How Do I Apply Parseval's Theorem to a Modulated Signal in the Frequency Domain?

[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 followingI'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

 

[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.

 

[Mene]

I understand your confusion and can offer some guidance on how to approach this problem. First, it is important to understand the concept of a Fourier Transform, which is a mathematical tool used to analyze signals in the frequency domain. In this case, your signal is modulated and already in the frequency domain, so you can apply Parseval's theorem to find the power of the signal.

To do this, you will need to square the signal, which in this case is represented by the function z(t). This means that you will need to square the expression for W(w), which is given by:

W(w) = (1/2)[p2(w+6)+p2(w-6)]

To square this expression, you will need to expand it using the binomial theorem and then integrate using Parseval's theorem, which states that the integral of the squared signal in the time domain is equal to the integral of the squared signal in the frequency domain divided by 2π.

So, to solve this problem, you will need to expand the expression for W(w) and then integrate it using Parseval's theorem. This will give you the power of the signal, which is represented by Z(ω). From there, you can use the definition of power to find the width of your pulse.

I hope this helps clarify the process for solving this problem. Remember, the key is to understand the concept of Fourier Transform and how it can be applied to analyze signals in the frequency domain. Good luck!