## Gaussians Closed under Multiplication

Define

where are arbitrary complex numbers. Then by direct calculation, we have

Completing the square, we obtain

(D.2) |

with

Note that this result holds for *Gaussian-windowed chirps*
(
and
complex).

### Product of Two Gaussian PDFs

For the special case of two Gaussian probability densities,

the product density has mean and variance given by

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