# An Enhanced Precoder for Multi User Multiple-Input Multiple-Output Downlink Systems
## 4 MGBD Scheme and Adaptive MGBD Scheme  
### Modified GBD Precoder  
The MGBD precoder transmits a signal with some IUIs to increase degrees of freedom and residual IUI is mitigated by postcoding. Therefore, the postcoding matrix for the proposed scheme consists of left singular vectors for \(\mathbf{H}_{i}\). The postcoding matrix \(\mathbf{G}_{i}\) is given by,  
\[\mathbf{G}_{i}=[\begin{array}{cccc}\mathbf{u}_{i,1},&\mathbf{u}_{i,2},& \cdots,&\mathbf{u}_{i,r_{i}}\end{array}]^{H}. \tag{21}\]  
On the other hand, the precoding matrix for the proposed scheme is divided into two parts as follows,  
\[\mathbf{F}=\gamma\mathbf{F}_{\text{MU}}\mathbf{F}_{\text{SU}}, \tag{22}\]where \(\gamma\) is scaling factor, \(\mathbf{F}_{\text{MU}}\) is precoding matrix to achieve MMSE and \(\mathbf{F}_{\text{SU}}\) is precoding matrix to optimize the SU MIMO system. \(\mathbf{F}_{\text{MU}}\) is determined by the equation that minimizes mean square error (MSE) as follows,  
\[\mathbf{F}_{\text{MU}}=\operatorname*{arg\,min}_{\mathbf{F}_{\text{MU}}}E \left\{\sum_{i=1}^{K}\left\|\mathbf{\tilde{T}}_{i}\mathbf{F}_{\text{MU},i} \right\|^{2}+\frac{\left\|\mathbf{G}\mathbf{W}\right\|^{2}}{\gamma^{2}}\right\}, \tag{23}\]  
where \(\mathbf{\tilde{T}}_{i}\) is expressed as follows,  
\[\mathbf{\tilde{T}}_{i}=\Big{[}\begin{array}{cccccc}\mathbf{T}_{1}^{T}& \cdots&\mathbf{T}_{i-1}^{T}&\mathbf{T}_{i+1}^{T}&\cdots&\mathbf{T}_{K}^{T} \end{array}\Big{]}^{T}, \tag{24}\]  
where \(\mathbf{T}_{i}\) is multiplication of \(\mathbf{G}_{i}\) and \(\mathbf{H}_{i}\) as follows,  
\[\mathbf{T}_{i}=\mathbf{G}_{i}\mathbf{H}_{i}, \tag{25}\]  
and SVD for \(\mathbf{\tilde{T}}_{i}\) is \(\mathbf{\tilde{T}}_{i}=\mathbf{\tilde{U}}_{i}\mathbf{\tilde{\Sigma}}_{i} \mathbf{\tilde{V}}_{i}^{H}\), where \(\mathbf{\tilde{U}}_{i}\), \(\mathbf{\tilde{\Sigma}}_{i}\) and \(\mathbf{\tilde{V}}_{i}\) are left singular vector matrix, singular value matrix, right singular vector matrix. The \(\mathbf{F}_{\text{MU}}\) is obtained by calculating Equation (23). According to Appendix A, the obtained \(\mathbf{F}_{\text{MU}}\) is as follows,  
\[\mathbf{F}_{\text{MU}}=\mathbf{\tilde{V}}_{i}\Bigg{(}\mathbf{\tilde{\Sigma}}_ {i}^{T}\mathbf{\tilde{\Sigma}}_{i}+\frac{r\sigma_{W}^{2}}{P_{T}}\mathbf{I}_{N_ {T}}\Bigg{)}^{-\frac{1}{2}}. \tag{26}\]  
Although remaining IUI exists, the channel can be regarded as parallel SU channels by precoding with \(\mathbf{F}_{\text{MU}}\). The SVD of channel precoded with \(\mathbf{F}_{MU}\) is given by,  
\[\mathbf{H}_{i}\mathbf{F}_{\text{MU},i}=\mathbf{\tilde{U}}_{i}\mathbf{\tilde{ \Sigma}}_{i}\Big{[}\begin{array}{cc}\mathbf{\tilde{V}}_{i}^{(1)}&\mathbf{ \tilde{V}}_{i}^{(0)}\end{array}\Big{]}^{H}, \tag{27}\]  
where \(\mathbf{\tilde{U}}_{i}\) is left singular vectors and \(\mathbf{\tilde{\Sigma}}_{i}\) is singular value matrix. \(\mathbf{\tilde{V}}_{i}^{(1)}\) and \(\mathbf{\tilde{V}}_{i}^{(0)}\) are right singular vectors correlated with non zero singular values and right singular vectors correlated with zero singular values. Therefore, \(\mathbf{\tilde{V}}_{i}^{(1)}\) is used as \(\mathbf{F}_{\text{SU},i}\)[16; 20] and precoding matrix \(\mathbf{F}\) for proposed scheme is as follows,  
\[\mathbf{F}=\gamma\mathbf{\tilde{V}}_{i}\Bigg{(}\mathbf{\tilde{\Sigma}}_{i}^{T }\mathbf{\tilde{\Sigma}}_{i}+\frac{r\sigma_{W}^{2}}{P_{T}}\mathbf{I}_{N_{T}} \Bigg{)}^{-\frac{1}{2}}\mathbf{\tilde{V}}_{i}^{(1)}. \tag{28}\]  
The rest of this subsection discusses the effects of the relationship between \(r\) and rank of \(\mathbf{H}\). In the following classification, both cases assume that the number of data streams for each UE is equal to \(r_{i}\) and each UE has \(N_{R_{i}}\) receive antennas.