Sinusoidal Frequency Estimation

The form of the least-squares estimator (4.13) in the known-frequency case immediately suggests the following frequency estimator for the unknown-frequency case:

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 (4.13) evaluated at that frequency.

It can be shown [115] that (4.14) 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,

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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About the Author: Julius Orion Smith III
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.