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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:

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

Applying a Gaussian window to this chirp yields

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

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University