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) if $2int x(t) g (t) dt + int int [x (s) - m_1 (s)] h(s,t) [x (t) - m_1 (t)] ds dt lt; c,$ choose 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 $int r_0 (s,t) g (s) ds = m_1 (t) - m_0 (t),$ 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., m0(t) &equal; m1(t) &equal; 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) &equal; 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 gen- ralization of their results.