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Embedded SystemsFPGAElectronics

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

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Exponential Distribution

Among probability distributions $ p(x)$ which are nonzero over a semi-infinite range of values $ x\in[0,\infty]$ and having a finite mean $ \mu$, the exponential distribution has maximum entropy.

To the previous case, we add the new constraint

$\displaystyle \int_{-\infty}^\infty x\,p(x)\,dx = \mu < \infty $

resulting in the objective function

\begin{eqnarray*} J(p) &\isdef & -\int_0^\infty p(x) \, \ln p(x)\,dx + \lambda... ...\\ & & + \lambda_1\left(\int_0^\infty x\,p(x)\,dx - \mu\right) \end{eqnarray*}

Now the partials with respect to $ p(x)$ are

\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*}

and $ p(x)$ is of the form $ p(x) = e^{(\lambda_0-1)+\lambda_1x}$. The unit-area and finite-mean constraints result in $ \exp(\lambda_0-1) = 1/\mu$ and $ \lambda_1=-1/\mu$, yielding

$\displaystyle p(x) = \left\{\begin{array}{ll} \frac{1}{\mu} e^{-x/\mu}, & x\geq 0 \\ [5pt] 0, & \hbox{otherwise}. \\ \end{array}\right. $

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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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