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CONTENTS PREFACE PART I BASIC METHODS INTRODUCTION The Spectral Estimation Problem and its Application On the Uses o f Spectral Estimation, 6 Principal Approaches to Spectral Estimation, 7 Comparison o f Spectral Estimators, 8 A Common Test Case, 11 References, 13 Problems，14 Appendix 1A Listing o f Test Data Sets. 15 2 REVIEW OF LINEAR AND MATRIX ALGEBRA Introduction, 17 2.1 2.2 Definitions 17 2.3 Special Matrices, 19 2.4 M atrix Manipulations and Formulas. 23 2.5 Important Theorems, 25

2.6 Eigendecompo5>ition o f Matrices, 26 2.7 Solutions o f Linear Equations, 28 2.8 Minimization o f Quadratic and Hcrmitian Functions, 31 References, 33 Problems, 34 Appendix 2A Computer Program for FFT. 36 Appendix 2B Computer Program for Cholesky Solution o f Simultaneous Linear Equations, 38 3 REVIEW OF PROBABILITY, STATISTICS, AND RANDOM PROCESSES Introduction, 41 3.1 3.2 Some Useful Probability Density Functions，41 3.3 Estimation Theory, 45 3.4 Random Process Characlcrization, 51 3.5 Some Important Random Processes, 54 3.6 Hrgodicity o f the Autocorrelation Function, 58 3.7 An Alternative Definition o f the Power Spectral Density, 59 References, 60 Problems, 60 4 CLASSICAL SPECTRAL ESTIMATION Introduction, 63 4.1 4.2 Summary. 64 4.3 Pcriodogram, 65 4.4 Averaged Periodogram, 72 4.5 Blackman-Tukey Spectral Estimation. 77 4.6 Computer Simulation Examples, 82 References. 94 Problems, 94 Appendix 4A Bias and Variance o f Periodogram, % Appendix 4B Bias and Variance of Blackman-Tukey Spectral Estimator. 98 Appendix 4C Computer Program for the Pcriodogram. 100 Appendix 4D Computer Program for the Correlation Estimate, 102 Appendix 4E Computer Program for the Blackman- Tukey Spcctral Estimator，103 41 63 vi Contents

PARAMETRIC MODELING 106 Estimator, 140 5.8 Model Parameter Determination Based on PSD or ACF, Introduction, 106 5.1 5.2 Summary, 108 5.3 Rational Transfer Function Models, 109 5.4 Model Parameter Relationships to Autocorrelation, 114 5.5 Examples o f AR M A ，AR. and M A Processes, 118 5.6 Model Fitting, 131 5.7 MA Modeling and the Blackman-Tukey Spectral 141 Rcfcrences, 143 Problems • 143 Appendix 5A Computer Program to Generate Real White Gaussian Noise，145 Appendix 5B Computer Program to Generate Time Scries, 147 Appendix 5C Computer Program to Compute PSD Values, 150 6 AUTOREGRESSIVE SPECTRAL ESTIMATION: GENERAL 153 Introduction, 153 Summary. 154 Properties of AR Processes, 156 Properties o f the AR Spectral Estimator, 178 Estimation o f AR Paramclcrs and Reflection Coefficients, 185 Estimation o f the AR Power Spectral Dcnsily. 193 Effect o f Noise on the AR Spectral Estimator, 195 Considerations in Model Order Selection, 206 Rcfcrcnces, 207 Problems, 209 Appendix 6A Derivation o f Cramer- Rao Lower Bounds for AR Parameter Estimators, 211 Appendix 6B Computer Program for the Ixvinson Recursion. 213 Appendix 6C Computer Program for Step-Down Procedure，214 7 AUTOREGRESSIVE SPECTRAL ESTIMATION: METHODS 217 Introduction. 217 7.1 7.2 Summary, 218 7.3 Autocorrelation Method, 221 Contents vii

