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,

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