Proving the Energy of Even and Odd Signals

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Discussion Overview

The discussion revolves around a signals assignment that involves proving relationships between the energy of even and odd components of a continuous time, real-valued signal x(t). Participants explore the mathematical expressions and assumptions related to the energy calculations.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant proposes that the energy of even and odd components of a signal should satisfy the equation Ex_even = Ex_odd = 0.5 * Ex.
  • Another participant challenges this assumption by providing a counter-example using the function x(t) = t^2, suggesting that the initial claim does not hold true for arbitrary signals.
  • A different participant argues that the energy distribution between even and odd components can vary significantly, depending on the specific signal, and that the proposed equation may not be valid in general.
  • One participant provides a specific counter-example with the function x(t) = e^{-|t|}, demonstrating that the even component can have finite energy while the odd component has zero energy, indicating a flaw in the original question.
  • Another participant expresses confidence in proving part (b) of the question, which involves showing that the integral of the product of the even and odd components equals zero.
  • Clarification is sought regarding the notation "Ex" and its meaning in the context of the question.
  • After further discussion, a participant reveals that the professor confirmed the original question contained a mistake, and the correct relationship should be Ex_even + Ex_odd = Ex.

Areas of Agreement / Disagreement

Participants generally disagree on the validity of the initial claim regarding the energy distribution of even and odd components. The discussion remains unresolved until the professor's clarification is provided, which ultimately indicates a mistake in the question.

Contextual Notes

The discussion highlights limitations in the original question's assumptions and the dependence on specific definitions of energy in the context of signal processing. The initial misunderstanding led to various interpretations and counter-examples.

MrPacane
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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!
 
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It doesn't.

Just use x(t) = t^2 to see why.
 
Alright, thanks for your reply.

But then, does anyone know how to show that Ex_even = Ex_odd = 0.5 * Ex ?
 
MrPacane said:
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.
 
Last edited:
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 :) !
 
Ok, just take a counter-example. Let [itex]x(t) = e^{-|t|}[/itex].

[itex]x_e(t) = e^{-|t|}[/itex]

[itex]x_o(t) = 0[/itex].

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.
 
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?
 
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.
 
Last edited by a moderator:
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".
 

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