# Analysis of Scintillation Effects on Free Space Optical Communication Links in South Africa
## 5 Intensity Distribution
Therefore, the PDF of the Lognormal distribution, considering the effect of pointing errors, is derived in [17; 18; 58] as:  
\[f_{I}(I)=\frac{\xi^{2}}{\left(A_{o}\right)^{\xi^{2}}}I^{\xi^{2}-1}\frac{1}{2} \exp(q)erfc\left(\frac{In\left(\frac{I}{A_{o}}\right)+p}{\sigma_{SI}\sqrt{2}}\right) \tag{39}\]  
where  
\[p=0.5\sigma_{SI}^{2}+\xi^{2}\sigma_{SI}^{2} \tag{40}\]  
and  
\[q=\frac{\xi^{2}\sigma_{SI}^{2}\left(1+\xi^{2}\right)}{2} \tag{41}\]  
Additionally, the PDF of the Gamma-gamma distribution model, taking into account the effects of misalignment, is derived in [6; 16; 56] as:  
\[f_{I}(I)=\frac{\alpha\beta\xi^{2}}{A_{o}\Gamma(\alpha)\Gamma(\beta)}G_{1,3}^{ 3,0}\left[\frac{\alpha\beta\xi^{2}}{A_{o}}I\bigg{|}\xi^{2}-1,\alpha-1,\beta-1 \right],I>0 \tag{42}\]  
After some mathematical manipulations, the PDF can be further simplified as [6; 56]:  
\[f_{I}(I)=\frac{I^{-1}\xi^{2}}{\Gamma(\alpha)\Gamma(\beta)}G_{1,3}^{3,0}\left[ \frac{\alpha\beta\xi^{2}}{\xi^{2}+1}I\bigg{|}\xi^{2}+1\right],I\geq 0 \tag{43}\]  
The expression for the CDF of the Gamma-gamma distribution model, considering pointing error effects, is derived in [6; 56] as:  
\[F_{I}(I)=\frac{\xi^{2}}{\Gamma(\alpha)\Gamma(\beta)}G_{2,4}^{3,1}\left[\frac{ \alpha\beta\xi^{2}}{\xi^{2}+1}I\bigg{|}\xi^{2},\alpha,\beta,0\right] \tag{44}\]  
For commercial FSOC links employing the use of intensity modulation/direct detection (IM/DD) schemes and avalanche photodiode (APD) detectors, the instantaneous signal-to-noise ratio (SNR) at the receiver is defined as [42; 56; 59; 60]:\[W_{Leq}=\left(\frac{W_{L}^{2}\sqrt{\pi}erf(v)}{2v\exp\left(-v^{2}\right)} \right)^{0.5} \tag{36}\]0  
where \(\Re\) is the responsivity, \(g_{a}\) is the APD gain, \(\sigma_{n}^{2}\) is the total noise at the APD receiver, and \(P_{x}\) is the average optical power detected at the receiver. \(P_{x}\) is well defined in Equation (9) of [42]. The average SNR at the receiver is defined as [42; 56]:  
\[W_{Leq}=\left(\frac{W_{L}^{2}\sqrt{\pi}erf(v)}{2v\exp\left(-v^{2}\right)} \right)^{0.5} \tag{36}\]1  
The total noise at the APD receiver comprises the thermal and shot noise. It is given as [42; 51; 59; 60]:  
\[W_{Leq}=\left(\frac{W_{L}^{2}\sqrt{\pi}erf(v)}{2v\exp\left(-v^{2}\right)} \right)^{0.5} \tag{36}\]2  
where \(T\) is the temperature of the receiver, \(K_{b}\) is the Boltzmann constant, \(R_{b}\) is the bit rate, \(F_{n}\) is the noise figure of the amplifier, \(R\) is the APD load resistance, \(q\) is the electron charge, and \(F_{a}\) is excess noise factor. The excess noise factor is expressed as [42; 51; 59; 60]:  
\[W_{Leq}=\left(\frac{W_{L}^{2}\sqrt{\pi}erf(v)}{2v\exp\left(-v^{2}\right)} \right)^{0.5} \tag{36}\]3  
where \(k_{a}\) is the ionization factor. The PDF of SNRs for weak atmospheric turbulence using the Lognormal distribution model with pointing errors is derived by substituting Equation (45) into Equation (39), and is given below as [18]:  
\[W_{Leq}=\left(\frac{W_{L}^{2}\sqrt{\pi}erf(v)}{2v\exp\left(-v^{2}\right)} \right)^{0.5} \tag{36}\]4  
Applying the relation in [18; 31] where:  
\[W_{Leq}=\left(\frac{W_{L}^{2}\sqrt{\pi}erf(v)}{2v\exp\left(-v^{2}\right)} \right)^{0.5} \tag{36}\]5  
Equation (49) becomes:  
\[W_{Leq}=\left(\frac{W_{L}^{2}\sqrt{\pi}erf(v)}{2v\exp\left(-v^{2}\right)} \right)^{0.5} \tag{36}\]6  
The PDF of SNRs for moderate to strong atmospheric turbulence using the Gamma-gamma model with pointing errors, as derived in [55; 56], is given as:  
\[W_{Leq}=\left(\frac{W_{L}^{2}\sqrt{\pi}erf(v)}{2v\exp\left(-v^{2}\right)} \right)^{0.5} \tag{36}\]7  
The \(\sigma_{SI}^{2}(0,L)\) is used in characterizing the atmospheric turbulence strength due to the effect of scintillation. In a weak turbulence regime, \(\sigma_{SI}^{2}(0,L)<1\), and the Lognormal distribution model is employed. For moderate to strong fluctuations, \(\sigma_{SI}^{2}(0,L)\geq 1\), and the Gamma-gamma turbulence distribution is used [38; 44].