A First Course in Wavelets with Fourier Analysis (Hardcover, 2)

· 제목 : A First Course in Wavelets with Fourier Analysis (Hardcover, 2) 
· 분류 : 외국도서 > 과학/수학/생태 > 수학 > 응용수학
· ISBN : 9780470431177
· 쪽수 : 336쪽
· 출판일 : 2009-09-01

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