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Systematic and statistical uncertainties of the hilbert-transform based high-precision FID frequency extraction
Ran Hong1, Simon Corrodi2, Saskia Charity3
1Argonne National Laboratory, Lemont, IL, USA; University of Kentucky, Lexington, KY, USA.
Journal of Magnetic Resonance (San Diego, Calif. : 1997)
|July 12, 2021
Summary
This study details a Hilbert transform method for extracting magnetic field frequencies from nuclear magnetic resonance (NMR) signals. A novel down-sampling technique improves statistical uncertainty calculations for high-precision magnetic field measurements.
Area of Science:
- Physics
- Analytical Chemistry
Background:
- Pulsed nuclear magnetic resonance (NMR) is crucial for high-precision magnetic field measurements.
- Extracting signal frequency from free induction decay (FID) is key to NMR measurements.
- The Hilbert transform is a known method for phase and frequency extraction from FID signals.
Purpose of the Study:
- To provide a detailed implementation of a Hilbert-transform based FID frequency extraction method.
- To analyze the impact of artifacts and noise on phase function extraction.
- To develop a method for improving statistical uncertainty estimation in frequency extraction.
Main Methods:
- Detailed implementation of Hilbert-transform based FID frequency extraction.
- Analytical derivation of artifact and noise effects on the phase function.
- Study of noise correlations in phase function samples.
- Development of a down-sampling method to address singular covariance matrices.
Main Results:
- Artifacts and noise significantly affect the extracted phase function.
- The error covariance matrix for the extracted phase function is often singular.
- A down-sampling method effectively resolves the singular covariance matrix issue.
- The minimum chi-squared fit with the proposed method yields accurate statistical uncertainty.
Conclusions:
- The Hilbert transform method, with the proposed down-sampling technique, offers a robust approach for accurate magnetic field frequency extraction in NMR.
- Understanding and mitigating noise and artifacts are critical for reliable FID signal analysis.
- The developed method improves the statistical uncertainty estimation, crucial for high-precision measurements.
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