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A First Course in Wavelets with Fourier Analysis, 2nd Edition

ISBN: 978-1-118-62626-9
336 pages
December 2015
A First Course in Wavelets with Fourier Analysis, 2nd Edition (1118626265) cover image


A comprehensive, self-contained treatment of Fourier analysis and wavelets—now in a new edition

Through expansive coverage and easy-to-follow explanations, A First Course in Wavelets with Fourier Analysis, Second Edition provides a self-contained mathematical treatment of Fourier analysis and wavelets, while uniquely presenting signal analysis applications and problems. Essential and fundamental ideas are presented in an effort to make the book accessible to a broad audience, and, in addition, their applications to signal processing are kept at an elementary level.

The book begins with an introduction to vector spaces, inner product spaces, and other preliminary topics in analysis. Subsequent chapters feature:

  • The development of a Fourier series, Fourier transform, and discrete Fourier analysis

  • Improved sections devoted to continuous wavelets and two-dimensional wavelets

  • The analysis of Haar, Shannon, and linear spline wavelets

  • The general theory of multi-resolution analysis

  • Updated MATLAB code and expanded applications to signal processing

  • The construction, smoothness, and computation of Daubechies' wavelets

  • Advanced topics such as wavelets in higher dimensions, decomposition and reconstruction, and wavelet transform

Applications to signal processing are provided throughout the book, most involving the filtering and compression of signals from audio or video. Some of these applications are presented first in the context of Fourier analysis and are later explored in the chapters on wavelets. New exercises introduce additional applications, and complete proofs accompany the discussion of each presented theory. Extensive appendices outline more advanced proofs and partial solutions to exercises as well as updated MATLAB routines that supplement the presented examples.

A First Course in Wavelets with Fourier Analysis, Second Edition is an excellent book for courses in mathematics and engineering at the upper-undergraduate and graduate levels. It is also a valuable resource for mathematicians, signal processing engineers, and scientists who wish to learn about wavelet theory and Fourier analysis on an elementary level.

