### Stretch/Repeat (Scaling) Theorem

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

(3.31) |

*Proof:*

*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).

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Downsampling and Aliasing

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Repeat (Scaling) Operator