  We consider the continuous-time ARMA(1,1) Gaussian channel and derive its
feedback capacity in closed form. More specifically, the channel is given by
$\boldsymbol{y}(t) =\boldsymbol{x}(t) +\boldsymbol{z}(t)$, where the channel
input $\{\boldsymbol{x}(t) \}$ satisfies average power constraint $P$ and the
noise $\{\boldsymbol{z}(t)\}$ is a first-order {\em autoregressive moving
average} (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 a
white Gaussian process with unit double-sided spectral density.
  We show that the feedback capacity of this channel is equal to the unique
positive 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 a
discrete-time additive Gaussian channel, feedback may not increase the capacity
of a continuous-time additive Gaussian channel even if the noise process is
colored. The formula enables us to conduct a thorough analysis of the effect of
feedback on the capacity for such a channel. We characterize when the feedback
capacity equals or doubles the non-feedback capacity; moreover, we disprove
continuous-time analogues of the half-bit bound and Cover's $2P$ conjecture for
discrete-time additive Gaussian channels.
