transform signal

Question

Hello everyone I need help solving a Fourier transform for the given signal, I know it will be a frequency convolution for the first function it will be a window function and for the second function I do not know how to convert it (I do not know how to turn it into a sinc) I would love if someone could help me with this And within the boundaries of the integration of the convolution between them what will be the boundaries? This is the exercise

$$x_6(t) = \frac{\sin(\pi t)}{\pi t}\cdot \frac{\sin(2\pi (t-1))}{\pi (t-1)}$$

I know the sinc is $\operatorname{sinc}(t):=\frac{\sin(\pi t)}{\pi t}$.

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This is what I wrote I want to know how to find X 2

Answer 1

First you need to remember these 3 facts about Fourier Transform:

1. FT of sinc is rect function

$FT[sinc(t)]=rect(w)$

2. FT of a shifted signal is the FT of the original signal multiplied by exponent:

$FT[x(t-t_0)]=FT[x(t)]e^{-2jwt_0\pi}$

3. FT of a scaled signal is FT of the original divided by the scaling:

$FT[x(at)]=\frac{1}{a}FT[x(\frac{w}{a})]$

Now just use 1 and 3 to find the FT of $x_1(t)$, where $a=\pi$ in this case.

Then use 1, 2 and 3 to find the FT of $x_2(t)$, where $a=2\pi$ and $t_0=1$ in this case.

Finally, substitute them in the convolution expression you wrote in the first line: $\frac{1}{2\pi}FT[x_1(t)]*FT[x_2(t)]$