Shift Theorem Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

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).}


\hbox{\sc DFT}_k[\hbox{\sc Shift}_\Delta(x)] &\isdef & \sum_{n...
...}x(m) e^{-j 2\pi mk/N} \\
&\isdef & e^{-j \omega_k \Delta} X(k)

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$. (To consider $ \omega_k$ as radians per second instead of radians per sample, just replace $ \Delta$ by $ \Delta T$ so that the delay is in seconds instead of samples.) Note that spectral magnitude is unaffected by a linear phase term. That is, $ \left\vert e^{-j
\Delta}X(k)\right\vert =
\left\vert X(k)\right\vert$.

Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work] [Order a printed hardcopy]

``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]