fast algorithms for the digital computation of linear canonical transforms a dissertation submitted...

FAST ALGORITHMS

FOR THE DIGITAL COMPUTATION

OF

LINEAR CANONICAL TRANSFORMS

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF ELECTRICAL

ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Aykut Koc

March 2011

This dissertation is online at: http://purl.stanford.edu/fq782pt6225

© 2011 by Aykut Koc. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

ii

http://purl.stanford.edu/fq782pt6225

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Lambertus Hesselink, Primary Adviser


Shanhui Fan


R Pease

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

ALTHOUGH IT IS straightforward to determine the relationship between thein-focus

image and the object of a simple optical system such as a lens,it is far more

challenging to compute the input/output relationships of general first-order and astigmatic

optical systems. Such optical systems are known as quadratic-phase systems (QPS) and

they include the Fresnel propagation in free space, propagation in graded-index media,

passage through thin lenses, and arbitrary concatenationsof any number of these, including

anamorphic, astigmatic, nonorthogonal elements. Such computation is accomplished by

representing the physical system with a general mathematical framework of integrations

against kernels and then distilling the entire system into one input-output relationship that

can be represented by a linear integral transform. The underlying mathematical integral

transforms can be applied to a wider field of signal processing where they are known as the

linear canonical transform (LCT) of a signal. Conventionalnumerical integration methods

have a computational complexity ofO(N2) where N is the space-bandwidth product of the

sampling scheme, e.g. the number of pixels in the field for an optical system.

The algorithms described here yield a complexity of onlyO(N log N). The key is

the use of different decompositions (or factorizations) ofa given input/output relationship

into simpler ones. Instead of following the general physical subparts in cascaded systems

and computing input-output relations separately, these algorithms use the simplest possible

decompositions to represent the entire system in terms of least possible number of steps.

The algorithms are Fast Fourier Transform (FFT) based methods and the only essential

v

deviation from exactness arises from approximating a continuous Fourier transform (FT)

with the discrete Fourier transform (DFT). Thus the algorithms work with a performance

similar to that of the fast Fourier transform algorithm in computing the Fourier transform,

both in terms of speed and accuracy. Unlike conventional techniques these algorithms also

track and control the space-bandwidth products, in order toachieve information that is

theoretically sufficient but not wastefully redundant.

vi

to my late grandmother,

Fatma

vii

Acknowledgments

IWAITED FOR writing the acknowledgments section until I got all three signatures and

hence made my Ph.D thesisde facto approved. I did not realize that this very small

part of the thesis would end up being by far the hardest one to write, even after the relief

of knowing that my Ph.D story has come to a happy ending. Five and a half years have

passed at Stanford with their joys and sorrows. During theseyears, I worked on my Ph.D, a

study which is in general deceptively seen as an individual accomplishment. However, after

all these years, I believe that I did not do it alone but with the contributions from several

people. Therefore, now I will do my best to deliver the rightsto those who collectively

made this thesis possible. I am proud of having them in my lifemore than I am proud of

earning a Ph.D at Stanford.

First of all, I want to send my thanks to my advisor Professor Lambertus ‘Bert’ Hes-

selink. I am indebted most of my Stanford experience to him and I cannot express the full

extent of my gratitude to him. I sincerely think that he is a truly ideal advisor in all aspects.

Professionally, academically and personally, he has thought me lots. His guidance, expe-

rience, support and vast knowledge were always there for me.Lastly, a final and special

thank you goes to Bert for the hospitality he showed during the great weekend retreats at

his estate in Tahoe that we enjoyed as a research group. Thosewere wonderful times that I

will always remember.

My oral defense and thesis reading committee members, Professor Shanhui Fan and

Professor R. Fabian Pease helped me through the hardships ofmy orals exam and thesis

ix

writing stages. I thank them for their feedback, insights and inputs that contributed to my

thesis. I would also like to thank Professor Martin M. Fejer for agreeing to serve as the

chair of my oral defense committee.

I would like to thank Professor Haldun M. Ozaktas of Bilkent University for all his

contributions to my academic development. I started research in an undergraduate senior

year project under his supervision. With his passion for research and his vast knowledge,

he was a role model for me and greatly affected my decision to pursue a Ph.D.

I also need to thank Professor Peter Peumans for his support,especially during the

early hard times. My Master’s Program Advisor, Professor Stephen Boyd should also be

acknowledged for his help during my early years of studies.

Dr. Yuzuru Takashima and Ludwig Galambos, senior researchers in our department,

have also contributed to my development at Stanford. I wouldlike to especially acknowl-

edge Dr. Yuzuru Takashima for his support, guidance and helpin the projects that I was

involved in.

I would like to thank for all the help, companionship and discussions provided by my

past and present research group members: Paul, Yao-Te, Yin,Xiaobo, Brian, Yuxin, Eu-

gene and Toan. Particularly, I cannot forget the friendships of Paul and Yao-Te. Thank you

guys, for listening my absurd business ideas during luncheswe enjoyed.

Our research group’s administrative assistants Ms. LilyanSequeira and Ms. Ann

Guerra deserve to be acknowledged for the administrative issues that have superbly been

taken care of for our group. I also send my thanks and gratitude to Ms. Natasha Newson,

the Student Services Officer of Electrical Engineering Department, for her helpful attitude

that have made lots of official issues easy.

I couldn’t complete this long process without the support and enjoyment provided by

great companions and friends. I had the chance to befriend great people during my years

at Stanford. I have countless good moments and memoirs with them. I am grateful for

their support and friendship. Although I cannot list every friend of mine, I still want to

x

name at least some of them: Berk Atikoglu, DuyguOzuysal, YusufOzuysal, Ayse Turker

Cınar, Murat Cınar, Bihter Padak, Erdem Sasmaz, GurerKıratlı, Turev Dara Acar, Hakan

Baba, Emel Tasyurek,OzgeIslegen, Onur Kılıc, Uygar Sumbul, AliOzer Ercan, Murat

Aksoy, EmineUlku Sarıtas, Tolga Cukur,Ipek Kasımoglu,Ismail Kasımoglu, Vinay Ma-

jjigi, Maryam Etezadi-Amoli, Michelle Hewlett, Thomas Sushil John, and Zuley Rivera-

Alvidrez.

I owe my deepest gratitude to my mother for raising me and being there always at all

costs. I am indebted everything to her, who gave me the first key steps of my education.

Her dedication to education was, in my opinion, key to my future educational success. I

would not be here without her unconditional love and support. I am also grateful to my

family, especially to my grandparents who helped raise me while I was a little kid and who

gave me their love.

Finally, I want to acknowledge my greatest support since thelast three years,Ozlem.

Without her love, I would not finish, even dare to continue my studies. She is always happy,

cheerful, optimistic and every such good thing that one can imagine. Most importantly, she

has offered these to me generously and I have found the necessary encouragement even

when I am feeling ‘down’. Thank you very much my love...

March 2011

Stanford, CA, USA.

xi

Don’t be proud of your knowledge,

Consult the ignorant and the wise;

The limits of art are not reached,

No artist’s skills are perfect. †

†the Vizier Ptahhotep, from the Papyrus Prisse, dating back the Middle Kingdom of Ancient Egypt

xii

Contents

Abstract v

Acknowledgments ix

1 Introduction 1

1.1 Motivation and Previous Work . . . . . . . . . . . . . . . . . . . . . . .. 9

1.2 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . .. 15

2 Fundamentals 17

2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 Quadratic-Phase Systems and Matrix Optics . . . . . . . . .. . . . 17

2.1.2 Linear Canonical Transforms - 1D . . . . . . . . . . . . . . . . . .18

2.1.3 Linear Canonical Transforms - 2D . . . . . . . . . . . . . . . . . .19

2.1.4 Linear Canonical Transforms - Complex . . . . . . . . . . . . .. . 23

2.1.5 Relation of LCTs to the Wigner distribution . . . . . . . . .. . . . 24

2.2 Special Linear Canonical Transforms . . . . . . . . . . . . . . . .. . . . . 26

2.2.1 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.2 Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.3 Chirp Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.4 Chirp Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . 28

xiii

2.2.5 Fractional Fourier transformation . . . . . . . . . . . . . . .. . . 30

3 The Algorithm for 1D Quadratic-Phase Systems 33

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Analysis of Decompositions and Algorithm I . . . . . . . . . . .. . . . . 35

3.3 Algorithm II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Results and Verification of the Algorithms . . . . . . . . . . . .. . . . . . 46

3.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 The Algorithm for 2D Quadratic-Phase Systems 55

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.1 A 3-Sphere for Space-Bandwidth Control, Wigner Distribution and

Dimensional Normalization for 2D Functions . . . . . . . . . . . . 57

4.3 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4 Space-Bandwidth and Sampling Rate Control . . . . . . . . . . .. . . . . 66

4.4.1 The First Coordinate Rotator . . . . . . . . . . . . . . . . . . . . .67

4.4.2 2D Separable Fractional Fourier Transform . . . . . . . . .. . . . 69

4.4.3 The Second Coordinate Rotator . . . . . . . . . . . . . . . . . . . 70

4.4.4 2D Scaling Operation . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.4.5 2D Chirp Multiplication . . . . . . . . . . . . . . . . . . . . . . . 73

4.4.6 Summary of the Algorithm . . . . . . . . . . . . . . . . . . . . . . 74

4.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75


5 The Algorithm for Complex Quadratic-Phase Systems 87

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

xiv

5.2.1 Wigner Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2.2 CLCTs in Optics and Special CLCTs . . . . . . . . . . . . . . . . 93

5.2.2.1 Complex Scaling (Magnification) . . . . . . . . . . . . . 93

5.2.2.2 Gaussian Apertures (Complex Chirp Multiplication) . . . 94

5.2.2.3 Gauss-Weierstrass Transform . . . . . . . . . . . . . . . 95

5.2.2.4 Complex-ordered Fractional Fourier Transform . . .. . 97

5.3 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.3.1 b = 0 case: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.3.2 b 6= 0 case: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105


6 Application to the Beam Propagation Method 113

6.1 Basics of BPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.2 The Relation between BPM and ABCD-Systems . . . . . . . . . . . .. . 118

6.3 Numerical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121


7 Conclusion 129

7.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

A Simpson’s Rule for 1D and 2D Functions 133

Bibliography 135

xv

List of Tables

3.1 Restrictions and oversampling factors . . . . . . . . . . . . . .. . . . . . 40

3.2 Computational Complexities.I(x, y) stands for the cost to interpolatex

samples by a factor ofy to obtainxy samples andS(z) stands for the cost

of the scaling operation onz samples. . . . . . . . . . . . . . . . . . . . . 41

3.3 Percentage errors for different functions F, transforms T, and algorithms A. 48

4.1 Percentage errors for different functions F and transforms T. . . . . . . . . 77

4.2 Percentage errors for different interpolation methodsand functions F for T1. 84

4.3 Percentage errors for different interpolation methodsand functions F for T2. 85

5.1 Summary of the conditions to have bounded,R → R CLCTs . . . . . . . . 103

5.2 Percentage errors for different functions F and transforms T. . . . . . . . . 107

xvii

List of Figures

1.1 A simple optical system . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

1.2 Free space propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . .2

1.3 Passage through thin lens . . . . . . . . . . . . . . . . . . . . . . . . . .. 3

1.4 Propagation through QGRIN . . . . . . . . . . . . . . . . . . . . . . . . .4

1.5 Example input and kernel . . . . . . . . . . . . . . . . . . . . . . . . . . .11

1.6 An example general system . . . . . . . . . . . . . . . . . . . . . . . . . .14

2.1 Effect of scaling on the Wigner distribution. . . . . . . . . .. . . . . . . . 27

2.2 Effect of Fourier transformation on the Wigner distribution. . . . . . . . . . 28

2.3 Effect of chirp multiplication on the Wigner distribution. . . . . . . . . . . 29

2.4 Effect of fractional Fourier transformation on the Wigner distribution. . . . 31

3.1 Sequence of geometrical distortions for the decomposition in Eq. 3.1. The

parallelogram in (c) is obtained by shearing the dashed rectangle in (b) in

order to cover the worst case. . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Sequence of geometrical distortions for the decomposition in Eq. 3.18. . . . 44

3.3 Example function F4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4 Transforms (T1) of F1, F2, F3, F4. The results obtained with Methods I

and II and the reference result have been plotted with dotted, dashed, and

solid lines respectively. However, in most cases these lines are indistin-

guishable since the results are very close. . . . . . . . . . . . . . .. . . . 49

xix

3.5 Percentage errors versusN for selected functions and transforms. . . . . . 50


4.2 T1 of F1 (our algorithm and reference) . . . . . . . . . . . . . . . .. . . . 78





5.1 The effect of the CCM operation on the WD. . . . . . . . . . . . . . .. . 96



5.4 Transform (T1) of F1, F2, F3, F4, F5. The results obtainedwith the pre-

sented algorithm and the reference result have been plottedwith dotted and

solid lines, respectively. However, the two types of lines are almost indis-

tinguishable since the results are very close. . . . . . . . . . . .. . . . . . 109

5.5 Transform (T3) of F1, F2, F3, F4, F5. The results obtainedwith the pre-

sented algorithm and the reference result have been plottedwith dotted and

solid lines, respectively. However, the two types of lines are almost indis-

tinguishable since the results are very close. . . . . . . . . . . .. . . . . . 110

5.6 CFRT with order0.8 − i0.2 of F1, F2, F3, F4, F5. The results obtained

with the presented algorithm and the reference result have been plotted

with dotted and solid lines, respectively. However, the twotypes of lines

are almost indistinguishable since the results are very close. . . . . . . . . . 111

6.1 Test System 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.2 Amplitudes for Test System 1 . . . . . . . . . . . . . . . . . . . . . . . .. 123

6.3 Phases for Test System 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

xx

6.4 Index distribution for Test System 2 . . . . . . . . . . . . . . . . .. . . . 125

6.5 Average index for Test System 2 . . . . . . . . . . . . . . . . . . . . . .. 126

6.6 Amplitudes for Test System 2 . . . . . . . . . . . . . . . . . . . . . . . .. 126

6.7 Phases for Test System 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

xxi

Chapter 1

Introduction

SIMULATION AND COMPUTATIONAL study of optical and signal processing systems

are of prominent importance as our information age give way to bigger and bigger

systems with enormous numbers of parameters to adjust and study. Therefore, computa-

tional methods and fast/efficient algorithms to digitize and digitally compute the optical

systems are needed. In Fig. 1.1, a very simple optical systemmodel is shown where we are

interested in relating the samples of the input field to the samples of the output field.

Figure 1.1: A simple optical system

There are several well-known computational methods for modeling these input-output

1

2 CHAPTER 1. INTRODUCTION

relations such as Finite-Difference Time-Domain (FDTD), Rigorous Coupled-Wave Analy-

sis (RCWA), Beam Propagation Method (BPM), etc. Each of these tools and methods have

their advantages and disadvantages as well as certain situations and systems that they are

suitable for. A broad class of optical systems including Fresnel propagation in free space,

propagation in graded-index media, passage through thin lenses, and arbitrary concatena-

tions of any number of these can also be modeled and represented by Quadratic-Phase

Systems (QPS). To get a better understanding about the QPSs,one can start studying very

basic cases. First of all, consider the input-output relation of a free-space propagation as

given in Fig. 1.2.

UiUo

xi

yi

x

z

y

Figure 1.2: Free space propagation

When the incoming light as described by complex field function Ui(xi, yi) propagates a

distancez in free space under the paraxial approximation, the output complex field function

Uo(x, y) is obtained. The relation betweenUi andUo is given by Eq. 1.1, which is also

called the Fresnel transform. Eq. 1.1 is an integration against a kernel, where the kernel in

this case represents the Fresnel free-space propagation.

Uo(x, y) = − i

λ

eikz

z

∫ ∞

−∞

∫ ∞

−∞

ei k2z

[(x−xi)2+(y−yi)2]Ui(xi, yi)dxidyi (1.1)

Secondly, consider a light beam passing through a thin lens as given in Fig. 1.3. The

3

Ui

zf

Uo

Figure 1.3: Passage through thin lens

input-output relation of this simple system is given by

Uo(x, y) = e−i k2f

(x2+y2)Ui(xi, yi). (1.2)

Eq. 1.2 is an equation of multiplication with a quadratic-phase and it can be seen as a

special case of an integration against a kernel by using impulse functions, though there is

not an explicit integration.

Lastly, consider a light beam propagates through a Quadratic Graded-Index media

(QGRIN) as shown in Fig. 1.4. A QGRIN is a media whose transverse index profile,n(x),

is given by the quadratic equationn(x) =√

n21(1 − (n2/n1)x2), wheren1 andn2 are the

parameters of the media. As light propagates through this media, the light rays bend and

for a normalized distanced0 =√

n1/n2, the light is focused. For any distanced before

focus, the input-output relation of the light fields are given by

Ud(x) =

∫ ∞

−∞

eiπ(cot θx2−2 csc θxxi+cot θx2i )Ui(xi)dxi. (1.3)


d

x

Ui

d0

Uo

Figure 1.4: Propagation through QGRIN

whereθ = dπ/do2.

General optical systems that are any arbitrary combinationof these three basic systems,

namely propagation, thin lens effect and Quadratic-GRIN media propagation are known

as Quadratic-Phase systems. A Quadratic-Phase system (QPS) is a unitary system, with

parameter matrixM, whose outputg(u) is related to its inputf(u) through a quadratic-

phase integral:

g(u) =√

β e−iπ/4

∫ ∞

−∞

exp[iπ(αu2 − 2βuu′ + γu′2)

]f(u′) du′, (1.4)

whereα, β, γ are real parameters. The parameter matrixM (also called ABCD-matrix),

which is defined for the general case as

M =

A B

C D

=

γ/β 1/β

−β + αγ/β α/β

, (1.5)

5

is another way of representing the QPSs in phase-space.M operates on the vectors repre-

senting the space and bandwidth pairs for the input fields. Then the output space/bandwidth

pair can be simply found by a simple matrix multiplication. This is also analogous to the

well-known ray-transfer matrices. For the aforementionedexamples, the free-space propa-

gation has the matrix representation

M =

A B

C D

=

1 λz

0 1

(1.6)

and the passage through the thin lens has the matrix

M =

A B

C D

=

1 0

−1/λf 1

(1.7)

whereλ is the wavelength andf is the focal length.

When we study cascades of these system, we can simply use the matrix multiplication

method to find the overall input-output relation. To do this,consider a cascade of systems

S1,2,...,m, each represented by a matrixM1,2,...,m. If these systems are cascaded, the overall

matrix is simplyM = MmMm−1...M2M1. For example, consider a system consists of

propagation ofz1, passage through a lens of focal lengthf and again a propagation ofz2.

These three steps have the following matrices, respectively:

M1 =

1 λz1

0 1

(1.8)

M2 =

1 0

−1/λf 1

(1.9)


M3 =

1 λz2

0 1

(1.10)

When we cascade them, we get:

M =

γ/β 1/β


= M3M2M1 =

(f − z2)/f λz1(f − z2)/f + λz2

−1/λf −z1/f + 1

(1.11)

Eq. 1.11 can be solved to find the QPS parametersα, β andγ so that we can find the re-

sulting integral equation that links the input to the outputinstead of manipulating cascades

of integrals.

QPSs are identical to the Linear Canonical Transforms (LCTs). LCT is the name given

to the same input-output relationships used in signal processing. LCTs supply a general

framework and mathematical model to study QPSs and other application areas in signal

processing. Since the literature that uses the notation of LCTs is larger than that of QPSs

and LCTs allow us to study QPSs and other systems in a more streamlined way, we will

use the name LCT from now on to refer to QPSs. Linear CanonicalTransforms (LCTs),

which are commonly referred to as quadratic-phase integrals or quadratic-phase systems

in optics [1], have also been referred to by different names such as generalized Huygens

integrals [2], generalized Fresnel transforms [3, 4], special affine Fourier transforms [5, 6],

extended fractional Fourier transforms [7], and Moshinsky-Quesne transforms [8], among

other names. More importantly, the ABCD systems widely usedin optics, [9], is also

represented by linear canonical transforms or quadratic-phase systems.

There are four main classes of LCTs: one dimensional LCTs (1D-LCTs), two dimen-

sional separable LCTs (2D-S-LCTs), two-dimensional non-separable LCTs (2D-NS-LCTs)

and complex LCTs (CLCTs).

The class of 1D-LCTs [8, 10] is a three-parameter class of linear integral transforma-

tions [1,11,12] which includes among its many special cases, the one-parameter subclasses

7

of fractional Fourier transforms (FRTs)1, scaling operations, and chirp multiplication (CM)

and chirp convolution (CC) operations, the latter also known as Fresnel transforms.

The class of two-dimensional non-separable linear canonical transforms (2D-NS-LCTs)

is the class of linear integral transforms [1, 11, 12] that includes among its several special

cases non-separable two-dimensional fractional Fourier transforms (2D-NS-FRTs) [13],

two-dimensional versions of chirp multiplication (2D-CM)and chirp convolution (2D-

CC) operations, the two-dimensional Fourier transform (2D-FT), and generalized astig-

matic scaling (magnification) operations, as well as their separable special cases. The

class of non-separable transforms is significantly more general than 2D separable lin-

ear canonical transforms (2D-S-LCTs) since it can represent a wide variety of anamor-

phic/astigmatic/nonorthogonal systems as well. The systems these integrals represent are

also known asABCD systems, which are also known as lossless first-order optical sys-

tems [8,14–20]. Classification of first-order optical systems and their representation through

linear canonical transforms are studied in [21] and [17, 22–24] for one-dimensional and

two-dimensional cases, respectively.

Two-dimensionalseparable LCTs or symmetrical transforms that do not include the

general non-separable case are addressed in [8, 10, 15, 25–28]. The most special case

possible are the isotropic 2D-LCTs in which the system is fully symmetric, orthogonal

and the parameters for both of the dimensions are identical.This case can be represented

by only three parameters as in a 1D-LCT [22]. When the system is still orthogonal but

the parameters for the orthogonal dimensions differ, the system becomes a 2D-S-LCT,

which is represented by six parameters [22]. This case is also termed as axially symmet-

ric [24]. The separable 2D transforms do not pose much difficulty because the separable

transform is essentially two independent one-dimensionaltransforms along the two di-

mensions and the dimensions can be treated independently. However, thenon-separable

1Not to confuse with the famous fast Fourier transform’s abbreviation FFT, the general convention is touse FRT or frFT for fractional Fourier transforms instead ofusing again FFT.


transform (2D-NS-LCT) is the most general case of this classof integrals where the two

dimensions are coupled to each other by four additional cross-parameters, increasing the

total number of parameters to ten. This general case is non-separable, non-axially sym-

metric, non-orthogonal, and anamorphic/astigmatic [2, 19, 22, 24, 29]. 2D-NS-LCTs are

able to represent not only systems involving anamorphic/astigmatic components and refer-

ence surfaces, but other interesting systems such as optical mode convertors and resonators

since they can represent the coupling between the dimensions [22,30–32]. Another promi-

nent feature of 2D-NS-LCTs is their ability to represent systems with rotations between

any arbitrary planes in phase-space, like rotations and gyrations [22, 24]. These systems

are collected under the general name of gyrators and are useful in two-dimensional image

processing, signal processing, mode transformation, etc.[24,33–36]. The efficient and ac-

curate digital computation of 2D-NS-LCTs is of importance in many areas of optics, optical

signal processing and general digital image processing.

Finally, Bilateral Laplace transforms, Bargmann transforms, Gauss-Weierstrass trans-

forms, [8, 37, 38], fractional Laplace transforms, [39, 40], and complex-ordered fractional

Fourier transforms [41–44] are all special cases of complexlinear canonical transforms

(CLCTs).

An important special case of the CLCTs is the family of complex-ordered fractional

Fourier transforms (CFRTs). The CFRT is the generalizationof the fractional Fourier trans-

form (FRT) where the order of the transformation is allowed to be a complex number, and

consequently the abcd matrix elements are in general complex. The optical interpretation

of the CFRT, its properties and optical realizations can be found in [41–45]. An interesting

property of CFRTs is that, in some restricted cases, they canbe optically realized by real

LCTs [46].

To avoid confusion, we note that a number of publications have used the termcomplex

Fractional Fourier transformation to refer to a particular generalization of the FRT [47,48],

which isnot a complex FRT in the sense of the order parameter being a complex number.

1.1. MOTIVATION AND PREVIOUS WORK 9

The entity referred to as a complex FRT in these publicationsis distinct from what we

refer to as a complex FRT, and is actually a special case of real two-dimensional (2D)

non-separable (NS) or non-symmetrical FRTs. Since such transforms are a special case of

two-dimensional non-separable LCTs, their digital computation is covered by the algorithm

proposed in [49]. To avoid confusion with this important butdistinct entity, we will use the

term complex-ordered to refer to complex FRTs belonging to the class of CLCTs. By

developing a general algorithm for CLCTs, we also obtain an algorithm for the important

special case of CFRTs.

Linear canonical transforms (LCTs) appear widely in optics[2,10,11], electromagnet-

ics, classical and quantum mechanics [8, 50, 51], as well as in computational and applied

mathematics [52]. The application areas of LCTs include, among others, the study of scat-

tering from periodic potentials [53–55], laser cavities [2,56,57], and multilayered structures

in optics and electromagnetics [58]. They can also be used for fast and efficient realization

of filtering in linear canonical transform domains [59].

1.1 Motivation and Previous Work

These integral transforms are of great importance in electromagnetic, acoustic, and other

wave propagation problems since they represent the solution of the wave equation under a

variety of circumstances. At optical frequencies, LCTs canmodel a broad class of optical

systems including thin lenses, sections of free space in theFresnel approximation, sections

of quadratic graded-index media, and arbitrary concatenations of any number of these,

sometimes referred to as first-order optical systems or QPSs[1, 5, 6, 10, 12]. Therefore,

given its ubiquitous nature and numerous applications, thediscrezitation, sampling and the

fast/efficient digital computation of LCTs is of considerable interest. Their fast and accurate

digital computation is of vital importance to utilize thesetools in applications in a digital

domain. Many works have addressed the problem of sampling ofreal and continuous


LCTs and some computation issues, using both decomposition-based and discrete-LCT-

based methods [60–68].

The recent advances in the fractional Fourier transform andlinear canonical transform

areas can be found in the review [69]. A review on real one-dimensional (1D) and real

symmetrical (separable) two-dimensional (2D) LCTs are given in [70].

When we want to find the input-output relation of LCTs in a digital domain, we need

to start with the general integral transform that takes the input function and transforms it to

the output function by performing an integration against a kernel as the following:

g(u)︸︷︷︸output

=√

β e−jπ/4

∫ ∞

−∞


]

︸︷︷︸kernel

f(u′)︸︷︷︸input

du′, (1.12)

The conventional method to solve the problem of digitally compute this integral is the brute

force numerical integration. First, one need to take the samples of the input and those of

the kernel and then to find each sample of the output, one need to multiply and then sum

through the input and kernel samples. This kind of computation obviously takesO(N2)

time whereN is the biggest of the number of samples necessary to properlysample the in-

put and the kernel. This method cannot be practically applied to the large problems because

of two reasons. Firstly, the algorithm hasO(N2) computational cost, which means that the

computation time increases with the number of samples very rapidly. Secondly, the value of

N should generally be taken extremely larger than the value necessary for representing the

input function because of the highly oscillatory kernel forces much bigger values forN . For

instance, we can consider a simple propagation of a Gaussian(represented byexp(−πx2))

in free-space with the following system parameters: wavelengthλ = 500nm, propagation

distancez = 100mm. We sample the input Gaussian withN = 256 samples and plot in

Fig. 1.5. On the other hand, a sample value of the kernel is also plotted in Fig. 1.5 but

proper sampling of the kernel can only be achieved by usingN = 64 × 256 samples due

to its oscillatory nature. Therefore, if one uses this bruteforce method, the biggest of these


-0.5 0 0.5 1 1.5 2

-0.5

0

0.5

1

-10 -5 0 5 100

0.2

0.4

0.6

0.8

1

Input Kernel

Figure 1.5: Example input and kernel

N values should be taken, which increases the computation cost dramatically.

In this thesis, we develop algorithms for digitally computing continuous LCTs with

careful attention to sampling issues. There has been a certain amount of work on defin-

ing discrete/finite fractional Fourier transforms and, to amuch lesser degree, discrete/finite

linear canonical transforms [71–90]. While definitions of the discrete fractional Fourier

transform (DFRT) may be considered satisfactory and well recognized [80, 88, 90], defi-

nition of the DLCT is far from being established. Further work on the definition and fast

computation of discrete transforms, and their relationship to their continuous counterparts

is desirable.

The Fourier transform is the most popular and prominent special case of LCTs and the

algorithm for its fast computation, namely the fast Fouriertransform (FFT) algorithm [91]

is a breakthrough in science and applied mathematics because it allows the application of

FT by using digital tools inO(N log N) time in parallel with the developments in com-

puter technology that makes the computational power cheaper and easier to use. Later,

Fourier transform has been generalized to fractional Fourier transform (FRT), which is

also another important special case of LCTs. A fast algorithm for the digital computation

of FRT is developed in [62]. Computation of the Fresnel diffraction integral, which is a


special case of LCTs has also received the greatest attention since it describes the prop-

agation of light in free space (see Refs. [92] and [93] and thereferences therein). Since

the input-output relationship represented by the Fresnel integral is time-invariant and takes

the form of a convolution, it can be computed inO(N log N) time. The algorithms we

present can compute the most general case of LCTs inO(N log N) time, despite the fact

that the relationship represented by the more general LCTs is not time-invariant and is not

a convolution.

A fast and accurate algorithm for numerical computation of two-dimensional non-

separable linear canonical transforms (2D-NS-LCTs) is also developed in this thesis. The

general two-dimensional non-separable case poses severalchallenges which do not exist

in the one-dimensional (1D) case and the separable two-dimensional case. The algorithm

takesO(N log N) time whereN is the two-dimensional space-bandwidth product of the

signal. the family of complex linear canonical transforms (CLCTs), which represent the

input-output relationship of complex quadratic-phase systems (CQPS). Allowing the linear

canonical transform (LCT) parameters to be complex numbersmakes it possible to repre-

sent paraxial optical systems that involve complex parameters. These include lossy systems

such as Gaussian apertures, Gaussian ducts or complex graded-index media (CGRIN), as

well as lossless thin lenses and sections of free space and any arbitrary combinations of

them. Complex-ordered fractional Fourier transforms (CFRTs) are a special case of CLCTs

and therefore a fast and accurate algorithm to compute CFRTsis included as a special case

of the presented algorithm.

In other words, these fast algorithms opens the path of general usage and easy appli-

cation of this general class of integral transform in a same way that fast Fourier transform

(FFT) algorithm opens the way for FT/DFT to become widely-used tools.

It is also important to underline that hereN is chosen close to the time-bandwidth prod-

uct of the set of input signals, which is usually the smallestpossible value ofN that can


be chosen in terms of information-theoretic considerations. Therefore, the presented al-

gorithms are highly efficient. Indeed, the distinguishing feature of the present approach is

the care with which sampling and space-bandwidth product issues are handled. Straight-

forward use of conventional numerical methods can result ininefficiencies either because

their complexity is larger thanO(N log N) and/or because the highly oscillatory quadratic-

phase kernel in these systems forcesN to be chosen much larger than the time-bandwidth

product of the signals. In other words, the straightforwardmethod of sampling the input

field and the kernel, and then calculating the output field is not suitable for several reasons.

First of all, due to the highly oscillatory nature of the integral kernel, a naive application

of the Nyquist sampling theorem to determine the sampling rate would result in an exces-

sively large number of samples and inefficient computation.On the other hand, ignoring

the oscillations of the kernel and determining the samplingrate according to the input field

alone may cause under-representation of the output field in the Nyquist-Shannon sense.

This unacceptable situation arises due to the fact that the particular 2D-LCT that we are

calculating may increase the space-bandwidth product in one or both of the dimensions. If

we do not increase the number of samples that we are working with so as to compensate

for this increase, there will be information loss and we willnot be able to recover the true

transformed output from our computed samples.

The methods developed in this thesis, in general, uses different decompositions (or

factorizations) of the given LCT into other simpler LCTs with the purpose of fast and

accurate calculation of the LCT integral. Different decompositions may be advantageous

for LCTs with different parameters. The use of matrices willgreatly facilitate our study of

different decompositions, since dealing directly with thecorresponding integral expressions

is quite cumbersome. We will study all these decompositionsin later Chapters of the thesis.

If we turn back to the example of our initial first-order optics scheme, we can practically

exemplify and summarize the general methods and algorithmsdeveloped in this thesis as


the following. Consider a general cascade of several propagations under Fresnel approx-

imation, passage through thin lenses and propagation through QGRIN media as shown in

Fig. 1.6. To find the relation between the inputU1 to the overall outputUK , if the con-

U1 UK

xi

yix

y

z…

U2 U3 U4 U5 U8 U9 UK-2 UK-1

QGRINL1

Input ( ) Output

U6 U7

L2L3 LT

Figure 1.6: An example general system

ventional methods are used, one need to work through the input-output relations of every

physical subpart of the system, do necessary calculations and operations to handle sam-

pling issues and need to work through cumbersome integral equation within other integral

equations. This way is both computationally inefficient andinformation-theoretically in-

correct if one cannot follow the space-bandwidth of the light fields along the way. Instead

of doing this, one can:

1. CalculateMsystem by simple matrix multiplications

2. Do not follow the actual physical system steps but decomposeMsystem into simplest

possible decomposition

3. Solve for the parameters of the decomposition in terms ofMsystem

4. Carry out the decomposition steps by paying attention to the space-bandwidth prod-

uct issues. Start by the minimum amount ofN required for the input field and in-

crease the sampling rate only when necessary for each step

5. Computation reduces to O(NlogN) time

1.2. ORGANIZATION OF THE THESIS 15

1.2 Organization of the Thesis

The thesis is organized as follows.

In Chapter 2 the fundamentals, mathematical preliminariesand background informa-

tion for the general topic of LCTs are given. First, the definitions of the kinds of LCTs are

given. The one-dimensional, two-dimensional non-separable and complex LCTs are dis-

cussed. Secondly, the ABCD-system in geometrical optics are briefly explained. Thirdly,

the Wigner distribution (WD), which is a method of space-frequency distribution to track

down the signal’s space-frequency extents, is introduced and its relationship to the LCTs is

given. Then the important special cases of the LCTs that arise in several application areas

are summarized. Finally, the summary of some applications of LCTs are covered.

In Chapter 3, one-dimensional LCTs (1D-LCTs) and the fast computation algorithms

for their digital computation are addressed. 1D-LCTs are important because they serve as

the basic and most fundamental class of LCTs. If the systems that needed to be analyzed

and computed digitally are one-dimensional or two-dimensional with the two-dimensions

are symmetric to each other, then the 1D-LCTs are sufficient to model and compute these

systems. Additionally, there may be systems of interest in which the modeling can be

made only in one-dimension, because the system may simply beone-dimensional or it may

be a two-dimensional system with perfect symmetry so that itcan be modeled by a one-

dimensional system. Two algorithms are presented, compared and tested with respect to

brute force reference calculations. The first is based on decomposition of the LCT into

chirp multiplication, Fourier transformation, and scaling operations. The second is based

on decomposition of the LCT into a fractional Fourier transform followed by scaling and

chirp multiplication. Then, we present the characterization and tests performed on the

algorithms to verify their accuracy and performance. Finally, the concluding remarks are

given.

In Chapter 4, two-dimensional LCTs (2D-LCTs) are addressed. An algorithm for their


fast and accurate digital computation is designed and presented. Some tools that are used

in the development of this particular algorithm is also explained in this Chapter. Similarly,

the numerical test are performed and presented to demonstrate the proper functioning of

the algorithm.

Chapter 5 covers the complex LCTs in detail. This class of LCTs is the most general and

sophisticated of the entire family and their mathematical foundations are more involved.

There are some restrictions on the transform parameters to ensure stability of the systems

that are represented by these transforms. All these issues addressed and conditions are

derived. Then a fast algorithm is designed for CLCTs. Numerical tests are presented and

the Chapter conclusion is given.

In Chapter 6, the relationship between the so-called Beam Propagation Method (BPM)

and LCTs are studied. Usually, BPM is used to model system in inhomegenous media

and LCTs are capable of representing light propagation in homogeneous media as well

as some special inhomogeneous media. We investigated the possible ways to find a link

between these two tools in an effort to make the implementation of BPM faster for certain

systems. First, the conditions and required properties of systems are found and presented

to make such a link possible. Then, we present how a BPM can be implemented in a

series of cascades of LCT systems under certain conditions.We present some test cases,

characterize and draw the limitations of such a link.

Finally, Chapter 7 gives the conclusions.

Chapter 2

Fundamentals

IN THIS CHAPTER, the preliminaries and the details of the fundamentals of LCTs will

be given. These include the main definitions and properties of different types of LCTs,

important special cases of LCTs, and some other mathematical tools that will be used to

develop the algorithms.

2.1 Preliminaries

2.1.1 Quadratic-Phase Systems and Matrix Optics

A quadratic-phase system (QPS) is a unitary system, with parameter matrixM, whose

outputfM(u) is related to its inputf(u) through a quadratic-phase integral:

fM(u) =√

β e−jπ/4

∫ ∞

−∞


]f(u′) du′, (2.1)

whereα, β, γ are real parameters. This relationship is also known under other names

including linear canonical transforms and ABCD-systems. [1,6,8,10,12,14].

The2 × 2 matrixM, whose elements areA, B, C, D, represents the same information

17

18 CHAPTER 2. FUNDAMENTALS

as the three parametersα, β, andγ which uniquely define the QPS:

M =

A B

C D

=

γ/β 1/β


=

α/β −1/β

β − αγ/β γ/β

−1

(2.2)

The unit-determinant matrixM is in the class of unimodular matrices. More on the group-

theoretical structure of QPSs may be found in [8,10].

The result of repeated application (concatenation) of QPSscan be handled easily with

the above-defined matrix. When two or more QPSs are cascaded,the resulting system is

again a QPS whose matrix is given by multiplying the matrix ofeach QPS in the cascade

structure. That is, if two QPSs with matricesM1 andM2 operate in a successive manner,

then the equivalent system is a QPS with the matrixM3 = M2M1. QPSs are not com-

mutative. The matrix of the inverse of an QPS is simply another QPS whose matrix is the

inverse of the matrix of the original QPS [8,10].

2.1.2 Linear Canonical Transforms - 1D

The one-dimensional linear canonical transform (1D-LCT) of f(u) with parameter matrix

M is denoted asfM(u) = (CMf)(u):

(CMf)(u) =√

βe−iπ/4

∫ ∞

−∞


]f(u′) du′, (2.3)

whereα, β, andγ are real parameters independent ofu andu′ and whereCM is the LCT

operator. The transform is unitary. The2 × 2 matrix M whose elements areA, B, C, D,

represents the same information as the three parametersα, β, andγ which uniquely define

the LCT:

M =

A B

C D

=

γ/β 1/β


=

α/β −1/β

β − αγ/β γ/β

−1

(2.4)

2.1. PRELIMINARIES 19

The unit-determinant matrixM belongs to the class of unimodular matrices. More on the

group-theoretical structure of LCTs may be found in [8,10].

The result of repeated application (concatenation) of LCTscan be handled easily with

the above-defined matrix. When two or more LCTs are cascaded,the resulting transform is

again an LCT whose matrix is given by multiplying the matrix of each LCT in the cascade

structure. That is, if two LCTs with matricesM1 andM2 operate in a successive manner,

then the equivalent transform is an LCT with the matrixM3 = M2M1. LCTs are not

commutative. The matrix of the inverse of an LCT is simply another LCT whose matrix is

the inverse of the matrix of the original LCT [8,10].

2.1.3 Linear Canonical Transforms - 2D

Two-dimensional non-separable linear canonical transforms (2D-NS-LCTs) are the most

general possible case of LCTs. In this section, the definition of 2D-NS-LCTs is given.

An explicit-kernel definition with the least possible number of independent variables is

provided and the forward and backward relations between theparameters of this defini-

tion and the parameters of conventionalABCD-matrices are derived. This derivation is

important because it allows the usage of LCTs with more ease by explicitly exposing the

parameters of the transform. It also makes the link between two main definitions of LCTs

by giving the forwards- and backwards-relations between the two different, but equivalent,

sets of transform parameters.

The 2D-NS-LCT with parameter matrixM, of an input functionf(u), can be denoted

and defined as [94,95]

g(u) = fM(u) = (CMf)(u) =1√

det iB

∫ ∞

−∞

∫ ∞

−∞

exp[iπ(u′TB−1Au′ − 2u′TB−1u

+uTDB−1u)]f(u′) du′ (2.5)

whereu = [ux uy]T, u′ = [u′

x u′y]

T with T denoting the transpose operation.A,B,C,D


are2 × 2 submatrices defining the transformation matrixM of the system that represents

the 2D-LCT,B being nonsingular. The matrixM, which is given as

M =

A B

C D

, (2.6)

is real and symplectic so that the following hold (I stands for the2 × 2 identity matrix)

[23,95]:

ABT = BAT, CDT = DCT, ADT − BCT = I,

ATC = CTA, BTD = DTB, ATD − CTB = I. (2.7)

From a group-theoretical point of view, 2D-NS-LCTs form theten-parameter symplec-

tic groupSp(4, R). (M has16 parameters with six constraints leaving 10 independent

parameters). More on group-theoretical properties of LCTscan be found in [8,10].

We will write the integral relationship between the input function f(ux, uy) and the

output functiong(ux, uy) more explicitly as

g(ux, uy) = e−iπ/2√

βxβy − ηxηy

∫ ∞

−∞

∫ ∞

−∞

K(ux, uy, u′x, u

′y)f(u′

x, u′y) du′

xdu′y,

K(ux, uy, u′x, u

′y) = exp[iπ(αxu

2x − 2βxuxu

′x + 2ηxuxu

′y + ηαuxuy + γxu

′x2

+αyu2y − 2βyuyu

′y + 2ηyu

′xuy + ηγu

′xu

′y + γyu

′y2)]

(2.8)

whereαx, βx, γx, αy, βy, γy, ηα, ηx, ηy, ηγ are the10 independent parameters defining

the 2D-NS-LCT (we will refer to them as the “scalar parameters”). These parameters also

uniquely define the LCT. We will use this set of parameters fortwo reasons. First, although

the definition using matrices gives us a compact and streamlined representation, the kernel

and coefficients are not seen easily and explicitly in this case. When one needs to restrict the


parameters to obtain the kernel of any desired particular subclass of 2D-LCTs, it is not easy

to derive the elements of theABCD matrices directly, whereas this is straightforward with

Eq. 2.8. Secondly and more importantly, when theABCD submatrices are used directly,

we need to manipulate 16 parameters (four2×2 matrices with four elements each), despite

the fact that only 10 of them are independent. However, with the explicit definition we use

the least number of required parameters, namely 10, and match the corresponding ten-

parameter symplectic group with exactly these 10 parameters. In other words, we have

given the simple relations between the two different representations of the same LCT. This

is an issue of only convention in giving the definitions of theLCTs.

It is easy to convert from one set of parameters to the other. The 10 scalar parameters

are given in terms of the elements ofA, B, D as follows (only three of the submatrices are

independent):

αx =D11B22 − D12B21

detB(2.9)

βx =B22

detB(2.10)

ηx =B21

det B(2.11)

ηα =D12B11 + D21B22 − D11B12 − D22B21

detB(2.12)

γx =B22A11 − B12A21

detB(2.13)

αy =D22B11 − D21B12

detB(2.14)

βy =B11

detB(2.15)

ηy =B12

detB(2.16)

ηγ =A21B11 + A12B22 − A11B21 − B12A22

detB(2.17)


γy =B11A22 − A12B21

detB(2.18)

If we wish to obtain the submatricesA, B, D in terms of the scalar parameters, we can

use the following reverse formulas:

A =1

2(βxβy − ηxηy)

ηyηγ + 2βyγx ηγβy + 2ηyγy

ηγβx + 2ηxγx ηxηγ + 2βxγy

(2.19)

B =1

βxβy − ηxηy

βy ηy

ηx βx

(2.20)

D =1

2(βxβy − ηxηy)

ηxηα + 2βyαx ηαβx + 2ηyαx

ηαβy + 2ηxαy ηyηα + 2βxαy

(2.21)

As noted earlier, submatrixC is not independent and can be expressed in terms ofA, B,

D:

C11 = (A11D11B22 + A12D12B22 − B22 − B12A21D11 − B12A22D12)/ detB

C21 = (A12D22 + A11D21 − B12C22)/B11

C12 = (A21D11 + A22D12 − B21C11)/B22

C22 = (A22D22B11 + A21D21B11 − B11 − B21A12D22 − B21A11D21)/ detB

(2.22)

Eq. 2.22, along with the corresponding entries in Eqs. 2.19,2.20, 2.21, definesC in terms

of the scalar parameters. (Because the final expressions forC are cumbersome we do not

write them here explicitly.) Note that when we set the “cross” parametersηα, ηx, ηy, ηγ

to zero, the generalized 2D-NS transformation matrixM will reduce to the transformation

matrix of the 2D separable case studied in [25]. Also note that A, B, C, D as given in

Eqs. 2.19, 2.20, 2.21, 2.22 satisfy the required propertiesgiven in Eq. 2.7.


The utility of giving the relations between these two different but equivalent definitions

of the LCTs is that one can obtain an arbitrary LCT given in oneof the two definitions,

Eq. 2.5 and Eq. 2.8. Since the 2D LCTs are cumbersome with lotsof parameters, it is not

easy to directly transfer one definition to the other. So the above bidirectional relations are

useful to make the link between the two conventions.

2.1.4 Linear Canonical Transforms - Complex

The CLCT off(u) with complex parameter matrixMC is denoted asfMC(u) = (CMC

f)(u):

(CMCf)(u) =

∫ ∞

−∞

KC(u, u′)f(u′) du′,

KC(u, u′) = e−iπ/4

√β exp

[iπ(αu2 − 2βuu′ + γu′2)

], (2.23)

whereα, β, γ are complex parameters independent ofu andu′ and whereCMCis the CLCT

operator.MC again has unit-determinant and is given by

MC =

a b

c d

=

ar + iac br + ibc

cr + icc dr + idc

=

γ/β 1/β


(2.24)

wherear, ac, br, bc, cr, cc, dr, dc are real numbers. The overline over the parametersα, β,

γ is to emphasize that these parameters are now complex, corresponding to a total of 6 real

parameters:α = αr + iαc, β = βr + iβc, γ = γr + iγc. In terms of these parameters the

kernelKC can be rewritten as

KC(u, u′) = e−iπ/4√

βr + iβc eiπ(αru2−2βruu′+γru′2)e−π(αcu2−2βcuu′+γcu′2). (2.25)


2.1.5 Relation of LCTs to the Wigner distribution

Here we will review the relationship between LCTs and the Wigner distribution, which will

aid us in understanding the effects of the elementary blocksused in our decompositions.

The relationship between the first-order optical systems (quadratic-phase systems or linear

canonical transforms) and the Wigner distributions are studied in [1,11,12,96]. The Wigner

distributionWf (u, µ) of a signalf(u) can be defined as follows [97,98]:

Wf (u, µ) =

∫ ∞

−∞

f(u + u′/2)f ∗(u − u′/2)e−2πiµu′

du′. (2.26)

Roughly speaking,W (u, µ) is a function which gives the distribution of signal energy over

time and frequency. Its integral over time and frequency,∫∞

−∞

∫∞

−∞W (u, µ) du dµ, gives

the signal energy.

Let f denote a signal andfM be its LCT with parameter matrixM. Then, the Wigner

distribution (WD) offM can be expressed in terms of the WD off as [10]

WfM(u, µ) = Wf (Du − Bµ,−Cu + Aµ). (2.27)

This means that the WD of the transformed signal is a linearlydistorted version of the orig-

inal distribution. The Jacobian of this coordinate transformation is equal to the determinant

of the matrixM, which is unity. Therefore this coordinate transformationdoes not change

the support area of the Wigner distribution. (A precise definition of the support area is not

necessary for the purpose of this paper; it may be defined as the area of the region where

the values of the Wigner distribution are non-negligible, or the area of a region containing

a certain high percentage of the total energy.) The invariance of support area means that

LCTs do not concentrate or deconcentrate energy, ie. they donot carry energy in or out

of the defined support area, keeping the total energy with thesupport area constant. The

support area of the Wigner distribution can also be approximately interpreted as the number


of degrees of freedom of the signal. Therefore, the number ofsamples needed to represent

the signal does not change after an LCT operation.

