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First Course in Wavelets with Fourier Analysis

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ISBN-10: 0470431172

ISBN-13: 9780470431177

Edition: 2nd 2009

Authors: Albert Boggess, Francis J. Narcowich

List price: $105.00
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Description:

Presenting the subject from the point of view of signal analysis, A First Course in Wavelets with Fourier Analysis, 2nd Edition provides a self-contained mathematical treatment of the subject that is accessible to a broad audience. Expanded applications to signal processing, exercises, and complete proofs of the presented theory, as well as solutions to selected exercises in the back of the book, make this an accessible reference for mathematicians, signal processing engineers, scientists, and advanced undergraduates.
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Book details

List price: $105.00
Edition: 2nd
Copyright year: 2009
Publisher: John Wiley & Sons, Limited
Publication date: 10/6/2009
Binding: Hardcover
Pages: 336
Size: 6.20" wide x 9.30" long x 1.00" tall
Weight: 1.298
Language: English

Preface and Overview
Inner Product Spaces
Motivation
Definition of Inner Product
The Spaces L<sup>2</sup> and l<sup>2</sup>
Definitions
Convergence in L<sup>2</sup> 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
Fourier Series
Introduction
Historical Perspective
Signal Analysis
Partial Differential Equations
Computation of Fourier Series
On the Interval -� &leq; x &leq; �
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
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
Plancherel Theorem
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
Discrete Signals & Matlab
Exercises
Haar Wavelet Analysis
Why Wavelets?
Haar Wavelets
The Haar Scaling Function
Basic Properties of the Haar Scaling Function
The Haar Wavelet
Haar Decomposition and Reconstruction Algorithms
Decomposition
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' 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
Exercises on 2D Wavelets
Relating Decomposition and Reconstruction
Transfer Function Interpretation
Wavelet Transform
Definition of the Wavelet Transform
Inversion Formula for the Wavelet Transform
Technical Matters
Proof of the Fourier Inversion Formula
Technical Proofs from Chapter 5
Rigorous Proof of Theorem 5.17
Proof of Theorem 5.10
Proof of the Convergence Part of Theorem 5.23
Solutions to Selected Exercises
MATLAB&#8220; Routines
General Compression Routine
Use of MATLAB's FFT Routine for Filtering and Compression306
Sample Routines Using MATLAB's Wavelet Toolbox
MATLAB Code for the Algorithms in Section 5.2
Bibliography
Index