Wavelet Transforms and Their Applications

Lokenath Debnath

Wavelet Transforms and Their Applications

With 69 Figures

Springer Science+Business Media, LLC

Lokenath Debnath

Wavelet Transforms and Their Applications

With 69 Figures

Springer Science+Business Media, LLC

Lokenath Debnath Department of Mathematics University of Texas-Pan American Edinburg, TX 78539-2999 USA

Library of Congress Cataloging-in-Publication Data Debnath, Lokenath.

Wavelet transforms and their applications / Lokenath Debnath. p. cm.

Includes bibliographical references and index. ISBN 978-1-4612-6610-5 ISBN 978-1-4612-0097-0 (eBook) DOI 10.1007/978-1-4612-0097-0 1. Wavelets (Mathematics) 2. Signal processing-Mathematics. I. Title.

QA403.3 .D43 2001 621.382'2---dc21 2001035266

Printed on acid-free paper.

© 2002 Springer Science+Business Media New York Originally published by Birkhăuser Boston in 2002 Softcover reprint ofthe hardcover 1 st edition 2002 AII rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher, Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodol­ogy now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely byanyone.

ISBN 978-1-4612-6610-5 SPIN 10773883

Production managed by Louise Farkas; manufacturing supervised by Jerome Basma. Typeset by the author.

9 8 7 6 5 432 1

Lokenath Debnath Department of Mathematics University of Texas-Pan American Edinburg, TX 78539-2999 USA

Library of Congress Cataloging-in-Publication Data Debnath, Lokenath.

Wavelet transforms and their applications / Lokenath Debnath. p. cm.

Includes bibliographical references and index. ISBN 978-1-4612-6610-5 ISBN 978-1-4612-0097-0 (eBook) DOI 10.1007/978-1-4612-0097-0 1. Wavelets (Mathematics) 2. Signal processing-Mathematics. I. Title.

QA403.3 .D43 2001 621.382'2---dc21 2001035266

Printed on acid-free paper.

© 2002 Springer Science+Business Media New York Originally published by Birkhăuser Boston in 2002 Softcover reprint ofthe hardcover 1 st edition 2002 AII rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher, Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodol­ogy now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely byanyone.

ISBN 978-1-4612-6610-5 SPIN 10773883

Production managed by Louise Farkas; manufacturing supervised by Jerome Basma. Typeset by the author.

9 8 7 6 5 432 1

Contents

Preface xi

Chapter 1 Brief Historical Introduction 1

1.1 Fourier Series and Fourier Transfonns ................................................. .

1.2 Gabor Transforms ..... .......... ......... ........... ... ... ........ .......... ........ .... .... ........ 4

1.3 The Wigner-Ville Distribution and Time-Frequency

Signal Analysis.... ... ........... ......... ......... ....................... ...... .... ................ ... 7

1.4 Wavelet Transfonns ............................................................................... 12

1.5 Wavelet Bases and Multiresolution Analysis ........................................ 17

1.6 Applications of Wavelet Transfonns ..................................................... 20

Chapter 2 Hilbert Spaces and Orthonormal Systems 23

2.1 Introduction ............. .... ................... ............ ................... ........ ................. 23

2.2 Normed Spaces ....................................................................................... 25

2.3 The I! Spaces ........................................................................................ 28

2.4 Generalized Functions with Examples .................................................. 35

2.5 Definition and Examples of an Inner Product Space ............................ 46

2.6 Norm in an Inner Product Space .................. .......................................... 50

2.7 Definition and Examples of a Hilbert Space ......................................... 53

2.8 Strong and Weak Convergences ............................................................ 59

2.9 Orthogonal and Orthonormal Systems .................................................. 62

vi Contents

2.10 Properties of Orthonormal Systems ....................................................... 68

2.11 Trigonometric Fourier Series ......... .................... .......... ..... ... ... .... ........... 79

2.12 Orthogonal Complements and the Projection Theorem ........................ 83

2.13 Linear Funtionals and the Riesz Representation Theorem ................... 89

2.14 Separable Hilbert Spaces ........................................................................ 92

2.15 Linear Operators on Hilbert Spaces ....................................................... 95

2.16 Eigenvalues and Eigenvectors of an Operator ..... ........ ... .... .................. 117

2.17 Exercises ................................................................................................. 130

Chapter 3 Fourier Transforms and Their Applications 143

3.1 Introduction ............................................................................................ 143

3.2 Fourier Transforms in LI (~) ................................................................ 145

3.3 Basic Properties of Fourier Transforms ................................................. 150

3.4 Fourier Transforms in I! (~) ................................................................ 166

3.5 Poisson's Summation Formula .............................................................. 182

3.6 The Shannon Sampling Theorem and Gibbs's Phenomenon ............... 187

3.7 Heisenberg's Uncertainty Principle ....................................................... 200

3.8 Applications of Fourier Transforms in Mathematical Statistics ........... 202

3.9 Applications of Fourier Transforms to Ordinary

Differential Equations ............................................................................ 210

