The Muon g-2 Experiment (E989) at Fermilab measures the muon magnetic anomaly, and its first result has improved the uncertainty from 0.54 ppm (in the previous experiment E821) to 0.46 ppm. In this experiment, the storage ring magnetic field is measured using nuclear magnetic resonance (NMR) probes. The free induction decay (FID) signals from the NMR probes are analyzed using a high-accuracy frequency extraction method. The phase function of the FID is extracted via the Hilbert transform, and the FID frequency is then extracted by fitting the phase function to a polynomial. In this presentation, the development and implementation of this FID frequency extraction method will be described in detail. Systematic effects of artifacts and a method of mitigating these effects are discussed. A thorough investigation of the statistical uncertainties of this method will be presented as well. 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.