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The Bilateral z-Transform 105
Im{z}
0
a
Re{z}
FIGURE 4.2 The ROC in Example 4.1
□ EXAMPLE 4.1 Let x 1(n) =a n u(n), 0 < |a| < ∞. (This sequence is called a positive-time
sequence). Then
∞∑
∞∑ ( )
X 1(z) = a n z −n a n
∣ 1 ∣∣
=
=
z 1 − az ; if a
∣ < 1
−1 z
Note:
0
0
= z , |z| > |a| ⇒ROC1: |a| < |z| <
z − a }{{} }{{}
∞
R R x− x+
X 1(z) inthis example is a rational function; that is,
X 1(z) = △ B(z)
A(z) = z
z − a
where B(z) =z is the numerator polynomial and A(z) =z−a is the denominator
polynomial. The roots of B(z) are called the zeros of X(z), whereas the roots
of A(z) are called the poles of X(z). In this example X 1(z) has a zero at the
origin z =0andapole at z = a. Hence x 1(n) can also be represented by a
pole-zero diagram in the z-plane in which zeros are denoted by ◦ and poles by
× as shown in Figure 4.2. □
□ EXAMPLE 4.2 Let x 2(n) =−b n u(−n−1), 0 < |b| < ∞. (This sequence is called a negative-time
sequence.) Then
−1
−1
∑ ∑ ( )
X 2(z) =− b n z −n b n
∞∑ ( ) z n ∑ ∞ ( ) z n
= − = − =1−
z
b b
=1−
−∞
−∞
1
1 − z/b = z
z − b , ROC2: 0 }{{}
R x−
< |z| < |b|
}{{}
R x+
The ROC 2 and the pole-zero plot for this x 2(n) are shown in Figure 4.3.
1
0
Im{z}
FIGURE 4.3 The ROC in Example 4.2
0
b
Re{z}
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