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Index

acoustical application, 356adaptive detection, 77, 106asymptotic performance, 6, 69asymptotic relative efficiency, 6, 7

beta function, 28, 72, 101- incomplete, 236Bolzano-Weierstr ass theorem, 329Borel o -field, 321bounded sequence, 329breakpoint, 335

cardinality, 380cdf, see cumulative distribution functioncentral limit theorem, 210, 286, 344common deviation parameter, 207composite signal , 10conditional mean, 124conditional variance, 124confidence, 320convergence in law, 293convergent subsequence, 329convolution, 241correlated signal, 93, 112correl ation , 63cost , 326crisp information, 321crisp information space , 336, 349- ordered, 336crisp probability space, 323cross-correlation, 243cumulative distribut ion function , 29

decay parameter, 164DeMoivre-Laplace theorem, 210dependence, 63dependence parameter , 285detection probability, 4, 6detector- Fisher-Yates, 204- fuzzy sign , 350- generalized correlator, 125

- generalized correlator array, 126- linear rank, see Wilcoxon- locally opt imum, 1, 4- locally opt imum fuzzy, 2, 360- locally optimum quantizer, 334- locally optimum rank, 2, 29- - one sample, 32-- two sample, 32- locally optimum stochastic signal, 131- Mann-Whitney, 204- median-shift sign, 205- memoryless,290- midr ise-quantizer, 334- midtread-quantizer, 334- one-memory , 290- parametric quadratic, 245- polarity coincidence correlator, 204- quantizer, 334- sign, 204- sign correlator array, 69, 80- signed-rank, 245- square-law, 64- square-law array, 97- three- level midtread-quantizer, 346- two-level midri se-quantizer, 345- uniformly most powerful, 4- Wilcoxon, 204- Wilcoxon signed-rank, 245direct path, 193distribution- bivariate t, 17- bivariate Gaussian, 17, 24- Cauchy, 19- double exponential, 17- generalized Cauchy , 17,19- generalized Gaus sian, 17, 19- generalized logistic , 206- heavy-tailed, 24, 207, 293- Laplace, see double exponential- light-tailed, 207, 293- logistic, 17

394 Index

- multimodal, 212- multinomial, 343- Rayleigh, 193- Student's t, 17duality, 171

efficacy, 7entropy, 357even part, 193, 194event space , 321exact information, 321experiment, 321exponential tail model, 294

fading, 193false-alarm probability, 4, 6Fisher 's information, 12frequency response, 89fuzzy criterion- bayes, 326- maximum likelihood, 326- maximum a posteriori probability, 328- minimax, 327- Neyman-Pearson, 327fuzzy critical function, 325fuzzy dec ision , 319fuzz y event, 321fuzzy hypothesis test , 319fuzzy information, 319, 321fuzzy information space, 336fuzz y information system, 321fuzzy partition, 321fuzzy power function, 325fuzzy random sample, 321fuzzy set, 1fuzzy sign nonlinearity, 350fuzzy test, 321,325- locally most powerful, 332- locally optimum, 332fuzzy test function, 325

gamma function, 17gene ralized Neyman-Pearson lemma, 4, 5generalized observation model , 8geometrically decaying pdf, 294

iid, see independent and identicallydistributed

imprecise information, 319impulse funct ion, 67impulse response sequence, 89impul sive noise, 204, 207, 294incomplete expected value , 45incredibility, 335, 341, 350

independent and identically distributed, 9index of fuzziness, 357information function, 12inhomogeneity, 246, 356interpolating polynomial, 224

joint probability density function, 9

L'Hospital's rule, 174least favorable , 327Lebesgue-Stieltjes integral , 323level of significance, 327

m-ary representation, 370measurable space, 321median-shift value, 205membership function, 321most powerful test, 332moving average , 283multi path, 193

noise model , 8- additive , 1,60- dependent, 283- multiplicative, 8- non-additive, 8- purely-additive, 8- signal-dependent, 8- weakly-dependent, 285nonhomogeneous, 307

observation- prewhitened, 287, 301- transformed, 287, 301- weight-averaged, 287observation model , see noise modelodd part, 193, 194optimum quantization level, 362order statistic, 29, 32ordered fuzzy information space, 336ordered sample space , 379orthogonal system, 321

pdf, see probability density functionpercentile, 344performance- asymptotic, 5, 6- finite sample-size, 5, 6prnf, see probability mass functionpower function , 4power spectral density, 90preas signed size, 325, 347, 354primary signal , 193probability density function , see distribution

probability mass function , 188probability measure , 321

quantization error, 319quant ization level, 335quantizer, 319quantizer characteristic, 334

random dispers ion, 356randomization parameter , 205, 332rank statistic, 29, 30, 32receiver array, 3reference observation, 32regular observation, 32regularity condition, 67relative deviation , 362reparametrization, 14-16

sample fuzzy information, 321, 329score function, 29, 31, 42- even, 194,253- hybrid , 253- odd,194,253secondary path, 193self-noise, 319, 367sensitivity, 292sign statistic, 2, 29signal- colored, 89- correlated, 89, 138- deterministic, 4- known,4

Index 395

- random,4- stochasti c, 4- transform ed known, 289- transformed random , 300- uncorrelated,89, 137- white, 89signal strength , 6, 78, 107size of fuzzy test, 325size of the test, 4space diversity, 123strongly unimodal , 44

Taylor series expansion, 311test statistic, see detectortime correlation, 243transformation noise, 283trapezoidal membership function, 335turbulence, 356type-1 error, 327type-II error, 327

uniformly most powerful , 4unimodal , 17unimodal pdf, 17

vagueness , 319Volterra expansion, 283, 285Volterra kernels, 283, 285

whitened observation vector, 299

Zadeh's definition of probabil ity, 337