## 4 Enabling Technologies for FSO Communication
### Aperture Averaging
#### 4.5.3 Impact of Aperture Averaging  
In FSO systems, the impact of aperture averaging is examined by taking into account various facets of these systems. For aperture-averaged FSO Rxs, it is widely recognized that the \(\Gamma\Gamma\) distribution is valid for a point Rx under a bounded Gaussian wave model; however, it does not support aperture averaging phenomena [236; 237]. Also, experimental studies indicate that the LN model is valid for a point Rx in the weak turbulence regime and performs well in all turbulence regimes for aperture-averaged data [234; 238]. Conversely, while the \(\Gamma\Gamma\) model is considered valid for a point Rx across all turbulence regimes, this validity does not extend to scenarios involving aperture averaging [234; 238; 239]. Additionally, it has been demonstrated that the LN and \(\Gamma\Gamma\) models become virtually identical when \(D>6\rho_{0}\), where \(\rho_{0}\) is the spatial coherence radius [240]. To analyze the impact, consider the PDF of the irradiance, \(h_{\mathrm{a}}\), which follows the \(\Gamma\Gamma\) distribution as defined in Equation (22) [91; 94; 96; 97]:  
\[f_{h_{\mathrm{a}}}(h_{\mathrm{a}})=\frac{2(\alpha\beta)^{(\alpha+\beta)/2}}{ \Gamma(\alpha)\Gamma(\beta)}(h_{\mathrm{a}})^{\frac{(\alpha+\beta)}{2}-1}\mathrm{ K}_{\alpha-\beta}(2\sqrt{\alpha\beta h_{\mathrm{a}}}), \tag{38}\]  
Utilizing the power series expansion of the modified Bessel function around the origin, Equation (38) can be estimated as [241]  
\[f_{h_{\mathrm{a}}}(h_{\mathrm{a}})\approx a_{0}\,\mathrm{I}^{\beta-1}+a_{1} \,\mathrm{I}^{\beta}, \tag{39}\]  
In Equation (39), only the initial two terms of the power series expansion are taken into account, with \(a_{0}\) and \(a_{1}\) defined accordingly as [241]  
\[a_{0}= \frac{\Gamma(\alpha-\beta)}{\Gamma(\alpha)\Gamma(\beta)}(\alpha \beta)^{\beta}, \tag{40a}\] \[a_{1}= \frac{\Gamma(\alpha-\beta)}{\Gamma(\alpha)\Gamma(\beta)}\frac{( \alpha\beta)^{\beta+1}}{(1-\alpha+\beta)} \tag{40b}\]  
All the expressions provided for \(\Gamma\Gamma\) are valid for point Rxs, where the diameter of the Rx lens (\(D\)) is zero. Specifically, the channel parameters \(\alpha\) and \(\beta\) in Equation (24a) and (24b), as well as the power series expansions in (39), (40a) and (40b), apply in the case of \(D=0\). For Rxs that utilize aperture averaging, where \(D>0\), there is a reduction in the variance in intensity fluctuations. In such cases, the \(\Gamma\Gamma\) statistical model remains applicable but with modified parameters \(\alpha\) and \(\beta\)[233; 241]. The effect of aperture averaging is that Equations (39), (40a), and (40b) are no longer valid. For \(D>0\), the parameter \(\beta\) may exceed the parameter \(\alpha\), which is different from the situation with point Rxs. Nonetheless, this issue can be readily addressed, as when \(\beta>\alpha\), Equation (39) can be reformulated as  
\[f_{\gamma\gamma}(I)\approx a_{0}I^{\kappa-1}+a_{1}I^{\alpha} \tag{41}\]  
where (40a) and (40b) can be expressed as  
\[a_{0}= \frac{\Gamma(\beta-\alpha)}{\Gamma(\alpha)\Gamma(\beta)}(\alpha \beta)^{\alpha}, \tag{42a}\] \[a_{1}= \frac{\Gamma(\beta-\alpha)}{\Gamma(\alpha)\Gamma(\beta)}\frac{( \alpha\beta)^{\alpha+1}}{(1-\beta+\alpha)} \tag{42b}\]  
It is important to mention that these expressions are akin to (39), (40a), and (40b), where the roles of \(\alpha\) and \(\beta\) should be reversed when \(\beta>\alpha\). Furthermore, examine the scintillation index, \(\sigma_{N}^{2}(D)\), associated with a Rx lens of diameter \(D\). The impact of aperture averaging is to reduce this variance when \(D>0\). Thus, the effectiveness of aperture averaging is influenced by the propagation conditions [233]. The degree of intensity fluctuation is dependent on the size of the Rx aperture [235]. The extent to which aperture averaging diminishes the variance in intensity fluctuations is measured by a parameter known as the _aperture averaging factor_[242].