3.10 Solutions ofIntegral Equations .............................................................. 214

3.11 Solutions of Partial Differential Equations ........................................... 218

3.12 Applications of Multiple Fourier Transforms to

Partial Differential Equations ........... ............ ..... ... ........ ............... .......... 230

3.13 Construction of Green's Functions by the Fourier

Transform Method ...... ................................................ ..................... ....... 236

3.14 Exercises ................................................................................................. 249

Contents

Chapter 4 The Gabor Transform and Time-Frequency

Signal Analysis

vii

257

4.1 Introduction ............................................................................................ 257

4.2 Classification of Signals and the Joint Time-Frequency

Analysis of Signals ................................................................................. 258

4.3 Definition and Examples of the Gabor Transforms .......................... .... 264

4.4 Basic Properties of Gabor Transforms .................................................. 269

4.5 Frames and Frame Operators ................................................................. 274

4.6 Discrete Gabor Transforms and the Gabor

Representation Problem ........... ................... ...... ................... ....... ........... 284

4.7 The Zak Transform and Time-Frequency Signal Analysis .................. 287

4.8 Basic Properties of Zak Transforms ...................................................... 290

4.9 Applications ofZak Transforms and the Balian-Low Theorem ........... 295

4.10 Exercises... .... .... ..... ............. ........................................ ......... ............ ....... 304

Chapter 5 The Wigner-Ville Distribution and

Time-Frequency Signal Analysis 307

5.1 Introduction ............................................................................................ 307

5.2 Definitions and Examples of the Wigner-Ville Distribution ................ 308

5.3 Basic Properties of the Wigner-Ville Distribution ................................ 319

5.4 The Wigner-Ville Distribution of Analytic Signals and

Band-Limited Signals ............................................................................. 328

5.5 Definitions and Examples of the Woodward

Ambiguity Functions .............................................................................. 331

5.6 Basic Properties of Ambiguity Functions .............................................. 339

5.7 The Ambiguity Transformation and Its Properties ............................... 346

5.8 Discrete Wigner-Ville Distributions ...................................................... 350

5.9 Cohen's Class of Time-Frequency Distributions .................................. 354

5.10 Exercises ................................................................................................. 357

viii Contents

Chapter 6 Wavelet Transforms and Basic Properties 361

6.1 Introduction ............................................................................................ 361

6.2 Continuous Wavelet Transforms and Examples ................................... 365

6.3 Basic Properties of Wavelet Transforms ............................................... 378

6.4 The Discrete Wavelet Transforms ......................................................... 382

6.5 Orthonormal Wavelets ........................................................................... 392

6.6 Exercises ................................................................................................. 399

Chapter 7 Multiresolution Analysis and Construction

of Wavelets

403

7.1 Introduction .............. ........... ........ ... ..................... ... .......... ... .......... ...... ... 403

7.2 Definition of Multiresolution Analysis and Examples ......................... 405

7.3 Properties of Scaling Functions and Orthonormal Wavelet Bases ....... 412

7.4 Construction of Orthonormal Wavelets ................................................. 431

7.5 Daubechies' Wavelets and Algorithms ................................................. 447

7.6 Discrete Wavelet Transforms and Mallat's Pyramid Algorithm .......... 466

7.7 Exercises .................................................................................................... 471

Chapter 8 Newland's Harmonic Wavelets 475

8.1 Introduction ............................................................................................ 475

8.2 Harmonic Wavelets ................................................................................ 475

8.3 Properties of Harmonic Scaling Functions ............................................ 482

8.4 Wavelet Expansions and Parseval's Formula ....................................... 485

8.5 Concluding Remarks ............. .......................... ...... ........ ................ .... ..... 487

8.6 Exercises ................................................................................................. 487

Chapter 9 Wavelet Transform Analysis of Turbulence 491

9.1 Introduction ............................................................................................ 492

Contents ix

9.2 Fourier Transforms in Turbulence and the

Navier-Stokes Equations ........................................................................ 495

9.3 Fractals, Multifractals, and Singularities in Turbulence ....................... 505

9.4 Farge's Wavelet Transform Analysis of Turbulence ............................ 512

9.5 Adaptive Wavelet Method for Analysis of Turbulent Flows ............... 515

9.6 Meneveau's Wavelet Analysis of Turbulence ...................................... 519

Answers and Hints for Selected Exercises

Bibliography

Index

525

539

555

Preface

Overview Historically, the concept of "ondelettes" or "wavelets" originated from the

study of time-frequency signal analysis, wave propagation, and sampling theory.

