Exam1 question 3.

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


Solution

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

$ =\sum_{k=-\infty}^{\infty} x[k]h[n-k] $

$ =\sum_{k=-\infty}^{\infty} 2^{k}u[-k]u[-(n-k)] $

$ =\sum_{k=-\infty}^{0}2^{k}u[-u+k] $

$ r = -k \! $

$ =\sum_{r=0}^{\infty}(\frac{1}{2})^{r}u[-n-r] $

$ -n-r \geqq 0 $

$ -n\geqq r $

$ =(\sum_{r=0}^{-n}(\frac{1}{2})^{n}) $

if -n>= 0

else , 0

therefore,

$ =\frac{1-\frac{1}{2}^{-n+1}}{1-\frac{1}{2}} $

if n=<0

else, 0

Alumni Liaison

Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett