# Practical 3-D Beam Pattern Based Channel Modeling for Multi-Polarized Massive MIMO Systems +
## 4 Received Spatial Correlation
\begin{array}{l}\rho_{d,\varphi_{0},\varphi_{0}}^{V}(t+\Delta t)=E\left\{h_ {a_{V}}(\vartheta,\varphi)\right\}\left\{h_{b_{V}}^{*}(\vartheta,\varphi) \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}\\ \sum_{n=1}^{N}e^{\left(\varphi_{n}\
u^{\prime}(l^{\prime}k^{\prime}-1).d_{V}^ {\text{J}E}\left\|\varphi\right\|\Delta t^{\prime}\cos(\vartheta)_{AAoA}-( \vartheta)_{n}\right)}\end{array}  \tag{43}\]  
\[ \begin{array}{l}\rho_{d,\varphi_{0},\varphi_{0}}^{H}(t+\Delta t)=E\left\{h_ {a_{H}}(\vartheta,\varphi)\right\}\left\{h_{b_{H}}^{*}(\vartheta,\varphi) \right\}=\int_{-\pi}^{\pi}\int_{\frac{\pi}{2}}^{\frac{\pi}{2}}h_{a}(\varphi_{ H},\vartheta_{H})h_{b}^{*}(\varphi_{H},\vartheta_{H})P(\varphi-\varphi_{0})P( \vartheta-\vartheta_{0})d_{\varphi}d_{\vartheta}\\ \sum_{n=1}^{N}e^{\left(\varphi_{n}\
u^{\prime}(l^{\prime}k^{\prime}-1).d_{H}^ {\text{J}E}\left\|\varphi\right\|\Delta t^{\prime}\cos(\vartheta)_{AAoA}-( \vartheta)_{n}\right)}\end{array}  \tag{44}\]  
\[ \begin{array}{l}\rho_{d,\varphi_{0},\varphi_{0}}^{V/H}(t+\Delta t)=E\left\{h_ {a_{V/H}}(\vartheta,\varphi)\right\}\left\{h_{b_{V/H}}^{*}(\vartheta,\varphi) \right\}=\int_{-\pi}^{\pi}\int_{\frac{\pi}{2}}^{\frac{\pi}{2}}h_{a}(\varphi_{V/ H},\vartheta_{V/H})h_{b}^{*}(\varphi_{V/H},\vartheta_{V/H})\\ \hskip 14.226378 ptP(\varphi-\varphi_{0})P(\vartheta-\vartheta_{0})d_{\varphi}d_{ \vartheta}\sum_{n=1}^{N}e^{\left(\varphi_{n}\
u^{\prime}(l^{\prime}k^{\prime}-1 ).d_{V/H}^{\text{J}E}\left\|\varphi\right\|\Delta t^{\prime}\cos(\vartheta)_{AAoA} -(\vartheta)_{n}\right)}\end{array}  \tag{45}\]  
where \((\vartheta,\varphi)_{v}\) and \(\parallel v\parallel\) denote the DoT and mobile speed for different polarization, respectively. In case when DoT=90\({}^{\circ}\), the Equations (43)-(45) can be re-organized to  
\[\rho_{d,\varphi_{0},\varphi_{0}}^{V}(t+\Delta t) =\sum_{n=1}^{N}e^{\left(\varphi_{n}\
u^{\prime}(l^{\prime}k^{ \prime}-1).d_{V}^{\text{J}E}\left\|\varphi\right\|\sin(\vartheta)_{AA}\right)} \tag{46}\]  
\[\rho_{d,\varphi_{0},\varphi_{0}}^{H}(t+\Delta t) =\sum_{n=1}^{N}e^{\left(\varphi_{n}\
u^{\prime}(l^{\prime}k^{\prime }-1).d_{H}^{\text{J}E}\left\|\varphi\right\|\sin(\vartheta)_{AAoA}\right)} \tag{47}\] \[\rho_{d,\varphi_{0},\varphi_{0}}^{V/H}(t+\Delta t) =\sum_{n=1}^{N}e^{\left(\varphi_{n}\
u^{\prime}(l^{\prime}k^{ \prime}-1).d_{V/H}^{\text{J}E}\left\|\varphi\right\|\sin(\vartheta)_{AAoA}\right)} \tag{48}\]According to the above procedure of channel modeling, generating the realistic channel coefficient to evaluate the three aforementioned scenarios V, H and V/H polarized antenna elements in (11.1)-(11.3) can be extend to Equations (49)-(51) when DoT = 0\({}^{\circ}\) and Equations (52)-(54) when DoT = 90\({}^{\circ}\). \[h_{lk,Pk^{\prime},n}^{V}(t+\Delta t,\tau)=\left( \begin{array}{l} \delta(n-1)\sqrt{\frac{\mathbb{C}}{\mathbb{C}+\uparrow}}D_{DOS_{i},lk}^{BS}(t+ \Delta t)\times\bar{a}(\phi,\theta)_{lk}\times\begin{bmatrix}\exp(j\Phi_{ LOS}^{p,D})&0\\ 0&\exp(j\Phi_{LOS}^{h,h})\end{bmatrix}\times D_{DOS_{i},Pk^{\prime}}^{IE}(t+ \Delta t)\\ \times\exp\left(j\mathfrak{s}_{Pk^{\prime}}(l^{\prime}k^{\prime}-1).A_{V}^{UE} \parallel v\parallel\Delta t^{\prime}\cos(\theta)_{AAA}-(\theta)_{v}\right)+ \sqrt{\frac{A_{v}}{\mathbb{C}+\uparrow}}\sum_{i=1}^{N}D_{n,i,k}^{BS}(t+ \Delta t)\times\bar{a}(\phi,\theta)_{lk}\times\\ \begin{bmatrix}\exp(j\Phi_{n}^{(v,D)})&\sqrt{\lambda^{k}}\exp(j\Phi_{n}^{(v,h)} )\\ \sqrt{\lambda^{k}}\exp(j\Phi_{n}^{(v,h)})&\exp(j\Phi_{n}^{(v,h)})\\ \exp\left(j\mathfrak{s}_{Pk^{\prime}}(l^{\prime}k^{\prime}-1).A_{V}^{UE} \parallel v\parallel\Delta t^{\prime}\cos(\theta)_{AAA}-(\theta)_{v}\right)\end{bmatrix} \times D_{DOS_{i},Pk^{\prime}}^{UE}(t+\Delta t)\\ \end{array} \right) \tag{49}\]  
\[h_{lk,Pk^{\prime},n}^{H}(t+\Delta t,\tau)=\left(