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
.

A
frequencydomain 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 rightrear 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)
.

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
.
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