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

Abstract
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 covarianceR(s, t)has a continuous second partial derivative and the sample functionx(t)is almost surely differentiable. The result can be generalized to the case of higher-order differentiation. Namely, if(\delta^{2n}/\delta s^{n} \deltat^{n}) R(s, t)is continuous andx(t)has thenth derivativex^{(n)}(t)almost surely, then the series can be differentiated term by termntimes, and the resultant series converges in the stochastic mean tox^{(n)}(t)uniformly int. 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 arem_{1}(t)andm_{2}(t)and the noise covariance isR(s, t). Then we prove the well-known conjecture that the optimum receiver correlates the observable waveform with the solutiong(t)of the integral equation\int R(s, t)g(s) ds = m_{2}( t) - m_{1}(t)even if the solution contains\delta-functions and their derivatives. This result can be generalized to the case ofM-ary sure signals.

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