A First Course in Wavelets with Fourier Analysis

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 -p = x = p, 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