## 3 System Model and Problem Formulation
### Interference Model
#### 3.4.2 Interference from LEO Beams to GEO Ground Stations  
Countries around the world have gradually recognized the strategic significance and important values of satellite frequency and orbit resources in the development of space economy, and declared a large amount of satellite network applications to ITU to seize the frequency and orbit resources. At present, emerging NGSO communication constellation systems represented by O3b, Starlink, OneWeb, Telesat, Kuiper, etc., plan to use very concentrated frequencies, mainly Ku/Ka/Q/V. Traditional Geostationary Satellite Orbit (GSO) satellite communication systems also mainly use Ku/Ka frequency band resources, and a large number of systems are already operating in orbit [38]. According to Article 22 of the ITU \"Radio Regulations\" [39], when using the Ku/Ka frequency bands, the communication systems of the NGSO fixed-satellite service and the satellite broadcasting service shall not cause unacceptable interference to the GSO satellite communication system, and shall not seek the protection of the GSO satellite communication system.  
The interference power from LEO spotheam serving cell \(i\) to a GEO earth station \(e\) is denoted by  
\[I_{i,e}=\delta_{i,e}P_{i}G_{l}^{i}(\theta_{i,e})G_{l}^{S}(\theta_{e})/L_{e}^{l} \tag{10}\]  
where \(\delta_{i,e}\in\{0,1\}\) is a binary variable that indicates whether the frequency band occupied by the LEO spot beam and the GEO ground station overlaps, and \(\delta_{i,e}=1\) means there is overlap, otherwise \(\delta_{i,e}=0\). \(P_{i}\) represents the transmitting power of the LEO spotheam \(i\), \(G_{l}^{i}(\theta_{i,e})\) is the transmitting antenna gain of the LEO satellite, and \(G_{l}^{S}(\theta_{e})\) denotes the receiving antenna gain of the GEO earth station.  
To simplify the model, we only consider the free space loss in the transmission loss, which is \(L_{e}^{l}=4\pi d_{e}^{l}/\lambda\), where \(d_{e}^{l}\) is the distance between LEO satellite and the GEO earth station and \(\lambda\) is the carrier wavelength.  
The interference threshold of the LEO satellite beam to the GEO ground station is set as \(I_{th}\). On the one hand, when \(I_{i,e}>I_{th}\), it is considered that the LEO beam produces unacceptable interference to the ground station, and the beam should be turned off at this time, and other LEO satellites should be selected to serve the cell in the case of multiple coverage. On the other hand, the protection of GEO ground stations limits the maximum power transmitted in each subband.  
Figure 4: Illustration diagram of interference coefficient calculation.  
More specifically, for each sub-band \(\beta_{m}\), it corresponds to a set of ground stations that share the same frequency band, which is denoted as \(E_{m}=\left\{e_{1},e_{2},\cdots e_{\xi}\right\}\). For a certain cell (beam), the off-axis angles of the ground stations in the set \(E\) regarding to the beam are different, but they all meet the same threshold \(I_{th}\). According to Equation (10), the maximum transmitting power of the beam with regard to each ground station can be obtained, and the set of them is denoted as \(P_{i}^{\beta m}=\left\{P_{1},P_{2},\cdots P_{\xi}\right\}\), in which the minimum power is the corresponding threshold transmitting power when the beam \(i\) uses the subband \(\beta_{m}\), that is \(P{th}_{i}^{\beta m}=\min P_{i}^{\beta m}\). In this paper, we define this constraint relationship as the Constraint Map of the cell, which is shown in Figure 5.