Whether the z-transform of a signal exists depends on the
complex variable as well as the signal itself. exists if
and only if the argument is inside the *region of convergence
(ROC)* in the z-plane, which is composed of all values for the
summation of the Z-transform to converge. The ROC of the Z-transform is
determined by
(a circle), the magnitude of variable
, while the ROC for the Laplace transform is determined by ,
(a vertical line), the real part of . This formula is always needed in
the examples:

**Example 1:** The Z transform of a right sided signal is

For this summation to converge, i.e., for to exist, it is necessary to have , i.e., the ROC is . As a special case when , and we have

**Example 2:** The Z-transform of a left sided signal
is:

For the summation above to converge, it is required that , i.e., the ROC is . Comparing the two examples above we see that two different signals can have identical z-transform, but with different ROCs.

**Example 3: ** Find the inverse of the given z-transform
.
Comparing this with the definition of z-transform:

we get

In general, we can use the time shifting property

to inverse transform the given above to directly.

**Example 4: ** Sometimes the inverse transform of a given can be
obtained by long division.

By a long division, we get

which converges if the ROC is , i.e., and we get

. Alternatively, the long division can also be carried out as:

which converges if the ROC is , i.e., and we get