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Spectral Audio Signal Processing
    Fourier Transforms for Continuous/Discrete Time/Frequency
       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,

$\displaystyle \zbox {x(t-\tau)\longleftrightarrow e^{-j\omega \tau}X(\omega).}$ (3.3)

For the inverse Fourier transform, we have

\begin{eqnarray*} \hbox{\sc IFT}_\omega(\hbox{\sc Shift}_\nu(X)) &\isdef & \fra... ...nfty X(\sigma) e^{j\sigma t}d\sigma\\ &\isdef & e^{j\nu t} x(t) \end{eqnarray*}

or,

$\displaystyle \zbox {e^{j\nu t} x(t) \longleftrightarrow X(\omega-\nu).}$ (3.4)

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Previous: Scaling Theorem
Next: Convolution Theorem
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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