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The z-Transform
主講人:虞台文
Content
Introduction z-Transform Zeros and Poles Region of Convergence Important z-Transform Pairs Inverse z-Transform z-Transform Theorems and Properties System Function
The z-Transform
Introduction
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
The z-Transform
z-Transform
Definition The z-transform of sequence x(n) is defined by
n
nznxzX )()(
Let z = ej.
( ) ( )j j n
n
X e x n e
Fourier Transform
z-Plane
Re
Im
z = ej
n
nznxzX )()(
( ) ( )j j n
n
X e x n e
Fourier Transform is to evaluate z-transform on a unit circle.
Fourier Transform is to evaluate z-transform on a unit circle.
z-Plane
Re
Im
X(z)
Re
Im
z = ej
Periodic Property of FT
Re
Im
X(z)
X(ej)
Can you say why Fourier Transform is a periodic function with period 2?
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 znxznxzX |||)(|)(|)(|
ROC is centered on origin and consists of a set of rings.
ROC is centered on origin and consists of a set of rings.
Example: Region of Convergence
Re
Im
n
n
n
n znxznxzX |||)(|)(|)(|
ROC is an annual ring centered on the origin.
ROC is an annual ring centered on the origin.
xx RzR ||
r
}|{ xx
j RrRrezROC
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
)()( nuanx n )()( nuanx n
1 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8
n
x(n)
. . .
Example: A right sided Sequence
)()( nuanx n )()( nuanx n
n
n
n znuazX
)()(
0n
nn za
0
1)(n
naz
For convergence of X(z), we require that
0
1 ||n
az 1|| 1 az
|||| az
az
z
azazzX
n
n
10
1
1
1)()(
|||| az
aa
Example: A right sided Sequence ROC for x(n)=anu(n)
|||| ,)( azaz
zzX
|||| ,)( az
az
zzX
Re
Im
1aa
Re
Im
1
Which one is stable?Which one is stable?
Example: A left sided Sequence
)1()( nuanx n )1()( nuanx n
1 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8n
x(n)
. . .
Example: A left sided Sequence
)1()( nuanx n )1()( nuanx n
n
n
n znuazX
)1()(
For convergence of X(z), we require that
0
1 ||n
za 1|| 1 za
|||| az
az
z
zazazX
n
n
10
1
1
11)(1)(
|||| az
n
n
n za
1
n
n
n za
1
n
n
n za
0
1
aa
Example: A left sided Sequence ROC for x(n)=anu( n1)
|||| ,)( azaz
zzX
|||| ,)( az
az
zzX
Re
Im
1aa
Re
Im
1
Which one is stable?Which one is stable?
The z-Transform
Region of Convergence
Represent z-transform as a Rational Function
)(
)()(
zQ
zPzX 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) =
Example: A right sided Sequence
)()( nuanx n |||| ,)( azaz
zzX
Re
Im
a
ROC is bounded by the pole and is the exterior of a circle.
Example: A left sided Sequence
)1()( nuanx n|||| ,)( az
az
zzX
Re
Im
a
ROC is bounded by the pole and is the interior of a circle.
Example: Sum of Two Right Sided Sequences
)()()()()( 31
21 nununx nn
31
21
)(
z
z
z
zzX
Re
Im
1/2
))((
)(2
31
21
121
zz
zz
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()()()()( 21
31 nununx nn
21
31
)(
z
z
z
zzX
Re
Im
1/2
))((
)(2
21
31
121
zz
zz
1/3
1/12
ROC is bounded by poles and is a ring.
ROC does not include any pole.
Example: A Finite Sequence
10 ,)( Nnanx n
nN
n
nN
n
n zazazX )()( 11
0
1
0
Re
ImROC: 0 < z <
ROC does not include any pole.
1
1
1
)(1
az
az N
az
az
z
NN
N
1
1
N-1 poles
N-1 zeros
Always StableAlways 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 includ
es the unit circle. The ROC cannot include any poles Finite Duration Sequences: The ROC is the entire z-plane except possibl
y z=0 or z=. Right sided sequences: The ROC extends outward from the outermost fi
nite pole in X(z) to z=. Left sided sequences: The ROC extends inward from the innermost nonz
ero 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
Find the possible ROC’s
More on Rational z-Transform
Re
Im
a b c
Consider the rational z-transform with the pole pattern:
Case 1: A right sided Sequence.
More on Rational z-Transform
Re
Im
a b c
Consider the rational z-transform with the pole pattern:
Case 2: A left sided Sequence.
More on Rational z-Transform
Re
Im
a b c
Consider the rational z-transform with the pole pattern:
Case 3: A two sided Sequence.
More on Rational z-Transform
Re
Im
a b c
Consider the rational z-transform with the pole pattern:
Case 4: Another two sided Sequence.
