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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 $\operatorname{sinc}(t):=\frac{\sin(\pi t)}{\pi t}$.


enter image description here

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

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  • $\begingroup$ "I can't convert it to a sinc" uh, can you write down the formula of a sinc in your question, please? $\endgroup$ Apr 22 at 9:40
  • $\begingroup$ sinc(x)=sin(pix)/pix $\endgroup$ Apr 22 at 12:47
  • $\begingroup$ that's not what I meant with "in your question", so I quickly added the formula to your question. OK; so, I think you need to express all that you can in your $x_6$ with sinc(t) and modifications of that yourself first. That really doesn't seem very hard, so please do that, or explain where the hardness is! $\endgroup$ Apr 22 at 13:50
  • $\begingroup$ Hi Orel! (Do) you know the multiplication-convolution property of the Fourier transform? Have you tried it? It will turn out to be a convolution integral in frequency, but have you actually written it? Which part is hard for you? Please write down your steps either in handwriting (picture) or in better LaTex. $\endgroup$
    – Fat32
    Apr 22 at 13:57
  • $\begingroup$ Hi thank you very much for the answer I understand the exercise, I have a multiplication between functions while it is a convolution at this frequency I understand I know that the transformation of the first function would actually be a rectangle but regarding the second transformation I can not convert it I would be happy to help I ask for help $\endgroup$ Apr 22 at 14:40
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First you need to remember these 3 facts about Fourier Transform:

  1. FT of sinc is rect function

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

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

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

  1. 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: $\frac{1}{2\pi}FT[x_1(t)]*FT[x_2(t)]$

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  • $\begingroup$ Thank you very much! I really owed the feature 2 you helped me a lot. $\endgroup$ Apr 25 at 6:18

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