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

TL;DR

This paper analyzes the uncertainties in high-precision FID frequency extraction using the Hilbert transform in NMR, proposing methods to mitigate artifacts and accurately estimate statistical errors.

Contribution

It provides a detailed implementation, analytical derivation of noise effects, and a novel down-sampling method to improve frequency uncertainty estimation in Hilbert-transform based FID analysis.

Findings

Artifacts and noise impact phase extraction analyzed

A down-sampling method fixes covariance matrix issues

Proper statistical uncertainty estimation achieved

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.