Shift Theorem

01 Mar 1998

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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

Theorem: For any $$x\$ and any integer $$\$ ,

Proof:

The shift theorem says that a delay in the time domain corresponds to a linear phase term in the frequency domain. More specifically, a delay of $$\$ samples in the time waveform corresponds to the linear phase term $$e^{-j \$ multiplying the spectrum, where $$\$ . (To consider $$\$ as radians per second instead of radians per sample, just replace $$\$ by $$\$ so that the delay is in seconds instead of samples.) Note that spectral magnitude is unaffected by a linear phase term. That is, $$\$ .

Subsections

Linear Phase Terms

Application of the Shift Theorem to FFT Windows

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

Shift Theorem