Editing Z-transform (section)

===Example 3 (anti causal ROC)===
[[Image:Region of convergence 0.5 anticausal.svg|thumb|250px|ROC (blue), {{pipe}}''z''{{pipe}}&nbsp;=&nbsp;.5 (dashed black circle), and the unit circle (dotted grey circle).]]

Let <math>x[n] = -(.5)^n \, u[-n-1] </math> (where <math>u</math> is the [[Heaviside step function]]). Expanding <math>x[n]</math> on the interval <math>(-\infty, \infty)</math> it becomes

:<math>x[n] = \left \{ \dots, -(.5)^{-3}, -(.5)^{-2}, -(.5)^{-1}, 0, 0, 0, 0, \dots \right \}.</math>

Looking at the sum

:<math>\begin{align}
\sum_{n=-\infty}^{\infty}x[n] \, z^{-n} &= -\sum_{n=-\infty}^{-1}(.5)^n \, z^{-n} \\
&= -\sum_{m=1}^{\infty}\left(\frac{z}{.5}\right)^{m} \\
&= -\frac{(.5)^{-1}z}{1 - (.5)^{-1}z} \\
&= -\frac{1}{(.5)z^{-1}-1} \\
&= \frac{1}{1 - (.5)z^{-1}} \\
\end{align}</math>

and using the infinite [[geometric series]] again, the equality only holds if <math>|(.5)^{-1} z| < 1</math> which can be rewritten in terms of <math>z</math> as <math>|z| < (.5).</math> Thus, the ROC is <math>|z| < (.5).</math> In this case the ROC is a disc centered at the origin and of radius 0.5.

What differentiates this example from the previous example is ''only'' the ROC. This is intentional to demonstrate that the transform result alone is insufficient.
{{Clear}}