Accurate temporal resolution of harmonic content in both amplitude and phase

Abstract

Frequency content of a periodic signal is easily performed using FFT. While magnitude is well predicted, phase information is usually meaningless and temporal changes are difficult to resolve accurately over short time periods, even with techniques such as DTFT or wavelets. These problems arise in the analysis of musical sounds and should be solvable since a well trained human ear can detect subtle and rapid changes to timbre and pitch that occur with expressiveness, techniques such as `vibrato', and lack of tone control exhibited by music students. Starting with the premise that a musical sound is truly periodic, we obtain much more accurate information from a Fourier \emph{series} than from a Fourier \emph{transform}. In this paper the Fourier series of a periodic signal is evaluated using a least squares fit, as was done before the popularisation of the FFT algorithm, but the difference is that we precisely define the frequency before fitting the coefficients, which can be done over as few as 3 or 4 cycles. The proposed technique achieves the above objectives and opens up the possibility of exploring the role of phase in the quantification of musical sound, a critical component that has been traditionally ignored.