Optimum Reception of Binary Sure and Gaussian Signals

The problem of optimum reception of binary sure and Gaussian signals is to specify, in terms of the received waveform, a scheme for deciding between two alternative mean and covariance functions with minimum error probability. In the context of a general treatment of the problem, this article presents a solution which is both mathematically rigorous and convenient for physical application. The optimum decision scheme obtained consists in comparing, with a predetermined threshold c, the sum of a linear and a quadratic form in the received waveform x(t); namely, choose m0(t) and r0(s,t) if2∫ x(t) g (t)\ dt + ∫ ∫ [x (s) - m₁ (s)] h(s,t) [x (t) - m₁ (t)]\ ds\ dt <; c,choose m1(t) and r1(s,t) if otherwise, where m0(t), m1(t), r0(s,t) and r1(s,t) are the two mean and covariance functions, and g(t) is the square-integrable solution of∫ r₀ (s,t) g (s)\ ds = m₁ (t) - m₀ (t),while h(s,t) is the symmetric and square-integrable solution of∫ ∫ r₀ (s,u) h (u,v) r₁ (v,t)\ du\ dv = r₁ (s,t) - r₀ (s,t).Note that under the assumption of zero mean functions, i.e., m0(t) m1(t) 0, the above result is reduced to the one in a previous article by this author, while with the assumption of identical covariance functions, i.e., r0(s,t) r1(s,t), it is reduced to the classical result essentially obtained by Grenander. Sections I and II introduce the problem and summarize the main results with certain pertinent remarks, while a detailed mathematical treatment is given in Section III. Although Appendices A–D are not directly required for solution of the problem, they are added to provide a tutorial background for the results on equivalance and singularity of two Gaussian measures obtained by Grenander, Root and Pitcher as well as some generalization of their results.