A VMD-PE-SG denoising method based on K–L divergence for satellite atomic clock

The time scale of the global navigation satellite system (GNSS) is the core element for its position, navigation and timing services. A highly stable atomic clock is essential to ensure the reliability of the GNSS time scale. This study proposed a novel hybrid denoising model combining variational mode decomposition (VMD), K–L divergence, permutation entropy (PE), and Savitzky–Golay (SG) filter for satellite atomic clocks. Firstly, the key parameter of VMD is solved efficiently by taking the minimum sum of K–L divergence of decomposed modes as the constraint condition, and the optimised parameters are applied to the decomposition process. On this basis, the PE algorithm is used to determine the modes decomposed by VMD into signal-dominant and noise-dominant components by searching for the mutation of PE value at two adjacent points. Finally, the noise-dominant components are denoised by the SG filter and then reconstructed with the signal-dominant components to form the denoised signal. The analysis of the simulated signal shows that the method can effectively remove noise from the simulated signal, and the resulting denoised signal is similar to the pure signal. Compared with commonly used ensemble empirical mode decomposition and wavelet denoising methods, the signal-noise ratio of the proposed method is improved by 21.2% and 28.9%, and the root mean square error is improved by 24.1% and 29.8%, respectively. The results of experimental data testify that the K–L VMD-PE-SG-based denoising method can significantly reduce the dominant noise within one day, thus effectively improving the short to medium-term frequency stability. Compared with the original signal, the stability of the smoothing time within 76 800 s is generally improved, and the degree of improvement depends on the type of atomic clock and the smoothing time.