7.4 Covariance Method, 222 7.5 Modified Covariance Method, 225 7.6 Burg Method. 228 7.7 Recursive M LE . 232 7.8 Model Order Selection，234 7.9 Spectral Estimation o f Noisy AR Processes, 237 7.10 Computer Simulation Examples, 240 References, 253 Problems, 256 Appendix 7A Development o f Akaike Information Criterion, 258 Appendix 7B Computer Program for Autocorrelation Method, 260 Appendix 7C Computer Program for Covariance and Modified Covariance Methods, 262 Appendix 7D Computer Program for Burg Method，265 Appendix 7E Computer Progrdm for Recursive M LE Method, 267 8 MOVING AVERAGE SPECTRAL ESTIMATION 271 Introduction, 271 8.1 8.2 Summary, 271 8.3 The M A Spectral Estimator, 272 8.4 Maximum Likelihood Estimation: Durbin's Method, 273 8.5 Statistics o f the M A Parameter and Spcctral Estimators, 277 8.6 Mode! Order Selection, 279 8.7 Other MA Estimators, 280 8.8 Computer Simulation Examples, 282 References, 287 Problems, 287 Appendix 8A Computer Program for Durbin Method, 288 AUTOREGRESSIVE MOVING AVERAGE SPECTRAL ESTIMATION: GENERAL 9 290 Introduction, 290 Summary, 290 Maximum Likelihood Estimation, 291 Statistics o f the Maximum Likelihood Estimator, 293 Numerator Determination for Known Autoregressive Parameters，2% Model Order Selection, 297 A Special ARM A Model. 299 References, 300

Problems, 301 Appendix 9A Derivation o f the Cramer- Rao Bounds for ARMA Parameter Estimators, 302 AUTOREGRESSIVE MOVING AVERAGE SPECTRAL ESTIMATION: METHODS Introduction, 306 Summary, 307 Akaike Approximate M LE , 309 Modified Yule-W alkcr Equations，312 Least Squares Modified Yule-W alker Equations, 316 Input-O utput Identification Approaches, 318 Computer Simulation Examples, 322 References, 342 Problems, 344 Appendix 10A Evaluation o f Partial Derivatives for Akaike M LE , 346 Appendix 10B Positive Definite Property o f Approximate Hessian, 350 Appendix 10C Computer Program fo r Akaike M LE , 351 Appendix 10D Computer Program fo r Modified Y ule- Walker Equations Method. 358 Appendix 10E Computer Program for Least Squares Modified Yule-W alker Equations Method, 361 Appendix 10F Computer Program fo r Mayne-Firoozan Method, 364 11 M INIM UM VARIANCE SPECTRAL ESTIMATION 370 I n I . I n 1 1 1 1 1 n Introduction, 370 Summary, 370 Maximum Likelihood Estimation o f Signal Amplitude, 372 Filtering Interpretation o f the Linear Minimum Variance Unbiased Estimator，374 The Minimum Variance Spectral Estimator, 378 Comparison o f the MVSE and AR Spectral Estimators, 380 Computer Simulation Examples，383 References. 391 Problems, 392 Appendix 11A Computer Program for Minimum Variance Spcctral Estimator, 393 Contents 1 1 1

