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Gaussian Window and Transform

The Gaussian ``bell curve'' is possibly the only smooth, nonzero function, known in closed form, that transforms to itself.4.15

$\displaystyle \frac{1}{\sigma\sqrt{2\pi}}e^{-t^2 \left / 2\sigma^2\right.} \;\longleftrightarrow\; e^{-\omega^2 \left/ 2\left(1/\sigma\right)^2\right.}$ (4.55)

It also achieves the minimum time-bandwidth product

$\displaystyle \sigma_t\sigma_\omega = \sigma\times (1/\sigma) = 1$ (4.56)

when ``width'' is defined as the square root of its second central moment. For even functions $ w(t)$ ,

$\displaystyle \sigma_t \isdefs \sqrt{\int_{-\infty}^\infty t^2 w(t) dt}.$ (4.57)

Since the true Gaussian function has infinite duration, in practice we must window it with some usual finite window, or truncate it.

Depalle [58] suggests using a triangular window raised to some power $ \alpha $ for this purpose, which preserves the absence of side lobes for sufficiently large $ \alpha $ . It also preserves non-negativity of the transform.

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University