  The capacity of stationary additive Gaussian noise channels with feedback 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, resolving a
long-standing open problem studied by Butman, Schalkwijk--Tiernan, Wolfowitz,
Ozarow, Ordentlich, Yang--Kavcic--Tatikonda, and others.
