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 x(t), namely, ${\eqlignno{\rm choose} \quad r_{0}(s, t) \quad if \int\int x(s)h(s, t)x(t) \quad ds dt \quad \lt; k, \cr {\rm choose} \quad r_{1}(st) \quad if \int\int x(s)h(s, t)x(t) \quad ds dt \geqq k},$ where r 0 (s, t) and r 1 (s, t) are the covariance functions while h(s, t) is given as a solution of the integral equation, $\int\int r_{0}(s, u)h(u, v)r_{1}(v, t) du dv = r_{1}(s, t) - r_{0}(s, t).$ This may be regarded as a generalization of the “correlation detection” in the case of binary sure signals in noise. Section I defines the problem, reviews the literature, and, together with certain pertinent remarks, summarizes principal results. A detailed mathematical treatment follows in Section II and the Appendices.