## 5 Conclusions  
To address the problem that MRS signals are difficult to reliably extract in the application process, noise suppression and MRS signal extraction methods based on matching pursuit with an oscillating atomic library were studied in reference to the sparse decomposition principle. In accordance with the oscillation characteristics shared by the MRS signal and power-line harmonics, an oscillating atomic library dictionary was constructed, and the parameters \(\rho=0\) and \(\rho>0\) were used to distinguish the power-line harmonics and MRS signals. In the process of MP algorithm implementation, a two-step denoising approach involving extracting the harmonics from the original signal and then extracting the MRS signal to suppress random noise was adopted to realize the effective removal of noise in the MRS data. The following conclusions were obtained by comparing the simulation test results of noisy MRS data under different SNRs, \(T_{2}^{*}\), and \(f_{1}\). The proposed algorithm achieved good results in SNR enhancement, time-domain and frequency-domain signals before and after denoising, and parameter estimation of extracted signals. Moreover, SNR could be improved by up to 36 dB, and the uncertainty of the initial amplitude and average relaxation time was small. The processing results of the measured data further verified the effectiveness and practicability of the algorithm. Compared with the SSA and HMC algorithms, the proposed MP algorithm can remove power-line harmonics and random noise effectively, and it has advantages over SSA and HMC in terms of SNR improvement and parameter estimation.  
Conceptualization, B.T. and H.L.; methodology, X.L.; software, X.L.; validation, H.D. and L.W.; formal analysis, L.W. and H.Z.; investigation, X.L.; resources, L.W. and H.Z.; data curation, H.D. and L.W.; writing--original draft preparation, X.L., L.W. and B.T.; writing--review and editing, X.L., H.D. and B.T.; visualization, X.L. and H.D.; supervision, H.L.; project administration, B.T.; funding acquisition, B.T. All authors have read and agreed to the published version of the manuscript.  
This study was supported by the National Key Research and Development Program: Research on Observation Technology of Airborne Magnetic Vector Gradients of Low Temperature Superconductors (2021 YFB3900201).  
This study does not involve humans or animals.  
This study does not involve humans.  
The data used in this study involves the privacy of the project. If necessary, contact the corresponding author to obtain.  
Figure 8: Processing results of field-measured data: (**a**) the results processing with the MP, HMC, and SSA algorithms under the fifth pulse moment and the time domain waveform of \(\text{SNR}=-8.42\) dB data, with (**c**) the corresponding spectrum, and (**b**) under the seventh pulse moment and the time domain waveform of \(\text{SNR}=-4.71\) dB data, with (**d**) the corresponding spectrum.  
**Acknowledgments:** This work was supported in part by the National Science Foundation of China (Grants No. 41604083 and 41704103), the Natural Science Foundation of Jilin Province (Grants No. 20190201111 JC and 20160101281 JC), and the Fundamental Research Funds for the Central Universities.