Digital Signal Processing and the z-transform

EC 2314 Digital Signal Processing
By
Dr. K. Udhayakumar

The z-Transform
Dr. K. Udhayakumar

Content
 Introduction
 z-Transform
 Zeros and Poles
 Region of Convergence
 Important z-Transform Pairs
 Inverse z-Transform
 z-Transform Theorems and Properties
 System Function

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

Definition
 The z-transform of sequence x(n) is defined by






n
n
z
n
x
z
X )
(
)
(
 Let z = ej.
( ) ( )
j j n
n
X e x n e
 



 
Fourier
Transform

z-Plane
Re
Im
z = ej







n
n
z
n
x
z
X )
(
)
(
( ) ( )
j j n
n
X e x n e
 



 
Fourier Transform is to evaluate z-transform
on a unit circle.

z-Plane
Re
Im
X(z)
Re
Im
z = ej


Periodic Property of FT
Re
Im
X(z)

 
X(ej)
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.



 









n
n
n
n
z
n
x
z
n
x
z
X |
||
)
(
|
)
(
|
)
(
|
ROC is centered on origin and
consists of a set of rings.

Example: Region of Convergence
Re
Im



 









n
n
n
n
z
n
x
z
n
x
z
X |
||
)
(
|
)
(
|
)
(
|
ROC is an annual ring centered
on the origin.

 
 x
x R
z
R |
|
r
}
|
{ 





 x
x
j
R
r
R
re
z
ROC

Stable Systems
Re
Im
1
 A stable system requires that its Fourier transform is
uniformly convergent.
 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
)
(
)
( n
u
a
n
x n

1 2 3 4 5 6 7 8 9 10
-1
-2
-3
-4
-5
-6
-7
-8
n
x(n)
. . .

)
(
)
( n
u
a
n
x n

n
n
n
z
n
u
a
z
X 




 )
(
)
(





0
n
n
n
z
a





0
1
)
(
n
n
az
For convergence of X(z), we
require that






0
1
|
|
n
az 1
|
| 1


az
|
|
|
| a
z 
a
z
z
az
az
z
X
n
n




 



 1
0
1
1
1
)
(
)
(
|
|
|
| a
z 

a
a
ROC for x(n)=anu(n)
|
|
|
|
,
)
( a
z
a
z
z
z
X 


Re
Im
1
a
a
Re
Im
1
Which one is stable?

Example: A left sided Sequence
)
1
(
)
( 


 n
u
a
n
x n
1 2 3 4 5 6 7 8 9 10
-1
-2
-3
-4
-5
-6
-7
-8
n
x(n)
.
.
.

)
1
(
)
( 


 n
u
a
n
x n
n
n
n
z
n
u
a
z
X 



 


 )
1
(
)
(
For convergence of X(z), we
require that






0
1
|
|
n
z
a 1
|
| 1


z
a
|
|
|
| a
z 
a
z
z
z
a
z
a
z
X
n
n






 



 1
0
1
1
1
1
)
(
1
)
(
|
|
|
| a
z 
n
n
n
z
a 






1
n
n
n
z
a






1
n
n
n
z
a






0
1

a
a
ROC for x(n)=anu( n1)
|
|
|
|
,
)
( a
z
a
z
z
z
X 


Re
Im
1
a
a
Re
Im
1
Which one is stable?

The z-Transform
Region of
Convergence

Represent z-transform as a
Rational Function
)
(
)
(
)
(
z
Q
z
P
z
X 
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) = 

)
(
)
( n
u
a
n
x n
 |
|
|
|
,
)
( a
z
a
z
z
z
X 


Re
Im
a
ROC is bounded by the
pole and is the exterior
of a circle.

)
1
(
)
( 


 n
u
a
n
x n
|
|
|
|
,
)
( a
z
a
z
z
z
X 


Re
Im
a
ROC is bounded by the
pole and is the interior
of a circle.

