# Research on Polarization Coding Cooperative Communication Scheme for FSO System
## 2 Cooperative Communication Model  
The system model of the FSO cooperative communication transmission scheme under system polarization coding is shown in Figure 1. A half-duplex communication mode is used between the nodes, and the transmission link conforms to the gamma-gamma distribution model. The system polarization code is used at the source node \(S\) and relay node \(R\), and the fast serial offset list decoding method is used at the destination node \(D\). In half-duplex mode, each node sends and receives signals at different time slots. In this system, the system polarization code is only used to encode the information bits of the original information \(M\). Assuming that the information sequence encoded by the source node is \(x_{s}\), during the first time slot of the FSO cooperative communication system, the source node \(S\) sends signals to the relay node \(R\) and the destination node \(D\), respectively. After receiving it, the relay node decodes it and calculates the bit-error rate. If the bit-error rate is lower than the threshold value, the intersection of the freeze bit set of the direct transmission \(S-D\) link and the \(S-R\) link information bit set is extracted, the intersection is processed using polarization code, and then the information is sent to the destination node. The channel coefficient of the \(S-R\) link is \(h_{sr}\) and the channel coefficient of the \(S-D\) link is \(h_{sd}\). Therefore, the received signals of the relay node \(R\) and the destination node \(D\) can be expressed as:  
\[y_{sr}=\sqrt{P_{S}}\eta_{R}h_{sr}x_{s}+n_{sr} \tag{1}\]  
\[y_{sd}=\sqrt{P_{S}}\eta_{D}h_{sd}x_{s}+n_{sd} \tag{2}\]  
where \(P_{S}\) is the transmit power at the source node, \(\eta_{R}\) and \(\eta_{D}\) are photoelectric conversion efficiency at relay node and destination node, respectively, and \(n_{sr}\) and \(n_{sd}\) are the complex additive Gaussian white noise of the \(S-R\) link and the \(S-D\) link, with a mean of zero and a variance per dimension of \(\sigma_{sr}^{2}\) and \(\sigma_{sd}^{2}\), respectively. During the second time slot, the relay node \(R\) transmits the system polarization-encoded signal to the destination node \(D\). The destination node \(D\) recovers the information bit set of the \(S-R\) link using the frozen bit set of the signal and the polarization code of the \(S-R\) link system. Finally, the received signal is combined with EGC and decoded. The channel coefficient of the \(R-D\) link is \(h_{rd}\), so the signal \(y_{rd}\) received by the destination node \(D\) can be expressed as:  
\[y_{rd}=\sqrt{P_{R}}\eta_{D}h_{rd}x_{r}+n_{rd} \tag{3}\]  
Figure 1: Basic system block diagram of cooperative communication system. where \(n_{nd}\) is the complex additive Gaussian white noise with zero mean and \(\sigma_{rd}^{2}\) variance per dimension. Since FSO links obey gamma-gamma distribution, the probability density function (PDF) of the channel state \(h\) of each branch can be expressed as:  
\[f_{h}(h)=\frac{2(\alpha\beta)^{(\alpha+\beta)/2}}{\Gamma(\alpha)\Gamma(\beta)}h^ {(\alpha+\beta)/2-1}\times K_{\alpha-\beta}\left(2\sqrt{\alpha\beta h}\right) \tag{4}\]  
\(\Gamma(\cdot)\) is the gamma function, and \(K_{v}(\cdot)\) is the second kind of Bessel function. \(\alpha\) and \(\beta\) characterize the effective number of vortices in large and small regions, respectively.