Is the system y[n] = x[n] + x[n-1] invertible?

Question

Is the system y[n] = x[n] + x[n-1] invertible? If yes, what's the value of x[n]? If no, could you please introduce a counter example?

I couldn't find any counter example so I assumed the system is invertible and then tried to generate x[n] in terms of the output y but it always lead to a signal depending on a previous input.

x[n] = y[n] - x[n-1]

x[0] = y[0] - x[-1] --- 1

x[-1] = y[-1] - x[-2] --- substituting into 1:

x[0] = y[0] - y[-1] + x[-2],

and so on.. the resulting signal will always depend on another input. Does this imply non-invertibility of the system? if yes, do I not need a counter example?

Note: this isn't a homework question.

I'm voting to close this question as off-topic because it's homework with no attempt shown. — Hearth, Mar 18 '19 at 20:17
This question is not actually a homework question "the question in homework asked if the system is causal" and I did solve that. I didn't know I should post my attempts though, will update the question. — Moustafa Elsayed, Mar 18 '19 at 20:20
Sorry about that, I tried searching on electronics stackexchange only. Thanks! — Moustafa Elsayed, Mar 18 '19 at 20:37
+1 for wanting to know more than the homework assignment asks for — Huisman, Mar 18 '19 at 20:47
As you've added your reasoning, I've retracted my close vote. — Hearth, Mar 18 '19 at 22:06

score 0 · Accepted Answer · answered Mar 18 '19 at 20:44

Adjustment to https://math.stackexchange.com/questions/1453938/is-yn-xn-xn-1-invertible-system/1700642#1700642

Take $$ x_1[2n] = 1 \text{ and } x_1[2n-1] = -1 $$ and $$ x_2[2n] = 2 \text{ and } x_2[2n-1] = -2 $$ for all \$n\in \mathbb{N}\$.

Both have output \$y[n]=0 \$. The system cannot be inverted: one output cannot distinguish at least two different inputs.

Is the system y[n] = x[n] + x[n-1] invertible?

1 Answers1