The authors consider transformation noise processes, which are those that can be generated by passing an underlying noise process through a memoryless invertible nonlinearity. They point out the close relationship between the dependency structure of the transformation noise and that of the underlying noise. The use of the class of transformation noise generated from multivariate Gaussian background noise to approximate other noise processes is proposed. This class has tractable, closed-form joint densities. Some ad hoc ways to find the parameters in the model are suggested. It is shown that the nonlinearities in the optimal detector, in the case of transformation noise with a mixture marginal density are approximately the sum of the nominal term and a term proportional to the parameter in .>

The asymptotically robust detection of signals and the analogous problem of estimation of location are considered for noise distributions with heavier than exponential tails. The solution to both problems is identical and is obtained when using efficacy or asymptotic variance as the performance measure. The density of the noise is assumed to belong to a E-contamination class where the contaminant is either (1) from the class of unimodal densities, (2) from the class of densities limited in support to an interval about the mode of the nominal density, or (3) a mixture of a member of (1) with a member of (2). The robust detector or estimator is similar to a censored version of the nonlinearity used for maximum-likelihood estimation or locally optimal detection in the presence of noise with the nominal distribution.>

The bispectrum cancels certain types of noise effects and detects certain types of nonlinear interactions. Two detectors based on the bicovariance, the Fourier inverse of the bispectrum, have been proposed for these reasons. The performance of these detectors is simulated and compared to the performance of the linear and the sign detectors for Gaussian noise, Gaussian noise corrupted by a sinusoid with random phase, and Arctic under-ice data. Sample bispectra and bicovariance functions for actual acoustic data are presented.>

Motivated by the recurring use of the generalized Gaussian family to model different underwater noise sources and the asymptotic performance levels of some commonly used detectors for this family, the authors examined the performance of these detectors for several different underwater noise sources. The sources considered are non-Gaussian, highly correlated, and generally nonstationary, and they vary from being lighter- to heavier-tailed than Gaussian. Although the linear detector had the best performance, the linear rank detector and the L-level uniform quantizer consistently had similar performance levels. Because of their simplicity and robustness, either of these detectors would seem to be the best choice for these environments.>

A method is proposed for conflict-free access of a broadcast channel. The method uses a variable-length coding scheme to determine which user gains access to the channel. For an idle channel, an equation for optimal expected overhead is derived and a coding scheme that produces optimal codes is presented. Algorithms for generating optimal codes for access on a busy channel are discussed. Suboptimal schemes are found that perform in a nearly optimal fashion. The method is shown to be superior in performance to previously developed conflict-free channel access schemes.>

Suboptimal nonlinear detectors for known small signals in non-Gaussian noise are investigated. It is assumed that either the locally optimal nonlinearity is too complex to use or that the noise density is not known precisely. A memoryless suboptimal nonlinearity (ZNL) can be chosen, and the family of densities for which it is optimal is found. A member of this family is then fitted to the observed noise, and the corresponding detector is used. When a rational function is chosen for the nonlinearity, the Pearson family is the set of solution densities. This is not only a general family which contains many common univariate densities, but for nearly Gaussian noise the method of moments can be used efficiently to fit a member density to the noise. The coefficients of a ZNL are estimated for several (non-Pearson) densities using the first four noise moments.>

The robust detection of signals in additive dependent noise is considered. The solution to the finite-sample problem is obtained when the Bayes risk is used as the performance measure. For the multivariate densities involved we assume that they belong to an e-contamination model. The robust detection structure is shown to be optimum for the least-favorable density and is a censored version of the nominal likelihood ratio.

The optimum nonlinearity is defined for detection of a weak signal when minimal knowledge of the dependency structure of the observations is available. Specifically, it is assumed that the observations form a one-dependent strictly stationary sequence of random variables and that only a finite number of moments of the marginal density and the correlation coefficient between consecutive observations are known. It is assumed that the bivariate densities involved can be represented as diagonal series, using orthonormal polynomials. Using efficacy as a performance measure, the optimum nonlinearity is required to satisfy a saddle-point condition over this class of bivariate densities.

The design of signal quantizers when the statistical description of the source is a histogram on a finite domain is considered. Minimum mean and minimax error criteria are discussed, both leading to the design of piecewise linear compressor functions. The problem of allocating the histogram intervals in the symmetric source density case is addressed through the use of Chebyshev probability inequalities. Finally, an adaptive implementation of the proposed system is described.

