Optimum, linear, discrete filtering of signals containing a nonrandom component

The problem of filtering nonrandom signals from stationary random noise has recently received considerable attention. The filter design procedure developed by Wiener is not applicable in this case since that procedure is predicted on the assumption that the signal to be filtered is stationary and random. Lately, both Booton and the team of Zadeh and Ragazzini have developed optimum filters for the smoothing of nonrandom signals; however, both of these filters are of the continuous type, whereas in many applications in which discontinuous control is used there is need for discrete filters for such signals. This paper presents equations governing the design of a discrete version of the Zadeh-Ragazzini filter. The input signal is assumed to be the sum of a nonrandom polynomial and a stationary random component and is assumed to be obscured by stationary random noise. An approximate formula for the output noise power of an optimum filter designed to make a zero-lag estimate of either its input function or one of the derivatives thereof is derived for the important special case in which the noise is white and the signal is a nonrandom polynomial. A brief discussion is given of the use of the filter with nonrandom, nonpolynomial signals.