## 3 Performance Analysis
### Average Bit Error Rate  
The ABER is another metric commonly used to measure the performance of communication systems. It reflects the average accuracy of the data transmitted by the considered wireless communication system under all possible channel states. For the considered system, the end-to-end ABER is related to the ABER of each path, which is their minimum bit error value. In particular, when all paths are independent and identically distributed, the end-to-end ABER of the system is equal to the ABER of a single path. In this case, the ABER of the system can be derived. Assuming that the receiver uses OOK modulation and direct detection, the end-to-end BER conditioned on the channel gain \(h_{m,j}\) for the \(m\)th path is expressed as  
\[P_{\text{e}|h_{m,j}}(j)=\Pr(1)\Pr\left(\text{e}|1,h_{m,j}\right)+\Pr(0)\Pr \left(\text{e}|0,h_{m,j}\right),j=1,2 \tag{26}\]  
where \(\Pr(0)\) and \(\Pr(1)\) are the probabilities that the transmitted power is \(0\) and \(2P_{j}\), respectively. Since \(\Pr(0)=\Pr(1)=1/2\) and \(\Pr\left(\text{e}|0,h_{m,j}\right)=\Pr\left(\text{e}|1,h_{m,j}\right)\), Equation (26) can be further written as  
\[P_{\text{e}|h_{m,j}}(j)=\frac{1}{2}\text{erfc}\left(\frac{P_{j}Th_{m,j}}{ \sqrt{2}\sigma_{j}}\right),j=1,2 \tag{27}\]  
where \(P_{j}\) is the average transmit optical power of node S or each relay and \(\sigma_{j}\) is the standard variance of noise at each relay or node D. Specifically, when \(j=1\), \(P_{j}=P_{\text{t}}\) and \(\sigma_{j}=\sigma_{\text{r}}\); when \(j=2\), \(P_{j}=P_{\text{o}}\) and \(\sigma_{j}=\sigma_{\text{d}}\). According to the following equality [29]  
\[\text{erfc}(x)=\frac{1}{\sqrt{\pi}}G_{1,2}^{2,0}\left[x^{2}\left| \begin{array} []{cc}1\\ 0,\frac{1}{2}\end{array} \right.\right] \tag{28}\]  
Equation (27) can be rewritten as \[P_{\rm e}|_{h_{m,j}}(j)={\rm erfc}\left(\frac{P_{j}Th_{m,j}}{\sqrt{2}\sigma_{j}} \right)\frac{1}{2\sqrt{\pi}}G_{1,2}^{2,0}\left[\frac{P_{j}^{2}T^{2}h_{m,j}^{2}} {2\sigma_{j}^{2}}\left| \begin{array}{c}1\\ 0,\frac{1}{2}\end{array} \right.\right],j=1,2 \tag{29}\]  
Due to the effect of channel fading, the \(j\)th hop's ABER can be expressed as  
\[P_{\rm e}(j)=\int_{0}^{\infty}f_{h_{m,j}}(h_{m,j})P_{\rm e}|_{h_{m,j}}(j){\rm d }h_{m,j} \tag{30}\]  
where \(P_{\rm e}(1)\) denotes the ABER of the first hop (i.e., the source-to-relay link) and \(P_{\rm e}(2)\) denotes the ABER of the second hop (i.e., the relay-to-destination link). Substituting (14) and (29) into (30), we can obtain the expression for the \(j\)th hop ABER, which is as follows:  
\[P_{\rm e}(j)=\frac{1}{2}\exp\left(-\frac{\theta_{\rm FoV,m,j}^{2 }}{2\sigma_{0}^{2}}\right)+\frac{2^{\alpha_{m,j}+\beta_{m,j}-4}\zeta_{m,j}^{2 }}{\sqrt{\pi^{3}}\Gamma(\alpha_{m,j})\Gamma(\beta_{m,j})}\left[1-\exp\left(- \frac{\theta_{\rm FoV,m,j}^{2}}{2\sigma_{0}^{2}}\right)\right]\] \[\times G_{6,3}^{2,5}\left[\frac{8\left(A_{m,j}h_{m,j}^{1}P_{j}T \right)^{2}}{\alpha_{m,j}^{2}\beta_{m,j}^{2}\sigma_{j}^{2}}\right]^{\frac{2- \zeta_{m,j}^{2}}{2}},\frac{1-\alpha_{m,j}}{2},\frac{2-\alpha_{m,j}}{2},\frac{ 1-\beta_{m,j}}{2},\frac{2-\beta_{m,j}}{2},1}{0,\frac{1}{2},\frac{-\zeta_{m,j}^ {2}}{2}},\frac{1}{2}\left.\right],j=1,2 \tag{31}\]  
According to [30], the ABER of the dual-hop system meets the following conditions:  
\[\left\{ \begin{array}{l}P_{\rm e,j}=\left(1-P_{\rm e,j-1}\right)P_{\rm e}(j) +P_{\rm e,j-1}(1-P_{\rm e}(j)),j=1,2\\ P_{\rm e,0}=0\end{array} \right. \tag{32}\]  
where \(P_{\rm e,j}\) is the ABER from the source node to the relay node \((j=1)\) or destination node \((j=2)\).