n=

1 2 3 4 5 6 7 81

2

3

4

5

6

7

8

9

10

Number of antennas − nT

Average

Capacity

− bits/s/Hz

= nR

φc = 0 rad; κ = 2,...,8;φc = π/2 rad; κ = 2,...,8;

Isotropic scattering (κ = 0);

κ increases

T Rn n= 5 dBγ = 0, /2cφ π=

2, ..., 8κ =

0 radcφ = cφ π=

κ


results quantified a significant capacity reduction when correlation was present, and pre-dominantly when the scattering was not isotropic. In particular, it was shown that capacitystagnates very easily when transmitter-sided correlation is present, and that fixed-lengthcorrelated antenna arrays greatly refrain the unbounded capacity potential of independentfading scenarios. It was also observed that in a typical cellular mobile communication sys-tem, the most logical configuration option (in what concerns capacity) is to set the numberof mobile unit antennas larger or equal than the number of base station antennas, irrespec-tive of any length constraints and/or correlation applicable to the antenna arrays of mo-bile units.

CHAPTER 5

OVERSPREAD TRANSMISSION OVER WIRELESS LTV MIMO SYSTEMS

5.1 Introduction

In common urban or sub-urban propagation environments, the typical delay spread of thewireless mobile channel is much smaller than its coherence time, which means that thechannel may be viewed as almost time-invariant during consecutive time-frames spanningless than the coherence time. Since the coherence time is in the order of milliseconds, oneusual approach for designing the receiver is to first consider the channel as time-invariant,devise an equalizer to the received signal (e.g. a linear or decision-feedback equalizer) andthen use an adaptive technique (e.g. least mean square (LMS), recursive least squares(RLS), or Kalman) to track the channel variations and update the equalizer parameters(viz. [77], [78], [79],[80], and [81]). This procedure finds its most important application infrequency-selective slowly fading channels, and, under mild conditions, channel trackingmay be replaced by blind or decision directed techniques. There are, however, severaldrawbacks to the adaptive process, namely: 1) it is not a smooth process, as abrupt changesbetween consecutive intervals are required [78]; 2) no pilot-aided transmission (PAT) is ex-ploited; 3) it may be unstable under certain channel variations [82]; 4) it does not take ad-vantage of the time variations in the channel to improve the receiver performance.Introducing “vertical Bell Labs layered space-time” (V-BLAST) [83] architecture into thepicture also has its own problems [84], [85].

Recent research has shown that not incorporating the channel variations into the re-ceiver design penalizes its performance. It has been shown that the RAKE receiver is in-efficient when the channel is time-varying [86], and attempts have been made to correctthe problem by exploiting Doppler diversity [87], [88], [89] and [90], using techniques suchas time-frequency (canonical) representations (TFRs) and the basis-expansion model(BEM). These developments encourage the study of wireless channels in a more generalsetting, as the inherent deficiencies of receiver implementation are the consequence of

119

120 CHAPTER 5 OVERSPREAD TRANSMISSION OVER WIRELESS LTV MIMO SYSTEMS

considering the channel as interval-wise invariant. The primary barrier for such a study isthe analysis of linear time-varying (LTV) filters in a system-wide perspective, which ismuch more involved than analysing conventional linear time-invariant systems. Designingthe receiver structures when all its units are time-varying may seem an overwhelming task,at least without a tractable mathematical description, but if it proves to be possible then itwill lead to results of the utmost importance and generality. Aiming at a possible solution,this chapter will dwell into the more practical aspects of MIMO transmission, receptionand detection. The idea is to study MIMO systems in a more generic framework, with noconstrictive assumptions being presumed from the start except those of a time-varyingwireless propagation medium and accurate channel estimation at the receiver (viz. [91],[92], [93] and [94]). This will allow for more general conclusions that may me made morespecific by simple deduction.

The wireless channel between each pair of antennas will be considered linear andtime-varying. Also, depending on the coherence-time , coherence-band-width , signal bandwidth , and a signalling interval , it can be of twotypes: 1) underspread if

or 2) overspread if

When the signal bandwidth is the reciprocal of the signalling interval (half roll-off signal-ling), the underspread and overspread conditions are simply and

, respectively. Both and depend on the stochastic properties ofthe wireless channel, which in turn are a function of the relative distance and motion be-tween transmitter and receiver, and also of the propagation conditions within the wirelessmedium. If one is to devise a slowly fading, frequency non-selective channel, by concur-rently choosing a signal bandwidth and a signalling interval , then anecessary condition is that it be underspread. In an underspread channel (such as the typ-ical mobile propagation channel), frequency selectivity ( ) induces slow fading(time-flatness, ), and time selectivity (fast-fading, ) induces a frequen-cy-flat response ( ). If we are forced to have selectivity in both time and frequen-cy then the channel is necessarily overspread.

The typical wideband channel (e.g. CDMA2000, W-CDMA) is overspread in delay( ), hence frequency-selective, yet slowly fading ( ). Onthe contrary, to capitalize on capacity, some multicarrier systems (e.g. OFDM) with verylow carrier separations (not uncommonly lower than 10 kHz, viz. 3GPP LTE eMBMS,1 kHz in DAB, a.k.a. Eureka 147) may be frequency-flat ( ),hence delay-underspread, but their signalling rate may be too low to permit a slow fadingbehavior. These narrowband sub-channels may be fast-fading, hence Doppler-overspread( ), which means that the induced Doppler spread is not negligi-ble from the receiver perspective, as inter-carrier interference (ICI) will be introduced.Channels such as the ones invoked self-dictate the need for a thorough analysis of trans-mission overspreading, let alone their blending with MIMO technology. Within the gener-

(5.1)

(5.2)

( ) 1/2c Dt fπΔ ≈

( ) 1/2cf τπσΔ ≈ W T

[ ][ ] 2( ) ( ) 1 ( /2 )( /2 ) (2 )c c Df T W t f T W WTτσ π π π −Δ Δ > < =∼

[ ][ ] 2( ) ( ) 1 ( /2 )( /2 ) (2 )c c Df T W t f T W WTτσ π π π −Δ Δ < > =∼

2(2 )Dfτσ π −<2(2 )Dfτσ π −> ( )ctΔ ( )cfΔ

( )cW f< Δ ( )cT t< Δ

( )cW f> Δ

( )cT t< Δ ( )cT t> Δ

( )cW f< Δ

1/ ( )cW W fτσ → Δ ( )cT tΔ

( ) /2cW f Tτσ π< Δ → <

( ) /2c DT t f W π> Δ → >

5.2 Input/Output Model 121

ic framework of this chapter, all types of wireless channels will be studied withoutpreliminary distinction.

5.2 Input/Output Model

The time-varying channel impulse response between the -th input and the -th output isdenoted by , and it is conveniently chosen as the response at time of an impulseapplied at time . It can legitimately be thought of as the impulse response at a speci-fied time , since the time-scale at which the wireless channel varies is typically much long-er than the delay spread of the channel. For a time-varying multipath wireless channel, thechannel impulse response is given by

accounting for the attenuations and delays of multipath replicas. Dueto the random nature of the wireless medium, we must interpret the time-varying param-eters of (5.3) as sample functions of statistically independent stochastic processes.

Denoting the number of inputs by , the transmitted signal from the -th input by, and the additive noise at the -th output by , the signal received by the -th out-

put is given by the superposition

In a more compact vector-matrix notation, we can write

where we define the column vector-valued signals , ,, and the matrix-valued channel response ( being

the number of outputs).

5.2.1 BASEBAND MIMO CHANNEL MODEL

In the model of (5.5), is the transmitted vector-valued signal which, in a typical wire-less application, is limited to an approximate passband of bandwidth

and center frequency . The “up-conversion”/”down-conversion” to/from thecarrier frequency are performed solely to permit transmission over the wireless medi-um, and are always the last and first stages of transmitter and receiver operation, respec-tively. Since most of the other processing stages are performed at the baseband, itbecomes important to derive an equivalent baseband model from (5.5).

Defining the Fourier transform of as and noting that, since is

(5.3)

(5.4)

(5.5)

j i

( , )ijc tτ t

t τ−

t

( )( ) ( )

1

( , ) ( ) ( ( )),ijN t

ij ijij k k

k

c t t tτ α δ τ τ=

= −∑( )( )ijk tα τ( )( )ij

k t ( )ijN t

Tn j

( )js t i ( )in t i

1

( ) ( , ) ( ) ( ).Tn

i ij j i

j

r t c t s t d n tτ τ τ∞

−∞=

= − +∑∫

( ) ( , ) ( ) ( ),t t t d tτ τ τ∞

−∞= − +∫r C s n

[ ] 1( ) ( )Ri nt r t ×=r [ ] 1( ) ( )

Ri nt n t ×=n

[ ] 1( ) ( )T

j nt s t ×=s [ ]( , ) ( , )R T

ij n nt c tτ τ ×=C Rn

( )ts

[ ]/2, /2c cf B f B− += 2B W cf

cf

( )ts [ ]( ) ( )jf S f=s ( )ts


real-valued, the Fourier transform is conjugate symmetric (i.e. ), we have

If the baseband signal is denoted by and its Fourier transform by , then the lattermay be defined as two times the positive frequency part of shifted to the origin, that is

which substituted into (5.6) yields

We should have in mind that due to the conjugate symmetry of , no loss of informa-tion occurs if one chooses for the purpose of a signal model the mathematical conve-nience of relative to . In fact, as shown in Figure 5.1, is easily obtained from

using carrier multiplication followed by real part extraction, and is recovered from using carrier multiplication followed by ideal lowpass filtering. Although, for simplic-

ity, these operations are best seen in the complex domain, the real domain equivalent hasa similar construction.

The communication designer can then concentrate its energies on defining the basebandsignal that should be used for transmission.

To unravel the baseband model for the wireless MIMO channel we start by taking theFourier transform of relative to the response delay and denote it by . Clearly

Now, if is the output of a noiseless channel, it is related to the input by

which, interchanging the order of integration, easily leads to

(5.6)

(5.7)

(5.8)

FIGURE 5.1 Illustration of up-conversion and down-conversion operations in the complex domain.

(5.9)

(5.10)

(5.11)

( ) ( )f f∗= −s s

/22 2 2

/2

/2 /22 2 2

/2 /2

( ) ( ) ( ) ( )

2 e ( ) e 2 ( ) .

c

c

cc

c

f Bj ft j ft j ft

f B

f B Bj ft j ft j f t

cf B B

t f e df f e f e df

f e df f f e df e

π π π

π π π

∞ +−

−∞ −

+ +

− −

⎡ ⎤= = + −⎢ ⎥⎣ ⎦

⎧ ⎡ ⎤ ⎫⎪ ⎪⎪ ⎪⎢ ⎥= ℜ = ℜ +⎨ ⎬⎢ ⎥⎪ ⎪⎪ ⎪⎩ ⎣ ⎦ ⎭

∫ ∫

∫ ∫

s s s s

s s

( )ts ( )fs

( )fs

2 ( ), /2,( )

, /2,

cf f f Bf

f B

⎧ + ≤⎪⎪= ⎨⎪ >⎪⎩

ss

0

/2

2 2 2

/2( ) e ( ) e ( ) .c c

Bj ft j f t j f t

Bt f e df e t eπ π π

+

−

⎧ ⎡ ⎤ ⎫⎪ ⎪⎪ ⎪⎢ ⎥= ℜ = ℜ⎨ ⎬⎢ ⎥⎪ ⎪⎪ ⎪⎩ ⎣ ⎦ ⎭∫s s s

( )fs

( )ts ( )ts ( )ts

( )ts ( )ts

( )ts

( )ts

2 cj f te π

eℜ i ( )ts( )ts

Lowpass Filter Bank

22 cj f te π−

( )tz/2B/2B−

1

( )ts

( , )tτC τ ( , )f tC

2( , ) ( , ) .j ft f t e dfπ ττ∞

−∞= ∫C C

( )tu ( )ts

2( ) ( , ) ( ) ( , ) ( ) ,j ft t t d f t e t df dπ ττ τ τ τ τ∞ ∞ ∞

−∞ −∞ −∞= − = −∫ ∫ ∫u C s C s

2

2

( ) ( , ) ( )

( , ) ( ) .

j f

j ft

t f t e t d df

f t f e df

π τ

π

τ τ∞ ∞

−∞ −∞∞

−∞

⎡ ⎤= −⎢ ⎥

⎢ ⎥⎣ ⎦

=

∫ ∫∫

u C s

C s


It now becomes clear that is the inverse Fourier transform of , which is amatrix-vector product and, by virtue of the band-limited nature of , is also band-lim-ited, except that the passband will be slightly larger. This propertyalso follows from the Fourier transform of (5.11), i.e. the convolution

where the variable is associated with the rate of change of the wireless channel and iscalled the Doppler frequency. One infers that, despite the fact that MIMO channel responses

or are not explicitly band-limited in the variable, the only frequencies thatare required to compute the output or lie in the passband ,and thus it is possible to define a baseband representation for that is very similar to(5.7). More specifically, we are able to relax the band-limited condition by letting ,which leads to

or equivalently

It now follows from (5.10) that the noiseless output is given by

Comparing with (5.8), we find that (5.15) provides the relation between and ,which using (5.14) is found to be

which is simply a convolution scaled by . From the Fourier transform

we are led to conclude that, due to the convolution in the variable the bandwidth of will be somewhat larger than that of , which is an obvious consequence of the Dopplerspread of the channel.

Continuing, the complete (noisy) baseband model for the MIMO wireless channel is

(5.12)

(5.13)

(5.14)

(5.15)

(5.16)

(5.17)

( )tu ( , ) ( )f t fC s

( )fs

[ ]/2, /2c u c uf B f B− +

( ) ( , ) ( )

( , ) ( )

f f f d

f fν

ν ν ν ν

ν ν

∞

−∞= − −

= − ⊗

∫u C s

C s

ν

( , )f tC ν( , )fC f

( )tu ( )fu [ ]/2, /2c cf B f B− +( , )f tC

B → ∞

( , ) 2 ( , ),cf t f f t= +C C

2( , ) 2 ( , ) .cj ft t e π ττ τ −=C C

( )tu

2 ( )

2 2

( ) ( , ) ( ) ( , ) e ( )

e ( , ) ( )

c

c c

j f t

j f j f t

t t t d t t e d

t t e d e

π τ

π τ π

τ τ τ τ τ τ

τ τ τ

∞ ∞−

−∞ −∞∞

−

−∞

= − = ℜ −

⎡ ⎤= ℜ −⎢ ⎥

⎢ ⎥⎣ ⎦

∫ ∫∫

u C s C s

C s

( )tu ( )tu

2( ) ( , ) ( )

1( , ) ( ) ,

2

cj ft t t e d

t t d

π ττ τ τ

τ τ τ

∞−

−∞∞

−∞

= −

= −

∫∫

u C s

C s

1/2

1( ) ( , ) ( )

21

( , ) ( )2

f f f d

f fν

ν ν ν ν

ν ν

∞

−∞= − −

= − ⊗

∫u C s

C s

ν ( )tu

( )ts


thence given by

Equation (5.18) is the final channel model because it accounts for the noise that is inevi-tably present at the channel outputs. In the context of the current presentation, ismodelled as a wide-sense stationary (WSS) vector random process with zero mean, auto-correlation

and power density spectrum

In other words, the noise process is both spatially and temporally white, hence white. Since(in theory) the noise process spans the entire spectrum, one must be more careful on char-acterizing the baseband representation of . To that end, we shall firstly require thefollowing result from the theory of stochastic processes.

LINEAR FILTERING OF VECTOR STOCHASTIC PROCESSES: If and are, respectively,input and output (complex) WSS vector random processes of a linear time-invariant (LTI)system with transfer function , then the relation between input and output power den-sity spectrums is

where is the conjugate transpose of .This result is easily derived from the cor-relation functions and the convolution integral. We have

and thus

where is the matrix-valued cross-correlation function between input and output. is related to by

From (5.23) and (5.24) it follows that , and taking the Fou-

(5.18)

(5.19)

(5.20)

(5.21)

(5.22)

(5.23)

(5.24)

2

1( ) ( , ) ( ) ( )

2

1( , ) ( ) ( ).

2j ft

t t t d t

f t f e df tπ

τ τ τ∞

−∞∞

−∞

= − +

= +

∫∫

r C s n

C s n

( )tn

[ ] 0( ) ( ) ( ) ( ) ,2

T NE t tτ τ δ τ= + =nnR n n I

0( ) .2

Nf =nnS I

( )tn ( )tn

( )tx ( )ty

( )fL

( ) ( ) ( ) ( ),Hf f f f=yy xxS L S L

( )H fL ( )fL

( ) ( ) ( ) ,t t dα α α∞

−∞= −∫y L x

[ ] [ ]1 1

( ) ( ) ( ) ( ) ( ) ( )2 2

( ) ( ) ( ) ( ),

H HE t t E t t d

d

τ τ α α τ τ

α τ α τ τ τ

∞

−∞∞

−∞

= + = − +

= − = ∗

∫∫

yy

xy xy

R y y L x y

L R L R

( )τxyR

( )τxyR ( )τxxR

[ ] [ ]1 1

( ) ( ) ( ) ( ) ( ) ( )2 2

( ) ( ) ( ) ( ).

H H H

H H

E t t E t t d

d

τ τ τ α α α

τ α α α τ τ

∞

−∞∞

−∞

= + = + −

= + = ∗ −

∫∫

xy

xx xx

R x y x x L

R L R L

( ) ( ) ( ) ( )Hτ τ τ τ= ∗ ∗ −yy xxR L R L


rier transform reveals (5.21).

Recalling that, as illustrated in Figure 5.1, is determined from by down-con-version, we first modulate by the carrier and obtain a vector-valued signal

with autocorrelation

Taking the Fourier transform, the power density spectrum of is

which, for an arbitrary noise bandwidth , corresponds to a translation of the passband to the baseband . Due to the white nature of ,

this spectral translation has no effect on the noise process, but the same does not happenin the lowpass filtering stage of the down-conversion. In effect, with the aid of (5.21) andthe definition of the ideal lowpass filter

we would derive the power density spectrum of as , and otherwise, which is clearly a lowpass filtered version of the white noise. However, since

is (in theory) not band-limited in spectrum, we are prompted to choose a basebandnoise model with that leads to the following characterization of :

In short, we characterize as a WSS process with a white power density spectrum thatis two times that of . It must be ascertained that, even though white noise is not phys-ically realizable, this theoretical characterization is neither unrealistic nor fallacious from adesign point of view. In fact, before down-conversion, a receiver is expected to pass thereceived signal through a bandpass filter with passband much smaller than the center fre-quency, but wider than the signal bandwidth. As a consequence the noise will have indeeda baseband representation, one that we interpret as spectrum-flat for mere mathematicalsimplicity. In addition, in the optimum receiver design, the spectral components of noiseoutside the received signal band will always have to be removed before detection, andhence the “white” assumption does not adulterate the design process.

Figure 5.2 depicts how the passband model of (5.5) reduces to the baseband modelof (5.18). Based on this equivalence, the study that will follow will focus on the basebandchannel model of the wireless MIMO system.

(5.25)

(5.26)

(5.27)

(5.28)

( )tn ( )tn

( )tn 22 cj f te π−

2( ) 2 ( ) cj f tt t e π−=z n

[ ] [ ] 2

2

1( ) ( ) ( ) 2 ( ) ( )

22 ( ) .

H c

c

T j f

j f

E t t E t t e

e

π τ

π τ

τ τ τ

τ

−

−

= + = +

=

zz

nn

R z z n n

R

( )tz

0( ) 2 ( ) ,cf f f N= + =zz nnS S I

NB

[ ]− +/2, /2c N c Nf B f B [ ]− /2, + /2N NB B ( )tn

⎧ ≤⎪⎪= ⎨⎪ >⎪⎩

, /2,( )

, /2,

f Bf

f B

IL

0

( )tn = ≤0( ) , /2f N f BnnS I 0

( )tn

→ ∞B ( )tn

2

20

0

( ) 2 ( ) ,

( ) 2 ( ) ( ) ,

( ) 2 ( ) .

c

c

j f t

j f

c

t t e

e N

f f f N

π

π ττ τ δ τ

−

−

=

= =

= + =

nn nn

nn nn

n n

R R I

S S I

and

( )tn

( )tn


5.2.2 BASEBAND MIMO INPUT DESIGN

The goal of a digital communication system is to convey digital information between twodistinct points in space, separated by a communication channel. One of the points will actas the transmitter of information and the other as the receiver of information, thoughboth can change roles. This information has usually been formatted as a sequence of binarydigits (bits) that may contain some added redundancy to improve the fidelity ofsignal transmission. Implicitly, the binary sequence has a binary rate and a binary signallinginterval , which means that the digital communication system shall convey a bitevery seconds throughout the channel.

To reduce the effective rate of transmission, the transmitter typically maps each blockof bits from the sequence to a symbol of the complex set , yieldingthe discrete-time sequence with a symbol rate and signalling interval . Ina MIMO system there are inputs to the channel, meaning that distinct sequences ofsymbols can be transmitted at the same time. We denote this set of sequences as a vector-valued sequence . To convey to the receiver, it must be converted in some wayto a vector-valued continuous waveform , appropriate for transmission over the base-band channel . Recalling the sampling theorem, we notice that may be regardedas a sequence of vector-valued samples from some signal with Fourier transform .Ideal sampling with an impulse train leads to

which has the Fourier transform

Since is periodic with period , in the worst case scenario ( ) the sam-ples can be conveyed by a signal with bandwidth . One concludes at once thatthe transmitter may pass through a lowpass filter before transmission withoutthe risk of losing any information required to reconstruct the samples. Letting be the

FIGURE 5.2 Equivalent passband and baseband models for the MIMO wireless channel.

