Medical uses of statistics

Through expansive coverage and easy-to-follow explanations, A First Course in Wavelets with Fourier Analysis , Second Edition provides a self-contained mathematical treatment of Fourier analysis and wavelets, while uniquely presenting signal analysis applications and problems. Essential and fundamental ideas are presented in an effort to make the book accessible to a broad audience, and, in addition, their applications to signal processing are kept at an elementary level.

The book begins with an introduction to vector spaces, inner product spaces, and other preliminary topics in analysis. Subsequent chapters feature:

The development of a Fourier series, Fourier transform, and discrete Fourier analysis

Improved sections devoted to continuous wavelets and two-dimensional wavelets

The analysis of Haar, Shannon, and linear spline wavelets

The general theory of multi-resolution analysis

Updated MATLAB code and expanded applications to signal processing

The construction, smoothness, and computation of Daubechies' wavelets

Advanced topics such as wavelets in higher dimensions, decomposition and reconstruction, and wavelet transform

Applications to signal processing are provided throughout the book, most involving the filtering and compression of signals from audio or video. Some of these applications are presented first in the context of Fourier analysis and are later explored in the chapters on wavelets. New exercises introduce additional applications, and complete proofs accompany the discussion of each presented theory. Extensive appendices outline more advanced proofs and partial solutions to exercises as well as updated MATLAB routines that supplement the presented examples.

A First Course in Wavelets with Fourier Analysis , Second Edition is an excellent book for courses in mathematics and engineering at the upper-undergraduate and graduate levels. It is also a valuable resource for mathematicians, signal processing engineers, and scientists who wish to learn about wavelet theory and Fourier analysis on an elementary level.

Author Notes

Albert Boggess, PhD. is Professor of Mathematics at Texas AM University. Dr. Boggess has over twenty-five years of academic experience and has authored numerous publications in his areas of research interest, which include overdetermined systems of partial differential equations, several complex variables, and harmonic analysis.
Francis J. Narcowich, PhD. is Professor of Mathematics and Director of the Center for Approximation Theory at Texas AM University. Dr. Narcowich serves as an Associate Editor of both the SIAM Journal on Numerical Analysis and Mathematics of Computation, and he has written more than eighty papers on a variety of topics in pure and applied mathematics. He currently focuses his research on applied harmonic analysis and approximation theory.

