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This book is organized into sections and chapters, each somewhat self-contained. The earlier chapters give a fairly complete developmentof the discrete wavelet transform (DWT) as a series expansion of signals in terms of wavelets and scaling functions. The later chapters are shortdescriptions of generalizations of the DWT and of applications. They give references to other works, and serve as a sort of annotated bibliography.Because we intend this book as an introduction to wavelets which already have an extensive literature, we have included a rather long bibliography.However, it will soon be incomplete because of the large number of papers that are currently being published. Nevertheless, a guide to the otherliterature is essential to our goal of an introduction.

A good sketch of the philosophy of wavelet analysis and the history of its development can be found in a book published by the NationalAcademy of Science in the chapter by Barbara Burke [link] . She has written an excellent expanded version in [link] , which should be read by anyone interested in wavelets. Daubechies gives a brief history ofthe early research in [link] .

Many of the results and relationships presented in this book are in the form of theorems and proofs or derivations. A real effort has been madeto ensure the correctness of the statements of theorems but the proofs are often only outlines of derivations intended to give insight into theresult rather than to be a formal proof. Indeed, many of the derivations are put in the Appendix in order not to clutter the presentation. We hopethis style will help the reader gain insight into this very interesting but sometimes obscure new mathematical signal processing tool.

We use a notation that is a mixture of that used in the signal processing literature and that in the mathematical literature. We hope this willmake the ideas and results more accessible, but some uniformity and cleanness is lost.

The authors acknowledge AFOSR, ARPA, NSF, Nortel, Inc., Texas Instruments, Inc. and Aware, Inc. for their support of this work. We specificallythank H. L. Resnikoff, who first introduced us to wavelets and who proved remarkably accurate in predicting their power and success. We also thankW. M. Lawton, R. O. Wells, Jr., R. G. Baraniuk, J. E. Odegard, I. W. Selesnick, M. Lang, J. Tian, and members of the Rice ComputationalMathematics Laboratory for many of the ideas and results presented in this book. The first named author would like to thank the Maxfield and Oshmanfamilies for their generous support. The students in EE-531 and EE-696 at Rice University provided valuable feedback as did Bruce Francis, StrelaVasily, Hans Schüssler, Peter Steffen, Gary Sitton, Jim Lewis, Yves Angel, Curt Michel, J. H. Husoy, Kjersti Engan, Ken Castleman, Jeff Trinkle,Katherine Jones, and other colleagues at Rice and elsewhere.

We also particularly want to thank Tom Robbins and his colleagues at Prentice Hall for their support and help. Their reviewers addedsignificantly to the book.

We would appreciate learning of any errors or misleading statements that any readers discover. Indeed, any suggestions for improvement of the bookwould be most welcome. Send suggestions or comments via email to csb@rice.edu. Software, articles, errata for this book, and otherinformation on the wavelet research at Rice can be found on the world-wide-web URL: http: / / dsp.rice.edu/ with links to other sites where wavelet research is being done.

C. Sidney Burrus, Ramesh A. Gopinath, and Haitao Guo

Houston, Texas; Yorktown Heights, New York; and Cuppertino, California

Instructions to the reader

Although this book in arranged in a somewhat progressive order, starting with basic ideas and definitions, moving to a rather complete discussionof the basic wavelet system, and then on to generalizations, one should skip around when reading or studying from it. Depending on the backgroundof the reader, he or she should skim over most of the book first, then go back and study parts in detail. The Introduction at the beginning and theSummary at the end should be continually consulted to gain or keep a perspective; similarly for the Table of Contents and Index. The Matlab programs in the Appendix or the Wavelet Toolbox from Mathworks or other wavelet software should be used for continual experimentation. Thelist of references should be used to find proofs or detail not included here or to pursue research topics or applications. The theory andapplication of wavelets are still developing and in a state of rapid growth. We hope this book will help open the door to this fascinating newsubject.

Openstax-connexions edition

We thank Pearson, Inc. for permission (given in 2012) to put this content (originally published in 1998 with Prentice Hall) into the OpenStax Cnx system online under the Creative Commons attribution only (cc-by) copyright license. We also thank Daniel Williamson at OpenStax for his contributions. This edition has some minor errors corrected and some more recent references added. In particular, Stéphane Mallat latest book, a Wavelet Tour of Signal Processing [link] also available in OpenStax at https://legacy.cnx.org/content/col10711/latest/ and Kovačević, Goyal, and Vetterli's new book, Fourier and Wavelet Signal Processing [link] online at http://www.fourierandwavelets.org/ A valuable collection of basic papers has been published [link] and a book on Frames [link] .

If one starts with Louis Scharf's book, A First Course in Electrical and Computer Engineering , which is in OpenStax at https://legacy.cnx.org/content/col10685/latest/ followed by Richard Baraniuk's book, Signals and Systems , at https://legacy.cnx.org/content/col10064/latest/ and Martin Vetterli et al book, Foundations of Signal Processing at http://www.fourierandwavelets.org/ one has an excellent set of signal processing resources, all online.

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Source:  OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
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