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Fourier Transform (FT) and Inverse

The Fourier transform of a signal $ x(t)\in{\bf C}$ , $ t\in(-\infty,\infty)$ , is defined as

$\displaystyle X(\omega) \isdef \int_{-\infty}^\infty x(t) e^{-j\omega t} dt, \protect$ (B.1)

and its inverse is given by

$\displaystyle x(t) = \frac{1}{2\pi}\int_{-\infty}^\infty X(\omega) e^{j\omega t} d\omega. \protect$ (B.2)



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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA