Noisy feedback communications with side information at the decoder

In this paper, we introduce a new framework of communication channels with noisy feedback where the feedback information received by the transmitter is also available to the decoder with some finite delays. The merits of this new framework are demonstrated by two aspects: 1) Its (Shannon) capacity can be characterized by the causal conditional directed information. As an illustration, we characterize the n-block capacity of Gaussian noisy feedback channels under our new framework and propose an iteration algorithm to obtain a lower bound; 2) By constructing a specific linear coding scheme for the first-order moving average Gaussian channels with intermittent(erasure) feedback, we show that the new framework allows linear feedback coding schemes with positive transmission rate, which is (in certain scenario) much larger than the non-feedback Gaussian channel capacity.