## 4 Results  
In this section, we consider the dense LEO satellite communication scenario consisting of 6372 satellites in the Ka-band and multiple UTs. The orbital parameters of the LEO constellation are listed in Table 2[37]. The positions of multiple users are randomly generated in a selected ground area, and the number of users is 16. Moreover, the detailed simulation parameters are shown in Table 3. Note that the peak gain and 3 dB beamwidth of the satellite antenna in Table 3 are applied for a single beam. The simulation environment is built on a PC with an Intel Core i7-10700F, 16 GB of RAM, and a Windows 10 (64 bit) system. In this paper, the data of satellite position are exported from Satellite Tool Kit (STK) 11.2.0, which provides a physics-based modeling environment for analyzing platforms and payloads in a realistic mission context [38]. Due to the movement of the satellites, the overlapping area appears first and then disappears. Thus, the entire time of users covered by the overlapping coverage area is limited, and the maximum duration time is 21 s. During the entire simulation time, the maximum number of interfering satellites is \(Q=28\), and the minimum time slot is 1 s [19]. We take the exhaustive search (ES) as the reference to verify the effectiveness of beam pointing optimization based on PSO. The ES traverses all directions with \(grid=0.1^{\circ}\) and finds the maximum SINR. The PSO algorithm is implemented on each UT and runs independently. To compare the performance of difference schemes clearly, we select a user located at (39.56\({}^{\circ}\) N, 116.20\({}^{\circ}\) E) to show the simulations in Figures 8-10. For clarity, the scheme without a beam pointing optimization for interference avoidance is defined as BPO-WIA, while the schemes with a beam pointing optimization using ES and PSO are denoted as BPO-ES and BPO-PSO, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Altitude** & **Inclination** & **Planes** & **Satellites per Plane** \\ \hline
1200 km & 87.9\({}^{\circ}\) & 36 & 49 \\
1200 km & 55\({}^{\circ}\) & 32 & 72 \\
1200 km & 40\({}^{\circ}\) & 32 & 72 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Orbital parameters [37]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Scenario Parameters** & **Symbol** & **Value** \\ \hline Earth radius & \(R_{e}\) & 6378 km \\ Satellite orbital radius & \(R\) & 7578 km \\ Satellite orbital altitude & \(h\) & 1200 km \\ Carrier frequency & \(f_{c}\) & 20 GHz [39] \\ Bandwidth & \(B_{w}\) & 400 MHz [39] \\ Minimum visibility elevation angle of the UT & \(ell_{th}\) & 55\({}^{\circ}\)[3] \\ Number of satellite beams & \(N_{b}\) & 16 [3] \\ Satellite EIRP density & \(E_{d}\) & 10 dBW/MHz [39] \\ Satellite beam radius & \(r\) & 20 km [39] \\ Peak gain of satellite antenna & \(G_{t}\) & 38.5 dBi [39] \\ Peak gain of UT antenna & \(G_{r}\) & 39.7 dBi [39] \\ Vertical 3 dB beamwidth of UT antenna & \(\Theta_{\rm 3 dB}\) & 1.48\({}^{\circ}\) \\ Horizontal 3 dB beamwidth of UT antenna & \(\Phi_{\rm 3 dB}\) & 1.48\({}^{\circ}\) \\ System temperature & \(T\) & 290 K \\ \hline PSO Parameters & Symbol & Value \\ \hline Number of particles & \(J\) & 10 \\ Maximum iterations & \(M\) & 30 \\ Minimum velocity & \(v_{\rm min}\) & \(-0.3\) \\ Maximum velocity & \(v_{\rm max}\) & 0.3 \\ Minimum inertia weight & \(w_{\rm min}\) & 0.4 [33] \\ Maximum inertia weight & \(w_{\rm max}\) & 0.9 [33] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simulation parameters. In Figure 8, the SINR using BPO-WIA initially decreases and then increases. This is because the motion of the satellites causes the overlapping area to first increase and then decrease. Thus, the interference also increases first and then decreases. The most severe interference occurs when the user is covered by the center of the maximum beam overlapping area, so the SINR is low and even close to 0 dB at time index \(n=7\) and \(n=8\). The lowest and average SINR is 0.02 dB (time index \(n=8\)) and 7.59 dB, respectively.