High signal-to-noise ratio asymptotics of entropy-constrained Gaussian channel capacity

TL;DR

This paper analyzes the asymptotic behavior of the capacity of Gaussian channels under input entropy constraints at high SNR, revealing the optimal distribution and exponential decay of entropy gap.

Contribution

It characterizes the capacity-achieving distribution as a discrete Gaussian on a scaled lattice and quantifies the exponential decay of the entropy gap at high SNR.

Findings

Capacity-achieving distribution is a discrete Gaussian on a scaled lattice.

The entropy gap decreases exponentially with SNR.

The exponential decay rate of the entropy gap is explicitly characterized.

Abstract

We study the input-entropy-constrained Gaussian channel capacity problem in the asymptotic high signal-to-noise ratio (SNR) regime. We show that the capacity-achieving distribution as SNR goes to infinity is given by a discrete Gaussian distribution supported on a scaled integer lattice. Further, we show that the gap between the input entropy and the capacity decreases to zero exponentially in SNR, and characterize this exponent.