Editing Z-transform (section)

==Region of convergence==
{{See also|Pole–zero_plot#Discrete-time systems}}
The [[Radius of convergence|region of convergence]] (ROC) is the set of points in the complex plane for which the Z-transform summation [[Convergent series|converges]] (i.e. doesn't blow up in magnitude to infinity):

:<math>\mathrm{ROC} = \left\{ z : \left|\sum_{n=-\infty}^{\infty}x[n]z^{-n}\right| < \infty \right\} </math>

===Example 1 (no ROC)===
Let <math>x[n] = (.5)^n\ . </math> 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}, 1, (.5), (.5)^2, (.5)^3, \dots \right \} = \left \{\dots, 2^3, 2^2, 2, 1, (.5), (.5)^2, (.5)^3, \dots \right\}.</math>

Looking at the sum

:<math>\sum_{n=-\infty}^{\infty}x[n]z^{-n} \to \infty.</math>

Therefore, there are no values of <math>z</math> that satisfy this condition.

===Example 2 (causal ROC)===
[[Image:Region of convergence 0.5 causal.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] </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, 0, 0, 0, 1, (.5), (.5)^2, (.5)^3, \dots \right \}.</math>

Looking at the sum

:<math>\sum_{n=-\infty}^{\infty}x[n]z^{-n} = \sum_{n=0}^{\infty}(.5)^nz^{-n} = \sum_{n=0}^{\infty}\left(\frac{.5}{z}\right)^n = \frac{1}{1 - (.5)z^{-1}}.</math>

The last equality arises from the infinite [[geometric series]] and the equality only holds if <math>|(.5)z^{-1}| < 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 the complex plane with a disc of radius 0.5 at the origin "punched out".{{clear}}

===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}}

===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.