Improved Bounds on the Number of Support Points of the Capacity-Achieving Input for Amplitude Constrained Poisson Channels

TL;DR

This paper analyzes the capacity-achieving input for the amplitude-constrained discrete-time Poisson channel with dark current $\lambda$. It develops a streamlined, analytic framework based on KKT conditions, estimation-theoretic identities, and Tijdeman's zero-counting lemma to bound both the location and the cardinality of the optimal input's support. The authors provide explicit upper bounds on the number of mass points, improving the $\lambda=0$ case from $\mathsf{A}\log^2(\mathsf{A})$ to $\mathsf{A}$, extend bounds to nonzero $\lambda$, and give a lower bound $|\mathrm{supp}(P_{X^{\star}})| \ge \lceil\max\{2, e^{C(\mathsf{A},\lambda)}\}\rceil$, with capacity lower bounds showing $e^{C(\mathsf{A},\lambda)}$ grows at least like $\sqrt{\mathsf{A}}$. The approach yields tighter, more transparent bounds than previous oscillation-based methods and applies uniformly to nonzero dark current. The results enhance understanding of the structure of capacity-achieving distributions in Poisson channels and offer a foundation for refined design and analysis in optical and particle-counting communication systems.

Abstract

This work considers a discrete-time Poisson noise channel with an input amplitude constraint $\mathsf{A}$ and a dark current parameter $λ$. It is known that the capacity-achieving distribution for this channel is discrete with finitely many points. Recently, for $λ=0$, a lower bound of order $\sqrt{\mathsf{A}}$ and an upper bound of order $\mathsf{A} \log^2(\mathsf{A})$ have been demonstrated on the cardinality of the support of the optimal input distribution. In this work, we improve these results in several ways. First, we provide upper and lower bounds that hold for non-zero dark current. Second, we produce a sharper upper bound with a far simpler technique. In particular, for $λ=0$, we sharpen the upper bound from the order of $\mathsf{A} \log^2(\mathsf{A})$ to the order of $\mathsf{A}$. Finally, some other additional information about the location of the support is provided.

Figures (1)

• Figure 1: Capacity $C(\mathsf{A},0)$ vs size of the optimal input distribution $N=|{\mathsf{supp}}(P_{X^\star})|$ for the case $\lambda=0$. Each circle denotes one value of $\mathsf{A}$. Solid line: interpolation of the values of $\mathsf{A}$ for which the number of support points increases.

Theorems & Definitions (13)

• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4: Tijdeman's Number of Zeros Lemma Tijdeman1971number
• Theorem 1
• Proposition 1
• proof
• Lemma 5