Time Reversal

Free Books Spectral Audio Signal Processing

For any complex signal , $n\in(-\infty,\infty)$ , we have

$\displaystyle \zbox {\hbox{\sc Flip}(x) \;\longleftrightarrow\;\hbox{\sc Flip}(X)}$

(3.11)

where $\hbox{\sc Flip}_n(x)\isdef x(-n)$ .

Proof:

$\begin{eqnarray*} \hbox{\sc DTFT}_\omega(\hbox{\sc Flip}(x)) &\isdef & \sum_{n=-\infty}^{\infty} x(-n)e^{-j\omega n} \eqsp \sum_{m=\infty}^{-\infty} x(m)e^{-j(-\omega) m} \eqsp X(-\omega) \\ [5pt] &\isdef & \hbox{\sc Flip}_\omega(X) \end{eqnarray*}$

Arguably, $\hbox{\sc Flip}(x)$ should include complex conjugation. Let

$\displaystyle \hbox{\sc Flip}_n'(x)\isdefs \overline{\hbox{\sc Flip}_n(x)}\,\mathrel{\mathop=}\,\overline{x(-n)}$

(3.12)

denote such a definition. Then in this case we have

$\displaystyle \zbox {\hbox{\sc Flip}'(x) \;\longleftrightarrow\;\overline{X}}$

(3.13)

Proof:

$\displaystyle \hbox{\sc DTFT}_\omega(\hbox{\sc Flip}'(x)) \isdefs \sum_{n=-\infty}^{\infty} \overline{x(-n)}e^{-j\omega n} \eqsp \sum_{m=\infty}^{-\infty} \overline{x(m)e^{-j\omega m}} \isdefs \overline{X(\omega)}$