Signal processing - The z – Transform
1. The z – Transform
2. The Inverse z – Transform
3. Properties of the z – Transform
4. System Function of LTI Systems
5. LTI Systems Characterized by Linear
Constant – Coefficient Difference Equations
6. Connections between Pole – Zero Locations
and Time – Domain Behavior
7. The One – Sided z – Transform
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Nguyễn Công Phương
SIGNAL PROCESSING
The z – Transform
Contents
I. Introduction
II. Discrete – Time Signals and Systems
III.The z – Transform
IV. Fourier Representation of Signals
V. Transform Analysis of LTI Systems
VI. Sampling of Continuous – Time Signals
VII.The Discrete Fourier Transform
VIII.Structures for Discrete – Time Systems
IX. Design of FIR Filters
X. Design of IIR Filters
XI. Random Signal Processing
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The z – Transform
1. The z – Transform
2. The Inverse z – Transform
3. Properties of the z – Transform
4. System Function of LTI Systems
5. LTI Systems Characterized by Linear
Constant – Coefficient Difference Equations
6. Connections between Pole – Zero Locations
and Time – Domain Behavior
7. The One – Sided z – Transform
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The z – Transform (1)
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]
k k k
x n x k n k y n x k h n k h k x n kδ
∞ ∞ ∞
=−∞ =−∞ =−∞
= − → = − = −∑ ∑ ∑
for all[ ] ,
Re( ) Im( )
nx n z n
z z j z
=
= +
for all[ ] [ ] [ ] ,n k k n
k k
y n h k z h k z z n
∞ ∞
− −
=−∞ =−∞
→ = =
∑ ∑
( ) [ ]
k
k
H z h k z
∞
−
=−∞
= ∑
for all[ ] ( ) ,ny n H z z n→ =
for all for all[ ] , [ ] ( ) ,n nk k k k k
k k
x n c z n y n c H z z n= → =∑ ∑
The z – Transform (2)
• ROC (region of convergence):
the set of values of z for which
X(z) converges
• Zeros: the values of z for which
X(z) = 0
• Poles: the values of z for which
X(z) is infinite
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( ) [ ]
n
n
X z x n z
∞
−
=−∞
= ∑
0
ω
jz e ω=
1
Im( )z
Re( )z
Unit circle
z – plane
0
ω
jz re ω=
cosr ω
Im( )z
Re( )z
z – plane
sinr ω
r
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The z – Transform (3)
Ex. 1
1 21 2 3 4 5 1 2 3 4 5[ ] { }, [ ] { }.x n x n↑ ↑= =Given
Determine their z – transforms?
( ) [ ] n
n
X z x n z
∞
−
=−∞
= ∑
0 1 2 3 4
1 1 1 1 1 10 1 2 3 4( ) [ ] [ ] [ ] [ ] [ ]X z x z x z x z x z x z
− − − −
= + + + +
1 2 3 41 2 3 4 5z z z z− − − −= + + + +
ROC: entire z – plane except z = 0
2 1 0 1 2
2 2 2 2 2 22 1 0 1 2
( ) ( )( ) [ ] [ ] [ ] [ ] [ ]X z x z x z x z x z x z− − − − − −= − + − + + +
2 0 1 21 2 3 4 5z z z z z− −= + + + +
2 1 22 3 4 5z z z z− −= + + + +
ROC: entire z – plane except z = 0 & z = ∞
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The z – Transform (4)
Ex. 2
1 2 3 0[ ] [ ], [ ] [ ], [ ] [ ],x n n x n n k x n n k kδ δ δ= = − = + >
Determine their z – transforms?
( ) [ ]
n
n
X z x n z
∞
−
=−∞
= ∑
1 0 1 2
1 1 1 1 11 0 1 2
( )( ) ... [ ] [ ] [ ] [ ] ...X z x z x z x z x z− − − −= + − + + + +
1 0 1 20 1 0 0 1( )... ...z z z z− − − −= + + + + + =
ROC: entire z – plane
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The z – Transform (5)
Ex. 3
1 0
0 otherwise
,
[ ]
,
n M
x n
≤ ≤
=
Find the z – transforms of the square – pulse sequence
( ) [ ]
n
n
X z x n z
∞
−
=−∞
= ∑
ROC: |z| > 1
0
1
M
n
n
z−
=
=∑
1
2 3 11 if 1
1
... ,
M
N AA A A A A
A
+
−
+ + + + + = <
−
1
1
1
1
( )
( )
MzX z
z
− +
−
−
→ =
−
1 1 1z z−
0 1
Im
Re
ROC
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The z – Transform (6)
Ex. 4
Find the z – transforms of the sequence x[n] = anu[n]?
( ) [ ]
n
n
X z x n z
∞
−
=−∞
= ∑ 1
0 0
( )
n n n
n n
a z az
∞ ∞
− −
= =
= =∑ ∑
2 3 11 if 1
1
... ,A A A A
A
+ + + + = <
−
1
1
1
( )
zX z
az z a−
→ = =
− −
Zero: z = 0
Pole: z = a
1 1az z a− ROC: |z| > a
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The z – Transform (7)
Ex. 4
Find the z – transforms of the sequence x[n] = anu[n]? 1
1
1
( )
zX z
az z a−
= =
− −
Zero: z = 0; pole: z = a; ROC: |z| > a
0
1
n
0 1a< <
0
1
n
1a =
0
1
n
1a >
0
1
Im
Re
ROC
a
0
1
Im
Re
ROC
0
1
Im
Re
ROC
a
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The z – Transform (8)
Ex. 5
0 0
1
0
,
[ ] [ ]
,
n
n
n
x n a u n
a n
≥
= − − − =
− <
Find the z – transforms of the sequence
( ) [ ]
n
n
X z x n z
∞
−
=−∞
= ∑
1 1
1
( )
n n n
n n
a z az
− −
− −
=−∞ =−∞
= − = −∑ ∑
2 3 11 if 1
1
... ,A A A A
A
+ + + + = <
−
1
1
1
1
( )
zX z a z
a z z a
−
−
→ = − =
− −
Zero: z = 0
Pole: z = a
1 1a z z a− < → < ROC: |z| < a
1 1 2 21( ...)a z a z a z− − −= − + + +
[ ] [ ] ( )n
z
x n a u n X z
z a
= → =
−
Zero: z = 0; pole: z = a; ROC: |z| > a
0
n
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The z – Transform (9)
Ex. 6
0
0
,
[ ]
,
n
n
a n
x n
b n
≥
=
− <
Find the z – transforms of the sequence
( ) [ ]
n
n
X z x n z
∞
−
=−∞
= ∑
1
0
n n n n
n n
b z a z
− ∞
− −
=−∞ =
= − +∑ ∑
1
0
If 1 n n
n
z
az z a a z
z a
∞
− −
=
→ =
−
∑
( )
z zX z
z b z a
→ = +
− −
1
1If 1 n n
n
zb z z b b z
z b
−
− −
=−∞
< → < → − =
−
∑
Zero: z = 0
Pole: z = a, b
ROC: a < |z| < b
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The z – Transform (10)
0 n
0
Im
Re
ROC
a
0
n
0
Im
Re
ROC
b
b
a
Im
Re
ROC
b0 a
0 n
ROC: |z| > a ROC: |z| < a ROC: a < |z| < b
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The z – Transform (11)
Ex. 7
0 00 0 2[ ] (cos ) [ ], ,
nx n r n u n rω ω pi= > ≤ ≤Find the z – transforms of the sequence
0
0
(cos )
n n
n
r n zω
∞
−
=
=∑( ) [ ] n
n
X z x n z
∞
−
=−∞
= ∑
1 1
2 2
cos sin cos
j j j
e j e eθ θ θθ θ θ −= + → = +
0 01 1
0 0
1 1
2 2
( ) ( ) ( )
j jn n
n n
X z re z re zω ω
∞ ∞
−− −
= =
→ = +∑ ∑
2 2 1cos sin cos sinje jθ θ θ θ θ= + = + =
0 01 1 11 1 1j jre z re z rz z rω ω−− − −&
0 01 1
1 1 1 1 ROC
2 1 2 1
( ) , :j jX z z r
re z re zω ω−− −
→ = + >
− −
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The z – Transform (12)
Ex. 7
0 00 0 2[ ] (cos ) [ ], ,
nx n r n u n rω ω pi= > ≤ ≤Find the z – transforms of the sequence
0 0 0 0
0 0
1 1 1
1 1 1 2 2
0
1 1 2
2 1 1 2 1 2
( ) ( ) ( )
( )( ) [ ( cos ) ]
j j j j
j j
re z re z rz e e
re z re z r z r z
ω ω ω ω
ω ω ω
− −− − −
−− − − −
− + − − +
= =
− − − +
0 01 1
1 1 1 1
2 1 2 1
( ) j jX z
re z re zω ω−− −
= +
− −
2cos sin cosj j je j e eθ θ θθ θ θ−= + → + =
1
0
1 2 2
0
1
1 2
(cos )
( )
( cos )
r zX z
r z r z
ω
ω
−
− −
−
→ =
− +
0 0
0( cos )
( )( )
j j
z z r
z re z reω ω
ω
−
−
=
− −
Zero: z1 = 0; z2 = rcosω0
ROC: |z| > r
0 0
1 2Pole : ;
j jp re p reω ω−= =
0
r
Im
Re
ROC
1z 2z
1p
2p
The z – Transform (13)
• ROC
– There is no pole inside a ROC
– The ROC is a connected region
– For finite duration sequences, the ROC is the
entire z – plane, sometimes except for z = 0 and
z = ∞
• The z – transform
–We need both X(z) and its ROC
– X(z) is not defined outside the ROC
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The z – Transform (14)
The z – Transform
1. The z – Transform
2. The Inverse z – Transform
3. Properties of the z – Transform
4. System Function of LTI Systems
5. LTI Systems Characterized by Linear
Constant – Coefficient Difference Equations
6. Connections between Pole – Zero Locations
and Time – Domain Behavior
7. The One – Sided z – Transform
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The Inverse z – Transform (1)
11
2
[ ] ( ) n
C
x n X z z dzjpi
−
= ∫
1 2 1
0 1 2 1
1 2
1 21
( )...
( )
...
N
N
N
N
b b z b z b zX z
a z a z a z
− − − −
−
− − −
+ + +
=
+ + +
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The Inverse z – Transform (2)
Ex. 1
1
1 1
1
1 1 0 2
( )
( )( . )
zX z
z z
−
− −
+
=
− −
1
1 2
1 1 1 1
1
1 1 0 2 1 1 0 2( )( . ) .
z K K
z z z z
−
− − − −
+
= +
− − − −
1 1 1
1 21 1 0 2 1( . ) ( )z K z K z
− − −→ + = − + −
1 21 1 1 1 0 2 1 1 1( . ) ( )z K K= → + = − × + − 1 2 5.K→ =
2 1 5.K→ = −1 20 2 1 5 1 0 2 5 1 5. ( . ) ( )z K K= → + = − × + −
1 1
2 5 1 5
1 1 0 2
. .
( )
.
X z
z z− −
→ = −
− −
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The Inverse z – Transform (3)
Ex. 1
1
1
1
[ ] , ROC :
n
a u n z a
az−
→ >
−
If 1z >
1
1
2 5 2 5
1
1 5 1 5 0 2
1 0 2
.
. [ ]
.
. ( . ) [ ]
.
n
u n
z
u n
z
−
−
→
−→
− → −
−
1
1 1
1
1 1 0 2
( )
( )( . )
zX z
z z
−
− −
+
=
− −
1 1
2 5 1 5
1 1 0 2
. .
.z z− −
= −
− −
2 5 1 5 0 2[ ] . [ ] . ( . ) [ ]nx n u n u n→ = −
1
11
1
[ ] , ROC :
n
a u n z a
az−
− − − → <
−
If 0 2.z <
1
1
2 5 2 5 1
1
1 5 1 5 0 2 1
1 0 2
.
. [ ]
.
. ( . ) [ ]
.
n
u n
z
u n
z
−
−
→ − − −
−→
− → − −
−
2 5 1 1 5 0 2 1[ ] . [ ] . ( . ) [ ]nx n u n u n→ = − − − + − −
If 0 2 1. z< <
1
1
2 5 2 5 1
1
1 5 1 5 0 2
1 0 2
.
. [ ]
.
. ( . ) [ ]
.
n
u n
z
u n
z
−
−
→ − − −
−→
− → −
−
2 5 1 1 5 0 2[ ] . [ ] . ( . ) [ ]nx n u n u n→ = − − − −
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The Inverse z – Transform (4)
Ex. 2
1
1 2
1
1 2 5
( )
.
zX z
z z
−
− −
+
=
− +
1 2 1 25
1 21 2 5 0 0 5 1 5 1 58
.
