**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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 OperatorUnlike all previous operators, the operator maps a length signal to a length signal, where and are integers. We use “” instead of “” as the time index to underscore this fact.

Definition:Astretch by factoris defined by

Thus, to stretch a signal by the factor , insert 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.