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

Rzz

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