It is well known that a non-zero function and its FT cannot both be confined to finite

intervals. However, in practice we always work with samplesof finite duration signals.

We assume that a large percentage of the signal energy, as represented by the WD, is con-

fined to an ellipse with diameters∆T in the time dimension and∆B in the frequency

dimension, which can be ensured by choosing∆T and∆B suitably. This implies that the

time-domain representation is approximately confined to the interval[−∆T/2, ∆T/2] and

that the frequency-domain representation is approximately confined to[−∆B/2, ∆B/2].

We then define the time-bandwidth product∆T∆B, which is always≥ 1, because of

the uncertainty relation. Let us now introduce the scaling parameters and scaled coordi-

nates, such that the time- and frequency-domain representations are confined to intervals of

length∆T/s and∆Bs. Let s =√

∆T/∆B so that the lengths of both intervals become

equal to the dimensionless quantity√

∆B∆T which we denote by∆u, and the ellipse

becomes a circle with diameter∆u. In the new coordinates, signals can be represented in

both domains with∆u2 samples spaced∆u−1 apart. We will assume that this dimensional

normalization has been performed and that the coordinatesu andµ are dimensionless. The

space-bandwidth product concept and its importance for theoptical signals and systems are

given in more detail in [99]. Those interested in its furthergeneralization to bicanonical

width products may see [100].

For a signal with rectangular space-frequency support, thespace-bandwidth product is

equal to the number of degrees of freedom. This is not true forsignals with other support

shapes. While we have observed that LCTs do not change the number of degrees of freedom

of a signal, they may change its time-bandwidth product. This will be illustrated in what

follows.


2.2 Special Linear Canonical Transforms

Here we discuss the effects of certain operations, all special cases of LCTs, on the Wigner

distribution of a signal. These are of interest since we willdecompose general LCTs in

terms of these operations.

2.2.1 Scaling

The scaling operation is a special case of the LCT defined as:

CMMf(u) = MMf(u) =

√1/M f(u/M), (2.28)

MM =

M 0

0 1/M

=

1/M 0

0 M

−1

. (2.29)

Its effect on the WD is given by

WMM f(u, µ) = Wf(u/M, Mµ), (2.30)

whereM > 0. The scaling operation does not change the support area, time-bandwidth

product, or required number of samples (Fig. 2.1), but it changes the sampling intervals in

both the time and frequency domains by factors ofM and1/M respectively.

2.2.2 Fourier Transformation

The ordinary Fourier transform operation is also a special case of the LCT:

CFlcf(u) = Flcf(u) = e−iπ/4

∫ ∞

−∞

f(u′)e−i2πuu′

du′, (2.31)

2.2. SPECIAL LINEAR CANONICAL TRANSFORMS 27

µ

u

d1

d2

(a) Before scaling operation

µ

u

d2/M

Md1

(b) After scaling operation (parameter M)

Figure 2.1: Effect of scaling on the Wigner distribution.

Flc =

0 1

−1 0

=

0 −1

1 0

−1

. (2.32)

The subscript “lc” reminds us that the definition of the Fourier transform as a special case

of LCTs differs from the conventional definition by the factor e−iπ/4. Readers wishing

to understand the technical reason behind this inconsequential discrepancy may consult

[8,10]. The effect of Fourier transformation on the WD is given by

WFlcf (u, µ) = Wf(−µ, u), (2.33)

which is a rotation ofπ/2 in the clockwise direction (Fig. 2.2), which again does not change

the time-bandwidth product.

2.2.3 Chirp Multiplication

The chirp multiplication (CM) operation is another specialcase of the LCT:

CQqf(u) = Qqf(u) = e−iπqu2

f(u), (2.34)


µ

u

d1

d2

(a) Before Fourier transformation

µ

u

d2

d1

(b) After Fourier transformation

Figure 2.2: Effect of Fourier transformation on the Wigner distribution.

Qq =

1 0

−q 1

=

1 0

q 1

−1

. (2.35)

Its effect on the WD is

WQqf (u, µ) = Wf(u, µ + qu). (2.36)

Although the support area and therefore the number of degrees of freedom are pre-

served after chirp multiplication, the time-bandwidth product increases. This is due to the

increase in signal bandwidth as a result of vertical shearing of the WD (Fig. 2.3). The new

time-bandwidth product isd1(d2 + |q|d1). If we wish a signal with such a support to be

recoverable from its samples in the conventional manner, this is the number of samples we

need. (The sampling interval in the time domain is1/(d2 + |q|d1) and that in the frequency

domain is1/d1.)

2.2.4 Chirp Convolution

The chirp convolution (CC) operation is indeed the Fresnel propagation integral and it is

the dual of the chirp multiplication and corresponds to a horizontal, rather than a vertical


u

µ

d1

d2

(a) Before chirp multiplication

u

µ

d2 + |q|d1

d1

(b) After chirp multiplication

Figure 2.3: Effect of chirp multiplication on the Wigner distribution.

shear in the time-frequency plane:

CRrf(u) = Rrf(u) = f(u) ∗ e−iπ/4

√1/r exp(iπu2/r), (2.37)

Rr =

1 r

0 1

=

1 −r

0 1

−1

. (2.38)

Its effect on the WD is

WRrf (u, µ) = Wf(u − rµ, µ). (2.39)

The Fresnel propagation integral for a distancez along the propagation direction and

for a wavelength ofλ is denoted byF(z, λ) and is given by

F(z, λ)(x′) =eiπz/λ

√iλz

∫ ∞

−∞

ei πλz

(x′−x)2f(x) dx (2.40)

Its ABCD matrix (the propagation matrix) is given by

1 zλ

2π

0 1

. (2.41)


Therefore, Fresnel propagation is exactly a chirp convolution with parameterr =

zλ/2π. In terms ofα-β-γ parameters of LCTs, the Fresnel propagation special case can be

obtained by settingα = β = γ = 2π/zλ.

2.2.5 Fractional Fourier transformation

Theath order fractional Fourier transform (FRT) [10,101–107] of a functionf(u), denoted

fa(u), is most commonly defined as

Faf(u) = fa(u) =

∫ ∞

−∞

Ka(u, u′)f(u′) du′, (2.42)

Ka(u, u′) = Aθ exp[iπ(cot θ u2 − 2 csc θ uu′ + cot θ u′2)

],

Aθ =√

1 − i cot θ, θ =aπ

2

whena 6= 2j andKa(u, u′) = δ(u − u′) whena = 4j andKa(u, u′) = δ(u + u′) when

a = 4j ± 2, wherej is an integer. The square root is defined such that the argument of the

result lies in the interval(−π/2, π/2]. For0 < |a| < 2 (0 < |θ| < π), Aθ can be rewritten

without ambiguity as

Aθ =e−i[πsgn(θ)/4−θ/2]

√| sin θ|

, (2.43)

wheresgn(·) is the sign function. Whena is outside the interval0 ≤ |a| ≤ 2, we need

simply replacea by its modulo4 equivalent lying in this interval and use this value in

Eq. (2.43).

The FRT is also a special case of the LCT with matrix

Falc =

cos(aπ/2) sin(aπ/2)

− sin(aπ/2) cos(aπ/2)

=

cos(aπ/2) − sin(aπ/2)

sin(aπ/2) cos(aπ/2)

−1

, (2.44)


differing only by the factore−iaπ/4:

CFalcf(u) = Fa

lcf(u) = e−iaπ/4Faf(u) (2.45)

Again the subscript “lc” denotes this inconsequential discrepancy between the definition of

the FRT given by Eq. 2.42 and the FRT defined as a special case ofthe LCT [8, 10]. The

FRT rotates the WD in the clockwise direction with an angle ofθ = aπ/2, as shown in

Fig. 2.4:

WFalcf

= Wf [cos(aπ/2)u − sin(aπ/2)µ, sin(aπ/2)u + cos(aπ/2)µ] . (2.46)

µ

u

d1

d2

(a) Before FRT operation

µ

u

d2

d1

(b) After FRT operation

Figure 2.4: Effect of fractional Fourier transformation onthe Wigner distribution.

QPSs/LCTs are capable of exactly representing quadratic-graded index (GRIN) media,

[108], in which the inhomogeneous refractive index distribution is given byn2(x) = n21[1−

(n2/n1)x2] wheren1 andn2 are the medium parameters andx is the transverse coordinate.

The parametersn1 andn2 are assumed to be constants along the propagation direction.

FRTs are directly giving the propagation through these GRINmedia. When the propagation

distance through a GRIN media with parametersn1 andn2 is d, and if the order of FRT,a,


is set toa = d/d0, whered0 = π2

√n1

n2, then the result of FRT gives the output field after

GRIN propagation.

Chapter 3

The Algorithm for 1D Quadratic-Phase

Systems

3.1 Introduction1

IN THIS CHAPTER we discuss two approaches for the digital computation of LCTs.

The first algorithm decomposes an LCT with arbitrary transform parameters into some

combinations of three simpler operations: scaling, Fourier transformation, and chirp multi-

plication. The second method decomposes the LCT into fractional Fourier transformation,

chirp multiplication and scaling. Both are fast algorithmswhich takeO(N log N) time,

whereN is the time-bandwidth product of the input signal. Despite the highly oscillatory

nature of the integral kernel, special care is taken to carefully manage the sampling rate so

as to ensure that the number of samplesN is not chosen much larger than the time(space)-

bandwidth product of the input signal, so that the algorithms are as efficient as possible. A

naive application of the Nyquist sampling theorem to determine the sampling rate, on the

other hand, would result in an excessively large value ofN and inefficient computation.

The only deviation from exactness arises from the approximation of a continuous Fourier

1This chapter is taken from [64]. Copyright 2008 by IEEE.

33

34 CHAPTER 3. THE ALGORITHM FOR 1D QUADRATIC-PHASE SYSTEMS

transform with the discrete Fourier transform (DFT). Thus the algorithms compute LCT

integrals with a performance similar to that of the fast Fourier transform (FFT) algorithm

in digitally computing the continuous Fourier transform (FT), both in terms of speed and

accuracy.

The proposed algorithms use matrix factorizations to decompose LCTs into cascade

combinations of the elementary LCT blocks discussed above.Since each stage can be com-

puted inO(N log N) time, the overall LCT can also be. Numerous such decompositions are

possible [10, 29], but they are not equally suited for numerical purposes. For instance, di-

rect naive application of the decomposition of chirp multiplication, Fourier transformation,

scaling (magnification), and again chirp multiplication, which suggests itself upon inspec-

tion of Eq. (2.3) will in general lead to very high sampling rates. This is because the early

appearance of the chirp multiplication in the cascade and the lack of sampling rate manage-

ment that controls the first chirp multiplication. If one directly uses this decomposition for

every parameter set, there may be two large increases in the number of samples because of

the two chirp multiplications. These combined increases inthe number of samples results

in unnecessary high sampling rates. We have carried out a systematic exhaustive analysis

of all possible decompositions of arbitrary LCTs into the three basic operations of scal-

ing, chirp multiplication (CM), and Fourier transformation (FT). We have considered all

possible decompositions with three, four, and five cascade blocks. Every permutation has

been checked to see if that decomposition is capable of expressing an LCT with arbitrary

parameters. We have also considered the required sampling rates for each decomposition

and pick the ones that require the least possible number of samples.

This chapter will start by studying the different decompositions (or factorizations) of

the given LCT into other LCTs with the purpose of fast and accurate calculation of the LCT

integral. Different decompositions may be advantageous for LCTs with different parame-

ters. The use of matrices will greatly facilitate our study of different decompositions, since

dealing directly with the corresponding integral expressions is quite cumbersome.

3.2. ANALYSIS OF DECOMPOSITIONS AND ALGORITHM I 35

The chapter is organized as follows: Section 3.2 presents a systematic analysis of all

possible decompositions of an LCT into the three basic operations of scaling, Fourier trans-

formation and chirp multiplication. Based on this, the firstalgorithm is also presented in

Section 3.2. The second approach involving the fractional Fourier transform is given in

Section 3.3. Numerical results, verification of the performance of the developed algorithms

and the comparison of the two approaches are presented in Section 3.4. In Section 3.5, the

conclusion is given. In this section, by deriving the 1D algorithms, we also show the main

essence of the algorithms developed by demonstrating the general approach taken in the

fundamental 1D case. Then, the same main approach of divide and conquer will be applied

to more general cases of two-dimensional LCTs and complex LCTs in later chapters of the

thesis.

3.2 Analysis of Decompositions and Algorithm I

It is well known that arbitrary LCTs can be decomposed in either the formQq1RrQq2 or

Rr1QqRr2 , whereQq andRr are the chirp multiplication and convolution operations re-

spectively [10]. Chirp convolutions can be realized as a Fourier transform followed by a

chirp multiplication followed by an inverse Fourier transform (the Fourier transform of a

chirp is also a chirp). Since we already consider all permutations involving chirp multi-

plication and Fourier transformation, approaches involving chirp convolution are also in-

cluded in our development.

While generating all possible decompositions through permutations, duplicate decom-

positions arise. For example, two consecutive scaling or two consecutive chirp multiplica-

tion operations can both be collapsed into one. Or, for instance, in the case of five-stage

decompositions with more free parameters than the three free parameters of LCTs, we

have the freedom of choosing the additional parameters. When we select the scaling pa-

rameter as1, this reduces to an equivalent four stage decomposition. Inother words, some


decompositions turn out to be equivalent to others, reducing the total number of possible

decompositions.

Careful consideration shows that there are a total of sixteen distinct decompositions.

Four of these have 4 stages and twelve of them have 5 stages; decompositions with 3 stages

are not flexible enough to match arbitrary LCTs. However, some decompositions among

these sixteen are equivalent in implementation. For example a scaling and FT cascade can

be replaced with a FT and inverse scaling cascade, a trivial and inconsequential difference.

When such trivial replacements are deducted from the set of sixteen decompositions, we

end up with twelve decompositions. We will immediately eliminate two of these decom-

positions. These two decompositions involve significantlymore computational load than

the others. As will be seen, the CM stages, especially when they occur early in the cas-

cade, require us to increase the sampling rate and thus the complexity. Generally speaking

it is desirable to have as few CM stages as possible and to havethem appear as late in

the cascade as possible. Therefore we eliminate the two decompositions having three CM

stages. Finally, our exhaustive consideration and carefulsorting out of all decompositions

with three, four, and five stages leaves us with ten distinct decompositions to consider:

M =

1/β 0

0 β

1 0

α/β2 1

0 1

−1 0

1 0

γ 1

(3.1)

M =

1 0

α 1

0 1

−1 0

1 0

γ/β2 1

β 0

0 1/β

(3.2)

M =

1 0

α 1

1/β 0

0 β

0 1

−1 0

1 0

γ 1

(3.3)

M =

1 0

α − β2/γ 1

−γ/β 0

0 −β/γ

0 1

−1 0

1 0

−1/γ 1

0 1

−1 0

(3.4)

M =

1 0

α − β2/γ 1

0 1

−1 0

1 0

−γ/β2 1

0 1

−1 0

−γ/β 0

0 −β/γ

(3.5)


M =

−γ/β 0

0 −β/γ

1 0

−γ + αγ2/β2 1

0 1

−1 0

1 0

−1/γ 1

0 1

−1 0

(3.6)

M =

0 1

−1 0

−α/β 0

0 −β/α

1 0

−α/β2 1

0 1

−1 0

1 0

γ − β2/α 1

(3.7)

M =

0 1

−1 0

1 0

−1/α 1

0 1

−1 0

−β/α 0

0 −α/β

1 0

γ − β2/α 1

(3.8)

M =

0 1

−1 0

1 0

−1/α 1

0 1

−1 0

1 0

−α + γα2/β2 1

−β/α 0

0 −α/β

(3.9)

M =

0 1

−1 0

1 0

−1/α 1

−α/β 0

0 −β/α

0 1

−1 0

1 0

γ − β2/α 1

(3.10)

Recall that the diagonal matrices correspond to scaling, the skew-diagonal ones to FT,

and the lower-triangular ones to CM. The parametersα, β, γ of these operations are chosen

in terms of the elementsA, B, C, D of the matrixM so as to equate the left and right hands

of the above equations (see Eq. (2.4)).

Scaling and FT do not require an increase in the number of samples. However, since

CM will change the time-bandwidth product, we introduce×2 oversampling when con-

fronted with the first CM operation, to allow us room to maneuver. We will however try

to avoid oversampling beyond this, as much as possible. Specifically, we will impose any

necessary restrictions so as to avoid further oversamplinguntil the stage that involves the

very last CM.

We will make sure that after each stage, the number of samplesis sufficient (in the

Nyquist sense) for recovery of the continuous signal. In order to illustrate how we deal

with each decomposition, we consider the decomposition given in Eq. 3.1 as an example

(Fig. 3.1). For graphical purposes, here we assumed that theinitial time-frequency support

is a square of edge length∆u rather than a circle. Since we require that2N samples be


µ

u

∆u(1 + |γ|)

∆u

(a) After the first stage: CM I

µ

u

∆u(1 + |γ|)

∆u

(b) After the second stage: FT

µ

u

∆u(1 + |γ|)

∆u[1 + (1 + |γ|) |α|β2 ]

(c) After the third stage: CM II

µ

u

∆u|β|[1 + (1 + |γ|) |α|β2 ]

∆u(1 + |γ|)/|β|

(d) After the fourth stage: scaling

Figure 3.1: Sequence of geometrical distortions for the decomposition in Eq. 3.1. Theparallelogram in (c) is obtained by shearing the dashed rectangle in (b) in order to coverthe worst case.

sufficient, we must ensure that the time-bandwidth product does not exceed2N :

∆u×∆u(1+ |γ|) ≤ 2N = 2∆u2 ⇒ |γ| ≤ 1 ⇒ −1 ≤ γ ≤ 1. (3.11)

Therefore, we obtain the restriction−1 ≤ γ ≤ 1 for the parameterγ appearing in the first

chirp multiplication operation. This restriction ensuresthat oversampling by two is suffi-

cient, by ensuring that the bandwidth following the geometric distortion has not increased

by more than a factor of two. After the FT, the second chirp multiplication operation, and


the scaling operation, the time-bandwidth product becomes:

∆u (1 + |γ|)∆u

[1 +

|α|β2

(1 + |γ|)]

. (3.12)

Note that the parallelogram in Fig. 3.1(c) is obtained by shearing the dashed rectangle in

Fig. 3.1(b) in order to cover the worst case. Is the above expression greater than2N =

2∆u2 and if so, how much greater is it? Letk/2 denote the additional oversampling factor

required. Equating the above expression to the number of samplesk∆u2 corresponding to

a total ofk oversampling, leads us to the minimum value ofk as

k ≥ 1 + |γ| + |α|(1 + |γ|)2

β2. (3.13)

If the right hand side of this expression turns out to be≤ 2, that means that we do not need

any additional oversampling, in which case we simply takek = 2. Before continuing, we

also note that the scaling operation merely changes the sampling interval in the sense of

reinterpretation of the same samples with a scaled samplinginterval, in a manner which

corresponds to scaling of the underlying continuous signal.

Thus, by carefully following the evolution of the time-frequency support region through

each stage of the decomposition, we have obtained (i) any necessary restrictions on the

parameters of the stages appearing in the decomposition, sothat oversampling beyond×2

is not required until the last CM stage, (ii) the additional oversampling factork/2 which

may be needed before the last CM stage to fully represent the continuous output signal. We

underline that these considerations are guided by our goal to accommodate arbitrary input

signals. In those cases where there exists somea priori knowledge of the input signal or

the signal is restricted to a particular class, it may be possible to customize the approach

here with benefit.

The same procedure has been repeated for the decompositionsgiven in Eqs. 3.2–3.10

and the results are given in Table 3.1. It will be convenient to choosek as the smallest


integer≥ 2 satisfying the applicable inequality.

We have also added together the computational complexity ofeach stage to obtain the

overall computational complexity of each decomposition and presented these in Table 3.2.

Decomposition Restriction Oversampling factor

Eq. 3.1 |γ| ≤ 1 k ≥ 1 + |γ| + |α|(1+|γ|)2

β2

Eq. 3.2 |γ| ≤ 1 k ≥ 1 + |γ| + |α|(1+|γ|)2

β2

Eq. 3.3 |γ| ≤ 1 k ≥ 1 + |γ| + |α|(1+|γ|)2

β2

Eq. 3.4 |γ| ≥ 1 k ≥ 1 + 1|γ|

+ (1+|γ|)2

β2 |α − β2

γ|

Eq. 3.5 |γ| ≥ 1 k ≥ 1 + 1|γ|

+ (1+|γ|)2

β2 |α − β2

γ|

Eq. 3.6 |γ| ≥ 1 k ≥ 1 + 1|γ|

+ (1+|γ|)2

β2 |α − β2

γ|

Eq. 3.7 |γ − β2

α| ≤ 1 k ≥ 1 + |γ − β2

α| + | α

β2 |[1 + |γ − β2

α|]2

Eq. 3.8 |γ − β2

α| ≤ 1 k ≥ 1 + |γ − β2

α| + | α

β2 |[1 + |γ − β2

α|]2

Eq. 3.9 |γ − β2

α| ≤ 1 k ≥ 1 + |γ − β2

α| + | α

β2 |[1 + |γ − β2

α|]2

Eq. 3.10 |γ − β2

α| ≤ 1 k ≥ 1 + |γ − β2

α| + | α

β2 |[1 + |γ − β2

α|]2

Table 3.1: Restrictions and oversampling factors

Observation of Table 3.1 reveals a natural grouping of the ten decompositions. The

decompositions given in Eqs. 3.1, 3.2, 3.3 exhibit the same restriction and oversampling

factors. Likewise, the decompositions given in Eqs. 3.4, 3.5, 3.6 share the same restriction

and oversampling factors. Moreover, the restrictions of these two groups are complemen-

tary, the first three can be used for|γ| ≤ 1 and the last three can be used for|γ| ≥ 1,

spanning the whole parameter space of LCTs.

Finally, we observe that the computational complexity of the third and last group of

four decompositions is larger than the others. This group has a term with complexity

(Nk) log(Nk) due to the second CM operation being followed by a DFT operation. There-

fore we discard the last four decompositions belonging to the third group. It should be

noted that our elimination procedure is primarily based on complexity as opposed to error.

However, as we will see in Table 3.3, the algorithm obtained produces results with errors as


Decomposition Computational ComplexityEq. 3.1 2N + (2N) log(2N) + Nk + I(N, 2) + I(2N, k/2) + S(Nk)Eq. 3.2 2N + (2N) log(2N) + Nk + I(N, 2) + I(2N, k/2) + S(N)Eq. 3.3 2N + (2N) log(2N) + Nk + I(N, 2) + I(2N, k/2) + S(2N)Eq. 3.4 2N + (2N) log(2N) + N log N + Nk + I(N, 2) + I(2N, k/2)

+S(2N)Eq. 3.5 2N + (2N) log(2N) + N log N + Nk + I(N, 2) + I(2N, k/2)

+S(N)Eq. 3.6 2N + (2N) log(2N) + N log N + Nk + I(N, 2) + I(2N, k/2)

+S(Nk)Eq. 3.7 2N + (2N) log(2N) + (Nk) log(Nk) + Nk + I(N, 2)

+I(2N, k/2) + S(Nk)Eq. 3.8 2N + (2N) log(2N) + (Nk) log(Nk) + Nk + I(N, 2)

+I(2N, k/2) + S(2N)Eq. 3.9 2N + (2N) log(2N) + (Nk) log(Nk) + Nk + I(N, 2)

+I(2N, k/2) + S(N)Eq. 3.10 2N + (2N) log(2N) + (Nk) log(Nk) + Nk + I(N, 2)

+I(2N, k/2) + S(2N)

Table 3.2: Computational Complexities.I(x, y) stands for the cost to interpolatex samplesby a factor ofy to obtainxy samples andS(z) stands for the cost of the scaling operationonz samples.

small as can be reasonably expected; therefore we have not really lost anything by basing

our procedure primarily on complexity.

Use of the information summarized in the two tables finally leads us to choose to work

with two decompositions, one from the first and one from the second group of three. To

ensure that oversampling by two is sufficient, we ended up using two complementary de-

compositions for different regions of the parameter space.Since they have a slightly lower

complexity, we will prefer to work with the second decomposition in both groups, whose

advantage arises primarily from the relative positions of the scaling and CM operations.

The algorithm can now be outlined as follows:


• If |γ| ≤ 1, use the decomposition:

M =

1 0

α 1

0 1

−1 0

1 0

γ/β2 1

β 0

0 1/β

.

