## 3 Challenges in FSO Communication
### FSO System and Channel Models
#### 3.2.2 FSO Channel Models
\begin{split} C_{n}^{2}(h_{n})&=0.00594(v_{w}/27)^{2} (10^{-5}h_{n})^{10}\exp(-h_{n}/1000)\\ &+2.7\times 10^{-16}\exp(-h_{n}/1500)+\hat{A}\exp(-h_{n}/100),\end{split}  \tag{19}\]  
where \(h_{n}\) represents the altitude in meters (m) and \(\hat{A}\) denotes the nominal value of \(C_{n}^{2}(0)\) at ground level in m\({}^{-2/3}\). The value of \(C_{n}^{2}\) for FSO links at ground level is approximately \(1.7\times 10^{-14}\) m\({}^{-2/3}\) during the daytime and \(8.4\times 10^{-15}\) m\({}^{-2/3}\) at night. Generally, \(C_{n}^{2}\) ranges from \(10^{-13}\) m\({}^{-2/3}\) in strong turbulence to \(10^{-17}\) m\({}^{-2/3}\) in weak turbulence, with an average typical value of \(10^{-15}\) m\({}^{-2/3}\)[94; 97]. The variable \(v_{w}\) is the root mean square wind speed (pseudo wind) in meters per second (m/s), typically valued at 21 m/s, is given as [94]  
\[w=\left[\frac{1}{15\times 10^{3}}\int_{5\times 10^{3}}^{20\times 10^{3}}V^{2}(h_ {n})dh_{n}\right]^{1/2}, \tag{20}\]  
where \(V(h_{n})\) is generally characterized by the Bufton wind model as described by [94]  
\[V(h_{n})=\omega_{s}h_{n}+V_{g}+30\exp\left[-\left(\frac{h_{n}-9400}{4800} \right)^{2}\right], \tag{21}\]  
where \(V_{g}\) denotes the ground wind speed, and \(\omega_{s}\) represents the _skew rate_, which is related to the satellite's movement in relation to the observer on the ground. Gamma-Gamma Distribution: In many cases, when the LN distribution characterization is invalid due to strong turbulence regimes, the IT distribution is typically used to model scintillation effects. Furthermore, the IT model can characterize fading gains across both weak and strong turbulence scenarios. The PDF of \(h_{\text{a}}\) based on the IT distribution is defined as [6; 91; 94; 96; 97]  
\[f_{h_{\text{a}}}(h_{\text{a}})=\frac{2(\alpha\beta)^{(\alpha+\beta)/2}}{\Gamma (\alpha)\Gamma(\beta)}(h_{\text{a}})^{\frac{(\alpha+\beta)}{2}-1}\text{K}_{ \alpha-\beta}(2\sqrt{\alpha\beta h_{\text{a}}}), \tag{22}\]  
where \(\Gamma(\cdot)\) denotes the gamma function, while \(K_{\
u}(\cdot)\) represents the modified Bessel function of the second kind of order \(\
u\). The parameters \(\alpha\) and \(\beta\) refer to the effective number of large-scale and small-scale eddies involved in the scattering process, respectively. These parameters are defined for the plane wave as [91; 94; 96; 97]  
\[\alpha= \left[\exp\left(\frac{0.49\sigma_{R}^{2}}{(1+1.11\sigma_{R}^{12/5 })^{7/6}}\right)-1\right]^{-1}, \tag{23a}\] \[\beta= \left[\exp\left(\frac{0.51\sigma_{R}^{2}}{(1+0.69\sigma_{R}^{12/5 })^{5/6}}\right)-1\right]^{-1},\]  
where \(d\triangleq\left(\frac{kD^{2}}{4L}\right)^{1/2}\), with \(D\) denoting the diameter of the Rx aperture, and \(\sigma_{R}^{2}\) denoting the Rytov variance, a metric for the strength of turbulence fluctuations. The Rytov variance \(\sigma_{R}^{2}\) is expressed for both plane and spherical waves, respectively, as [91; 94; 97]  
\[\sigma_{R}^{2}\big{|}_{\text{plane}}=1.23\ C_{n}^{2}\ k^{7/6}\ L^{11/6}, \tag{25a}\] \[\sigma_{R}^{2}\big{|}_{\text{spherical}}=0.492\ C_{n}^{2}\ k^{7/6}\ L^{11/6}, \tag{25b}\]  
Additionally, the normalized variance in the irradiance, commonly known as the scintillation index (\(\sigma_{N}^{2}\)), can be expressed in relation to \(\sigma_{x}^{2}\) and the eddies involved in the scattering process (\(\alpha\) and \(\beta\)), respectively, as [94; 98; 6; 91]  
\[\sigma_{N}^{2} \triangleq\frac{\langle\mathsf{h}_{\text{a}}^{2}\rangle-\langle \mathsf{h}_{\text{a}}\rangle^{2}}{\langle\mathsf{h}_{\text{a}}\rangle^{2}} \tag{26a}\] \[=\] (26b) \[= exp\Big{(}4\sigma_{x}^{2}\Big{)}-1\] (26c) \[V(h_{n})=\omega_{s}h_{n}+V_{g}+30\exp\left[-\left(\frac{h_{n}-9400}{4800} \right)^{2}\right], \tag{21}\]0  
Malaga (\(\mathcal{M}\))-Distribution: The \(\mathcal{M}\)-distribution serves as a versatile model suitable for characterizing the entire range of turbulent regimes. Table 2 outlines the methods for deriving current AT models from the \(\mathcal{M}\)-distribution model [99].