Conjugation and Reversal
Theorem: For any
,
![$\displaystyle \zbox {\overline{x} \;\longleftrightarrow\;\hbox{\sc Flip}(\overline{X}).}
$](http://www.dsprelated.com/josimages_new/mdft/img1320.png)
Proof:
![\begin{eqnarray*}
\hbox{\sc DFT}_k(\overline{x})
&\isdef & \sum_{n=0}^{N-1}\ov...
...n) e^{-j 2\pi n(-k)/N}}
\isdef \hbox{\sc Flip}_k(\overline{X})
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img1321.png)
Theorem: For any
,
![$\displaystyle \zbox {\hbox{\sc Flip}(\overline{x}) \;\longleftrightarrow\;\overline{X}.}
$](http://www.dsprelated.com/josimages_new/mdft/img1322.png)
Proof: Making the change of summation variable
, we get
![\begin{eqnarray*}
\hbox{\sc DFT}_k(\hbox{\sc Flip}(\overline{x}))
&\isdef & \s...
...sum_{m=0}^{N-1}x(m) e^{-j 2\pi m k/N}}
\isdef \overline{X(k)}.
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img1324.png)
Theorem: For any
,
![$\displaystyle \zbox {\hbox{\sc Flip}(x) \;\longleftrightarrow\;\hbox{\sc Flip}(X).}
$](http://www.dsprelated.com/josimages_new/mdft/img1325.png)
Proof:
![\begin{eqnarray*}
\hbox{\sc DFT}_k[\hbox{\sc Flip}(x)] &\isdef & \sum_{n=0}^{N-1...
...-1}x(m) e^{j 2\pi mk/N} \isdef X(-k) \isdef \hbox{\sc Flip}_k(X)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img1326.png)
Corollary:
For any
,
![$\displaystyle \zbox {\hbox{\sc Flip}(x) \;\longleftrightarrow\;\overline{X}}$](http://www.dsprelated.com/josimages_new/mdft/img1327.png)
![$\displaystyle \mbox{($x$\ real).}$](http://www.dsprelated.com/josimages_new/mdft/img1328.png)
Proof: Picking up the previous proof at the third formula, remembering that is real,
![$\displaystyle \sum_{m=0}^{N-1}x(m) e^{j 2\pi mk/N}
= \overline{\sum_{m=0}^{N-1}...
.../N}}
= \overline{\sum_{m=0}^{N-1}x(m) e^{-j 2\pi mk/N}}
\isdef \overline{X(k)}
$](http://www.dsprelated.com/josimages_new/mdft/img1329.png)
![$ x(m)$](http://www.dsprelated.com/josimages_new/mdft/img1330.png)
Thus, conjugation in the frequency domain corresponds to reversal in the time domain. Another way to say it is that negating spectral phase flips the signal around backwards in time.
Corollary:
For any
,
![$\displaystyle \zbox {\hbox{\sc Flip}(X) = \overline{X}}$](http://www.dsprelated.com/josimages_new/mdft/img1331.png)
![$\displaystyle \mbox{($x$\ real).}$](http://www.dsprelated.com/josimages_new/mdft/img1328.png)
Proof: This follows from the previous two cases.
Definition: The property
is called Hermitian symmetry
or ``conjugate symmetry.'' If
, it may be called
skew-Hermitian.
Another way to state the preceding corollary is
![$\displaystyle \zbox {x\in{\bf R}^N\;\longleftrightarrow\;X\;\mbox{is Hermitian}.}
$](http://www.dsprelated.com/josimages_new/mdft/img1334.png)
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Symmetry
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Linearity