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).