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Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Gaussian Function Properties
       Integral of a Complex Gaussian

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Integral of a Complex Gaussian



Theorem:

$\displaystyle \zbox {\int_{-\infty}^\infty e^{-p t^2}dt = \sqrt{\frac{\pi}{p}}, \quad \forall p\in {\bf C}: \; \mbox{re}\left\{p\right\}>0} $



Proof: Let $ I(p)$ denote the integral. Then

\begin{eqnarray*} I^2(p) &=& \left(\int_{-\infty}^\infty e^{-p x^2}dx\right) \le... ...2}\right\vert _0^\infty = \frac{\pi}{-p} (0 - 1) = \frac{\pi}{p} \end{eqnarray*}

where we needed re$ \left\{p\right\}>0$ to have $ e^{-p r^2}\to 0$ as $ r\to\infty$. Thus,

$\displaystyle I(p) = \sqrt{\frac{\pi}{p}} $

as claimed.


Subsections
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Next: Area Under a Real Gaussian

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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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