Introduction to Time-Frequency Laws in Continuum Physics

This book can be viewed in two parts. While Chapters Two through Six focus on linear transforms, mainly the Gabor expansion and the wavelet transform, Chapters Seven through Nine are dedicated to bilinear time-frequency representations. Chapter Ten can be thought of as a combination of time-frequency and time-scale (that is, wavelets) decomposition. The presentation of the wavelet transform in this book is aimed at readers who need to know only the basics and perhaps apply these new techniques to solve problems with existing commercial software. It may not be sufficient for academic researchers interested in creating their own set of basic functions by techniques other than the elementary filter banks introduced here.
All chapters start with the discussion of basic concepts and motivation, then provide theoretical analysis and, finally, numerical implementation. Most algorithms introduced in this book are a part of the software package, Signal Processing Toolset, a National Instruments product. Visit www. nj. com for more information about this software.
This book is neither a research monograph nor an encyclopedia, and the materials presented here are believed to be the most basic fundamentals of time-frequency and wavelet analysis. Many theoretically excellent results, which are not practical for digital implementation, have been omitted. The contents of this book should provide a strong foundation for the time-frequency and wavelet analysis neophyte, as well as a good review tutorial for the more experienced signal-processing reader.

【Main Contents】

Preface
Chapter 1 Introduction
Chapter 2 Fourier Transform
A Mathematical Prism
2.1 Frame
2.2 Fourier Transform
2.3 Relationship between Time and Frequency Representations
2.4 Characterization of Time Waveform and Power Spectrum
2.5 Uncertainty Principle
2.6 Discrete Poisson-Sum Formula
Chapter 3 Short-Time Fourier Transform and Gabor Expansion
3.1 Short-Time Fourier Transform
3.2 Gabor Expansion
3.3 Periodic Discrete Gabor Expansion
3.4 Orthogonal-Like Gabor Expansion
3.5 A Fast Algorithm for Computing Dual Functions
3.6 Discrete Gabor Expansion
Chapter 4 Linear Time-Variant Filters
4.1 LMSE Method
4.2 Iterative Method
4.3 Selection of Window Functions
Chapter 5 Fundamentals of the Wavelet Transform
5.1 Continuous Wavelet Transform
5.2 Piecewise Approximation
5.3 Multiresolution Analysis
5.4 Wavelet Transformation and Digital Filter Banks
5.5 Applications of the Wavelet Transform
Chapter 6 Digital Filter Banks and the Wavelet Transform
6.1 Two-Channel Perfect Reconstruction Filter Banks
6.2 Orthogonal Filter Banks
6.3 General Tree-Structure Filter Banks and Wavelet Packets
Chapter 7 Wigner-Ville Distribution
7.1 Wigner-Ville Distribution
7.2 General Properties of the Wigner-Ville Distribution
7.3 Wigner-Ville Distribution for the Sum of Multiple Signals
7.4 Smoothed Wigner-Ville Distribution
7.5 Wigner-Ville Distribution of Analytic Signals
7.6 Discrete Wigner-Ville Distribution
Chapter 8 Other Time-Dependent Power Spectra
8.1 Ambiguity Function
8.2 Cohen‘s Class
8.3 Some Members of Cohen‘s Class
8.4 Reassignment
Chapter 9 Decomposition of the Wigner-Vi!le Distribution
9.1 Decomposition of the Wigner-Ville Distribution
9.2 Time-Frequency Distribution Series
9.3 Selection of Dual Functions
9.4 Mean Instantaneous Frequency and Instantaneous Bandwidth
9.5 Application for Earthquake Engineering
Chapter I0 Adaptive Gabor Expansion and Matching Pursuit
10.1 Matching Pursuit
10.2 Adaptive Gabor Expansion
10.3 Fast Refinement
10.4 Applications of the Adaptive Gabor Expansion
10.5 Adaptive Gaussian Chirplet Decomposition
Appendix Optimal Dual Functions
Bibliography
Index