# An Enhanced Precoder for Multi User Multiple-Input Multiple-Output Downlink Systems
## 3 Conventional Precoding Schemes
### GBD Precoder  
The key point for the GBD scheme which was proposed in [17] is partial nulling increasing degrees of freedom for a wireless communication system [21]. Therefore, the GBD precoder does not completely eliminate channels for other UEs and the remaining IUI is removed by using postcoding.  
The SVD of the \(i\)-th UE channel is \(\mathbf{H}_{i}=\mathbf{U}_{i}\mathbf{\Sigma}_{i}\mathbf{V}_{i}^{H}\). GBD precoder selects dominant components for \(\mathbf{H}_{i}\) as follows,  
\[\mathbf{D}_{i}=\left[\begin{array}{cccc}\mathbf{v}_{i,1},&\mathbf{v}_{i,2},& \cdots,&\mathbf{v}_{i,r_{i}}\end{array}\right], \tag{16}\]  
where \(\mathbf{V}_{i}=\left[\begin{array}{cccc}\mathbf{v}_{i,1},&\mathbf{v}_{i,2},& \cdots,&\mathbf{v}_{i,r_{i}},&\cdots,&\mathbf{v}_{i,N_{T}}\end{array}\right]\) is right singular vector matrix and \(\mathbf{v}_{i,j}\) is the \(j\)-th right singular vector for \(\mathbf{V}_{i}\). The matrix of dominant components for other UEs except for the \(i\)-th UE is \(\mathbf{\tilde{D}}_{i}=\left[\begin{array}{cccc}\mathbf{D}_{1},&\cdots,& \mathbf{D}_{i-1},&\mathbf{D}_{i+1},&\cdots,&\mathbf{D}_{K}\end{array}\right]\). Then, \(\mathbf{D}_{i}\) is projected onto orthogonal space of \(\mathbf{\tilde{D}}_{i}\) by using complementary projection in order to mitigate IUI. The orthogonal vectors with \(\mathbf{\tilde{D}}_{i}\) in \(\mathbf{D}_{i}\) are obtained as follows,  
\[\mathbf{P}_{i}=\left\{\mathbf{I}-\mathbf{\tilde{D}}_{i}\Big{(}\mathbf{\tilde{ D}}_{i}^{H}\mathbf{\tilde{D}}_{i}\Big{)}^{-1}\mathbf{\tilde{D}}_{i}^{H} \right\}\mathbf{D}_{i}, \tag{17}\]  
where \(\mathbf{P}_{i}\) is transmit weight matrix and GBD precoder uses \(\mathbf{P}_{i}\) as precoding matrix \(\mathbf{F}_{i}^{\text{GBD}}\) for the \(i\)-th UE.  
The GBD precoder suppresses the remaining IUI by utilizing postcoding matrix \(\mathbf{G}^{\text{GBD}}\). The postcoding matrix \(\mathbf{G}_{i}^{\text{GBD}}=[\begin{array}{cccc}\mathbf{u}_{i,1},&\mathbf{u }_{i,2},&\cdots,&\mathbf{u}_{i,r_{i}}\end{array}]^{H}\) consists of left singular vectors for \(\mathbf{H}_{i}\). The IUI elements for the \(i\)-th UE are represented as follows,  
\[\sum_{k=1,k\
eq i}^{K}\mathbf{G}_{k}^{\text{GBD}}\mathbf{H}_{k}\mathbf{F}_{i}^ {\text{GBD}}=\sum_{k=1,k\
eq i}^{K}\left[\begin{array}{cccc}\mathbf{I}_{r_{ i}}&\mathbf{0}\end{array}\right]\mathbf{\Sigma}_{k}\left[\begin{array}{cccc} \mathbf{0}_{r_{i}}\\ \Delta\end{array}\right], \tag{18}\]  
where \(\Delta\) and \(\mathbf{0}\) are a matrix with small-valued elements and a matrix with zero elements. Therefore, the effective channel \(\mathbf{H}_{\text{eff}}\) is given by,  
\[\mathbf{H}_{\text{eff}}=\left[\begin{array}{cccc}\mathbf{G}_{1}^{\text{GBD }}\mathbf{H}_{1}\mathbf{F}_{1}^{\text{GBD}}&\cdots&\mathbf{0}\\ \vdots&\ddots&\vdots\\ \mathbf{0}&\cdots&\mathbf{G}_{K}^{\text{GBD}}\mathbf{H}_{K}\mathbf{F}_{K}^{ \text{GBD}}\end{array}\right], \tag{19}\]  
and essentially effective channel \(\mathbf{G}_{i}^{\text{GBD}}\mathbf{H}_{i}\mathbf{F}_{i}^{\text{GBD}}\) is given by,  
\[\mathbf{G}_{i}^{\text{GBD}}\mathbf{H}_{i}\mathbf{F}_{i}^{\text{GBD}}=\left[ \begin{array}{cccc}\mathbf{I}_{r_{i}}&\mathbf{0}\end{array}\right]\mathbf{ \Sigma}_{i}\mathbf{V}_{i}^{H}\mathbf{F}_{i}^{\text{GBD}}. \tag{20}\]  
Consequently, the number of singular values for desired channel is reduced from \(N_{R_{i}}\) to \(r_{i}\) and IUI channel is converged to zero.  
The GBD scheme increases diversity gain for total systems but is not optimized with MMSE criterion. Therefore, the goal of proposed scheme is that the precoder is optimized with modified MMSE criteria by considering noise entries and provides diversity gain.