### Identifying Chirp Rate

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

(11.33) |

Setting gives

(11.34) |

The

*log magnitude*Fourier transform is given by

(11.35) |

and the phase is

(11.36) |

Note that

*both log-magnitude and (unwrapped) phase are parabolas in*.

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

We can write

Note that the window ``amplitude-rate''
is always positive.
The ``chirp rate''
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
. 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
from the known FFT
analysis window (typically ``close to Gaussian'').

#### Chirplet Frequency-Rate Estimation

The chirp rate may be estimated from the relation as follows:

- Let
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
estimate may be estimated from the
spectral peak by

#### Simulation Results

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.

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Audio Filter Banks

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Modulated Gaussian-Windowed Chirp