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The z-Transform
Introduction
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Content Introduction
z-Transform Zeros and Poles
Region of Convergence
Importantz
-Transform Pairs Inverse z-Transform
z-Transform Theorems and Properties
System Function
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Why z-Transform? A generalization of Fourier transform
Why generalize it? FT does not convergeon all sequence
Notationgood for analysis
Bring the power of complex variable theorydeal with
the discrete-time signals and systems
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The z-Transform
z-Transform
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Definition Thez-transform of sequencex(n) is defined by
n
nznxzX )()(
Letz = ej.
( ) ( )j j n
n
X e x n e
Fourier
Transform
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z-Plane
Re
Im
z = ej
n
nznxzX )()(
( ) ( )
j j n
nX e x n e
Fourier Transform is to evaluate z-transform
on a unit circle.
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z-Plane
Re
Im
X(z)
Re
Im
z = ej
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Periodic Property of FT
Re
Im
X(z)
X(ej)
Can you say why Fourier Transform is
a periodic function with period 2?
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The z-Transform
Zeros and Poles
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Definition Give a sequence, the set of values ofzfor which the
z-transform converges, i.e., |X(z)|
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Example: Region of Convergence
Re
Im
n
n
n
n znxznxzX |||)(|)(|)(|
ROC is an annual ring centered
on the origin.
xx RzR ||r
}|{ xx
j RrRrezROC
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Stable Systems
Re
Im
1
A stable system requires that its Fourier transformis
uniformly convergent.
Fact: Fourier transform is to
evaluatez-transform on a unit
circle.
A stable system requires the
ROC ofz-transform to include
the unit circle.
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Example: A right sided Sequence
)()( nuanx n
1 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8
n
x(n)
. . .
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Example: A right sided Sequence
)()( nuanx n
n
n
n znuazX
)()(
0n
nnza
0
1)(
n
naz
For convergence ofX(z), we
require that
0
1 ||n
az 1|| 1 az
|||| az
az
z
azazzX
n
n
10
1
1
1)()(
|||| az
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aa
Example: A right sided Sequence
ROC forx(n)=anu(n)
||||,)( azaz
zzX
Re
Im
1 aaRe
Im
1
Which one is stable?
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Example: A left sided Sequence
)1()( nuanx n
1 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8
n
x(n)
...
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Example: A left sided Sequence
)1()( nuanx n
n
n
n znuazX
)1()(
For convergence ofX(z), we
require that
0
1 ||n
za 1|| 1 za
|||| az
az
z
zazazX
n
n
10
1
1
11)(1)(
|||| az
n
n
nza
1
n
n
nza
1
n
n
nza
0
1
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aa
Example: A left sided Sequence
ROC forx(n)=anu(n1)
||||,)( azaz
zzX
Re
Im
1 aaRe
Im
1
Which one is stable?
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The z-Transform
Region of
Convergence
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Represent z-transform as a
Rational Function
)(
)(
)( zQ
zP
zX
whereP(z)and Q(z)are
polynomials inz.
Zeros:The values ofzs such thatX(z) = 0
Poles:The values ofzs such thatX(z) =
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Example: A right sided Sequence
)()( nuanx n ||||,)( az
az
zzX
Re
Im
a
ROC isbounded by the
poleand is the exteriorof a circle.
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Example: A left sided Sequence
)1()( nuanx n ||||,)( az
az
zzX
Re
Im
a
ROC isbounded by the
poleand is the interiorof a circle.
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Example: Sum of Two Right Sided Sequences
)()()()()(31
21 nununx nn
31
21
)(
z
z
z
zzX
Re
Im
1/2
))((
)(2
31
21
121
zz
zz
1/3
1/12
ROC isbounded by poles
and is the exterior of a circle.
ROC does not include any pole.
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Example: A Two Sided Sequence
)1()()()()(21
31 nununx nn
21
31
)(
z
z
z
zzX
Re
Im
1/2
))((
)(2
21
31
121
zz
zz
1/3
1/12
ROC isbounded by poles
and is a ring.
ROC does not include any pole.
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Example: A Finite Sequence
10,)( Nnanx n
nN
n
nN
n
n zazazX )()( 11
0
1
0
Re
Im
ROC: 0
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Properties of ROC
A ringor diskin the z-plane centered at the origin.
The Fourier Transform ofx(n) is converge absolutely iff the ROC
includes the unit circle. The ROC cannot include any poles
Finite Duration Sequences:The ROC is the entirez-plane except
possiblyz=0 orz=.
Right sided sequences:The ROC extends outward from the outermost
finite pole inX(z) toz=.
Left sided sequences:The ROC extends inward from the innermost
nonzero pole inX(z) toz=0.
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More on Rational z-Transform
Re
Im
a b c
Consider the rationalz-transform
with the pole pattern:
Find the possibleROCs
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More on Rational z-Transform
Re
Im
a b c
Consider the rationalz-transform
with the pole pattern:
Case 1: A right sided Sequence.
