## 1 Introduction
### Channel Modeling  
In order to establish an FSO communications link with high reliability and availability, reasonable estimations of several statistical parameters related to the phenomenon of scintillation should be made. Signal fading can significantly degrade the link's performance and increase the bit error rate (BER). Therefore, an accurate formulation of the PDF of these random fluctuations would allow a better prediction of the link's reliability and thus it should be taken into consideration during system design. Apparently, there is no PDF that could sufficiently model all kind of turbulence conditions. It is important to notice the importance of the tails of a PDF model, since this region yields important information regarding the detection and fading probability. The moments of the irradiance fluctuations for a given PDF is given as [14],  
\[m_{k}=\int\limits_{0}^{\infty}I^{k}p(I)dI \tag{3}\]  
where \(I\) is the received irradiance and \(p(I)\) the corresponding PDF. Higher-order coefficients also exist and provide a useful mean of model comparison, considering the PDF tail contribution for non-Gaussian distributions. For practical reasons, we are looking for controllable models that allow for accurate prediction of the channel's performance. Apart from turbulence effects, a PDF distribution is also affected by the aperture averaging effect that tends to reduce the deleterious effects of scintillation. Therefore, whenever modeling an optical channel and estimating the received power, we need to clarify whether a point receiver or an aperture with certain dimensions is assumed [1]. Due to its relatively simple form, the LN distribution is the most widely used turbulence model. Moreover, this model is assumed valid for a weak turbulence regime in case of a point receiver and all turbulence regimes for aperture averaged measurements [1]. The PDF of the LN distribution is given by [4],  
\[f_{I}(I)=\frac{1}{l\sigma\sqrt{2\pi}}\exp\left(-\frac{\left(\ln(I)+\sigma^{2}/2 \right)^{2}}{2\sigma^{2}}\right) \tag{4}\]  
where \(\sigma^{2}\) is the log-irradiance, which depends on the aperture diameter, the wavelength and turbulence strength. On the other hand, the GG distribution is obtained as the product of two independent gamma modeled distributions, modelling small and large scale fluctuations. The great advantage of the GG distribution is that its two parameters, \(a\) and \(b\), are directly related to \(C_{n}\)2. The PDF of the GG distribution is given as [4],  
\[f_{I}(I)=\frac{2(ab)^{\frac{a+b}{2}}}{\Gamma(a)\Gamma(b)}I^{\frac{a+b}{2}-1}K_{a -b}\left(2\sqrt{abI}\right) \tag{5}\]  
where \(K_{q}(. )\) is the modified Bessel function of the second kind of order \(q\). The parameters \(a\) and \(b\) are directly related to turbulence and system parameters for a spherical wave as follows [4],  
\[a=\left[\exp\left(\frac{0.49\delta_{s}^{2}}{\left(1+0.1d^{2}+0.5\delta_{s}^{1 2/5}\right)^{7/6}}\right)-1\right]^{-1}\text{ and }b=\left[\exp\left(\frac{0.5\delta_{s}^{2}(1+0.69\delta_{s}^{12/5})^{-5/6}}{ 1+0.9d^{2}+0.62d^{2}\delta_{s}^{12/5}}\right)-1\right]^{-1} \tag{6}\]  
where \(\delta_{s}^{2}=0.49C_{n}^{2}k^{2/6}L^{11/6}\) the Rytov variance for a spherical wave and \(d=\sqrt{kD^{2}4L}\), where \(k\) is the optical wavenumber, \(L\) the link's length and \(D\) the diameter of the aperture. As mentioned above, an accurate distribution model is necessary to estimate the performance of an optical link. A critical metric related to system's availability is the system's outage probability, defined as the probability that the receiver's signal-to-noise (SNR) ratio \(\gamma\) falls below a critical threshold \(\gamma_{th}\)[4]. Mathematically, this is expressed as,  
\[P_{out}=\Pr(\gamma\leq\gamma_{th})=F_{\gamma}(\gamma_{th}) \tag{7}\]  
where \(F_{\gamma}\) is the corresponding cumulative distribution function (CDF) of the SNR. The outage probability can now be expressed in accordance with the model that has been used to model the turbulent channel.