### Fundamental Frequency Estimation

As mentioned in §6.11.2 above, it is advisable to
estimate the fundamental frequency of vibration (often called
``*F0*'') in order that the partial overtones are well resolved
while maintaining maximum time resolution for estimating the decay
time-constant.

Below is a summary of the F0 estimation method used in calibrating loop filters with good results [471]:

- Take an FFT of the middle third of a recorded plucked string tone.
- Find the frequencies and amplitudes of the largest peaks, where is chosen so that the retained peaks all have a reasonable signal-to-noise ratio.
- Form a histogram of peak spacing
- The pitch estimate is defined as the most common spacing in the histogram.

#### Approximate Maximum Likelihood F0 Estimation

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 :

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 ( 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 to a range that does not include any
spectral nulls (or simply omit index when
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.

#### References on F0 Estimation

An often-cited book on classical methods for pitch detection,
particularly for voice, is that by Hess [192]. The harmonic
comb method can be considered an approximate maximum-likelihood pitch
estimator, and more accurate maximum-likelihood methods have been
worked out
[114,547,376,377].
More recently, Klapuri has been developing some promising methods for
multiple pitch estimation
[254,253,252].^{7.12}A comparison of real-time pitch-tracking algorithms applied to guitar
is given in [260], with consideration of
latency (time delay).

#### Extension to Stiff Strings

An advantage of the harmonic-comb method, as well as other frequency-domain maximum-likelihood pitch-estimation methods, is that it is easily extended to accommodate stiff strings. For this, the stretch-factor in the spectral-peak center-frequencies can be estimated--the so-called coefficient of inharmonicity, and then the harmonic-comb (or other maximum-likelihood spectral-matching template) can be stretched by the same amount, so that when set to the correct pitch, the template matches the data spectrum more accurately than if harmonicity is assumed.

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EDR-Based Loop-Filter Design

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Dispersion Filter Design