Communication over the Gaussian channel with rate-limited feedback

We consider communication over an AWGN discrete time memoryless channel with noiseless delay-less rate-limited feedback. For the case where the feedback rate is lower than the data rate transmitted over the forward channel, we show that the decay of the probability of error is at most linearly exponential in block-length and obtain an upper bound for the error exponent. For the case where the feedback rate exceeds the forward rate, we propose a simple iterative scheme that achieves an error probability decaying L-fold exponentially (i.e. in general form of exp(-exp (...(exp(O(n)))...))/L as a function of the block-length when the feedback rate is at least L times the forward rate, for some positive integer L. Our results show that the error exponent as a function of the feedback rate has a discontinuity at the point where the feedback rate is equal to the forward rate.