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114 Chapter 4 THE z-TRANSFORM
Im{z}
Im{z}
Im{z}
0 1/3 1
Re{z}
1/3 1
Re{z}
1/3 1
Re{z}
ROC 1
ROC 2
ROC 3
FIGURE 4.5 The ROCs in Example 4.7
a. ROC 1:1< |z| < ∞. Here both poles are on the interior side of the ROC 1;
that is, |z 1|≤R x− =1and |z 2|≤1. Hence from (4.15)
x 1(n) = 1 2 u(n) − 1 ( ) 1 n
u(n)
2 3
which is a right-sided sequence.
b. ROC 2:0< |z| < 1 . Here both poles are on the exterior side of the ROC2;
3
that is, |z 1|≥R x+ = 1 and |z2| ≥ 1 . Hence from (4.15)
3 3
x 2(n) = 1 2 {−u(−n − 1)}− 1 { ( − 1
) n }
3 u(−n − 1)
2
= 1 2
( 1
3
) n
u(−n − 1) − 1 u(−n − 1)
2
which is a left-sided sequence.
c. ROC 3: 1 < |z| < 1. Here pole z1 is on the exterior side of the ROC3—that
3
is, |z 1|≥R x+ = 1—while pole z 2 is on the interior side—that is, |z 2|≤ 1 . 3
Hence from (4.15)
x 3(n) =− 1 2 u(−n − 1) − 1 2
( 1
3
) n
u(n)
which is a two-sided sequence.
□
4.3.1 MATLAB IMPLEMENTATION
AMATLAB function residuez is available to compute the residue part
and the direct (or polynomial) terms of a rational function in z −1 . Let
X(z) = b 0 + b 1 z −1 + ···+ b M z −M B(z)
a 0 + a 1 z −1 =
+ ···+ a N z−N A(z)
=
N∑
M−N
R k
1 − p k z −1 + ∑
C k z −k
k=0
} {{ }
M≥N
k=1
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