On Gaussian Channels with Feedback under Expected Power Constraints and with Non-Vanishing Error Probabilities

TL;DR

This paper investigates the impact of feedback on Gaussian channels with expected power constraints, establishing the $ ext{ε}$-capacity, second-order bounds, and capacity regions for single- and multi-user scenarios, highlighting feedback's benefits.

Contribution

It provides new asymptotic expansions for feedback channels, bounds on second-order terms, and characterizes the $ ext{ε}$-capacity region for Gaussian MACs with feedback.

Findings

Feedback improves maximal achievable rates.

Second-order term bounds involve nested logarithms and square roots.

$ ext{ε}$-capacity depends on error probability, indicating the failure of the strong converse.

Abstract

In this paper, we consider single- and multi-user Gaussian channels with feedback under expected power constraints and with non-vanishing error probabilities. In the first of two contributions, we study asymptotic expansions for the additive white Gaussian noise (AWGN) channel with feedback under the average error probability formalism. By drawing ideas from Gallager and Nakibo\u{g}lu's work for the direct part and the meta-converse for the converse part, we establish the $\varepsilon$-capacity and show that it depends on $\varepsilon$ in general and so the strong converse fails to hold. Furthermore, we provide bounds on the second-order term in the asymptotic expansion. We show that for any positive integer $L$, the second-order term is bounded between a term proportional to $-\ln_{(L)} n$ (where $\ln_{(L)}(\cdot)$ is the $L$-fold nested logarithm function) and a term proportional to $+\sqrt{n\ln n}$ where $n$ is the blocklength. The lower bound on the second-order term shows that feedback does provide an improvement in the maximal achievable rate over the case where no feedback is available. In our second contribution, we establish the $\varepsilon$-capacity region for the AWGN multiple access channel (MAC) with feedback under the expected power constraint by combining ideas from hypothesis testing, information spectrum analysis, Ozarow's coding scheme, and power control.