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Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Gaussian Function Properties
       Gaussians Closed under Multiplication
          Product of Two Gaussian PDFs

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Product of Two Gaussian PDFs

For the special case of two Gaussian probability densities,

\begin{eqnarray*} x_1(t) &\isdef & \frac{1}{\sqrt{2\pi\sigma_1^2}}e^{-\frac{(t-\... ...ac{1}{\sqrt{2\pi\sigma_2^2}}e^{-\frac{(t-\mu_2)^2}{2\sigma_2^2}} \end{eqnarray*}

the product density has mean and variance given by

\begin{eqnarray*} \mu &=& \frac{\frac{\mu_1}{2\sigma_1^2} + \frac{\mu_2}{2\sigm... ...;\eqsp \; \frac{\sigma_1^2\sigma_2^2}{\sigma_1^2 + \sigma_2^2}. \end{eqnarray*}

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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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