Identifying Chirp Rate

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

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

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

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

The log magnitude Fourier transform is given by

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

and the phase is

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