Capacity-achieving discrete signaling over additive noise channels

Discrete input distributions are capacity-achieving for a variety of noise distributions whenever the input is subject to peak power or other bounding constraints. In this paper, we consider additive noise with arbitrary absolutely-continuous distribution and ask the question what the optimal input distribution over a set of fixed signaling points would be. The capacity-achieving distribution is characterized by constant Kullback Leibler distance between the shifted noise distribution and a certain mixture hereof. As an application, the optimal input distribution for binary symmetric signaling over exponential noise channels is determined. It further follows that in certain symmetric cases the uniform distribution over all signaling points is capacity-achieving.