# Research on Polarization Coding Cooperative Communication Scheme for FSO System
## 4 Outage Probability and Bit-Error Rate Analysis
### System Bit-Error Rate Analysis  
Command \(\varepsilon\), \(\varepsilon_{SR}\), and \(\varepsilon_{RD}\) respectively represent the event \(\{\hat{M}\
eq M\}\), \(\{\hat{M}_{R}\
eq M\}\) and the error transmission of the link \(R-D\), where \(M\) is the original information sent by the source node, \(\hat{M}\) is the system output information, and \(\hat{M}_{R}\) is the decoded information at the relay node. Order \(\varepsilon^{C}\), \(\varepsilon_{SR}^{C}\), and \(\varepsilon_{RD}^{C}\) are complementary events. The bit-error rate of a cooperative FSO communication system with system polarization coding is expressed as:  
\[\begin{array}{c}\Pr(\varepsilon)=\Pr(\varepsilon|\varepsilon_{SR})\Pr( \varepsilon_{SR})+\Pr(\varepsilon|\varepsilon_{SR}^{C})\Pr(\varepsilon_{SR}^{C}) \\ \leq\Pr(\varepsilon_{SR})+\Pr(\varepsilon|\varepsilon_{SR}^{C})\end{array} \tag{25}\]At the source node, the code rate using system polarization coding is \(R<I(W_{SR})\), and the upper bound of \(\Pr(\varepsilon_{SR})\) is \(O\left(2^{-N^{\beta}}\right)\). The second term in Equation (25) can be expressed as:  
\[\begin{array}{ll}\Pr\left(\varepsilon|\varepsilon_{SR}^{C}\right)&=\Pr\left( \varepsilon|\varepsilon_{SR}^{C},\varepsilon_{RD}\right)\Pr\left(\varepsilon_ {RD}|\varepsilon_{SR}^{C}\right)+\Pr\left(\varepsilon|\varepsilon_{SR}^{C}, \varepsilon_{RD}^{C}\right)\Pr\left(\varepsilon_{RD}^{C}|\varepsilon_{SR}^{C}\right) \\ &=\Pr\left(\varepsilon|\varepsilon_{SR}^{C},\varepsilon_{RD}\right)\Pr\left( \varepsilon_{RD}\right)+\Pr\left(\varepsilon|\varepsilon_{SR}^{C}, \varepsilon_{RD}^{C}\right)\Pr\left(\varepsilon_{RD}^{C}\right)\\ &\leq\Pr\left(\varepsilon_{RD}\right)+\Pr\left(\varepsilon|\varepsilon_{SR}^{C },\varepsilon_{RD}^{C}\right)\end{array} \tag{26}\]  
In the above formula, \(\varepsilon_{SR}\) and \(\varepsilon_{RD}\) are independent of each other. All terms are obtained under the construction of the system polarization code, so the upper limit of all terms is \(O\left(2^{-N^{\beta}}\right)\). By substituting Equation (26) into Equation (25), the upper limit of the bit-error rate of the cooperative FSO communication system with system polarization coding can be obtained.  
At the receiving end of the FSO communication system using BPSK modulation, if the influence of atmospheric turbulence on the communication system is not considered, the bit-error rate under BPSK modulation can be expressed as:  
\[P_{BPSK}=\frac{1}{2}erfc\left(\sqrt{SNR}\right) \tag{27}\]  
In Equation (27), \(SNR\) is the instantaneous signal-to-noise ratio at the destination node, and \(erfc(\cdot)\) is the error function, which can be defined as:  
\[erfc(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-\eta^{2}}d\eta \tag{28}\]  
In the FSO cooperative communication system involved in this paper, considering the influence of atmospheric turbulence on the FSO system, combined with the gamma-gamma atmospheric turbulence channel model, the system bit-error rate modulated by BPSK can be expressed as:  
\[P_{e-BPSK}=\int_{0}^{\infty}P_{BPSK}f_{h}(h)dh=\int_{0}^{\infty} erfc\left(\sqrt{SNR}\right)\frac{\alpha\beta^{(\alpha+\beta)/2}}{\Gamma( \alpha)\Gamma(\beta)}h^{(\alpha+\beta)/2-1}\times K_{\alpha-\beta}\left(2 \sqrt{\alpha\beta l}\right)dh \tag{29}\]  
where \(h\) is channel status information. Finally, according to the relationship between \(h\) and the instantaneous signal-to-noise ratio of the channel, the bit-error rate \(P_{e-BPSK}\) at the destination node of the system can be solved, which meets the conditions of Equation (26).