On optimal input distribution and capacity limit of Bernoulli-Gaussian impulsive noise channels

In this paper, we rigorously analyze the optimal input distribution and capacity of an additive Bernoulli-Gaussian (BG) impulsive noise (IN) channel in high and low input power regimes. First, we obtain an input distribution for which the channel output is Gaussian distributed. This distribution, if valid, shall result in the capacity of the channel. At an asymptotically high input power level, we then show that the derived input is always valid and in fact, it resembles a Gaussian distribution. As such, the Gaussian channel input is considered approximately optimal. Using the monotonicity property of the characteristic function (CF), we then develop a necessary condition for the existence of the derived optimal input for a finite level of input power. The condition indicates that a sufficiently high input power is usually required. Then focusing on the low power region, we first derive an upper bound on the channel capacity assuming full knowledge of noise state. A closed-form expression of the mutual information (MI) achieved by Gaussian inputs, which is considered as a lower bound on the channel capacity, is then developed. By comparing these two bounds, it is shown that a Gaussian input asymptotically results in the capacity. Interestingly, it is also demonstrated that such a capacity is the same as the capacity of an erasure channel in low power regimes.