Wiener/LMMSE Interpolation

Wiener filtering method requires knowledge of channel properties, fading correlation function and noise power, to estimate the unknown channel transfer function at non-pilot locations. These properties are assumed to be known at the receiver for the estimator to perform optimally.

The LMMSE estimate $\hat{H_k}$ at k-th instance is calculated by filtering the pilot channel estimate vector $\hat{H} = [\hat{H_0} \hat{H_p} ... \hat{H_{p(N_p-1)}}]^T$ by a Wiener filter $c_{LMMSE} = [c_0 c_1 .... c_{N_p-1} ]$ as follows: $$\hat{H_k} = {c_{LMMSE}}^H \cdot \hat{H} $$ where $c_i$ is the i-th filter coefficient and $\hat{H_{pi}}$ is the i-th pilot's noisy fading coefficient estimate.

The Wiener filter coefficients are calculated as: $$c_{LMMSE} = (R_h + \sigma ^2I)^{-1}r $$
where

$R_h$ is the autocorrelation matrix of the channel at pilot locations,

$\sigma ^2$ is the noise-variance,

$r$ is the cross-correlation vector of the channel at k-th instance and the channel at pilot locations

The autocorrelation function of a Rayleigh faded channel is a zeroth-order Bessel function of the first kind. $$R_h = J_0(2\pi f_Dt)$$ at delay t when the maximum doppler shift is $f_D$.