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Index

Absolutely continuous distribution, 206 Adaptive filter, 435 Adjoint matrix, 90 Aliasing, 60 Aliasing theorem, 59 All

pole filter, 191 zero filter, 191

Alternative hypothesis (see Hypothesis) Amplitude modulated signal, 11 Amplitude shift keying (see ASK) Analytic function, 168 Analytic signal, 51 Antenna gain, 645 Antipodal signal (see Signal) A periodic convolution, 26 A posteriori density, 357 A priori probability, 355, 492 ARMA,321 AR process, 321 ASK, 12 ASN, 578, 586-589 Asymptotically stable, 144-145 Asymptotic relative efficiency, 491, 599 Autocorrelation

coefficient, 31 function, 23,309

Autocovariance,309 Autogressive process, 321 Autoregressive-moving average process, 321 Average sample number, 578, 586-589 Average value, 16,212 (see Mean, Expected

value, First moment) ensemble (see Expected value), 212 statistical (see Mean), 212 time (see Sample mean), 16

Backward prediction error, 447-448 Bandlimited signal, 21, 22 Bandpass sampling theorem, 153

Bandwidth, 21 comparison, 628 efficiency, 573, 627, 633

Bartlett window, 445 Baseband signal, 423 Basis, 17, 156 Baud Rate (see Symbol rate) Bayes risk, 479 Bayes Rule, 248 Bayes theorem, 248 Bernoulli random variable, 207 Beta distribution, 267 BFSK,514 BIBO (bounded input-bounded output), 83,

143 Bilinear z transform, 71 Binary detection (see Detection) Binary frequency shift keying (see BFSK) Binary hypothesis (see Hypothesis) Binary phase shift keying (see BPSK) Binomial

coefficient, 207 distribution, 207

Binomial theorem, 217 Bit Rate, 568 Bivarate Gaussian density, 243 Blackwell-Rao theorem (see Rao-Blackwell) Boolean field, 201 Borel-Cantelli Lemma, 276 Bound

Chebyshev, 226 Chernoff, 276 exponential, 276

BPSK, 500, 515 Brownian motion, 296 Burg method (see Maximum entropy

method) Butterworth filter, 51

Capacity theorem (see Channel capacity) Carrier frequency, 11

655

656 INDEX

Cascade system, 195-196 Cauchy

density, 267 integral theorem, 171 Riemann equation, 168 Schwartz inequality, 160, 277

Causal filter, 46, 409 Cayley-Hamilton theorem, 99 Central limit theorem, 274 Channel

bandlimited, 21-22,500 capacity, 628 fading, 516-517, 535 multipath,516

Chapman-Kolmogorov Equation, 428 Characteristic

equation, 38, 95 function, 212 joint, 280 polynomial, 95 roots, 95 value, 95

Chebyshev's inequality, 226 Chernoff bound, 276 Chi square density, 255 Chi square test (see N onparametric) Cholesky decomposition, 106,546 Circular convolution, 69 Coherent signal, 457 Colored noise, 314, 537 Compatibility condition, 279 Complementary error function, 219,463 Complementary solution, 136 Completely

controllable, 162 observable, 163

Complex conjugate, 13 envelope, 425 Gamisian r.v., 634 Gaussian vector, 426

Composite hypothesis (see Hypothesis) Conditional

density, 250 distribution, 250 expectation, 251 probability, 247

Confidence coefficient, 433 interval,433

Confluence hypergeometric function, 526 Consistent estimator, 352

Consistent test, 491 Continuous process, 429 Contour integration, 171 Controllable, 162 Convergence

almost surely, 273 mean square, 273 probability, 273

Convex function, 273 Convolution

a periodic, 28 circular, 69 continuous, 28 discrete, 66 integral, 21, 41 periodic, 69 theorem, 28, 37

Correlation coefficient, 31 function, 23, 309

Correlator-detector, 497 Cost function, 355 Covariance, 30, 241, 289

matrix 241 stationary process, 308

Cramer-Rao bound (see Cramer-Rao inequality)

Cramer-Rao inequality, 376, 437 Critical

level, 465 region, 465, 467

Cross correlation, 32 correlation coefficient, 312 covariance, 31, 63,241,311 power spectral density, 32

Curve fitting (see Linear regression) Cyclo-stationary process, 425

Data rate, 644 Decision space, 479 Delta function, 20 De Moivre-Laplace theorem, 276, 596 Demodulation (see Detection) DeMorgan's law, 257 Density function

