Transmission of a Bit over a Discrete Poisson Channel with Memory

A coding scheme for transmission of a bit maps a given bit to a sequence of channel inputs (called the codeword associated to the transmitted bit). In this paper, we study the problem of designing the best code for a discrete Poisson channel with memory (under peak-power and total-power constraints). The outputs of a discrete Poisson channel with memory are Poisson distributed random variables with a mean comprising of a fixed additive noise and a linear combination of past input symbols. Assuming a maximum-likelihood (ML) decoder, we search for a codebook that has the smallest possible error probability. This problem is challenging because error probability of a code does not have a closed-form analytical expression. For the case of having only a total-power constraint, the optimal code structure is obtained, provided that the blocklength is greater than the memory length of the channel. For the case of having only a peak-power constraint, the optimal code is derived for arbitrary memory and blocklength in the high-power regime. For the case of having both the peak-power and total-power constraints, the optimal code is derived for memoryless Poisson channels when both the total-power and the peak-power bounds are large.