# Analysis of Scintillation Effects on Free Space Optical Communication Links in South Africa
## 7 Average Bit Error Rate (BER) Analysis
### Return-to-Zero On-Off Keying (RZ-OOK) FSOC Links  
The conditional BER for RZ-OOK SISO FSOC links in the absence of atmospheric turbulence is given as [56]:  
\[P_{RZ-OOK}(e/\gamma)=Q\bigg{(}\frac{\sqrt{2\gamma}}{2}\bigg{)}=0.5 erfc\bigg{(} 0.5\gamma^{0.5}\bigg{)} \tag{57}\]  
1. Weak Atmospheric Turbulence  
In order to derive the expression for the BER of RZ-OOK SISO FSOC links in the presence of weak atmospheric turbulence, inserting Equations (51) and (57) into Equation (56) yields:  
\[P_{RZ-OOK}=\int\limits_{0}^{\infty}\Bigg{(}0.5 erfc\bigg{(}0.5\gamma^{0.5} \bigg{)}\bullet\frac{\varrho^{2}}{(A_{o})^{\beta^{2}}}\frac{\gamma^{(0.5\xi^{2 })-1}}{\overline{\gamma}^{0.5\xi^{2}}}\frac{\exp(q)}{\sqrt{\pi}}\bullet\Bigg{[} \frac{1}{12}\exp\left(-\Bigg{(}\frac{0.5 In\left(\frac{\gamma}{\overline{ \gamma}A_{S}}\right)+p}{4\sigma_{SI}}\right)^{2}\right)+\frac{1}{4}\exp\left(- \Bigg{(}\frac{2 In\left(\frac{\gamma}{\overline{\gamma}A_{S}}\right)+4p}{9 \sigma_{SI}}\Bigg{)}^{2}\right)\Bigg{]}\Bigg{)}d_{\gamma} \tag{58}\]  
Variable substitutions where:  
\[z_{1}=\frac{0.5 In\left(\frac{\gamma}{\overline{\gamma}A_{S}^{2}}\right)+p}{4 \sigma_{SI}} \tag{59}\]  
and  
\[z_{2}=\frac{2 In\left(\frac{\gamma}{\overline{\gamma}A_{S}^{2}}\right)+4p}{9 \sigma_{SI}} \tag{60}\]  
are employed. Making \(\gamma\) the subject of (59) and (60) yields:  
\[\gamma=\overline{\gamma}A_{o}^{2}\exp(8z_{1}\sigma_{SI}-2p) \tag{61}\]  
and  
\[\gamma=\overline{\gamma}A_{o}^{2}\exp(4.5z_{2}\sigma_{SI}-2p) \tag{62}\]  
Differentiating \(\gamma\) with respect to \(z_{1}\) and \(z_{2}\) in (61) and (62), respectively, gives:  
\[d\gamma=8\overline{\gamma}A_{o}^{2}\sigma_{SI}\exp(8z_{1}\sigma_{SI}-2p)dz_{1} \tag{63}\]  
and  
\[d\gamma=4.5\overline{\gamma}A_{o}^{2}\sigma_{SI}\exp(4.5z_{2}\sigma_{SI}-2p)dz _{2} \tag{64}\]  
Therefore, substituting Equations (59)-(63) and (64) in Equation (58) generates:  
\[ \begin{array}{l}P_{RZ-OOK}=\int\limits_{0}^{\infty}\bigg{(}\frac{\varepsilon^ {2}\sigma_{SI}\exp(q)}{3\sqrt{\pi}}\bigg{(}\left(\exp(8z_{1}\sigma_{SI}-2p) \right)^{(0.5\xi^{2})}\bullet erfc\bigg{(}\frac{\sqrt{7A_{o}^{2}\exp(8z_{1} \sigma_{SI}-2p)}}{2}\bigg{)}\bigg{)}\bullet\exp\left(-z_{1}^{2}\right)\bigg{)} dz_{1}\\ +\\ \int\limits_{0}^{\infty}\bigg{(}\frac{9\xi^{2}\sigma_{SI}\exp(q)}{16\sqrt{ \pi}}\bigg{(}\left(\exp(4.5z_{2}\sigma_{SI}-2p)\right)^{(0.5\xi^{2})}\bullet erfc \bigg{(}\frac{\sqrt{7A_{o}^{2}\exp(4.5z_{2}\sigma_{SI}-2p)}}{2}\bigg{)}\bigg{)} \bullet\exp\left(-z_{2}^{2}\right)\bigg{)}dz_{2}\end{array}  \tag{65}\]Since a closed form solution does not exist for the above integration, the Gauss-Hermite quadrature approximation, as presented in [62; 63], is used to evaluate the integration, where:  
\[\int\limits_{-\infty}^{\infty}\exp\Big{(}-x^{2}\Big{)}f(x)dx\cong\sum\limits_{i= 1}^{N}W_{i}f(z_{i}) \tag{66}\]  
Therefore, the BER of RZ-OOK SISO FSOC links in the presence of weak atmospheric turbulence and pointing errors is:  
\[P_{RZ-\mathrm{OOK}}=\frac{\xi^{2}\sigma_{SI}\exp(q)}{2\sqrt{\pi}}\bullet\left( \begin{array}{c}\left(\frac{1}{3}\sum\limits_{i=1}^{N}W_{i}\bigg{(}\left(\exp(8 z_{1}\sigma_{SI}-2p)\right)^{(0.5\xi^{2})}\bullet erfc\bigg{(}\frac{\sqrt{7A_{0}^{2} \exp(8z_{1}\sigma_{SI}-2p)}}{2}\bigg{)}\bigg{)}\right)\\ +\\ \left(\frac{9}{16}\sum\limits_{i=1}^{N}W_{i}\bigg{(}\left(\exp(4.5z_{2}\sigma_{ SI}-2p)\right)^{(0.5\xi^{2})}\bullet erfc\bigg{(}\frac{\sqrt{7A_{0}^{2}\exp(4 5z_{2}\sigma_{SI}-2p)}}{2}\bigg{)}\bigg{)}\bigg{)}\end{array} \right)\right) \tag{67}\]  
where the values of the weights, \(W_{i}\), and zeros of the Hermite polynomial, \(z_{i}\), are given in [62; 63]. 2. Moderate to Strong Atmospheric Turbulence Applying equation (07.34.03.0619.01) in [53] to Equation (57), we obtain: \[G_{1,2}^{2,0}\bigg{[}0.25\gamma\bigg{|} \begin{matrix}1\\ 0,0.5\end{matrix} \bigg{]}=\pi^{0.5}erfc\bigg{(}0.5\gamma^{0.5}\bigg{)}\] (68) Therefore, combining Equations (57) and (68) yields: \[P_{RZ-\mathrm{OOK}}(e|\gamma)=\frac{0.5}{\pi^{0.5}}G_{1,2}^{2,0}\bigg{[}0.25 \gamma\bigg{|} \begin{matrix}1\\ 0,0.5\end{matrix}