Nth derivative Fourier transform property

[ElijahRockers]

Homework Statement

I am given f(t) = e^-|t| and I found that F(w) = ##\sqrt{\frac{2}{\pi}}\frac{1}{w^2 + 1}##

The question says to use the nth derivative property of the Fourier transform to find the Fourier transform of sgn(t)f(t), and gives a hint: "take the derivative of e^-|t|"

I also found the Fourier transform for t*f(t) using another property, but this part has me stumped.

Homework Equations

sgn(t) =
1 for t>0
0 for t=0
-1 for t<0

The Attempt at a Solution

I took the derivative of e^-|t|, and got ##\frac{-te^{-|t|}}{|t|}##

But I'm not quite sure how I can use that result, combined with the nth derivative property, to find the F.T. of sgn(t)f(t). I plotted f(t), f'(t) and sgn(t)f(t), but I'm struggling to see the link between them that can help me solve this one... any guidance would be welcome.

 

[Dick]

Homework Statement

I am given f(t) = e^-|t| and I found that F(w) = ##\sqrt{\frac{2}{\pi}}\frac{1}{w^2 + 1}##

The question says to use the nth derivative property of the Fourier transform to find the Fourier transform of sgn(t)f(t), and gives a hint: "take the derivative of e^-|t|"

I also found the Fourier transform for t*f(t) using another property, but this part has me stumped.

Homework Equations

sgn(t) =
1 for t>0
0 for t=0
-1 for t<0

The Attempt at a Solution

I took the derivative of e^-|t|, and got ##\frac{-te^{-|t|}}{|t|}##

But I'm not quite sure how I can use that result, combined with the nth derivative property, to find the F.T. of sgn(t)f(t). I plotted f(t), f'(t) and sgn(t)f(t), but I'm struggling to see the link between them that can help me solve this one... any guidance would be welcome.

You do know that ##\frac{t}{|t|}=sgn(t)##, right? Aside from the issue of the left hand side not being defined at ##t=0##, but that ambiguity doesn't matter for a Fourier transform.

 

[ElijahRockers]

I definitely did not realize that... derp. That simplifies things. Thank you!