On the Feedback Capacity of Stationary Gaussian Channels

The capacity of stationary additive Gaussian noise channels with feedback is characterized as 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–Butman coding scheme achieves the feedback capacity for the first-order autoregressive moving-average Gaussian channel, resolving a long-standing open problem studied by Butman, Schalkwijk–Tiernan, Wolfowitz, Ozarow, Ordentlich, Yang–Kavčić–Tatikonda, and others.