A Friendly Guide to Wavelets, by Gerald Kaiser
Sixth Printing, 1999

Birkhauser-Boston, 1994 * ISBN 0-8176-3711-7 * Hardcover * 300 pages

"I loved ' Friendly Guide to Wavelets' I advised it to my graduate students."

- Yves Meyer, Univ. Paris-Dauphine

"Outstanding! Written for broad audience of potential users!
I bought this book when I was particularly interested in continuous wavelet transforms. I found myself flying through the exposition. I felt confident about the knowledge I was acquiring and I was quickly able to apply it. Although I come at this book as a mathematician, I think that it is ideal for engineers and physical scientists who usually have far better grounding in signal processing and related issues in Fourier analysis than do mathematicians. I have recommended this book to students, friends and colleagues with high praise."

- Norman Bleistein, Amazon, March 1999

"...this book is in my judgement by far the best choice available for a course on wavelets."

- Edmund Christiansen, Univ. Odense, Denmark

"I wholeheartedly recommend this book for a solid and friendly introduction to wavelets, for anyone who is comfortable with the mathematics required of undergraduate electrical engineers. The book' appeal is that it covers all the fundamental concepts of wavelets in an elegant, straightforward way. It offers truly enjoyable (friendly!) mathematical exposition that is rich in intuitive explanations, as well as clean, direct, and clear in its theoretical developments. I found Kaiser' straightforward end-of-chapter exercises excellent...Kaiser has written an excellent introduction to the fundamental concepts of wavelets. For a book of its length and purpose, I think it should be essentially unbeatable for a long time."

- W. Kilmer, Proceedings of the IEEE, November 1998

"The book starts with reminding the basics of linear algebra and gradually, with amazing tact and feel of measure, introduces the reader to the notion of Hilbert and function spaces ... The scheme '' accepted in the mathematical literature is safely changed, for the reader' profit, by explanations revealing the essence of the introduced concepts for a potential user ... attracting his attention to some subtleties important for applications. Excercises at the end of each chapter help in active learning of the material ... Electromagnetic and acoustic wavelets constructed by the author reveal a deep relation between physical wavelets and ... signal analysis in the theory of communications."

- Yu. A. Danilov, Physics Uspechi (Russian Academy of Sciences), October 1997

"Kaiser has written a book that deserves its title: the book is not only expository at the advanced undergraduate level, but also thoroughly pedagogical. It is therefore a good text for newcomers to the subject. The author, a mathematical physicis, develops an eminently readable style in motivating and explaining wavelets, requiring only some linear algebra, Fourier series and Fourier integrals as prerequisites."

- Hans van den Berg, Notices of the Dutch Mathematical Society, February 1997

"This volume is probably the most gentle introduction to wavelet theory on the market. As such, it responds to a significant need. The intended audience will profit from the motivation and common-sense explanations in the text. Ultimately, it may lead many readers, who may not otherwise have been able to do so, to go further into wavelet theory, Fourier analysis, and signal processing."

- M. Frazier, SIAM Review, December 1995

"The first half of the book is appropriately named. It is a well-written, nicely organized exposition...a welcome addition to the literature. The second part of the book introduces the concept of electromagnetic wavelets...This theory promises to have many other applications and could well lead to new ways of studying these topics. This book has a number of unique features which...makes it particularly valuable for newcomers to the field."

- W. Gilbert, Mathematical Reviews, September 1995

"The book is indeed what its title promises: A friendly guide to wavelets...In short, Kaiser' book is excellently written and can be considered as one of the best textbooks on this topic presently available...it will enjoy wide distribution among mathematicians and physicists interested in wavelet analysis."

- E. Werner, Internation Mathematical News, Austrian Math. Soc., August 1995

"I found this to be an excellent book. It is eminently more readable than the books...which might be considered the principal alternatives for textbooks on wavelets."

- L. Hudgins, Physics Today, July 1995

"It is well produced and certainly readable...This material should present no difficulty for fourth-year undergraduates...It also will be useful to advanced workers in that it presents a different approach to wavelet theory from the usual one."

- A. D. Booth, Computing Reviews, June 1995

"Why another book on this hot topic? Gerald Kaiser found that most of his students taking wavelet analysis found the mathematics of existing books too difficult. In "A Friendly Guide to Wavelets" Kaiser develops the needed analytical machinery at the beginning. He also develops a general notation system based on linear algebra and he offers straightforward exercises at the ends of chapters. All this makes the subject available to anyone with a basic knowledge of matrix theory, Fourier series and Fourier integral transforms."

-Library of Science Book of the Month Club, January 1995

This volume consists of two parts. Chapters 1-8, Basic Wavelet Analysis, are aimed at graduate students or advanced undergraduates in science, engineering, and mathematics. They are designed for an introductory one-semester course on wavelets and time frequency analysis, and can also be used for self-study or reference by practicing researchers in signal analysis and related areas. The reader is not presumed to have a sophisticated mathematical background; therefore, much of the needed analytical machinery is developed from the beginning. The only prerequisite is a knowledge of matrix theory, Fourier series, and Fourier integral transforms. Notation is introduced which facilitates the formulation of signal analysis in a modern and general mathematical language, and the illustrations should further ease comprehension. Each chapter ends with a set of straightforward exercises designed to drive home the concepts.

Chapters 9-11, Physical Wavelets, are at a more advanced level and represent original research. They can be used as a text for a second-semester course or, when combined with Chapters 1 and 3, as a reference for a research seminar. Whereas the wavelets of Part I can be any functions of "time," physical wavelets are functions of space-time constrained by differential equations. In Chapter 9, wavelets specifically dedicated to Maxwell$#039s equations are constructed. These wavelets are electromagnetic pulses parameterized by their point and time of emission or absorption, their duration, their helicity, and the velocity of the emitter or absorber. The duration also acts as a scale parameter. We show that every electromagnetic wave can be composed from such wavelets. This fact is used in Chapter 10 to give a new formulation of electromagnetic imaging, such as radar, accompanied by a geometrical model for scattering based on conformal transformations. In Chapter 11, a similar set of wavelets is developed for acoustics. A relation is established at the fundamental level of differential equations between physical waves and time signals. This gives a one-to-one correspondence between physical wavelets and a particular family of time wavelets.


  1. Preface
    Suggestions to the Reader
    Symbols, Conventions, and Transforms

    Part I: Basic Wavelet Analysis

  2. Preliminaries: Background and Notation
  3. Windowed Fourier Transforms
  4. Continuous Wavelet Transforms
  5. Generalized Frames: Key to Analysis and Synthesis
  6. Discrete Time-Frequency Analysis and Sampling
  7. Discrete Time-Scale Analysis
  8. Multiresolution Analysis
  9. Daubechies' Orthonormal Wavelet Bases

    Part II: Physical Wavelets

  10. Introduction to Wavelet Electromagnetics
  11. Applications to Radar and Scattering
  12. Wavelet Acoustics