,
. . . .
jz z p j e− − ±− + = → = ± =
1
1 2
1 2 1 1
1 2
1
1 2 5 1 1.
z K K
z z p z p z
−
− − − −
+
= +
− + − −
1 1 1
1 2 2 11 1 1( ) ( )z K p z K p z
− − −→ + = − + −
1 1 1 2 1 21 1 1 1 1/ ( / ) ( )z p p K p p K= → + = − + −
0 93
1 0 5 0 67 0 83
.. . . jK j e−→ = − =
0 93
2 0 5 0 67 0 83
.. . . jK j e→ = + =2 2 1 2 1 21 1 1 1 1/ ( ) ( / )z p p K K p p= → + = − + −
0 93 1 25 0 93 1 250 83 1 58 0 83 1 58. . . .[ ] . ( . ) [ ] . ( . ) [ ]j j n j j nx n e e u n e e u n− −→ = +
1 25 0 93 1 25 0 930 83 1 58 ( . . ) ( . . ). ( . ) ( )n j n j ne e− − −= +
1 25 0 93 1 25 0 93 2 1 25 0 93( . . ) ( . . ) cos( . . )j n j ne e n− − −+ = −
1 67 1 58 1 25 0 93 1 67 1 58 1 25 53 13o[ ] . ( . ) cos( . . ) [ ] . ( . ) cos( . . ) [ ]n nx n n u n n u n→ = − = −
The z – Transform
1. The z – Transform
2. The Inverse z – Transform
3. Properties of the z – Transform
4. System Function of LTI Systems
5. LTI Systems Characterized by Linear
Constant – Coefficient Difference Equations
6. Connections between Pole – Zero Locations
and Time – Domain Behavior
7. The One – Sided z – Transform
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Properties of the z – Transform
The z – Transform
1. The z – Transform
2. The Inverse z – Transform
3. Properties of the z – Transform
4. System Function of LTI Systems
5. LTI Systems Characterized by Linear
Constant – Coefficient Difference Equations
6. Connections between Pole – Zero Locations
and Time – Domain Behavior
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System Function of LTI Systems
(1)
[ ] [ ]* [ ] [ ] [ ]
k
y n x n h n x k h n k
∞
=−∞
= = −∑
z – transform
[ ]x n
[ ]y n( ) ( ) ( )Y z X z H z=
z – transform
[ ]h n
Inverse
z – transform
( )X z
( )H z
x n h n X z H z→[ ]* [ ] ( ) ( )
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System Function of LTI Systems
(2)Ex.
Given a system with impulse response h[n] = anu[n], |a| < 1, and an input
x[n] = u[n]. Find the output?
[ ] [ ]* [ ]y n h n x n= ( ) ( ) ( )Y z H z X z→ =
1
0
1
1
( ) ,n n
n
H z a z z a
az
∞
−
−
=
= = >
−
∑
1
0
1 1
1
( ) ,n
n
X z z z
z
∞
−
−
=
= = >
−
∑
1 1
1 1 1
1 1
( ) , max{ , }
( )( )
Y z z a
az z− −
= > =
− −
1 1
1 1 1
1 1 1
,
a
z
a z az− −
= − >
− − −
1
11 1
1 1
[ ] ( [ ] [ ]) [ ]
n
n ay n u n a u n u n
a a
+
+ −
= − =
− −
System Function of LTI Systems
(3)
• A system function H(z) with the ROC that is the
exterior of a circle, extending to infinity, is a
necessary condition for a discrete – time LTI
system to be causal, but not a sufficient one
• An LTI system is stable if and only if the ROC of
the system function H(z) includes the unit circle
|z| = 1
• An LTI system with rational H(z) is both causal
and stable if and only if all poles of H(z) are
inside the unit circle and its ROC is on the