A modified version of sequential testing is proposed and applied to discrete-time signal detection. The resulting detection procedure exhibits simplicity in structure and in analysis and retains most of the optimal features of sequential detection. Also, two Simple and efficient truncation schemes are suggested for the proposed detection procedure which is unrealizable without truncation.

The detection of a vanishingly small, known signal in multi-variate noise is considered. Efficacy is used as a criterion of detector performance, and the locally optimal detector (LOD) for multivariate noise is derived. It is shown that this is a generalization of the well-known LOD for independent, identically distributed (i.i.d.) noise. Several characterizations of multivariate noise are used as examples; these include specific examples and some general methods of density generation. In particular, the class of multivariate densities generated by a zero-memory nonlinear transformation of a correlated Gaussian source is discussed in some detail. The detector structure is derived and practical aspects of obtaining detector subsystems are considered. Through the use of Monte Carlo simulations, the performance of this system if compared to that of the matched filter and of the i.i.d. LOD. Finally, the class of multivariate densities generated by a linear transformation of an i.i.d, noise source is described, and its LOD is shown to be a form frequently suggested to deal with multivariate, non-Gaussian noise: a linear filter followed by a memoryless nonlinearity and a correlator.

Detection of weak signals in a special\varphi-mixing noise class is considered. The detector structure is restricted to sums of memoryless nonlinear transformations of the observations, correlated with the data sequence and compared to a fixed threshold. Using the efficacy to measure performance, the nonlinearity that has min-max performance is derived.

Several investigators have considered polar coordinates quantization of a circularly symmetric source; in particular, the independent bivariate Gaussian source. Their schemes quantize the polar coordinates independently in an attempt to reduce the mean-square error below that of an analogous rectangular coordinates quantizer yet retain an implementation simpler than that of the optimal bivariate quantizer. The design of a spherical coordinates quantizer inkdimensions withk>2(k=2matches published results) is considered. Examples are presented along with comparisons to the rectangular (one-dimensional) and optimal schemes.

Mehr and McFadden's conditionally Markov processes and Jamison's reciprocal processes are two independent extensions of Slepian's work on a particular stationary Gaussian process. The relationship between them and their relationship with Gauss-Markov processes are discussed.

A class of nonparametric sequential tests is considered for testing a symmetric density under the hypothesis against a one-sided shift alternative. The test statistic at each observation is the sum of intermediate statistics obtained from the ranks within the most recentmobservations, where tn is a fixed ranking size. Excessive ranking of the data can be avoided with a proper choice of tn so that real-time implementation of the sequential rank test is feasible. Approximate expressions for the power and average sample number functions are given. Comparison with existing nonparametric tests is studied. Results show that the ranking sizemneed not be very large in order that the performance of the proposed test be almost as good as sequential rank tests which require ranking of all the data.

This paper considers the effect of the carrier phase error on the probability of error in a binary quadrature amplitude modulation (QAM) data transmission system in the presence of intersymbol interference and additive Gaussian noise. The analysis shows that there exists a threshold value of the carrier phase error such that the error probability remains almost unchanged as long as the carrier phase error is less than that threshold value, but increases rapidly once the threshold is exceeded. Good agreement between theory and computer simulations is obtained. Some practical implications of this threshold effect are discussed.

The performance and design of data quantizers for use in memoryless discrete-time detection systems operating with dependent samples is considered. This problem is approached on a large-sample-size small-signal basis, andm-dependence is used to model the dependence structure among data samples. By considering the particular situation of detecting known constant signals in additive noise, previous results for memoryless detector analysis are applied to derive expressions for the evaluation of quantizer-detector performance for this situation. Design criteria are established for the optimum selection of quantizer parameters, and these are seen to be generalizations of earlier designs for the corresponding independent sampling case.

The problem of designing memoryless detectors for known signals in stationary m-dependent noise processes is considered. Applying the criterion of asymptotic relative efficiency, the optimal such detector is shown to be characterized by the solution to a Fredholm integral equation whose kernel depends only on the second-order probability distributions of the noise. General expressions are derived for this solution and for the asymptotic efficiency of the optimal detector relative to other memoryless detectors. To illustrate the analysis, specific results are given for the particular case where the noise process is derived by memoryless nonlinear transformation of a Gaussian process. In addition, an extension of the analytical results to the more general case of\phi-mixing noise processes is discussed.

