## Design Criteria for the Quadratically Interpolated FFT Method (III):
Bias due to Amplitude and Frequency Modulation

### Mototsugu Abe and Julius O. Smith III

#### Abstract

Quadratic peak interpolation in a zero-padded Fast Fourier Transform
(FFT) has been widely used for sinusoidal parameter estimation in
audio applications. In this series of papers, we investigate the
estimation errors associated with various choices of analysis
parameters, and provide precise criteria for designing the
estimator. In this third paper, we focus on the bias caused by
amplitude and frequency modulations of sinusoidal components. We
investigate the first order modulations and derive a theory to predict
the bias with a Gaussian window. We then extend it to other often-used
windows, numerically confirm the results, and provide a criterion to
determine the maximum window length as functions of prescribed maximum
error bound and AM/FM rates of a signal. The results show, for
example, if we are to bound the frequency bias below 1Hz using a Hann
window for a sinusoid whose magnification rate is 100 times per second
and chirp rate is 1kHz/s, we need to use a window whose length is less
than 31ms.
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