( )ts UpConverter

( )tsτ( , )tC

( ),tn

( )tr DownConverter

( )tu ( )tr

= 0( )2

NfnnS I

( )ts ( )trτ

1( , )

2tC

( )tu

( ),tn = 0( )f NnnS I

⇔

(5.29)

(5.30)

= 0,1kb

bR

= 1/b bT R

bT

= 2log ( )r M kb 1M

i ia =

ka sR = 1/s sT R

Tn Tn

ka ka

( )ts

( , )f tC ka

( )tx ( )fx

( ) ( ) ( ) ( ) ,s ss kk k

t t t kT t kTδ δ∞ ∞

=−∞ =−∞

= − = −∑ ∑x x a

( )2 2 1( ) .s sj fT j fkT

s ks sk k

kf e e f

T Tπ π

∞ ∞−

=−∞ =−∞

⎛ ⎞⎟⎜= = = − ⎟⎜ ⎟⎜⎝ ⎠∑ ∑x a a x

( )s fx sR ( ) 0, s f f> ∀x

( )tx /2sR

( )s fx ( )fG

τ( )G


impulse response of that filter, then is obtained by convolving with , yielding

The channel output is given by

where is the convolution between and , i.e.

which induces the frequency response

For mathematical tractability, neither nor are yet being restricted to causal im-pulse responses. Based on the MIMO channel model, the input model of the MIMO sys-tem is now completely defined, as shown in Figure 5.3. If no noise was present and the

condition was met, it is apparent that the input sequence could be easily recov-ered by sampling the noiseless output. The sampling operation would yield

which, for a delay of symbol intervals, would require

or equivalently

(5.31)

(5.32)

(5.33)

(5.34)

FIGURE 5.3 Transmission model with discrete-time vector-valued input sequence.

(5.35)

(5.36)

(5.37)

( )ts ( )s tx

( ) ( ) ( ) ( ) .ss kk

t t d t kTτ τ τ∞∞

−∞ =−∞

= − = −∑∫s G x G a

0

1( ) ( , ) ( ) ( ) ( , ) ( ) ( )

2

( , ) ( ),

s

s kk

t t t d t t t d t

t kT t t

α α α τ τ τ∞ ∞

−∞∞

=−∞

= − + = − +

= − +

∫ ∫

∑

r C s n H x n

H a n

τ( , )tH α( , )tC τ( )G

0

1 1( , ) ( , ) ( ) ( , ) ( ) ,

2 2t t t d

ατ α τ α τ α α

∞

= ⊗ = −∫H C G C G

=1

( , ) ( , ) ( ).2

f t f t fH C G

τ( )G τ( , )tH

( )ts ( )trτ

1( , )

2tC

( )tu

( )tn

⇔

τ( )G

( )tr( )tu

( )tn

τ( , )tH

ka

ka

≥R Tn n

( ) (( ) , ) ,s s s kk

lT l k T lT∞

=−∞

= −∑u H a

D

( , ) for all s s k DkT lT kδ −=H I

( )2 21, , .s sj fT j fDT

s ss sk

ke lT f lT e

T Tπ π

∞−

=−∞

⎛ ⎞⎟⎜= − =⎟⎜ ⎟⎜⎝ ⎠∑H H I


These equations define the generalization of the Nyquist criterion to MIMO time-varyingchannels. In other words, they mean that to achieve linear separability of the samples in thevector-valued input sequence, a should be chosen in (5.34) so that satisfies(5.37). Such will exist only when ( is a column matrix) and the upper

matrix of is nonsingular, hence invertible ( is full rank). Besides thecorruption by channel noise, there are two naked-eye difficulties with this approach. Thefirst is that may (and generally will) be distorted by a frequency-selective thatthe transmitter does not know, as it typically has no channel state information (CSI) avail-able. The second is that, even if the transmitter knew , since the latter is time-depen-dent, the spectral shaping function would also have to exhibit a time dependence (i.e.

), with the inconvenience of (5.34) being no longer valid. In fact, by the same rea-soning one concludes that

and

where is the maximum delay spread of the wireless channel at time and

It becomes self-evident that, because depends on the past of by a convolu-tion (either in time or in Doppler frequency), it is much harder to design a transmissionfilter that satisfies (5.37) at every sampling instant, especially if the channel varies very rap-idly. Yet sub-optimal, a practical alternative to achieve (near-)separability of samples wouldbe to consider the channel response interval-wise invariant. More specifically, onecould consider as approximately invariant during (at most) the coherence-time

of the wireless channel, divide the time frame into equispaced intervals, and thenspecify a different function for every set of samples within each interval. If eachinterval was seconds in duration we could roughly write

and (5.37) would become

for all . The feasibility of this approach will naturally depend on the rate ofvariation of the channel response, and may be practical only for slowly-fading channelsthat satisfy the condition , which is typically the case.

(5.38)

(5.39)

(5.40)

(5.41)

(5.42)

( )fG ( , )f tH

( )fG ≥R Tn n τ( , )tH

×T Tn n ( , )f tC ( , )f tC

( )ts ( , )f tC

( , )f tC

( )fG

( , )f tG

τ

τ α τ α α α= − −∫max( )

0

1( , ) ( , ) ( , ) ,

2

t

t t t dH C G

max( )2

0

2

1( , ) ( , ) ( , )

2

1( , ) ( , ) ,

2

tj f

j tt

f t t f t e d

f t f e d

τπ τ

πν

τ τ τ

ν ν ν

−

∞−

−∞

= −

= +

∫∫

H C G

C G

τmax( )t t

πνξ

τν ξ ξ−

−=∫

max

2

( )( , ) ( , ) .

tj

tt t

f f e dG G

( , )f tH ( , )f tG

( , )f tC

( , )f tC

( )ctΔ

( , )f tG slT

( )ctΔ

max1

( ) ( ) ( ) , .( ) ( .( ) ) ( 1).( ) ,2m m m c c cf f f m t m t t m tτ= Δ + Δ < ≤ + ΔH C G

π∞

−

=−∞

⎛ ⎞⎟⎜ − =⎟⎜ ⎟⎜⎝ ⎠∑ 21,sj fDT

s s mk

kf e

T TH I

⎣ ⎦= Δ/( )cm t t

τΔ max( ) ( )ct t


5.2.3 BASEBAND MIMO OUTPUT DESIGN

After defining the vector-valued signal to be transmitted based on the sequence ofsamples , our interest diverges to signal reception and output MIMO design. Irre-spective of the shaping filter used at the transmitter, the discussion shall focus pri-marily on the global transfer function and the discrete-input transmission model of(5.32). The optimum criterion for designing the MIMO output is the minimization of theerror probability of symbol detection, that is, given the received signal

the receiver should make a decision about which sequence of symbols was effectivelytransmitted, and this decision should be performed with minimum probability of error.

Since the input is discrete, it makes sense to search for a model with a discrete outputthat lacks no relevant information for the decision process. The first model that may cometo mind is based on the sampling theorem. Due to the lowpass characteristic of , thenoiseless output will be bandlimited and have a finite energy given by

Its bandwidth is (finitely) larger than that of because of the (limited) time-variabil-ity of the channel response. From the sampling theorem it follows that is uniquely de-termined by its samples taken in intervals of seconds, and hence maybe expanded into a series of orthogonal functions as

The samples may be recovered either by pure sampling or by projecting ontothe basis in the usual manner:

Recalling from (5.32) that

we conclude that the orthogonal set spans the signal space of theset .

Trying to perform detection using the samples in (5.43) is obviously possible becausethey carry all the information present in , but brute estimation of the original symbolvector samples from the samples is another issue. Unfortunately, there aresome serious drawbacks: 1) signal processing at a sampling rate higher than the input sym-bol rate is required, something that even the optimal detector may not need; 2) pre-filter-ing is necessary to reject out-of-band noise and, since no filter is ideal, may provoke signaldamage; 3) the samples possess redundant information that is not important for detection,

(5.43)

(5.44)

(5.45)

(5.46)

(5.47)

( )ts

ka

( , )f tG

( , )f tH

( ) ( , ) ( ) ( ) ( ),s kk

t t kT t t t t∞

=−∞

= − + = +∑r H a n u n

ka

( )fG

( )tu

∞

−∞ −=∫ ∫

/22 2

/2( ) ( ) .

u

u

B

Bt dt f dfu U

uB ( )ts

( )tu

( )ukTu = 1/u uT B

( ) −sinc ( )/u ut kT T I

( )[ ]∞

=−∞

= −∑( ) sinc ( )/ ( ) .u u u

k

t t kT T kTu I u

( )ukTu ( )tu


( )[ ]1

( ) sinc ( )/ ( ) .u u uu

lT lT T dT

τ τ τ∞

−∞= −∫u I u

( ) ( , ) ,s kk

t t kT t∞

=−∞

= −∑u H a


−( , )st kT tH

( )tu

ka ( )ukTr


so further filtering/processing is required to avoid overburdening the detector; 4) thephysically unrealizable ideal functions (i.e. the Nyquist Kernel func-tions) lack full-orthogonality

a fact that may impair detection for certain types of channels; and 5) they fail to capturethe time-varying nature of the MIMO channel, which is not our intention from the verybeginning.

We thus search for an alternative orthogonal set to discretize the output, and to thisend we shall resort to: 1) the theory of general linear time-varying operators (see [95], [96]and [97] for insight); and 2) the classical method of Maximum Likelihood Sequence Esti-mation (MLSE) extended to a time-varying environment.

5.3 Discretizing the MIMO Input-Output Model

We have seen that the output of the MIMO channel is given by

where we remark that the matrix valued functions are withdrawn from parallellines in the domain of the graph of , as depicted in Figure 5.4. This observation con-

trasts with the time-invariant scenario, where the shaping functions are simply shifted rep-licas of one another. Here, each symbol is “carried over” to the output by a differentfunction, making it harder for the receiver to understand what has happened since trans-mission. The channel is fast-fading if the response varies between consecutive sig-nalling intervals , which defaults to a significantly different being used every

seconds.Moving on, we try to devise a model for the system that avoids the intricate convolu-

tion notation. After some thought, we may realize that one possibility that makes reason-

(5.48)

( )−sinc ( )/u ut kT T

( ) ( )1

sinc ( )/ sinc ( ' )/ ( ' )u u u uu k

t kT T t kT T t tT

δ∞

=−∞

− − ≠ −∑

(5.49)

FIGURE 5.4 Transmission domains of a time-varying linear channel as seen by the receiver.

( ) ( ) ( ) ( , ) ( ) ( )

( , ) ( ) ( , ) ( )sl ll

t t t t t t

t t t lT t t

τ

τ

τ

τ

= + = ⊗ +

= ⊗ + = − +∑r u n H x n

H a n H a n

( , )st lT t−H

( , )tτH

sT−2 sT− sT 2 sT0 τ

t

( , )st kT t−

( , )tτH

slT ( , )st lT t−H

slT

5.3 Discretizing the MIMO Input-Output Model 131

able sense is to use operator theory to describe the channel. In fact, the expression for can be rewritten as

where

denotes a two-sided-continuous time-varying matrix operator (or a matrix of operators).The operator can be interpreted as a matrix of continuous-time matrices (the co-efficients) and as a vector of continuous-time vectors (functions). Indeed, we areprompted to use the standard matrix notation and algebraic conventions with no limita-tions, as long as we remind (5.51) and acknowledge that operator multiplication is an infi-nite sum of infinitesimal parts (the integral). The product of operators and is definedas

and the hermitian adjoint product given by

Other rules may follow and will be noted further ahead. We thus rewrite (5.49) as simply as

Moreover, we can also give life to a one-sided-discrete time-varying matrix operator andreassemble (5.49) as

where the upper dot above the operator informs that it is discrete on one of its sides (im-plicit from the construction) and continuous on the other, and the dot above the vectorinforms that it behaves as a vector of discrete-time vectors. Although the output is stillcontinuous, the integral in (5.50) has been replaced by an infinite sum.

5.3.1 DEFINING A BASIS FOR MIMO RESPONSE EXPANSION

Now, what we would like to do is to approximate as function of generic orthonormalbasis functions , i.e.

(5.50)

(5.51)

(5.52)

(5.53)

(5.54)

(5.55)

( )tu

' '

( ) ( , ) ( ) ( , ) ( )

( , ') ( ') '

s s

st t t

t t t t t d

t t t dt

τ τ

τ

τ τ τ τ

= −

= ⊗ = −

=

∫∫

u H x H x

H x

( , ') ( ', ) ( , ') ( ', ' )t t t t t t t t t t= − ⇔ = −H H H H

( , ')t tH

( ')s tx

A B

''( , ') ( , '') ( '', ') ''

tt t t t t t dt= = ∫C AB C A B

''( , ') ( '', ) ( '', ') ''

H H

tt t t t t t dt= = ∫C A B C A B

s= + = +r u n Hx n

( ) ( , ) ( )

( , ) ( )

s ll

s ll

t t lT t t

t lT t

= − +

+ → = +

∑∑

r H a n

H a n r Ha n

(5.56)

( )tu

( , )st lT t−Φ

( ) ( , ) ( , )s l ll

t t lT t tτ

τ= = − = ⊗∑u b u b bΦ Φ Φ


such that the following semi-unitarity condition is satisfied

We start by setting a finite energy requirement on , that is

The energy of the estimator equals the energy of , as shown by

At this point, we would like to determine the optimal vector . To do so, and for sim-plicity, we shall use the widely accepted minimum squared error (MSE) criterion, i.e.

which will guarantee that the energy in the difference between and will be minimum.We let be the orthogonal projection of onto the range of , , and write

where is the nullspace of the hermitian adjoint . Hence, applying the Pythago-rean theorem, we have

which leads to the conclusion that setting minimizes the energy of the errorvector. Letting , from (5.49) we know that

and consequently

As expected, the optimal coefficients are obtained by orthogonally projecting onto

(5.57)

(5.58)

(5.59)

(5.60)

(5.61)

(5.62)

(5.63)

(5.64)

( , ) ( , )H H

s s k lt

t kT t t lT t dt δ −= − − =∫I IΦ Φ Φ Φ

( )tu

22 ( ) ( ) ( )H H

tt t t dt= < ∞ = < ∞∫u u u u u u

u b

22

2

( ) ( ) ( , ) ( , )

H H HH

H H Hs sl m

t tl m

Hl l l

l

t t dt t lT t t mT t dt

= = = =

⎡ ⎤= − −⎢ ⎥

⎢ ⎥⎣ ⎦

= =

∑∑∫ ∫∑

u u u b b b b b

u u b b

b b b

Φ Φ

Φ Φ

b

22argmin argminopt = − = −b b

b u u u bΦ

u u

ou u Φ ( )R Φ

[ ] ( ) ( )( )

2arg minopt o o

Ho

o

⊥

= − + −

− ∈ =

⎡ ⎤− ∈⎣ ⎦

bb u u u b

u u

u b

R N

R

Φ

Φ Φ

Φ Φ

( )HΦN

HΦ

( )2 22argmin argminopt o o o= − + − = −b b

b u u u b u bΦ Φ

o= =u b uΦ

[ ]o o= + −u u u u

[ ] ( ) [ ]

[ ]( , ) ( ) ( )H

H Ho o

s ot

t lT t t t dt

− ∈ → − =

− − =∫

u u u u 0

u u 0

N Φ Φ

Φ

( ) ( , ) ( ) ( , ) ( )

H Ho

H H Hopt opt

Hopt Hs sk

t tt lT t dt t lT t t dt

=

= → = =

= = −∫ ∫

u u

b u b u u

b u u

Φ

Φ Φ

Φ Φ Φ Φ

Φ Φ

( )tu


the basis functions , or equivalently, onto , and as the second line estab-lishes, are least square solutions of .

We realize that after selecting we automatically have , but we would also like toknow if there is a choice of that sets the error energy to zero. From (5.49) we know that

leading us to the conclusion that imposing the condition , and will havethe same energy. More interestingly, and a consequence of orthogonal projection, whenthey both have the same energy, they will also have zero energy difference, as can be seenby

We conclude that, besides the condition (5.57), the orthonormal set shouldalso satisfy the additional sufficient restriction

which renders as a full-orthonormal (i.e. full-unitary) matrix operator and the set a complete one. Moreover,

and thus the estimate of has finite energy, .In short, conditions (5.57), (5.58) and (5.67) guarantee an approximation almost eve-

rywhere (a.e.) between and

and since in theory the optimal detection is always energy-based, and in practice isalways a “well-behaved” real-world vector-valued signal, one shall assume that everywhere. The optimal coefficients, , can be rewritten in terms of the original coef-ficients (the sequence of transmitted symbols), , as follows

where is a two-sided-discrete (two upper dots) matrix operator that denotesthe projection of onto , and

is the correlation (inner-product) between and . In turn, this allows

(5.65)

(5.66)

(5.67)

(5.68)

(5.69)

(5.70)

(5.71)

( , )st lT t−Φ ( )R Φ

=b uΦ

Φ b

Φ

22 H HH

o opt opt opt= = =u b b b u uΦΦ

H= IΦΦ ou u

2 22

2 2 0H

HHo o o o

H HH

opt opt=

− = − − +

= − + =I

u u u u u uu u

u b b u uΦΦ

ΦΦ

( , )st lT t−Φ

( , ) ( ' , ') ( ' )H H

s s

l

t lT t t lT t t tδ∞

=−∞

⎡ ⎤⎢ ⎥= − − = −⎢ ⎥⎢ ⎥⎣ ⎦∑I IΦΦ Φ Φ

Φ

( , )st lT t−Φ

2 2HH

opt = = < ∞b u u uΦΦ

( )tu 2o < ∞u

( )tu ( )tu

2 ( ( ) ( )) ( ( ) ( )) 0 ( ) ( ) . .H

tt t t t dt t t a e− − − = → =∫u u u u u u u u

( )tu

( ) ( )t t=u u

optb

a

( ), ,

H Hopt

optk l l l k k lk

l l−

= = =

= =∑ ∑b u Ha H a

b H a H a

Φ

Φ Φ

Φ Φ

H=H HΦ Φ

H Φ

, ,( , ) ( , ) ( , ) ( , )H H

s s s sk l k l kt t

t kT t lT dt t kT t t lT t dt−= = = − −∫ ∫H H H HΦ ΦΦ Φ

( , )st lT t−H ( , )st kT t−Φ


us to express in terms of and using (5.67), as

Using (5.70) and the Cauchy-Schwarz inequality, the finite energy condition in (5.68) canalso be expressed in terms of the energy of the input vector and the Frobenius norm ofthe operator , , as follows

which means that for the expansion to converge in energy it suffices that we guarantee

The first equality in (5.72) resembles the familiar QR factorization from linear algebrawith matrices replaced by linear operators, except that, in order to be causal, is neces-sarily a lower-triangular operator, that is . To determine we shall as-sume, with no loss of generality, that the first symbol to be transmitted was and the last

, so that we have

The concept behind the factorization is a backward, block-wise Gram-Schmidt orthogo-nalization procedure. After is received (and is known) we compute the Lth

normal basis matrix as follows

where

The subsequent basis matrix, , is found by projecting onto the previous one,computing the difference and normalizing the outcome, which yields

(5.72)

(5.73)

(5.74)

(5.75)

(5.76)

(5.77)

(5.78)

H Φ HΦ

,

,

( , ) ( , )

( , ) ( , )

s s k lk

s s k l kk

t lT t kT

t lT t t kT t −

=

=

− = −

∑∑

H H

H H

H H

Φ

Φ

Φ

Φ

Φ

Φ

a

HΦ ( )2 H

Ftr=H H HΦ Φ Φ

( )

( )

22 ( , ) ( , ) ( , ) ( , ) ( , )( ) 2

, , , , ,

22 ( , ) ( , )( ), ,

i i i H i i Hi Hl m mk k l k l k l k l k l

l l m l

i i Hiopt k k l k l

i k k l

∗ ∗ ∗ ∗ ∗

∗ ∗

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜= ≤ =⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜⎟ ⎟ ⎟ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎛⎜⎜= ≤⎝

∑ ∑ ∑ ∑

∑∑ ∑∑

b H a H H a a H H a

b b H H

Φ Φ Φ Φ Φ

Φ Φ22 2F

i

⎞⎟⎟ =⎟⎜ ⎟⎜ ⎟⎠∑ a H aΦ

2 22 2o opt F

= ≤ < ∞u b H aΦ

HΦ

, ,k l k l= ∀ <H 0Φ Φ

0a

La

,( , ) ( , )L

s s k lk l

t lT t kT≥

= ∑H HΦΦ

La ( , )st LTH

, ,

1

( , ) ( , ) ( , )

( , ) ( , ) ( , )

s s sL L L L

s s s

t LT t LT t LT

t LT t LT t LT−

= → =

=

H H H H

H H

Φ ΦΦ

Φ

1/21/2( , ) ( , ) ( , )

H Hs s s

LL tt LT t LT t LT dt

⎡ ⎤⎡ ⎤= = ⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥⎣ ⎦∫H H H H H

( ,( 1) )st L T−Φ

1

( ,( 1) ) ( ,( 1) ) ( , ) ( , ) ( ,( 1) )

( ,( 1) ) ( , ) ( , ) ( ,( 1) )

Hs s s s s

t

Hs s s s

t

t L T t L T t LT t LT t L T dt

t L T t LT t LT t L T dt−

⎡ ⎤− = − − − ⋅⎢ ⎥

⎢ ⎥⎣ ⎦

⋅ − − −

∫

∫

H H

H H

Φ Φ Φ

Φ Φ


and

The same reasoning applies to any general , which is derived by projectingonto to the previous space, finding the difference and normalizing, i.e.

and also

As it is, the process of computing the basis and involves a greatdeal of processing, but provided that the matrix-valued functions are independ-ent of one another, it truly constructs an orthonormal basis for the received signal and, ofno less importance, guarantees a causal . Another interesting fact is that if the channelvaries slowly so that the coherence time spans several signalling intervals, the factorizationin (5.72) is always possible, even if it results in a trivial solution as in (5.76). Whenever thechannel remains constant, the next basis function that is computed will be related to theprevious one by , showing that it should still be independentor otherwise the channel would not be causal. Even if the channel varies in such a mannerthat forces dependence between some channel functions, that is not an issue because theonly difference is that some basis functions may be derived as null matrices.

