lecture 15 (chapter10)

7/31/2019 Lecture 15 (Chapter10)

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Chapter 10: The Z-Transform

Adapted from: Lecture notes from MIT, Binghamton University

Hamid R. Rabiee

Arman Sepehr

Fall 2010


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Lecture 15 (Chapter 10)

OutlineIntroduction to the z-Transform

-

Inverse z-Transform

-

System Functions of DT LTI Systemso Causality

Geometric Evaluation of z-Transforms and DT Frequency Responses

- -

System Function Algebra and Block Diagrams

-

2Sharif University of Technology, Department of Computer Engineering, Signals & Systems


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The z-Transform

Motivation: Analogous to Laplace Transform in CT

[ ] ( ) n y n H z z=[ ] x n {Eigen function

( ) H z

restrict ourselves

just to z = e j

= n

The (Bilateral) z-Transform

=n

[ ] ( ) [ ] { [ ]} Z nn

x n X z x n z Z x n

=

= =



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The ROC and the Relation Between T and DTFT jre z = ||, zr =

==

==nn

er n xren xre

[ ] n x n r = F


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Exam le #1 [ ] [ ] sided-right- nuan xn=

Thisform for

=

=-n

)( nn znua z X

PFE andinverse z-

transform

=

=0n

az

z1

i.e.1If 1 aa > | a |,outside a circle

This form to find pole and zerolocations



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Example #2:[ ] sided-left-]1[ = nuan x n

[ ]{ }

=

= 1 )(n

nn znua z X

=

=

1

1

n

nn za

n

=

=

==0

1 nn

nn za za

1 1a

,

111 11

z za za

=

=

=

Same X(z) as in Ex #1, but different ROC.

||||.,.,1 If 1 a zei za


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The z-Transform

[ ] [ ] [ ]{ }( ) Z n x n X z x n z Z x n

=

= =

[ ]


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Some Intuition on the Relation between zT and LT

( ) ( ) ( ) { ( )} Z st x t X s x t e dt L x t

= =

[ ]T enT x nsT

n n xT

=

= )()(lim 0 321Let t=nT

[ ]

=

=

n

nsT

T en xT )(lim

0

-

[ ] [ ] [ ]}{)( n x z zn x z X n xn

n ==

=

Can think of z-transform as DT version of Laplacetransform with

sT e=



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More intuition on zT- LT, s-plane - z-plane relationship ze sT =

=lane-sinaxis s lan-zincircleunita1 == T je z

LHP in s-plane, Re(s) < 0 |z| = | e sT | < 1, inside the |z| = 1 circle.Special case, Re(s) = - | z| = 0.

RHP in s-plane, Re(s) > 0 |z| = | e sT | > 1, outside the |z| = 1 circle.Special case, Re(s) = + | z| = .

A vertical line in s-plane, Re(s) = constant | e sT | = constant, a circle in z-plane.



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Properties of the ROCs of z-Transforms

(1) The ROC of X ( z) consists of a ring in the z-plane centered aboutthe ori in e uivalent to a vertical stri in the s- lane

(2) The ROC does not contain any poles (same as in LT).



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More ROC Pro erties

(3) If x[n] is of finite duration, then the ROC is the entire z-= = , .

Why?

2 N

n

=

=1 N n

zn x z



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ROC Properties Continued(4) If x[n] is a right-sided sequence, and if | z| = r o is in the ROC, then all

finite values of z for which | z| > r o are also in the ROC.

= 1

1

nfaster thaconverges

n

N n

r n x

[ ]

=

1

0

N n

nr n x



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Side by Side

(5) If x[n] is a left-sided sequence, and if | z| = r o is in the ROC, then all finite< < o

(6) If x[n] is two-sided, and if |z| = r o is in the ROC , then the ROC consistsof a ring in the z-plane including the circle | z| = r .

What types of signals do the following ROC correspond to?

right-sided left-sided two-sided



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Example #1[ ] 0 ,|| >= bbn x n

+= nunun x

n 1

From

[ ] znubbznu

n 1 ,1

1

,1

11

1

singnalsided-right- 31

singnalsided-two- 11


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Inversion by Identifying Coefficients in the Power Series

[ ] n zn x of tcoefficien-[ ]

= n zn x z X )(

+= 23 )( 43 z-z z z X Example #3:

=n

[ ]

= 3 x 3

[ ]=

=

4

n x

x-

2

A finite-duration DT sequence

20 Sharif University of Technology, Department of Computer Engineering, Signals & Systems


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Example #4:

(a) azaz z X +++== )(11

)( 2111 L

a zaz >


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Properties of z-Transforms

(1) Time Shifting [ ] ),(00 z X znn x n

The rationality of X(z) unchanged , different from LT. ROCunchanged except for the possible addition or deletion of the originor infinity

no z may eno< 0 ROC z (maybe)

(2) z-Domain Differentiation same ROC[ ] zdX znnx )(Derivation: z[ ]

=

=

n

n

n

zdX

zn x z X

1)(

)(

=

ndz

[ ]

= n znnx zdX

z)(

=n



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Convolution Property and System Functions

Y ( z) = H ( z) X ( z) , ROC at least the intersection of the ROCs of H ( z) and X ( z),can e gger ere s po e zero cance a on. e.g.

a z z H ,1

)( >

=

z zY

za z z X

allROC 1)(

,)(=

=

[ ] FunctionSystemThe )( = =

n

n znh z H

H(z) + ROC te s us everyt ing about system



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CAUSALITY(1) h[n] right-sided ROC is the exterior of a circle possibly

including z = : [ ]

=

=1

)( N n

n znh z H

.includedoes but,circleaoutsideROC

atrermt en t e, 111

=0=>z= ROC

A DT LTI system with system function H ( z) is causal the ROC of H ( z) is the exterior of a circle including z =



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Causality for Systems with Rational System Functionsb zb zb zb

z H N N M

M M

M ++++=

)( 1

011

1

LL

A DT LTI s stem with rational s stem function H z is causal

N M

if ,at poles No

(a) the ROC is the exterior of a circle outside the outermost pole;

and b if we write H z as a ratio of ol nomials

)()(

)( z D z N

z H =

)(degree)(degree z D z N



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Stability

LTI System Stable ROC of H ( z) includesthe unit circle | z| = 1

[ ]


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Geometric Evaluation of a Rational z-Transform

Example #1:

Example #2:

--1

1 po eor er -rs-2 a z

z

=

)()(X ,1

)( 122 z X z z X ==

Example #3:

1

)()(

)(1

1

jP j

i Ri

z z

M z X

=

=

=

All same asin s- lane

jP j

i Ri

z z M z X

=

=

=

1

1)(

+= R P

z z M z X = =i j1 1



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Geometric Evaluation of DT Fre uenc Res onsesFirst-Order System

one real pole ,

11

)( 1 >=

=

a za z z

az z H

,


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Second-Order System

Two poles that are a complex conjugate pair ( z1= re j = z2*)12 z


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Demo: DT pole-zero diagrams, frequency response



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DT LTI Systems Described by LCCDEs

[ ] [ ]==

= M

k

k

N

k

k k n xbk n ya00

Use the time-shift property

= N M

k k

k k z X zb zY za )()(

= =k k 0 0

=

z X z H zY )()()(

=

= N

k

M

k

k k zb

z H 0)( Rational

ROC: Depends on Boundary Conditions, left-, right-, or two-sided.For Causal SystemsROC is outside the outermost pole

=k 0



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S stem Function Al ebra and Block Dia rams

Feedback System(causal systems)

negative feedback configuration

)()( 1 z H zY

Example #1: )()(1)( 21 z H z H z X +

==

11

)(

= z H

4 z

n xnn +=

11

z-1 DDelay



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Example #2:

Cascade of two systems

( )111111

211

11

21)(

=

= z

z z z

z H



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Unilateral z-Transform

= n zn x z

Note:=0n

(1) If x[n] = 0 for n < 0, then z z =

(2)UZT of x[n] = BZT of x[n]u[n] ROC always outside a circleand includes z =

(3) For causal LTI systems, )()( z H z =H



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Properties of Unilateral z-Transform

Convolution property (for x1[n


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Use of UZTs in Solving Difference Equations with Initial Conditions

[ ] ][]1[2 n xn yn y =+

UZT of Difference Equation

1

1

,]1[

==

z

nun x y

=

++

11

]}1[{

)(12)( z

n y

zY z z

4 4 84 4 76UZ

Y

z11121121

2)( +

++

= Y

ZIR Output purely due to the initial conditions,

4 4 4 34 4 4 2143421 ZSR ZIR

.


L t 15 (Ch t 10)


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Example (continued)

= 0System is initially at rest:

ZSR 11121

1)()()(

+==

z z z z z

XH

XHY4342143421

121

1)()(

+

==

z

z H zH

= 0 Get response to initial conditions

ZIR 2=

121 + z

22 nun n=


lecture 15 (chapter10)

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