Digital Signal Processing
Z-transform
dftwave
z-Transform
Background-Definition
- Fourier transform
n j
n
j
e n x e X
e e
= ] [ ) (
extracts the essence of x[n]
but is limited in the sense that it can handle stable systems only.
) (
e j
e X
converges if
< | ] [ | n x
i.e., stable system Fourier Transform converges
- So, we want to extend it such that it can be used as a tool to
analyze digital systems in general.
Let
n j
n
n j
r
e r n x e X
e e
= ) ] [ ( ) (
then it converges if
The condition for convergence is relaxed!
<
| ] [ |
n
r n x
(e.g.)
| |
2
1
1
2 | 2 | | ) ( |
| 2 | | ) ( |
] [ 2 ] [
r
r e r e X
e e X
n u n x
n n n j n n j
r
n j n j
n
= < =
=
=
e e
e e
converges if 2 | | > r
- This implies that can handle some systems that
cannot due to divergence.
- Therefore we define z-transform to be
) (
e j
r
e X
) (
e j
e X
=
= =
n
n
z re
j
r
z n x e X z X
jw
] [ | ) ( ) (
e
Representing the condition for convergence of
in terms of region of convergence RoC.
) (
e j
r
e X
(e.g.) in case x[n] = 2
n
u[n]
) (
e j
r
e X exists for |r|>2.
So, RoC is |z| = |re
je
|>2.
In general, if ] [ ] [ n u a n x
n
=
a z >
C R
o
is
- In terms of ,
) (z X
) (
jw
e X
is a special case
Where , or 1 = z 1 = r
2
causal
(e.g.) ] 1 [ ] [ = n u a n x
n
1
1
1
1
0
1
1
1
1
1
1 1
1
1
) ( 1
] 1 [ ) (
=
= =
= =
az
z a
z a
z a
z a z a
z a z n u a z X
n
n
n
n n
n
n n
n
n n
C R
o
1
1
<
z a
a z <
: , or
2=|a|
(e.g.) Two - sided sequence
1 1
1
0
2
1
1
1
3
1
1
1
)
2
1
( )
3
1
( ) (
+
=
=
z z
z z z X
n
n n
n
n n
] 1 [ )
2
1
( ] [ )
3
1
( ] [ = n u n u n x
n n
3
1
> z
2
1
< z
,
1/2
1/3
Some Common z-Transforms
(1)
(2)
(3)
(4)
(5)
(6)
(7)
] [ 1 ] [ z all n o
1
1
[ ] [ 1]
1
u n z
z
>
] 1 [
1
1
] 1 [
1
<
z
z
n u
] [
1
1
] [
1
a z
az
n u a
n
>
] 0 ,
, 0 , 0 [ ] [
<
>
m if except z all
m if except z all z m n
m
o
] [
1
1
] 1 [
1
a z
az
n u a
n
<
] [
) 1 (
] [
2 1
1
a z
az
az
n u na
n
>
(8)
(9)
(10)
(11)
(12)
(13)
] [
) 1 (
] 1 [
2 1
1
a z
az
az
n u na
n
<
] 1 [
cos 2 1
cos 1
] [ ] [cos
2 1
1
>
+
z
z z w
z w
n u n w
o
o
o
] [
cos 2 1
cos 1
] [ cos
2 2 1
1
r z
z r z w
rz w
n u n w r
o
o
o
n
>
+
] [
cos 2 1
sin
] [ sin
2 2 1
1
r z
z r z w
rz w
n u n w r
o
o
o
n
<
+
] 1 [
cos 2 1
sin
] [ ] [sin
2 1
1
<
+
z
z z w
z w
n u n w
o
o
o
] 0 [
1
1
]] [ ] [ [
1
>
z
az
z a
N n u n u a
N N
n
Properties of
C R
o
(1) in general
(2) absolutely converges
(3) cannot contain a pole
(4) FIR sequence entire z plane, may be except for 0 or
(5) Right-sided sequence outward of the outermost pole
(6) Left-sided sequence inward from the innermost pole
(7) Two-sided sequence a ring in between two adjacent rings
(8) is a connected region
C R
o
C R
o
s < < s
L o R
r C R r 0
) (
jw
e X
C R UC
o
c
a b c
(e.g.) If x[n] is a sum of 3 sequences whose poles
are a, b, c respectively,
There exist A possible s as shown below
C R
o
All right-sided
All left-sided
two left-sided
two right-sided
a b c
z-Transform Properties
(1) Linearity
(2) Time shifting
(e.g.)
