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 m 0 (t) and r 0 (s,t) if $2\int x(t) g (t)\ dt + \int \int [x (s) - m_1 (s)] h(s,t) [x (t) - m_1 (t)]\ ds\ dt \lt; c,$ choose m 1 (t) and r 1 (s,t) if otherwise, where m 0 (t), m 1 (t), r 0 (s,t) and r 1 (s,t) are the two mean and covariance functions, and g(t) is the square-integrable solution of $\int r_0 (s,t) g (s)\ ds = m_1 (t) - m_0 (t),$ while h(s,t) is the symmetric and square-integrable solution of $\int \int r_0 (s,u) h (u,v) r_1 (v,t)\ du\ dv = r_1 (s,t) - r_0 (s,t).$ Note that under the assumption of zero mean functions, i.e., m 0 (t) e m 1 (t) e 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., r 0 (s,t) e r 1 (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.