# Performance Analysis of FSO-UWOC Mixed Dual-Hop Relay System with Decode-and-Forward Protocol
## 2 System and Channel Modeling
### Derivation of the PDF and CDF of Instantaneous SNR PDF and CDF for the R-D Link
\begin{array}{rl}f_{h_{2}}(h_{2})&=\frac{\xi_{2}^{2}}{h_{2}\Gamma(a)}G_{c, 1+c}^{1+c0}\left(\left(\frac{h_{2}}{B\Delta_{0}h_{l2}}\right)^{c}\right|& \left[\begin{array}{c}\left.\left[\right.\right];\frac{1+\xi_{2}^{2}}{c}, \ldots,\frac{\xi_{2}^{2}+c-1}{c},\frac{c+\xi_{2}^{2}}{c}\\ \frac{\xi_{2}^{2}}{c},\frac{1+\xi_{2}^{2}}{c},\ldots,\frac{\xi_{2}^{2}+c-1}{c},a;\left[\begin{array}{c}\left.\right.\right]\\ \end{array}\right)\\ &=\frac{\xi_{2}^{2}}{h_{2}\Gamma(a)}G_{1,2}^{2,0}\left[\left(\frac{h_{2}}{B \Delta_{0}h_{l2}}\right)^{c}\right|&\frac{c+\xi_{2}^{2}}{\frac{\xi_{2}^{2}}{c},a}\left.\end{array}\right]\end{array}  \tag{20}\]  
where the value of \(c\) is taken as an integer (it can be rounded to the nearest integer if non-integer). The second equality in the equation arises from the equivalence simplification of the G-function, as presented in [29] (Equation 8.2.2.9). Based on the derived PDF of the hybrid fading channel coefficient \(h_{2}\), we now proceed to derive the PDF of the instantaneous received SNR \(\gamma_{2}\) for the UWOC link. 1. Similarly, in HD, the instantaneous received SNR of the UWOC link is \(\gamma_{2}=\eta\sqrt{P_{2}}h_{2}/\sigma_{2}^{2}\), and the average electronic SNR is \(\rho_{\rm heterodyne}=\eta\sqrt{P_{2}}/\sigma_{2}^{2}\). Clearly, \(h_{2}=\gamma_{2}/\rho_{\rm heterodyne}\), so the PDF of the instantaneous received SNR \(\gamma_{2}\) is given by: \[ \begin{array}{ll}f_{\gamma_{2}}(\gamma)&=f(h_{2})\Bigg{|}_{h_{2}=\frac{ \gamma}{\rho_{\rm heterodyne}}}\Big{|}\frac{\partial h_{2}}{\partial\gamma_{ 1}}\Big{|}\\ &=\frac{\xi_{2}^{2}}{\Gamma(a)\gamma}G_{1,2}^{2,0}\Big{[}\frac{1}{(bA_{0})^{ \gamma}}\Big{(}\frac{\gamma}{h_{2}\rho_{\rm heterodyne}}\Big{)}^{c}\Bigg{|} \begin{array}{c}\frac{c+\xi_{2}^{2}}{\xi^{2}}\\ \frac{\xi_{2}^{2}}{c},a\end{array}\Bigg{]}\end{array} \] (21)
2. In IM/DD, the instantaneous received SNR of the UWOC link is \(\gamma_{2}=\eta^{2}P_{2}h_{2}^{2}/\sigma_{2}^{2}\), and the average electronic SNR is \(\rho_{IM/DD}=\eta^{2}P_{2}/\sigma_{2}^{2}\). Therefore, \(h_{2}=\sqrt{\gamma_{2}/\rho_{IM/DD}}\), and based on this, the PDF of the instantaneous received SNR \(\gamma_{2}\) can be derived as: \[ \begin{array}{ll}f_{\gamma_{2}}(\gamma)&=f(h_{2})\Bigg{|}_{h_{2}=\sqrt{\frac {\gamma}{\rho_{IM/DD}}}}\Big{|}\frac{\partial h_{2}}{\partial\gamma_{1}}\Big{|} \\ &=\frac{\xi_{2}^{2}}{2\Gamma(a)\gamma}G_{1,2}^{2,0}\Bigg{[}\frac{1}{(bA_{0})^{ \gamma}}\bigg{(}\frac{\gamma}{h_{2}^{2}\rho_{IM/DD}}\bigg{)}^{\frac{c}{2}} \Bigg{|}\begin{array}{c}\frac{c+\xi_{2}^{2}}{\xi^{2}}\\ \frac{\xi_{2}^{2}}{c},a\end{array}\Bigg{]}\end{array} \] (22)  
By combining the derivations from (1) and (2) above, the unified expression for the PDF of \(\gamma_{2}\) can be obtained as:  
\[f_{\gamma_{2}}(\gamma)=\frac{\xi_{2}^{2}}{r\Gamma(a)\gamma}G_{1,2}^{2,0}\Bigg{[} \frac{1}{(bA_{0})^{\varepsilon}}\bigg{(}\frac{\gamma}{\rho_{r}}\bigg{)}^{ \xi}\Bigg{|} \begin{array}{c}\frac{c+\xi_{2}^{2}}{\xi^{2}}\\ \frac{\xi_{2}^{2}}{c},a\end{array} \Bigg{]} \tag{23}\]  
where \(r=1\) for HD and \(r=2\) for IM/DD; \(\rho_{1}=h_{I2}\rho_{\rm heterodyne}\) and \(\rho_{2}=h_{I2}^{2}\rho_{IM/DD}\). Substituting Equation (23) into the CDF definition of \(\gamma_{2}\) (given by \(F_{\gamma_{2}}(\gamma)=\int_{0}^{\gamma}f_{\gamma_{2}}(x)dx\)) yields:  
\[F_{\gamma_{2}}(\gamma)=\frac{\xi_{2}^{2}}{r\Gamma(a)}\int_{0}^{\gamma}x^{-1}G_{ 1,2}^{2,0}\Bigg{[}\bigg{(}\frac{1}{bA_{0}}\bigg{)}^{c}\rho_{r}^{-}\frac{c}{r} \frac{c}{x}\Bigg{|} \begin{array}{c}\frac{c+\xi_{2}^{2}}{\xi^{2}}\\ \frac{\xi_{2}^{2}}{c},a\end{array} \Bigg{]}dx \tag{24}\]  
Equation (24) can be derived and simplified using a process similar to that of Equation (14), leading to the following formulation:  
\[F_{\gamma_{2}}(\gamma)=\frac{\xi_{2}^{2}}{c\Gamma(a)}G_{2,3}^{2,1}\Bigg{[} \frac{1}{(bA_{0})^{\varepsilon}}\bigg{(}\frac{\gamma}{\rho_{r}}\bigg{)}^{ \xi}\Bigg{|} \begin{array}{c}\frac{\xi_{2}^{2}}{\xi^{2}}+1\\ \frac{\xi_{2}^{2}}{c},a,0\end{array} \Bigg{]} \tag{25}\]