## 2 System Model
\right]}_{\triangleq\widehat{\mathbf{ y}}}, \tag{8}\]  
where \(x_{k-\Delta}^{(i)}\) is the transmit signal at time (\(k-\Delta\)) by the \(i\)-th input antenna, and \(\Delta\) is an integer representing the decision delay, usually \(0\leq\Delta\leq N_{f}+\
u-1\). Using \(N=ln_{o}N_{f}+n_{i}N_{b}\) to represent the number of DFE taps for simplicity, so \(\widetilde{w}_{i}\) is an \(N\times 1\) vector. Then the equalizer mean square error (MSE) for the \(i\)-th input antenna is expressed as  
\[\xi_{i} \triangleq\mathrm{E}\bigg{[}\big{|}\varepsilon_{k}^{(i)}\big{|}^{2} \bigg{]}=\mathrm{E}\bigg{[}\big{|}\chi_{k-\Delta}^{(i)}-\widetilde{w}_{i}^{ \mathrm{H}}\widetilde{y}\big{|}^{2}\bigg{]} \tag{9}\] \[=\varepsilon_{x}-\widetilde{w}_{i}^{\mathrm{H}}\widetilde{r}_{ \Delta,i}-\widetilde{r}_{\Delta,i}^{\mathrm{H}}\widetilde{w}_{i}+\widetilde{w }_{i}^{\mathrm{H}}R_{\widetilde{y}\widetilde{y}}\widetilde{w}_{i}\]  
where \(\mathbf{R}_{\widetilde{y}\widetilde{y}}\triangleq\mathrm{E}\Big{[}\widetilde{ \widetilde{y}}\widetilde{y}^{\mathrm{H}}\Big{]}\), \(\widetilde{r}_{\Delta,i}=\mathbf{R}_{\widetilde{y}\widetilde{x}}\mathbf{1}_{\Delta,i}\), \(\mathbf{R}_{\widetilde{y}\widetilde{x}}\triangleq\mathrm{E}\Big{[}\widetilde{ \widetilde{y}}\widetilde{x}_{kk-N_{f}-v+1}^{\mathrm{H}}\Big{]}\) and \(\mathbf{1}_{\Delta,i}\) is the \((n_{i}\Delta+i)\)-th column of \(\mathbf{I}_{n_{i}(N_{f}+v)}\). Performing the Cholesky decomposition \(\mathbf{R}_{\widetilde{y}\widetilde{y}}=\widetilde{L}\widetilde{L}^{\mathrm{H}}\), where \(\widetilde{L}\) is an \(N\times N\) lower triangular matrix, (9) can be rewritten as  
\[\xi_{i}=\underbrace{\varepsilon_{x}-\widetilde{r}_{\Delta,i}^{\mathrm{H}} \widetilde{L}^{-\mathrm{H}}\widetilde{L}^{-1}\widetilde{r}_{\Delta,i}}_{ \triangleq\xi_{\min,i}}+\underbrace{\big{\|}\widetilde{L}^{\mathrm{H}} \widetilde{w}_{i}-\widetilde{L}^{-1}\widetilde{r}_{\Delta,i}\big{\|}_{2}^{2}}_ {\triangleq\xi_{\mathrm{excess},i}}. \tag{10}\]  
From (10) it shows that the different choice of \(\widetilde{w}_{i}\) can only change \(\xi_{\mathrm{excess},i}\), and \(\xi_{\min,i}\) remains unchanged. Obviously, \(\xi_{\mathrm{excess},i}\geq 0\). Therefore, when \(\xi_{\mathrm{excess},i}=0\), \(\widetilde{w}_{i}\) achieves the optimum solution under the MMSE criterion as  
\[\widetilde{w}_{\mathrm{opt},i}=\widetilde{L}^{-H}\widetilde{L}^{-1}\widetilde {r}_{\Delta,i}=\mathbf{R}_{\widetilde{y}\widetilde{y}}^{-1}\widetilde{r}_{\Delta, i}. \tag{11}\]  
\(\widetilde{w}_{\mathrm{opt},i}\) is usually non-sparse with too much complexity to compute directly. Therefore, in order to reduce the computational complexity, considering to reduce the number of the DFE nonzero taps at the cost of a small increase in the equalizer MSE, i.e., finding the sparse solution \(\widetilde{w}_{s,i}\) for \(\widetilde{w}_{i}\) when \(\xi_{\mathrm{excess},i}\leq\theta\) (\(\theta>0\)), the sparse FIR MIMO DFE design problem can be transformed into the \(l_{0}\) norm minimization problem as follows [15]  
\[\widetilde{w}_{s,i}=\underset{\widetilde{w}_{i}}{\mathrm{argmin}}\|\widetilde{ w}_{i}\|_{0}\text{ subject to }\|\widetilde{L}^{\mathrm{H}}\widetilde{w}_{i}-\widetilde{L}^{-1} \widetilde{r}_{\Delta,i}\|_{2}^{2}\leq\theta. \tag{12}\]  
In general, finding the optimal solution of the above \(l_{0}\) norm minimization problem is NP-hard. One of the approaches to computing the suboptimal solution for the approximation of (12) is greedy algorithms. To design the sparse solution for the FIR MIMO DFE, in this paper, we propose the scheme based on the OMP algorithm which is a widely-used greedy algorithm. The OMP algorithm selects an atom from the matrix \(\widetilde{L}^{\mathrm{H}}\) (i.e., the column vector of the matrix \(\widetilde{L}^{\mathrm{H}}\)) that is most correlated with the current residual, then solves a constraint least squares problem to obtain all the selected atoms coefficients and updates the residual in each iteration. There is no correction for the selected nonzero taps positions in OMP algorithm, which will reduce the design accuracy for the sparse FIR DFE.