Coding for Additive White Noise Channels with Feedback Corrupted by   Uniform Quantization or Bounded Noise
  We present simple coding strategies, which are variants of the Schalkwijk-Kailath scheme, for communicating reliably over additive white noise channels in the presence of corrupted feedback. More specifically, we consider a framework comprising an additive white forward channel and a backward link which is used for feedback. We consider two types of corruption mechanisms in the backward link. The first is quantization noise, i.e., the encoder receives the quantized values of the past outputs of the forward channel. The quantization is uniform, memoryless and time invariant (that is, symbol-by-symbol scalar quantization), with bounded quantization error. The second corruption mechanism is an arbitrarily distributed additive bounded noise in the backward link. Here we allow symbol-by-symbol encoding at the input to the backward channel. We propose simple explicit schemes that guarantee positive information rate, in bits per channel use, with positive error exponent. If the forward channel is additive white Gaussian then our schemes achieve capacity, in the limit of diminishing amplitude of the noise components at the backward link, while guaranteeing that the probability of error converges to zero as a doubly exponential function of the block length. Furthermore, if the forward channel is additive white Gaussian and the backward link consists of an additive bounded noise channel, with signal-to-noise ratio (SNR) constrained symbol-by-symbol encoding, then our schemes are also capacity-achieving in the limit of high SNR. 
Nuno C Martins, Tsachy Weissman