Systematic and statistical uncertainties of the hilbert-transform based high-precision FID frequency extraction method

Ran Hong; Simon Corrodi; Saskia Charity; Stefan Baeßler; Jason Bono; Timothy Chupp; Martin Fertl; David Flay; Alejandro García; Jimin George; Kevin Louis Giovanetti; Timothy Gorringe; Joseph Grange; Kyun Woo Hong; David Kawall; Brendan Kiburg; Bingzhi Li; Liang Li; Rachel Osofsky; Dinko Počanić; Suvarna Ramachandran; Matthias Smith; Herbert Erik Swanson; Alec Tewsley-Booth; Peter Winter; Tianyu Yang; Kai Zheng

doi:10.1016/j.jmr.2021.107020

Systematic and statistical uncertainties of the hilbert-transform based high-precision FID frequency extraction method

J Magn Reson. 2021 Aug:329:107020. doi: 10.1016/j.jmr.2021.107020. Epub 2021 Jun 15.

Authors

Ran Hong¹, Simon Corrodi², Saskia Charity³, Stefan Baeßler⁴, Jason Bono³, Timothy Chupp⁵, Martin Fertl⁶, David Flay⁷, Alejandro García⁸, Jimin George⁷, Kevin Louis Giovanetti⁹, Timothy Gorringe¹⁰, Joseph Grange¹¹, Kyun Woo Hong¹², David Kawall⁷, Brendan Kiburg³, Bingzhi Li¹³, Liang Li¹⁴, Rachel Osofsky⁸, Dinko Počanić¹², Suvarna Ramachandran², Matthias Smith⁸, Herbert Erik Swanson⁸, Alec Tewsley-Booth⁵, Peter Winter², Tianyu Yang⁵, Kai Zheng²

Affiliations

¹ Argonne National Laboratory, Lemont, IL, USA; University of Kentucky, Lexington, KY, USA. Electronic address: rhong@anl.gov.
² Argonne National Laboratory, Lemont, IL, USA.
³ Fermi National Accelerator Laboratory, Batavia, IL, USA.
⁴ University of Virginia, Charlottesville, VA, USA; Oak Ridge National Lab, Oak Ridge, TN, USA.
⁵ University of Michigan, Ann Arbor, MI, USA.
⁶ University of Washington, Seattle, WA, USA; Johannes Gutenberg-Universität Mainz, Mainz, Germany.
⁷ University of Massachusetts, Amherst, MA, USA.
⁸ University of Washington, Seattle, WA, USA.
⁹ James Madison University, Harrisonburg, VA, USA.
¹⁰ University of Kentucky, Lexington, KY, USA.
¹¹ Argonne National Laboratory, Lemont, IL, USA; University of Michigan, Ann Arbor, MI, USA.
¹² University of Virginia, Charlottesville, VA, USA.
¹³ Argonne National Laboratory, Lemont, IL, USA; Shanghai Jiao Tong University, Shanghai, China.
¹⁴ Shanghai Jiao Tong University, Shanghai, China.

PMID: 34252841
DOI: 10.1016/j.jmr.2021.107020

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.

Keywords: FID; Frequency extraction; High-precision magnetometer; Hilbert transform; Uncertainty analysis.

Publication types

Research Support, Non-U.S. Gov't
Research Support, U.S. Gov't, Non-P.H.S.