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Identifying Chirp Rate

Consider again the Fourier transform of a complex Gaussian in (10.27):

$\displaystyle e^{-pt^2} \;\longleftrightarrow\;\sqrt{\frac{\pi}{p}} \, e^{-\frac{\omega^2}{4p}}\;\isdef \;F(\omega)$ (11.33)

Setting $ p=\alpha - j\beta$ gives

$\displaystyle e^{-\alpha t^2} e^{j\beta t^2} \;\longleftrightarrow\; \sqrt{\frac{\pi}{\alpha-j\beta}} \, e^{-\frac{\alpha}{4(\alpha^2+\beta^2)}\omega^2} e^{-j\frac{\beta}{4(\alpha^2+\beta^2)}\omega^2}.$ (11.34)

The log magnitude Fourier transform is given by

$\displaystyle \ln\left\vert F(\omega)\right\vert \eqsp \hbox{constant} -\frac{\alpha}{4(\alpha^2+\beta^2)}\omega^2$ (11.35)

and the phase is

$\displaystyle \angle F(\omega) \eqsp \hbox{constant} -\frac{\beta}{4(\alpha^2+\beta^2)}\omega^2.$ (11.36)

Note that both log-magnitude and (unwrapped) phase are parabolas in $ \omega$ .

In practice, it is simple to estimate the curvature at a spectral peak using parabolic interpolation:

\begin{eqnarray*} c_m &\isdef & \frac{d^2}{d\omega^2} \ln\vert F(\omega)\vert \eqsp - \frac{\alpha}{2(\alpha^2+\beta^2)}\\ [5pt] c_p &\isdef & \frac{d^2}{d\omega^2} \angle F(\omega) \eqsp - \frac{\beta}{2(\alpha^2+\beta^2)} \end{eqnarray*}

We can write

\begin{eqnarray*} \zbox {\frac{d^2}{d\omega^2} \ln F(\omega) \eqsp c_m + jc_p \eqsp - \frac{\alpha}{2(\alpha^2+\beta^2)} - j\frac{\beta}{2(\alpha^2+\beta^2)}.} \end{eqnarray*}

Note that the window ``amplitude-rate'' $ \alpha $ is always positive. The ``chirp rate'' $ \beta $ may be positive (increasing frequency) or negative (downgoing chirps). For purposes of chirp-rate estimation, there is no need to find the true spectral peak because the curvature is the same for all $ \omega$ . However, curvature estimates are generally more reliable near spectral peaks, where the signal-to-noise ratio is typically maximum. In practice, we can form an estimate of $ \alpha $ from the known FFT analysis window (typically ``close to Gaussian'').


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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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