The Gaussian Channel with Noisy Feedback

Upper and lower bounds are derived on the reliability function of the additive white Gaussian noise channel with output fed back to the transmitter over an independent additive white Gaussian noise channel. Special attention is paid to the regime of very low feedback noise variance and it is shown that the reliability function is asymptotically inversely proportional to the feedback noise variance. This result shows that the noise in the feedback link, however small, renders the communication with noisy feedback fundamentally different from the perfect feedback case. For example, it is demonstrated that with noisy feedback, linear coding schemes fail to achieve any positive rate. In contrast, an asymptotically optimal coding scheme is devised, based on a three-phase detection/retransmission protocol, which achieves an error exponent inversely proportional to the feedback noise variance for any rate less than capacity.