On quantization of channel output feedback for the Gaussian channel

The availability of a noiseless feedback channel can allow for simplistic transmission schemes with very high reliability; such advantages were solidified by a linear feedback scheme developed by Schalkwijk and Kailath (S-K) [1], [2]. However, the S-K scheme and related techniques rely on the presence of a noiseless feedback channel which is not practical. In this paper, we look at introducing the notion of quantization into linear feedback coding. To accomplish this, we look at adapting a recent linear feedback scheme [3] built on a noisy feedback channel to a system where the feedback is now quantized to be in the form of a binary message. This form of feedback gives insight into the relationship between bits of feedback and received signal-to-noise ratio given a blocklength, and, also, how the precision of feedback relates to average transmit power.