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An Introduction to Statistical Signal Processing

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Index

Φ function, 64 δ response, 102

a.m.s., 431 abstract space, 389 additive

finite, 42 additivity, 18, 42

countable, 43 finite, 18

a ne, 226 algebra, 24 alphabet, 104

continuous, 116 discrete, 116 mixed, 116

amplitude continuous, 116 discrete, 116

area, 11

ARMA random process, 348 asymptotically mean stationary, 431 asymptotically uncorrelated, 256,

258 autocorrelation matrix, 227 autoregressive, 160, 295 autoregressive filter, 346

autoregressive random process, 348 average

probabilistic, 47 statistical, 47

axioms, 18

axioms of probability, 25

439

Balakrishnan A.V., 305 Bayes risk, 136

Bayes’ rule, 131, 133, 136 Bernoulli process, 94, 158 binomial, 427

binomial coe cient, 427 Binomial counting process, 162,

349

binomial counting process, 163 bit, 184

Bonferoni inequality, 78 Borel field, 37

Borel sets, 37 Borel space, 56

Borel-Cantelli lemma, 247

categorical, 390 Cauchy-Schwarz inequality, 206 causal, 410

cdf, 81, 107, 119

central limit theorem, 199, 235 chain rule, 131, 162

channel noisy, 135

characteristic function, 148, 150, 197

Cherno inequality, 249 chi-squared, 113 collectively exhaustive, 401 complement, 394 complementation, 394 complete, 77

440

complete the square, 126, 140, 152 completion, 77

conditional di erential entropy, 270 conditional expectation, 210 conditional mean, 142

conditional pmf, 130 conditional probability, 71

nonelementary, 168, 169 conditional variance, 142 consistency, 16, 121 continuity, 43

continuity from above, 45 continuity from below, 45 continuous time, 116 convergence

almost everywhere, 240 almost surely, 240 pointwise, 240

w.p. 1, 240

with probability one, 240 convergence in distribution, 236 convergence in mean square, 240 convergence in probability, 240 convolution, 406

discrete, 138 modulo 2, 138 sum, 138

coordinate function, 100 correlation, 203 correlation coe cient, 133 countable, 400

counting process, 163 covariance, 205 cross-correlation, 317 cross-covariance, 214 cross-spectral density, 317

cumalitive distribitution function, 119

cumulative distribution function, 81, 107

decision rule, 135

INDEX

decreasing sets, 33 DeMorgan’s law, 399 density

mass, 11 dependent, 92

derived distribution, 21, 88 detection, 135

di erence symmetric, 396

di erential entropy, 209 Dirac delta, 66 directly given, 22 discrete spaces, 28 discrete time, 116 disjoint, 394, 401 distance, 78

distribution, 87, 105, 117 convergence in, 236 joint, 122

marginal, 122 domain, 401

domain of definition, 21 dominated convergence theorom,

202 dot product, 403

doubly exponential, 428 doubly stochastic, 375

eigenvalue, 404 eigenvector, 404 element, 389 elementary events, 23 elementary outcomes, 23 empty set, 393 eq:2ndorder, 252

equivalent random variables, 89 ergodic decomposition, 376 ergodic theorem, 190

ergodic theorems, 187 ergodicity, 373

error, mean squared, 216 estimate

INDEX

minimum mean squared error, 219

estimation, 146

maximum a posteriori, 147 event, 23

event space, 12, 23, 31 trivial, 27

events

elementary, 23 expectation, 46, 56, 187, 189

conditional, 210 fundamental theorem of, 195 iterated, 211

nested, 211 expected value, 190 experiment, 12, 26 exponential, 428

field, 24 filter

autoregressive, 346 linear, 406

moving average, 344 transversal, 345

FIR, 344

Fourier transform, 148, 151, 308 function, 401

identity, 47 measurable, 98

fundamental theorem of expectation, 195

Gamma, 428 Gaussian, 158, 428

jointly, 213

Gaussian random vectors, 152 geometric, 427

Hamming weight, 55, 160 hard limiter, 99

hidden Markov model, 167

identically distributed, 89

441

identity function, 47 identity mapping, 88 iid, 94, 128, 158 IIR, 344

image, 402

impulse response, 406 increasing sets, 32 increments, 351

independent, 351 stationary, 351

independence, 70, 127 linear, 204

independent, 127

independent and stationary increments, 350

independent identically distributed, 94, 128

independent increments, 351 independent random variables, 127 indicator function, 46, 196 induction, 50

inequality Tchebychev, 242

infinite

countably, 400 inner product, 403 integral

Lebesgue, 58, 423 intersection, 394 interval, 393

closed, xv, 393 half-closed, 393 half-open, 393 open, xv, 393 inverse image, 87, 402

inverse image formula, 88 isi, 351

iterated expectation, 211

Jacobian, 114

joint distribution, 122 jointly Gaussian, 153

442

Kronecker delta response, 102 Kronecker delta response, 408

Laplace transform, 151 Laplacian, 428

law of large numbers, 187, 190 Lebesgue integral, 423

linear, 406

linear models, 348 logistic, 428

MAP, 136

MAP estimation, 147 mapping, 401 marginal pmf, 91 Markov chain, 162 Markov inequality, 242

