## 4 Numerical Results and Discussion
### Signal Quality
Figure 9 demonstrates a decrease in the angular difference between interfering UEs and the center beam over time, accompanied by increased received antenna gain. Particularly for the UE located at \((x,y)=(70,0)\) kilometers, the angular difference decreases from \(6.65^{\circ}\) to \(2.3^{\circ}\), increasing antenna gain from  
Figure 6: The SINR results for 3 GPP setting, FRF3, and the proposed scheme (target SINR \(-5.6\) dB) in NTN Urban scenario at (**a**) t = 0, (**b**) t = 60, (**c**) t = 100. Figure 7: The interference results for 3 GPP setting, FRF3, and the proposed scheme (target SINR \(-5.6\) dB) in NTN Rural scenario at (**a**) t = 0, (**b**) t = 60, (**c**) t = 100. 12 dBi to 26.7 dBi. This increase in antenna gain ultimately contributes to the escalating interference levels. As shown in Figure 10, the average interference rises from \(0.64\times 10^{-14}\) to \(2.08\times 10^{-14}\) watts for the 3 GPP setting. Despite the FRF3 aims to mitigate interference, the average interference still rises from \(0.06\times 10^{-14}\) to \(0.66\times 10^{-14}\) watts. Figure 4 illustrates that the satellite forms a narrower beam with the proposed scheme to mitigate interference as time progresses. Furthermore, as shown in Figure 11, the proposed scheme effectively reduces the overlap of beam footprints by adjusting the beamwidth. Specifically, the half-power beamwidth is adjusted to only \(1.212^{\circ}\) at \(t=100\), significantly reducing the overlap of the beam footprints. Although the average interference power for all three schemes increases as the elevation angle decreases and FRF3 exhibits lower interference power than the proposed scheme, the proposed scheme demonstrates a minor increment owing to its narrower beamwidth. Specifically, the 3 GPP setting and FRF3 experience an increase in average interference power of \(1.44\times 10^{-14}\) (\(225\%\)) and \(0.6\times 10^{-14}\) (\(1000\%\)) watts, respectively, while the proposed scheme only sees an increase of \(0.5\times 10^{-14}\) (\(123\%\)) watts. This difference can be attributed to the proposed scheme's narrower beamwidth, effectively reducing interference overlap. The proposed scheme also has a significant impact on serving power. As explained in [16; 18], FSPL increases with decreasing elevation angle, leading to a decline in the average serving power for both the  
Figure 8: Beam topology and footprints with elevation angle (**a**) \(\epsilon_{0}(0)=90^{\circ}\) and (**b**) \(\epsilon_{0}(100)=36.53^{\circ}\) for the 3 GPP setting and FRF3. Black dots represent the position of the beam center, and red circles represent the footprints of half-power beamwidth (\(4.4127^{\circ}\)). Figure 9: Angular difference \(\varphi_{0}^{s}(t)\) and received antenna gain \(G(\varphi_{0}^{s}(t),2)\) between interfering UE and the center beam. 3 GPP setting and FRF3, as shown in Figure 10. Conversely, the average serving power for the proposed scheme increases from \(0.77\times 10^{-14}\) to \(2.16\times 10^{-14}\) watts due to higher antenna gain. This higher antenna gain compensates for the growing FSPL and mitigates the impact of growing interference.