## 3 Statistics of the SNR of an Equivalent End-to-End Communication Link  
### CDF of the End-to-End SNR  
From Equation (12) in [7] and the definition of the Fox-H function in Equation (1.1) in [12], the PDF of the received SNRs for both detection techniques can be expressed by  
\[f_{\gamma_{i}}(\gamma)=\Delta_{i}\gamma^{-1}H_{2,2}^{2,1}\left[\frac{\theta_{ i}\gamma}{\bar{\gamma}_{i}}\left| \begin{array}{c}(1-b_{i},r_{i}),\left(1+z_{i}^ {2},r_{i}\right)\\ (a_{i},r_{i}),\left(z_{i}^{2},r_{i}\right)\end{array} \right.\right], \tag{6}\]  
where \(\Delta_{i}=\frac{z_{i}^{2}}{\Gamma(a)\Gamma(b_{i})}\), \(\theta_{i}=\frac{\left(a_{i}z_{i}^{2}\right)^{\gamma_{i}}}{\left(b_{i}-1\right) ^{\gamma_{i}}\left(1+z_{i}^{2}\right)^{\gamma_{i}}}\), and \(H_{p,q}^{m,n}[\cdot]\) denotes the Fox-H function. Here, \(r=1\) represents HD and \(r=2\) represents IM/DD. Furthermore, \(\bar{\gamma}_{i}\) is the average electrical SNR of the \(i\)-th link, which can be defined as \(\bar{\gamma}_{i}=\bar{\mu}_{i}\) for \(r_{i}=1\). For \(r_{i}=2\), \(\bar{\gamma}_{i}=\bar{\mu}_{i}a_{i}b_{i}z_{i}^{2}\left(z_{i}^{2}+1\right)/\left[ (a_{i}+1)(b_{i}+1)\left(z_{i}^{2}+1\right)^{2}\right]\), where the average SNR is \(\bar{\mu}_{i}=\mathbb{E}[\gamma_{i}]\) and \(\mathbb{E}\) is the expectation operator. Moreover, by integrating (6), the CDF of SNR of the \(i\)-th FSO link can be expressed by  
\[F_{\gamma_{i}}(\gamma)=r_{i}\Delta_{i}H_{3,3}^{2,2}\left[\frac{\theta_{i}\gamma }{\bar{\gamma}_{i}}\left| \begin{array}{c}(1-b_{i},r_{i}),(1,r_{i}),\left(1+ z_{i}^{2},r_{i}\right)\\ (a_{i},r_{i}),\left(z_{i}^{2},r_{i}\right),\left(0,r_{i}\right)\end{array} \right.\right]. \tag{7}\]For the proposed AF relaying dual-hop system, the CDF of the end-to-end SNR, \(\gamma_{c2e}\), can be calculated as  
\[F_{\gamma_{c2e}}(\gamma)=\int_{0}^{\infty}\text{Pr}\bigg{[}\frac{ \gamma_{1}\gamma_{2}}{\gamma_{2}+\mathcal{C}}<\gamma\bigg{|}\gamma_{2}\bigg{]}f _{\gamma_{2}}(\gamma_{2})d\gamma_{2} \tag{8}\] \[=F_{\gamma_{1}}(\gamma)+\int_{0}^{\infty}f_{\gamma_{1}}(x+\gamma )F_{\gamma_{2}}\bigg{(}\frac{\mathcal{C}\gamma}{x}\bigg{)}dx.\]  
By using the definition of the univariate Fox-\(H\) function Equation (1.1) in [12], we can write the integral in (8) as  
\[F_{\gamma_{c2e}}(\gamma)=F_{\gamma_{1}}(\gamma)-\frac{r_{2} \Delta_{1}\Delta_{2}}{4\pi^{2}}\int_{\mathcal{C}_{1}}\int_{\mathcal{C}_{2}} \int_{0}^{\infty}\ \ (x+\gamma)^{-1}\Big{(}\frac{\theta_{1}(x+\gamma)}{\bar{ \tau}_{1}}\Big{)}^{-\tau_{1}}\] \[\times\frac{\Gamma(a_{1}+r_{1}\tau_{1})\Gamma\big{(}z_{1}^{2}+r_ {1}\tau_{1}\big{)}\Gamma(b_{1}-r_{1}\tau_{1})}{\Gamma\big{(}1+z_{1}^{2}+r_{1} \tau_{1}\big{)}}\bigg{(}\frac{C\theta_{2}\gamma}{\bar{\gamma}_{2}x}\bigg{)}^{- \tau_{2}}\] \[\times\frac{\Gamma(a_{2}+r_{2}\tau_{2})\Gamma\big{(}z_{2}^{2}+r_{ 2}\tau_{2}\big{)}\Gamma(b_{2}-r_{2}\tau_{2})\Gamma(-r_{2}\tau_{2})}{\Gamma(1- r_{2}\tau_{2})\Gamma\big{(}1+z_{2}^{2}+r_{2}\tau_{2}\big{)}}dxd\tau_{1}d\tau_{2}, \tag{9}\]  
where \(\mathcal{C}_{1}\) and \(\mathcal{C}_{2}\) represent the \(\tau_{1}\)-plane and \(\tau_{2}\)-plane contours, respectively. Moreover, by integrating \(\int_{0}^{\infty}\frac{x^{2}}{(x+\gamma)^{1+\tau_{1}}}dx=\gamma^{-\tau_{1}+ \frac{\Gamma(\tau_{1}-\tau_{2})\Gamma(1+\tau_{2})}{\Gamma(1+\tau_{1})}}\), changing the integral variable \(\tau_{1}\rightarrow-\tau_{1}\), utilizing Equation (A.1) in [12], and performing some manipulations, we can obtain the CDF of the end-to-end SNR, \(\gamma_{c2e}\), as  
\[F_{\gamma_{c2e}}(\gamma)=F_{\gamma_{1}}(\gamma)+r_{2}\Delta_{1} \Delta_{2}H_{1,0,2,3,3,4}^{0,1;1,2;3,2}\bigg{[}\frac{\bar{\gamma}_{1}}{\bar{ \theta}_{1}\gamma},\frac{C\theta_{2}}{\bar{\gamma}_{2}}\big{|}(1;1,1)\bigg{|} \begin{matrix}\psi_{1}\\ \psi_{2}\end{matrix} \bigg{|} \begin{matrix}\psi_{3}\\ \psi_{4}\end{matrix}