## 3 Channel Statistics with Gamma-Gamma Model  
In FSO system, one of the main impairments is atmospheric optical turbulence, which can cause scintillation. Scintillation-induced fading adversely affects FSO links and damages its communication performance. The scintillation results from the index of refraction fluctuations in the atmosphere, which can cause random fluctuations of the received signal intensity and severely reduce the level of the optical signal. The statistics of the scintillation strength is usually regarded as following the gamma-gamma distribution, which is applicable to all turbulence situations from weak to strong. This model, proposed in [18], is based on the modulation process, in which the optical radiation fluctuation through turbulent atmosphere is assumed to be composed of small-scale (scattering) and large-scale (refraction) effects. Therefore, the normalized received irradiance \(I_{t}\) is defined as the product of two statistically independent random processes \(I_{x}\) and \(I_{y}\). \[I_{t}=I_{x}I_{y} \tag{1}\]  
Figure 1: Typical fully connected CNN network architecture. \(I_{x}\) and \(I_{y}\) are generated by the large-scale and small-scale turbulent eddies, respectively, and both obey the gamma distribution given by [19]. Consequently, their probability density functions (PDF) are given by  
\[p(I_{x})=\frac{\alpha(\alpha I_{x})^{\alpha-1}}{\Gamma(\alpha)}\exp(-\alpha I_{x });I_{x}>0;\alpha>0 \tag{2}\]  
\[p(I_{y})=\frac{\beta\big{(}\beta I_{y}\big{)}^{\alpha-1}}{\Gamma(\beta)}\exp(- \beta I_{y});I_{y}>0;\beta>0 \tag{3}\]  
By fixing \(I_{x}\) and using the change of variable, \(I_{y}=I_{t}/I_{x}\), the conditional PDF given by Equation (4) is obtained, in which \(I_{x}\) is the (conditional) mean value of \(I_{t}\). \[p(I_{t}/I_{x})=\frac{\beta\big{(}\beta I_{t}/I_{y}\big{)}^{\beta-1}}{I_{x} \Gamma(\beta)}\exp(-\beta I_{t}/I_{x});I_{t}>0 \tag{4}\]  
To obtain the unconditional irradiance distribution, the conditional probability \(p(I_{t}/I_{x})\) is averaged over the statistical distribution of \(I_{x}\) given by Equation (2) to obtain the following gamma-gamma irradiance distribution function. \[ \begin{array}{ll}p(I_{t})&=\int_{\alpha}^{\infty}p(I_{t}/I_{x})p(I_{x})dI_{ x}\\ &=\frac{2(\alpha\beta)^{(\alpha+\beta)/2}}{\Gamma(\alpha)\Gamma(\beta)}I_{t}( \frac{\alpha+\beta}{2})-1K_{\alpha-\beta}\big{(}2\sqrt{\alpha\beta I_{t}}\big{)},I_{t}>0\end{array}  \tag{5}\]  
where \(\alpha\) and \(\beta\) represent the effective number of large-scale and small-scale eddies in the scattering process, respectively. \(\Gamma(\cdot)\) represents the gamma function, and \(K_{n}(\cdot)\) is the modified Bessel function of the second kind of order \(n\). If the optical radiation at the receiver is assumed to be a plane wave, then the two parameters \(\alpha\) and \(\beta\) that characterize the irradiance fluctuation PDF are related to the atmospheric conditions [18], as follows:  
\[\alpha=\left[\exp\left(\frac{0.49\sigma_{l}^{2}}{\big{(}1+1.11\sigma_{l}^{12/5 }\big{)}^{7/6}}\right)-1\right]^{-1} \tag{6}\]  
\[\beta=\left[\exp\left(\frac{0.51\sigma_{l}^{2}}{\big{(}1+0.69\sigma_{l}^{12/5 }\big{)}^{5/6}}\right)-1\right]^{-1} \tag{7}\]  
while the scintillation index is given by  
\[\sigma_{N}^{2}=\exp\left[\frac{0.49\sigma_{l}^{2}}{\big{(}1+1.11\sigma_{l}^{1 2/5}\big{)}^{7/6}}+\frac{0.51\sigma_{l}^{2}}{\big{(}1+0.69\sigma_{l}^{12/5} \big{)}^{5/6}}\right]-1 \tag{8}\]  
Often, \(\sigma_{l}\) is the log irradiance variance for a plane wave [19], which is defined as  
\[\sigma_{l}^{2}=1.23C_{n}^{2}k^{7/6}L^{11/6} \tag{9}\]  
In Equation (9), \(C_{n}^{2}\) is the refractive index structure parameter, which characterizes the atmospheric optical turbulence effect. Generally, the \(C_{n}^{2}\) range varies from \(10^{-15}\) m\({}^{-2/3}\) to \(10^{-12}\) m\({}^{-2/3}\) for the weak to strong turbulence regime, respectively. The gamma-gamma turbulence model given by Equation (5), which is applicable to all turbulence strengths from weak to strong.