Some Results on the Computation of Feedback Capacity of Gaussian Channels with Memory

We study the problem of computing the capacity of channels with feedback for the class of Gaussian channels with linear state-space models (possibly with hidden states) under the presence of quadratic input constraints. We first show that the input-output directed information is maximized by a Gaussian feedback policy. A few special cases are considered where such an optimization can be performed in a computationally tractable manner. In such cases, we show that the supremum of directed information admits a single-letter expression involving the convex log-determinant program as the block length tends to infinity.