The problem of designing quantizer-detectors whose performance is insensitive to small deviations in noise statistics is considered. The problem is approached on a small-signal asymptotic basis using the Huber-Tukey [1] contaminated density class to model the noise. By applying a formulation originally noted by Martin and Schwartz, it is shown that the proposed robust quantizer has properties exhibited by established robust procedures. As an example, results are presented for the case of contaminated Gaussian noise for various degrees of contamination.

The relative efficiency of a sequential hypothesis test compared to a fixed sample size test is defined as the ratio of the expected sample size of the sequential test to the sample size of the fixed sample size test with the same size and power. Asymptotic behavior of the relative efficiency is studied for the detection of a constant signal in additive noise. With some regularity conditions imposed on the noise density, the asymptotic relative efficiency of the sequential probability ratio test with respect to the corresponding fixed sample size likelihood ratio test is a function of the size and the power. As the size a approaches zero and the power approaches unity, this asymptotic relative efficiency has a limiting value depending on the functional relationship of a and1 - \betaas they approach zero. Comparison of the power functions is also studied.

This paper considers the design of a tapped delay line (TDL) equalizer from an error probability point of view. We define a new class of TDL equalizers, which includes the zero-forcing equalizer (ZFE) as a special case. Some optimal properties of the new equalizers are presented and their relationships to mean-square equalizers (MSE's) and ZFE are discussed. It is shown that, for a certain class of additive noise, some of the new equalizers result in a probability of error lower than that of the MSE.

The robustness of suboptimal nonlinear detectors for known discrete-time signals in non-Gaussian noise is investigated. The measure of robustness used is the degradation of asymptotic relative efficiency (compared to a linear detector) from that achieved by an optimal nonlinear detector. The first order density of non-Gaussian noise is modeled as a mixture of a small variance Gaussian background noise pdf and a large variance impulsive pdf.

The Ali-Silvey class of distance measures is applied to the problem of designing quantizers for use in binary detection systems. To find optimal solutions the notion of quantization must be generalized slightly, and necessary conditions are established for the selection of quantizer parameters in this context. Local, or small-signal, conditions are also derived and these are seen to agree with asymptotic results based on Pitman efficiency. As an example, four-level generalized quantization for detecting a constant signal in additive generalizedGaussian noise is investigated.

The quantization of the observed data for sequential signal detection is studied. The criteria used are the minimizations of the average sample number under the hypothesis, the average sample number under the alternative, and the maximum average sample number. Numerical results show that the performance is not very sensitive to different criteria. Using a sequential probability ratio test (SPRT), the performances of optimum quantizers are compared to systems with unquantized data. The asymptotic relative efficiencies of the quantizerSPRT's with respect to the SPRT for unquantized data are derived for symmetric noise densities. The relation between these asymptotic relative efficiencies and those of fixed-sample-size detectors is noted.

The effect of intersymbol interference and additive noise on the performance of a digital data transmission system is considered. The data sequence is assumed to be independent and equiprobable. The additive noise is independent of the signal but is not restricted to be Gaussian. A simple lower bound, an upper bound, and a simple approximation to the upper bound, on the probability of error are derived. The approximation to the upper bound is twice the lower bound; hence, either can be taken as an approximation to the actual error probability.

In fixed sample size signal detection, the sign detector is easier to implement and requires much less knowledge of the noise density than does the optimum fixed sample size detector. These advantages accrue at a cost of more observations needed to achieve the same performance. By utilizing the sign detector sequentially, it is shown that considerable saving in the (average) number of observations is obtained over the corresponding optimal fixed sample size detector. The asymptotic relative efficiencies, as the signal-to-noise ratio and both probabilities of error approach zero, of the sequential sign detector compared to fixed sample size detectors are derived. The possibility of truncating the test to avoid excessively large numbers of observations is considered.

It is shown that, for a Gaussian process and for some non-Gaussian processes, any memoryless nonlinearity has a whitening effect in the sense that the output spectrum is smoother and occupies a greater bandwidth than the input spectrum. The ratio between input and output bandwidths is investigated by using several measures of bandwidth. Also, it is shown that classes of nonlinearities exist that are equivalent in the sense of producing the same spectral transformations.

