## 2 System and Channel Model
### One-Hop Hybrid FSO/RF Link Based on Selective Combination Scheme  
In the one-hop hybrid FSO/RF sub-link with the selective combination scheme, the receiver detects two branch signals at the same time and selects the signal with the maximum SNR to demodulate. Then, the output SNR \(\gamma^{\text{SC}}\) of the selective combiner on the one-hop hybrid link can be expressed as [15]:  
\[\gamma^{\text{SC}}=\text{max}(\gamma^{\text{FSO}},\gamma^{\text{RF}}). \tag{8}\]  
The CDF of the SNR \(\gamma^{\text{SC}}\) can be expressed as [15]:  
\[F_{\gamma^{\text{SC}}}(\gamma)=\text{Pr}(\text{max}(\gamma^{\text{FSO}}, \gamma^{\text{RF}})\leq\gamma)=\text{Pr}(\gamma^{\text{FSO}}\leq\gamma,\gamma ^{\text{RF}}\leq\gamma)= F_{\gamma^{\text{FSO}}}(\gamma)F_{\gamma^{\text{RF}}}(\gamma). \tag{9}\]  
Adding subscripts \((x,y)\) to all variables in the Malaga turbulence model and Nakagami fading channel, when the subscript is \((x,y)=(i,j)\), it represents the \(j\)-th hop hybrid FSO/RF link of the \(i\)-th path, when the subscript is \((x,y)=(s,d)\), it represents the direct hybrid FSO/RF link. Substituting Equations (5) and (7) into Equation (9), the CDF of the selective combiner output SNR \(\gamma_{s,d}^{\text{SC}}\) of the direct channel and the CDF of the selective combiner output SNR \(\gamma_{i,j}^{\text{SC}}\) of the \(j\)-th hop of the \(i\)-th path can be obtained as follows:  
\[F_{\gamma_{i,g}^{\text{SC}}}(\gamma)=\frac{D_{x,y}}{\Gamma(m_{x,y})}\sum_{k=1} ^{\beta_{x,y}}c_{x,y}^{k}G_{1,2}^{1,1}\left(\frac{m_{x,y}\gamma}{\overline{ \gamma}_{x,y}^{\text{RF}}}\middle| \begin{array}{c}1\\ m_{x,y},0\end{array} \right)G_{r+1,3r+1}^{3r,1}\left(E_{x,y}\frac{\gamma}{ \mu_{x,y}^{\text{FSO},r}}\middle| \begin{array}{c}1,\kappa_{x,y}^{1}\\ \kappa_{x,y}^{2},0\end{array} \right). \tag{10}\]  
According to formula (07.34.16.0003.01) in Ref. [21], the product of two Meijer-G functions in Equation (10) can be replaced by the extended generalized bivariate [PERSON]'s G-function (EGBMGF),  
\[F_{\gamma_{i,g}^{\text{SC}}}(\gamma)=\frac{D_{x,y}}{\Gamma(m_{x,y})}\left[ \sum_{k=1}^{\beta_{x,y}}c_{x,y}^{k}G_{0,0,1,2r+1,3r+1}^{0,1;1;3r,1}\!\!\left( \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \  
According to formula (3.6.1) in Refs. [24; 25],\(\int_{a}^{\infty}(x-a)\tau\exp(-b(x-a))f(x)dx\approx\sum\limits_{\tau=1}^{n}w_{ \tau}f(x_{\tau})\) is the approximate expression of the generalized Gauss-Laguerre integral, where \(w_{\tau}\) is the weight function and \(x_{\tau}\) is the special point of abscissa. Therefore, Equation (18) can be simplified as:  
\[\begin{split} P_{\text{b}}^{\text{multi}}&=\frac{q^{p }}{2\Gamma(p)}\frac{D_{s,d}}{\Gamma(m_{s,d})}\sum\limits_{\tau=1}^{n}w_{\tau} \Biggl{[}\sum\limits_{k=1}^{  
calculating the approximate value of the generalized Gauss-Laguerre quadrature function with \(\tau=30\), the ABER of the relay-aided hybrid system can be obtained according to Equation (19). To simplify the simulation analysis, it is assumed in this section that the atmospheric turbulence intensity is consistent for all FSO links in the relay-aided system (but the atmospheric scintillation index also varies depending on the transmission distance), and the fading parameters of all RF links are also the same. The structure parameters (\(N=2\), \(M=3\)), (\(N=2\), \(M=5\)), and (\(N=4\), \(M=3\)) have been selected to avoid entanglement. According to the meaning of the scintillation index, the turbulence parameters set for the 1 km FSO link under the Malaga turbulence are selected as (\(\alpha=5\), \(\beta=3\), \(\rho=0.25\)), (\(\alpha=3\), \(\beta=2\), \(\rho=0.75\)), and (\(\alpha=2\), \(\beta=1\), \(\rho=0.9\)), respectively, for weak, moderate, and strong turbulence conditions. For weak turbulence, the parameters of the Malaga turbulence are (\(\alpha=0.36\), \(\beta=1\), \(\rho=0.95\)) and (\(\alpha=0.0291\), \(\beta=1\), \(\rho=0.95\)), respectively, for 2 km and 3 km FSO links. Other parameters for the Malaga turbulence include \(\Omega=1.3265\), \(b_{0}=0.1079\), and \(\varphi_{A}-\varphi_{B}=\pi/2\).