Correlation detection of signals perturbed by a random channel

We show that the concept of correlation detection of deterministic signals in additive Gaussian noise can be extended in a natural manner to the detection of signals that are transmitted through a "Gaussian" random channel besides being corrupted by additive Gaussian noise. Such situations are typical in communication over scatter-multipath channels (with or without a specular component). In the deterministic case, the receiver essentially crosscorrelates the received signal with the signal before the additive noise was introduced. When a random channel is present, however, this latter signal, i.e., the output of the random channel, is not known at the receiver, However, knowing the statistics of the channel and the noise, the receiver can make an estimate of it from the received signal on the hypothesis that a particular signal was transmitted. The optimum receiver then crosscorrelates this estimate with the received signal.