Post on 05-Apr-2018
7/31/2019 z Transform (1)
1/30
5/21
The z-Transform
Content
z Introduction
z z-Transform
z Zeros and Poles
z Region of Convergence
z mportant z- rans orm a rs
z Inverse z-Transform
z z-Transform Theorems and Properties
z System Function
7/31/2019 z Transform (1)
2/30
5/21
The z-Transform
Introduction
Why z-Transform?
z A generalization of Fourier transform
z Why generalize it?
FT does not converge on all sequence
Notation good for analysis
the discrete-time signals and systems
7/31/2019 z Transform (1)
3/30
5/21
The z-Transform
z-Transform
Definition
z Thez-transform of sequencex(n) is defined by
=
=n
nznxzX )()(
Fourier
Trans orme z = e .
( ) ( )j j n
n
X e x n e
=
=
7/31/2019 z Transform (1)
4/30
5/21
z-Plane
Re
Im
z = ej
=
=n
nznxzX )()(
j j n
n=
Fourier Transform is to evaluate z-transform
on a unit circle.
z-Plane
X(z)
Re
Im
z = ej
Re
Im
7/31/2019 z Transform (1)
5/30
5/21
Periodic Property of FT
X(z)
X(ej)
Re
Im
Can you say why Fourier Transform isa periodic function with period 2?
The z-Transform
7/31/2019 z Transform (1)
6/30
5/21
Definitionz Give a sequence, the set of values ofz for which the
z-transform converges, i.e., |X(z)|
7/31/2019 z Transform (1)
7/30
5/21
Stable Systems
Im
z A stable system requires that its Fourier transform is
uniformly convergent.
z Fact: Fourier transform is to
evaluatez-transform on a unit
Re
1.
z A stable system requires the
ROC ofz-transform to includethe unit circle.
Example: A right sided Sequence
)()( nuanx n=
x(n)
1 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8
n. . .
7/31/2019 z Transform (1)
8/30
5/21
Example: A right sided Sequence
)()( nuanx n=
n
n
n znuazX
== )()(
For convergence ofX(z), we
require that
=
Im Im
Which one is stable?
aaRe
1aa
Re
1
7/31/2019 z Transform (1)
9/30
5/21
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)
...
Example: A left sided Sequence
)1()( = nuanx n
n
n
nznuazX
= = )1()(
For convergence ofX(z), we
require that
7/31/2019 z Transform (1)
10/30
5/21
Example: A left sided Sequence
ROC forx(n)=anu( n1)
||||,)( azaz
zzX
=
Im
ROC is bounded b the
Rea
pole and is the exterior
of a circle.
7/31/2019 z Transform (1)
12/30
5/21
Example: A left sided Sequence
)1()( = nuanx n ||||,)( azaz
zzX 0)
or (ifmz
)1( nu 11 z1||
)1( nuan 111
az|||| az
7/31/2019 z Transform (1)
18/30
5/21
Z-Transform Pairs
Sequence z-Transform ROC
)(][cos0 nun 21
0
1
0
]cos2[1
][cos1
+
zz
z1|| >z
)(][sin 0 nun 210
1
0
]cos2[1
][sin
+
zz
z1|| >z
1]cos[1 zr
)(]cos[0 nunr
n 2210 ]cos2[1
+ zrzrrz >
)(]sin[ 0 nunr
n
2210
1
0
]cos2[1
]sin[
+
zrzr
zrrz >||
otherwise0
10 Nnan
11
1
az
zaNN
0|| >z
The z-Transform
-
