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


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_{-\infty}^\infty x(t-\tau) e^{-j\omega t}dt\qquad\mbox{(define $\sigma=t-\tau$)}\\
&=& \int_{-\infty}^\infty x(\sigma) e^{-j\omega (\sigma+\tau)}d\sigma\\
&=& e^{-j\omega \tau}\int_{-\infty}^\infty x(\sigma) e^{-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.12)


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

[How to cite this work]  [Order a printed hardcopy]  [Comment on this page via email]

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2016-07-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA