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 a n u   
1 
1 az  1 z  a
– Example: The inverse z-transform of
X
 

1 
1
1
2 z  1 z 
1
2
 x
 



1
2


 n u
 
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Inverse Z-Transform by Partial Fraction Expansion
• Assume that a given z-transform can be expressed as
X
 
 k
M

 0 k
N

 0 b k z  k a k z  k
• Apply partial fractional expansion
X
 

M  r

 0
N
B r z  r  k
N

 1 , k  i
1 
A k d k z  1
 m s

 1

1 
C m d i z  1
 m
• First term exist only if M>N
– B r 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
 

M  r

 0
N
B r z  r  k
N

 1 , k  i
1 
A k d k z  1
 m s

 1

1 
C m d i z  1
 m
• Coefficients are given as
A k


1  d k z  1

X
  z  d k
C m

 s  m
1
   s  m


 d s  m dw s  m


1  d i w
 s X
   

 w  d i
 1
• Easier to understand with examples
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X
Example: 2 nd Order Z-Transform
 



1
1
1
4 z  1


 

1
1
2 z  1


ROC : z 
1
2
– Order of nominator is smaller than denominator (in terms of z -1 )
– No higher order pole
X
 



A
1
1
4 z  1





A
1
2
2 z  1


A
1
 

1
A
2
 

1
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1
4 z  1


X
  z 
1
4


1 
1
2
1
1
4
1
2 z  1


X z 
1
2


1 
1
4
1


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1
2
 1


 1
  1
 2
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X
 



1
Example Continued
 1
1
4 z  1





1
2
1
2 z  1

 z 
1
2
• ROC extends to infinity
– Indicates right sided sequence x
 
 2


1
2


 n u
 
-


1
4


 n u
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X 
1
1


3
2
2 z z  1
 1


Example #2
1
2 z  z
2
 2



1

1
1
2 z

 1 z


 1

1
2

 z  1
 z  1
• Long division to obtain B o
1
2 z  2 
3
2 z  1  1 z  2 z  2
 2 z  1
 3 z  1

2
1
 2
5 z  1  1
X
X
 2 


1

1
2
1  z  1
5 z  1



1  z  1

 2 
1 
A
1
1
2 z  1

1
A

2 z  1
A
1
  1
1
2 z  1


X
  z 
1
2
  9 A
2


1  z  1

X
  z  1
 8
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Example #2 Continued
X
 
 2 
1 
9
1
2 z  1

1 
8 z  1 z  1
• ROC extends to infinity
– Indicates right-sides sequence x
 
 2 
 
 9


1
2


 n u
 
8u
 
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Inverse Z-Transform by Power Series Expansion
• The z-transform is power series
X
 
 n 


 x
  z  n
• In expanded form
X
 
   x
  z 2  x
  z 1  x
    z  1  x
  z  2  
• Z-transforms of this form can generally be inversed easily
• Especially useful for finite-length series
• Example
X  z 2 

 z 2 
1
2 z
1
2
 z  1



1  z  1

1  z  1

1
1  z  1
2 x
   n  2


1
2

 n  1
  

1
2

 n  1
 x
 









1
1

1
2
1
2
0 n n n n n



 1

 2
 1
0
2
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Z-Transform Properties: Linearity
• Notation x
 
   X
 
ROC  R x
• Linearity ax
1
 
 bx
2
 
   aX
1
 
 bX
2
 
ROC  R x
1
 R x
2
– Note that the ROC of combined sequence may be larger than either ROC
– This would happen if some pole/zero cancellation occurs
– Example: x
 
 a n u
 
a n u
 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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Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing

Z-Transform Properties: Time Shifting x
 n  n o

   z  n o
X
 
ROC  R x
• Here n o 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
X
 
 z  1


1 
1
1
4 z  1

 z 
1
4 x
 



1
4


 n 1 u
 n 1

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Z-Transform Properties: Multiplication by Exponential z n o x
 
   X
 z / z o

ROC  z o
R x
• ROC is scaled by |z o
|
• All pole/zero locations are scaled
• If z o
• If z o is a positive real number: z-plane shrinks or expands is a complex number with unit magnitude it rotates
• Example: We know the z-transform pair u
 
  
1 -
1 z 1
• Let’s find the z-transform of
ROC : x
 
 r n cos
  u 
1
2 z  1
  n u
 

1
2
 re  j  o
 n u
 
X
 

1 
1 / 2 re j  o z  1

1 
1 / 2 re  j  o z  1 z  r
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Z-Transform Properties: Differentiation nx
 
    z dX dz
ROC  R x
• Example: We want the inverse z-transform of
X
 
 log

1  az  1
 z  a
• Let’s differentiate to obtain rational expression dX dz
 

1

 az  2 az  1
  z dX
  dz
 az  1
1 
1 az  1
• Making use of z-transform properties and ROC nx
 
 a
  n  1 u
 n  1
 x
 

  n  1 a n n u
 n  1

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Z-Transform Properties: Conjugation x *    X *
 
ROC  R x
• Example
X
 
X 

 n 


 x
  z  n

 n 


 x
  z  n



X    n 


 x 
   

 n 


 x 
 n 


 x  z n z  n  Z
 
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Z-Transform Properties: Time Reversal x
 
   X

1 / z

ROC 
1
R x
• ROC is inverted
• Example: x
 
 a  n u
 
• Time reversed version of a n u
 
X
 

1 
1 az

1
-
a -1 z  1 a 1 z  1 z  a  1
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x
1
Z-Transform Properties: Convolution
 
 x
2
 
   X
1
   
ROC : R x 1
 R x
2
• Convolution in time domain is multiplication in z-domain
• Example:Let’s calculate the convolution of
X
1

1 
1 az  1 x
1
 
ROC :
 z a n u
 a
  and x
2
X
2
   

1 
1 z  1
• Multiplications of z-transforms is
Y
 
 X
1
   
 
1  az  1
1

1  z  1

ROC :
• ROC: if |a|<1 ROC is |z|>1 if |a|>1 ROC is |z|>|a| z
• Partial fractional expansion of Y(z)
Y 
1
1
 a

 1 
1 z  1 y
 


1 
1
1 
1 az  1 a
 u
 


 asume ROC
 a n  1 u

: z  1
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