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:

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