Mathematics for Electrical Engineering and Computing

Using partial fractions to find the inverse transform

Example 15.29

Find

$Z^{-1}\left\{\frac{z^2}{(z-1)(z-0.5)}\right\}.$

Solution Notice that most of the values of the transform in Table 15.2 have a factor of z in the numerator. We write

$\frac{z^2}{(z-1)(z-0.5)}=z\left(\frac{z}{(z-1)(z-0.5)}\right)$

We use the 'cover up’ rule to write

$\begin{array}{c}\frac{z}{(z-1)(z-0.5)}=\frac{1}{0.5(z-1)}-\frac{1}{z-0.5}\\ =\frac{2}{z-1}-\frac{1}{z-0.5}\end{array}$

So

$\frac{z^2}{(z-1)(z-0.5)}=\frac{2z}{z-1}-\frac{z}{z-0.5}$

and using ...