One of the main reasons for the discovery of wavelets and wavelet transforms is

that the Fourier transform analysis does not contain the local information of

signals. So the Fourier transform cannot be used for analyzing signals in a joint

time and frequency domain. In 1982, Jean MorIet, in collaboration with a group

of French engineers, first introduced the idea of wavelets as a family of

functions constructed by using translation and dilation of a single function,

called the mother wavelet, for the analysis of nonstationary signals. However,

this new concept can be viewed as the synthesis of various ideas originating

from different disciplines including mathematics (Calder6n-Zygmund operators

and Littlewood-Paley theory), physics (coherent states in quantum mechanics

and the renormalization group), and engineering (quadratic mirror filters,

sideband coding in signal processing, and pyramidal algorithms in image

processing).

Wavelet analysis is an exciting new method for solving difficult problems in

mathematics, physics, and engineering, with modern applications as diverse as

wave propagation, data compression, image processing, pattern recognition,

computer graphics, the detection of aircraft and submarines, and improvement in

CAT scans and other medical image technology. Wavelets allow complex

information such as music, speech, images, and patterns to be decomposed into

elementary forms, called the fundamental building blocks, at different positions

and scales and subsequently reconstructed with high precision. With ever greater

demand for mathematical tools to provide both theory and applications for

xii Preface

science and engineering, the utility and interest of wavelet analysis seem more

clearly established than ever. Keeping these things in mind, our main goal in this

modest book has been to provide both a systematic exposition of the basic ideas

and results of wavelet transforms and some applications in time-frequency

signal analysis and turbulence.

Audience and Organization

This book is appropriate for a one-semester course in wavelet transforms

with applications. There are two basic prerequisites for this course: Fourier

transforms and Hilbert spaces and orthonormal systems. The book is also

intended to serve as a ready reference for the reader interested in advanced study

and research in various areas of mathematics, physics, and engineering to which

wavelet analysis can be applied with advantage. While teaching courses on

integral transforms and wavelet transforms, the author has had difficulty

choosing textbooks to accompany lectures on wavelet transforms at the senior undergraduate and/or graduate levels. Parts of this book have also been used to

accompany lectures on special topics in wavelet transform analysis at U.S. and

Indian universities. I believe that wavelet transforms can be best approached

through a sound knowledge of Fourier transforms and some elementary ideas of

Hilbert spaces and orthonormal systems. In order to make the book self­contained, Chapters 2 and 3 deal with Hilbert spaces and orthonormal systems

and Fourier transforms with examples of applications. It is not essential for the

reader to know everything about these topics, but limited knowledge of at least

some of them would be sufficient. There is plenty of material in this book for a one-semester graduate-level course for mathematics, science, and engineering

students. Many examples of applications to problems in time-frequency signal

analysis and turbulence are included. The first chapter gives a brief historical introduction and basic ideas of

Fourier series and Fourier transforms, Gabor transforms, and the Wigner-Ville

distribution with time-frequency signal analysis, wavelet transforms, wavelet

bases, and multiresolution analysis. Some applications of wavelet transforms are

also mentioned.

Chapter 2 deals with Hilbert spaces and orthonormal systems. Special

attention is given to the theory of linear operators on Hilbert spaces, with some

emphasis on different kinds of operators and their basic properties. The

fundamental ideas and results are discussed, with special attention given to

orthonormal systems, linear functionals, and the Riesz representation theorem.

Preface xiii

The third chapter is devoted to the theory of Fourier transforms and their

applications to signal processing, differential and integral equations, and

mathematical statistics. Several important results including the approximate

identity theorem, convolution theorem, various summability kernels, general

Parseval relation, and Plancherel' s theorem are discussed in some detail.

Included are Poisson's summation formula, Gibbs's phenomenon, the Shannon

sampling theorem, and Heisenberg's uncertainty principle.

