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| Preface and Overview | |
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| Inner Product Spaces | |
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| Motivation | |
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| Definition of Inner Product | |
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| The Spaces L<sup>2</sup> and l<sup>2</sup> | |
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| Definitions | |
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| Convergence in L<sup>2</sup> Versus Uniform Convergence | |
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| Schwarz and Triangle Inequalities | |
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| Orthogonality | |
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| Definitions and Examples | |
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| Orthogonal Projections | |
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| Gram-Schmidt Orthogonalization | |
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| Linear Operators and Their Adjoints | |
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| Linear Operators | |
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| Adjoints | |
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| Least Squares and Linear Predictive Coding | |
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| Best-Fit Line for Data | |
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| General Least Squares Algorithm | |
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| Linear Predictive Coding | |
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| Exercises | |
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| Fourier Series | |
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| Introduction | |
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| Historical Perspective | |
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| Signal Analysis | |
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| Partial Differential Equations | |
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| Computation of Fourier Series | |
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| On the Interval -� ≤ x ≤ � | |
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| Other Intervals | |
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| Cosine and Sine Expansions | |
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| Examples | |
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| The Complex Form of Fourier Series | |
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| Convergence Theorems for Fourier Series | |
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| The Riemann-Lebesgue Lemma | |
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| Convergence at a Point of Continuity | |
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| Convergence at a Point of Discontinuity | |
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| Uniform Convergence | |
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| Convergence in the Mean | |
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| Exercises | |
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| The Fourier Transform | |
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| Informal Development of the Fourier Transform | |
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| The Fourier Inversion Theorem | |
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| Examples | |
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| Properties of the Fourier Transform | |
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| Basic Properties | |
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| Fourier Transform of a Convolution | |
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| Adjoint of the Fourier Transform | |
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| Plancherel Theorem | |
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| Linear Filters | |
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| Time-Invariant Filters | |
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| Causality and the Design of Filters | |
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| The Sampling Theorem | |
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| The Uncertainty Principle | |
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| Exercises | |
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| Discrete Fourier Analysis | |
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| The Discrete Fourier Transform | |
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| Definition of Discrete Fourier Transform | |
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| Properties of the Discrete Fourier Transform | |
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| The Fast Fourier Transform | |
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| The FFT Approximation to the Fourier Transform | |
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| Application: Parameter Identification | |
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| Application: Discretizations of Ordinary Differential Equations | |
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| Discrete Signals | |
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| Time-Invariant, Discrete Linear Filters | |
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| Z-Transform and Transfer Functions | |
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| Discrete Signals & Matlab | |
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| Exercises | |
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| Haar Wavelet Analysis | |
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| Why Wavelets? | |
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| Haar Wavelets | |
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| The Haar Scaling Function | |
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| Basic Properties of the Haar Scaling Function | |
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| The Haar Wavelet | |
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| Haar Decomposition and Reconstruction Algorithms | |
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| Decomposition | |
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| Filters and Diagrams | |
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| Summary | |
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| Exercises | |
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| Multiresolution Analysis | |
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| The Multiresolution Framework | |
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| Definition | |
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| The Scaling Relation | |
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| The Associated Wavelet and Wavelet Spaces | |
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| Decomposition and Reconstruction Formulas: A Tale of Two Bases | |
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| Summary | |
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| Implementing Decomposition and Reconstruction | |
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| The Decomposition Algorithm | |
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| The Reconstruction Algorithm | |
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| Processing a Signal | |
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| Fourier Transform Criteria | |
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| The Scaling Function | |
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| Orthogonality via the Fourier Transform | |
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| The Scaling Equation via the Fourier Transform | |
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| Iterative Procedure for Constructing the Scaling Function | |
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| Exercises | |
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| The Daubechies Wavelets | |
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| Daubechies' Construction | |
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| Classification, Moments, and Smoothness | |
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| Computational Issues | |
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| The Scaling Function at Dyadic Points | |
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| Exercises | |
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| Other Wavelet Topics | |
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| Computational Complexity | |
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| Wavelet Algorithm | |
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| Wavelet Packets | |
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| Wavelets in Higher Dimensions | |
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| Exercises on 2D Wavelets | |
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| Relating Decomposition and Reconstruction | |
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| Transfer Function Interpretation | |
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| Wavelet Transform | |
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| Definition of the Wavelet Transform | |
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| Inversion Formula for the Wavelet Transform | |
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| Technical Matters | |
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| Proof of the Fourier Inversion Formula | |
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| Technical Proofs from Chapter 5 | |
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| Rigorous Proof of Theorem 5.17 | |
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| Proof of Theorem 5.10 | |
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| Proof of the Convergence Part of Theorem 5.23 | |
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| Solutions to Selected Exercises | |
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| MATLAB“ Routines | |
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| General Compression Routine | |
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| Use of MATLAB's FFT Routine for Filtering and Compression306 | |
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| Sample Routines Using MATLAB's Wavelet Toolbox | |
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| MATLAB Code for the Algorithms in Section 5.2 | |
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| Bibliography | |
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| Index | |