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Spectral Analysis of Signals Petre Stoica Uppsala University and Randolph Moses The Ohio State University PEARSON --- Pn'llti('t' lIall Upper Saddle River, New Jersey 07458

Library of Congress Cataloging-in-Publication Data Spectral Analysis of SignalslPetre Stoica and Randolph Moses p. cm. Includes bibliographical references index. ISBN 0-13-113956-8 1. Spectral theory I. Moses, Randolph n. Title 512'-dc21 QA814.G27 2005 00-055035 CIP Vice President and Editorial Director, ECS: Marcia J. Horton Executive Managing Editor: Vince O'Brien Managing Editor: David A. George Production Editor: Scott Disanno Director of Creative Services: Paul Belfanti Creative Director: Jayne Conte Cover Designer: Bruce Kenselaar Art Editor: Greg Dulles Manufacturing Buyer: Lisa McDowell Senior Marketing Manager: Holly Stark © 2005 Pearson Education, Inc. Pearson Prentice Hall Pearson Education, Inc. Upper Saddle River, New Jersey 07458 .---. PEARSON Pit»))! /( '(' Ittll 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. Pearson Prentice BalInt is a trademark of Pearson Education, Inc. MATLAB is a registered trademark of The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098. The author and publisher of this book have used their best efforts in preparing this book. These efforts include the development, research, and testing of the theories and programs to determine their effectiveness. The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book. The author and publisher shall not be liable in any event for incidental or consequential damages in connection with. or arising out of, the furnishing. performance, or use of these programs. Printed in the United States of America 10 5 4 1':;-: 6 3 2 9 8 7 ISBN 0-13-113956-8 Pearson Education Ltd., London Pearson Education Australia Ply. Ltd., Sydney Pearson Education Singapore, Pte. Ltd. Pearson Education North Asia Ltd., Hong Kong Pearson Education Canada Inc., Toronto Pearson Educaci6n de Mexico, S.A. de C.V. Pearson Education-Japan, Tokyo Pearson Education Malaysia, Pte. Ltd. Pearson Education, Inc., Upper Saddle River. New Jersey -

Contents List of Exercises Preface Notational Conventions Abbreviations 1 Basic Concepts Introduction 1.1 1.2 Energy Spectral Density of Detenninistic Signals 1.3 Power Spectral Density of Random Signals First Definition of Power Spectral Density 1.3.1 1.3.2 Second Definition of Power Spectral Density 1.4 Properties of Power Spectral Densities 1.5 The Spectral Estimation Problem 1.6 Complements 1.6.1 Coherence Spectrum 1.7 Exercises 2 Nonparametric Methods Introduction 2.1 2.2 Periodogram and Correlogram Methods 2.2.1 Periodogram 2.2.2 Correlogram xi xv xix xxi 1 1 3 5 6 7 9 13 13 13 16 23 23 24 24 24 v i I \' H \ , 1 ·1 ~',l "-i 11 Jr.• i;~ Ie:~ n /'.: ,., ·1 ;~ ~1 .~ 11 ~ :. I ••

Contents vi 2.3 2.4 2.5 Radix-2 FFT Zero Padding Bias Analysis of the Periodogram Periodogram Computation via FFT 2.3.1 2.3.2 Properties of the Periodogram Method 2.4.1 2.4.2 Variance Analysis of the Periodogram The Blackman-Tukey Method The Blackman-Tukey Spectral Estimate 2.5.2 Nonnegativeness of the Blackman-Tukey Spectral Estimate 2.5.1 Time-Bandwidth Product and Resolution-Variance Tradeoffs in Wmdow Design Some Common Lag Windows Temporal Wmdows and Lag Windows 2.6.2 2.6.3 Wmdow Design Example 2.6.4 2.6.1 2.6 Window Design Considerations 2.7 Other Refined Periodogram Methods 2.8 Complements Bartlett Method 2.7.1 2.7.2 Welch Method Daniell Method 2.7.3 Sample Covariance Computation via FFf FFf-Based Computation of Windowed Blackman-Tukey Periodograms Data- and Frequency-Dependent Temporal Windows: The Apodization Approach Estimation of Cross-Spectra and Coherence Spectra 2.8.1 2.8.2 2.8.3 2.8.4 2.8.5 More Time-Bandwidth Product Results Exercises 2.9 3 Parametric Methods for Rational Spectra Introduction Signals with Rational Spectra 3.1 3.2 3.3 Covariance Structure of ARMA Processes 3.4 AR Signals 3.5 Order-Recursive Solutions to the Yule-Walker Equations 3.4.1 3.4.2 Yule-Walker Method Least-Squares Method Levinson-Durbin Algorithm 3.5.1 3.5.2 Delsarte-Genin Algorithm 3.6 MA Signals 3.7 ARMA Signals 3.7.1 Modified Yule-Walker Method 3.7.2 Two-Stage Least-Squares Method ARMA State-Space Equations 3.8 Multivariate ARMA Signals 3.8.1 I JI • :: :! 27 27 29 30 30 34 39 40 41 42 42 44 45 50 52 52 53 55 58 58 60 62 67 70 75 90 90 91 93 94 94 96 99 100 102 • 106 107 108 111 113 114

