Dept. of Electronics Eng. DH26029 Signals and Systems
-1-
)or 0for expect (ROC with ),(][
ROC with ),(][
0
0
RzXznnx
RzXnx
nZ
Z
212121
222
111
contaning ROC with ),()(][][
ROC with ),(][
ROC with ),(][
RRzbXzaXnbxnax
RzXnx
RzXnx
Z
Z
Z
)(Fig.10.15 )(][
: case Special
ROC with ,][
ROC with ),(][
00
0
0
0
0
0
zeXnxe
ez
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zXnxz
RzXnx
jnj
j
n
Z
Z
Z
10.5 Properties of the z-Transform
10.5.3 Scaling in the z-Domain
10.5.2 Time Shifting
10.5.1 Linearity
Dept. of Electronics Eng. DH26029 Signals and Systems
-2-
RzXnx
RzXnx
1 ROC with ,
1][
ROC with ),(][
Z
Z
kk
k
k
RzXnx
RzXnx
kn
knknxnx
1 ROC with ),(][
ROC with ),(][
of multiple anot is if ,0
of multiple a is if ],[][
Z
Z
10.5.4 Time Reversal
10.5.5 Time Expansion
Dept. of Electronics Eng. DH26029 Signals and Systems
-3-
?at ,at zero)(or ploe a has )( If
)()( real is ][ Note)
ROC with ,][
ROC with ),(][
00
zzzzzX
zXzXnx
RzXnx
RzXnx
Z
Z
10.56) Problem :n (Derivatio
contaning ROC with ),()(][][
ROC with ),(][
ROC with ),(][
212121
222
111
RRzXzXnxnx
RzXnx
RzXnx
Z
Z
Z
10.5.6 Conjugation
10.5.7 The Convolution Property
Dept. of Electronics Eng. DH26029 Signals and Systems
-4-
Rdz
zdXznnx
RzXnx
ROC with ,)(
][
ROC with ),(][
Z
Z
]1[)(
][
shifting) Time( ,1
]1[)(
Linearity)( ,1
][)(
10.1) Example( ,1
1][ From
, 1
)(][
1
11
1
1
1
1
nun
anx
azaz
aznuaa
azaz
anuaa
azaz
nua
azaz
az
dz
zdXznnx
n
n
n
n
Z
Z
Z
Z
azazzX ),1log()( 1Ex. 10.27)
10.5.8 Differentiation in the z-Domain
Dept. of Electronics Eng. DH26029 Signals and Systems
-5-
mean?) thisdoes(What finite. is )(lim
finite:]0[ ],[ causal aFor Note)
0for 1
0for 0,
][ causalfor ][)( pf)
)(lim]0[0,0][
0
zX
xnx
nz
nzz
nxznxzX
zXxnnx
z
n
n
n
n
z
Checking the correctness of the z-transform calculation for a signal
10.3) (Example
11
1)(
1
211
31
1
23
zz
zzX
consistent :1]0[1)(lim
xzXz
][2
16][
3
17][ nununx
nn
Ex. 10.19)
10.5.9 The Initial-Value Theorem Causal Sequence
10.5.10 Summary of Properties (Table 10.1, p. 775)
r)denominato(Onumerator)(O
poles finite of # zeros finite of #
Dept. of Electronics Eng. DH26029 Signals and Systems
-6-
For discrete-time LTI systems,
)()()( zXzHzY
system theof responsefrequency :)( jeH
system theoffunction r or transfefunction system :)(zH
jez
10.7 Analysis and Characterization of LTI Systems
Using z-Transforms
Dept. of Electronics Eng. DH26029 Signals and Systems
-7-
Property 8 : X(z) : rational, x[n] : right sided
=> ROC : the region in the z-plane outside the outermost pole.
For a causal system
0
][)(n
nznhzH
A discrete-time LTI system is causal
iff the ROC is the exterior of a circle, including infinity.
A discrete-time LTI system with rational system function H(z)
is causal iff
(a) The ROC is the exterior of a circle outside
the outermost pole (property 8)
(b) O (numerator) ≤ O (denominator)
10.7.1 Causality
Dept. of Electronics Eng. DH26029 Signals and Systems
-8-
k
kh ][ Absolutely Summable
(Sufficient and Necessary)
An LTI system is stable iff the ROC of its system function
H(z) includes the unit circle.
A causal LTI system with rational system function H(z) is stable
iff all of the poles of H(z) lie inside the unit circle -
i.e., they must all have magnitude smaller than 1.
