# 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  
This section describes the 3-dimensional (3D) channel model proposed by the 3 rd generation partnership project (3 GPP). This section describes the 3D channel model to show that the proposed relay selection scheme can select the appropriate relay in the 3D channel environment. Figure 4 shows the 3D channel model. The 3D channel model includes several parameters. The large-scale parameters of the 3D channel model are shadow fading, delay spread, and angular spread. The small-scale parameters of the 3D channel model are delays, cluster power, angle-of-arrival (AOA), and angle-of-departure (AOD). In addition, the 3D channel model includes parameters related to polarization and antenna configuration. The coefficients of the 3D channel model reflecting the above parameters are as follows [8]:  
\[H_{u,s\mu}(t)=\sqrt{\frac{R_{s}}{M}}\sum\limits_{m=1}^{M}\left[ \begin{array}{cc}F_{UE,u\mu}(\theta_{m,u,OAOA},\ \phi_{m,u,OAOA})\\ F_{UE,u\mu}(\theta_{m,u,OAOA},\ \phi_{m,u,OAOA})\\ \times\end{array} \right]^{T}\times\left[ \begin{array}{cc}e^{i\phi_{m,u}^{ \prime}}&\sqrt{K_{s,1}^{-1}e^{i\phi_{m,u}^{\prime}}}\\ \sqrt{K_{s,m}^{-1}e^{i\phi_{m,u}^{\prime}}}&e^{i\phi_{m,u}^{\prime}}\\ \end{array} \right]\ \times\left[ \begin{array}{cc}F_{BS,s\mu}(\theta_{m,u, AOD,\ \phi_{m,u, AOD})}\\ F_{BS,s\mu}(\theta_{m,u, AOD,\ \phi_{m,u, AOD})}\\ \end{array} \right] \tag{16}\] \[\times\ e^{2\pi\lambda_{0}^{-1}(\
u_{UE,u}^{\prime}2u_{UE,u})}\ \times\ e^{2\pi\lambda_{0}^{-1}(\
u_{ BS,u}^{\prime}2u_{BS,u})}\ \times\ e^{2\pi\
u_{0},u^{\prime}}.\]  
Since this paper does not consider the parameter related to the antenna polarization in Equation (16), the 3D channel coefficient used in this paper is as follows:  
\[H_{u,s,\mu}(t) = \sqrt{\frac{R_{s}}{M}}\sum\limits_{m=1}^{M}\left[ \begin{array}{ cc}F_{UE,u\mu}(\theta_{n,m,uOA},\ \phi_{n,m,uOA})\\ F_{UE,u\mu}(\theta_{n,m,uOA},\ \theta_{n,m,uOA})\\ \times\ e^{|2\pi\lambda_{0}^{-1}(\sigma_{UE,u}^{\prime}\bar{a}_{UE,u})}\ \times\ e^{|2\pi\lambda_{0}^{-1}(\sigma_{BS,u}^{\prime}\bar{a}_{BS,s})}\ \times\ e^{|2\pi\
u_{0},u^{\prime}},\end{array} \right]^{T}\times\left[ \begin{array}{cc}F_{BS,s\mu}(\theta_{n,m,uOD},\ \phi_{n,m,uOD})\\ F_{BS,s\mu}(\theta_{n,m,uOD},\ \phi_{n,m,uOD})\end{array} \right] \tag{17}\] \[\times\ e^{|2\pi\lambda_{0}^{-1}(\sigma_{UE,u}^{\prime}\bar{a}_{UE,u})}\ \times\ e^{|2\pi\lambda_{0}^{-1}(\sigma_{BS,u}^{\prime}\bar{a}_{BS,s})}\ \times\ e^{|2\pi\
u_{0},u^{\prime}},\]  
where the \(u\) and \(s\) refer to the UE and BS antenna element, respectively. The \(n\) is the path index. The \(P_{n}\) and \(M\) denote the power of path and total number of paths, respectively. The \(F_{UE,u,\theta}\), \(F_{UE,u,\phi}\) are the UE antenna pattern in direction of the spherical basis vectors, \(\theta\) in the zenith direction, \(\phi\) in the azimuth direction. The \(F_{BS,\phi}\) and \(F_{BS,s\mu}\) are the BS antenna pattern in direction of the spherical basis vectors, \(\theta\) in the elevation direction, \(\phi\) in the azimuth direction. In addition, the \(\hat{r}_{UE,u,m}\), \(\hat{r}_{BS,u,m}\) are the UE and BS spherical unit vectors expressed in Cartesian coordinates. The \(\hat{r}_{n,m}\) is defined as follows [7]:  
\[\hat{r}_{n,m}=\left( \begin{array}{cc}\sin\theta\ \cos\phi\\ \sin\theta\ \sin\phi\\ \cos\theta\end{array} \right). \tag{18}\]  
The \(\overline{d}_{UE,u}\) and \(\overline{d}_{BS,s}\) are the location vectors of the UE and BS antenna elements, respectively. The \(\overline{d}\) is as follows [7]:  
\[\overline{d}=\left( \begin{array}{cc}0\\ (p-1)d_{H}\\ (q-1)d_{V}\end{array} \right), \tag{19}\]  
where the \(p\) and \(q\) are number of antenna element in the horizontal and vertical direction, respectively. The \(d_{H}\) is element spacing in the horizontal direction and the \(d_{V}\) is element spacing in the vertical direction. The last parameter of the 3D channel model, \(v_{n,m}\), refers to the Doppler frequency of the UE moving at velocity \(v\).