Identifying Chirp Rate

Consider again the Fourier transform of a complex Gaussian in (9.8):

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

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

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

The log magnitude Fourier transform is given by

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

and the phase is

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

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:

$\begin{eqnarray*} c_m &\isdef & \frac{d^2}{d\omega^2} \ln\vert F(\omega)\vert = ... ...\omega^2} \angle F(\omega) = - \frac{\beta}{2(\alpha^2+\beta^2)} \end{eqnarray*}$

We can write

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

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'').

Subsections

Previous: Modulated Gaussian Windowed Chirp
Next: Chirplet Frequency-Rate Estimation

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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.

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Spectral Audio Signal Processing
    Applications of the STFT
       Gaussian Windowed Chirps
          Identifying Chirp Rate