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