Optimum detection of random signals in noise, with application to scatter-multipath communication--I

Solutions are obtained in open form for the optimum, probability-computing detector of either Gaussian signals, or known signals transmitted via scatter-paths, where the signals have been further perturbed by additive white Gaussian noise. The optimum receiver operates on the received waveforms with filter-functions and biasing constants determined by pairs of inhomogeneous and homogeneous integral equations, respectively. General solution in closed form has not been obtained, but it is possible to draw a few broad conclusions, among them that the filter-functions can be physically realizable. Approximate solution (for the optimum scatter-path receiver) at small signal-to-noise ratios yields a block diagram having interesting implications. For a single-scatter-path, the optimum receiver may be interpreted as the combination of a correlator with an optimum estimator of the Wiener type. Certain special cases in which complete solution is possible have been investigated in detail, and appropriate curves are presented. The role and performance of the probability-computing detector in an optimum decision-making receiver, for the types of channels considered, is deferred to a companion paper.