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

Document Type

Article

Publication Date

11-1-2021

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 \mathsf {A}\log {2}(\mathsf {A}) and a 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. 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.

Identifier

85114731502 (Scopus)

Publication Title

IEEE Transactions on Information Theory

External Full Text Location

https://doi.org/10.1109/TIT.2021.3111836

e-ISSN

15579654

ISSN

00189448

First Page

7050

Last Page

7066

Issue

11

Volume

67

Grant

694630

Fund Ref

Horizon 2020 Framework Programme

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