Capacity Achieving Distribution of Amplitude-Constrained AWGN Channel

The capacity-achieving distribution of such a channel is discrete, and the size of its support can be polynomially bounded in terms of the maximum amplitude of the channel. Using the fact that the Gaussian kernel is strictly positive, the problem essentially reduces to bounding the number of modes of a constrained Gaussian mixture. The final bound is achieved using Tijdeman’s Number of Zeros Lemma, which bounds the number of zeros of an analytic function in a bounded disc in terms of its maximum value. We carried out some experiments to observe the nature of capacity-achieving distribution for a discrete channel, and made an attempt to generalize the existing proof to bound the support size for the discrete case.