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Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Applications of the STFT
       Gaussian Windowed Chirps
          Modulated Gaussian Windowed Chirp

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Modulated Gaussian Windowed Chirp

By the modulation theorem for Fourier transforms,

$\displaystyle \zbox {x(t)e^{-j\omega_0t}\;\longleftrightarrow\;X(\omega+\omega_0).} $

This is proved in §B.1.6 as the dual of the shift-theorem. It is also evident from inspection of the Fourier transform:

$\displaystyle \int_{-\infty}^\infty \left[x(t)e^{-j\omega_0 t}\right] e^{-j\ome... ...t_{-\infty}^\infty x(t)e^{-j(\omega+\omega_0) t} dt \isdefs X(\omega+\omega_0) $

Applying the modulation theorem to the Gaussian transform pair above yields

$\displaystyle \zbox {e^{-pt^2} e^{-j\omega_0 t} \;\longleftrightarrow\;\sqrt{\f... ...+\omega_0)^2}{4p}},\quad \forall p\in {\bf C}: \; \mbox{re}\left\{p\right\}>0.}$ (10.9)

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).

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Next: Identifying Chirp Rate

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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