Feedback Capacity of Stationary Gaussian Channels

The feedback capacity of additive stationary Gaussian noise channels is characterized as the solution to a variational problem in the noise power spectral density. 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 achieves the feedback capacity for the first-order autoregressive moving-average Gaussian channel, positively answering a long-standing open problem studied by Butman, Tiernan–Schalkwijk, Wolfowitz, Ozarow, Ordentlich, Yang–KavČiĆ–Tatikonda, and others. More generally, it is shown that a $k$-dimensional generalization of the Schalkwijk–Kailath coding 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. This development reveals intriguing connections between estimation, control, and feedback communication.