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Determine if the following systems are time-invariant or time-variant. (i) $y(n)=x(n)+x(n-1)$ (ii) $y(n)=x(-n)$
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Solution:

(i) $y(n)=x(n)+x(n-1)$

Given:

output of the system $y(n)=x(n)+x(n-1)$

If the input is delayed by ' $\mathrm{k}$ ' units in time, we have

$ y(n, k)=x(n-k)+x(n-k-1) $

If the output is delayed by ' k ' units in time, then $(n-\gtn-k)$

$ y(n-k)=x(n-k)+x(n-k-1) $

Here, $y(n, k)=y(n-k)$

Therefore, the system is time-invariant.

(ii) $y(n)=x(-n)$

Given:

output of the system $y(n)=x(-n)$

If the input is delayed by ' $\mathrm{k}$ ' units in time, we have,

$ y(n, k)=x(-n-k) $

If the output is delayed by ' k ' units in time, then (n->n-k)

$ y(n-k)=x(-n+k) $

Here, $y(n, k) \neq y(n-k)$

Therefore, the system is time-variant.

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