DFE for QPSK in MATLAB
If we focus on finite-duration impulse response filters and apply the
MSE criterion to optimize the coefficients of the feedback and feedforward filters. 


The equalizer output can be expressed as:
\begin{equation}\hat{I}_{k}=\sum_{j=-K_{1}}^{0} c_{j} v_{k-j}+\sum_{j=1}^{K_{2}} c_{j} \tilde{I}_{k-j}\end{equation}

where $\hat{I}_{k}$ is an estimate of the $k$ th information symbol, $\left\{c_{j}\right\}$ are the tap coefficients of the filter, and $\left\{\tilde{I}_{k-1}, \ldots, \tilde{I}_{k-K_{2}}\right\}$ are previously detected symbols. The equalizer is assumed to have ( $K_{1}+1$ ) taps in its feedforward section and $K_{2}$ in its feedback section.

The MSE criterion result in a mathematically tractable optimization of the equalizer coefficients; based on the assumption that previously detected symbols in the feedback filter are correct, the minimization of MSE:
\begin{equation}J\left(K_{1}, K_{2}\right)=E\left|I_{k}-\hat{I}_{k}\right|^{2}\end{equation}
This leads to the following set of linear equations for the coefficients of the feedforward filter:

\begin{equation}\sum_{j=-K_{1}}^{0} \psi_{l j} c_{j}=f_{-l}^{*}, \quad l=-K_{1}, \ldots,-1,0\end{equation}
where
\begin{equation}\psi_{l j}=\sum_{m=0}^{-l} f_{m}^{*} f_{m+l-j}+N_{0} \delta_{l j}, \quad l, j=-K_{1}, \ldots,-1,0\end{equation}

The coefficients of the feedback filter of the equalizer are given in terms of the coefficients of the feedforward section by the following expression:

\begin{equation}c_{k}=-\sum_{j=-K_{1}}^{0} c_{j} f_{k-j}, \quad k=1,2, \ldots, K_{2}\end{equation}

The values of the feedback coefficients result in complete elimination of intersymbol
interference from previously detected symbols, provided that previous decisions are
correct and that $K_2 &gt; L$