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Fourier Transform of Complex Gaussian



Theorem:

$\displaystyle \zbox {e^{-pt^2} \;\longleftrightarrow\;\sqrt{\frac{\pi}{p}} \, e^{-\frac{\omega^2}{4p}},\quad \forall p\in \mathbb{C}: \; \mbox{re}\left\{p\right\}>0}$ (D.16)



Proof: [202, p. 211] The Fourier transform of $ e^{-pt^2}$ is defined as

$\displaystyle \int_{-\infty}^\infty e^{-pt^2} e^{-j\omega t}dt \eqsp \int_{-\infty}^\infty e^{-(pt^2+j\omega t)} dt.$ (D.17)

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

$\displaystyle e^{-\frac{\omega^2}{4p}} \int_{-\infty}^\infty e^{-p\left(t+j\frac{\omega}{2p}\right)^2} dt \eqsp \sqrt{\frac{\pi}{p}}\, e^{-\frac{\omega^2}{4p}}$ (D.18)

using our previous result.



Subsections
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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2016-07-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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