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Determine whether the following systems are stable or not $y(n)=3 x(n)$.

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Determine whether the following systems are stable or not $y(n)=3 x(n)$.

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written 20 months ago by |

**Solution:**

Let $x(n)=\delta(n), y(n)=h(n)$

$ h(n)=3 \delta(n) $

Condition for stability,

$\sum_{k=-\infty}^{\infty}|h(k)|\lt\infty$

$ \begin{gathered}\\ \sum_{k=-\infty}^{\infty}|h(k)|=\sum_{k=0}^{\infty}|3 \delta(k)|=\sum_{k=0}^{\infty} 3 \delta(k)=3 \\\\ \because \delta(k)=0 \text { for } k \neq 0 \text { and } \delta(k)=1 \text { for } k=0 \\\\ \because \sum_{k=-\infty}^{\infty}|h(k)|\lt\infty \text { the given system is stable}\\ \end{gathered}\\ $

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