Discrete Signals/Systems Even/Odd Problem

[LongApple]

Homework Statement

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

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Homework Equations

None[/B]

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?

 

[STEMucator]

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.