What is F{y(t)} if y(ω) = F{x(t)}?

[homad2000]

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)}

 

[I like Serena]

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!

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.

 

[homad2000]

Hi homad2000!

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?

 

[I like Serena]

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?

 

[homad2000]

I believe i should add the 2 pi, because we use w = 2 * pi * f

Thank you for your help, I appreciate it :)

 

[I like Serena]

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.