# Performance of UAV-to-Ground FSO Communications with APD and Pointing Errors
## 2 System and Channel Modeling
### Channel Modeling
#### 2.2.2 Pointing Error Loss  
We consider the Gaussian beam at the FSO Tx, and the pointing error-induced geometrical loss can be modeled as [24, 25]  
\[X_{pu}\simeq A_{0}\exp\left(-\frac{2r_{tr}^{2}}{\omega_{z_{eq}}^{2}}\right), \tag{18}\]  
where \(r_{tr}=\sqrt{x_{d}^{2}+y_{d}^{2}}\) is the distance of the beam sport shifted to the position \([x_{d},y_{d}]\) of the receiver lens (see Figure 3). The parameter \(A_{0}=\left(\text{erf}(v)\right)^{2}\) represents the maximal fraction of the collected intensity with \(v=\frac{\sqrt{\pi}r}{\sqrt{2}\omega_{y}}\), and \(r\) denotes the Rx lens radius, \(\text{erf}(\cdot)\) is the error function, \(\omega_{y_{eq}}^{2}=\omega_{y}^{2}\frac{\sqrt{\pi}\text{ erf}(v)}{y_{2}\exp(-v^{2})}\) denotes the equivalent beam waist at the distance \(y\), \(\omega_{y}=\omega_{0}[1+\epsilon(\lambda L\pi\omega_{0}^{2})^{2}]^{1/2}\) with \(\omega_{0}\) represents the beam waist radius of the Tx at \(y=0\), \(e=1+2\omega_{0}^{2}/\rho_{0}^{2}\), and \(\rho_{0}=(0.55\text{C}_{n}^{2}k^{2}L)^{-3/5}\) is the coherence length. The center of the incoming optical beam is deviated from the center of the FSO Rx lens due to the random displacement of FSO Tx caused by the hovering UAV. On the other hand, the angle-of-arrival of the Tx optical signal can be explained as follows. As described in Figure 2, the Rx lens focus the incoming light onto the PD area using either APD or PIN photo-detectors in the (\(x\), \(z\)) plane. The incidence angle relative to the Rx detector axis is denoted by \(\theta_{AoA}\). Hence, it is expressed as \(\theta_{AoA}\!=\!\sqrt{\theta_{AoA,x}^{2}\!+\!\theta_{AoA,z}^{2}}\). Furthermore, it can be approximated as \(\theta_{AoA}\!\simeq\!\sqrt{\left(\theta_{AoA,tx}\!+\!\theta_{AoA,rx}\right)^{ 2}\!+\!\left(\theta_{AoA,tx}\!+\!\theta_{AoA,rx}\right)^{2}}\), which has a Rayleigh distribution as [18]  
\[f_{\theta_{AoA}}\left(\theta_{AoA}\right)\!=\!\frac{\theta_{AoA}}{\sigma_{AoA }^{2}}\exp\!\left(-\frac{\theta_{AoA}^{2}}{2\sigma_{AoA}^{2}}\right),\quad \theta_{AoA}\geq 0. \tag{19}\]  
In Equation (19), \(\sigma_{\theta_{AoA}}^{2}\!=\!\sigma_{10}^{2}\!+\!\sigma_{10}^{2}\) denotes the variance of the \(\theta_{AoA}\), \(\sigma_{10}^{2}\) and \(\sigma_{10}^{2}\) represent the variances of the orientation deviations of the Tx and Rx, respectively. Due to the limitations of Rx's FOV and AoA fluctuations, a link interruption occurs for \(\theta_{AoA}\geq\theta_{FOV}\). We assume that the link loss \(X_{pa}\) takes two discrete values of \"1\" and \"0\", corresponding to one of the \(\{2^{\log_{2}\left(M_{l}\times M_{Q}\right)}\}\) symbols for the determination that the incoming optical signal was projected on the Rx's FOV or not. Therefore, because of AoA fluctuations, the corresponding lost \(X_{pa}\) can be obtained by [15]  
\[X_{pa}=\Pi(\theta_{AoA}/\theta_{FOV}). \tag{20}\]  
In Equation (20), \(\Pi(x)=1\) if \(x\leq 1\), otherwise \(\Pi(x)=0\). Moreover, the RV \(\theta_{AoA}\) conditioned on \(\theta_{AoA,tx}\) and \(\theta_{AoA,tx}\) has a Rician distribution that can be represented as [18]  
\[f_{\theta_{AoA}|\theta_{xz}}(\theta_{AoA})=\frac{\theta_{AoA}}{\sigma_{\theta_ {AoA}}^{2}}\exp\!\left(-\frac{\theta_{AoA}^{2}+\theta_{xz}^{2}}{2\sigma_{ \theta_{AoA}}^{2}}\right)l_{0}\!\left(\frac{\theta_{AoA}\theta_{xz}}{\sigma_{ \theta_{AoA}}^{2}}\right), \tag{21}\]  
Figure 3: Modeling the received beam at the FSO Rx station in the presence of fluctuations due to UAV hovering. where \(I_{0}(\cdot)\) is the modified Bessel function of the first kind with order zero. From (20) and (21), the distribution of \(X_{pa}\) can be derived as  
\[ \begin{split} f_{X_{pa}|\theta_{xz}}\big{(}X_{pa}\big{)}& =F_{\theta_{AoA|\theta_{xz}}}(\theta_{FOV})\delta\big{(}X_{pa}-1 \big{)}\\ &+\Big{(}1{-}F_{\theta_{AoA|\theta_{xz}}}(\theta_{FOV})\Big{)} \delta\big{(}X_{pa}\big{)},\end{split}