Cover
A First Course in Wavelets with Fourier Analysis, 2nd Edition
Contents
Preface and Overview
0 Inner Product Spaces
0.1 Motivation
0.2 Definition of Inner Product
0.3 The Spaces Lยฒ and lยฒ
0.3.1 Definitions
Lยฒ Inner Product
The Space lยฒ
Relative error
0.3.2 Convergence in Lยฒ Versus Uniform Convergence
0.4 Schwarz and Triangle Inequalities
0.5 Orthogonality
0.5.1 Definitions and Examples
0.5.2 Orthogonal Projections
0.5.3 Gram-Schmidt Orthogonalization
0.6 Linear Operators and Their Adjoints
0.6.1 Linear Operators
0.6.2 Adjoints
0.7 Least Squares and Linear Predictive Coding
0.7.1 Best-Fit Line for Data
0.7.2 General Least Squares Algorithm
0.7.3 Linear Predictive Coding
Main Idea
Role of Least Squares
Summary of Linear Predictive Coding
Exercises
1 Fourier Series
1.1 Introduction
1.1.1 Historical Perspective
1.1.2 Signal Analysis
1.1.3 Partial Differential Equations
Separation of Variables
1.2 Computation of Fourier Series
1.2.1 On the Interval -ฯ โค x โค ฯ
1.2.2 Other Intervals
Intervals of General Length
1.2.3 Cosine and Sine Expansions
Even and Odd Functions
Fourier Cosine and Sine Series on a Half Interval
1.2.4 Examples
1.2.5 The Complex Form of Fourier Series
Relation Between the Real and Complex Fourier Series
1.3 Convergence Theorems for Fourier Series
1.3.1 The Riemann-Lebesgue Lemma
1.3.2 Convergence at a Point of Continuity
Step 1. Substituting the Fourier Coefficients
Step 2. Evaluating the Sum on the Right Side
Step 3. Evaluation of the Partial Sum of Fourier Series
Step 4. Integrating the Fourier Kernel
Step 5. The End of the Proof of Theorem 1.22
1.3.3 Convergence at a Point of Discontinuity
1.3.4 Uniform Convergence
1.3.5 Convergence in the Mean
Exercises
2 The Fourier Transform
2.1 Informal Development of the Fourier Transform
2.1.1 The Fourier Inversion Theorem
Comparison with Fourier Series
2.1.2 Examples
2.2 Properties of the Fourier Transform
2.2.1 Basic Properties
2.2.2 Fourier Transform of a Convolution
2.2.3 Adjoint of the Fourier Transform
2.2.4 Plancherel Theorem
2.3 Linear Filters
2.3.1 Time-Invariant Filters
Physical Interpretation
2.3.2 Causality and the Design of Filters
A Faulty Filter
Causal Filters
2.4 The Sampling Theorem
2.5 The Uncertainty Principle
Proof of Uncertainty Principle
Exercises
3 Discrete Fourier Analysis
3.1 The Discrete Fourier Transform
3.1.1 Definition of Discrete Fourier Transform
3.1.2 Properties of the Discrete Fourier Transform
3.1.3 The Fast Fourier Transform
3.1.4 The FFT Approximation to the Fourier Transform
3.1.5 Application: Parameter Identification
3.1.6 Application: Discretizations of Ordinary Differential Equations
3.2 Discrete Signals
3.2.1 Time-Invariant, Discrete Linear Filters
3.2.2 Z-Transform and Transfer Functions
Connection with Fourier Series
Isometry Between Lยฒ and lยฒ
Convolutions
Adjoint of Convolution Operators.
3.3 Discrete Signals & Matlab
Exercises
4 Haar Wavelet Analysis
4.1 Why Wavelets?
4.2 Haar Wavelets
4.2.1 The Haar Scaling Function
4.2.2 Basic Properties of the Haar Scaling Function
4.2.3 The Haar Wavelet
4.3 Haar Decomposition and Reconstruction Algorithms
4.3.1 Decomposition
4.3.2 Reconstruction
4.3.3 Filters and Diagrams
4.4 Summary
Exercises
5 Multiresolution Analysis
5.1 The Multiresolution Framework
5.1.1 Definition
5.1.2 The Scaling Relation
5.1.3 The Associated Wavelet and Wavelet Spaces
5.1.4 Decomposition and Reconstruction Formulas: A Tale of Two Bases
5.1.5 Summary
5.2 Implementing Decomposition and Reconstruction
5.2.1 The Decomposition Algorithm
Initialization
Iteration
Termination
5.2.2 The Reconstruction Algorithm
Initialization
Iteration
Termination
5.2.3 Processing a Signal
5.3 Fourier Transform Criteria
5.3.1 The Scaling Function
5.3.2 Orthogonality via the Fourier Transform
An Interesting Identity
5.3.3 The Scaling Equation via the Fourier Transform
5.3.4 Iterative Procedure for Constructing the Scaling Function
Exercises
6 The Daubechies Wavelets
6.1 Daubechies' Construction
6.2 Classification, Moments, and Smoothness
Singularity Detection
6.3 Computational Issues
Zero-Padding
Periodic Extension
Smooth Padding
Symmetric Extensions
6.4 The Scaling Function at Dyadic Points
Exercises
7 Other Wavelet Topics
7.1 Computational Complexity
7.1.1 Wavelet Algorithm
7.1.2 Wavelet Packets
7.2 Wavelets in Higher Dimensions
Exercises on 2D Wavelets
7.3 Relating Decomposition and Reconstruction
7.3.1 Transfer Function Interpretation
7.4 Wavelet Transform
7.4.1 Definition of the Wavelet Transform
7.4.2 Inversion Formula for the Wavelet Transform
Proof of the Wavelet Transform Theorem
Appendix A: Technical Matters
A.1 Proof of the Fourier Inversion Formula
A.2 Technical Proofs from Chapter 5
A.2.1 Rigorous Proof of Theorem 5.17
A.2.2 Proof of Theorem 5.10
A.2.3 Proof of the Convergence Part of Theorem 5.23
Appendix B: Solutions to Selected Exercises
Chapter 0
Chapter 1
Chapter 2
Chapter 4
Chapter 5
Chapter 6
Appendix C: MATLABยฎ Routines
C.1 General Compression Routine
C.2 Use of MATLAB's FFT Routine for Filtering and Compression
C.3 Sample Routines Using MATLAB's Wavelet Toolbox
C.4 MATLAB Code for the Algorithms in Section 5.2
Bibliography
Index