# Practice Question on Computing the Output of an LTI system by Convolution

The unit impulse response h[n] of a DT LTI system is

$h[n]= \frac{1}{3^n} \$

Use convolution to compute the system's response to the input

$x[n]= \delta[n+2]+\delta[n+1]+\delta[n]+\delta[n-1]. \$

$y[n]=h[n]*x[n]=\sum_{k=-\infty}^\infty \frac{1}{3^k}\delta[n+2-k]+\sum_{k=-\infty}^\infty \frac{1}{3^k}\delta[n+1-k]+\sum_{k=-\infty}^\infty \frac{1}{3^k}\delta[n-k]+\sum_{k=-\infty}^\infty \frac{1}{3^k}\delta[n-1-k]$

$y[n]=\frac{1}{3^{n+2}}+\frac{1}{3^{n+1}}+\frac{1}{3^{n}}+\frac{1}{3^{n-1}}$

--Cmcmican 20:25, 31 January 2011 (UTC)

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BSEE 2004, current Ph.D. student researching signal and image processing.

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