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Table of Contents

Preface and Overview ix

0 Inner Product Spaces 1

0.1 Motivation 1

0.2 Definition of Inner Product 2

0.3 The Spaces L2 and l2 4

0.3.1 Definitions 4

0.3.2 Convergence in L2 Versus Uniform Convergence 8

0.4 Schwarz and Triangle Inequalities 11

0.5 Orthogonality 13

0.5.1 Definitions and Examples 13

0.5.2 Orthogonal Projections 15

0.5.3 Gram–Schmidt Orthogonalization 20

0.6 Linear Operators and Their Adjoints 21

0.6.1 Linear Operators 21

0.6.2 Adjoints 23

0.7 Least Squares and Linear Predictive Coding 25

0.7.1 Best-Fit Line for Data 25

0.7.2 General Least Squares Algorithm 29

0.7.3 Linear Predictive Coding 31

Exercises 34

1 Fourier Series 38

1.1 Introduction 38

1.1.1 Historical Perspective 38

1.1.2 Signal Analysis 39

1.1.3 Partial Differential Equations 40

1.2 Computation of Fourier Series 42

1.2.1 On the Interval −π ≤ x ≤ π 42

1.2.2 Other Intervals 44

1.2.3 Cosine and Sine Expansions 47

1.2.4 Examples 50

1.2.5 The Complex Form of Fourier Series 58

1.3 Convergence Theorems for Fourier Series 62

1.3.1 The Riemann–Lebesgue Lemma 62

1.3.2 Convergence at a Point of Continuity 64

1.3.3 Convergence at a Point of Discontinuity 69

1.3.4 Uniform Convergence 72

1.3.5 Convergence in the Mean 76

Exercises 83

2 The Fourier Transform 92

2.1 Informal Development of the Fourier Transform 92

2.1.1 The Fourier Inversion Theorem 92

2.1.2 Examples 95

2.2 Properties of the Fourier Transform 101

2.2.1 Basic Properties 101

2.2.2 Fourier Transform of a Convolution 107

2.2.3 Adjoint of the Fourier Transform 109

2.2.4 Plancherel Theorem 109

2.3 Linear Filters 110

2.3.1 Time-Invariant Filters 110

2.3.2 Causality and the Design of Filters 115

2.4 The Sampling Theorem 120

2.5 The Uncertainty Principle 123

Exercises 127

3 Discrete Fourier Analysis 132

3.1 The Discrete Fourier Transform 132

3.1.1 Definition of Discrete Fourier Transform 134

3.1.2 Properties of the Discrete Fourier Transform 135

3.1.3 The Fast Fourier Transform 138

3.1.4 The FFT Approximation to the Fourier Transform 143

3.1.5 Application: Parameter Identification 144

3.1.6 Application: Discretizations of Ordinary Differential Equations 146

3.2 Discrete Signals 147

3.2.1 Time-Invariant Discrete Linear Filters 147

3.2.2 Z-Transform and Transfer Functions 149

3.3 Discrete Signals & Matlab 153

Exercises 156

4 Haar Wavelet Analysis 160

4.1 Why Wavelets? 160

4.2 Haar Wavelets 161

4.2.1 The Haar Scaling Function 161

4.2.2 Basic Properties of the Haar Scaling Function 167

4.2.3 The Haar Wavelet 168

4.3 Haar Decomposition and Reconstruction Algorithms 172

4.3.1 Decomposition 172

4.3.2 Reconstruction 176

4.3.3 Filters and Diagrams 182

4.4 Summary 185

Exercises 186

5 Multiresolution Analysis 190

5.1 The Multiresolution Framework 190

5.1.1 Definition 190

5.1.2 The Scaling Relation 194

5.1.3 The Associated Wavelet and Wavelet Spaces 197

5.1.4 Decomposition and Reconstruction Formulas: A Tale of Two Bases 201

5.1.5 Summary 203

5.2 Implementing Decomposition and Reconstruction 204

5.2.1 The Decomposition Algorithm 204

5.2.2 The Reconstruction Algorithm 209

5.2.3 Processing a Signal 213

5.3 Fourier Transform Criteria 214

5.3.1 The Scaling Function 215

5.3.2 Orthogonality via the Fourier Transform 217

5.3.3 The Scaling Equation via the Fourier Transform 221

5.3.4 Iterative Procedure for Constructing the Scaling Function 225

Exercises 228

6 The Daubechies Wavelets 234

6.1 Daubechies’ Construction 234

6.2 Classification Moments and Smoothness 238

6.3 Computational Issues 242

6.4 The Scaling Function at Dyadic Points 244

Exercises 248

7 Other Wavelet Topics 250

7.1 Computational Complexity 250

7.1.1 Wavelet Algorithm 250

7.1.2 Wavelet Packets 251

7.2 Wavelets in Higher Dimensions 253

Exercises on 2D Wavelets 258

7.3 Relating Decomposition and Reconstruction 259

7.3.1 Transfer Function Interpretation 263

7.4 Wavelet Transform 266

7.4.1 Definition of the Wavelet Transform 266

7.4.2 Inversion Formula for the Wavelet Transform 268

Appendix A: Technical Matters 273

A.1 Proof of the Fourier Inversion Formula 273

A.2 Technical Proofs from Chapter 5 277

A.2.1 Rigorous Proof of Theorem 5.17 277

A.2.2 Proof of Theorem 5.10 281

A.2.3 Proof of the Convergence Part of Theorem 5.23 283

Appendix B: Solutions to Selected Exercises 287

Appendix C: MATLAB® Routines 305

C.1 General Compression Routine 305

C.2 Use of MATLAB’s FFT Routine for Filtering and Compression 306

C.3 Sample Routines Using MATLAB’s Wavelet Toolbox 307

C.4 MATLAB Code for the Algorithms in Section 5.2 308

Bibliography 311

Index 313

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Author Information

ALBERT BOGGESS, PhD, is Professor of Mathematics at Texas A&M University. Dr. Boggess has over twenty-five years of academic experience and has authored numerous publications in his areas of research interest, which include overdetermined systems of partial differential equations, several complex variables, and harmonic analysis.

FRANCIS J. NARCOWICH, PhD, is Professor of Mathematics and Director of the Center for Approximation Theory at Texas A&M University. Dr. Narcowich serves as an Associate Editor of both the SIAM Journal on Numerical Analysis and Mathematics of Computation, and he has written more than eighty papers on a variety of topics in pure and applied mathematics. He currently focuses his research on applied harmonic analysis and approximation theory.

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The Wiley Advantage

  • Offers self-contained exposition of Fourier analysis and wavelets
  • Contains expanded applications to signal processing and additional exercises throughout the book
  • Provides complete proofs of the presented theory in addition to solutions to selected exercises in the back of the book
  • Features improved sections devoted to continuous wavelets and two-dimensional wavelets
  • Presents updated Matlab code in the appendix and online via a related website


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"A first course in wavelets with Fourier analysis, second edition is an excellent book for courses in mathematics and engineering at the upper-undergraduate and graduate levels. It is also a valuable resource for mathematicians, signal processing engineers, and scientists who wish to learn about wavelet theory and Fourier analysis on an elementary level." (Mathematical Reviews, 2011)

"The discussions of applications avoid the deep jargon of signal processing … accessible to a wider audience." (Book News, December 2009)

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