Feedback Capacity of Stationary Gaussian Channels

The feedback capacity of additive stationary Gaussian noise channels is characterized as the solution to a variational problem. Toward this end, it is proved that the optimal feedback coding scheme is stationary. When specialized to the first-order autoregressive moving average noise spectrum, this variational characterization yields a closed-form expression for the feedback capacity. In particular, this result shows that the celebrated Schalkwijk-Kailath coding scheme achieves the feedback capacity for the first-order autoregressive moving average Gaussian channel, positively answering a long-standing open problem studied by Butman, Schalkwijk-Tiernan, Wolfowitz, Ozarow, Ordentlich, Yang-Kavcic-Tatikonda, and others. More generally, it is shown that a k-dimensional generalization of the Schalkwijk-Kailath coding scheme achieves the feedback capacity for any autoregressive moving average noise spectrum of order k. Simply put, the optimal transmitter iteratively refines the receiver's knowledge of the intended message.