This paper generalizes and proves the discrete and finite nature of the capacity-achieving signaling schemes for general classes of non-Gaussian point-to-point and multiple-access channels (MACs) under peak power constraints. Specifically, we first investigate the detailed characteristics of capacity-achieving 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) on a capacity-achieving input 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 a capacity-achieving input is continuous if and only if the KTC function 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 function must be bounded away from zero. As such, any capacity-achieving input must be discrete with a finite number of mass points. Finally, we exploit $U$ having an arbitrary distribution to show that the optimal input distributions that achieve the sum-capacity of an $M$ -user MAC under GM noise are discrete and finite. We also prove that there exist at least two distinct points that achieve-
the sum capacity on the rate region.
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