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Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Gaussian Function Properties
       Gaussians Closed under Multiplication

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Gaussians Closed under Multiplication

Define

\begin{eqnarray*} x_1(t) &\isdef & e^{-p_1(t+c_1)^2}\\ x_2(t) &\isdef & e^{-p_2(t+c_2)^2}\\ \end{eqnarray*}

where $ p_1,p_2,c_1,c_2$ are arbitrary complex numbers. Then by direct calculation, we have

\begin{eqnarray*} x_1(t)\cdot x_2(t) &=& e^{-p_1(t+c_1)^2} e^{-p_2(t+c_2)^2}\\ ... ...{p_1 + p_2} t + \frac{p_1 c_1^2 + p_2 c_2^2}{p_1 + p_2}\right]} \end{eqnarray*}

Completing the square, we obtain

$\displaystyle x_1(t)\cdot x_2(t) = g \cdot e^{-p(t+c)^2} $

with

\begin{eqnarray*} p &=& p_1+p_2\\ [5pt] c &=& \frac{p_1 c_1 + p_2 c_2}{p_1 + p_2}\\ [5pt] g &=& e^{-p_1 p_2 \frac{(c_1 - c_2)^2}{p_1 + p_2}} \end{eqnarray*}

Note that this result holds for Gaussian-windowed chirps ($ p$ and $ c$ complex).


Subsections
Previous: Gaussian Window and Transform
Next: Product of Two Gaussian PDFs

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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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