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Repeat Operator

Like the $ \hbox{\sc Stretch}_L()$ and $ \hbox{\sc Interp}_L()$ operators, the $ \hbox{\sc Repeat}_L()$ operator maps a length $ N$ signal to a length $ M\isdeftext LN$ signal:

Definition: The repeat $ L$ times operator is defined for any $ x\in\mathbb{C}^N$ by

$\displaystyle \hbox{\sc Repeat}_{L,m}(x) \isdef x(m), \qquad m=0,1,2,\ldots,M-1,

where $ M\isdef LN$ , and indexing of $ x$ is modulo $ N$ (periodic extension). Thus, the $ \hbox{\sc Repeat}_L()$ operator simply repeats its input signal $ L$ times.7.11 An example of $ \hbox{\sc Repeat}_2(x)$ is shown in Fig.7.8. The example is

$\displaystyle \hbox{\sc Repeat}_2([0,2,1,4,3,1]) = [0,2,1,4,3,1,0,2,1,4,3,1].

Figure: Illustration of $ \hbox{\sc Repeat}_2(x)$ .

A frequency-domain example is shown in Fig.7.9. Figure 7.9a shows the original spectrum $ X$ , Fig.7.9b shows the same spectrum plotted over the unit circle in the $ z$ plane, and Fig.7.9c shows $ \hbox{\sc Repeat}_3(X)$ . The $ z=1$ point (dc) is on the right-rear face of the enclosing box. Note that when viewed as centered about $ k=0$ , $ X$ is a somewhat ``triangularly shaped'' spectrum. We see three copies of this shape in $ \hbox{\sc Repeat}_3(X)$ .

Figure: Illustration of $ \hbox{\sc Repeat}_3(X)$ . a) Conventional plot of $ X$ . b) Plot of $ X$ over the unit circle in the $ z$ plane. c) $ \hbox{\sc Repeat}_3(X)$ .

The repeat operator is used to state the Fourier theorem

$\displaystyle \hbox{\sc Stretch}_L \;\longleftrightarrow\;\hbox{\sc Repeat}_L,

where $ \hbox{\sc Stretch}_L$ is defined in §7.2.6. That is, when you stretch a signal by the factor $ L$ (inserting zeros between the original samples), its spectrum is repeated $ L$ times around the unit circle. The simple proof is given on page [*].

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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 © 2021-02-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University