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

We investigate the achievable error probability in communication over an AWGN discrete time memoryless channel with noiseless delayless rate-limited feedback. For the case where the feedback rate
$R_{\scriptscriptstyle FB}$
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
$R_{\scriptscriptstyle FB}$
higher than the error exponent in the absence of feedback. For the case where the feedback rate exceeds the forward rate (
$R_{\scriptscriptstyle FB}\geq R$
), 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
$R_{\scriptscriptstyle FB}\geq (L-1)R$
. While the above results are proved under an average feedback rate constraint, we show that all the achievability results for
$R_{\scriptscriptstyle FB}\geq R$
hold in a more restrictive case where the feedback constraint is expressed in terms of the per-channel-use feedback rate. Our results show that the error exponent as a function of
$R_{\scriptscriptstyle FB}$
has a strong discontinuity at
$R$
, where it jumps from a finite value to infinity.