## 2 Scenario Description
### Antenna Parameters
#### 2.2.2 UT Antenna and Array Model  
Once all UTs locations are generated, each UT is given a random array orientation. The orientation of an array with respect to the GCS is defined by a set of three angles \((\alpha,\beta,\) and \(\gamma)\) that fully describe an arbitrary 3-D rotation, where \(\alpha\) represents the bearing angle, \(\beta\) the downtilt angle, and \(\gamma\) the slant angle. The following distributions are assumed:  
\[ \begin{array}{l}\alpha\sim\mathcal{U}(-180^{\circ},+180^{\circ})\\ \beta\sim\mathcal{N}(\mu_{\beta},\sigma_{\beta}^{2})\\ \gamma\sim\mathcal{N}(\mu_{\gamma},\sigma_{\gamma}^{2})\end{array}  \tag{19}\]  
where \(N\) denotes a normal distribution with \(\mu_{\beta}=\mu_{\gamma}=0\) and \(\sigma_{\beta}^{2}=\sigma_{\gamma}^{2}=36^{\circ}\). We assume that each UT can also be equipped with a Uniform Planar Array (UPA) with \(N_{\tau z}\) antennas along the \(z\)-axis and \(N_{\tau y}\) along the \(y\)-axis when \(\alpha=\beta=\gamma=0^{\circ}\), i.e., \(N_{\tau}=N_{\tau z}\times N_{\tau y}\) is also the number of antenna with the same polarization. Antenna elements have the same \(\lambda/2\) spacing of the BS along both directions and can be single or dual polarized. Given a Zenith of Departure (ZoA) \(\theta\) and an Azimuth of Departure (\(AoA\)) \(\phi\) in the GCS, we need first to convert into LCS through the following relationship:  
\[\theta^{\prime}(\alpha,\beta,\gamma;\theta,\phi)=\mathrm{acos}(\cos\beta\cos \gamma\cos\theta+(\sin\beta\cos\gamma\cos(\phi-\alpha)-\sin\gamma\sin(\phi- \alpha))sin\theta) \tag{20}\]  
\[\begin{array}{l}\phi^{\prime}(\alpha,\beta,\gamma;\theta,\phi)=\\ =\arg\biggl{(} \begin{array}{c}(\cos\beta\sin\theta\cos(\phi-\alpha)-\sin \beta\cos\theta)+\\ +j(\cos\beta\sin\gamma\cos\theta+(\sin\beta\sin\gamma\cos(\phi-\alpha))+\cos \gamma\sin(\phi-\alpha))\sin\theta)\end{array}  \tag{21}\]  
the \(N_{r}\times 1\) SV for the UPA at the UT can be expressed as:  
\[\mathbf{a}_{rx}(\theta^{\prime},\phi^{\prime})=\mathbf{a}_{z}(\theta^{\prime} )\otimes\mathbf{a}_{y}(\theta^{\prime},\phi^{\prime}) \tag{22}\]  
where \(\mathbf{a}_{z}(\theta^{\prime})\) and \(\mathbf{a}_{y}(\theta^{\prime},\phi^{\prime})\) are computed according to (10) and (11) with \(\theta^{\prime}\) computed as in (20) and \(\phi^{\prime}\) as in (21). We assume that the antenna elements of the array of each UT are half-wave dipoles that are aligned along the \(z\)-axis when \(\alpha=\beta=\gamma=0\). The radiation power pattern can be expressed as:  
Figure 3: 3 GPP antenna element radiation pattern. \[B^{\prime}(\theta^{\prime})=\frac{\cos^{2}\left(\frac{\pi}{2}\cos\theta^{\prime} \right)}{\sin^{2}\theta^{\prime}} \tag{23}\]  
\[G_{E,max}=5\quad\text{dBi}. \tag{24}\]  
Due to the fact that the UT array experiences a random 3-D rotation, we need to relate the polarized field components \(F_{rx,\theta}(\theta)\), \(F_{rx,\phi}(\theta)\) in the GCS, with the components \(F_{rx,\theta^{\prime}}(\theta^{\prime})\), \(F_{rx,\phi^{\prime}}(\theta^{\prime})\) in the LCS through:  
\[ \begin{pmatrix}F_{rx,\theta}(\theta,\phi)\\ F_{rx,\phi}(\theta,\phi)\end{pmatrix} = \begin{pmatrix}+cos\psi&-sin\psi\\ +sin\psi&+cos\psi\end{pmatrix}\begin{pmatrix}F^{\prime}_{rx,\theta^{\prime}}( \theta^{\prime})\\ F_{rx,\phi^{\prime}}(\theta^{\prime})\end{pmatrix}  \tag{25}\]  
where the angle \(\psi\) can be computed as:  
\[\psi=\arg\biggl{(} \begin{pmatrix}\sin\gamma\cos\theta\sin(\phi-\alpha)+\cos \gamma(\cos\beta\sin\theta-\sin\beta\cos\theta\cos(\phi-\alpha))\\ +j(\sin\gamma\cos(\phi-\alpha)+\sin\beta\cos\gamma\sin(\phi-\alpha))\end{pmatrix}  \biggr{)}. \tag{26}\]  
The same \"Model-2\" of the polarized antenna modelling of [21] is applied for the UT. In case of single polarized antenna (purely vertically polarized antenna) we have:  
\[ \begin{array}{l}F^{\prime}_{rx,\theta^{\prime}}(\theta^{\prime})=\sqrt{B^{ \prime}(\theta^{\prime})}\\ F^{\prime}_{rx,\phi^{\prime}}(\theta^{\prime})=0\end{array}  \tag{27}\]  
while for the case of dual cross-polarization  
\[