Coding for Additive White Noise Channels with Feedback Corrupted by Uniform Quantization or Bounded Noise

摘要

We present simple coding strategies, which are variants of theSchalkwijk-Kailath scheme, for communicating reliably over additive white noisechannels in the presence of corrupted feedback. More specifically, we considera framework comprising an additive white forward channel and a backward linkwhich is used for feedback. We consider two types of corruption mechanisms inthe backward link. The first is quantization noise, i.e., the encoder receivesthe quantized values of the past outputs of the forward channel. Thequantization is uniform, memoryless and time invariant (that is,symbol-by-symbol scalar quantization), with bounded quantization error. Thesecond corruption mechanism is an arbitrarily distributed additive boundednoise in the backward link. Here we allow symbol-by-symbol encoding at theinput to the backward channel. We propose simple explicit schemes thatguarantee positive information rate, in bits per channel use, with positiveerror exponent. If the forward channel is additive white Gaussian then ourschemes achieve capacity, in the limit of diminishing amplitude of the noisecomponents at the backward link, while guaranteeing that the probability oferror converges to zero as a doubly exponential function of the block length.Furthermore, if the forward channel is additive white Gaussian and the backwardlink consists of an additive bounded noise channel, with signal-to-noise ratio(SNR) constrained symbol-by-symbol encoding, then our schemes are alsocapacity-achieving in the limit of high SNR.