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We consider the continuous-time ARMA(1,1) Gaussian channel and derive itsfeedback capacity in closed form. More specifically, the channel is given by$\boldsymbol{y}(t) =\boldsymbol{x}(t) +\boldsymbol{z}(t)$, where the channelinput $\{\boldsymbol{x}(t) \}$ satisfies average power constraint $P$ and thenoise $\{\boldsymbol{z}(t)\}$ is a first-order {\em autoregressive movingaverage} (ARMA(1,1)) Gaussian process satisfying $$\boldsymbol{z}^\prime(t)+\kappa\boldsymbol{z}(t)=(\kappa+\lambda)\boldsymbol{w}(t)+\boldsymbol{w}^\prime(t),$$ where $\kappa>0,~\lambda\in\mathbb{R}$ and $\{\boldsymbol{w}(t) \}$ is awhite Gaussian process with unit double-sided spectral density.  We show that the feedback capacity of this channel is equal to the uniquepositive root of the equation $$ P(x+\kappa)^2 = 2x(x+\vert\kappa+\lambda\vert)^2 $$ when $-2\kappa<\lambda<0$ and is equal to $P/2$otherwise. Among many others, this result shows that, as opposed to adiscrete-time additive Gaussian channel, feedback may not increase the capacityof a continuous-time additive Gaussian channel even if the noise process iscolored. The formula enables us to conduct a thorough analysis of the effect offeedback on the capacity for such a channel. We characterize when the feedbackcapacity equals or doubles the non-feedback capacity; moreover, we disprovecontinuous-time analogues of the half-bit bound and Cover's $2P$ conjecture fordiscrete-time additive Gaussian channels.

Feedback Capacity of the Continuous-Time ARMA(1,1) Gaussian Channel