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

Mototsugu Abe and Julius O. Smith III


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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