Define a DT LTI System
Let the DT LTI system be: $ y[n]=x[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} $
$ F[z]=\sum_{m= - \infty}^{\infty}\delta{[m-5]}z^{-m} $
since $ x[m]\delta{[m-5]}=x[-5] $ by the sifting property then:
$ F[z]=z^{-5} $
where z is an input into our system.
So when z^n is input into our system, we should get $ F[z]z^n=z^{-5}z^n $ back out.
Response of the system to the signal defined in Question 1
$ x(t)=(5+3j)cos(4t)+(1+2j)sin(3t) $
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