The z-Transform - Digital Signal Processing: Fundamentals and Applications - page 150

5.3.1 Partial Fraction Expansion Using MATLAB
The MATLAB function residue() can be applied to perform the partial fraction expansion of a
z-transform function XðzÞ=z . The syntax is given as

½ R ; P ; K ¼ residue ð B ; A Þ

Here, B and A are the vectors consisting of coefficients for the numerator and denominator
polynomials, BðzÞ

and AðzÞ , respectively. Notice that BðzÞ and AðzÞ are the polynomials with

increasing positive powers of z.

AðzÞ ¼ b 0 z M þ b 1 z M 1

þ b 2 z M 2 BðzÞ þ/þ b M

þ/þ a N
The function returns the residues in vector R , corresponding poles in vector P , and polynomial
coefficients (if any) in vector K . The expansion format is shown as

z N þ a 1 z N 1 þ a 2 z 2

BðzÞ
AðzÞ ¼

r 1
z p 1 þ

r 2

z p 2 þ/þ k 0 þ k 1 z 1

þ/

For a pole p j of multiplicity m , the partial fraction includes the following terms:
BðzÞ
AðzÞ ¼ /þ

r jþ 1
ðz p j Þ

r j
z p j þ

r jþm

ðz p j Þ m þ/þ k 0 þ k 1 z 1

2 þ/þ

þ/

EXAMPLE 5.12
Find the partial expansion for each of the following z-transform functions:

1
ð1 z 1 Þð1 0:5z 1 Þ

a. X ðzÞ¼

z 2 ðz þ 1Þ
ðz 1Þðz 2 z þ 0:5Þ

b. Y ðzÞ¼

z 2
ðz 1Þðz 0:5Þ

c. X ðzÞ¼

2

Solution:
a. From MATLAB, we can show the denominator polynomial as
» conv([1 L 1],[1 L 0.5])
D [
1.0000 L 1.5000 0.5000

This leads to

z 2
z 2 1:5z þ 0:5

1
1 z 1 1 0:5z 1 ¼

1
1 1:5z 1 þ 0:5 2 ¼

X ðzÞ¼

and

X ð z z ¼ z

z 2 1:5z þ 0:5