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Spectral Audio Signal Processing
Gaussian Function Properties
Gaussian Moments
Higher Order Moments Revisited
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Higher Order Moments Revisited
Theorem:
The 
th central moment of the Gaussian pdf 
with mean 
and variance 
is given by
![$\displaystyle m_n \isdef {\cal E}_p\{(x-\mu)^n\} = \left\{\begin{array}{ll}
(n-...
...a^n, & \hbox{$n$\ even} \\ [5pt]
$0$, & \hbox{$n$\ odd} \\
\end{array}\right.
$](http://www.dsprelated.com/josimages_new/sasp/img2573.png)
Proof: [191, p. 148]
The Gaussian integral
times with respect to 
to obtain
![\begin{eqnarray*}
\int_{-\infty}^\infty (-x^2) e^{-\alpha x^2} dx &=& \sqrt{\pi}...
...-\alpha x^2} dx &=& \sqrt{\pi}[(2k-1)!]2^{-k}\alpha^{-(2k+1)}\\
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp/img2575.png)
Setting
and 
yields
![$\displaystyle \int_{-\infty}^\infty x^{n} e^{-\frac{x^2}{2\sigma^2}} dx = \sqrt{\pi}[(n-1)!]2^{-n/2}(2\sigma^2)^{n+1}.
$](http://www.dsprelated.com/josimages_new/sasp/img2578.png)
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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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