The Feedback Capacity of the First-Order Moving Average Gaussian Channel

The feedback capacity of the stationary Gaussian additive noise channel has been open, except for the case where the noise is white. Here we obtain the closed-form feedback capacity of the first-order moving average additive Gaussian noise channel. Specifically, the channel is given by Yi = Xi +Zi, i = 1, 2, . . . , where the input {Xi} satisfies a power constraint and the noise {Zi} is a first-order moving average Gaussian process defined by Zi = αUi−1+Ui, |α| < 1, with white Gaussian innovation {Ui}i=0. We show that the feedback capacity of this channel is − log x0, where x0 is the unique positive root of the equation ρ x2 = (1 − x2)(1 − |α|x)2, and ρ is the ratio of the average input power per transmission to the variance of the noise innovation Ui. Paralleling the simple linear signalling scheme by Schalkwijk and Kailath for the additive white Gaussian noise channel, the optimal transmitter sends a real-valued information-bearing signal at the beginning of communication, then subsequently processes the feedback noise process through a simple linear stationary first-order autoregressive filter to help the receiver decode the information. The resulting decoding error decays doublyexponentially in the duration of the communication. This feedback capacity of the first-order moving average Gaussian channel is very similar, in form, to the best known achievable rate for the first-order autoregressive Gaussian noise channel studied by Butman, Wolfowitz, and Tiernan, although the optimality of the latter is yet to be established.