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Spectral Audio Signal Processing
Applications of the STFT
Fundamental Frequency Estimation from Sinusoidal Peaks
Getting Closer to Maximum Likelihood
Search Spectral Audio Signal Processing
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Getting Closer to Maximum Likelihood
In applications for which the fundamental frequency F0 must be
measured very accurately in a periodic signal, the estimate 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 
:
![$\displaystyle \arg\max_{{\hat f}_0} \sum_{k=1}^K
\log\left[\left\vert X(k{\hat f}_0)\right\vert+\epsilon\right]$](http://www.dsprelated.com/josimages_new/sasp/img1482.png)
![$\displaystyle \arg\max_{{\hat f}_0} \prod_{k=1}^K \left[\left\vert X(k{\hat f}_0)\right\vert+\epsilon\right]
\protect$](http://www.dsprelated.com/josimages_new/sasp/img1483.png)
where
The purpose of
is an insurance against multiplying the
whole expression by zero due to a missing partial (e.g., due to a
comb-filtering null). If
in (9.1), it is
advisable to omit indices 
for which
is too close to a
spectral null, since even one spectral null can push the product of
peak amplitudes to a very small value. At the same time, the product
should be penalized in some way to reflect the fact that it has fewer
terms (
is one way to accomplish this).
As a practical matter, it is important to inspect the magnitude spectra of the data frame manually to ensure that a robust row of peaks is being matched by the harmonic comb. For example, it is typical to look at a display of the frame magnitude spectrum overlaid with vertical lines at the optimized harmonic-comb frequencies. This provides an effective picture of the F0 estimate in which typical problems (such as octave errors) are readily seen.
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Next: References on Estimation
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Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.
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