Test the stability of the system whose impulse response , $ h(n)=\left(\frac{1}{2}\right)^n u(n) $

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Test the stability of the system whose impulse response ,

$h(n)=\left(\frac{1}{2}\right)^n u(n)$

written 2.3 years ago by

prajapatijaimin • 3.3k

• modified 2.3 years ago

discrete time signal processing

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1 Answer

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written 2.3 years ago by

prajapatijaimin • 3.3k

Solution:

$\begin{aligned}\\ & \sum_{n=-\infty}^{\infty}|h(n)|\lt\infty \\\\ & =\sum_{n=-\infty}^{\infty}\left|\left(\frac{1}{2}\right)^n u(n)\right| \\\\ & =\sum_{n=0}^{\infty}\left(\frac{1}{2}\right)^n \\\\ & =1+\frac{1}{2}+\frac{1}{4} \ldots \ldots \ldots . \infty \\\\ & =\frac{1}{1-\frac{1}{2}} \\\\ & =2\lt\infty\\ \end{aligned}\\$ $

Hence the system is stable.

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