This paper is concerned with the problem of obtaining the optimal linear vector coding (transformation) method that matches anr-dimensional vector signal and ak-dimensional channel under a given channel power constraint and mean-squared-error criterion. The encoder converts thercorrelated random variables intorindependent random variables and selects at mostkindependent random variables which correspond to theklargest eigenvaiues of the signal covariance matrixQ. The encoder reinserts cross correlation into thekrandom variables in such a way that the largest eigenvalue ofQis assigned to the smallest eigenvalue of the channel noise covariance matrixRand the second largest eigenvalue ofQto the second smallest eigenvalue ofR, etc. When only the total power for allkchannels is prescribed, the optimal individual channel power assignments are obtained in terms of the total power, the eigenvalues ofQ, and the eigenvalues ofR. When the individual channel power limits are constrained byP_{1}, ..., P_{k}andRis a diagonal matrix, the necessary conditions of an inverse eigenvalue problem must be satisfied to optimize the vector signal transmission system. An iterative numerical method has been developed for the case of correlated channel noise.

A random signal observed in additive noise is to be linearly preprocessed for transmission via a channel of dimensionality less than that of either the observations or the signal. At the receiver, the signal is to be linearly reconstructed under a weighted-mean-square-error criterion. It is shown that there is no loss of overall optimality if the operation on the raw observations is partitioned into an optimal (least mean-square-error) prefilter followed by a linear coder.

The reliability of digital computing systems is a significant problem. A scheme using multiply redundant computing devices with a majority voting device to determine the output has been shown to improve the reliability of that output. In this paper, a devices called a dissent detector is suggested as an addition to the system. The function of this device is that of an alarm to indicate the quality of the information (likelihood that the information is correct). Dissent is defined as a family of logic functions that compare the outputs of the voters to the output of the majority gate and report the disagreement or degree of disagreement that takes place. The operation of a dissent-detector that reports any disagreement in a triplicated majority voting system is demonstrated by example. The example illustrates the importance of the dissent information to the conditional reliability of the output of the majority gate. The results show that, with very little additional equipment, the conditional reliability of a majority voting system can be greatly enhanced. In fact, the conditional reliability of the example system exceeds the reliability of any of the system components, including the majority gate and the dissent detector.

Optimal (least-mean-square linear) filtering of random signals prior to sampling may be represented as a Wiener-Kalman state-variable filter followed by a coder that synthesizes the signal to be sampled as a linear combination of the estimated states. Although the prefilter doubles the number of states of the overall presampling signal process, the postsampling reconstruction filter need only model the original signal generator and the coder. Overall optimization involves selecting the parameters of the coder to minimize a weighted time-averaged error criterion.

Formulas are derived for linear (least-square) reconstruction of multidimensional ({\em e.g.}, space-time) random fields from sample measurements taken over a limited region of observation. Data may or may not be contaminated with additive noise, and the sampling points may or may not be constrained to lie on a periodic The solution of the optimum filter problem in wave-number space is possible under certain restrictive conditions: 1) that the sampling locations be periodic and occupy a sector of the Euclidean sampling space, and 2) that the wave-number spectrum be factorable into two components, one of which represents a function nonzero only within the data space, the other only within the sector imaging the data space through the origin. If the values of the continuous field are accessible before sampling, a prefiltering operation can, in general, reduce the subsequent error of reconstruction. However, the determination of the optimum filter functions is exceedingly difficult, except under very special circumstances. A one-dimensional second-order Butterworth process is used to illustrate the effects of various postulated constraints on the sampling and filtering configuration.