7/31/2019 z Transform (1)
19/30
5/21
The z-Transform
z-Transform Theorems
Linearity
xRzzXnx = ),()]([Z
yRzzYny = ),()]([Z
yx RRzzbYzaXnbynax +=+ ),()()]()([Z
Overlay of
the above two
ROCs
7/31/2019 z Transform (1)
20/30
5/21
ShiftxRzzXnx = ),()]([Z
x
n
RzzXznnx =+ )()]([0
0Z
Multiplication by an Exponential Sequence
+
7/31/2019 z Transform (1)
21/30
5/21
Differentiation of X(z)
xRzzXnx = ),()]([Z
xRzdz
zdXznnx = )()]([Z
Conjugation
xRzzXnx = ),()]([Z
xRzzXnx = *)(*)](*[Z
7/31/2019 z Transform (1)
22/30
5/21
ReversalxRzzXnx = ),()]([Z
xRzzXnx /1)()]([1 = Z
Real and Imaginary Parts
xRzzXnx = ),()]([Z
xRzzXzXnxe += *)](*)([)]([ 21R
xjRzzXzXnx = *)](*)([)]([
21Im
7/31/2019 z Transform (1)
23/30
5/21
Initial Value Theorem
0for,0)(
7/31/2019 z Transform (1)
24/30
5/21
Convolution of Sequences
=
=k
knykxnynx )()()(*)(
=
=
=n
n
k
zknykxnynx )()()](*)([Z
=
=
=k
n
n
zknykx )()(
=
=
=k
n
n
kznyzkx )()(
)()( zYzX=
The z-Transform
S stem Function
7/31/2019 z Transform (1)
25/30
5/21
Shift-Invariant System
h(n)
x(n) y(n)=x(n)*h(n)
X(z) Y(z)=X(z)H(z)H(z)
Shift-Invariant System
X(z) Y(z)
)(
)()(
zX
zYzH =
7/31/2019 z Transform (1)
26/30
5/21
Nth
-Order Difference Equation
==
=M
r
r
N
k
k rnxbknya00
)()(
=M
r
r
Nk
k zbzXzazY )()(== rk 00
==
=
N
k
k
k
M
r
r
r zazbzH00
)(
Representation in Factored Form
=
=
M
r
rzcA
1
1)1(
Contributespoles at 0 andzeros at cr
=
N
k
rzd1
1)1(
Contributeszeros at 0 andpoles at dr
7/31/2019 z Transform (1)
27/30
5/21
Stable and Causal Systems
=
=
M
r
rzcA1
1)1(
Im
Causal Systems : ROC extends outward from the outermost pole.
=
N
k
rzd
1
1)1(Re
Stable and Causal Systems
=
=
M
r
rzcA
1
1)1(
Im
Stable Systems : ROC includes the unit circle.
1
=
N
k
rzd1
1)1(e
7/31/2019 z Transform (1)
28/30
5/21
ExampleConsider the causal system characterized by
)()1()( nxnayny +=
1=
Im
1
11 azRea
)()( nuanh n=
Determination of Frequency Responsefrom pole-zero pattern
z A LTI system is completely characterized by itspole-zero pattern.
1zz
Example:
0j
Im
p1
))(( 21 pzpzz
=
))(()(
21
1
00
0
0
pepe
zeeH
jj
jj
=
Re
z1
p2
7/31/2019 z Transform (1)
29/30
5/21
Determination of Frequency Response
from pole-zero pattern
z A LTI system is completely characterized by itspole-zero pattern.
1zz
Example:
0j
Im
p1
|H(ej)|=? H(ej)=?
))(( 21 pzpzz
=
))(()(
21
1
00
0
0
pepe
zeeH
jj
jj
=
Re
z1
p2
Determination of Frequency Responsefrom pole-zero pattern
z A LTI system is completely characterized by itspole-zero pattern.
Example:
0j
Im
p1
|H(ej)|=? H(ej)=?
| | 2
Re
z1
p2
e =| | | | 1 3
H(ej) = 1(2+ 3 )
7/31/2019 z Transform (1)
30/30
5/21
Example11
1)(
=
azzH
Im
0 2 4 6 8
-10
0
10
20
dB
Re
a0 2 4 6 8
-2
-1
0
1
2