# Analysis of Scintillation Effects on Free Space Optical Communication Links in South Africa
## 5 Intensity Distribution  
In this section, the statistical analysis of the irradiance fluctuations and the channel characteristics for the weak and moderate to strong turbulence regimes are carried out using the Lognormal and Gamma-gamma turbulence distributions, respectively. The PDF of the lognormal distribution is given in [14; 15; 17; 18] as:  
\[f_{I}(I)=\frac{I^{-1}}{\sigma_{SI}\sqrt{2\pi}}\exp\left\{-\left(\frac{In(I)+ \left(0.5\sigma_{SI}^{2}\right)}{\sigma_{SI}\sqrt{2}}\right)^{2}\right\} \tag{28}\]  
While the PDF of the Gamma-gamma turbulence distribution can be expressed as [17; 39; 51; 52]:  
\[f_{I}(I)=\frac{2I^{-1}(\alpha\beta I)^{0.5(\alpha+\beta)}}{\Gamma(\alpha) \Gamma(\beta)}K_{\alpha-\beta}\left(2\sqrt{\alpha\beta I}\right) \tag{29}\]  
where \(I\) is the normalized irradiance, \(\Gamma(\cdot)\) represents the Gamma function, and \(K_{\alpha-\beta}(\cdot)\) is the modified Bessel function of the second kind and of order \(\alpha-\beta\). \(\alpha\) is the effective number of large-scale turbulence eddies. It is defined as [43]:  
\[\alpha=\left(\sigma_{x}^{2}\right)^{-1}=\left(\exp\left(\sigma_{\ln x}^{2} \right)-1\right)^{-1} \tag{30}\]  
where \(\sigma_{x}^{2}\) is the normalized large-scale (refractive) variance. \(\beta\) is the effective number of small-scale turbulence eddies. It is given as [43]:  
\[\beta=\left(\sigma_{y}^{2}\right)^{-1}=\left(\exp\left(\sigma_{\ln y}^{2} \right)-1\right)^{-1} \tag{31}\]  
where \(\sigma_{y}^{2}\) is the normalized small-scale (diffractive) variance. Using equation (07.34.03.0605.01) in [53], where:  
\[G_{0,2}^{2,0}[\alpha\beta I|\alpha,\beta]=2(\alpha\beta I)^{0.5(\alpha+\beta) }K_{\alpha-\beta}\left(2\sqrt{\alpha\beta I}\right) \tag{32}\]  
Equation (29) can be rewritten as:  
\[f_{I}(I)=\frac{I^{-1}}{\Gamma(\alpha)\Gamma(\beta)}G_{0,2}^{2,0}\left[\alpha \beta I\right]^{-}_{\alpha,\beta}\right],I>0 \tag{33}\]  
where \(G_{u,v}^{s,I}\left(z \begin{bmatrix}a_{1},\ldots,a_{p}\\ c_{1},\ldots,c_{p}\end{bmatrix} \right)\) is the Meijer \(G\) function, which is well defined in [54]. Integrating \(f_{I}(I)\) in Equation (33) gives the CDF of \(I\). This is derived by using equation (07.34.21.0003.01) in [53]. Thus, we have:  
\[F_{I}(I)=\frac{1}{\Gamma(\alpha)\Gamma(\beta)}G_{1,3}^{2,1}\left[\alpha\beta I \begin{bmatrix}1\\ \alpha,\beta,0\end{bmatrix} ,I>0 \tag{34}\]  
In this work, pointing errors represent the misalignment between the transmitter and receiver caused by the laser beam being displaced horizontally or vertically, i.e., a two-dimensional configuration is being considered. The transmitter and receiver planes are assumed to be parallel, and the laser beam is perpendicular to the receiver area. The pointing error parameter, \(\xi\), is defined as the ratio between the equivalent beam waist or radius at the receiver (\(W_{Leq}\)) and the standard deviation of the jitter or pointing error at the receiver (\(\sigma_{s}\)). It can be expressed as [6; 16; 55; 56]:  
\[\xi=\frac{W_{Leq}}{2\sigma_{s}} \tag{35}\]  
The beam waist \(W_{L}\) of a Gaussian beam, which is the radius calculated at \(e^{2}\), determines the value of the parameter, \(W_{Leq}\) at distance, \(L\). \(W_{Leq}\) is given as [6; 16; 18; 55; 56; 57]:  
\[W_{Leq}=\left(\frac{W_{L}^{2}\sqrt{\pi}erf(v)}{2v\exp\left(-v^{2}\right)} \right)^{0.5} \tag{36}\]  
where \(W_{L}=W_{o}\sqrt{(\Theta_{c}+\Lambda_{o})(1+(1.63\Lambda_{1}\sigma_{s}^{2}))}\), \(erf(. )\) is the error function and parameter \(v\) is expressed as [6; 16; 55; 56]:  
\[v=\frac{r\sqrt{\pi}}{W_{L}\sqrt{2}} \tag{37}\]  
where \(r\) represents the radius of a circular detector aperture. At distance \(L=0\), the fraction of the collected power is represented by parameter \(A_{o}\). It is expressed as [6; 16; 55; 56]:  
\[A_{o}=\left[erf(v)\right]^{2}=\left[1-erfc(v)\right]^{2} \tag{38}\]  
where \(erfc(. )\) is the complementary error function.