The z-Transform
Important
z-Transform Pairs
Z-Transform Pairs
Sequence z-Transform ROC
)(n 1 All z
)( mn mz All z except 0 (if m>0)or (if m<0)
)(nu 11
1 z
1|| z
)1( nu 11
1 z
1|| z
)(nuan 11
1 az
|||| az
)1( nuan 11
1 az
|||| az
Z-Transform Pairs
Sequence z-Transform ROC
)(][cos 0 nun 210
10
]cos2[1
][cos1
zz
z1|| z
)(][sin 0 nun 210
10
]cos2[1
][sin
zz
z1|| z
)(]cos[ 0 nunr n 2210
10
]cos2[1
]cos[1
zrzr
zrrz ||
)(]sin[ 0 nunr n 2210
10
]cos2[1
]sin[
zrzr
zrrz ||
otherwise0
10 Nnan
11
1
az
za NN
0|| z
The z-Transform
Inverse z-Transform
The z-Transform
z-Transform Theorems and Properties
Linearity
xRzzXnx ),()]([Z
yRzzYny ),()]([Z
yx RRzzbYzaXnbynax ),()()]()([Z
Overlay of the above two
ROC’s
Shift
xRzzXnx ),()]([Z
xn RzzXznnx )()]([ 0
0Z
Multiplication by an Exponential Sequence
xx- RzRzXnx || ),()]([Z
xn RazzaXnxa || )()]([ 1Z
Differentiation of X(z)
xRzzXnx ),()]([Z
xRzdz
zdXznnx
)()]([Z
Conjugation
xRzzXnx ),()]([Z
xRzzXnx *)(*)](*[Z
Reversal
xRzzXnx ),()]([Z
xRzzXnx /1 )()]([ 1 Z
Real and Imaginary Parts
xRzzXnx ),()]([Z
xRzzXzXnxe *)](*)([)]([ 21R
xj RzzXzXnx *)](*)([)]([ 21Im
Initial Value Theorem
0for ,0)( nnx
)(lim)0( zXxz
Convolution of Sequences
xRzzXnx ),()]([Z
yRzzYny ),()]([Z
yx RRzzYzXnynx )()()](*)([Z
Convolution of Sequences
k
knykxnynx )()()(*)(
n
n
k
zknykxnynx )()()](*)([Z
k
n
n
zknykx )()(
k
n
n
k znyzkx )()(
)()( zYzX
The z-Transform
System Function
Shift-Invariant System
h(n)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)H(z)X(z) Y(z)
)(
)()(
zX
zYzH
)(
)()(
zX
zYzH
Nth-Order Difference Equation
M
rr
N
kk rnxbknya
00
)()(
M
rr
N
kk rnxbknya
00
)()(
M
r
rr
N
k
kk zbzXzazY
00
)()(
N
k
kk
M
r
rr zazbzH
00)(
N
k
kk
M
r
rr zazbzH
00)(
Representation in Factored Form
N
kr
M
rr
zd
zcAzH
1
1
1
1
)1(
)1()(
N
kr
M
rr
zd
zcAzH
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
kr
M
rr
zd
zcAzH
1
1
1
1
)1(
)1()(
N
kr
M
rr
zd
zcAzH
1
1
1
1
)1(
)1()( Re
ImCausal Systems : ROC extends outward from the outermost pole.
Stable and Causal Systems
N
kr
M
rr
zd
zcAzH
1
1
1
1
)1(
)1()(
N
kr
M
rr
zd
zcAzH
1
1
1
1
)1(
)1()( Re
ImStable Systems : ROC includes the unit circle.
1
Example
Consider the causal system characterized by
)()1()( nxnayny
11
1)(
azzH 11
1)(
azzH
Re
Im
1
a
)()( nuanh n
Determination of Frequency Response from pole-zero pattern
A LTI system is completely characterized by its pole-zero pattern.
))(()(
21
1
pzpz
zzzH
))(()(
21
1
pzpz
zzzH
Example:
))(()(
21
1
00
0
0
pepe
zeeH jj
jj
))(()(
21
1
00
0
0
pepe
zeeH jj
jj
0je
Re
Im
z1
p1
p2
Determination of Frequency Response from pole-zero pattern
A LTI system is completely characterized by its pole-zero pattern.
))(()(
21
1
pzpz
zzzH
))(()(
21
1
pzpz
zzzH
Example:
))(()(
21
1
00
0
0
pepe
zeeH jj
jj
))(()(
21
1
00
0
0
pepe
zeeH jj
jj
0je
Re
Im
z1
p1
p2
|H(ej)|=?|H(ej)|=? H(ej)=?H(ej)=?
Determination of Frequency Response from pole-zero pattern
A LTI system is completely characterized by its pole-zero pattern.
Example:
0je
Re
Im
z1
p1
p2
|H(ej)|=?|H(ej)|=? H(ej)=?H(ej)=?
|H(ej)| =| |
| | | | 1
2
3
H(ej) = 1(2+ 3 )
Example
11
1)(
azzH 11
1)(
azzH
Re
Im
a
0 2 4 6 8-10
0
10
20
0 2 4 6 8-2
-1
0
1
2d
B
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