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Stretch Operator

Unlike all previous operators, the $ \hbox{\sc Stretch}_L()$ operator maps a length $ N$ signal to a length $ M\isdeftext LN$ signal, where $ L$ and $ N$ are integers. We use ``$ m$ '' instead of ``$ n$ '' as the time index to underscore this fact.

Figure: Illustration of $ \protect\hbox{\sc Stretch}_3(x)$ .

A stretch by factor $ L$ is defined by

$\displaystyle \hbox{\sc Stretch}_{L,m}(x) \isdef
x(m/L), & m/L\mbox{ an integer} \\ [5pt]
0, & m/L\mbox{ non-integer.} \\
\end{array} \right.

Thus, to stretch a signal by the factor $ L$ , insert $ L-1$ zeros between each pair of samples. An example of a stretch by factor three is shown in Fig.7.6. The example is

$\displaystyle \hbox{\sc Stretch}_3([4,1,2]) = [4,0,0,1,0,0,2,0,0].

The stretch operator is used to describe and analyze upsampling, that is, increasing the sampling rate by an integer factor. A stretch by $ K$ followed by lowpass filtering to the frequency band $ \omega\in(-\pi/K,\pi/K)$ implements ideal bandlimited interpolation (introduced in Appendix D).

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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 © 2018-02-13 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University