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 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 [248].

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 4.1.

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Subsections

• • Minimum Zero-Padding for High-Frequency Peaks
  • Minimum Zero-Padding for Low-Frequency Peaks
  • Matlab for Computing Minimum Zero-Padding Factors
  
  
• Least Squares Sinusoidal Parameter Estimation
  • Sinusoidal Amplitude Estimation
  • Sinusoidal Amplitude and Phase Estimation
  • Sinusoidal Frequency Estimation
  
  
• Maximum Likelihood Sinusoid Estimation
• Likelihood Function
  • Multiple Sinusoids in Additive Gaussian White Noise
  • Non-White Noise
  
  
• Generality of Maximum Likelihood Least Squares

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Order a Hardcopy of Spectral Audio Signal Processing

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Next: Minimum Zero-Padding for High-Frequency Peaks

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written by 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.