In the context of memoryless channels, the traditional information theoretic approach to studying feedback is to consider ideal noise-free instantaneous feedback of the channel outputs to the encoder. This was acceptable because the classical results were essentially negative: Shannon pointed out that feedback does not improve capacity and in the context of symmetric DMCs, Dobrushin (following earlier work by Berlekamp) showed that it does not improve the fixed block-coding error exponents either, at least in the interesting high rate regime. The celebrated Schalkwijk-Kailath scheme is a seeming exception to this, but Wyner showed that the resulting reliability boosts relied crucially on the average nature of the power constraint.
However, posing the communication problem in different ways does allow noiseless feedback to dramatically improve reliability, even for DMCs. If improvements are claimed with ideal instantaneous feedback, then it is natural to wonder whether such improvements remain with noisy or otherwise limited feedback. In this talk, I consider the bitwise average-delay (or soft deadline) setting that is closely related to the work of Forney, Burnashev, Yamamoto & Itoh, Kudryashov, Horstein, and others.
After a brief review, we will deal with a truly noisy feedback link and establish a somewhat surprising result: Not only can the Burnashev bound be beaten, but the resulting scheme can also be made to function with asymptotically no loss in reliability using only a noisy DMC on the feedback path, as long as it has sufficiently high capacity and we are allowed to code appropriately over it. In this sense, the gains due to feedback are shown to be robust to noise in the feedback link.
[Parts of this talk are the result of joint work with my former student Tunc Simsek and former postdoc Stark Draper.]