# Discrete Signals/Systems Even/Odd Problem

• LongApple
In summary, the question is asking if it is possible to re-construct a signal from the odd part of the signal and the part of the signal to the left of n=0. This is not possible because the even part of the signal would be excluded from the re-construction. If the equality was instead x[n] = x_r[n] + x_l[n], it would be possible to re-construct the signal.
LongApple

## Homework Statement

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

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## 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?

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.

## 1. What is a discrete signal/system?

A discrete signal/system is a type of signal or system that is represented by a sequence of values at distinct time intervals. This means that the signal or system can only take on specific values at specific points in time, rather than being continuously varying.

## 2. What is the difference between a discrete signal and a continuous signal?

A discrete signal is a signal that is represented by a sequence of values at distinct time intervals, while a continuous signal is one that is represented by a continuous function. This means that a continuous signal can take on any value at any point in time, while a discrete signal can only take on specific values at specific times.

## 3. What does it mean for a signal/system to be even?

A signal or system is considered even if it satisfies the property of symmetry, where the signal or system is unchanged when reflected about the Y-axis. In other words, if the signal or system is mirrored about the Y-axis, the two halves will be identical.

## 4. What does it mean for a signal/system to be odd?

A signal or system is considered odd if it satisfies the property of antisymmetry, where the signal or system is flipped and inverted when reflected about the Y-axis. In other words, if the signal or system is mirrored about the Y-axis, the two halves will be identical but with opposite signs.

## 5. How can I determine if a signal/system is even or odd?

The even/odd property of a signal or system can be determined by checking if the signal or system satisfies the respective symmetry or antisymmetry property. This can be done by reflecting the signal or system about the Y-axis and comparing the two halves. If they are identical, the signal or system is even. If they are identical but with opposite signs, the signal or system is odd.

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