Linear feedback rate bounds for regressive channels (Corresp.)

Abstract

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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\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}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\rho = P/N_{0}W, \alpha_{l}, \cdots, \alpha_{m}</tex> are regression coefficients, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</tex> is power, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W</tex> is bandwidth, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N_{0}</tex> is the one-sided innovation spectrum, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> is a root of the polynomial <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(X^{2} - 1)x^{2m} - \rho \left( x^{m} + \sum^{m}_{k=1} \alpha_{k} x^{m - k} \right)^{2} = 0.</tex> It is conjectured that the lower bound is the feedback capacity.