The authors consider the robust detection of a known constant signal in additive, mutually independent and identically distributed noise which is nominally Laplace. Employing the classical saddlepoint methodology of Huber (1951), they specify the form of the robust detector and obtain closed form results which allow for the analysis of the performance of the robust detector. They show via examples that the robustness can entail a considerable cost in detector performance. In addition, they compare the performance of the robust detector to that of the Neyman-Pearson detector for examples where the noise is Laplace and also where the noise is least favorable for the given detector.>

Several aspects of the mean square estimation of a second-order random variable based upon elements from a random field are considered. The role of the underlying probability space is stressed. Numerous examples are presented that point out many of the subtleties associated with this endeavor.>

An observation is made concerning the average zero-crossing rate for some stationary random processes considered by Barnett and Kedem (see ibid., vol.37, no.4, p.1188-94, 1991).>

For any positive integer k and any integer N > 1, a probability space, a Gaussian vector X taking values in R/sup k/ defined on the space with a positive definite covariance matrix, and an N-level k-dimensional quantizer Q are presented such that the random vector Q(X) takes on each of the N values in its range with equal probability and such that X and Q(X) are independent. It is pointed out that the results obtained can be used to answer many questions which arise when quantization is employed. Examples of this are provided.>

It is shown that multidimensional convolution need not be associative. Further, for any positive integer k, it is shown that the multidimensional convolution of two real valued, bounded, integrable, nowhere zero functions defined in R/sup k/ can be identically equal to zero. These results are discussed in an algebraic setting, and a consequence involving random fields is briefly considered. Several aspects of multidimensional convolution that would be of interest to the signal processing community are included.>

The authors consider the problem of optimal estimation of a random variable X based on an observation denoted by a random vector Y. A commonly encountered problem involves estimating X via h(Y) so as to minimize E( Phi (X-h(Y))), where h is Borel measurable and Phi is a Borel measurable cost function chosen to adequately reflect the fidelity demands of the problem under consideration. The authors place a mild condition on the regular conditional distribution of X given sigma (Y) that ensures that E( Phi (X-h(Y))) is minimized for any cost function Phi that is nonnegative, even and convex. In addition, it is shown that given any Borel measurable function g: R to R, there exist random variables X and Y possessing a joint density function such that E(X mod Y=y)=g(y) almost everywhere with respect to Lebesgue measure.>

The problem of fusing or combining various estimates to obtain a single good estimate is investigated. The problem of fusion in estimation theory is addressed, and several examples using common distributions in which virtually any method of fusion would be useless in approximating the random variable of interest are presented. A theorem which, for a very general situation, shows that fusion resulting in an almost surely exact approximation is always possible is presented. In particular, this result addresses the situation in which the data consists of the random variable of interest corrupted by additive Gaussian noise and the random variable of interest could be any second-order random variable. Finally, an example which illustrates the utility of this result is presented.>

In this paper we present a performance analysis of a CSMA/CD protocol designed to effectively resolve collisions in a local area network such as Ethernet. We give expressions for the average time required to resolve a collision involvingkstations and the average time for a particular station involved in ak-way collision to send its packet successfully. We also consider the efficiency of this protocol and give a lower bound for the maximum efficiency. The above analysis is independent of any particular packet arrival distribution other than the assumptions that in a successful transmission each station is equally likely to be the one transmitting, that in a collision each station is equally likely to be a participant, and that the probability of the simultaneous arrival of packets is zero.

The discrete-time detection of a constant signal in corrupting noise is considered. First the case where the noise is independent and identically distributed is considered, and the criterion of asymptotic relative efficiency is employed to investigate the effect on the detector's performance induced by altering the form of the detector nonlinearity from that of the locally optimal nonlinearity. The results show that the resultant degradation in performance can be bounded in terms of theL_{2}distance between the locally optimal nonlinearity and the nonlinearity of interest. We then extend our results to the case of weakly dependent\phi-mixing noise and see that, in particular, asymptotic relative efficiency can be viewed as a mapping between metric spaces that is continuous at the point of interest.

Although Max's algorithm is frequently thought of as a simple and adequate method for the design of scalar minimum mean squared error quantizers, there are two tactical problems associated with this iterative scheme: the initialization and the updating of the algorithm. This paper proposes easy and efficient solutions to these problems. A straightforward method for obtaining good initial conditions for three common input densities is presented. Also, an efficient technique for updating the algorithm for a wide class of input densities is given.

It is shown how a modified sign detector may be designed for the nonparametric detection of a constant signal in strong mixing noise. An upper bound on the asymptotic performance of the detector is established, and the form of a detector whose performance achieves this bound is specified. The optimal design of the detector under a finite sample criterion is also considered, and it is shown that there is a marked difference in the detector designs resulting from the two different criteria.

In the context of minimum mean-square error symmetric uniform quantization, we show that for several different distributions on the input signals, log-log plots of step size versus number of output levels and mean-square error versus number of output levels both exhibit nearly linear behavior. This observation results in a straightforward design procedure for symmetric uniform quantization.

