Achievable Error Exponents in the Gaussian Channel with Rate-Limited Feedback

TL;DR

This paper analyzes the error decay rates in Gaussian channels with rate-limited feedback, showing exponential and doubly exponential error decay depending on feedback rate, with a focus on the impact of feedback constraints.

Contribution

It introduces new bounds and schemes for error exponents in Gaussian channels with limited feedback, including a simple iterative scheme for high feedback rates.

Findings

Error probability decays exponentially when feedback rate R_FB < R.

Error probability decays doubly exponentially when R_FB ≥ R.

Error exponent exhibits a discontinuity at R_FB = R.

Abstract

We investigate the achievable error probability in communication over an AWGN discrete time memoryless channel with noiseless delay-less rate-limited feedback. For the case where the feedback rate R_FB is lower than the data rate R transmitted over the forward channel, we show that the decay of the probability of error is at most exponential in blocklength, and obtain an upper bound for increase in the error exponent due to feedback. Furthermore, we show that the use of feedback in this case results in an error exponent that is at least RF B higher than the error exponent in the absence of feedback. For the case where the feedback rate exceeds the forward rate (R_FB \geq R), we propose a simple iterative scheme that achieves a probability of error that decays doubly exponentially with the codeword blocklength n. More generally, for some positive integer L, we show that a L-th order exponential error decay is achievable if R_FB \geq (L-1)R. We prove that the above results hold whether the feedback constraint is expressed in terms of the average feedback rate or per channel use feedback rate. Our results show that the error exponent as a function of R_FB has a strong discontinuity at R, where it jumps from a finite value to infinity.