## 2 System Model and Conventional Precoding Schemes
### Modified Generalized Block Diagonalization  
The recent study which is related with GBD was proposed in [23]. The MGBD was proposed with enhanced performance compared to GBD. Most conventional schemes use all antenna resources to remove IUI perfectly. However, MGBD uses part of the antenna resources to obtain diversity gain like GBD. Since the MGBD also uses the partial nulling concept to achieve diversity gain, the MGBD is chosen as the bench-marked scheme. The difference between the GBD and MGBD is that the MGBD is optimized with a modified minimum mean square error (MMSE) criterion in order to improve BER gain. The modified MMSE criterion is designed to minimize the Frobenius norm for non-diagonal components of equivalent channel that cause IUI. The precoding matrix of MGBD is presented as follows,  
\[\mathbf{P}_{k}^{\text{MGBD}}=\mathbf{P}_{k}^{\text{MU}}\mathbf{P}_{k}^{\text{ SU}}. \tag{20}\]  
To calculate the \(\mathbf{P}_{k}^{\text{MU}}\), the effective channel which is the multiplication of post-coder (18) and the channel matrix of the \(k\)th user is used. The effective channel is expressed as follows,  
\[\mathbf{T}_{k}=\mathbf{R}_{k}^{H}\mathbf{H}_{k}. \tag{21}\]  
The effective channels of all users except the \(k\)th user is expressed as follows,  
\[\mathbf{\tilde{T}}_{k}= \begin{bmatrix}\mathbf{T}_{1}^{T}&\cdots&\mathbf{T}_{k -1}^{T}&\mathbf{T}_{k+1}^{T}&\cdots&\mathbf{T}_{k}^{T}\end{bmatrix} ^{T}. \tag{22}\]  
The SVD of \(\mathbf{\tilde{T}}_{k}\) is written as follows,  
\[\mathbf{\tilde{T}}_{k}=\mathbf{\tilde{U}}_{k}\mathbf{\tilde{\Sigma}}_{k} \mathbf{\tilde{V}}_{k}^{H}, \tag{23}\]  
where \(\mathbf{\tilde{U}}_{k}\), the diagonal elements of \(\mathbf{\tilde{\Sigma}}_{k}\) and \(\mathbf{\tilde{V}}_{k}\) are the set of left singular vectors, the singular values and the set of right singular vectors, respectively. The first precoder \(\mathbf{P}_{k}^{\text{MU}}\) can be expressed as follows,  
\[\mathbf{P}_{k}^{\text{MU}}=\mathbf{\tilde{V}}_{k}\bigg{(}\mathbf{\tilde{ \Sigma}}_{k}^{H}\mathbf{\tilde{\Sigma}}_{k}+\alpha\mathbf{I}\bigg{)}^{-1/2}. \tag{24}\]  
Since MGBD uses the effective channel for making the precoding matrix, MGBD can achieve performance enhancement compared to GBD. The SVD of the channel precoded with \(\mathbf{P}_{k}^{\text{MU}}\) is as follows,  
\[\mathbf{H}_{k}\mathbf{P}_{k}^{\text{MU}}=\mathbf{\tilde{U}}_{k}\mathbf{\tilde {\Sigma}}_{k}\Big{[}\mathbf{\hat{V}}_{k}^{(1)}\mathbf{\hat{V}}_{k}^{(0)} \Big{]}^{H}, \tag{25}\]  
where \(\mathbf{\tilde{U}}_{k}\), the diagonal elements of \(\mathbf{\tilde{\Sigma}}_{k}\), \(\mathbf{\hat{V}}_{k}^{(1)}\) and \(\mathbf{\hat{V}}_{k}^{(0)}\) are the set of left singular vectors, the singular values, the set of right singular vectors corresponding to non-zero singular values and the set of right singular vectors corresponding to zero singular values, respectively. The second precoder \(\mathbf{P}_{k}^{\text{SU}}\) is as follows,  
\[\mathbf{P}_{k}^{\text{SU}}=\mathbf{\hat{V}}_{k}^{(1)}. \tag{26}\]  
Therefore, final precoding matrix of MGBD is expressed as follows,  
\[\mathbf{P}_{k}^{\text{MGBD}}=\mathbf{\tilde{V}}_{k}\bigg{(}\mathbf{\tilde{ \Sigma}}_{k}^{H}\mathbf{\tilde{\Sigma}}_{k}+\alpha\mathbf{I}\bigg{)}^{-1/2} \mathbf{\hat{V}}_{k}^{(1)}. \tag{27}\]  
The way to remove residual IUI at the receiver is exactly same as GBD. The residual IUI is eliminated by multiplying post-coder at the receiver. The process for obtaining a precoding matrix of the MGBD is very similar with RBD. Therefore, when the number of data streams is equal to the number of the transmit antennas, the precoding matrix of the MGBD is same as the precoding matrix of the RBD. It means that the performance of the MGBD with full data streams is same as the performance of the RBD. However, when the number of data streams is reduced, the MGBD can obtain the transmit diversity gain. Thus, the performance of the MGBD is better than RBD when the MGBD has the reduced data streams.