# Analysis of Scintillation Effects on Free Space Optical Communication Links in South Africa
## 7 Average Bit Error Rate (BER) Analysis
### Binary Phase Shift Keying (BPSK) FSOC Links  
The conditional BER for BPSK SISO FSOC links in the absence of atmospheric turbulence is presented in [57; 64] as:  
\[P_{BPSK}(e/\gamma)=Q(\sqrt{\gamma})=0.5 erfc\bigg{(}\frac{\gamma^{0.5}}{\sqrt{2 }}\bigg{)} \tag{73}\]  
1. Weak Atmospheric Turbulence Inserting Equations (51) and (73) into Equation (56) yields the expression for the BER of BPSK SISO FSOC links in the presence of weak atmospheric turbulence and pointing errors:  
\[P_{BPSK}=\int\limits_{0}^{\infty}\Bigg{(}0.5 erfc\bigg{(}\frac{\gamma^{0.5}}{ \sqrt{2}}\bigg{)}\bullet\frac{\xi^{2}}{(A_{o})^{\xi^{2}}}\frac{\gamma^{(0.5 \xi^{2})-1}}{\gamma^{0.5\xi^{2}}}\frac{\exp(q)}{\sqrt{\pi}}\bullet\Bigg{[} \frac{1}{12}\exp\left(-\left(\frac{0.5 In\big{(}\frac{\gamma}{7.4\xi}\big{)}+p }{4\sigma_{SI}}\right)^{2}\right)+\frac{1}{4}\exp\left(-\left(\frac{2 In\big{(} \frac{\gamma}{7.4\xi}\big{)}+4p}{9\sigma_{SI}}\right)^{2}\right)\Bigg{]}\Bigg{)} d\gamma \tag{74}\]  
Further simplification of Equation (74), by substituting Equations (59)-(63) and (64) into it, gives an expression in the form:  
\[P_{BPSK}=\int\limits_{0}^{\infty}\Bigg{(}\frac{\xi^{2}\sigma_{SI} \exp(q)}{3\sqrt{\pi}}\bigg{(}\left(\exp(8z_{1}\sigma_{SI}-2p)\right)^{(0.5\xi ^{2})}\bullet erfc\bigg{(}\frac{\sqrt{\pi A_{o}^{2}\exp(8z_{1}\sigma_{SI}-2p) }}{\sqrt{2}}\bigg{)}\bigg{)}\bullet\exp\big{(}-z_{1}^{2}\big{)}\bigg{)}dz_{1} \tag{75}\] \[+\] \[\int\limits_{0}^{\infty}\Bigg{(}\frac{9\xi^{2}\sigma_{SI}\exp(q)}{ 16\sqrt{\pi}}\bigg{(}\left(\exp(4.5z_{2}\sigma_{SI}-2p)\right)^{(0.5\xi^{2})} \bullet erfc\bigg{(}\frac{\sqrt{\pi A_{o}^{2}\exp(4.5z_{2}\sigma_{SI}-2p)}}{ \sqrt{2}}\bigg{)}\bigg{)}\bullet\exp\big{(}-z_{2}^{2}\big{)}\bigg{)}dz_{2}\]  
Therefore, evaluating Equation (75) using the Gauss-Hermite quadrature approximation in Equation (66) yields the BER of BPSK SISO FSOC links in the presence of weak atmospheric turbulence and pointing errors:  
\[P_{BPSK}=\frac{\xi^{2}\sigma_{SI}\exp(q)}{2\sqrt{\pi}}\bullet\Bigg{(} \begin{array} []{c}\bigg{(}\frac{1}{3}\sum\limits_{i=1}^{N}W_{i}\bigg{(}\left(\exp(8z_{1} \sigma_{SI}-2p)\right)^{(0.5\xi^{2})}\bullet erfc\bigg{(}\frac{\sqrt{\pi A_{ o}^{2}\exp(8z_{1}\sigma_{SI}-2p)}}{\sqrt{2}}\bigg{)}\bigg{)}\bigg{)}\\ +\\ \bigg{(}\frac{9}{16}\sum\limits_{i=1}^{N}W_{i}\bigg{(}\left(\exp(4.5z_{2} \sigma_{SI}-2p)\right)^{(0.5\xi^{2})}\bullet erfc\bigg{(}\frac{\sqrt{\pi A_{ o}^{2}\exp(4.5z_{2}\sigma_{SI}-2p)}}{\sqrt{2}}\bigg{)}\bigg{)}\bigg{)}\end{array} \Bigg{)} \tag{76}\]  
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 The conditional BER for BPSK SISO FSOC links in terms of the Meijer G function and in the absence of atmospheric turbulence is presented in [65] as:  
\[P_{BPSK}(e|\gamma)=\frac{0.5}{\pi^{0.5}}G_{1,2}^{2,0}\bigg{[}0.5\gamma\bigg{|} \begin{matrix}1\\ 0,0.5\end{matrix} \bigg{]} \tag{77}\]  
Substituting Equations (52) and (77) into Equation (56) gives:  
\[P_{BPSK}=\frac{\xi^{2}}{4\Gamma(\alpha)\Gamma(\beta)\pi^{0.5}}\bullet\int\limits _{0}^{\infty}\gamma^{-1}\bullet G_{1,2}^{2,0}\bigg{[}0.5\gamma\bigg{|} \begin{matrix}1\\ 0,0.5\end{matrix} \bigg{]}\bullet G_{1,3}^{3,0}\Bigg{[}\frac{\alpha\beta\xi^{2}} {\xi^{2}+1}\bigg{(}\frac{\gamma}{\gamma}\bigg{)}^{0.5}\bigg{|} \begin{matrix} \xi^{2}+1\\ \xi^{2},\alpha,\beta\end{matrix} \bigg{]}d\gamma \tag{78}\]  
Applying equation (07.34.21.0013.01) in [53] to evaluate Equation (78) yields:\[P_{BPSK}=\frac{\xi^{2}2^{\alpha+\beta-4}}{\pi^{1.5}\Gamma(\alpha)\Gamma(\beta)}G_{4 }^{6}\mathcal{I}\Bigg{[}\left(\frac{\alpha\beta\xi^{2}}{\xi^{2}+1}\right)^{2} \frac{1}{87}\Bigg{|}\frac{1,0.5,\frac{\xi^{2}+1}{2},\frac{\xi^{2}+2}{2}}{\xi^{2 }+1}, \tag{79}\]  
The BERs estimated for the zero inner scale and infinite outer scale model and finite inner and finite outer scale model for different receive SNRs are presented in Figure 7a-d, and are based on the numerical values in Tables 2-5, respectively.