Repeat Operator
Like the
and
operators, the
operator maps a length
signal to a length
signal:
Definition: The repeat times operator is defined for any
by
![$\displaystyle \hbox{\sc Repeat}_{L,m}(x) \isdef x(m), \qquad m=0,1,2,\ldots,M-1,
$](http://www.dsprelated.com/josimages_new/mdft/img1253.png)
![$ M\isdef LN$](http://www.dsprelated.com/josimages_new/mdft/img1254.png)
![$ x$](http://www.dsprelated.com/josimages_new/mdft/img25.png)
![$ N$](http://www.dsprelated.com/josimages_new/mdft/img35.png)
![$ \hbox{\sc Repeat}_L()$](http://www.dsprelated.com/josimages_new/mdft/img1252.png)
![$ L$](http://www.dsprelated.com/josimages_new/mdft/img1214.png)
![$ \hbox{\sc Repeat}_2(x)$](http://www.dsprelated.com/josimages_new/mdft/img1259.png)
![$\displaystyle \hbox{\sc Repeat}_2([0,2,1,4,3,1]) = [0,2,1,4,3,1,0,2,1,4,3,1].
$](http://www.dsprelated.com/josimages_new/mdft/img1260.png)
A frequency-domain example is shown in Fig.7.9.
Figure 7.9a shows the original spectrum , Fig.7.9b
shows the same spectrum plotted over the unit circle in the
plane,
and Fig.7.9c shows
. The
point (dc) is on
the right-rear face of the enclosing box. Note that when viewed as
centered about
,
is a somewhat ``triangularly shaped''
spectrum. We see three copies of this shape in
.
![]() |
The repeat operator is used to state the Fourier theorem
![$\displaystyle \hbox{\sc Stretch}_L \;\longleftrightarrow\;\hbox{\sc Repeat}_L,
$](http://www.dsprelated.com/josimages_new/mdft/img1264.png)
![$ \hbox{\sc Stretch}_L$](http://www.dsprelated.com/josimages_new/mdft/img1265.png)
![$ L$](http://www.dsprelated.com/josimages_new/mdft/img1214.png)
![$ L$](http://www.dsprelated.com/josimages_new/mdft/img1214.png)
![[*]](../icons/crossref.png)
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Downsampling Operator
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Interpolation Operator