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

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

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

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Answer 1

$ y[n]=x[n]*h[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 20:23, 31 January 2011 (UTC)

Answer 2

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Answer 3

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