  The feedback capacity of the stationary Gaussian additive noise channel has
been open, except for the case where the noise is white. Here we find the
feedback capacity of the stationary first-order moving average additive
Gaussian noise channel in closed form. Specifically, the channel is given by
$Y_i = X_i + Z_i,$ $i = 1, 2, ...,$ where the input $\{X_i\}$ satisfies a power
constraint and the noise $\{Z_i\}$ is a first-order moving average Gaussian
process defined by $Z_i = \alpha U_{i-1} + U_i,$ $|\alpha| \le 1,$ with white
Gaussian innovations $U_i,$ $i = 0,1,....$
  We show that the feedback capacity of this channel is $-\log x_0,$ where
$x_0$ is the unique positive root of the equation $ \rho x^2 = (1-x^2) (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 and
subsequently refines the receiver's error by processing the feedback noise
signal through a linear stationary first-order autoregressive filter. The
resulting error probability of the maximum likelihood decoding 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
\emph{autoregressive} Gaussian noise channel studied by Butman, Wolfowitz, and
Tiernan, although the optimality of the latter is yet to be established.
