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8/11/2019 Z Transform and It's Applications
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Chapter 4
Z Transform
8/11/2019 Z Transform and It's Applications
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Analog signal is sampled at every T sec with an ON period of t
Definition
Since s = +j, and a system become stable for 0,z =esT= e(+j)T= eT ej
So, the magnitude of z is eT and the angle is
Hence in terms of z-transform the system will be stable for |z| 1
n
tnTtnxtx )()()(
n
snT
n
snT enxtdttenTtnxsX )()()()(0
sTez
)()()( ztXznxtsXn
n
n
nznxzX )()(
8/11/2019 Z Transform and It's Applications
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Elementary signals
Unit step function 01)( nnx
1;11
1.............11)(
1
21
0
zz
z
zzzzzX
n
n
Power function 0)( nanx n
azaz
zaz
zaazzazX nn
n
;
11...........1)( 1
221
0
Ramp function 0)( nnnx
1||;)1()1(
.........320)(221
1321
0
z
z
z
z
zzzzznzX n
n
The values ofz for whichX(z)is finite are known as region of
convergence (ROC)
Try other signals: impulse function, .
8/11/2019 Z Transform and It's Applications
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Significance of ROC
For causal sequence
azaz
zaz
zazazX n
n
n
n
n
;1
1)()(1
1
00
000)( nforandnforanx n
For Anti causal
sequence
000)( nforaandnfornx n
azaz
zzazazX n
n
n
n
n
;)()(
1 11
The two sequences have sameX(z)but their ROC is different. Without ROC we can
not uniquely determine the sequencex(n). Generally, for causal sequence, the ROCis exterior of the circle having radius a and for anti causal sequence it is interior of
the circle.
FindX(z)and ROC for x(n) = nu(n) + nu(-n-1)
Answer
zROCzzzX :)( 1
8/11/2019 Z Transform and It's Applications
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Properties
)()()(
)()()()()()( 2211221
zHzXzY
zXzknxzXazXanxanax
k
8/11/2019 Z Transform and It's Applications
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LTI System
)(......)1()()(......)1()(1010
MnxbnxbnxbNnyanyanyaMN
N
N
M
M
zazaa
zbzbb
zX
zYzH
..........
..........
)(
)()(
1
10
1
10
N
NN
M
MMMN
azaza
bzbzbz
..........
..........1
10
1
10
kN
k
k
M
k
k
k
za
zb
zH
0
0)(
)(......)()()(......)()( 1
10
1
10 zXzbzXzbzXbzYzazYzazYa M
M
N
N
)(]......[)(]......[ 1101
10 zXzbzbbzYzazaa M
M
N
N
).().........)((
).().........)((
21
21
N
MMN
pzpzpz
zzzzzzz
z1, z2, . . . zM zeros
p1, p2, . . . pN poles
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LTI System Stability
LTI DT causal system is BIBO stable provided that all poles of
the system transfer function lie inside the unit circle in Z-plane
8/11/2019 Z Transform and It's Applications
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Inverse z-transform
Values of x(n) are the coefficients of z-nand can be obtained by direct inspection of
X(z). Normally X(z) is often expressed as a ratio of two polynomials in z-1or in z
..........)2()1()0()()( 21
0
zxzxxznxzXn
n
MM
NN
zbzbb
zazaazX
_110
1
10
............
...........)(
Three Methods
Power series expansion method
Partial fraction expansion methodResidue method / Contour integration
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IZT: Partial Fraction Expansion Method
M < NProper
rational function
M NImproperrational function
1,...........1
............
)(
)()( 01
1
1
10
azaza
zbzbb
zD
zNzH
N
N
M
M
N
N
pz
A
pz
A
pz
A
z
zH
..............
)(
2
2
1
1Let all poles are
distinct
Nkz
zHpzAKpz
KK ..........3,2,1,
)()(
21 5.05.11
1)(
zzz
)(})5.0()1(2{)(
5.0
1
1
2)(
1)5.0)(1(
)5.0(2
)5.0)(1(
)1(
5.01)5.0)(1(
)(
5.
2
1
1
21
nunh
zzz
zH
zz
zzA
zz
zzA
z
A
z
A
zz
z
z
zH
nn
zz
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IZT: Multiple-order poles
2
2
211 )1)(1(
)(
)1)(1(
1)(
zz
z
z
zH
zzzH
)1..(....................)1(11)1)(1(
)( 2321
2
2
zA
zA
zA
zzz
zzH
4
1)()1(
1
1 zz
zHzA
2
1)()1(
1
2
3 zz
zHzA
)2(....................)1()1(
)1()()1( 321
22 AAzA
z
z
z
zHz
4
3
1
)()1(
1
2
1
2
2
zz z
z
dz
d
z
zHz
dz
dATo find A2, differentiate (2)
with respect to z and put z=1
2)1(2
1
14
3
14
1)(
z
z
z
z
z
zzH
)(
2
1)1(
4
3)1(
4
1)( nunnh nn
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Thank you