Cauchy,267 chi square, 255 definition, 206 Erlang, 266 exponential, 261 Gamma, 266

Gaussian, 211 joint, 280 laplace, 266 log normal, 256 marginal, 233-234 Maxwell, 267 normal (see Gaussian) Poisson, 209, 220 Rayleigh, 266 Rice, 272 uniform, 210, 268

Detection binary, 457-559 integrate and dump, 404, 622 M-ary, 559-574 matched, 497 maximum a-posteriori, 493-494, 561 maximum likelihood, 486 non-parametric,590-673 sequential,574-590

Determinant of matrix, 87 Differential

approximation, 554 decoding, 532 encoding, 532

Differential Phase Shift Keying (see DPSK) Differentiation of matrix, 103 Digital filter, 90 Digital modulation

ASK,12 FSK, 13 PSK,12

Digital signal, 2 Dirichlet's condition, 24 Discrete

convolution, 78, 328 Fourier transforms, 68 parameter process, 278 time systems, 328 transforms, 61-84

Discrete matched filter, 401, 439 Discrete Wiener filter, 416-420 Dispersion matrix, 241 Distortionless, 47 Distribution-free tests,S 90 Distribution function (see also Probability

distribu lion) Bernoulli, 207 Beta, 267 binomial, 267 definition, 205 exponential, 261

geometric, 208 joint, 227 marginal, 233-234 Poisson, 209, 220 properties, 305 student's t, 255

DPSK,560 Dynamic system, 39

INDEX 657

Effective isotropic radiated power, 645 Efficient estimator, 353 Eigen

function, 542 value, 95, 542 vector, 95

EIRP, 645 Energy signal, 16 Entropy, 363,443 Envelope

definition, 342 density (see Rayleigh) detector, 523-524

Erf x (see Error function), 219 Ergodic random process, 444 Erlang density, 266 Error

function, 219,463,621 of 1st kind, 459, 465, 486 of 2nd kind, 460, 465,486

Error probability ASK, 515, 633 bit, 514-515 DPSK, 530, 574 FSK,514,519,530,565 MSK,514 PCM,624 PSK,514-515 QAM,627 QPSK,632 symbol, 573-574, 633

Estimation colored noise, 375, 387 Estimation criteria, 355 Estimator

biased, 380 consistent, 352 efficient, 353, 379 Gauss-Markov, 437 least squares, 374-375 linear, 376 maximum a posteriori, 357 maximum entropy method, 452 maximum likelihood, 353, 365

658 INDEX

Estimator (continued) mean square, 356 minimum variance, 352, 438 Moment method, 432 nonlinear, 370 recursive, 390,453 unbiased, 351

Event, 199 Expansion

binomial,3 cosine, 3 exponential, 3 Fourier, 24 Laurent series, 173 logarithmic, 3 Taylor's, 3

Expectation operator, 281 Expected value (see Mean) Exponential

bound, 276 density, 261 distribu tion, 261

Fading, 516-517, 535 Failure rate, 350 False alarm probability (see Probability) Fast Fourier transform, 166 Feedback system, 195,426 Filter

adaptive, 435 all pole, 191 all zero, 191 bandpass, 50 Bu tterworth, 51 cascaded, 195 causal, 192 digital, 190 finite impulse response, 191 high pass, 50 infinite impulse response, 191 Kalman, 453 low pass, 49 matched, 393-404 minimum phase, 192 non recursive, 191 parallel, 195 quadrature, 51 realizable (see Causal) recursive, 191 Wiener, 404-420

Finite energy, 16 power, 16

Finite impulse response, 191 First moment (see Mean) Fisher Information, 376 Fourier series, 24 Fourier transforms, 26 Fredholm

integral equation, 544 resolvent kernel, 554 of second kind, 554

Frequency deviation, 626 diversity, 631 response function (see Response

function) shift keying (see FSK)

FSK, l3 Function

basis, 17 Bessel,521 complementary error, 6 confluent hypergeometric, 526 Dirac delta, 20 error, 6,219,621 Gamma, 6 impulse response, 6 Legender,6 Marcum Q, 529 Modified Bessel, 6 orthonormal, 16 sine, 26 transfer, 47,83 triangular, 2 unit step, 2

Functions of R.V., 221 Fundamental Identity, 487, 583 Fundamental matrix, l38

Gamma density, 266 function, 6

Gauss-Markov estimation, 437 Gauss method, 123 Gaussian

bivariate r.v., 243 characteristic function, 214. 240 density, 211 entropy, 363 joint density. 242 Markov process, 289,301 random variable, 211, 240 vector, 242

