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.

The object of this paper is to give some novel calculations of noise statistics. The zero crossing problem is analyzed for nonstationary noise, and some features of the envelope of noise are established. The theory underlying the formulation of the average number of times per second a noise (possibly nonstationary) takes on a specified value is given in some detail. The crossing rate is found in terms of a rather simple integral for the nonstationary noise consisting of the sum of stationary Gaussian noise and a determinate signal. The integral is evaluated in closed form for any determinate signal having a graph composed of straight line segments. In turn the sine wave case is approximated by a triangular wave case. The following two properties of the envelope of noise are proved: (1) the mean square output of an envelope detector is twice the mean square input (independent of the statistics of the input), and (2) the derivative of the envelope of stationary Gaussian noise has a Gaussian first probability distribution. The first property is easy to derive, but it seems not to have been mentioned in the literature; of eourse, it is well known for Gaussian noise and for sine waves. The general version given here is useful in calculations of signal to noise ratios when the interference has non-Gaussian statistics. A heuristic derivation of the second property was included in reference (3). This second result has-application in certain radars, and it is also of general interest in that it displays a nonlinear transformation of Gaussian noise which leaves the first distribution Gaussian. It is rather burdensome to analyze rigorously noise problems that cover a wide front; however, since the scope of this paper is rather narrow, the analysis hera has been carried through on a rigorous basis. It is felt that additional insight is gained when at least some of the problems are so analyzed.