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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}\\
&=& e^{-p_1 t^2 - 2 p_1 c_1 t - p_1 c_1^2 -p_2 t^2 - 2 p_2 c_2 t - p_2 c_2^2}\\
&=& e^{-(p_1+p_2) t^2 - 2 (p_1 c_1 + p_2 c_2) t - (p_1 c_1^2 + p_2 c_2^2)}\\
&=& e^{-(p_1+p_2)\left[t^2 + 2\frac{p_1 c_1 + p_2 c_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

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



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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2016-07-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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