## 4 Minimum Mean Square Error (MMSE) Based Equalizer  
Once an adequate estimate of the channel vector \(\mathbf{h}_{n_{r}n_{t}}\) has been obtained, the effect of the training sequence must be removed from the received signal before equalization. This is achieved by using TER block, as shown in Figure 1. Since the training sequence for each UE is also known at the BS, the effect of this superimposed training sequence can be removed as below  
\[\tilde{y}_{n_{r}}(k)=y_{n_{r}}(k)-\sum_{n_{t}=1}^{N_{t}}\ \sum_{\ell=0}^{L}\ \hat{h}_{n_{r}n_{t}}^{\ell}\ c_{n_{t}}(k-\ell), \tag{23}\]  
where \(\hat{h}_{n_{r}n_{t}}^{\ell}\) is the estimate of \(\ell\)th tap CIR from \(n_{t}\)-th UE to \(n_{r}\)-th receive antenna. The CIR estimate \(\mathbf{\hat{h}}_{n_{r}n_{t}}\) may be taken from any of the estimation techniques discussed previously, i.e., \(\mathbf{\hat{h}}_{n_{r}n_{t}}^{\text{SIT}-\text{LS}}\), \(\mathbf{\hat{h}}_{n_{r}n_{t}}^{\text{SIT}-\text{SGMP}}\) or \(\mathbf{\hat{h}}_{n_{r}n_{t}}^{\text{SIT}-\text{GP}}\).  
The optimal weights,\(\mathbf{w}_{n_{t}}\), for the equalizer at \(n_{r}\)-th receive antenna, can be computed as in [52], and are given below  
\[\mathbf{w}_{n_{r}n_{t}}=\left(\mathbf{\hat{H}}\mathbf{\hat{H}}^{*}\ +\ 2\sigma_{n_{r}}^{2}\mathbf{I}\right)^{-1}\ \mathbf{\hat{H}}\ |_{(n_{r}-1)(L_{e}+L-1)+(\tau_{d}+1)}, \tag{24}\]  
where \(n_{r}\) is the receiver index such that \(1\leq n_{r}\leq N_{r}\), \(L_{e}\) is the length of equalizer, \(\tau_{d}\) represents the symbol mappers' decision delay at equalizer's output, \(\mathbf{I}\) denotes the \((N_{t}\times L_{e})\times(N_{t}\times L_{e})\) identitymatrix and \(\hat{\mathbf{H}}|_{i}\) is the \(i\)th column of \(\hat{\mathbf{H}}\). The estimate of the noise variance \(\sigma_{n_{r}}^{2}\) is obtained as in [26]. The \(L_{e}\times(L_{e}+L-1)\) convolutional matrix \(\hat{\mathbf{H}}\) is given by  
\[\hat{\mathbf{H}}_{n_{r}n_{t}}=\left[\begin{array}{ccccc}\hat{h}_{n_{r}n_{t}}^ {0}&\hat{h}_{n_{t}n_{t}}^{1}&\cdots&\hat{h}_{n_{r}n_{t}}^{L-1}&0&\cdots&0\\ 0&\hat{h}_{n_{r}n_{t}}^{0}&\hat{h}_{n_{r}n_{t}}^{1}&\cdots&\hat{h}_{n_{r}n_{t}}^ {L-1}&\ddots&\vdots\\ \vdots&\ddots&\ddots&\ddots&\ddots&\ddots&0\\ 0&\cdots&0&\hat{h}_{n_{r}n_{t}}^{0}&\hat{h}_{n_{r}n_{t}}^{1}&\cdots&\hat{h}_{ n_{r}n_{t}}^{L-1}\end{array}\right]. \tag{25}\]  
The overall convolutional matrix \(\hat{\mathbf{H}}\) is given below  
\[\hat{\mathbf{H}}=\left[\begin{array}{ccccc}\hat{\mathbf{H}}_{1,1}&\hat{ \mathbf{H}}_{1,2}&\cdots&\hat{\mathbf{H}}_{1,N_{t}}\\ \hat{\mathbf{H}}_{2,1}&\mathbf{H}_{2,2}&\cdots&\hat{\mathbf{H}}_{2,N_{t}}\\ \vdots&\vdots&\cdots&\vdots\\ \hat{\mathbf{H}}_{N_{t},1}&\hat{\mathbf{H}}_{N_{t},2}&\cdots&\hat{\mathbf{H}} _{N_{t},N_{t}}\end{array}\right]. \tag{26}\]  
The information symbols transmitted from the \(n_{t}\)-th transmitter can thus be estimated as below, followed by a symbol mapper, as shown in Figure 1:  
\[\hat{b}_{n_{t}}(k)=\sum_{n_{r}=1}^{N_{r}}\sum_{i=0}^{L_{e}-1}w_{n_{r}n_{t}}^{ i}g_{n_{r}}(k-i). \tag{27}\]  
The estimated symbol vector, \(\mathbf{\hat{b}}_{n}\), is then fed as an input to a symbol mapper, as shown in Figure 1. The symbol mapper performs mapping of the estimated symbols as per used modulation scheme at the UEs. The output of the symbol mapper is the decoded symbol vector \(\mathbf{\hat{b}}_{n}\).