Repeat (Scaling) Operator
We define the repeat operator in the frequency domain as
a scaling of frequency axis by some integer factor
:
![]() |
(3.30) |
where

The repeat operator maps the entire unit circle (taken as
to
) to a segment of itself
, centered about
, and repeated
times. This is illustrated in Fig.2.2
for
.
Since the frequency axis is continuous and
-periodic for DTFTs,
the repeat operator is precisely equivalent to a scaling operator for
the Fourier transform case (§B.4). We call it ``repeat''
rather than ``scale'' because we are restricting the scale factor to
positive integers, and because the name ``repeat'' describes more
vividly what happens to a periodic spectrum that is compressively
frequency-scaled over the unit circle by an integer factor.
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Stretch/Repeat (Scaling) Theorem
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Stretch Operator