### Getting Closer to Maximum Likelihood

In applications for which the fundamental frequency
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
:

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 (10.1), 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 estimate in which typical problems (such as octave errors) are readily seen.

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References on Estimation

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Useful Preprocessing