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Optimal Peak-Finding in the Spectrum

Based on the preceding sections, an ``obvious'' method for deducing sinusoidal parameters from data is to find the amplitude, phase, and frequency of each peak in a zero-padded FFT of the data. We have considered so far the following issues:

  1. Make sure the data length (or window length) is long enough so that all sinusoids in the data are resolved.
  2. Use enough zero padding so that the spectrum is heavily oversampled, making the peaks easier to interpolate.
  3. Use quadratic interpolation of the three samples surrounding a dB-magnitude peak in the heavily oversampled spectrum.
  4. Evaluate the fitted parabola at its extremum to obtain the interpolated amplitude and frequency estimates for each sinusoidal component.
  5. Similarly compute a phase estimate at each peak frequency using quadratic or even linear interpolation on the unwrapped phase samples about the peak.
For future reference, we will call this the quadratically interpolated FFT (QIFFT) method [271].

The question naturally arises as to how good is the QIFFT method for spectral peak estimation? Is it optimal in any sense? Are there better methods? Are there faster methods that are almost as good? These are questions that generally fall under the topic of sinusoidal parameter estimation.

We will show that the QIFFT method is a fast, ``approximate maximum-likelihood method.'' When properly configured, it is in fact extremely close to the true maximum-likelihood estimator for a single sinusoid in white noise. It is also close to the maximum likelihood estimator for multiple sinusoids that are well separated in frequency (i.e., side-lobe overlap can be neglected). Finally, the QIFFT method can be considered optimal perceptually in the sense that any errors induced by the suboptimality of the QIFFT method are inaudible when the zero-padding factor is a factor of 5 or more. While a zero-padding factor of 5 is sufficient for all window types, including the rectangular window, less zero-padding is needed with windows having flatter main-lobe peaks, as summarized in Table 5.3.

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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