# Interference Avoidance through Periodic UAV Scheduling in RIS-Aided UAV Cluster Communications
## 2 System Models and Problem Formulations
Interference among multiple UAV clusters can be intricate due to the dynamic positions of the UAVs. Building upon the concept of collision beams introduced in previous work [23], we propose the UAV interference matrix (UIM) to quantify the interference between UAVs belonging to different clusters. We assume there are \(N_{c}\) UAV clusters, each comprising \(M\) UAVs to be scheduled within a single scheduling frame. We denote the UAV \(j\) in cluster \(i\) as UAV \((i,j)\), which is served by the beam \((i,j)\). Notably, beam \((i^{\prime},j^{\prime})\) is directed to UAV \((i^{\prime},j^{\prime})\), but it can cause interference with UAV \((i,j)\). We define the UAV interference matrix \(\mathbf{C}_{i,i^{\prime}}\) with dimensions of \(M\times M\), to represent the interference received by cluster \(i\) originating from cluster \(i^{\prime}\). The element \(c_{i,i^{\prime}}(j_{i},j^{\prime})\) represents the interference level that UAV \((i,j)\) experiences from beam \((i^{\prime},j^{\prime})\). Interference alone is insufficient to determine the system's performance. Therefore, we employ the estimated signal-to-interference ratio (SIR) to derive the UAV interference matrix (UIM). Given the unknown nature of the small-scale channel \(\alpha_{RU}\), we utilize the transmission power and beamforming gain to estimate the SIR, and calculate the UIM elements. The beamforming gain function is determined based on factors such as the relative position of the RIS and UAVs, distance, beam width, and beam direction. The gain function is given by [31]  
\[g(w_{h},w_{v},\Delta\theta,\Delta\phi)=\frac{\pi}{w_{h}}e^{\eta\left(\frac{\Delta \theta}{w_{h}}\right)^{2}}\frac{\pi}{w_{v}}e^{\eta\left(\frac{\Delta\theta}{w_ {v}}\right)^{2}}, \tag{5}\]  
where \(\eta\) is \(-4\log 2\), \(w_{h}\) and \(w_{v}\) are the horizontal and vertical beam widths, respectively. \(\Delta\theta\) is the angular difference between the beam direction and the receiver. The variables \(z_{i,j}\), \(\phi_{i,j}\), and \(\theta_{i,j}\) are used to represent the distance, relative horizontal angle, and relative vertical angle between UAV \(j\) and RIS-UAV \(i\). It is assumed that the beam is perfectly aligned with the UAV without deviations caused by coefficient quantification. The angular deviation between UAV \(j\) and the interference beam is calculated as \(\Delta\theta=\left|\phi_{i^{\prime},j^{\prime}}-\theta_{i^{\prime},j}\right|\), and \(\Delta\phi=\left|\phi_{i^{\prime},j^{\prime}}-\phi_{i^{\prime},j}\right|\). The SIR of UAV \(j\) under the interference of beam \((i^{\prime},j^{\prime})\) can be estimated by  
\[ \begin{split}\eta\left(i,j,i^{\prime},j^{\prime}\right)& =\frac{S}{I}=\frac{g_{U}(w_{h},w_{v},0,0)L\left(z_{i,j}\right)}{g_ {I}(w_{h},w_{v},\Delta\theta,\Delta\phi)L\left(z_{i^{\prime},j}\right)}\\ &=\frac{f\left(z_{i,j}\right)}{f\left(z_{i^{\prime},j}\right)e^{ \eta\left(\frac{\Delta\theta}{w_{h}}\right)^{2}+\eta\left(\frac{\Delta\theta}{ w_{v}}\right)^{2}}},\end{split}  \tag{6}\]  
where the function \(f(z)\) is the path loss function. Without loss of generality, we assume the elements of UIM can only take on the values of 0 or 1, which, respectively, indicate the absence or presence of strong interference. The SIR in (6) is compared to a predefined threshold in order to determine the value of the UIM. Given the transmission power, the beam width, and locations of UAVs, the signal-to-interference ratio (SIR) is calculated for \((i,j)\). Specifically, the element of UIM \(c_{i,j^{\prime}}(i,j^{\prime})\) is set to 1 if the estimated SIR \(\eta(i,j,i^{\prime},j^{\prime})\) is greater than a preset threshold \(\eta th\). It indicates that the beam \((i^{\prime},j^{\prime})\) causes strong interference to UAV \((i,j)\), as shown in (7). \[c_{i,j^{\prime}}\left(j,j^{\prime}\right)=\left\{ \begin{array}{ll}1&\quad \eta(i,j,i^{\prime},j^{\prime})<\eta th\\ 0&\quad\eta(i,j,i^{\prime},j^{\prime})\geqslant\eta th.\end{array} \right. \tag{7}\]  
In a scheduling frame, the UAVs within each cluster are scheduled periodically with their associated beams.