The Gaussian Channel with Noisy Feedback: Near-Capacity Performance via Simple Interaction

Consider a pair of terminals connected by two independent additive white Gaussian noise channels, and limited by individual power constraints. The first terminal would like to reliably send information to the second terminal, within a given error probability. We construct an explicit interactive scheme consisting of only (non-linear) scalar operations, by endowing the Schalkwijk-Kailath noiseless feedback scheme with modulo arithmetic. Our scheme achieves a communication rate close to the Shannon limit, in a small number of rounds. For example, for an error probability of $10^{-6}$, if the Signal to Noise Ratio ($\mathrm{SNR}$) of the feedback channel exceeds the $\mathrm{SNR}$ of the forward channel by $20\mathrm{dB}$, our scheme operates $0.8\mathrm{dB}$ from the Shannon limit with only $19$ rounds of interaction. In comparison, attaining the same performance using state of the art Forward Error Correction (FEC) codes requires two orders of magnitude increase in delay and complexity. On the other extreme, a minimal delay uncoded system with the same error probability is bounded away by $9\mathrm{dB}$ from the Shannon limit.