Bayesian inference of linear sine-fitting parameters from integrating digital voltmeter data

Algorithms based on the discrete Fourier transform and the least-squares three-parameter sine fit have been used for accurate low-frequency voltage measurement using integrating digital voltmeters. Bayesian statistics is here applied to the common problem of fitting a truncated Fourier series with constant and known fundamental frequency to a time series created by the readings of a high-resolution digital voltmeter used to digitize the output of a waveform generator. The sufficient conditions for regarding the discrete Fourier transform or the least-squares as the best estimate of the fitting parameters are presented. Expressions for the estimates of the noise variance are also derived. Two design of experiments that yield fast estimates that are less sensitive to frequency accuracy are analysed. The uncertainties associated with the amplitudes of the harmonics of the truncated Fourier series are evaluated. The uncertainty evaluation takes into account the systematic effects introduced by commercial digital voltmeters. It is shown that it is possible to measure the amplitude of the fundamental harmonic of a 100 Hz low-distortion waveform with a relative uncertainty of 2.5 x 10(super -6) and the amplitudes of the corresponding next five harmonics with an uncertainty relative to the fundamental of 1.5 x 10(super -6) using commercial stable waveform generators and high-resolution digital voltmeters.