### 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

*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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