Signals and Systems - Electrical Engineering

9.5 One-Sided Z-Transform Inverse 549

− 1 −0.5^0 0.5^1

−1.5

− 1

−0.5

0

0.5

1

1.5

Real part

Imaginary part

0 5 10

(a) (b)

15 20

0

0.2

0.4

0.6

0.8

1

n

x[

n],

x^1

[n

]

FIGURE 9.9
(a) Poles and zeros ofX(z)and (b) inverse Z-transformsx[n]andx 1 [n]found usingfilterand the residues.

(2) Multiple Poles
Whenever multiple poles are present one has to be careful in interpreting the MATLAB results. First,
usehelpto get more information onresiduezand how the partial fraction expansion for multiple
poles is done. Notice from the help file that the residues are ordered the same way the poles are.
Furthermore, the residues corresponding to the multiple poles are ordered from the lowest to the
highest order. Also notice the difference between the partial fraction expansion of MATLAB and ours.
For instance, consider the Z-transform

X(z)=

az−^1

( 1 −az−^1 )^2

|z|>a

with inversex[n]=nanu[n]. Writing the partial fraction expansion as MATLAB does gives

X(z)=

r 1

1 −az−^1

+

r 2

( 1 −az−^1 )^2

r 1 =−1,r 2 = 1 (9.33)