Linear feedback rate bounds for regressive channels (Corresp.)

This article presents new tighter upper bounds on the rate of Gaussian autoregressive channels with linear feedback. The separation between the upper and lower bounds is small. We have
\frac{1}{2} \ln \left( 1 + \rho \left( 1+ \sum_{k=1}^{m} \alpha_{k} x^{- k} \right)^{2} \right) \leq C_{L} \leq \frac{1}{2} \ln \left( 1+ \rho \left( 1+ \sum_{k = 1}^{m} \alpha_{k} / \sqrt{1 + \rho} \right)^{2} \right), \mbox{all \rho}
, where
\rho = P/N_{0}W, \alpha_{l}, \cdots, \alpha_{m}
are regression coefficients,
P
is power,
W
is bandwidth,
N_{0}
is the one-sided innovation spectrum, and
x
is a root of the polynomial
(X^{2} - 1)x^{2m} - \rho \left( x^{m} + \sum^{m}_{k=1} \alpha_{k} x^{m - k} \right)^{2} = 0.
It is conjectured that the lower bound is the feedback capacity.