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

Abstract

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. Concretely, an upper bound of order <inline-formula> <tex-math notation= LaTeX >$\\mathsf {A}\\log ^{2}(\\mathsf {A})$ </tex-math></inline-formula> and a lower bound of order <inline-formula> <tex-math notation= LaTeX >$\\sqrt { \\mathsf {A}}$ </tex-math></inline-formula> are established where <inline-formula> <tex-math notation= LaTeX >$\\mathsf {A}$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation= LaTeX >$- \\log P_{Y^\\star }(0)$ </tex-math></inline-formula> where <inline-formula> <tex-math notation= LaTeX >$P_{Y^\\star }$ </tex-math></inline-formula> 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. Furthermore, on the per-symbol basis, a nonvanishing lower bound on the probability of error for detecting the capacity-achieving distribution is established under the maximum a posteriori rule.