Chapter 3 introduce the z-transform. This chapter presents the following content: z-transform, properties of the ROC, inverse z-transform, properties of z-transform.
Trang 1Digital Signal Processing, III, Zheng-Hua Tan, 2006
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Digital Signal Processing, Fall 2006
Zheng-Hua Tan
Department of Electronic Systems Aalborg University, Denmark
zt@kom.aau.dk
Lecture 3: The z-transform
Digital Signal Processing, III, Zheng-Hua Tan, 2006
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Course at a glance
Discrete-time signals and systems
Fourier-domain
representation
DFT/FFT
System structures
Filter structures Filter design
Filter
z-transform
MM1
MM2
MM9,MM10
MM3
MM6
MM4
Sampling and
System analysis System
Trang 2Digital Signal Processing, III, Zheng-Hua Tan, 2006
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Part I: z-transform
z-transform
Limitation of Fourier transform
Fourier transform
Condition for the convergence of the infinite sum
If x[n] is absolutely summable, its Fourier transform exists
(sufficient condition)
)
( 2
1 ] [
] [ ) (
∫
∑
−
−
∞
−∞
=
=
=
π π
ω ω
ω ω
ω
n x
e n x e
X
n j j
n j
n j
∞
<
≤
≤
−∞
=
−
∞
−∞
=
−
∞
−∞
n j n
n j n
e
X ( ) | | [ ] | | [ ] || | | [ ] |
: 1
|
|
) 2 ( 1
1 ) ( : 1
1
1 ) ( : 1
|
| ] [ ] [
>
+ +
−
=
=
−
=
<
=
∑∞
−∞
=
−
−
a
k e
e X a
ae e
X a n u a n x
k j j
j j
n
π ω πδ ω ω
ω ω
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z-transform
Fourier transform
z-transform
The complex variable z in polar form
) ( ) (
r
|z| = = =
jω
re
z =
n j
n
e
∞
−∞
=
∑
=
) ( ] [
] [ )
n
↔
−∞
=
∑
n j n n
j
re X
z
∞
−
−
∞
∞
=
)
(
Digital Signal Processing, III, Zheng-Hua Tan, 2006
6
z-plane
z-transform is a function of a complex
variable Æ using the complex z-plane
Z-transform on unit circle
<-> Fourier transform Linear frequency axis in Fourier transform ÆUnit circle in z-transform (periodicity in freq of Fourier transform)
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Region of convergence – ROC
Fourier transform does not converge for all
sequences
z-transform does not converge for all sequences or
for all values of z.
ROC – for any given seq., the set of values of z for
which the z-transform converges
n j n
j x n e e
X ω ∞ −ω
−∞
=
∑
) (
∞
<
∑∞
−∞
=
−
|
|
| ] [
|
n
n
z n x
∑∞
−∞
=
− < ∞
n
n
r n
x [ ] |
|
n j n
n n
n
re
−∞
=
−
−
∞
−∞
=
X(z)
ROC is ring!
ROC
Outer boundary is a circle (may extend to infinity)
Inner boundary is a circle (may extend to include the
origin)
If ROC includes unit circle, Fourier transform
converges
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Zeros and poles
The most important and useful z-transforms –
rational function:
Zeros: values of z for which X(z)=0.
