## 3 Interference Situational Aware Beam Pointing Optimization
### Beam Pointing Optimization  
Traditionally, the beam center of the UT antenna should point to the communication link AoA. As shown in Figure 7, the beam colored as black points to its serving satellite with \(\mathbf{\theta}_{r}^{\text{C}u}(n)=[\bar{\theta}_{r}^{\text{C}u}(n),\bar{\varphi} _{r}^{\text{C}u}(n)]=[\theta_{r}^{S}(n),\varphi_{r}^{S}(n)]\)). However, co-channel interference occurs inevitably due to the same frequency reuse among satellites. Therefore, the beam center pointing of the UT needs to be optimized for interference mitigation. In Figure 7, the green beam direction denotes the optimized beam center pointing of the UT antenna (i.e., \(\hat{\mathbf{\theta}}_{r}^{\text{C}u}(n))=[\bar{\theta}_{r}^{\text{C}u}(n),\hat{ \varphi}_{r}^{\text{C}u}(n)]\)). During the service time of the serving satellite, the interfering satellites vary dynamically. To characterize the time-varying interference and communication links, the AoAand AoD of interfering and communication links have been divided into discrete status for each time slot in Section 3.1. Thus, the interference is fixed for each time slot. We intend to use the PSO algorithm to maximize the SINR at each time slot to obtain the corresponding optimal beam center pointing. We take the beam pointing optimization process at time index \(n\) as an example. The interference situation at the single time slot can be obtained via (15) and (16). Then, the PSO optimization method can be adopted to achieve the optimal beam pointing of each UT. The advantage and detailed description of the PSO algorithm can be found in [33; 35; 36]. In this paper, we explain how to implement the PSO algorithm to solve the problem defined in (7). Let the number of the particle swarm of PSO optimization be \(J\), and the maximum iteration is \(M\). The upper and lower bounds of the particle swarm two-dimensional search space are defined as follows: \(L_{\theta,\max}=\theta_{r}^{S}(n)+\Theta_{\rm 3 dB}\), \(L_{\varphi,\max}=\varphi_{r}^{S}(n)+\Phi_{\rm 3 dB}\), \(L_{\theta,\min}=\theta_{r}^{S}(n)-\Theta_{\rm 3 dB}\), and \(L_{\varphi,\min}=\varphi_{r}^{S}(n)-\Phi_{\rm 3 dB}\), where \(\theta_{r}^{S}(n)\) and \(\varphi_{r}^{S}(n)\) are the communication link AoA estimated by (11) and (12), \(\Theta_{\rm 3 dB}\) and \(\Phi_{\rm 3 dB}\) are the horizontal and vertical 3 dB beamwidth of the UT antenna. Specifically, the position and velocity of the \(j\)th particle are represented as follows:  
\[l_{j}(n,m) =(\bar{\theta}_{r_{j}}^{C_{U}}(n,m),\bar{\varphi}_{r_{j}}^{C_{U}} (n,m)), \tag{17}\] \[v_{j}(n,m) =(v_{\theta_{j}}^{C_{U}}(n,m),v_{\phi_{r_{j}}^{C_{U}}}^{C_{U}}(n,m )), \tag{18}\]  
where \((n,m)\) denotes the \(m\)th iteration at the time index \(n\), \(j=1,2,\ldots,J\), and \(m=1,2,\ldots,M\). Particle position \(l_{j}\) represents the elevation and azimuth angles of beam center pointing of the UT, \(\bar{\theta}_{r_{j}}^{C_{U}}\) and \(\bar{\varphi}_{r_{j}}^{C_{U}}\) are restricted in \([L_{\theta,\min},L_{\theta,\max}]\) and \([L_{\varphi,\min},L_{\varphi,\max}]\), \(v_{j}\) denotes the particle velocity and is limited between \(v_{\min}\) and \(v_{\max}\). The fitness function is as follows:  
\[f_{j}(n,m)=\mathrm{SINR}_{j}(n,m). \tag{19}\]  
Based on the fitness function, the individual and group position optimal value of the particle swarm at the \(m\)th iteration are expressed as follows: \(p_{\rm best}(n,m)\) and \(g_{\rm best}(n,m)\)  
Figure 7: Beam pointing optimization under interference. Then, with the guidance of \(p_{\text{best}}(n,m)\) and \(g_{\text{best}}(n,m)\), the velocity and position of the \(j\)th particle at \((m+1)\)th iteration can be updated by the following:  
\[v_{j}(n,m+1) =w(n,m)v_{j}(n,m)+c_{1}r_{1}(p_{\text{best}}(n,m)-l_{j}(n,m))\] \[+c_{2}r_{2}(g_{\text{best}}(n,m)-l_{j}(n,m)), \tag{20}\] \[l_{j}(n,m+1) =l_{j}(n,m)+v_{j}(n,m+1), \tag{21}\]  
where \(c_{1}\) and \(c_{2}\) are learning factors, and \(r_{1}\) and \(r_{2}\) are random numbers in \([0,1]\) to increase search randomness.