(New page: Category:2010 Fall ECE 438 Boutin ---- == Solution to Q1 of Week 10 Quiz Pool == ---- a. The difference equation for this system is :<math>\begin{align} & Y(z) = az^{-1}Y(z)+X(z)-z^{...) |
|||
| Line 24: | Line 24: | ||
---- | ---- | ||
| + | Credit: Prof. Charles Bouman | ||
Back to [[ECE438_Week10_Quiz|Lab Week 10 Quiz Pool]] | Back to [[ECE438_Week10_Quiz|Lab Week 10 Quiz Pool]] | ||
Latest revision as of 18:00, 26 October 2010
Solution to Q1 of Week 10 Quiz Pool
a. The difference equation for this system is
- $ \begin{align} & Y(z) = az^{-1}Y(z)+X(z)-z^{-1}X(z) \\ & H(z) = \frac{Y(z)}{X(z)} = \frac{1-z^{-1}}{1-az^{-1}} \\ \end{align}\,\! $
- poles at $ z=a $ and zeros at $ z=1 $.
b. ROC $ |z|>a $
- $ H(z)=\frac{1}{1-az^{-1}}-\frac{z^{-1}}{1-az^{-1}} $
- $ \Rightarrow h[n]=a^{n}u[n]-a^{n-1}u[n-1] $
- The system is stable if ROC contains the unit circle ($ |z|=1 $), therefore $ |a|<1 $.
c. ROC $ |z|<a $
- $ H(z)=\frac{1}{1-az^{-1}}-\frac{z^{-1}}{1-az^{-1}} $
- $ \Rightarrow h[n]=-a^{n}u[-n-1]+a^{n-1}u[-(n-1)-1] $
- $ \Rightarrow h[n]=-a^{n}u[-n-1]+a^{n-1}u[-n] $
- The system is stable if ROC contains the unit circle ($ |z|=1 $), therefore $ |a|>1 $.
Credit: Prof. Charles Bouman
Back to Lab Week 10 Quiz Pool
Back to ECE 438 Fall 2010 Lab Wiki Page
Back to ECE 438 Fall 2010
