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Spectral Audio Signal Processing
    Gaussian Function Properties
       Fourier Transform of Complex Gaussian

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Fourier Transform of Complex Gaussian



Theorem:

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



Proof: [192, p. 211] The Fourier transform of $ e^{-pt^2}$ is defined as

$\displaystyle \int_{-\infty}^\infty e^{-pt^2} e^{-j\omega t}dt \eqsp \int_{-\infty}^\infty e^{-(pt^2+j\omega t)} dt. $

Completing the square of the exponent gives

\begin{eqnarray*} pt^2 + j\omega t - \frac{\omega^2}{4p} + \frac{\omega^2}{4p} &=& p\left(t+j\frac{\omega}{2p}\right)^2 + \frac{\omega^2}{4p} \end{eqnarray*}

Thus, the Fourier transform can be written as

$\displaystyle e^{-\frac{\omega^2}{4p}} \int_{-\infty}^\infty e^{-p\left(t+j\frac{\omega}{2p}\right)^2} dt \eqsp \sqrt{\frac{\pi}{p}}\, e^{-\frac{\omega^2}{4p}} $

using our previous result.


Subsections
Previous: Gaussian Integral with Complex Offset
Next: Alternate Proof

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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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