Low Complexity Shalvi-Weinstein Algorithm

Abstract

Blind equalization can effectively reduce the intersymbol interference introduced by the frequency selective channel in the absence of the training sequence. The Shalvi-Weinstein Algorithm (SWA) performs better under most channels, especially for highly distorted ones compared with constant modulus algorithm (CMA) or its modified versions. The disadvantage of the SWA is the high complexity resulting from the computation of the inverse matrix. A low complexity SWA based on dichotomous coordinate descent algorithm is proposed in the paper, whose computation complexity is on the same order of magnitude as the CMA. Besides the low complexity, the proposed algorithm also avoids the possible numerical error resulting from the computation of the matrix inversion. Moreover, a low complexity of decision directed algorithm based on RLS is presented for a dual mode blind equalization. Simulations verify the effectiveness of the algorithm.

Appendix

$$\begin{aligned} \Delta {\mathbf{R}}\left( n \right) * {\mathbf{w}}\left( {n - 1} \right) = & \left[ {{\mathbf{R}}\left( n \right) - {\mathbf{R}}\left( {n - 1} \right)} \right] * {\mathbf{w}}\left( {n - 1} \right) \\ = & \left[ {\lambda * {\mathbf{R}}\left( {n - 1} \right) + {\mathbf{x}}\left( n \right) * {\mathbf{x}}^{H} \left( n \right) - {\mathbf{R}}\left( {n - 1} \right)} \right] * {\mathbf{w}}\left( {n - 1} \right) \\ = & \left( {\lambda - 1} \right) * {\mathbf{R}}\left( {n - 1} \right) * {\mathbf{w}}\left( {n - 1} \right) + {\mathbf{x}}\left( n \right) * {\mathbf{x}}^{H} \left( n \right) * {\mathbf{w}}\left( {n - 1} \right) \\ = & \left( {\lambda - 1} \right) * \left[ {{\mathbf{q}}\left( {n - 1} \right) - {\mathbf{r}}\left( {n - 1} \right)} \right] + y\left( n \right)^{*} * {\mathbf{x}}\left( n \right) \\ \end{aligned}$$

Substituting the above equation into the following equation.

$$\begin{aligned} {\mathbf{q}}_{0} \left( n \right) = & {\mathbf{r}}\left( {n - 1} \right) + \Delta {\mathbf{q}}\left( n \right) - \Delta {\mathbf{R}}\left( n \right) * {\mathbf{\hat{w}}}\left( {n - 1} \right) \\ = & {\mathbf{r}}\left( {n - 1} \right) + \Delta {\mathbf{q}}\left( n \right) - \left( {\lambda - 1} \right) * \left[ {{\mathbf{q}}\left( {n - 1} \right) - {\mathbf{r}}\left( {n - 1} \right)} \right] - y\left( n \right)^{*} * {\mathbf{x}}\left( n \right) \\ = & \lambda * {\mathbf{r}}\left( {n - 1} \right) + \left[ {\Delta {\mathbf{q}}\left( n \right) + {\mathbf{q}}\left( {n - 1} \right)} \right] - \lambda * {\mathbf{q}}\left( {n - 1} \right) - y\left( n \right)^{*} * {\mathbf{x}}\left( n \right) \\ = & \lambda * {\mathbf{r}}\left( {n - 1} \right) + {\mathbf{q}}\left( n \right) - \lambda * {\mathbf{q}}\left( {n - 1} \right) - y\left( n \right)^{*} * {\mathbf{x}}\left( n \right) \\ = & \lambda * {\mathbf{r}}\left( {n - 1} \right) + \left| {y\left( n \right)} \right|^{2} * y^{*} \left( n \right) * {\mathbf{x}}\left( n \right)/\beta + {\mathbf{\rho }}\left( n \right)/\beta - y\left( n \right)^{*} * {\mathbf{x}}\left( n \right) \\ = & \lambda * {\mathbf{r}}\left( {n - 1} \right) + \left[ {\left( {\left| {y\left( n \right)} \right|^{2} - \beta } \right) * y^{*} \left( n \right) * {\mathbf{x}}\left( n \right) + {\mathbf{\rho }}\left( n \right)} \right]/\beta \\ = & \lambda * {\mathbf{r}}\left( {n - 1} \right) + \left[ {e_{{SWA}}^{*} \left( n \right) * {\mathbf{x}}\left( n \right) + {\mathbf{\rho }}\left( n \right)} \right]/\beta \\ = & \lambda * {\mathbf{r}}\left( {n - 1} \right) + \left[ {e_{{SWA}}^{*} \left( n \right) * {\mathbf{x}}\left( n \right) + c_{{SWA}} * {\kern 1pt} e_{{SWA}}^{*} \left( n \right) * {\mathbf{x}}\left( n \right)} \right]/\beta {\kern 1pt} \\ = & \lambda * {\mathbf{r}}\left( {n - 1} \right) + \left( {1 + c_{{SWA}} } \right) * {\kern 1pt} e_{{SWA}}^{*} \left( n \right) * {\mathbf{x}}\left( n \right)/\beta \\ \end{aligned}$$