I have the following equation :
$X(z) = \frac{1-z^{-1}}{1-0.25z^{-1}}$
The question I must answer is pretty simple : I need to find the inverted z transform and then create a graph for that.
The only additional information I have is that I "inspire" myself (actual translated words) from the z transform equation $X(z) = \sum_{-\infty}^\infty x[n]z^{-n}$
I do not know how to start from there and any help in the right direction would be appreciated.
Thank you for your help.
You can find the inverse $z$-transform by using linearity. Note that (writing $a = 0.25$), we have $$X(z) = \frac{1}{1- az^{-1}} - \frac{z^{-1}}{1 - az^{-1}}.$$ So if $x_{1}[n]$ is the inverse $z$-transform of $\frac{1}{1- az^{-1}}$ and $x_{2}[n]$ is that of $\frac{z^{-1}}{1- az^{-1}}$, then your answer is $x[n] = x_{1}[n] - x_{2}[n]$. You should either know or be able to look up the inverse $z$-transform of $\frac{1}{1- az^{-1}} $ (it can be found in this table). To get the inverse $z$-transform of $\frac{z^{-1}}{1- az^{-1}} $, use the "time delay" property displayed in the table here: https://en.wikipedia.org/wiki/Z-transform#Properties.