The role of weak dependence between observed samples is investigated in the detection of time varying signals in noise, with the goal of establishing quantitative conditions for when the dependency can be ignored. A result is presented that allows bounding the variation in false-alarm rate and detection probability induced by ignoring the dependency. This result is applied to the case of stationary Gaussian noise, and it is shown that the dependency can be ignored if the noise autocorrelation decreases sufficiently fast. In fact, a bound on the variation in false-alarm rate and detection probability is linked to the rate of descent of the noise autocorrelation through an expression that can be easily evaluated.

In this correspondence we consider the design of memoryless discrete-time detectors for a\phi-mixing signal in\phi-mixing noise, where a large degree of dependency may exist between the signal and the noise. The fidelity criterion is taken to be the asymptotic relative efficiency, and it is shown that with this fidelity criterion the nonlinearity in the optimal detector can be found as the solution of an integral equation in which knowledge of only the second-order statistics of the random processes involved is required.

We consider discrete time memoryless detection of random signals in dependent non-Gaussian noise, where both the signal and noise belong to a large class of strong mixing processes, and where a large amount of dependency may occur between the signal and noise. The problem of approximating the optimal nonlinearity under the criterion of asymptotic relative efficiency is considered, and sufficient conditions are presented to insure that the loss in performance of the approximation can be made arbitrarily small. Particular applications are then made to extensions of existing results where the approximating nonlinearity is a quantizer or a polynomial.

The twin tools of asymptotic quantization theory and rate-distortion theory are used to find an expression for the approximate total mean-squared distortion of the series connection of a nonuniform quantizer and a noisy channel. The nonuniform quantizer is modeled as a compressor followed by a uniform quantizer. The approximate total mean-squared error distortion is minimized over a class of parameterized compressor characteristics for input processes whose univariate probability density functions are members of the generalized Gaussian family.

We show that the width of the center spectral lobe for a wide class of MSK-type signals is lower bounded by that for conventional PSK, and we also provide an expression that can be used to determine the width of the center spectral lobe for MSK.

The existence of optimalN-level quantizers is demonstrated for several classes of quantizers including block quantizers, memoryless quantizers, and uniform step-size quantizers. No assumptions whatsoever are made regarding the input probability distribution. The distortion measure considered generalizes many of the commonly used error-based distortion measures, including mean squared-error distortion andrth law distortion.

Asymptotic properties of therth power distortion measure associated with quantizedk-dimensional random variables are considered. Subject only to a moment condition, it is shown that the infimum over allNlevel quantizers of the quantityN^{r/k}times therth power average distortion converges to a finite constant asN \rightarrow \infty.

A nonparametric scheme for detecting a constant signal embedded in m-dependent noise is cousidered, aud the performance of the detector in both the asymptotic case and the case of a finite number of samples is investigated. It is shown that the design of the detector with a finite amount of data can be radically different from the design based upon the asymptotic limit.

The problem of minimum mean-squared error prediction of a discrete-time random process using a nonlinear filter consisting of a zero-memory nonlinearity followed by a linear filter is studied. Classes of random processes for which the best predictor is realizable using a nonlinear filter of the above form are discussed. For those random processes for which the best predictor is not realizable using the above nonlinear filter, an iterative procedure is presented for finding a suboptimal nonlinear filter; special attention is directed to the case where the nonlinearity is a polynomial. Also, a noniterative approach based on nonlinear regression is presented.

In this paper we present the derivation of the bivariate distribution function for median filtered sequences of independent, arbitrary, second-order random variables. This result is then used to qualitatively analyze second moment properties of median filtered data. The results hint towards a low-pass characteristic of the median filters and a low dependency of the output spectrum on input alphabet size and distribution.

The design of detectors for known signals in non-Gaussian\phi-mixing noise is considered. The class of\phi-mixing processes considered is seen to be quite general and allows flexible modeling of a variety of dependent noises. Applying the criterion of asymptotic relative efficiency, the design of the optimal memoryless detector is specified and is seen to depend only on second-order statistical knowledge of the noise. It is then shown that in many cases this design reduces to approximating the noise process with anm-dependent process, finding the corresponding nonlinearity as a solution to a Fredholm integral equation of the second kind, and obtaining the optimal nonlinearity through a limiting process. In addition, conditions are given for the existence of a unique optimal nonlinearity. A bound on the performance of the optimal\phi-mixing detector relative to that of the detector designed under anm-dependent assumption is given. Extensions to the ease of detectors with memory are considered.

The bandlimitedness of the output of a time invariant zero memory nonlinearity whose input is the sum of a Gaussian process and another independent random process is discussed. It is shown that, if the nonlinearity is not a polynomlal, or if the Gaussian input component is not bandlimted, then the output is not bandlimited.

It is shown that a random process is spherically invariant if and only if it is equivalent to a zero-mean Gaussian process multiplied by an independent random variable. Several properties of spherically invariant random processes follow in a simple and direct fashion from this representation.

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.