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Stretch/Repeat (Scaling) Theorem

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

$\displaystyle \zbox {\hbox{\sc Stretch}_L(x) \;\longleftrightarrow\;\hbox{\sc Repeat}_L(X)}$ (3.31)


\hbox{\sc DTFT}_\omega[\hbox{\sc Stretch}_L(x)]
&\isdef & \sum_{n=-\infty}^{\infty}\hbox{\sc Stretch}_{L,n}(x)e^{-j\omega n}\\
&=& \sum_{m=-\infty}^{\infty}x(m)e^{-j\omega m L}\qquad \hbox{($m\isdef n/L$)}\\
&\isdef & X(\omega L)

As $ \omega$ traverses the interval $ [-\pi,\pi)$ , $ X(\omega L)$ traverses the unit circle $ L$ times, thus implementing the repeat operation on the unit circle. Note also that when $ \omega
= 0$ , we have $ \omega L = 0$ , so that dc always maps to dc. At half the sampling rate $ \omega=\pm\pi$ , on the other hand, after the mapping, we may have either $ Y(\pi)=X(-\pi)$ ($ L$ odd), or $ X(0)$ ($ L$ even), where $ Y(\omega)
\isdeftext X(\omega L)$ .

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

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

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2015-02-01 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University