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) $

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