Error Exponents for the Gaussian Channel with Active Noisy Feedback

TL;DR

This paper investigates the maximum exponential decay rate of error probability in transmitting a single bit over a Gaussian channel with noisy feedback, considering different power constraints and extending some results to general channels.

Contribution

It characterizes the optimal error exponents under various power constraints in Gaussian channels with noisy feedback, revealing their dependence on signal-to-noise ratios.

Findings

Error exponents are finite and grow with the larger SNR between forward and feedback links.

Almost-sure power constraints yield smaller error exponents than expected constraints.

Results extend to arbitrary rate communication and general discrete memoryless channels.

Abstract

We study the best exponential decay in the blocklength of the probability of error that can be achieved in the transmission of a single bit over the Gaussian channel with an active noisy Gaussian feedback link. We impose an \emph{expected} block power constraint on the forward link and study both \emph{almost-sure} and \emph{expected} block power constraints on the feedback link. In both cases the best achievable error exponents are finite and grow approximately proportionally to the larger between the signal-to-noise ratios on the forward and feedback links. The error exponents under almost-sure block power constraints are typically strictly smaller than under expected constraints. Some of the results extend to communication at arbitrary rates below capacity and to general discrete memoryless channels.