### 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 denotes the radian frequency variable after applying the repeat operator.

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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