Example: Sum of Two Right Sided Sequences
)
(
)
(
)
(
)
(
)
( 3
1
2
1
n
u
n
u
n
x n
n



3
1
2
1
)
(




z
z
z
z
z
X
Re
Im
1/2
)
)(
(
)
(
2
3
1
2
1
12
1




z
z
z
z
1/3
1/12
ROC is bounded by poles
and is the exterior of a circle.
ROC does not include any pole.

Example: A Two Sided Sequence
)
1
(
)
(
)
(
)
(
)
( 2
1
3
1




 n
u
n
u
n
x n
n
2
1
3
1
)
(




z
z
z
z
z
X
Re
Im
1/2
)
)(
(
)
(
2
2
1
3
1
12
1




z
z
z
z
1/3
1/12
ROC is bounded by poles
and is a ring.

Example: A Finite Sequence
1
0
,
)
( 


 N
n
a
n
x n
n
N
n
n
N
n
n
z
a
z
a
z
X )
(
)
( 1
1
0
1
0







 

Re
Im
ROC: 0 < z < 
1
1
1
)
(
1





az
az N
a
z
a
z
z
N
N
N


 1
1
N-1 poles
N-1 zeros
Always Stable

Properties of ROC
 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
Re
Im
a b c
Consider the rational z-transform
with the pole pattern:
Find the possible
ROC’s

Re
Im
a b c
Case 1: A right sided Sequence.

Re
Im
a b c
Case 2: A left sided Sequence.

Re
Im
a b c
Case 3: A two sided Sequence.

Re
Im
a b c
Case 4: Another two sided Sequence.

0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Bounded Signals
-5
0
5
a=0.4
-5
0
5
a=0.9
-5
0
5
a=1.2
0 5 10
-5
0
5
a=-0.4
0 5 10
-5
0
5
a=-0.9
0 5 10
-5
0
5
a=-1.2
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
0 2 4 6 8
-1
-0.5
0
0.5
1

BIBO Stability
 Bounded Input Bounded Output Stability
– If the Input is bounded, we want the Output is
bounded, too
– If the Input is unbounded, it’s okay for the Output to
be unbounded
 For some computing systems, the output is
intrinsically bounded (constrained), but limit
cycle may happen

The z-Transform
Important
z-Transform Pairs

Z-Transform Pairs
Sequence z-Transform ROC
)
(n
 1 All z
)
( m
n 
 m
z All z except 0 (if m>0)
or  (if m<0)
)
(n
u 1
1
1

 z
1
|
| 
z
)
1
( 

 n
u 1
1
1

 z
1
|
| 
z
)
(n
u
an 1
1
1

 az
|
|
|
| a
z 
)
1
( 

 n
u
an 1
1
1

 az
|
|
|
| a
z 

Z-Transform Pairs
Sequence z-Transform ROC
)
(
]
[cos 0 n
u
n
 2
1
0
1
0
]
cos
2
[
1
]
[cos
1








z
z
z
1
|
| 
z
)
(
]
[sin 0 n
u
n
 2
1
0
1
0
]
cos
2
[
1
]
[sin







z
z
z
1
|
| 
z
)
(
]
cos
[ 0 n
u
n
rn
 2
2
1
0
1
0
]
cos
2
[
1
]
cos
[
1








z
r
z
r
z
r
r
z 
|
|
)
(
]
sin
[ 0 n
u
n
rn
 2
2
1
0
1
0
]
cos
2
[
1
]
sin
[







z
r
z
r
z
r
r
z 
|
|


 


otherwise
0
1
0 N
n
an
1
1
1




az
z
a N
N
0
|
| 
z

Signal Type ROC
Finite-Duration Signals
Infinite-Duration Signals
Causal
Anticausal
Two-sided
Causal
Anticausal
Two-sided
Entire z-plane
Except z = 0
Entire z-plane
Except z = infinity
Entire z-plane
Except z = 0
And z = infinity
|z| < r1
|z| > r2
r2 < |z| < r1

