## 3 Channel Model
### FSO Channel
#### 3.1.3 Pointing Error  
The primary factors contributing to PE in this system are variations in the velocity of the AWACS and mobile GS, as well as fluctuations in the relative position and orientation of the UAV caused by random air disturbances and internal vibrations within the surrounding atmosphere. It is worth noting that the Pointing-Acquisition-Tracking (PAT) system on UAVs and mobile GS scan dynamically adjust the orientation of the transmitter and receiver lenses to maintain beam alignment. However, the PAT system is insufficient to compensate for random position fluctuations caused by sudden changes in the speed of the AWACS and mobile ground station. The change in height on the z-axis for AWACS, UAVs, and mobile GSs is not taken into account as this is negligible compared to their distance. 1. AWACS-to-UAV link:  
For these links, the combined radial displacement vector from the \(i\) th UAV detector center to the center of the beam footprint at the receiver aperture on the UAV is expressed as \(\mathbf{r_{SR_{i}}}\)=\(\mathbf{r_{R}}\)+\(\mathbf{r_{R}}=\big{[}x_{SR_{i}},y_{SR_{i}}\big{]}\), where \(\mathbf{r_{S}}=[x_{s},y_{s}]\) is the displacement vector of the AWACS and \(\mathbf{r_{R}}=[x_{R},y_{R}]\) is the displacement vector of the UAV, as shown in Figure 2a. This fluctuation, caused by a large number of random events, follows a Gaussian distribution; we have \(x_{SR_{i}}\sim\mathcal{N}\left(0,\sigma_{SR_{i}x}^{2}\right)\) and \(y_{SR_{i}}\sim\mathcal{N}\left(0,\sigma_{SR_{i}y}^{2}\right)\). The position deviation of the AWACS is caused by sudden changes in its velocity. The velocity variation is denoted as \(\mathbf{v_{S}}=\big{[}v_{Sx},v_{Sy}\big{]}\), where \(v_{Sx}\) and \(v_{Sy}\) follow the Gaussian distribution \(\mathcal{N}\left(0,\sigma_{Sy}^{2}\right)\) with mean zero and the velocity variance of the AWACS \(\sigma_{Sv}^{2}\). Assuming that there is a slight change in the AWACS's speed within a short time interval \(\Delta t\), the displacement distances can be expressed as \(x_{S}=v_{Sx}\Delta t/2\) and \(y_{S}=v_{Sy}\Delta t/2\). The hovering UAV's position fluctuation is denoted by \(\mathbf{r_{R}}=[x_{R},y_{R}]\), where \(x_{R}\sim\mathcal{N}\big{(}0,\sigma_{Rx}^{2}\big{)}\) and \(y_{R}\sim\mathcal{N}\big{(}0,\sigma_{Ry}^{2}\big{)}\). Mathematically speaking, the variance of \(x_{SR_{i}}\) and \(y_{SR_{i}}\) can be expressed as  
\[\sigma_{SR_{i}x}^{2}=\sigma_{Sx}^{2}+\sigma_{Rx}^{2}=\sigma_{Sv}^{2}(\Delta t )^{2}/4+\sigma_{Rx}^{2} \tag{16}\]  
\[\sigma_{SR_{i}y}^{2}=\sigma_{Sy}^{2}+\sigma_{Ry}^{2}=\sigma_{Sv}^{2}(\Delta t )^{2}/4+\sigma_{Ry}^{2} \tag{17}\]  
2. UAV-to-GS link:  
For these links, the combined radial displacement vector from the mobile GS detector center to the center of the \(i\) th beam footprint at the receiver aperture on the GS is expressed as \(\mathbf{r_{R,D}}\)=\(\mathbf{r_{R+}}\)+\(\mathbf{r_{0}}\) = \(\big{[}x_{R_{i}D},y_{R_{i}D}\big{]}\), where \(x_{R_{i}D}\sim\mathcal{N}\big{(}0,\sigma_{R,Dx}^{2}\big{)}\), \(y_{R_{i}D}\sim\mathcal{N}\Big{(}0,\sigma_{R,Dy}^{2}\big{)}\), and \(\mathbf{r_{0}}=\big{[}x_{\theta x},x_{\theta y}\big{]}\) are the displacement vectors caused by the orientation fluctuation of hovering UAVs, and \(\mathbf{r_{D}}=[x_{D},y_{D}]\) is the displacement vector of the mobile GS, as shown in Figure 2b. Assuming that the UAV jitter angle in the \(x-\)axis \(\theta_{x}\) and \(y-\)axis \(\theta_{y}\) is sufficiently small, the values of \(x_{\theta x}\) and \(x_{\theta y}\) can be approximated as \(x_{\theta x}\simeq L_{RD}\theta_{x}\) and \(x_{\theta y}\simeq L_{RD}\theta_{y}\), where \(L_{RD}\) is the transmitted distance, and \(\theta_{x},\theta_{y}\sim\mathcal{N}\big{(}0,\sigma_{\theta}^{2}\big{)}\). The position deviation of the mobile GS with an unsteady velocity is caused by sudden changes in its velocity.