How do you take the Fourier transform of sin(t)/t using Parseval's Theorem?

[Moomax]

Homework Statement

Evaluate INT(|X(t)|^2) dt using parsevals theorem

where x(t) = (sin(t)cos(10t))/(pi*t)

Homework Equations

parsevals theorem: int(|f(t)|^2 dt = (1/2*pi)INT(|F(W)|^2 dw

The Attempt at a Solution

So I've tried several attempts at this problem and this is my latest:

first I use the fact that sin(x)*cos(y) = (sin(x+y)+sin(x-y)) /2
to get sin(t)cos(10t)/pi*t = (sin(t+10t) + sin(t - 10t))/(2*pi*t)

then I split it up into : sin(11t)/2t*pi + sin(-9t)/2t*pi

then what I was going to do was take the Fourier transform of each function here however, I can't figure out how in the world to take the Fourier transform of sin(t)/t

anyone have any ideas? thanks!

 

[jbunniii]

Do you know a function whose Fourier transform is $\sin(\omega)/\omega$?

 

[Moomax]

I checked on my transform table and looked around online a little and didn't see any transform that equals sin(w)/w. If there was i'd use the f(w) <-> F(t) rule and then it could work for me. Does that transform exist?

 

[Count Iblis]

Hint:

Compute the (inverse) Fourier transform of f(x), defined as:

f(x) = 1 for -L < x < L,

otherwise f(x) = 0

 

[jbunniii]

Yes, it exists and is one of the most fundamental transforms! If your table doesn't have it, I would get another table (seriously). Count Iblis' hint is right on the money.

By the way, $\sin(x)/x$ is sometimes called $sinc(x)$, so look in your table for that. Caution: some authors define $sinc(x) = \sin(\pi x)/(\pi x)$.

 

[Moomax]

ooohhhh now it makes sense! thank you! XD