# 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 S-R Ink
\begin{array}{ll}f_{h_{1}}(h_{1})&=\int\frac{+\infty}{h_{1}}f_{h_{1}|h_{a}}( h_{1}|h_{a})f_{h_{a}}(h_{a})dh_{a}\\ &=\frac{2(\alpha\beta)\frac{\alpha+\beta}{2}}{\Gamma(\alpha)\Gamma(\beta)} \frac{\xi_{1}^{2}}{(A_{0}h_{l1})^{5}_{1}}h_{1}^{2}\int\frac{+\infty}{h_{1}} \frac{\alpha+\beta}{2}-\xi_{1}^{2}-1}K_{\alpha-\beta}(2\sqrt{\alpha\beta h_{ a}})dh_{a}\end{array}  \tag{7}\]  
Applying the identity \(K_{v}(2\sqrt{x})=\frac{1}{2}G_{0,2}^{2,0}\bigg{(}x\bigg{|}-\bigg{)}\)[29] (Equation 8.4.23.1), along with [28] (Equations 2.24.2.3 and 8.2.2.15), Equation (7) can be further simplified as follows:  
\[f_{h_{1}}(h_{1})=\frac{\xi_{1}^{2}}{\Gamma(\alpha)\Gamma(\beta)h_{1}}G_{1,3}^{ 3,0}\bigg{[}\frac{\alpha\beta h_{1}}{A_{0}h_{l1}}\bigg{|}\frac{1+\xi_{1}^{2}} {\xi_{1}^{2},\alpha,\beta}\bigg{]} \tag{8}\]  
The symbol \(G_{p,q}^{m,n}(\cdot)\) represents the Meijer-G function [30] (Equation 9.301). Based on the definition of the moment-generating function for a random variable and referring to [30] (Equation 7.811.4), the \(n\)-th order raw moment is derived as:  
\[E[h_{1}^{n}]=\frac{\xi_{1}^{2}}{\Gamma(\alpha)\Gamma(\beta)}\frac{\Gamma(\xi_{ 1}^{2}+n)\Gamma(\alpha+n)\Gamma(\beta+n)}{\Gamma\big{(}1+\xi_{1}^{2}+n\big{)} }\bigg{(}\frac{\alpha\beta}{A_{0}h_{l1}}\bigg{)}^{-n} \tag{9}\]  
1. The HD [31], i.e., coherent detection, combines a weak optical signal with a strong local oscillator signal at a photodetector. Based on the formulae in [31], in the case of the HD scheme, the instantaneous received SNR for the S-R link is denoted as \(\gamma_{1}=\sqrt{P_{1}}h_{1}/\sigma_{1}^{2}\), the average electronic received SNR is \(\mu_{\text{heterodyne}}=\frac{\sqrt{P_{1}}E[h_{1}]}{\sigma_{1}^{2}}=\frac{ \sqrt{P_{1}}A_{0}\xi_{1}^{2}h_{1}}{\big{(}1+\xi_{1}^{2}\big{)}\sigma_{1}^{2}}\). Let \(\eta_{1}=\frac{\xi_{1}^{2}}{\big{(}1+\xi_{1}^{2}\big{)}}\), then \(h_{1}=\frac{A_{0}h_{l1}\eta_{1}\gamma_{1}}{\mu_{\text{heterodyne}}}\), and the PDF of the instantaneous SNR \(\gamma\) is given by: \[ \begin{array}{ll}f_{\gamma_{1}}(\gamma)&=f(h_{1})\bigg{|}_{h_{1}=\frac{A_{0}h_ {1}h_{1}\gamma_{1}}{\mu_{\text{heterodyne}}}}\bigg{|}\frac{\partial h_{1}}{ \partial\gamma}\bigg{|}\\ &=\frac{\xi_{1}^{2}}{\Gamma(\alpha)\Gamma(\beta)\gamma}G_{1,3}^{3,0}\bigg{[} \eta_{1}\alpha\beta\frac{\gamma}{\mu_{\text{heterodyne}}}\bigg{|}\begin{array}[ ]{c}1+\xi_{1}^{2}\\ \xi_{1}^{2},\alpha,\beta\end{array}\bigg{]}\end{array} \] (10)
2. In the case of the IM/DD scheme [31], the instantaneous received SNR for the S-R link is denoted as \(\gamma_{1}=P_{1}h_{1}^{2}/\sigma_{1}^{2}\), and the average electronic received SNR is \[\mu_{IM/DD}=\frac{P_{1}E^{2}[h_{1}]}{\sigma_{1}^{2}}=\frac{P_{1}A_{2}\sigma_{1}^{2 }\sigma_{1}^{4}h_{1}^{2}}{\left(1+\varsigma_{1}^{2}\right)\sigma_{1}^{2}}.\]  
Let \(\eta_{1}=\frac{\varsigma_{1}^{2}}{\left(1+\varsigma_{1}^{2}\right)}\), then \(h_{1}^{2}=\frac{A_{2}^{2}h_{1}^{2}\eta_{1}^{2}\tau_{1}}{\mu_{IM/DD}}\), and the PDF of the instantaneous SNR \(\gamma\) is given by:  
\[ \begin{array}{ll}f_{\gamma_{1}}(\gamma)&=f(h_{1})\left|_{h_{1}=\frac{A_{0}h_ {1}\gamma_{1}\gamma_{1}}{\sqrt{\mu_{IM/DD}}}}\right|\frac{\left|\frac{\partial h _{1}}{\partial\tau}\right|}{\partial\tau}\\ &=\frac{\varsigma_{1}^{2}}{2\Gamma(\alpha)\Gamma(\beta)\gamma}G_{1,3}^{3,0} \left[\eta_{1}\alpha\beta\sqrt{\frac{\gamma}{\mu_{IM/DD}}}\right|\begin{array} []{c}1+\varsigma_{1}^{2}\\ \varsigma_{1}^{2},\alpha,\beta\end{array}\right]\end{array}  \tag{11}\]  
Combining the conclusions from (1) and (2), the following unified expression is obtained:  
\[f_{\gamma_{1}}(\gamma)=\frac{\varsigma_{1}^{2}}{r\Gamma(\alpha)\Gamma(\beta) \gamma}G_{1,3}^{3,0}\left[\eta_{1}\alpha\beta\left(\frac{\gamma}{\mu_{r}} \right)^{\frac{1}{r}}\right| \begin{array}{c}1+\varsigma_{1}^{2}\\ \varsigma_{1}^{2},\alpha,\beta\end{array} \right] \tag{12}\]  
In the above expression, \(r\) is a constant related to the detection type, where \(r=1\) represents HD and \(r=2\) represents IM/DD; \(\mu_{1}=\mu_{\text{heterodyne}}\), \(\mu_{2}=\mu_{\text{IM/DD}}\).