Differentiation of Karhunen-Loève expansion and application to optimum reception of sure signals in noise

The first part of this paper is concerned with differentiation of the Karhunen-Loève expansion of a stochastic process. In particular, we establish that the expansion series can be differentiated term by term while retaining the same sense of convergence, ff the covariance has a continuous second partial derivative and the sample function is almost surely differentiable. The result can be generalized to the case of higher-order differentiation. Namely, if is continuous and has the th derivative almost surely, then the series can be differentiated term by term times, and the resultant series converges in the stochastic mean to uniformly in . In the second half, the above result is applied to the problem of optimum reception of binary signals in Gaussian noise. Suppose the binary sure signals are and and the noise covariance is . Then we prove the well-known conjecture that the optimum receiver correlates the observable waveform with the solution of the integral equation even if the solution contains -functions and their derivatives. This result can be generalized to the case of -ary sure signals.