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Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Applications of the STFT
       Gaussian Windowed Chirps

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Gaussian Windowed Chirps

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

$\displaystyle \zbox {e^{-pt^2} \;\longleftrightarrow\;\sqrt{\frac{\pi}{p}} e^{-... ...ga^2}{4p}},\quad \forall p\in {\bf C}: \; \mbox{re}\left\{p\right\}>0} \protect$ (10.8)

This result is valid when $ p$ is complex. Writing $ p$ in terms of real variables $ \alpha $ and $ \beta $ as

$\displaystyle p = \alpha - j\beta, $

we have

$\displaystyle x(t) = e^{-p t^2} = e^{-\alpha t^2} e^{j\beta t^2} = e^{-\alpha t^2} \left[\cos(\beta t^2) + j\sin(\beta t^2)\right] $

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


Subsections
Previous: Sines + Noise + Transients Models
Next: Modulated Gaussian Windowed Chirp

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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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