Generalized-N eyman-Pearson lemma, 616 Geometric distribution, 208

Goodness-of-fit test, 590 Gram-Charlier Expansion, 264 Gram-Schmidt orthonomoralization, 156,

364 Gray coding, 560 Grou p delay, 47

Hard limiter, 271 Hazard function, 349 Hilbert-Schmidt method, 557 Hilbert transform, 51-53 Holder's inequality, 161 Homogenous Markov chain, 427 Hurwitz polynomial, 450 Hypothesis

alternative, 459 binary, 464 composite, 464 M-ary,464 multiple, 464 null, 459 simple, 464

Ideal observer, 486 III conditioning, 119 Impulse response function, 46 Indempotent matrix, 93 Independent

increment process, 284 random variable, 234, 235

Inequality Chebyshev's, 226 Chernoff, 276 Cramer-Rao,437 Gauchy-Schwartz, 277 Hadamard, 7 Holder, 7, 273 information, 8 Jensensen,273 Minkowski, 161

Infinite im pulse response filter, 191

Innovation filter, 330 process, 331 sequence, 365

lnphase component, 522 Integrate and dump filter, 404, 622 Interarrival time, 283 Intersection of sets (see Product of sets) Inverse

Fourier transform, 26 Laplace transform, 36

of matrix, 90 z transform, 76, 80

Jacobian, 270 Joint

INDEX 659

characteristic function, 280 distribution, 234, 270 Gaussian density (see Gaussian vector) probability density, 279

Kalman filtering, 453-456 Kalman gain, 456 Karhunen-Loeve expansion, 334,542 Karhunen-Loeve transformation, 337 Kolomogorov equations, 429 Kolomogorov-Smirnov test (see Non-

Parametric) Kronecker

delta function, 65 product, 158

Kruskal-Wallis test (see Nonparametric)

Laplace transform convolution theorem, 37 differentiation, 36 final value theorem, 37 initial value theorem, 36 integral theorem, 37 inverse transform, 36 linearity, 36 time-delay, 36 transform pair's, 35-36

Laplacian random variable, 266, 600 Laurent series expansion, 173 Law of large numbers, 274 Least squares estimator, 374 Levinson-Durbin algorithm, 456 Likelihood

function, 468--469, 494 ratio, 468--469, 494

Limit theorems central limit theorem, 274 Kinchin's theorem, 274 Kolmogorov theorem, 274 large number, 274 Lindberg-Levy, 275

Linear estimation (see Estimator) filter, 45 independence, 104, 105 mean square estimator, 356 minimum variance estimator, 352 phase, 47

660 INDEX

Linear (continued)

prediction (see Prediction filter) regression, 374 system, 45-50

Linearly dependent, 104, 155 independent, 104, 155

Link calculation, 641-650 Locally uniformly powerful test, 478 Log likelihood ratio function, 465,

469 Log-normal density, 256, 618

Manchester coding, 622 Mann-Whitney test (see Nonparametric) MAP (see Maximum a posteriori estimator

and detection) MA process, 320 Marcum Q function, 529 Marginal

density, 234 distribution, 233

Markov chain, 286,427 process, 280

Martingale, 285 M -ary Detection (see Detection) MASK,632 Matched filter, 393-404,439,497 Matrix

addition, 85 adjoint, 90 cofactor, 88 diagonal, 85 diagonalization, 97, 545 differentiation, 103 dispersion, 343 Hermitian, 93, 106 Householder, 115 identity, 85 indempotent, 93 integration, 103 inverse, 90 inverse lemma, 115 minor, 85 modal,165 multiplication, 85 nonsingular,92 norm, 119 orthogonal, 93 positive definite, 112 rank, 104 singular, 90

SKEW symmetric, 93 square, 85 square root (see Cholesky decomposition) symmetric, 85 trace, 102 transpose, 85 unitary, 93 Van der Mode, 164

Matrix decomposition (see Cholesky) Maximum a posteriori

detection, 493-494, 561 estimation, 357

Maximum entropy method, 451-452 Maximum likelihood

detection, 486 estimation, 353, 365

Maxwell density, ,267 Mean, 212 Measurement update, 456 Median, 256 MEM (see Maximum entropy method) Memory, 261 Memoryless (see Memory), 423 Mercer's theorem, 334 Method of moment, 432 MFSK,565 Minimax

criterion, 487 risk function, 487

Minimum error probability, 486 mean square estimate, 356 phase, 192

Minkowaski inequality, 161 Miss probability (see Probability) ML (see Maximum likelihood

estimation) Modal matrix, 165 Mode, 256 Modulation (see Signal) Moment, 212

estimator, 432 generating function, 221 theorem, 214

Moving average process, 320 MPSK,570-574 Multiple

amplitude shift keying (see MASK) frequency shift keying (see MFSK) phase shifting keying (see MPSK)

