The z-Transform
主講人：虞台文
Content
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Introduction
z-Transform
Zeros and Poles
Region of Convergence
Important z-Transform Pairs
Inverse z-Transform
z-Transform Theorems and Properties
System Function
The z-Transform
Introduction
Why z-Transform?


A generalization of Fourier transform
Why generalize it?
–
–
–
FT does not converge on all sequence
Notation good for analysis
Bring the power of complex variable theory deal with
the discrete-time signals and systems
The z-Transform
z-Transform
Definition

The z-transform of sequence x(n) is defined by
X ( z) 

 x ( n) z
n
n  

Let z = ej.
j
X (e ) 

 x ( n )e
n 
Fourier
Transform
 j n
z-Plane
X ( z) 

 x ( n) z
n
Im
z = ej
n  
j
X (e ) 

 x ( n )e

 j n
Re
n 
Fourier Transform is to evaluate z-transform
on a unit circle.
z-Plane
Im
X(z)
z = ej

Re
Im
Re
Periodic Property of FT
X(ej)
X(z)


Im
Re
Can you say why Fourier Transform is
a periodic function with period 2?

The z-Transform
Zeros and Poles
Definition

Give a sequence, the set of values of z for which the
z-transform converges, i.e., |X(z)|<, is called the
region of convergence.
| X ( z ) |

n
x
(
n
)
z


n  

n
|
x
(
n
)
||
z
|


n  
ROC is centered on origin and
consists of a set of rings.
Example: Region of Convergence
| X ( z ) |

n
x
(
n
)
z


n  
Im

n
|
x
(
n
)
||
z
|


n  
ROC is an annual ring centered
on the origin.
r
Re
Rx  | z | Rx 
j
ROC  {z  re | Rx   r  Rx  }
Stable Systems

A stable system requires that its Fourier transform is
uniformly convergent.
Im

1

Re
Fact: Fourier transform is to
evaluate z-transform on a unit
circle.
A stable system requires the
ROC of z-transform to include
the unit circle.
Example: A right sided Sequence
x ( n)  a n u ( n )
x(n)
...
-8 -7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7 8 9 10
n
Example: A right sided Sequence
For convergence of X(z), we
require that
x ( n)  a u ( n )
n
X ( z) 


 a u (n)z
n
n  
n
1
|
az
|

n 0
| z || a |

  a n z n
n 0

  (az 1 ) n
n 0
| az 1 | 1

1
z
X ( z )   (az ) 

1
1  az
za
n 0
1 n
| z || a |
Example: A right sided Sequence
ROC for x(n)=anu(n)
z
X ( z) 
,
za
a
| z || a |
Which one is stable?
Im
Im
1
1
a
a
Re
a
Re
Example: A left sided Sequence
x(n)  a nu(n  1)
-8 -7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7 8 9 10
n
...
x(n)
Example: A left sided Sequence
x(n)  a u(n  1)
n

X ( z )    a u (n  1)z
n
n  
1
   a n z n
n
For convergence of X(z), we
require that

1
|
a
 z|
| a 1 z | 1
n 0
| z || a |
n  

  a  n z n
n 1

 1   a n z n
n 0

1
z
X ( z )  1   (a z )  1 

1
1 a z z  a
n 0
1
n
| z || a |
Example: A left sided Sequence
ROC for x(n)=anu( n1)
z
X ( z) 
,
za
a
| z || a |
Which one is stable?
Im
Im
1
1
a
a
Re
a
Re
The z-Transform
Region of
Convergence
Represent z-transform as a
Rational Function
P( z )
X ( z) 
Q( z )
where P(z) and Q(z) are
polynomials in z.
Zeros: The values of z’s such that X(z) = 0
Poles: The values of z’s such that X(z) = 
Example: A right sided Sequence
z
X ( z) 
,
za
x ( n)  a n u ( n )
| z || a |
Im
a
Re
ROC is bounded by the
pole and is the exterior
of a circle.
Example: A left sided Sequence
z
X ( z) 
,
za
x(n)  a nu(n  1)
| z || a |
Im
a
Re
ROC is bounded by the
pole and is the interior
of a circle.
Example: Sum of Two Right Sided Sequences
x(n)  ( 12 ) n u (n)  ( 13 ) n u (n)
z
z
2 z ( z  121 )
X ( z) 


1
1
z2 z3
( z  12 )( z  13 )
Im
ROC is bounded by poles
and is the exterior of a circle.
1/12
1/3
1/2
Re
ROC does not include any pole.
Example: A Two Sided Sequence
x(n)  ( 13 ) n u (n)  ( 12 ) n u (n  1)
z
z
2 z ( z  121 )
X ( z) 


1
1
z3 z2
( z  13 )( z  12 )
Im
ROC is bounded by poles
and is a ring.
1/12
1/3
1/2
Re
ROC does not include any pole.
Example: A Finite Sequence
x(n)  a n ,
0  n  N 1
N 1
X ( z)   a z
n
n 0
n
Im
N 1
  ( az )
1 n
n 0
N-1 zeros
1  (az 1 ) N

1  az 1
1 zN  aN
 N 1
z
za
ROC: 0 < z < 
ROC does not include any pole.
N-1 poles
Re
Always Stable
Properties of ROC

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


A ring or disk in the z-plane centered at the origin.
The Fourier Transform of x(n) is converge absolutely iff the ROC
includes the unit circle.
The ROC cannot include any poles
Finite Duration Sequences: The ROC is the entire z-plane except
possibly z=0 or z=.
Right sided sequences: The ROC extends outward from the outermost
finite pole in X(z) to z=.
Left sided sequences: The ROC extends inward from the innermost
nonzero pole in X(z) to z=0.
More on Rational z-Transform
Consider the rational z-transform
with the pole pattern:
Im
Find the possible
ROC’s
a b
c
Re
More on Rational z-Transform
Consider the rational z-transform
with the pole pattern:
Im
Case 1: A right sided Sequence.
a b
c
Re
More on Rational z-Transform
Consider the rational z-transform
with the pole pattern:
Im
Case 2: A left sided Sequence.
a b
c
Re
More on Rational z-Transform
Consider the rational z-transform
with the pole pattern:
Im
Case 3: A two sided Sequence.
a b
c
Re
More on Rational z-Transform
Consider the rational z-transform
with the pole pattern:
Im
Case 4: Another two sided Sequence.
a b
c
Re
The z-Transform
Important
z-Transform Pairs
Z-Transform Pairs
Sequence
z-Transform
(n)
1
(n  m)
z m
ROC
All z
All z except 0 (if m>0)
or  (if m<0)
u (n)
1
1  z 1
| z | 1
 u (n  1)
1
1  z 1
| z | 1
a u (n)
1
1  az 1
| z || a |
 a nu (n  1)
1
1  az 1
| z || a |
n
Z-Transform Pairs
Sequence
z-Transform
[cos 0 n]u(n)
1  [cos 0 ]z 1
1  [2 cos 0 ]z 1  z 2
| z | 1
[sin 0 n]u (n)
[sin 0 ]z 1
1  [2 cos 0 ]z 1  z 2
| z | 1
[r n cos 0 n]u (n)
1  [r cos 0 ]z 1
1  [2r cos 0 ]z 1  r 2 z 2
| z | r
[r n sin 0 n]u (n)
[r sin 0 ]z 1
1  [2r cos 0 ]z 1  r 2 z 2
| z | r
1 aN zN
1  az 1
| z | 0
a n

0
0  n  N 1
otherwise
ROC
The z-Transform
Inverse z-Transform
The z-Transform
z-Transform Theorems
and Properties
Linearity
Z[ x(n)]  X ( z ),
z  Rx
Z [ y (n)]  Y ( z ),
z  Ry
Z [ax(n)  by (n)]  aX ( z )  bY ( z ),
z  Rx  R y
Overlay of
the above two
ROC’s
Shift
Z[ x(n)]  X ( z ),
z  Rx
Z[ x(n  n0 )]  z X ( z )
n0
z  Rx
Multiplication by an Exponential Sequence
Z[ x(n)]  X ( z ),
Rx- | z | Rx 
1
Z [a x(n)]  X (a z )
n
z | a | Rx
Differentiation of X(z)
Z[ x(n)]  X ( z ),
dX ( z )
Z [nx(n)]   z
dz
z  Rx
z  Rx
Conjugation
Z[ x(n)]  X ( z ),
Z[ x * (n)]  X * ( z*)
z  Rx
z  Rx
Reversal
Z[ x(n)]  X ( z ),
1
Z[ x(n)]  X ( z )
z  Rx
z  1 / Rx
Real and Imaginary Parts
Z[ x(n)]  X ( z ),
z  Rx
Re[ x(n)]  12 [ X ( z )  X * ( z*)]
Im[ x(n)]  21j [ X ( z )  X * ( z*)]
z  Rx
z  Rx
Initial Value Theorem
x(n)  0,
for n  0
x(0)  lim X ( z )
z 
Convolution of Sequences
Z[ x(n)]  X ( z ),
Z [ y (n)]  Y ( z ),
z  Rx
z  Ry
Z[ x(n) * y (n)]  X ( z )Y ( z )
z  Rx  R y
Convolution of Sequences
x ( n) * y ( n) 

 x(k ) y (n  k )
k  
 
 n
Z[ x(n) * y (n)]     x(k ) y (n  k )  z
n    k  





k  
n  
 x(k )  y(n  k )z
 X ( z )Y ( z )
n



k  
x(k ) z  k

n
y
(
n
)
z

n  
The z-Transform
System Function
Shift-Invariant System
y(n)=x(n)*h(n)
x(n)
h(n)
X(z)
H(z)
Y(z)=X(z)H(z)
Shift-Invariant System
X(z)
Y(z)
H(z)
Y ( z)
H ( z) 
X ( z)
Nth-Order Difference Equation
N
a
k 0
M
k
y (n  k )   br x(n  r )
r 0
N
M
k 0
r 0
Y ( z ) ak z  k  X ( z ) br z  r
M
H ( z )   br z
r 0
r
N
 ak z
k 0
k
Representation in Factored Form
Contributes poles at 0 and zeros at cr
M
H ( z) 
A (1  cr z 1 )
r 1
N
1
(
1

d
z
 r )
k 1
Contributes zeros at 0 and poles at dr
Stable and Causal Systems
Causal Systems : ROC extends outward from the outermost pole.
Im
M
H ( z) 
A (1  cr z 1 )
r 1
N
1
(
1

d
z
 r )
k 1
Re
Stable and Causal Systems
Stable Systems : ROC includes the unit circle.
Im
M
H ( z) 
A (1  cr z 1 )
r 1
N
1
(
1

d
z
 r )
k 1
1
Re
Example
Consider the causal system characterized by
y (n)  ay (n  1)  x(n)
1
H ( z) 
1
1  az
h ( n)  a n u ( n )
Im
1
a
Re
Determination of Frequency Response
from pole-zero pattern

A LTI system is completely characterized by its
pole-zero pattern.
Im
Example:
z  z1
H ( z) 
( z  p1 )( z  p2 )
H (e j0 ) 
(e j0
e j0  z1
 p1 )(e j0  p2 )
p1
e j 0
z1
Re
p2
Determination of Frequency Response
from pole-zero pattern

H(e )=?
A LTI jsystem
is completely characterized
by its

j
pole-zero pattern.
|H(e )|=?
Im
Example:
z  z1
H ( z) 
( z  p1 )( z  p2 )
H (e j0 ) 
(e j0
e j0  z1
 p1 )(e j0  p2 )
p1
e j 0
z1
Re
p2
Determination of Frequency Response
from pole-zero pattern

H(e )=?
A LTI jsystem
is completely characterized
by its

j
pole-zero pattern.
|H(e )|=?
Im
Example:
|H(ej)|
=
|
|
2
|
||
|
H(ej) = 1(2+ 3 )
z1
p1
e j 0
1
p2
3
Re
Example
1
H ( z) 
1
1  az
dB
20
Im
10
0
-10
0
2
4
6
8
0
2
4
6
8
2
1
a
Re
0
-1
-2