## 4 Performance Analysis of an FSO RIS-ACs Link over F-S \(\mathcal{F}\) TI fading Channels
### Level Crossing Rate (\(N_{\gamma}\))  
The level crossing rate (\(N_{\gamma}\)) is defined as a time rate of change of the TI-faded signal in time-variant TI fading channels. For the predetermined SNR threshold \(\gamma_{th}\), the \(N_{\gamma}(\gamma_{th})\) can be written as ([32], Equation (14)):  
\[N_{\gamma}(\gamma_{th})=\int_{0}^{\infty}\dot{\gamma}p_{\gamma\dot{\gamma}}( \gamma_{th},\dot{\gamma})d\dot{\gamma} \tag{33}\]  
where, \(p_{\gamma\dot{\gamma}}(\gamma,\dot{\gamma})\) is the joint distribution of the received SNR for the considered FSO RIS-ACs transmission system, \(\gamma\) and its first derivative \(\dot{\gamma}\). Since we could express the received SNR as \(\gamma=\gamma_{h_{1}}\gamma_{h_{2}}\), where S-RIS and RIS-D channel coefficients \(\gamma_{h_{1}}\) and \(\gamma_{h_{2}}\) of the Fisher-Snedecor \(\mathcal{F}\) TI fading model as already shown in Section 1 can be further expressed as \(\gamma_{h_{1}}=\frac{\gamma_{c_{1}}}{\gamma_{g_{C_{1}}}}\) and \(\gamma_{h_{2}}=\frac{\gamma_{g_{C_{2}}}}{\gamma_{g_{C_{2}}}}\), respectively, the \(p_{\gamma\dot{\gamma}}(\gamma,\dot{\gamma})\) can be written as an integral-form (I-F) expression of a joint PDF of i.i.d RVs, \(\gamma,\dot{\gamma},\gamma_{g_{C_{1}}},\gamma_{g_{C_{2}}}\) and \(\gamma_{g_{C_{2}}}\), as follows from ([45], Equation (12)):  
\[p_{\gamma\dot{\gamma}}(\gamma,\dot{\gamma})=\int_{0}^{\infty}d\gamma_{g_{C_{1} }}\int_{0}^{\infty}d\gamma_{g_{C_{2}}}\int_{0}^{\infty}p_{\gamma\dot{\gamma} \gamma_{g_{C_{1}}}}\gamma_{g_{C_{2}}}(\  
The conditional distribution \(p_{\gamma|\gamma_{y_{G_{1}}}\gamma_{x_{G_{2}}}\gamma_{y_{G_{2}}}}(\gamma_{y_{G_{1}}} \gamma_{x_{G_{2}}}\gamma_{y_{G_{2}}})\) was then transformed into:  
\[p_{\gamma|\gamma_{y_{G_{1}}}\gamma_{x_{G_{2}}}\gamma_{y_{G_{2}}}} \left(\gamma|\gamma_{y_{G_{1}}}\gamma_{x_{G_{2}}}\gamma_{y_{G_{2}}}\right)= \left|\frac{d\gamma_{x_{G_{1}}}}{d\gamma}\right|p_{\gamma_{x_{G_{1}}}}\left( \frac{\gamma\gamma_{y_{G_{1}}}\gamma_{y_{G_{2}}}}{\gamma_{x_{G_{2}}}}\right) \tag{36}\]  
From (33) to (36), the \(N_{\gamma}(\gamma_{th})\) of the FSO RIS-ACs transmission system in F-S \(\mathcal{F}\) TI fading propagation environments was expressed as:  
\[N_{\gamma}(\gamma_{th}) = \int_{0}^{\infty}d\gamma_{y_{G_{1}}}\int_{0}^{\infty}d\gamma_{x_{ G_{2}}}\int_{0}^{\infty}\left|\frac{d\gamma_{x_{G_{1}}}}{d\gamma}\right|p_{ \gamma_{x_{G_{1}}}}\left(\frac{\gamma\gamma_{y_{G_{1}}}\gamma_{y_{G_{2}}}}{ \gamma_{x_{G_{2}}}}\right) \tag{37}\] \[\times\ p_{\gamma_{y_{G_{1}}}}\Big{(}\gamma_{y_{G_{1}}}\Big{)}p_{ \gamma_{x_{G_{2}}}}\Big{(}\gamma_{x_{G_{2}}}\Big{)}p_{\gamma_{y_{G_{2}}}}\Big{(} \gamma_{y_{G_{2}}}\Big{)}d\gamma_{y_{G_{2}}}\] \[\times\ \int_{0}^{\infty}\gamma p_{\dot{\gamma}|\gamma\gamma_{y_{G_{1} }}\gamma_{x_{G_{2}}}\gamma_{y_{G_{2}}}}\Big{(}\dot{\gamma}|\gamma\gamma_{y_{G_ {1}}}\gamma_{x_{G_{2}}}\gamma_{y_{G_{2}}}\Big{)}d\dot{\gamma}\]  
where,  
\[\int_{0}^{\infty}\dot{\gamma}p_{\dot{\gamma}|\gamma\gamma_{y_{G_{1}}}\gamma_{ x_{G_{2}}}\gamma_{y_{G_{2}}}}\Big{(}\dot{\gamma}|\gamma\gamma_{y_{G_{1}}}\gamma_{ x_{G_{2}}}\gamma_{y_{G_{2}}}\Big{)}d\dot{\gamma}=\frac{1}{\sqrt{2\pi}}\sigma_{\dot{ \gamma}} \tag{38}\]  
The parameter \(\sigma_{\dot{\gamma}}^{2}\) is the variance of \(\dot{\gamma}\). Furthermore, \(\dot{\gamma}\) is the first derivative of \(\gamma=\gamma_{h_{1}}\gamma_{h_{2}}\) and can be written as:  
\[\dot{\gamma} = \gamma_{h_{2}}\dot{\gamma}_{h_{1}}+\gamma_{h_{1}}\dot{\gamma}_{h_ {2}} \tag{39}\]  
where \(\dot{\gamma}_{h_{1}}\) and \(\dot{\gamma}_{h_{2}}\) are the first derivatives of \(\gamma_{h_{1}}\) and \(\gamma_{h_{2}}\), respectively.