## 4 Results
The SINR with BPO-ES has the optimal SINR performance. However, the computation complexity of BPO-ES is highest, i.e., \(\mathcal{O}(\frac{(2\Theta_{\text{sub}})}{grid})\times\frac{(2\Theta_{\text{ sub}})}{grid})\), where \(grid\) is \(0.1^{\circ}\). The time consumption for BPO-ES at each time slot is 1.11 s, as shown in Table 4.  
In Figure 8, the optimized SINR of different particles and maximum iterations are shown. Moreover, the corresponding convergence curve at the moment of the strongest interference (time index \(n=8\)) is depicted in Figure 9. In Figures 8 and 9, when the number of particles is set as \(J=10\), the obtained SINR using \(M=30\) or \(M=40\) outperforms that using \(M=20\). In addition, when the maximum iterations are set to \(M=10\), the optimized SINR using \(J=10\) or \(J=15\) is better than that using \(J=5\). It can be concluded that the optimal value can be found with proper iterative times or particles. Furthermore, the time consumption is related to the number of particles and iterations. As shown in Table 4, the computation complexity of BPO-PSO is \(\mathcal{O}(M)\). To balance optimization performance and time consumption, we choose \(M=30\) and \(J=10\) in this paper. Its runtime for the one-shot optimization is approximately 0.39 s. According to Figure 8, BPO-PSO (\(M=30,J=10\)) and BPO-ES can achieve almost the same performance and are better than BPO-WIA during the simulation time. The SINR achieved by BP-ES and BP-PSO at the strongest interference moment is 8.23 dB (time index \(n=8\)), and the average SINR is 10.05 dB. Thus, BPO-PSO is more suitable for the high dynamic LEO satellite scenario than BPO-ES due to its lower time consumption. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Computation Complexity** & **Setting** & **Time Consumption** \\ \hline BPO-ES = \(\mathcal{O}(\frac{(2\Theta_{\text{sub}})}{grid}\cdot\frac{(2\Phi_{\text{ sub}})}{grid})\) & \(grid=0.1^{\circ}\) & 1.11 s \\  & \(M=30\), \(J=10\) & 0.39 s \\ BPO-PSO = \(\mathcal{O}(M)\) & \(M=40\), \(J=10\) & 0.51 s \\  & \(M=30\), \(J=5\) & 0.20 s \\  & \(M=30\), \(J=15\) & 0.56 s \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Complexity comparison. Figure 8: The variation in SINR over time. The SINR with different numbers of beams is shown in Figure 10. The original SINR with BPO-WIA using \(N_{b}=36\) is lower than \(N_{b}=16\) due to the influence of co-channel interference. This is because the total beam coverage area and interfering links are increased due to the increase in the number of beams, which leads to a higher interference level. Moreover, at the time index \(n=2\), the improvement of SINR obtained by BPO-PSO with \(N_{b}=36\) and \(N_{b}=16\) is 3.5 dB and 3.0 dB, respectively. At time index \(n=8\), the improvement of SINR obtained by BPO-PSO with \(N_{b}=36\) and \(N_{b}=16\) is 10.0 dB and 8.2 dB, respectively. Thus, the BPO-PSO proposed in this paper can improve SINR with more gains for stronger interference scenarios, and it can also be applied to scenarios with various numbers of beams. Furthermore, the simulation results of the multi-user scenario are provided in Figures 11-13. The receiving antenna pointing of each user can be optimized separately and achieve the maximum SINR. Based on the above optimized SINR, the complementary cumulative distribution function (CCDF) of the SINR is depicted in Figure 11. The minimum SINR of BPO-WIA is \(-7.89\) dB, while the minimum SINR of BPO-ES and BPO-PSO is \(-0.63\) dB and \(-1.1\) dB. Compared to BPO-WIA, the probability of BPO-PSO can achieve \(17.27\%\) and \(8.93\%\) improvement at \(\text{SINR}=5\) dB and \(10\) dB; thus, the SINR is effectively improved for stronger interference. The multi-user average SINR of BPO-WIA, BPO-ES, and BPO-PSO is depicted in Figure 12. The maximum, minimum, and mean SINR of  
Figure 10: The SINR with different number of beams. Figure 9: The convergence curve of BPO-PSO for time index \(n=8\). multi-user with BPO-WIA are 10.94 dB, 2.84 dB, and 7.85 dB.