## 2 System and Channel Models
### Channel Models
Similar to the OGS-to-satellite FSO link, the PDF of \(\gamma_{f}\) for HAP-to-AP FSO link is written as  
\[f_{\gamma_{f}}(\gamma)=\frac{\xi_{3}^{2}}{b_{\gamma}\Gamma(\alpha_{3})\Gamma( \beta_{3})}G_{1,3}^{3,0}\bigg{[}\frac{\alpha_{3}\beta_{3}\xi_{3}^{2}}{\xi_{3} ^{2}+1}\bigg{(}\frac{\gamma}{\mu_{3}}\bigg{)}^{\frac{1}{2}}\bigg{|} \begin{array} []{c}\xi_{3}^{2}+1\\ \xi_{3}^{2},\ \alpha_{3},\ \beta_{3}\end{array} \bigg{]} \tag{13}\]  
where \(\mu_{3}=\frac{\alpha_{3}\beta_{3}\xi_{3}^{2}(\zeta_{3}^{2}+2)}{(\alpha_{3}+1)( \beta_{3}+1)(\zeta_{3}^{2}+1)^{2}}\overline{\gamma}_{f}\) for IM/DD technique, \(\mu_{3}=\overline{\gamma}_{f}\) for HD technique. Similar to (10), the CDF of \(\gamma_{f}\) can be obtained from (13) and is given by  
\[F_{\gamma_{f}}(\gamma)=F^{3}G_{b+1,3b+1}^{3b,\ 1}\Bigg{(}\frac{D_{3}^{b} \gamma}{b^{2b}\mu_{3}}\bigg{|} \begin{array}{cc}1&E_{5}\\ 6&0\end{array} \bigg{)} \tag{14}\]  
where \(F^{3}=\frac{\xi_{2}^{2}\mu_{3}^{2}+\beta_{3}-2}{(2\pi)^{-1}\Gamma(\alpha_{3}) \Gamma(\beta_{3})}\), \(D_{3}=\frac{\alpha_{3}\beta_{3}\xi_{3}^{2}}{\xi_{3}^{2}+1}\), \(E_{5}=\left\{\frac{\xi_{3}^{2}+1}{b},\ldots,\frac{\xi_{3}^{2}+b}{b}\right\}\), \(E_{6}=\left\{\frac{\xi_{3}^{2}}{b},\ldots,\frac{\xi_{3}^{2}+b-1}{b},\ldots, \frac{\alpha_{3}}{b},\ldots,\frac{\alpha_{3}+b-1}{b},\frac{\beta_{3}}{b},\ldots,\frac{\beta_{3}+b-1}{b}\right\}\). The HAP-to-users RF link is assumed to follow Nakagami-_m_ fading with \(N_{t}\) antennas. the PDF and CDF of \(\gamma_{RF}\) are given by ([13], Equation (11))  
\[f_{\gamma_{RF}}(\gamma)=\frac{1}{\Gamma(mN_{t})}\left(\frac{m}{\overline{\gamma}_ {RF}}\right)^{mN_{t}}\gamma^{mN_{t}-1}\exp\left(-\frac{m\gamma}{\overline{ \gamma}_{RF}}\right) \tag{15}\]  
\[F_{\gamma_{RF}}(\gamma)=1-\frac{1}{\Gamma(mN_{t})}\Gamma\left(mN_{t},\frac{m \gamma}{\overline{\gamma}_{RF}}\right) \tag{16}\]  
where \(m\) is the fading severity parameter. The instantaneous capacity of HAP-to-AP FSO link can be calculated by  
\[C_{f}=W_{f}\log_{2}\left(1+\tau\gamma_{f}\right) \tag{17}\]  
where \(W_{f}\) is the bandwidth of FSO link, \(\tau=e/2\pi\) for IM/DD technique and \(\tau=1\) for HD technique. Based on (13) and (17), and using the power transformation of random variables, the PDF of instantaneous capacity for HAP-to-AP FSO link can be obtained as  
\[f_{C_{f}}(c)=\frac{\xi_{3}^{2}2^{c/W_{f}}\ln 2}{bW_{f}\Gamma(\alpha_{3})\Gamma( \beta_{3})}\left(2^{c/W_{f}}-1\right)^{-1}G_{1,3}^{30}\Bigg{[}\frac{\alpha_{3} \beta_{3}\xi_{3}^{2}}{\xi_{3}^{2}+1}\bigg{(}\frac{2^{c/W_{f}}-1}{\tau\mu_{3}} \bigg{)}^{\frac{1}{2}}\bigg{|} \begin{array}{c}\xi_{3}^{2}+1\\ \xi_{3}^{2},\ \alpha_{3},\ \beta_{3}\end{array} \bigg{]} \tag{18}\]Applying series expansion to \(\exp(\cdot)\) function in Equation (15), the PDF of \(\gamma_{RF}\) is further expressed by  
\[f_{\gamma_{RF}}(\gamma)=\frac{1}{\Gamma(mN_{t})}A^{mN_{t}}\sum\limits_{i=0}^{ \infty}\frac{\left(-A\right)^{i}}{i!}\gamma^{mN_{t}+i-1} \tag{19}\]  
where \(A=m/\gamma_{RF}\). The instantaneous capacity of HAP-to-user RF link is calculated by  
\[C_{RF}=W_{r}\log_{2}(1+\gamma_{RF}) \tag{20}\]  
where \(W_{r}\) is the bandwidth of the HAP-to-users RF link. According to (19) and (20), the PDF of instantaneous capacity for HAP-to-users RF link is given by  
\[f_{C_{RF}}(c)=\frac{2^{c/W_{r}}\ln 2}{W_{r}\Gamma(mN_{t})}A^{mN_{t}}\sum\limits_{i =0}^{\infty}\frac{\left(-A\right)^{i}}{i!}\left(2^{c/W_{r}}-1\right)^{mN_{t}+ i-1} \tag{21}\]  
The CDF of instantaneous capacity of HAP-to-users RF link is obtained by  
\[F_{C_{RF}}(c)=\int_{0}^{c}f_{C_{RF}}(x)dx \tag{22}\]  
Substituting (21) into (22), \(F_{C_{RF}}(c)\) can be further obtained as  
\[F_{C_{RF}}(c)=\frac{A^{mN_{t}}}{\Gamma(mN_{t})}\sum\limits_{i=0}^{\infty}\frac {\left(-A\right)^{i}\left(2^{c/W_{r}}-1\right)^{mN_{t}+i}}{i!\left(mN_{t}+i \right)} \tag{23}\]  
The instantaneous sum capacity of parallel FSO/RF links is defined as \(C_{t}=C_{f}+C_{RF}\).