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:

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