Poles : values of z for which X(z) is infinite
Close relation between poles and ROC
z z
Q z
P
z Q
z P z X
in s polynomial are
) ( and ) (
) (
) ( )
Digital Signal Processing, III, Zheng-Hua Tan, 2006
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Right-sided exponential sequence
z-transform
∑
∑
∞
=
−
∞
−∞
=
−
=
=
=
0
1) (
] [ )
(
] [
]
[
n
n n
n n n
az
z n u a z
X
n u
a
n
x
∞
<
∑∞
=
−
0
1
|
|
n
n
az
1
|
| , 1
1 ]
−
z n
u Z
|
|
| , 1
1 ) ( )
0
a z
z az az
z
X
n
−
=
−
=
=
−
∑
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Left-sided exponential sequence
z-transform
∑
∑
∑
∞
=
−
−
−∞
=
−
∞
−∞
=
−
−
=
−
=
−
−
−
=
−
−
−
=
0 1 1
) ( 1
] 1 [ )
(
] 1 [
]
[
n
n n
n n n
n n
n
z a z
a
z n u a z
X
n u
a
n
x
∞
<
∑∞
=
−
0
1
|
|
n
n
z
a
|
|
| , 1
1
)
a z
z az
z
−
=
−
Sum of two exponential sequence
) 3
1 )(
2
1 (
) 12
1 ( 2
3
1 1 1 2
1
1
1
) 3
1 ( ) 2
1
(
)
(
] [ ) 3
1 ( ]
[
)
2
1
(
]
[
1 1
0 1 0
1
+
−
−
= +
+
−
=
− +
=
− +
=
−
−
∞
=
−
∞
=
− ∑
∑
z z
z z z
z
z z
z
X
n u n
u
n
x
n
n n
n
n n
2
1
|
|
3
1
|
| and
2
1
|
|
1
| ) 3
1 (
| and 1
|
2
1
>
>
>
<
−
−
z
z z
z z
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Sum of two exponential sequence
Another way to calculate:
2
1
|
| 3
1 1 1 2
1 1
1 ]
[ ) 3
1 ( ]
[
)
2
1
(
3
1
|
| , 3
1 1
1 ]
[
)
3
1
(
2
1
|
| , 2
1 1
1 ]
[
)
2
1
(
] [ ) 3
1 ( ] [ ) 2
1
(
]
[
1 1
1 1
>
+
+
−
↔
− +
>
+
↔
−
>
−
↔
− +
=
−
−
−
−
z z
z
n u n
u
z z
n
u
z z
n
u
n u n
u n
x
Z n n
Z n
Z n
n n
Digital Signal Processing, III, Zheng-Hua Tan, 2006
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Two-sided exponential sequence
2
1
|
| , 3
1
|
| , 2
1 1 1 3
1
1
1
)
(
2
1
|
| , 2
1 1
1 ] 1
[
)
2
1
(
3
1
|
| , 3
1 1
1 ]
[
)
3
1
(
1 1
1 1
<
>
−
+ +
=
<
−
↔
−
−
−
>
+
↔
−
−
−
−
−
z z
z z
z
X
z z
n
u
z z
n
u
Z n
Z
n
] 1 [ ) 2
1 ( ] [
)
3
1
(
]
) 3
1 )(
2
1
(
) 12
1 (
2
+
−
−
=
z z
z
z
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Finite-length sequence
⎩
⎨
=
otherwise.
,
0
1 0
,
]
n
x
n
a z
a z z az
az
z a z
a
z
X
N N
N N
n N
n n N
n
n
−
−
=
−
−
=
=
=
−
−
−
−
−
=
−
−
∑
1 1
1
1 1
0 1
0
1 1
) (
1
) ( )
(
0 and
|
|
|
|
ROC
1
0
1
≠
∞
<
∞
<
∑−
=
−
z a
az
N
n
n
1 , , 1 ,
at
pole
1 , , 1 , 0 ,
) / 2
(
) / 2
(
−
=
=
=
−
=
=
N k
ae
z
a z
N k
ae
z
N k
j
k
N k
j
k
π
π
Some common z-transform pairs
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Part II: Properties of the ROC
Properties of the ROC
Digital Signal Processing, III, Zheng-Hua Tan, 2006
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Properties of the ROC
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Properties of the ROC
The algebraic expression
or pole-zero pattern does
not completely specify
the z-transform of a
must be specified!
Stability, causality and
the ROC
Part III: Inverse z-transform
Inverse z-transform
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Inverse z-transform
Needed for system analysis: 1) z-transform,
2) manipulation, 3) inverse z-transform.
Approaches:
Inspection method
Partial fraction expansion
Power series expansion
Digital Signal Processing, III, Zheng-Hua Tan, 2006
22
Inspection method
By inspection, e.g.
Make use of
2
1
|
| ), 2
1 1
1 (
)
(
1
>
−
=
z z
X
? 2
1
|
|
if z <
] [ ) 2
1 ( ]
[
x n = nu n
∴
|
|
|
| ), 1
1 (
)
az n
u
−
] 1 [ ) 2
1 ( ] [ course,
Trang 12Digital Signal Processing, III, Zheng-Hua Tan, 2006
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Partial fraction expansion
For rational function, get the format of a sum
of simpler terms, and then use the inspection
method.
