Optimum discrete signaling over channels with arbitrary noise distribution

Abstract

General channels with arbitrary noise distributions and a finite set of signaling points are considered in this paper. We aim at finding the capacity-achieving input distribution. As a structural result we first demonstrate that mutual information is a concave function of the input distribution and a convex function of the channel transfer densities. Using the Karush-Kuhn-Tucker theory, capacity achieving distributions are then characterized by constant Kullback-Leibler divergence between each channel transfer density and the mixture hereof built by using the probabilities as weights. If, as a special case, the noise distribution and the signaling points are rotationally symmetric, then the uniform input distribution is optimal. For 2-PAM modulation and certain types of asymmetric noise distributions, including exponential, gamma and Rayleigh, we present extensive numerical evaluations of the optimal input. Furthermore, for 4-QAM we determine the optimal input from a restricted symmetric class of distributions for correlated Gaussian noise.