DT LTI System: $ y[n]=\frac{4x[n+7]}{3}+3x[n-5]+6x[n]\! $
Part A
$ h[n]=\frac{4\delta[n+7]}{3}+3\delta[n-5]+6\delta[n]\! $
The following functions can be used to find the system function:
$ H(z)=\sum_{m=-\infty}^{\infty}h[m]z^{-m} $
$ H(z)=\sum_{m=-\infty}^{\infty}(\delta[m]+\delta[m-1])z^{-m} $
$ H(z)=\frac{4}{3}e^{jwn}+3e^{j5wn}+6\! $
Part B
The signal used in Q2 was $ x[n]=3+sin\bigg(\frac{\pi}{2}n\bigg) $
which is equal to:
$ x[n]=3+ \frac{1}{2i} e^{i\frac{\pi}{2}n}-\frac{1}{2i}e^{-i\frac{\pi}{2}n} $ and we know that $ w=4\! $
So, the response of this system to the signal I defined in Q2 is:
$ y[n]=\bigg(\frac{4}{3}e^{j4n}+3e^{j20n}+6\bigg)\bigg(3+ \frac{1}{2i} e^{i\frac{\pi}{2}n}-\frac{1}{2i}e^{-i\frac{\pi}{2}n}\bigg) $
