时频变换与小波变换导论：英文版

　　本书从工程师的角度，而不是从抽象的数学方程的角度，用启发性的观点介绍了"时频和小波分析"问题。内容丰富、结构简明是本书的一大特点。本书每一章都从背景简介开始，进而转向理论推导，最后得到具有高度实用性的数字工具。书中所有的算法都可以在商业软件中找到，如美国国家仪器公司（National Instruments，NI）的Signal Processing Toolset，各种应用的例子都可以从NI网站上直接下载得到。本书包含了从NI用户中收集到的多种真实应用，其中很多都是首次公开发表。本书的主要内容：●短时傅里叶变换●离散和正交Gabor展开式●线性时变滤波器●小波分析基础●双线性时频表示法，包括Wigner-Ville分布和分解●小波变换用的数字滤波器库●混合时频和时间标度（小波）分解 钱世锷是美国国家仪器公司的DSP总工程师， 拥有8项专利. 他致力子时频和小波分析的理论和应用研究多年. 在应用方面， 他主要研究旋转电机. 通信. 雷达. 地震学. 生物医学等. 他是IEEE Signal Processing MagazJne关于时频分析问题的特邀编辑， 还是Joint Time-Frequency Analysis一书的主要作者. For a long time， I wondered if the recently popularized time-frequency and wavelet transforms were merely academic exercises. Do applied engineers and scientists really need signal processing tools other than the FFT？ After 10 years of working with engineers and scientists from a wide variety of disciplines， I have finally come to the conclusion that， so far， neither the time-frequency nor wavelet transform appear to have had the revolutionary impact upon physics and pure mathematics that the Fourier transform has had. Nevertheless， they can be used to solve many real-world problems that the classical Fourier transform cannot. As James Kaiser once said， "The most widely used signal processing tool is the FFT; the most widely misused signal processing tool is also the FFT." Fourier transform-based techniques are effective as long as the frequency contents of the signal do not change with time. However， when the frequency contents of the data samples evolve over an observation period， time-frequency or wavelet transforms should be considered. Specifically， the time-frequency transform is suited for signals with slow frequency changes （narrow instantaneous bandwidth）， such as sounds heard during an engine mn-up or run-down， whereas the wavelet transform is suited for signals with rapid changes （wide instantaneous frequency bandwidth）， such as sounds associated with engine knocking. The success of applications of the time-frequency and wavelet transforms largely hinges on understanding their fundamentals. It is the goal of this book to provide a brief introduction to time-frequency and wavelet transforms for those engineers and scientists who want to use these techniques in their applications， and for students who are new to these topics. Keeping this goal in mind， I have included the two related subjects， time-frequency and wavelet transforms， under a single cover so that readers can grasp the necessary information and come up to speed in a short time. Professors can cover these topics in a single semester. The coexistence of the time-frequency and wavelet approaches in one book， I believe， will help comparative understanding and make complementary use easier. 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.I wrote this book to appeal to the reader‘s intuition rather than to rely on abstract mathematical equations and wanted the material to be easily understood by a reader with an engineering or science undergraduate education. To achieve this， mathematical rigor and lengthy derivation have been sacrificed in many places. Hopefully， this style will not unduly offend purists.On the other hand， "Formulas were not invented simply as weapons of intimidation" [22]. In many cases， mathematical language， I feel， is much more effective than plain English. Words are sometimes clumsy and ambiguous. For me， it is always a joy to refresh my knowledge of what I learned in school but have not used since.Some of the material presented in this book is the result of collaborative work which so greatly profited from the contributions of friends and colleagues that I must mention them. It was my graduate advisor， Professor Joel M. Morris， who led me into such a fascinating field. Motivated by the suppression of cross-term interference， in the early 90S the idea of the decomposition of the Wigner-Ville distribution emerged， which led to a series of interesting results. With Shidong Li and Kai Chen， the relationship of the most similar dual and the pseudo inverse was discovered. Based on Wexler and Raz‘s periodic discrete Gabor expansion [225]， Dapang Chen and I obtained its infinite counterpart which resulted in an interesting time-dependent spectrum， the time-frequency distribution series， also known as the Gabor spectrogram. To improve the time-frequency resolution， we also proposed the adaptive Gabor expansion