# How to compute the DFE based MMSE equalizer FF and FB coefficients in a non adaptative way?

I am reading the Proakis 5th edition and I came across a paragraph explaining the tuning of DFE filters, i.e, how to compute the FF(FeedForward) and FB (Feedback Filter) coefficients.

Here a sumary of the DFE equalizer principle.

The DFE equalizer consists of two filters, a feedforward filter and a feedback filter, arranged as shown in Figure below. The input to the feedforward filter is the received signal sequence. The feedback filter has as its input the sequence of decisions on previously detected symbols.

Functionally, the feedback filter is used to remove that part of the ISI from the present estimated symbol caused by previously detected symbols. In our treatment, we apply the MSE criterion to optimize the filter' coefficients. we can see that the DFE ouptut 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 $$k \mathrm{th}$$ 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.

Based on the assumption that previously detected symbols in the feedback filter are correct, the minimization of MSE: $$\begin{equation}J\left(K_{1}, K_{2}\right)=E\left|I_{k}-\hat{I}_{k}\right|^{2}\end{equation}$$

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 $$K_2 > L$$.

Here my questions:

1. how to compute the $$c_{j} \quad for \quad l=-K_{1}, \ldots,-1,0$$
2. how to compute the $$c_{j} \quad for \quad k=1,2, \ldots, K_{2}$$
• I am still struggling to understand the filter updating process, the computation of the demodulated result and the computation of the error. Attached you can find the simulation I've done so far, the BER ~ 0.5 for all any range of SNR. That means I am missing something ... Anyway, Thank you for your help again. Kind regards – Abby_DSP Mar 19 at 12:43