View source for HW4.4 Caleb Tan ECE301Fall2008mboutin - Rhea

==Defining the DT LTI system==

<math>x[n] \rightarrow system \rightarrow y[n] = 5x[n]</math>

==a) Finding the unit impulse response h[n] and the system function F(z).==

<math>x[n] = \delta [n] \rightarrow system \rightarrow y[n]=5\delta [n]</math>

Therefore the unit impulse response, <big><math>h[n] = 5\delta [n]</math></big>

For a DT LTI system, 

<math>Z^n \rightarrow system \rightarrow F(z)Z^n</math>

Output of the system, <math>F(z)Z^n = h[n]*Z^n = \sum_{m=-\infty}^{\infty} h[m]Z^{n-m} = Z^n\sum_{-\infty}^{\infty}h[m]Z^{-m}</math>

Therefore, <math>F(z) = \sum_{m = -\infty}^{\infty}h[m]Z^{-m} = \sum_{m = -\infty}^{\infty}5\delta [m] Z^{-m}</math>

==b) Computing the response to the system when x[n] is the input from Question 2==

<big><math>x[n] = cos(5\pi n) = e^{j\pi n}</math></big>

Therefore, <big><math>Z^n = e^{j\pi n}</math></big>

<big><math>Z = e^{j\pi}</math></big>

<math>F(e^{j\pi}) = \sum_{m = -\infty}^{\infty}5\delta [m] (e^{j\pi})^{-m}</math>