The ZT doesn’t converge for all sequences. When it does converge, it’s only over a region of the z-plane. The values in the z-plane for which the ZT converges are known as the region of convergence (ROC).
Convergence of the ZT requires that
![image0.jpg](https://www.dummies.com/wp-content/uploads/405298.image0.jpg)
The right side of this equation shows that x[n]r-n is absolutely summable (the sum of all terms |x[n]r–n| is less than infinity). This condition is consistent with the absolute summability condition for the DTFT to converge to a continuous function of
![image1.jpg](https://www.dummies.com/wp-content/uploads/405299.image1.jpg)
Convergence depends only on |z| = r, so if the series converges for z = z1, then the ROC also contains the circle |z| = |z1|. In this case, the general ROC is an annular region in the z-plane, as shown. If the ROC contains the unit circle, the DTFT exists because the DTFT is the ZT evaluated on the unit circle.
![image2.jpg](https://www.dummies.com/wp-content/uploads/405300.image2.jpg)
The ROC has important implications when you’re working with the ZT, especially the two-sided ZT. When the ZT produces a rational function, for instance, the roots of the denominator polynomial are related to the ROC. And for LTI systems having a rational ZT, the ROC is related to a system’s bounded-input bounded-output (BIBO) stability. The uniqueness of the ZT is also ensured by the ROC.
Consider the right-sided sequence
![image3.jpg](https://www.dummies.com/wp-content/uploads/405301.image3.jpg)
The term right-sided means that the sequence is 0 for
![image4.jpg](https://www.dummies.com/wp-content/uploads/405302.image4.jpg)
and the nonzero values extend from n0 to infinity. The value of n0 may be positive or negative.
To find X(z) and the ROC, follow these steps:
Reference the definition to determine the sum:
Find the condition for convergence by summing the infinite geometric series:
Thus, the ROC is |z| > |a|.
To find the sum of Step 1 in closed form, use the finite geometric series sum formula:
The geometric series convergence condition corresponds with the ROC.
Consider the left-sided sequence xb[n] = –anu[–n–1]. The term left-sided means that the sequence is 0 for
![image8.jpg](https://www.dummies.com/wp-content/uploads/405306.image8.jpg)
To find X(z) and the ROC, write the definition and the infinite geometric series:
![image9.jpg](https://www.dummies.com/wp-content/uploads/405307.image9.jpg)
If the sum looks unfamiliar in its present form, you can change variables in the sum, an action known as re-indexing the sum, by letting m = –n, then in the limits
![image10.jpg](https://www.dummies.com/wp-content/uploads/405308.image10.jpg)
![image11.jpg](https://www.dummies.com/wp-content/uploads/405309.image11.jpg)