  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.
