# Analysis of Scintillation Effects on Free Space Optical Communication Links in South Africa
## 6 Outage Probability Analysis  
Outage probability is a critical performance indicator that defines the likelihood of the instantaneous SNR going below the threshold SNR. Once this occurs, the link's communication will fail. It is expressed below as [45; 61]:  
\[\Pr_{out}=\text{Probability}[\gamma\leq\gamma_{th}]=F_{\gamma}[\gamma_{th}] \tag{53}\]  
where \(\gamma_{th}\) is the threshold SNR and \(F_{\gamma}[\gamma_{th}]\) is the CDF of the instantaneous SNR.  
From Equation (45), the normalized irradiance at the receiver can be expressed as [6; 57]:  
\[I=\left(\frac{\gamma}{\overline{\gamma}}\right)^{0.5}=\left(\frac{\gamma_{th} }{\overline{\gamma}}\right)^{0.5} \tag{54}\]  
Therefore, substituting for \(I\) in Equation (44) presents the expression for estimating the outage probability of the FSO link over the turbulent atmospheric channel while considering the effect of pointing errors [6; 56]:  
\[\Pr_{out}=\frac{\xi^{2}}{\Gamma(\alpha)\Gamma(\beta)}G_{2,4}^{3,1}\left[\frac{ \alpha\beta\xi^{2}}{\xi^{2}+1}\left(\frac{\gamma_{th}}{\overline{\gamma}} \right)^{0.5}\left|\begin{matrix}1,\xi^{2}+1\\ \xi^{2},\alpha,\beta,0\end{matrix}\right.\right] \tag{55}\]  
Figure 5a-d are based on the numerical values in Tables 2-5, respectively. These figures are generated using the parameters in Table 6, while computing with Equations (1)-(3), (15), (30), (31), (35), (46) and (55). The average receiver SNRs needed to achieve various outage probabilities over different link distances while considering the effect of pointing errors in the presence of turbulent eddies are presented in Figure 5a-d for the locations of interest. In these figures, the average receiver SNRs required to attain different outage probabilities are quite similar. It is also evident from these figures that the impact of the normalized jitter standard deviation on the outage probability is significant (when comparing Figure 5c,d where \(\frac{\sigma_{s}}{r}=1\) to Figure 5a,b where \(\frac{\sigma_{s}}{r}=2\)). This implies that the lower the value of \(\frac{\sigma_{s}}{r}\), the better the overall system performance. Additionally, the higher the value of \(\xi\), the better the outage probability performance of the FSOC links. In the presence of finite inner and outer scales of turbulence, that is, where \(l_{o}\) = 0.005 m and \(L_{o}\) = 10 m, the outage probabilities of the FSOC links are quite similar to when these turbulent eddies have sizes of zero and infinity in the Kolmogorov model with an infinitely large inertial range.  
\begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{1}{c}{**FSOC Link Parameters**} \\
**Light Source** & **Laser** \\ \hline Wavelength \((\lambda)\) & 1550 nm \\ Transmit Power \((P_{T})\) & 20 dBm \\ Receiver Sensitivity \((R_{S})\) & \(-\)40 dBm \\ Receiver Aperture Diameter \((D)\) & 10 cm \\ Eye Safety & Class 1M \\ Transmit Beam Divergence Angle \((\phi)\) & 1.75 mrad \\ Responsivity \((\Re)\) & 0.5 A/W \\ Bit Rate \((R_{b})\) & 10 Gb/s \\ Detector & Avalanche Photodiode (APD) \\ Boltzmann’s Constant \((K_{b})\) & 1.381 \(\times\) 10\({}^{-23}\) J/K \\ Temperature \((T)\) & 298 K \\ Planck’s Constant \((h)\) & 6.626 \(\times\) 10\({}^{-34}\) Js \\ Speed of Light \((c)\) & \(3\times 10^{8}\) m/s \\ APD Load Resistance \((R)\) & 100\(\Omega\) \\ APD Gain \((g_{a})\) & 50 \\ Amplifier Noise Figure \((F_{n})\) & 2 \\ Charge of an Electron & 1.602 \(\times\) 10\({}^{-19}\) C \\ Ionization factor for InGaAs APD \((k_{a})\) & 0.7 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Commercial FSOC link parameters used in computations.  
Figure 5: Outage probability vs. Average SNR (dB) based on zero inner scale and infinite outer scale model and finite inner and finite outer scale model for weak and moderate to strong atmospheric turbulence periods in different locations of South Africa.