## 3 Performance Evaluation
### Weighted Average BER  
The average BER of our proposed three-hop hybrid FSO/RF system can be calculated by ([8], Equation (36))  
\[\overline{P}_{eSRD}=\overline{P}_{eSR}+\overline{P}_{eRD}-\overline{P}_{eSR} \overline{P}_{eRD} \tag{32}\]  
where \(\overline{P}_{eSR}\) is the average BER of the dual-hop FSO link from OGS to HAP, \(\overline{P}_{eRD}\) means the weighted average BER of parallel FSO/RF links for HAP-to-ground users. Here, we propose the weighted average BER to evaluate the average BER performance of parallel FSO/RF links, which is calculated by using the weighted ergodic capacity of parallel FSOand RF links. The generalized expression for average BER under different modulation schemes is shown as ([2], Equation (46))  
\[\overline{P}_{e}=\frac{\sigma}{2\Gamma(p)}\sum_{u=1}^{n_{0}}q_{u}^{p}\int_{0}^{ \infty}x^{p-1}e^{-q_{u}x}F(x)dx \tag{33}\]  
where \(n_{0}\), \(\sigma\), \(p\), and \(q_{u}\) are parameters depending on the type of detection and modulation techniques shown in Table 2[3].  
The average BER of the dual-hop FSO link from OGS to HAP is obtained by substituting (28) into (33) shown as  
\[\begin{split}&\overline{P}_{eSR}=\exp\left(-\frac{\theta_{FoV}^{2 }}{2\sigma_{angle,u}^{2}}\right)+\left(1-\exp\left(-\frac{\theta_{FoV}^{2}}{2 \sigma_{angle,u}^{2}}\right)\right)\left(\frac{\sigma F_{p}}{2\Gamma(p)}- \frac{\sigma\xi_{1}^{2}F^{2}}{2b\Gamma(\alpha_{1})\Gamma(\beta_{1})\Gamma(p)} \right.\\ &\times\sum_{u=1}^{n_{0}}G_{1,0,3,3b,\,3b+1}^{0,1:1:3.3b+1}\left[ \begin{array}{cc}1&-\zeta_{1}^{2},1-\alpha_{1},1-\beta_{1}\\ -&\zeta_{1}^{2},0\end{array}\right]\left|\begin{array}{cc}E_{3}\\ E_{4},0\end{array}\right.\left|\frac{1}{D_{1}}(q_{u}\mu_{1})\frac{1}{b},\frac{ D_{2}^{b}Q}{b^{2b}\mu_{2}}\right|\right)\\ \\ \end{split} \tag{34}\]  
where  
\[F_{p}=F^{1}\sum_{u=1}^{n_{0}}G_{b+2,3b+1}^{3b,\,2}\left(\frac{D_{1}^{b}}{b^{2 b}\mu_{1}q_{u}}\left|\begin{array}{cc}1&-p&E_{1}\\ E_{2}&0\end{array}\right.\right) \tag{35}\]  
The average BER of the HAP-to-AP single FSO link can be derived by substituting (14) into (33) and applying ([28], Equation (07.34.21.0013.01)) shown as  
\[\overline{P}_{ef}=\frac{\sigma}{2\Gamma(p)}F^{3}\sum_{u=1}^{n_{0}}G_{b+2,3b+1 }^{3b,\,2}\left(\frac{D_{3}^{b}}{b^{2b}\mu_{3}q_{u}}\left|\begin{array}{cc} 1,1-p,E_{5}\\ E_{6},0\end{array}\right.\right) \tag{36}\]  
Similarly, the average BER of HAP-to-users single RF link is obtained by substituting (16) into (33) and using ([28], Equation (07.34.03.0613.01)), then ([28], Equation (07.34.21.0013.01)) given by  
\[\overline{P}_{\sigma}=\frac{\sigma n_{0}}{2}-\frac{\sigma}{2\Gamma(p)\Gamma( mN_{t})}\sum_{u=1}^{n_{0}}G_{2,2}^{2,1}\left(\frac{m}{\overline{\gamma}_{RF}q_{u}} \left|\begin{array}{cc}1,&1-p\\ mN_{t},&0\end{array}\right.\right) \tag{37}\]  
Since it is difficult to evaluate the average BER performance of parallel FSO/RF links, the weighted average BER is adopted, which is defined as  
\[\overline{P}_{eRD}=\frac{C_{f}}{C_{t}}\overline{P}_{ef}+\frac{C_{RF}}{C_{t}} \overline{P}_{\sigma} \tag{38}\]  
From (38), we can find that the weighted average BER is determined by the average BER of the FSO link, the average BER of the RF link, and their weighted instantaneous link capacity. Substituting (34) and (38) into (32), the average BER of our proposed system can be obtained.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Modulation** & \(\sigma\) & \(p\) & \(q_{u}\) & \(n_{0}\) & **Detection** \\ \hline OOK & 1 & 0.5 & 0.5 & 1 & IM/DD \\ BPSK & 1 & 0.5 & 1 & 1 & HD \\ M-PSK & \(\frac{2}{\max\left(\log_{2}M,2\right)}\) & 0.5 & \(\sin^{2}\left(\frac{(2n-1)\pi}{M}\right)\) & \(\max\left(\frac{M}{4},1\right)\) & HD \\ M-QAM & \(\frac{4}{\log_{2}M}\left(1-\frac{1}{\sqrt{M}}\right)\) & 0.5 & \(\frac{3(2n-1)^{2}}{2(M-1)}\) & \(\frac{\sqrt{M}}{2}\) & HD \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters for Different Modulation Schemes.