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Spectral Audio Signal Processing
    Fourier Transforms for Continuous/Discrete Time/Frequency
       Fourier Theorems for the DTFT
          Shift Theorem for the DTFT

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Shift Theorem for the DTFT

We define the shift operator for sampled signals $ x(n)$ by

$\displaystyle \hbox{\sc Shift}_{l,n}(x) \isdef x(n-l) $

where $ l$ is any integer ( $ l\in{\bf Z}$). Thus, $ \hbox{\sc Shift}_l(x)$ is a right-shift or delay by $ l$ samples.

The shift theorem states3.3

$\displaystyle \zbox {\hbox{\sc Shift}_l(x) \leftrightarrow e^{-j(\cdot)l}X}, $

or, in operator notation,

$\displaystyle \hbox{\sc DTFT}_\omega[\hbox{\sc Shift}_l(x)] = \left( e^{-j\omega l} \right) X(\omega) $



Proof:

\begin{eqnarray*} \hbox{\sc DTFT}_\omega[\hbox{\sc Shift}_l(x)] &\isdef & \sum_{... ...fty}x(m) e^{-j \omega m} \\ &\isdef & e^{-j \omega l} X(\omega) \end{eqnarray*}

Note that $ e^{-j\omega l}$ is a linear phase term, so called because it is a linear function of frequency with slope equal to $ -l$:

$\displaystyle \angle \left(e^{-j \omega l}\right) = -\omega l $

The shift theorem gives us that multiplying a spectrum $ X(\omega)$ by a linear phase term $ e^{-j\omega l}$ corresponds to a delay in the time domain by $ l$ samples. If $ l<0$, it is called a time advance by $ \vert l\vert$ samples.

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Previous: Real Even (or Odd) Signals
Next: Convolution Theorem for the DTFT
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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