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More on Rational z-Transform
Re
Im
a b c
Consider the rationalz-transform
with the pole pattern:
Case 2: A left sided Sequence.
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More on Rational z-Transform
Re
Im
a b c
Consider the rationalz-transform
with the pole pattern:
Case 3: A two sided Sequence.
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More on Rational z-Transform
Re
Im
a b c
Consider the rationalz-transform
with the pole pattern:
Case 4: Another two sided Sequence.
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The z-Transform
Important
z-Transform Pairs
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Z-Transform Pairs
Sequence z-Transform ROC
)(n 1 Allz
)( mn mz Allz except 0 (if m>0)or (if m
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Z-Transform Pairs
Sequence z-Transform ROC
)(][cos 0 nun 210
1
0
]cos2[1
][cos1
zz
z1|| z
)(][sin 0 nun 210
1
0
]cos2[1
][sin
zz
z1|| z
)(]cos[ 0 nunrn 221
0
1
0
]cos2[1
]cos[1
zrzr
zrrz||
)(]sin[ 0 nunrn 221
0
1
0
]cos2[1
]sin[
zrzr
zrrz||
otherwise0
10 Nnan
11
1
az
za NN
0|| z
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The z-Transform
Inverse z-Transform
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The z-Transform
z-Transform Theorems
and Properties
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LinearityxRzzXnx ),()]([Z
yRzzYny ),()]([Z
yx RRzzbYzaXnbynax ),()()]()([Z
Overlay of
the above two
ROCs
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ShiftxRzzXnx ),()]([Z
x
nRzzXznnx )()]([ 00Z
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Multiplication by an Exponential Sequence
xx- RzRzXnx ||),()]([Z
x
n RazzaXnxa ||)()]([ 1Z
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Differentiation of X(z)
xRzzXnx ),()]([Z
xRzdz
zdXznnx
)()]([Z
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ConjugationxRzzXnx ),()]([Z
xRzzXnx *)(*)](*[Z
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ReversalxRzzXnx ),()]([Z
xRzzXnx /1)()]([1 Z
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Real and Imaginary Parts
xRzzXnx ),()]([Z
xRzzXzXnxe *)](*)([)]([ 21R
xj RzzXzXnx *)](*)([)]([
21Im
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Initial Value Theorem
0for,0)( nnx
)(lim)0( zXxz
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Convolution of Sequences
xRzzXnx ),()]([Z
yRzzYny ),()]([Z
yx RRzzYzXnynx )()()](*)([Z
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Convolution of Sequences
k
knykxnynx )()()(*)(
n
n
k
zknykxnynx )()()](*)([Z
k
n
n
zknykx )()(
k
n
n
k znyzkx )()(
)()( zYzX
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The z-Transform
System Function
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Shift-Invariant System
h(n)
x(n) y(n)=x(n)*h(n)
X(z) Y(z)=X(z)H(z)H(z)
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Shift-Invariant System
H(z)
X(z) Y(z)
)(
)()(
zX
zYzH
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N
th
-Order Difference Equation
M
r
r
N
k
k rnxbknya00
)()(
M
r
r
r
N
k
k
k zbzXzazY00
)()(
N
k
k
k
M
r
r
r zazbzH00
)(
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Representation in Factored Form
N
k
r
M
r
r
zd
zcA
zH
1
1
1
1
)1(
)1(
)(
Contributespolesat 0 andzerosat cr
Contributeszerosat 0 andpolesat dr
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Stable and Causal Systems
N
k
r
M
r
r
zd
zcA
zH
1
1
1
1
)1(
)1(
)( Re
Im
Causal Systems : ROC extends outward from the outermost pole.
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Stable and Causal Systems
N
k
r
M
r
r
zd
zcA
zH
1
1
1
1
)1(
)1(
)( Re
Im
Stable Systems : ROC includes the unit circle.
1
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ExampleConsider the causal system characterized by
)()1()( nxnayny
1
1
1)(
az
zH
Re
Im
1
a
)()( nuanh n
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Determination of Frequency Response
from pole-zero pattern
A LTI system is completely characterized by its
pole-zero pattern.
))((
)(21
1
pzpz
zzzH
Example:
))(()(
21
1
00
0
0
pepe
zeeH
jj
jj
0je
Re
Im
z1
p1
p2
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Determination of Frequency Response
from pole-zero pattern
A LTI system is completely characterized by its
pole-zero pattern.
))((
)(21
1
pzpz
zzzH
Example:
))(()(
21
1
00
0
0
pepe
zeeH
jj
jj
0je
Re
Im
z1
p1
p2
|H(ej)|=? H(ej)=?
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Determination of Frequency Response
from pole-zero pattern
A LTI system is completely characterized by its
pole-zero pattern.
Example:
0je
Re
Im
z1
p1
p2
|H(ej)|=? H(ej)=?
|H(ej)| =| |
| | | | 1
2
3
H(ej) = 1(2+ 3 )
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Example1
1
1)(
az
zH
Re
Im
a
0 2 4 6 8-10
0
10
20
0 2 4 6 8-2
-1
0
1
2
dB
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