exterior fo a circle, extending to infinity
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System Function of LTI Systems
(4)
H1(z)
[ ]x n y n[ ]
+
H2(z)
H1(z)+H2(z)
[ ]x n [ ]y n
H1(z)
[ ]x n [ ]y n
H2(z) H1(z)H2(z)
[ ]x n [ ]y n
The z – Transform
1. The z – Transform
2. The Inverse z – Transform
3. Properties of the z – Transform
4. System Function of LTI Systems
5. LTI Systems Characterized by Linear
Constant – Coefficient Difference Equations
6. Connections between Pole – Zero Locations and
Time – Domain Behavior
7. The One – Sided z – Transform
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LTI Systems Characterized by
LCCDE (1)
1 1
[ ] [ ] [ ]
N M
k k
k k
y n a y n k b x n k
= =
= − − + −∑ ∑
1 1
[ ] [ ] [ ]
N M
k k
k k
y n a y n k b x n k
= =
→ + − = −∑ ∑
[ ] ( )y n Y z→
1 1
[ ] ( ) : [ ] ( )
N N
k k
k k
k k
y n k z Y z a y n k a z Y z− −
= =
− → − →
∑ ∑
1 1
[ ] ( ) : [ ] ( )
M M
k k
k k
k k
x n k z X z b x n k b z X z− −
= =
− → − →
∑ ∑
1 1
1 ( ) ( )
N M
k k
k k
k k
a z Y z b z X z− −
= =
→ + =
∑ ∑
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LTI Systems Characterized by
LCCDE (2)
1 1
[ ] [ ] [ ]
N M
k k
k k
y n a y n k b x n k
= =
= − − + −∑ ∑
1 1
1 ( ) ( )
N M
k k
k k
k k
a z Y z b z X z− −
= =
→ + =
∑ ∑
1
1
1
( )
( )
( )
M
k
k
k
N
k
k
k
b z
Y z H z
X z
a z
−
=
−
=
→ = =
+
∑
∑
11
0 0 1
1 0 0
( ) ... ( )...( )
M
k M M M MM
k M
k
b bB z b z b z z z b z z z z z
b b
− − − −
=
= = + + + = − −
∑
1
1 1
1
1( ) ( ... ) ( )...( )
N
k N N N N
k N N
k
A z a z z z a z a z z p z p− − − −
=
= + = + + + = − −∑
1
1 1
0 0
1
1 1
1
1
( ) ( )
( )
( )
( )
( ) ( )
M M
M k k
k k
N NN
k k
k k
z z z z
B z zH z b b
A z z
z p p z
−
−
= =
−
−
= =
− −
→ = = =
− −
∏ ∏
∏ ∏
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LTI Systems Characterized by
LCCDE (3)
1 2 1 21 4 3 5 4 3( ) ( ) ( ) ( )z z Y z z z X z− − − −− + = − +
Ex.
1 2
1 2
5 4 3
1 4 3
( )
( ) .
( )
Y z z zH z
X z z z
− −
− −
− +
= =
− +
Consider a system function
Find its corresponding difference equation?
1
2
1
2
( ) [ ]
( ) [ ]
z X z x n
z X z x n
−
−
→ −
→ −
4 1 3 2 5 4 1 3 2[ ] [ ] [ ] [ ] [ ] [ ]y n y n y n x n x n x n→ − − + − = − − + −
4 1 3 2 5 4 1 3 2[ ] [ ] [ ] [ ] [ ] [ ]y n y n y n x n x n x n→ = − − − + − − + −
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LTI Systems Characterized by
LCCDE (4)
1
1
1
( )
M
k
k
k
N
k
k
k
b z
H z
a z
−
=
−
=
=
+
∑
∑
1
1 11
M N N
k k
k
k k k
AC z
p z
−
−
−
= =
= +
−
∑ ∑
1 1
[ ] [ ] ( ) [ ]
M N N
n
k k k
k k
h n C n k A p u nδ
−
= =
→ = − +∑ ∑
0 1 1 0
Stable : [ ]
M N N
n
k k k
n k k n
h n C A p
∞ − ∞
= = = =
= + < ∞∑ ∑ ∑ ∑
1 for 1,...,kp k N→ < =
A causal LTI with a rational system function is stable if and only if all poles
of H(z) are inside the unit circle in the z – plane. The zeros can be anywhere.
