## 2 System Model of NGSO Satellites Equipped with Multi-Beam Antennas
### Interference Evaluation Index and Antenna Radiation Patterns  
Interference evaluation indexes are utilized to indicate how seriously the target system is interfered by other systems. We choose to use different evaluation indexes in different situations. Commonly used indexes consist of the carrier to interference ratio \(C/I\), interference to noise ratio \(I/N\), signal to noise ratio SNR, and signal to interference plus noise ratio SINR, among other indexes. Based on the aforementioned equations, the four indexes can be calculated by the following equations. \[\frac{C}{I}=\frac{P_{\text{tw}}G_{\text{tw}}(\theta_{1})G_{\text{r}}(\theta_{2 })(\lambda_{1}/d_{\text{w}})^{2}}{F_{\text{R}}P_{\text{ti}}G_{\text{t}}\left( \overline{\theta}_{1}\right)G_{\text{r}}\left(\overline{\theta}_{2}\right)( \lambda_{2}/d_{\text{i}})^{2}} \tag{5}\]  
\[\frac{I}{N}=\frac{F_{\text{R}}P_{\text{ti}}G_{\text{t}}\left(\overline{\theta} _{1}\right)G_{\text{r}}\left(\overline{\theta}_{2}\right)\left(\frac{\lambda _{2}}{4\pi d_{\text{i}}}\right)^{2}}{\kappa TW} \tag{6}\]  
\[\text{SNR}=\frac{P_{\text{tw}}G_{\text{tw}}(\theta_{1})G_{\text{r}}(\theta_{ 2})}{\kappa TW}\left(\frac{\lambda_{1}}{4\pi d_{\text{w}}}\right)^{2} \tag{7}\]  
\[\text{SINR}=\frac{P_{\text{tw}}G_{\text{tw}}(\theta_{1})G_{\text{r}}(\theta_{ 2})(\lambda_{1}/4\pi d_{\text{w}})^{2}}{F_{\text{R}}P_{\text{ti}}G_{\text{ti} }\left(\overline{\theta}_{1}\right)G_{\text{r}}\left(\overline{\theta}_{2} \right)(\lambda_{2}/4\pi d_{\text{i}})^{2}+\kappa TW} \tag{8}\]  
where \(\lambda_{1}\) and \(\lambda_{2}\) are wavelengths of the wanted and interfering satellites, \(T\) is the receiver's noise temperature, and \(\kappa\) and \(W\) represent the Boltzmann constant and communication link bandwidth, respectively. Here we choose SNR and SINR to illustrate how well our target systems perform with and without interference from other satellites. The index SINR signifies very clearly the system performance of a single link. As mentioned previously, the HTS is equipped with multi-beam antennas, which means multiple links are built in just one satellite. To take the performance of all the links into consideration and evaluate how much information a satellite can transmit, the capacity \(R\) are presented as follows on the basis of the SINR of each link. Supposing that Gaussian coding is adopted in the communication systems, the capacity \(R\) for the total number of links in a satellite can be calculated as  
\[R=\sum_{i=1}^{N_{\text{t}}}W_{i}\cdot\log_{2}(1+\text{ SINR }_{i}) \tag{9}\]  
Figure 2: Utilization of overlaped frequency bands. where \(N_{i}\) is the total number of links in a satellite, and \(W_{i}\) and \(\text{SINR}_{i}\) are the bandwidth and SINR of the \(i\)th link, respectively. This expression is a generalization of Shannon-Hartley theorem, and it provides the theoretical maximum for a link capacity. Due to the keep-moving characteristic of the NGSO satellites, the off-axis angles of the transmitter and receiver vary continuously. We make use of the radiation patterns to obtain the gain of different antennas based on the off-axis angles at a specific time. The ITU has recommended several patterns for different scenarios. The radiation patterns described in the following are chosen for simulations in the later section. According to ITU-R S.1528 [32], the reference radiation pattern for low Earth orbit satellite antenna, whose antenna aperture diameter to wavelength ratio \((D/\lambda)<35\), is given by  
\[G(\psi)=\left\{ \begin{array}{ll}G_{m}&\text{dBi}&0\leq\psi<\psi_{b}\\ G_{m}-3(\psi/\psi_{b})^{2}&\text{dBi}&\psi_{b}\leq\psi<Y\\ G_{m}+L_{s}-25\text{log}(\psi/Y)&\text{dBi}&Y\leq\psi<Z\\ L_{F}&\text{dBi}&Z\leq\psi<180^{\circ}\end{array} \right.