Feedback Capacity of the Continuous-Time ARMA(1,1) Gaussian Channel
Jun Su, Guangyue Han, Shlomo Shitz Shamai · IEEE Transactions on Information Theory · 2024
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 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$\kappa \gt 0,~\lambda \in \mathbb {R}$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$-2\kappa \lt \lambda \lt 0$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.