Transmission of a Bit over a Discrete Poisson Channel with Memory

TL;DR

This paper investigates optimal coding strategies for transmitting a bit over a discrete Poisson channel with memory, considering various power constraints, and derives optimal codes in different regimes.

Contribution

It provides the first comprehensive analysis of optimal code structures for Poisson channels with memory under peak and total power constraints.

Findings

Optimal code structure identified for total-power constraint with blocklength > memory length.

Optimal code derived for high-power regime with peak-power constraint for arbitrary memory.

Optimal code for memoryless Poisson channel with both constraints when bounds are large.

Abstract

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.