First Course in Wavelets with Fourier Analysis [Hardcover]

Albert Boggess (Author), Francis J. Narcowich (Author)

Book Description

ISBN-10: 0130228095 | ISBN-13: 978-0130228093 | Publication Date: January 16, 2001 | Edition: 1

This book fills the gap between volumes on wavelets that are either too advanced (in terms of mathematical background required) or that contain too little mathematical theory underlying wavelets. It presents most of the theory underlying Fourier analysis and wavelets in a clear and comprehensive fashion-- without requiring advanced background in real analysis. Provides a careful balance between theory and practical algorithms, and features a clear presentation of applications to digital signal processing--e.g., data compression, digital filtering and singularity detection. Includes illustrations and MATLAB code used in many of the algorithms. Inner Product Spaces. Fourier Series. The Fourier Transform. Discrete Fourier Analysis. Wavelet Analysis. Multiresolution Analysis. The Daubechies Wavelets. For anyone interested in Wavelets and Fourier Analysis.

Editorial Reviews

Review

"A first course in wavelets with Fourier analysis, second edition is an excellent book for courses in mathematics and engineering at the upper-undergraduate and graduate levels. It is also a valuable resource for mathematicians, signal processing engineers, and scientists who wish to learn about wavelet theory and Fourier analysis on an elementary level." (Mathematical Reviews, 2011)

"The discussions of applications avoid the deep jargon of signal processing … accessible to a wider audience." (Book News, December 2009)

--This text refers to an alternate Hardcover edition.

From the Back Cover

A comprehensive, self-contained treatment of Fourier analysis and wavelets—now in a new edition

Through expansive coverage and easy-to-follow explanations, A First Course in Wavelets with Fourier Analysis, Second Edition provides a self-contained mathematical treatment of Fourier analysis and wavelets, while uniquely presenting signal analysis applications and problems. Essential and fundamental ideas are presented in an effort to make the book accessible to a broad audience, and, in addition, their applications to signal processing are kept at an elementary level.

The book begins with an introduction to vector spaces, inner product spaces, and other preliminary topics in analysis. Subsequent chapters feature:

• The development of a Fourier series, Fourier transform, and discrete Fourier analysis

• Improved sections devoted to continuous wavelets and two-dimensional wavelets

• The analysis of Haar, Shannon, and linear spline wavelets

• The general theory of multi-resolution analysis

• Updated MATLAB code and expanded applications to signal processing

• The construction, smoothness, and computation of Daubechies' wavelets

• Advanced topics such as wavelets in higher dimensions, decomposition and reconstruction, and wavelet transform

Applications to signal processing are provided throughout the book, most involving the filtering and compression of signals from audio or video. Some of these applications are presented first in the context of Fourier analysis and are later explored in the chapters on wavelets. New exercises introduce additional applications, and complete proofs accompany the discussion of each presented theory. Extensive appendices outline more advanced proofs and partial solutions to exercises as well as updated MATLAB routines that supplement the presented examples.

A First Course in Wavelets with Fourier Analysis, Second Edition is an excellent book for courses in mathematics and engineering at the upper-undergraduate and graduate levels. It is also a valuable resource for mathematicians, signal processing engineers, and scientists who wish to learn about wavelet theory and Fourier analysis on an elementary level. --This text refers to an alternate Hardcover edition.

See all Editorial Reviews

Product Details

• Hardcover: 260 pages
• Publisher: Prentice Hall; 1 edition (January 16, 2001)
• Language: English
• ISBN-10: 0130228095
• ISBN-13: 978-0130228093
• Product Dimensions: 9.3 x 7.2 x 0.8 inches
• Shipping Weight: 1.4 pounds
• Average Customer Review: 4.2 out of 5 stars  See all reviews (5 customer reviews)
• Amazon Best Sellers Rank: #697,542 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

28 of 28 people found the following review helpful:

4.0 out of 5 stars A good Starter on Wavelets, October 21, 2001

By 

J.C. van den Berg (Amsterdam The Netherlands) - See all my reviews

This review is from: First Course in Wavelets with Fourier Analysis (Hardcover)

