Revision as of 13:30, 10 October 2008 by Park1 (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.)


Answer

$ y[n] = x[n] * h[n] , where * is convolution/, $

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

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

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

$ = \sum^{-n}_{0} \frac{1}{2}^{k}, $ for n=<0
$ = 0 , $for n>0

$ =\frac{1-\frac{1}{2}^{-n+1}}{1-\frac{1}{2}}/, $
$ = 0 $for n>0

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

Alumni Liaison

Prof. Math. Ohio State and Associate Dean
Outstanding Alumnus Purdue Math 2008

Jeff McNeal