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Introduction to Time-Frequency and Wavelet Transforms

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Introduction to Time-Frequency and Wavelet Transforms


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  • Copyright 2002
  • Dimensions: K
  • Pages: 304
  • Edition: 1st
  • Book
  • ISBN-10: 0-13-030360-7
  • ISBN-13: 978-0-13-030360-8

The practical, heuristic introduction to time-frequency and wavelet analysis.

  • Heuristic approach focuses on numerical implementation and real-world applications
  • Presents algorithms found in NI's Signal Processing Toolset and other commercial software
  • Gabor expansions, linear time-variant filters, and key wavelet transform concepts
  • Bilinear time-frequency representation
  • Combining time-frequency and wavelet decomposition

In Introduction to Time-Frequency and Wavelet Transforms, Shie Qian takes a heuristic approach to time-frequency and wavelet analysis, drawing upon the engineer's intuition—not abstract equations. Qian presents the essence of the subject: the information needed to identify applications, choose approaches, and apply time-frequency and wavelet analysis successfully.

Each chapter starts with introductory background, moves to theoretical derivation, and concludes with practical numerical implementation. All algorithms can be found in commercial software, such as the Signal Processing Toolset from National Instruments, and all examples are available for download at NI's Web site. The book presents multiple real-world applications collected from NI's customers—many published here for the first time. Coverage includes:

  • Discrete, period discrete, and orthogonal-like Gabor expansions
  • Short-time Fourier transforms
  • Fast algorithms for computing dual functions
  • Linear time-variant filters
  • Fundamental wavelet transform concepts
  • Bilinear time-frequency representations, including Wigner-Ville distribution and decomposition
  • Cohen's Class and other time-dependent power spectra
  • Combining time-frequency and time-scale (wavelet) decomposition

If you've wanted to utilize time-frequency and wavelet analysis, but you've been deterred by highly mathematical treatments, Introduction to Time-Frequency and Wavelet Transforms is the accessible, practical guide you've been searching for.

Sample Content

Online Sample Chapter

Digital Filter Banks and the Wavelet Transform

Table of Contents

1. Introduction.

2. Fourier Transform A Mathematical Prism.

Frame. Fourier Transform. Relationship between Time and Frequency Representations. Characterization of Time Waveform and Power Spectrum. Uncertainty Principle. Discrete Poisson-Sum Formula. Short-Time Fourier Transform and Gabor Expansion.

3. Short-Time Fourier Transform.

Gabor Expansion. Periodic Discrete Gabor Expansion. Orthogonal-Like Gabor Expansion. A Fast Algorithm for Computing Dual Functions. Discrete Gabor Expansion.

4. Linear Time-Variant Filters.

LMSE Method. Iterative Method. Selection of Window Functions.

5. Fundamentals of theWavelet Transform.

Continuous Wavelet Transform. Piecewise Approximation. Multiresolution Analysis. Wavelet Transformation and Digital Filter Banks. Applications of the Wavelet Transform.

6. Digital Filter Banks andtheWavelet Transform.

Two-Channel Perfect Reconstruction Filter Banks. Orthogonal Filter Banks. General Tree-Structure Filter Banks and Wavelet Packets.

7. Wigner-Ville Distribution.

Wigner-Ville Distribution. General Properties of the Wigner-Ville Distribution. Wigner-Ville Distribution for the Sum of Multiple Signals. Smoothed Wigner-Ville Distribution. Wigner-Ville Distribution of Analytic Signals. Discrete Wigner-Ville Distribution.

8. Other Time-Dependent Power Spectra.

Ambiguity Function.

Cohens Class. Some Members of Cohens Class. Reassignment.

9. Decomposition of the Wigner-Ville Distribution.

Decomposition of the Wigner-Ville Distribution. Time-Frequency Distribution Series. Selection of Dual Functions. Mean Instantaneous Frequency and Instantaneous Bandwidth. Application for Earthquake Engineering.

10. Adaptive Gabor Expansion and Matching Pursuit.

Matching Pursuit. Adaptive Gabor Expansion. Fast Refinement. Applications of the Adaptive Gabor Expansion. Adaptive Gaussian Chirplet Decomposition.

Optimal Dual Functions.





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 run-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 co-existence 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.ni.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.


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