(→b) Computing the response to the system when x[n] is the input from Question 2) |
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==b) Computing the response to the system when x[n] is the input from Question 2== | ==b) Computing the response to the system when x[n] is the input from Question 2== | ||
| − | <big><math>x[n] = cos(5\pi n)</math></big> | + | <big><math>x[n] = cos(5\pi n) = e^{j\pi n}</math></big> |
Revision as of 14:14, 26 September 2008
Defining the DT LTI system
$ x[n] \rightarrow system \rightarrow y[n] = 5x[n] $
a) Finding the unit impulse response h[n] and the system function F(z).
$ x[n] = \delta [n] \rightarrow system \rightarrow y[n]=5\delta [n] $
Therefore the unit impulse response, $ h[n] = 5\delta [n] $
For a DT LTI system,
$ Z^n \rightarrow system \rightarrow F(z)Z^n $
Output of the system, $ 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} $
Therefore, $ F(z) = \sum_{-\infty}^{\infty}h[m]Z^{-m} = \sum_{-\infty}^{\infty}5\delta [m] Z^{-m} $
b) Computing the response to the system when x[n] is the input from Question 2
$ x[n] = cos(5\pi n) = e^{j\pi n} $
