Gaussian Central Moments
The characteristic function of a zero-mean Gaussian is
![]() |
(D.53) |
Since a zero-mean Gaussian




![]() |
(D.54) |
In particular,
![\begin{eqnarray*}
\Phi^\prime(\omega) &=& -\frac{1}{2}\sigma^2 2\omega\Phi(\omega)\\ [5pt]
\Phi^{\prime\prime}(\omega) &=& -\frac{1}{2}\sigma^2 2\omega\Phi^\prime(\omega)
-\frac{1}{2}\sigma^2 2\Phi(\omega)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img2865.png)
Since
and
, we see
,
, as expected.
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