Z Transform, Inverse Z Transform

1. 1
The Inverse z-Transform
IIIB.ScMathematics
Transforms
Ms.S.Swathi Sundari, M.Sc.,M.Phil.,

2. 2
z-transforms Table -1

3. 3
z-transforms Table -2

4. 4
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
– Example: The inverse z-transform of
z  aan
unZ

1
1 az1
2
1 1
1  
1
n
xn 
2
 unz 
2

1 
Xz
z1

5. 5
◆Find the inverse z-transform of:
◆Step 1: Divide both sides by z:
◆Step 2: Perform partial fraction:
◆Step 3: Multiply both sides by z:
◆Step 4: Obtain inverse z-transform of each term from table (#1 & #6):
The Inverse Z-Transform

6. 6
• Apply partial fractional expansion
• 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
k0
Inverse Z-Transform by Partial Fraction Expansion
• Assume that a given z-transform can be expressed as
M
akzk
bkzk
k 0
N
Xz
s
m1  m1
ik
MN
r
 r
r0 1 dz
Cm
k 1,ki 1  d z1
Ak
N
B z 

Xz

7. 7
Inverse Z-Transform by Power Series Expansion
• The z-transform is power series


n
n
xnzXz
• 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
2
2
1 11
 z2

1
z  1 
1
z1
2 2
1  z 1  z 



 1 1  z
2
1
2
1
n 1    n  1 nn n 2x    

0
 2

1
 1
n  2

2
n  1
n  0
n  1
n  2
 1
xn 
1

8. 8
Z-Transform Properties: Linearity
• Notation
8
– Note that the ROC of combined sequence may be larger than
either ROC
– This would happen if some pole/zero cancellation occurs
– Example:
• 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?
ROC  RxxnZ
 Xz
ROC  Rx  R
1 x2
• Linearity
ax1n bx2nZ
 aX1z bX2 z
xn an
un- an
un - N

9. 9
Find inverse z-transform – long division
◆Consider this example:
◆Perform long division:
◆Thus:
◆Therefore

10. 10