Capacity-Achieving Distributions for the Discrete-Time Poisson Channel—Part II: Binary Inputs

Discrete-time Poisson (DTP) channels exist in many scenarios including space laser communication systems which operate over long distances and which can be corrupted by reflected and scattered light. Through simulation, binary-input distributions have been observed to be optimal in many cases, however, little analytical work exists on conditions for optimality or the form of optimal signalling. In this second part, the general properties of Part I are extended to the case of DTP channels where binary-inputs are optimal. Necessary and sufficient conditions on the optimality of binary (i.e. two mass point) distributions are presented by leveraging the general properties of DTP capacity-achieving distributions. Closed-form expressions of the capacity-achieving distributions are derived in several important special cases including zero dark current and for high dark current. Numerical results are presented to elucidate the developed analytical work.