Preface |
Acknowledgments |
Inner Product Spaces / 0: |
Motivation / 0.1: |
Definition of Inner Product / 0.2: |
The Spaces L[superscript 2] and l[superscript 2] / 0.3: |
Definitions / 0.3.1: |
Convergence in L[superscript 2] versus Uniform Convergence / 0.3.2: |
Schwarz and Triangle Inequalities / 0.4: |
Orthogonality / 0.5: |
Definitions and Examples / 0.5.1: |
Orthogonal Projections / 0.5.2: |
Gram-Schmidt Orthogonalization / 0.5.3: |
Linear Operators and Their Adjoints / 0.6: |
Linear Operators / 0.6.1: |
Adjoints / 0.6.2: |
Least Squares and Linear Predictive Coding / 0.7: |
Best Fit Line for Data / 0.7.1: |
General Least Squares Algorithm / 0.7.2: |
Linear Predictive Coding / 0.7.3: |
Exercises / 0.8: |
Fourier Series / 1: |
Introduction / 1.1: |
Historical Perspective / 1.1.1: |
Signal Analysis / 1.1.2: |
Partial Differential Equations / 1.1.3: |
Computation of Fourier Series / 1.2: |
On the Interval -[pi] [less than or equal] x [less than or equal] [pi] / 1.2.1: |
Other Intervals / 1.2.2: |
Cosine and Sine Expansions / 1.2.3: |
Examples / 1.2.4: |
The Complex Form of Fourier Series / 1.2.5: |
Convergence Theorems for Fourier Series / 1.3: |
The Riemann-Lebesgue Lemma / 1.3.1: |
Convergence at a Point of Continuity / 1.3.2: |
Convergence at a Point of Discontinuity / 1.3.3: |
Uniform Convergence / 1.3.4: |
Convergence in the Mean / 1.3.5: |
The Fourier Transform / 1.4: |
Informal Development of the Fourier Transform / 2.1: |
The Fourier Inversion Theorem / 2.1.1: |
Properties of the Fourier Transform / 2.1.2: |
Basic Properties / 2.2.1: |
Fourier Transform of a Convolution / 2.2.2: |
Adjoint of the Fourier Transform / 2.2.3: |
Plancherel Formula / 2.2.4: |
Linear Filters / 2.3: |
Time Invariant Filters / 2.3.1: |
Causality and the Design of Filters / 2.3.2: |
The Sampling Theorem / 2.4: |
The Uncertainty Principle / 2.5: |
Discrete Fourier Analysis / 2.6: |
The Discrete Fourier Transform / 3.1: |
Definition of Discrete Fourier Transform / 3.1.1: |
Properties of the Discrete Fourier Transform / 3.1.2: |
The Fast Fourier Transform / 3.1.3: |
The FFT Approximation to the Fourier Transform / 3.1.4: |
Application--Parameter Identification / 3.1.5: |
Application--Discretizations of Ordinary Differential Equations / 3.1.6: |
Discrete Signals / 3.2: |
Time Invariant, Discrete Linear Filters / 3.2.1: |
Z-Transform and Transfer Functions / 3.2.2: |
Haar Wavelet Analysis / 3.3: |
Why Wavelets? / 4.1: |
Haar Wavelets / 4.2: |
The Haar Scaling Function / 4.2.1: |
Basic Properties of the Haar Scaling Function / 4.2.2: |
The Haar Wavelet / 4.2.3: |
Haar Decomposition and Reconstruction Algorithms / 4.3: |
Decomposition / 4.3.1: |
Reconstruction / 4.3.2: |
Filters and Diagrams / 4.3.3: |
Summary / 4.4: |
Multiresolution Analysis / 4.5: |
The Multiresolution Framework / 5.1: |
Definition / 5.1.1: |
The Scaling Relation / 5.1.2: |
The Associated Wavelet and Wavelet Spaces / 5.1.3: |
Decomposition and Reconstruction Formulas: A Tale of Two Bases / 5.1.4: |
Implementing Decomposition and Reconstruction / 5.1.5: |
The Decomposition Algorithm / 5.2.1: |
The Reconstruction Algorithm / 5.2.2: |
Processing a Signal / 5.2.3: |
Fourier Transform Criteria / 5.3: |
The Scaling Function / 5.3.1: |
Orthogonality via the Fourier Transform / 5.3.2: |
The Scaling Equation via the Fourier Transform / 5.3.3: |
Iterative Procedure for Constructing the Scaling Function / 5.3.4: |
The Daubechies Wavelets / 5.4: |
Daubechies's Construction / 6.1: |
Classification, Moments, and Smoothness / 6.2: |
Computational Issues / 6.3: |
The Scaling Function at Dyadic Points / 6.4: |
Other Wavelet Topics / 6.5: |
Computational Complexity / 7.1: |
Wavelet Algorithm / 7.1.1: |
Wavelet Packets / 7.1.2: |
Wavelets in Higher Dimensions / 7.2: |
Relating Decomposition and Reconstruction / 7.3: |
Transfer Function Interpretation / 7.3.1: |
Wavelet Transform / 7.4: |
Definition of the Wavelet Transform / 7.4.1: |
Inversion Formula for the Wavelet Transform / 7.4.2: |
Technical Matters / Appendix A: |
Proof of the Fourier Inversion Formula / A.1: |
Rigorous Proof of Theorem 5.17 / A.2: |
Proof of Theorem 5.10 / A.2.1: |
Proof of the Convergence Part of Theorem 5.23 / A.2.2: |
Matlab Routines / Appendix B: |
General Compression Routine / B.1: |
Use of MATLAB's FFT Routine for Filtering and Compression / B.2: |
Sample Routines Using MATLAB's Wavelet Toolbox / B.3: |
MATLAB Code for the Algorithms in Section 5.2 / B.4: |
Bibliography |
Index |
Preface |
Acknowledgments |
Inner Product Spaces / 0: |
Motivation / 0.1: |
Definition of Inner Product / 0.2: |
The Spaces L[superscript 2] and l[superscript 2] / 0.3: |