Whether the z-transform of a function exists depends on whether
or not the transform summation converges

which in turn depends on the duration and magnitude of as well as the magnitude (the phase of determines the frequency of a sinusoid which is bounded and has no effect on the convergence of the integral).

**Right sided signals: **
may have infinite duration for
, and a value with
tends to attenuate as
.

**Left sided signals: ** may have infinite duration for ,
and a value with
tends to attenuate as
.

Based on these observations, we can get the following properties for the ROC:

- If is of finite duration, then the ROC is the entire z-plane (the
z-transform summation converges, i.e., exists, for any ) except
possibly and/or .
- The ROC of consists of a ring centered about the origin in the z-plane.
The inner boundary can extend inward to the origin in some cases, and the
outer can extend to infinity in other cases.
- If is right sided and the circle is in the ROC, then any
finite for which is also in the ROC.
- If is left sided and the circle is in the ROC, then any
for which is also in the ROC.
- If is two-sided, then the ROC is the intersection of the two one-sided
ROCs corresponding to the two one-sided parts of . This intersection
can be either a ring or an empty set.
- If is rational, then its ROC does not contain any poles (by definition
dose not exist). The ROC is bounded by the poles or
extends to infinity.
- If is a rational z-transform of a right sided function , then
the ROC is the region outside the out-most pole. If for
(causal), then the ROC includes .
- If is a rational z-transform of a left sided function , then
the ROC is inside the innermost pole. If for
(anti-causal), then the ROC includes .
- Fourier transform
of discrete signal exists if the
ROC of the corresponding z-transform contains the unit circle
or .

**Example 1: **

When , for , when , for . Therefore neither nor are included in the ROC.

**Example 2:**

The Z-transform is linear, and is the sum of the transforms for the two terms:

If , i.e., decays when , the intersection of the two ROCs is , and we have:

However, if , i.e., grows without a bound when , the intersection of the two ROCs is a empty set, the Z-transform does not exist.

**Example 3: ** Given the following z-transform, find the corresponding signal:

The two poles are and , respectively. The has three possible ROCs associated with three different time signals :

- The region outside the out-most pole , with the corresponding
right sided time function

- The region inside the innermost pole , with the corresponding
left sided time function

- The ring between the two poles , with the corresponding two
sided time function