Editing Z-transform (section)

===Examples conclusion===
Examples 2 & 3 clearly show that the Z-transform <math>X(z)</math> of <math>x[n]</math> is unique when and only when specifying the ROC. Creating the [[pole–zero plot]] for the causal and anticausal case show that the ROC for either case does not include the pole that is at 0.5. This extends to cases with multiple poles: the ROC will ''never'' contain poles.

In example 2, the causal system yields a ROC that includes <math>|z| = \infty</math> while the anticausal system in example 3 yields an ROC that includes <math>|z| = 0 .</math>

[[Image:Region of convergence 0.5 0.75 mixed-causal.svg|thumb|250px|ROC shown as a blue ring 0.5&nbsp;<&nbsp;{{pipe}}''z''{{pipe}}&nbsp;<&nbsp;0.75]]
In systems with multiple poles it is possible to have a ROC that includes neither <math>|z| = \infty</math> nor <math>|z| = 0 .</math> The ROC creates a circular band. For example,

:<math>x[n] = (.5)^n \, u[n] - (.75)^n \, u[-n-1]</math>

has poles at 0.5 and 0.75. The ROC will be 0.5 < {{abs|''z''}} < 0.75, which includes neither the origin nor infinity. Such a system is called a mixed-causality system as it contains a causal term <math>(.5)^n \, u[n]</math> and an anticausal term <math>-(.75)^n \, u[-n-1] .</math>

The [[Control theory#Stability|stability]] of a system can also be determined by knowing the ROC alone. If the ROC contains the unit circle (i.e., {{abs|''z''}} = 1) then the system is stable. In the above systems the causal system (Example 2) is stable because {{abs|''z''}} > 0.5 contains the unit circle.

Let us assume we are provided a Z-transform of a system without a ROC (i.e., an ambiguous <math>x[n]</math>). We can determine a unique <math>x[n]</math> provided we desire the following:
* Stability
* Causality

For stability the ROC must contain the unit circle. If we need a causal system then the ROC must contain infinity and the system function will be a right-sided sequence. If we need an anticausal system then the ROC must contain the origin and the system function will be a left-sided sequence. If we need both stability and causality, all the poles of the system function must be inside the unit circle.

The unique <math>x[n]</math> can then be found.