The MAC Poisson Channel: Capacity and Optimal Power Allocation

Abstract

The majority of worldwide data and voice traffic is transported using optical communication channels. As the demand for bandwidth continues to increase, it is of great importance to find closed form expressions of the information capacity for the optical communications applications at the backbone as well as the access networks. In particular, we introduce an information-theoretic derivation of the capacity expressions of Poisson channels that model the application. The closed form expression for the capacity of the single input single output (SISO) Poisson channel-derived by Kabanov in 1978, and Davis in 1980 will be revisited. Similarly, we will derive closed form expressions for the capacity of the multiple accesses Poisson channel (MAC) under the assumption of constant shot noise. This provides a framework for an empirical form of the k-users MAC Poisson channel capacity with average powers that are not necessarily equal. Moreover, we interestingly observed that the capacity of the MAC Poisson channel is a function of the SISO Poisson channel and upper bounded by this capacity plus some quadratic non-linear terms. We have also observed that the optimum power allocation in the case of Poisson channels follows a waterfilling alike interpretation to the one in Gaussian channels, where power is allotted to less noisy channels. Therefore, we establish a comparison between Gaussian channels and Poisson optical channels in the context of information theory and optical communications.