Fourier Transform Question

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: