Gaussian Channel with Noisy Feedback and Peak Energy Constraint

TL;DR

This paper investigates optimal coding strategies over Gaussian channels with noisy feedback under peak energy constraints, demonstrating that small feedback noise can improve error exponents beyond no-feedback scenarios.

Contribution

It extends Pinsker's results by showing noisy feedback can enhance error exponents, introducing two coding schemes and analyzing their performance relative to feedback noise levels.

Findings

Linear feedback coding outperforms nonlinear schemes when feedback noise is very small.

Both schemes improve error exponents compared to no-feedback case under certain conditions.

Performance depends on feedback noise power, with different schemes optimal at different noise levels.

Abstract

Optimal coding over the additive white Gaussian noise channel under the peak energy constraint is studied when there is noisy feedback over an orthogonal additive white Gaussian noise channel. As shown by Pinsker, under the peak energy constraint, the best error exponent for communicating an M-ary message, M >= 3, with noise-free feedback is strictly larger than the one without feedback. This paper extends Pinsker's result and shows that if the noise power in the feedback link is sufficiently small, the best error exponent for conmmunicating an M-ary message can be strictly larger than the one without feedback. The proof involves two feedback coding schemes. One is motivated by a two-stage noisy feedback coding scheme of Burnashev and Yamamoto for binary symmetric channels, while the other is a linear noisy feedback coding scheme that extends Pinsker's noise-free feedback coding scheme. When the feedback noise power $\alpha$ is sufficiently small, the linear coding scheme outperforms the two-stage (nonlinear) coding scheme, and is asymptotically optimal as $\alpha$ tends to zero. By contrast, when $\alpha$ is relatively larger, the two-stage coding scheme performs better.