Optimal inputs of single-user and multi-user non-gaussian aggregate interference channels

This paper generalizes and proves the discrete and finite nature of the optimal signaling schemes for general classes of non-Gaussian aggregate interference point-to-point and multiple-access channels (MACs) under peak power constraints. The considered channels are general enough to represent all non-Gaussian communication channels of engineering interest. Specifically, we first investigate the detailed characteristics of optimal inputs for a single- user channel that is impaired by two types of noise: a Gaussian mixture (GM) noise Z consisting of Gaussian elements with arbitrary means, and the interference U with an arbitrary distribution. The only very mild condition imposed on U is that its second moment is finite. To this end, one of the important results is the establishment of the Kuhn-Tucker condition (KTC) and the proof of analyticity of the KTC using Fubini-Tonelli's and Morera's theorems. Using the Bolzano- Weierstrass's and Identity's theorems, we then show that an optimal input is continuous if only if the KTC is zero on the entire real line. However, by examining an upper bound on the tail of the output PDF, it is demonstrated that the KTC must be bounded away from zero. As such, any optimal input must be discrete with finite number of mass points. Finally, we exploit the arbitrary property of U to show that the optimal input distributions that achieve the sum-capacity of an M-user MAC under GM noise are finite discrete. We also prove that there exist at least two distinct points that achieve the sum capacity on the rate region.