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DTFT of Real Signals

The previous section established that the spectrum $ X$ of every real signal $ x$ satisfies

$\displaystyle \hbox{\sc Flip}(X)\eqsp \overline{X}.$ (3.16)

I.e.,

$\displaystyle \zbox {x(n)\in\mathbb{R}\;\longleftrightarrow\;X(-\omega) = \overline{X(\omega)}.}$ (3.17)

In other terms, if a signal $ x(n)$ is real, then its spectrum is Hermitian (``conjugate symmetric''). Hermitian spectra have the following equivalent characterizations: Note that an even function is symmetric about argument zero while an odd function is antisymmetric about argument zero.


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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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