Transcript The Inverse z-Transform - Embedded Signal Processing

The Inverse z-Transform
In science one tries to tell people, in such a way
as to be understood by everyone, something
that no one ever knew before.
But in poetry, it's the exact opposite.
Paul Dirac
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, ©1999-2000 Prentice Hall
Inc.
The Inverse Z-Transform
• Formal inverse z-transform is based on a Cauchy integral
• Less formal ways sufficient most of the time
– Inspection method
– Partial fraction expansion
– Power series expansion
• Inspection Method
– Make use of known z-transform pairs such as
Z
anun 


1
1  az1
z  a
– Example: The inverse z-transform of
1
Xz  
1
1  z 1
2
Copyright (C) 2005 Güner Arslan
1
z 
2
n

1
xn    un
2
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Inverse Z-Transform by Partial Fraction Expansion
• Assume that a given z-transform can be expressed as
M
Xz  
b z
k 0
N
k
k
k
a
z
 k
k 0
• Apply partial fractional expansion
Xz 
M N
B z
r
r 0
r
s
Ak
Cm
 

1
1
k 1,k  i 1  dk z
m 1 1  d z
i
N


m
• First term exist only if M>N
– Br is obtained by long division
• Second term represents all first order poles
• Third term represents an order s pole
– There will be a similar term for every high-order pole
• Each term can be inverse transformed by inspection
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Partial Fractional Expression
Xz 
M N
B z
r
r
r 0
s
Ak
Cm
 


1
1
k 1,k  i 1  dk z
m 1 1  d z
i
N


m
• Coefficients are given as


Ak  1  dk z 1 Xz  z  d
Cm 
1
s  m!  di 
s m
k

 
 ds m
s
1 
1  diw X w 

s m
 dw
w  di1
• Easier to understand with examples
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Example: 2nd Order Z-Transform
Xz 
1
1 1 
1 1 

1

z
1

z 


4
2



ROC : z 
1
2
– Order of nominator is smaller than denominator (in terms of z-1)
– No higher order pole
A1
A2
Xz  

1
1 1 


1 
1  z   1  z 
4
2

 

1 1 
1

A1  1  z Xz

 1
1
1
4



1  1  
z

4
1  

2  4  

1
1


A2  1  z 1 Xz

2
1
1
2



1  1  
z

2
1  

4  2  

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Example Continued
Xz 
1
2

1 1  
1 1 

1

z
1

z 

 
4
2

 

z 
1
2
• ROC extends to infinity
– Indicates right sided sequence
n
n
1
1
xn  2  un -   un
2
 4
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Example #2


2
1  2z 1  z 2
1  z 1
Xz 

3
1
1
1  z 1  z 2 1  z 1  1  z 1
2
2
2




z 1
• Long division to obtain Bo
2
1
 1  5z 1
Xz  2 
1 1 

1
1  z  1  z
2


5z 1  1
A1
A2
Xz   2 

1 1 1  z 1
1 z
2
1 2 3 1
2
1
z  z  1 z  2z
2
2
z 2  3z 1  2
1


A1  1  z1 Xz
 9
1
2


z



A2  1  z 1 Xz
z 1

8
2
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Example #2 Continued
Xz   2 
9
8

1 1 1  z 1
1 z
2
z 1
• ROC extends to infinity
– Indicates right-sides sequence
n
1
xn  2n  9  un - 8un
2
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Inverse Z-Transform by Power Series Expansion
• The z-transform is power series
Xz  

 xn z
n
n  
• In expanded form
Xz    x 2 z2  x 1 z1  x0  x1 z1  x2 z2  
• Z-transforms of this form can generally be inversed easily
• Especially useful for finite-length series
• Example
1


 1 n  2
Xz   z2 1  z 1  1  z 1 1  z 1
 1
2


  2 n  1

1
1 1
2


x
n

 1 n  0
 z  z 1 z
2
2
 1
n1

2
1
1
 0
xn  n  2  n  1  n  n  1
n2

2
2

Copyright (C) 2005 Güner Arslan


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Z-Transform Properties: Linearity
• Notation
Z
xn 

 Xz
ROC  Rx
• Linearity
Z
ax1n  bx2 n

 aX1 z  bX2 z
ROC  Rx1  Rx2
– Note that the ROC of combined sequence may be larger than
either ROC
– This would happen if some pole/zero cancellation occurs
– Example:
xn  anun - anun - N
•
•
•
•
•
Both sequences are right-sided
Both sequences have a pole z=a
Both have a ROC defined as |z|>|a|
In the combined sequence the pole at z=a cancels with a zero at z=a
The combined ROC is the entire z plane except z=0
• We did make use of this property already, where?
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Z-Transform Properties: Time Shifting
Z
xn  no  

 zno Xz
ROC  Rx
• Here no is an integer
– If positive the sequence is shifted right
– If negative the sequence is shifted left
• The ROC can change the new term may
– Add or remove poles at z=0 or z=
• Example




1
1

Xz  z 
 1  1 z 1 


4


1
z 
4
n-1
1
xn    un - 1
 4
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351M Digital Signal Processing
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Z-Transform Properties: Multiplication by Exponential
Z
znoxn 

 Xz / zo 
•
•
•
•
•
ROC  zo Rx
ROC is scaled by |zo|
All pole/zero locations are scaled
If zo is a positive real number: z-plane shrinks or expands
If zo is a complex number with unit magnitude it rotates
Example: We know the z-transform pair
1
un 

1 - z-1
• Let’s find the z-transform of
Z
xn  rn cosonun 
Xz 
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ROC : z  1




n
1 jo n
1
re
un  re jo un
2
2
1/2
1/2

1  rejo z 1 1  re jo z 1
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z r
12
Z-Transform Properties: Differentiation
dXz 
nxn 
 z
dz
Z
ROC  R x
• Example: We want the inverse z-transform of

Xz  log1  az1

z  a
• Let’s differentiate to obtain rational expression
dXz
 az2
dXz
1
1

 z
 az
1
dz
dz
1  az
1  az1
• Making use of z-transform properties and ROC
nxn  a a un  1
n1
xn   1
n 1
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an
un  1
n
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Z-Transform Properties: Conjugation
 
Z
x* n 

 X* z*
ROC  R x
• Example
Xz  

n


x
n
z

n  



X z     xn z n  
 n  





 x n z

n  
   x n z    x n z

X z 

n  
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
n 
n


n


 Z x n
n  
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Z-Transform Properties: Time Reversal
Z
x n 

 X1 / z
ROC 
1
Rx
• ROC is inverted
• Example:
xn  anu n
• Time reversed version of anun
1
- a-1z 1
Xz 

1  az 1 - a-1z 1
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351M Digital Signal Processing
z  a1
15
Z-Transform Properties: Convolution
Z
x1n  x2 n 

 X1 zX2 z
ROC: Rx1  Rx2
• Convolution in time domain is multiplication in z-domain
• Example:Let’s calculate the convolution of
x1 n  anun and x2 n  un
1
1


X1 z 
ROC
:
z

a
X
z

2
1  az1
1  z 1
• Multiplications of z-transforms is
1
Y z  X1 zX2 z 
1  az1 1  z1


ROC : z  1

• ROC: if |a|<1 ROC is |z|>1 if |a|>1 ROC is |z|>|a|
• Partial fractional expansion of Y(z)
Y z 
1  1
1


asumeROC : z  1

1
1 
1  a 1  z
1  az 
1
yn 
un  an1un
1a

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351M Digital Signal Processing

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