# Analysis of Scintillation Effects on Free Space Optical Communication Links in South Africa
## 7 Average Bit Error Rate (BER) Analysis
### Quadrature Amplitude Modulation (SIM-QAM) FSOC Links
1 \(\times\) J Rectangular QAM FSOC Links  
The conditional probability of error for I \(\times\) J rectangular SIM-QAM signals in the absence of atmospheric turbulence is given in [66] as:  
\[P_{(I\times I)QAM}(e/\gamma)=\frac{1}{\log_{2}(I\bullet I)}\left(\frac{I-1}{I} erfc\left(\sqrt{\frac{3\gamma\log_{2}(I\bullet I)}{I^{2}+J^{2}-2}}\right)+\frac{I-1}{J} erfc\left(\sqrt{\frac{3\gamma\log_{2}(I\bullet I)}{I^{2}+J^{2}-2}} \right)\right) \tag{84}\]  
where \(I\) and \(J\) are the dimensions of the in-phase and quadrature signals [24; 66]. 1. Weak Atmospheric Turbulence  
Similarly, inserting Equations (53), (59)-(63), (64) and (84) into Equation (56) gives:  
\[ \begin{array}{l}P_{(I\times I)QAM}=\frac{2\delta^{2}\sigma_{S \mathrm{I}}\exp(q)}{\sqrt{\pi}\log_{2}(I\bullet I)}\bullet\left(\left(\frac{I -1}{I}\right)+\left(\frac{I-1}{I}\right)\right)\times\\ \left(\begin{array}{l}\int\limits_{0}^{\infty}\left(\frac{1}{3}\left(\left( \exp(8z_{1}\sigma_{SI}-2p)\right)^{(0.5\xi^{2})}\bullet erfc\left(\sqrt{\frac {3\gamma A_{b}^{2}\exp(8z_{1}\sigma_{SI}-2p)\bullet\log_{2}(I\bullet I)}{I^{2} +I^{2}-2}}\right)\right)\bullet\exp(-z_{1}^{2})\right)dz_{1}\\ +\\ \int\limits_{0}^{\infty}\left(\frac{9}{16}\left(\left(\exp(4.5z_{2}\sigma_{SI }-2p)\right)^{(0.5\xi^{2})}\bullet erfc\left(\sqrt{\frac{3\gamma A_{b}^{2}\exp (4.5z_{1}\sigma_{SI}-2p)\bullet\log_{2}(I\bullet I)}{I^{2}+I^{2}-2}}\right) \right)\bullet\exp(-z_{2}^{2})\right)dz_{2}\end{array}\right)\end{array}  \tag{85}\]  
Employing the Gauss-Hermite quadrature approximation in Equation (66) to evaluate Equation (85) generates the expression for the BER of I \(\times\) J rectangular SIM-QAM SISO FSOC links in the presence of weak atmospheric turbulence and pointing errors:  
\[ \begin{array}{l}P_{(I\times I)QAM}=\frac{2\delta^{2}\sigma_{S \mathrm{I}}\exp(q)}{\sqrt{\pi}\log_{2}(I\bullet I)}\bullet\left(\left(\frac{I -1}{I}\right)+\left(\frac{I-1}{I}\right)\right)\times\\ \left(\begin{array}{l}\left(\frac{1}{3}\sum\limits_{i=1}^{N}W_{i}\left(\left( \exp(8z_{1}\sigma_{SI}-2p)\right)^{(0.5\xi^{2})}\bullet erfc\left(\sqrt{\frac {3\gamma A_{b}^{2}\exp(8z_{1}\sigma_{SI}-2p)\bullet\log_{2}(I\bullet I)}{I^{2 }+I^{2}-2}}\right)\right)\right)\\ +\\ \left(\frac{9}{16}\sum\limits_{i=1}^{N}W_{i}\left(\left(\exp(4.5z_{2}\sigma_{ SI}-2p)\right)^{(0.5\xi^{2})}\bullet erfc\left(\sqrt{\frac{3\gamma A_{b}^{2}\exp(4.5z_{1} \sigma_{SI}-2p)\bullet\log_{2}(I\bullet I)}{I^{2}+I^{2}-2}}\right)\right) \right)\end{array}\right)\end{array}  \tag{86}\]  
2. Moderate to Strong Atmospheric Turbulence  
The average BER for I \(\times\) J Rectangular SIM-QAM SISO FSOC links in the presence of moderate to strong atmospheric turbulence while considering the effect pointing errors is presented in [67] as:  
\[P_{(I\times I)QAM}=\frac{\delta^{2}2^{\alpha+\beta-3}}{\pi^{1.5}\Gamma(\alpha )\Gamma(\beta)}\frac{1}{\log_{2}(I\bullet I)}\left(\frac{I-1}{I}+\frac{I-1}{J} \right)G_{3,6}^{5,2}\Bigg{[}\left(\frac{\alpha\beta\delta^{2}_{2}}{\delta^{2 }+1}\right)^{2}\frac{\left(I^{2}+J^{2}-2\right)}{48\gamma\log_{2}(I\bullet I)} \Bigg{|}_{\frac{\delta^{2}}{\gamma},\frac{\alpha}{2},\frac{\alpha+1}{2},\frac {\beta+1}{2},0}^{\frac{\alpha^{2}+2}{2}}\Bigg{]} \tag{87}\]  
The BER performance of SIM rectangular 8 \(\times\) 4 or 32-QAM FSOC links is presented for zero inner scale and infinite outer scale model and finite inner and finite outer scale model for different receive SNRs in Figure 9a-d. In Figure 9b-d, the BER results of the FSOC links deployed in all the investigated cities are plotted using Equations (1)-(3), (15), (35), (38), (40), (41), (46) and (86), while the BER results in Figure 9a are computed using Equations (1)-(3), (15), (30), (31), (35), (46) and (87). Similarly, the results in Figure 9a-d are based on the numerical values in Tables 2-5, respectively. As expected, the BER performance of SIM 32-QAM FSOC links is poorer than SIM 16-QAM FSOC links in Figure 8a-d, but the 32-QAM links transmits a greater amount of information.