Real-time digital signal processing: implementations, ... changes in the input signal is limited by its internal clock rate, so that it may be slow to
Document source : notes.ump.edu.my
g
j
1
m À j3
d
mÀj
dz
mÀj
z À p
l
m
Xz
z
!
zp
l
:
4:2:14
Example 4.8: Consider the function
Xz
z
2
z
z À 1
2
:
We first express X(z) as
Xz
g
1
z À 1
g
2
z À 1
2
:
From (4.2.14),we have
g
1
d
dz
z À 1
2
Xz
z
4
5
z1
d
dz
z 1
z1
1,
g
2
z À 1
2
Xz
z
z1
z 1
j
z1
2:
Thus
Xz
z
z À 1
2z
z À 1
2
:
From Table 4.3,we obtain
xn ZT
À1
z
z À 1
h
i
ZT
À1
2z
z À 1
2
4
5
1 2n, n ! 0:
The residue method is based on Cauchy's integral theorem expressed as
1
2pj
c
z
kÀmÀ1
dz 1 if k m
0 if k T m.
&
4:2:15
Thus the inversion integral in (4.2.8) can be easily evaluated using Cauchy's residue
theorem expressed as
xn
1
2pj
c
Xzz
nÀ1
dz
residues of Xzz
nÀ1
at poles of Xzz
nÀ1
within C:
4:2:16
THE Z-TRANSFORM
139
Summary :
zp l : 4:2:14 Example 4.8: Consider the function Xz z 2 z z À 1 2 : We first express X(z) as Xz g 1 z À 1 g 2 z À 1 2 : From (4.2.14),we have g 1 d dz z À 1 2 Xz z 4 5 z1 d dz z 1 z1 1, g 2 z À 1 2 Xz z z1 z 1 j z1 2: Thus Xz z z À 1 2z z À 1 2 : From Table 4.3,we obtain xn ZT À1 z z À 1 h i ZT À1 2z z À 1 2 4 5 1 2n, n !
Tags :
nà1,xzz,2pj,màj,thus,cauchys,integral,residue,expressed,theorem,kàmà1,amp,43we