  It is well known that the problem of computing the feedback capacity of a
stationary Gaussian channel can be recast as an infinite-dimensional
optimization problem; moreover, necessary and sufficient conditions for the
optimality of a solution to this optimization problem have been characterized,
and based on this characterization, an explicit formula for the feedback
capacity has been given for the case that the noise is a first-order
autoregressive moving-average Gaussian process. In this paper, we further
examine the above-mentioned infinite-dimensional optimization problem. We prove
that unless the Gaussian noise is white, its optimal solution is unique, and we
propose an algorithm to recursively compute the unique optimal solution, which
is guaranteed to converge in theory and features an efficient implementation
for a suboptimal solution in practice. Furthermore, for the case that the noise
is a k-th order autoregressive moving-average Gaussian process, we give a
relatively more explicit formula for the feedback capacity; more specifically,
the feedback capacity is expressed as a simple function evaluated at a solution
to a system of polynomial equations, which is amenable to numerical computation
for the cases k=1, 2 and possibly beyond.
