#### 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:

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,

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

**Next Section:**

Multiple Sinusoids in Additive Gaussian White Noise

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Sinusoidal Amplitude and Phase Estimation