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Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Gaussian Function Properties
       Gaussian Moments
          Moment Theorem

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Moment Theorem



Theorem: For a random variable $ x$,

$\displaystyle {\cal E}\{x^n\} = \left.\frac{1}{j^n}\frac{d^n}{d\omega^n}\Phi(\omega)\right\vert _{\omega=0} $

where $ \Phi(\omega)$ is the characteristic function of the PDF $ p(x)$ of $ x$:

$\displaystyle \Phi(\omega) \isdef {\cal E}_p\{ e^{j\omega x} \} = \int_{-\infty}^\infty p(x)e^{j\omega x}dx $

(Note that $ \Phi(\omega)$ is the complex conjugate of the Fourier transform of $ p(x)$.)



Proof: [191, p. 157] Let $ m_i$ denote the $ i$th moment of $ x$, i.e.,

$\displaystyle m_i \isdef {\cal E}_p\{x^{m_i}\} \isdef \int_{-\infty}^\infty x^{m_i} p(x)dx $

Then

\begin{eqnarray*} \Phi(\omega) &=& \int_{-\infty}^\infty p(x)e^{j\omega x} dx \\... ...{(j\omega)^2}{2} m_2 + \cdots + \frac{(j\omega)^n}{n!}m_n+\cdots \end{eqnarray*}

where the term-by-term integration is valid when all moments $ m_i$ are finite.

Previous: Higher Order Moments Revisited
Next: Gaussian Characteristic Function

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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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