In operator notation:

CM = Q−α Jk/2 Flc Q−γ/β2 J2 Mβ, (3.14)

whereJx represents the×x oversampling operation. The minimum value ofk is:

k ≥ 1 + |γ| + |α|(1 + |γ|)2

β2. (3.15)

• If |γ| > 1, use the decomposition:

M =

1 0

α − β2/γ 1

0 1

−1 0

1 0

−γ/β2 1

0 1

−1 0

−γ/β 0

0 −β/γ

.

In operator notation:

CM = Q−α+β2/γ Jk/2 Flc Qγ/β2 J2 Flc M−γ/β, (3.16)

The minimum value ofk is:

k ≥ 1 +1

|γ| +(1 + |γ|)2

β2|α − β2

γ| (3.17)

We have chosen to avoid unnecessary increases in the time-bandwidth product in the

early stages to avoid increasing the number of samples untilthe last CM stage, where the

major and unavoidable increase in sampling rate occurs.

3.3. ALGORITHM II 43

3.3 Algorithm II

The second approach we discuss is based on the following decomposition involving the

FRT, scaling, and chirp multiplication:

M =

A B

C D

=

1 0

−q 1

M 0

0 1/M

cos θ sin θ

− sin θ cos θ

. (3.18)

Hereθ = aπ/2 wherea is the order of the FRT,q is the chirp multiplication parameter,

andM is the scaling factor. As we will see, these three parametersare sufficient to satisfy

the above equality for arbitrary ABCD matrices, so that thisdecomposition is capable of

representing arbitrary LCTs. Since the fast method proposed in [62] can be used for the

computation of the FRT, this decomposition leads to a fast algorithm for LCTs. This de-

composition was inspired by the optical interpretation in Ref. [107] and is also a special

case of the widely-known Iwasawa decomposition [19, 23, 95]. It was also proposed later

in [61,63]. Fig. 3.2 illustrates the sequence of geometrical distortions corresponding to this

decomposition, where the initial time-frequency support is a circle of diameter∆u.

If we multiply out the right hand side of Eq. 3.18 and replace the matrix entriesA, B,

C, D with α, β, γ, we obtain:

γ/β 1/β


=

M cos θ M sin θ

−qM cos θ − sin θ/M −qM sin θ + cos θ/M

, (3.19)

which is equivalent to four equations which we can solve fora, q, M :

a = (2/π)arccotγ, (3.20)

M =

√1 + γ2/β, γ ≥ 0,

−√

1 + γ2/β, γ < 0,(3.21)

q = γβ2/(1 + γ2) − α. (3.22)


µ

u

θ = aπ/2 = cot−1(γ)

∆u

∆u

(a) After the first stage: FRT

µ

u

M∆u

∆u/M

(b) After the second stage: scaling

µ

u

M∆u

∆u/M + |q|M∆u

(c) After the third stage: CM

Figure 3.2: Sequence of geometrical distortions for the decomposition in Eq. 3.18.

3.3. ALGORITHM II 45

The ranges of the square root and the arccotangent both lie in(−π/2, π/2]. Figure 3.2

shows the geometric effect of the decomposition stages on the WD of a function, which

is rotation, scaling, and shearing, respectively. In operator notation this algorithm can be

expressed as:

CM = Qq Jk MM Falc (3.23)

In this method, the first operation is a FRT, whose fast computation in O(N log N)

time is presented in Refs. [62,109]. (Other works dealing with fast computation of the FRT

include [110, 111].) The algorithm presented in Ref. [62] was based on decomposing the

FRT into a CM followed by a CC followed by a final CM, and computed the samples of

the continuous FRT in terms of the samples of the original signal. Just as in the present

work, care was taken to ensure that the output samples represented the continuous FRT in

the Nyquist-Shannon sense. The presently discussed algorithm employs the algorithm in

Ref. [62] as a subroutine. The only approximation in this subroutine comes from the step

involving chirp convolution in which a DFT/FFT is used to approximate the samples of

the continuous FT. No other approximation need be made, either in this subroutine or in

any of the other operations that we employ. Thus the only source of approximation can be

traced to the evaluation of a continuous FT by use of a DFT (implemented with a FFT),

which is a consequence of the fundamental fact that the signal energy cannot be confined to

finite intervals in both domains. The second operation in this method is scaling, which only

involves a reinterpretation of the same samples with a scaled sampling interval. The final

operation is CM which takesO(N) time, leading to an overall complexity ofO(N log N).

(Detailed expressions will be given in Section 3.4.) As in the first method, it is again

necessary to ensure that the output samples are sufficient torepresent the transformed signal

in the Nyquist-Shannon sense. Since LCTs distort the original time-frequency support,

both the time and frequency extent of the signal, as well as its time-bandwidth product may

increase, despite the fact that the area of the support remains the same. Therefore, a greater


number of samples than∆u2 may be needed to represent the transformed signal as we have

showed in an example in Fig. 3.1.

Delaying confrontation with the necessity to deal with thisgreater number of samples

until the very last step is a significant advantage of this method. Since the FRT corresponds

to rotation and scaling only to reinterpretation of the samples, these steps do not require

us to increase the number of samples. At the last CM step, if wemultiply the samples of

the intermediate result with the samples of the chirp, the samples obtained will be good

approximations of the true samples of the transformed signal at that sampling interval. If

these samples are sufficient for our purposes, nothing further needs be done. However, in

general these samples will be below the Nyquist rate for the transformed signal and will

not be sufficient for full recovery of the continuous function. To obtain a sufficient number

of samples that will allow full recovery, we must interpolate the intermediate result at least

by a factork corresponding to the increase in time-bandwidth product:

k ≥ 1 +∣∣γ − α(1 + γ2)/β2

∣∣ . (3.24)

Again, for convenience we choosek to be the smallest integer satisfying this inequality.

3.4 Results and Verification of the Algorithms

We have considered several examples to illustrate and compare the presented methods. We

refer to the first algorithm involving Fourier transformation, scaling, and chirp multiplica-

tion as A1, and the second algorithm involving the fractional Fourier transform as A2. We

consider the chirped pulse functionexp(−πu2 − iπu2), denoted F1, and the trapezoidal

function1.5tri(u/3)− 0.5tri(u), denoted F2 (tri(u) = rect(u) ∗ rect(u)). Since these two

functions are well confined to a circle with diameter∆u = 8 we takeN = 82. We also

consider the binary sequence 01101010 occupying[−8, 8] with each bit 2 units in length,

3.4. RESULTS AND VERIFICATION OF THE ALGORITHMS 47

so thatN = 162. This binary sequence is denoted by F3 and the function shownin Fig. 5.2

is denoted by F4, again withN = 162. These choices for∆u result in∼ 0 %, 0.0002 %,

0.47 %, 0.03 % of the energies of F1, F2, F3, F4 respectively, to fall outside the chosen

frequency extents. The chosen time extents include all of the energies of F2, F3, F4 and

virtually all of the energy of F1. We consider two transforms, the first (T1) with parameters

(α, β, γ) = (−3,−2,−1), and the second (T2) with parameters(−4/5, 1, 2). The LCTs

T1 and T2 of the functions F1, F2, F3, F4 have been computed both by the presented fast

methods A1 and A2 and by a highly inefficient brute force numerical approach based on

composite Simpson’s rule with extensive number of intervals that can handle highly oscil-

latory functions which is here taken as a reference. The details of the Simpson’s method is

given in Appendix A.

The results for all functions (F1, F2, F3, F4) and both algorithms (A1, A2) are plotted in

Figure 3.4 for transform T1, and tabulated in Table 3.3 for both transforms (T1, T2). Also

shown are the errors that arise when using the DFT in approximating the FT of the same

functions, which serves as a reference. (The error is definedas the energy of the difference

normalized by the energy of the reference, expressed as a percentage.)

The key observations that can be made from this table are as follows. The errors ob-

tained depend on the function, since different functions have different amounts of energy

contained in their tails which fall outside the chosen time and frequency extents (or as-

sumed time-frequency region). For those cases in which the error is large, such as F3, this

means that we have determined the time-bandwidth product less conservatively than the

other examples, and the error can be reduced by increasingN . Generally speaking, the

errors obtained depend very little on the transform parameters or which method we use,

and are comparable to the error arising when we use the DFT to approximate the FT. Since

a DFT lies at the heart of both methods, this is the smallest error we could have hoped to

achieve to begin with.

Fig. 3.5 shows the error versus number of sample pointsN for selected functions and


−8 −6 −4 −2 0 2 4 6 80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Figure 3.3: Example function F4

A1 T1 A1 T2 A2 T1 A2 T2 DFTF1 3.2 × 10−22 9.5 × 10−22 2.7 × 10−17 6.6 × 10−17 2.0 × 10−21

F2 7.8 × 10−4 8.1 × 10−4 11 × 10−4 9.9 × 10−4 6.2 × 10−4

F3 1.5 1.6 1.4 1.5 1.2F4 9.7 × 10−2 11 × 10−2 8.9 × 10−2 9.9 × 10−2 8.3 × 10−2

Table 3.3: Percentage errors for different functions F, transforms T, and algorithms A.

transforms. We observe that the error decreases steeply at first with increasingN as ex-

pected, but saturates when we approach and exceed the time-bandwidth product of the sig-

nals (here64). This demonstrates that the number of samplesN can be chosen comparable

to the time-bandwidth product, which is the smallest numberwe can expect to work with,

and need not be chosen larger. A2 was used to obtain this plot for illustration purposes but


−3 −2 −1 0 1 2 3−0.5

0

0.5

1Re of transform of F1

−3 −2 −1 0 1 2 3−0.5

0

0.5

1Im of transform of F1

−3 −2 −1 0 1 2 3−1

−0.5

0

0.5

1

1.5Re of transform of F2

−3 −2 −1 0 1 2 3−1

−0.5

0

0.5

1

1.5Im of transform of F2

−6 −4 −2 0 2 4 6−2

−1

0

1


−6 −4 −2 0 2 4 6−2

−1

0

1


−6 −4 −2 0 2 4 6−2

−1

0

1

2


−6 −4 −2 0 2 4 6−2

−1

0

1

2


Figure 3.4: Transforms (T1) of F1, F2, F3, F4. The results obtained with Methods I and IIand the reference result have been plotted with dotted, dashed, and solid lines respectively.However, in most cases these lines are indistinguishable since the results are very close.


similar results can also be obtained when we use A1.

100

101

102

103

0

2

4

6

8

10

12

14

16

18

20

N

Err

or %

F1 T1F1 T2F2 T1F2 T2

Figure 3.5: Percentage errors versusN for selected functions and transforms.

We now turn our attention to discussing the complexity (cost) of the algorithms as a

function ofN , the number of sample points. Based on the preceding paragraph,N can be

chosen comparable to the time-bandwidth product so that theexpressions given below can

also be interpreted as functions of time-bandwidth product.

The computational complexity of the first method is given by either of the following

expressions, depending on which decomposition is used (which was determined by whether

|γ| ≤ 1 or not, respectively):

• 2N + 2N log 2N + Nk + I(N, 2) + I(2N, k/2) + S(N)

• 2N + 2N log 2N + N log N + Nk + I(N, 2) + I(2N, k/2) + S(N)


On the other hand, for the second method, one of the followingapplies depending on the

branch of the FRT algorithm used, which depends on whether0.5 ≤ |a| ≤ 1 or |a| < 0.5,

respectively. Not surprisingly, this turns out to be the same as the condition|γ| ≤ 1 or

|γ| > 1, respectively:

• 6N + 6N log 2N + Nk + I(N, 2) + I(2N, k/2) + S(2N)

• 6N + 6N log 2N + N log N + Nk + I(N, 2) + I(2N, k/2) + S(2N)

The above expressions are derived by calculating the complexity of the FRT algorithm

of [62] in its most efficient implementation (6N + 6N log 2N + I(N, 2) [+N log N ]),

and adding the cost of the other operations.

Although included for completeness, the costS(·) of the scaling operation is minimal

and not of much consequence, since it amounts only to a reinterpretation of samples. In

the above expressions,I(x, y) stands for the cost to interpolatex samples by a factor ofy

to obtainxy samples. There are several efficient approximate interpolation methods which

have complexities of orderO(xy) [112]. We will write this cost ascxy wherec is a constant.

Taking the difference of the costs of the two methods, MethodII will have lower cost if

(1 + c)(kI − kII) > 4[1 + log 2N ], (3.25)

wherekI andkII are associated with Method I and Method II respectively. If we go back

to the steps of each method, we observe that it is usually possible to choosekII much

more tightly thankI . Numerical simulations also confirm thatkII is usually smaller than

kI . Therefore, either method may turn out to have lower cost depending on the values of

α, β, γ, N and it is not possible in general to declare one method superior over the other in

this respect. If need be, both methods can be incorporated inthe same code and the more

efficient one invoked based on the parameters, but in many cases the difference may not be

very significant. However, apart from its effect on cost of computation, having the lowest


oversampling factor is desirable in itself since it produces an output represented with the

least number of samples. This is a plus for Method II, which wealso favor for its elegant

construction.

Finally, we compare Method I and Method II, with the direct use of the well-known

CM-CC-CM decomposition [2] without the kind of sampling rate management undertaken

in this paper (“Method III”):

M =

1 0

α − β 1

1 1/β

0 1

1 0

γ − β 1

, (3.26)

which is valid as long asβ 6= 0. The CC stage has been implemented in the Fourier domain

as a CM operation. At each step of the process, the sampling rate has been chosen as the

minimum compatible with the shear-induced increases in thetime and frequency extents.

Considering T1, the minimum oversampling ratek is 5, 2, 8 for Methods I, II, and III, re-

spectively. With the more demanding T2, the minimum value ofk is 14, 7, 24, respectively.

Larger values ofk and thus larger total numbers of samples not only result in representa-

tional redundancy, they translate into greater computational time. For instance, for F4 and

T2, the time of computation in seconds is0.0910, 0.0396, 0.347 for Methods I, II, and III,

respectively (obtained with MATLAB running on a personal computer), demonstrating that

Method III is significantly slower. The corresponding percentage errors are comparable as

expected:0.11, 0.099, 0.10, respectively.

3.5 Concluding Remarks

In this chapter, two algorithms for the computation of linear canonical transforms (LCTs)

from theN samples of the input signal inO(N log N) time are discussed. Our approach is

based on concepts from signal analysis and processing rather than conventional numerical

analysis. With careful consideration of sampling issues,N can be chosen very close to the

3.5. CONCLUDING REMARKS 53

time-bandwidth product of the signals, and need not be much larger. The transform output

may have a higher time-bandwidth product due to the nature ofthe transform family. This is

accomplished by doing a careful study of the deformations ofthe time-bandwidth products

of the signals along the steps of the algorithm. As can be seenin Fig. 3.1, the effects of

the every step of a decomposition that is used in the algorithm is carefully studied and

the time and frequency extents of the signals and the resulting time-bandwidth products

are calculated. From these calculation, one can judge the minimum required number of

samples that should be taken to represent the signal in an information theoretically proper

way. As a result, the number of samplesN is kept at the minimum possible values to match

the resulting time-bandwidth product of the output signal.

Both algorithms relate the samples of the input function to the samples of the continuous

LCT of this function in the same sense that the fast Fourier transform (FFT) implementation

of the discrete Fourier transform (DFT) computes the samples of the continuous FT of a

function. Since the sampling rates are carefully controlled, the output samples obtained

are accurate approximations to the true ones and the continuous LCT can be recovered

via interpolation of these samples. The only inevitable source of deviation from exactness

arises from the fundamental fact that a signal and its transform cannot both be of finite

extent. This is the same source of deviation encountered when using the DFT/FFT to

compute the continuous FT. Thus the algorithms compute LCTswith a performance similar

to the DFT/FFT in computing the Fourier transform, both in terms of speed and accuracy.

The fact that the two methods, although being arrived at fromconsiderably different

starting points, both exhibit similar limits in performance, strongly suggests that the per-

formance achieved is close to the best achievable. Indeed, as already noted, it is difficult

to expect an accuracy which is better than that of the DFT in approximating the FT, and

a cost which is less thanO(N log N) with N being close to the time-bandwidth product.

Despite the different decompositions employed, an interesting structural similarity emerges

between the two methods in their optimized forms. Both methods have two branches, the


first one determined by whether|γ| > 1 or not, and the second one determined by whether

|a| < 0.5 or not. If a is expressed in terms of the LCT parametersα, β, γ, we see that

the two conditions are the same. Therefore, the separation of the LCT parameter space

into two regions is most likely not a characteristic of the algorithm chosen, but an intrinsic

structural property of the LCT parameter space.

Compared to earlier approaches, these algorithms not only handle a much more general

family of integrals, but also effectively address certain difficulties, limitations, or tradeoffs

that arise in other approaches to computing the Fresnel integral, which is of importance in

the theory of diffraction (see Ref. [61] for a systematic comparison of several approaches).

These algorithms can also be used for efficient realization of filtering in linear canonical

transform domains [59].

Chapter 4

The Algorithm for 2D Quadratic-Phase

Systems

4.1 Introduction1

THIS CHAPTER PRESENTSthe fast algorithms for the two-dimensional QPSs, or

2D-LCTs. Given an algorithm for efficiently computing 1D-LCTs [60, 63, 64], the

efficient computation of separable 2D transforms is straightforward because the kernel can

be separated and the 2D transform can be reduced to two successive 1D-LCTs. Much

work has been done on 1D and 2D separable LCTs in terms of sampling issues and fast

algorithms for their digital computation, [65–68]. On the other hand, in the non-separable

case, the two dimensions are coupled. Handling this case requires special attention and

to the best of our knowledge has not been addressed before. The current established LCT

computation algorithms presented in Chapter 3 are not able to compute 2D-NS-LCTs.

An alternative representation of LCTs is presented and studied in [95]. This decom-

position is based on the well-known Iwasawa decomposition [113]. In [95], the authors

1This chapter is taken from [49]. Copyright 2010 by OSA.

55


further decompose the first matrix of the Iwasawa decomposition into a 2D separable frac-

tional Fourier transform that is sandwiched between two coordinate rotators. We had earlier

employed a 1D Iwasawa decomposition to develop a fast and efficient algorithm for 1D-

LCTs [63, 64]. In this chapter, we use the 2D version of this Iwasawa-type decomposition

to derive our efficient algorithm. As in the 1D case, the distinguishing feature of our ap-

proach is the way our algorithm carefully addresses sampling and space-bandwidth product

issues from an information-theoretical perspective. Special care is taken to ensure that the

output samples represent the continuous transform in the Nyquist-Shannon sense during

every stage of the algorithm, so that the continuous transform can be fully recovered from

the samples.

To our knowledge, there is no algorithm in the literature that efficiently calculates 2D-

NS-LCTs. Furthermore, despite the highly oscillatory nature of the integral kernel, the

sampling rate is managed so as to ensure that the number of samples used is sufficient, but

not much larger than the space-bandwidth product of the input signal, so that the algorithms

are as efficient as possible. Straightforward sampled numerical integral computation takes

O(N2) time whereN = MN for a 2D signal sampled on aM × N grid. In contrast,

the complexity of our algorithm isO(N log N). This efficiency is even more crucial in

the 2D case than in the 1D case since the number of points is much larger. By choosing

the number of samplesN equal to the 2D space-bandwidth product of the signals, we

ensure that the efficiency is near the best that is theoretically possible. More generally,

through each stage of the algorithm, we carefully manage thesampling rate to maintain the

information theoretically sufficient, but not wastefully redundant, sampling required for

reconstruction of the underlying continuous functions at any stage of the algorithm.

In Chapter 2, the definition of 2D-NS-LCTs is given. An explicit-kernel definition

with the least possible number of independent variables is provided and the forward and

backward relations between the parameters of this definition and the parameters of conven-

tional ABCD-matrices are derived. Section 5.2 provides the preliminary mathematical


background and the tools that we use in the algorithm. In Section 4.3, our algorithm is

presented. Section 4.4 addresses the issue of sampling rateand space-bandwidth product

control in order to ensure the necessary sampling rates sufficient for proper reconstruction

in the Nyquist-Shannon sense at each step of the algorithm. Next, numerical results are

reported in Section 4.5. We conclude in Section 5.5.

4.2 Preliminaries

4.2.1 A 3-Sphere for Space-Bandwidth Control, Wigner Distribution

and Dimensional Normalization for 2D Functions

When we study one-dimensional input functions and one-dimensional LCTs, the corre-

sponding Wigner distribution is two-dimensional. One dimension represents the space

extent and the other represents the spatial frequency extent of the signal. However, for two-

dimensional signals, there exist two space extents and two corresponding spatial frequency

extents resulting in a four-dimensional Wigner distribution. The definition of Wigner distri-

bution and its effects on linear canonical transforms for one-dimensional signals were given

in 2.1.5. The corresponding Wigner distributionWf(ux, uy, µx, µy) of a two-dimensional

signalf(ux, uy) can be defined as follows [1]:

Wf (ux, uy, µx, µy) =

∫ ∞

−∞

∫ ∞

−∞

f(ux + u′x/2, uy + u′

y/2)f ∗(ux − u′x/2, uy − u′

y/2)

×e−2πi(µxu′

x+µyu′

y) du′xdu′

y. (4.1)

We call this four-dimensional Wigner distribution the “4D Wigner distribution” whereas

the usual two-dimensional Wigner distribution used for one-dimensional functions will

be referred to as the “2D Wigner distribution”. Its integralover time and frequency,∫∞

−∞

∫∞

−∞W (u, µ) du dµ, gives the signal energy. Letf denote a function andfM be its


2D-LCT with parameter matrixM. Then, the relation between the Wigner distribution

(WD) of fM and the WD off can be expressed as [1]:

WfM(Ms) = Wf(s), (4.2)

where the vectors = [ux uy µx µy]T is used for the sake of notational simplicity. An

example of the use of the Wigner distribution in sampling issues from another perspective

can be found in [114].

In Chapter 3, we used 2D Wigner distributions for tracking and control of the space-

bandwidth products of signals through the stages of our algorithms. 2D Wigner distribu-

tions are easy to visualize and therefore easy to understand. However, for two-dimensional

signals we cannot graphically show the Wigner distributionbecause it is a four-dimensional

function. Therefore we will develop and use a more abstract and rigorous approach to

space-bandwidth tracking and control in order to achieve the information-theoretic mini-

mum sampling rate for lossless reconstruction of the continuous output function from the

output samples.

First, we need to recall the geometrical object known as a “3-sphere”. In general, for a

natural numbern, an “n-sphere” is the generalization of the ordinary “2-sphere” in common

three-dimensional Euclidian space to any dimension. Explicitly, an n-sphere, denoted as

Sn and centered at the origin, is the analogue of a sphere in(n + 1)-dimensional Euclidian

space and is defined as:

Sn = {x ∈ Rn+1 : ‖x‖ = r} (4.3)

where the positive real numberr is the radius of the n-sphere andRn+1 is an (n + 1)-

dimensional vector space overR. More onn-spheres can be found in [115,116]. Then, we

can provide the generic definition of the 3-sphere,S3, centered at the origin explicitly as:

S3 = {(x1, x2, x3, x4) ∈ R4 : x2

1 + x22 + x2

3 + x24 = r2}. (4.4)


Now, let us turn our attention to our two-dimensional input functions. It is well known

that a non-zero function and its Fourier transform (FT) cannot both be confined to finite

regions. However, in practice, we always work with samples of finite extent functions

by assuming that the energy of the signal falling outside of some region is negligible.

In general, the signal will exhibit some distribution of energy in the two-dimensional

space-frequency hypervolume (which is four dimensional).We will assume that a finite

hyperellipsoidal boundary inR4 is chosen so as to confine most of the energy of the

signal. This hyperellipsoidal boundary will imply finite extents in the two space dimen-

sions and the two spatial-frequency dimensions. The intervals of confinement thus defined

will be denoted by[−∆Sx/2, ∆Sx/2] and[−∆Sy/2, ∆Sy/2] in the space dimensions, and

[−∆Bx/2, ∆Bx/2] and[−∆By/2, ∆By/2] in the spatial-frequency dimensions. The space

and spatial-frequency representations of the signal will be approximately confined within

these intervals. Given these, it also follows that both space-domain extents are confined

within the worst case interval[−∆Smax/2, ∆Smax/2], where∆Smax = max{∆Sx, ∆Sy},

and both frequency-domain extents are confined within the interval[−∆Bmax/2, ∆Bmax/2],

where∆Bmax = max{∆Bx, ∆By}. Under these conditions, the Wigner distribution of the

function is confined within the boundaryO in R4 (note that this is not defining a 3-Sphere

yet):

O = {(sx, sy, bx, by) ∈ R4 :

s2x

(∆Smax/2)2+

b2x

(∆Bmax/2)2

+s2

y

(∆Smax/2)2+

b2y

(∆Bmax/2)2= 1} (4.5)

wheresx andsy are temporary space variables andbx andby are temporary spatial-frequency

variables of the Wigner distribution of the signal.

