On Properties of the Support of Capacity-Achieving Distributions for Additive Noise Channel Models with Input Cost Constraints

We study the classical problem of characterizing the channel capacity and its achieving distribution in a generic fashion. We derive a simple relation between three parameters: the input-output function, the input cost function and the noise probability density function, one which dictates the type of the optimal input. In Layman terms we prove that the support of the optimal input is bounded whenever the cost grows faster than a cut-off rate equal to the logarithm of the noise PDF evaluated at the input-output function. Furthermore, we prove a converse statement that says whenever the cost grows slower than the cut-off rate, the optimal input has necessarily an unbounded support. In addition, we show how the discreteness of the optimal input is guaranteed whenever the triplet satisfy some analyticity properties. We argue that a suitable cost function to be imposed on the channel input is one that grows similarly to the cut-off rate. Our results are valid for any cost function that is super-logarithmic. They summarize a large number of previous channel capacity results and give new ones for a wide range of communication channel models, such as Gaussian mixtures, generalized-Gaussians and heavy-tailed noise models, that we state along with numerical computations.