On the reception of binary signals in the presence of a small random delay

Receiver design for a binary communication system which operates over a linear channel with a random delay is considered. It is assumed that the variance of the random delay is very small and that the rate of growth of its moments is restricted. Under certain smoothness requirements on the received signal an approximation to the test statistic, which is optimum in the Neyman-Pearson sense, is derived for the case of gaussian receiver noise with covariance R(τ) &equal; R(0)e
^β$|τ|
. It is found that the test statistic, which in general is nonlinear, assumes the linear form of a crosscorrelator when phase reversal signaling is employed. The case where the noise is white and phase reversal signaling is used is investigated. The correlation waveform in this case is found to consist of the expected value of the received signal plus a term dependent on the slope of the signal when the delay is equal to its mean value.