Let us now introduce the scaling parameterP and scaled dimensionless coordinates

ux = sx/P,


uy = sy/P,

µx = bxP,

µy = byP, (4.6)

such that the two space- and two frequency-domain representations are confined to in-

tervals of length∆Smax/P and ∆BmaxP , respectively. LetP =√

∆Smax/∆Bmax

so that the lengths of all the four intervals become equal to the dimensionless quantity√

∆Smax∆Bmax , which we denote by∆u. Expressed in dimensionless coordinates, the

boundaryO defined in Eq. 4.5 reduces to the desired 3-sphere, denoted byOsp:

Osp = {(ux, uy, µx, µy) ∈ R4 : u2

x + u2y + µ2

x + µ2y =

(√∆Smax∆Bmax

2

)2

=

(∆u

2

)2

}.(4.7)

To summarize, after the dimensional normalization procedure given above has been per-

formed, the 4D Wigner distribution of our two-dimensional input function can be assumed

to be confined within a 3-sphereOsp of diameter∆u.

4.3 The Algorithm

As noted before, one of the most important features of our method is to control the sampling

rate of the function with the goal of having enough samples tobe able to reconstruct the

continuous function without information loss, and at the same time without needlessly

increasing the number of samples to maintain efficiency. In this Section, we present our

algorithm, discuss the stages in the decomposition and derive the parameters of each stage

from the parameters of the 2D-NS-LCT that is being computed.The effects of each stage of

the decomposition on the Wigner distribution of our function (thus on the space-bandwidth

products) and associated sampling rate issues will be addressed in Section 4.4.

4.3. THE ALGORITHM 61

The Iwasawa decomposition is the core of our algorithm. After the dimensional nor-

malization explained in Section 5.2.4.2.1, any transformation matrixM can be written in

the following Iwasawa form [95,113]:

M =

A B

C D

=

I 0

−G I

S 0

0 S−1

X Y

−Y X

(4.8)

where

G = −(CAT + DBT)(AAT + BBT)−1 (4.9)

S = (AAT + BBT)1/2 (4.10)

X = (AAT + BBT)−1/2A (4.11)

Y = (AAT + BBT)−1/2B (4.12)

Given the4 × 4 matrix M, we can determine2 × 2 matricesG, S, X, Y by using Eqs.

4.9, 4.10, 4.11, 4.12. If we are able to develop a fast algorithm to compute the three stages

in O(N log N) time, the overall transform can also be calculated inO(N log N) time. In

this decomposition, the first operation is an orthosymplectic system, followed by a scaling

(magnification) system, finally followed by a two-dimensional chirp multiplication (2D-

CM). (Note that each of the stages of the algorithm are special cases of 2D-NS-LCTs.)

We begin with the first and the most sophisticated stage of thedecomposition, the or-

thosymplectic system. This stage of the decomposition can be further decomposed into

a two-dimensionalseparable fractional Fourier transform (2D-S-FRT) that is sandwiched

between two coordinate rotators [95]:

X Y

−Y X

= Rr2Fax,ay

Rr1 (4.13)


where the4 × 4 matricesRr1, Fax,ay, Rr1 are defined as:

Rr1 =

cos(r1) sin(r1) 0 0

− sin(r1) cos(r1) 0 0

0 0 cos(r1) sin(r1)

0 0 − sin(r1) cos(r1)

(4.14)

Rr2 =

cos(r2) sin(r2) 0 0

− sin(r2) cos(r2) 0 0

0 0 cos(r2) sin(r2)

0 0 − sin(r2) cos(r2)

(4.15)

Fax,ay=

cos(axπ/2) 0 sin(axπ/2) 0

0 cos(ayπ/2) 0 sin(ayπ/2)

− sin(axπ/2) 0 cos(axπ/2) 0

0 − sin(ayπ/2) 0 cos(ayπ/2)

(4.16)

Rr1 andRr2 are rotation matrices that impose rotations of anglesr1 andr2, respectively,

through the spatial variables(ux, uy) and through their frequency variables(µx, µy). Un-

like these traditional rotators which rotate within space and spatial frequency separately, the

fractional Fourier transform rotates within the space-frequency planes of each dimension.

Fax,aystands for 2D-S-FRT that makes separable rotations of angleaxπ/2 over the vari-

ables(ux, µx) and of angleayπ/2 over the variables(uy, µy). Since this two-dimensional

FRT operation is separable, it corresponds to two one-dimensional fractional Fourier trans-

formation operations performed over each of the dimensions. Explicitly, this means first

performing 1D-FRTs with the fractional orderax for each of the rows (or columns) and

then performing 1D-FRTs with the fractional orderay for each of the columns (or rows)

of the sampling grid. It is this observation that enables us to implement this stage of the

decomposition efficiently inO(N log N) time. There are fast and established algorithms to


compute one-dimensional fractional Fourier transforms [61–64], so that this stage can be

calculated inO(N log N) time easily.

The interpretation of the coordinate rotators requires care. When we are working with

sampled functions, we know the value and coordinates (the location where the particular

sample is taken) of all the samples we have. A coordinate rotation can be interpreted in

this situation as a rotation of the locations of the samples resulting in a new sampling

grid, rather than a change in the sample values. If we assume we start with a regular

rectangular grid, after the coordinate rotation, the grid would no longer coincide with the

original grid unless the rotation is an integer multiple ofπ/2. Unfortunately, in order to

perform FRT operations along the horizontal and vertical directions, we need the samples

to be on a regular rectangular grid in order to employ available fast algorithms. Therefore,

we must carry out an interpolation operation to determine the values of the function on

a regular rectangular grid. There are several techniques and algorithms to perform this

interpolation efficiently. We have chosen to use in our numerical simulations fast and

standard implementations of nearest neighbor, bilinear and cubic interpolations, [117,118],

but any other efficient method may also be used. This interpolation step and its performance

can be a major source of error in our algorithm, as we will further discuss later.

We now turn our attention to determining the coordinate rotation anglesr1 andr2, and

the FRT fractional ordersax anday. When we plug Eqs. 4.14, 4.15, 4.16 in Eq. 4.13, carry

out the matrix multiplications and equate the entries of both sides of Eq. 4.13, we get the

following equations in the four unknownsr1, r2, ax anday:

X11 = cos r1 cos r2 cos(axπ/2) − sin r1 sin r2 cos(ayπ/2)

X12 = sin r1 cos r2 cos(axπ/2) + cos r1 sin r2 cos(ayπ/2)

X21 = − cos r1 sin r2 cos(axπ/2) − sin r1 cos r2 cos(ayπ/2)

X22 = − sin r1 sin r2 cos(axπ/2) + cos r1 cos r2 cos(ayπ/2)


Y11 = cos r1 cos r2 sin(axπ/2) − sin r1 sin r2 sin(ayπ/2)

Y12 = sin r1 cos r2 sin(axπ/2) + cos r1 sin r2 sin(ayπ/2)

Y21 = − cos r1 sin r2 sin(axπ/2) − sin r1 cos r2 sin(ayπ/2)

Y22 = − sin r1 sin r2 sin(axπ/2) + cos r1 cos r2 sin(ayπ/2) (4.17)

These equations are sufficient to solve for and unambiguously determine the rotation and

fractional Fourier angles of the decomposition in a straightforward manner, provided one

pays proper attention to sign considerations when inverting the trigonometric functions.

To summarize, the first stage of our algorithm involves determining the angles from the

above equations, performing the first coordinate rotation,following this by two 1D-FRTs

over each of the dimensions, and then finishing with the second coordinate rotation. All

these steps can be calculated inO(N log N) time.

The second stage is the scaling operation and it seems to be the simplest of the three

stages. It is not, however, as trivial as in the one-dimensional case [64]. In one dimension,

it corresponds to only a reinterpretation of the spacing between the samples. The sampling

interval scales with the scaling parameter. Intuitively, it squeezes in or stretches out the

total number of samples as the word scaling implies. This means there is no change in the

total number of samples and thus no need to oversample the input samples. The analogue of

the one-dimensional scalar scaling parameter in the two-dimensional case is the matrixS.

WhenS is diagonal, which means there is no coupling between the twodimensions of the

function for scaling purposes, the scaling is separable. Due to this separability, this situation

does not impose an increase in the space-bandwidth productsand thus does not require

oversampling, just as in the one-dimensional case. But whenthe off-diagonal elements of

S are non-zero, the scaling operation is no longer so trivial.Although the total number

of degrees of freedom of the signal remains the same, the space-bandwidth products may

increase and an oversampling to match this increase may be necessary. Readers wishing to

better understand how the space-bandwidth product may increase despite the fact that the


number of degrees of freedom remains the same are referred to[64], where these issues

are studied graphically for the 1D case. An analogous, though not visually demonstrable,

situation exists for 2D signals. The sampling rate control mechanism for such 2D scaling

operations will be developed in detail in Section 4.4. At this point, we note that in those

cases where the number of samples needs to be increased, the oversampling should be

performed first, prior to scaling. Afterwards, scaling is achieved by mere reinterpretation of

the locations of the samples without changing the samples themselves (other than a constant

multiplicative factor). Computationally, such a scaling operation amounts to modifying the

information that tells us which coordinates the samples belong to. Since it requires only

the reinterpretation of the coordinates of the samples plusa possible oversampling, it does

not impose much computational load. Eq. 4.10 gives us the scaling parameters. The matrix

S can be easily used to determine the output samples by using the input-output relation of

the scaling operation:

fsc(u) =1√

detSf(S−1u) (4.18)

wheref is the function to be scaled andfsc is the scaled function, andu = [ux uy]T.

The last stage of our main Iwasawa decomposition is the 2D-CMoperation whose pa-

rameters are given by the matrixG as defined in Eq. 4.9. The input-output relation of this

2D-CM is given as:

fch(u) = e−iπ(G11x2+(G12+G21)xy+G22y2)f(u) (4.19)

wherefch stands for the chirp-multiplied function. The 2D-CM operation is the stage that is

mainly burdened with any shears inherent in the 2D-NS-LCT tobe computed. Such shears

may considerably increase the space-bandwidth products ofthe function. Thus, before

the 2D-CM operation, the space-bandwidth products of the function should be calculated

carefully and any necessary oversampling should be performed. This CM operation may

turn out to be non-separable or separable for particular 2D-NS-LCTs but regardless, it


requires only one multiplication for each sample, resulting in O(N) time computation.

As a result, we see that we can perform all of the stages of the decomposition in

O(N log N) time or faster, which makes the computational complexity (cost) of our overall

algorithmO(N log N).

4.4 Space-Bandwidth and Sampling Rate Control

In this section, we develop a method to track the space-bandwidth products of our functions

as we perform the consecutive operations in our decomposition and focus on how to control

the number of samples efficiently. We need to calculate the necessary sampling intervals

and sampling rates for both dimensions that are necessary torepresent the continuous signal

without information loss for each of the stages. Oversampling should be undertaken prior

to any stage that increases either of the space-bandwidth products.

As given in Section 5.2.4.2.1, the Wigner distribution of the input signal is assumed to

be confined within a 3-sphere with radius∆u/2, which also means that the signal is as-

sumed to be almost space- and band-limited in both dimensions. The 4D Wigner represen-

tation gives us two space extents, two spatial-frequency extents and two space-bandwidth

products, one for each dimension of the function. Let us denote the space-bandwidth prod-

uct along theux direction byNx and that along theuy direction byNy. These extents define

the minimum required number of samples along the corresponding direction, with the to-

tal number of samples beingNx × Ny. Since the Wigner distribution is confined within a

3-sphere of diameter∆u, all the extents of the function (space and spatial-frequency) are

equal to∆u at the beginning. Thus, the function should be sampled on aNx × Ny grid,

where theux-coordinate of the function spans the intervalux = (−∆u/2, ∆u/2) and the

uy-coordinate spans the intervaluy = (−∆u/2, ∆u/2). The distance between two adja-

cent samples is equal to∆u−1 along both dimensions. As a result, the space-bandwidth

products are initiallyNx = Ny = ∆u2.

4.4. SPACE-BANDWIDTH AND SAMPLING RATE CONTROL 67

We will track the effects of each stage in our algorithm to theWigner distribution

boundary to which the original function is confined, and calculate the extents and the two

space-bandwidth products and eventually the required structure of the sampling grid before

each stage is performed. We will address each of the three stages (the first with 3 steps)

given in Section 4.3 in sequence. To start, we first write downthe 3-sphere boundary in

hyper-spherical coordinates. The 3-sphereOsp given in Eq. 4.7 can be transformed to the

equivalent hyper-spherical coordinates with the following coordinate transformation:

Osp =

ux

uy

µx

µy

=∆u

2

cos φ1

sin φ1 cos φ2

sin φ1 sin φ2 cos φ3

sin φ1 sin φ2 sin φ3

(4.20)

where angular hyperspherical coordinatesφ1 andφ2 range over[0, π], and angular hyper-

spherical coordinateφ3 ranges over[0, 2π]. (Note that this coordinate system transforma-

tion is not unique.) The sum of the squares of the elements of the vector on the right-hand

side of Eq. 4.20 again equals(∆u/2)2 as expected. Eq. 4.2 allows us to calculate the new

boundarysout of the Wigner distribution after any operation from the boundarysin before

the operation. Just as the old boundary confined most of the energy of the signal repre-

sented by the Wigner distribution, so does the new boundary.This is because the mapping

in Eq. 4.2 merely maps values of the Wigner distribution to new space-frequency points,

and values which were confined within the old boundary remainconfined within the new

boundary.

4.4.1 The First Coordinate Rotator

At the very beginning of the algorithm, we start with the boundary vectorsin = Osp.

In other words, the input boundary vectorsin before the first coordinate rotator is given


by Eq. 4.20. Thensout is found by multiplyingsin with the transformation matrix of the

coordinate rotator as

sout = Rr1sin

=∆u

2

cos(r1) sin(r1) 0 0

− sin(r1) cos(r1) 0 0

0 0 cos(r1) sin(r1)

0 0 − sin(r1) cos(r1)

cos φ1

sin φ1 cos φ2



=∆u

2

cos(r1) cos φ1 + sin(r1) sin φ1 cos φ2

− sin(r1) cos φ1 + cos(r1) sin φ1 cos φ2

cos(r1) sin φ1 sin φ2 cos φ3 + sin(r1) sin φ1 sin φ2 sin φ3

− sin(r1) sinφ1 sin φ2 cos φ3 + cos(r1) sin φ1 sin φ2 sin φ3

(4.21)

With φ1 andφ2 ranging over[0, π], andφ3 ranging over[0, 2π], sout represents the boundary

of the output Wigner distribution. We can show that this boundary remains a 3-sphere of

radius∆u2

by writing:

u2x + u2

y + µ2x + µ2

y = (∆u

2)2[(cos(r1) cos φ1 + sin(r1) sin φ1 cos φ2)

2

+(− sin(r1) cos φ1 + cos(r1) sin φ1 cos φ2)2

+(cos(r1) sin φ1 sin φ2 cos φ3 + sin(r1) sin φ1 sin φ2 sin φ3)2

+(− sin(r1) sin φ1 sin φ2 cos φ3 + cos(r1) sin φ1 sin φ2 sin φ3)2]

= (∆u

2)2 (4.22)

as can be verified after some algebra with trigonometric functions. This result means that

the coordinate rotation operation does not change the 3-sphere nature of the confining

boundary of the Wigner distribution, and since rotating ann-sphere (just like an ordi-

nary sphere) does not change its extent along any direction,does not have any effect on


the space-bandwidth products. Therefore, no matter what the angles are, the coordinate

rotation operation does not require a change in the number ofsamples and the sampling

grid.

4.4.2 2D Separable Fractional Fourier Transform

Since the previous rotation operation left the Wigner distribution confined within the orig-

inal 3-sphere, and since we are interested only in the worst-case boundary,sin (sout of the

preceding operation) can still be expressed as in Eq. 4.20. Then, newsout is found as:

sout = Fax,aysin

=∆u

2

cos(axπ/2) 0 sin(axπ/2) 0

0 cos(ayπ/2) 0 sin(ayπ/2)

− sin(axπ/2) 0 cos(axπ/2) 0

0 − sin(ayπ/2) 0 cos(ayπ/2)

×

cos φ1

sin φ1 cos φ2



=∆u

2

cos(axπ/2) cos φ1 + sin(axπ/2) sinφ1 sin φ2 cos φ3

cos(ayπ/2) sinφ1 cos φ2 + sin(ayπ/2) sinφ1 sin φ2 sin φ3

− sin(axπ/2) cosφ1 + cos(axπ/2) sinφ1 sin φ2 cos φ3

− sin(ayπ/2) sinφ1 cos φ2 + cos(ayπ/2) sinφ1 sin φ2 sin φ3

(4.23)

As in the coordinate rotator step,sout again defines the boundary of the output Wigner

distribution. Once again it defines a 3-sphere sinceu2x + u2

y + µ2x + µ2

y = (∆u/2)2. This

too can be easily shown by using simple algebra and trigonometric function properties.

This is an expected result since FRT corresponds to rotationin joint space-frequency; if


the original confinement region is ann-sphere, it remains ann-sphere after a 2D-S-FRT.

Therefore, we again need not change the number of samples andsampling grid before the

FRT step.

4.4.3 The Second Coordinate Rotator

Similar considerations as with the first coordinate rotation apply so that an increase in the

number of samples or a change in the sampling grid is not needed.

4.4.4 2D Scaling Operation

Up to the scaling stage, we do not have to worry at all about thesampling rate. The three

steps which constitute the first stage have the effect of rotating the original 3-sphere, and

the extent of the 4D Wigner distribution remains unchanged in all directions. We are able

to track the confinement boundary through the steps precisely since we are able to write

down the entire boundary parametrically by using hyper-spherical coordinates and since

after each step, the transformed points still form a 3-sphere. In fact, the Wigner distribution

of the signal is confined within the same 3-sphere as at the beginning. However, the scaling

operation does not preserve the 3-sphere and thus it is very difficult to track all the points

on the boundary since they may not constitute an easily trackable geometrical object by

analytical and parametric means. Therefore, instead of tracking the infinite number of

boundary points of our 3-sphere, we will use a tesseract (a 4-cube), which is basically the

counterpart of an ordinary cube inR4, just as the 3-sphere is the counterpart of the ordinary

sphere [115]. The unit tesseract is defined as

{(x1, x2, x3, x4) ∈ R4 : −1 ≤ xi ≤ 1}. (4.24)


It has 16 vertices and we will use these 16 points to track the Wigner distribution after

the scaling operation. We take the smallest tesseract that contains the 3-Sphere within it-

self and use its 16 vertices to find the 16 vertices of the output. These 16 vertices define

the maximum extents of the distribution and by employing them, we can safely define the

worst-case boundary confining the Wigner distribution after the operation. Then the two

space-bandwidth products can be calculated by finding, separately for each of the four

coordinates, the maximum distances between the corresponding coordinates of the 16 ver-

tices. Readers wishing to find a simpler example of such a streamlined procedure in a

one-dimensional setting can refer to [61].

Let us represent, inR4, the coordinates of the 16 vertices of the tesseract of edge length

∆u (which is the smallest one confining the 3-sphere with diameter ∆u) with columns of

the matrixV:

V =∆u

2

1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1

1 1 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1 −1 −1

1 1 −1 −1 1 1 −1 −1 1 1 −1 −1 1 1 −1 −1

1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1

.

(4.25)

After the scaling operation is performed, these coordinates of the 16 input vertices are

mapped to 16 new vertices, which we will hold in the columns ofV as follows:

V = [v1 v2 v3 ... v15 v16] =

S 0

0 S−1

V (4.26)

wherevi (i = 1, 2, ..., 16) are vectors inR4 that hold the coordinates of the scaled vertices.

Then, we need to find the coordinate-wise distances for everypossible combination of pairs

of vertices, for each of the four coordinates separately. There are 120 possible combinations

of pairs out of 16 vectors. We calculate the distances between their coordinates and denote


this withdi,j as

di,j =

|vi(1) − vj(1)||vi(2) − vj(2)||vi(3) − vj(3)||vi(4) − vj(4)|

(4.27)

and then construct the4 × 120 distance matrixD

D = [d1,2 d1,3 ... d1,16 d2,3 d2,4 .... d2,16 .... d15,16] (4.28)

wherei = 1, 2, ..., 15 andj = i + 1, ...., 16. By usingD, we can define the sampling grid

and sampling rates that are necessary to represent the function without any information

loss. The extent of the function alongux anduy should bemax(D1,1, D1,2, D1,3, ..., D1,120)

andmax(D2,1, D2,2, D2,3, ..., D2,120), respectively. On the intervals alongux anduy given

above, the samples should be taken with intersample spacings of (max(D3,1, D3,2, D3,3,

..., D3,120))−1 and (max(D4,1, D4,2, D4,3, ..., D4,120))

−1, respectively. The corresponding

space-bandwidth products are then equal to

NSx = max(D1,1, D1,2, D1,3, ..., D1,120) × max(D3,1, D3,2, D3,3, ..., D3,120)(4.29)

NSy = max(D2,1, D2,2, D2,3, ..., D2,120) × max(D4,1, D4,2, D4,3, ..., D4,120)(4.30)

and the total necessary number of samples after the scaling is given byN = NSxNSy.

Remember that the number of samples should be increased to this numberN = NSxNSy

before the scaling operation is performed (the minimum appropriate integer number of

samples greater than the calculated values may be used for simplicity). The determined

number of samples should be uniformly spread so as to snugly fit the original extents (thus

they will be spaced closer than the original samples). Afterthe scaling operation is per-

formed by using the matrixS, these samples are transformed to new and extended positions


as predicted by the above calculations. Finally, though nota necessity, a similar simple in-

terpolation may be employed (as done after the coordinate rotations) to carry the samples

from these transformed locations to the regular grid withinthe predicted extents. This may

facilitate implementation of the next operation.

4.4.5 2D Chirp Multiplication

The 2D-CM operation is the stage that is mainly burdened withany shears which may

be inherent in the 2D-NS-LCT to be computed. Such shears may considerably increase

the space-bandwidth products of the function. These increases are unavoidable if these

elongating distortions in space-frequency are part of the 2D-NS-LCT which we wish to

compute. This will in turn require an increase in the number of samples if we wish to be

able to reconstruct the continuous output function withoutany information loss. Therefore,

as in the previous subsection, we must increase the number ofsamples prior to the chirp

multiplication operation. The vertices obtained as a result of the scaling operation are

taken as the starting vertices for the 2D-CM operation. We begin with the coordinates of

these vertices, denoted byV, and determine what happens to them as a result of the 2D-CM

operation, and calculate the new difference matrixD by using the following equation along

with Eqs. 4.27, 4.28, 4.29, and 4.30:

V = [v1 v2 v3 ... v15 v16] =

I 0

−G I

V. (4.31)

Finally, the sampling extents, rates, and locations can be determined similarly as in the

scaling stage. After the number of samples has been increased, the 2D-CM stage can be

safely performed to complete the entire transformation.


4.4.6 Summary of the Algorithm

Having explained all the stages in detail, we summarize the entire algorithm stage by stage.

The algorithm can be compactly stated in operator notation as follows:

CM = QG KG MS KS Rr2 Fax,ay J Rr1 (4.32)

where the operatorsQG, MS, Rr2 , Fax,ay , Rr1 respectively represent: 2D-CM with pa-

rameter matrixG, 2D scaling with parameter matrixS, coordinate rotation with angler2,

2D-S-FRT with ordersax anday, and coordinate rotation with angler1. J stands for a

simple interpolation without oversampling that is performed to obtain the function on a

regular rectangular grid from the rotated samples.KS andKG stand for the interpolation

operations before the scaling and chirp multiplication operations, respectively. Beyond the

task ofJ , these also increase the number of samples as explained in Section 4.4.4.4.4 and

Section 4.4.4.4.5.

The algorithm can be summarized as follows:

1. Normalize the input field (function) as explained in Section 5.2.4.2.1 and obtain the

input samples.

