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Time Reversal
For any complex signal
,
, we have
![$\displaystyle \zbox {\hbox{\sc Flip}(x) \;\longleftrightarrow\;\hbox{\sc Flip}(X)}$](img126.png) |
(3.11) |
where
.
Proof:
Arguably,
should include complex conjugation. Let
![$\displaystyle \hbox{\sc Flip}_n'(x)\isdefs \overline{\hbox{\sc Flip}_n(x)}\,\mathrel{\mathop=}\,\overline{x(-n)}$](img130.png) |
(3.12) |
denote such a definition. Then in this case we have
![$\displaystyle \zbox {\hbox{\sc Flip}'(x) \;\longleftrightarrow\;\overline{X}}$](img131.png) |
(3.13) |
Proof:
![$\displaystyle \hbox{\sc DTFT}_\omega(\hbox{\sc Flip}'(x)) \isdefs \sum_{n=-\infty}^{\infty} \overline{x(-n)}e^{-j\omega n} \eqsp \sum_{m=\infty}^{-\infty} \overline{x(m)e^{-j\omega m}} \isdefs \overline{X(\omega)}$](img132.png) |
(3.14) |
In the typical special case of real signals (
), we have
so that
![$\displaystyle \zbox {\hbox{\sc Flip}(x) \;\longleftrightarrow\;\overline{X}.}$](img135.png) |
(3.15) |
That is, time-reversing a real signal conjugates its spectrum.
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