Markov process, 162, 165, 167 mass, 11

matrix, 403

maximum a posteriori, 136 maximum a posteriori estimation,

147

maximum likelihood estimation, 147 mean, 47, 57, 190

conditional, 142 mean function, 158

mean squared error, 216, 218 mean vector, 206 measurable, 98

measurable space, 25 measure

probability, 42 measure theory, 11 memoryless, 159 metric, 78

minimum distance, 145 mixture, 69, 375

ML estimator, 147 MMSE, 217

modulo 2 arithmetic, 134 moment, 47, 57

INDEX

centralized, 194 second, 206

moment generating function, 198 moments, 194

monotone convergence theorem, 202 moving average, 275, 295, 344 moving average filter, 344 moving-average random process,

348 MSE, 216

mutually exclusive, 394 mutually independent, 94

nested expectation, 211 noise

white, 302 noisy channel, 135 nonempty, 390

nonnegative definite, 405 numeric, 390

one-sided, 29, 405 one-step prediction, 221 one-to-one, 402

onto, 402 operation, 146 orthogonal, 230

orthogonality principle, 226, 230, 231

outer product, 404

Paley-Wiener criteria, 304 partition, 401

pdf, 17, 61 k-dimensional, 68 chi-squared, 113

doubly exponential, 62, 428 elementary conditional, 74 exponential, 62, 428 Gamma, 428

Gaussian, 62, 428

Laplacian, 62, 428 logistic, 428

INDEX

Rayleigh, 428 uniform, 17, 61, 428 Weibull, 428

Phi function, 64 pmf, 20, 48

binary, 48, 427 binomial, 48, 427 conditional, 73, 130 Poisson, 428 product, 124 uniform, 48, 427

pmf:geometric, 427 point, 389

pointwise convergence, 240 Poisson, 428

Poisson counting process, 350, 362 positive definite, 405

power set, 35

power spectral density, 289, 291 prediction

one-step, 221 predictor

one-step, 219 optimal, 218

preimage, 402 probabilistic average, 190 probability

a posteriori, 71 a priori, 71 conditional, 71 unconditional, 71

probability density function, 17, 61

probability mass function, 20, 48 probability measure, 12, 18, 25,

42

probability of error, 135 probability space, 11, 12, 23, 26

complete, 77 trivial, 26

probability theory, 11 product pmf, 124

443

product space, 392 projection, 100

quantizer, 21, 99, 412, 424

random object, 85 random process, 93, 115

ARMA, 348 autoregressive, 348 Bernoulli, 158 counting, 163 Gaussian, 158

iid, 158 isi, 351

Markov, 167 moving-average, 348

random variable, 21, 46, 56, 85, 87

continuous, 110 discrete, 110 Gaussian, 98 mixture, 110

random variable, 97 random variables

equivalent, 89 independent, 127

random vector, 89, 90, 115 Gaussian, 152

random walk, 275 range, 21, 401 range space, 402 Rayleigh, 428 rectangles, 41 regression, 146

regression coe cients, 346 relative frequency, 189

sample autocorrelation, 293 sample points, 23

sample space, 12, 23 sampling function, 100 sequence space, 29 set, 392

444

empty, 393 one-point, 393 singleton, 393 universal, 389

set di erence, 396 set theory, 389 shift, 251 sigma-algebra, 23 sigma-field, 12, 23 sigma-field

generated, 37 signal, 14, 405

continuous time, 30 discrete time, 29

signal processing, 14, 21, 46 simple function, 424 single-sided, 405

space, 389 empty, 390 product, 392 trivial, 390

spectrum, 308 stable, 283, 407, 409

standard deviation, 206 stationarity property, 251 stationary, 253–255

first order, 251 strict sense, 253 strictly, 253 weakly, 252

stationary increments, 351 Stieltjes ingegral, 108 stochastic process, 116 subset, 392

superposition, 406 symmetric, 403 symmetric di erence, 396 system, 405

continuous time, 405 discrete time, 405 linear, 406

INDEX

tapped delay line, 345 Tchebychev inequality, 242 telescoping sum, 44 threshold detector, 145 time

continuous, 116 discrete, 116

time series, 116 time-invariant, 406 Toeplitz, 252 Toeplitz matrix, 420 transform

Laplace, 151 transversal filter, 345 trivial probability space, 26 two-sided, 29, 406

uncorrelated, 204 asymptotically, 256, 258

uncountable, 400 uniform, 427 union, 394 union bound, 78 unit impulse, 66

unit pulse response, 102 universal set, 389 univorm, 428

variance, 47, 57, 194, 206 conditional, 142

vector, 402 random, 115

volume, 11

weakly stationary, 252 Weibull, 428

weight, 11 white noise, 302

Wiener process, 297, 349, 350 discrete time, 166

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