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Chapter 10
PARTIAL FRACTIONS
Copyright c
°
1996 by Ali H. Sayed. All rights reserved. These notes are distributed only to the
students attending the undergraduate DSP course EE113 in the Electrical Engineering Department
at UCLA. The notes cannot be reproduced without written consent from the instructor: Prof. A.
H. Sayed, Electrical Engineering Department, UCLA, CA 90095, sayed@ee.ucla.edu.
The
z

transform of a sequence can be a rational function of
z
or not. In this chapter,
we describe a useful method for inverse transforming
rational
z

transforms. The procedure
is known as the partial fraction method. The method is best explained by examples. But
let us Frst present some general facts.
Partial Fractions
Given a rational
z

transform
X
(
z
), the method of partial fraction expansion expresses it
as a sum of simpler transforms whose inversions are immediate. The simple transforms are
usually terms of the forms:
A,
A
z

p
,
A
(
z

p
)
2
,...,
for some constants (
A,p
). These terms can also appear multiplied by powers of
z
or
z

1
.
The main reason for choosing to express
X
(
z
) as a sum of such terms is that the inverse
transforms of these terms are readily available.
1. ±or example, for a region of convergence of the form

z

>

p

we can invert every term
as follows:
A
←→
Aδ
(
n
)
A
z

p
←→
Ap
n

1
u
(
n

1)
A
(
z

p
)
2
←→
A
·
(
n

1)
p
n

2
u
(
n

1)
Likewise, for a region of convergence of the form

z

<

p

we can invert every term as
follows:
A
←→
Aδ
(
n
)
A
z

p
←→ 
Ap
n

1
u
(

n
)
A
(
z

p
)
2
←→ 
A
·
(
n

1)
p
n

2
u
(

n
)
94
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2.
Complex terms
. Sometimes, a complex conjugate pair (
p,p
*
) occurs that contributes
with a sum of the form
A
z

p
+
A
*
z

p
*
.
Its inverse
z
transform for a region of convergence of the form

z

>

p

is
[
Ap
n

1
+
A
*
p
*
(
n

1)
]
u
(
n

1)
.
If we express
{
A,p
}
in polar forms, say
A
=

A

e
jα
,
p
=

p

e
jω
then the above result leads to the transform relation
A
z

p
+
A
*
z

p
*
←→
2
·
A
·
p

n

1
·
cos[
ω
(
n

1) +
α
]
u
(
n

1)
,

z

>

p

For a region of convergence of the form

z

<

p

, we obtain
A
z

p
+
A
*
z

p
*
←→ 
2
·
A
·
p

n

1
·
cos[
ω
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