# 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  
7.3.1. M-ary Square SIM-QAM FSOC Links  
The conditional probability of error for M-ary square SIM-QAM signals in the absence of atmospheric turbulence as derived in [64; 66] is given as:  
\[P_{MQAM}(e/\gamma)=\Bigg{(}\frac{\sqrt{M}-1}{\sqrt{M}\log_{2}\sqrt{M}}\Bigg{)} erfc\Bigg{(}\sqrt{\frac{3\gamma\log_{2}M}{2(M-1)}}\Bigg{)} \tag{80}\]  
where \(M\) is the even number of bits per symbol for square constellations. 1. Weak Atmospheric Turbulence  
Similar to the previous computations of the BER expressions of RZ-OOK and BPSK modulated signals, substituting Equations (53), (59)-(64) and (80) into Equation (56) gives:  
\[\begin{array}{l}P_{MQAM}=\Bigg{(}\frac{2\xi^{2}\sigma_{S1}\exp(d)}{\sqrt{ \pi}}\bullet\frac{\sqrt{M}-1}{\sqrt{M}\log_{2}\sqrt{M}}\Bigg{)}\times\\ \Bigg{(} \begin{array}{l}\int\limits_{0}^{\infty}\Bigg{(}\frac{1}{3}\bigg{(} \left(\exp(8z_{1}\sigma_{SI}-2p)\right)^{(0.55^{2})}\bullet erfc\Bigg{(} \sqrt{\frac{3\log_{2}(M)\bullet\gamma A_{o}^{2}\exp(8z_{1}\sigma_{SI}-2p)}{2( M-1)}}\Bigg{)}\Bigg{)}\bullet\exp(-z_{1}^{2})\Bigg{)}dz_{1}\\ +\\ \int\limits_{0}^{\infty}\Bigg{(}\frac{9}{16}\Bigg{(}\left(\exp(4.5z_{2}\sigma_ {SI}-2p)\right)^{(0.55^{2})}\bullet erfc\Bigg{(}\sqrt{\frac{3\log_{2}(M) \bullet\gamma A_{o}^{2}\exp(4z_{2}\sigma_{SI}-2p)}{2(M-1)}}\Bigg{)}\Bigg{)} \bullet\exp(-z_{2}^{2})\Bigg{)}dz_{2}\end{array} \Bigg{)} \tag{81}\]  
where further similar mathematical calculations produce the expression for the BER of M-ary square SIM-QAM SISO FSOC links in the presence of weak atmospheric turbulence and pointing errors:  
\[ \begin{array}{l}P_{MQAM}=\Bigg{(}\frac{\xi^{2}\sigma_{S1}\exp(d)}{\sqrt{\pi }}\bullet\frac{\sqrt{M}-1}{\sqrt{M}\log_{2}\sqrt{M}}\Big{)}\times\\ \Bigg{(}\begin{array}{l}\Bigg{(}\frac{1}{3}\sum\limits_{i=1}^{N}W_{i}\Bigg{(} (\exp(8z_{1}\sigma_{SI}-2p))^{(0.55^{2})}\bullet erfc\Bigg{(}\sqrt{\frac{3 \log_{2}(M)\bullet\gamma A_{o}^{2}\exp(8z_{1}\sigma_{SI}-2p)}{2(M-1)}}\Bigg{)} \Bigg{)}\Bigg{)}\\ +\\ \Bigg{(}\frac{9}{16}\sum\limits_{i=1}^{N}W_{i}\Bigg{(}(\exp(4.5z_{2}\sigma_{SI} -2p))^{(0.55^{2})}\bullet erfc\Bigg{(}\sqrt{\frac{3\log_{2}(M)\bullet\gamma A_ {o}^{2}\exp(4.5z_{1}\sigma_{SI}-2p)}{2(M-1)}}\Bigg{)}\Bigg{)}\Bigg{)}\end{array} \Bigg{)}\end{array}  \tag{82}\]  
2. Moderate to Strong Atmospheric Turbulence  
The average BER for M-ary Square SIM-QAM SISO FSOC links in the presence of moderate to strong atmospheric turbulence, while taking in account the effect of pointing errors, is derived in [67] as:  
\[P_{MQAM}=\frac{\xi^{2}2^{\pm i-3}}{\pi^{15}\Gamma(\pi)\Gamma(\beta)}\Bigg{(} \frac{\sqrt{M}-1}{\sqrt{M}\log_{2}\sqrt{M}}\Bigg{)}G_{3,6}^{5,2}\Bigg{[}\left( \frac{\alpha\beta\xi^{2}}{\xi^{2}+1}\right)^{2}\frac{(M-1)}{24\overline{\gamma }\log_{2}M}\bigg{|}\frac{z_{1}^{2}}{2},\frac{\alpha}{2},\frac{\alpha+1}{2}, \frac{\beta+1}{2},\frac{\beta+1}{2},0\Bigg{]} \tag{83}\]  
Equations (1)-(3), (15), (35), (38), (40), (41), (46) and (82) are used in plotting the BER results for the FSOC links in all the investigated cities based on the zero inner scale and infinite outer scale model and finite inner and finite outer scale model for different receive SNRs in Figure 8b-d. The results in Figure 8a are generated using Equations (1)-(3), (15), (30), (31), (35), (46) and (83). Additionally, the numerical values in Tables 2-5 determine the BERs for the SIM 16-QAM FSOC links in Figure 8a-d. In the instance when turbulence eddies have sizes of zero and infinity in the Kolmogorov model with infinitely large inertial range, receive SNRs of \(\sim\)70 dB are required to obtain a BER of \(10^{-5}\) for 16-QAM FSOC links deployed in all the investigated cities, over a link distance of 1 km, as shown in Figure 8c. Figure 8: BER of SIM 16-QAM FSOC links vs. average SNR (dB) over weak and moderate to strong atmospheric turbulence channels for various cities in South Africa. The BERs of 16-QAM FSOC links in Figure 8a-d surpass the BER results of conventional OOK FSOC links in Figure 6a-d.  
7.3.2.