Optimum Reception of Binary Gaussian Signals

The problem of optimum reception of binary Gaussian signals is to specify, in terms of the received waveform, a scheme for deciding between two alternative covariance functions with minimum error probability. Although a considerable literature already exists on the problem, an optimum decision scheme has yet to appear which is both mathematically rigorous and convenient for physical application. In the context of a general treatment of the problem, this article presents such a solution. The optimum decision scheme obtained consists in comparing, with a predetermined threshold k, a quadratic form (of function space) in the received waveform xit), namely, choose r0(s,t) if x(s)h(s,t)x(t) ds dt k, where r0(s,t) and ri(s,t) are the covariance functions while h(s,t) as a solution of the integral equation, Jj r0(s,u)h(u,v)ri(v,t) du dv = n{s,t) -- r0(s,t).