Achieving positive rates over AWGN channels with quantized feedback and linear processing

It is well known that feedback does not increase the capacity of the AWGN channel. Schalkwijk and Kailath presented a feedback scheme that dramatically improves the error probability for finite block lengths. Unfortunately, the Schalkwijk-Kailath scheme requires perfect feedback, which is often not feasible. We study the performance of linear transmission schemes for AWGN channels with separate quantization of the feedback signal and the receiver input signal. The quantizers are designed in a rate-information-optimal manner using the Gaussian information bottleneck. We provide expressions for achievable rates and error probabilities in terms of the so-called information-rate function. It turns out the exploiting knowledge of the quantized feedback signal at the receiver is crucial for achieving strictly positive rates.