# Relay Selection Scheme for Multi-Hop Transmission of MU-MIMO System
## 4 Proposed Relay Selection Scheme  
### Singular Value Decomposition (SVD)-Based Relay Selection in 2-Dimensional (2D) Channel Model  
This section describes the proposed relay selection scheme in the conventional 2-dimensional (2D) channel environment. Unlike the conventional relay selection schemes, the proposed relay selection scheme uses only the channel information between the BS and relays. When the channel matrix between the BS and relays is \(\mathbf{H}_{S,i}\in\mathbb{C}^{N_{r}\times N_{t}}\), the singular value decomposition (SVD) of \(\mathbf{H}_{S,i}\) is as follows:  
\[\mathbf{H}_{S,i}=\mathbf{U}_{i}\underset{i}{\sum}\mathbf{V}_{i}^{H},\quad \mathbf{U}_{i}\in\mathbb{C}^{N_{r}\times N_{r}},\ \ \ \sum_{i}\in\mathbb{R}^{N_{r}\times N_{t}},\ \ \ \mathbf{V}_{i}\in\mathbb{C}^{N_{t}\times N_{t}}, \tag{8}\]  
where the \(\mathbf{U}_{i}\) and \(\mathbf{V}_{i}\) are orthogonal matrices and the main diagonal entries of the \(\sum_{i}\) are the singular values of the \(\mathbf{H}_{S,i}\) and other entries are zero. In the proposed relay selection scheme, the BS calculates the following equation for each relay to select the appropriate relay,  
\[\sum_{tr,i}=\text{tr}\Big{(}\underset{i}{\sum}\Big{)}=\sum_{k=1}^{N_{t}}\sigma _{i,k}, \tag{9}\]  
where the \(\sigma_{i,k}\) is the singular value of the \(\mathbf{H}_{S,i}\). In other words, the BS calculates the sum of the singular value of the channel matrix between the BS and relays. Then, the relay having the largest value of Equation (9) is selected. As a result, the proposed relay selection scheme can select an appropriate relay using only the channel information between the BS and relays. After the relay is selected, the BStransmits the signal to be transmitted to the UEs through the MIMO channel between the BS and selected relay. The received signal of the selected relay is as follows:  
\[\mathbf{y}_{R}=\mathbf{H}_{S,R}\mathbf{x}^{S}+\mathbf{N}_{R}. \tag{10}\]  
The selected relay detects the signal using the zero forcing (ZF) detection scheme [17]. The filter matrix used in the ZF detection scheme is as follows:  
\[\mathbf{G}_{R}=\left(\mathbf{H}_{S,R}^{H}\cdot\mathbf{H}_{S,R}\right)^{-1} \cdot\mathbf{H}_{S,R}^{H}. \tag{11}\]  
The relay detects the signal by multiplying the received signal by the filter matrix \(\mathbf{G}_{R}\)[17]. The detected signal at the relay is as follows:  
\[\hat{\mathbf{x}}_{R} =\mathbf{G}_{R}\mathbf{y}_{R}=\left(\mathbf{H}_{S,R}^{H}\cdot \mathbf{H}_{S,R}\right)^{-1}\cdot\mathbf{H}_{S,R}^{H}\cdot\left(\mathbf{H}_{S,R}\mathbf{x}^{S}+\mathbf{N}_{R}\right)=\left(\mathbf{H}_{S,R}^{H}\cdot \mathbf{H}_{S,R}\right)^{-1}\cdot\left(\mathbf{H}_{S,R}^{H}\cdot\mathbf{H}_{S,R}\right)\mathbf{x}^{S}+\mathbf{G}_{R}\mathbf{N}_{R} \tag{12}\] \[=\mathbf{x}^{S}+\mathbf{G}_{R}\mathbf{N}_{R}.\]  
Equation (12) represents the process in which the selected relay through the proposed relay selection scheme detects the signal transmitted from the base station through the filter matrix of the ZF detection scheme. When the selected relay transmits the signal to UEs, it transmits the signal by applying the BD-based MU-MIMO pre-coding to cancel inter-user interference. As described in Section 2.3, when the relay transmits the signal with a pre-coding matrix applied, the received signal of the \(j\)-th UE is as follows:  
\[\mathbf{y}_{UE,j}=\mathbf{H}_{R,UE\;j}\mathbf{W}_{j}\hat{\mathbf{x}}_{R,UE\;j }+\mathbf{N}_{j}, \tag{13}\]  
where the \(\mathbf{W}_{j}\) denotes the pre-coding matrix of the \(j\)-th UE. The \(j\)-th UE detects the received signal using the ZF detection scheme. The filter matrix used in the \(j\)-th UE to detect the signal is as follows:  
\[\mathbf{G}_{UE,j}=\left(\left[\mathbf{H}_{R,UE\;j}\mathbf{W}_{j}\right]^{H} \left[\mathbf{H}_{R,UE\;j}\mathbf{W}_{j}\right]\right)^{-1}\left[\mathbf{H}_{ R,UE\;j}\mathbf{W}_{j}\right]^{H}. \tag{14}\]  
The \(j\)-th UE detects the signal by multiplying the received signal by the filter matrix \(\mathbf{G}_{UE,j}\).