region of convergence (ROC) in the z-plane

1. ROC of Z-transform
PREPARED BY
JAMI VENKATA SUMAN
ECE DEPARTMENT
GMRIT

2. The z-transform and the DTFT
• The z-transform is a function of the complex z variable
• Convenient to describe on the complex z-plane
• If we plot z=ej for =0 to 2 we get the unit circle
Re
Im
Unit Circle

r=1
0
2 0 2

 j
eX
2

3. Stability and the ROC
For a > 0: az
az
X(z)nuanx n


 
for
1
1
][][ 1
• If the ROC is
outside the unit
circle, the signal is
unstable.
1for
1
1
][][
1





z
z
X(z)
nunx • If the ROC includes
the unit circle, the
signal is stable.
3

4. For a < 0: az
az
X(z)nuanx n


 
for
1
1
][][ 1
• If the ROC is
outside the unit
circle, the signal is
unstable.
1for
1
1
][][
1





z
z
X(z)
nunx • If the ROC includes
the unit circle, the
signal is stable.
4

5. Example:
If:
The z-Transform is the same, but the region of convergence is different.
   
?
1
)(1
1
]1[)(
signal)sided(left]1[][
1
1
1
1















za
za
zaznuazX
nuanx
n
n
n
nn
n
azza 
or,,11
az
z
az
za
za
zaza
za
za
X(z)




















1
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
5

6. For: az
az
X(z)nuanx n


 
for
1
1
]1[][ 1
• If the ROC includes
the unit circle, the
signal is stable. 1for
1
1
][][
1





z
z
X(z)
nunx • If the ROC includes
the unit circle, the
signal is unstable.
6

7. Two-Sided Exponential Sequence
     1-n-u
2
1
-nu
3
1
nx
nn













11
1
0
1
0n
n
1
z
3
1
1
1
z
3
1
1
z
3
1
z
3
1
z
3
1





























11
0
11
1
n
n
1
z
2
1
1
1
z
2
1
1
z
2
1
z
2
1
z
2
1






























z
3
1
1z
3
1
:ROC 1

 
z
2
1
1z
2
1
:ROC 1


 


























2
1
z
3
1
z
12
1
zz2
z
2
1
1
1
z
3
1
1
1
zX
11
Re
Im
2
1
oo
12
1
xx3
1

7

8. Properties of the ROC
8

9. 1. The ROC is an annular ring in the z-plane centered
about the origin (which is equivalent to a vertical
strip in the s-plane).
2. DTFT exists if and only if the ROC includes the unit circle
3. The ROC does not contain any poles (similar to
the Laplace transform).
4. If x[n] is of finite duration, then the ROC is the entire
z-plane except possibly z = 0 and/or z = :
5. If x[n] is a right-sided sequence, and if |z| = r0 is in the ROC, then all finite values
of z for which |z| > r0 are also in the ROC.
6. If x[n] is a left-sided sequence, and if |z| = r0 is in the ROC, then all finite values of
z for which |z| < r0 are also in the ROC.









)Re()()()(][]1[][
)Re()()()(0][]1[][
1)()()(1][][][
][
1
sesXTttxzROCzzXnnx
sesXTttxzROCzzXnnx
sXttxzallROCzXnnx
znxX(z)
sT
sT
n
n



:CT:DT
9

10. 7. If x[n] is a two-sided sequence, and if |z| = r0 is in the ROC, then the ROC consists
of a ring in the z-plane including |z| = r0.
• Example:
right-sided left-sided two-sided
0][  bbnx
n
bz
bzbbz
zX
b
z
zb
nub
bz
bz
nub
nubnubnx
n
n
nn



















1
1
1
1
1
)(
1
1
1
]1[
1
1
][
][][][
111
11
1
10

11. 8. The ROC must be a connected region
9. If the z-transform of x(n) is rational then its ROC is bounded by poles
or extended to infinity.
10. If the z-transform of x(n) is rational and right sided then ROC is z-
plane outside the outermost pole.
11. If the z-transform of x(n) is rational and left sided then ROC is the
region in z-plane inside the innermost pole.
11

12. Z-Transform
Region of Convergence
|z|<a b<|z| b<|z|<a all |z|
12

13. 13