Multichannel detectors, 603 Multidimensional

random variable, 227 Wiener filtering, 441

Multiple hypothesis (see Hypothesis) Mutually exclusive events, 201

Narrowband process, 338 Narrowband signal, 21 Neyman-Pearson

criterion, 466-467 lemma, 466

Noncoherent detection, 515-535 signals, 457

N onparametric tests chi-square tests, 590 Kruskal-Wallis tests, 606 Kolomogorov-Smirnov test 592 Mann-Whitney test, 605-606 methods, 590-613 Sign test, 596 Spearman-Hottling-Pabst test, 605 Spearman rank correlation, 605, 611 t-test,601 Wilcoxon test, 600

Nonstationary process, 309 Norm of

matrix, 119 vector, 63

Normal density (see Gaussian) Normal equation, 118, 372 Nth moment, 290 Null hypothesis (see Hypothesis) Nyquist

interval, 57 rate, 57

Observable, 163 One sigma, 220 One step predictor, 442 Operating characteristic function, 584 Optimum decision function, 468 Optimum linear filter

maximum signal to noise ratio, 393 minimum mean square (see Wiener)

Orthogonal LV., 237 Orthogonality principle, 371 Orthonormal

basis, 156 expansion, 332, 334 function, 16

Over-determined linear system, 118

Paley-Wiener criterion (see Wiener-Paley) Parseval's theorem, 25, 68, 82 Partial fraction method, 38

Parzen window, 446 Path loss, 642 PCM,623 PDF (see Density function) Periodic convolution, 67 Periodic process, 424 Periodogram, 428

INDEX 661

Phase shift keying (see BPSK) Physical realizability (see Causal) Point process, 286 Poisson distribution, 220, 301 Poisson process, 288, 301 Polynomial curve filtering (see Curve

fitting) Positive definite, 101, 112 Power

series (see Taylor's) signal, 26

Power spectral density BPSK,625 CPM,626 FSK,626 MSK,625 QAM,626 QPSK,625

Power test, 460 Pre-envelope, 425 Prediction filter, 442, 447-448 Predictor, 442, 453 Pre-whitening filter, 402, 545 Principal part, 174 Pro bability

a priori, 355,492 axioms, 201-202 conditional, 247 density function (see Density function) of detection, 465 of error, 460 of false alarm, 460 of miss, 460 space, 201

Probability distribution function beta, 267 binomial, 267 Cauchy, 267 chi-square, 255 Erlang, 266 exponential, 261 gamma, 266 Laplace, 266 log-normal, 256 Maxwell,267 normal,346 normal-bivariate, 233

662 INDEX

Probability distribution function (continued) poisson, 267 Rayleigh, 266 student's, 255 triangular, 268 uniform, 268

PSD (see Power spectral density) Pseudo-inverse matrix, 109 PSK,571 Pulse Code Modulation (see PCM)

QPSK,571 Quadratic form, 100 Quadrature component, 522 Quadra ture filter, 51 Quadrature phase shift keying (see QPSK) Quantization error, 623

Rain attenuation, 646--650 Random process

continuous-time, 282 covariance stationary, 308 discrete-time, 282 Gaussian, 297 independent increment, 289 Markov, 296 narrow band, 338 nonstationary, 286 point, 286 Poisson, 288 stationary, 308 strict sense stationary, 249 weakly stationary (see Wide sense) wide sense stationary, 308 Wiener, 297

Random Signal (see Random process) Random variables

definition, 205 independent, 234 orthogonal, 237 uncorrelated,236

Rank, 157 Rao-Blackwell Theorem, 434 Rational function, 169 Rayleigh density, 266, 343 Receiver operating characteristic (see ROC) Rectifier

full wave, 271 half wave, 271

Recursive estimation, 387-393 least square method, 390, 438

Reflection coefficient, 449 Region of covergence, 71 Relative efficiency, 353 Reliability function, 349 Residue theorem, 175 Rician density, 343, 526 Riesz-Fejer theorem, 179 Risk function, 356 RLC circuit, 43 ROC, 473 Round-off errors, 121 Row operation, 127

