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

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 y(t) =x(t) +z(t), where the channel input {x(t) } satisfies average power constraint P and the noise {z(t)} is a first-order autoregressive moving average (ARMA(1,1)) Gaussian process satisfying z^'(t)+κz(t)=(κ+λ)w(t)+w^'(t), where κ>0, λ∈ℝ and {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+κ)^2 = 2x(x+|κ+λ|)^2 when -2κ<λ<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.