Identifying Chirp Rate

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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'').

Chirplet Frequency-Rate Estimation

The chirp rate $\beta$ may be estimated from the relation $\beta = \alpha c_m/c_p$ as follows:

Let ${\hat \alpha}$ denote the measured (or known) curvature at the midpoint of the analysis window .
Let and denote weighted averages of the measured curvatures and along the log-magnitude and phase of a spectral peak, respectively.
Then the chirp-rate $\beta$ estimate may be estimated from the spectral peak by

$\displaystyle \zbox {{\hat \beta}\isdefs {\hat \alpha}\frac{[c_p]}{[c_m]}}$

Simulation Results

**Figure 10.24:** Real Gaussian-windowed chirp (time domain).
$\includegraphics[width=\twidth]{eps/gwchirp}$

Figure 10.24 shows the waveform of a Gaussian-windowed chirp (``chirplet'') generated by the following matlab code:

fs = 8000;
x = chirp([0:1/fs:0.1],1000,1,2000);
M = length(x);
n=(-(M-1)/2:(M-1)/2)';
w = exp(-n.*n./(2*sigma.*sigma));
xw = w(:) .* x(:);

Figure 10.25 shows the same chirplet in a time-frequency plot. Figure 10.26 shows the spectrum of the example chirplet. Note the parabolic fits to dB magnitude and unwrapped phase. We see that phase modeling is most accurate where magnitude is substantial. If the signal were not truncated in the time domain, the parabolic fits would be perfect. Figure 10.27 shows the spectrum of a Gaussian-windowed chirp in which frequency decreases from 1 kHz to 500 Hz. Note how the curvature of the phase at the peak has changed sign.

**Figure 10.25:** Real Gaussian-windowed chirp (spectrogram).
$\includegraphics[width=\twidth]{eps/gwchirpsgC}$

**Figure 10.26:** Gaussian-windowed chirp (frequency domain).
$\includegraphics[width=\twidth]{eps/gwchirpxform}$

**Figure 10.27:** Downgoing chirp.
$\includegraphics[width=\twidth]{eps/gwchirpdownxform}$

**Figure 10.28:** Short chirp--time waveform.
$\includegraphics[width=\twidth,height=3in]{eps/gwchirpshort}$

**Figure 10.29:** Short chirp--spectrum.
$\includegraphics[width=\twidth]{eps/gwchirpshortxform}$

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Audio Filter Banks
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Modulated Gaussian-Windowed Chirp