LTI Systems Characterized by
LCCDE (5)
• If N > 0: an Infinite Impulse Response (IIR) system
• If N = 0:
a Finite Impulse Response (FIR) system
• If N ≥ 1: a recursive system
• If N = 0: a nonrecursive system
• If ak = 0, for k = 1, , N: all – zero system
• If bk = 0, for k = 1, , M: all – pole system
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1
1
1 1 1 1
1
11
( ) [ ] [ ] ( ) [ ]
M
k
k M N N M N N
k nk k
k k k kN
k k k k kk
k
k
b z
AH z C z h n C n k A p u n
p z
a z
δ
−
− −
−=
−
− = = = =
=
= = + → = − +
−
+
∑
∑ ∑ ∑ ∑
∑
0
0
0
,
[ ] [ ]
, otherwise
M
n
k
k
b n M
h n b n kδ
=
≤ ≤
= − =
∑
The z – Transform
1. The z – Transform
2. The Inverse z – Transform
3. Properties of the z – Transform
4. System Function of LTI Systems
5. LTI Systems Characterized by Linear
Constant – Coefficient Difference Equations
6. Connections between Pole – Zero
Locations and Time – Domain Behavior
7. The One – Sided z – Transform
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Connections between Pole – Zero
Locations and Time – Domain Behavior (1)
th
th
1
1
1 1
first-order systemorder FIR1
order system
11
( )
( )
M
k
k M N N
kk k
kN
k k k k
k
NM Nk
N
b z
AH z C z
p z
a z
−
−
−=
−
− = =
−=
= = +
−
+
∑
∑ ∑
∑
1
1 0
N
k
k
k
a z−
=
+ =∑The equation has K1 real roots & 2K2 complex conjugate roots
1
0 1
1 1 1 2
1 21 1 1
*
*
b b zA A
pz p z a z a z
−
− − − −
+
+ =
− − + +
1 2 1
0 1
1 1 2
1 1 1 1 21 1
( )
K KM N
k k k k
k
k k kk k k
A b b zH z C z
p z a z a z
−
−
−
− − −
= = =
+
= + +
− + +
∑ ∑ ∑
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Connections between Pole – Zero
Locations and Time – Domain Behavior (2)
11
( ) , , real [ ] [ ]n
bH z a b h n ba u n
az−
= → =
−
0
1
Im
Re
0
1
n
Decaying alternating
exponential
0
1
n
Unit alternating step
0
1
n
Growing alternating
exponential
0
1
n
Decaying
exponential
0
1
n
Growing exponential
0
1
n
Unit step
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Connections between Pole – Zero
Locations and Time – Domain Behavior (3)
1
0 1 0 1
1 2 2
1 2 1 21
( )
( ) k k
k k
b b z z b z bH z
a z a z z a z a
−
− −
+ +
= =
+ + + +
2
1 1 21
1 2 1 2
0
4
0
2,
Zero : ; , pole :
a a ab
z z p
b
− ± −
−
= = =
02[ ] cos( ) [ ]
nh n A r n u nω θ= +
0
n
n
r
[ ]h n
0
1
Im
Rer1r <
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Connections between Pole – Zero
Locations and Time – Domain Behavior (4)
0
n
n
r
[ ]h n
0
1
Im
Re1r >
0
1
Im
Re1r = 0
n
1r =[ ]h n
The z – Transform
1. The z – Transform
2. The Inverse z – Transform
3. Properties of the z – Transform
4. System Function of LTI Systems
5. LTI Systems Characterized by Linear
Constant – Coefficient Difference Equations
6. Connections between Pole – Zero Locations
and Time – Domain Behavior
7. The One – Sided z – Transform
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The One – Sided z – Transform
two-sided/bilateral -transform( ) [ ] ( )n
n
X z x n z z
∞
−
=−∞
= ∑
0
one-sided/unilateral -transform( ) { [ ]} [ ] ( )n
n
X z Z x n x n z z
∞
+ + −
=
= =∑
1 2 31 1 0 1 2{ [ ]} [ ] [ ] [ ] [ ] ...Z x n x x z x z x z+ − − −− = − + + + +
1 1 21 0 1 2[ ] ( [ ] [ ] [ ] ...)x z x x z x z− − −= − + + + +
11[ ] ( )x z X z− += − +
1 21 2 1{ [ ]} [ ] [ ] ( )Z x n x x z z X z+ − − +− = − + − +
1
{ [ ]} ( ) [ ]
k
k m k
m
Z x n k z X z x m z+ − + −
=
− = + −∑
Các file đính kèm theo tài liệu này:
- z_transform_8914.pdf