Structural Results and Improved Upper Bounds on the Capacity of the Discrete-Time Poisson Channel

TL;DR

This paper introduces new upper bounds on the capacity of the discrete-time Poisson channel, improves existing bounds, and characterizes the structure of capacity-achieving distributions under various constraints.

Contribution

It develops a novel framework for upper bounding the channel capacity, re-derives and improves upon previous bounds, and proves that the optimal input distribution has a countably infinite support.

Findings

New capacity upper bounds that improve previous results.

Re-derivation of Martinez's bounds as a special case.

Proof that the optimal distribution's support is countably infinite.

Abstract

New capacity upper bounds are presented for the discrete-time Poisson channel with no dark current and an average-power constraint. These bounds are a simple consequence of techniques developed for the seemingly unrelated problem of upper bounding the capacity of binary deletion and repetition channels. Previously, the best known capacity upper bound in the regime where the average-power constraint does not approach zero was due to Martinez (JOSA B, 2007), which is re-derived as a special case of the framework developed in this paper. Furthermore, this framework is carefully instantiated in order to obtain a closed-form bound that noticeably improves the result of Martinez everywhere. Finally, capacity-achieving distributions for the discrete-time Poisson channel are studied under an average-power constraint and/or a peak-power constraint and arbitrary dark current. In particular, it is shown that the support of the capacity-achieving distribution under an average-power constraint only must be countably infinite. This settles a conjecture of Shamai (IEE Proceedings I, 1990) in the affirmative. Previously, it was only known that the support must be unbounded.