# Fourier Transform Question

1. Oct 29, 2007

### verd

Alright. So... Conceptually I completely understand what I'm doing. I'm just a bit confused about how, mechanically, to solve this.

I basically have the following:
W(w)=(1/2)[p2(w+6)+p2(w-6)]

Where p2(w) is a pulse of width 2. Of course, w is omega, and the +/-6 is the shift. This is a modulated signal and it's in the frequency domain.

Now this is my problem...

w(t) is put into a squaring function, which produces z(t).

In this problem, the frequency domain is infinitely easier to work with than the time domain. So I know I can use Parseval's theorem, the special case for squaring a function. My problem is that I am then presented with the following

I'm using:

$$\int_{-\infty}^{\infty}x^2(t)dt=\frac{1}{2\pi}\int_{-\infty}^{\infty}|X(\omega)|^2d\omega$$

Can I say that

$$Z(\omega)=\frac{1}{2\pi}\int_{-\infty}^{\infty} | \frac{1}{2} [ p_{2}(\omega + 4)+p_{2}(\omega - 4)] |^2 d \omega$$?

How do I deal with this integral? Do I just simply take the integral of the height of W(w) from -7 to -5 and then from 5 to 7 and just add them?

Am I supposed to end up with T/2, where T=the width of my pulse??

Any suggestions would be greatly appreciated. Thanks

Last edited: Oct 29, 2007
2. Nov 18, 2009

### bhupala

The Parseval's relation gives you the concept of law of conservation of energy. Both RHS and LHS are just real numbers. You can use frequency domain stuff to calculate Z(\omega) but not by using Paarseval's relation.