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INDEX

a-fJ tracker. 305 OI-{3-y tracker. 308 a posteriori probabilities. 11 a priori probabilities. 9 Absolute continuity of

measures. 370 Acceptance region. 8 Affl-nity. 127 Alternative hypothesis. 8 Amplifier-limiter. 70 Arrival time. estimation of. 457 Asymptotic efficiency of an

estimate. 245 Asymptotic normality

of least-squares. 264. 267 of MLE·s. 255. 455

Asymptotic relative efficiency (ARE). 128

Autoregressive (AR) sequence. 349 prediction of. 352

Autoregressive/moving-average (ARMA) sequence. 349

Bayes ~timate. 197 Bayes risk. 9. 44. 197 Bayes rule. 9 Benes class of nonlinear

diffusions. 521 Berry-Eseen bound. 134 Bessel function. 49 Bhattacharyya bound. 127 Bias. 246 Binary channel. 12.29. 39 Binary integrator. 165

Cameron-Martin formuill. 404

Cauchy's criterion for mean-square convergence. 389

Cauchy noise. 73. 118 Cepstrum.345 Channel equalization. 217

Characteristic function. 105. 117 Chernoff bounds. 124 Cholesky decomposition. 84.

340 Coherent detection. 66. 399

locally optimum. 71 Completeness. 225

of a sufficient statistic. 227 for exponential families. 229

Conditional density in nonlinear filtering. 497 evolution equation. 497 unnormalized.516

Conditional-mean estimate. 199

Conditional-median estimate. 201

Conditional-mode estimate. 203

Conditional risk. 8. 44. 197 Consistency

in hypothesis testing. 129 of least-squares. 262. 267 of MLE·s. 252. 455

Constant-false-alarm-rate (CFAR) detection. 158

Correlation detector. 67 Correlator.67

bank. 391 nonlinear. 72

Correlator-limiter. 186 Costs

absolute error. 199 posterior. 12 squared error. 198 uniform. 11.201

Counting measure. 372 Covariance matrix. 74 Cubic sensor problem. 520 Cumulant generating

function. 121 Cramer-Rao bound. 238 Critical region. 8

Decision rule. 8 Bayes. 10. 44 minimax. 22 Neyman-Pearson. 34 randomized. 32 sequential. 137 terminal. 137

Deflection. 425 Detection probability. 32 Differential phase-shift

keying (DPSK). 190 Diffusions. 490

evolution of the densities. 492 evolution of the moments. 493

Dominated convergence theorem. 254

Drift. of a diffusion. 490 Dynamical system. 287

E-contaminated mixture. 179. 268

Efficacy of a test. 133 Eigenfunctions. 379 Eigenvalues

of a matrix. 82 of a covariance function. 379

Eigenvectors. 82 Empirical distribution

function. 174 Energy detector. 103. 169 Equivalence of measures. 376 Estimator-correlator

discrete time. 111 continuous time -

diffusion signals. 519 continuous time -

Gaussian signals. 425. 429 continuous time -

nonGaussian signals. 444 Exponential distribution.

204.244 Exponential family. 228 Extended Kalman filter. 524

phase-locked loop. 526

S4S

Factorization theorem. 221 False-alarm probability. 31 Filterin,g.290 Fisher's information. 238

continuous time signals. 453 matrix parameters. 258

Fisher-Yates test. 167 Fokker-Planck equation. 492 Fredholm determinant. 437 Fredholm integral equation.

407.488 Fredholm resolvent. 435. 466 Fundamental identity of

sequential analysis. 149

Gamma density. 105 Gamma function. 106

incomplete. 107 Gaussian noise. 66. 399 Gaussian random process. 382 Gaussian random vector. 74 Girsanov's theorem. 445 Grenander's dichotomy. 387 Grenander's theorem. 377

Hard limiter. 73 Hellinger integral. 127 Hypothesis testing

Bayesian. 7 composite. 43 locally optimum. 54 M-ary.7 minimax. 19 Neyman-Pearson. 31 nonparametric. 157 simple. 43 UMP.50

Homogeneity. tests of. 175

Information inequality. 235 for exponential families. 238 for vector parameters. 258

Innovations process continuous time. 441. 445 discrete time. 110.298 linear. 464 vector. 469

Innovation theorem Gaussian processes. 438 linear. 464 nonGaussian processes. 445

In-phase channel. 94 Ito correction term. 431 Ito differentiation rule. 502

vector version. 508 Ito processes. 504 Ito stochastic integral. 431.

444

J -divergence. 127 Jensen's inequality. 124

Kalman-Bucy filter as an approximate nonlinear

filter. 534 continuous time. 482 discrete time. 292. 323 measurement update. 297 time update. 297 with correlated state and

measurement noises. 361 with feedback. 361

Kalman gain matrix. 296. 483 Karhunen-Loeve expansion. 380 Kolmogorov-Smirnov tests. 174 Kolmogorov-Szego-Krein

formula. 341 Kolmogorov's forward

equation. 492 Kushner's equation. 498

Laplacian noise. 67 Least-favorable distribution

for detection. 179 for estimation. 270

Least-favorable priors. 22 Least-squares estimate

of signal parameters. 260 consistency of. 262. 267 asymptotic normality of. 264. 267

Lebesgue decomposition. 448 Lebesgue measure. 371 Lebesgue-Stieltjes integral.

370.444

546

Level of a test. 32 Levinson algorithm. 321 Levy-Doob theorem. 445 Lie algebra. 520 Likelihood equation. 243 Likelihood-ratio test. 10

generalized. 55 Linear detector. 104 Linear growth condition. 490 Linear MMSE estimation.

309.460 Linear observation model.

216.471 Lipschitz condition. 490 Locally most powerful

(LMP) testing. 54 Location testing. 15.27.36.51

M -estimate. 269 Martingale. 445 Matched filter. 67 Maximum a posteriori

probability (MAP) decision rule. 12

MAP estimation. 203 MAP equation. 204 Mann-Whitney test. 174 Marcum's Q -function. 98 Markov inequality. 121 Markov process. 490

transition densities. 491 Maximum-likelihood

estimates (MLE·s). 242 asymptotic normality of. 255.455 consistency of. 252. 455 of signal parameters. 259. 451 of vector parameters. 257

Maximum-likelihood test. 55 Mean-square continuity. 381 Mean-square integral. 378

Stieltjes. 379 Measurability. 367 Measure. 367

absolute continuity. 370 probability. 367 CT-finite. 369 singularity. 375

Median. 201 Mercer's theorem, 380 Minimal sufficiency. 220 Minim um-mean-absolute-error

(MMAE) estimation. 199 Minimum-mean-squared-error

(MMSE) estimation. 198. 309 Minimum-probability-of-error

decision rule. 11 Minimum-variance unbiased

estimate (MVUE). 218 Miss probability. 31 Moment-space bounds. 128

Neyman-Pearson criterion. 32

Neyman-Pearson lemma. 33 Noise blanker. 73 Noncoherent detection. 92. 407 Nonlinear filtering

approximations. 524. 530. 534 basic equation of. 506 conditional density. 498 evolution of the conditional

mean. 501 evolution of the conditional

variance. 502 unnormalized conditional

density. 516 Nonparametric test. 158 Normal scores test. 167 Null hypothesis. 8

547

On-off keying (OOK). 92 Orthogonality principle. 313.462 Outliers. 269

Paley-Wiener condition discrete time. 337 continuous time. 487

Parameter estimation Bayesian. 197 maximum-likelihood. 242 recursive. 271 robust. 268 signal parameters. 259 unbiased. 219 vector parameters. 212. 257

Partial correlation (PARCOR) coefficients. 321

Periodogram. 114 Phase-locked loop. 526 Pitcher's equation. 398 Pitcher's theorem. 395 Pitman-Noether theorem. 132 Poisson distribution. 275 Polarity coincidence correlator

(PCC).170 Power of a test. 32 Power function. 38 Power spectrum. 114.328 Prediction. 290. 471

Kalman-Bucy.292 Levinson. 319 Wiener-Kolmogorov.341

Prior probabilities. 9 least favorable. 22

Probability ratio test. 10 Projection

onto an eigenvector. 82 theorem. 314

Pseudosignal. 76. 398

Quadratic detector. 103. 125.424

Quadratic variation. 434. 510 Quadrature channel. 94 Quaternary phase-shift keying

(QPSK).189

Radar range estimation. 457 Radiometer. 103 Radon-Nikodym derivative. 371

on (Rn .En ). 372 on discrete sets. 373

Radon-Nikodym theorem. 371 Rao-Blackwell theorem. 223 Rank tests. 166 Receiver operating

characteristics (ROCs). 38 Recursive estimation

of fixed parameters. 271 of discrete-time signals. 292 of continuous-time signals. 482.499.524

Reflection coefficients. 321

Rejection region. 8 Relative efficiency. 129 Relative entropy. 127 Residual. 298 Resolvent kernel. 466 Riccati equation

discrete time. 305 continuous time. 483

Robust detection. 175 Robust estimation. 268

Saddle-point approximation. 120

Sample number of a sequential test. 145

Schwarz inequality. 135 Second-order filter. 528 Sequential detection. 136 Sequential probability ratio

test (SPRT). 143 truncated. 156

Shepp's theorem. 428 Signal-amplitude estimation

Bayesian. 209 maximum-likelihood. 246 minimum-variance unbiased. 230 robust. 268

Signal detection. 63 deterministic signals. 64. 384 nonparametric. 158 robust. 175 signals with random parameters. 90. 407 sequential. 136 stochastic signals. 101. 413

Signal estimation. 286 Kalman-Bucy filtering. 292. 470 Levinson filtering. 319 nonlinear filtering. 488 phase tracking. 524 Wiener-Kolmogorov filtering. 325.485

Signal-parameter estimation. 259 (See also. Signal-amplitude estimation)

548

arrival time. 457 continuous time. 451 robust. 268

Signal selection. 86 Signal-to-noise ratio (SNR).

81. 392 generalized. 425

Significance level. 32 Sign test. 162 Singular detection. 375.

390.395 for rational signals and noise. 427

Singularity of measures. 375 Gaussian measures. 426 with respect to Wiener measure. 428

Smoothing. 290.471 Soft limiter. 70 Spectral decomposition

of a covariance function. 380 of a matrix. 82

Spectral factorization continuous time. 486 discrete time. 337 of rational spectra. 345

Square root of a matrix. 84 State estimation. 290 Stochastic approximation. 273 Stochastic differential

equation. 489 Stochastic integral

Ito. 431. 444 mean-square. 378 Stratonovich.431

Stochastic system. 287 linear. 291. 471 state equation. 287. 471 measurement equation. 290. 471

Stopping rule. 13 7 Sufficiency. 220 System identification. 265

persistence of excitation in. 267 sufficient richness in. 267

t -test. 163 Thermal noise. 66.382

Toeplitz matrix. 320 Tracking. 286 Track-while-scan (TWS)

radar. 290. 303. 306 Transition densities of

a diffusion. 491 Two-channel tests. 168 Types I and II errors. 31

Unbiasedness asymptotic. 245 of a test. 53 of an estimate. 219

Uniformly most powerful (UMP) tests. 50

Unnormalized conditional density in nonlinear filtering. 516

Wald's approximations. 147 Wald's identity. 150 Wald-Wolfowitz theorem. 145 White noise

continuous time. 400. 419 discrete time. 85. 336

549

Whitening filter in detection. 85 in estimation. 299. 340 in continuous time. 417. 486

Wide-sense Markov model. 350

Wiener process. 401 properties. 414 representation. 416

Wiener-Hopf equation continuous time. 435. 485 general. 318 for causal filtering. 486 for noncausal filtering. 327. 486

Wiener-Kolmogorov filtering. 325.485 causal. 333. 344 noncausal. 326 MSE error in. 330. 332

Wilcoxon test. 166. 174

Yule-Walker equations. 320

Zakai equation. 517