## 3 Hierarchical Modulation and Physical Layer Network Code Scheme  
To solve the asymmetric problem in the collaborative FSO communication system, considering the channel characteristics of the FSO communication system, the hierarchical modulation-combined physical-layer network coding scheme was used in the cooperative FSO communication system to achieve the improvement of the performance of the cooperative FSO communication system. In the system shown in Figure 1, hierarchical modulation was used at the source nodes \(S_{1}\), \(S_{2}\), \(S_{3}\), and \(S_{4}\), and hierarchical modulation was used at the relay node in a manner that combined the physical-layer network encoding. To further suppress the interference of atmospheric turbulence on the FSO communication system and obtain higher power utilization, the hierarchical modulation process used 4/16-QAM modulation. The hierarchical modulation and physical layer network code (HM-PNC: hierarchical modulation and physical layer network code) scheme was implemented as follows: Each source node divided the information to be sent into two levels based on the relative importance of the data, namely high-priority data (\(S_{ih}\)) and low-priority data (\(S_{il}\)), where \(i\in(1,2,3,4)\). Each source node contained 4 bits of information, assuming that the first two bits were high-priority data and the last two bits were low-priority information. Both \(S_{ih}\) and \(S_{il}\) used 4 QAM modulation. We assumed that the transmit power distribution factor between high and low priorities was \(\alpha\), and the final message sent at the source node was:  
\[S_{i}=\sqrt{\alpha}S_{ih}+\sqrt{1-\alpha}S_{il} \tag{5}\]  
The specific principle of 4/16-QAM hierarchical modulation is shown in Figure 2. Figure 2a is a constellation diagram of the high priority data \(S_{ih}\), and the arrow refers to the mapping point of data 00 after 4 QAM modulation; Figure 2b is a constellation diagram of low priority data \(S_{il}\), and the arrow refers to the mapping point of data 01 after 4-QAM modulation; Figure 2c is the constellation chart of the data to be sent \(S_{i}\), which is consistent with the constellation chart modulated by 16 QAM. In Figure 2, \(2d_{2}\) represents the minimum distance between constellation points of high-priority data, with \(d_{1}=\sqrt{\alpha/2}\); \(2d_{2}\) represents the minimum distance between constellation points of low-priority data within the same quadrant, with \(d_{2}=\sqrt{(1-\alpha)/2}\)[15]; and \(2d_{2}\) represents the minimum distance between constellation points for low-priority data in adjacent quadrants. In a collaborative FSO communication system, since the channel conditions between the relay node and each source node are better, each source node in Figure 1 can obtain all transmission information, including high priority and low priority data. In the system shown in Figure 1, the four source nodes in the first stage send information to the relay node; the second stage relay node sends information to the four source nodes. Each source node can use a direct-through link to obtain high-priority information for neighbor nodes, so only the low-priority information of neighbors needs to be obtained through the relay node; however, the complete information of the remote node must be obtained through the relay node. In the Figure 1 system, the encoding codeword encoded by the physical-layer network at the relay node has the following structure (\(\xi_{i}\)):  
\[\xi_{1} =S_{1h}\oplus S_{3l} \tag{6}\] \[\xi_{2} =S_{1l}\oplus S_{3h}\] (7) \[\xi_{3} =S_{2h}\oplus S_{4l}\] (8) \[\xi_{4} =S_{2l}\oplus S_{4h} \tag{9}\]  
According to the hierarchical modulation scheme used in this system, the relay node can convert four network-encoded codewords into two mixed messages (\(J_{i}\)), and the specific implementation process is as follows:  
\[J_{i}=\sqrt{\alpha}\phi_{i}+\sqrt{1-\alpha}\phi_{i},\quad i=1,2 \tag{10}\]  
Thereinto, \(\phi\), \(\varphi\in\xi=\{\xi_{1},\xi_{2},\xi_{3},\xi_{4}\}\) and \(\phi_{1}\
eq\phi_{1}\
eq\phi_{2}\
eq\phi_{2}\).