## 2 System Model and Problem Formulation
### Antenna Model  
In this paper, planar URA is employed for the satellite and the UT. According to 3 GPP TR 37.840 [22], the LEO satellite and UT antenna pattern can be derived. The composite array radiation pattern of the satellite is expressed as follows:  
\[G_{t}^{\mathcal{S}}=G_{E}(\theta_{t}^{\mathcal{S}},\varphi_{t}^{\mathcal{S}})+ 10\log_{10}(|\tilde{W}_{t}(\theta_{t}^{\mathcal{S}},\varphi_{t}^{\mathcal{S}}, \varphi_{t}^{\mathcal{S}})|^{2}), \tag{1}\]  
where \(G_{t}^{\mathcal{S}}\) is the satellite antenna gain at the transmitter, which is composed of single element gain \(G_{E}(\theta_{t}^{\mathcal{S}},\varphi_{t}^{\mathcal{S}})\) and array gain \(10\log_{10}(|\tilde{W}_{t}(\theta_{t}^{\mathcal{S}},\varphi_{t}^{\mathcal{S}}, \widetilde{\theta}_{t}^{\mathcal{S}},\widetilde{\varphi}_{t}^{\mathcal{S}})|^ {2}),\theta_{t}^{\mathcal{S}}\) and \(\varphi_{t}^{\mathcal{S}}\) are the elevation and azimuth angle of AoD, \(\widetilde{\theta}_{t}^{\mathcal{S}}\) and \(\varphi_{t}^{\mathcal{S}}\) denote the electrical down-tilt steering  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Definition** \\ \hline \(G_{t}^{\mathcal{S}},G_{t}^{\mathcal{S}}\) & Antenna gain of transmitter and receiver \\ \(\theta_{t}^{\mathcal{S}},\varphi_{t}^{\mathcal{S}}\) & Elevation and azimuth angles of AoD \\ \(\theta_{t}^{\mathcal{S}},\varphi_{t}^{\mathcal{S}}\) & Beam center pointing of the satellite antenna \\ \(\theta_{t}^{\mathcal{S}},\varphi_{t}^{\mathcal{S}}\) & Elevation and azimuth angles of AoA \\ \(\hat{\mathbf{\sigma}}_{t}^{\mathcal{C}u}=[\theta_{t}^{\mathcal{C}u},\varphi_{t}^{ \mathcal{C}u}]\) & Beam center pointing of UT antenna \\ \(\theta_{t}^{\mathcal{S}},\varphi_{t}^{\mathcal{S}}\) & Elevation and azimuth angles of AoA of communication link \\ \(\Theta_{3 dB},\Phi_{3 dB}\) & Horizontal and vertical 3 dB beamwidth of the UT antenna \\ \(N_{\text{tr}},N_{\text{ty}}\) & Number of arrays placed along x-axis and y-axis of the satellite antenna \\ \(d_{x},d_{y}\) & Separation of two array elements on x-axis and y-axis of the satellite antenna \\ \(N_{\text{rr}},N_{\text{ry}}\) & Number of array elements placed along x-axis and y-axis of the UT antenna \\ \(d_{rx},d_{ry}\) & Separation of two array elements on x-axis and y-axis of the UT antenna \\ \([\theta_{t}^{\mathcal{S}},\varphi_{t}^{\mathcal{S}}],[\theta_{t}^{\mathcal{S}}, \varphi_{t}^{\mathcal{S}}]\) & AoD and AoA of communication link \\ \([\theta_{t}^{\mathcal{U}},\varphi_{t}^{\mathcal{U}}][\theta_{t}^{\mathcal{U}}, \varphi_{t}^{\mathcal{U}}]\) & AoD and AoA of \(d\)th interfering link from \(q\)th interfering satellite \\ \([\hat{\theta}_{t}^{\mathcal{S}},\hat{\varphi}_{t}^{\mathcal{S}}]\) & Beam center pointing of serving satellite antenna \\ \(\hat{\mathbf{\sigma}}_{t}^{\mathcal{C}u}=[\theta_{t}^{\mathcal{C}u},\varphi_{t}^{ \mathcal{C}u}]\) & Optimized beam center pointing of UT antenna \\ \hline \hline \end{tabular}
\end{table}
Table 1: Symbol list.  
and the electrical horizontal steering, which represent the beam center pointing of the satellite antenna.  
Similarly, the antenna model of the UT is as follows:  
\[G_{r}^{S}=G_{E}(\theta_{r}^{S},\varphi_{r}^{S})+10\log_{10}(|\tilde{W}_{r}(\theta_ {r}^{S},\varphi_{r}^{S},\tilde{\theta}_{r}^{C_{U}},\tilde{\theta}_{r}^{C_{U}})|^ {2}), \tag{2}\]  
where \(G_{r}^{S}\) is the UT antenna receiving gain, \(\theta_{r}^{S}\) and \(\varphi_{r}^{S}\) are the elevation and azimuth angle of AoA, \(\tilde{\theta}_{r}^{C_{U}}\) and \(\tilde{\varphi}_{r}^{C_{U}}\) express the electrical down-tilt steering and the electrical horizontal steering, which denote the beam center pointing of UT antenna.