Some Common z-Transform Pairs
Sequence Transform ROC
1. [n] 1 all z
2. u[n] z/(z-1) |z|>1
3. -u[-n-1] z/(z-1) |z|<1
4. [n-m] z-m all z except 0 if m>0 or ฅ if m<0
5. anu[n] z/(z-a) |z|>|a|
6. -anu[-n-1] z/(z-a) |z|<|a|
7. nanu[n] az/(z-a)2 |z|>|a|
8. -nanu[-n-1] az/(z-a)2 |z|<|a|
9. [cos0n]u[n] (z2-[cos0]z)/(z2-[2cos0]z+1) |z|>1
10. [sin0n]u[n] [sin0]z)/(z2-[2cos0]z+1) |z|>1
11. [rncos0n]u[n] (z2-[rcos0]z)/(z2-[2rcos0]z+r2) |z|>r
12. [rnsin0n]u[n] [rsin0]z)/(z2-[2rcos0]z+r2) |z|>r
13. anu[n] - anu[n-N] (zN-aN)/zN-1(z-a) |z|>0

The z-Transform
Inverse z-Transform

Inverse Z-Transform by Partial Fraction
Expansion
 Assume that a given z-transform can be expressed as
 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
 






 N
k
k
k
M
k
k
k
z
a
z
b
z
X
0
0
 
 


 












s
1
m
m
1
i
m
N
i
k
,
1
k
1
k
k
N
M
0
r
r
r
z
d
1
C
z
d
1
A
z
B
z
X

Partial Fractional Expression
 Coefficients are given as
 Easier to understand with examples
 
 


 












s
1
m
m
1
i
m
N
i
k
,
1
k
1
k
k
N
M
0
r
r
r
z
d
1
C
z
d
1
A
z
B
z
X
    k
d
z
1
k
k z
X
z
d
1
A 



   
   
  1
i
d
w
1
s
i
m
s
m
s
m
s
i
m w
X
w
d
1
dw
d
d
!
m
s
1
C

















Example: 2nd Order Z-Transform
– Order of nominator is smaller than denominator (in terms of z-1)
– No higher order pole
 
2
1
z
:
ROC
2
1
1
4
1
1
1
1
1


















z
z
z
X
 

















 1
2
1
1
z
2
1
1
A
z
4
1
1
A
z
X
  1
4
1
2
1
1
1
z
X
z
4
1
1
A 1
4
1
z
1
1 
























 


  2
2
1
4
1
1
1
z
X
z
2
1
1
A 1
2
1
z
1
2 























 



Example Continued
 ROC extends to infinity
– Indicates right sided sequence
 
2
1
z
z
2
1
1
2
z
4
1
1
1
z
X
1
1




















     
n
u
4
1
-
n
u
2
1
2
n
x
n
n














Example #2
 Long division to obtain Bo
   
 
1
z
z
1
z
2
1
1
z
1
z
2
1
z
2
3
1
z
z
2
1
z
X
1
1
2
1
2
1
2
1























1
z
5
2
z
3
z
2
1
z
2
z
1
z
2
3
z
2
1
1
1
2
1
2
1
2














 
 
1
1
1
z
1
z
2
1
1
z
5
1
2
z
X















  1
2
1
1
z
1
A
z
2
1
1
A
2
z
X 
 




  9
z
X
z
2
1
1
A
2
1
z
1
1 











    8
z
X
z
1
A
1
z
1
2 





Example #2 Continued
 ROC extends to infinity
– Indicates right-sides sequence
  1
z
z
1
8
z
2
1
1
9
2
z
X 1
1





 

       
n
8u
-
n
u
2
1
9
n
2
n
x
n










An Example – Complete Solution
3
8
6z
z
14
14z
3z
lim
U(z)
lim
c 2
2
z
z
0 










4
-
z
14
14z
3z
8
6z
z
14
14z
3z
2)
(z
(z)
U
2
2
2
2









2
-
z
14
14z
3z
8
6z
z
14
14z
3z
4)
(z
(z)
U
2
2
2
4









8
6z
z
14
14z
3z
U(z) 2
2





4
z
c
2
z
c
c
U(z) 2
1
0





1
4
-
2
14
2
14
2
3
(2)
U
c
2
2
1 






3
2
-
4
14
4
14
4
3
(4)
U
c
2
4
2 






4
z
3
2
z
1
3
U(z)












