## 3 System Performance under Both Weak Turbulence and Moderate-to-Strong Turbulence
### Average BER and Channel Capacity under Moderate-to-Strong Turbulence  
Here, we further investigate the average BER and channel capacity of the communication system under moderate-to-strong turbulence, which is modeled by Gamma-Gamma distribution as we have mentioned in Section 2.  
For moderate-to-strong turbulence, by inserting Equations (5) and (8) into Equation (7), the average BER can be written as  
\[P_{e}=\int_{0}^{\infty}Q\left(\frac{RAI}{\sqrt{2}\sigma}\right)\frac{2(\alpha \beta)^{\frac{\alpha+\beta}{2}}}{\Gamma(\alpha)\Gamma(\beta)}\,t^{\frac{\alpha +\beta}{2}}-1K_{\alpha-\beta}\left(2\sqrt{\alpha\beta I}\right)dI. \tag{13}\]Note that the Bessel function in Equation (13) can be further recast with [PERSON]'s G function from [29] as  
\[K_{\wp}(x)=\frac{1}{2}G_{0,2}^{2,0}\Bigg{[}\frac{x^{2}}{4}\Bigg{|}\begin{array}[ ]{c}\frac{v}{2},-\frac{v}{2}\\ \end{array}\Bigg{]}. \tag{14}\]  
By substituting the \(Q\)-function and Bessel function expressed by Meijer's G function into Equation (13), the average BER can be written as  
\[P_{e}=\int_{0}^{\infty}\frac{(\alpha\beta)^{\frac{\alpha+\beta}{2}}}{2\sqrt{ \pi}\Gamma(\alpha)\Gamma(\beta)}G_{0,2}^{2,0}\Bigg{[}\frac{\gamma I^{2}}{4} \Bigg{|}\begin{array}{c}1\\ 0,\frac{1}{2}\\ \end{array}\Bigg{]}G_{0,2}^{2,0}\Bigg{[}\alpha\beta I\Bigg{|}\begin{array}{c} 1\\ \frac{\alpha-\beta}{2},\frac{\beta-\alpha}{2}\\ \end{array}\Bigg{]}\,dI. \tag{15}\]  
Based on the properties of Meijer's G function [30], the average BER for an FSO communication system under moderate-to-strong turbulence can be finally simplified as  
\[P_{e}=\frac{2^{\alpha+\beta-3}}{\pi^{\frac{3}{2}}\Gamma(\alpha)\Gamma(\beta)}G _{5,2}^{2,4}\Bigg{[}\frac{4\gamma}{(\alpha\beta)^{2}}\Bigg{|}\begin{array}{c }\frac{2-\alpha}{2},\frac{1-\alpha}{2},\frac{2-\beta}{2},\frac{1-\beta}{2},1 \\ 0,\frac{1}{2}\\ \end{array}\Bigg{]}. \tag{16}\]  
By substituting Equation (5) into Equation (10), the average channel capacity of the FSO communication system under moderate-to-strong turbulence can be expressed as  
\[C=\int_{0}^{\infty}B\text{log}_{2}\Big{(}1+\gamma I^{2}\Big{)}\frac{2(\alpha \beta)^{\frac{\alpha+\beta}{2}}}{\Gamma(\alpha)\Gamma(\beta)}\Big{.}^{\frac{ \alpha+\beta}{2}-1}K_{\alpha-\beta}\Big{(}2\sqrt{\alpha\beta I}\Big{)}\,dI. \tag{17}\]  
With the aid of the identical equations in Equation (14) and \(\text{log}_{2}(1+x)=\frac{1}{\ln 2}G_{2,2}^{1,2}\Bigg{[}\begin{array}{c} \begin{array}{c}1,1\\ 1,0\\ \end{array}\Bigg{]},\) we have  
\[C=\int_{0}^{\infty}\frac{B(\alpha\beta)^{\frac{\alpha+\beta}{2}}}{\ln 2 \Gamma(\alpha)\Gamma(\beta)}I^{\frac{\alpha+\beta}{2}-1}G_{2,2}^{1,2}\Bigg{[} \gamma I^{2}\Big{|}\begin{array}{c}1,1\\ 1,0\\ \end{array}\Bigg{]}G_{0,2}^{2,0}\Bigg{[}\alpha\beta I\Bigg{|}\begin{array}{c }-\\ \frac{\alpha-\beta}{2},\frac{\beta-\alpha}{2}\\ \end{array}\Bigg{]}\,dI. \tag{18}\]  
Finally, the average channel capacity of the FSO system under Gamma-Gamma channel turbulence can be simplified with the aid of the integration property of Meijer's G function [30] as  
\[C=\frac{B2^{\alpha+\beta-2}}{\pi\ln 2\Gamma(\alpha)\Gamma(\beta)}G_{6,2}^{1,6} \Bigg{[}\frac{16\gamma}{(\alpha\beta)^{2}}\Bigg{|}\begin{array}{c}1,1,\frac {1-\alpha}{2},\frac{2-\alpha}{2},\frac{1-\beta}{2},\frac{2-\beta}{2}\\ 1,0\\ \end{array}\Bigg{]}. \tag{19}\]