## 2 System Model of NGSO Satellites Equipped with Multi-Beam Antennas
### Analysis of Interference Based on Off-Axis Angle and Beam Distance  
Here it is assumed that the interfered ground stations locate at the center of each beam of the corresponding communication satellite. Taking the GSO satellite communication system as an interfered instance, the illustrated scenarios are depicted in Figures 4 and 5. According to the interference evaluation index given in Equation (8), the augmentation of interference \(I\) would result in the decline of SINR. To increase the SINR level of ground stations, it is apparent that the signal intensity from the interfering satellites should be as weak as possible, which means that two off-axis angles of the interfering link should be as large as possible. In addition, the off-axis angles \(\theta_{1}\) and \(\theta_{2}\) are closely related to the distance \(l\) between different beam centers as illustrated in Figure 4. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Downlink carrier frequency, \(f\) & 20 GHz \\
**GSO satellite** & \\ GSO maximum transmit gain, \(G_{\text{mG}}\) & 45.39 dBi \\ GSO 3 dB beamwidth & 1.259\({}^{\circ}\) \\ Radio of major axis to minor axis, \(z\) & 1.5 \\ near-in-side-lobe level, \(L_{N}\) & \(-20\) dB \\
**NGSO satellite** & \\ NGSO maximum transmit gain, \(G_{\text{mL}}\) & 37.65 dBi \\ NGSO 3 dB beamwidth & 11.58\({}^{\circ}\) \\
**GSO ground station** & \\ Antenna diameter, \(D\) & 1.2 m \\ Maximum receive gain, \(G_{\text{mr}}\) & 45.76 dBi \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters of antenna gains. Figure 3: Variation of gain of antennas with off-axis angle for different antenna radiation patterns. From Equation (2), one can find that the denominator depends only on the communication link distance \(d_{\rm i}\). Considering the period of time when the NGSO satellite first approaches and then moves away from the GSO ground station, the variation of link distance \(\Delta d_{\rm i}\) between the NGSO satellite and GSO ground station appear slightly different, which could be ignored compared with more significant variation of the off-axis angles between the satellite and ground station antennas. Therefore, theoretically, the variations of the link distance and off-axis angle are coupled, but the link distance has only a negligible influence on the interference \(I\). Here, we assume that the path loss stays constant during the observation period. Let  
\[F_{\rm u}=G_{\rm ii}(\theta_{1})\ {\rm dBi}\ +G_{\rm r}(\theta_{2})\ {\rm dBi} \tag{13}\]  
This parameter largely represents the change of interference \(I\). Let the distance between the beam centers of the NGSO and GSO be \(l\). It should be noted that the distance \(l\) is measured over the surface of the Earth, and thereupon we obtain the geocentric angle \(\theta_{0}=1/R_{\rm e}\). In the light of the law of cosines, the link distance between the NGSO satellite and GSO ground station can be calculated as  
\[d_{\rm i}=\sqrt{R_{\rm e}^{2}+(R_{\rm e}+h)^{2}-2\rm cos(\theta_{0})R_{\rm e}( R_{\rm e}+h)} \tag{14}\]  
where \(R_{\rm e}\) is the Earth's equatorial radius and \(h\) is the orbital altitude of interfering satellite. Then we find the off-axis angle \(\theta_{1}\)  
\[\theta_{1}=\arcsin\left[\frac{R_{\rm e}\cdot\sin(\theta_{0})}{\sqrt{R_{\rm e }^{2}+(R_{\rm e}+h)^{2}-2\rm cos(\theta_{0})R_{\rm e}(R_{\rm e}+h)}}\right] \tag{15}\]  
Through the relationship \(\theta_{2}=\theta_{0}+\theta_{1}\), the expression of \(\theta_{2}\) could be easily obtained  
\[\theta_{2}=\theta_{0}+\arcsin\left[\frac{R_{\rm e}\cdot\sin(\theta_{0})}{ \sqrt{R_{\rm e}^{2}+(R_{\rm e}+h)^{2}-2\rm cos(\theta_{0})R_{\rm e}(R_{\rm e}+ h)}}\right] \tag{16}\]  
Figure 4: The downlink scenarios where beams have no tilting angle and only NGSO satellite beam have a tilting angle. (**a**) Pointing of NGSO and GSO satellites beams without tilting angle; (**b**) Pointing of NGSO satellite beams with tilting angle \(\beta_{1}\).