The Region of Convergence has a number of properties that are
dependent on the characteristics of the signal,
xn
x
n
.

The ROC cannot contain any poles.
By definition a pole is a where
Xz
X
z
is infinite.
Since
Xz
X
z
must be finite for all zz for
convergence, there cannot be a pole in the ROC.

If
xn
x
n
is a finiteduration sequence, then the ROC is the
entire zplane, except possibly
z=0
z
0
or
z=∞
z
.
A finiteduration sequence is a sequence that
is nonzero in a finite interval
n
1
≤n≤
n
2
n
1
n
n
2
.
As long as each value of
xn
x
n
is finite then the sequence will be absolutely summable.
When
n
2
>0
n
2
0
there will be a
z1
z
term and thus the ROC will not include
z=0
z
0
.
When
n
1
<0
n
1
0
then the sum will be infinite and thus the ROC will not
include
z=∞
z
.
On the other hand, when
n
2
≤0
n
2
0
then the ROC will include
z=0
z
0
,
and when
n
1
≥0
n
1
0
the ROC will include
z=∞
z
.
With these constraints, the only signal, then, whose ROC
is the entire zplane is
xn=cδn
x
n
c
δ
n
.
The next properties apply to infinite duration sequences. As
noted above, the ztransform converges when
Xz<∞
X
z
.
So we can write
Xz=∑
n
=−∞∞xnz−n≤∑
n
=−∞∞xnz−n=∑
n
=−∞∞xnz−n
X
z
n
x
n
z
n
n
x
n
z
n
n
x
n
z
n
(3)
We can then split the infinite sum into positivetime and
negativetime portions. So
Xz≤Nz+Pz
X
z
N
z
P
z
(4)
where
Nz=∑
n
=−∞1xnz−n
N
z
n
1
x
n
z
n
(5)
and
Pz=∑
n
=0∞xnz−n
P
z
n
0
x
n
z
n
(6)
In order for
Xz
X
z
to be finite,
xn
x
n
must be bounded. Let us then set
xn≤
C
1
r
1
n
x
n
C
1
r
1
n
(7)
for
n<0
n
0
and
xn≤
C
2
r
2
n
x
n
C
2
r
2
n
(8)
for
n≥0
n
0
From this some further properties can be derived:

If
xn
x
n
is a leftsided sequence, then the ROC extends inward
from the innermost pole in
Xz
X
z
.
A leftsided sequence is a sequence where
xn=0
x
n
0
for
n>
n
2
>−∞
n
n
2
.
Looking at the negativetime portion from the above
derivation, it follows that
Nz≤
C
1
∑
n
=−∞1
r
1
nz−n=
C
1
∑
n
=−∞1
r
1
zn=
C
1
∑
k
=1∞z
r
1
k
N
z
C
1
n
1
r
1
n
z
n
C
1
n
1
r
1
z
n
C
1
k
1
z
r
1
k
(10)
Thus in order for this sum to converge,
z<
r
1
z
r
1
,
and therefore the ROC of a leftsided sequence is of the
form
z<
r
1
z
r
1
.

If
xn
x
n
is a twosided sequence, the ROC will be a ring in the
zplane that is bounded on the interior and exterior by
a pole.
A twosided sequence is an sequence with
infinite duration in the positive and negative
directions. From the derivation of the above two
properties, it follows that if
r
2
<z<
r
2
r
2
z
r
2
converges, then both the positivetime and negativetime
portions converge and thus
Xz
X
z
converges as well. Therefore the ROC of a twosided
sequence is of the form
r
2
<z<
r
2
r
2
z
r
2
.
"My introduction to signal processing course at Rice University."