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] $