More importantly, since in practice the channel has finite memory, it follows that and are non-overlapping for (where and is the maximum delay spread of ), and hence it will only be

necessary to compute for . Accordingly, the expression for the op-timal coefficients is simplified as

Also, the upper limit of the sums in (5.80) and (5.81) is in practice and not ,thus greatly streamlining the orthonormalization procedure. This is better seen by expand-

(5.79)

(5.80)

(5.81)

(5.82)

1, 1

, 1

( ,( 1) ) ( , ) ( , ) ( ,( 1) )

( , ) ( ,( 1) )

Hs s s sL L

t

Hs sL L

t

t L T t LT t LT t L T dt

t LT t L T dt

− −

−

= − − −

= −

∫∫

H H H

H H

Φ

Φ

Φ Φ

Φ

( ,( ) )st L m T−Φ

1

1

1

( ,( ) ) ( ,( ) ) ( , ) ( , ) ( ,( ) )

( ,( ) ) ( , ) ( , ) ( ,( ) )

LH

s s s s sti L m

LH

s s s sti L m

t L m T t L m T t iT t iT t L m T dt

t L m T t iT t iT t L m T dt

= − +

−

= − +

⎡ ⎤⎢ ⎥− = − − − ⋅⎢ ⎥⎢ ⎥⎣ ⎦

⋅ − − −

∑ ∫

∑ ∫

H H

H H

Φ Φ Φ

Φ Φ

,1

,

( ,( ) ) ( , ) ( , ) ( ,( ) )

( , ) ( ,( ) )

LH

s s s sL m L mti L m

Hs si L m

t

t L m T t iT t iT t L m T dt

t iT t L m T dt L m i L

− −= − +

−

= − − −

= − − < ≤

∑ ∫

∫

H H H

H H

Φ

Φ

Φ Φ

Φ

( ,( ) )st L m T−Φ ,i L m−HΦ

( , )st lTH

HΦ

( ,( 1) ) ( , )s s st l T t lT lT+ = +H H

( ,( ) )st L m T−H ( , )st iTΦ ( )i L m D− − >

⎡ ⎤max / sD Tτ= maxτ ( , )tτH

,i L m−HΦ i L m D≤ − +

optb

( ),

optk l lk

k D l k− < ≤

= ∑b H aΦ

L m D− + L


ing the decomposition onto its matrix form

from which it is clear that is not only lower triangular but band-based sparse with low-er bandwidth . Additionally, if ( ) then the channel is underspread intime and flat in frequency, so that there is no need for operator orthonormalization be-cause (5.75) becomes , so that is diagonal and a plain QR fac-torization (or simply (5.76)) is plenty. The main drawback for achieving causality is that wehave to start the orthonormalization backwards from , something that may notbe feasible if is too large. An alternative approach may be to keep sufficiently low andperform the orthonormalization for every vector symbols, even though this will havesome impact on the detection stages of the receiver.

All in all, we have decomposed the original operator into terms and , as de-picted in Figure 5.5, where and . is any function that satisfies

the definition

The output of is given by

indicating that the system is in fact consistent ( has in its range), and its ex-act solutions are the least squares solutions. Additionally, since , we have

which must imply the following

(5.83)

FIGURE 5.5 Decomposition of the MIMO transfer matrix into discrete and orthonormal operators.

(5.84)

(5.85)

(5.86)

(5.87)

( , 0 ) ( , 0 )

( , 1 ) ( , 1 )

( , ( ) ) ( , ( ) )

( , ( 1) ) ( , ( 1) )

( , ) ( , )

Ts s

s s

s s

s s

s s

t T t T

t T t T

t L D T t L D T

t L T t L T

t LT t LT

− −=

− −

⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟⎜ ⎜⎟⎟⎜ ⎜⎟⎜ ⎜⎟⎜ ⎜⎟⎜ ⎜⎟⎜ ⎜⎟⎜ ⎜⎟⎟⎜ ⎜⎝ ⎠ ⎝ ⎠

H

H

H

H

H

Φ

Φ

Φ

Φ

Φ

0,0

,0 ,

, 2 ,

, ,

T

D D D

L D L D L D L D

L L D L L

− − − −

−

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟ ⎜ ⎟⎟ ⎜ ⎟⎟ ⎟⎜⎟ ⎟⎜⎟ ⎟⎜⎟ ⎟⎜⎟ ⎟⎜⎟ ⎟⎜⎟ ⎟⎟ ⎝ ⎠

H 0 0 0

H H 0 0

0 H H 0

0 0 H H

… …

…

…

Φ

Φ Φ

Φ Φ

Φ Φ

HΦ

D max sTτ < 1D =

,( , ) ( , )s s l lt lT t lT=H HΦΦ HΦ

( , )st LTH

L L

L

H Φ HΦH

= IΦ ΦH

=H HΦ Φ ( , )tτΦ

( , )tτH

( , )tτΦ

( , )st lTH

,k lH Φ

,l kHΦ

/ ( )tu u

/ ( )tn n

/ ( )tr r

( )/ optopt kb b ( , )st lTΦ

/ ka a

( , ) ( , )s st kT t t kT− =Φ Φ

H

opt o= = = =u Ha H a b uΦΦ Φ

=b uΦ Φ ( )tuH

=H HΦ Φ

Hopt = =b H a HaΦ Φ

H H

o opt= = = ⇒u b Ha uΦ ΦΦ ΦΦ H

= IΦΦ


We conclude that an exact approximation to has been found without requiring that, that is, without the need of a full-orthonormal .

Our intention at the beginning of this section was to discretize the MIMO input-out-put model in (5.49) or, equivalently, (5.53). Up to this point we have determined a “loss-less” representation of in terms of a semi-orthonormal extracted from the channeltransfer operator . The input-output relation is rewritten as

and now we apply the adjoint operator on the left, yielding

One should note that applying the adjoint operator is equivalent to passing the receivedsignal through a filter (one that satisfies (5.84)) and then sampling at

, because

so that the model in Figure 5.5 is extended as shown in Figure 5.6. By the cascading, we

find that a discrete output has been recovered from the discrete input by applying the two-sided-discrete operator or discrete matrix filter , as illustrated in Figure 5.7.

Even though this discrete model is obtained by simple projection onto the spacespanned by the channel operator , one does not know yet if it conveys all the informa-tion that the original continuous model conveyed. If it does, we are in good company, butif it does not, then some important information that could be useful to the detector’s de-cisions may have been lost. We already know that can be constructed from be-cause spans the entire space of the channel’s noiseless output, so there should be noproblem here. However, does not span the entire noise space because the condition

is not met, and as a result some information from the output is lost after

(5.88)

(5.89)

(5.90)

FIGURE 5.6 Discretizing the MIMO output with the semi-orthonormal channel operator .

( )tuH

= IΦΦ Φ

( )tu Φ

H

( ) ( , ) ( )

opt

s ll

t t lT t t

= + = + = +

= − +∑r u n b n u n

r u n

Φ

Φ

Φ Φ

Φ

( )( , ) ( )

H H H Hopt

optHsk kk

tt kT t t dt

= = + = +

= +

= − = +∫

r r u n b n

u n

r r b n

Φ

Φ Φ

Φ Φ

Φ Φ Φ Φ

Φ

( )tr ( , )H tτ τ− −Φ

st kT=

( , ) ( ) ( , ) ( )

( , ) ( ) ( , ) ( )

s

ss

H Hs s sk t kTt

H H

t kTt kT

t kT t t dt kT kT dt

t t dt t t

τ τ

ττ

τ τ τ

τ τ τ τ τ

= −

==

= − = − − −

⎤ ⎤= − − − = − − ⊗⎥ ⎥⎦⎥⎦

∫ ∫∫

r r r

r r

Φ Φ Φ

Φ Φ

/ ( )tr r( , )H tτ τ− −Φ

skT

opt= +r b n Φ Φ

( , )H

st kTΦ

Φ

HΦ HΦ

Φ

( )tu optb

Φ

ΦH

= IΦΦ ( )tr


the projection. We shall now determine whether or not this information is relevant froman optimal input estimation perspective and, in case it is deemed irrelevant, the model ofFigure 5.7 can be regarded as equivalent to the original.

5.3.2 TIME-VARYING MAXIMUM LIKELIHOOD SEQUENCE ESTIMATION (TV-MLSE)

Let us idealize a full-orthonormal matrix operator that satisfies both conditions and . As discussed in the previous section, this operator spans the en-

tire space of the MIMO noisy output , because if its estimate is

then it equals at least almost everywhere

Developing on this, we orthogonally project onto the range of and obtain an equiv-alent discrete representation that losslessly describes . It is given by

The sole purpose of doing this is to simplify the mathematical treatment that will follow.From (5.28) we statistically model the input baseband noise as zero-mean space-timewhite gaussian noise with autocorrelation

which leads to the following autocorrelation of the discretized noise

and its multivariate normal distribution

FIGURE 5.7 Discrete MIMO model after projecting the output onto .

( , )tτH

( , )st lTH

/ ( )tu u

/ ( )tn n

/ ( )tr r

( )/ optopt kb b

( , )H tτ τ− −ΦskT

opt= +r b n Φ Φ

n Φ

, ( , ) ( , )H H

s sk lt

t kT t lT dt= = ∫H H H H Φ ΦΦ Φ

( , )H

st kTΦ

/ ka a

( )R Φ

(5.91)

(5.92)

(5.93)

(5.94)

(5.95)

(5.96)

ΨH

= IΨ ΨH

= IΨΨ

( )tr

H= =r r r rΨ ΨΨ Ψ

( )tr

. .H

a e= =r r rΨΨ

( )tr Ψ

( )tr

( , ) ( ) ( , ) ( )

H H H

H Hs sk k k

t tt kT t t dt t kT t t dt

= = + = +

= − + − = +∫ ∫r r u n u n

r u n u n

Φ Ψ Ψ

Ψ Ψ Ψ

Ψ Ψ Ψ

Ψ Ψ

[ ]

[ ]

0

0( ) ( ')

12

1( ') ( ) ( ') ( ')

2

H

Ht t

E N

t t E t t N t tδ

= =

− = = −

nn

n n

R nn I

R n n I

[ ] 0

0

1 12 2

( )k l

HH H

k l

E E N

k l N δ −

⎡ ⎤= = =⎢ ⎥⎣ ⎦− =

n n

n n

R n n nn I

R I

Ψ Ψ

Ψ Ψ

Ψ ΨΨ Ψ

( ) ( )0[ ], ,k kk kE N≡n nn n R 0 I∼

Ψ ΨΨ ΨN N


Denoting the number of MIMO outputs as , its pdf is swiftly derived as

We also recognize that, due to its gaussian nature, the discrete vector components and are independent for , which means that the pdf of is the product of (5.97)

for every , that is

The importance of the pdf in (5.98) lies in the fact that it sets the ground to the max-imum likelihood (ML) detection of the transmitted symbols. ML detection guarantees thelowest average error probability of the input symbol sequence, so it is optimal from thisperspective. In fact, if the symbols of the input set are equally likely to be transmitted, thenthe classical Bayes rule permits the following

where the conditional pdf can be retrieved from the noise pdf by a simple trans-formation , yielding

which is minimized when the quadratic form is minimized, andthus (5.99) is rewritten in the form

But, from (5.93), , and accordingly (5.101) becomes

In words, the input vector sequence of maximum likelihood is the one that minimizes theenergy between the noisy and noiseless outputs of the MIMO channel, a conclusion thatextends the SISO time-invariant result with evident similarity.

(5.97)

(5.98)

(5.99)

(5.100)

(5.101)

(5.102)

Rn

( ) ( ) ( )[ ]( )( )

( )

10

10

11 2

0

12

0

2 det

2

Hk kR

k

HkkR

tr Nnk

Nn

f N e

N e

π

π

−

−

−−−

−−

=

=

I n nn

n n

n I Ψ Ψ

Ψ

ΨΨ

Ψ

knΨ

lnΨ k l≠ nΨk

( ) ( ) ( )1

012

02H

Rk

Nnk

k k

f f N eπ−−−

⎡ ⎤⎢ ⎥= = ⎢ ⎥⎢ ⎥⎣ ⎦

∏ ∏ n nn nn n ΨΨΨ ΨΨ Ψ

( ) ( )argmax | argmax |ML P f= =a a

a u r r uΨ Ψ Ψ Ψ

( )fn nΨ Ψ

= −n r uΨ Ψ Ψ

( ) ( )( )

( )( )

( )( ) ( )1

012

0

||

= 2H

RNn

k

df f f

d

N eπ−

∧

∧

− − −−

= − = −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦∏

n n

r u r u

nr u r u r u

r uΨ Ψ

Ψ Ψ Ψ Ψ

ΨΨ Ψ Ψ Ψ Ψ Ψ

Ψ Ψ

( ) ( )H− −r u r uΨ Ψ Ψ Ψ

( ) ( ) ( ) 2argmax | argmin argmin

HML f == = − − −

a a aa r u r u r u r uΨ Ψ Ψ Ψ Ψ Ψ Ψ Ψ

( )H

− = −r u r uΨ Ψ Ψ

( ) ( ) ( ) ( )

( ) ( )

2

( )

argmin argmin argmin

argmin ( ) ( , ) ( ) ( , )

argmin ( ) ( , ) ( ) ( , )

k

k

HHML

HML

s sk l lt l l

H

l lt

t t lT t t t lT t dt

t t t t dtτ τ

τ τ

= == − − − − −

⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜= − − − −⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎝ ⎠ ⎝ ⎠

= − ⊗ − ⊗

∑ ∑∫

∫

a a a

a

a

a r u r u r Ha r Ha r Ha

a r H a r H a

r H a r H a


5.3.3 TIME-SHEAR MATRIX MATCHED FILTER (TS-MMF)

Expanding the norm in (5.102), we reach

where is the output of the adjoint operator and the only output information re-quired for ML decision. Again, recalling the definition in (5.51), the adjoint channel oper-ator expands as

yielding as the matrix matched filter (MMF) for the time-varying MIMO sys-tem. This is the optimal filter for signal detection. It is non-causal, non-predictive, and ineffect translates to the adjoint operator in the operator domain, because

so that the optimum receiver operator is always the hermitian adjoint operator of thetransmitter operator. It is important to note that is not necessarily obtainedfrom the entire as long as it satisfies

or, in other words, equals the operator in the transmission domains of Figure 5.4. Thematched filter is solely required to be matched to the transmission domains, not the entire

. This fact comes as no surprise in that, since the input symbols are carried by ma-trix-valued functions , the best receiver should correlate with each of them, andit indeed does, as shown by

or, more simply, as

(5.103)

(5.104)

(5.105)

(5.106)

(5.107)

(5.108)

( ) ( )

2

2

argmin argmin 2Re

argmin 2Re

H HHML

H

⎡ ⎤= − − = − +⎢ ⎥⎣ ⎦⎡ ⎤= − +⎢ ⎥⎣ ⎦

a a

a

a r Ha r Ha a H r Ha

a y Ha

H=y H r

( ) ( , ) ( ) ( , ) ( )

( , ) ( )s

H H Hs s s

t t

H

t kT

kT t kT t dt t kT t t dt

t tτ

τ τ=

= = = −

⎤= − − ⊗ ⎥⎦

∫ ∫y H r y H r H r

H r

( , )H tτ τ− −H

( ', )H

t tH

' '

'

( ) ( , ) ( ) ( ' , ') ( ') '

( ', ) ( ') '

H H

t t t

H

t

t t t d t t t t dt

t t t dt

τττ τ τ τ

= −= − − − = −

=

∫ ∫∫

y H r H r

H r

( , )H tτ τ− −H

( , )tτH

( , ) ( , )HH

s s skT kT kTτ τ τ− − = −H H

HH

( , )tτH

( , )st lT t−H

( ) ( , ) ( ) ( , ) ( )

( , ) ( , ) ( , ) ( )

s

H Hs s

t kT t

H Hs s sl

t tl

kT t t t kT t t dt

t kT t t lT t dt t kT t t dt

ττ τ

=

⎤= − − ⊗ = −⎥⎦⎡ ⎤

= − − + −⎢ ⎥⎢ ⎥⎣ ⎦

∫∑ ∫ ∫

y H r H r

H H a H n

( ) (( ) , ) ( , )s s s s sl k k l kl l

kT k l T kT lT kT −= − + = +

= + = +

∑ ∑H H

H H H

y S a n S a n

y Sa n u n


where

is the correlation between the different . Contrary to the LTI scenario, is non-stationary because it depends on each particular and the difference

, hence not Toeplitz. One may also check that

meaning that is non-negative definite hermitian, hence a valid covariance operator. Thecontinuous MMF output before sampling is given by

and as a result the transfer function between the MIMO input and the matched filter out-put is the continuous correlation function

This matrix valued bi-dimensional function is not causal due to the non-causality of thematrix matched filter, and gathers a lot of redundant information that can be safely ig-nored (by sampling) without penalty.

The sampled correlation matrix can be seen as the correlation between thedifferent transmission domains in Figure 5.4, a fact that leads us to another interpretationof the matrix matched filter. Writing , we see that there is an im-plicit linear transformation of coordinates

where are the old coordinates. This transformation is visualized in Figure 5.8, fromwhich it can be perceived that the matched filter has the form of a -skewed mirroringof the channel transfer-function (TF) . In other words, experiences a vertical(time) shear of unit slope followed by an horizontal (delay) reflection. The delay reflectionis already present in the classical time-invariant matched filter ( ), so thepeculiar aspect in this case is the time shearing of the MIMO time-varying response. Inlight of this, we shall distinguish this new filter with the name time-shear matrix matched filter(TS-MMF). The system diagram of the TS-MMF is shown in Figure 5.9.

(5.109)

(5.110)

(5.111)

(5.112)

(5.113)

,

,

(( ) , ) ( , ) ( , )

( , ) ( , ) ( , )

Hs s s sk l k

t

H Hs s s sk l

t

k l T kT t kT t t lT t dt

kT lT t kT t lT dt

−− = = − −

= = = =

∫∫

H

H H

S H H H

S H H H S H H H

( , )st lT t−H

( , )s slT kTS k

k l−

2

2

0

( ) ( , ) ( , )

(( ) , ) 0

HH HH

HHs s lk

tk l

Hs s lk

k l

t t kT t t lT t dt

k l T kT

= = = ≥

⎡ ⎤= − −⎢ ⎥

⎢ ⎥⎣ ⎦

= − ≥

∑ ∑ ∫∑ ∑

u u u a H Ha a Sa

u a H H a

a S a

S

( , )

( ) ( , ) ( ) ( , ) ( , ) ( )

Hs

Hs l l

l

t lT

t t lT t t t t tτ τ

τ τ τ

= +

= − + = ⊗ + − − ⊗∑ H

y S a H n

y S a n S a H n

( , ) ( , ) ( , )

( , ) ( , ) ( , ) ( ( ), )

H

H H

t t t

t t d t t d

α

α α

τ α α τ

α α τ α α α α α α τ α α

= − − ⊗

= − − − − = − − −∫ ∫

S H H

H H H H

( , )s slT kTS

( , ) ( , )Ht tτ τ τ= − −MMF H

1 0 '( , ) ( , )

1 1 't t t t

τ ττ τ τ

− ⎛ ⎞⎛ ⎞⎛ ⎞ ⎟⎟⎜⎜⎟⎜ ⎟⎟⎜⎜→ − − ⇔ =⎟⎜ ⎟⎟⎜⎜⎟⎟⎜ ⎟⎟− ⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠⎝ ⎠

( ', ')tτ

/4π

( , )tτH ( , )tτH

( ) ( )Hτ τ= −MMF H


5.3.4 DRAWBACKS OF THE TS-MMF

Despite being the filter that minimizes the error rate at the detector output, the TS-MMFintroduces some difficulties into the system, mainly implementation related. First of all,the implementation of the time domain samples of the TS-MMF derivedfrom the TF requires a perfect knowledge of the time-varying nature of the chan-nel. While this is possible for slowly varying channels, it is totally prohibitive for channelswhere because effectively changes before it has the chance to bemeasured. The only solution in such an adverse scenario may be channel prediction. Onthe other hand, whenever the channel is fast-fading such that , a possiblecandidate to fill estimation gaps may be interpolation [98], [99].

Another difficulty with the TS-MMF is its non-causality, due to the mirroring effectapplied to . Consequently, implementation of is only possible if a suf-ficient delay is introduced into the filter, resulting

as depicted in Figure 5.10. Inconveniently, this means that the output at time requiresknowledge of the channel at time , so the TS-MMF trades its causality by predictiv-ity. This knowledge is possible by sending a pilot at time and waiting seconds

FIGURE 5.8 Mapping of the MIMO channel response onto its associated matched filter.

FIGURE 5.9 Matched filtering of the MIMO time-varying channel model.

sT−2 sT− sT 2 sT0

'τ

't

sT

sT−

2 sT

sT

sT−

τ

t

( , ) ( , )t tτ τ τ→ − −

2 sT−

( , )tτH

( , )st lTH

/ ( )tu u

/ ( )tn n

/ ( )tr r

skT( , )H tτ τ− −H

( , )H

st kTH

= + Hy Sa n

Hn Hu

,( , ) ( , ) ( , )H H

s s s sk lt

kT lT t kT t lT dt= = = = ∫H HS H H H S H H H

/ ka a

(5.114)

( , )HskTτ τ− −H

( , )tτH

max( )c st T τΔ < < ( , )tτH

max ( )c st Tτ < Δ <

( , )tτH ( , )H tτ τ− −H

0 maxτ τ>

0 0 0( , ) ( , )Ht tτ τ τ τ τ τ− = − + −MMF H

t

0t τ+

't t= 0τ


to get the full response , and then perform some variable changes

Again, this is a problem when because even if one decided to wait

seconds for the response, it would vary during that time, meaning that it would not be pos-sible to do the same for a if .

Finally, another important problem of the TS-MMF concerns to detection. FromFigure 5.9 we see that the noise output after sampling is , which leads to thefollowing non-stationary autocorrelation

Thus, the output noise vector samples are not uncorrelated, hence the noise is not whiteand the decision process is more complicated because the distribution

is much harder to work with. Since is a sufficient statistic, the ML criterion would be

This is caused by the lack of semi-orthonormality of the channel operator , so adds upas an extra motivation for (losslessly) orthogonalizing the channel.

5.3.5 OPTIMALITY OF THE DISCRETE MIMO MODEL IN

At this point we no longer need the operator , introduced in Section 5.3.2, so we return

(5.115)

FIGURE 5.10 Delayed matched filtering of the MIMO time-varying channel model.

(5.116)

(5.117)

(5.118)

( ', )H t t t−

0'0 0( ', ) ( , ' ) ( , ' ) ( , ' )

tH t t t H t t t H t t t H t t t

ττ τ

+ − −− → + → − − → − + −

max( )c st T τΔ < < 0τ

sT−2 sT− sT 2 sT

0

'τ

't

sT

sT−

2 sT

τ

t

0τ

0 0( , ) ( , )t tτ τ τ τ τ→ − + −

0sT τ+

0sT τ− +

02 sT τ− +

0τ0t τ+

t

''t t= max'' 't t τ− <

H=Hn H n

[ ] 0 0

0 0

1 12 2

( , ) ( , ) ( , ) (( ) , )k l

H HH H

Hs s s s

t

E E N N

k l k N t kT t t lT t dt N k l T kT

⎡ ⎤= = = =⎢ ⎥⎣ ⎦

− = − − = −∫H H

H H

n n H H

n n

R n n H nn H H H S

R H H S

( ) ( ) ( ) 111

2det 2Htr

f eπ−⎛ ⎞⎟⎜−− ⎟⎜ ⎟⎜⎝ ⎠⎡ ⎤= ⎢ ⎥⎣ ⎦

n n H HH H

H H H

R n nn H n nn R

Hr

( ) ( )

( ) ( )

1

1

argmin

argmin

HML

H

−

−

= − −

= − −

H HH H n n H Ha

H Ha

a r u R r u

r Sa S r Sa

H

( )R Φ

Ψ


to the semi-orthonormal channel operator from Section 5.3.1 and try to express the MLcriterion in (5.102) within , checking for its equivalence. The projection of the noise

onto is denoted by , so the MIMO input-output relation after thechannel decomposition equals

where is the part of the noise out of . Furthermore, because of orthogonalprojection we note that it is orthogonal to , i.e.

Based on these facts, one may revise (5.102) by adding and subtracting ,

because and is independent of . And, finally, we ex-press the maximum likelihood estimates as a sole function of the discretized MIMOmodel of Figure 5.7 or, equivalently, the space ,

The exclusive use of the identity should be noticed from this deduction, and alsohow elegantly the latter has been accomplished due to its operator-based nature. One con-cludes that, after projecting the MIMO output onto the operator extracted from thechannel transfer matrix , all the information required for optimal detection is stillavailable, so one can confidently rely on the time-varying causal discrete model

where , for any ensuing analysis. is taken from (5.81). Notonly the model is attractively discrete and causal, but also the noise is still space-time whitebecause

which is a highly desirable characteristic since it simplifies the statistical treatment of themodel. Capitalizing on this, (5.122) has the simple form

(5.119)

(5.120)

(5.121)

(5.122)

(5.123)

(5.124)

(5.125)

Φ

( )R Φ

( )tn ( )R ΦH

=n nΦ Φ

=H HΦΦ

( )= + = + + = + +r Ha n H a n H a nΦ Φ Φ ΦΦ Φ ε Φ ε

( )tε ε ( )R Φ

( )R Φ

( ) ( )H H ⊥= − = − → = → ∈ =n n r r 0Φ Φε Φ Φ Φ ε ε Φ ΦN R

rΦΦ

( ) ( )

( ) ( )

22

2 22

argmin argmin

argmin argmin

ML = − = − + −

⎡ ⎤= − + − = −⎢ ⎥⎣ ⎦

a a

a a

a r H a r r r H a

r r r H a r H a

Φ Φ Φ Φ

Φ Φ Φ Φ Φ

Φ Φ Φ

Φ Φ Φ

( )H H− =r H a 0Φ Φ Φ ε −r rΦΦ a

MLa

( )R Φ

( ) ( ) ( )

( ) ( )

2

2

argmin argmin

argmin

argmin

HML

H H

⎡ ⎤= − = − −⎢ ⎥⎣ ⎦

⎡ ⎤= − −⎢ ⎥⎣ ⎦

= −

a a

a

a

a r H a r H a r H a

r H a r H a

r H a

Φ Φ Φ Φ Φ Φ

Φ Φ Φ Φ

Φ Φ

Φ Φ Φ Φ Φ

Φ Φ

H= IΦ Φ

Φ

( , )tτH

, ,0

k D

k k l l k l k k l kl k D l

−= − =

= +

= + = +∑ ∑

r H a n

r H a n H a n

Φ Φ Φ

Φ Φ Φ Φ Φ

⎡ ⎤max / sD Tτ= , ,l k k k l−=H HΦ Φ

[ ] 01 12 2

HH HE E N⎡ ⎤= = =⎢ ⎥⎣ ⎦n nR n n nn IΦ Φ Φ Φ

Φ Φ

21

, ,argmink

ML k k k k k l lk l k D

−

= −

= − −∑ ∑a

a r H a H aΦ Φ Φ


where is the current channel input and is the set of previous inputs (orcurrent state) that accounts for the channel memory. Thus a trellis of states can be builtand the Viterbi algorithm applied with confluent path elimination at each stage and sur-viving distance metrics

5.3.6 DECOMPOSING THE TS-MMF INTO DISCRETE AND SEMI-ORTHONORMAL FACTORS

We gather from Figure 5.5 that can be successfully decomposed into factors and , still, up to this point, we only have the expression that decomposes intoterms and ,

It is not possible to retrieve from and , because they only possess essentialinformation that is conveyed from the channel, that is, the transmission domains ofFigure 5.4. However, it is possible to build a simple function that equals the operator on these domains and is null otherwise

where is the usual Kronecker delta. Its sampled version is

The function in (5.128) can adequately replace as

so it must be the case that

which must be interpreted using the two-dimensional Dirac impulse function inthe following manner

(5.126)

ka 1,...,k D k− −a a

m

2

1 , 1,..., min ,...,m D

m

surv survm m D m m l l m m Dl m D

DM DM−

− + − −= −

= − +∑aa a r H a a aΦ Φ

(5.127)

(5.128)

(5.129)

(5.130)

(5.131)

(5.132)

( , )tτH ,l kHΦ

( , )tτΦ H

Φ HΦ =H HΦΦ

,

,

( , ) ( , )

( , ) ( , )

s s k lk

s s k l kk

t lT t kT

t lT t t kT t −

= =

= − = −

∑∑

H H H H

H H

Φ Φ

Φ

Φ Φ

Φ

( , )tτH Φ HΦ

H

( ( )) ( ( ))( )( , ) ( , ) ( , )s st kT s t kTtd

k k

t t t kT tτ ττ τ δ δ− − − −= = −∑ ∑H H H

a bδ −

( )( , ) ( , ) ( ( )) ( , ) ( ( ))s s stdsk k

t t t kT t kT t t kTτ τ δ τ δ τ= − − = − − −∑ ∑H H H

( , )tτH

( ), ( )( ) ( , ) ( , ) ( , ) ( , )optl k k m td mk

lt t t t t

α α τ τα α τ τ= ⊗ = ⊗ ⊗ = ⊗ = ⊗u b H a H a H aΦΦ Φ

( ) ( ) ,( , ) ( , ) ( ( )) ( , )stds td l kk

t t t kT tα

τ τ δ τ α= − − = ⊗∑H H HΦΦ

2( , )x yδ

2( ) ,

,

( , ) ( , ) ( , )s stds l kl k

t t lT t kTα

τ α δ τ= ⊗ − −∑H HΦΦ


Convolving, we find

and, naturally

matching (5.127). So, (5.133) seems to be the expression we were looking for. To dispelany remaining doubts we double-check by substitution into (5.130)

and set and , yielding

which is the expected expansion of in , utterly confirming our conjecture.Now, one may wonder if the TS-MMF can be decomposed similarly to

(5.131). With operator conventions we simply find the hermitian adjoint of (5.127) and get

but its relation with may seem obscure. So what we do is to start from(5.132) as follows

(5.133)

(5.134)

(5.135)

(5.136)

(5.137)

(5.138)

2( ) ,

,

,

( , ) ( , ) ( , )

( , ) ( ( ))

s stds l kl k

s sk l kl k

t t lT t kT d

t kT t t lT

ατ α δ τ α α α

δ τ−

⎡ ⎤= − − − −⎢ ⎥

⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥= − − −⎢ ⎥⎢ ⎥⎣ ⎦

∑ ∫

∑ ∑

H H

H

Φ

Φ

Φ

Φ

( ) ,

,

( , ) ( , ) ( , )

( , )

s s s l mtd k l kl k

l ms k m k

k

t mT t t mT t t kT t

t kT t

δ −−

=−

⎡ ⎤⎢ ⎥− = − = −⎢ ⎥⎢ ⎥⎣ ⎦

= −

∑ ∑∑

H H H

H

Φ

Φ

Φ

Φ

( ) ( )

( ( )),

( ) ( , ) ( , ) ( )

( , ) ( )s

std m td mm

s t lT sk l k ml k m

t t t t mT

t kT t t mT d

τ τ

ττ

τ τ δ

δ δ τ τ− −−

= ⊗ = ⊗ −

⎡ ⎤⎢ ⎥= − − −⎢ ⎥⎢ ⎥⎣ ⎦

∑

∑ ∑ ∑ ∫

u H a H a

H aΦΦ

st mTτ = − m l=

,

( ),

( ) ( , )

( , ) ( , )

s k l k ll k

l k l opts sl k k l k

k l k

t t kT t

t kT t t kT t

−

→ −−

⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥⎣ ⎦

= − = −

∑ ∑∑ ∑ ∑

u H a

H a b

Φ

Φ

Φ

Φ Φ

( )tu ( )R Φ

( , )H tτ τ− −H

,( , ) ( , )H H H H H H

s sk lk

t lT t kT= = ∑H H H HΦ ΦΦ Φ

( , )H tτ τ− −H

2( ) ,

,

2( ) ,

,

( , ) ( , ) ( , )

( , ) ( , ) ( , )

s stds l kl k

H H Hs stds l k

l k

t t lT t kT

t lT t kT t d

α

α

τ α δ τ

τ δ τ α α α α

= ⊗ − −

→ = − − − −

∑

∑∫

H H

H H

Φ

Φ

Φ

Φ


and now perform appropriate variable changes to reach at

Comparing this expression with (5.132) we deduce that the TS-MMF is decomposed as

which is the convolution between a discrete matched filter and a continuous or-thonormalized matched filter (from this point on to be designated OR-THO-TS-MMF). Indeed the decomposition is possible and it is strongly consistent with(5.131).

To eliminate the doubts that the TS-MMF decomposition could have been inferredfrom (5.137) we recall (5.104), where it is found that behaves as a matched filter fol-lowed by sampling

Equivalently,

and

from which we confirm (5.140).Moreover, it is straightforward to show that the full transfer function given by (5.112)

can also be decomposed into discrete and orthogonal terms. Using the identity

we are led to

(5.139)

(5.140)

(5.141)

(5.142)

(5.143)

(5.144)

(5.145)

2( ) ,

,

2,

,

2,

,

( , ) ( , ) ( , )

( , ) ( , )

( , ) ( , )

H H Hs stds l k

l k

H Hs s sl k l

l k

H Hs sl k l

l k

t lT t kT t d

lT t kT lT t d

lT t kT t

α

α

α

τ τ δ τ α τ α α τ α

δ α α τ α τ α

δ α τ τ

− −

− −

− − = − − − − − − −

= − − − + − + −

= − − ⊗ − −

∑∫

∑∫∑

H H

H

H

Φ

Φ

Φ

Φ

Φ

Φ

( ) ,( , ) ( , ) ( , )H H H Htds l k lt t t

ατ τ τ τ τ τ− −− − ≡ − − = ⊗ − −H H HΦ Φ

,H

l k l− −HΦ

( , )H tτ τ− −Φ

HH

( )

( ) ( , ) ( )

( , ) ( ) ( , ) ( )s s

Hs sk

t

H Htd

t kT t kT

kT t kT t dt

t t t tτ τ

τ τ τ τ= =

= =

⎤ ⎤= − − ⊗ ≡ − − ⊗⎥ ⎥⎦ ⎦

∫Hy r H r

H r H r

( , ) ( ) ( , ) ( )s

H Hsk

t kTtt kT t dt t t

ττ τ

=

⎤= = − − ⊗ ⎥⎦∫r r rΦ Φ Φ

, , ,

, , ,

( )

( , ) ( )s

H H Hs k l k l l k l l l k l k l

l l l

H H H Hl k l k l r l k l r l

l t kT

kT

t tα α τ

τ τ

− − − −

− − − − − −=

= = = =

⎤= ⊗ = ⊗ = ⊗ − − ⊗ ⎥⎦

∑ ∑ ∑Hy r H r H r H r

H r H r H r

Φ Φ Φ Φ Φ Φ

Φ Φ Φ Φ Φ Φ

]' ''

( , ) ( , ) ( , )s

s

H

t t t t Ht kT

t kTt t t

ρ αρ

αρ α α ρ δ

= − → −

==

= =

⎤= − − ⊗ =⎥⎦

I

I

Ω Φ Φ

Ω Φ Φ

( ) ,

, , ,

( , ) ( ) ( , ) ( , )

( ) ( )

s s

H Hstds r p rt kT t kT

k k

Hs sl m r p r l m

k

t t kT t t

t kT t kT

β α

ρ β

τ δ α α ρ

δ δ

− −= =

− −

⎡ ⎤⎢⎤ − = ⊗ − − ⊗ ⋅⎥⎢⎦ ⎦⎢⎣⎤⋅ − ⊗ = ⊗ −⎥⎦

∑ ∑∑

S H

H H H

Φ

Φ Φ Φ

Φ Φ


and the last member of (5.145) can be simplified as follows

Finally, from the definition in (5.129) we have the result

which equated with (5.146) shows that the time-sampled version of behaves as adiscrete matrix filter with response

We should observe the remarkable resemblance of (5.148) with (5.112). Since the role of is to discretize the domains of transmission, the role of is to match the re-

ceiver to these discrete domains. This result could also have been easily determined usingmatrix operators. From (5.72) and (5.109) we have

which after some clear-cut manipulations yields

One realizes once more that working with operators is much more intuitive and leanthan working with integrals, sums and convolutions. After completing the deductions inthe operator “world”, the time domain results are derived with the utmost simplicity. Thesystem diagram pertaining to these decompositions is shown in Figure 5.11.

5.4 Frequency-Domain Description

Describing the MIMO LTI system in the frequency domain is as easy as taking Fouriertransforms of all vectors and matrices and then performing simple matrix-vector multipli-cations. The inherent simplicity results from three facts: 1) MIMO LTI systems are de-

(5.146)

(5.147)

(5.148)

(5.149)

(5.150)

( ) , ,, ,

2, ,

,

2 2, ,

( , ) ( ) ( ) ( )

( , )

( , ) (

s

Hs s stds r p r l mt kT

k r p l m k

k p m k r Hs s s s r k r l r k r

k l r

Hs s r k r l k

r

t t kT t kT t pT

rT lT t rT mT

lT t kT

τ δ δ δ

δ τ

δ τ δ τ

− −=

= ∧ = −− − − −

− −

⎤ − = − − ⋅⎦

⎡ ⎤⎢ ⎥⋅ − − − − = ⋅⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤⋅ − − = ⊗⎢ ⎥⎣ ⎦

∑ ∑ ∑ ∑

∑ ∑

S H H

H H

H H

Φ Φ

Φ Φ

Φ Φ,

, )s s

k l

lT t kT− −∑

( ),

2

,

( , ) ( ) ( , ) ( ( )) ( )

( , ) ( , )

ss s stds t kT

k k l

s s s s

k l

t t kT t t lT t kT

lT kT lT t kT

τ δ τ δ τ δ

δ τ

=⎤ − = − − −⎦

= − −

∑ ∑∑

S S

S

( )( , )tds tτS

, ,( , ) Hs s r k r l k

rlT kT − −= ⊗S H HΦ Φ

,l kHΦ ,H

r k r− −HΦ

, ,( , )

H H H H H

Hs s r k r l

r

kT lT

= = = =

= ∑S H H H H H H H H

S H H

Φ Φ Φ Φ Φ Φ

Φ Φ

Φ Φ Ω

, , , ,

, , , ,

( , ) (( ) , )

( , )

H Hs s s sr k r l r k r r l r

r rl k l H H

s s r k r l r k r r k r l kr

r

kT lT k l T kT

lT kT

− −

→ −

− − − − − −

= ⇒ − =

⇒ = = ⊗

∑ ∑

∑

S H H S H H

S H H H H

Φ Φ Φ Φ

Φ Φ Φ Φ

5.4 Frequency-Domain Description 149

scribed by Toeplitz matrix operators, i.e.

2) the matrix valued exponentials are eigenfunctions of Toeplitz operators,i.e.

where the following identities have been used,

and 3) these eigenfunctions are fully orthonormal, i.e.

This last property means that they losslessly expand any finite energy vector signal as

and provide the means to transform operator multiplication into plain multiplication

FIGURE 5.11 Decomposition of the TS-MMF and global function into discrete and semi-orthonormal filters.

( , )tτH/ ( )tu u

/ ( )tn n

/ ( )tr r

( , )H tτ τ− −ΦskT

skT( , )H tτ τ− −H

= + Hy Sa n

,H

l k l− −HΦr Φ

,l kHΦ ( , )tτΦu Φ

, ,( , )H H

s s r k r l krlT kT − −= = ⊗S H H S H H Φ Φ Φ Φ

Hn Sa

/ ka a

(5.151)

(5.152)

(5.153)

(5.154)

(5.155)

(5.156)

' '( ) ( , ') ( ') ' ( ') ( ') '

Toeplitz

t tt t t t dt t t t dt

= =

= = −∫ ∫y Hx H x

y H x H x

2( , ) j ftt f e π=K I

' '( ') ( ', ) ' ( ') ( ', ) ' ( , ) ( ) ( , )

( )

H

t tH

Toeplitz col const

t t t f dt t t f dt t f f t f

f−

⎡ ⎤− = =⎢ ⎥

⎢ ⎥⎣ ⎦⎡ ⎤= ⋅ = ⋅⎢ ⎥⎣ ⎦

∫ ∫H K H K K K

H K H K K K

Λ

Λ

( ', ) ( ' , ) ( , )

( ', ) ( ', )H

t f t t f t f

t f t f

= −

− =

K K K

K K

( , ) ( , ') ( ' )

( , ) ( ', ) ( ' )

H H

t

H H

f

t f t f dt f f

t f t f df t t

δ

δ

= = −

= = −

∫∫

K K I K K I

KK I K K I

( ) ( , ) ( )

( ) ( , ) ( )

f

H H

t

t t f f df

f t f t dt

= =

= =

∫∫

x Kx x K x

x K x x K x

F F

F F

( )

( ) ( ) ( ) ( ) ( )

Toeplitz Toeplitz

HH

f

f f f f f

= = = ⋅

= = ⋅ ⋅ =

y H x H Kx Kx

y K y K Kx x y x

Λ

Λ = Λ Λ

F F

F F FF F


Unfortunately, and to the best of the author’s knowledge, when the operators are notToeplitz there are no universal eigenfunctions known to guarantee a property equivalentto (5.156). There are some results available in the literature, namely chirp eigenfunctions[100] and time-frequency distributions [101], [102], that try to circumvent this problembut fail to do so in a satisfying manner. In the first case, chirp eigenfunctions are madeavailable to a class of very simple, two-ray flat fading channel models, and in the secondcase it is found that the time-frequency version of (5.156) is only a good approximationfor underspread channels. The difficulty surrounding LTV channels is that they alsospread the input in frequency, not only in time, resulting in a convolution in both domains.Accounting for these limitations, the frequency-domain characterization of the LTVMIMO channel will be addressed by trying to extend the operator approach to the two-dimensional channel Fourier transform

where the underscript denotes that we are working in the Fourier domain.

5.4.1 FREQUENCY-DOMAIN OPERATORS

From Section 5.2 it is known that delay convolution with a generic has the follow-ing frequency-domain counterpart

where the convolution is now in the Doppler variable . If the channels’ time variationsare statistically modelled, then the Fourier transform in the Doppler variable must also bemodelled in statistical terms. The random matrix ,

is such that and

It is informative to graphically depict the convolution in (5.158) to understand how itworks. This can be done by noticing that for a fixed the graph of varies alonglines of equation , as shown in Figure 5.12. The interval of convolution is atmost , which is the maximum span of intersection of the lines with the domain of

. We see that whenever the output at will be the contribution of all inputfrequencies within Doppler range, effectively embedding the input with inter-frequencyinterference (IFI) that is responsible for time-fading. Due to the skewed nature of the con-volution, the bandwidth of the output signal will always be higher than that of the input,

(5.157)2 ( )( , ) ( , ) ( , ) j f t

tt f t e dtdπ τ ν

ττ ν τ τ− +→ = ∫ ∫H H H

F

F

F

(5.158)

(5.159)

(5.160)

( , )tτH

( ) ( , ) ( ) ( , ) ( )f f f d f fνν

ν ν ν ν ν ν= − − = − ⊗∫y H x H xF F F FF

ν

( , )f νHF

2( , ) ( , ) j t

tf f t e dtπνν −= ∫H H

F F

.

2( , )ijE f t

⎡ ⎤< ∞⎢ ⎥

⎢ ⎥⎣ ⎦HF

. . . .

22( , ) ( , ) ( , ) ( , ) 0j t

ij ij ij ijf t f e d E f t f tπν

νν ν ⎡ ⎤

= → − =⎢ ⎥⎢ ⎥⎣ ⎦∫H H H H

F F F F

f ( , )f ν ν−HF

( )g fν ν= − +

max2 2ν( , )f νH

Fmax 0ν > f


and at the most

negatively impacting the fidelity of the original signal (specially if it is narrowband). Whenthe TF is time-invariant ( ) the Doppler frequency response is an impulse function

so that the frequency output is the typical product

Setting in (5.158) and defining the counterpart of (5.51) as

the first can be restated as follows

Using the time-domain definition in (5.51) one can easily show that is the Fouriertransform of ,

Using the two-sided-continuous frequency operator we have effectively replacedfrequency convolutions by linear operator products. In fact, owing to the continuity of thediscrete-time fourier transform of discrete-time signals, this approach can be easily ex-

FIGURE 5.12 Skewed Doppler convolution induced by LTV MIMO channels.

(5.161)

(5.162)

(5.163)

(5.164)

(5.165)

(5.166)

ν

0

maxνBx

BH

f

f

maxν−

maxν

'f

'ν

f

()

g

f

ν

ν=

−+

( , ) ( , )f fν ν ν→ −

( , )f νHF

( )fxF

( ) maxmin , 2B B B ν≤ +y x H

max 0ν =

( ) ( ) ( )( , ) ( ) ( ) ( ) ( )LTI LTI LTIf f fν ν ν δ ν δ ν⋅ ⋅ ⋅− = − =H H HF F F

( ) ( ) ( ) ( ) ( ) ( )LTI LTIf f f d f fν

δ ν ν ν⋅ ⋅= − =∫y H x H xF F F FF

'f fν = −

( , ') ( ', ') ( , ') ( ', )f f f f f f f f f f− = − ⇔ − = +H H H HF F F F

' '( ) ( ', ') ( ') ' ( , ') ( ') '

f ff f f f f df f f f df= − = −

=

∫ ∫y H x H x

y Hx

F F F FF

F FF

( , ')f fHF

( , ')t tH

2 ( ' ') 2 (( ') ' ' )

' '( , ') ( ', ) ' ( , ') '

( ', ') ( , ') ( , ') ( , ')

j ft f t j f f t f t

t t t tt t t t t e dtdt t t e dtdt

f f f f f t t f f

π π− + − + −= − =

= − + = ⇒ →

∫ ∫ ∫ ∫H H H

H H H HF F F

F

F

HF


tended to one-sided-discrete operators: 1)

and 2)

and two-sided-discrete operators

where the integration in the first and latter case is over one period . The latter case isintuitive because it is the discrete counterpart of two-sided-continuous operators, howeverthe first and second cases are a bit harder because they involve hybrid operators. We shallprove the first. Starting with the system

and the inverse discrete-time Fourier transform (IDTFT)

we have

Also, from (5.51), (5.166) and the IDTFT, we arrive at

which finally leads to the definition in (5.167)

completing the proof. Again, and since the DTFT of a constant is given by

(5.167)

(5.168)

(5.169)

(5.170)

(5.171)

(5.172)

(5.173)

(5.174)

(5.175)

( )

( ) ( ) ( )

1

'/

2 ( ') 2 '

( ) ( , ') ' '

( , ') ', '

s

s s

sf f

j f f T j f T

f f f f f df

f f f e f eπ π

−

−

= = = −

− = =

∫y Hx y Hx y H x

H H x x

F F F FF F

F F F F

F

( )

( ) ( ) ( )'

2 ' 2 '

( ) ( , ') ' '

( , ') , ' 's s

f

j f T j f T

f f f f df

f f e f f f eπ π

= = = −

− = − =

∫y Hx y Hx y H x

H H y y

F F F FF F

F F F F

F

( )

( )

1

'/

2 ' 2 ( ')

( ) ( , ') ' '

( , ') ,

s

s s

sf f

j f T j f f T

f f f f f df

f f e eπ π

−

−

= = = −

− =

∫y Hx y Hx y H x

H H

F F F FF F

F F

F

sf

( ) ( , ) ( , )s sl ll l

t t lT t t lT= − =∑ ∑y H x H x

( )1 2 2

'/

s s

s

j T j lTsl

f ff e e dπν πν ν−= ∫x x

( )

( )

1 2 ' 2 ' 1

'/ '/

1

'/

( ) ( , ) ' ( , ') ( ') '

( , ') ( ') '

s s

s s

s

j f lT j f Ts s s

f f f fl

sf f

t f t lT e e df f t f f df

f f f f f df

π π− −

−

= = −

→ = − =

∑∫ ∫

∫

y H x H x

y H x y Hx

F F

F F F FF F

1 2 ( ' ')

'/

1 2 ( ' ')

'/

( ', ) ( , ') ( , ') '

( , ') ( ', ' ) ( ', ) '

s

s

j ft f ts

f f f

j ft f ts

f f f

t t t t t f f f e df df

t t t t t f f f f e df df

π

π

− +

− +

− = =

→ = − = + −

∫ ∫∫ ∫

H H H

H H H

F

F

( ) ( )2 ' 2 ( ')( ', ) , ( , ') ',s sj f T j f f Tf f f f e f f f eπ π −+ − = → − =H H H HF F F F

2 ( )sj lTs s

l k

e f kfπν δ ν− = −∑ ∑


a time-invariant TF is expressed as

which substituted into (5.167) yields the anticipated frequency product

5.4.2 THE DISCRETE MIMO MODEL IN THE SPACE

It has been shown in Section 5.3 that the MIMO noiseless output can be losslessly ex-panded in the space by the following time-domain pair

which, at the same time, decomposes the MIMO channel operator as . Eventhough this operator product is undertaken in the time domain, it may also be carried outin the frequency domain by taking into account the rules established in Section 5.4.1. Wethus restate (5.178) as follows

In effect, we are expressing the frequency output in the space . The channel de-composition is expanded as

and we must have the orthogonality condition (identity operator)

or, from (5.167), equivalently

(5.176)

(5.177)

( ) ( )

( )

( ) ( )

( )

2 ( ')', ( ) ( ' )

' ( ( ' ))

sLTI LTI

LTI

j f f Ts s

k

s s s

k

f e f f f kf

f f kf f f kf

π

ζζ δ ζ δ

δ

⋅ ⋅

⋅

− = − ⊗ −

= − − −

∑∑

H H I

H

F F

F

( ) ( )

( ) ( ) ( ) ( )

( )

( ) ( )

1 2 ( ') 2 '

'/

' 02 ' 2

'/

( ) ', '

' ( ' ) '

s sLTI

s

ss s

LTI LTI

s

j f f T j f Ts

f f

f f kfj f T j fTs s

f fk

f f f e e df

f kf f f kf e df f e

π π

π πδ

⋅

⋅ ⋅

− −

− = =

=

= − − − =

∫∑∫

y H x

H x H x

F FF

F F F F

( )ΦF

R

(5.178)

(5.179)

(5.180)

(5.181)

(5.182)

( )R Φ

H Hopt opt= = = = = =u Ha H a b b u Ha H aΦ ΦΦ Φ Φ Φ

=H HΦΦ

H Hopt opt⋅ ⋅ ⋅ ⋅= = = = = =u Ha H a b b u Ha H aΦ ΦΦ Φ Φ Φ

F F F F F F F F F F F F F F F FF F

uF

( )ΦF

R

( ) ( ) ( )

( ) ( ) ( )

1

''/

''

, ' , '' '', ' ''

, ' '', '', ' ''

s

sf f

H H

f

f f f f f f f df

f f f f f f df

⋅ ⋅

⋅ ⋅

−= − = − −

= − = − −

∫∫

H H H H

H H H H

Φ Φ

Φ Φ

Φ Φ

Φ Φ

F F F F F FF F

F F F F F FF F

( ) ( ) ( )''

, ' '', '', ' ''

( ) ( ' ) ( ' )

H H

f

s s s s

k k

f f f f f f df

f f f kf f f f kfζ

δ ζ δ δ

= = − = − −

= − ⊗ − = − −

∫∑ ∑

I

I I

Ω Φ Φ Ω Φ ΦF F F F F F

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

2 ' 2 ( ') 2 ( '' ) 2 ( '' ')

''

' ''2 ( ) 2 2 2 ( )

2 2

, , ', ''

, , ,

, , ( )

s s s s

s s s s

s s

Hj f T j f f T j f f T j f f T

f

f f f f Hj f T j T j T j T

H j T j Ts s

k

e e f e f e df

e e f e f e d

f e f e f kf

π π π π

ν ζπ ν πν πζ π ν ζ

ζ

πζ πν

ζ

ν ζ

ν δ ν

− − −

= − → −− − −

−

=

→ = −

= ⊗ − = −

∫∫

∑ I

Ω Φ Φ

Ω Φ Φ

Φ Φ

F F F

F F F

F F


One infers that if orthonormality is present in the time domain it must also be present inthe frequency domain, and operators smooth the path to the orthogonality conditions. Itis interesting to narrow (5.182) to the LTI scenario and see if the result is the one expected.Applying (5.176)

one recovers

which, comparing with (5.182), implies

This expression is no less than the sampled version of , andresembles the Nyquist condition for transmission without intersymbol interference (ISI).It may also be easily retrieved from the initial orthogonality relationship in (5.57), restrict-ed to time-invariancy

Now, taking the Fourier transform and sampling we find

which is exactly (5.185). In light of these deductions one confirms that the time-invariantcase can be obtained directly from the time-varying formulas by setting some fundamentalrestrictions such as (5.176).

It may be finally checked by similar derivations that the channel decomposes as

(5.183)

(5.184)

(5.185)

(5.186)

(5.187)

(5.188)

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

2

2 ( )

2 ( ) 2 2

, ( )

, ( )

, ( )

sLTI LTI

sLTI LTI

s s sLTI LTI

H j T Hs s s

k

j Ts s s

m

j f T j T j fTs s

k

f e f f kf kf

f e f f mf mf

e e f e kf

πζ

π ν ζ

π ν πν π

δ ζ

ν ν δ ν ζ

δ ν

⋅ ⋅

⋅ ⋅

⋅ ⋅

−

−

−

= − − +

− = − − − +

= −

∑∑∑

Φ Φ

Φ Φ

Ω Ω

F F

F F

F F

( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( )

2 2 2 ( )

/

( ) , ,

( ) ( )

(

s s sLTI LTI LTI

LTI LTI

s s s

LTI LTI

j fT H j T j Ts s

k

Hs s s s s s

k m

kf kf mf Hs s s s s

m

f e kf f e f e d

f f f kf f mf kf mf d

f f f mf f mf kf

π πζ π ν ζ

ζ

ζ

ζ ν

δ ν ν ζ

ν δ ζ δ ν ζ ζ

δ ν

⋅ ⋅ ⋅

⋅ ⋅

⋅ ⋅

− −

= = −

− = −

= − − − − − +

⎡ ⎤⎢ ⎥= − − −⎢ ⎥⎢ ⎥⎣ ⎦

∑ ∫∑ ∑ ∫

∑

Ω Φ Φ

Φ Φ

Φ Φ

F F F

F F

F F)

k∑

( ) ( ) ( )( ) ( ) ( )2 s

LTI LTI LTIj fT H

s s s

m

e f f mf f mfπ⋅ ⋅ ⋅= − − =∑ IΩ Φ Φ

F F F

( ) ( ) ( )( ) ( ) ( )LTI LTI LTIHf f f⋅ ⋅ ⋅=Ω Φ Φ

F F F

( )( ) ( )( )

( ) ( ) ( )( ) ( )

( ) ( ( ) )

( ) ( ) ( ) ( ) ( )

LTI LTILTI

LTI LTI LTILTI LTI

Hs s s l

H H

lT kT k l T d

t t d t

τ

ττ

τ τ τ δ

τ τ τ τ

= − − − =

→ = − − = − ⊗

∫∫

IΩ Φ Φ

Ω Φ Φ Φ Φ

( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )2

( ) ( )LTI LTI LTI

sLTI LTI LTI

H

j fT Hs s s

m

f f f

e f f mf f mfπ

⋅ ⋅ ⋅

⋅ ⋅ ⋅

=

→ = − − =∑ I

Ω Φ Φ

Ω Φ Φ

F F F

F F F

( ) ( ) ( )2 2 2 ( ) 2

/, , ,s s s s

s

j T j T j f T j T

ff e f e e eπν πζ π ν πν

ζν ζ ⋅

−− = − ⊗H HΦΦF F F F


and its discrete version equals

supplying us with sufficient information to construct a diagram (Figure 5.13) for the time-varying (Doppler-spread) MIMO discretization in the frequency domain.

5.4.3 THE TS-MMF IN THE FREQUENCY DOMAIN

It was found in Section 5.3.3 that the continuous matrix matched filter to the channel was the filter . To find its frequency-domain counterpart

one can proceed in one of two manners. The first is to begin with the two-dimensional inverse Fourier transform

and make some variable changes

from which we retrieve

The alternative method is to remember that the MMF in the time-domain corresponds tothe Hermitian adjoint operator in the operator domain, and the same happens in the fre-quency domain. If denotes the two-sided-continuous channel frequency operator, then

(5.189)( ) ( ) ( )2 ( ) 2 2 2, , ,s s s sHj f T j T j T j Te e f e f eπ ν πν πζ πν

ζν⋅

− −= ⊗ −H HΦ ΦF F FF

FIGURE 5.13 Frequency perspective of the discrete MIMO model after channel decomposition and output projection onto .

/ ( )tu u

/ ( )tn n

/ ( )tr r opt= +r b n Φ Φ

n Φ

( )2, sj Tf e πνν−H H F F

( , )st lT t−H H

/ ( )fu uF F

( )2, sH H j Tf e πζ− Φ Φ

F F

( ) ( ) ( )2 ( ) 2 2 2, , ,s s s sHj f T j T j T j Te e f e f eπ ν πν πζ πν

ζν⋅ ⋅

− −= ⊗ −H H H ΦΦ ΦF F F FFF

,k l k−H H Φ Φ

⋅H ΦF F opt⋅b

F

optb

( )2, sj Tf e πζζ− Φ ΦF F

( , )st lT t− Φ Φ

( )( ) 2 sopt j fTe π⋅bF

( )/ optopt kb b

( , )H H

slTτ τ− − Φ Φ/ ka a

( )2/ sj fTe πa aF F

( )ΦF

R

(5.190)

(5.191)

(5.192)

(5.193)

( , )tτH ( , ) ( , )Ht tτ τ τ= − −MMF H

( , )f ν ν−MMFF

2 ( )( , ) ( , ) j f t

ft f e d dfπ τ ν

ντ ν ν+= ∫ ∫H H

F

2 ( )

2 ( )

( , ) ( , )

( , )

H H j f t

f

H j f t

f

t f e d df

f e d df

π τ ν ντ

ν

π τ ν

ν

τ τ ν ν

ν ν ν

− +

+

− − =

= + −

∫ ∫∫ ∫

H H

H

F

F

( , ) ( , )Hf fν ν ν− = −MMF HF F

HF

'( ) ( ', ) ( ') '

H H

ff f f f df= = −∫y H r y H r

F F F FF F


Now we set and recall (5.164) to get

which, in light of (5.158), gives the frequency expression for the TS-MMF as in (5.192),and yields a (skewless) convolution that is very similar to the delay convolution in (5.50).This means that the TS-MMF has the desirable properties of not introducing IFI and notexpanding the system bandwidth (either input or channel limited). After sampling the out-put the TS-MMF behaves as

a result that might also have been directly retrieved from

To decompose the TS-MMF we start from

apply (5.167) and (5.169), and set and , finding

From (5.198) one realizes that the frequency-domain TS-MMF decomposes into twomatched filters, one matched to the discrete filter and the other matched to the or-thonormalized version of the channel , as already obtained in (5.189). This is in conso-nance with the time-domain results.

To conclude the frequency-domain interpretation of the TS-MMF approach to dis-cretization, we set the aim of finding the global input-output transfer function. Again,working with operators and using the orthogonality condition in (5.181) one easily attains

which expands as

and leaves the operator domain in the following form

(5.194)

(5.195)

(5.196)

(5.197)

(5.198)

(5.199)

(5.200)

(5.201)

'f f ν= −

' '

'

( ) ( ', ) ( ') ' ( , ' ) ( ') '

( , ) ( ) ( , ) ( )

H H

f f

f f H H

f f f f df f f f f df

f f d f fν

ννν ν ν ν

→ −

= − = −

= − − = − ⊗

∫ ∫∫

y H r H r

H r H r

F F F FF

F F F F

( )2( , ) ( , ( )) , sH H H j Ts s s

k

lT f f kf f e πντ τ ν −− − → − − =∑H H HF F

( )2

'( ) ( ', ) ( ') ' , ( )s

H H j T

ff f f f df f e f dπν

νν ν−= − = −∫ ∫y H r H r

F F F FF

( ) ( ) ( )1

''/ ', '', ', '' ''

s

H H H HH Hs

f ff f f f f f f df⋅ ⋅

−= − = − −∫H H H HΦ ΦΦ Φ

F F F F F FF F

'f f ν= − ''f f ζ= −

( ) ( ) ( )

( ) ( )

2 1 2 2 2 ( )

/

2 2 2

/

, , ,

, ,

s s s s

s

s s s

s

H Hj T H j fT j T j Ts

f

HH j fT j T j T

f

f e f e e f e d

e e f e

πν π πζ π ν ζ

ζ

π πζ πν

ζ

ζ ζ⋅

⋅

− − − − −

− −

= −

= ⊗

∫H H

H

Φ

Φ

Φ

Φ

F F FF

F FF

⋅HΦF F

ΦF

H HH H H⋅ ⋅ ⋅ ⋅ ⋅ ⋅= = = =S H H H H H H H HΦ Φ ΦΦ Φ ΦΦ Φ Ω

F F F F F F F F F F F FF F FF F F

1

''/( , ') ( '', ) ( '', ') ''

s

Hs

f ff f f f f f f df⋅ ⋅

−− = − −∫S H HΦΦF F F FF

( ) ( ) ( )

( ) ( )

2 ( ) 2 1 2 2 2 ( ) 2 ( )

/

2 2 2 ( ) 2

/

, , ,

, ,

s s s s s s

s

s s s s

s

j f T j T H j fT j T j f T j Ts

f

H j fT j T j f T j T

f

e e f e e e e d

e e e e

π ν πν π πζ π ν π ν ζ

ζ

π πζ π ν πν

ζ

ζ⋅ ⋅

⋅ ⋅

− − − − −

− −

=

= ⊗

∫S H H

H H

Φ Φ

Φ Φ

F F FF F

F FF F


The MIMO system after matched filtering and sampling is thus decomposed into a dis-crete, Doppler-spread filter and its respective discrete matrix matched filter, noticeably re-calling the identical time-domain counterpart

Whenever the condition

is met, then there is linear separability of vector samples, so that there is neither ISI norDoppler spread levied on the input symbols, only additive noise.

Let us inquire what happens when the channel is time-invariant. In light of (5.176) oneinfers

so that (5.201) is reexpressed as

leading to the spectral product

which is exactly the DTFT of the time-invariant instance of (5.202)

or, from the z-Transform,

From the previous discussion in Section 5.3.3 it was acknowledged that is hermitian andnon-negative definite, yet non-stationary. For the stationary case, multivariate spectral de-compositions such as (5.206) (or (5.208)) are not really unexpected, and in fact have beenaround for quite a while since the time of Norbert Wiener [103], [104]. The accomplish-ment has been to extend the decomposition to the realm of non-stationary discrete oper-ators, and the result is (5.201), not a product, but a convolution. The system diagram afterthese decompositions is depicted in Figure 5.14, where , and are nec-

(5.202)

(5.203)

(5.204)

(5.205)

(5.206)

(5.207)

(5.208)

, ,( , ) Hs s r k r l k

rlT kT − −= ⊗S H HΦ Φ

( )2 ( ) 2( , ) , ( )s sj f T j Ts s l s s

k

lT kT e e f kfπ ν πνδ δ ν−= = −∑S I S I∼F

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

2 ( ) 2 2

2 2 2 ( )

2 ( ) 2 ( ) 2 ( )

, ( )

, ( )

, ( )

s s sLTI LTI

s s s sLTI LTI

s s s sLTI LTI

j f T j T j fTs s

m

H j fT j T H j f kf Ts s

k

j f T j T j f mf Ts s

m

e e f e mf

e e f e kf

e e f e mf

π ν πν π

π πζ π

π ν π ν ζ π ν

δ ν

δ ζ

δ ν ζ

⋅ ⋅

⋅ ⋅

⋅ ⋅

−

− −

− − − −

= −

= − +

= − +

∑∑∑

S S

H H

H H

Φ Φ

Φ Φ

F F

F FF F

F FF F

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( )

2 2 ( ) 2 ( )

/

0 2 2

( )

( ) ( )

( )

s s s s sLTI LTI LTI

s

ss s

LTI LTI

j fT H j f kf T j f mf Ts

m k m

s sf

kf H j fT j fTs

m

e mf e e

kf mf d

e e mf

π π π ν

ζ

ζ π π

δ ν

δ ζ δ ν ζ ζ

δ ν

⋅ ⋅ ⋅

⋅ ⋅

− − −

= =

− =

⋅ − + − +

= −

∑ ∑∑

∫∑

S H H

H H

Φ Φ

Φ Φ

F F FF F

F FF F

( ) ( ) ( )( ) ))

2 2 2((

s s sLTI LTILTI

j fT H j fT j fTe e eπ π π⋅ ⋅ ⋅=S H HΦΦF F F FF

( ) ( ) ( )( ) )( ) ) ) )

2 2 2(( ( (( ) s s s

LTI LTILTI LTI LTI LTI

H j fT H j fT j fTs r l

rlT e e eπ π π

⋅ ⋅ ⋅−= ⊗ → =S H H S H HΦΦ Φ ΦF F F FF

F

( )( ) )( ) ) ) ) (( ( (( ) ( ) 1/ ( )LTI LTILTI LTI LTI LTI

H Hs r l

rlT z z z⋅ ⋅ ⋅

∗−= ⊗ → =S H H S H HΦΦ Φ Φ

F F F FF

Z

S

( )f⋅rF

2( )sj fTe π⋅y

F

( )f⋅rΦF F


essarily stochastic spectral representations of their time-domain counterparts.

5.5 Noise Whitening and Full Discretization

Two approaches to optimal detection in LTV MIMO, namely 1) the TS-MMF and 2) theorthonormalized TS-MMF, have been presented in Section 5.3, having the latter been fa-voured over the first. In the following sections we shall study the possibilities of 1) orthon-ormalizing the TS-MMF using a fully discrete matrix filter and 2) fully discretizing boththe TS-MMF and the ORTHO-TS-MMF.

5.5.1 WHITENING THE NOISE AT THE TS-MMF OUTPUT

We have seen that the advantage of the ORTHO-TS-MMF was the uncorrelatedness ofthe noise samples at its output, because this meant a reduction in detection complexity. Itwas also shown that the ORTHO-TS-MMF originated from the channel decomposi-tion into a causal (that may or may not be invertible) and a semi-orthonormal thatassured the white noise condition

At the output of the TS-MMF the autocorrelation is

so it becomes natural to ask if there exists a filter that, applied at the TS-MMF output,renders the noise white, i.e.

FIGURE 5.14 Discrete MIMO model after matched filtering and spectral decomposition.

/ ( )tn n

skT

Hn

( )2, sj Tf e πνν−HF

( )2 ( ) 2,s sj f T j Te eπ ν πν⋅

−HΦF F

( )2, sj Tf e πζζ−ΦF

opt⋅bF

( )2, sj TH f e πν−HF

( , )H f ν−HF

( )2 sj fTe πaF

( )2 sj fTe πyF

( )fuF

( )frF

( )2, sH j Tf e πν−ΦF

skT

( )2 2,s sH j fT j Te eπ πζ⋅

−HΦF F⋅r Φ

F F

( ),H f ν−ΦF

( )2 ( ) 2,s sj f T j Te eπ ν πν⋅

−HΦF F

( )2 2,s sH j fT j Te eπ πζ⋅

−HΦF F

opt⋅bF

( )2 ( ) 2,s sj f T j Te eπ ν πν−SF

Sa F F

(5.209)

(5.210)

(5.211)

HΦ

HΦ Φ

0 0H

N N= =n nR IΦ Ψ

Φ Φ

HH

0 0H H

N N= =H Hn nR H H SΦ ΦΦ Φ

ϒ

0 0 0H H

H H H H HN N N= = =

H Hn nR H H H H Iϒ ϒ Φ Φ Φ Φϒ Φ Φ ϒ ϒ ϒ

5.5 Noise Whitening and Full Discretization 159

If such a filter indeed exists, it is fully discrete and must be such that

We know, however, that, whenever is defined, the solution of minimum Frobeniusnorm of the system is obtained from the Moore-Penrose pseudoinverse of

, , so we may test the choice of a filter that satisfies

To make things clearer, we now expand in a URV factorization, for instance, its sin-gular value decomposition (SVD)

from which we decompose the pseudoinverse as and rewrite (5.213) in theform

Consequently, a sufficient condition for the fulfilment of (5.212) is , which is sat-isfied if is square and non-singular, hence necessarily a full row-rank operator. Thiscan also be seen from the full row-rank formula

that shows that the pseudoinverse is a right inverse of . In other words, the hermitianadjoint of the pseudoinverse is a left inverse of and whitens the noise at theTS-MMF output

We should keep in mind that is the band-based sparse L factor resulting from ablock-wise QL operator factorization, and its dimensions are , where is thenumber of MIMO inputs, is the number of transmitted vector symbols and is thenumber of independent functions that carry those symbols to the output. is always smaller or equal than , so that has a lower triangular row-echelon form. For(5.212) to be valid the columns of must be linearly independent, i.e.

Being the channel statistically determined, so are the functions and so is theoperator . Moreover, from the statistical independence of a set of domain functions

one assesses the statistical independence of the matrix coefficients within eachcolumn of , and thence the condition in (5.218) can be deemed satisfied with highprobability. The whitening approach to the TS-MMF orthonormalization is depicted inthe system diagram of Figure 5.15.

(5.212)

(5.213)

(5.214)

(5.215)

(5.216)

(5.217)

(5.218)

ϒ

( ) ( )H HH H H H H H H H= = =H H H H I HΦ Φ Φϒ ϒ = ϒ ϒ Θ Θ Θ ϒ Φ

HΦH H H

=H HΦΦHΦ

†HΦ

†

† †

H

H H H H H H H

== = =

HH H H H H

Φ

Φ Φ Φϒ

Θ ϒ Φ

HΦ

( ) ( ) ( ) ( );H H H H

= = = = =H UDV U U V V I U H V HΦ Φ ΦR R R R

† 1 H−=H VD UΦ

†H H H H H H H H H H H−= = =H H UD V VD U UUΦ ΦΘ Φ Φ Φ

H=UU I

U HΦ

( ) ( )1 1† †H H H H− −= → = =H H H H H H H H H H IΦ Φ Φ Φ Φ Φ Φ Φ Φ Φ

HΦHHΦ

†H= HΦϒ

† †0 0 0

H H H H H HN N N= = = =n nn nR H H H H R I

Φ ΦΘ Θ Φ Φ Φ ΦΦ Φ Φ Φ

HΦ

T L Tn N n L× Tn

L LN

( , )st lT t−H LN

L HΦHHΦ

, H H

kl k l ll

= → = = ∀ → =∑H c 0 c 0 H c 0 c 0Φ Φ

( , )st lT t−H

HΦ LN

( , )st lT t−H

HΦ


5.5.2 FULLY DISCRETIZING THE TS-MMF AND ORTHO-TS-MMF

We have seen that the MIMO channel acts on the input symbols by means of a semi-dis-crete matrix operator, producing at its output a continuous vector signal. The operatorworks by sampling the channel TF at equispaced parallel transmission domains, asseen in Figure 5.4, scaling the matrix functions thus obtained by the input vector samplesand then superposing each of them in an overall weighted sum . The sum is continu-ous at any time instant and, because of the bandlimited character of the operator , it isalso bandlimited, hence discretizable by sampling. The possibility of discretizing the re-ceived vector signal prior to matched filtering would be rather convenient, in the sensethat not continuous but discrete filters TS-MMF or ORTHO-TS-MMF would be requiredfor optimal detection.

It was shown in Section 5.2 that the symbols are previously sent through a time-invariant pulse shaping filter that outputs the signal for transmission. The input-output relation has the following two equivalent forms

On the other hand, the channel noiseless output is related to by

where the global frequency-domain TF is

We shall assume that the shaping filter is the squared-root raised cosine (SRRC) of lessthan unity roll-off factor , Nyquist bandwidth , and total bandwidth ,i.e.

FIGURE 5.15 Orthonormalizing the TS-MMF using the Moore-Penrose pseudoinverse of the discrete channel filter.

( , )st lTH/ ( )tu u

/ ( )tn n

/ ( )tr r

( )/ optopt kb b

opt= +r b n Φ Φ

n Φ

( , )H

st kTΦ

( , )H

st lTH

†H H H= =H H H H H

Φ ΦΦ

,k lH Φ

Hr

( , )st kTΦ

†,

Hl kH Φ

/ ka a

(5.219)

(5.220)

(5.221)

(5.222)

( , )tτH

( )tu

H

( )tr

ka

τ( )G ( )ts

( )2( ) ( ) ( ) ( ) ( ) ( ) sj fTss k

k

t t d t kT f f e π

ττ τ τ= − = − → =∑∫s G x G a s G a

F F F

F

( )fsF

( )

( )

2 ( )

2 ( )

1 1( ) ( , ) ( ) ( , ) ( )

2 2

( , )

s

s

j f T

j f T

f f f d f f e d

f e d

π ν

ν ν

π ν

ν

ν ν ν ν ν ν ν ν

ν ν ν

−

−

= − − = − −

= −

∫ ∫∫

u C s C G a

H a

F F F F F F

F F

1( , ) ( , ) ( )

2f f fν ν=H C G

F F F

roll sB f= (1 )B roll+

,( ) ( )sf rollf RC f=G I

F


where

and the following condition is met

Since is bandlimited and has limited Doppler bandwidth, one concludesthat is bandlimited on both arguments, that is

and, in light of Figure 5.12, its output’s bandwidth is limited to

Consequently

and it is assumed that the signalling rate is high enough to accommodate both the roll-offand the Doppler spread. We will be sampling before applying the matched filter, andto avoid aliasing we must sample at a rate higher than (5.226). Restricting the roll-off factorto , then an adequate sampling rate is , and so we write

Now we apply the TS-MMF as given in (5.195), and, knowing that it does not intro-duce extra bandwidth, we are free to sample its output also at twice the signalling rate, ob-taining

To simplify (5.229) we must look closely to the following bandwidth limits

from which it is possible to conclude that there is no sampling-induced overlapping on theinner convolution and, consequently, that the sum of the products in (5.229) can be ex-

(5.223)

(5.224)

(5.225)

(5.226)

(5.227)

(5.228)

(5.229)

(5.230)

( )( ),

1/ (1 )2

1( ) 1 cos (1 ) / (1 ) (1 )

2 2 2 2

0 (1 )2

B roll

BB f roll

B B BRC f f roll Broll roll f roll

BB

f roll

π

< −

= + − − − ≤ < +

≥ +

⎧⎪⎪⎪⎪⎪⎪ ⎡ ⎤⎪⎨ ⎢ ⎥⎪ ⎣ ⎦⎪⎪⎪⎪⎪⎪⎩

,( ) ( ) ( )s

Hs s s s f roll s

k k

f f kf f kf f RC f kf− − = − =∑ ∑G G I IF F

( )fGF

( , )f νCF

( , )f νHF

max

(1 )/2( , ) 0

s sf f roll ff forν

ν ν

⎧ > + <⎪⎪= ⎨ >⎪⎪⎩H

max max2 (1 ) 2 2s sB B f roll fν ν≤ + = + + <u H

( )

max

2( ) ( , )

0 (1 )/2

sj fT

s s

f f e

for f f roll f

π

νν ν

ν

= − ⊗

= > + + <

u H aF F F

( )tu

max1 2 / sroll fν< − 2 sf

( ) .

2 /22 ( ) 2 ( 2 )sj fTs s s

k

e f f f k fπ = = −∑u u uF F F

( )

( )

.

( )( ) ( )2 /22

2 /2

( ) 2 ( 2 )

2 2 , ( 2 )

MFsMF MF

s

j fTs ss

k

H j Ts s s

k

e f f f k f

f f k f e f k f d

π

πν

νν ν−

= = −

= − − −

∑

∑∫

u u u

H u

F F F

F F

( )max

max

2 /2(1 )/2

, 0 2

( ) 0 (1 )/2

ss sH j T

s

s s

f f roll ff e

k f k

f f f roll f

πν

ν ν

ν ν ν

−⎧ < + <⎪⎪≠ → ⎨⎪ − < ∀⎪⎩

− ≠ → − < + + <

H

u

F

F


changed by a product of sums, and the integration be confined to one period of length. The outcome is

which is not unexpected because it sums up to the periodic convolution resulting fromfiltering with a discrete time-varying filter, namely

The fully discrete TS-MMF is thus

and its time-domain counterpart is

which is easily obtained from (5.167) as follows

It becomes clear that discretizing the TS-MMF has the drawback of demanding the esti-mation of twice the original transmission domains. Continuity is traded by redundanttransmission data. As shown in Figure 5.16, if the receiver is willing to sample the received

vector signal, it must at the same time gather information in the additional time domains

(5.231)

(5.232)

(5.233)

(5.234)

(5.235)

FIGURE 5.16 Effect of sampling the received signal at twice the signalling rate.

2 sf

( ) ( )

( ) ( )

( ) ( ) ( )

( ) 2 /2 2 /2

/2

2 /2 2 ( ) /2

1 2 /2 2 /2 2 ( ) /2

/2

2 , 2 ( 2 )

2 ,

2 ,

s sMF

s

s s

s s s

s

Hj fT j Ts s s

f k l

H j T j f Ts

k

H j fT j T j f Ts

f

e f k f e f f l f d

f k f e e d

f e e e d

π πν

ν

πν π ν

ν

π πν π ν

ν

ν ν

ν

ν

−

− −

− − −

= − − −

= −

=

∑ ∑∫∑∫

∫

u H u

H u

H u

F F F

F F

F F

( ) ( ) ( )( ) 2 /2 2 /2 2 /2 2 /2

/2,s s s sMF

s

Hj fT j fT j T j fT

fe e e eπ π πν π

ν

−= ⊗u H uF F F

( ) ( )2 ( ) /2 2 /2 2 /2 2 /2, ,s s s sHj f T j T j fT j Te e e eπ ν πν π πν− −=MMF HF F

( )( /2, /2) /2,( ) /2Hs s s slT kT lT k l T= − −MMF H

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

.( )

( ) ( )

12 /2 2 ' /2(2 )

'/2

(2 ) (2 ) (2 )(2 )

2 ', '

/2 /2, /2 /2

/2,( ) /2 /2

s sMF

s

MF MF

j fT H j f Ts s

f f

H Hs s s ss s ss

lH

s s sl

e f f f e df

kT lT kT lT

lT k l T kT

π π−= −

→ = =

= − − ⊗

∫∑

u H u

u H u u H u

H u

F F F

sT 2 sTsT−2 sT− 0

sT

2 sT

sT−

2 sT−

k

l

τ

t

(

)

(/2

,/2

)

/2,(

)/2

s

s

s

s

lT

kT

lT

kl T

→−

−


, something that may not be acceptable and/or feasible depending on the sys-tem design. If the receiver chooses the sampling path, it must then perform sample deci-mation at the TS-MMF output

Aliasing will necessarily occur as part of decimation, but in this case it is a necessary in-gredient of optimal detection.

Another trait of the sampling approach is related to the noise. As explained inSection 5.2, the noise introduced by the system is theoretically modelled as space-timewhite, and for the purpose of detection with the TS-MMF it is a valid model because theTS-MMF effectively blocks the noise for . However, losslessly discre-tizing the matched filter will require that the noise be eliminated for with (prefer-ably) an ideal filter bank, otherwise aliasing and distortion will occur. Even though notexplicit, this pre-filtering stage is already present in the down-conversion process of thereceiver front-end (see Figure 5.1), but since the baseband noise was deemed white, this“implicit” filtering should be made explicit in the respective system diagram. Be that as itmay, spectral noise filtering will always be present (e.g. as part of multiuser separation), soour only impair concern should be the ideal desideratum that such a filter never possessesin practice.

Even though the full discretization was derived for the TS-MMF, it directly extendsto the ORTHO-TS-MMF

and implies that the orthonormalization described in Section 5.3.1 can be discretized, yetit has to be done for twice the number of functions. The alternative is to discretize the TS-MMF instead and then commit the discrete output to the whitening approach devised inSection 5.5.1. The receiver diagram after full filter discretization is depicted in Figure 5.17.

(5.236)

(5.237)

( /2, )st kT t−

.

. . . .

( ) ( ) ( )

( ) ( ) ( )( )

11

01

1 12 2 2

0

( ) ( ) ( )

( ) ( ) 2 ( ) ( )

MF MF MF

MF MF MFMF

M

s s s s s s

k i kM

s s ss s si

f f f kf M Mf f if kf

M f if f f f f

−−

=−

− −

=

= − = − −

⎡ ⎤= − → = + −⎢ ⎥⎣ ⎦

∑ ∑ ∑

∑

u u u

u u u u

F F F

F F F F

(1 )/2sf f roll> +

sf f>

( ) ( ) ( )

( ) ( ) ( )

( )

( )

( )

2 /2 2 /2 2 /2 2 /2

/2

(2 )

(2 ) (2 )(2 )

,

/2 /2, /2 /2

s s s sO MF

s

O MF

O MF

Hj fT j fT j T j fT

f

Hs s s ss

lH

s ss

e e e e

kT lT kT lT

π π πν π

ν−

−

−

−= ⊗

→ =

=

∑u u

u u

u u

Φ

Φ

Φ

F F F

FIGURE 5.17 Optimal detection based, fully discretized receiver model for the MIMO LTV channel.

/ ( )tu u

/ ( )tn n

/ ( )tr r opt= +r b n Φ ΦHr 1

sfsf− /2smT ( ).

2 /2 2 /2(2 ) ,s sHH j fT j T

s e eπ πν−H H F F

( )2 /2sj fTe πΠr

( )(2 ) /2,( ) /2H H

s ss mT l m T− −H H

( ) ( ) ( )/2 /2 /2s s slT lT lTΠ Π= +r u n

†,

Hl kH Φ

( )( ) /2MFslTΠr

2

( ) ( ) ( )2 2 2 2

/ , , ,s s s s

s

H H H HH j T H j fT j T j Tf

f e e e f eπν π πζ πν

ζ⋅ ⋅

− − −= = ⊗H H H H ΦΦ

Φ ΦF F F F F FFF

( )2( , ) , sH HH H j T

st kT t f e πζ−− ∼ Φ Φ Φ ΦF F

,( , ) ( , )H H H H H H

s sk lkt lT t kT= = ∑H H H H Φ ΦΦ Φ


5.6 Linear Detection for the Overspread MIMO Channel

Optimal detection of the sequence of vector samples is accomplished by employing theViterbi algorithm extended to the MIMO set-up, as it yields a maximum likelihood esti-mate of . As usual, the penalty incurred by optimal detection is computational complex-ity, and in the MIMO case it becomes overwhelming. If is the symbol constellation sizeand is the channel memory span, then at each stage of detection (every sig-nalling interval) the number of distance metrics to be computed is , of which

survive. Thus there is an exponential growth in the detection complexity that maybecome unbearable for certain channel conditions and MIMO configurations. To flankthis situation we turn to time-varying linear filtering as an alternative method of detection,mainly because linear complexity scaling is guaranteed. The idea behind the following de-velopments is an extension of standard methods of linear estimation (viz. [105], [106],[107], [108]) to time-varying linear operators.

5.6.1 CONSTRAINED LINEAR DETECTION

The first detector that we derive is based on the assumption that the MIMO response canbe cropped (or constrained) to an extent of vector samples. This is equiv-alent to say that, after applying the ORTHO-TS-MMF, is structured approximately asgiven by (5.83) and, accordingly, the matched filtered response is expressed as

From this expression one readily finds that all the information about is conveyed bythe vector samples , which means that a linear detector may be based solely onthese to determine an estimate of , . Denoting the linear detector operator by , theestimate that it provides at time instant is obtained as follows

from which we understand that is necessarily anticausal, even though this is not an issuebecause a sufficient lag may be introduced.

The error resulting from (5.239) is the difference between and its estimate , i.e., in terms of which may be written as

At this point we must seek for the operator that provides the optimal linear estimate of, but, since is statistically determined, must be obtained in the mean sense. To this

end we shall use the minimum mean squared error (MMSE) criterion, which will guarantee thatthe average energy in the error of linear estimation is minimized. Mathematically, we de-

a

a

M

⎡ ⎤max / sD Tτ=( 1) TD nM +

TDnM

(5.238)

(5.239)

(5.240)

⎡ ⎤max / sD Tτ=

HΦ

, ,0

k D

k k l l k l k k l kl k D l

−= − =

= +

= + = +∑ ∑

r H a n

r H a n H a n

Φ Φ Φ

Φ Φ Φ Φ Φ

ka

...k k D+r rΦ Φ

ka ka D

k

, , , ,

k D n k D k D

k k n n k n n l l k n nn k l n D n k n k

+ + +

= = − = =

⎡ ⎤⎢ ⎥= = +⎢ ⎥⎢ ⎥⎣ ⎦

∑ ∑ ∑ ∑a D r D H a D nΦ Φ Φ

D

ka ka

k k k= −e a a ka

0

, ,

k D

k k k k n n k n k k n kn k n D

+

−= =−

= + = + = +∑ ∑a a e D r e D r eΦ Φ

D

ka rΦ D

5.6 Linear Detection for the Overspread MIMO Channel 165

scribe the MMSE criterion as

where the following notation is used for simplicity

Since it is evident that the mean error energy is minimal when isthe orthogonal projection of onto , that is, , and thus

Expanding the error into (5.243) we are led to

The left expectation can be simplified by expanding with (5.238), yielding

which, assuming that the transmitted vector samples are independent and zero-mean,hence uncorrelated and orthogonal, simplifies to

Equivalently, the right expectation in (5.244) expands as

where complete independence between the noise and the transmitted samples was also as-sumed. Inserting (5.246) and (5.247) into (5.244) we retrieve

where

(5.241)

(5.242)

(5.243)

(5.244)

(5.245)

(5.246)

(5.247)

(5.248)

(5.249)

[ ] , : , :

| , : argmin argmink k

k k k D k k k D

Hopt k k k D kkE tr

+ +

+ = = e eD D

D e e R

:k n k D k k D≤ ≤ + → +

( ):k k k D+∈a rΦR ka

ka ( ):k k D+rΦR ( ):k k k D⊥

+∈e rΦR

| , :H

k opt k k k DnE k n k D +⎡ ⎤ = ≤ ≤ + →⎣ ⎦e r 0 DΦ

|k k opt k= −e a a

| | ,

k D

H H Hk opt k opt k p pn n n

p k

E E E k n k D+

=

⎡ ⎤⎡ ⎤ ⎡ ⎤= = ≤ ≤ +⎣ ⎦ ⎣ ⎦⎣ ⎦ ∑a r a r D r rΦΦ Φ Φ

nrΦ

[ ] , ,2k l

n nH HH H

k k n l n ln ll n D l n D

E E= − = −

⎡ ⎤ = =⎣ ⎦ ∑ ∑ a aa r a a H R HΦ ΦΦ

ka

, 2k l k k k k

HHk l k n knE k n k Dδ − ⎡ ⎤= → = ≤ ≤ +⎣ ⎦a a a a a aR R a r R HΦΦ

[ ]

, ,

min ,

0, ,max ,

2 2l l

p nHH HH

p p l l m n m pn nl p D m n D

n pH

p np l n ll n p D

E E E

N δ

= − = −

−= −

⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣ ⎦

= +

∑ ∑

∑ a a

r r H a a H n n

H R H I

Φ Φ Φ ΦΦ Φ

Φ Φ

min ,

0, | , , ,max ,

| , ,

k k l l

k D n pH H

p nn k opt k p p l n lp k l n p D

k D

opt k p p np k

N k n k D

k n k D

δ+

−= = −

+

=

⎡ ⎤⎢ ⎥= + ≤ ≤ +⎢ ⎥⎢ ⎥⎣ ⎦

= ≤ ≤ +

∑ ∑

∑

a a a aR H D H R H I

D T

Φ Φ Φ

min ,

0, , ,max ,

l l

n pH

p np n p l n ll n p D

N δ −= −

= +∑ a aT H R H IΦ Φ


In matrix notation, (5.248) is equivalent to

or, more compactly, to

Determining can now be done by standard Gaussian elimination or computingthe inverse of , which, being singular, still has a generalized inverse (e.g.Drazin inverse or Moore-Penrose pseudoinverse), i.e.

The optimal estimate so determined is

which is finally applied to a symbol-by-symbol slicer bank that compares each distance toeach constellation symbol and chooses the ones with the lowest distance.

The optimal linear detector can be further simplified when the symbols are mutuallyorthogonal, , where is the average power of the symbol constellation,and the result is

where

and is the average branch SNR with reference to the MIMO input (where is the branch noise power at the output of the ORTHO-TS-MMF).

5.6.2 UNCONSTRAINED LINEAR DETECTION

The constrained linear detector was built upon a sample by sample approach, and each es-timate was based on a truncated sequence of received vector samples . Yetmore expensive, a better detector can be derived by including in the process the entire se-quence of vector samples or, equivalently, , which means that all the informa-tion available in is used to build the detector.

The transmission model is unchanged

(5.250)

(5.251)

(5.252)

(5.253)

(5.254)

(5.255)

, | , , ,

, ,| ,,

k k

T THk k opt k k k k k k D

Hk D k k D k Dopt k k Dk D k

+

+ + +++

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎟⎜ ⎟ ⎟⎜ ⎜⎟⎜ ⎟ ⎟⎜ ⎜⎟⎜ ⎟ ⎟⎟ ⎜ ⎜⎟⎜ ⎟⎟ ⎜ ⎜⎟ ⎟⎜ ⎟ ⎜ ⎜= ⎟ ⎟⎜ ⎟ ⎜ ⎜⎟ ⎟⎟⎜ ⎟⎜ ⎟⎜⎟⎜ ⎟ ⎟⎜ ⎜⎟⎜ ⎟ ⎟⎜ ⎜⎟⎜ ⎟ ⎟⎜ ⎟⎟ ⎟ ⎜⎜ ⎟ ⎝ ⎠⎝ ⎠⎝ ⎠

a a

H D T T

R

T TDH

…Φ

Φ

, : | , : : , :k k

Hk k k D opt k k D k k D k k D+ + +=a aR H D TΦ

| , :opt k k DD

: , :k k D k k D+ +T

†| , : , : : , :k k

Hopt k k k D k k k D k k D k k D+ + + += a aD R H TΦ

( )| | , | , | ,

| , : :

kk D

opt k opt k n n opt k k opt k k Dn k

k D

opt k k k D k k D

+

+=

+

+ +

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜= = ⎟⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜⎝ ⎠

=

∑r

a D r D Dr

D r

Φ

Φ

Φ

Φ

( )/2k k

P= aa aR I aP

†| , : , : : , :

Hopt k k k D k k k D k k D k k D+ + + +=D H XΦ

( )min ,

, , ,max ,

1/n p

Ht p np n p l n l

l n p D

γ δ −= −

= +∑X H H IΦ Φ

2/t Pγ σ= a nΦ2

02Nσ =nΦ

(5.256)

...k k D+r rΦ Φ

0 ... Lr rΦ Φ rΦHΦ

( )H H= = + = +r r Ha n H a nΦ Φ ΦΦ Φ

5.6 Linear Detection for the Overspread MIMO Channel 167

and accounts for the channel and orthogonal matched filtering as well. The linear estima-tor of is

and is still the linear detector operator. The error induced by the estimator is the sumof two independent entities: the symbol distortion and the residual noise, i.e.

which amounts to the divergence of the estimator from the original sequence of vectorsamples. The error is once more minimized when it is orthogonal to the matched filter’soutput, that is

Expanding the error in (5.259) we have

which means that two statistical expectations need to be taken in order to proceed, and . Because the noise is independent from the transmitted symbols,both are trivially obtained. The first one is

and the second one

which, since

simplifies to

Equating (5.261) and (5.264) we get

and after applying the generalized inverse (or Gaussian elimination), yields the linear de-tector operator sought for

A further simplification is achieved for independently transmitted zero-mean symbols

(5.257)

(5.258)

(5.259)

(5.260)

(5.261)

(5.262)

(5.263)

(5.264)

(5.265)

(5.266)

(5.267)

a

( )= = + = + − +a Dr DH a Dn a DH I a DnΦ Φ Φ Φ Φ

D

distortion residual

= + = + → = − = − −a n

a a e Dr e e a a a DH a DnΦ Φ Φ

argmin argminH HoptE E tr⎡ ⎤ ⎡ ⎤= → = =⎢ ⎥ ⎣ ⎦⎣ ⎦ ee

D Der 0 D e e R

Φ

( ) ( )H H H H Hopt opt optE E E E E⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= − = − = → =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦er a a r a D r r 0 ar D r rΦ ΦΦ Φ Φ Φ Φ

HE ⎡ ⎤⎢ ⎥⎣ ⎦ar

ΦHE ⎡ ⎤= ⎢ ⎥⎣ ⎦r rR r r

Φ Φ Φ Φ

2H HH HE E⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎣ ⎦⎣ ⎦ aaar aa H R HΦ ΦΦ

2 2H HH H HE E E⎡ ⎤ ⎡ ⎤⎡ ⎤= + = +⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦ aa n nr r H aa H n n H R H R

Φ ΦΦ Φ Φ Φ Φ ΦΦ Φ

[ ] 01 12 2

HH HE E N⎡ ⎤= = =⎢ ⎥⎣ ⎦n nR n n nn IΦ Φ Φ Φ

Φ Φ

0HHE N⎡ ⎤= = +⎢ ⎥⎣ ⎦r r aaR r r H R H I

Φ Φ Φ Φ ΦΦ

( )0H H

opt N= +aa aaR H D H R H IΦ Φ Φ

( )†0H H

opt N= +aa aaD R H H R H IΦ Φ Φ

( )†

21 1/2

2H H

optt

E a Pγ

⎛ ⎞⎟⎡ ⎤ ⎜= = → = + ⎟⎜⎣ ⎦ ⎟⎜⎝ ⎠aaaR I I D H H H IΦ Φ Φ


where again , and is somewhat resemblant of (5.254) and (5.255).The average energy in the error produced by the optimal linear detector is minimal,

and provides a symbol-by-symbol low complexity alternative to the TV-MLSE detector.The minimum average error energy can be determined as follows. Since the estimator isorthogonal to the error, the latter’s autocorrelation is given by

and consequently

However, from (5.265) we deduce that

which can be related to (5.269) by introducing the generalized inverse of , so that weget the following solution

and, ultimately

5.6.3 ZERO-FORCING LINEAR DETECTION

The zero-forcing detector is derived by the same steps as the two previous detectors, ex-cept that the presence of channel noise is entirely ignored. The constrained zero-forcinglinear detector is found to be given by

and, by the same reasoning, the unconstrained zero-forcing linear detector by

which, whenever is invertible, reduces to . The zero-forcing detector per-formance is inferior to the optimal linear detector, in particular because if containslarge entries the noise will be greatly enhanced while the input symbols will not. Never-theless, if the channel is indeed noiseless, the zero-forcing detector utterly removes thechannel distortion provided that is not badly conditioned and the channel state infor-mation is sufficiently accurate.

(5.268)

(5.269)

(5.270)

(5.271)

(5.272)

2/t Pγ σ= a nΦ

( )

( )

( ) 1 1 12 2 2

opt H H Hopt opt

opt opt

E E E⎡ ⎤ ⎡ ⎤⎡ ⎤= = − = −⎣ ⎦ ⎣ ⎦⎣ ⎦

= − = −

ee aa

aa aa aa

R ea a D r a R D r a

R D H R I D H R

Φ Φ

Φ Φ

( ) ( )

min2 2

optHoptE tr tr⎡ ⎤ = = −⎣ ⎦ ee aae e R I D H RΦ

( ) 0H

opt optN− =aaI D H R H DΦ Φ

HHΦ

( ) ( )† ( ) †0 0

H opt Hopt opt opt optN N− = → = − =aa ee aaI D H R D H R I D H R D HΦ Φ Φ Φ

( ) †0min

2 2HH

opt optE tr tr N⎡ ⎤ = − =⎣ ⎦ aae e I D H R D HΦ Φ

(5.273)

(5.274)

†min ,

| , : , : , ,max ,

n pH H

opt k k k D k k k D p l n ll n p D

+ += −

⎛ ⎞⎟⎜ ⎟⎜= ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠∑D H H HΦ Φ Φ

( )†H Hopt =D H H HΦ Φ Φ

HΦ1

opt−

=D HΦ1−

HΦ

HΦ

5.7 Error Performance of the ORTHO-TS-MMF with Linear Detection 169

5.7 Error Performance of the ORTHO-TS-MMF with Linear Detection

Having derived a detection oriented, TV-MLSE based, optimal discrete model for theoverspread, ORTHO-TS-MMF filtered MIMO system, and the corresponding optimallinear TV-MMSE detector, the aim of this section is to study its performance in terms ofthe average error rate of the transmitted sequence of symbols. To accomplish this, we shalldivert from the pure theoretical standpoint and engage in Monte-Carlo based numericalsimulation of the MIMO system.

5.7.1 CHANNEL MODEL FOR SIMULATION

We begin with the continuous passband channel model from the -th input to the -thoutput, as given by (5.3) and repeated here for convenience

Its baseband version is

where the multipath delays and associated phases are

The carrier frequency is and the angular Doppler shift is given by

where is the relative velocity between transmitter and receiver, is the wavelength, are the different angles of arrival relative to the direction of motion, and is the maxi-mum Doppler shift.

To simplify the model some assumptions need to be made. The sole purpose is tostreamline the simulation by invoking some typical channel characteristics, meaning theydo not induce any loss of generality whatsoever. The imposed assumptions are:1. the carrier frequency is much larger than the Doppler shift, so that the multipath

delays may be considered time-invariant, i.e. ;2. the relative motion is small in scale, which guarantees that the statistical properties of

the channel remain unchanged during the simulation; also, the number of multipathreplicas and respective attenuations do not vary with time;

3. the distance between transmitter and receiver is sufficiently large so that each multi-path component arrives at different receiver antennas by parallel paths; this meansthat ; moreover, the multipath attenuations are equal, i.e. ;

4. each transmitter produces the same number of multipath components, and each oneimpinges on each receiver antenna, i.e. ; additionally, it is assumed that the

(5.275)

(5.276)

(5.277)

(5.278)

j i

( )

( ) ( )

1

( , ) ( ) ( ( ))ijN t

ij ijij n n

n

c t t tτ α δ τ τ=

= −∑

( )( )

2 ( ) ( ) ( )

1

( , ) 2 ( , ) 2 ( ) ( ( ))ij

ijc n

N t

j f ij j t ijij n nij

n

c t c t e t e tπ τ ϕτ τ α δ τ τ−

=

= = −∑

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( / )

( ) 2 ( ) 2 ( ( / ) )

ij ij ijn n n c

ij ij ij ij ij ijn c n c n n c n n

t t

t f t f t t

τ τ ω ω

ϕ π τ π τ ω ω ϕ ω

= −

= − = − − = +

cω

( ) ( ) ( ) ( )2 2 cos 2 cosij ij ij ijn n n D nf

υω πν π φ π φ

λ= = =

υ λ ( )ijnφ

Df

( ) ( )( )ij ijn ntτ τ=

( ) (1 )ij jn nφ φ= ( ) (1 )ij j

n nα α=

ijN N=


multipath components coming from different transmitters are completely independ-ent;

5. the receiver moves in parallel to the axis of the receiver array, as shown in Figure 5.18;

this means that the different phases are related by

where , is the antenna separation, and is the speed of light. There willbe two situation for the antenna separation, one in which all the antennas will be eq-uispacedly fit within , and other in which the array will scale freely as moreantennas are added, i.e. .

The chosen statistical model for the multipath delays is the exponential model, whichis completely defined by the RMS delay spread as

where is uniformly distributed in the interval . The delays to the other antennasare computed from

Equivalently, the attenuations are exponentially decaying , simulating thecharacteristic of the multipath power delay profile. The angles of arrival will be con-sidered uniformly distributed in the interval , simulating the isotropic scatteringmodel. The RMS delay spread will be the rule-of-thumb of a typical mobile propagationenvironment, i.e. , and unless otherwise noted, the number of multipath compo-nents will be . Since it is a natural candidate for 4G LTE licensing, the frequency ofoperation will be chosen for the simulation. At this carrier frequency andmobile speeds of about , the maximum Doppler shift is , whichwill also be the simulation default.

FIGURE 5.18 Illustrating the different multipath phases of a moving array with colinear axis.

(5.279)

(5.280)

(5.281)

i

Wavefront of then -th path from the

j -th transmitter antennaDire

ction of

motio

n

(1 )jnϕ (2 )j

nϕ ( )ijnϕ

( )ijnφ

d

1

2

Receiver Array

( )

( ) ( 1, ) (1 ) (1 ) (1 )

(1 ) (1 )

cos( ) ( 1)cos( )

2 ( / )( 1)cos( )

ij i j j j jn n n n n

j jc n n

d d i

f d c i

ϕ ϕ β φ ϕ β φ

π τ φ

−= + = + −

= − − −

2 /β π λ= d c

max 0.5d λ=

max 0.5R Rd dn nλ= =

τσ

(1 )(1 ) (1 ) (1 ) /1ln(1 ) ( )

jnj j j

n n nu f e ττ στ τ

ττ σ τ

σ−= − − → =

(1 )jnu (0,1)

( ) (1 ) (1 )( / )( 1)cos( )ij j jn n nd c iτ τ φ= − −

(1 )(1 ) /2jnj

n e ττ σα −=(1 )jnφ

(0,2 )π

3 sτσ μ=

8N =

2.6cf GHz=

85 /km hυ = 200Df Hz≈


The pulse shaping filter is the 1/3 roll-off SRRC with impulse response

delayed an integer number of symbols and truncated on both sides

which, cascaded with the wireless channel yields the global input-output response

5.7.2 DISCRETE CHANNEL MODEL FOR SIMULATION

We know that the noiseless continuous output of the MIMO channel is given by

and it was shown in section Section 5.5.2 that a sampling rate of was required to avoidaliasing with a comfortable margin. Consequently, we will be sampling (5.285) at a timerate that is twice the signalling rate, i.e.

where we have to pay attention to the fact that the operator has two different discrete rateson each of its sides. If the noise is confined to a bandwidth , then the global noisydiscrete channel model is

The entries of , , are one-dimensional operators that definethe response between the -th input to the -th output of the MIMO channel. They canbe related to (5.284) as follows

and provide all the information required to build the fully discrete channel operator forsimulation.

5.7.3 BUILDING THE ORTHO-TS-MMF WITH BLOCK-WISE HOUSEHOLDER REFLECTIONS

The ORTHO-TS-MMF may be computed using a discretized version of the orthogonal-ization procedure described in section Section 5.3.1. Unfortunately, experience has shown

(5.282)

(5.283)

(5.284)

( )( ) ( ) ( )( )

( ) ( )( )

1

1 2

cos 1 4 sin 1( )

4 1 4s s s

s s

roll f rollf roll fsrrc

roll f rollf

π τ τ π ττ

π τ

−

−+ + −

=−

delaysrrc

( / ) 0 2 /( )

0

delay s delay ssrrc srrc f srrc fg

otherwise

τ ττ

⎧ − ≤ ≤⎪⎪= ⎨⎪⎪⎩

( )( ) (1 )(1 ) ( )

1

1( , ) ( , ) ( ) ( )

2

ij jn n

Nj tj ij

n nij ijn

h t c t g d e gϕ ω

ατ α τ α α α τ τ+

=

= − = −∑∫

(5.285)

(5.286)

(5.287)

(5.288)

( ) ( , )s ll

t t lT= = ∑u Ha u H a

2 sf

2 2( /2) ( /2, )s s s ll

kT kT lT= =∑u H a u Ha

sB f=

2 2 2

( /2) ( /2, ) ( /2)s s s sll

kT kT lT kT= +

= +

∑r H a n

r Ha n

( /2, )s skT lTH ( /2, )s sijh kT lT

j i

( ) ( )( ) (1 ) /2(1 ) ( )

1

( /2, ) ( /2 , /2)

/2ij j

n n s

s s s s sij ij

Nj kTj ij

n s s n

n

h kT lT h kT lT kT

e g kT lTϕ ωα τ+

=

= −

= − −∑


that it is rather slow to complete and even worst, has poor numerical stability. This rendersthe orthonormalization very time consuming if one is to address stability with variableprecision arithmetic. An alternative version of the orthogonalization based on the modi-fied Gram-Schmidt orthogonalization was implemented, but despite being more stable itwas still slower than desired.

To address both issues we now explain a novel method based on backward block-wiseHouseholder reflections that proved to be incredibly fast and stable. Our goal is to per-form the following decomposition

where is block lower triangular with block size and is block orthogonal,each block consisting of columns. We remember the reader that the indexes in (5.289)are indexing the blocks and not the entries themselves. The idea is to sequentially eliminatethe entries above each block in the lower right diagonal of , where the lower right diag-onal is interpreted as the diagonal that starts with the utmost lower right block of . Toeliminate the entries above the utmost lower right block we begin by finding the QR fac-torization of the utmost right columns of , i.e.

where is lower triangular. Next we retrieve the utmost lower right block of and compute

which lets as define

where is the same size of and is all zeros except on the lower right diagonal, where itis all ones. The block can be used to build the first Householder reflector

which multiplied by , , will have a very interesting propertythat we now demonstrate. The products and can be expanded as

Moreover, from (5.291) we know that

which inserted into (5.294) reveals that

(5.289)

(5.290)

(5.291)

(5.292)

(5.293)

(5.294)

(5.295)

(5.296)

2 2 ,( /2, ) ( /2, )s s s s m lm

kT lT kT mT= = ∑H H H HΦ ΦΦ Φ

HΦ T Tn n× 2Φ

Tn

2H

2H

Tn 2H

(:, )sLT =H QR

R T Tn n× Q

( ) 1/2( , :) Hlast

−= → =B Q P B B B

( ) 1/2H −= − = −v Q eP Q Q Q eP

e Q

v

12 ( )H Hr

−= −H I v v v v

Q 12 ( )H Hr

−= −H Q Q v v v v QHv v Hv Q

H H H H

H H

= + − −

= −

v v I P P B P P B

v Q I P B

( ) ( )

( )

1/2 1/2

1/2

H H H H

H H H

− −= =

= =

P P B B B B B B I

P B B P B B

( )1/22 2 2H H H= − =v v I B B v Q


This condition means that the product of the Householder reflector by is in fact

and we immediately conclude that the entries above the utmost lower right block of have been eliminated. Since the purpose of all of this is to perform the elimination in ,we multiply (5.297) by , which yields the same elimination

We thus apply the Householder reflector to the entire ,

and then normalize the bottom block row with

The first step of the decomposition ends here, yet further steps are required to eliminatethe rest of the entries above the block diagonal. The procedure described so far is still val-id, except that it is applied to excluding the block columns that have already been elim-inated and corresponding block rows. The process ends when the first block column of

undergoes elimination, and at this point the lower entries of are in fact in its block lower triangular form.There is still one subtlety with the method just described, and it concerns the product in (5.291). To avoid the inherent loss of numerical information in computing this

particular product, we use the following trick: we perform the singular value decomposi-tion of and then compute as

which circumvents the problem and replaces the formula in (5.291).To this point, we have described how to obtain the operator for simulation, but

the ORTHO-TS-MMF is yet to be determined. One obvious option is to apply theprocedure just described to the identity matrix instead of , which after normalizationwith (5.300) yields . What this approach has in obviousness, it also has in cost. It isslow. Another approach was tested in the forward direction, but was also slow, so will notbe described. Experimentation revealed that the swiftest approach was to perform Gaus-sian elimination on the system , since and are already known. will not be strictly required for the simulation since completely defines the noiselesschannel after matched filtering, and also because the average noise power at the input of

remains unchanged by it

5.7.4 MONTE CARLO SIMULATION, NUMERICAL RESULTS AND DISCUSSION

Monte Carlo simulation was performed using the Matlab® programming language, and

(5.297)

(5.298)

(5.299)

(5.300)

(5.301)

(5.302)

Q

12 ( )H Hr

−= − = − =H Q Q v v v v Q Q v eP

Q

2H

R

(:, )r r sLT= =H QR H H ePR

2H

2 2 2 21 2 ( )

set H Hr

−= = −H H H H v v v v H

( ) 1/2 ( , :) ( , :)

setH H H last last−

= → =E R R R P H EH

2H

2H T TLn Ln× 2H

HΦ

HB B

B P

( ) 1/2H H H−= → = =B USV P B B B UV

HΦ

2

HΦ

2H

2

HΦ

2 2

H H H=H HΦ Φ

2H HΦ 2Φ

HΦ

2Φ

( ) ( )( ) ( )2 2 2 2 2 2

H H

H H

size P E E tr

tr E size P P P

⎡ ⎤⎡ ⎤= = ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎡ ⎤= = → =⎣ ⎦

n

n n n

n n n n n

n n n

Φ

Φ

Φ Φ ΦΦ Φ

ΦΦ Φ


the implemented code is made available in Appendix C. Simulation starts by randomlygenerating and modulating a message of 64 complex symbols, either using BPSK, QPSK,8-QAM or 16-QAM, yet the default modulation for simulation will be 16-QAM. Afterthat, the channel operator is built using (5.288) and the ensemble of random parame-ters described previously, namely delays, phases, Doppler shifts and attenuations. The ran-dom message is applied to and noise is added, simulating the MIMO transmission.Then the ORTHO-TS-MMF is computed and applied to the noisy output of thechannel operator. The output is finally filtered by the unconstrained linear detector de-rived in Section 5.6.2, and applied to a slicer that makes the decisions on which were thetransmitted constellation symbols. To yield reliable error rates this process is repeated sothat at least 100000 symbol transmissions are simulated. Several plots were obtained fortypical, randomly chosen channel realizations, and each plot expresses the error rate (orestimate of the average probability of symbol error) as a function of the SNR (stepped at2 dB) at the output of the MIMO channel, before matched filtering. The plots will now bedescribed.

Figure 5.19 depicts the error rate for the different types of modulation and for two

signalling intervals for a MIMO arrangement of and with equispaced fitwithin . We conclude that enlarging the constellation has the expected conse-quence of increasing the average error probability, as for a given SNR the symbols will becloser to one another as the constellation enlarges. In addition, slightly increasing the sig-nalling interval from to also increases the error rate. This happensbecause for , such an increase reduces the ability of the detector to resolve themultipath components, which means that taking advantage of delay diversity will be lesspossible.

FIGURE 5.19 Average error rates for several symbol constellation sizes and two signalling intervals, for

and with equispaced fit within .

2H

2H

2

HΦ

0 5 10 15 2010

-4

10-3

10-2

10-1

100

Average SNR at the ORTHO-TS-MMF input (dB)

Average

probability of

symbol error

BPSKQPSK8-QAM16-QAM

black - 1

gr

0

- 20ay

s

s

T s

T s

μ

μ

=

=

2Tn = 4Rn = max 0.5d λ=

2Tn = 4Rn =

max 0.5d λ=

10sT sμ= 20sT sμ=

3 sτσ μ=


Plots for the average error rate as the number of transmitter antennas increases areshown in Figure 5.20. We see that increasing the number of transmitter antennas has the

effect of increasing the error probability, which is legitimate because the symbols trans-mitted from different transmitter antennas will interfere with one another, causing confu-sion at the receiver. The penalty for higher data rates (using the same bandwidth and thusimproving the spectral efficiency) is an increase in the probability of symbol error. Never-theless, for the error rates are indeed acceptable, becoming even lower as thenumber of receiver antennas increases.

A different situation appears in Figure 5.21. One transmitter antenna is assumed, andthe number of receiver antennas increases. We readily gather that increasing the size of thereceiver array always has the consequence of decreasing the error probability, which meansthat the receiver is able to use reception diversity to improve the error performance. Thisimprovement is even larger when free scaling of the receiver array is permitted, which isnatural because the receiver antennas will be positioned further apart, therefore decreasingthe cross-correlation between the received signals and increasing diversity. Equivalentplots but for the case and are shown in Figure 5.22, and we infer thesame behaviour. The best performance takes place for larger , but when the receiver an-tennas’ placement is restricted to a maximum length it becomes harder to attain lower er-ror rates. Finally, plots for are displayed in Figure 5.23, but this time for 10 timesthe signalling rate, , which allows for much lower error rates gained by multi-path resolution. Again, we observe the expected behaviour of lower error rates for largerreceiver arrays, confirming the previous results and showing that it is the same regardlessof the number of transmitter antennas.

