Systematic and Statistical Uncertainties of the Hilbert-Transform Based High-precision FID Frequency Extraction Method

Publication type:
Journal article
Metadata:
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 widely 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. 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 $\chi^2$ used in the fitting routine. A down-sampling method for fixing the singular covariance matrix has been developed, so that the minimum $\chi^2$-fit yields properly the statistical uncertainty of the extracted frequency. Other practical methods of obtaining the statistical uncertainty are also discussed.
Autoren
  • Ran Hong
  • Simon Corrodi
  • Saskia Charity
  • Stefan Baessler
  • Jason Bono
  • Timothy Chupp
  • Martin Fertl
  • David Flay
  • Alejandro Garcia
  • Jimin George
  • Kevin Louis Giovanetti
  • Timothy Gorringe
  • Joseph Grange
  • Kyun Woo Hong
  • David Kawall
  • Brendan Kiburg
  • Bingzhi Li
  • Rachel Osofsky
  • Dinko Pocanic
  • Suvarna Ramachandran
  • Matthias Smith
  • Herbert Erik Swanson
  • Alec Tewsley-Booth
  • Peter Winter
  • Tianyu Yang
  • Kai Zheng
Autoren-URL
http://arxiv.org/abs/2101.08412v2
Schlüsselwörter
  • physics.ins-det
  • physics.ins-det
Datum der Veröffentlichung
2021
Herausgeber URL
http://dx.doi.org/10.1016/j.jmr.2021.107020
Datum der Datenerfassung
2021
Datum, an dem der Datensatz öffentlich gemacht wurde
2021
Titel
Systematic and Statistical Uncertainties of the Hilbert-Transform Based High-precision FID Frequency Extraction Method

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2101.08412v2.pdf

Data source: arXiv

Other metadata sources:
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 widely 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. 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 $\chi^2$ used in the fitting routine. A down-sampling method for fixing the singular covariance matrix has been developed, so that the minimum $\chi^2$-fit yields properly the statistical uncertainty of the extracted frequency. Other practical methods of obtaining the statistical uncertainty are also discussed.
Autoren
  • Ran Hong
  • Simon Corrodi
  • Saskia Charity
  • Stefan Baessler
  • Jason Bono
  • Timothy Chupp
  • Martin Fertl
  • David Flay
  • Alejandro Garcia
  • Jimin George
  • Kevin Louis Giovanetti
  • Timothy Gorringe
  • Joseph Grange
  • Kyun Woo Hong
  • David Kawall
  • Brendan Kiburg
  • Bingzhi Li
  • Rachel Osofsky
  • Dinko Pocanic
  • Suvarna Ramachandran
  • Matthias Smith
  • Herbert Erik Swanson
  • Alec Tewsley-Booth
  • Peter Winter
  • Tianyu Yang
  • Kai Zheng
Autoren-URL
http://arxiv.org/abs/2101.08412v2
ISSN
1090-7807
Zeitschrift
Journal of Magnetic Resonance
Schlüsselwörter
  • physics.ins-det
  • physics.ins-det
Online publication date
2021
Datum der Veröffentlichung
2021
Status
Published
Herausgeber
Elsevier
Herausgeber URL
https://www.sciencedirect.com/science/article/pii/S1090780721001099
Datum der Datenerfassung
2021
Titel
Systematic and Statistical Uncertainties of the Hilbert-Transform Based High-precision FID Frequency Extraction Method
Sub types
  • Article
Ausgabe der Zeitschrift
329

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