# Performance Analysis of FSO-UWOC Mixed Dual-Hop Relay System with Decode-and-Forward Protocol
## 4 System Performance Analysis
### Average Channel Capacity  
The average channel capacity is determined by the expectation of the instantaneous mutual information between the source and destination nodes [37]:  
\[C=\frac{1}{2}E[\log_{2}(1+\tau\gamma)]=\frac{1}{2\ln 2}\int_{0}^{\infty}\ln(1+ \tau\gamma)f_{\gamma^{DF}}(\gamma)d\gamma \tag{44}\]  
where \(\gamma\) represents the end-to-end SNR of the hybrid dual-hop system, and \(\tau\) is a constant whose value is determined by the specific receiver detection method. When \(r=2\), corresponding to the IM/DD scheme, the derivation in [38] shows that \(\tau=e/2\pi\), and the channel capacity calculated by the above formula is actually the lower bound of the system's capacity. On the other hand, when \(\tau=1\), the analysis in [39] indicates that the formula corresponds to the exact value of the system's channel capacity under HD, rather than the lower bound. Substituting Equation (28) into Equation (44) and performing transformations and manipulations using [29] (Equations 8.4.6.5, 8.3.2.21), as well as [40] (Equation 2.1.4), the average channel capacity of the mixed dual-hop transmission system under the DF relaying protocol can be obtained as:  
\[C =\frac{1}{2\ln 2}\int_{0}^{\infty}H_{22}^{2}\bigg{(}\tau\bigg{|} \begin{array}{c}(1,1),(1,1)\\ (1,1),(0,1)\end{array} \bigg{)}f_{\gamma}^{DF}(\gamma)d\gamma \tag{45}\] \[=\underbrace{\frac{1}{2\ln 2}\frac{\delta_{11}^{2}}{\Gamma(a) \Gamma(\beta)}\int_{0}^{\infty}\gamma^{-1}H_{1,3}^{20}\bigg{[}\frac{(\eta_{1} \alpha\beta)^{\prime}}{\mu_{r}}\gamma\bigg{|} \begin{array}{c}(1+\delta_{1}^{ 2},r)\\ (\delta_{1}^{2},r),(\alpha,r),(\beta,r)\end{array} \bigg{]}}_{\text{$\ell_{1} $}}H_{22}^{12}\bigg{(}\tau\gamma\bigg{|} \begin{array}{c}(1,1),(1,1)\\ (1,1),(0,1)\end{array} \bigg{)}d\gamma\] \[+\underbrace{\frac{1}{2\ln 2}\frac{\delta_{21}^{2}}{\varepsilon \Gamma(a)}\int_{0}^{\infty}\gamma^{-1}H_{1,2}^{20}\Bigg{[}\frac{1}{(\frac{ \delta_{21}^{2}}{\varepsilon}+1,\frac{r}{\varepsilon})}{(\frac{\delta_{21}^{2} }{\varepsilon},\frac{r}{\varepsilon})}\Bigg{]}}_{\text{$\ell_{2}$}}H_{22}^{12} \bigg{(}\tau\gamma\bigg{|} \begin{array}{c}(1,1),(1,1)\\ (1,1),(0,1)\end{array} \bigg{)}d\gamma\] \[-\underbrace{\frac{1}{2\ln 2}\frac{\delta_{11}^{2}\delta_{22}^{2}}{ \varepsilon\Gamma(a)\Gamma(a)\Gamma(\beta)}\int_{0}^{\infty}\gamma^{-1}G_{22}^ {12}\bigg{(}\tau\gamma\bigg{|} \begin{array}{c}1,1\\ 1,0\end{array} \bigg{)}G_{1,3}^{30}\Bigg{[}\eta_{1}\alpha\beta\Big{(}\frac{ \gamma}{\mu_{r}}\Big{)}\frac{1}{\tau}\bigg{|} \begin{array}{c}1+\delta_{1}^{ 2}\\ \delta_{1}^{2},\alpha,\beta\\ \delta_{1}^{2},\alpha,0\end{array} \bigg{]}}_{\text{$\ell_{2}$}}G_{23}^{23} \Bigg{[}\frac{1}{(\frac{1}{\rho_{0}})^{\varepsilon}}\bigg{(}\frac{\gamma}{ \mu_{r}}\Big{)}\frac{1}{\tau}\bigg{|} \begin{array}{c}1+\delta_{1}^{2}\\ \delta_{1}^{2},\alpha,\beta\\ \delta_{1}^{2},\alpha,\beta\end{array} \bigg{]}}_{\text{$\ell_{2}$}}\] \[-\underbrace{\frac{1}{2\ln 2}\frac{\delta_{11}^{2}\delta_{22}^{2}}{ \Gamma(a)\Gamma(a)\Gamma(\beta)}\int_{0}^{\infty}\gamma^{-1}G_{22}^{12} \bigg{(}\tau\gamma\bigg{|} \begin{array}{c}1,1\\ 1,0\end{array} \bigg{)}G_{24}^{31}\Bigg{[}\eta_{1}\alpha\beta\Big{(}\frac{\gamma }{\mu_{r}}\Big{)}\frac{1}{\tau}\bigg{|} \begin{array}{c}1,\delta_{1}^{2}+1 \\ \delta_{1}^{2},\alpha,\beta,0\end{array} \bigg{]}G_{1,2}^{20}\Bigg{[}\frac{1}{( \frac{\gamma}{\mu_{r}})\frac{\varepsilon}{\tau}\bigg{|} \begin{array}{c}\frac{ \varepsilon+\delta_{1}^{2}}{\varepsilon}\\ \delta_{1}^{2},\alpha,\beta}\end{array} \bigg{]}}_{\text{$\ell_{2}$}}d\gamma\] \[=I_{\text{$\ell_{1}$}}+I_{\text{$\ell_{2}$}}-I_{\text{$\ell_{3}$}}-I_ {\text{$\ell_{4}$}}.\]  
Applying ([41], Equation 2.8.4), \(I_{\text{$\ell_{1}$}}\) can be expressed as:  
\[I_{\text{$\ell_{1}$}}=\frac{\delta_{1}^{2}}{2\ln 2\Gamma(\alpha)\Gamma(\beta)}H_{3,5}^{5,1 }\Bigg{[}\frac{(\eta_{1}\alpha\beta)^{\prime}}{\mu_{r}\tau}\bigg{|} \begin{array} []{c}(0,1);(1,1),(1+\delta_{1}^{2},r)\\ (\delta_{1}^{2},r),(\alpha,r),(\beta,r),(0,1),(0,1)\end{array}