The estimation of a probability density function from measurements corrupted by independent additive Poisson noise is considered. An estimate is derived that is asymptotically unbiased and consistent in the quadratic mean. Also, a practical realization of the estimator is given.

Generalizations of the simple sign detector are considered for nonparametric detection. Nonparametric operation is obtained with only a symmetry assumption on the noise density functions, by basing the detectors on conditional statistical tests. The performance (both asymptotic and finite-sample, finite-signal) of the generalized sign detector is shown to be very good compared to the Wilcoxon detector and to the linear detector, whereas the detector structure remains simple.

A first-order mixture noise density is considered as a model for impulsive noise channels. It consists of a mixture of a small variance, probably Gaussian, background noise pdf and a large variance impulsive pdf. Optimal nonlinear detector structures for known discretetime signals in such noise are derived. Their large-sample performance compared to that of a linear detector is studied using asymptotic relative efficiency.

A formula for the continuous part of the power spectral density of a block coded digital PAM signal is well known for the case of independent blocks. Based on this formula, a procedure is proposed for the synthesis of block codes yielding a polynomial approximation to a desired signal power spectral density. The code is defined by a matrix multiplication over the set of real numbers. An expression is also found for the length of the codewords and source words in terms of the degree of the approximating polynomial and the cardinality of a particular subset of its real zeros.

The Tukey-Huber contaminated noise model is used toobtain rain-max detectors in the asymptotic case for known signals inadditive noise. According to this model, the noise densityf(x)is defined byf(x) = (1 - \ )g(x) + \varepsilon h(x)for a given\varepsilonand densityg(x), withh(x)an arbitrary density from a large class. A general theorem is obtainedspecifying the most robust detector for additive contaminated noisewithg(x)satisfying certain regularity conditions. As an example,detector structures are derived by the application of the theorem for the case whereg(x)belongs to the class of generalized Gaussian densities(parameterized by their rates of exponential decay). The sign detector is shown to be the asymptotically most robust detector wheng(x)is a double-exponential density.

This paper finds an expression for the spectral density of an error-correcting coded digital sequence after modulation. This expression is valid for dependent source words and error-correcting linear block codes. Several examples are given of the ratio of the spectral density of the coded sequence to an uncoded sequence with identical source word rates for both independent and dependent source words.

A class of nonparametric detectors is formulated for dependent input sequences of sampled data. Detectors in this class are nonadaptive modifications of standard nonparametric detectors designed for independent inputs and operate on multivariate samples derived by grouping the dependent univariate input samples. These groups are transformed to single variables and processed according to the corresponding standard nonparametric detection scheme. Performance analysis is initially based on assumptions implying independence of the multivariate samples, and results are then shown to remain valid under weaker conditions. Two modified one-channel detectors are analyzed to illustrate the improved performance achieved over corresponding standard non-parametric detectors.

A wide sense stationary (WSS) processX(t), t \in I, has shift operatorsT_h: X(t) \rightarrow X(t + h), h \in I, which are unitary operators in the Hilbert spaceH(X)generated in the usual way byX(t). We study the class of uniformly hounded linearly stationary (UBLS) processes; This is the class of processes having shift operatorsT_hthat are linear and bounded with\parallel T_h \parallel ^ 2 \leq M, for some constantM. Examples are given of UBLS processes resulting from linear transformations on non-stationary white noise. The notion of an UBLS almost white noise process is defined, and some special cases are studied. Also, possible applications to time series modeling are indicated. The canonical structure of a finite-dimensional deterministic UBLS process is obtained. Theorems for superposition and multiplication of UBLS processes are presented. Finally, continuous-time white noise is given a rigorous treatment in terms of generalized processes, and conditions for UBLS are given.

The structure and performance of a class of nonlinear detectors for discrete-time signals in additive white noise are investigated. The detectors considered consist of a zero-memory nonlinearity (ZNL) followed by a linear filter whose output is compared with a threshold. That this class of detectors is a reasonable one to study is apparent from the fact that both the Neyman-Pearson optimum and the locally optimum (i.e., weak-signal optimum) detectors for statistically independent noise samples can be put into this form. The measure of detector performance used is the asymptotic relative efficiency (ARE) of the nonlinear detector under study with respect to a linear detector appropriate for the same detection problem. A general expression for this ARE is given along with the result that the non-linearity maximizing this expression is any linear function of the nonlinearity in the appropriate constant-signal locally optimum detector. To illustrate the structure and performance of these nonlinear detectors for a wide range of non-Gaussian noise distributions, three general classes of symmetric, unimodal, univariate probability density functions are introduced that are generalizations of the Gaussian, Cauchy, and beta distributions.

