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
where

, and indexing of

is modulo

(
periodic extension).
Thus, the

operator simply repeats
its input signal

times.
7.10 An example of

is shown in
Fig.
7.8. The example is
Figure:
Illustration of
.
![\includegraphics[width=\twidth]{eps/repeat}](http://www.dsprelated.com/josimages_new/mdft/img1261.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

.
Figure:
Illustration of
.
a) Conventional plot of
.
b) Plot of
over the unit circle in the
plane.
c)
.
![\includegraphics[width=4in]{eps/repeat3d}](http://www.dsprelated.com/josimages_new/mdft/img1263.png) |
The repeat operator is used to state the
Fourier theorem
where

is defined in §
7.2.6. That is, when you
stretch a signal by the factor

(inserting zeros between the
original samples), its spectrum is repeated

times around the unit
circle. The simple proof is given on page
![[*]](../icons/crossref.png)
.
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