Contents 3.8.2 3.8.3 Subspace Parameter Estimation-Theoretical Aspects Subspace Parameter Estimation-Implementation Aspects 3.9 Complements The Partial Autocorrelation Sequence Some Properties of Covariance Extensions 3.9.1 3.9.2 3.9.3 The Burg Method for AR Parameter Estimation 3.9.4 The Gohberg-Semencul Formula 3.9.5 MA Parameter Estimation in Polynomial Time ", 3.10 Exercises 4 Parametric Methods for Line Spectra Introduction 4.1 4.2 Models of Sinusoidal Signals in Noise 4.2.1 Nonlinear Regression Model 4.2.2 ARMA Model 4.2.3 Covariance Matrix Model 4.3 Nonlinear Least-Squares Method 4.4 High-Order Yule-Walker Method 4.5 Pisarenko and MUSIC Methods 4.6 Min-Norm Method 4.7 ESPRIT Method 4.8 4.9 Complements Forward-Backward Approach 4.9.1 Mean-Square Convergence of Sample Covariances for Line Spectral Processes 4.9.2 The Caratheodory Parameterization of a Covariance Matrix 4.9.3 Using the Unwindowed Periodogram for Sine Wave Detection in White Noise 4.9.4 NLS Frequency Estimation for a Sinusoidal Signal with Time-Varying Amplitude 4.9.5 Monotonically Descending Techniques for Function Minimization 4.9.6 4.9.7 A Useful Result for Two-Dimensional (2D) Sinusoidal Signals Frequency-Selective ESPRIT-Based Method 4.10 Exercises 5 Filter-Bank Methods Introduction 5.1 5.2 Filter-Bank Interpretation of the Periodogram 5.3 Refined Filter-Bank Method Slepian Baseband Filters 5.3.1 5.3.2 RFB Method for High-Resolution Spectral Analysis 5.3.3 RFB Method for Statistically Stable Spectral Analysis 5.4 Capon Method 5.4.1 Derivation of the Capon Method 5.4.2 Relationship between Capon and AR Methods vii 117 120 122 122 123 125 128 131 135 150 150 155 155 155 156 157 162 166 171 174 175 178 178 181 183 186 188 194 203 209 217 217 220 222 223 226 228 232 232 238

If .~I iii Contents 5.5 Filter-Bank Reinterpretation of the Periodogram 5.6 Complements 5.6.1 Another Relationship between the Capon and AR Methods 5.6.2 Multiwindow Interpretation of Daniell and Blackman-Tukey Periodograms 5.6.3 Capon Method for Exponentially Damped Sinusoidal Signals 5.6.4 Amplitude and Phase Estimation Method (APES) 5.6.5 Amplitude and Phase Estimation Method for Gapped Data (GAPES) 5.6.6 Extensions of Filter-Bank Approaches to Two-Dimensional Signals 5.7 Exercises ~ Spatial Methods 6.1 Introduction 6.2 Array Model The Modulation-Transmission-Demodulation Process 6.2.1 6.2.2 Derivation of the Model Equation 6.3 Nonparametric Methods 6.3.1 Beamforming 6.3.2 Capon Method 6.4 Parametric Methods 6.4.1 Nonlinear Least-Squares Method 6.4.2 Yule-Walker Method 6.4.3 6.4.4 Min-Norm Method 6.4.5 ESPRIT Method Pisarenko and MUSIC Methods 6.5 Complements 6.5.1 On the Minimum-Norm Constraint 6.5.2 NLS Direction-of-Arrival Estimation for a Constant-Modulus Signal 6.5.3 Capon Method: Further Insights and Derivations 6.5.4 Capon Method for Uncertain Direction Vectors 6.5.5 Capon Method with Noise-Gain Constraint 6.5.6 6.5.7 The CLEAN Algorithm 6.5.8 Unstructured and Persymmetric ML Estimates of the Covariance Matrix Spatial Amplitude and Phase Estimation (APES) 6.6 Exercises APPENDICES A Linear Algebra and Matrix Analysis Tools Introduction A.l A.2 Range Space, Null Space, and Matrix Rank A.3 Eigenvalue Decomposition A.3.1 General Matrices A.3.2 Hermitian Matrices A.4 Singular Value Decomposition and Projection Operators 242 246 246 250 253 256 259 262 269 275 275 277 278 280 285 288 291 293 294 295 297 297 297 298 298 300 302 306 311 319 326 331 334 342 342 342 344 345 348 351

Contents A.5 Positive (Semi)Definite Matrices A.6 Matrices with Special Structure A.7 Matrix Inversion Lemmas A.8 Systems of Linear Equations A.8.1 Consistent Systems A.8.2 Inconsistent Systems A.9 Quadratic Minimization B Cramer-Rao Bound Tools B.1 Introduction B.2 The CRE for General Distributions B.3 The CRE for Gaussian Distributions B.4 The CRB for Line Spectra B.5 The CRE for Rational Spectra B.6 The CRE for Spatial Spectra ", C Model Order Selection Tools C.1 Introduction C.2 Maximum Likelihood Parameter Estimation C.3 Useful Mathematical Preliminaries and Outlook C.3.1 Maximum A Posteriori (MAP) Selection Rule C.3.2 Kullback-Leibler Information e.3.3 Outlook: Theoretical and Practical Perspectives C.4 Direct Kullback-Leibler (KL) Approach: No-Name Rule C.5 Cross-Validatory KL Approach: The AlC Rule C.6 Generalized Cross-Validatory KL Approach: the GIC Rule C.7 Bayesian Approach: The Bre Rule e.8 Summary and the Multimodel Approach C.8.1 Summary C.8.2 The Multimodel Approach D Answers to Selected Exercises Bibliography References Grouped by Subject Index ix 357 361 363 364 364 366 371 373 373 376 378 383 384 387 397 397 398 402 402 405 406 407 409 412 413 417 417 418 420 423 435 443

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