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Approximate Maximum Likelihood F0 Estimation

In applications for which the fundamental frequency F0 must be measured very accurately in a periodic signal, the estimate $ {\hat f}_0$ obtained by the above algorithm can be refined using a gradient search which matches a so-called ``harmonic comb'' to the magnitude spectrum of an interpolated FFT $ X(\omega)$ :

$\displaystyle {\hat f}_0 \isdefs \arg\max_{{\hat f}_0} \sum_{k=1}^K \log\left[\left\vert X(k{\hat f}_0)\right\vert+\epsilon\right]
\eqsp \arg\max_{{\hat f}_0} \prod_{k=1}^K \left[\left\vert X(k{\hat f}_0)\right\vert+\epsilon\right]


K &=& \mbox{number of peaks}\\
k &=& \mbox{harmonic number of $k$th peak (valid method for non-stiff strings)}\\
\epsilon &=& \mbox{small value on the order of spectral magnitude \emph{noise floor level}}

Note that freely vibrating strings are not exactly periodic due to exponenential decay, coupling effects, and stiffness (which stretches harmonics into quasiharmonic overtones, as explained in §6.9). However, non-stiff strings can often be analyzed as having approximately harmonic spectra ( $ \leftrightarrow$ periodic time waveform) over a limited time frame.

Since string spectra typically exhibit harmonically spaced nulls associated with the excitation and/or observation points, as well as from other phenomena such as recording multipath and/or reverberation, it is advisable to restrict $ K$ to a range that does not include any spectral nulls (or simply omit index $ k$ when $ k{\hat f}_0$ is too close to a spectral null), since even one spectral null can push the product of peak amplitudes to a very small value. As a practical matter, it is important to inspect the magnitude spectra of the data manually to ensure that a robust row of peaks is being matched by the harmonic comb. For example, a display of the frame magnitude spectrum overlaid with vertical lines at the optimized harmonic-comb frequencies yields an effective picture of the F0 estimate in which typical problems (such as octave errors) are readily seen.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University