A Friendly Guide to Wavelets

317 Pages·2011·67.334 MB·English

by  Gerald Kaiser (auth.)

Modern Birkhäuser Classics

Many of the original research and survey monographs in pure and
applied mathematics published by Birkhäuser in recent decades have
been groundbreaking and have come to be regarded as foundational
to the subject. Through the MBC Series, a select number of these
modern classics, entirely uncorrected, are being re-released in
paperback (and as eBooks) to ensure that these treasures remain
accessible to new generations of students, scholars, and researchers.
To Yadwiga-Wanda and Stanisiaw Wiodek,
to Teofila and Franciszek Kowalik,
and to their children Janusz, Krystyn, Mirek, and Renia.
Risking their collective lives, they took a strangers' infant
into their homes and hearts. Who among us would have
the heart and the courage of these simple country people?
To my mother Cesia, who survived the Nazi horrors,
and to my father Bernard, who did not.
And to the uncounted multitude of others, less fortunate than I,
who did not live to sing their song.
A Friendly Guide
to Wavelets
Gerald Kaiser
Reprint of the 1994 Edition
§? Birkhauser
Gerald Kaiser
Center for Signals and Waves
3608 Govalle Avenue
Austin, TX 78702
U.S.A.
[email protected]
www.wavelets.com
Originally published as a hard cover edition.
ISBN 978-0-8176-8110-4 e-ISBN 978-0-8176-8111-1
DOI 10.1007/978-0-8176-8111-1
ISBN 978-0-8176-8110-4 e-ISBN 978-0-8176-8111-1
DLiObrIa r1y0 o.1f 0C0o7n/g9re7s8s- 0C-o8n1tr7o6l -N8u1m11be-1r: PCN applied for
Mathematics Subject Classification (2010): 15-XX, 30-XX, 31-XX, 32-XX, 35-XX, 41-XX, 42-XX,
44-XX, 46-XX, 65-XX, 78-XX, 94-XX
© Gerald Kaiser 2011
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY
10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection
with any form of information storage and retrieval, electronic adaptation, computer software, or by similar
or dissimilar methodology now known or hereafter developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are
not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject
to proprietary rights.
Printed on acid-free paper
§? Birkhauser
birkhauser-science.com
G E R A L D K A I S E R
A Friendly Guide
to Wavelets
Birkhauser
Boston • Basel • Berlin
1994
Gerald Kaiser
The Virginia Center for Signals and Waves
1921 Kings Road
Glen Allen, VA 23060
Library of Congress Cataloging-in-Publication Data
Kaiser, Gerald.
A friendly guide to wavelets / Gerald Kaiser.
p. cm.
Includes bibliographical references and index.
ISBN 0-8176-3711-7 (acid-free). - ISBN 3-7643-3711-7 (acid-free)
1. Wavelets. I. Title.
OA403.3.K35 1994 94-29118
515'.2433-dc20 CIP
Printed on acid-free paper _,. ,, .. WEP®
v v Birkhauser flp
© Gerald Kaiser 1994
Sixth printing, 1999
All rights reserved. This work may not be translated or copied in whole or in part without
the written permission of the publisher (Birkhauser Boston, c/o Springer-Verlag New
York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in
connection with reviews or scholarly analysis. Use in connection with any form of
information storage and retrieval, electronic adaptation, computer software, or by similar
or dissimilar methodology now known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication,
even if the former are not especially identified, is not to be taken as a sign that such names,
as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used
freely by anyone.
ISBN 0-8176-3711-7
ISBN 3-7643-3711-7
Typeset by Martin Stock, Cambridge, MA.
Printed and bound by Quinn-Woodbine, Woodbine, NJ.
Printed in the U.S.A.
9 8 7 6
Contents
Preface xi
Suggestions to the Reader xvii
Symbols, Conventions, and Transforms xix
Part I: Basic Wavelet Analysis
1. Preliminaries: Background and Notation
1.1 Linear Algebra and Dual Bases 3
1.2 Inner Products, Star Notation, and Reciprocal Bases 12
1.3 Function Spaces and Hilbert Spaces 19
1.4 Fourier Series, Fourier Integrals, and Signal Processing 26
1.5 Appendix: Measures and Integrals 34
Exercises 41
2. Windowed Fourier Transforms
2.1 Motivation and Definition of the WFT 44
2.2 Time-Frequency Localization 50
2.3 The Reconstruction Formula 53
2.4 Signal Processing in the Time-Frequency Domain 56
Exercises 58
3. Continuous Wavelet Transforms
3.1 Motivation and Definition of the Wavelet Transform 60
3.2 The Reconstruction Formula 64
3.3 Frequency Localization and Analytic Signals 70
3.4 Wavelets, Probability Distributions, and Detail 73
Exercises 77
viii A Friendly Guide to Wavelets
4. Generalized Frames: Key to Analysis and Synthesis
4.1 From Resolutions of Unity to Frames 78
4.2 Reconstruction Formula and Consistency Condition 85
4.3 Discussion: Frames versus Bases 88
4.4 Recursive Reconstruction 89
4.5 Discrete Frames 91
4.6 Roots of Unity: A Finite-Dimensional Example 95
Exercises 96
5. Discrete Time-Frequency Analysis and Sampling
5.1 The Shannon Sampling Theorem 99
5.2 Sampling in the Time-Frequency Domain 102
5.3 Time Sampling versus Time-Frequency Sampling 110
5.4 Other Discrete Time-Frequency Subframes 112
5.5 The Question of Orthogonality 119
Exercises 121
6. Discrete Time-Scale Analysis
6.1 Sampling in the Time-Scale Domain 123
6.2 Discrete Frames with Band-Limited Wavelets 124
6.3 Other Discrete Wavelet Frames 133
Exercises 137
7. Multiresolution Analysis
7.1 The Idea 139
7.2 Operational Calculus for Subband Filtering Schemes 147
7.3 The View in the Frequency Domain 160
7.4 Frequency Inversion Symmetry and Complex Structure 169
7.5 Wavelet Packets 172
Exercises 174
8. Daubechies' Orthonormal Wavelet Bases
8.1 Construction of FIR Filters 176
8.2 Recursive Construction of <f> in the Frequency Domain 183
3.3 Recursion in Time: Diadic Interpolation 189
8.4 The Method of Cumulants 191
8.5 Plots of Daubechies Wavelets 197
Exercises 197
Part II: Physical Wavelets
9. Introduction to Wavelet Electromagnetics
9.1 Signals and Waves 203
9.2 Fourier Electromagnetics 206
9.3 The Analytic-Signal Transform 215
9.4 Electromagnetic Wavelets 224
9.5 Computing the Wavelets and Reproducing Kernel 229
9.6 Atomic Composition of Electromagnetic Waves 233
9.7 Interpretation of the Wavelet Parameters 236
9.8 Conclusions 244
10. Applications to Radar and Scattering
10.1 Ambiguity Functions for Time Signals 245
10.2 The Scattering of Electromagnetic Wavelets 256
11. Wavelet Acoustics
11.1 Construction of Acoustic Wavelets 271
11.2 Emission and Absorption of Wavelets 275
11.3 Nonunitarity and the Rest Frame of the Medium 281
11.4 Examples of Acoustic Wavelets 283
References 291
Index 297