Feedback plays an interesting role in communication systems. While not essential to reliable communication, feedback can increase the capacity, improve the probability of error, reduce the delay in communication, and simplify the system design. Nonetheless, the role of feedback in communication is not completely understood. For example, the feedback capacity of nonwhite Gaussian channels, despite numerous bounds and partial results, has been open even for the simplest case.
In this talk, we characterize the Gaussian feedback capacity as the solution to a variational problem. In the special case of the first-order autoregressive moving average noise, this variational characterization gives a closed-form expression for the feedback capacity, answering a long-standing open question studied by Butman, Schalkwijk--Tiernan, Wolfowitz, Ozarow, Ordentlich, Yang--Kavcic--Tatikonda, and many others. More generally, we can show that a variant of the celebrated Schalkwijk--Kailath coding scheme achieves the feedback capacity for the general finite-order autoregressive moving-average Gaussian channel. Simply put, the optimal transmitter iteratively refines the receiver's knowledge of the intended message.