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I came across a question while studying signals and systems, and I am not able to prove it. The question wants me to show that the following equation is shift invariant and linear. Showing that it is linear is not too difficult, but I am having a hard time showing that it is shift invariant.

$y(n)=\sum_{m=-\infty}^{n} x(m)$

Thank you.

big tree
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1 Answers1

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If the input $x(n)$ is shifted by $k$ units, we have $$ y(n,k)=\sum^n_{m=-\infty}x(m-k) $$ From the definition of shift invariant systems, we must have $$ y(n-k)=\sum^{n-k}_{m=-\infty}x(m)=\sum^{n}_{v=-\infty}x(v-k) $$

The right hand sides are equal therefore the system is shift invariant.

wd violet
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