### Stretch/Repeat (Scaling) Theorem

Using these definitions, we can compactly state the *stretch
theorem*:

(3.31) |

*Proof: *

As traverses the interval , traverses the unit circle times, thus implementing the repeat operation on the unit circle. Note also that when , we have , so that dc always maps to dc. At half the sampling rate , on the other hand, after the mapping, we may have either ( odd), or ( even), where .

The stretch theorem makes it clear how to do
*ideal sampling-rate conversion* for integer upsampling ratios
:
We first stretch the signal by the factor
(introducing
zeros
between each pair of samples), followed by an *ideal lowpass
filter* cutting off at
. That is, the filter has a gain of 1
for
, and a gain of 0 for
. Such a system (if it were realizable) implements *ideal bandlimited interpolation* of the original signal by the factor
.

The stretch theorem is analogous to the *scaling theorem* for
continuous Fourier transforms (introduced in §2.4.1
below).

**Next Section:**

Downsampling and Aliasing

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Repeat (Scaling) Operator