What is F{y(t)} if y(ω) = F{x(t)}?

homad2000
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Homework Statement



if y(ω) = F{x(t)}, what is F{y(t)} (F is the Fourier transform operation)

Homework Equations



non

The Attempt at a Solution



I tried finding F^-1{y(ω)}, which is equal too x(t), but I could not go on with finding F{y(t)}
 
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homad2000 said:

Homework Statement



if y(ω) = F{x(t)}, what is F{y(t)} (F is the Fourier transform operation)


Homework Equations



non

The Attempt at a Solution



I tried finding F^-1{y(ω)}, which is equal too x(t), but I could not go on with finding F{y(t)}

Hi homad2000! :smile:

Check the section on "duality" on the wiki page: http://en.wikipedia.org/wiki/Fourier_transform
It says what the transform is of a transform with the domain swapped.
 
I like Serena said:
Hi homad2000! :smile:

Check the section on "duality" on the wiki page: http://en.wikipedia.org/wiki/Fourier_transform
It says what the transform is of a transform with the domain swapped.

Ok, correct me if I'm wrong:

I got F{y(t)} = x(-ω) ? or should I add the 2π to that?
 
homad2000 said:
Ok, correct me if I'm wrong:

I got F{y(t)} = x(-ω) ? or should I add the 2π to that?

Yep. That's it.

Whether or not 2π should be added depends on the definition of your Fourier transform.
As you can see on the wiki page, there are 3 different common definitions.
Which of the 3 does your textbook use?
 
I believe i should add the 2 pi, because we use w = 2 * pi * f

Thank you for your help, I appreciate it :)
 
homad2000 said:
I believe i should add the 2 pi, because we use w = 2 * pi * f

Thank you for your help, I appreciate it :)

That would not be the reason.

Your Fourier transform would be defined as either:
$$F(\omega)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} f(t) e^{-i \omega t}\, dt$$
or
$$F(\omega)=\int_{-\infty}^{\infty} f(t) e^{-i \omega t}\, dt$$

In the first case you would not have a factor 2pi, while in the second case you would have a factor 2pi.
 
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