## 3 Performance Analysis
\right)\\ -\sum\limits_{m=1}^{n}H_{m}\frac{K_{CN}Q\big{(}\sqrt{\delta a_{m}}\big{)}}{ \sqrt{\gamma}}\left(\begin{array}{c}\frac{\alpha_{1}\beta_{1}}{\eta_{1}} \big{(}\frac{\sqrt{a_{m}}}{\sqrt{l<l>\eta_{1}}}\big{)}^{\beta_{1}-1}e^{a_{m}-[ -(\frac{\sqrt{a_{m}}}{\sqrt{l<l>\eta_{1}}})]^{\beta_{1}}}\left(1-e^{-(\frac{ \sqrt{a_{m}}}{\sqrt{l<l>\eta_{1}}})^{\beta_{1}}}\right)^{\alpha_{1}-1}\times\\ \frac{\alpha_{2}\betaFigure 2a,b show the variation in the SNR of HQAM, XQAM, RQAM, and QAM with BER at different anisotropy factors of u = 1 and u = 3, respectively. The results show that the BER decreases not only with the increase in SNR but also with the increase in the anisotropy factor. As can be seen below, the bit error rate of the G-H-S system under four modulation schemes gradually decreases. When the SNR is the same, the G-H-S system with HQAM modulation has the lowest BER. When the BER is the same, the SNR of the G-H-S system modulated by HQAM is the lowest. Moreover, the BER of G-H-S and G-S systems decrease monotonically with increased SNR. When the SNR is small, the difference between the BER of G-H-S and G-S systems is small. As the SNR increases, the difference between them increases. The lower the SNR, the more severe the interference the signal is subjected to and the larger the BER of the system. The G-H-S and G-S lines in the figure have a slow downward trend when the SNR is low. The higher the SNR, the less interference the signal is subjected to and the lower the error rate of the system. Therefore, the G-H-S and G-S lines in the figure show a significant downward trend when the SNR is high. The G-H-S system with a HAP is less affected by turbulence and weather, so its bit error rate drops faster. Therefore, the two lines of the G-H-S and G-S systems using QAM are relatively close when the SNR is low, and the difference between them gradually increases when the SNR is large. The G-S system has a higher BER. For example, when the SNR is 22 dB and u = 1, the BER of the G-H-S and G-S systems are 0.47 and 0.83, respectively. When the SNR is 70 dB and u = 1, the BERs of G-H-S and G-S systems are 2.17 \(\times\) 10\({}^{-6}\) and 3.23 \(\times\) 10\({}^{-5}\), respectively. However, with the increase in anisotropy factor u, the influence of turbulence on the propagating laser beam will be weakened, and the BER of the system will be further reduced. To achieve the same BER 10\({}^{-7}\), when the anisotropy factor u = 1, the SNR ratio of the G-H-S system modulated by HQAM is 70 dB. Compared with the G-H-S system with HQAM modulation when u = 3, the required SNR is increased by 20 dB. Therefore, the G-H-S system with HQAM modulation can achieve better communication performance in the ground-satellite uplink. Figure 3a,b show the variation in the zenith angle of HQAM, XQAM, RQAM, and QAM with BER at different anisotropy factors of u = 1 and u = 3, respectively. The results show that the BER of the system not only increases with the increase in the zenith angle but also decreases with the increase in the anisotropy factor. In ground-satellite laser communication, the zenith angle is an essential physical quantity that can change atmospheric turbulence parameters. As can be seen below, the BER of the four modulation schemes all increase with the increase in the zenith angle and decrease with the increase in the anisotropy factor at the same zenith angle. For example, when u = 1 and the zenith angle is 22\({}^{\circ}\), the BERs of HQAM, XQAM, RQAM, and QAM are 2.97 \(\times\) 10\({}^{-4}\), 7.67 \(\times\) 10\({}^{-4}\), 2.02 \(\times\) 10\({}^{-3}\), and 3.65 \(\times\) 10\({}^{-3}\), respectively. When u = 3, and the zenith angle is 22\({}^{\circ}\), the BERs of HQAM, XQAM, RQAM, and QAM are 2.05 \(\times\) 10\({}^{-5}\), 6.09 \(\times\) 10\({}^{-5}\), 2.