Digital Signal Processing Reference
In-Depth Information
Here we see that for the positive-time sequence, the ROC includes
z
having magnitude greater
than 0.9, whereas for the negative-time sequence,
z
must have magnitude less than 0.8. In other words,
there are no
z
that can satisfy both criteria, and thus the transform is undefined.
Note that the z-transform alone cannot uniquely define the underlying time domain sequence
since, for example, the positive time sequence
0
.
9
n
u
0
.
9
n
u
[
n
]
and the negative-time sequence
−
[−
n
−
1
]
have the same z-transform
1
0
.
9
z
−
1
Therefore, it is necessary to specify both the z-transform and the ROC to uniquely define the
underlying time domain sequence.
1
−
Finite Length Sequence
When the sequence
x
is bounded in magnitude, the ROC is generally the
entire complex plane, possibly excludingz=0and/or z =
[
n
]
is finite in length and
x
[
n
]
∞
.
Example 2.8.
Evaluate the z-transform and state the ROC for the following three sequences:
(a) x[n] = [1,0,-1,0,1] with time indices [0,1,2,3,4].
(b) x[n] = [1,0,-1,0,1] with time indices [-5,-4,-3,-2,-1].
(c) x[n] = [1,0,-1,0,1] with time indices [-2,-1,0,1,2].
For (a) we get
z
−
4
which has its ROC as the entire complex plane except for
z
= 0. For (b) we get
z
0
z
−
2
−
+
z
1
which has its ROC as the entire complex plane except for z =
z
5
z
3
−
+
∞
. For (c) we get
z
−
2
which has its ROC as the entire complex plane except for
z
= 0 and
z
=
z
2
z
0
−
+
∞
.
2.3.4 SUMMARY OF ROC FACTS
•
Since convergence is determined by the magnitude of
z
, ROCs are bounded by circles.
•
For finite sequences that are zero-valued for all
n<
0, the ROC is the entire
z
-plane except
for
z
=0.
•
For finite sequences that are zero-valued for all
n>
0, the ROC is the entire
z
-plane except
for
z
=
∞
.
•
For infinite length sequences that are causal (positive-time or right-handed), the ROC lies
outside a circle having radius equal to the pole of largest magnitude.
•
For infinite length sequences that are anti-causal (negative-time or left-handed), the ROC lies
inside a circle having radius equal to the pole of smallest magnitude.
