Approximate Maximum Likelihood F0 Estimation

In applications for which the fundamental frequency F0 must be measured very accurately, 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 $X(\omega)$:

$$\begin{eqnarray*} {\hat f}_0 &\isdef & \arg\max_{{\hat f}_0} \sum_{i=1}^K \log\l... ...}^K \left[\left\vert X(k_i{\hat f}_0)\right\vert+\epsilon\right] \end{eqnarray*}$$

where

$$\begin{eqnarray*} K &=& \mbox{number of peaks, and}\\ k_i &=& \mbox{estimated h... ...on the order of the spectral magnitude \emph{noise floor level}} \end{eqnarray*}$$

Since spectra of freely vibrating strings 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, 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.