z transform and it's applications

8/11/2019 Z Transform and It's Applications

1/11

Chapter 4

Z Transform


2/11

Analog signal is sampled at every T sec with an ON period of t

Definition

Since s = +j, and a system become stable for 0,z =esT= e(+j)T= eT ej

So, the magnitude of z is eT and the angle is

Hence in terms of z-transform the system will be stable for |z| 1

n

tnTtnxtx )()()(

n

snT

n

snT enxtdttenTtnxsX )()()()(0

sTez

)()()( ztXznxtsXn

n

n

nznxzX )()(


3/11

Elementary signals

Unit step function 01)( nnx

1;11

1.............11)(

1

21

0

zz

z

zzzzzX

n

n

Power function 0)( nanx n

azaz

zaz

zaazzazX nn

n

;

11...........1)( 1

221

0

Ramp function 0)( nnnx

1||;)1()1(

.........320)(221

1321

0

z

z

z

z

zzzzznzX n

n

The values ofz for whichX(z)is finite are known as region of

convergence (ROC)

Try other signals: impulse function, .


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Significance of ROC

For causal sequence

azaz

zaz

zazazX n

n

n

n

n

;1

1)()(1

1

00

000)( nforandnforanx n

For Anti causal

sequence

000)( nforaandnfornx n

azaz

zzazazX n

n

n

n

n

;)()(

1 11

The two sequences have sameX(z)but their ROC is different. Without ROC we can

not uniquely determine the sequencex(n). Generally, for causal sequence, the ROCis exterior of the circle having radius a and for anti causal sequence it is interior of

the circle.

FindX(z)and ROC for x(n) = nu(n) + nu(-n-1)

Answer

zROCzzzX :)( 1


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Properties

)()()(

)()()()()()( 2211221

zHzXzY

zXzknxzXazXanxanax

k


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LTI System

)(......)1()()(......)1()(1010

MnxbnxbnxbNnyanyanyaMN

N

N

M

M

zazaa

zbzbb

zX

zYzH

..........

..........

)(

)()(

1

10

1

10

N

NN

M

MMMN

azaza

bzbzbz

..........

..........1

10

1

10

kN

k

k

M

k

k

k

za

zb

zH

0

0)(

)(......)()()(......)()( 1

10

1

10 zXzbzXzbzXbzYzazYzazYa M

M

N

N

)(]......[)(]......[ 1101

10 zXzbzbbzYzazaa M

M

N

N

).().........)((

).().........)((

21

21

N

MMN

pzpzpz

zzzzzzz

z1, z2, . . . zM zeros

p1, p2, . . . pN poles


7/11

LTI System Stability

LTI DT causal system is BIBO stable provided that all poles of

the system transfer function lie inside the unit circle in Z-plane


8/11

Inverse z-transform

Values of x(n) are the coefficients of z-nand can be obtained by direct inspection of

X(z). Normally X(z) is often expressed as a ratio of two polynomials in z-1or in z

..........)2()1()0()()( 21

0

zxzxxznxzXn

n

MM

NN

zbzbb

zazaazX

_110

1

10

............

...........)(

Three Methods

Power series expansion method

Partial fraction expansion methodResidue method / Contour integration


9/11

IZT: Partial Fraction Expansion Method

M < NProper

rational function

M NImproperrational function

1,...........1

............

)(

)()( 01

1

1

10

azaza

zbzbb

zD

zNzH

N

N

M

M

N

N

pz

A

pz

A

pz

A

z

zH

..............

)(

2

2

1

1Let all poles are

distinct

Nkz

zHpzAKpz

KK ..........3,2,1,

)()(

21 5.05.11

1)(

zzz

)(})5.0()1(2{)(

5.0

1

1

2)(

1)5.0)(1(

)5.0(2

)5.0)(1(

)1(

5.01)5.0)(1(

)(

5.

2

1

1

21

nunh

zzz

zH

zz

zzA

zz

zzA

z

A

z

A

zz

z

z

zH

nn

zz


10/11

IZT: Multiple-order poles

2

2

211 )1)(1(

)(

)1)(1(

1)(

zz

z

z

zH

zzzH

)1..(....................)1(11)1)(1(

)( 2321

2

2

zA

zA

zA

zzz

zzH

4

1)()1(

1

1 zz

zHzA

2

1)()1(

1

2

3 zz

zHzA

)2(....................)1()1(

)1()()1( 321

22 AAzA

z

z

z

zHz

4

3

1

)()1(

1

2

1

2

2

zz z

z

dz

d

z

zHz

dz

dATo find A2, differentiate (2)

with respect to z and put z=1

2)1(2

1

14

3

14

1)(

z

z

z

z

z

zzH

)(

2

1)1(

4

3)1(

4

1)( nunnh nn


11/11

Thank you

z transform and it's applications

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