frequency domain transformation

1. Chapter 6
The z-Transform

2. Outline
o Introduction
o Z-Transform
o Zeros and Poles
o Region of Convergence
o Inverse z-Transform
o Geometric Evaluation of FT from P-Z plot
o Z-Transform Theorems and Properties
o System Function

3. The z-Transform
Introduction

4. Why z-Transform?
 A generalization of Fourier transform
 Why generalize it?
– FT does not converge on all sequence
– Bring the complex variable theory deal with the discrete-time
signals and systems

5. The z-Transform
z-Transform

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

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

8. z-Plane
Re
Im
X(z)
Im
z = ej

Re

9. Periodic Property of FT
Re
Im
X(z)

 
X(ej)

10. The z-Transform
Zeros and Poles

11. 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(n) || z |n
 
n

| X (z) | x(n)zn
n
ROC is centered on origin and
consists of a set of rings.

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

13. 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.

14. Example: A right sided Sequence
x(n)  an
u(n)
-8 -7 -6 -5 -4 -3 -2 -1
n
x(n)
. . .
1 2 3 4 5 6 7 8 9 10

15. Example: A right sided Sequence
x(n)  an
u(n)

X (z)  an
u(n)zn
n

 an
zn
n0

 (az1
)n
n0
For convergence of X(z), we
require that

| az1
|  
n0
| az1
|1
| z || a |

1

z
1 az1
z  a

n0
1 n
X (z)  (az )
| z || a |

16. a
a
Example: A right sided Sequence
ROC for x(n)=anu(n)
, | z || a |
z
z  a
X (z) 
Re
Im
1
a
a
Re
Im
1
Which one is stable?

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

18. Example: A left sided Sequence
x(n)  an
u(n 1)

X (z)   an
u(n 1)zn
n
For convergence of X(z), we
require that

| a1
z |  
n0
| a1
z |1
| z || a |
z


1
1 a1
z z  a
X (z) 1 (a1
z)n
1
n0
| z || a |
  an
zn
n
1

 an
zn
n1

1 an
zn
n0

19. a
a
Example: A left sided Sequence
ROC for x(n)=anu( n1)
, | z || a |
z
z  a
X (z) 
Re
Im
1
a
a
Re
Im
1
Which one is stable?

20. The z-Transform
Region of
Convergence

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

22. Example: A right sided Sequence
x(n)  an
u(n) , | z || a |
X (z) 
z  a
z
Re
Im
a
ROC is bounded by the
pole and is the exterior
of a circle.

23. Example: A left sided Sequence
x(n)  an
u(n 1) , | z || a |
X (z) 
z  a
z
Re
Im
a
ROC is bounded by the
pole and is the interior
of a circle.

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

25. Example: A Two Sided Sequence
x(n)  ( 1
)n
u(n) (1
)n
u(n 1)
3 2
2
z  1
3
z  1

Re
X (z) 
Im
1/2
2
3
 12
(z  1 )(z  1 )
z z 2z(z  1 )
1/3
1/12
ROC is bounded by poles
and is a ring.
ROC does not include any pole.

26. Example: A Finite Sequence
n n 1 n
N 1 N 1
n0 n0
x(n)  an
, 0  n  N 1
X (z)  a z  (az )
Re
Im
ROC: 0 < z < 
ROC does not include any pole.
1 (az1
)N

1 az z z  a
 aN
zN

1 N 1
1
N-1 poles
N-1 zeros
Always Stable

27. 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.

28. More on Rational z-Transform
Re
a b c
Consider the rational z-transform
with the pole pattern:
Im
Find the possible
ROC’s

29. More on Rational z-Transform
Consider the rational z-transform
with the pole pattern:
Im
Case 1:Aright sided Sequence.
a b c
Re

30. More on Rational z-Transform
Consider the rational z-transform
with the pole pattern:
Im
Case 2:Aleft sided Sequence.
a b c
Re

31. More on Rational z-Transform
Consider the rational z-transform
with the pole pattern:
Im
Case 3:Atwo sided Sequence.
a b c
Re

32. 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

33. The z-Transform
Important
z-Transform Pairs

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

35. Z-Transform Pairs
Sequence z-Transform ROC
| z |1
0
[cos n]u(n)
0
0
1[2cos ]z1
 z2
1[cos ]z1
0
[sin  n]u(n)
0
0
1[2cos ]z1
 z2
[sin  ]z1
| z |1
0
[rn
cos n]u(n)
0
0
1[2r cos ]z1
 r2
z2
1[r cos ]z1
| z | r
0
[rn
sin  n]u(n)
0
0
1[2r cos ]z1
 r2
z2
[r sin ]z1
| z | r

0  n  N 1
0 otherwise
an
1 aN
zN
1 az1
| z | 0

36. The z-Transform
Inverse z-Transform

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

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

39. Determination of Frequency Response
from pole-zero pattern
e j0
Re
 A LTI system is completely characterized by its
pole-zero pattern.
Im
Example:
z1
p1
p2
|H(ej)| =
| |
| | | | 1
2
3
H(ej) = 1(2+ 3 )

40. The z-Transform
z-Transform Theorems
and Properties

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

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

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

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

45. Conjugation
z  Rx
Z[x(n)]  X (z),
z  Rx
Z[x*(n)]  X *(z*)

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

47. Real and Imaginary Parts
Z[x(n)]  X (z), z  Rx
z  Rx
zRx
[X (z)  X *(z*)]
Re[x(n)] 
Im[x(n)]
1
2
1
2 j
[X (z)  X *(z*)]

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

49. Convolution of Sequences
z  Rx
z  Ry
Z[x(n)]  X (z),
Z[y(n)]  Y(z),
z Rx  Ry
Z[x(n)* y(n)] X (z)Y(z)

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

51. The z-Transform
System Function

52. Shift-Invariant System
x(n) y(n)=x(n)*h(n)
X(z) Y(z)=X(z)H(z)
h(n)
H(z)

53. Shift-Invariant System
H(z)
X(z) Y(z)
X (z)
H(z) 
Y(z)

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

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

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

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

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