# An Enhanced Precoder for Multi User Multiple-Input Multiple-Output Downlink Systems
## 3 Conventional Precoding Schemes
### RBD Precoder  
The RBD scheme which was proposed in [16] was designed to optimize the BD scheme in MMSE criterion. The author in [16] divides the precoding matrix into two parts as follows,  
\[\mathbf{F}^{\text{RBD}}=\gamma_{\text{RBD}}\mathbf{F}_{\text{MU}}^{\text{RBD }}\mathbf{F}_{\text{SU}}^{\text{RBD}}, \tag{10}\]  
where \(\gamma_{\text{RBD}}\) is scaling factor for RBD, \(\mathbf{F}_{\text{MU}}^{\text{RBD}}\) converts MU channel into parallel SU channels guaranteeing MMSE and \(\mathbf{F}_{\text{SU}}^{\text{RBD}}\) optimizes performance for the SU channel. The expression for MMSE criterion to obtain \(\mathbf{F}_{\text{MU}}^{\text{RBD}}\) is as follows,  
\[\mathbf{F}_{\text{MU}}^{\text{RBD}}=\operatorname*{arg\,min}_{\mathbf{F}_{ \text{MU}}^{\text{RBD}}}E\left\{\left\|\left(\mathbf{I}_{N_{R}}-\mathbf{H} \mathbf{F}_{\text{MU}}^{\text{RBD}}\right)\mathbf{X}\right\|^{2}+\frac{\| \mathbf{W}\|^{2}}{\gamma^{2}}\right\}. \tag{11}\]  
However, the RBD algorithm does not solve Equation (11) to find \(\mathbf{F}_{\text{MU}}^{\text{RBD}}\). Instead, \(\mathbf{F}_{\text{MU}}^{\text{RBD}}\) is optimized with modified MMSE criterion that minimizes IUI. The expression for modified MMSE criterion is as follows,  
\[\mathbf{F}_{\text{MU}}^{\text{RBD}}=\operatorname*{arg\,min}_{\mathbf{F}_{ \text{MU}}^{\text{RBD}}}E\left\{\sum_{i=1}^{K}\left\|\tilde{\mathbf{H}}_{i} \mathbf{F}_{\text{MU}}^{\text{RBD}}\right\|^{2}+\frac{\|\mathbf{W}\|^{2}}{ \gamma^{2}}\right\}. \tag{12}\]  
\(\mathbf{F}_{\text{MU}}^{\text{RBD}}\) satisfying Equation (12) is as follows [16; 20],  
\[\mathbf{F}_{\text{MU},i}^{\text{RBD}}=\tilde{\mathbf{V}}_{i}\left(\tilde{ \mathbf{\Sigma}}_{i}^{T}\tilde{\mathbf{\Sigma}}_{i}+\frac{N_{R}\sigma_{W}^{2}}{ P_{T}}\mathbf{I}_{N_{T}}\right)^{-\frac{1}{2}}, \tag{13}\]  
where \(\sigma_{W}^{2}\) and \(P_{T}\) are noise power and transmit power. The channel \(\mathbf{H}\mathbf{F}_{\text{MU}}^{\text{RBD}}\) is changed to parallel SU MIMO channels. The SVD for \(\mathbf{H}_{i}\mathbf{F}_{\text{MU},i}^{\text{RBD}}\) is defined as \(\mathbf{H}_{i}\mathbf{F}_{\text{MU},i}^{\text{RBD}}=\mathbf{\Omega}_{i}^{ \text{RBD}}\mathbf{\Sigma}_{i}^{\text{RBD}}\left[\begin{array}{cc}\tilde{ \mathbf{V}}_{i}^{\text{RBD}(1)}&\tilde{\mathbf{V}}_{i}^{\text{RBD}(0)}\\ \end{array}\right]^{H}\), where \(\mathbf{\Omega}_{i}^{\text{RBD}}\), \(\mathbf{\Sigma}_{i}^{\text{RBD}}\), \(\tilde{\mathbf{V}}_{i}^{\text{RBD}(1)}\) and \(\tilde{\mathbf{V}}_{i}^{\text{RBD}(0)}\) are left singular vector matrix, singular value matrix, right singular vector matrix correlated with non zero singular values and right singular vector matrix correlated with zero singular values. The solution for performance optimization of SU MIMO is as follows [16; 20],  
\[\mathbf{F}_{\text{SU},i}^{\text{RBD}}=\tilde{\mathbf{V}}_{i}^{\text{RBD}(1)}. \tag{14}\]  
Consequently, the \(i\)-th UE precoding matrix for RBD is given by,  
\[\mathbf{F}^{\text{RBD}}=\gamma_{\text{RBD}}\tilde{\mathbf{V}}_{i}\left( \tilde{\mathbf{\Sigma}}_{i}^{T}\tilde{\mathbf{\Sigma}}_{i}+\frac{N_{R}\sigma_{W }^{2}}{P_{T}}\mathbf{I}_{N_{T}}\right)^{-\frac{1}{2}}\tilde{\mathbf{V}}_{i}^{ \text{RBD}(1)}. \tag{15}\]  
Although RBD scheme provides improved performance compared to the BD scheme, the RBD precoder does not offer additional diversity gains, which the BD precoder does.