# Relay Selection Scheme for Multi-Hop Transmission of MU-MIMO System
## 2 System Model
### Block Diagonalization (BD) Scheme  
The MU-MIMO scheme can increase the channel capacity and throughput without additional bandwidth allocation by simultaneously transmitting signals to multiple UEs. Therefore, the MU-MIMO scheme plays an important role in increasing the cell average throughput and cell edge user throughput of the LTE-A communication system. Figure 3 shows the system model of the MU-MIMO scheme. In the MU-MIMO scheme, the BS transmits the signal multiplied by the pre-coding matrix so that the UEs can distinguish their signal even if the BS transmits signals to multiple UEs at the same time. The block diagonalization (BD) scheme is a typical MU-MIMO pre-coding scheme. In the BD scheme-based MU-MIMO system, the received signal of the \(j\)-th UE is as follows:  
\[\mathbf{y}_{j}=\mathbf{H}_{S,UE\;j}\mathbf{W}_{j}\mathbf{x}_{j}+\sum_{k=1,\;k \
eq j}^{K}\mathbf{H}_{S,UE\;j}\mathbf{W}_{k}\mathbf{x}_{k}+\mathbf{N}_{j}, \tag{1}\]  
Figure 3: The system model of the MU-MIMO scheme. Figure 2: The system model of the DF relaying scheme. where the \(K\) denotes the total number of UEs, and the \(j\) refers to the index of the UE. \(\mathbf{H}_{S,UE\;j}\in\mathbb{C}^{N_{R}\times N_{S}}\) means the MIMO channel matrix between the BS and the \(j\)-th UE and the \(\mathbf{W}_{j}\in\mathbb{C}^{N_{R}\times N_{R}}\) denotes the pre-coding matrix of the \(j\)-th UE. \(\mathbf{x}_{j}\in\mathbb{C}^{N_{R}\times 1}\) refers to the transmitted signal to the \(j\)-th UE, and \(\mathbf{N}_{j}\) is the additive white Gaussian noise (AWGN). In Equation (1), \(\mathbf{H}_{S,UE\;j}\mathbf{W}_{k}\) is the interference to the \(j\)-th UE. Therefore, the \(\mathbf{H}_{S,UE\;j}\mathbf{W}_{k}\) must satisfy the following condition. The condition is as follows:  
\[\mathbf{H}_{S,UE\;j}\mathbf{W}_{k}=\mathbf{0}^{N_{R}\times N_{S}},\;\;\;\forall \;j\
eq k. \tag{2}\]  
In order to satisfy the total transmission power constraint, the pre-coding matrix \(\mathbf{W}_{j}\) must be unitary. If the condition of Equation (2) is satisfied, the received signal of the \(j\)-th UE is as follows:  
\[\mathbf{y}_{j}=\mathbf{H}_{S,UE\;j}\mathbf{W}_{j}\mathbf{x}_{j}+\mathbf{N}_{j}. \tag{3}\]  
In order to generate the pre-coding matrix satisfying the condition of the Equation (2), \(\widetilde{\mathbf{H}}_{S,UE\;j}\) is defined as follows:  
\[\widetilde{\mathbf{H}}_{S,UE\;j}=\Big{[}\left(\mathbf{H}_{S,UE\;j}\right)^{H} \cdots\left(\mathbf{H}_{S,UE\;j-1}\right)^{H}\left(\mathbf{H}_{S,UE\;j+1} \right)^{H}\cdots\left(\mathbf{H}_{S,UE\;K}\right)^{H}\Big{]}, \tag{4}\]  
where the \(\widetilde{\mathbf{H}}_{S,UE\;j}\) is defined as a channel matrix of all UEs excluding the channel matrix of the \(j\)-th UE, and the singular value decomposition (SVD) of the \(\widetilde{\mathbf{H}}_{S,UE\;j}\) is as follows:  
\[\widetilde{\mathbf{H}}_{S,UE\;j}=\widetilde{\mathbf{U}}_{j}\widetilde{\sum}_{ j}\Big{[}\widetilde{\mathbf{V}}_{j}^{non-zero}\;\;\;\widetilde{\mathbf{V}}_{j}^{ zero}\Big{]}^{H}, \tag{5}\]  
where the \(\widetilde{\mathbf{V}}_{j}^{non-zero}\) and \(\widetilde{\mathbf{V}}_{j}^{zero}\) are composed of right vectors that correspond to non-zero singular values and zero singular values, respectively. The \(\widetilde{\mathbf{V}}_{j}^{zero}\) becomes a null space orthogonal basis matrix of the \(\widetilde{\mathbf{H}}_{S,UE\;j}\). The \(\widetilde{\mathbf{H}}_{S,UE\;j}\widetilde{\mathbf{V}}_{j}^{zero}\) is as follows:  
\[\widetilde{\mathbf{H}}_{S,UE\;j}\widetilde{\mathbf{V}}_{j}^{ zero} =\widetilde{\mathbf{U}}_{j}\Big{[}\widetilde{\sum}_{j}^{non- zero}\;\;\;\mathbf{0}\Big{]}\left[ \begin{array}{c}\left(\widetilde{\mathbf{V}}_{j}^{non- zero}\right)^{H}\\ \left(\widetilde{\mathbf{V}}_{j}^{zero}\right)^{H}\end{array}