Capacity-achieving distributions of impulsive ambient noise channels

This paper studies the characterization of the optimal input for impulsive ambient noise channels under average power constraint. Our focus is on the two-term Gaussian mixture complex noise model, which has been widely used to model impulsive noise arising in various communication channels. We first demonstrate that there exists a unique input distribution that achieves the channel capacity and the capacity-achieving input distribution has a uniformly distributed phase. By examining the Kuhn-Tucker conditions (KTC), we further show that if the optimal amplitude input distribution contains an infinite number of mass points on a bounded interval, the channel output must be Gaussian distributed. However, by using Bernstein's theorem to examine the completely monotonic condition, it is shown that the assumption of a Gaussian distributed output is not valid. As a result, there is always a finite number of mass points on any bounded interval in the optimal amplitude distribution. In addition, by applying a novel bounding technique on the KTC and using the Envelop Theorem, we demonstrate that the optimal amplitude distribution cannot have an infinite number of mass points. That gives us a unique solution of the optimal input having discrete amplitude with a finite number of mass points. Given such interesting results, we also develop an efficient way to compute the discrete optimal input and the corresponding capacity.