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Discrete Signals/Systems Even/Odd Problem

  1. Jun 8, 2015 #1
    1. The problem statement, all variables and given/known data

    http://ocw.mit.edu/courses/electric...s-fall-2011/assignments/MIT6_003F11_sol01.pdf


    upload_2015-6-8_4-12-12.png
    upload_2015-6-8_4-12-24.png

    2. Relevant equations
    None


    3. The attempt at a solution
    To be honest, I don't think you can prove something is impossible with just a sentence or two.

    I don't see why we check x odd [n] and x left[n] at 0 and notice they are both 0.

    1. Why do we think to examine those numbers?

    2. What do them both being zero tell us?
     
  2. jcsd
  3. Jun 8, 2015 #2

    Zondrina

    User Avatar
    Homework Helper

    In a mathematical sense, the question is asking if this equality holds:

    $$x[n] = x_o[n] + x_l[n], \quad \forall n \in \mathbb{Z}$$

    In words: Can we re-construct ##x[n]## from the odd part of the signal plus the part of the signal to the left of ##n = 0##, ##\forall n \in \mathbb{Z}##?

    Intuitively this cannot be possible, because the even part of the signal for ##n \geq 0## will be excluded from the re-construction of ##x[n]##.

    If you can imagine it, there would be a continuous line to the left of ##n = 0## because of ##x_l[n]##, but for ##n \geq 0##, there would be many holes in the signal because only ##x_o[n]## is accounted for.

    If the equality given was instead ##x[n] = x_r[n] + x_l[n]##, it would be possible to re-construct the signal.
     
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