## 4 Simulation Results  
In this section, the performance of the UD-LSC system to cooperatively serve multiple LBCs while sharing spectrum with the GEO satellite communication system is simulated by MATLAB. In the simulation scenario, there are 19 LBCs and seven GBCs on the ground, as shown in Figure 1. Actually, the footprint of satellites on the Earth's surface is not circular, especially the beam far away from the center of the satellite's coverage area. The shape of the footprint and different EIRPs are given parameters to calculate the interfered link and the interfering link; then, the SINR of each link can be further calculated. It has an effect on the calculation result of SINR but has no effect on the implementation of the algorithm. Each LEO satellite forms 7 beams to serve ground cells and cooperate with other LEO satellites. The FFR scheme for multiple GEO and LEO beams is assumed to pursue high-spectrum efficiency. The GEO satellite system only shares the beam-hopping parameters with the LEO satellite system, so that the wide coverage of the GEO satellite has little impact with the cooperation with LEO satellites. The uneven traffic demands of different LBCs are assumed. The main parameters of the two systems are listed in Table 1, referring to 3 GPP TR 38.821 [17]. To verify the effectiveness of the proposed MGBA-SPA scheme solving the joint beam association and power allocation problem, as a contrast, we give four baseline schemes as follows. * Match game-based beam association with uniform power allocation (MGBA-UPA). In the MGBA-UPA scheme, the proposed Algorithm 1 is adopted to finish the beam association. Then, a uniform power allocation between \(L\) LEO beams of each LEO satellite is used. * Match game-based beam association with traffic-based power allocation (MGBA-TPA). In the MGBA-TPA scheme, after Algorithm 1 is implemented, the transmit power is allocated as \(P_{k,m}=P_{s,max}\cdot\left(T(m)/T_{total}\right)\). * Random beam association with uniform power allocation (RBA-UPA). In the RBA-UPA scheme, the LBCs and LEO satellites are associated randomly. Then, the UPA method is adopted. * Random beam association with traffic-based power allocation (RBA-TPA). In the RBA-TPA scheme, the random association between LEOs and LBCs is adopted. Then, the transmit power of each LEO satellite is allocated based on the traffic request of each beam cell. Figures 3-5 simulate the sum of satisfactory performance of the proposed MGBA-SPA scheme and four related baseline schemes. The sum of satisfaction can be calculated as \(\Phi_{sum}=\sum_{m=1}^{M}\Phi_{(}m)\). To evaluate the cooperation between LEO satellites, a performance comparison against the number of LEO satellites is simulated in Figure 3, where the number of active beams of the GEO system is three and the average traffic per LBC is 0.5 Gbps. Simulation results show that the proposed MGBA-SPA scheme outperforms other baseline schemes with a different number of cooperative satellites. As the number of cooperative satellites increases, the performance of the MGBA-SPA scheme tends to stabilize, \(\Phi_{sum}\approx 16\), since \(four\) LEO satellites can bring enough degrees of freedom in the angle domain to mitigate the inter-system interference and inter-beam interference. As for four baseline schemes, the sum of satisfaction is improved by consuming more power, which comes from more LEO satellites. In addition, the match game-based beam association scheme obviously has better performance than the random scheme.