## 4 Numerical Results and Discussion
### Uplink Coverage Probability  
Figures 12 and 13 show the uplink coverage probability for different target SINR in NTN rural and NTN urban scenarios. The simulations presented in Figures 12 and 13 show the proposed scheme's effectiveness in maximizing uplink coverage probability. The results demonstrate that the scheme's performance closely approximates the upper bound established by exhaustive antenna aperture search methods. Specifically, the figures illustrate that the proposed scheme achieves notable proximity to the upper bound, with a maximum deviation of around 0.9% in the NTN rural scenario and around 5% in the NTN urban scenario. This indicates the robustness and efficacy of the proposed scheme in optimizing beamwidth. Furthermore, the proposed optimal beamwidth maximizing uplink coverage probability scheme is advantageous over the 3 GPP setting. As shown in Figures 7 and 10,  
Figure 11: Beam topology and footprints with elevation angle \(\epsilon_{0}(100)=36.53^{\circ}\) for the proposed optimal beamwidth scheme. Black dots represent the position of the beam center, and red circles represent the footprints of half-power beamwidth (\(1.212^{\circ}\)). Figure 10: Average serving power and average interference power for QEFC LEO satellite in NTN rural scenario. The target SINR for the proposed scheme is \(-5.6\) dB. the interference increases at low elevation angles, which degrades the uplink coverage probability. In the NTN rural scenario, with a target SINR of \(-8.61\) dB, the uplink coverage probability for the 3 GPP setting starts above 0.95 but sharply declines to less than 0.36 by \(t=100\). Similarly, in the scenario with a target SINR of \(-5.6\) dB, the uplink coverage probability for the 3 GPP setting nearly diminishes to zero by \(t=100\). In contrast, the proposed scheme maintains an uplink coverage probability of at least 0.93 throughout all simulated times in the NTN rural scenario, irrespective of the target SINR. In the NTN urban scenario, where the low LOS probability impacts the uplink coverage probability, the proposed scheme still significantly improves compared to the 3 GPP setting. Despite FRF3 being considered a helpful interference mitigation scheme, the proposed scheme demonstrates improved uplink coverage probability through increased serving power, as shown in Figure 10. In the NTN rural scenario, the proposed scheme and FRF3 exhibit similar uplink coverage probability performance within \(t=0\) and \(t=80\). Beyond \(t=80\), the proposed scheme outperforms FRF3. In the NTN urban scenario, the uplink coverage probability of FRF3 surpasses that of the proposed scheme by approximately 0.085 and 0.03 at the beginning, with target SINR values of \(-5.6\) and \(-8.61\), respectively. However, as time progresses, the uplink coverage probability of FRF3 decreases, eventually falling below that of the proposed scheme around \(t=55\) and falling below by approximately 0.176 and 0.08 at \(t=100\), with target SINR values of \(-5.6\) and \(-8.61\), respectively. The simulation results and analysis show the proposed scheme's advantages. The proposed scheme significantly enhances uplink coverage probability, SINR, and interference mitigation. Furthermore, it closely matches the results obtained from exhaustive search methods. These improvements indicate that the proposed scheme effectively addresses the low SINR problem in the QEFC scenario. However, it is essential to acknowledge its limitations. Typically, satellites are equipped with two types of antennas: phased array antennas or parabolic antennas. Since the proposed scheme focuses on adjustable beamwidth, it may not be suitable for satellites equipped with parabolic antennas. Moreover, as illustrated in Figure 4, the optimal equivalent antenna aperture is substantial, three times larger than that of the 3 GPP setting. Achieving such high antenna gain and narrow beam design requires a significant number of antenna elements. This affects the satellite's weight and, consequently, the deployment cost.