Optimum Reception of Binary Sure and Gaussian Signals

Suppose the received waveform x(t) observed during the interval 0 ^ t ^ 1 is the sample function of a Gaussian process, whose mean and covariance functions are either mo(t) and ro(s,t) or mi{t) and fi(s,/). We assume that m0(t) and mi(t) are continuous while r0(s,t.) and ri(s,t) are positive-definite as well as continuous. Denote by Hk , k = 0,1, the hypothesis that mk{t) and rk(s,t) are the mean and covariance functions of the Gaussian process [xt, 0 ^ t ^ 1}. Suppose further that a, 0 a 1, and 1 -- a are the a priori probabilities associated with the two hypotheses H 0 and Hi respectively. Then, reception of binary sure and Gaussian signals may be regarded as a problem of deciding between two hypotheses H 0 and Hi upon observation of the sample function x(I). Thus, the problem of optimum reception of binary sure and Gaussian signals is to specify a decision scheme in terms of x{t) such that its error probability is minimum.* In the previous article, 1 a general treatment of the problem was made under the assumption that m0(t) = m}(t) = 0, and several forms of the optimum decision schemes were given under additional conditions with varying degrees of restriction. The following is most restrictive but most convenient for physical application: