# 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  
The R-D link (i.e., the UWOC link) also experiences hybrid channel fading due to path loss \(h_{l2}\) caused by seawater absorption and scattering effects, oceanic turbulence \(h_{f}\), and zero-boresight pointing errors \(h_{po}\). The PDF of oceanic turbulence fading under the GGD model is given as follows [16]:  
\[f_{h_{f}}\left(h_{f}\right)=\frac{c\cdot h_{f}^{ac-1}}{b^{ac}}\frac{\exp\left( -\left(\frac{h_{f}}{b}\right)^{c}\right)}{\Gamma(a)},h_{f}>0 \tag{15}\]  
where \(b\) is a scale factor; \(a\) and \(c\) are shape parameters. Defining the scintillation index \(\sigma_{1}^{2}\) for turbulence as the normalized variance of the received light intensity fluctuations [21], and, furthermore, with the constraint of expectation \(E[h_{f}]=1\), the relationships among the three parameters \(a\), \(b\), \(c\) in GGD expression above and the scintillation index \(\sigma_{1}^{2}\) can be calculated by:  
\[\sigma_{1}^{2}=\frac{\Gamma(a)\Gamma(a+2/c)}{\Gamma^{2}(a+1/c)}-1 \tag{16}\]  
\[b=\frac{\Gamma(a)}{\Gamma(a+1/c)} \tag{17}\]Clearly, by using Equations (16) and (17), we can determine the three parameters of the GGD model by predefining the scintillation index \(\sigma_{l}^{2}\) that characterizes the turbulence intensity and the shape parameter \(c\) in subsequent simulations. Assume that the hybrid channel fading coefficient of the R-D link is \(h_{2}=h_{f}h_{p}h_{l2}\), where \(h_{l2}\) represents the UWOC link path loss. According to the research in [19], the path loss considering the absorption and scattering effects of seawater as well as geometric losses in the UWOC link can be accurately expressed by the Elamassie closed-form path-loss formula, i.e., \(h_{l2}=\left(\frac{D_{R}}{\theta d}\right)^{2}e^{-c_{0}d\left(\frac{D_{R}}{ \theta d}\right)^{T}}\). This is a function of the receiver aperture diameter \(D_{R}\), beam divergence angle \(\theta\), link transmission distance \(d\), and extinction coefficient \(c_{0}\). Clearly, once variables such as \(D_{R}\), \(\theta\), and \(c_{0}\) are fixed, \(h_{l2}\) is a variable related to the transmission distance \(d\) between the transmitter and receiver nodes, which can be used to describe the path loss in the UWOC system. From the proof process of Equation (6), it can be concluded that the conditional PDF of \(h_{2}\) is given by:  
\[ \begin{array}{rl}f_{h_{2}|h_{f}}\left(h_{2}\middle|h_{f}\right)&=f_{h_{p}} \left(h_{p}\right)\middle|_{h_{p}=\frac{h_{2}}{h_{p}h_{2}}}\left|\frac{\partial h _{p}}{\partial h_{2}}\right|&=\frac{1}{h_{f}h_{2}}f_{h_{p}}\left(\frac{h_{2}} {h_{f}h_{l2}}\right)\\ &=\frac{\xi_{2}^{2}}{\left(A_{0}h_{f}h_{2}\right)^{2}}\frac{h_{2}^{2}-1}{2} \left(\frac{h_{2}}{A_{0}h_{l2}}\leq h_{f}\leq+\infty\right)\end{array}  \tag{18}\]  
where \(\xi_{2}^{2}\) represents the ratio of the equivalent beam waist radius to the standard deviation of the pointing error displacement in the UWOC link. From the proof of Equation (7), the PDF of the composite fading channel \(h_{2}\) for the UWOC link can be expressed as:  
\[ \begin{array}{rl}f_{h_{2}}(h_{2})&=\frac{\int_{h_{2}}^{+\infty} \left.f_{h_{2}}\right|h_{f}}{A_{0}h_{l2}}\left(h_{2}\middle|h_{f}\right)f_{h_{ f}}\left(h_{f}\right)dh_{f}\\ &=\frac{c\xi_{2}^{2}h_{2}^{\xi_{2}^{2}-1}}{b^{ac}(A_{0}h_{l2})^{\xi_{2}^{2}} \Gamma(a)}\frac{\int_{h_{2}}^{+\infty}\left.h_{f}^{\infty-1-\xi_{2}^{2}}\exp \left(-\left(\frac{h_{f}}{b}\right)^{c}\right)dh_{f}\right.}\end{array}  \tag{19}\]  
Using Equation \(e^{-x}=C_{0,1}^{1,0}\left(x\middle| \begin{array}{c}-\\ 0\end{array} \right)\)[29] (Equation 8.4.3.1), and applying [29] (Equations 2.24.2.3, 8.2.2.15), Equation (19) can be further rewritten as:  
\[