Preface and Overview	p. ix
0 Inner Product Spaces	p. 1
0.1 Motivation	p. 1
0.2 Definition of Inner Product	p. 2
0.3 The Spaces L 2 and l 2	p. 4
0.3.1 Definitions	p. 4
0.3.2 Convergence in L 2 Versus Uniform Convergence	p. 8
0.4 Schwarz and Triangle Inequalities	p. 11
0.5 Orthogonality	p. 13
0.5.1 Definitions and Examples	p. 13
0.5.2 Orthogonal Projections	p. 15
0.5.3 Gram-Schmidt Orthogonalization	p. 20
0.6 Linear Operators and Their Adjoints	p. 21
0.6.1 Linear Operators	p. 21
0.6.2 Adjoints	p. 23
0.7 Least Squares and Linear Predictive Coding	p. 25
0.7.1 Best-Fit Line for Data	p. 25
0.7.2 General Least Squares Algorithm	p. 29
0.7.3 Linear Predictive Coding	p. 31
Exercises	p. 34
1 Fourier Series	p. 38
1.1 Introduction	p. 38
1.1.1 Historical Perspective	p. 38
1.1.2 Signal Analysis	p. 39
1.1.3 Partial Differential Equations	p. 40
1.2 Computation of Fourier Series	p. 42
1.2.1 On the Interval -¿ ≤ x ≤ ¿	p. 42
1.2.2 Other Intervals	p. 44
1.2.3 Cosine and Sine Expansions	p. 47
1.2.4 Examples	p. 50
1.2.5 The Complex Form of Fourier Series	p. 58
1.3 Convergence Theorems for Fourier Series	p. 62
1.3.1 The Riemann-Lebesgue Lemma	p. 62
1.3.2 Convergence at a Point of Continuity	p. 64
1.3.3 Convergence at a Point of Discontinuity	p. 69
1.3.4 Uniform Convergence	p. 72
1.3.5 Convergence in the Mean	p. 76
Exercises	p. 83
2 The Fourier Transform	p. 92
2.1 Informal Development of the Fourier Transform	p. 92
2.1.1 The Fourier Inversion Theorem	p. 92
2.1.2 Examples	p. 95
2.2 Properties of the Fourier Transform	p. 101
2.2.1 Basic Properties	p. 101
2.2.2 Fourier Transform of a Convolution	p. 107
2.2.3 Adjoint of the Fourier Transform	p. 109
2.2.4 Plancherel Theorem	p. 109
2.3 Linear Filters	p. 110
2.3.1 Time-Invariant Filters	p. 110
2.3.2 Causality and the Design of Filters	p. 115
2.4 The Sampling Theorem	p. 120
2.5 The Uncertainty Principle	p. 123
Exercises	p. 127
3 Discrete Fourier Analysis	p. 132
3.1 The Discrete Fourier Transform	p. 132
3.1.1 Definition of Discrete Fourier Transform	p. 134
3.1.2 Properties of the Discrete Fourier Transform	p. 135
3.1.3 The Fast Fourier Transform	p. 138
3.1.4 The FFT Approximation to the Fourier Transform	p. 143
3.1.5 Application: Parameter Identification	p. 144
3.1.6 Application: Discretizations of Ordinary Differential Equations	p. 146
3.2 Discrete Signals	p. 147
3.2.1 Time-Invariant, Discrete Linear Filters	p. 147
3.2.2 Z-Transform and Transfer Functions	p. 149
3.3 Discrete Signals & Matlab	p. 153
Exercises	p. 156
4 Haar Wavelet Analysis	p. 160
4.1 Why Wavelets?	p. 160
4.2 Haar Wavelets	p. 161
4.2.1 The Haar Scaling Function	p. 161
4.2.2 Basic Properties of the Haar Scaling Function	p. 167
4.2.3 The Haar Wavelet	p. 168
4.3 Haar Decomposition and Reconstruction Algorithms	p. 172
4.3.1 Decomposition	p. 172
4.3.2 p. 176
4.3.3 Filters and Diagrams	p. 182
4.4 Summary	p. 185
Exercises	p. 186
5 Multiresolution Analysis	p. 190
5.1 The Multiresolution Framework	p. 190
5.1.1 Definition	p. 190
5.1.2 The Scaling Relation	p. 194
5.1.3 The Associated Wavelet and Wavelet Spaces	p. 197
5.1.4 Decomposition and Reconstruction Formulas: A Tale of Two Bases	p. 201
5.1.5 Summary	p. 203
5.2 Implementing Decomposition and Reconstruction	p. 204
5.2.1 The Decomposition Algorithm	p. 204
5.2.2 The Reconstruction Algorithm	p. 209
5.2.3 Processing a Signal	p. 213
5.3 Fourier Transform Criteria	p. 214
5.3.1 The Scaling Function	p. 215
5.3.2 Orthogonality via the Fourier Transform	p. 217
5.3.3 The Scaling Equation via the Fourier Transform	p. 221
5.3.4 Iterative Procedure for Constructing the Scaling Function	p. 225
Exercises	p. 228
6 The Daubechies Wavelets	p. 234
6.1 Daubechies' Construction	p. 234
6.2 Classification, Moments, and Smoothness	p. 238
6.3 Computational Issues	p. 242
6.4 The Scaling Function at Dyadic Points	p. 244
Exercises	p. 248
7 Other Wavelet Topics	p. 250
7.1 Computational Complexity	p. 250
7.1.1 Wavelet Algorithm	p. 250
7.1.2 Wavelet Packets	p. 251
7.2 Wavelets in Higher Dimensions	p. 253
Exercises on 2D Wavelets	p. 258
7.3 Relating Decomposition and Reconstruction	p. 259
7.3.1 Transfer Function Interpretation	p. 263
7.4 Wavelet Transform	p. 266
7.4.1 Definition of the Wavelet Transform	p. 266
7.4.2 Inversion Formula for the Wavelet Transform	p. 268
Appendix A Technical Matters	p. 273
A.l Proof of the Fourier Inversion Formula	p. 273
A.2 Technical Proofs from Chapter 5	p. 277
A.2.1 Rigorous Proof of Theorem 5.17	p. 277
A.2.2 Proof of Theorem 5.10	p. 281
A.2.3 Proof of the Convergence Part of Theorem 5.23	p. 283
Appendix B Solutions to Selected Exercises	p. 287
Appendix C MATLAB" Routines	p. 305
C.1 General Compression Routine	p. 305
C.2 Use of MATLAB's FFT Routine for Filtering and Compression\|306
C.3 Sample Routines Using MATLAB's Wavelet Toolbox	p. 307
C.4 MATLAB Code for the Algorithms in Section 5.2	p. 308
Bibliography	p. 311
Index	p. 313

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