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

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