CT Fourier Transform

Question

I need to find the Fourier Transform of the given signal below;

$$ x(t) = \frac{\sin(\pi t)}{\pi t} \frac{\sin(2\pi t)}{\pi t}.$$

I know that if $ x(t) = \frac{\sin(Wt)}{\pi t} $ , then $ X(w) = \begin{cases} 1, & |w| < W,\\ 0, & |w|>W. \end{cases} $

How can I apply this to multiplication? I need help there, thanks.

Answer 1

I want to share the answer.

$$ x(t) = \frac{\sin(\pi t)}{\pi t} \frac{\sin(2\pi t)}{\pi t}$$

Let $ x_{1}(t) = \frac{\sin(\pi t)}{\pi t}$ , then $ X_{1}(w) = \begin{cases} 1, & |w| < \pi\\ 0, & |w|>\pi \end{cases} $

Let $ x_{2}(t) = \frac{\sin(2\pi t)}{\pi t}$ , then $ X_{2}(w) = \begin{cases} 1, & |w| <2\pi\\ 0, & |w|>2\pi. \end{cases} $

$x(t) = x_{1}(t) x_{2}(t)$

By using multiplication property;

$ X(w) = \frac{1}{2\pi} (X_{1}(w)*X_{2}(w) )$

Clearly, convolution of $X_{1}(w)$ and $X_{2}(w)$ gives us the result.