Problem 5
An LTI system has unit impulse response $ h[n] = u[n] - u[n-2]\, $.
a) Compute the system function $ H(z)\, $.
$ H(z)=\sum_{k=-\infty}^{\infty}h[k]z^{-k} $
$ H(z)=\sum_{k=-\infty}^{\infty}(u[k]-u[k-2])z^{-k} $
$ H(z)=\sum_{k=0}^{1}z^{-k} $
$ H(z)= 1 + \frac{1}{z} $
b) What is the system response to the input $ x[n]=\cos(\pi n)\, $.
Note that $ \cos(\pi n)=(-1)^n\, $
The output $ Y[n]= F(z)z^{n} = (1 + \frac{1}{-1})(-1)^{n} = 0\, $
