An upper bound to the capacity of the band-limited Gaussian autoregressive channel with noiseless feedback

Upper bounds to the capacity of band-limited Gaussian th-order autoregressive channels with feedback and average energy constraint are derived. These are the only known hounds on one- and two-way autoregressive channels of order greater than one. They are the tightest known for the first-order case. In this case let be the regression coefficient, the innovation variance, the number of channel iterations per source symbol, and ; then the first-order capacity is bounded by begin{equation} C^1 leq begin{cases} frac{1}{2} ln [frac{e}{sigma^2}(1+ mid alpha_1 mid ) ^ 2 +1], & frac{e}{sigma^2} leq frac{1}{1- alpha_1^2} \\ frac{1}{2} ln [frac{e}{sigma^2} + frac{2mid alpha_1 mid}{sqrt{1-alpha_1^2}} sqrt{frac{e}{simga^2}} + frac{1}{1-alpha_1^2}], & text{elsewhere}.\\ end{cases} end{equation} This is equal to capacity without feedback for very low and very high and is less than twice this one-way capacity everywhere.