Second-order z-transform
) 2
1 1 )(
4
1
1
(
1 )
(
1
1 −
− −
−
=
z z
z
X
] [ ) 4
1 ( ] [ )
2
1
(
2
]
) 2
1 1 (
2 ) 4
1
1
(
1 )
(
2
| ) ( ) 2
1
1
(
1
| ) ( ) 4
1
1
(
) 2
1 1 ( ) 4
1
1
(
)
(
1 1
2 / 1 1
2
4 / 1 1
1
1 2 1
1
−
−
=
−
=
−
−
−
−
+
−
−
=
=
−
=
−
=
−
=
−
+
−
=
z z
z
X
z X z
A
z X z
A
z
A z
A
z
X
z z
∑
∑
=
−
=
−
= N
k
k k
M
k
k k
z a
z b z
X
0
0
) (
∑
= N
k z d
A z
X
1
1
1 ) (
∑
∑
=
−
=
−
−
−
k
k
M
k k
z d
z c a
b z X
1
1 1
1
0 0
) 1
(
) 1 ( )
(
k
d z k
A = ( )( 1 − −1) |=
2
1
|
| , 8 1 4
3
1
1
)
(
2 1
>
+
−
=
−
z z
z
X
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25
What about M>=N?
) 1 )(
2
1 1 (
) 1 ( )
(
1 1
2 1
−
−
−
−
−
+
=
z z
z z
X
] [ 8 ] [ ) 2
1 ( 9 ] [ 2 ]
x = δ − n +
) 1 (
8 ) 2
1 1
(
9 2
)
(
8
| ) ( )
1
(
9
| ) ( ) 2
1
1
(
division.
long
by Found
2
) 1 ( ) 2
1 1 (
)
(
1 1
1 1
2
2 / 1 1
1
0
1 2 1
1 0
−
−
=
−
=
−
−
−
−
+
−
−
=
=
−
=
−
=
−
=
=
−
+
−
+
=
z z
z
X
z X z
A
z X z
A
B
z
A z
A B
z
X
z z
1
|
| , 2
1 2
3
1
2
1
)
(
2 1
2 1
>
+
−
+ +
=
−
−
−
−
z z z
z z z
X
1 5
2 3
2 1 2 1 2
3 2 1
1
1 2
1 2 1 2
−
−
+
−
+ + +
−
−
−
−
−
−
−
−
z
z z
z z z z
Digital Signal Processing, III, Zheng-Hua Tan, 2006
26
Power series expansion
By long division
] [ ]
x = n
|
|
|
| , 1
1 )
az z
−
1 1
−
−
−
az
1
1
1 1
2 2
2 2 1 1 1
2 2 1 1
−
−
−
−
−
−
−
−
−
−
+ + +
−
z a
z a az az az
z a az az
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Finite-length sequence
1 2
1 1 1
2
2
1 1 2
1 )
(
) 1 )(
1 )(
2
1 1 (
)
(
−
−
−
−
+
−
−
=
− +
−
=
z z
z
z
X
z z z
z
z
X
] 1 [ 2
1 ] [ ] 1 [ 2
1 ] 2 [
]
[ n = n + − n + − n + n −
Part IV: Properties of z-transform
Properties of z-transform
Trang 15Digital Signal Processing, III, Zheng-Hua Tan, 2006
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Linearity
Linearity
2 1 contains ROC
), ( )
( ]
[ ]
Z
R R z
bX z aX n bx
n
z z
z n
n
u
n
u
z z
z n
u
z z n
u
Z Z
Z
All , 1 1
1 ] [ ] 1 [
]
[
1
|
| , 1
]
1
[
1
|
| , 1
1
]
[
1 1 1
1 1
=
−
−
↔
=
−
−
>
−
↔
−
>
−
↔
−
−
−
−
−
δ
2
1
ROC ), ( ]
[
ROC ), ( ]
[
2 2
1 1
x Z
x Z
R z
X
n
x
R z
X
n
x
=
↔
=
n
z n x z
∞
−∞
=
∑
=
at least
Digital Signal Processing, III, Zheng-Hua Tan, 2006
30
Time shifting
Example
) 4
1 1
1 ( ] 1 [ ) 4
1 (
4
1 1
1 ] [ ) 4
1 (
) 4
1 1
1 ( 4
1 1 ) (
4
1
|
| , 4 1
1 ) (
1 1
1
1
1 1
1 1
−
−
−
−
−
−
−
−
−
↔
−
−
↔
−
=
−
=
>
−
=
z z
n u
z n
u
z
z z
z z X
z z
z X
Z n
Z n
)
or 0 for (except ROC
), ( ]
[
1
0 0
∞
=
↔
x
n Z
R
z X z n
n
n
z n x z
−∞
=
∑
=
Trang 16Digital Signal Processing, III, Zheng-Hua Tan, 2006
