Summary form only. Provides an overview of the technical articles and features of interest to readers that appear outside of this publication.

Summary form only. Provides an overview of the technical articles and features of interest to readers that appear outside of this publication.

Summary form only. Provides an overview of the technical articles and features of interest to readers that appear outside of this publication.

Summary form only. Provides an overview of the technical articles and features of interest to readers that appear outside of this publication.

Summary form only. Provides an overview of the technical articles and features of interest to readers that appear outside of this publication.

Summary form only. Provides an overview of the technical articles and features of interest to readers that appear outside of this publication.

Summary form only. Provides an overview of the technical articles and features of interest to readers that appear outside of this publication.

Summary form only. Provides an overview of the technical articles and features of interest to readers that appear outside of this publication.

Summary form only. Provides an overview of the technical articles and features of interest to readers that appear outside of this publication.

Summary form only. Provides an overview of the technical articles and features of interest to readers that appear outside of this publication.

Summary form only. Provides an overview of the technical articles and features of interest to readers that appear outside of this publication.

Classical Gaussian white noise in communications and signal processing is viewed as the limit of zero-mean second-order Gaussian processes with a compactly supported flat spectral density as the support goes to infinity. The difficulty of developing a theory to deal with nonlinear transformations of white noise has been to interpret the corresponding limits. In this paper, we show that a renormalization and centering of powers of band-limited Gaussian processes is Gaussian white noise, and, as a consequence, homogeneous polynomials under suitable renormalization remain white noises.

The 'noise-in-noise' problem is viewed as an estimation problem rather than a detection problem. Specifically, this is the problem of estimating the random scale parameter 'a' from observationsx(t), wherex(t) = aS(t) + N(t) \mbox{0 \leq t \leq T \leq \infty}. Here,S(t)andN(t)are Gaussian processes with known covariances. The optimal mean-square estimator is nonlinear, and the bulk of the paper is concerned with methods for determining it. In particular, a computer algorithm based on steepest descent, is developed. Also, the relationship to the detection problem, particularly the so-called singular cases, is examined.

There are many communication as well as control systems--in fact, an increasing number of them--in which at some stage a continuous data source is sampled "periodically," at the nominal rate of2 Wsamples a second,Wbeing the highest frequency component of the data. There are, generally speaking, two kinds of errors introduced by the sampling mechanism: errors in amplitude and errors in timing, or "time jitter." This paper is concerned with the latter. We assume a random model for the jitter. We begin with a study of the properties of the jittered samples for both deterministic and stochastic signals. Depending on the stochastic properties of the jitter, the presence of a discrete component in the signal may give rise to new discrete components as a result of jitter. Generally speaking, however, the effect of jitter is to produce a (frequency) selective attenuation as well as a uniform spectral density component. The more correlation in the jitter, the less the spectral distribution is affected. Various measures of the "error" due to jitter are estimated. Thus the error may be the mean square in the fitted samples or some linear or nonlinear operation thereof. Also, weighted mean-square errors are considered, and general methods of estimating these errors are developed. The problem of optimal use of the jittered samples is next considered. Interpreting the optimality to be in the mean-square sense, an explicit solution for optimal linear operation is obtained. Also for a wide class of signals it is shown that jitter does not affect the nature of the optimal operations; linear operations, for instance, remain linear, although with different weights. To illustrate the methods an example drawn from telemetry is given, where the timing is derived from the zero crossings of a sine wave and the time jitter is taken as due to noise. The jitter is highly correlated and the results involve some lengthy calculations.

This paper presents the first results in an unconventional approach to the problem of mean-square optimization. Instead of obtaining a representation for the optimal operator for a process, we seek to characterize the class of processes for which the optimal operator is of specified form. If the processes are given, so that the multivariate characteristic functions are known, then our results can be used to tell whether it is possible for the optimal operator to have a specified form. The bulk of the paper pertains to the signal extraction problem where the signal and noise are independent and additive, and it is desired to estimate some function of the signal. Here, with a slight shift in viewpoint, we phrase the characterization problem in the following way: Given, for example, a noise process, determine the class of signal processes for which the optimal extraction system is of specified form. The case where the noise process is Gaussian comes in for special attention.

Two sampling (integral interpolation) theorems for continuous signals (continuous parameter stochastic processes) are proved. The first of these is the sampling principle introduced by Shannon, precise formulation or proof of which has not appeared hitherto. Obtained as a secondary result in this connection is a generalization of a result on the spectra of sampled signals given by Bennet. The second theorem is a stochastic version of the Newton-Gauss interpolation formula as representative of a different class of sampling theorems.

The problem of determining optimal non-linear least-square filters is solved for a class of stationary time series. This theory is then used as the basis for developing a band-width reduction scheme using non-linear encoding and decoding filters, for the same class of signals. A simple illustrative example is included.