Properties of the Support of the Capacity-Achieving Distribution of the Amplitude-Constrained Poisson Noise Channel

TL;DR

This paper analyzes the properties of the capacity-achieving input distribution for an amplitude-constrained Poisson noise channel, providing bounds on the number of mass points and other distribution characteristics.

Contribution

It introduces new upper and lower bounds on the number of mass points in the optimal input distribution for this channel.

Findings

Upper bound of order A log^2(A) on the number of mass points

Lower bound of order sqrt(A) on the number of mass points

Capacity equals -log of the optimal output probability at zero

Abstract

This work considers a Poisson noise channel with an amplitude constraint. It is well-known that the capacity-achieving input distribution for this channel is discrete with finitely many points. We sharpen this result by introducing upper and lower bounds on the number of mass points. Concretely, an upper bound of order $\mathsf{A} \log^2(\mathsf{A})$ and a lower bound of order $\sqrt{\mathsf{A}}$ are established where $\mathsf{A}$ is the constraint on the input amplitude. In addition, along the way, we show several other properties of the capacity and capacity-achieving distribution. For example, it is shown that the capacity is equal to $ - \log P_{Y^\star}(0)$ where $P_{Y^\star}$ is the optimal output distribution. Moreover, an upper bound on the values of the probability masses of the capacity-achieving distribution and a lower bound on the probability of the largest mass point are established. Furthermore, on the per-symbol basis, a nonvanishing lower bound on the probability of error for detecting the capacity-achieving distribution is established under the maximum a posteriori rule.