which turned out to be the same scheme as that employed by the matching pursuit method independently developed by Stephane Mallat and Zhifeng Zhang during the same time period [172]. With Qinye Yin， such an adaptive decomposition scheme was generalized into the Gaussian chirplet cases. The fast-refinement algorithm initially appeared when Qinye Yin visited Austin， Texas. As a result of his insightful observation， the computation of the adaptive Gaussian chirplet approximation has been sigfiificantly improved. All these years later， I clearly remember a discussion at Xiang-Geng Xia‘s office in Malibu， California， in front of the magnificent beach there. The subject was the Gabor expansion-based time-varying filter. As a result of that discussion， a few days later Xiang-Geng called me and said， "With a tight frame， the iteration of the time-varying filter converges！" That memory is indelible. I would also like to thank Professors Xiang-Geng Xia and Richard G. Baraniuk for their contributions in Chapter 5 and Section 8.3， respectively. In a larger sense， this book is the result of the enthusiasm and support from numerous customers， colleagues， and friends. I want to take this opportunity to express my sincere thanks to all of them. Particularly， I would like to thank Dr. James Truchard and Jeff Kodosky， the founders of National Instruments Corporation. It is their great enthusiasm and continuous support that keep such a "non-profitable" project evolving and making all those interesting applications take place. This book has been an on-off project for almost three years and I extend my thanks to Bernard Goodwin at Prentice Hall for his endless patience and generous assistance. There were so many errors in the original draft that I dare not look at it again. Mahesh Chugani carefully read the entire manuscript. His numerous comments and suggestions improved the book significantly. My deepest thanks are reserved for my mother， Yuzhen Wu， and my family： my wife， Jun， and daughter， Nancy. I am very grateful for their understanding， support， and patience during this formidable project. Shie Qian Austin， Texas Preface Chapter 1Introduction Chapter 2Fourier Transform A Mathematical Prism 2.1Frame 2.2Fourier Transform 2.3Relationship between Time and Frequency Representations 2.4Characterization of Time Waveform and Power Spectrum 2.5Uncertainty Principle 2.6Discrete Poisson-Sum Formula Chapter 3Short-Time Fourier Transform and Gabor Expansion 3.1Short-Time Fourier Transform 3.2Gabor Expansion 3.3Periodic Discrete Gabor Expansion 3.4Orthogonal-Like Gabor Expansion 3.5A Fast Algorithm for Computing Dual Functions 3.6Discrete Gabor Expansion Chapter 4 Linear Time-Variant Filters 4.1LMSE Method 4.2Iterative Method 4.3Selection of Window Functions Chapter 5 Fundamentals of the Wavelet Transform 5.1Continuous Wavelet Transform 5.2Piecewise Approximation 5.3Multiresolution Analysis 5.4Wavelet Transformation and Digital Filter Banks 5.5Applications of the Wavelet Transform Chapter 6 Digital Filter Banks and the Wavelet Transform 6.1Two-Channel Perfect Reconstruction Filter Banks 6.2Orthogonal Filter Banks 6.3General Tree-Structure Filter Banks and Wavelet Packets Chapter 7 Wigner-Ville Distribution 7.1Wigner-Ville Distribution 7.2General Properties of the Wigner-Ville Distribution 7.3Wigner-Ville Distribution for the Sum of Multiple Signals 7.4Smoothed Wigner-Ville Distribution 7.5Wigner-Ville Distribution of Analytic Signals 7.6Discrete Wigner-Ville Distribution Chapter 8 Other Time-Dependent Power Spectra 8.1Ambiguity Function 8.2Cohen‘s Class 8.3Some Members of Cohen‘s Class 8.4Reassignment Chapter 9 Decomposition of the Wigner-Vi！le Distribution 9.1Decomposition of the Wigner-Ville Distribution 9.2 Time-Frequency Distribution Series 9.3Selection of Dual Functions 9.4Mean Instantaneous Frequency and Instantaneous Bandwidth 9.5Application for Earthquake Engineering Chapter I0 Adaptive Gabor Expansion and Matching Pursuit 10.1Matching Pursuit 10.2Adaptive Gabor Expansion 10.3Fast Refinement 10.4 Applications of the Adaptive Gabor Expansion 10.5 Adaptive Gaussian Chirplet Decomposition AppendixOptimal Dual Functions Bibliography Index

　　钱世锷：是美国国家仪器公司的DSP总工程师，拥有8项专利。他致力于时频和小波分析的理论和应用研究多年。在应用方面，他主要研究旋转电机、通信、雷达、地震学、生物医学等。他是IEEESignalProcessingMagazine关于时频分析问题的特邀编辑，还是JointTime-FrequencyAnalysis一书的主要作者。

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