## Gaussian Windowed Chirps (Chirplets)

As discussed in §G.8.2, an interesting generalization of
sinusoidal modeling is *chirplet modeling*. A *chirplet* is
defined as a Gaussian-windowed sinusoid, where the sinusoid has a
constant amplitude, but its frequency may be linearly ``sweeping.''
This definition arises naturally from the mathematical fact that the
Fourier transform of a Gaussian-windowed chirp signal is a complex Gaussian
pulse, where a *chirp signal* is defined as a sinusoid having linearly
modulated frequency, *i.e.*, quadratic phase:

(11.25) |

Applying a Gaussian window to this chirp yields

(11.26) |

where . It is thus clear how naturally Gaussian amplitude envelopes and linearly frequency-sweeping sinusoids (chirps) belong together in a unified form called a chirplet.

The basic chirplet
can be regarded as an
*exponential polynomial signal* in which the polynomial is of
order 2. Exponential polynomials of higher order have also been
explored [89,90,91].
(See also §G.8.2.)

### Chirplet Fourier Transform

The Fourier transform of a complex Gaussian pulse is derived in §D.8 of Appendix D:

This result is valid when is complex. Writing in terms of real variables and as

(11.28) |

we have

(11.29) |

That is, for complex, is a chirplet (Gaussian-windowed chirp). We see that the chirp oscillation frequency is zero at time . Therefore, for signal modeling applications, we typically add in an arbitrary frequency offset at time 0, as described in the next section.

### Modulated Gaussian-Windowed Chirp

By the *modulation theorem* for Fourier transforms,

(11.30) |

This is proved in §B.6 as the dual of the shift-theorem. It is also evident from inspection of the Fourier transform:

(11.31) |

Applying the modulation theorem to the Gaussian transform pair above yields

(11.32) |

Thus, we frequency-shift a Gaussian chirp in the same way we frequency-shift any signal--by complex modulation (multiplication by a complex sinusoid at the shift-frequency).

### 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.

**Next Section:**

FFT Filter Banks

**Previous Section:**

Time Scale Modification