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Outline

Intro ductionRelation Between LT and ZTAnalyzing L TISystem swith ZT Geometric EvaluationUnilateral ZT Signals and Systems

Lecture 8: Z Transform

Farzaneh Abdollahi

Department of Electrical Engineering

Amirkabir University of Technology

Winter 2012

Farzaneh Abdollahi Signal and Systems Lecture 8 1/29

OutlineIntro ductionRelation Between LT and ZTAnalyzing L TISystem swith ZT Geometric EvaluationUnilateral ZT

Introduction

Relation Between LT and ZT

ROC Properties

The Inverse of ZT

ZT Properties

Analyzing LTI Systems with ZT

Geometric Evaluation

LTI Systems Description

Unilateral ZT

Farzaneh Abdollahi Signal and Systems Lecture 8 2/29

OutlineIntro ductionRelation Between LT and ZTAnalyzing L TISystem swith ZT Geometric EvaluationUnilateral ZT

I

Z transform (ZT) is extension of DTFT

I Like CTFT and DTFT, ZT and LT have similarities and dierences. I We had denedx[n] =znas a basic function for DT LTI systems,s.t. z n!H(z)zn I

In Fourier transformz=ej!, in other words,jzj= 1

I

In Z transformz=rej!

I By ZT we can analyze wider range of systems comparing to Fourier

Transform

I The bilateral ZT is dened:

X(z)= 1X

1x[n]zn

)X(rej!) =1X

1x[n](rej!)n=1X

1fx[n]rngej!n

=Ffx[n]rngFarzaneh Abdollahi Signal and Systems Lecture 8 3/29

OutlineIntro ductionRelation Between LT and ZTAnalyzing L TISystem swith ZT Geometric EvaluationUnilateral ZT

Region of Convergence (ROC)

I Note that:X(z) exists only for a specic region ofzwhich is called

Region of Convergence (ROC)

I ROC: is the z=rej!by whichx[n]rnconverges:

ROC:fz=rej!s:t:P1

n=1jx[n]rnj<1g I

Roc does not depend on!

IRoc is absolute summability condition ofx[n]rn

I

Ifr= 1, i,e,z=ej! X(z) =Ffx[n]g

I ROC is shown in z-planeFarzaneh Abdollahi Signal and Systems Lecture 8 4/29

OutlineIntro ductionRelation Between LT and ZTAnalyzing L TISystem swith ZT Geometric EvaluationUnilateral ZT

Example

I

Considerx[n] =anu[n]

I

X(z) =P1

n=1anu[n]zn=P1 n=0(az1)n I

Ifjzj>jaj X(z) is bounded

I )X(z) =zza;ROC:jzj>jajFarzaneh Abdollahi Signal and Systems Lecture 8 5/29

OutlineIntro ductionRelation Between LT and ZTAnalyzing L TISystem swith ZT Geometric EvaluationUnilateral ZT

Example

I

Considerx[n] =anu[n1]

I

X(z) =P1

n=1anu[n1]zn1P1 n=0(a1z)n I

Ifja1zj<1 jzj I )X(z) =zza;ROC:jzjOutlineIntro ductionRelation Between LT and ZTAnalyzing L TISystem swith ZT Geometric EvaluationUnilateral ZT

I In the recent two examples two dierent signals had similar ZT but with dierent Roc I To obtain uniquex[n] bothX(z) and ROC are required I

IfX(z) =N(z)D(z)

IRoots ofN(z) zeros of X(z); They make X(z) equal to zero.

IRoots ofD(z) poles of X(z); They make X(z) to be unbounded.Farzaneh Abdollahi Signal and Systems Lecture 8 7/29

OutlineIntro ductionRelation Between LT and ZTAnalyzing L TISystem swith ZT Geometric EvaluationUnilateral ZT

Relation Between LT and ZT

I

In LT:x(t)L$X(s) =R1

1x(t)estdt=Lfx(t)g

I

Now denet=nT:

X(s) = limT!0P1

n=1x(nT)(esT)n:T= lim

T!0TP1

n=1x[n](esT)n I

In ZT:x[n]Z$X(z) =P1

n=1x[n]zn=Zfx[n]g I )by takingz=esTZT is obtained from LT. I

j!axis in s-plane is changed to unite circle in z-planeFarzaneh Abdollahi Signal and Systems Lecture 8 8/29

