## 4 Performance Evaluation of Mixed Plc-Fso System
### Bit Error Rate Analysis  
For a variety of binary and non-binary modulation schemes, a unified expression of the BER is given as in [1; 6; 9]  
\[\bar{P}_{b}=\frac{\delta}{2\Gamma(p)}\sum_{k=1}^{n^{\prime}}q_{k} ^{p}\int_{0}^{\infty}\gamma^{p-1}e^{-q_{k}\gamma}F_{\gamma_{e}}(\gamma)d\gamma\] \[=\frac{\delta n^{\prime}}{2}-\frac{\delta}{2\Gamma(p)}\sum_{k=1}^ {n^{\prime}}\big{(}\Omega\mathcal{I}_{1}-\Delta_{1}\Omega\mathcal{I}_{2}- \Delta_{2}\Omega\mathcal{I}_{3}\big{)}, \tag{13}\]  
where \(n^{\prime},\delta,p\), and \(q_{k}\) are used to denote the parameters for different modulation schemes, as defined in [1; 6; 9]. Furthermore, we have terms \(\mathcal{I}_{1}\), \(\mathcal{I}_{2}\), and \(\mathcal{I}_{3}\) which can be expressed as  
\[\mathcal{I}_{1} =q_{k}^{p}\int_{0}^{\infty}\gamma^{p-1}e^{-q_{k}\gamma}H_{2,4}^{4, 0}\bigg{[}\frac{t}{\psi}\bigg{|}\begin{array}{c}(1,r)(\xi^{2}+1,r)\\ (\xi^{2},r)(\alpha,r)(\beta,r)(0,r)\end{array}\bigg{]}dt, \tag{14}\] \[\mathcal{I}_{2} =q_{k}^{p}\int_{0}^{\infty}\gamma^{p-1}e^{-q_{k}\gamma}H_{1,2}^{1,1}\bigg{[}\frac{\varphi t}{\Phi}\bigg{|}_{(m_{t},1)\ (0,1)}^{(1,1)}\bigg{]}\times H_{2,4}^{4,0}\bigg{[}\frac{t}{\psi}\bigg{|} \begin{array}{c}(1,r)(\xi^{2}+1,r)\\ (\xi^{2},r)(\alpha,r)(\beta,r)(0,r)\end{array}\bigg{]}dt,\] (15) \[\mathcal{I}_{3} =q_{k}^{p}\int_{0}^{\infty}\gamma^{p-1}e^{-q_{k}\gamma}H_{1,2}^{1,1}\bigg{[}\frac{\varphi^{\prime}t}{\Phi^{\prime}}\bigg{|}_{(m_{t}^{\prime},1) \ (0,1)}^{(1,1)}\bigg{]}\times H_{2,4}^{4,0}\bigg{[}\frac{t}{\psi}\bigg{|} \begin{array}{c}(1,r)(\xi^{2}+1,r)\\ (\xi^{2},r)(\alpha,r)(\beta,r)(0,r)\end{array}\bigg{]}dt. \tag{16}\]  
Then, by expanding the Fox H-function ([16], Equation (1.2)), we have \(\mathcal{I}_{1}\) as  
\[\mathcal{I}_{1}=\frac{q_{k}^{p}}{2\pi j}\int_{0}^{\infty}\int_{C}\frac{\Gamma( \xi^{2}+rs)\Gamma(\alpha+rs)\Gamma(\beta+rs)\Gamma(0+rs)}{\Gamma(1+rs)\Gamma(1+ \xi^{2}+rs)}\bigg{(}\frac{t}{\psi}\bigg{)}^{-s}t^{p-1}e^{-q_{k}t}dsdt \tag{17}\]  
\[P^{\text{out}}(\gamma_{th})\approx 1+\frac{\Omega}{r}\times\Bigg{[}\frac{\Gamma\big{(}\alpha-\xi^{2} \big{)}\Gamma\big{(}\beta-\xi^{2}\big{)}\Gamma\big{(}-\xi^{2}\big{)}}{\Gamma(1- \xi^{2})}\bigg{(}\frac{\gamma}{\psi}\bigg{)}^{\frac{\xi^{2}}{r}}+\frac{\Gamma \big{(}\xi^{2}-\alpha\big{)}\Gamma\beta-\dot{\alpha}\Gamma(-\alpha)}{\Gamma(1- \alpha)\Gamma(1+\xi^{2}-\alpha)}\bigg{(}\frac{\gamma}{\psi}\bigg{)}^{\frac{ \alpha}{r}}\] \[+\frac{\Gamma\big{(}\xi^{2}-\beta\big{)}\Gamma(\alpha-\beta) \Gamma(-\beta)}{\Gamma(1-\beta)\Gamma(1+\xi^{2}-\beta)}\bigg{(}\frac{\gamma}{ \psi}\bigg{)}^{\beta/r}+\frac{\Gamma\big{(}\xi^{2}\big{)}\Gamma(\alpha)\Gamma (\beta)}{\Gamma(1+\xi^{2})}\Bigg{]}\] \[\times\Bigg{[}1-\Delta_{1}\times\frac{\Gamma(\varphi)}{\Gamma(1 +\varphi)}\Big{(}\frac{\varphi\gamma}{\phi}\Big{)}^{\varphi}-\Delta_{2} \times\frac{\Gamma(\varphi\prime)}{\Gamma(1+\varphi\prime)}\bigg{(}\frac{ \varphi\prime\gamma}{\phi\prime}\bigg{)}^{\varphi\prime}\Bigg{]}. \tag{23}\]  
Consequently, the diversity order of the dual-hop system is defined as  
\[d=-\lim_{\hat{\gamma}\rightarrow\infty}\frac{\log\big{(}P^{\text{out}}\big{)} }{\log\left(\hat{\gamma}\right)}, \tag{24}\]  
and can be given by  
\[d=\min\bigg{\{}\frac{\xi^{2}}{r},\frac{\alpha}{r},\frac{\beta}{r},\varphi, \varphi\prime\bigg{\}}. \tag{25}\]