## 2 System and Channel Models
### FSO Link  
The relay R converts the RF signal into the optical signal utilizing subcarrier intensity modulation (SIM) and forwards it to D. A direct current (DC) can be added to the signal to avoid the signal having negative values [18]. The optical signal at R can be written as:  
\[S_{opt}=G\left(1+\eta_{e}^{\frac{t}{2}}y_{R}\right) \tag{4}\]  
where \(\eta_{e}\) represents the photoelectric conversion ratio, and \(G\) is the amplification gain. The value of \(t\) depends on the type of detection method (i.e., \(t=1\) corresponds to HD and \(t=2\) corresponds to IM/DD). Through the FSO link, the signal at D can be written as:  
\[y_{D}=I^{\frac{t}{2}}G\left[1+(\eta_{e})^{\frac{t}{2}}(h_{RF}x+n_{SR})\right]+ n_{RD} \tag{5}\]  
where \(I\) denotes the channel coefficient of the FSO link. \(n_{RD}\) represents AWGN with the zero mean and \(N_{02}\) variance. Filtering out the DC bias, the signal at D can be derived as:  
\[y_{D}=(\eta_{e}I)^{\frac{t}{2}}G(h_{RF}x+n_{SR})+n_{RD} \tag{6}\]  
It is assumed that the FSO link is impaired by \(\mathcal{F}\)-distributed atmospheric turbulence \((I_{a})\), pointing errors \((I_{p})\), and path loss \((I_{l})\), in which the channel coefficient \(I=I_{a}I_{p}I_{l}\). The path loss \(I_{l}\) is given by the Beer-Lambert Law as:  
\[I_{l}=\text{exp}(-\sigma L) \tag{7}\]  
Figure 1: Block diagram of the RF-FSO system. where \(\sigma\) is the attenuation coefficient and \(L\) is the propagation distance. \(\mathcal{F}\) distribution consists of Gamma distribution and inverse Gamma distribution. The PDF of \(I_{a}\) is written as [25]:  
\[f_{I_{a}}(x)=\frac{a^{a}(b-1)^{b}x^{a-1}}{\mathcal{B}(a,b)(ax+b-1)^{a+b}} \tag{8}\]  
where \(\mathcal{B}(\cdot,\cdot)\) is the beta function. The parameters \(a\) and \(b\) describe the atmospheric turbulence conditions. Parameters \(a\) and \(b\) are computed as [25]:  
\[a=\frac{1}{\exp\left(\sigma_{\ln S}^{2}\right)-1}\quad\text{and}\quad b=\frac{ 1}{\exp\left(\sigma_{\ln L}^{2}\right)-1}+2 \tag{9}\]  
where \(\sigma_{\ln S}^{2}\) is the small-scale log-irradiance variance and \(\sigma_{\ln L}^{2}\) is the large-scale log-irradiance variance. The infinite plane waves, spherical waves, and Gaussian-beam waves can be used in the corresponding analysis. Supposing spherical wave propagation, the \(\sigma_{\ln S}^{2}\) is given by [28]:  
\[\sigma_{\ln S}^{2}=\frac{0.51\delta_{SP}^{2}\left(1+0.69\delta_{SP}^{12/5} \right)^{-5/6}}{1+0.90d^{2}\left(\sigma_{1}/\delta_{SP}\right)^{12/5}+0.62d^{ 2}\sigma_{1}^{12/5}} \tag{10}\]  
where \(\sigma_{SP}^{2}\) is the spherical wave scintillation index with weak irradiance fluctuations, which is obtained by [29]:  
\[\sigma_{SP}^{2} =9.65\sigma_{1}^{2}\bigg{\{}0.4\left(1+9/Q_{l}^{2}\right)^{11/12} \left[\sin\left(\frac{11}{6}\arctan\frac{Q_{l}}{3}\right)\right.\] \[+\left.\frac{2.61}{(9+Q_{l}^{2})^{1/4}}\sin\left(\frac{4}{3} \arctan\frac{Q_{l}}{3}\right)-\frac{0.52}{\left(9+Q_{l}^{2}\right)^{7/24}}\right. \tag{11}\] \[y_{D}=I^{\frac{t}{2}}G\left[1+(\eta_{e})^{\frac{t}{2}}(h_{RF}x+n_{SR})\right]+ n_{RD} \tag{5}\]0  
where \(\sigma_{1}^{2}=0.5C_{n}^{2}\beta^{7/6}L^{11/6}\) is the Rytov variance. \(C_{n}^{2}\) is the index of the refraction structure parameter. \(Q_{l}=10.89L/\left(\beta l_{0}^{2}\right)\). \(l_{0}\) (mm) indicates the inner scale. \(d=\sqrt{\beta D^{2}/4L}\) is the equivalent aperture diameter. \(\beta=2\pi/\lambda\) is the optical wave number and \(\lambda\) is the wavelength. \(\sigma_{\ln L}^{2}\) can be derived by \(\sigma_{\ln L}^{2}=\sigma_{\ln L}^{2}(l_{0})-\sigma_{\ln L}^{2}(L_{0})\)[28]. \(l_{0}\) and \(L_{0}\) denote the inner scale and outer scale.