Time Reversal
For any complex signal
,
, we have
![]() |
(3.11) |
where

Proof:
![\begin{eqnarray*}
\hbox{\sc DTFT}_\omega(\hbox{\sc Flip}(x))
&\isdef & \sum_{n=-\infty}^{\infty} x(-n)e^{-j\omega n}
\eqsp \sum_{m=\infty}^{-\infty} x(m)e^{-j(-\omega) m}
\eqsp X(-\omega) \\ [5pt]
&\isdef & \hbox{\sc Flip}_\omega(X)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img130.png)
Arguably,
should include complex conjugation. Let
![]() |
(3.12) |
denote such a definition. Then in this case we have
![]() |
(3.13) |
Proof:
![]() |
(3.14) |
In the typical special case of real signals (
), we have
so that
![]() |
(3.15) |
That is, time-reversing a real signal conjugates its spectrum.
Next Section:
Symmetry of the DTFT for Real Signals
Previous Section:
Linearity of the DTFT