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. In particular, the upper bound of order $\mathsf{A} \log^2(\mathsf{A})$ and 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.

翻译：这项工作考虑了带有振幅限制的 Poisson 噪声频道。 众所周知, 此频道的容量实现输入分布是受输入振幅限制的。 我们通过引入质量点数的上下界限来放大这个结果。 特别是, $\ mathsf{ A}\ log2\\\ mathsf{ A} 的上限值和 $\ sqrt\ mathsf{ A} 的下限值。 此外, $\ mathsf{ A} 的上限值是受输入振幅限制的。 此外, 我们展示了能力及能力实现分布的若干其他属性。 例如, 显示能力等于$ -\ log P ⁇ \\\\\\\\\\\\\\\\\\ star} (0) 美元, 其中$$是最佳输出分布的值。 此外,, 能力实现分布的概率的上限的上限是受值的上限约束值的上限 。