Proving the Energy of Even and Odd Signals

[MrPacane]

Hi everyone,

In my signals assignment, I'm asked to show that, for a continuous time, real-valued signal x(t):

Ex_even = Ex_odd = 0.5 * Ex

So here's what I've done:

Ex_even = ∫|(x(t) + x(-t))/2|²dt
Ex_even = 0.5 * ∫|(x(t)² + 2x(t)x(-t) + x(-t)²)/2|dt
Ex_even = 0.5 * [ 0.5 * ∫x(t)²dt + ∫x(t)x(-t)dt + 0.5 * ∫x(-t)²dt ]

Now, I assume that ∫x(t)x(-t)dt must go to zero (when integrated from -∞ to +∞), but I don't understand why. Could someone explain it to me?

Thanks!

 

[Antiphon]

It doesn't.

Just use x(t) = t^2 to see why.

 

[MrPacane]

Alright, thanks for your reply.

But then, does anyone know how to show that Ex_even = Ex_odd = 0.5 * Ex ?

 

[uart]

Alright, thanks for your reply.

But then, does anyone know how to show that Ex_even = Ex_odd = 0.5 * Ex ?

As Antiphon already pointed it is not true, not for an arbitrary signal anyway. So it's pointless trying to "show it" if it is false!

As you know, an arbitrary signal may be decomposed into odd and even components. For some signals the odd component will be zero, so all the energy is in the even component. For some signals the even competent will be zero, so all the energy is in the odd component. Other signals will have the energy distributed between the odd and even components, but in general they won't have an equal distribution of the total energy.

Please go back and check the exact question that you were asked. You may have missed something or somehow misinterpreted the question. As it stands, what you have asked makes no sense.

 

[MrPacane]

Here's the exact question:

Let x_even and x_odd be the even and odd parts of a continuous time, real-valued signal x. Show that
a) Ex_even = Ex_odd = 0.5 * Ex
b) ∫ x_even*x_odd = 0 (integral is from -∞ to +∞)

What am I missing?

Again, thanks for your time :) !

 

[uart]

Ok, just take a counter-example. Let $x(t) = e^{-|t|}$.

$x_e(t) = e^{-|t|}$

$x_o(t) = 0$.

The energy in the even component is finite and the energy in the odd component is zero. Clearly there is something wrong with the question.

 

[uart]

BTW. I've got no issue with part (b). That part is easily proved by spiting the integral into two parts (-inf to 0) and (0 to inf).

Are you certain that "Ex" denotes "energy in x(t)" in this question?

 

[MrPacane]

Yes, I'm pretty sure... Here's a screenshot of the question:

http://imageshack.us/a/img689/3365/21602169.jpg

EDIT: I just sent an e-mail to my prof asking him to clarify this... I'll get back to you as soon as I get his answer.

 

[MrPacane]

Ok guys, I'm sorry for wasting your time, the prof said it's a mistake.

The question should have been:

Let x_even and x_odd be the even and odd parts of a continuous time, real-valued signal x. Show that
a) Ex_even + Ex_odd = Ex

Which is quiet easy to prove. Problem "solved".