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 $.
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