12 SUMMARY OF SPECTRAL ESTIMATORS 396 12.1 丨mroduction, 3% 12.2 Test Case Data Comparison, 3% 12.3 General Comparison, 401 References, 402 Problems, 403 PART II ADVANCED CONCEPTS 13 SINUSOIDAL PARAMETER ESTIMATION Iniroduction. 407 Summary. 408 Maximum Likelihood Estimation, 408 Cramer-Rao Bounds, 413 Approximate M LE Methods, 416 Frequency Estimation by Spectral Estimation, 420 Properties o f the Autocorrelation M atrix, 422 Principal Component Frequency Estimation，425 Noise Sub^pacc Frequency Estimation, 429 Model Order Selection, 434 Computer Simulation Examples, 436 References • 438 Problems, 440 Appendix 13A Proof o f Spanning Property o f Principal Eigenveclon>, 442 Appendix 13B Proof o f Pisarenko Property, 443 3 3 3 _ 3 1 ■ • « 2 • « 3 • H - I 4 • 5 • 14 MULTICHANNEL SPECTRAL ESTIMATION 3 Introduction, 446 Summary, 447 Review o f Linear Systems and Fourier Transforms, 447 Review o f Random Processes• 452 Classical Spectral Estimation, 455 Rational Transfer Function Models, 457 Autoregressive Spectral Estimation, 460 Autoregressive Moving Average Spectral Estimation. 465 Minimum Variance Spectral Estimation, 468 Computer Simulation Examples. 469 References, 471 6 • 7 • 8 • 9 3 3 9 14 14 比• 癱 3 405 407 446 Contents

Problems, 473 Appendix 14A Derivaiion o f Levinson Algorithm for Solution o f the Multichannel Yule-W alkcr Equations, 475 15 TWO-DIMENSIONAL SPECTRAL ESTIMATION 479 5. 1 IX i .si 1 f l/^5. lrl 1 5 5 5. 1 1 3, 1^ 2. 4- 6- ^ 8- 7 Inlroduction. 479 Summary，480 Review o f Linear Systems and Fourier Transforms, 480 Review o f Random Processes, 487 Classical Spectral Estimation, 489 Autoregressive Spectral Estimation. 492 Maximum Entropy Spectral Eslimation, 505 Minimum Variance Spectral Estimation, 506 Sinusoidal Parameter Estimation, 508 Computer Simulation Examples, 509 o References, 512 Problems. 514 16 OTHER APPLICATIONS OF SPECTRAL ESTIMATION METHODS 1 1 518 6 参 • • 6 •參 I Introduction，518 Time Series Extrapolation and Interpolation, 518 Signal Detection, 519 Bandwidth Compression, 520 Spectral Smoothing and Modeling, 521 Beamforming/Direction Finding. 522 Lattice Filters, 524 Other Applications, 524 References. 525 1 2 3 4 APPENDIX 1 SUMMARY OF COMPUTER PROGRAMS 6 APPENDIX 2 GLOSSARY OF SYMBOLS, n ABBREVIATIONS, AND NOTATIONAL CONVENTIONS APPENDIX 3 DESCRIPTION OF FLOPPY DISK 6 CONTENTS 1 APPENDIX 4 DESCRIPTION OF MENU-DRIVEN SPECTRAL ESTIMATION SOFTWARE 6 INDEX 5 6 ••秦 6 1 7 Contents 6 8 527 533 537 539 541 xi

Chapter 1 Introduction THE SPECTRAL ESTIMATION PROBLEM AND ITS APPLICATION The general problem o f spcctral estimation is that o f determining the spectral contcnt o f a random process based on a finite set o f observations from that pro cess. Form ally, the power spectral density (PSD)，which w ill be denoted by 尸„(/)， o f a complex wide sense stationary (WSS) random process x[n] is defined as P x x if ) = 2 ^xx[^l exp k--- * f (1.1) where r „fA l is the autocorrelation function (ACF) o f x[/i] defined as r«[A：】= f (x^[n]x[n 十 A:J) (1.2) and t is the cxpcclalion operator. The frequency f may either be thought o f as the fraction o f the sampling frequency used in obtaining the data samples from a continuous random process or as the number of cycles/sample. The PSD function describes the distribution o f power with frequency o f the random process. Phys ically. we could determine this distribution by filtering the random process with a bandpass filte r that is centered at / = / 0 and has a sufficiently narrow filte r bandwidth, and then measuring the power at its output. The power is then divided by the filte r bandwidth. This procedure would be repeated for all center fre quencies - I s： f 0 ^ This mclhod. however, presupposes that the obsened random process w ill be o f sufficient duration to allow the filte r transients to decay. The narrower the filte r bandwidth, the longer the observation interval must be.

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