## 4 Phase Denoise Experiment and Result
### Result of Synchronization Phase Denoising
After phase synchronization, the peak position shift is eliminated and the peak phase error is greatly reduced, but the side lobes are still slightly asymmetric and the peak amplitude is attenuated. Finally, all indexes are improved after denoising, and the results of the proposed method are closer to the theoretical values. In addition, Figure (a)a shows the frequency deviation and the residual frequency, which are derived by differentiating the imaging phase difference in Figure 9 and the residual phases in Figure 10, respectively. It can be seen that the frequency deviation between LT-1A and LT-1B is about \(-0.03\) Hz before phase synchronization. After phase compensation, the center of residual frequency is moved to \(0\) Hz but still occupies a wide range. After phase denoising by CS method, the range of residual frequency is significantly narrowed. Figure (b)b shows the Allan deviations of the corresponding frequency in Figure (a)a. All the Allan deviations are better than \(10^{-11}\) in the time interval of interest. It should be noted that white phase noise with a slope of \(-1\) is the dominant noise in Allan deviation rather than the common white frequency noise in oscillators. This is because the phase noise is not directly sampled in the rubidium clock, but measured through the whole synchronization link. The broadband white phase noise is introduced by the devices in the link. Therefore, Figure (b)b does not directly represent the frequency stability of the rubidium clock but reflects the accuracy of synchronization compensation. Due to the application of the proposed method, the Allan deviation is further improved and is \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Theoretical** & **With Phase Error** & **Before Denoising** & **Denoising by KF** & **Denoising by CS** \\ \hline IRW & \(4.90\) m & \(4.90\) m & \(4.90\) m & \(4.90\) m & \(4.90\) m \\ PSLR (L) & \(-13.26\) dB & \(-13.33\) dB & \(-13.25\) dB & \(-13.25\) dB & \(-13.26\) dB \\ PSLR (R) & \(-13.26\) dB & \(-13.19\) dB & \(-13.27\) dB & \(-13.27\) dB & \(-13.26\) dB \\ ISLR & \(-10.07\) dB & \(-10.07\) dB & \(-10.07\) dB & \(-10.07\) dB & \(-10.07\) dB \\ Peak amplitude & \(1.0000\) & \(0.9861\) & \(0.9999\) & \(1.0000\) & \(1.0000\) \\ Peak position & \(0.0000\) m & \(0.5441\) m & \(0.0000\) m & \(0.0000\) m & \(0.0000\) m \\ Peak phase & \(0.0000^{\circ}\) & \(-87.2629^{\circ}\) & \(-0.2182^{\circ}\) & \(-0.2176^{\circ}\) & \(-0.1964^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the response after azimuth compression. Figure 9: Result of phase denoising. (**a**) Synchronization phases (the phases are shifted for clear display). (**b**) Local zooming of synchronization phases. superior to that of KF method. Thus, the performance of the proposed synchronization phase accuracy improvement method is verified. Figure 10: Residual phases. (**a**) Residual phase before denoising. (**b**) Statistical histogram of (**a**). (**c**) Residual phase after denoising by KF. (**d**) Statistical histogram of (**c**). (**e**) Residual phase after denoising by CS. (**f**) Statistical histogram of (**e**). The synchronization accuracy under different SNRs are also evaluated and listed in Table 4. In any case, the denoising effect of CS is no less than that of KF. However, similar to KF method, the synchronization accuracy can be significantly improved in the case of low SNR. For example, when the SNR is 38 dB, the accuracy is improved by 63.11%. When SNR is up to 60 dB, it only increases by 1.65%. This is because the noise is already at a low level, in which case the STD of the residual error has been lower than 0.1\({}^{\circ}\). Therefore, when the SNR of synchronization signal is high, it can be considered that the obtained synchronization phase is credible. When the satellite is in the long baseline mode or the transmission channel is not ideal, the proposed method can play a big role in improving synchronization phase accuracy.