مرکز منطقه ای اطلاع رساني علوم و فناوري - 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 covariance $R(s, t)$ has a continuous second partial derivative and the sample function $x(t)$ is almost surely differentiable. The result can be generalized to the case of higher-order differentiation. Namely, if $(\\delta ^{2n}/\\delta s^{n} \\delta t^{n}) R(s, t)$ is continuous and $x(t)$ has the $n$ th derivative $x^{(n)}(t)$ almost surely, then the series can be differentiated term by term $n$ times, and the resultant series converges in the stochastic mean to $x^{(n)}(t)$ uniformly in $t$ . 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 $m_{1}(t)$ and $m_{2}(t)$ and the noise covariance is $R(s, t)$ . Then we prove the well-known conjecture that the optimum receiver correlates the observable waveform with the solution $g(t)$ 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 of $M$ -ary sure signals.