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Shift Theorem



Theorem: For any $ x\in{\bf C}^N$ and any integer $ \Delta$ ,

$\displaystyle \zbox {\hbox{\sc DFT}_k[\hbox{\sc Shift}_\Delta(x)] = e^{-j\omega_k\Delta} X(k).}
$



Proof:

\begin{eqnarray*}
\hbox{\sc DFT}_k[\hbox{\sc Shift}_\Delta(x)] &\isdef & \sum_{n=0}^{N-1}x(n-\Delta) e^{-j 2\pi nk/N} \\
&=& \sum_{m=-\Delta}^{N-1-\Delta} x(m) e^{-j 2\pi (m+\Delta)k/N}
\qquad(m\isdef n-\Delta) \\
&=& \sum_{m=0}^{N-1}x(m) e^{-j 2\pi mk/N} e^{-j 2\pi k\Delta/N} \\
&=& e^{-j 2\pi \Delta k/N} \sum_{m=0}^{N-1}x(m) e^{-j 2\pi mk/N} \\
&\isdef & e^{-j \omega_k \Delta} X(k)
\end{eqnarray*}

The shift theorem is often expressed in shorthand as

$\displaystyle \zbox {x(n-\Delta) \longleftrightarrow e^{-j\omega_k\Delta}X(\omega_k).}
$

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 $ \Delta$ samples in the time waveform corresponds to the linear phase term $ e^{-j \omega_k \Delta}$ multiplying the spectrum, where $ \omega_k\isdeftext 2\pi k/N$ .7.14Note that spectral magnitude is unaffected by a linear phase term. That is, $ \left\vert e^{-j
\omega_k
\Delta}X(k)\right\vert =
\left\vert X(k)\right\vert$ .



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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 © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA