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, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">choose r₀(s, t) if ∫∫ x(s)h(s, t)x(t) ds dt <; k, choose r₁(st) if ∫∫ x(s)h(s, t)x(t) ds dt k,</tex> where r <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> (s, t) and r <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> (s, t) are the covariance functions while h(s, t) is given as a solution of the integral equation, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∫∫ r₀(s, u)h(u, v)r₁(v, t) du dv = r₁(s, t) - r₀(s, t).</tex> 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.