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Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    More Fourier Theorems
       Selected Continuous Fourier Theorems
          Shift Theorem

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Shift Theorem

The shift theorem for Fourier transforms states that delaying a signal $ x(t)$ by $ \tau$ seconds multiplies its Fourier transform by $ e^{-j\omega\tau}$.



Proof:

\begin{eqnarray*} \hbox{\sc FT}_\omega(\hbox{\sc Shift}_\tau(x)) &\isdef & \int... ...{-j\omega \sigma}d\sigma\\ &\isdef & e^{-j\omega \tau}X(\omega) \end{eqnarray*}

Thus,

$\displaystyle \zbox {x(t-\tau)\;\longleftrightarrow\;e^{-j\omega \tau}X(\omega).}$ (B.1)

Previous: Scaling Theorem
Next: Modulation Theorem (Shift Theorem Dual)

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About the Author: 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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