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106 Chapter 4 THE z-TRANSFORM
Note: If b = a in this example, then X 2(z) =X 1(z) except for their respective
ROCs; that is, ROC 1 ≠ROC 2. This implies that the ROC is a distinguishing
feature that guarantees the uniqueness of the z-transform. Hence it plays a very
important role in system analysis.
□
□ EXAMPLE 4.3 Let x 3(n) =x 1(n)+x 2(n) =a n u(n) − b n u(−n − 1) (This sequence is called a
two-sided sequence.) Then using the preceding two examples,
∞∑
−1
∑
X 3(z) = a n z −n − b n z −n
=
n=0
−∞
{ z
z − a , ROC1: |z| > |a| }
+
= z
z − a +
z
z − b ;
{ z
z − b , ROC1: |z| < |b| }
ROC3: ROC1 ∩ ROC2
If |b| < |a|, than ROC 3 is a null space, and X 3(z) does not exist. If |a| < |b|,
then the ROC 3 is |a| < |z| < |b|, and X 3(z) exists in this region as shown in
Figure 4.4.
□
4.1.1 PROPERTIES OF THE ROC
From the observation of the ROCs in the preceding three examples, we
state the following properties.
1. The ROC is always bounded by a circle since the convergence
condition is on the magnitude |z|.
2. The sequence x 1 (n) =a n u(n)inExample 4.1 is a special case of a rightsided
sequence, defined as a sequence x(n) that is zero for some n<
n 0 .From Example 4.1, the ROC for right-sided sequences is always
outside of a circle of radius R x− .Ifn 0 ≥ 0, then the right-sided
sequence is also called a causal sequence.
3. The sequence x 2 (n) =−b n u(−n−1) in Example 4.2 is a special case of a
left-sided sequence, defined as a sequence x(n) that is zero for some n>
n 0 .Ifn 0 ≤ 0, the resulting sequence is called an anticausal sequence.
From Example 4.2, the ROC for left-sided sequences is always inside
of a circle of radius R x+ .
Im{z}
Im{z}
a
0
b
a > b
Re{z}
0
a
b
a < b
Re{z}
FIGURE 4.4 The ROC in Example 4.3
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