Define a DT LTI System
Let the DT LTI system be: $ y[n]=u[n-5] $
Obtain the Unit Impulse Response h[n] and the System Function F[z] of the system
First to obtain the unit impulse response h[n] we plug in $ \delta{[n]} $ into our y[n].
$ h[n]=\delta{[n-5]} $
Then the system function F[z] is obtained by
$ F[z]=\sum_{m= - \infty}^{\infty}h[m]z^{-m} $
where z is an input into our system. Let $ z = e^{jk\omega_o} $
So when z^n is input into our system, we should get $ F[z]z^n $ back out.
$ F[z]=\sum_{m= - \infty}^{\infty}\delta{[m-5]}e^{-mjk\omega_o} $
