Time Reversal

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For any complex signal , $n\in(-\infty,\infty)$ , we have

$\displaystyle \zbox {\hbox{\sc Flip}(\overline{x}) \leftrightarrow \overline{X}}$

where $\hbox{\sc Flip}_n(x)\isdef x(-n)$ , and $\overline{x(n)}$ denotes the complex conjugate of

. In the typical special case of real signals ( $x(n)\in{\bf R}$ ), we have

$\displaystyle \zbox {\hbox{\sc Flip}(x) \leftrightarrow \overline{X}}$

Thus, reversing a signal in time congugates its spectrum.

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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