Frequency shifting property of CTFT

Statement

If $x(t) \xleftrightarrow{F} X(\omega)$, then

$$e^{j\omega_0 t}x(t) \xleftrightarrow{F} X(\omega - \omega_0)$$

Multiplying a signal by a complex exponential shifts its spectrum by $\omega_0$.

Proof

$$F\{e^{j\omega_0 t}x(t)\} = \int_{-\infty}^{\infty}x(t)e^{j\omega_0 t}e^{-j\omega t}dt = \int_{-\infty}^{\infty}x(t)e^{-j(\omega - \omega_0)t}dt = X(\omega - \omega_0)$$

The last integral is the definition of $X$, evaluated at $\omega - \omega_0$.

Application: modulation

Since $\cos\omega_0 t = \frac{1}{2}(e^{j\omega_0 t} + e^{-j\omega_0 t})$:

$$x(t)\cos\omega_0 t \xleftrightarrow{F} \frac{1}{2}\left[X(\omega - \omega_0) + X(\omega + \omega_0)\right]$$

This is amplitude modulation: the baseband spectrum is copied to $\pm\omega_0$ with half the height.

  X(w)                 FT of x(t) cos(w0 t)
    1                      1/2         1/2
   /\                      /\          /\
  /  \                    /  \        /  \
 -+--+-- w          -----+----+--0--+----+---- w
 -W  W                  -w0            w0

Examples

• With $x(t) = 1$ (so $X(\omega) = 2\pi\delta(\omega)$): $e^{j\omega_0 t} \leftrightarrow 2\pi\delta(\omega - \omega_0)$, and therefore $\cos\omega_0 t \leftrightarrow \pi[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)]$.
• With $x(t) = e^{-at}u(t)$: $e^{-at}e^{j\omega_0 t}u(t) \leftrightarrow \frac{1}{a + j(\omega - \omega_0)}$, a spectrum whose peak has moved from $\omega = 0$ to $\omega = \omega_0$.

Dual of time shifting: a shift in time multiplies the spectrum by $e^{-j\omega t_0}$. A shift in frequency multiplies the signal by $e^{j\omega_0 t}$.