2. Given the transform matrixM, obtain the chirp multiplication (G) and scaling (S)

matrices, the coordinate rotation angles (r1 andr2) and FRT orders (ax anday) by

using Eqs. 4.8, 4.9, 4.10, 4.11, 4.12, 4.13, 4.14, 4.15, 4.16, and 4.17.

3. Perform the first coordinate rotation and obtain the samples on a regular grid by

simple interpolation.

4. Use the fast algorithm for 1D-FRTs to implement the 2D-S-FRT by successively

applying 1D-FRTs along the two dimensions.

4.5. NUMERICAL RESULTS 75

5. Perform the second coordinate rotation and obtain the samples on a regular grid by

simple interpolation.

6. Use the method given in Section 4.4.4.4.4 to obtain the necessary number of samples

before the 2D scaling operation, perform the oversampling and then apply the scaling

operation. Optionally, go back to a regular rectangular grid by simple interpolation

after scaling has changed the locations of the samples to a non-rectangular grid.

7. Use the method given in Section 4.4.4.4.5 to obtain the necessary number of samples

before the 2D chirp multiplication operation, perform the oversampling and then

apply the chirp multiplication operation.

4.5 Numerical Results

Here we report numerical results for some example functionsand transforms in order to

demonstrate the performance and accuracy of our algorithm.We also discuss sources of

error in our algorithm and the effect of interpolation methods on the error. As example input

functions, we consider the 2D Gaussian fieldexp(−π(x2 + y2)) and denote it with F1, the

2D Chirped-Gaussian fieldexp(−π(x2 + y2))× exp(−iπ(x2 + y2)) and denote it with F2,

a 2D Non-Symmetric Chirped Gaussian fieldexp(−π(3x2 + y2)) × exp(−iπ(x2 + 2y2))

and denote it with F3. All these first three input fields are sampled on a64 × 64 grid.

Additionally, we also consider a more challenging functionexhibiting discontinuities and

larger frequency extents depicted in Fig. 4.1. This S-shaped function is denoted with F4,

and is sampled on a256 × 256 grid. We consider two different arbitrarily chosen 2D-

NS-LCTs, the first one (T1) has a parameter set(αx, βx, γx, αy, βy, γy, ηx, ηy, ηα, ηγ) =

(−3,−2,−1, 2, 3, 4, 0.1, 0.2, 1,−0.1) and the second one (T2) has a parameter set(1, 2, 3,

−2,−1,−0.8, 0.6,−0.5, 0.3,−0.4). As a result of the space-bandwidth and sampling rate

control procedure presented in Section 4.4 and for the givennumber of initial samples, the


output fields are obtained by the algorithm on141× 166 and740× 211 sampling grids for

T1 and T2 respectively, for the input functions F1, F2, and F3. For the input function F4,

the output grids are563 × 663 and2958 × 842 for T1 and T2, respectively. T1 is of such

a nature that it requires a relatively small amount of oversampling whereas T2 is of such a

nature that it requires a relatively large amount of oversampling. These oversamplings are

necessary to be able to recover the continuous output from the output samples produced by

the algorithm.

−8 −6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


The 2D-NS-LCTs (T1 and T2) of the functions F1, F2, F3, F4 havebeen calculated both

by the presented fast algorithm and by an extremely finely tuned and inefficient brute force

numerical approach based on the 2D Simpson’s method [119] which we use as an accurate

reference. See Appendix A for more details of the Simpson’s method. The results for


T1 T2 DFTF1 2.25 × 10−3 3.82 × 10−4 2.12 × 10−23

F2 1.12 × 10−2 1.09 × 10−3 2.02 × 10−21

F3 7.17 × 10−2 3.21 × 10−3 2.58 × 10−8

F4 2.07 1.92 1.05

Table 4.1: Percentage errors for different functions F and transforms T.

T1 of (F1, F2, F4) and T2 of (F3, F4) along with the corresponding brute force reference

results are plotted in Figures 4.2 - 4.6. The error percentages for all functions (F1, F2,

F3, F4) are tabulated in Table 4.1, for both transforms T1 andT2. There are no visible

differences for F1, F2, F3 and a very small visible difference for F4. We define the error as

the energy of the difference of the two results normalized bythe energy of the reference,

expressed as a percentage. The tabulated error percentagesshow that the presented fast

algorithm is very accurate. Another important observationfrom Table 4.1 is that the error

does not depend so much on the transform parameter set as it does on the transformed

function; the error percentages for T1 and T2 are close to each other. In general, our

algorithm maintains approximately the same performance over different transforms. A

similar conclusion was reached for the 1D case [63, 64]. To the best of our knowledge the

presented algorithm is the first fast and accurate algorithmthat is capable of computing the

very general class of 2D-NS-LCTs and the first generalization of the one-dimensional fast

algorithms for LCTs to two dimensions. Moreover, it also deals with the space-bandwidth

and sampling rate issues very carefully so that the output samples—indeed the samples at

any stage—are sufficient to accurately reconstruct the underlying continuous function, but

are not wastefully redundant either. Therefore our algorithm is able to effectively obtain a

continuous transform from a continuous input function.

In Table 4.1, we also show the errors that arise when the DFT isused to approximately

compute the ordinary 2D Fourier transform (2D-FT) of the same functions. (The DFT

would most likely be implemented with the FFT algorithm but how the DFT is imple-

mented does not effect the error comparison.) The same reference method that we use in


Re of T1 of F1 − Our Algorithm

x

y

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(a) Real part (fast algorithm)

Re of T1 of F1 − Reference

x

y

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(b) Real part (reference)

Im of T1 of F1 − Our Algorithm

x

y

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

−0.3

−0.2

−0.1

0

0.1

0.2

(c) Imaginary part (fast algorithm))

Im of T1 of F1 − Reference

x

y

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

−0.3

−0.2

−0.1

0

0.1

0.2

(d) Imaginary part (reference)

Figure 4.2: T1 of F1 (our algorithm and reference)



x

y

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8



x

y

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8



x

y

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.4

−0.3

−0.2

−0.1

0

0.1

(c) Imaginary part (fast algorithm)


x

y

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.4

−0.3

−0.2

−0.1

0

0.1





x

y

−1 −0.5 0 0.5 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−0.3

−0.2

−0.1

0

0.1

0.2

0.3



x

y

−1 −0.5 0 0.5 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−0.3

−0.2

−0.1

0

0.1

0.2

0.3



x

y

−1 −0.5 0 0.5 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

(c) Imaginary part (fast algorithm)


x

y

−1 −0.5 0 0.5 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−0.3

−0.2

−0.1

0

0.1

0.2

0.3




Abs of T1 of F4 − Our Algorithm

x

y

−5 0 5

−10

−5

0

5

10

0

0.5

1

1.5

(a) Magnitude (fast algorithm)

Abs of T1 of F4 − Reference

x

y

−5 0 5

−10

−5

0

5

10

0

0.5

1

1.5

(b) Magnitude (reference)

Phase of T1 of F4 − Our Algorithm

x

y

−5 0 5

−10

−5

0

5

10

−3

−2

−1

0

1

2

3

(c) Phase (fast algorithm)

Phase of T1 of F4 − Reference

x

y

−5 0 5

−10

−5

0

5

10

−3

−2

−1

0

1

2

3

(d) Phase (reference)



Abs of T2 of F4 − Our Algorithm

x

y

−10 −5 0 5 10−15

−10

−5

0

5

10

15

0

0.5

1

1.5

(a) Magnitude (fast algorithm)

Abs of T2 of F4 − Reference

x

y

−10 −5 0 5 10−15

−10

−5

0

5

10

15

0

0.5

1

1.5

(b) Magnitude (reference)

Phase of T2 of F4 − Our Algorithm

x

y

−10 −5 0 5 10−15

−10

−5

0

5

10

15

−3

−2

−1

0

1

2

3

(c) Phase (fast algorithm)

Phase of T2 of F4 − Reference

x

y

−10 −5 0 5 10−15

−10

−5

0

5

10

15

−3

−2

−1

0

1

2

3

(d) Phase (reference)



calculating the error percentages for our algorithm is usedto numerically calculate “exact”

continuous Fourier transforms of the example functions. The DFT serves as an ultimate

benchmark for comparing our results. Theoretically, our algorithm cannot reduce the error

below that value which results from computing a Fourier transform with the DFT because

they share the same inevitable source of error that arises from the fundamental fact that a

signal and its transform cannot both be of finite extent. In the one-dimensional version of

our algorithm, as well as the separable two-dimensional case, it is possible to achieve errors

which approach that for the DFT, and which are thus the best which one may ever hope to

obtain [63, 64]. Unfortunately, the necessity of interpolation in the two-dimensional case

does not allow this, but still it is possible to achieve very low errors that would be acceptable

in most applications.

The key observations that can be made from this table are as follows. The resulting er-

rors depend strongly on the function and the assumed space and spatial-frequency extents.

Indeed, this is the main determinant of the error for a given interpolation method. Different

functions have differing degrees of decay rates of their tails, and for given assumed extents,

different amounts of energy left out of the extents. Since a function cannot be made to

contain 100% of its energy in both the space and spatial-frequency domains, a compro-

mise between error and computational complexity is necessary. If we choose the extents

within which we assume the function and its Fourier transform to be mostly contained in

a conservative manner, the extents will be relatively largeand the number of samples will

be relatively large. If we economize on the extents and the number of samples, a relatively

large fraction of the energy will be left outside and the resulting error will be large. Among

our examples, F4 is an example where the space-bandwidth product has been chosen less

conservatively than the other examples, and therefore the error is relatively large around

2%. The error can be reduced by increasing the number of samples taken.

From a fundamental perspective, our algorithm is supposed to compute 2D-NS-LCTs

with a performance similar to the DFT in computing the Fourier transform. As noted, this


F1 F2 F3 F4Nearest 3.4 × 10−3 1.75 × 10−1 6.18 × 10−1 11.3Bilinear 1.02 × 10−2 5.04 × 10−2 2.66 × 10−1 4.33

Cubic 2.25 × 10−3 1.12 × 10−2 7.17 × 10−2 2.07

Table 4.2: Percentage errors for different interpolation methods and functions F for T1.

is achieved for separable transforms which reduce to one-dimensional transforms. How-

ever, in the general non-separable case, although our algorithm is quite accurate, for the

first three functions, its accuracy is quite below that for the DFT. This degradation is due

to the complex and challenging nature of non-separable LCTswhich forces us to employ

interpolation operations from irregular grids to regular grids. This is an important source

of error. If needed, to reduce this interpolation error, onecan use and implement more

advanced, sophisticated and accurate interpolation methods. However, this is totally dif-

ferent area of research and beyond the scope of this thesis. For F4, our algorithm has a

comparable accuracy with the DFT. This is due to the fact thatin this case, the error is a

result of the significant amount of signal energy that lies outside the assumed space and

spatial-frequency extents. This source of error, which affects both our algorithm and the

DFT in the same way, dominates the error arising from interpolation (which affects only

the non-separable LCT computation), so that the results aresimilar. On the other hand, for

the other functions, the interpolation error (which does not affect the DFT) results in higher

errors for the LCT computations as compared to the DFT.

To be more confident in the above claims, we also studied the effects of the method of

interpolation on our algorithm and studied how they change the accuracy of the algorithm.

We employed in our algorithmnearest neighbor, bilinear, andcubic interpolation methods

because they are among the most standard, mainstream, and efficient methods, [117, 118].

Different versions of the algorithm have been implemented by using each of the above

methods. The error percentages resulting from the use of different interpolation methods

are tabulated in Tables 4.2 and 4.3 for T1 and T2, respectively.


F1 F2 F3 F4Nearest 1.72 × 10−1 3.28 × 10−1 4.59 × 10−1 11.24

Bilinear 1.78 × 10−2 5.58 × 10−2 8.4 × 10−2 6.24

Cubic 3.82 × 10−4 1.09 × 10−3 3.21 × 10−3 1.92

Table 4.3: Percentage errors for different interpolation methods and functions F for T2.

As can be seen from the tabulated data, the error values are affected considerably by the

interpolation method chosen. The best results are obtainedwhen we use the cubic interpo-

lation method, which is the most advanced among the three. Since there are essentially two

sources of error, the one that is fundamental equally affecting LCTs and the DFT, we are

not surprised to observe that as the quality of the interpolation is increased, the accuracy of

the algorithm improves and approaches to the DFT benchmark.

The results of our fast algorithm were obtained within a couple of seconds by using

MATLAB code running on a standard personal computer. The calculation of the brute

force reference results took several days.


We presented an algorithm for the fast digital computation of the most general family of

two-dimensional non-separable linear canonical transforms. This family of transform inte-

grals represents a quite general class of two-dimensional quadratic-phase systems in optics.

Our approach is based on concepts from signal analysis and processing rather than conven-

tional numerical analysis. With careful consideration of sampling issues, the number of

samplesM × N of the sampling grid can be chosen very close to the space-bandwidth

product of the functions. A naive approach based on examination of the frequency content

of the integral kernels would, on the other hand, result in anunnecessarily high number

of samples being taken due to the highly oscillatory nature of the kernels, which would

not only be representationally inefficient but also increase computation time and storage


requirements. The transform output may have a higher space-bandwidth product than the

input due to the nature of the transform family. Through careful space-bandwidth tracking

and control, we can assure that the output samples obtained are accurate approximations to

the true ones and that they are sufficient (but not unnecessarily redundant) in the Nyquist-

Shannon sense, allowing full reconstruction of the underlying continuous function.

The algorithm takes the samples of the input function and maps them to the samples of

the continuous 2D-NS-LCT of this function in the same sense that the fast Fourier transform

(FFT) implementation of the discrete Fourier transform (DFT) computes the samples of the

continuous FT of a function.

A second source of error which was not of substantial impact in the 1D case or the sepa-

rable 2D case but which is significant in the non-separable 2Dcase arises from the necessity

to carry out interpolations to revert samples on rotated grids to the original rectangular grid.

This error depends on how accurately the interpolation operation is handled. We have used

well-established and standard methods for interpolation that are readily available, since

advancing methods of interpolation is beyond the scope of this thesis. While we believe

the levels of accuracy attained with these interpolation methods will be sufficient for most

applications, in those cases where they are not, more efficient and customized interpolation

methods for non-rectangular grids can be utilized to further improve accuracy.

Chapter 5

The Algorithm for Complex

Quadratic-Phase Systems

5.1 Introduction1

LINEAR CANONICAL TRANSFORMS with real parameters have received consider-

ably more attention than LCTs with complex parameters [8]. Real linear canonical

transforms (RLCTs) are unitary mappings between the elements of Hilbert space of square

integrable functions of a variable inR. RLCTs are represented by2 × 2 unimodular real

matrices

MR =

a b

c d

, (5.1)

with determinant equal to 1, wherea,b,c,d are real. The parameter matricesMR form the

real symplectic groupSp(2, R) with three independent parameters, [26]. RLCTs are of

great importance in electromagnetic, acoustic, and other wave propagation problems since

they represent the solution of the wave equation under a variety of circumstances. At optical

frequencies, RLCTs can model a broad class of lossless optical systems including thin

1This chapter is taken from [120]. Copyright 2010 by OSA.

87

88CHAPTER 5. THE ALGORITHM FOR COMPLEX QUADRATIC-PHASE SYSTEMS

lenses, sections of free space in the Fresnel approximation, sections of quadratic graded-

index media, and arbitrary concatenations of any number of these, sometimes referred to

as first-order optical systems or quadratic-phase systems (QPS) [1,5,6,10,12,17,19,20,22,

23,25,33,34].

Extension of RLCTs to complex linear canonical transforms (CLCTs) is rather in-

volved [8,37,121,122]. The extension is very briefly summarized as follows. When we let

the entries of the unimodular transform matrices be complexnumbers, we obtain the unit

determinant matrices

MC =

a b

c d

, (5.2)

wherea,b,c,d are complex parameters. The matricesMC form the complex symplectic

groupSp(2, C) with six independent parameters [122]. However, CLCTs represented by

this symplectic group can no longer be established as a unitary mapping between the Hilbert

space of square integrable functions inR. Instead, we have a mapping from the Hilbert

space of square integrable functions of a real variable, to analytic functions of a complex

variable on the complex plane in the Bargmann-Hilbert spaceof square integrable func-

tions, [123,124], as established in [8,37,121,122]. The CLCTs are required to be bounded

but not necessarily unitary, in which case we need to represent CLCTs with a semigroup

HSp(2, C) within the groupSp(2, C). More on the mathematical foundations and theory

of CLCTs can be found in [8,37,121,122].

The distinguishing feature of our approach is the way our algorithm carefully addresses

sampling and space-bandwidth product issues from an information-theoretical perspective.

Special care is taken to ensure that the output samples represent the continuous transform

in the Nyquist-Shannon sense during every stage of the algorithm, so that the continuous

transform can be fully recovered from the samples. Despite the highly oscillatory nature of

the integral kernel, we carefully manage the sampling rate so as to ensure that the number

of samples used is sufficient, but not much larger than the space-bandwidth product of

5.1. INTRODUCTION 89

the input signal, so that the algorithms are as efficient as possible. The straightforward

method of sampling the input field and the kernel, and then calculating the output field is not

suitable for several reasons. First of all, due to the highlyoscillatory nature of the integral

kernel, a naive application of the Nyquist sampling theoremto determine the sampling

rate would result in an excessively large number of samples and inefficient computation.

On the other hand, ignoring the oscillations of the kernel and determining the sampling

rate according to the input field alone may cause under-representation of the output field

in the Nyquist-Shannon sense. This unacceptable situationarises due to the fact that the

particular 2D-LCT that we are calculating may increase the space-bandwidth product in

one or both of the dimensions. If we do not increase the numberof samples that we are

working with so as to compensate for this increase, there will be information loss and we

will not be able to recover the true transformed output from our computed samples. The

computation of complex LCTs involves a number of issues which do not arise in the case

of real LCTs. The decompositions employ complex chirp multiplications whose effect on

the Wigner distribution must be clarified to ensure proper space-bandwidth tracking and

control.

Complex-parametered LCTs allow several kinds of optical systems to be represented,

including lossy as well as lossless ones. When complex parameters are involved, LCTs

may no longer be unitary and boundedness issues may arise. The decomposition of general

CLCTs into Fourier transforms and real and imaginary chirp multiplications allows us to

derive conditions on the transform parameters that ensure boundedness.

We also need to find the conditions under whichHSp(2, C) can be constructed as a

mapping fromR → R. This is because we are interested in optical applications of CLCTs

where the inputs and outputs are functions of real spatial variables. Such CLCTs which map

functions over Hilbert spaces from the real line to the real line are calledpassive CLCTs

in [37], whereas CLCTs that map functions from the real line to analytic functions on

complex Bargmann-Hilbert spaces are calledactive CLCTs. Thus the eligible parameters


also depend on how theHSp(2, C) semigroup is constructed forR → R. Wolf derived

the parameter spaces for which the CLCTs represent a mappingfrom R → R and the

output is bounded [37]. However, this construction excludes some optically important

special cases like Gaussian apertures. The decompositionswe use allow the derivation of

conditions which do not exclude Gaussian apertures. Specification of such conditions were

not necessary in the RLCT case.

The chapter is organized as follows. Section 5.2 presents some mathematical prelimi-

naries that we use in the derivation of our algorithm for the complex case and review some

special and important CLCTs in optics. In Section 5.3, we present a careful analysis of

every possible case the complex transform matrix may assumeand present fast algorithms

based on decompositions into generalized chirp multiplications, real scalings, and FTs.

We also determine whether a given CLCT represents an optically possible bounded input-

output relationship from the real line to the real line. Numerical examples to demonstrate

the accuracy of the algorithm are given in Section 5.4. Finally we conclude in Section 5.5.

The CLCT of f(u) with complex parameter matrixMC is denoted asfMC(u) =

(CMCf)(u):

(CMCf)(u) =

∫ ∞

−∞

KC(u, u′)f(u′) du′,

KC(u, u′) = e−iπ/4

√β exp

[iπ(αu2 − 2βuu′ + γu′2)

], (5.3)

whereα, β, γ are complex parameters independent ofu andu′ and whereCMCis the CLCT

operator.MC again has unit-determinant and is given by

MC =

a b

c d

=

ar + iac br + ibc

cr + icc dr + idc

=

γ/β 1/β


(5.4)

wherear, ac, br, bc, cr, cc, dr, dc are real numbers. The overline over the parametersα, β,


γ is to emphasize that these parameters are now complex, corresponding to a total of 6 real

parameters:α = αr + iαc, β = βr + iβc, γ = γr + iγc. In terms of these parameters the

kernelKC can be rewritten as

KC(u, u′) = e−iπ/4√

βr + iβc eiπ(αru2−2βruu′+γru′2)e−π(αcu2−2βcuu′+γcu′2). (5.5)

The bidirectional relationship between theα, β, γ parameters and the matrix entries are

given as follows:

αr =drbr + dcbc

b2r + b2

c

, αc =dcbr − drbc

b2r + b2

c

βr =br

b2r + b2

c

, βc =−bc

b2r + b2

c

γr =arbr + acbc

b2r + b2

c

, γc =acbr − arbc

b2r + b2

c

(5.6)

ar =βcγc + βrγr

β2r + β2

c

, ac =βrγc − βcγr

β2r + β2

c

br =βr

β2r + β2

c

, bc =−βc

β2r + β2

c

dr =βcαc + βrαr

β2r + β2

c

, dc =αcβr − αrβc

β2r + β2

c

(5.7)

5.2 Preliminaries

5.2.1 Wigner Distributions

Here we will review the relationship between LCTs and the Wigner distribution, which will

aid us in understanding the effects of the elementary blocksused in our decompositions.


The Wigner distributionWf(u, µ) of a signalf(u) can be defined as follows [97,98]:

Wf (u, µ) =

∫ ∞

−∞

f(u + u′/2)f ∗(u − u′/2)e−2πiµu′

du′. (5.8)

Roughly speaking,W (u, µ) is a function which gives the distribution of signal energy over

space and frequency. Its integral over space and frequency,∫∞

−∞

∫∞

−∞W (u, µ) du dµ, gives

the signal energy.

Let f denote a signal andfM be its LCT with parameter matrixM. Then, the Wigner

distribution (WD) offM can be expressed in terms of the WD off as [10]

WfM(u, µ) = Wf (du − bµ,−cu + aµ). (5.9)

This means that the WD of the transformed signal is a linearlydistorted version of the orig-

inal distribution. The Jacobian of this coordinate transformation is equal to the determinant

of the matrixM, which is unity. Therefore this coordinate transformationdoes not change

the support area of the Wigner distribution. The invarianceof the support area means that

LCTs do not concentrate or deconcentrate energy. The support area of the Wigner distri-

bution can also be approximately interpreted as the number of degrees of freedom of the

signal. Therefore, the number of samples needed to represent the signal does not change

after a real LCT operation.

For the purpose of space-bandwidth tracking as employed in our algorithm, we do not

require a full characterization of the effects of CLCTs on the WD. However, we do need

to know the effect of multiplying a function with another function on the WD to derive a

space-bandwidth product tracking method for CLCTs. This multiplication property is not

required in deriving our previous algorithms for real LCTs,[49,64] but will be necessary in

our CLCT algorithm. The WD has the following multiplicationproperty [10]: leth(u) and

f(u) be two functions and letWh(u, µ) andWf(u, µ) be their corresponding WDs. Then


h(u)f(u) has the WD given by

∫Wh(u, µ − µ′)Wf(u, µ′) dµ′. (5.10)

In other words, when two functions are multiplied, the WD of the resulting function is given

by the convolution of the WDs of the initial two functions along the frequency dimension.

5.2.2 CLCTs in Optics and Special CLCTs

Magnification (scaling), Fourier transformation (FT), real fractional Fourier transforma-

tion (RFRT), real chirp multiplication (CM), complex chirpmultiplication (CCM), Gauss-

Weierstrass Transform, complex-ordered fractional Fourier transformation (CFRT) are all

special cases of CLCTs that have optical realizations. Scaling, FT, RFRT, and CM, which

have real parameters, belong to the narrower class of RLCTs and have been reviewed

in [64]. In this Section, we only review complex parameteredcases that are essential for

our development.

5.2.2.1 Complex Scaling (Magnification)

Simple scaling with a real parameterM is an operation which corresponds to optical mag-

nification [64]. If the parameterM is allowed to be complex, we obtain the complex scaling

operation, which is a special case of CLCTs. With complex scaling, the real axis on which

the input function is defined is mapped to a straight line in the complex plane passing

through the origin and making an anglearg(M) with the real axis. The mapping becomes

R → C. The interpretation of complex scaling in quantum mechanics has been discussed

in [125, 126], while an interpretation from a signal processing perspective can be found

in [127]. However, we are not aware of the realization and application of complex scaling

from a purely optical point of view.


5.2.2.2 Gaussian Apertures (Complex Chirp Multiplication)

Gaussian apertures, also calledsoft apertures, are a special case of CLCTs. They are ac-

tually chirp multiplications with a complex parameter and are the complex counterparts of

CM operations. We will hereafter refer to them as complex chirp multiplication (CCM).

The definition of CCM is similar to the definition of real CM, where we replace the chirp

parameter with a purely imaginary complex parameter:

CQiqf(u) = Qiqf(u) = eπqu2

f(u), (5.11)

Qiq =

1 0

−iq 1

=

1 0

iq 1

−1

. (5.12)

whereq is real and soiq is a purely imaginary parameter. It essentially behaves like a mul-

tiplicative filter where the transmission is dependent on the transverse dimension quadratic-

exponentially. To exclude the unbounded case we requireq ≤ 0.

We now discuss the effect of complex chirp multiplication (CCM) on the WD. We need

this result in order to track and control the space-bandwidth product of CLCTs. This result

is not needed in algorithms for RLCTs because there are no CCMstages in RLCTs, and is

of a considerably different nature than the operations employed there. We use the property

given in Eq. 5.10 withh(u) = eπqu2. The WD ofh(u), denotedWh(u, µ), can be obtained

by directly using the definition of the WD (Eq. 5.8):

Wh(u, µ) =

√2

−qe2πqu2

e2πq

µ2

, q < 0. (5.13)

The WD of the Gaussian function is a 2D Gaussian function in the space-frequency plane.

Sinceq < 0, this function decays with increasingu andµ. Therefore, we can specify

a rectangular region which contains almost all of the energyof the function. We will

choose the extents of this rectangle to correspond to plus/minus four standard deviations of


the Gaussian in both the space and frequency dimensions, which defines a rectangle with

extents

g1 =√

16/π|q|

g2 =√

16|q|/π (5.14)

in the space dimension and the frequency dimension, respectively. When the WD of the

input function and the WD of the Gaussian function are convolved along theµ direction to

find the resulting WD of the output function (as illustrated in Fig. 5.1), the resulting space

extent of the support of the output WD will be given bymin(d1, g1) and the resulting fre-

quency extent will be given byd2 + g2 whered1 andd2 are the space and spatial-frequency

extents of the input function.

5.2.2.3 Gauss-Weierstrass Transform

The Gauss-Weierstrass transform with parametert is given by the integral transform [8],

Gtf(u) =√

1/t

∫ ∞

−∞

e−π(u−u′)2

t f(u′) du′. (5.15)

It gives the solution of the heat equation. The complex chirpconvolution (CCC) operation,

which is a special case of CLCTs, is represented by the transform matrix

Rir =

1 ir

0 1

(5.16)

and is equivalent to convolution by a Gaussian function:

CRirf(u) = Rirf(u) = f(u) ∗ eiπ/4

√1/r exp(πu2/r). (5.17)


(a) The input function (b) Rectangle containing WD of Gaussian func-tion

(c) Convolution along spatial frequency direction (d) Resulting WD

Figure 5.1: The effect of the CCM operation on the WD.

We observe that CCC is the same as the Gauss-Weierstrass transform when we choose the

CCC parameterr = −t. As in the case of the FT, there is again the inconsequential constant

phase factore−iπ/4 difference between the two definitions. CCC operations are covered by

our algorithm since they are a special case of CLTs. CCC operations are most conveniently

calculated by expressing them as an FT followed by a CCM operation followed by an


inverse FT.

The combined effect of two CCM (or CCC) operations followingeach other is again a

CCM (or CCC) operation, whose parameter is found by summing the parameters of the two

constituent operations. If two CCM operations with real or complex parametersq1 andq2

follow each other, the equivalent operation is a new CCM operation with parameterq1 +q2.

If two CCC operations with real or complex parametersr1 andr2 follow each other, the

equivalent operation is a new CCC operation with parameterr1 + r2.

5.2.2.4 Complex-ordered Fractional Fourier Transform

Theath order real fractional Fourier transform (RFRT, or simplyFRT) is well studied in

the literature, [10, 13, 27, 101–107]. Complex FRTs are FRTswhose order parameter is

complex [41–45].

When the order is an imaginary numberib, then we obtain the following special case

of CLCTs with the transform matrix

Fiblc =

cosh(bπ/2) i sinh(bπ/2)

−i sinh(bπ/2) cosh(bπ/2)

, (5.18)

which again differs only by the factorebπ/4 from fractional Fourier transforms as commonly

defined:

CFiblcf(u) = F ib

lc f(u) = ebπ/4F ibf(u). (5.19)

Since FRTs are additive in index, a real-ordered and a purelyimaginary-ordered FRT can be

combined asFa+ib = FaF ib to yield a general complex-ordered FRT, where the complex

order may be denoted byac = a+ ib. CFRTs can be optically realized by using thin lenses,

free-space propagations, and Gaussian apertures, or by combination of Gauss-Weierstrass

transforms with Gaussian apertures [41].


5.3 The Algorithm

We now show how givenABCD matrices can be decomposed in a manner that leads to

a fast algorithm for computation of CLCTs. In the most general case, the matrixMC is

composed of the four complex parametersa,b,c,d, whose real and imaginary parts add up

to a total of 8 parameters. These 8 parameters are restrictedby the unimodularity condition

onMC, which requires the real part of the determinant to be1 and the imaginary part to be

0. Because of these two equations, the total number of independent parameters of a general

CLCT is 6. These 6 parameters correspond to the 6 parameters of the groupHSp(2, C),

which is a 6 parameter semigroup of the complex symplectic groupSp(2, C).

Before giving the main decomposition which covers the general case, we start with a

special case whose treatment is straightforward:

5.3.1 b = 0 case:

Whenb = 0, the unimodularity requirement requiresa 6= 0 and the transform output can

be written as

(CMCf)(u) =

1√a

ejcy2

2a f(y/a). (5.20)

In this case, the output is given by a scaling operation with parametera followed by a

chirp multiplication operation with parameter−c/2a. We will restrict ourselves to the

case wherea is real, since only in this case will the scaling operation result in aR → R

mapping. The case wherea is complex produces complex scaling operations and therefore

causes mappings from functions on the real line to functionson the complex plane. This

case would require special treatment, which we do not attempt since we are not aware of

any optical realization or application of such transforms.Also necessary is the condition

Im(c/a), which is necessary to ensure boundedness. In order to have abounded andR → R

mapping, it becomes necessary fora to be real andIm(c/a) ≥ 0. Together with the unit


determinant condition, these conditions can be explicitlysummarized as follows:

ar 6= 0

d = 1/a

ac = 0

arcc ≥ 0 (5.21)

where the first two are intrinsically required to define any LCT (detMC = 1) and the last

two are required to obtain a boundedR → R mapping. When the conditions in 5.21 are

satisfied, the matrixMC can be decomposed as:

MC =

ar 0

c 1/ar

=

1 0

c/ar 1

ar 0

0 1/ar

=

1 0

cr/ar 1

1 0

icc/ar 1

ar 0

0 1/ar

. (5.22)

The above decomposition can be used for the fast computationof the special caseb = 0.

5.3.2 b 6= 0 case:

Now, we turn our attention to the more general case in which the following decomposition

will be the basis of our fast algorithm:

MC =

1 0

−q3r 1

1 0

−iq3c 1

0 −1

1 0

1 0

−q2r 1

1 0

−iq2c 1


×

0 1

−1 0

1 0

−q1r 1

1 0

−iq1c 1

. (5.23)

This decomposition consists of three imaginary CM and real CM pairs with Fourier/Inverse-

Fourier transform operations in between. The imaginary CM and real CM pairs can also be

viewed as complex CM (CCM) operations:

MC =

1 0

−(q3r + iq3c) 1

0 −1

1 0

1 0

−(q2r + iq2c) 1

×

0 1

−1 0

1 0

−(q1r + iq1c) 1

. (5.24)

The three matrices in the center can also be expressed as a CCCoperation:

MC =

1 0

−(q3r + iq3c) 1

1 (q2r + iq2c)

0 1

1 0

−(q1r + iq1c) 1

(5.25)

which is nothing but the complex version of the well-known CM-CC-CM decomposition

[10].

When we multiply out the matrices on the right-hand side of Eq. 5.23, equate the result

to the general CLCT matrix given in Eq. 5.4 and solve for our decomposition parameters

in terms of the CLCT parameters, we get the following:

q1r =br − brar − acbc

b2r + b2

c

q1c =bcar − bc − brac

b2r + b2

c

q2r = br

q2c = bc

q3r =br − brdr − dcbc

b2r + b2

c


q3c =bcdr − bc − brdc

b2r + b2

c

(5.26)

Thus, all 6 parameters of our decomposition have been expressed in terms of the 6 parame-

ters of the CLCT which we desire to compute. By using Eq. 5.7, we can also easily calculate

the decomposition parameters in terms of the complexα,β,γ parameters if needed.

The decomposition in Eq. 5.23 can also be expressed in operator notation:

CM = Qq3rQiq3c

F−1lc Qq2r

Qiq2cFlc Qq1r

Qiq1c(5.27)

We now discuss the various cases that arise depending on the values of the parameters.

Whenb 6= 0, separate treatment is required depending on whethera is zero or not. First,

consider the case whena = 0. The decomposition parameters given in Eq. 5.26 become

q1r =br

b2r + b2

c

q1c =−bc

b2r + b2

c

q2r = br

q2c = bc

q3r =br − brdr − dcbc

b2r + b2

c

q3c =bcdr − bc − brdc

b2r + b2

c

(5.28)

As discussed in Section 5.2. 5.2.2. 5.2.2.2, the CCM parametersq1c, q2c, q3c should be≤ 0

leading to the conditions

−bc

b2r + b2

c

≤ 0 (5.29)

bc ≤ 0 (5.30)bcdr − bc − brdc

b2r + b2

c

≤ 0 (5.31)


Eqs. 5.29 and 5.30 implybc = 0 and Eq. 5.31 becomesbrdc ≥ 0. When we setbc = 0 in

Eq. 5.28, we obtain the decomposition parameters:

q1r = 1/br

q1c = 0

q2r = br

q2c = 0

q3r = (1 − dr)/b2r

q3c = −dc/br (5.32)

with the conditionbrdc ≥ 0. The decomposition we should use in this case therefore can

be expressed as

MC =

1 0

−(1 − dr)/br 1

1 0

idc/br 1

0 −1

1 0

×

1 0

−br 1

0 1

−1 0

1 0

−1/br 1

. (5.33)

We now turn our attention to the caseb 6= 0 anda 6= 0. The decomposition given in

Eq. 5.23 and the decomposition parameters given in Eq. 5.26 are applicable. The below

three conditions should be satisfied to have a bounded andR → R mapping:

bc ≤ 0

bcar − brac ≤ bc

bcdr − brdc ≤ bc (5.34)


which can be equivalently expressed in terms of theα,β,γ parameters:

βc ≥ 0

αc ≥ βc

γc ≥ βc (5.35)

which depends only on the imaginary parts. This is expected since real LCTs are always

bounded and unitary, and it is the imaginary parts that are involved in issues of bounded-

ness. These conditions are derived by restricting the parameters of the Gaussian aperture

steps in the CLCT decompositions we employ. There are no suchconditions required for

RLCTs. However, these constraints are crucial for computation of CLCTs. To better illus-

trate these conditions, we summarize them in Table 5.1.

Case 1 Case 2 Case 3b = 0 b 6= 0 & a = 0 b 6= 0 & a 6= 0ar 6= 0 bc = 0 bc ≤ 0d = 1/a brdc ≥ 0 bcar − brac ≤ bc

ac = 0 bcdr − brdc ≤ bc

arcc ≥ 0

Table 5.1: Summary of the conditions to have bounded,R → R CLCTs

The special caseb = 0 requires the computation of only real and complex chirp mul-

tiplications and a real scaling operation. The decomposition for the general case includes

chirp multiplications and Fourier transformations. Chirpmultiplications require onlyN

multiplications and can be done inO(N) time. The FT and inverse FT can be computed in

O(N log N) time by using the fast Fourier transform algorithm (FFT). Wealso note that the

scaling operation merely changes the sampling interval in the sense of reinterpretation of

the same samples with a scaled sampling interval, in a mannerwhich corresponds to scal-

ing of the underlying continuous signal. Thus the cost of thescaling operation is minimal


and not of consequence, since it amounts only to a reinterpretation of the samples. There-

fore, the overall CLCT can be computed inO(N log N) time. To ensure that the number

of samples required to represent the function is sufficient in the Nyquist-Shannon sense at

each step of the decomposition, we will track the space-bandwidth representation of the

function by using the WD and increase the sampling rate when necessary. We will do this

by the help of the procedures summarized in [64] for real steps and by the help of Fig. 5.1

and the discussion given in Sec. 5.2. 5.2.2. 5.2.2.2 for the complex components. Since the

FRT corresponds to rotation and the scaling operation only to a reinterpretation of the sam-

ples, these steps never require us to increase the number of samples. Chirp multiplications,

however, require careful handling of the space-bandwidth and sampling issues.

Finally, we summarize our algorithm and the associated space-bandwidth product track-

ing and sampling control methodology for the most general case. (For theb = 0 and

b 6= 0, a = 0 special cases, this procedure can be easily simplified to correspond to the sim-

pler decompositions in Eq. 5.22 and Eq. 5.33, respectively.) Whenever the current number

of samples will not be sufficient to fully represent the operated-on signal in the Nyquist-

Shannon sense, an increase in the number of samples is required prior to performing the

operation.

1. We will useEs andEf to denote the spatial and frequency extents of the function as

we go through the stages of the algorithm. We assume that the initial space-frequency

support is a square of edge length∆u so that at the beginningEs = Ef = ∆u, and

the signal can be represented withEsEf = ∆u2 samples.

2. The first step of the decomposition is the first CCM with parameterq1c. We use

Eq. 5.14 to obtain the space and frequency extents of the Gaussian function, which

we denote byGs1 andGf1, respectively.Es andEf are changed according toEs →min(Es, Gs1) andEf → Es + Gf1. The required number of samples then becomes

Es × Ef which are taken in the interval[−∆Es/2, ∆Es/2] with a spacing of1/Ef

5.4. NUMERICAL EXAMPLES 105

apart from each other. This may or may not require an increasein the number of

samples depending on whether the newEs × Ef product is bigger than the starting

number of samples∆u2. If an increase in the number of samples is required, we

oversample the signal using an appropriate interpolation scheme and then the CCM

operation is performed on the input samples.

3. The second step is a CM operation with parameterq1r. We see that the extents must

now becomeEs → Es andEf → Ef + |q1r|Es. The number of samples required

becomesEs × (Ef + |q1r|Es) which will require oversampling with a factork = 1+

|q1r|Es/Ef . After this oversampling is performed, the CM operation is performed.

4. We now take the FT of the samples by using the FFT algorithm.We haveEs → Ef

andEf → Es since FT only switches the spatial variable and its spatial-frequency

variable. The FT operation does not change the space-bandwidth product of the

signal, so oversampling is not required at this stage.

5. Repeat Steps 2 and 3 with the parametersq2c andq2r corresponding to subsequent

stages of the decomposition.

6. Repeat Step 4, this time with an inverse FT operation instead of a forward FT opera-

tion.

7. Repeat Steps 2 and 3 with the parametersq3c andq3r corresponding to the final stages

of the decomposition, to get the final output samples.

5.4 Numerical Examples

We have considered several examples to illustrate the performance of the presented al-

gorithm. We consider the chirped pulse functionexp(−πu2 − iπu2), denoted F1, and

the trapezoidal function1.5tri(u/3) − 0.5tri(u), denoted F2 (tri(u) = rect(u) ∗ rect(u)).


Since these two functions are well confined to a circle in the space-frequency plane with

diameter∆u = 8, we takeN = 82. We also consider the binary sequence 01101010

occupying[−8, 8] with each bit 2 units in length, so thatN = 162. This binary se-

quence is denoted by F3 and the function shown in Fig. 5.2 is denoted by F4, again with

N = 162. F4 is chosen to test the algorithm with an information-theoretically challeng-

ing function who has several sharp rises and falls at arbitrary positions. Additionally, we

also test the example function given in Fig. 5.3, withN = 82, that has complex values

(ie. amplitude and phase). These choices for∆u result in∼ 0 %, 0.0002 %, 0.47 %,

0.03 %, 0.25 % of the energies of F1, F2, F3, F4, F5 respectively, to fall outside the cho-

sen frequency extents. The chosen space extents include allof the energies of F2, F3,

F4, F5 and virtually all of the energy of F1. We consider threetransforms, the first (T1)

with parameters(αr, βr, γr; αc, βc, γc) = (−2, 1.2,−0.9; 0.04, 0.02, 0.12), the second (T2)

with parameters(1.15,−0.14,−0.1; 0.003, 0.001, 0.002), and the third (T3) with parame-

ters(−1.2,−0.3, 0.1; 0.6, 0.5, 1). The CLCTs T1, T2 and T3 of the functions F1, F2, F3,

F4, F5 have been computed both by the presented fast algorithm and by a highly inefficient

brute force numerical approach based on Simpson’s numerical integration, which is here

taken as a reference.(See Appendix A.)

The results for all functions (F1, F2, F3, F4, F5) are plottedin Fig. 5.4 and Fig. 5.5 for

transform T1, and T2, respectively and tabulated in Table 5.2 for all transforms T1, T2 and

T3. Also shown are the errors that arise when using the DFT in approximating the FT of

the same functions, which serves as a reference. (The error is defined as the energy of the

difference normalized by the energy of the reference, expressed as a percentage.)

We also tested our algorithm for the CFRT, which is an important special case of

CLCTs and the complex extension of the real-parametered FRT. A CFRT with order0.8 −i0.2 is calculated with our algorithm and the reference method and the results are plot-

ted for all functions (F1, F2, F3, F4, F5) in Fig. 5.6, and again tabulated in Table 5.2.

The CFRT order0.8 − i0.2 corresponds to CLCT parameters(αr, βr, γr; αc, βc, γc) =

5.4. NUMERICAL EXAMPLES 107

−8 −6 −4 −2 0 2 4 6 80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


(0.292, 0.9919, 0.292; 0.3331, 0.098, 0.3331).

T1 T2 T3 CFRT DFTF1 4.12 × 10−6 2.19 × 10−6 2.8 × 10−3 1.24 × 10−5 2.0 × 10−21

F2 3.73 × 10−4 7.1 × 10−3 1.4 × 10−3 1.2 × 10−3 6.2 × 10−4

F3 0.53 0.35 0.26 0.22 1.2F4 1.2 × 10−3 4.96 × 10−2 2.0 × 10−3 2.2 × 10−3 7.1 × 10−2

F5 0.11 0.2 8.0 × 10−3 6.4 × 10−3 1.7

Table 5.2: Percentage errors for different functions F and transforms T.

Examination of the table shows that our algorithm can accurately compute CLCTs for

a variety of transforms and functions. We observe that the main determinant of the error is

not the transform, but the function, and more specifically the energy of the function lying

outside of the assumed extents. If we require the error to be further reduced, we can reduce

the excluded energy by increasing the extents and the numberof samples involved.


−8 −6 −4 −2 0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Amplitude

−8 −6 −4 −2 0 2 4 6 8−1.5

−1

−0.5

0

0.5

1

1.5Phase



We presented an algorithm for the fast and accurate digital computation of the general fam-

ily of complex-parametered linear canonical transforms. This family of transform integrals

can represent a general class of complex quadratic-phase systems in optics. Our approach

is based on concepts from signal analysis and processing rather than conventional numeri-

cal analysis. With careful consideration of sampling issues, the number of samplesN can

be chosen very close to the space-bandwidth product of the functions. A naive approach

based on examination of the frequency content of the integral kernels would, on the other


−3 −2 −1 0 1 2 3−1

−0.5

0

0.5


−3 −2 −1 0 1 2 3−1

−0.5

0

0.5


−2 −1 0 1 2

−1

−0.5

0


−2 −1 0 1 2

−1

−0.5

0


−3 −2 −1 0 1 2 3−1

−0.5

0


−3 −2 −1 0 1 2 3−1

−0.5

0


−2 −1 0 1 2

−1.5

−1

−0.5

0

0.5


−2 −1 0 1 2

−1.5

−1

−0.5

0

0.5


−2 −1 0 1 2−1

−0.5

0

Re of transform of F5

−2 −1 0 1 2−1

−0.5

0

Im of transform of F5

Figure 5.4: Transform (T1) of F1, F2, F3, F4, F5. The results obtained with the presentedalgorithm and the reference result have been plotted with dotted and solid lines, respec-tively. However, the two types of lines are almost indistinguishable since the results arevery close.


−2 −1 0 1 2−0.5

0

0.5


−2 −1 0 1 2−0.5

0

0.5


−2 −1 0 1 2−0.5

0

0.5


−2 −1 0 1 2−0.5

0

0.5


−2 −1 0 1 2−0.5

0


−2 −1 0 1 2−0.5

0


−2 −1 0 1 2−0.5

0

0.5


−2 −1 0 1 2−0.5

0

0.5


−2 −1 0 1 2

−0.2

−0.1

0

0.1

0.2


−2 −1 0 1 2

−0.2

−0.1

0

0.1

0.2


Figure 5.5: Transform (T3) of F1, F2, F3, F4, F5. The results obtained with the presentedalgorithm and the reference result have been plotted with dotted and solid lines, respec-tively. However, the two types of lines are almost indistinguishable since the results arevery close.


−2 −1 0 1 2

0

0.2

0.4

Re of CFRT of F1

−2 −1 0 1 2−0.8

−0.6

−0.4

−0.2

0

Im of CFRT of F1

−2 −1 0 1 2

0

0.5

1

1.5Re of CFRT of F2

−2 −1 0 1 2−0.8

−0.6

−0.4

−0.2

0

0.2

Im of CFRT of F2

−2 −1 0 1 2

−0.2

0

0.2

0.4

0.6

Re of CFRT of F3

−2 −1 0 1 2−1

−0.5

0

0.5Im of CFRT of F3

−2 −1 0 1 2−0.5

0

0.5

1

1.5

2Re of CFRT of F4

−2 −1 0 1 2−1.5

−1

−0.5

0

0.5Im of CFRT of F4

−2 −1 0 1 2−0.5

0

0.5

1Re of CFRT of F5

−2 −1 0 1 2

−0.4

−0.2

0

0.2

Im of CFRT of F5

Figure 5.6: CFRT with order0.8− i0.2 of F1, F2, F3, F4, F5. The results obtained with thepresented algorithm and the reference result have been plotted with dotted and solid lines,respectively. However, the two types of lines are almost indistinguishable since the resultsare very close.


hand, result in an unnecessarily high number of samples being taken due to the highly os-

cillatory nature of the kernels, which would not only be representationally inefficient but

also increase computation time and storage requirements. The transform output may have

a higher space-bandwidth product than the input due to the nature of the transform family.

Through careful space-bandwidth tracking and control, we can assure that the output sam-

ples obtained are accurate approximations to the true ones and that they are sufficient (but

not unnecessarily redundant) in the Nyquist-Shannon sense, allowing full reconstruction of

the underlying continuous output functions.

The algorithm takes the samples of the input function and maps them to the samples

of the continuous CLCT of this function in the same sense thatthe fast Fourier transform

(FFT) implementation of the discrete Fourier transform (DFT) computes the samples of the

continuous FT of a function.

Chapter 6

Application to the Beam Propagation

Method

THE BEAM PROPAGATION METHOD (BPM) is an important computational method

in electromagnetics to solve the time-harmonic Helmholtz equation under the slowly

varying envelope approximation (SVEA), [128–131]. It predicts the propagation of beams

in inhomogeneous media in which the refractive index changes are small relative to the

average index such that the SVEA can hold. It is mostly used tosimulate and study optical

waveguides and other optical devices with an inhomogeneousrefractive index distribution.

It has also been modified to be used in analysis of diffractiongratings and in anisotropic

media, [132,133].

On the other hand, quadratic-phase systems (QPSs) can modela broad class of optical

systems including thin lenses, sections of free space in theFresnel approximation, sections

of quadratic graded-index media, and arbitrary concatenations of any number of these,

sometimes referred to as first-order optical systems [10,12]. Fractional Fourier transforms

(FRTs) [101,102], scaling operations, and chirp multiplication (CM) and chirp convolution

(CC) operations, the latter also known as Fresnel transforms, are special cases of QPSs,

[10].

113

114 CHAPTER 6. APPLICATION TO THE BEAM PROPAGATION METHOD

The QPSs are capable of exactly representing so-called quadratic-graded index me-

dia (GRIN), [108], in which the inhomogeneous refractive index distribution is given by

n2(x) = n21[1− (n2/n1)x

2] wheren1 andn2 are the medium parameters andx is the trans-

verse coordinate. However, in GRIN the parameters are assumed to be the same along

the propagation coordinate and the change along the transverse direction is limited to a

quadratic dependence. Therefore, this index distributionis currently themost inhomoge-

neous case QPSs are capable of representing to the best of our knowledge. In this chapter,

we study the link of QPSs to BPM in an attempt to extent this limit in the degree of inho-

mogeneity further. Also, our contribution allows BPM to become extremely efficient under

certain conditions.

In our method, each layer of BPM is represented by a cascade oftwo fundamental

ABCD-steps. One is the chirp convolution and the other is thechirp multiplication. The

former stands for the propagation in an homogenous medium (Fresnel transform), which

is the first step of the corresponding single BPM layer, and the latter gives the approxi-

mated phase correction, which is the second step of the BPM layer. Our idea is to find, for

every BPM layer, the corresponding chirp convolution sub-systems for each BPM layer, ap-

proximate the phase correction with chirp multiplication sub-systems, cascade all of these

sub-systems to find a single equivalent QPS. In other words, the overall ABCD matrix is

the multiplication of the matrices of the subsystems.

The chapter is organized as follows. In Sections 6.1 and 2.1.1, we summarize the basics

of BPM and QPS/ABCD-Systems, respectively. In Section 6.2,we give the main procedure

to derive the relation between BPM and QPSs. In Section 6.3, we present the examples of

our method and Section 6.4 is devoted to the concluding remarks.

6.1. BASICS OF BPM 115

6.1 Basics of BPM

In this section, we summarize the basic structure of BPM. We assume the 2D case with

propagation in thez direction and an inhomogeneous medium with refractive index distri-

butionn(x, z), x being the transverse direction. We start with a time-harmonic monochro-

matic wave fieldE(x, z, t) = Re{U(x, z)e−jωt} that propagates in an inhomogeneous

medium.U(x, z) is the complex amplitude andω is the angular frequency. Then the scalar

wave equation is given by∂2E

∂t2=

c2

n2∇2E (6.1)

wherec is the speed of light in free-space and∇2 = ∂2/∂x2 + ∂2/∂z2. By using the

time-harmonic assumption, the wave-equation is simplifiedto the Helmholtz Equation in

an inhomogeneous medium:

(∇2 + n2k20)U(x, z) = 0 (6.2)

wherek0 = 2π/λ0 andλ0 is the free-space wavelength.

Next, we make the first key assumption of BPM that we have a refractive index distribu-

tion in the formn(x, z) = n + ∆n(x, z), where∆n ≪ n meaning that the inhomogeneous

medium is such that the refractive index varies within a small neighborhood of an average

valuen, ie. a weak index modulation. Additionally, when we make theparaxial approxi-

mation and assume that the wave-propagation is along directions making very small angles

with thez direction,U(x, z) can be written as

U(x, z) = U(x, z)e−jnk0z (6.3)

whereU(x, z) part is a slowly varying function ofz. This so-called slowly-varying enve-

lope approximation (SVEA) is justified by BPM’s assumptionsof weak index modulation


and of paraxial propagation. By substituting Eq. 6.3 into Eq. 6.2 we get

(∇2 − 2jnk0∂

∂z+ n2 − n2)U = 0 (6.4)

By SVEA, we can neglect the second order partial derivative of U with respect toz in

Eq. 6.4 to get the final “paraxial wave equation for inhomogeneous media”, [128],

∂

∂zU =

−j

2nk0[∂2U

∂x2+ (n2 − n2)k0U ] (6.5)

Eq. 6.5 is the fundamental BPM equation that provides its theoretical foundation. This

equation can be solved through iteration and starting from an initial value. The implemen-

tation of Eq. 6.5 is done by dividing thez axis into N slices of length∆z and by treating

each slice as a combination of propagation of∆z in a homogeneous medium of refractive

indexni wherei = 1, 2, ..., N are the average refractive index values along the correspond-

ing slice and a virtual lens effect for phase correction. Thelens effect is the multiplication

of the phaseexp(−j(n(x, zi) − ni)k0∆z) due to the slowly varying index distribution on

top of the average value, which models the inhomogeneous medium. Note thatzi = i∆z

and z0 is the input plane. Then, starting from an initial wave field at z = 0, the BPM

method iterates this initial value for each slice and uses the output of i’th slice as the in-

put of (i+1)’th. By letting N be large and∆z be very small, the above assumptions in the

derivation of Eq. 6.5 are justified. This way the slowly-varying inhomogeneous index vari-

ation is separated from the homogenous propagation by virtue of Eq. 6.5 and after iterating

the entire set of slices, one can get the output wave field.

For the homogeneous propagation steps, the common method touse is the Angular

Spectrum Method (ASM) that relies on the Angular Spectrum ofPlanes Waves, [16, 130].

If ASM is used to implement the homogenous propagation part one can get the iteration

equation for one slice betweenzi andzi+1 as:

6.1. BASICS OF BPM 117

U(x, zi+1) = F−1

{F(

U(x, zi) exp

(j2π∆z

√n2

i

λ20

− f 2x

))}

× exp (−j (n(x, zi) − ni) k0∆z) (6.6)

wherezi+1 = zi + ∆z andni is the average refractive index fori’th slice.F stands for the

Fourier transformation andfx is the spatial-frequency of variablex. Also note that, Eq. 6.6

assumesfx < ni/λ0, which means that evanescent waves are ignored.

We can write this in operator notation to be used later as

U(x, zi+1) = Ci PniU(x, zi) (6.7)

wherePnistands for the free-space propagation in a homogeneous region of length∆z and

with refractive indexni and whereCi is the operator for the multiplicative phase compen-

sation ofCi = exp (−j (n(x, zi) − ni) k0∆z) for the slicei.

The advantages of BPM is that it is a relatively simple, straightforward and fast method.

On the other hand, it has some drawbacks that it cannot handlereflections and non-paraxial

propagation problems as well as works only for small variations in the refractive index.

Reflections cannot be handled because we ignored the effect of the second partial derivative

with respect toz in the above derivation. However, more advanced BPM algorithms have

been developed to work in non-paraxial propagation cases aswell. The version of BPM that

we consider in this chapter is a version of the BPM as given in [130,134]. We also consider

two-dimensional BPM (2D-BPM) where we have one transverse dimension,x and the

propagation dimension,z. Generalization of BPM to three dimensions is straightforward.


6.2 The Relation between BPM and ABCD-Systems

As given in Section 6.1, BPM should run the ASM method or any other method for ho-

mogeneous propagation and then make a phase compensation asa multiplicative filter for

each slice. Typically, BPM is implemented with large amounts of slices to better satisfy the

assumptions made in its derivation. Although faster than some other computational tools,

it still carries significant amounts of burden due to its iterative nature and the repetitive cal-

culations on the large number of data points of the wavefunction. In this chapter, we show

that, under a quadratic-phase approximation, each step of BPM can be written as a QPS.

By representing eachPniandCi as a QPS defined by ABCD matrices and then combining

them by using the cascade rule for QPSs as given in Section 2.1.1, we can obtain a single

QPS. This overall QPS, which is an approximation to BPM, is defined by a single ABCD

matrix. Given the input wave field, there are fast algorithmsto digitally calculate the output

field in O(N log N) time, [60, 63, 64]. Thus, the iterative nature of BPM can be by-passed

and computational savings by several factors can be achieved.

The homogenous propagation part of BPM,Pni, can also be implemented by using

Fresnel transform/propagation instead of ASM, [16]. The Fresnel transform gives the out-

put field,E(x, z), after a propagation of lengthz in an homogenous medium of refractive

indexn from the input fieldE(x, 0) with the relation:

E(x, z) =e

jn 2πλ0

z

√jλz

∫E(x′, 0)e

jnπ

λ0z(x−x′)2

dx′ (6.8)

It is a special case of QPS systems and the corresponding ABCDmatrix, denoted byMF ,

is given by

MF =

1 λ0zn

0 1

(6.9)

whereα = β = γ = nλ0z

. If one needs to care about the constant phase terms preceding the

6.2. THE RELATION BETWEEN BPM AND ABCD-SYSTEMS 119

above equation, it should be taken into account that the ABCDmatrix formalism for the

Fresnel transform does not take into account the preceding constant phase-term ofejn 2πλ0

z,

which should be added to the resulting field at the end.

So far, we did not make any approximation to the BPM. The crucial operation to relate

the BPM and QPSs is the implementation of the multiplicativephase correctionCi. Since

this phase correction operates as a multiplicative filter inthe BPM implementation, it pre-

vents us from obtaining an overall calculation but requiresus to stick to iteration. If we

approximate this multiplicative phase filter with a quadratic-phase filter we can represent

this step as a special case of QPSs as well. The quadratic-phase filter is actually a chirp-

multiplication operation given byE ′(x, z) = e−jπqx2E(x, z). The corresponding ABCD

matrix is

MC =

1 0

−q 1

. (6.10)

Recall the phase correction term

Ci = exp (−j (n(x, zi) − ni) k0∆z) = exp (−j∆n(x, zi)k0∆z) (6.11)

in Eq. 6.6. We want to approximate it with a formexp (−j(ax2 + b)k0∆z). To find the best

approximation we use the Mean-squared error (MSE) approximation such that we solve the

following optimization problems:

minai,bi

(aix2 + bi − ∆n(x, zi))

2 (6.12)

to get thea andb parameters of the quadratic approximation for each layer, where (i =

1, 2, ..., N) represents the BPM slices alongz. Because we work on digital computational

problems, a discritization and then practical solution of the above optimization problem


can be carried out as follows. We discritize the problem in Eq. 6.12 such that

minai,bi

Nx∑

k=1

(aix2k + bi − ∆n(xk, zi))

2 (6.13)

where (k = 1, 2, ..., Nx) represents the samples taken along the transverse direction. Solv-

ing this optimization problem yields the following two dimensional system for eachi:

ai

bi

=

∑Nx

k=1 x4k

∑Nx

k=1 x2k

∑Nx

k=1 x2k Nx

−1

∑Nx

k=1 x2k∆n(xk, zi)

∑Nx

k=1 ∆n(xk, zi)

. (6.14)

Once these optimization parameters are determined, we can write the ABCD matrix repre-

sentationSi of a single BPM iteration step as follows:

Si =

1 0

−2ai∆z/λ0 1

1 λ0∆z/ni

0 1

(6.15)

with the constant phase residueexp(j 2π

λ0∆z(ni − bi)

). Note that thebi terms in this con-

stant phase residue are due to the quadratic approximation and ni terms are due to the

constant phase compensation of the Fresnel transform. Thenthe overall ABCD matrix that

approximates the inhomogeneous media with a QPS can be obtained by simple matrix mul-

tiplication requiring manipulation of only2 × 2 matrices. The overall matrixMoverall is

given by

Moverall = SNSN−1 · · · S2S1 (6.16)

and the constant phase residues can be combined to add at the very end altogether as

exp(j 2π

λ0∆z∑

i(ni − bi))

. Once theMoverall is calculated, it can be computed fast by al-

gorithms given in [60,63,64] for only one single wave field given at the input side, instead

of iterating the intermediate wavefunctions in the classical BPM. The cost of optimization,

which only includes a matrix multiplication and summations, and the cost of calculating

6.3. NUMERICAL TESTS 121

2×2 matrixMoverall are considerably smaller than the savings done by not performing the

BPM iterations on the wavefield that are typically sampled atlarge amounts of data points.

6.3 Numerical Tests

To test and characterize the reported relation between BPM and QPS, we set up some

example systems. We implemented both BPM and the QPS implementation of it, calculate

the output fields from both methods and then compare them. Thefirst system we consider is

the propagation along a medium with an arbitrary refractiveindex distribution. The system

can be seen in Fig. 6.1. Here the input field propagates along the positivez-axis. The media

is also divided intoN = 25 layers of∆z width along the propagation direction. Each layer

has a different average refractive indexni (wherei = 1, 2, ..., N) chosen randomly between

2.5 and 3.5. Each layer is also divided into 4 sub-regions with arbitrarily chosen transverse

widths as shown in Fig. 6.1. Each of these subregions has a random index modulation

where the maximum deviation from the average is2.5 × 10−3. These variations are also

different from those of the other layers. As∆n ≪ n, this ensures the validity of the

SVEA of BPM. A square-wave input field, which is defined from−10 mm to10 mm with

a support of2 mm width, is propagated along thez-axis for 0.1 mm. The input field is

sampled with 256 samples and 10,000 BPM layers are used. The wavelength is650 nm.

The error between the output fields of the conventional BPM and our QPS-based BPM

is 0.9%. The error is defined as the energy of the difference normalized by the energy of

the reference (conventional BPM), expressed as a percentage. In addition to this quite high

accuracy, our QPS-based BPM is 72 times faster than conventional BPM. The computa-

tional times are measured usingMATLAB’s tic andtoc methods. The fields at the output of

the system are plotted in Figs. 6.2 and 6.3.

As a second example system, we consider the generalization of GRIN systems to show

the extension of the limits of QPS to represent more general GRIN media. We assume the


Figure 6.1: Test System 1

second test media has GRIN distribution along the transverse direction but a different GRIN

profile is present for slices of every 10 nm. Basically, for each of these slices, we have

different parameters of GRIN. In this way, the media obtainssome kind of inhomogeneous

distribution. The index distribution of this medium is given in Fig. 6.4 and the average

index distribution along thez direction is plotted in Fig. 6.5. The wavelength of the light,

sampling extensions and input function are same as those of the first test system.

The error between the two methods for this second test case is1.64% and the speed im-

provement is now by a factor of74. The results are plotted in Figs. 6.6 and 6.7. This second

example particularly shows the extension of the ability of QPSs to representgeneralized

GRIN media.


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 10−3

0

0.2

0.4

0.6

0.8

1

1.2

1.4BPM and QPS−BPM outputs (magnitudes)

x in meters

QPS−BPMBPM

Figure 6.2: Amplitudes for Test System 1


Our aim is to make the link between two important topics in optics and to give new in-

sights to both of these important methods. By showing the conditions in which the BPM

can be represented as a QPS and then can be computed in a more efficient way, we can

contribute to both of these techniques. For BPM, exploitingthis relationship can lead to

faster implementations for certain systems. For the QPS point of view, this relationship

shows that the limits of the capabilities of QPSs can be stretched to the point that they

can, in a certain extent, represent the wave-propagation insome inhomogeneous medium

satisfying the aforementioned conditions. Especially, QPSs are shown to be capable of


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 10−3

−3

−2

−1

0

1

2

3BPM and QPS−BPM outputs (phases)

x in meters

Pha

se in

rad

ians

QPS−BPMBPM

Figure 6.3: Phases for Test System 1

representing a generalized GRIN medium, which has a more inhomogeneous index profile

than that of GRIN media. This definitely expands the theory ofQPS and ABCD-systems.

On top of these, we think that the demonstration of such a relationship is itself interesting

and can make two different important areas of research converge towards each other for

likely future developments and contributions.

BPM is generally used to find only the intensity within the structure. This also strength-

ens the applicability of our method since its main source of error comes from the phase

approximations. Also, even if there are some sub-parts of the entire region, in which our


Index distribution for Test System 2

z in meters

x in

met

ers

0 2 4 6 8 10

x 10−5

−10

−8

−6

−4

−2

0

2

4

6

8

x 10−3

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

1.95

Figure 6.4: Index distribution for Test System 2

QPS-based BPM method’s performance degrades because the quadratic-phase approxima-

tion does not hold, conventional BPM can be used for these sub-parts and for the other

sub-parts our fast method can be used instead of iteration. Using this hybrid method in-

stead of using BPM for the entire region, the speed of the calculation can still be partially

increased.


0 0.2 0.4 0.6 0.8 1

x 10−4

1.4

1.5

1.6

1.7

1.8

1.9

2

z in meters

nAverage index along the z axis for Test System 2

Figure 6.5: Average index for Test System 2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 10−3

0

0.2

0.4

0.6

0.8

1

1.2

1.4BPM and QPS−BPM outputs (magnitudes)

x in meters

QPS−BPMBPM

Figure 6.6: Amplitudes for Test System 2


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 10−3

−4

−3

−2

−1

0

1

2

3

4BPM and QPS−BPM outputs (phases)

x in meters

Pha

se in

rad

ians

QPS−BPMBPM

Figure 6.7: Phases for Test System 2

Chapter 7

Conclusion

IN THIS THESIS, four algorithms for the computation of linear canonical transforms

(LCTs) from theN samples of the input signal inO(N log N) time are discussed. Our

approach is based on concepts from signal analysis and processing rather than conventional

numerical analysis. With careful consideration of sampling issues,N can be chosen very

close to the time-bandwidth product of the signals, and neednot be much larger. The trans-

form output may have a higher space-bandwidth product due tothe nature of the transform

family.

All algorithms relate the samples of the input function to the samples of the continuous

LCT of this function in the same sense that the fast Fourier transform (FFT) implementation

of the discrete Fourier transform (DFT) computes the samples of the continuous FT of a

function. Since the sampling rates are carefully controlled, the output samples obtained

are accurate approximations to the true ones and the continuous LCT can be recovered

via interpolation of these samples. The only inevitable source of deviation from exactness

arises from the fundamental fact that a signal and its transform cannot both be of finite

extent. This is the same source of deviation encountered when using the DFT/FFT to

compute the continuous FT. Thus the algorithms compute LCTswith a performance similar

to the DFT/FFT in computing the Fourier transform, both in terms of speed and accuracy.

129

130 CHAPTER 7. CONCLUSION

This limitation affects not only the separable and one-dimensional versions of the algo-

rithm reported earlier, but also the computation of Fouriertransforms using the DFT. Thus

this is a source of error we cannot hope to overcome.

Compared to earlier approaches, these algorithms not only handle a much more general

family of integrals, but also effectively address certain difficulties, limitations, or tradeoffs

that arise in other approaches to computing the Fresnel integral, which is of importance in

the theory of diffraction.

We have also developed the link between the compact, matrix-based 16-parameter def-

inition of two-dimensional non-separable LCTs and the 10-parameter explicit kernel defi-

nition.

Complex-parametered LCTs allow several kinds of optical systems to be represented,

including lossy as well as lossless ones. When complex parameters are involved, LCTs may

no longer be unitary and boundedness issues may arise. We have identified the conditions

for a CLCT to constitute a bounded map from functions on the real axis to functions on

the real axis. As a special case of our general CLCT algorithm, we have also obtained an

efficient and accurate algorithm for complex-ordered fractional Fourier transforms.

7.1 Future Work

In this part, several possible future extensions and research directions for the algorithms

and approaches developed in this thesis will be summarized.

In this thesis algorithms up to 2D-LCTs are covered. 2D-LCTsare sufficient to repre-

sent a realistic 3D optical system where we have two transverse directions and a direction

of propagation. However, from a more theoretical and abstract point of view, there are

LCTs with more than two dimensions. The fundamental definitions of the LCTs can be ex-

tended straightforwardly to several dimensions. Then, thefast algorithms and other issues

can be studied in these high dimensional domains.

7.1. FUTURE WORK 131

As the name implies, LCTs are ’linear’ transforms. In addition to the linear versions of

them, the other related class is the Radial canonical transforms, [39,121]. Radial transforms

are defined in multidimensions and assume spherical symmetry. For example, the Hankel

transform, which is defined in terms of the famous Bessel functions of the first kind and

which is important in telecommunications, is a prominent member of the radial canonical

transforms. The question of whether similar fast and efficient algorithms for computation

of these radial transforms can be found is another future extension of the research in this

thesis.

LCTs are also closely associated to the underlying mathematical theory and opera-

tions like the Radon transform and the Projection-Slice Theorem used in medical imaging,

tomography and magnetic resonance, [135–138]. Researching the ways to extend these

algorithms and/or the key decomposition based approaches to the concepts used in medical

imaging can be done to see whether the decomposition approaches can improve the image

processing algorithms in those fields.

The relationship of the LCTs to the Beam Propagation Method (BPM) is a very in-

teresting finding. Currently, the relationship derived in Chapter 6 of the thesis has some

restrictions and it is revealed that the relationship holdssubject to some conditions about

the refractive index distribution of the medium. More research can be carried out to further

investigate this relationship and to look for possible potential new findings and methods to

enlarge the types of systems that can be modeled by using LCTs(or equivalently QPSs).

Another future extension to the thesis may be the development of new simulation and

modeling software based on the developed algorithms with all graphical user interfaces

and modeling interfaces. Polishing the code and developinghigh-level software additions

to allow users to define their optical systems and to set parameters in a graphical tool can

be done. Then the computational fast algorithms are used underneath to quickly simulate

the system and obtain the output. Ultimately, it may be possible to release this as some

stand alone simulation software for first-order optical system design and analysis.

132 CHAPTER 7. CONCLUSION

Appendix A

Simpson’s Rule for 1D and 2D Functions

In this Section, Simpson’s rule will be summarized. Simpson’s rule is a very-well known

method for numerical integration in numerical analysis, [139, 140]. The Simpson’s rule is

credited to the famous mathematician Thomas Simpson of the 18th Century.

Simpson’s rule approximates the definite integral of a function f by using quadratic

polynomials. For one-dimensional functions the basic approximation equation is the fol-

lowing:

∫ b

a

f(x) dx ≈ b − a

6

[f(a) + 4f

(a + b

2

)+ f(b)

](A.1)

where the corresponding error termE is given by

E =1

90(b − a)5f (4)(x) (A.2)

wherex is an arbitrary value in the interval[a, b]. The accuracy of the Simpson’s method is

quite high if the function to be approximated is smooth in theinterval[a, b]. To satisfy this,

usually a composite Simpson’s rule should be used. In this method, the main interval[a, b]

is divided intom number of subintervals, wherem is even. Then, let us divide[a, b] into

m subintervals denoted by{[xk−1, xk]}m/2k=1 where the length of each subinterval is given by

133

134 APPENDIX A. SIMPSON’S RULE FOR 1D AND 2D FUNCTIONS

(b − a)/m. Finally,

∫ b

a

f(x) dx ≈ b − a

3(f(a)+f(b))+

2(b − a)

3

m/2−1∑

k=1

f(x2k)+4(b − a)

3

m/2∑

k=1

f(x2k−1) (A.3)

The error term for the composite Simpson’s rule is

E = −(b − a)5f (4)(x)

180m4(A.4)

wherex is an arbitrary value in the interval[a, b].

By takingm very large enough, one can obtain very short intervals in which the function

f is almost constant. Therefore, the composite Simpson’s method can be used to find very

accurate results at the cost of computational time.

For two-dimensional functions,f(x, y), there are also Simpson’s rule based meth-

ods, [119]. The generalization to two dimensions are as follows:

The double integral that needs to be numerically calculatedis:

∫ d

c

∫ b

a

f(x, y) dx dy (A.5)

The main interval alongx-direction[a, b] is divided byn number of subintervals and along

y-direction[c, d] is divided bym number of subintervals wheren andm are even. Then the

two-dimensional integral can be approximated as

∫ d

c

∫ b

a

f(x, y) dx dy ≈ (b − a)(d − c)

36nm

n∑

i=1

m∑

j=1

Qi,jf(a + i(b − a)/n, c + j(d − c)/m)

(A.6)

whereQi,j is following the one-dimensional Simpson’s rule coefficient pattern1, 4, 2, 4,

2, 4, ..., 2, 4, 1 along the boundaries of the integration region (ie. for at least one ofi andj

is zero) and following the ruleQi,j = Qi,0Q0,j in the interior regions.

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Aykut Koc

I certify that I have read this dissertation and that, in my opinion, it is fully

adequate in scope and quality as a dissertation for the degree of Doctor of

Philosophy.

(Lambertus Hesselink) Principal Advisor



Philosophy.

(Shanhui Fan)



Philosophy.

(R. Fabian Pease)

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fast algorithms for the digital computation of linear canonical transforms a dissertation submitted...

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