On the capacity of linear additive channels with the noise spanning Hermite functions

Abstract

We consider a linear additive noise channel where the input is average-power constrained and the noise probability law is not necessarily Gaussian, but is rather in the finite span of even Hermite Functions. We study the nature of the capacity achieving input distribution of such a channel. It's well known, by Shannon's Theorem, that the capacity achieving distribution of the described channel is of a continuous type, namely Gaussian, whenever the noise is Gaussian. In our study, we present some sample case analysis and develop a general procedure that proves the discreteness of the capacity achieving distribution whenever the noise is in the finite span of even Hermite Functions with the exception of the Gaussian.