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Fourier Transform (FT) and Inverse

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

$\displaystyle X(\omega) \isdefs \int_{-\infty}^\infty x(t) e^{-j\omega t} dt \protect$ (3.4)

and its inverse is given by

$\displaystyle x(t) \eqsp \frac{1}{2\pi}\int_{-\infty}^\infty X(\omega) e^{j\omega t} d\omega \protect$ (3.5)

Thus, the Fourier transform is defined for continuous time and continuous frequency, both unbounded. As a result, mathematical questions such as existence and invertibility are most difficult for this case. In fact, such questions fueled decades of confusion in the history of harmonic analysis (see Appendix G).



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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2016-07-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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