Region of convergence
The region of convergence (ROC) is the set of points in the complex plane for which the Z-transform summation converges.
Example 1 (no roc)
Let
.
Expanding
on
the interval
it
becomes
Looking at the sum
Therefore, there are no values of that satisfy this condition.
Example 2 (causal roc)
ROC shown in blue, the unit circle as a dotted grey
circle (appears reddish to the eye) and the circle
is
shown as a dashed black circle
Let
(where
is
the Heaviside
step function). Expanding
on
the interval
it
becomes
Looking at the sum
The
last equality arises from the infinite geometric
series and the equality only
holds if
which
can be rewritten in terms of
as
.
Thus, the ROC is
.
In this case the ROC is the complex plane with a disc of radius 0.5
at the origin "punched out".
Example 3 (anticausal roc)
ROC shown in blue, the unit circle as a dotted grey circle and the circle is shown as a dashed black circle
Let
(where
is
the Heaviside
step function). Expanding
on
the interval
it
becomes
Looking at the sum
Using
the infinite geometric
series, again, the equality
only holds if
which
can be rewritten in terms of
as
.
Thus, the ROC is
.
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.
Examples conclusion
Examples 2 & 3 clearly show that the Z-transform of 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 an ROC that includes
while
the anticausal system in example 3 yields an ROC that includes
.
ROC shown as a blue ring
In
systems with multiple poles it is possible to have an ROC that
includes neither
nor
.
The ROC creates a circular band. For example,
has
poles at 0.5 and 0.75. The ROC will be
,
which includes neither the origin nor infinity. Such a system is
called a mixed-causality
system as it contains a causal
term
and
an anticausal term
.
The
stability
of a system can also be determined by knowing the ROC alone. If the
ROC contains the unit circle (i.e.,
)
then the system is stable. In the above systems the causal system
(Example 2) is stable because
contains
the unit circle.
If you are provided a Z-transform of a system without an ROC (i.e., an ambiguous ) you can determine a unique provided you desire the following:
Stability
Causality
If you need stability then the ROC must contain the unit circle. If you need a causal system then the ROC must contain infinity and the system function will be a right-sided sequence. If you need an anticausal system then the ROC must contain the origin and the system function will be a left-sided sequence. If you need both, stability and causality, all the poles of the system function must be inside the unit circle.
The unique can then be found.