# Relay Selection Scheme for Multi-Hop Transmission of MU-MIMO System
## 4 Proposed Relay Selection Scheme
### Singular Value Decomposition (SVD)-Based Relay Selection in 3-Dimensional (3D) Channel Model
Figure 4: The 3D channel model. The proposed relay selection scheme can select the appropriate relay in the 3D channel environment. When the 3D channel matrix between the BS and relays is \(\mathbf{H}_{S,i}^{3D}\in\mathbb{C}^{N_{r}\times N_{t}}\), the SVD of \(\mathbf{H}_{S,i}^{3D}\) is as follows:  
\[\mathbf{H}_{S,i}^{3D}=\mathbf{U}_{i}^{3D}\mathbf{\sum}_{i}^{3D}\Big{(}\mathbf{V }_{i}^{3D}\Big{)}^{H},\ \ \ \ \ \ \mathbf{U}_{i}^{3D}\in\mathbb{C}^{N_{r}\times N_{t}},\ \ \ \sum_{i}^{3D}\in\mathbb{R}^{N_{t}\times N_{t}},\ \ \ \mathbf{V}_{i}^{3D}\in\mathbb{C}^{N_{t}\times N_{t}}, \tag{20}\]  
where the \(\mathbf{U}_{i}^{3D}\) and \(\mathbf{V}_{i}^{3D}\) are orthogonal matrices and the main diagonal entries of the \(\sum_{i}^{3D}\) are the singular values of the \(\mathbf{H}_{S,i}^{3D}\) and other entries are zero. In the proposed relay selection scheme, the BS calculates the following equation to select the appropriate relay,  
\[\sum_{tr,j}^{3D}=\text{tr}\Big{(}\mathbf{\sum}_{i}^{3D}\Big{)}=\sum_{k=1}^{N_ {t}}\sigma_{ijk}^{3D}, \tag{21}\]  
where the \(\sigma_{ijk}^{3D}\) represents the singular values of the \(\mathbf{H}_{S,i}^{3D}\). In other words, the BS calculates the sum of the singular values of the 3D channel matrix between the BS and relays. Then, the relay having the largest value is selected. As a result, the proposed relay selection scheme selects the relay using only 3D channel information between the BS and relays. After the relay is selected, the BS transmits the signal to the selected relay through the 3D channel. The received signal of the selected relay is as follows:  
\[\mathbf{y}_{R}^{3D}=\mathbf{H}_{S,R}^{3D}\mathbf{x}^{S}+\mathbf{N}_{j}. \tag{22}\]  
The selected relay uses the ZF detection scheme to detect the signal. The detected signal by the ZF detection scheme is as follows:  
\[\hat{\mathbf{x}}_{R}^{3D} =\mathbf{G}_{R}^{3D}\mathbf{y}_{R}^{3D}=\bigg{(}\left[\mathbf{H} _{S,R}^{3D}\right]^{H}\mathbf{H}_{S,R}^{3D}\bigg{)}^{-1}\Big{[}\mathbf{H}_{S,R }^{3D}\Big{]}^{H}\Big{(}\mathbf{H}_{S,R}^{3D}\mathbf{x}^{S}+\mathbf{N}_{R} \Big{)} \tag{23}\] \[=\bigg{(}\Big{[}\mathbf{H}_{S,R}^{3D}\Big{]}^{H}\mathbf{H}_{S,R}^ {3D}\bigg{)}^{-1}\bigg{(}\Big{[}\mathbf{H}_{S,R}^{3D}\Big{]}^{H}\mathbf{H}_{S, R}^{3D}\bigg{)}\mathbf{x}^{S}+\mathbf{G}_{R}^{3D}\mathbf{N}_{R}\] \[=\mathbf{x}^{S}+\mathbf{G}_{R}^{3D}\mathbf{N}_{R}.\]  
The selected relay transmits the signal applied to the BD-based MU-MIMO pre-coding to the UEs to remove the inter-user interference. The received signal of the \(j\)-th UE is as follows:  
\[\mathbf{y}_{UE,j}^{3D}=\mathbf{H}_{R,UE}^{3D}j\mathbf{W}_{j}^{3D}\hat{\mathbf{ x}}_{R,UE}^{3D}\ +\ \mathbf{N}_{j}, \tag{24}\]  
where the \(\mathbf{W}_{j}^{3D}\) is the pre-coding matrix of the \(j\)-th UE. The UE detects the received signal using the ZF detection scheme same as the relay. The detected signal at the \(j\)-th UE is as follows:  
\[\hat{\mathbf{x}}_{UE,j}^{3D} =\ \mathbf{G}_{UE,j}^{3D}\mathbf{y}_{UE,j}^{3D}=\bigg{(}\big{[} \mathbf{H}_{R,UE}^{3D}\mathbf{y}_{y}^{3D}\big{]}^{H}\big{[}\mathbf{H}_{R,UE}^ {3D}\mathbf{w}_{j}^{3D}\big{]}^{\bigg{)}^{-1}}\Big{[}\mathbf{H}_{R,UE}^{3D} \mathbf{w}_{j}^{3D}\Big{]}^{H}\Big{(}\mathbf{H}_{R,UE}^{3D}\mathbf{w}_{j}^{3D} \hat{\mathbf{x}}_{R,UE}^{3D}+\mathbf{N}_{j}\Big{)} \tag{25}\] \[=\ \Big{(}\big{[}\mathbf{H}_{R,UE}^{3D}\mathbf{w}_{j}^{3D}\big{]}^{H }\big{[}\mathbf{H}_{R,UE}^{3D}\mathbf{w}_{j}^{3D}\big{]}^{\bigg{)}^{-1}}\Big{(} \big{[}\mathbf{H}_{R,UE}^{3D}\mathbf{w}_{j}^{3D}\big{]}^{H}\big{[}\mathbf{H}_ {R,UE}^{3D}\mathbf{w}_{j}^{3D}\big{]}\Big{)}\hat{\mathbf{x}}_{R,UE}^{3D}+ \mathbf{G}_{UE,j}^{3D}\mathbf{N}_{j}\] \[\mathbf{H}_{S,i}^{3D}=\mathbf{U}_{i}^{3D}\mathbf{\sum}_{i}^{3D}\Big{(}\mathbf{V }_{i}^{3D}\Big{)}^{H},\ \ \ \ \ \ \mathbf{U}_{i}^{3D}\in\mathbb{C}^{N_{r}\times N_{t}},\ \ \ \sum_{i}^{3D}\in\mathbb{R}^{N_{t}\times N_{t}},\ \ \ \mathbf{V}_{i}^{3D}\in\mathbb{C}^{N_{t}\times N_{t}}, \tag{20}\]0  
The proposed relay selection scheme selects the relay by using the singular value of the channel matrix between the BS and relays in the 3D channel environment.