10.7.2 Stability
Dept. of Electronics Eng. DH26029 Signals and Systems
-9-
stable)nor causal(neither 122
1][ ,
2
1 : ROC
stable)but (noncausal ]1[2][2
1][
22
1 : ROC However,
unstable)but (causal ][22
1][2,
21
1
1
1)(
11
21
nunhz
nununh
z
nunhzzz
zH
n
n
n
n
n
n
Ex. 10.22)
Dept. of Electronics Eng. DH26029 Signals and Systems
-10-
]1[3
1][]1[
2
1][ nxnxnyny
)()()()( 1
311
21 zXzzXzYzzY
1
21
1
31
1
1)()(
z
zzXzY
1
21
1
31
1
1
)(
)()(
z
z
zX
zYzH
unstable) & l(anticausa
][2
1
3
1]1[
2
1][ ,
2
1 : ROC ii)
]1[2
1
3
1][
2
1][ ,
2
1 : ROC i)
1
1
nununhz
nununhz
nn
nn
10.7.3 LTI Systems Characterized by Linear Constant-Coefficient
Difference Equations
stability. or thecausality theof constraint additionalan and )( need We zH
Dept. of Electronics Eng. DH26029 Signals and Systems
-11-
)()(1
)()(
)(
)(
21
1
zHzH
zHzH
zX
zY
10.8 System function algebra and block diagram representations
Dept. of Electronics Eng. DH26029 Signals and Systems
-12-
1
411
1)(
zzH][]1[][
41 nxnyny Ex. 10.28)
Dept. of Electronics Eng. DH26029 Signals and Systems
-13-
1
1
411
41
1
211
1
1
21)(
z
zz
zzHEx. 10.29)
Dept. of Electronics Eng. DH26029 Signals and Systems
-14-
2
811
411
411
21 1
1
)1)(1(
1)(
zzzzzH
][]2[8
1]1[
4
1][ nxnynyny
1
411
21 1
1
1
1)(
zzzH
direct form
cascade form
Ex. 10.30)
Dept. of Electronics Eng. DH26029 Signals and Systems
-15-
1
41
31
1
21
32
2
811
411
411
21
11
1
1
11
1)(
zz
zzzzzH
(c) parallel form: Partial fraction
Dept. of Electronics Eng. DH26029 Signals and Systems
-16-
2
811
41
2
211
47
1
1
zz
zzzH
i) Direct-form representation
21
2
811
41 2
1
4
71
1
1)( zz
zzzH
ii) Cascade-form
iii) Parallel-form
1
41
1
1
21
1
41
1
21
1
1)(
z
z
z
zzH
1
411
21 1
314
1
354)(
zzzH
Ex. 10.31)
Dept. of Electronics Eng. DH26029 Signals and Systems
-17-
• Bilateral z-transform vs. unilateral z-transform
• Unilateral
- useful in analyzing causal systems specified by linear constant-
coefficient difference equations with nonzero initial conditions
(i.e., not initially at rest)
- Notation
- the bilateral transform of x[n]u[n]
- ROC : the exterior of a circle
][)(][
][)(0
nxznx
znxzn
n
UZUZ
X
X
10.9 The Unilateral z-Transform
Dept. of Electronics Eng. DH26029 Signals and Systems
-18-
azaz
a
za
znxz
azaz
zzX
n
nn
n
n
,1
][)(
,1
)(
1
0
1
0
1
X
]1[][ 1 nuanx n
10.9.1 Examples of Unilateral z-transform and Inverse Transforms
Ex. 10.33)
Dept. of Electronics Eng. DH26029 Signals and Systems
-19-
(compare Examples 10.9 ~ 10.11)
1
311
41
1
65
11
3)(
zz
zzX
- ROC must be the exterior of the circle
3
1 z
0for ][3
12][
4
1][
nnununx
nn
Ex. 10.34)
Dept. of Electronics Eng. DH26029 Signals and Systems
-20-
• Inverse unilateral z-transforms
- long division in the ROC az
221
11
1
1)( zaaz
azzX
• Rational function of
- for this to be unilateral transform,
Deg. (numerator) ≤ Deg. (denominator)
)(
)(:
zq
zpz
Dept. of Electronics Eng. DH26029 Signals and Systems
-21-
• Identical to the bilateral counterparts
- Linearity
- Scaling in the z-Domain
- Time expansion
- Conjugation
- Differentiation in the z-Domain
• Fundamentally a unilateral property
- Initial-value theorem (∵requirement : x[n]=0 for n < 0)
• No meaningful
- Time-reversal property
10.9.2 Properties of the Unilateral z-transform
Dept. of Electronics Eng. DH26029 Signals and Systems
-22-
• Identical in the convolution property
-
Ex. 10.36) Causal LTI system
][34
3
4
1][
1
41
31
43
131)()(][
][ of transform- bilateral) (andunilateral the],[][ If
31
1)(
rest initial ofcondition e with th][]1[3][
1111
1
nuny
zzzzzzz
nyznunx
zz
nxnyny
n
XHY
H
)()(][][
then,0 allfor 0][][ If
2121
21
zznxnx
nnxnx
XX
UZ
• Difference in the convolution property
-
][][][][
][][][][
,0for nonzero is ][or ][ If
2121
2121
21
nxnxnxnx
nxnxnxnx
nnxnx
UZUZUZ
ZZZ
Dept. of Electronics Eng. DH26029 Signals and Systems
-23-
• The shifting property for the unilateral transform
?][ Then,
)(]1[]2[)(]2[)(
]2[]1[][ ) ii
property)delay time(the )(]1[
][]1[][]1[
]1[]1[]1[)(
]1[][ ) i
211
1
0
1
0
)1(
10
mnx
zzzxxzzxz
nxnynw
zzx
znxzxznxx
znxxznxz
nxny
n
n
n
n
n
n
n
n
UZ
XYW
X
Y
• Time advance property for unilateral transforms
10.60 Problem pf)
]0[)(]1[ zxzznx XUZ
Dept. of Electronics Eng. DH26029 Signals and Systems
-24-
Ex. 10.37) causal LTI system
111
1
1
131
31
3)(
1)(33)(
]1[ ],[][
][]1[3][
zzzz
zzzz
ynunx
nxnyny
Y
YY
zero-input response zero-state response
0 ],[]2)3(3[][
)1 & 8( 1
2
31
3)(
11
nnuny
zzz
n
Y
10.9.3 Solving Difference Equations Using the Unilateral z-Transform