On the Reliability of Gaussian Channels with Noisy Feedback

We derive bounds on the reliability of the additive white Gaussian noise channel Y = X +Z with output fed back to the transmitter over independent additive white Gaussian noise channel Ỹ = Y + Z̃ . These bounds appear to be new even for transmission at zero rate. As a corollary, it is shown that linear feedback coding schemes of finite-dimensional constellation, such as the celebrated Schalkwijk–Kailath coding scheme, not only fail to achieve the capacity, but in fact cannot achieve any positive rate. Our approach is applicable to the derivation of upper bounds on the error exponents in various other scenarios involving channels with feedback.