# An Enhanced Precoder for Multi User Multiple-Input Multiple-Output Downlink Systems
## Appendix A Derivation of \(\mathbf{F}_{\text{MU}}\)  
This section explains proof that \(\mathbf{F}_{\text{MU}}\) is optimized with modified MMSE criterion. The \(\mathbf{F}_{\text{MU}}\) is designed that diversity gain is preserved and MSE is minimized. The postcoding matrix removes the remaining IUI and \(\mathbf{F}_{\text{MU}}\) is obtained as Equation (23). The Equation (23) can be rewritten as follows,  
\[ \begin{split}\mathbf{F}_{\text{MU}}&=\operatorname*{ arg\,min}_{\mathbf{F}_{\text{MU}}}E\left\{\sum\limits_{i=1}^{K}tr\left(\mathbf{ \tilde{T}}_{i}\mathbf{F}_{\text{MU},i}\mathbf{F}_{\text{MU},i}^{H}\mathbf{ \tilde{T}}_{i}^{H}\right)+\frac{tr(\mathbf{G}\mathbf{W}\mathbf{W}^{H}\mathbf{ G}^{H})}{\gamma^{2}}\right\}\\ &=\operatorname*{arg\,min}_{\mathbf{F}_{\text{MU}}}\left(\sum \limits_{i=1}^{K}tr\left(\mathbf{\tilde{T}}_{i}^{H}\mathbf{\tilde{T}}_{i} \mathbf{F}_{\text{MU},i}\mathbf{F}_{\text{MU},i}^{H}\right)+E\left\{\frac{tr (\mathbf{G}^{H}\mathbf{G}\mathbf{W}\mathbf{W}^{H})}{\gamma^{2}}\right\}\right) \\ &=\operatorname*{arg\,min}_{\mathbf{F}_{\text{MU}}}\left(\sum \limits_{i=1}^{K}tr\left(\mathbf{\tilde{T}}_{i}^{H}\mathbf{\tilde{T}}_{i} \mathbf{F}_{\text{MU},i}\mathbf{F}_{\text{MU},i}^{H}\right)+\frac{r_{\text{ MV}}^{2}}{\gamma^{2}}\right).\end{split}  \tag{1}\]The power for trasmitted signal vector is constrained to \(P_{T}\) as follows,  
\[ \begin{split} E\left\{\|\mathbf{\hat{X}}\|^{2}\right\}& =E\left\{\sum\limits_{i=1}^{K}\left\|\gamma_{\mathbf{F}\text{MU},i} \mathbf{F}_{\text{SU},i}\mathbf{X}_{i}\right\|^{2}\right\}\\ &= E\left\{\sum\limits_{i=1}^{K}tr\left(\gamma^{2}\mathbf{F}_{ \text{MU},i}\mathbf{F}_{\text{SU},i}\mathbf{X}_{i}\mathbf{X}_{i}^{H}\mathbf{F }_{\text{SU},i}^{H}\mathbf{F}_{\text{MU},i}^{H}\right)\right\}\\ &= \gamma^{2}tr\left(\sum\limits_{i=1}^{K}\mathbf{F}_{\text{MU},i} \mathbf{F}_{\text{SU},i}E\left\{\mathbf{X}_{i}\mathbf{X}_{i}^{H}\right\} \mathbf{F}_{\text{SU},i}^{H}\mathbf{F}_{\text{MU},i}^{H}\right)\\ &= \gamma^{2}tr\left(\sum\limits_{i=1}^{K}\mathbf{F}_{\text{MU},i} \mathbf{F}_{\text{SU},i}\mathbf{F}_{\text{SU},i}^{H}\mathbf{F}_{\text{MU},i}^{ H}\right)=P_{T}.\end{split}  \tag{10}\]  
\(\mathbf{F}_{\text{SU},i}\) is designed as unitary matrix. Therefore, the scaling factor \(\gamma\) satisfies as follows,  
\[\gamma^{2}=\frac{P_{T}}{tr\left(\sum\limits_{i=1}^{K}\mathbf{F}_{\text{MU},i} \mathbf{F}_{\text{MU},i}^{H}\right)}. \tag{11}\]  
By substituting Equation (11) into Equation (12), Equation (12) is given by,  
\[ \begin{split}\mathbf{F}_{\text{MU}}& =\operatorname*{arg\,min}\limits_{\mathbf{F}_{\text{MU}}}\left( \sum\limits_{i=1}^{K}tr\left(\mathbf{\tilde{T}}_{i}^{H}\mathbf{\tilde{T}}_{i} \mathbf{F}_{\text{MU},i}\mathbf{F}_{\text{MU},i}^{H}\right)+\frac{r_{WW}^{ 2}}{P_{T}}tr\left(\sum\limits_{i=1}^{K}\mathbf{F}_{\text{MU},i}\mathbf{F}_{ \text{MU},i}^{H}\right)\right)\\ &=\operatorname*{arg\,min}\limits_{\mathbf{F}_{\text{MU}}}\left( \sum\limits_{i=1}^{K}tr\left(\mathbf{\tilde{T}}_{i}^{H}\mathbf{\tilde{T}}_{i} \mathbf{F}_{\text{MU},i}\mathbf{F}_{\text{MU},i}^{H}+\frac{r_{WW}^{2}}{P_{T} }\mathbf{I}_{\text{MU},i}^{H}\right)\right)\\ &=\operatorname*{arg\,min}\limits_{\mathbf{F}_{\text{MU}}}\left( \sum\limits_{i=1}^{K}tr\left(\left(\mathbf{\tilde{T}}_{i}^{H}\mathbf{\tilde{T} }_{i}^{H}\mathbf{\tilde{T}}_{i}+\frac{r_{WW}^{2}}{P_{T}}\mathbf{I}_{T}\right) \mathbf{F}_{\text{MU},i}\right)\right).\end{split}  \tag{12}\]  
The SVD for \(\mathbf{\tilde{T}}_{i}\) is \(\mathbf{\tilde{T}}_{i}=\mathbf{\tilde{U}}_{i}\mathbf{\tilde{\Sigma}}_{i} \mathbf{\tilde{V}}_{i}^{H}\) and eigen decomposition for \(\mathbf{\tilde{T}}_{i}^{H}\mathbf{\tilde{T}}_{i}\) is \(\mathbf{\tilde{T}}_{i}^{H}\mathbf{\tilde{T}}_{i}=\mathbf{\tilde{V}}_{i} \mathbf{\tilde{\Sigma}}_{i}^{T}\mathbf{\tilde{\Sigma}}_{i}\mathbf{\tilde{V}}_ {i}^{H}\).