 

0
k
,
4
3
2
0
k
3,
u(k) 1
k
1
k

Inverse Z-Transform by Power Series
Expansion
 The z-transform is power series
 In expanded form
 Z-transforms of this form can generally be inversed easily
 Especially useful for finite-length series
 Example
   






n
n
z
n
x
z
X
            
 







 
 2
1
1
2
2
1
0
1
2 z
x
z
x
x
z
x
z
x
z
X
    
1
2
1
1
1
2
z
2
1
1
z
2
1
z
z
1
z
1
z
2
1
1
z
z
X


















         
1
n
2
1
n
1
n
2
1
2
n
n
x 










 



















2
n
0
1
n
2
1
0
n
1
1
n
2
1
2
n
1
n
x

Z-Transform Properties: Linearity
 Notation
 Linearity
– 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?
    x
Z
R
ROC
z
X
n
x 

 

        2
1 x
x
2
1
Z
2
1 R
R
ROC
z
bX
z
aX
n
bx
n
ax 



 


     
N
-
n
u
a
-
n
u
a
n
x n
n


Z-Transform Properties: Time Shifting
 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
    x
n
Z
o R
ROC
z
X
z
n
n
x o


 

 
 
4
1
z
z
4
1
1
1
z
z
X
1
1

















   
1
-
n
u
4
1
n
x
1
-
n








Z-Transform Properties: Multiplication by Exponential
 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
 Let’s find the z-transform of
    x
o
o
Z
n
o R
z
ROC
z
z
X
n
x
z 

 /
  1
z
:
ROC
z
-
1
1
n
u 1
-
Z


 

             
n
u
re
2
1
n
u
re
2
1
n
u
n
cos
r
n
x
n
j
n
j
o
n o
o 






  r
z
z
re
1
2
/
1
z
re
1
2
/
1
z
X 1
j
1
j o
o




 





Z-Transform Properties: Differentiation
 Example: We want the inverse z-transform of
 Let’s differentiate to obtain rational expression
 Making use of z-transform properties and ROC
   
x
Z
R
ROC
dz
z
dX
z
n
nx 


 

    a
z
az
1
log
z
X 1


 
   
1
1
1
2
az
1
1
az
dz
z
dX
z
az
1
az
dz
z
dX











     
1
n
u
a
a
n
nx
1
n




     
1
n
u
n
a
1
n
x
n
1
n





Z-Transform Properties: Conjugation
 Example
    x
*
*
Z
*
R
ROC
z
X
n
x 

 

   
     
        
 
n
x
Z
z
n
x
z
n
x
z
X
z
n
x
z
n
x
z
X
z
n
x
z
X
n
n
n
n
n
n
n
n
n
n















































Z-Transform Properties: Time Reversal
 ROC is inverted
 Example:
 Time reversed version of
   
x
Z
R
1
ROC
z
/
1
X
n
x 

 


   
n
u
a
n
x n

 
 
n
u
an
  1
1
1
-
1
-1
a
z
z
a
-
1
z
a
-
az
1
1
z
X 







Z-Transform Properties: Convolution
 Convolution in time domain is multiplication in z-domain
 Example:Let’s calculate the convolution of
 Multiplications of z-transforms is
 ROC: if |a|<1 ROC is |z|>1 if |a|>1 ROC is |z|>|a|
 Partial fractional expansion of Y(z)
        2
x
1
x
2
1
Z
2
1 R
R
:
ROC
z
X
z
X
n
x
n
x 

 


       
n
u
n
x
and
n
u
a
n
x 2
n
1 

  a
z
:
ROC
az
1
1
z
X 1
1 

    1
z
:
ROC
z
1
1
z
X 1
2 

 
     
  
1
1
2
1
z
1
az
1
1
z
X
z
X
z
Y 





  1
z
:
ROC
asume
az
1
1
z
1
1
a
1
1
z
Y 1
1











 

