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Embedded SystemsFPGAElectronics

Mathematics of the DFT
    Fourier Theorems for the DFT
       Fourier Theorems
          Shift Theorem

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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... ...}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$.7.13Note 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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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