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:

$$s(t) \eqsp e^{j\beta t^2}$$ (11.25)

Applying a Gaussian window to this chirp yields

$$x(t) \eqsp e^{-\alpha t^2} s(t) \eqsp e^{-(\alpha-j\beta) t^2} \isdefs e^{-p t^2}$$ (11.26)

where $p\isdef \alpha-j\beta$ . 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 $x(t)=e^{-p t^2}$ 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.)