## 2 System Model of NGSO Satellites Equipped with Multi-Beam Antennas
### Channel Allocation Method  
During the working life of satellite communication constellations, the spectrum resources used to carry information may unavoidably be idle in time and space. To make full utilization of the spectrum resources, channel allocation among different beams is an effective method. Based on the analysis above, we present a channel allocation method to mitigate the interference caused by NGSO satellites to ground stations in GSO systems. The precondition of proposed channel allocation is that the utilization situation of specific channels should be known to both systems, which means the way of working is cooperative. Given the initial orbital elements of a targeted satellite in the J2000 Earth centered inertial frame, the position and velocity vectors of the satellite in this frame can be obtained through orbit propagation. For a precise simulation instant, the transformation matrix between different coordinates can be achieved through classical orbital theory [35]. Based on the satellite position vectors, we can calculate the off-axis angles of interfering antenna beam \(\theta_{1}\) and that of the antenna of interfered receiver \(\theta_{2}\). As defined previously, we have a primary and a secondary systems in a scenario of interference. For the primary system with the GSO communication satellite and ground stations, the channel allocation method stays unaltered all along, while with the continuous motion of the satellite in the secondary system, the channel allocation method should be reconsidered at each time step. An illustration of 37 beams created by NGSO satellites is shown in Figure 8a. For ease of understanding, we mark the beams that work in the same channel by the same color. As shown in Figure 8b, seven beams are marked by seven colors, and they represent seven channels used. Figure 7: Downlink scenario where pointing of NGSO satellite beams have different tilting angles and relationship between off-axis angle and beam distance \(l\) with different tilting angles. **(a)** Pointing of NGSO satellite beams with different tilting angles; **(b)** Relationship between variation of off-axis angle and beam distance \(l\) when beams have different tilting angles. According to the analysis carried out above, we can see that if two beams that work in the same channel are too close to each other, severe interference would take place between the two communication systems. To avoid this situation, before employing channels to certain beams, those beams that are allocated to the same channel and will get too close to their co-channel beams should be excluded. This operation limits the intensity of interference to avoid a remarkable decline of the SINR, while it reduces the number of available channels for each beam and consequently diminishes the computational complexity. However, the minimum beam distance should not be too small, because there may be no available channel allocated to particular beams in that condition. Denote the channel allocation matrix by \(A\). It identifies which channel is employed by each beam, whose element \(a_{ij}\in\{0,1\}\) indicates whether the \(i\)th beam is allocated with the \(j\)th channel. \(l_{a}\) denotes the actual distance between the target beam and the beams that work in the identical channel. Note that, for beams with different tilting angles, perhaps the same distance matches along with different off-axis angles. As we hope to use beam distance to reflect the value of the off-axis angle, in order to treat each distance equally, which means the same distance can stand for the same off-axis angle, the revised distance \(l_{r}\) is added to the original data. The revised distance for different beams can be calculated through a similar analysis as shown in Figure 7b. Here, we only consider the revised distance for beams of NGSO satellites based on their tilting angles with respect to the center beam. The whole configuration of the beams can be optimized. The complete coverage of the HTS is an equilateral hexagon comprised of a certain number of beams.