# 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
Based on the Equation (10), the effective 3-D radio channel can be derived as [18]  
\[[\mathbf{H}]_{lk,l^{\prime}k^{\prime}}(t,\tau)=\alpha_{LOS}\sqrt{gt(\phi_{LOS}, \theta_{LOS},\theta^{tilt})}\sqrt{gr(\varphi_{LOS},\phi_{LOS},\theta^{tilt})} \mathbf{\tilde{a}}_{l^{\prime}k^{\prime}}(\varphi_{LOS},\vartheta_{LOS})\times \tag{11}\]  
\[\mathbf{\tilde{a}}_{ik}(\phi_{LOS},\theta_{LOS})+\sum_{i=1}^{I}\alpha_{i}(t,\tau) \sqrt{gt(\phi_{i},\theta_{i},\theta^{tilt})}\sqrt{gr(\varphi_{i},\vartheta_{i },\theta^{tilt})}\mathbf{\tilde{a}}_{l^{\prime}k^{\prime}}(\varphi_{i},\theta_{i })\times\mathbf{\tilde{a}}_{lk}(\phi_{i},\theta_{i})\]\[[\mathbf{H}]^{V}_{lk,l^{\prime}k^{\prime}}(t,\tau)= \underbrace{\alpha_{LOS}\sqrt{gt(\phi_{LOS},\theta_{LOS},\theta^{tilt})} \times\widetilde{\mathbf{a}}_{lk}(\phi,\theta)_{X_{Pol}}\sqrt{gr(\varphi_{LOS},\theta _{LOS},\phi^{tilt})}\times\widetilde{\mathbf{a}}_{l^{\prime}k^{\prime}}(\varphi, \theta)_{V}}_{LOS}+\] \[\underbrace{\sum_{i=1}^{I}\alpha_{i}(t,\tau)\sqrt{gt(\phi_{i}, \theta_{i},\theta^{tilt})}\times\widetilde{\mathbf{a}}_{lk}(\phi,\theta)_{X_{Pol}} \sqrt{gr(\varphi_{i},\theta_{i},\phi^{tilt})}\times\widetilde{\mathbf{a}}_{l^{ \prime}k^{\prime}}(\varphi,\theta)_{V}}_{NLOS} \tag{11.1}\] \[[\mathbf{H}]^{H}_{lk,l^{\prime}k^{\prime}}(t,\tau)= \underbrace{\alpha_{LOS}\sqrt{gt(\phi_{LOS},\theta_{LOS},\theta^{tilt})} \times\widetilde{\mathbf{a}}_{lk}(\phi,\theta)_{X_{Pol}}\sqrt{gr(\varphi_{LOS}, \theta_{LOS},\phi^{tilt})}\times\widetilde{\mathbf{a}}_{l^{\prime}k^{\prime}}( \varphi,\theta)_{H}}_{LOS}+\] \[\underbrace{\sum_{i=1}^{I}\alpha_{i}(t,\tau)\sqrt{gt(\phi_{i}, \theta_{i},\theta^{tilt})}\times\widetilde{\mathbf{a}}_{lk}(\phi,\theta)_{X_{Pol}} \sqrt{gr(\varphi_{i},\theta_{i},\phi^{tilt})}\times\widetilde{\mathbf{a}}_{l^{ \prime}k^{\prime}}(\varphi,\theta)_{H}}_{NLOS}\] (11.2) \[[\mathbf{H}]^{V/H}_{lk,l^{\prime}k^{\prime}}(t,\tau)= \underbrace{\alpha_{LOS}\sqrt{gt(\phi_{LOS},\theta_{LOS},\theta^{tilt})} \times\widetilde{\mathbf{a}}_{lk}(\phi,\theta)_{X_{Pol}}\sqrt{gr(\varphi_{LOS}, \theta_{LOS},\phi^{tilt})}\times\widetilde{\mathbf{a}}_{l^{\prime}k^{\prime}}( \varphi,\theta)_{V/H}}_{LOS}+\] \[\underbrace{\sum_{i=1}^{I}\alpha_{i}(t,\tau)\sqrt{gt(\phi_{i}, \theta_{i},\theta^{tilt})}\times\widetilde{\mathbf{a}}_{lk}(\phi,\theta)_{X_{Pol}} \sqrt{gr(\varphi_{i},\theta_{i},\phi^{tilt})}\times\widetilde{\mathbf{a}}_{l^{ \prime}k^{\prime}}(\varphi,\theta)_{V/H}}_{NLOS} \tag{11.3}\]  
where \((\phi,\,\theta)\) are the azimuth and elevation AoD and \((\varphi,\,\theta)\) are the azimuth and elevation AoA and \(\theta^{tilt},\theta^{tilt}\) are the elevation of the antenna boresight for both TX and RX, respectively. According to the 3-D antenna beam pattern and considering the signal direction of \(\phi\), \(\theta\) and \(\varphi\), \(\theta\) on the antenna arrays, the global patterns for the NLOS components can be represent as \(\sqrt{gt(\phi_{i},\theta_{i},\theta^{tilt})}\approx gt^{BS}_{H}(\phi_{i}),gt^{ BS}_{V}(\theta_{i},\theta^{tilt})\) and \(\sqrt{gr(\varphi_{i},\theta_{i},\theta^{tilt})}\approx g^{ILE}_{H}(\varphi_{i}), g^{ILE}_{V}(\theta_{i},\theta^{tilt})\) for the \(i_{th}\) path of horizontal and vertical pattern at the TX and RX, respectively. In addition, in similar way, the 3-D antenna beam pattern on the antenna arrays, the global patterns for the LOS component can be represent as \(\sqrt{gt(\phi_{LOS},\theta_{LOS},\theta^{tilt})}\approx gt^{BS}_{H}(\phi_{ LOS}),gt^{BS}_{V}(\theta_{LOS},\theta^{tilt})\) and \(\sqrt{gr(\varphi_{LOS},\theta_{LOS},\theta^{tilt})}\approx gr^{ILE}_{H}(\varphi _{LOS}),g^{ILE}_{V}(\theta_{LOS},\theta^{tilt})\) for the horizontal and vertical pattern at the TX and RX, respectively.