Taylor series approximation of semi-blind best linear unbiased channel estimates for the general linear model

Abstract

We present a low complexity approximate method for semi-blind best linear unbiased estimation (BLUE) of a channel impulse response vector (CIR) for a communication system, which utilizes a periodically transmitted training sequence, within a continuous stream of information symbols. The algorithm achieves slightly degraded results at a much lower complexity than directly computing the BLUE CIR estimate. In addition, the inverse matrix required to invert the weighted normal equations to solve the general least squares problem may be pre-computed and stored at the receiver. The BLUE estimate is obtained by solving the general linear model, y = Ah + w + n, for h, where w is correlated noise and the vector n is an AWGN process, which is uncorrelated with w. The Gauss - Markoff theorem gives the solution h = (A TC(h) -1A) -1A TC(h) -1y. In the present work we propose a Taylor series approximation for the function F(h) = (A TC(h) -1A) -1A TC(h) -1y where, F:R L → R L for each fixed vector of received symbols, y, and each fixed convolution matrix of known transmitted training symbols, A. We describe the full Taylor formula for this function, F(h) = F(h id) + ∑|α|≥|(h - h id) α(∂/∂h) αF(h id) and describe algorithms using, respectively, first, second and third order approximations. The algorithms give better performance than correlation channel estimates and previous approximations used, [15], at only a slight increase in complexity. The linearization procedure used is similar to that used in the linearization to obtain the extended Kaiman filter, and the higher order approximations are similar to those used in obtaining higher order Kaiman filter approximations,