| Abstract: |
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--Kav\v{c}i\'{c}--Tatikonda, and others. More generally, it is shown that a $k$-dimensional extension 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. |
