2 Shift properties of the Fourier transform

There are two basic shift properties of the Fourier transform:

(i) Time shift property: $\bullet \mathcal{F}\left\{f\left(t-t_0\right)\right\}=e^{-\text{i}\omega t_0}F\left(\omega \right)$

(ii) Frequency shift property $\bullet \mathcal{F}\left\{e^{\text{i}{\omega }_{0}t}f\left(t\right)\right\}=F\left(\omega -{\omega }_0\right).$

Here $t_0$ , ${\omega }_0$ are constants.

In words, shifting (or translating) a function in one domain corresponds to a multiplication by a complex exponential function in the other domain.

We omit the proofs of these properties which follow from the definition of the Fourier transform.

Example 2

Use the time-shifting property to find the Fourier transform of the function

$g\left(t\right)=\left\{\begin{array}{ccc}\hfill 1\hfill & \hfill \hfill & \hfill 3\le t\le 5\hfill \\ \hfill 0\hfill & \hfill \hfill & \hfill \text{otherwise}\hfill \end{array}\right.$

Figure 4

Solution

$g\left(t\right)$ is a pulse of width 2 and can be obtained by shifting the symmetrical rectangular pulse

$p_1\left(t\right)=\left\{\begin{array}{ccc}\hfill 1\hfill & \hfill \hfill & \hfill -1\le t\le 1\hfill \\ \hfill 0\hfill & \hfill \hfill & \hfill \text{otherwise}\hfill \end{array}\right.$

by 4 units to the right.

Hence by putting $t_0=4$ in the time shift theorem

$G\left(\omega \right)=\mathcal{F}\left\{g\left(t\right)\right\}=e^{-4\text{i}\omega }\frac{2}{\omega }sin\omega .$

Task!

Verify the result of Example 2 by direct integration.

Answer

Task!

Use the frequency shift property to obtain the Fourier transform of the

modulated wave

$g\left(t\right)=f\left(t\right)cos{\omega }_{0}t$

where $f\left(t\right)$ is an arbitrary signal whose Fourier transform is $F\left(\omega \right)$ .

First rewrite $g\left(t\right)$ in terms of complex exponentials:

Answer

Answer