Abstract
Pulsed nuclear magnetic resonance (NMR) is widely used in high-precision magnetic field measurements. The absolute value of the magnetic field is determined from the precession frequency of nuclear magnetic moments. The Hilbert transform is one of the methods that have been used to extract the phase function from the observed free induction decay (FID) signal and then its frequency. In this paper, a detailed implementation of a Hilbert-transform based FID frequency extraction method is described, and it is briefly compared with other commonly used frequency extraction methods. How artifacts and noise level in the FID signal affect the extracted phase function are derived analytically. A method of mitigating the artifacts in the extracted phase function of an FID is discussed. Correlations between noises of the phase function samples are studied for different noise spectra. We discovered that the error covariance matrix for the extracted phase function is nearly singular and improper for constructing the χ2 used in the fitting routine. A down-sampling method for fixing the singular covariance matrix has been developed, so that the minimum χ2-fit yields properly the statistical uncertainty of the extracted frequency. Other practical methods of obtaining the statistical uncertainty are also discussed.
Original language | English |
---|---|
Article number | 107020 |
Journal | Journal of Magnetic Resonance |
Volume | 329 |
DOIs | |
State | Published - Aug 2021 |
Bibliographical note
Funding Information:This work was supported by the U.S. Department of Energy (DOE), Office of Science under contracts DE-AC02-06CH11357 (Argonne National Laboratory), DE-FG02-88ER40415 (University of Massachussetts), and DE-FG02-97ER41020 (University of Washington), and National Science Foundation (NSF), Division of Physics Award No. 1807266 (University of Kentucky) and Award No. 1812314 (University of Michigan). This work was also supported in part by the National Natural Science Foundation of China Grant No. 11975153 (Shanghai Jiao Tong University). This document was prepared by the Muon g-2 collaboration using the resources of the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359.
Funding Information:
This work was supported by the U.S. Department of Energy (DOE), Office of Science under contracts DE-AC02-06CH11357 (Argonne National Laboratory), DE-FG02-88ER40415 (University of Massachussetts), and DE-FG02-97ER41020 (University of Washington), and National Science Foundation (NSF), Division of Physics Award No. 1807266 (University of Kentucky) and Award No. 1812314 (University of Michigan). This work was also supported in part by the National Natural Science Foundation of China Grant No. 11975153 (Shanghai Jiao Tong University).
Funding Information:
This document was prepared by the Muon collaboration using the resources of the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359.
Publisher Copyright:
© 2021 Elsevier Inc.
Keywords
- FID
- Frequency extraction
- High-precision magnetometer
- Hilbert transform
- Uncertainty analysis
ASJC Scopus subject areas
- Biophysics
- Biochemistry
- Nuclear and High Energy Physics
- Condensed Matter Physics