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 modolu (periodic extension).
Thus, the
operator simply repeats
its input signal times.^{7.7} An example of
is shown in Fig.7.8.
The example is

Figure 7.8:
Illustration of
.

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 7.9:
Illustration of
.
a) Conventional plot of .
b) Plot of over the unit circle in the plane.
c)
.

where
is defined in §7.2.6.
That is, when you stretch a signal by the factor , its spectrum is
repeated times around the unit circle.
The simple proof is given on page page .