     
 
n
u
a
n
u
a
1
1
n
y 1
n




The z-Transform
z-Transform Theorems
and Properties

Linearity
x
R
z
z
X
n
x 
 ),
(
)]
(
[
Z
y
R
z
z
Y
n
y 
 ),
(
)]
(
[
Z
y
x R
R
z
z
bY
z
aX
n
by
n
ax 



 ),
(
)
(
)]
(
)
(
[
Z
Overlay of
the above two
ROC’s

Shift
x
R
z
z
X
n
x 
 ),
(
)]
(
[
Z
x
n
R
z
z
X
z
n
n
x 

 )
(
)]
(
[ 0
0
Z

Multiplication by an Exponential Sequence



 x
x- R
z
R
z
X
n
x |
|
),
(
)]
(
[
Z
x
n
R
a
z
z
a
X
n
x
a 

 
|
|
)
(
)]
(
[ 1
Z

Differentiation of X(z)
x
R
z
z
X
n
x 
 ),
(
)]
(
[
Z
x
R
z
dz
z
dX
z
n
nx 


)
(
)]
(
[
Z

Conjugation
x
R
z
z
X
n
x 
 ),
(
)]
(
[
Z
x
R
z
z
X
n
x 
 *)
(
*
)]
(
*
[
Z

Reversal
x
R
z
z
X
n
x 
 ),
(
)]
(
[
Z
x
R
z
z
X
n
x /
1
)
(
)]
(
[ 1


 
Z

Real and Imaginary Parts
x
R
z
z
X
n
x 
 ),
(
)]
(
[
Z
x
R
z
z
X
z
X
n
x
e 

 *)]
(
*
)
(
[
)]
(
[ 2
1
R
x
j R
z
z
X
z
X
n
x 

 *)]
(
*
)
(
[
)]
(
[ 2
1
Im

Initial Value Theorem
0
for
,
0
)
( 
 n
n
x
)
(
lim
)
0
( z
X
x
z 



Convolution of Sequences
x
R
z
z
X
n
x 
 ),
(
)]
(
[
Z
y
R
z
z
Y
n
y 
 ),
(
)]
(
[
Z
y
x R
R
z
z
Y
z
X
n
y
n
x 

 )
(
)
(
)]
(
*
)
(
[
Z

Convolution of Sequences






k
k
n
y
k
x
n
y
n
x )
(
)
(
)
(
*
)
(
 















n
n
k
z
k
n
y
k
x
n
y
n
x )
(
)
(
)]
(
*
)
(
[
Z
 









k
n
n
z
k
n
y
k
x )
(
)
(  









k
n
n
k
z
n
y
z
k
x )
(
)
(
)
(
)
( z
Y
z
X


The z-Transform
System Function

Signal Characteristics from Z-
Transform
 If U(z) is a rational function, and
 Then Y(z) is a rational function, too
 Poles are more important – determine key
characteristics of y(k)
m)
u(k
b
...
1)
u(k
b
n)
y(k
a
...
1)
y(k
a
y(k) m
1
n
1 
















 m
1
j
j
n
1
i
i
)
p
(z
)
z
(z
D(z)
N(z)
Y(z)
zeros
poles

Why are poles important?













m
1
j j
j
0
m
1
j
j
n
1
i
i
p
z
c
c
)
p
(z
)
z
(z
D(z)
N(z)
Y(z)






m
1
j
1
-
k
j
j
impulse
0 p
c
(k)
u
c
Y(k)
Z-
1
Z domain
Time domain
poles
componen
ts

Various pole values (1)
-1 0 1 2 3 4 5 6 7 8 9
0
0.5
1
1.5
2
2.5
-1 0 1 2 3 4 5 6 7 8 9
0
0.2
0.4
0.6
0.8
1
-1 0 1 2 3 4 5 6 7 8 9
0
0.2
0.4
0.6
0.8
1
-1 0 1 2 3 4 5 6 7 8 9
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-1 0 1 2 3 4 5 6 7 8 9
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 0 1 2 3 4 5 6 7 8 9
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
p=1.1
p=1
p=0.9
p=-1.1
p=-1
p=-0.9

