# Practical 3-D Beam Pattern Based Channel Modeling for Multi-Polarized Massive MIMO Systems +
## 4 Received Spatial Correlation  
In practice, the channels between different antennas are often correlated, and therefore, the potential multi-antenna gains may not always be attainable [24]. We focus and analyze the spatial correlation of the receivers, including the effect of antenna polarization based on the practical multi-path wireless communication environment. We will then investigate and minimize such correlation at the RX by considering the AES to recognize the antenna element and its adjacent antenna elements from different polarization as well as changeable of the space \(d^{UE}\). It is because of this purpose, in this paper, we developed the RSC based in [25]. For the received signals with the mean AAoA (\(\varphi\)) and EAoA (\(\vartheta\)), the difference in their distance traveled is \(d^{UE}_{V},d^{UE}_{H}\), and \(d^{UE}_{V/H}\) for different polarizations. In addition, we are applying correlation for all possibilities antenna configuration for a given beam pattern. Let \(\alpha\) and \(\beta\) denote the amplitude and phase antenna. Therefore, their impulse response for V polarized elements can be represented as  
\[h_{a}(\varphi_{V},\vartheta_{V}) =\alpha e^{j\beta}\sqrt{P(\varphi_{V},\vartheta_{V})} \tag{34}\] \[h_{b}(\varphi_{V},\vartheta_{V}) =\alpha e^{j\beta+(\hat{\sigma}_{p/p^{\prime}}(l^{\prime}k^{ \prime}-1).d^{UE}_{V})}\sqrt{P(\varphi_{V},\vartheta_{V})} \tag{35}\]  
where \(\sqrt{P(\varphi_{V},\vartheta_{V})}\) denote the PAS for V polarized. In addition, their impulse response for H polarized elements can be given as  
\[h_{a}(\varphi_{H},\vartheta_{H}) =\alpha e^{j\beta}\sqrt{P(\varphi_{H},\vartheta_{H})} \tag{36}\] \[h_{b}(\varphi_{H},\vartheta_{H}) =\alpha e^{j\beta+(\varphi_{H}\
u^{\prime}(l^{\prime}k^{\prime}-1 ).d_{H}^{\text{J}E})}\sqrt{P(\varphi_{H},\vartheta_{H})} \tag{37}\]  
where \(\sqrt{P(\varphi_{H},\vartheta_{H})}\) denote the PAS for H polarized. Furthermore, their impulse response for V/H polarized elements can be represented as  
\[h_{a}(\varphi_{V/H},\vartheta_{V/H}) =\alpha e^{j\beta}\sqrt{P(\varphi_{V/H},\vartheta_{V/H})} \tag{38}\] \[h_{b}(\varphi_{V/H},\vartheta_{V/H}) =\alpha e^{j\beta+(\varphi_{H}\
u^{\prime}(l^{\prime}k^{\prime}-1 ).d_{V/H}^{\text{J}E})}\sqrt{P(\varphi_{V/H},\vartheta_{V/H})} \tag{39}\]  
where \(\sqrt{P(\varphi_{V/H},\vartheta_{V/H})}\) denote the PAS for V/H polarized. Next, let us define a RSC function with the mean EAoA and AAoA of \(\vartheta\) and \(\varphi\) in two antenna elements spaced apart by \(d^{\text{J}E}+i\) as  
\[\rho_{d,\varphi_{0},\varphi_{0}}^{V}=E\left\{h_{a}(\varphi_{V},\vartheta_{V} )\right\}\left\{h_{b}^{*}(\varphi_{V},\vartheta_{V})\right\}=\int_{-\pi}^{ \pi}\int_{\frac{\pi}{2}}^{\frac{\pi}{2}}h_{a}(\varphi_{V},\vartheta_{V})h_{b}^ {*}(\varphi_{V},\vartheta_{V})P(\varphi-\varphi_{0})P(\vartheta-\vartheta_{0} )d_{\varphi}d_{\vartheta} \tag{40}\]  
\[\rho_{d,\varphi_{0},\varphi_{0}}^{H}=E\left\{h_{a}(\varphi_{H},\vartheta_{H}) \right\}\left\{h_{b}^{*}(\varphi_{H},\vartheta_{H})\right\}=\int_{-\pi}^{ \pi}\int_{\frac{\pi}{2}}^{\frac{\pi}{2}}h_{a}(\varphi_{H},\vartheta_{H})h_{b}^ {*}(\varphi_{H},\vartheta_{H})P(\varphi-\vartheta_{0})d_{\varphi}d_{\vartheta} \tag{41}\]  
\[\rho_{d,\varphi_{0},\varphi_{0}}^{V/H}=E\left\{h_{a}(\varphi_{V/H},\vartheta_ {V/H})\right\}\left\{h_{b}^{*}(\varphi_{V/H},\vartheta_{V/H})\right\}=\int_{- \pi}^{\pi}\int_{\frac{\pi}{2}}^{\frac{\pi}{2}}h_{a}(\varphi_{V/H},\vartheta_{ V/H})h_{b}^{*}(\varphi_{V/H},\vartheta_{V/H}) \tag{42}\]  
However, in a non-stationary channel, EAoA, AAoA and angular spread (AS) are not equal to \(0^{\circ}\). There is a time difference between \(h_{a}(\varphi,\vartheta)\) and \(h_{b}(\varphi,\vartheta)\) for V, H, and V/H polarized elements due to the Doppler Spread. In Figure 3 shows the phase difference between antenna elements under the DoT variations. This yields the following spatial correlation function for the SULA as  
\[