To assess the variation of the error performance with the signalling rate, plots for sev-

FIGURE 5.20 Average error rates for two receiver sizes with equispaced fit within and increasing

number of transmitter antennas, for .

0 5 10 15 2010

-4

10-3

10-2

10-1

100


Average

probability of

symbol error

nR = 8, nT = 1nR = 8, nT = 2nR = 8, nT = 4nR = 8, nT = 8nR = 4, nT = 1nR = 4, nT = 2nR = 4, nT = 4

max 0.5d λ=100sT sμ=

2Tn =

2Tn = 1sT sμ=

Rn

4Tn =

0.1sT sμ=


eral were obtained as shown in Figure 5.24. We see that the lowest signalling interval isresponsible for the lowest error rates, which was already explained to be a consequence ofmultipath diversity resolution. As the signalling interval increases below , the er-ror rate also steadily increases, which is explained by an increasing difficulty of resolvingthe multipath components since these are now more and more concentrated and approacha situation of flat fading. When the RMS delay spread becomes much lower than the sig-

FIGURE 5.21 Average error rates for one transmitter antenna and several receiver sizes (equispaced fit within

and free scaling considered), for .

FIGURE 5.22 Average error rates for two transmitter antennas and several receiver sizes (equispaced fit within


0 5 10 15 2010

-4

10-3

10-2

10-1

100


Average

probability of

symbol error

nR = 1nR = 2nR = 4nR = 8nR = 1 (free)nR = 2 (free)nR = 4 (free)nR = 8 (free)


0 5 10 15 2010

-5

10-4

10-3

10-2

10-1

100


Average

probability of

symbol error

nR = 2nR = 4nR = 8nR = 2 (free)nR = 4 (free)nR = 8 (free)


sT

3 sτσ μ=


nalling rate the error performance severely degrades, as can be seen in the error rate curveof . In this situation there is no multipath diversity available for resolution andthe receiver cannot compensate for the ISI introduced. However, as the signalling ratesget lower and lower, the ISI will become negligible and, despite the fact that the channelmay enter fast-fading, the error rates will be significantly reduced. The curves for

and demonstrate this fact.

FIGURE 5.23 Average error rates for four transmitter antennas and several receiver sizes (equispaced fit within


FIGURE 5.24 Average error rates for several signalling rates, and a MIMO configuration of and

with equispaced receiver antenna fit within .

0 2 4 6 8 10 1210

-5

10-4

10-3

10-2

10-1

100


Average

probability of

symbol error

nR = 4nR = 8nR = 4 (free)nR = 8 (free)

max 0.5d λ= 0.1sT sμ=

0 5 10 15 2010

-4

10-3

10-2

10-1

100


Average

probability of

symbol error

0.1sT sμ=0.5sT sμ=1sT sμ=0.1sT ms=1sT ms=1sT s=

2Tn =4Rn = max 0.5d λ=

0.1sT ms=

1sT ms= 1sT s=


To study the influence of the Doppler shift on the error performance we have includ-ed the plots in Figure 5.25 and Figure 5.26. We see that when the channel is fast-fading

(the channel varies in a signalling interval basis), which happens for , the receivercan use the extra variability introduced by the Doppler shift to improve the detection, ir-respective of the number of receiver antennas. If this is not the case, i.e. if the signalling

FIGURE 5.25 Average error rates for several maximum Doppler shifts, and a MIMO configuration of

and several with equispaced receiver antenna fit within , for .

FIGURE 5.26 Average error rates for several maximum Doppler shifts, and a MIMO configuration of

and several with equispaced receiver antenna fit within , for .

0 5 10 15 2010

-4

10-3

10-2

10-1

100


Average

probability of

symbol error

fD = 0 HzfD = 200 HzfD = 1000 Hz

2Rn =

4Rn =8Rn =

2Tn =

Rn max 0.5d λ= 1sT s=

0 5 10 15 2010

-4

10-3

10-2

10-1

100


Average

probability of

symbol error

fD = 0 HzfD = 200 HzfD = 1000 Hz

8Rn =

4Rn = 2Rn =

2Tn =

Rn max 0.5d λ= 1sT sμ=

1sT s=

5.8 Conclusion 179

rate is much larger than the Doppler shift, which happens for , then the MIMOchannel is practically invariant during a large number of signalling intervals, making it dif-ficult for the receiver to accomplish a significant reduction in the error rate. This is a man-ifestation of Doppler diversity absence, or lack of time-selectivity. One realizes that one ofthe main advantages of the ORTHO-TS-MMF is its channel variability awareness, whichmakes it feasible to exploit the inherent Doppler diversity to provide better estimates ofthe transmitted symbols.

To complete the simulation results, plots for error rate variation with the maximumantenna separation are provided in Figure 5.27. It becomes clear that, as the maximum

separation decreases the error performance deteriorates. As the antennas become closerand closer to one another they become increasingly correlated to the extent of not permit-ting the compensation of both intra-sample ISI and time selectivity.

5.8 Conclusion

This chapter successfully solved a number of difficulties that persisted in the treatment ofthe overspread MIMO LTV channel. Using a baseband set-up it started by defining a con-tinuous input/output model of digital transmission that did not ignore the time-varyingnature of the wireless channel. Then it aimed for an optimal model discretization using achannel-extracted orthogonal basis. This was accomplished by orchestrating a theory ofcontinuous, discrete, and hybrid matrix operators that impressively simplified all MIMOcalculations. It was possible to build a semi-orthogonal version of the channel that, in turn,was sufficient to devise a simple discrete version of the input/output model that was in allrespects similar to a flat-fading MIMO channel. A semi-orthogonal matched filter wasalso derived which, above all: 1) maintained the white-nature of the input noise and 2)

FIGURE 5.27 Average error rates for several maximum receiver antenna separations, and a MIMO configuration

of and with equispaced receiver antenna fit within , for .

1sT sμ=

0 10 20 30 40 50 6010

-4

10-3

10-2

10-1

100


Average

probability of

symbol error

dmax = 10λdmax = λdmax = 0.5λdmax = 0.1λdmax = 0.01λdmax = 0.001λ

2Tn = 4Rn = maxd 1sT sμ=


maintained the optimality from a maximum likelihood detection perspective.In sequence, a full frequency description of the time-varying MIMO channel was giv-

en in terms of frequency operators, a description that was extended to the discrete input/output model just derived. Using this approach, novel formulas for the spectral factoriza-tion of time-varying MIMO filters were obtained. A section on noise whitening and fulldiscretization of the non-orthogonal matched filter was also presented, which made newincursions into the consequences of conventional sampling within the MIMO system.

Finally, to test the performance of the new model a new section on linear detectionwas introduced. Three linear detector operators were designed, one of these with the in-tent of optimal detection in the MMSE sense. The latter was tested with success usingMonte Carlo simulation, allowing for considerable insight into the error performance ofthe Doppler-aware MIMO matched filter.

CHAPTER 6

CONCLUSIONS AND FUTURE RESEARCH

6.1 Conclusions

This work has produced a number of significant results. The information-theoretic studyof MIMO systems under receiver-sided correlated flat-fading has revealed exact, closed-form formulas for the ergodic capacity of the MIMO channel. The more involved case ofmore receiver antennas than transmitter antennas was also successfully addressed. The re-sults were extended to situations of independent envelope fading, yielding new closed-form formulas. Abundant plots of the derived formulas show a substantial capacity reduc-tion when correlation and/or non-isotropic scattering is present, yet confirm the enor-mous advantages of using MIMO systems. For the case of a single transmitter antenna itwas confirmed that the maximal-ratio combiner attains capacity. Transmitter-sided corre-lation was studied by assuming no CSI at the transmitter, showing that a pronounced ca-pacity stagnation occurs for more than three transmitter antennas. In addition, it wasconcluded that the greatest advantage comes from equipping a mobile unit with more an-tennas than the base station, irrespective of the correlation. Fixed-length antenna arrayswithhold the capacity potential of independent fading, and it was shown that equispacedlinear arrays do not attain the maximum capacity. A method to improve capacity was de-vised and validated.

The study of overspreading in linear time-varying MIMO transmission was also a suc-cess. It is a general study that contemplates every single type of MIMO channel that onecan imagine. It studies delay overspreading and Doppler overspreading, the phenomenonsresponsible for frequency selectivity and time-selectivity, respectively, but also encompass-es flat-fading. An extremely valuable introduction of continuous, discrete and hybrid op-erators was responsible for attaining the results that were pursued. The time-varyingMIMO channel was factorized (both in time and frequency) into semi-orthogonal andcausal terms, creating the necessary momentum for the full discretization of the input/

181

182 CHAPTER 6 CONCLUSIONS AND FUTURE RESEARCH

output model of the MIMO system and the derivation of the optimal semi-orthogonaltime-shear matrix matched filter, the designated ORTHO-TS-MMF. With the purpose ofstudying the error performance of the model, time-varying linear detectors were deducedin the MMSE sense, and several Monte Carlo simulations were performed. Essentially,they showed that the error performance is boosted when the number of receiver antennasincreases, and that the penalty of achieving higher data rates (within the same bandwidth)with more transmitter antennas is an increase in error probability. In some cases the pen-alty can be overcome by adding more antennas to receiver. Additionally, they confirmedthat the error performance deteriorates as correlation increases, i.e. when the antenna sep-arations decrease. More importantly, they unveiled that the ORTHO-TS-MMF combinedwith the unconstrained linear detector can take advantage of both delay-diversity andDoppler diversity, significantly reducing the error probability. This means that frequencyselectivity and time-selective can be put to one’s favour because they are the effects of de-lay-Doppler diversity. Accordingly, the flat-fading channel has proven to dictate the worsterror performance overall. In effect, a form of time-frequency equalization in the contextof MIMO systems has been accomplished, all this without the need of an adaptive equal-ization algorithm: provided that channel tracking is accurate, the ORTHO-TS-MMF andthe detector are themselves already adapted to the channel variations.

6.2 Future Research

Despite the self-contained, all-inclusive nature of this thesis, it is clear that open issues re-main that point to new directions of research. With time they may lead to conclusions thatset new prospects on the performance of MIMO systems. The author has identified thefollowing lines of investigation:

1. Find closed-form formulas for the information-theoretic capacity of MIMO channelsunder transmitter-sided correlated fading, when channel state information is available atthe transmitter; or, more generically, under two-sided correlated fading;

2. Extend the derivations of the analytic expressions to the ergodic capacity of MIMOchannels under Nakagami-m distributed correlated fading and, eventually, doubly selectivewireless channels;

3. Test the performance of the ORTHO-TS-MMF when the Viterbi algorithm is used formaximum likelihood sequence estimation; study reduced-complexity ML detectors;

4. Compare the error performance of the ORTHO-TS-MMF to the error performanceof MIMO adaptive equalization techniques available in the literature;

5. Derive the minimum mean-square error decision-feedback (MMSE-DF) detector forthe discretized input/output model and test its performance; one should expect a superiorperformance as compared to that of the unconstrained linear detector of this thesis;

6. Study the performance of the discretized input/output model when applied to MIMO-OFDM, MIMO MC-CDMA and MIMO WCDMA systems;

7. Derive analytic formulas for the probability of symbol error of the discretized input/output MIMO model for different types of modulations;

6.2 Future Research 183

8. Derive the optimal time-varying pulse shaping MIMO filter when channel state infor-mation is available at the transmitter;

9. Study the impact of space-time coding (e.g. space-time turbo trellis codes) on the re-sults of the thesis and develop strategies of improvement.

184 CHAPTER 6 CONCLUSIONS AND FUTURE RESEARCH

APPENDIX APPENDIX A 185

APPENDIX

APPENDIX A

ALGORITHM IN THE MATHEMATICA® PROGRAMMING LANGUAGE TO

COMPUTE THE COEFFICIENTS IN (4.291):

coef[k_,n_]:= Block[l,i,coefsums=Range[k],

For[l=1,l<n-k,l++,For[i=k,i>0,i--,coefsums[[i]]=

i*Sum[coefsums[[j]],j,1,i]]];If[k==n,Return[1],Return[Sum[coefsums[[j]],j,1,k]

]]

]

,k na

186 APPENDIX APPENDIX B

APPENDIX B

ALGORITHM IN THE MATHEMATICA® PROGRAMMING LANGUAGE TO

COMPUTE THE ROW (COLUMN) INDICES (ANTENNA POSITIONS WITHIN

) OF THE PRINCIPAL SUBMATRIX OF WITH MINIMUM FROBE-NIUS NORM:

LeastCorrPos[covarmatrix_,p_]:=Block[i,currsum,minsum,minset,subsets=Subsets[Range[Length[covarmatrix]],p],

For[i=1,i<=Length[subsets],i++,currsum=Sum[Abs[covarmatrix[[subsets[[i]],

subsets[[i]]]]][[k,l]],k,1,p,l,1,p]; If[i==1,minsum=currsum;minset=subsets[[1]], If[currsum<minsum,minsum=currsum; minset=subsets[[i]] ] ] ]; Return[minset]]

maxd p p× Σ

APPENDIX APPENDIX C 187

APPENDIX C

CODE OF THE MONTE CARLO SIMULATION OF THE ORTHO-TS-MMFWITH UNCONSTRAINED LINEAR DETECTION, IN THE MATLAB® PRO-GRAMMING LANGUAGE:

mimo_channel_operator_generate.mfunction H=mimo_channel_operator_generate(L,nT,nR,Ts,Nmulti,dshift,rho,u,uaoa)%--------declare needed variablesfs=1/Ts; %signalling ratermsdspread=3e-6; %rms delay spreadfc=2.6e9; %center frequencyc=3e8; %speed of light;lambda=c/fc;d=rho*lambda; %antenna separationsrrcdelay=20; %srrc filter delayroll=1/3; %filter roll-off

Dopplerw=zeros(nR,nT,Nmulti);tau=zeros(nR,nT,Nmulti);phi=zeros(nR,nT,Nmulti);alpha=zeros(nR,nT,Nmulti);

for j=1:nT for n=1:Nmulti Dopplerw(1,j,n)=2*pi*dshift*cos(2*pi*uaoa(1,j,n)); tau(1,j,n)=-rmsdspread*log(1-u(1,j,n)); phi(1,j,n)=-mod(2*pi*fc*tau(1,j,n),2*pi); alpha(1,j,n)=exp(-tau(1,j,n)/(2*rmsdspread)); endend

for i=2:nR for j=1:nT for n=1:Nmulti Dopplerw(i,j,n)=Dopplerw(1,j,n); tau(i,j,n)=tau(1,j,n)+d/c*(i-1)*cos(2*pi*uaoa(1,j,n)); %phi(i,j,n)=-2*pi*fc*tau(i,j,n); phi(i,j,n)=phi(1,j,n)-2*pi*(rho)*(i-1)*cos(2*pi*uaoa(1,j,n)); alpha(i,j,n)=alpha(1,j,n); end endend

maxdelay=2*srrcdelay+ceil(fs*max(max(max(tau))));H=zeros(2*nR*(L+maxdelay),nT*L);for l=0:L-1 for k=2*l:2*(L+maxdelay)-1 H(k*nR+1:k*nR+nR,l*nT+1:l*nT+nT)=sum(alpha(1:nR,1:nT,1:Nmulti).*... exp(complex(0,phi(1:nR,1:nT,1:Nmulti)+Dopplerw(1:nR,1:nT,1:Nmulti)*k*Ts/2)).*... (4*roll*(cos((1+roll)*pi*1/2*(k-2*l-2*srrcdelay-2*fs*tau(1:nR,1:nT,1:Nmulti)))+... (sin((1-roll)*pi*1/2*(k-2*l-2*srrcdelay-2*fs*tau(1:nR,1:nT,1:Nmulti))))./... (4*roll*1/2*(k-2*l-2*srrcdelay-2*fs*tau(1:nR,1:nT,1:Nmulti))))./... (pi*sqrt(Ts)*(1-(4*roll*1/2*(k-2*l-2*srrcdelay-2*fs*tau(1:nR,1:nT,1:Nmulti))).^2))),3); endend

end

188 APPENDIX APPENDIX C

mimo_channel_operator_orthogonalize.mfunction [O,Ho]=mimo_channel_operator_orthogonalize(H,L,nT)Horiginal=H;H_height=size(H,1); H_width=size(H,2);

%------------------- O - solution 1 ---------------%EI=eye(H_width);%uMat=zeros(H_height,H_width);%O=[zeros(H_width,H_height-H_width) eye(H_width)];%---------------------------------------------%------------------- O - solution 2 ---------------%EI=eye(H_width);%O=eye(H_height);%---------------------------------------------

for j=L:-1:1 Hcol=H(1:H_height-(L-j)*nT,(j-1)*nT+1:j*nT); [Q,R]=qr(Hcol,0); %Q=Q(:,1:nT); R=R(1:nT,:);% Q=flipud(Q); B=Q(H_height-(L-j+1)*nT+1:H_height-(L-j)*nT,:); [U,S,V]=svd(B,0); P=U*V';

u=Q; u(H_height-(L-j+1)*nT+1:H_height-(L-j)*nT,:)=... u(H_height-(L-j+1)*nT+1:H_height-(L-j)*nT,:)-P;%*sqrtm(Q'*Q); E=sqrtm(R'*R)\R'*P';

H(1:H_height-(L-j)*nT,1:j*nT)=(H(1:H_height-(L-j)*nT,1:j*nT)... -2*(u/(u'*u)*u'*H(1:H_height-(L-j)*nT,1:j*nT))); H(H_height-(L-j+1)*nT+1:H_height-(L-j)*nT,1:j*nT)=... E*H(H_height-(L-j+1)*nT+1:H_height-(L-j)*nT,1:j*nT); %------------------- O - solution 1 --------------- %uo=[u;zeros((2*L-j)*nT,nT)]; uMat(:,H_width-(2*L-j+1)*nT+1:H_width-(2*L-j)*nT)=uo; %EI(H_width-(2*L-j+1)*nT+1:H_width-(2*L-j)*nT,H_width-(2*L-j+1)*nT+1:H_width-(2*L-j)*nT)=E; %--------------------------------------------- %------------------- O - solution 2 --------------- %EI(H_width-(2*L-j+1)*nT+1:H_width-(2*L-j)*nT,H_width-(2*L-j+1)*nT+1:H_width-(2*L-j)*nT)=E; %O(1:H_height-(2*L-j)*nT,1:H_height-(2*L-j)*nT)=(eye(H_height-(2*L-j)*nT)-... %2*(u/(u'*u)*u'))*O(1:H_height-(2*L-j)*nT,1:H_height-(2*L-j)*nT); %---------------------------------------------

endHo=H(H_height-L*nT+1:H_height,:);%------------------- O - solution 1 ---------------%for j=1:2*L% uo=uMat(:,(j-1)*nT+1:j*nT);% O=O-2*O*(uo/(uo'*uo)*uo'); %end%O=O'; O=O/EI;%---------------------------------------------%------------------- O - solution 2 ---------------%O=O'; O=O(:,H_height-H_width+1:end); O=O/EI;%---------------------------------------------%------------------- O - solution 3 ---------------O=Horiginal/Ho;%---------------------------------------------end

mimo_linear_detection_simulation.m%------------------------- detection simulation ----------

msg_length=64; Nmulti=8; snrdB_min=0; snrdB_max=60; snrdB_step=2;Pevec=zeros(1,floor((snrdB_max-snrdB_min)/snrdB_step)+1);

count=0; test_count=0;

APPENDIX APPENDIX C 189

for M=16 %constellation size

h = modem.qammod(M); modsym=modulate(h,0:M-1); %modulator object

u=rand(1,32,Nmulti); uaoa=rand(1,32,Nmulti); %random uniform samples

%--------------test routine--------------

niter=ceil(100000/msg_length); %number of iterations

msg_test=randint(msg_length,niter,M); %build messagea_test=modulate(h,msg_test); Pa=mean(mean(abs(a_test).^2)); PadB=10*log10(Pa);

dscount=0;for dshift=200 %maximum Doppler shift dscount=dscount+1;

nTcount=0; for nT=2 nR=4; L=msg_length/nT; if nR>1, rho=0.5/(nR-1); else rho=0; end %antenna separation %rho=0.5; %free length separation nTcount=nTcount+1;

Tscount=0; for Ts=100e-6 Tscount=Tscount+1;test_count=test_count+1;

H=mimo_channel_operator_generate(L,nT,nR,Ts,Nmulti,dshift,rho,u,uaoa); %generate channel [O,Ho]=mimo_channel_operator_orthogonalize(H,L,nT); %orthogonalize channel

rup_test=H*a_test; %simulate transmission nup_test_unit=(randn(size(rup_test,1),niter)+...

i*randn(size(rup_test,1),niter)); %generate noise Prup=mean(mean(abs(rup_test).^2)); PrupdB=10*log10(Prup);

%--------- compute error rate for several snr ------------- snr_count=0; for snr_rupdB=snrdB_min:snrdB_step:snrdB_max count=count+1; snr_count=snr_count+1; Pe=0; nerrors=0;

snr_rup=10^(snr_rupdB/10); PnupdB=PrupdB-snr_rupdB; Pnup=Prup/snr_rup; sqrthalfPnup=sqrt(Pnup/2); %noise standard deviation in each quadrature component

nup_test=sqrthalfPnup*nup_test_unit; %scale noise power no_test=O'*nup_test; Pno=mean(mean(abs(no_test).^2)); %compute noise power rupn_test=rup_test+nup_test; %add noise ron_test=O'*rupn_test; %simulate ortho-ts-mmf

%---------- build D ------------ Id=(Pno/Pa)*eye(length(Ho),length(Ho)); D=Ho'/(Ho*Ho'+Id); %detector operator

aest_test=D*ron_test; %simulate detector %-------------- slicer --------- for iter=1:niter for n=1:msg_length mindist=abs(aest_test(n,iter)-modsym(1)); adetect=modsym(1); for m=2:M dist=abs(aest_test(n,iter)-modsym(m)); if dist<mindist mindist=dist; adetect=modsym(m); end end if adetect~=a_test(n,iter) nerrors=nerrors+1; end

190 APPENDIX APPENDIX C

end end %waitbar(count/(4*(floor((snrdB_max-snrdB_min)/snrdB_step)+1))); Pevec(test_count,snr_count)=nerrors/(msg_length*niter); %error rates

end end endendendx=snrdB_min:snrdB_step:snrdB_max;semilogy(x,Pevec); axis([0 20 1e-4 1]) %plot

REFERENCES 191

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