Shift Operator

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Shift 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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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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Shift Operator



Definition: The shift operator is defined by

\

and $\ denotes the entire shifted signal. Note that since indexing is modulo $N$, the shift is circular. However, we normally use it to represent time delay by $\ samples. We often use the shift operator in conjunction with zero padding(appending zeros to the signal $x$) in order to avoid the “wrap-around” associated with a circular shift.
Figure 8.2:Successive one-sample shifts of a sampled periodic sawtooth waveform having first period $[0,1,2,3,4]$.
\

Figure 8.2 illustrates successive one-sample delays of a periodic signal having first period given by $[0,1,2,3,4]$.



Example: $\ (an impulse delayed one sample).



Example: $\ (a circular shift example).



Example: $\ (another circular shift example).

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