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Multiplication by exponential sequence
Examples
a z az
a z n
u
a
z z
n
u
Z
n
Z
>
−
=
−
↔
>
−
↔
−
−
−
|
| , 1
1 )
/ ( 1
1 ]
[
1
|
| , 1
1 ]
[
1 1
1
) (
]
[
ROC ), / ( ]
[
) (
0 0
0
0
=
↔
j F n
j
x
Z
n
e X n
x
e
|R
|z z
z X n
x
n
z n x z
∞
−∞
=
∑
=
1
|
| , cos
2 1
) cos 1 ( 1
2 1 1
2
1 )
(
] [ ) ( 2
1 ] [ ) ( 2
1 ] [ ) cos(
]
[
2 1 0
1 0 1
1 0
0 0
0 0
>
+
−
−
=
−
+
−
=
+
=
=
−
−
−
−
−
−
−
z z
z
z z
e z
e z
X
n u e n u e n u n n
x
j j
n j n
j
ω ω ω
ω ω
ω ω
Differentiation of X(z)
Example
] 1 [ ) ( ] [
] 1 [ ) ( ] [
|
|
|
| , 1
) ( ]
[ 1
) (
|
|
|
| ), 1
log(
) (
1 1
1 1 1
2 1
−
−
=
−
−
=
>
+
=
−
↔ +
−
=
>
+
=
−
−
−
−
−
−
−
n u n
a a n x
n u a a n nx
a z az
az dz
z dX z n nx
az
az dz
z dX
a z az
z X
n n Z
x
Z
R dz
z dX z
n
nx [ ] ↔ − ( ) , ROC = ( ) [ ]
n
n
z n x z
−∞
=
∑
=
Trang 17Digital Signal Processing, III, Zheng-Hua Tan, 2006
33
Conjugation of a complex sequence
x
Z
R z
X
n
x*[ ] ↔ *( *), ROC = ( ) [ ]
n
n
z n x z
∞
−∞
=
∑
=
Digital Signal Processing, III, Zheng-Hua Tan, 2006
34
Time reversal
Example
|
|
|
| , 1
1 )
(
] [ ]
[
1
−
−
<
−
=
−
=
a z az z
X
n u a
n
x
Z
R z
X
n
x [ − ] ↔ ( 1 / ), ROC = 1 / ( ) [ ]
n
n
z n x z
−∞
=
∑
=
|
|
|
| , 1
1 )
(
] [ ]
[
az z
X
n u a n
>
−
=
=
−
Trang 18Digital Signal Processing, III, Zheng-Hua Tan, 2006
35
Convolution of sequences
Example
?
]
[
]
[
]
[
] [ )
2
1
(
]
[
n
y
n
u
n
h
n u
n
=
=
2 1 contains
ROC
), ( ) ( ]
[
*
]
1
x x
Z
R R
z X z X n
x
n
x
∩
n
z n x z
∞
−∞
=
∑
=
) ) 2
1 1 ( 2 1 ) 1 (
1 ( 2
1 1 1
2
1
|
| , ) 1 )(
2
1 1 (
1 )
(
1
|
| , 1
1 ) (
2
1
|
| , 2
1 1
1 ) (
1 1
1 1 1 1
−
−
−
−
−
−
−
−
−
−
=
>
−
−
=
>
−
=
>
−
=
z z
z z z z
Y
z z z H
z z z
X
]) [ ) 2
1 ( ] [ ( 2
1
1
1
]
−
=
Initial-value theorem
) ( lim ] 0
x
z→∞
n
z n x z
−∞
=
∑
=
x[0]
lim ] [
] [ lim ) ( lim
=
=
=
−
∞
→
∞
−∞
=
−
∞
−∞
=
∞
→
∞
→
∑
∑
n z
n
n n
z z
z n x
z n x z
X
Trang 19Digital Signal Processing, III, Zheng-Hua Tan, 2006
37
Properties of z-transform
Digital Signal Processing, III, Zheng-Hua Tan, 2006
38
Summary
z-transform
Properties of the ROC
Inverse z-transform
Properties of z-transform
Trang 20Digital Signal Processing, III, Zheng-Hua Tan, 2006
39
Course at a glance
Discrete-time signals and systems
Fourier-domain
representation
DFT/FFT
System structures
Filter structures Filter design
Filter
z-transform
MM1
MM2
MM9,MM10
MM3
MM6
MM4
Sampling and reconstruction MM5
System analysis System