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Practical time-frequency analysis. Gabor and wavelet transforms with an implementation in S. With a preface by Ingrid Daubechies. (English) Zbl 1039.42504

Wavelet Analysis and Its Applications 9. San Diego, CA: Academic Press (ISBN 0-12-160170-6). 490 p. (1998).
The authors have crafted a well-organized, up-to-date and highly readable presentation of some of the major components of time-frequency/time-scale analysis techniques of 1-D signals using Gabor and wavelet transform methods. The development of the basic concepts is implemented with a useful set of S-tools in the form of S-code, a library of S-functions and a companion Swave toolbox which they make available as freeware. The book is intended, in part, as an outreach to the statistical community; this is one of the reasons that the S-language is included.
The book is organized in two parts. The first develops the central ideas of time-frequency analysis with emphasis on the analysis of noisy signals. The weight is on redundant time-frequency transforms. However, attention is given to nonredundant transforms in an applied setting. The issue of stationarity is discussed in detail and the statistical significance of spectral analysis and denoising methods addressed. The second part is primarily a reference manual for the library of S-tools and functions and is designed to implement computations relative to examples in the first part.
There is an ample supply of supporting examples and well-illustrated figures distributed throughout the text. Many examples are revisited several times with tools stressing the advantages of the method under discussion. Applications of the S-code are tied in with the examples. The chapters conclude with a Notes and Complements section giving additional references and discussion. The data sets were provided by colleagues at universities, federal and corporate laboratories and the Office of Naval Research.
The first part of the book is divided into three sections comprising a self-contained condensed course on signal analysis. In spite of the condensed nature, the authors have distilled the material which is relevant to their objectives and presented it in a way that is lucid, useful and interesting. This is accomplished by presenting the principal ideas and theory, sketching some of the important proofs with precise references, and giving explanations and examples to tie the development together. Thus, the sometimes “opaque” theorem-proof style found in books directed specifically to the mathematical community rather than the scientific community is avoided. A brief account of the contents follows.
The first section sets the stage for time-frequency analysis and provides classical background material which is presented in the interest of completeness. The material involves basic transform methods and stationary random processes and sets the stage for major issues considered in the rest of the book. Chapter one introduces the Fourier theory of deterministic signals. The Fourier transform is considered in the context of signals as functions of unbounded continuous variables. Fourier series and the discrete finite Fourier transform are developed, with special attention given to the issues of sampling and aliasing. The autocovariance and periodogram of a finite signal are discussed in the context of Wiener’s deterministic spectral theory. Correlogram-periodogram duality is explained. The chapter is concluded by introducing time-frequency representation of a signal via the Wigner-Ville transform and its association with the ambiguity function.
Chapter two is dedicated to time series analysis in the form of a primer on the spectral theory of random processes. The descriptive statistics introduced in the first chapter now appear as statistical estimators which are computed for a particular realization of a random signal. A systematic study of their statistical properties includes a comparison with Wiener’s spectral theory, spectral representations, nonparametric spectral estimation including the periodogram revisited, sampling and aliasing and practical estimation.
The second section is devoted to the continuous Gabor (CGT) and wavelet transforms and shows their versatility in the spectral analysis of nonstationary signals. The Swave library is introduced. The elements of classical spectral analysis of stationary random processes are revisited with these methods.
Chapter three is devoted to the CGT and its important properties. The authors consider redundancy and its consequences, invariance properties, commonly used windows, chirps and frequency modulated signals. They tie the CGT in with the analysis of data modelling seismic excitations of a structure under simulated earthquake conditions. This example is important in seismic damage assessment. Examples of speech analysis and underwater acoustics follow. The authors show how the S-commands provided the data analysis for the examples.
Chapter four introduces the continuous wavelet transform, emphasizing its capacity to perform very precise analyses of regularity properties of functions. In this context there is wavelet analysis of oscillating singularities and trigonometric chirps. The earthquake signals are revisited from the wavelets viewpoint. This and the previous chapter explain key features of the CGT and the wavelet transform which suit them to particular types of applications.
Chapter five is a study of the discrete Gabor and wavelet transforms and algorithms. The theory of frames is introduced and applied to wavelet and Gabor transforms. Several types of discrete wavelet transforms are linked to applications. Methods for discretizing the transforms are discussed, with emphasis on wavelets. The authors show how to take advantage of the flexibility of choice in the discretization of the time-frequency representation to control the level of redundancy. They discuss multiresolution analysis and orthonormal bases, computations of wavelet coefficients, coiflets and the filter-bank approach. The chapter concludes with discussions on how to implement computational algorithms for the transforms.
The third and final section of part one is a study of signal processing applications. The authors emphasize some of their recent work on ridge detection and statistical reconstruction of noisy signals. They review elements of the statistical theory of nonstationary stochastic processes in terms of time-frequency analysis and conclude with a chapter on frequency modulated signals with an emphasis on speech analysis.
Chapter six concerns the time-frequency analysis of stationary stochastic processes. A complete analysis of the CGT and continuous wavelet transform is given and connected with the spectral characteristics of these processes. The analysis reveals the first steps toward the spectral analysis of nonstationary processes. The methods are discussed for stationary and nonstationary processes and illustrated for white noise and Brownian motion as well as for fractional Brownian motion, which is an important example of a nonstationary process.
Chapter seven deals with analysis of frequency modulated signals. The essence of the chapter is the description of simple but powerful numerical methods for characterizing important properties of a signal from the so-called ridges. These ridges occur when the analyzing wavelet is sharply concentrated in a neighborhood of a fixed frequency. The reason is that the continuous wavelet transform tends to concentrate near a series of curves (called ridges). The Gabor transform also exhibits this phenomenon. The game plan is to associate the signal with a time-frequency representation, say, the continuous wavelet or Gabor transform, and thereby characterize the ridges in terms of time and frequency.
Chapter eight deals with reconstruction of a signal from the ridges. The authors review some of the methods that have recently been advocated. They pay special attention to thresholding methods and reconstruction using local extrema for the dyadic wavelet transform and ridges for the continuous wavelet and Gabor transforms. Their perspective is that of nonparametric regression and spline smoothing methods.
The second part of the book is a reference manual and library written by the authors to perform all of the calculations for the examples in the first part. Chapter nine explains the procedures for downloading the C-code, S-code and the Makefile required for installation of the Swave package. Chapter ten is a hard copy of the on-line help for Swave. Chapter eleven documents additional S-functions which either are simple utilities or called as subroutines in Chapter ten.
The references are easy to use. The bibliography has three parts. The first lists some 300 general references given in the text. The second deals with wavelets exclusively. It is a special list of books, monographs and other sources which discuss time-frequency and time-scale analysis of signals. The third is a list of books and monographs dealing with statistical applications based on the S-language. The index is divided into four parts dealing with notations, Swave functions and utilities, an author index and a subject index.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces
62G07 Density estimation
62G08 Nonparametric regression and quantile regression
62M15 Inference from stochastic processes and spectral analysis
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
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