The role of perfect feedback in communication is relatively well understood. Feedback can dramatically reduce the probability of error, decrease communication delay, and simplify the system design. For example, the celebrated Schalkwijk-Kailath coding scheme achieves the capacity of additive white Gaussian noise channel with exponentially improved reliability. On the other hand, when feedback is noisy, the role of feedback is largely unexplored, except for a vague optimism that engineering intuitions gained from the study of perfect feedback would carry over to communication scenarios such as space communication in which the noise in the feedback link is very small.
In this talk, we introduce a small Gaussian noise in the feedback link of the additive white Gaussian noise channel and ask how the noise in the feedback link affects the reliability of communication. First, we obtain upper bounds on the reliability using a change-of-measure and large deviations technique and a simple information theoretic argument. In particular, we show that having noisy feedback is no better than having the input power boosted by a constant factor. As a negative consequence of this result, the exponential boost of reliability under perfect feedback vanishes for any nonzero feedback noise. More negatively, we show that the Schalkwijk-Kailath coding scheme and its variants are very sensitive to the feedback noise and, in fact, fail to achieve any positive data rate.
On the positive side, we propose an alternative coding scheme based on a detection-retransmission protocol that is almost optimal and robust with respect to small feedback noise. For any rate less than capacity, the corresponding error exponent blows up as the feedback noise variance approaches zero. Thus, having noisy feedback is far better than having no feedback.
Joint work with Tsachy Weissman (Stanford) and Amos Lapidoth (ETH Zurich).