## 2 System and Channel Model  
We consider an end-to-end hybrid RF-FSO system, it mainly consists of three parts--a source and a destination with a relay between them, the system model is shown below in Figure 1. The signal is received by the relay from the source through the RF channel, which obeys Rayleigh distribution. Considering the relatively lower implementation complexity [16], we adopt a fixed-gain amplify-and-forward (AF) relay which amplifies the RF signals and forwards them to next node directly [12]. In other words, the received RF signal will be transformed into an optical signal by the relay using the SIM technique, and the optical signal will be amplified by a fixed gain, G. After that, it is transmitted to the destination through the FSO link, which is supposed to obey Gamma-Gamma distribution, and IM/DD is used as the detection method in the destination. The received signal of the relaying and the destination can be expressed \(y_{1}=h_{1}x+n_{1}\) and \(y_{2}=h_{2}y_{1}+n_{2}\), so the comprehensive expression is \(y_{2}=h_{2}(h_{1}x+n_{1})+n_{2}\), where \(y_{1}\) and \(y_{2}\) denote the received signal, \(x\) represents the normalized signal from signal source, \(h_{1}\) and \(h_{2}\) are the channel coefficient of the RF link and the FSO link, respectively. \(h_{2}=IG\eta\), where \(I\) represents the irradiance intensity of FSO channel, \(G\) denotes the fixed gain of relaying scheme, and \(\eta\) represents the conversion coefficient of electrical-to-optical. \(n_{1}\) and \(n_{2}\) are additive white Gaussian noise (AWGN) with \(\sigma_{1}\)\({}^{2}\) and \(\sigma_{2}\)\({}^{2}\) as variance and mean of zero. The RF link follows Rayleigh distribution in the hybrid RF-FSO system, the probability density function (PDF) of the RF link could be given as \(f_{\gamma_{1}}(\gamma_{1})=\frac{1}{\gamma_{1}}\exp\biggl{(}-\frac{\gamma_{1}}{ \gamma_{1}}\biggr{)}\), where \(\gamma_{1}\) represents the instantaneous signal to noise ratio (SNR) of the RF link and \(\overline{\gamma_{1}}\) denotes the average SNR in the RF channel. Using the definition of the cumulative distribution function (CDF), \(F_{1}=\int\limits_{0}^{\gamma_{1}}f_{\gamma_{1}}(\gamma)d\gamma_{1}\). Hence, the CDF of the Rayleigh distribution can be expressed as \(F_{1}(\gamma_{1})=1-\exp\bigl{(}\frac{\gamma_{1}}{\gamma_{1}}\bigr{)}\). The FSO link obeys Gamma-Gamma distribution with consideration of pointing error and atmospheric turbulence. The PDF of the SNR in FSO link can be given by [14; 17; 18; 19; 20; 21; 22]:  
\[f_{\gamma_{2}}(\gamma_{2})=\frac{\xi^{2}}{2\gamma_{2}\Gamma(\alpha)\Gamma( \beta)}\times G \begin{array}{c}3,0\\ 1,3\end{array} \biggl{(}\alpha\beta\sqrt{\frac{\gamma_{2}}{\gamma_{2}}}\biggr{|} \begin{array}{c}\xi^{2}+1\\ \xi^{2},\alpha,\beta\end{array} \biggr{)}, \tag{1}\]  
where \(\gamma_{2}\) represents the instantaneous SNR of the FSO hop, \(\overline{\gamma_{2}}\) is the average SNR in the FSO channel, \(\alpha\) and \(\beta\) are the fading parameters that can reflect turbulence conditions. \(\xi=\frac{\omega_{c}}{2\alpha_{s}}\), \(\xi\) is the pointing error parameter, which is determined by the pointing error displacement standard deviation at the destination and equivalent beam radius (i.e., when \(\xi\rightarrow\infty\), we get a case without influence of pointing error, the pointing error can be ignored). \(\Gamma(\cdot)\) is the Gamma function and \(G(\cdot)\) is the [PERSON]'s G function, and 3, 0, 1, 3 are the parameters associated with the [PERSON]'s G function, they are defined in reference [23]. Using the same method as in RF link, the CDF of the Gamma-Gamma distribution with the influence of pointing error can be given by [24; 25]:  
\[ \begin{array}{l}F_{2}(\gamma_{2})=\int\limits_{0}^{\gamma_{2}}f_{\gamma_{2} }(\gamma_{2})d\gamma_{2}\\ =\frac{\xi^{2}}{2\gamma_{2}\Gamma(\alpha)\Gamma(\beta)}G\begin{array}{c}3,1 \\ 2,4\end{array}\biggl{[}Z\biggr{|}\begin{array}{c}2,\xi^{2}+1\\ \xi^{2},\alpha,\beta,1\end{array}\biggr{]},\end{array}  \tag{2}\]  
where \(Z=\frac{\alpha\beta\gamma_{2}}{\sqrt{2}}\).