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Spectral Audio Signal Processing
    Fourier Transforms for Continuous/Discrete Time/Frequency
       Selected Continuous Fourier Theorems
          Flip Theorems

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Flip Theorems

Let the flip operator be denoted by

$\begin{eqnarray*} \hbox{\sc Flip}_t(x) &\isdef & x(-t)\\ \hbox{\sc Flip}_\omega(X) &\isdef & X(-\omega), \end{eqnarray*}$

where $t\in(-\infty,\infty)$ denotes time in seconds, and $\omega\in(-\infty,\infty)$ denotes frequency in radians per second. The following Fourier pairs are easily verified:

$\begin{eqnarray*} \hbox{\sc Flip}(x) &\longleftrightarrow& \hbox{\sc Flip}(X)\\ ... ...\overline{x} &\longleftrightarrow& \hbox{\sc Flip}(\overline{X}) \end{eqnarray*}$

The proof of the first relation is as follows:

$\begin{eqnarray*} \hbox{\sc FT}_{\omega}\left[\hbox{\sc Flip}(x)\right] &\isdef ... ...) \tau} d\tau\\ &=& X(-\omega) \isdef \hbox{\sc Flip}_\omega(X) \end{eqnarray*}$

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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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