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
m
th-order autoregressive channels with feedback and average energy constraint
E
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
\alpha_1
be the regression coefficient,
\sigma^2
the innovation variance,
N
the number of channel iterations per source symbol, and
e = E/N
; then the first-order capacity
C^1
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{2\mid \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
e/\sigma^2
and is less than twice this one-way capacity everywhere.