# Research on Polarization Coding Cooperative Communication Scheme for FSO System
## 4 Outage Probability and Bit-Error Rate Analysis  
### Outage Probability Analysis  
In the cooperative communication system, when the instantaneous transmission capacity of the channel is lower than the current transmission rate or the system signal-to-noise ratio is lower than the specified signal-to-noise ratio threshold, the system will be interrupted. The probability of the interruption is the interruption probability of the system. This section analyzes the outage probability of a cooperative system under an atmospheric turbulence channel, in which the link \(R-D\) and link \(S-D\) are independent of each other. Point-to-point transmission refers to the direct transmission link in the cooperative communication system. When the interruption occurs, the signal-to-noise ratio of the link is lower than the specified signal-to-noise ratio threshold, and the interruption probability can be expressed in the following form [16]:  
\[P_{o}=\Pr(\overline{\gamma}\leq\gamma_{th})=\Pr\left(h\leq\sqrt{\gamma_{th}/ \overline{\gamma}}\right)=\int_{0}^{\sqrt{\gamma_{th}/\overline{\gamma}}}f_{ h}(h)dh \tag{15}\]  
where \(\overline{\gamma}\) is the average signal-to-noise ratio of the current link, \(\gamma_{th}\) is the specified signal-to-noise ratio threshold. According to the literature [16], it can be obtained that:  
\[K_{o}(x)=\frac{1}{2}G \begin{array}{c}2,0\\ 0,2\end{array} \left( \begin{array}{c|c}\frac{x^{2}}{4}\end{array} \right| \begin{array}{c}-\\ \frac{v}{2},-\frac{v}{2}\end{array} \right) \tag{16}\]  
Therefore, Equation (4) can be written as:  
\[f_{h}(h)=\frac{(\alpha\beta)^{(\alpha+\beta)/2}h^{(\alpha+\beta)/2-1}}{\Gamma (\alpha)\Gamma(\beta)}\times G \begin{array}{c}2,0\\ 0,2\end{array} \left[\alpha\beta h\left| \begin{array}{c}-\\ \frac{\alpha-\beta}{2},\frac{\beta-\alpha}{2}\end{array} \right.\right] \tag{17}\]  
It is also obtained in the literature [17] that:  
\[(\alpha\beta h)^{(\alpha+\beta)/2}G \begin{array}{c}2,0\\ 0,2\end{array} \left(\alpha\beta h\left| \begin{array}{c}-\\ \frac{\alpha-\beta}{2},-\frac{\alpha-\beta}{2}\end{array} \right.\right)=( \alpha\beta h)^{(\alpha+\beta)/2}G \begin{array}{c}2,0\\ 0,2\end{array} \left(\alpha\beta h\left| \begin{array}{c}-\\ \alpha,\beta\end{array} \right.\right) \tag{18}\]  
Therefore, Equation (17) can be written in the following form:  
\[f_{h}(h)=\frac{1}{\Gamma(\alpha)\Gamma(\beta)h}\times G \begin{array}{c}2,0 \\ 0,2\end{array} \left[\alpha\beta h\left| \begin{array}{c}-\\ \alpha,\beta\end{array} \right.\right] \tag{19}\]By introducing Equation (19) into Equation (15), it can be obtained that the outage probability of the link is:  
\[P_{0}=\int_{0}^{\sqrt{\gamma_{th}/\overline{\gamma}}}\frac{1}{\Gamma(\alpha) \Gamma(\beta)h}\times G \begin{array}{c}2,0\\ 0,2\end{array} \left[\alpha gh\left| \begin{array}{c}-\\ \alpha,\beta\end{array} \right.\right]dh \tag{20}\]  
In the FSO system, the relationship between the average signal-to-noise ratio of the channel \(\overline{\gamma}\), the signal-to-noise ratio threshold \(\gamma_{th}\), and the channel state \(h\) is \(\gamma_{th}=\overline{\gamma}h^{2}\). Using this relationship to integrate and transform the integral on the right side of Equation (20), we get:  
\[\int_{0}^{\sqrt{\gamma_{th}/\overline{\gamma}}}\frac{1}{\Gamma(\alpha)\Gamma( \beta)h}\times G \begin{array}{c}2,0\\ 0,2\end{array} \left[\alpha gh\left| \begin{array}{c}-\\ \alpha,\beta\end{array} \right.\right]dh=\frac{1}{2}\int_{0}^{\gamma_{th}}\frac {1}{\gamma_{th}}G \begin{array}{c}2,0\\ 0,2\end{array} \left[\alpha\beta\sqrt{\gamma_{th}/\overline{\gamma}}\right| \begin{array}{c}-\\ \alpha,\beta\end{array} \right]d\overline{\gamma} \tag{21}\]  
According to the literature [17]:  
\[\int_{0}^{\gamma_{th}}\gamma_{th}^{-1}\times G \begin{array}{c}2,0\\ 0,2\end{array} \left[\alpha\beta\sqrt{\frac{\gamma_{th}}{\overline{\gamma}}} \left| \begin{array}{c}-\\ \alpha,\beta\end{array} \right.\right]d\gamma_{0}=2G \begin{array}{c}2,1\\ 1,3\end{array} \left[\alpha\beta\sqrt{\frac{\gamma_{th}}{\overline{\gamma_{0}}}} \left|