Stretch/Repeat (Scaling) Theorem

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

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).

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 (see §2.4.4).

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.