This paper considers a class of stochastic processes, called {\em spherically invariant},which have the property that all mean-square estimation problems on them have linear solutions. It is shown that their multivariate characteristic functions are univariate functions of a quadratic form. The corresponding densities are easily found by means of the Hankel transform. Relations between spherical invariance and normality are discussed. Properties relating to the linear estimation problem are given.

The determination of minimum-mean-squared-error (MMSE) nonlinear filters usually involves formidable mathematical difficulties. These difficulties may be bypassed by restricting attention to special classes of filters or special processes. One such class is Zadeh's classn_{1}, which for the general case also involves mathematical difficulties. In this work two realizations of classn_{1}are used for the MMSE reconstruction and filtering of a sampled signal. The cases where the filter reduces to a zero-memory nonlinearity followed by a linear filter are discussed. A suboptimum scheme composed of a zero-memory nonlinearity followed by a linear filter is considered for the reconstruction and filtering of a subclass of the separable process.

This paper presents an integral representation for the derivation of sampling expansions. The representation uses the theory of self-adjoint differential equations. Different methods of evaluating the resulting triple integral have different physical significances and yield the commonly used approaches to the derivation of sampling expansions. The first- and second-order differential operators are discussed, and the physical interpretation of the first-order case is emphasized.

The least mean square error predictor is shown to be linear for a class of processes that are generated by passing independent white noise through a minimum-phase linear filter. Many random processes generated in this way are non-Gaussian.

This paper treats some bandwidth properties of modulated signals of the formq(t) \cos [\omega_{c}t + \varphi(t)]where bothq(t)and\varphi(t)are modulating time functions. For such simultaneous amplitude and angle modulation (AAM), relationships are given connecting the bandwidth of the modulated signal with the properties of the modulating time functions. Some useful bounds on the bandwidth are found and a relationship is derived between the amplitude and the angle modulating functions which results in a minimum bandwidth. Several examples of such minimum bandwidth signals are given.

A procedure is obtained for modifying given sampled-data parametric detectors to make them asymptotically nonparametric. Unlike standard nonparametric devices, these detectors do not require the assumption of independent samples but only a knowledge of the input spectral shapes. As examples of this technique, two types of conventional array detectors are modified to produce nonparametric systems.

This paper is a survey and summary of some of the simpler one-input and two-input detection schemes. Particular emphasis is placed on those systems where less than a complete statistical description of the input is required. Such nonparametric detectors offer the advantages of ease of instrumentation and insensitivity to changes in the input statistics while yielding a fixed maximum false alarm rate. For both the one-input and two-input cases, the paper begins with a review of well-known parametric detectors, proceeds in the direction of increasing departure from the parametric noise model, and introduces corresponding nonparametric techniques.

The problem of detecting signals in noise is reviewed for the multiple input model, where each of the inputs can contain one of many possible signals. The detection procedure for this model becomes, in general, the testing of multiple hypotheses. Two detection criteria are examined for choosing among multiple hypotheses and it is found that, for both criteria, the decision is based on the likelihood functions for the various signals. Systems for computing likelihood ratios are examined in detail for the multiple input case. A multidimensional matched filter is considered and its relationship to the likelihood ratios is shown. Optimum signals are determined for the two-hypothesis problem.

It is shown that the polarity-coincidence correlator (PCC), a two-input detection device, is nonparametric in its false-alarm rate with respect to a wide class of signal and noise distributions. Operating in fixed distributions, the probability of detection is a nondecreasing function of input SNR. Although the PCC performs less well than an ordinary correlator or a Neyman-Pearson detector for Gaussian inputs, it can be markedly superior to both.

The multidimensional demodulation problem is considered from the point of view of statistical estimation theory and a posteriori most probable signal estimates are derived. Correlated signals and noises are treated. This formulation yields a set of two matrix integral equations which must be solved for the optimum estimates. For amplitude modulation, the problem reduces to that of finding a set of time varying filters which are, again, solutions to a matrix integral equation. Special cases such as two-receiver systems, quadrature modulation, and single-sideband have particularly simple representations and are considered in some detail.