  We consider the feedback capacity of a MIMO channel whose channel output is
given by a linear state-space model driven by the channel inputs and a Gaussian
process. The generality of our state-space model subsumes all previous studied
models such as additive channels with colored Gaussian noise, and channels with
an arbitrary dependence on previous channel inputs or outputs. The main result
is a computable feedback capacity expression that is given as a convex
optimization problem subject to a detectability condition. We demonstrate the
capacity result on the auto-regressive Gaussian noise channel, where we show
that even a single time-instance delay in the feedback reduces the feedback
capacity significantly in the stationary regime. On the other hand, for large
regression parameters (in the non-stationary regime), the feedback capacity can
be approached with delayed feedback. Finally, we show that the detectability
condition is satisfied for scalar models and conjecture that it is true for
MIMO models.
