# Fourier transform of sin(t)/t

1. Oct 4, 2009

### Moomax

1. The problem statement, all variables and given/known data

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

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

2. Relevant equations

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

3. 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!

2. Oct 4, 2009

### jbunniii

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

3. Oct 4, 2009

### 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?

4. Oct 4, 2009

### Count Iblis

Hint:

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

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

otherwise f(x) = 0

5. Oct 4, 2009

### 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 $sinc(x) = \sin(\pi x)/(\pi x)$.

Last edited: Oct 4, 2009
6. Oct 4, 2009

### Moomax

ooohhhh now it makes sense!!! thank you!! XD