# Achievable Error Exponents of One-Way and Two-Way AWGN Channels

## TL;DR

This paper derives achievable error exponents for one-way and two-way AWGN channels with noisy feedback under different power constraints, revealing how feedback and interaction can improve error performance and how message size relates to block length.

## Contribution

It introduces new error exponent bounds for one-way and two-way AWGN channels with noisy feedback under AS and EXP constraints, including the first derivation of error exponent regions for two-way channels.

## Key findings

- Active feedback under strong feedback links improves error exponents.
- Generalization of error exponents from two messages to multiple messages.
- Feedback and interaction can enhance error exponents in one direction but may reduce them in the other.

## Abstract

Achievable error exponents for the one-way with noisy feedback and two-way AWGN channels are derived for the transmission of a finite number of messages $M$ using fixed block length $n$, under the almost sure (AS) and the expected block (EXP) power constraints. In the one-way setting under noisy AWGN feedback, it is shown that under the AS constraint and when the feedback link is much stronger than the direct link, active feedback leads to a larger gain over the non-feedback error exponent than passive feedback. Under the EXP constraint, a previously known error exponent for the transmission of two messages is generalized to any arbitrary but finite number of messages $M$. In the two-way setting, where each user has its own message to send in addition to (possibly) aiding in the transmission of feedback for the opposite direction, error exponent regions are defined and derived for the first time for the AWGN two-way channel under both AS and EXP power constraints. It is shown that feedback or interaction may lead to error exponent gains in one direction, possibly at the expense of a decrease in the error exponents attained in the other direction. The relationship between $M$ and $n$ supported by our achievability strategies is explored.