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Revision as of 09:54, 15 October 2008 by Sje (Talk)

question

3. An LTI system has unit impulse response $ h[n]=u[-n] $ Compute the system's response to the input $ x[n]=2^{n}u[-n]. $(Simplify your answer until all \sum signs disappear.)

solution

$ y[n]=x[n]*h[n] $

$ y[n]=\sum^{\infty}_{k=-\infty}x[n]*h[n-k] $

$ y[n]=\sum^{\infty}_{k=-\infty}2^{k}u[-k]u[n-k] $

n[-k] = 1 ,-k>=0

k<=0

$ y[n]=\sum^{0}_{k=-\infty}2^{k}u[n-k] $

$ \sum^{0}_{k=n}2^{k}, n<=0 $

0, else

let r=-k, k=-r

$ \sum^{-n}_{r=0}2^{-r}, n<=0 $

0, else

$ \sum^{-n}_{r=0}(1/2)^{r}, n<=0 $

0, else

$ y[n] =2(1-(1/2)^{-n+1))u[-n] $

$ (2-2^{n})u[-n] $

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