A First Course in Wavelets with Fourier Analysis E-Book

Albert Boggess

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 398

Veröffentlichungsjahr: 2011

Beschreibung

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.

Leseprobe

CONTENTS

Preface and Overview

0 Inner Product Spaces

0.1 Motivation

0.2 Definition of Inner Product

0.3 The Spaces L2 and l2

0.4 Schwarz and Triangle Inequalities

0.5 Orthogonality

0.6 Linear Operators and Their Adjoints

0.7 Least Squares and Linear Predictive Coding

Exercises

1 Fourier Series

1.1 Introduction

1.2 Computation of Fourier Series

1.3 Convergence Theorems for Fourier Series

Exercises

2 The Fourier Transform

2.1 Informal Development of the Fourier Transform

2.2 Properties of the Fourier Transform

2.3 Linear Filters

2.4 The Sampling Theorem

2.5 The Uncertainty Principle

Exercises

3 Discrete Fourier Analysis

3.1 The Discrete Fourier Transform

3.2 Discrete Signals

3.3 Discrete Signals & Matlab

Exercises

4 Haar Wavelet Analysis

4.1 Why Wavelets?

4.2 Haar Wavelets

4.3 Haar Decomposition and Reconstruction Algorithms

4.4 Summary

Exercises

5 Multiresolution Analysis

5.1 The Multiresolution Framework

5.2 Implementing Decomposition and Reconstruction

5.3 Fourier Transform Criteria

Exercises

6The Daubechies Wavelets

6.1 Daubechies’ Construction

6.2 Classification, Moments, and Smoothness

6.3 Computational Issues

6.4 The Scaling Function at Dyadic Points

Exercises

7Other Wavelet Topics

7.1 Computational Complexity

7.2 Wavelets in Higher Dimensions

7.3 Relating Decomposition and Reconstruction

7.4 Wavelet Transform

Appendix A: Technical Matters

AppendixB: SolutionstoSelected Exercises

AppendixC: MATLAB®Routines

Bibliography

Index

Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc. Hoboken, New Jersey

Published simultaneously in Canada

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers MA 01923, [phone], fax [phone], or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, [phone], fax [phone], or online a http://www.wiley.com/go/permission.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

For general information on our other products and services or for technical support, please contac our Customer Care Department within the United States at [phone], outside the United States at [phone] or fax [phone].

Wiley also publishes its books in a variety of electronic formats. Some content that appears in prin may not be available in electronic format. For more information about Wiley products, visit our web site at www.wiley.com.

Library of Congress Cataloging-in-Publication Data:

Boggess, Albert.A first course in wavelets with fourier analysis / Albert Boggess, Francis J. Narcowich. – 2nd edp. cm.Includes bibliographical references and index.ISBN 978-0-470-43117-7 (cloth)1. Wavelets (Mathematics) 2. Fourier analysis. I. Narcowich, Francis J. II. Title.QA403.3.B64 2009515′.2433–dc222009013334