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Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Gaussian Function Properties
       Maximum Entropy Property of the Gaussian Distribution
          Maximum Entropy Distributions
             Gaussian Distribution

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Gaussian Distribution

The Gaussian distribution has maximum entropy relative to all probability distributions covering the entire real line $ x\in(-\infty,\infty)$ but having a finite mean $ \mu$ and finite variance $ \sigma^2$.

Proceeding as before, we obtain the objective function

\begin{eqnarray*} J(p) &\isdef & -\int_{-\infty}^\infty p(x) \, \ln p(x)\,dx +... ...bda_2\left(\int_{-\infty}^\infty x^2\,p(x)\,dx - \sigma^2\right) \end{eqnarray*}

and partial derivatives

\begin{eqnarray*} \frac{\partial}{\partial p(x)\,dx} J(p) &=& - \ln p(x) - 1 + \... ... \frac{\partial^2}{\partial p(x)^2 dx} J(p) &=& - \frac{1}{p(x)} \end{eqnarray*}

leading to

$\displaystyle p(x) = e^{(\lambda_0-1)+\lambda_1 x + \lambda_2 x^2} = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{x^2}{2\sigma^2}}. $

For more on entropy and maximum-entropy distributions, see [48].

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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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