Feedback Capacity of Gaussian Channels with Memory

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, the feedback capacity can be achieved with delayed feedback. Finally, we show that the detectability condition is satisfied for scalar models and conjecture that it is true for MIMO models.