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 Y_i = X_i + Z_i, i = 1,2,..., where the input {X_i} satisfies average power constraint and the noise {Z_i} is a first-order moving average Gaussian process defined by Z_i = alphaU_i-1 + U _i, |alpha| les 1, with white Gaussian innovation {U_i }_i=0 ^infin. We show that the feedback capacity of this channel is -log x₀, where x₀ is the unique positive root of the equation rhox² = (1 - x²)(1 - |alpha|x)², and rho is the ratio of the average input power per transmission to the variance of the noise innovation U_i. The optimal coding scheme parallels the simple linear signalling scheme by Schalkwijk and Kailath for the additive white Gaussian noise channel; the 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 recover the information by maximum likelihood decoding. The resulting probability of error decays doubly exponentially 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