## 2 System Model of NGSO Satellites Equipped with Multi-Beam Antennas  
### Downlink Analysis Model  
An interference scenario between two satellite communication systems with multi-beam antennas is illustrated in Figure 1. As the interference analysis of the uplink is quite similar to that of the downlink, here we take the downlink scenario as an example. For any satellite ground station or user terminal in the downlink scenarios, the signal intensity \(C\) that it receives from the corresponding satellite can be formulated as  
\[C=\frac{P_{\text{tw}}G_{\text{tw}}(\theta_{1})G_{\text{r}}(\theta_{2})}{\left( 4\pi d_{\text{w}}/\lambda\right)^{2}} \tag{1}\]  
where \(P_{\text{tw}}\) is the transmit power of the wanted satellite antenna, \(G_{\text{tw}}(\cdot)\) and \(G_{\text{r}}(\cdot)\) denote the gains of the wanted satellite antenna and ground station receiver, respectively, \(\theta_{1}\) and \(\theta_{2}\) are the off-axis angles between the main lobe axes and the lines of communication links, \(d_{\text{w}}\) is the distance between the wanted satellite and the ground station, and the wavelength of carrier signal \(\lambda\) is related to the light speed \(c\) and frequency \(f\) by the relationship \(c=\lambda\cdot f\). In most common cases, the ground station always keeps pointing to its corresponding satellite so as to achieve the maximum gain of receiving antenna. Thus, \(\theta_{2}\) is usually set to zero when calculating the wanted signal intensity \(C\). For any ground station on the Earth, it could receive signals that are sent from other satellites and carried in the identical frequency band rather than the wanted one. These signals would mainly contribute to the augmentation of the interference signal level. In Figure 1, let us assume that the interfered GSO ground station \(P\) is in the intersecting area of the coverage areas of the two beams. The downlink interference signal intensity \(I\) received by the ground station from a single interfering NGSO satellite antenna beam can be calculated as  
\[I=F_{\mathrm{R}}\frac{P_{\mathrm{H}}G_{\mathrm{H}}(\tilde{\theta}_{1})G_{ \mathrm{r}}(\tilde{\theta}_{2})}{\left(4\pi\mathrm{d}_{\mathrm{i}}/\lambda \right)^{2}} \tag{2}\]  
where \(F_{\mathrm{R}}\) is the frequency reuse factor, ranging from 0 to 1, depending on the carrier frequency the wanted and interfering satellites make use of. \(P_{\mathrm{H}}\) is the transmit power of the interfering satellite antenna, \(G_{\mathrm{H}}(\cdot)\) is the gain of interfering satellite antenna, \(d_{\mathrm{i}}\) is the distance between the interfering satellite and the ground station, and \(\tilde{\theta}_{1}\) and \(\tilde{\theta}_{2}\) represent the off-axis angles. As shown in Figure 2, assuming that \(W_{1}\) and \(W_{2}\) are the frequency bandwidths used by the wanted and interfering systems, respectively, and that \(W_{\mathrm{overlap}}\) represents the overlap bandwidth between the two frequency bands, the frequency reuse factor can be expressed as  
\[F_{R}=\frac{W_{\mathrm{overlap}}}{W_{2}} \tag{3}\]  
Nevertheless, in reality, the interference origin can hardly be onefold. With a large number of communication satellites launched into free space, the interfering objects for ground stations are usually multiple. Additionally, plenty of NGSO satellites are equipped with multi-beam antennas. Taking these factors into consideration, for a GSO ground station, the interference signal it receives possibly originates from various NGSO satellites  
Figure 1: An illustration of the downlink interference scenario. or various beams in a single satellite. It is therefore essential to take all these interference sources into account.