Systematic and Statistical Uncertainties of the Hilbert-Transform Based High-precision FID Frequency Extraction Method
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 used in the fitting routine. A down-sampling method for fixing the singular covariance matrix has been developed, so that the minimum -fit yields properly the statistical uncertainty of the extracted frequency. Other practical methods of obtaining the statistical uncertainty are also discussed.
Cite
@article{arxiv.2101.08412,
title = {Systematic and Statistical Uncertainties of the Hilbert-Transform Based High-precision FID Frequency Extraction Method},
author = {Ran Hong and Simon Corrodi and Saskia Charity and Stefan Baessler and Jason Bono and Timothy Chupp and Martin Fertl and David Flay and Alejandro Garcia and Jimin George and Kevin Louis Giovanetti and Timothy Gorringe and Joseph Grange and Kyun Woo Hong and David Kawall and Brendan Kiburg and Bingzhi Li and Rachel Osofsky and Dinko Pocanic and Suvarna Ramachandran and Matthias Smith and Herbert Erik Swanson and Alec Tewsley-Booth and Peter Winter and Tianyu Yang and Kai Zheng},
journal= {arXiv preprint arXiv:2101.08412},
year = {2021}
}