Weighted Least Squares Fit of a Real Tone to Discrete Data, by Means of an Efficient Fast Fourier Transform Search

摘要

the weighted least squares fit of a real tone with arbitrary amplitude, frequency, and phase, to a given set of real discrete data, is reduced to a one-dimensional maximization of a function of frequency only. This function is manipulated into a form that can be efficiently calculated by one FFT of a complex sequence that is related to the available real data and the arbitrary real weight sequence utilized. The decoupling of the complex FFT outputs, to yield the two functions that are necessary to conduct the coarse search in frequency, is accomplished in an extremely simple fashion. A refined interpolation procedure then fits a parabola in the region near the maximum and gives a fine-grained estimate of frequency. An explanation of the apparently anomalous behavior near zero and Nyquist frequencies is given, which shows that in the limit, a constant plus linear trend is being fitted to the discrete data. A program is presented for the complete procedure, including evaluation of the best frequency, amplitude, and phase of the fitted tone. The technique is applicable to short data records, without any approximations, and for arbitrary weight sequences.