Various pole values (2)
-1 0 1 2 3 4 5 6 7 8 9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-1 0 1 2 3 4 5 6 7 8 9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-1 0 1 2 3 4 5 6 7 8 9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
p=0.9
p=0.6
p=0.3
-1 0 1 2 3 4 5 6 7 8 9
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 0 1 2 3 4 5 6 7 8 9
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 0 1 2 3 4 5 6 7 8 9
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
p=-0.9
p=-0.6
p=-0.3

Conclusion for Real Poles
 If and only if all poles’ absolute values are
smaller than 1, y(k) converges to 0
 The smaller the poles are, the faster the
corresponding component in y(k) converges
 A negative pole’s corresponding component is
oscillating, while a positive pole’s
corresponding component is monotonous

How fast does it converge?
 U(k)=ak, consider u(k)≈0 when the absolute
value of u(k) is smaller than or equal to 2% of
u(0)’s absolute value
|
a
|
ln
4
k
3.912
ln0.02
|
a
|
kln
0.02
|
a
| k






11
0.36
4
|
0.7
|
ln
4
k
0.7
a







Rememb
er
This!
0 2 4 6 8 10 12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y(k)=0.7k
y(11)=0.0198

When There Are Complex Poles
…
U(z)
z
a
...
z
a
1
z
b
...
z
b
Y(z) n
n
1
1
m
m
1
1










c)...
bz
(az2


2a
4ac
b
b
z
2




0,
4ac
b2


)
2a
4ac
b
b
)(z
2a
4ac
b
b
a(z
c
bz
az
2
2
2 










0,
4ac
b2

 )
2a
b
4ac
i
b
)(z
2a
b
4ac
i
b
a(z
c
bz
az
2
2
2 










If
If
Or in polar coordinates,
)
ir
r
)(z
ir
r
a(z
c
bz
az2
θ
θ
θ
θ sin
cos
sin
cos 







What If Poles Are Complex
 If Y(z)=N(z)/D(z), and coefficients of both D(z) and N(z) are all real
numbers, if p is a pole, then p’s complex conjugate must also be a
pole
– Complex poles appear in pairs





















l
1
j
2
2
j
j
0
l
1
j j
j
0
r
)z
(2r
z
)
r
dz(z
bzr
p
z
c
c
ir
r
z
c'
ir
r
z
c
p
z
c
c
Y(z)
θ
θ
θ
θ
θ
θ
θ
cos
cos
sin
sin
cos
sin
cos
coskθ
dr
sinkθ
br
p
c
(k)
u
c
y(k) k
k
m
1
j
1
-
k
j
j
impulse
0 




 

Z-
1
Time domain

An Example
0 2 4 6 8 10 12 14 16 18 20
-1
-0.5
0
0.5
1
1.5
2
)
3
kπ
cos(
0.8
)
3
kπ
sin(
0.8
2
y(k)
0.64
0.8z
z
z
z
Y(z)
k
k
2
2









Z-Domain: Complex Poles
Time-Domain:
Exponentially Modulated Sin/C

Observations
 Using poles to characterize a signal
– The smaller is |r|, the faster converges the signal
 |r| < 1, converge
 |r| > 1, does not converge, unbounded
 |r|=1?
– When the angle increase from 0 to pi, the frequency of oscillation
increases
 Extremes – 0, does not oscillate, pi, oscillate at the maximum frequency

Change Angles
0.9
-0.9 Re
Im
0 5 10 15
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1

0 2 4 6 8 10 12 14
-6
-4
-2
0
2
4
6
8
10
12
Changing Absolute Value
Im
Re
1
0 5 10 15
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14
-3
-2
-1
0
1
2
3
4

Conclusion for Complex Poles
 A complex pole appears in pair with its
complex conjugate
 The Z-1-transform generates a combination of
exponentially modulated sin and cos terms
 The exponential base is the absolute value of
the complex pole
 The frequency of the sinusoid is the angle of
the complex pole (divided by 2π)

