Finite-Support Structure in i.i.d.-Constrained Capacity of Finite-Memory Poisson Channels

2026-07-24

AI Summary

This study addresses the capacity of discrete-time Poisson channels under peak- and average-intensity constraints with finite intersymbol interference, focusing on input distributions restricted to the class of independent and identically distributed (i.i.d.) sequences. By establishing a filtered-forgetting estimate for the entropy rate of the output process and combining Karush–Kuhn–Tucker (KKT) optimality conditions with a superlinear growth argument, the work rigorously proves that any capacity-achieving i.i.d. input distribution must have finite support. The analysis integrates tools from information theory, stochastic processes, complex analysis, and calculus of variations, constructing a holomorphic extension of the influence function to derive the first-order variation of the entropy rate. These findings provide a solid theoretical foundation for designing optimal input distributions in optical communication systems modeled by finite-memory Poisson channels.

Abstract

Discrete-time Poisson channels with finite intersymbol interference provide a natural model for direct-detection optical links in which multipath memory and signal-dependent shot noise appear simultaneously. Under peak and average optical-intensity constraints, we study the independent and identically distributed (i.i.d.)-constrained capacity problem of such channels. We prove that every input distribution maximizing the stationary mutual information rate within the i.i.d. input class has finite support. The proof is carried out directly on the entropy rate of the continuous-state hidden Markov output process induced by the finite-memory channel. We first establish a filtering-forgetting estimate whose constants are uniform over all admissible i.i.d. input laws. We then derive the entropy-rate first variation, construct a holomorphic extension of the corresponding influence function, and combine the Karush-Kuhn-Tucker condition with a supralinear growth argument.

Problem

• Poisson channels
• finite memory
• i.i.d.-constrained capacity
• finite support
• intersymbol interference

Innovation

• finite-support input
• Poisson channel
• i.i.d.-constrained capacity
• entropy rate
• hidden Markov process