# 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  
The S-R link is modeled as an FSO hybrid-fading channel, which incorporates three key components: atmospheric path loss \(h_{l1}\), GG atmospheric turbulence effects \(h_{a}\), and zero-boresight pointing error \(h_{pa}\). As established in prior studies [15; 26], atmospheric path loss \(h_{l1}\) is accurately described by the single-exponential Beer-Lambert law: \(h_{l1}=e^{-\sigma Z}\). Here, \(Z\) denotes the S-R link transmission distance, and \(\sigma\) represents the atmospheric attenuation coefficient, which is dependent on environmental factors such as visibility. For fixed values of \(Z\) and \(\sigma\), \(h_{l1}\) remains a deterministic constant. The PDF of the turbulence fading random variable following the GG distribution is given as [15; 26]:  
\[f(h_{a})=\frac{2(\alpha\beta)^{\frac{\alpha+\beta}{2}}}{\Gamma(\alpha)\Gamma( \beta)}h_{a}^{\frac{\alpha+\beta}{2}-1}K_{\alpha-\beta}\left(2\sqrt{\alpha \beta h_{a}}\right) \tag{1}\]  
in the above equation, \(\Gamma(\cdot)\) represents the Gamma function, while \(K_{\alpha-\beta}(\cdot)\) denotes the modified Bessel function of the second kind with an order of \((\alpha-\beta)\), where \(\alpha>0\), \(\beta>0\) are correlation coefficients associated with the Rytov variance \(\sigma_{R}^{2}\). In the case of a plane wave, the corresponding formula is provided in [26]:  
\[\alpha=\left[\exp\left(\frac{0.49\sigma_{R}^{2}}{\left(1+1.11\sigma_{R}^{12/5 }\right)^{7/6}}\right)-1\right]^{-1} \tag{2}\]  
\[\beta=\left[\exp\left(\frac{0.51\sigma_{R}^{2}}{\left(1+0.69\sigma_{R}^{12/5} \right)^{5/6}}\right)-1\right]^{-1} \tag{3}\]  
The zero-boresight pointing error random variable \(h_{pa}\) is used to describe the displacement of the laser source from the vertical distance \(Z\). Due to factors such as random platform jitter, the originally aligned transmitter and receiver experience random beam wandering \(R\), which causes signal attenuation at the receiver. Its mathematical description can be approximately expressed as [27]:  
\[f_{h_{pa}}\big{(}h_{pa}\big{)}=\frac{\zeta_{1}^{2}}{A_{0}^{\zeta_{1}}}h_{pa} \zeta_{1}^{2-1}\big{(}0\leq h_{pa}\leq A_{0}\big{)} \tag{4}\]  
where \(A_{0}=[erf(v)]^{2}\) represents the power ratio of the laser beam received by the receiver at the detection center \(R=0\), \(v=\sqrt{\pi}r_{a}/\sqrt{2}w_{z}\) denotes the ratio of the receiver aperture \(r_{a}\) to the beam width \(w_{z}\), \(\zeta_{1}^{2}=w_{z_{a}}^{2}/4\sigma_{s}^{2}\) is the ratio of the equivalent beam radius \(w_{seq}\) to the standard deviation \(\sigma_{s}\) of the pointing error displacement of the FSO link, and \(w_{seq}^{2}=w_{z}^{2}\sqrt{\pi}erf(v)/2 verf\big{(}-v^{2}\big{)}\) represents the equivalent beam width. The function \(erf(\cdot)\) refers to the error function. Therefore, the joint PDF of the hybrid channel fading \(h_{1}=h_{l1}h_{a}h_{pa}\) in the FSO link can be expressed as:  
\[f_{h_{1}}(h_{1})=\int_{0}^{\infty}f_{h_{1}|h_{a}}(h_{1}|h_{a})f_{h_{a}}(h_{a})dh_{a} \tag{5}\]  
where \(f_{h_{1}|h_{a}}(h_{1}|h_{a})\) represents the conditional PDF. Given the known values of \(h_{a}\) and \(h_{l1}\), the PDF can be calculated based on the random variable function value criterion [28], as follows:  
\[ \begin{array}{ll}f_{h_{1}|h_{a}}(h_{1}|h_{a})&=f_{h_{pa}}\left(h_{pa}\right) \bigg{|}_{h_{pa}=\frac{h_{1}}{h_{a}h_{l1}}}\bigg{|}\frac{\partial h_{pa}}{ \partial h_{l1}}\bigg{|}~{}=\frac{1}{h_{a}h_{l1}}f_{h_{pa}}\left(\frac{h_{1}}{ h_{a}h_{l1}}\right)\\ &=\frac{\xi_{1}^{2}}{(\alpha b_{a}h_{l1})^{2}}h_{1}^{2}-1\left(\frac{h_{1}}{h_ {l1}A_{0}}\leq h_{a}\leq+\infty\right)\end{array}  \tag{6}\]  
Substituting (6) and (1) into (4), the joint PDF of the hybrid-fading channel \(h_{1}\) for the FSO link with zero-boresight pointing error can be written as:  
\[