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Shift Operator

The shift operator is defined by

$\displaystyle \hbox{\sc Shift}_{\Delta,n}(x) \isdef x(n-\Delta), \quad \Delta\in\mathbb{Z},

and $ \hbox{\sc Shift}_{\Delta}(x)$ denotes the entire shifted signal. Note that since indexing is modulo $ N$ , the shift is circular (or ``cyclic''). However, we normally use it to represent time delay by $ \Delta$ samples. We often use the shift operator in conjunction with zero padding (appending zeros to the signal $ x$ , §7.2.7) in order to avoid the ``wrap-around'' associated with a circular shift.

Figure 7.2: Successive one-sample shifts of a sampled periodic sawtooth waveform having first period $ [0,1,2,3,4]$ .

Figure 7.2 illustrates successive one-sample delays of a periodic signal having first period given by $ [0,1,2,3,4]$ .

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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 © 2021-11-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University