OutlineIntro ductionRelation Between LT and ZTAnalyzing L TISystem swith ZT Geometric EvaluationUnilateral ZT

z-Planes-Plane

jzj<1 (insider the unit circle)Refsg<0 (LHP)special case:jzj= 0special case:Refsg=1jzj>1 (outsider the unit circle)Refsg>0 (RHP)special case:jzj=1special case:Refsg=1jzj=cte(a circle)Refsg=cte(a vertical line)Farzaneh Abdollahi Signal and Systems Lecture 8 9/29

OutlineIntro ductionRelation Between LT and ZTAnalyzing L TISystem swith ZT Geometric EvaluationUnilateral ZT

ROC Properties

I ROC ofX(z) is a ring in z-plane centered at origin I

ROC does not contain any pole

I Ifx[n] is of nite duration, then the ROC is the entire z-plane, except possiblyz= 0 and/orz=1 I

X(z) =PN2

n=N1x[n]zn I IfN1<0 x[n] has nonzero terms forn<0, whenjzj ! 1positive power ofzwill be unbounded IIfN2>0 x[n] has nonzero terms forn>0, whenjzj !0 negative power ofzwill be unbounded

IIfN10 only negative powers ofzexist z=1 2ROC

IIfN20 only positive powers ofzexist z= 02ROC

IExample:ZTLT

[n]$1$1 ROC: allz(t)$1 ROC: alls[n1]$z1ROC:z6= 0(tT)$esTROC:Refsg 6=1[n+ 1]$zROC:z6=1(t+T)$esTROC:Refsg 6=1Farzaneh Abdollahi Signal and Systems Lecture 8 10/29

OutlineIntro ductionRelation Between LT and ZTAnalyzing L TISystem swith ZT Geometric EvaluationUnilateral ZT

ROC Properties

I Ifx[n] is a right-sided sequence and if the circlejzj=r0is in the ROC, then all nite values ofzfor whichjzj>r0will also be in the ROC. I Ifx[n] is a left-sided sequence and if the circlejzj=r0is in the ROC, then all nite values ofzfor whichjzjIfX(z) is rational

I The ROC is bounded between poles or extends to innity,

Ino poles ofX(s) are contained in ROC

IIfx[n] is right sided, then ROC is in the out of the outermost pole I

Ifx[n] is causal and right sided thenz=1 2ROC

IIfx[n] is left sided, then ROC is in the inside of the innermost pole I

Ifx[n] is anticausal and left sided thenz= 02ROC

I If ROC includesjzj= 1 axis thenx[n] has FTFarzaneh Abdollahi Signal and Systems Lecture 8 11/29

OutlineIntro ductionRelation Between LT and ZTAnalyzing L TISystem swith ZT Geometric EvaluationUnilateral ZT

The Inverse of Z Transform (ZT)

I By consideringrxed, inverse of ZT can be obtained from inverse of FT: I x[n]rn=12R

2X(rej!|{z}

z)ej!nd! I x[n] =12R

2X(rej!)rne(j!)nd!

I assumingris xed dz=jzd! I )x[n] =12jHX(z)zn1dz I

Methods to obtain Inverse ZT:

1. If X(s) is rational , we can use expanding the rational algebraic into a linear combination of lower order terms and then one may use I X(z) =Ai1aiz1 x[n] =Aiaiu[n] if ROC is out of polez=ai I X(z) =Ai1aiz1 x[n] =Aiaiu[n1] if ROC is inside ofz=ai Do not forget to consider ROC in obtaining inverse of ZT! Farzaneh Abdollahi Signal and Systems Lecture 8 12/29

OutlineIntro ductionRelation Between LT and ZTAnalyzing L TISystem swith ZT Geometric EvaluationUnilateral ZT

Methods to obtain Inverse ZT:

2. If X is nonrational, use P owerseries expansion of X(z), then apply [n+n0],zn0 I

Example:X(z) = 5z2z+ 3z3

I x[n] = 5[n+ 2][n+ 1] + 3[n3] 3. If X is rational, p owerseries can b eobtained b ylong division I

Example:X(z) =11az1;jzj>jaj

1x1az11+az1+(az1)2+:::

1 +az1az

1 az1+a2z2. I x[n] =anu[n]Farzaneh Abdollahi Signal and Systems Lecture 8 13/29

OutlineIntro ductionRelation Between LT and ZTAnalyzing L TISystem swith ZT Geometric EvaluationUnilateral ZT

Methods to obtain Inverse ZT:

I

Example:X(z) =11az1;jzj I

X(z) =a1z(11a1z)

1x1a1z1+a1z+(a1z)2+:::

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