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Sinusoidal Frequency Estimation

The form of the least-squares estimator (5.41) in the known-frequency case immediately suggests the following frequency estimator for the unknown-frequency case:

$\displaystyle \hat{\omega}_0^\ast = \arg\{\max_{\hat{\omega}_0} \left\vert\hbox{\sc DTFT}_{\hat{\omega}_0}(x)\right\vert\}. \protect$ (6.43)

That is, the sinusoidal frequency estimate is defined as that frequency which maximizes the DTFT magnitude. Given this frequency, the least-squares sinusoidal amplitude and phase estimates are given by (5.41) evaluated at that frequency.

It can be shown [121] that (5.43) is in fact the optimal least-squares estimator for a single sinusoid in white noise. It is also the maximum likelihood estimator for a single sinusoid in Gaussian white noise, as discussed in the next section.

In summary,

$\textstyle \parbox{0.8\textwidth}{the least squares estimate for the sinusoidal parameters
of amplitude $A$, phase $\phi$, and frequency $\omega_0 $\ are obtained from
the complex amplitude and location of the magnitude peak in the DTFT
of the zero-padded observation data $x(n)$.}$

In practice, of course, the DTFT is implemented as an interpolated FFT, as described in the previous sections (e.g., QIFFT method).

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University