This work considers a discrete-time Poisson noise channel with an input amplitude constraint $\mathsf{A}$ and a dark current parameter $\lambda$. It is known that the capacity-achieving distribution for this channel is discrete with finitely many points. Recently, for $\lambda=0$, 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 $\lambda=0$, 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.

翻译：本文考虑具有输入幅度约束$\mathsf{A}$和暗电流参数$\lambda$的离散时间泊松噪声信道。已知该信道的容量可达分布是离散的且具有有限个支持点。近期，对于$\lambda=0$的情况，已证明了最优输入分布支持基数存在$\sqrt{\mathsf{A}}$阶的下界和$\mathsf{A} \log^2(\mathsf{A})$阶的上界。本文从多个方面改进了这些结果。首先，我们给出了适用于非零暗电流的上下界。其次，通过更简洁的技术得到了更精确的上界。特别地，对于$\lambda=0$的情形，我们将上界从$\mathsf{A} \log^2(\mathsf{A})$阶改进为$\mathsf{A}$阶。最后，还提供了关于支持点位置的其他补充信息。