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

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 A $\log^{2}$(A) and lower bound of order $\sqrt{\mathrm{A}}$ are established where 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 $P_{Y^{*}(0)}$ 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.