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).

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-\mu_1)^2}{2\sigma_1^2}}\\
x_2(t) &\isdef & \frac{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\sigma_2^2}}{\frac{1}{2\sigma_1^2} + \frac{1}{2\sigma_2^2}}
\;\eqsp \;
\frac{\mu_1\sigma_2^2 + \mu_2\sigma_1^2}{\sigma_2^2 + \sigma_1^2}\\ [5pt]
\sigma^2 &=& \left. \sigma_1^2 \right\Vert \sigma_2^2 \;\isdefs \;
\frac{1}{\frac{1}{\sigma_1^2} + \frac{1}{\sigma_2^2}} \;\eqsp \;
\frac{\sigma_1^2\sigma_2^2}{\sigma_1^2 + \sigma_2^2}.
\end{eqnarray*}


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Gaussians Closed under Convolution
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Gaussian Window and Transform