The fast Fourier transform and its applications

463 Pages·1988·8.507 MB·English

by  E Oran Brigham

THE FAST FOURIER
TRANSFORM
AND ITS APPLICATIONS
E. Oran Brigham
Avantek, Inc.
Prentice Hall
Englewood Cliffs, New Jersey 07632
Library of Congress Cataloging-in-Publication Data
Brigham, E. Oran
The fast Fourier transform and its applications I E. Oran Brigham.
p. cm. - (Prentice-Hall signal processing series)
Continues: The fast Fourier transform.
Bibliography: p.
Includes index.
ISBN 0-13-307505-2
I. Fourier transformations. I. Title. II. Series
QA403.B75 1988
515.7'23-dcI9
Editorial/production supervision and
interior design: Gertrude Szyferblatt
Cover design: Diane Saxe
Manufacturing buyer: Barbara Kittle/Cindy Grant
© 1988 by Prentice-Hall, Inc.
A Division of Simon & Schuster
Englewood Cliffs, New Jersey 07632
All rights reserved. No part of this book may be
reproduced, in any form or by any means,
without permission in writing from the publisher.
Printed in the United States of America
10 9 8 7 6 5 4 3
ISBN 0-13-307505-2
Prentice-Hall International (UK) Limited, LClIldon
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To Cami
A very special daughter

CONTENTS
PREFACE xiii
CHAPTER 1 INTRODUCTION 1
1.1 The Ubiquitous FFT
1.2 Interpreting the Fourier Transform 4
1.3 Digital Fourier Analysis 7
CHAPTER 2 THE FOURIER TRANSFORM 9
2.1 The Fourier Integral 9
2.2 The Inverse Fourier Transform II
2.3 Existence of the Fourier Integral 13
2.4 Alternate Fourier Transform
Definitions 22
2.5 Fourier Transform Pairs 23
CHAPTER 3 FOURIER TRANSFORM PROPERTIES 30
3.1 Linearity 30
3.2 Symmetry 32
3.3 Time and Frequency Scaling 32
3.4 Time and Frequency Shifting 35
vii
viii Contents
3.5 Alternate Inversion Formula 40
3.6 Even and Odd Functions 40
3.7 Waveform Decomposition 42
3.8 Complex Time Functions 44
3.9 Summary Table of Fourier Transform
Properties 46
CHAPTER 4 CONVOLUTION AND CORRELATION 50
4.1 Convolution Integral 50
4.2 Graphical Evaluation of the
Convolution Integral 51
4.3 Alternate Form of the
Convolution Integral 54
4.4 Convolution Involving Impulse
Functions 57
4.5 Time-Convolution Theorem 60
4.6 Frequency-Convolution Theorem 64
4.7 Correlation Theorem 65
CHAPTER 5 FOURIER SERIES AND SAMPLED WAVEFORMS 74
5.1 Fourier Series 74
5.2 Fourier Series as a Special Case of the
Fourier Integral 77
5.3 Waveform Sampling 79
5.4 Sampling Theorems 83
CHAPTER 6 THE DISCRETE FOURIER TRANSFORM 89
6.1 A Graphical Development 90
6.2 Theoretical Development 92
6.3 Discrete Inverse Fourier
Transform 97
6.4 Relationship Between the Discrete and
Continuous Fourier Transform 98
6.5 Discrete Fourier Transform
Properties 107
CHAPTER 7 DISCRETE CONVOLUTION AND CORRELATION 118
7.1 Discrete Convolution 118
7.2 Graphical Interpretation of Discrete
Convolution 119
7.3 Relationship Between Discrete and
Continuous Convolution 121
7.4 Graphical Interpretation of Discrete
Correlation 127
CHAPTER 8 THE FAST FOURIER TRANSFORM (FFT) 131
8.1 Matrix Formulation 131
8.2 Intuitive Development 132
8.3 Signal Flow Graph 136
8.4 Dual Nodes 138
8.5 WP Determination 140
8.6 Unscrambling the FFT 141
8.7 FFT Computation FlowChart 141
8.8 FFT BASIC and PASCAL Computer
Programs 145
8.9 Theoretical Development of the Base-2
FFT Algorithm 148
8.10 FFT Algorithms for Arbitrary
Factors 156
CHAPTER 9 FFT TRANSFORM APPLICATIONS 167
9.1 Fourier Transform Applications 167
9.2 FFT Data-Weighting Functions 178
9.3 FFT Algorithms for Real Data 188
9.4 Inverse Fourier Transform
Applications 195
9.5 Laplace Transform Applications 199
CHAPTER 10 FFT CONVOLUTION AND CORRELATION 204
10.1 FFT Convolution of Finite-Duration
Waveforms 204
x Contents
10.2 FFT Convolution of Infinite- and
Finite-Duration Waveforms 211
10.3 Efficient FFT Convolution 223
10.4 FFT Correlation of Finite-Duration
Waveforms 225
CHAPTER 11 TWO-DIMENSIONAL FFT ANALYSIS 232
11.1 Two-Dimensional Fourier
Transforms 232
11.2 Two-Dimensional FFTs 240
11.3 Two-Dimensional Convolution and
Correlation 255
11.4 Two-Dimensional FFT Convolution
and Correlation 260
CHAPTER 12 FFT DIGITAL FILTER DESIGN 272
12.1 FFT Time-Domain Digital Filter
Design 273
12.2 FFT Frequency-Domain Digital Filter
Design 280
CHAPTER 13 FFT MULTICHANNEL BAND-PASS FILTERING 291
13.1 FFT Band-Pass Integrate and Sample
Filters 291
13.2 FFT Band-Pass Filter Frequency
Response Characteristics 299
13.3 Multichannel Band-Pass Filtering by
Shifted FFTs 303
13.4 Sample Rate Considerations in FFT
Multichannel Filtering 313
13.5 FFT Multichannel Demultiplexing 315
CHAPTER 14 FFT SIGNAL PROCESSING AND SYSTEM
APPLICATIONS 320
14.1 Sampling Band-Pass Signals 320
14.2 Quadrature Sampling 327
14.3 FFT Signal Detection 337
Contents xi
14.4 FFT Cepstrum Analysis: Echo and
Multipath Removal 341
14.5 FFT Deconvolution 345
14.6 FFT Antenna Design Analysis 349
14.7 FFT Phase-Interferometer
Measurement System 355
14.8 FFT Time-Difference-of-Arrival
Measurement System 357
14.9 FFT System Simulation 360
14.10 FFT Power-Spectrum Analysis 365
14.11 FFT Beamforming 376
Appendix A The Impulse Function: A Distribution 386
A.l Impulse-Function Definitions 386
A.2 Distribution Concepts 388
A.3 Properties of Impulse Functions 390
A.4 Two-Dimensional Impulse
Functions 392
BIBLIOGRAPHY 394
INDEX 446