**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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 and any integer ,

Proof:

The shift theorem says that adelayin the time domain corresponds to alinear phase termin the frequency domain. More specifically, a delay of samples in the time waveform corresponds to the linear phase term 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, .

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