A First Course in Wavelets with Fourier Analysis

Transcription

1 * A First Course in Wavelets with Fourier Analysis Albert Boggess Francis J. Narcowich Texas A& M University, Texas PRENTICE HALL, Upper Saddle River, NJ 07458

2 Contents Preface Acknowledgments xi xix 0 Inner Product Spaces Motivation Definition of Inner Product The Spaces Iß and l Definitions Convergence in 1? versus Uniform Convergence Schwarz and Triangle Inequalities Orthogonality Definitions and Examples Orthogonal Projections Gram-Schmidt Orthogonalization Linear Operators and Their Adjoints Linear Operators Adjoints Least Squares and Linear Predictive Coding Best Fit Line for Data General Least Squares Algorithm Linear Predictive Coding Exercises 32 1 Fourier Series Introduction Historical Perspective Signal Analysis Partial Differential Equations 39 vn

3 viii Contents 1.2 Computation of Fourier Series On the Interval ir < x < w Other Intervals Cosine and Sine Expansions Examples The Complex Form of Fourier Series Convergence Theorems for Fourier Series The Riemann-Lebesgue Lemma Convergence at a Point of Continuity Convergence at a Point of Discontinuity Uniform Convergence Convergence in the Mean Exercises 82 2 The Fourier Transform Informal Development of the Fourier Transform The Fourier Inversion Theorem Examples Properties of the Fourier Transform Basic Properties Fourier Transform of a Convolution Adjoint of the Fourier Transform Plancherei Formula Linear Filters Time Invariant Filters Causality and the Design of Filters The Sampling Theorem The Uncertainty Principle Exercises Discrete Fourier Analysis The Discrete Fourier Transform Definition of Discrete Fourier Transform Properties of the Discrete Fourier Transform The Fast Fourier Transform The FFT Approximation to the Fourier Transform Application Parameter Identification Application Discretizations of Ordinary Differential Equations Discrete Signals Time Invariant, Discrete Linear Filters Z-Transform and Transfer Functions Exercises 151

4 Contents ix 4 Haar Wavelet Analysis Why Wavelets? Haar Wavelets The Haar Scaling Function Basic Properties of the Haar Scaling Function Basic Properties of the Haar Scaling Function The Haar Wavelet Haar Decomposition and Reconstruction Algorithms Decomposition Reconstruction Filters and Diagrams Summary Exercises Multiresolution Analysis The Multiresolution Framework Definition The Scaling Relation The Associated Wavelet and Wavelet Spaces Decomposition and Reconstruction Formulas: A Tale of Two Bases Summary Implementing Decomposition and Reconstruction The Decomposition Algorithm The Reconstruction Algorithm Processing a Signal Fourier Transform Criteria The Scaling Function Orthogonality via the Fourier Transform The Scaling Equation via the Fourier Transform Iterative Procedure for Constructing the Scaling Function Exercises The Daubechies Wavelets Daubechies's Construction Classification, Moments, and Smoothness Computational Issues The Scaling Function at Dyadic Points Exercises Other Wavelet Topics Computational Complexity Wavelet Algorithm Wavelet Packets Wavelets in Higher Dimensions 245

5 x Contents 7.3 Relating Decomposition and Reconstruction Transfer Function Interpretation Wavelet Transform Definition of the Wavelet Transform Inversion Formula for the Wavelet Transform 255 Appendix A Technical Matters 261 A.l Proof of the Fourier Inversion Formula 261 A.2 Rigorous Proof of Theorem A.2.1 Proof of Theorem A.2.2 Proof of the Convergence Part of Theorem Appendix B Matlab Routines 273 B.l General Compression Routine 273 B.2 Use of MATLAB'S FFT Routine for Filtering and Compression B.3 Sample Routines Using MATLAB'S Wavelet Toolbox 275 B.4 MATLAB Code for the Algorithms in Section Bibliography 279 Index 280