# Practical 3-D Beam Pattern Based Channel Modeling for Multi-Polarized Massive MIMO Systems +
## 4 Received Spatial Correlation
\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}}^{UE}(t+ \Delta t)\\ \times\exp\left(j\mathfrak{s}_{Pk^{\prime}}(l^{\prime}k^{\prime}-1).A_{H}^{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\left(j\mathfrak{s}_{Pk^{\prime}}(l^{\prime}k^{\prime}-1).4_{U}^{UE} \parallel v\parallel\Delta t^{\prime}\cos(\theta)_{AAA}-(\theta)_{v}\right)\end{bmatrix} \times D_{h,j,Pk^{\prime}}^{UE}(t+\Delta t)\times\\ \end{array} \right)\end{array}\right) \tag{50}\]  
\[h_{lk,Pk^{\prime},n}^{V/H}(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}}^{UE}(t+ \Delta t)\\ \times\exp\left(j\mathfrak{s}_{Pk^{\prime}}(l^{\prime}k^{\prime}-1).A_{V/H}^{UE }\parallel v\parallel\Delta t^{\prime}\cos(\theta)_{AAA}-(\theta,\theta)_{v} \right)+\sqrt{\frac{A_{v}}{\mathbb{C}+\uparrow}}\sum_{n=1}^{N}D_{n,i,k}^{BS} (t+\Delta t)\times\bar{a}(\phi,\theta)_{lk}\times\\ \begin{bmatrix}\exp(j\[PERSON]_{n}^{(v,D)})&\sqrt{\lambda^{k}}\exp(j\[PERSON]_{n}^{(v,h)} )\\ \sqrt{\lambda^{k}}\exp(j\Phi_{n}^{(v,h)})&\exp(j\Phi_{n}^{(v,h)})\\ \exp\left(j\mathfrak{s}_{Pk^{\prime}}(l^{\prime}k^{\prime}-1).4_{U/H}^{UE} \parallel v\parallel\Delta t^{\prime}\cos(\theta)_{AAA}-(\theta)_{v}\right)\end{bmatrix} \times D_{h,j,Pk^{\prime}}^{UE}(t+\Delta t)\times\\ \end{array} \right)\end{array}\right) \tag{51}\]  
Similarly, when DoT = 90\({}^{\circ}\), the realistic channel coefficients are presented as  
\[h_{lk,Pk^{\prime},n}^{V}(t+\Delta t,\tau)=\left( \begin{array}{l}\delta(n-1) \sqrt{\frac{\mathbb{C}}{\mathbb{C}+\uparrow}}D_{DOS_{i},lk}^{BS}(t+\Delta t) \times\bar{a}(\phi,\theta)_{lk}\times\begin{bmatrix}\exp(j\Phi_{ LOS}^{p,D})&0\\ 0&\exp(j\Phi_{LOS}^{h,h})\end{bmatrix}\times D_{DOS_{i},Pk^{\prime}}^{UE}(t+ \Delta t)\\ \times\exp\left(j\mathfrak{s}_{Pk^{\prime}}(l^{\prime}k^{\prime}-1).A_{V}^{UE} \parallel v\parallel\sin(\theta)_{AAA}\right)+\sqrt{\frac{A_{v}}{\mathbb{C}+ \uparrow}}\sum_{n=1}^{N}D_{n,i,k}^{BS}(t+\Delta t)\times\bar{a}(\phi,\theta)_{lk} \times\\ \begin{bmatrix}\exp(j\Phi_{n}^{(v,D)})&\sqrt{\lambda^{k}}\exp(j\Phi_{n}^{(v,h)} )\\ \sqrt{\lambda^{k}}\exp(j\Phi_{n}^{(v,h)})&\exp(j\Phi_{n}^{(v,h)})\end{bmatrix} \times D_{n,i,j,Pk^{\prime}}^{UE}(t+\Delta t)\times\exp\left(j\mathfrak{s}_{Pk^{ \prime}}(l^{\prime}k^{\prime}-1).A_{V}^{UE}\parallel v\parallel\sin(\theta)_{AAA }\right)\end{bmatrix}\right) \tag{52}\]  
\[h_{lk,Pk^{\prime},n}^{H}(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}^{UE}(t+\Delta t)\\ \times\exp\left(j\mathfrak{s}_{Pk^{\prime}}(l^{\prime}k^{\prime}-1).A_{H}^{UE} \parallel v\parallel\sin(\theta)_{AAA}\right)+\sqrt{\frac{A_{v}}{\mathbb{C}+ \uparrow}}\sum_{n=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)})\end{bmatrix} \times D_{n,i,j,Pk^{\prime}}^{UE}(t+\Delta t)\times\exp\left(j\mathfrak{s}_{Pk^{ \prime}}(l^{\prime}k^{\prime}-1).A_{H}^{UE}\parallel v\parallel\sin(\theta)_{AAA }\right)\end{array}\right)\end{array}