Table Of Content[SPECTRAL
Analysis
A NONPROBABIUSTIC
PRENTICE HALL INFORMATION AND SYSTEM SCIENCES SERIES
Thomas Kailath. Series Editor
STATISTICAL
SPECTRAL
ANALYSIS
A Nonprobabilistic Theory
Dr. William A. Gardner
Professor, Electrical Engineering and Computer Science
University of California, Davis
Davis, California 95616
President, Statistical Signal Processing, Inc.
Yountville, California 94599
PRENTICE HALL
Englewood Cliffs, New Jersey 07632
Library of Congress Cataloging-in-Publication Data
Gardner, William A.
Statistical spectral analysis.
Includes bibliographies and index.
1. Time-series analysis. 2. Signal processing.
3. Spectral theory (Mathematics) I. Title.
QA280.G37 1987 519.5'5 86-30566
ISBN 0-13-844572-9
© 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 987654321
ISBN 0-13-644572-^
Prentice-Hall International (UK) Limited, London
Prentice-Hall of Australia Pty. Limited, Sydney
Prentice-Hall Canada Inc., Toronto
Prentice-Hall Hispano American a, S.A., Mexico
Prentice-Hall of India Private Limited; New Delhi
Prentice-Hall of Japan, Inc., Tokyo
Simon & Schuster Asia Pte. Ltd., Singapore
Editora Prentice-Hall do Brasil, Ltda., Rio de Janeiro
To Nancy
In lieu of time
we might have spent together
CONTENTS
FOREWORD xiii
PREFACE xvii
ACKNOWLEDGMENTS xxi
GLOSSARIES xxiii
Part I Constant Phenomena 1
1. INTRODUCTION TO SPECTRAL ANALYSIS 3
A. Objectives and Motives, 3
B. Orientation, 5
1. What Is Spectral Analysis?, 5
2. Why Analyze Waveforms Into Sine Wave Components?, 7
C. Origins of Spectral Analysis, 12
D. Spectral Analysis and Periodicity, 20
E. Summary, 21
F. Overview of Part I, 22
Exercises, 23
Appendix 1-1: Linear Time-Invariant Transformations
and Fourier Transforms: A Review, 26
2. NONSTATISTICAL SPECTRAL ANALYSIS 34
A. Temporal and Spectral Resolution, 35
B. Data Tapering, 38
C. Time-frequency Uncertainty Principle, 42
D. Periodogram-Correlogram Relation, 42
E. Finite-Average Autocorrelation and Pseudo Spectrum, 43
F. Periodogram and Correlogram Relations for Filters, 46
G. Local Average Power Spectral Density, 48
H. Time Sampling and Aliasing, 49
I. Summary, 51
Exercises, 53
Appendix 2-1: Instantaneous Frequency, 63
vii
3. STATISTICAL SPECTRAL ANALYSIS 67
A. Motivating Example, 68
B. Temporal- and Spectral-Smoothing Equivalence, 72
C. The Limit Spectrum, 74
D. Examples of Spectral Density, 77
1. White Noise, 77
2. Sine Wave with Additive Noise, 78
3. Sine Wave with Multiplicative Noise (Amplitude Modulation), 78
4. Pulse-Amplitude Modulation, 79
5. Sine Wave with Amplitude and Phase Modulation, 80
E. Time-Sampling and Aliasing, 81
F. Time-Series Models, 83
1. The Moving Average Model, 84
2. The Autoregressive Model, 84
3. The ARMA Model, 85
G. Statistical Inference, 85
H. Summary, 87
Exercises, 88
Appendix 3-1: Band-pass Time-Series, 98
Appendix 3-2: Random-Signal Detection, 104
4. ANALOG METHODS 108
A. Temporal and Spectral Smoothing, 109
B. Fourier Transformation of Tapered Autocorrelation, 112
C. Spectral Leakage and Prewhitening, 113
D. Hopped Temporal Smoothing, 116
E. Wave Analysis, 118
1. Complex Implementation, 118
2. Real Implementation, 120
F. Demodulation, 120
1. Complex Implementation, 121
2. Real Implementation, 122
3. Swept-Frequency Implementation, 123
G. A General Representation, 125
H. Summary, 126
Exercises, 128
Appendix 4-1: Other Wave-Analysis Methods, 136
1. The Fano Identity, 136
2. The Schroeder-Atal Identity, 136
5. FRACTION-OF-TIME PROBABILISTIC ANALYSIS 138
A. Motivation, 138
B. Fraction-of-Time Probabilistic Model, 140
C. Bias and Variability, 143
7. The Finite-Time Complex Spectrum, 144
2. The Finite-Time Spectrum, 145
3. Statistical Spectra, 147
4. Time-Frequency Uncertainty Condition, 159
D. Resolution, Leakage, and Reliability: Design Trade-offs, 161
viii Contents
E. Summary, 169
Exercises, 170
0. DIGITAL METHODS 179
A. Introduction, 179
B. The DFT, 180
1. Resolution and Zero-Padding, 180
2. Circular Convolution, 185
3. The FST and CFT, 187
C. Methods Based on the DFT, 192
1. Bartlett-Welch Method, 193
2. Wiener-Daniell Method, 195
3. Blackman-Tukey Method, 196
4. Channelizer Methods, 197
5. Minimum-Leakage Method, 198
D. Fraction-of-time Probabilistic Analysis, 201
