ECE438 Lab Week10 Quiz Question 4 Solution
a. By computing X(z) and Y(z), we can obtain H(z)=Y(z)/X(z)
$ \begin{align} X(z)=\sum_{n=-\infty}^{\infty}x[n]z^{-n}&=\sum_{n=-\infty}^{\infty}(\frac{1}{2})^nu[n]z^{-n}+\sum_{n=-\infty}^{\infty}2^nu[-n-1]z^{-n} \\ &=\sum_{n=0}^{\infty}(\frac{1}{2})^nz^{-n}+\sum_{n=-\infty}^{-1}2^nz^{-n} \\ &=\sum_{n=0}^{\infty}(\frac{1}{2})^nz^{-n}+\sum_{n=1}^{\infty}2^{-n}z^{n} \\ &=\sum_{n=0}^{\infty}(\frac{1}{2})^nz^{-n}+\sum_{n=0}^{\infty}2^{-n}z^{n}-1 \\ &=\frac{1}{1-\frac{1}{2}z^{-1}}+\frac{1}{1-\frac{z}{2}}-1\text{ ,if }|z|>\frac{1}{2}\text{ and }|z|<2 &=\frac{\frac{3}{4}z^{-1}}{(1-\frac{1}{2}z^{-1})(z^{-1}-\frac{1}{2})}\text{ ,ROC: }\frac{1}{2}<|z|<2 \end{align} $