Steady-State Analysis
 If a signal finally converges, what value does it converge to?
 When it does not converge
– Any |pj| is greater than 1
– Any |r| is greater than or equal to 1
 When it does converge
– If all |pj|’s and |r|’s are smaller than 1, it converges to 0
– If only one pj is 1, then the signal converges to cj
 If more than one real pole is 1, the signal does not converge … (e.g. the ramp signal)

 k
dr
k
br k
k
cos
sin 




 

m
1
j
1
-
k
j
j
impulse
0 p
c
(k)
u
c
y(k) 2
1
-1
)
z
(1
z



An Example
k
k
0.9)
(
3
0.5
2
u(k)
0.9
z
3z
0.5
z
z
1
z
2z
U(z)











0 10 20 30 40 50 60
-1
0
1
2
3
4
5
6
converge to 2

Final Value Theorem
 Enable us to decide whether a system has a
steady state error (yss-rss)

Final Value Theorem
1
Theorem: If all of the poles of (1 ) ( ) lie within the unit circle, then
lim ( ) lim ( 1) ( )
k z
z Y z
y k z Y z


 
2
1 1
0.11 0.11
( )
1.6 0.6 ( 1)( 0.6)
0.11
( 1) ( ) | | 0.275
0.6
z z
z z
Y z
z z z z
z
z Y z
z
 
 
 
   

   
 0 5 10 15
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
k
y(k)
If any pole of (1-z)Y(z) lies out of or ON the
unit circle, y(k) does not converge!

What Can We Infer from TF?
 Almost everything we want to know
– Stability
– Steady-State
– Transients
 Settling time
 Overshoot
– …

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
H(z)
X(z) Y(z)
)
(
)
(
)
(
z
X
z
Y
z
H 

Nth-Order Difference Equation

 




M
r
r
N
k
k r
n
x
b
k
n
y
a
0
0
)
(
)
(

 




M
r
r
r
N
k
k
k z
b
z
X
z
a
z
Y
0
0
)
(
)
(







N
k
k
k
M
r
r
r z
a
z
b
z
H
0
0
)
(

Representation in Factored Form








 N
k
r
M
r
r
z
d
z
c
A
z
H
1
1
1
1
)
1
(
)
1
(
)
(
Contributes poles at 0 and zeros at cr
Contributes zeros at 0 and poles at dr

Stable and Causal Systems








 N
k
r
M
r
r
z
d
z
c
A
z
H
1
1
1
1
)
1
(
)
1
(
)
( Re
Im
Causal Systems : ROC extends outward from the outermost pole.

Stable and Causal Systems








 N
k
r
M
r
r
z
d
z
c
A
z
H
1
1
1
1
)
1
(
)
1
(
)
( Re
Im
Stable Systems : ROC includes the unit circle.
1

Example
Consider the causal system characterized by
)
(
)
1
(
)
( n
x
n
ay
n
y 


1
1
1
)
( 


az
z
H
Re
Im
1
a
)
(
)
( n
u
a
n
h n


Determination of Frequency Response
from pole-zero pattern
 A LTI system is completely characterized by its
pole-zero pattern.
)
)(
(
)
(
2
1
1
p
z
p
z
z
z
z
H




Example:
)
)(
(
)
(
2
1
1
0
0
0
0
p
e
p
e
z
e
e
H j
j
j
j



 



0

j
e
Re
Im
z1
p1
p2

pole-zero pattern.
)
)(
(
)
(
2
1
1
p
z
p
z
z
z
z
H




Example:
)
)(
(
)
(
2
1
1
0
0
0
0
p
e
p
e
z
e
e
H j
j
j
j



 



0

j
e
Re
Im
z1
p1
p2
|H(ej)|=? H(ej)=?

pole-zero pattern.
Example:
0

j
e
Re
Im
z1
p1
p2
|H(ej)|=? H(ej)=?
|H(ej)| =
| |
| | | | 1
2
3
H(ej) = 1(2+ 3 )

Example
1
1
1
)
( 


az
z
H
Re
Im
a
0 2 4 6 8
-10
0
10
20
0 2 4 6 8
-2
-1
0
1
2
dB

Digital Signal Processing and the z-transform

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