# Spectral Audio Signal ProcessingSpectrum Analysis of SinusoidsSinusoidal Peak InterpolationBias of Parabolic Peak Interpolation

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### Bias of Parabolic Peak Interpolation

Since the true window transform is not a parabola (except for the conceptual case of a Gaussian window transform expressed in dB), there is generally some error in the interpolated peak due to this mismatch. Such a systematic error in an estimated quantity (due to modeling error, not noise), is often called a bias. Parabolic interpolation is unbiased when the peak occurs at a spectral sample (FFT bin frequency), and also when the peak is exactly half-way between spectral samples (due to symmetry of the window transform about its midpoint). For other peak frequencies, quadratic interpolation yields a biased estimate of both peak frequency and peak amplitude. (Phase is essentially unbiased [1].)

Since zero-padding in the time domain gives ideal interpolation in the frequency domain, there is no bias introduced by this type of interpolation. Thus, if enough zero-padding is used so that a spectral sample appears at the peak frequency, simply finding the largest-magnitude spectral sample will give an unbiased peak-frequency estimator. (We will learn in §4.8.2 that this is also the maximum likelihood estimator for the frequency of a sinusoid in additive white Gaussian noise.)

While we could choose our zero-padding factor large enough to yield any desired degree of accuracy in peak frequency measurements, it is more efficient in practice to combine zero-padding with parabolic interpolation (or some other simple, low-order interpolator). In such hybrid schemes, the zero-padding is simply chosen large enough so that the bias due to parabolic interpolation is negligible. In §4.8 below, the Quadratically Interpolated FFT (QIFFT) method is described as one such hybrid scheme.

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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.