### 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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