What is the inverse Z transform of this:

Question

$X(z) = \displaystyle \frac{1}{z}{\left(1-\frac{z^2}{4}\right)\left(1+\frac{1}{z}\right)\left(1-z\right)}$

Using partial fractions expansion i came up to this:

$\displaystyle \frac{1}{X(z)} = \frac{-\frac{z}{3}}{1-\frac{z}{2}} + \frac{\frac{z}{3}}{1+\frac{z}{2}} + \frac{\frac{2z}{3}}{1-z}$

I tried but i can't go on any further.

Answer 1

You just need to multiply the terms and sort by powers of $z$. If you do so you should get

$$X(z)=\frac{z^2}{4}-\frac54+z^{-2}\tag{1}$$

So the work is in multiplying out the terms, not in actually figuring out the inverse $\mathcal{Z}$-transform, because from $(1)$ you can immediately write down $x[n]$:

$$x[n]=\frac14\delta[n+2]-\frac54\delta[n]+\delta[n-2]\tag{2}$$