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Stretch Theorem (Repeat Theorem)



Theorem: For all $ x\in{\bf C}^N$ ,

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



Proof: Recall the stretch operator:

$\displaystyle \hbox{\sc Stretch}_{L,m}(x) \isdef
\left\{\begin{array}{ll}
x(m/L), & m/L=\mbox{integer} \\ [5pt]
0, & m/L\neq \mbox{integer} \\
\end{array} \right.
$

Let $ y\isdeftext \hbox{\sc Stretch}_L(x)$ , where $ y\in{\bf C}^M$ , $ M=LN$ . Also define the new denser frequency grid associated with length $ M$ by $ \omega^\prime_k \isdeftext 2\pi k/M$ , and define $ \omega_k=2\pi k/N$ as usual. Then

$\displaystyle Y(k) \isdef \sum_{m=0}^{M-1} y(m) e^{-j\omega^\prime_k m}
= \sum_{n=0}^{N-1}x(n) e^{-j\omega^\prime_k nL}$   $\displaystyle \mbox{($n\isdef m/L$).}$

But

$\displaystyle \omega^\prime_k L \isdef \frac{2\pi k}{M} L = \frac{2\pi k}{N} = \omega_k .
$

Thus, $ Y(k)=X(k)$ , and by the modulo indexing of $ X$ , $ L$ copies of $ X$ are generated as $ k$ goes from 0 to $ M-1 = LN-1$ .


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-21 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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