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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)\\
\hbox{\sc Flip}(\overline{x}) &\longleftrightarrow& \overline{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 & \ensuremath{\int_{-\infty}^{\infty}}x(-t) e^{-j\omega t} dt\quad
\mbox{(set $\tau=-t$)}\\
&=& \int_{\infty}^{-\infty} x(\tau) e^{-j\omega (-\tau)} (-d\tau)\\
&=& \ensuremath{\int_{-\infty}^{\infty}}x(\tau) e^{-j(-\omega) \tau} d\tau\\
&=& X(-\omega) \isdef \hbox{\sc Flip}_\omega(X)
\end{eqnarray*}


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