On Linear Processing for an AWGN Channel with Noisy Feedback

Many communication systems can be modeled as having a noisy forward channel and a noisy or noiseless feedback channel. The use of the feedback channel is of great interest because it can greatly lower the complexity of the modulation scheme for the forward channel. In addition to complexity benefits, it can greatly increase the rate at which the probability of error decays. In this paper, we look at linear schemes and compare our results to the well-known Schalkwijk-Kailath coding scheme. Starting from a general linear coding scheme, a new linear feedback coding method is developed that is asymptotically optimal over all linear schemes. This new scheme is then used in a two-phase coding scheme that can achieve all rates below capacity with a probability of error that goes to zero.