# Practical 3-D Beam Pattern Based Channel Modeling for Multi-Polarized Massive MIMO Systems +
## 2 3-D Antenna Configuration
### 3-D AES and Antenna Element's Positioning
#### 2.1.2 Receive Antenna Configuration  
Similarly, in the user side, there are three types of antenna configurations using omnidirectional antenna with V, H, and V/H polarizations for SULA. The 3-D Cartesian coordinate systems \(\rho_{m^{\prime},n^{\prime}}\) with \(Y_{m^{\prime},n^{\prime}}=(x_{m^{\prime},n^{\prime}},y_{m^{\prime},n^{\prime }},z_{m^{\prime},n^{\prime}})\) spherical coordinates of a points, where \(x_{m^{\prime},n^{\prime}}=\rho_{m^{\prime},n^{\prime}}\sin\phi_{AAoA}\cos \theta_{EAoA},\quad y_{m^{\prime},n^{\prime}}=\rho_{m^{\prime},n^{\prime}} \sin\phi_{AAoA}\sin\theta_{EAoA}\), and \(z_{m^{\prime},n^{\prime}}=\rho_{m^{\prime},n^{\prime}}\cos\theta_{EAoA}\). Therefore, antenna element's point of SULA for V polarization can be obtained from its Cartesian coordinates \((x,y,z)\) by  
\[\rho^{\
u}_{m^{\prime},n^{\prime}}=\sqrt{0+y_{m^{\prime},n^{\prime}}^{2}+z_{m ^{\prime},n^{\prime}}^{2}} \tag{3}\]  
The position of two vertical antenna elements, one indexed by \(l^{\prime}k^{\prime}\) and the other one denote by \((m^{\prime}n^{\prime})\) at the RX can be computed as  
\[d^{UE}_{V}=\sqrt{0+(y_{m^{\prime}n^{\prime}}-y_{l^{\prime}k^{\prime}})^{2}+( z_{m^{\prime}n^{\prime}}-z_{l^{\prime}k^{\prime}})^{2}} \tag{4}\]In addition, antenna element's point of SULA for H polarization can be obtained from its Cartesian coordinates \((x,y,z)\) by  
\[\rho^{H}_{m^{\prime}n^{\prime}}=\sqrt{x^{2}_{m^{\prime}n^{\prime}}+y^{2}_{m^{ \prime}n^{\prime}}+0} \tag{5}\]  
The position of two horizontal antenna elements, one indexed by \(l^{\prime}k^{\prime}\) and the other one indexed by \((m^{\prime}n^{\prime})\) at the RX can be computed as  
\[d^{ILE}_{H}=\sqrt{(x_{m^{\prime}n^{\prime}}-x_{l^{\prime}k^{\prime}})^{2}+(y_{ m^{\prime}n^{\prime}}-y_{l^{\prime}k^{\prime}})^{2}+0} \tag{6}\]  
Furthermore, antenna element's point of SULA for V/H polarization can be obtained from its Cartesian coordinates \((x,y,z)\) by  
\[\rho^{V/H}_{m^{\prime}n^{\prime}}=\sqrt{x^{2}_{m^{\prime}n^{\prime}}+y^{2}_{m^ {\prime}n^{\prime}}+z^{2}_{m^{\prime}n^{\prime}}} \tag{7}\]  
The position of two vertical/horizontal antenna elements, one indexed by \(l^{\prime}k^{\prime}\) and the other one indexed by \((m^{\prime}n^{\prime})\) at the RX can be computed as  
\[d^{UE}_{V/H}=\sqrt{(x_{m^{\prime}n^{\prime}}-x_{l^{\prime}k^{\prime}})^{2}+(y_ {m^{\prime}n^{\prime}}-y_{l^{\prime}k^{\prime}})^{2}+(z_{m^{\prime}n^{\prime}} -z_{l^{\prime}k^{\prime}})^{2}} \tag{8}\]  
The polarization vector at angle \(a\) from the \(z\) axis in Figure 1e, has vertical and horizontal components of the antenna pattern, by expressing the response vector \(\chi\) in its \(\vartheta\) and \(\varphi\) components [4] which are proportional to  
\[\chi^{UE}_{l^{\prime}k^{\prime}}= \begin{bmatrix}\chi^{\vartheta}_{l^{\prime}k^ {\prime}}\\ \chi^{\varphi}_{l^{\prime}k^{\prime}}\end{bmatrix} = \begin{bmatrix}\cos a\sin \vartheta+\cos a\cos\vartheta\cos\varphi\\ \sin a\cos\phi\end{bmatrix}  \tag{9}\]  
where \(\chi^{\vartheta}\) and \(\chi^{\varphi}\) are the \(\vartheta\) and \(\varphi\) polarized responses of the antenna at the wave direction of \([\sin\varphi\cos\vartheta,\sin\varphi\sin\vartheta,\cos\vartheta]\) for the incoming wave in the response of the antenna. To elaborate the steps and methodology in generating a complete 3-D channel model for massive MIMO systems, considering the beam pattern for different polarizations. The channel impulse response between the \(lk_{th}\) and the \(l^{\prime}k^{\prime}_{th}\) antenna element both for line-of-sight (LOS) and Non-line-of-sight (NLOS) can be presented as  
\[[\mathbf{H}]_{lk,l^{\prime}k^{\prime}}(t,\tau)=\alpha_{LOS}h_{LOS,lk,l^{\prime}k^ {\prime}}(t)+\sum_{i=1}^{I}\alpha_{i}h_{i,lk,l^{\prime}k^{\prime}}(t,\tau) \tag{10}\]  
where \(\alpha\) is the complex amplitude of the LOS path of \(h_{LOS,lk,l^{\prime}k^{\prime}}(t)\), and NLOS components, path \(i\), where \(i=1,...,I\) of \(h_{i,lk,l^{\prime}k^{\prime}}(t,\tau)\).