) ( ) ( ] [ ] [
2 1 2 1
z bX z aX n bx n ax + +
) ( ] [ z X z n n x
o
n
o
and delay -1
], 1 [ )
4
1
(
] [ )
4
1
(
4
1
1
1
4
1
1
1
1
1
1
1
n u
n u
z
z
z
z
n
n
4
1
> z
z-Transform Properties..(cont.)
jw
e z =
(3) Multiplication by an Exponential Sequence
) ( ] [
o
n
o
z
z
X n x z
(e.g.)
(e.g.)
) ( ) ( ] [ ) (
) (
o o o
w w j jw n jw
e X z e X n x e
] [ ] ) ( ) [(
2
1
] [ ] ) ( ) [(
2
] [ cos
n u re re
n u e e
r
n u n w r
n jw n jw
n jw n jw
n
o
n
o o
o o
+ =
+ =
2 2 1
1
1 1
cos 2 1
cos 1
]
1
1
1
1
[
2
1
+
=
z r z w r
z w r
z re z re
o
o
jw jw
o
r z >
z-Transform Properties..(cont.)
(4) Differentiation of X(z)
(e.g.)
[ ] [ ]
o x
d
nx n z x z R C R
dz
=
a z az z X > + =
) 1 log( ) (
1
1
2
1
) (
=
az
az
dz
z dX
1
1
1
1
1
1
1
) (
+
=
+
=
az
z a
az
az
z X
dz
d
z
] 1 [ ) 1 ( ] [
] 1 [ ) ( ] [
1
1
=
=
n u
n
a
n x
n u a a n nx
n
n
n
z-Transform Properties..(cont.)
(5) Conjugation of Complex Sequence
(6) Time-Reversal
(7) Convolution-Integration
X o
R C R z X n x = ) ( ] [
* * *
)
1
( ] [
1
)
1
( ] [
*
* *
z
X n x
R
C R
z
X n x
X
o
=
) ( ) ( ] [ * ] [
2 1 2 1
z X z X n x n x
Inverse z-Transform
4-Ways:
Inversion by Contour Integration
Cauchy integral definition of the inverse z-
Transform
Example: Inverse DTFT
Implies.. Contour C is chosen as unit circle
Inversion Method -1
Concept of Partial Fraction Expansion-
Concept of Partial Fraction Expansion Inversion-
1. Find partial fraction expansion method in third
equivalent form
2. Invert by expansion
Inversion Method
Doing the Partial Fraction Expansion
Doing the Partial Fraction Expansion-2
Writing Down x[n]
X[n] depend on knowing the ROC
Example- ROC
a b c
If x[n] is a sum of 3 sequences whose poles
are a, b, c respectively,
There exist A possible s as shown below
C R
o
All right-sided
two left-sided
a b c
Example- Partial Fraction
Long Division
Finding the coefficients of Poles
Writing Down x[n]
Partial Fraction Expansion in MATLAB
Selected z-Transform Theorems
An IIR System
IIR Frequency Response
System Function Of a Difference Equation
H[z] and h[n]
Frequency Response of a DE
LTI System Characterization
Stability, Causality- illustration
(1)
Causal
. Stable
Outward
UC RoC
] [ )
2
1
( ] [ n u n x
n
=
n n
n
z z X
= )
2
1
( ) (
0
1
2
1
1
1
=
z
c
2
1
> z
2
1
1
2
1
1
RoC :
Stability, Causality- illustration..(cont)
Anti Causal
Unstable
Inward
UC RoC
] 1 [ )
2
1
( ] [ = n u n x
n
1
1
2
1
1
1
)
2
1
( ) (
=
= =
z
z z X
n n
n
2
1
< z
.
2
1
2
1
ROC :
(2)
Stability, Causality- illustration..(cont)
(3)
Causal
Unstable
RoC :
Outward
UC RoC
] [ ) 2 ( ] [ n u n x
n
=
1
0
2 1
1
2 ) (
=
= =
z
z z X
n
n
n
2 > z
.
1 2
1 2
Stability, Causality- illustration..(cont)
(4)
Anti Causal
Stable
Inward
UC RoC
What do you find?
] 1 [ 2 ] [ = n u n x
n
1
1
2 1
1
2 ) (
=
= =
z
z z X
n
n
n
2 < z
c
1
2
RoC :
