A Quadrature Signals Tutorial: Complex, But Not Complicated

Understanding the 'Phasing Method' of Single Sideband Demodulation

Complex Digital Signal Processing in Telecommunications

Introduction to Sound Processing

Introduction of C Programming for DSP Applications

Spectrum Analysis of Sinusoids

Sinusoidal Peak Interpolation

Bias of Parabolic Peak Interpolation

**Search Spectral Audio Signal Processing**

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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.

Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (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.

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