## 3 Average PER in Turbulent Channels
### Weak and Moderate Fading  
In weak and moderate fading, the OSNR fluctuations at each branch can be accurately described by the \(\gamma-\gamma\) distribution [15; 16; 17]  
\[f_{\lambda}(y)=\frac{2\,\left(m_{y}\,m_{x}\right)^{\frac{m_{y}+m_{x}}{2}}}{ \Gamma\left(m_{x}\right)\,\Gamma\left(m_{y}\right)\,y}\,\left(\frac{L\,y}{ \overline{\lambda}}\right)^{\frac{m_{y}+m_{x}}{2}}\,K_{m_{y}-m_{x}}\left(2\, \sqrt{m_{y}\,m_{x}\,\frac{L\,y}{\overline{\lambda}}}\right)\,, \tag{15}\]  
where \(m_{x}\) and \(m_{y}\) are related to the effective numbers of large and small-scale scatterers in the OWC link and \(\overline{\lambda}=\overline{E}/N_{0}=\overline{E_{b}}/N_{0}\,\log_{2}Q\) is the average OSNR. \(\Gamma\left(\cdot\right)\) and \(K_{v}\left(\cdot\right)\) denote the Gamma and second-kind modified Bessel functions, respectively. For a plane wave transmission [15], \(m_{x}\) and \(m_{y}\) are calculated from  
\[ \begin{split} m_{x}=&\frac{1}{\exp\left(\sigma_{ \ln x}^{2}\right)-1}\,,\\ m_{y}=&\frac{1}{\exp\left(\sigma_{\ln y}^{2}\right) -1}\,,\end{split}  \tag{16}\]  
where the log-intensity variances are given by  
\[ \begin{split}\sigma_{\ln x}^{2}=&\frac{0.49\, \sigma_{R}^{2}}{\left(1+1.11\,\sigma_{R}^{12/5}\right)^{7/6}}\,,\\ \sigma_{\ln y}^{2}=&\frac{0.51\,\sigma_{R}^{2}}{ \left(1+0.69\,\sigma_{R}^{12/5}\right)^{5/6}}\,,\end{split}  \tag{17}\]  
and the Rytov variance is equal to  
\[\sigma_{R}^{2}=1.23\,C_{n}^{2}\,k^{7/6}\,l^{11/6}\,. \tag{18}\]  
\(l\) is the link length, \(k\) is the wavenumber and \(C_{n}^{2}\) is the refractive index structure parameter. Table 1 summarizes the \(m_{x}\) and \(m_{y}\) values that have been obtained for a link that operates at 1550 nm and a structure constant of \(C_{n}^{2}=4.58\cdot 10^{-13}\) m\({}^{-2/3}\). The link length is either 100 m (weak turbulence) or 500 m (moderate turbulence). The EGC output fading distribution \(f_{\lambda_{egc}}(y)\) is not known, but it can be efficiently approximated by the \(\alpha-\mu\) distribution [18]  
\[f_{\lambda_{egc}}(y)=\frac{\alpha\mu^{\mu}\,y^{\alpha\mu-1}}{\hat{y}^{\alpha \mu}\,\Gamma\left(\mu\right)}\,\exp\left(-\mu\,\frac{y^{\alpha}}{\hat{y}^{ \alpha}}\right)\,. \tag{19}\]The parameters \(\alpha\), \(\mu\) and \(\hat{y}\) are calculated from \(m_{x}\), \(m_{y}\) and \(L\) using the methodology that is presented in [19] and Table 2 summarizes their values for the link and diversity arrangements under consideration. Figure 3 presents the average PER of the OWC system in weak \(\gamma-\gamma\) fading (\(l=100\) m) for both the exact and approximate instantaneous PER relations. The noise modes and the diversity order are equal to \(k=2,8\) and \(L=1,2,5,10\), respectively, resulting in 2-80 degrees of freedom in the \(\chi^{2}\) distributions. The results demonstrate that the approximation is in good agreement with the exact relation in weak fading. In addition, it becomes more accurate at higher OSNRs where the average PER improves. On the other hand, the modulation order, diversity order, and number of noise modes do not seem to affect the approximation accuracy in weak fading. The results also show that the PPM modulation order plays the most important role than diversity in weak fading. 16-PPM provides a 5 dB OSNR gain, while 256-PPM introduces 2 dB on top of that. The diversity gain is approximately 3 dB at high OSNRs, but the use of amplified diversity may actually degrade the performance in low OSNRs due to the fact that excess optical noise is introduced in the system. The conclusions are somewhat different in moderate \(\gamma-\gamma\) fading, which is also presented in Figure 3. Again, good agreement between approximate and exact results is observed, and the accuracy improves with the OSNR and with the diversity order. This is expected since fading becomes less intense when more branches are introduced, and it was demonstrated earlier that the approximation is highly accurate in weak fading. One the other hand, the accuracy is negatively affected from increasing the modulation order.