# Performance of UAV-to-Ground FSO Communications with APD and Pointing Errors
## 2 System and Channel Modeling
### Channel Modeling
#### 2.2.3 Combined Channel Models  
The AoA fluctuation and orientation deviations of the FSO transmitter cause deviations to the center of the received beam at the FSO receiver plane, and the dependence of the RVs \(X^{{}^{\prime}}\) and \(X_{pa}\) are conditioned on the RV of the \(\theta_{AoA}\) on the plane \((x,z)\). Therefore, the PDF of \(X\) can be obtained as  
\[f_{X}(X){=}\int_{0}^{\infty}\!\!\!f_{X|\theta_{AoA,xz}}f_{\theta_{AoA,xz}} \big{(}\theta_{AoA,xz}\big{)}d\theta_{AoA,xz}, \tag{25}\]  
where \(f_{x|y}(\cdot)\) is the PDF of the RV \(x\) conditioned on the RV \(y\), and:  
\[f_{X|\theta_{AoA,xz}}=\int_{0}^{\infty}\!\!\!\frac{f_{X_{pa}|\theta_{AoA,xz}}( X/X^{{}^{\prime}})}{X^{{}^{\prime}}}f_{X^{{}^{\prime}}|\theta_{AoA,xz}} \big{(}X^{{}^{\prime}}\big{)}dX^{{}^{\prime}}. \tag{26}\]  
Then, the completed statistical models of the UAV/FSO channel, considering the effect of link loss and atmospheric turbulence, the effective pointing error-induced geometrical loss can be derived.  
In the first case of weak to moderate turbulence conditions, the log-normal model of the air-to-ground FSO communication links-based UAV has a PDF obtained as  
\[\begin{split} f_{X}^{\text{LN}}(X)&=\exp\!\left({- }\frac{\theta_{FOV}^{2}}{2\sigma_{AoA}^{2}}\right)\delta(X)+\Xi_{1}\!\left[1{-} \exp\!\left({-}\frac{\theta_{FOV}^{2}}{2\sigma_{AoA}^{2}}\right)\right]\\ &\times X^{\Xi_{2}}Q\left(\frac{\ln\!\left({\frac{X}{A_{0}X_{i}}} \right)+\Xi_{3}}{\sigma_{R}}\right),\end{split} \tag{27}\]  
where \(\Xi_{1}{=}\frac{\Psi}{(A_{0}X_{i})^{\intercal}}\exp\left[0.5\sigma_{R}^{2} \Psi(1+\Psi)\right]\), \(\Xi_{2}{=}\Psi{-}1\), \(\Xi_{3}=0.5\sigma_{R}^{2}\Psi(1+2\Psi)\), and \(Q(\cdot)\) is the well-known \(Q\)-function, and \(\Psi{=}\frac{\omega_{FSO}^{2}}{4\left(L^{2}\sigma_{ho}^{2}+\sigma_{tp}^{2}+ \sigma_{rp}^{2}\right)}\). Moreover, \(\sigma_{tp}^{2}\) and \(\sigma_{rp}^{2}\) are the variances of the position deviations of the Tx and Rx, respectively.  
In the second case of moderate to strong atmospheric turbulence conditions, the GG model of the UAV-based air-to-ground FSO communication link has a PDF given as  
\[\begin{split} f_{X}^{\text{GG}}(X)&=\exp\left(-\frac{ \theta_{FOV}^{2}}{2\sigma_{AoA}^{2}}\right)\delta(X)\\ &+\ \Xi_{1}\left[1-\exp\left(-\frac{\theta_{FOV}^{2}}{2\sigma_{AoA}^{2}} \right)\right]\sum_{n=0}^{N}\left(c_{1}X^{\Psi-1}+c_{2}X^{n+a-1}+c_{3}X^{n+ \beta-1}\right).\end{split} \tag{28}\]  
In (28), \(0\ \leq\ X\ \leq\ X_{I}A_{0}X_{m}\), \(c_{1}=\left(c_{4}c_{5}X_{m}^{n+\beta-\Psi}-c_{4}c_{6}X_{m}^{n+a-\Psi}\right)\), \(c_{2}=\frac{c_{4}c_{6}}{(X_{I}A_{0})^{n+a-\Psi}}\), \(c_{3}=\frac{c_{4}c_{5}}{(X_{I}A_{0})^{n+b-\Psi}}\), \(c_{4}=\frac{\pi\Xi}{(X_{I}A_{0})^{\frac{n}{2}}\Gamma(a)\Gamma(\bar{\Gamma})sin (\pi(\pi(\alpha-\beta))})\), \(c_{5}=\frac{(a\beta)^{(n+\beta)}}{(n+\beta-\Xi)\Gamma(n-a+\bar{\beta}+1)n!}\), and \(c_{6}=\frac{(a\beta)^{(n+a)}}{(n+\beta-\Xi)\Gamma(n+a-\bar{\beta}+1)n!}\). It is noted that \(N\) is a natural number and \(X_{m}\) is the a coefficient. These parameters, however, depends on the Rytov variance [18].