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Energy of even & odd signals

  1. Sep 23, 2012 #1
    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?

  2. jcsd
  3. Sep 23, 2012 #2
    It doesn't.

    Just use x(t) = t^2 to see why.
  4. Sep 24, 2012 #3
    Alright, thanks for your reply.

    But then, does anyone know how to show that Ex_even = Ex_odd = 0.5 * Ex ?
  5. Sep 24, 2012 #4


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    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: Sep 24, 2012
  6. Sep 24, 2012 #5
    Here's the exact question:

    What am I missing?

    Again, thanks for your time :) !
  7. Sep 24, 2012 #6


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    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.
  8. Sep 24, 2012 #7


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    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?
  9. Sep 24, 2012 #8
    Yes, I'm pretty sure... Here's a screenshot of the question:

    http://imageshack.us/a/img689/3365/21602169.jpg [Broken]

    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: May 6, 2017
  10. Sep 24, 2012 #9
    Ok guys, I'm sorry for wasting your time, the prof said it's a mistake.

    The question should have been:

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