E. Summary, 202
Exercises, 202
7. CROSS-SPECTRAL ANALYSIS 211
A. Elements of Cross-Spectral Analysis, 211
B. Coherence, 215
C. Autocoherence and Periodicity, 220
D. Measurement Methods, 223
1. Temporal and Spectral Smoothing, 224
2. Fourier Transformation of Tapered Cross Correlation, 224
3. Cross-Wave Analysis, 224
4. Cross Demodulation, 227
E. Resolution, Leakage, and Reliability, 229
1. Cross periodogram, 229
2. Statistical Cross Spectra, 230
F. Summary, 234
Exercises, 234
Appendix 7-1: Propagation-Path Identification, 239
Appendix 7-2: Distant-Source Detection, 240
Appendix 7-3: Time- and Frequency-Difference-of-Arrival Estimation, 241
8. TIME-VARIANT SPECTRAL ANALYSIS 244
A. General Variation, 244
1. The Physical Spectrum, 244
2. Linear Time-Variant Systems, 246
3. Local Ergodicity, 249
B. Periodic Variation, 250
C. Summary, 251
Exercises, 252
9. PARAMETRIC METHODS 254
A. Introduction, 254
B. Autoregressive Modeling Theory, 255
Contents ix
1. Yule-Walker Equations, 256
2. Levinson-Durbin Algorithm, 257
3. Linear Prediction, 258
4. Wold-Cramer Decomposition, 259
5. Maximum-Entropy Model, 261
6. Lattice Filter, 263
7. Cholesky Factorization and Correlation Matrix Inversion, 265
C. Autoregressive Methods, 266
1. Introduction, 266
2. Least Squares Procedures, 273
3. Model-Order Determination, 281
4. Singular-Value Decomposition, 283
5. Maximum Likelihood Approach, 287
6. Discussion, 288
D. ARMA Methods, 290
1. Modified Yule-Walker Equations, 291
2. Estimation of the AR Parameters, 292
3. Estimation of the MA Parameters, 293
E. Experimental Study, 298
1. Periodogram Methods, 301
2. Minimum-Leakage Method, 306
3. Yule-Walker, Burg, and Forward-Backward Least-Squares AR
Methods, 306
4. Overdetermined-Normal-Equations AR Method, 307
5. Singular-Value-Decomposition Method, 318
6. Hybrid Method, 319
F. Summary, 328
Exercises, 329
Appendix 9-1: Table of Data, 343
Part II Periodic Phenomena 351
10. INTRODUCTION TO SECOND-ORDER PERIODICITY 355
A. Motivation and Overview, 355
B. Derivation of Fundamental Statistical Parameters, 359
7. Generation of Spectral Lines from Second-Order Periodicity, 359
2. Synchronized Averaging, 362
3. Cross-Spectral Analysis, 365
4. Optimum Generation of Spectral Lines, 367
C. Relationships to Woodward Radar Ambiguity and Wigner-Ville
Distribution, 369
D. Sine Waves and Principal Components, 373
7. Linear Periodically Time-Variant Transformations, 373
2. Cyclostationary Stochastic Processes, 375
E. The Link Between Deterministic and Probabilistic Theories, 376
F. Multiple Periodicities, 378
G. Summary, 380
Exercises, 381
x Contents
11. CYCLIC SPECTRAL ANALYSIS 384
A. Cyclic Periodogram and Cyclic Correlogram, 384
B. Temporal and Spectral Smoothing, Resolution, and Reliability, 386
C. The Limit Cyclic Spectrum, 389
1. Derivation, 389
2. Spectrum Types and Bandwidths, 390
3. Symmetries and Parseval Relations, 393
4. Cyclic Cross Spectra, 396
5. Spectral Autocoherence, 396
6. Filtering and Product Modulation, 398
D. Linear Periodically Time-Variant Transformations, 405
1. General Input-Output Relations, 405
2. Rice's Representation, 409
E. Summary, 414
Exercises, 414
12. EXAMPLES OF CYCLIC SPECTRA 419
A. Pulse and Carrier Amplitude Modulation, 420
B. Quadrature-Carrier Amplitude Modulation, 425
C. Phase and Frequency Carrier Modulation, 428
D. Digital Pulse Modulation, 434
E. Digital Carrier Modulation, 442
7. Amplitude-Shift Keying, 442
2. Phase-Shift Keying, 443
3. Frequency-Shift Keying, 448
F. Spread-Spectrum Modulation, 453
7. Direct Sequence PSK, 453
2. Frequency-Hopped FSK, 454
G. Summary, 457
Exercises, 457
13. MEASUREMENT METHODS 463
A. Temporal and Spectral Smoothing, 463
B. Fourier Transformation of Tapered Cyclic Autocorrelation or Ambiguity
Function, 467
C. Fourier Transformation of Spectrally Smoothed Wigner-Ville
Distribution, 470
D. Cyclic Wave Analysis, 470
E. Cyclic Demodulation, 475
F. Summary, 477
Exercises, 478
14. APPLICATIONS 481
A. Optimum Cyclic Filtering, 482
B. Adaptive Cyclic Filtering, 485
C. Cyclic System Identification, 488
D. Cyclic Parameter Estimation and Synchronization, 493
E. Cyclic Detection, 497
F. Cyclic Array Processing, 503
Contents xi
