We show how to find solutions to the problem considered by Mitryayev [l ] in the case where the loss power function is quadratic. This problem is to minimize mean-square error when digital data is represented by group code combinations and the a priori probability distribution is uniform. Furthermore, it is shown that there is always a solution which is linear in the sense of Mitryayev.

System performance in the presence of additive noise has been given considerable attention in the past. With the advent of sophisticated data-processing systems using high time-bandwidth products, it becomes necessary to consider the effects of stochastic delays. Depending on the particular application, the mathematical process to be considered is either a stochastic process of a stochastic process or a function of a stochastic process. Ifx(t), a stochastic process, represents the stochastic delay, andf(t)is either a stochastic process or a fixed functionf(t)represents the ideal signal) then the process to be studied isg(t) = f[t + x(t)]. This paper derives various properties ofg(t)and applies them to system performance. In the first section the correlation function and power spectrum ofg(t)are derived in terms of the characteristic function ofx(t)and the power density spectrum off(t)wheref(t)is a random process. It is shown that the effect of stochastic delays is to increase the bandwidth of the received data. In the second section, these results are applied to optimum least squares filtering, and optimum linear filters and expressions for the mean square error are derived. Important special cases are considered. In particular it is shown that with equal first-order statistics, high frequency delays cause less degradation than low frequency delays. Along somewhat similar lines, some results on random jitter in sampling are derived. The degradation due to high and low frequency jitter are compared. At high sampling rates, low frequency jitter causes more error; but when the sampling rate is matched to the bandwidth off(t), high frequency jitter causes greater degradation. In the final section the effect of stochastic delays on the ambiguity function are considered. Expressions are derived that show that the presence of stochastic delays degrades both time and doppler resolution.

Optical systems, which inherently possess two degrees of freedom rather than the single degree of freedom available in a single electronic channel, appear to offer some advantages over their electronic counterparts for certain applications. Coherent optical systems have the added property that one may easily obtain many successive two-dimensional Fourier transforms of any given light amplitude distribution, or, by use of astigmatic optics, one-dimensional transforms can be obtained. Therefore, most linear operations of an integral transform nature are easily implemented. The optical implementation of integral transforms which are of importance to communication theory is discussed; the general problems of optical filter synthesis and multichannel computation and data processing are introduced, followed by a discussion of potential applications. Astigmatic systems, which permit multichannel operation in lieu of two-dimensional processing, are treated as a special case of general two-dimensional processors. Complex input functions are discussed with relation to their role in coherent optical systems.