Sample au tocorrelation, 63 covariance, 63 function, 279 mean, 265 variance, 344

Sampling theorem, 58,153,154 Scintillation, 515, 517, 535 Set

complement, 200 disjoint, 201 empty, 201 intersection, 199 mu tually exclusive, 201 null (see Empty) product, 199 union, 199

Second moment, 212 Selective fading, 516-517 Sequential

detection, 574-590 estimation, 390-392 probability ratio test (see SPRT)

Sign test (see Non-parametric test) Signal-to-noise ratio, 394, 644 Signals

AM, 11 antipodal, 502, 504,621 ASK, 12,515,633 bandlimited, 21 bandpass, 50 biothogonal,568 BFSK,519 BPSK, 500, 515 continuous-time, 2 CPFSK (see CPM) CPM,626 Differential phase shift keying (see

DPSK)

discrete-time, 2 DPSK,530 energy, 16 FM, 11-12 FSK, 514, 519, 530 low-pass, 49 Manchester coded, 622 MSK, 514, 626 Narrow-band, 21 NRZ,622 OQPSK,625 orthogonal, 19,621 PCM,622 PM, 11 power, 16 QAM,627 QPSK, 570-571, 625 random (see Random process) sinusoidal, 2 SQPSK (see OQPSK) triangular, 2 unit step, 2

Simple hypothesis (see Hypothesis) Single-channel detector, 593 Singular matrix, 90 Singular-value decomposition, 108 Size of the test, 465 Skew symmetric, 93 Slave equation, 124 Slow Rayleigh fading, 535 Source coding, 559 Spearman-Rank

correlation coefficient (see nonparametric) test (see Nonparametric)

Specular, 516 Spectral

analy sis, 317 estimation, 444 factorization, 410 resolu tion, 446

Spectrum, 21 SPRT,574 Square law device, 424 Square root method, 106 Stable, 83, 143-145 Standard deviation, 212 Standard Wiener process, 298 State

space, 134 variable method, 128-147

Stationary process covariance, 308

INDEX 663

independent increment, 285 strict sense, 286 wide sense, 308

Statistically independent, 249 Steady state, 40 Step function (see Unit step function) Stochastic process (see Random process) Strictly stationary process, 286 Student-t-density,255 Sufficient statistics, 354 Sylvester's theorem, 101 Symbol rate, 568 Symmetric matrix, 85 Symmetry condition, 279 System

cascade, 426 causal, 46 dynamical, 39, l37, 141-142 feed-back, 195,426 impulse response function, 40, 46, 317 line~,46,65, 137, 141-142 time inv~iant, 46, 65 time variant, 46 transfer function, 47, l31

Tapped delay line (see Transversal filter) Taylor's series expansion, 3 Telegraph signal, 423 Temporarily homogeneous, 305 Timing error, 622 t-test (see nonparametric) Three sigma, 220 Threshold level, 460 Tiknov density, 619 Time update, 455 Total probability theorem, 248 Trace of matrix, 158 Transfer function, 46, 131 Transform

Fourier, 26,67-68 Hilbert, 51 Laplace, 35 Z-transform, 71

Transition matrix, 138 Transition probability, 305 Transversal filter, 134,436 Triangular density, 268 Trigonometric identities, 4 Tukey-Hanning Window, 446 Two sigma, 220

UMP, 469, 472,478 Unbiased estimator, 351

664 INDEX

Unbiased test, 491 Uncorrelated processes, 293 Uniform random variable, 268 Uniform ally most powerful (see UMP) Unit step function, 2

Vander Mode Matrix, 164 Variance, 212 Vector

addition, 62 column, 85 inner product, 63 norm, 63 orthogonal, 63 row, 85

Venn diagram, 200

Waiting time, 306 Wald equation, 577 Weakly stationary (see Wide sense) Well conditioning, 119 White noise, 314, 315

Whitening filter, 330, 545 Wide-sense stationary, 308 Wiener filter, 404-420, 441 Wiener-Hopf

equation, 363, 417 integral equation, 407, 409

Wiener-Kinchin theorem, 428 Wiener-Levy process, 297 Wiener-Paley condition, 330,409 Wiener process, 297-298 Wilcoxon test (see Nonparametric) Window

Bartlett, 445 Parzen,446 Tukey-Hanning, 446

Woodbury formula, 114 Word error (see Symbol error)

Yule-Walker equation, 372

Z-transform,71 Zeros, 167