## 4 Results and Discussion  
The simulations were performed with the commercial software tools Virtual Photonics Inc. (VPI) transmission maker and MATLAB to prove the proposed orthogonalization algorithm. Considering the complexity of the atmospheric channel and our practical application scenarios, we chose QPSK modulation format. The wavelength of the laser was 1550 nm, the data rate was 20 Gbit/s (corresponding to 10 Gsymbol/s), and 1 million symbols were simulated using a PRBS sequence. The modulated optical signal was launched into the EDFA, which could amplify the output power to 10 dBm. Then, laser beams \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Branch** & **Mean** & **Standard Variance** & **Q Value** \\ \hline \multirow{3}{*}{In-phase Quadrature} & \(\mu_{\text{Q1}}=0.5828\) & \(\sigma_{\text{I1}}=0.0853\) & \\  & \(\mu_{\text{Q0}}=-0.5976\) & \(\sigma_{\text{I0}}=0.0844\) & \(\text{Q}_{\text{I}}=6.9558\) \\ \cline{1-1}  & \(\mu_{\text{Q1}}=0.5543\) & \(\sigma_{\text{Q1}}=0.2144\) & \(\text{Q}_{\text{Q}}=2.5827\) \\ \cline{1-1}  & \(\mu_{\text{Q0}}=-0.5617\) & \(\sigma_{\text{Q0}}=0.2117\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of histogram statistics. Figure 5: (a) In-phase signal eye diagram; (b) quadrature signal eye diagram; (c) constellation diagram. Figure 6: (a) In-phase signal Level 0 histogram; (b) in-phase signal Level 1 histogram; (c) quadrature signal Level 0 histogram; (d) quadrature signal Level 1 histogram. were transmitted in free-space and detected by the PIN balance detector. The resolution of the analog-to-digital converter (ADC) is 8 bits. The optical turbulence parameters are presented in Table 2. As listed in Table 2, the laser beam suffers from a relatively weak scintillation effect. The received optical signal, after passing through the free-space link, was amplified by the EDFA, and the noise figure is 3.5. The samples of the received signals of the I branch and the Q branch are processed by the ASOP algorithm implemented in MATLAB. The constellation diagram and symbol error rate (SER) are estimated. Figure 7 shows the improvement of the IQ plot (constellation) with the orthogonalization compensation method, including phase noise resulting from the linewidth of the signal laser and LO. The linewidth is set to 100 kHz. We can see that the initial constellations with phase noise and nonorthogonality due to a phase mismatch of 20\({}^{\circ}\) (Figure 7a) are improved obviously after the GSOP (Figure 7b), the SYOP (Figure 7c), and the ASOP (Figure 7d). Here, we consider the GSOP as a conventional GSOP using a fixed reference vector (Q signal) and the same in the remainder of this article. The first constellation in the first line represents the situation that the quality of the in-phase signal is better than that of the quadrature signal, so the constellation is improved after the SYOP and further improved after the ASOP because the ASOP selects the right signal as the initial reference vector. Because the phase mismatch is of a small degree, the SYOP can also obtain a good compensation result. As for the GSOP, for the imperfect initial reference vector, the compensated constellation becomes worse. Similarly, the first constellation in the second line represents the situation that the quality of quadrature signal is better than that of the in-phase signal. In this situation, the three compensation methods can all obtain good results because the GSOP also selects the right initial reference vector. The first constellation in the third line represents the situation that the quality of quadrature signal, and the in-phase signal is about the same, but both are not very good. The three compensation methods can all preserve the orthonormal properties of the IQ signals. However, the constellation after the GSOP has rotated a large angle, which may cause performance deterioration. In this situation, the ASOP scheme selects the SYOP algorithm, and the rotated constellations can be further improved by the phase-rotating estimation algorithm.