At the time of writing of this review (October 2001), a standard academic search procedure
produces about twenty references per week of scientific papers using wavelet analysis in a very wide spectrum of sciences. More than 160 english language books have been published on wavelets since the first books appeared around 1990. Yet even now it is rare to find a book on this subject which is aiming at undergraduate students and yet is mathematically responsible, without being heavy going. Boggess and Narcovich have tried to do just that, and to my mind have admirably succeeded.
Assuming a standard background knowledge in calculus and linear algebra that many science and engineering students acquire in their first two years at university, they present the basics of Fourier analysis and wavelets in eight brief chapters. To prepare the way, they start in chapter 0 with an introduction to inner product spaces, without using advanced analysis, and building on the experience with ordinary vector spaces.
Also a sniff of linear operator theory is offered.
Chapter 1 introduces Fourier series in real and complex form. These originated in the eighteenth century study of vibrations and in the theory of heat, made famous by Fourier's classic book of 1808: Analytical Theory of Heat. The mathematical claims Fourier made, but which he could not all prove himself, gave the impetus to an enormous development of both mathematical theory and applications in all fields of natural science, which is still going on today. The applications briefly mentioned here are denoising and compression of signals, and finding the solution of partial differential equations. Various aspects of the convergence of Fourier series are dealt with. All concepts are illustrated with a good set of clear figures, and the chapter finishes with exercises that are going from very elementary to a little more ambitious, sometimes involving the use of simple computer algebra tasks. This format is maintained thorugh the entire text, except for the last chapter.
Chapter 2 proceeds with the Fourier Transform, including the important theory of linear time invariant filters. The existence of the impulse response function and its convolution character are shown. As an example the noise reducing Butterworth filter is presented. Sampling and the Nyquist frequency are touched upon, and a derivation of the uncertainty relations, originally coming from quantum mechanics, is given.
To analyse discrete data, one needs the discrete Fourier Transform, which is the subject of chapter 3, including of course the Fast Fourier Transform. Also the z-transform is introduced. Examples given are elementary cases of parameter identification in vibration, numerical solution of ordinary differential equations, as well as in the exercises: noise reduction and data compression.

These first 153 pages serve as a good undergraduate introduction to Fourier analysis.
The second half of the book is devoted to wavelets. Chapter 4 deals exclusively with Haar wavelets which are the oldest wavelets because they date from 1910! These wavelets constitute an orthonormal basis of functions, which makes for fast calucation, a very important aspect for many applications. The core ideas of the central concept of a "multiresolution analysis" of a signal, can be demonstrated with these simple wavelets. All of this is already understandable without the machinery of the preceding Fourier analysis, so you could jump into the book here and start reading about wavelets right away, picking up the Fourier analysis from the first part bit by bit as the need arises. As applications denoising and compression are mentioned again, as is the detection of a discontinuity in a signal.
The general case of a multiresolution analysis is the subject of chapter 5. Again a large part of the discussion can be swallowed without the need of the Fourier transform point of view. The explanation of the structure of a multiresolution analysis leading to an orhtonormal basis of wavelets is straightforward and clear. It is only when we want to go into more detail about the precise characteristics of the underlying wavelet and scaling function that the Fourier point of view is introduced. This then leads up to the presentation of the famous Daubechies wavelets in chapter 6. These wavelets revolutionised the field after their publication in 1988.
Chapter 7 which closes the book, gives several short remarks about various other topics among which are two-dimensional wavelets, and the continuous wavelet transform.
This chapter is more sketchy than the others, and left me much less satisfied. Also the motivation why these subjects are chosen was lacking almost completely, and there are no exercises. I was particularly disappointed not to find any discussion of the relative merits of the continuous versus the discrete wavelet transform, and there is no mention of any application of the continuous case. Yet the latter is also used frequently in many important scientific applications, and it started the modern wavelet endeavour in the early eighties in France.
That being said I still think this is a very useful book for anybody wanting to start with wavelets at an undergraduate level. A few helpful Matlab Codes are collected in an appendix as well as the more difficult parts of some proofs. The exercises make this good course material, but as a text for self study it will also be quite satisfactory for many newcomers that find most of the existing books too demanding.

2 of 3 people found the following review helpful:

5.0 out of 5 stars A necesary textboox, May 11, 2009

By 

Leonardo Patricio (USA) - See all my reviews
(REAL NAME)   

This review is from: First Course in Wavelets with Fourier Analysis (Hardcover)

A wonderful book. Here the different topics are treated carefully in a very straightforward way. It gives a clear path for the real understanding of the wavelet analysis.

3 of 5 people found the following review helpful:

5.0 out of 5 stars Informative Book for Wavelet Analysis!!!!, February 28, 2003

By 

Hsiao-Chun Wu (LSU, Baton Rouge, LA, USA) - See all my reviews

This review is from: First Course in Wavelets with Fourier Analysis (Hardcover)

...this book is one of most informative and legible books on wavelt theories and applications.

The author paves the theoretical development about wavelets and multi-resolution analysis EXCELLENTLY. With this book, you can construct wavelets for your own applications in engineering and science disciplines.

This book is very good for first year engineering-majored graduate students and all engineering scholars.