Stretch Operator

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Stretch Operator

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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Stretch Operator

Unlike all previous operators, the $\ operator maps a length $N$ signal to a length $M\ signal, where $L$ and $N$ are integers. We use ''$m$'' instead of ''$n$'' as the time index to underscore this fact.

Figure:Illustration of $\.
\



Definition: A stretch by factor $L$ is defined by

\

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. 8.4. The example is
\

The stretch operator is used to describe and analyze upsampling, i.e., increasing the sampling rate by an integer factor.

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