Problem 5
An LTI system has unit impulse response $ h[n]=u[n]-u[n-2] $. Compute (a) the system's function $ H(z) $ and (b) the system's response to the input $ x[n]=\cos(\pi n) $.
Part A
First, note this is discrete time. Doing the problem in continuous time gives a very different result, as I tragically learned on the test...
$ H(z)=\sum_{k=-\infty}^{\infty}h[k]z^{-k} $
$ H(z)=1+\frac{1}{z} $ due to the two step functions.
Part B
First, recognize that $ \cos(\pi n)=(-1)^n $.
The response to the input signal $ z^n $ is $ H(z)z^n $, giving
$ (1+\frac{1}{-1})(-1)^n=0 $.
