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.
Stretch/Repeat (Scaling) Theorem