Chapter 4 is concerned with classification of signals, joint time-frequency

analysis of signals, and the Gabor transform and its basic properties, including

the inversion formula. Special attention is given to frames and frame operators,

the discrete Gabor transform, and the Gabor representation problem. Included

are the Zak transform, its basic properties, including the Balian-Low theorem,

and applications for studying the orthogonality and completeness of Gabor

frames in the critical case.

The Wigner-Ville distribution and time-frequency signal analysis are the

main topics of Chapter 5. The basic structures and properties of the Wigner­

Ville distribution and the ambiguity function are discussed in some detail.

Special attention is paid to fairly exact mathematical treatment with examples

and applications in the time-frequency signal analysis. The relationship between

the Wigner-Ville distribution and ambiguity functions is examined with radar

signal analysis. Recent generalizations of the Wigner-Ville distribution are

briefly described.

Chapter 6 is devoted to wavelets and wavelet transforms with examples. The

basic ideas and properties of wavelet transforms are discussed with special

emphasis given to the use of different wavelets for resolution and synthesis of

signals. This is followed by the definition and properties of discrete wavelet

transforms. In Chapter 7, the idea of multiresolution analysis with examples and

construction of wavelets is described in some detail. This chapter includes

properties of scaling functions and orthonormal wavelet bases and construction

of orthonormal wavelets. Also included are treatments of Daubechies' wavelet

and algorithms, discrete wavelet transforms, and Mallat's pyramid algorithm.

Chapter 8 deals with Newland's harmonic wavelets and their basic

properties. Special attention is given to properties of harmonic scaling functions,

wavelet expansions, and Parseval's formula for harmonic wavelets.

The final chapter is devoted to a brief discussion of the Fourier transform

analysis and the wavelet transform analysis of turbulence based on the Navier-

xiv Preface

Stokes equations and the equation of continuity. Included are fractals,

multifractals, and singularities in turbulence. This is followed by Farge's and

Meneveau's wavelet transform analyses of turbulence in some detail. Special

attention is given to the adaptive wavelet method for computation and analysis

of turbulent flows.

Salient Features

The book contains a large number of worked examples, examples of

applications, and exercises which are either directly associated with applications

or phrased in terms of mathematical, physical, and engineering contexts in

which theory arises. It is hoped that they will serve as useful self-tests for

understanding of the theory and mastery of wavelets, wavelet transforms, and

other related topics covered in this book. A wide variety of examples,

applications, and exercises should provide something of interest for everyone.

The exercises truly complement the text and range from elementary to the

challenging.

This book is designed as a new source for modern topics dealing with

wavelets, wavelet transforms, Gabor transforms, the Wigner-Ville distribution,

multiresolution analysis, and harmonic wavelets and their applications for future

development of this important and useful subject. Its main features are listed

below:

1. A detailed and clear explanation of every concept and method which is

introduced, accompanied by carefully selected worked examples, with

special emphasis being given to those topics in which students experience

difficulty.

2. Special emphasis is given to the joint time-frequency signal analysis and the

ambiguity functions for the mathematical analysis of sonar and radar

systems.

3. Sufficient flexibility in the book's organization so as to enable instructors to

select chapters appropriate to courses of different lengths, emphases, and

levels of difficulty.

4. A wide spectrum of exercises has been carefully chosen and included at the

end of each chapter so that the reader may develop both manipulative skills

in the theory and applications of wavelet analysis and a deeper insight into

this most modern subject. Answers and hints for selected exercises are

provided at the end of the book for additional help to students.

Preface xv

5. The book provides important information that puts the reader at the forefront

of current research. An updated Bibliography is included to stimulate new

interest in future study and research.

Acknowledgments In preparing the book, the author has been encouraged by and has benefited

from the helpful comments and criticism of a number of faculty and postdoctoral

and doctoral students of several universities in the United States, Canada, and

India. The author expresses his grateful thanks go to these individuals for their

interest in the book. My special thanks go to Jackie Callahan and Ronee

Trantham who typed a manuscript with many diagrams and cheerfully put up

with constant changes and revisions. In spite of the best efforts of everyone

involved, some typographical errors doubtlessly remain. I do hope that these are

both few and obvious and will cause minimal confusion. Finally, the author

wishes to express his special thanks to Lauren Schultz, associate editor, Wayne

Yuhasz, executive editor, and the staff of Birkhauser for their help and

cooperation. I am deeply indebted to my wife, Sadhana, for her understanding

and tolerance while the book was being written.

Edinburg, Texas Lokenath Debnath