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The Gaussian pulse of width (second central moment) $\sigma$ centered on time 0 may be defined by

$\displaystyle g_\sigma(t) \frac{1}{\sigma\sqrt{2\pi}}\isdef e^{-\frac{t^2}{\sigma^2}}$

where the normalization scale factor is chosen to give unit area under the pulse. Its Fourier transform is derived in Appendix C to be

$\displaystyle G_\sigma(\omega) = e^{-\frac{\omega^2}{2(1/\sigma)^2}}.$

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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