Signal and Systems Prof. H. Sameti Chapter 10: Introduction to the z-Transform Properties of the ROC of the z-Transform Inverse z-Transform Examples Properties

Signal and SystemsProf. H. Sameti

Chapter 10:• Introduction to the z-Transform• Properties of the ROC of the z-Transform• Inverse z-Transform• Examples• Properties of the z-Transform• System Functions of DT LTI Systems

a. Causalityb. Stability

•Geometric Evaluation of z-Transforms and DT Frequency Responses• First- and Second-Order Systems• System Function Algebra and Block Diagrams• Unilateral z-Transforms

Book Chapter10: Section 1

2

The z-Transform

Computer Engineering Department, Signal and Systems

Motivation: Analogous to Laplace Transform in CT

We now do not restrict ourselves just to

z = ejω

The (Bilateral) z-Transform

][nx ][nh ][ny

nn zzHnyznx )(][ ][

LTI DTfor ionEigenfunct

convergesit assuming ][)(

n

nznhzH

]}[{][)(][ nxZznxzXnxn

n

3

The ROC and the Relation Between ZT and DTFT

Unit circle (r = 1) in the ROC⇒ DTFT X(ejω) exists


—depends only on r = |z|, just like the ROC in s-plane only depends on Re(s)

jrez ||, zr

n

njn

n

njj ernxrenxreX )][()]([)(

}][{ nrnx F

n

nj rnxrez |][|at which ROC


4

Example #1


This form for PFE and inverse z- transform

That is, ROC |z| > |a|, outside a circle

This form to findpole and zero locations

sided-right- nuanx n

-n

)( nn znuazX

0

1)(n

naz

||||,i.e.,1 If 1 azaz

az

z

az

11

1


5

Example #2:


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

sided-left-]1[ nuanx n

1

1 )(

n

nn

n

nn

za

znuazX

0

1

1

1 n

n

n

nn zaza

,

11

11

1

1

1

az

zza

za

za

||||.,.,1 If 1 azeiza


6

Rational z-Transforms


x[n] = linear combination of exponentials for n > 0 and for n < 0

— characterized (except for a gain) by its poles and zeros

)(

)(

rational is )(

zD

zNX(z)

zX

Polynomials in z


7

The z-Transform

• Last time:

• Unit circle (r = 1) in the ROC ⇒DTFT exists

• Rational transforms correspond to signals that are linear combinations of DT exponentials


-depends only on r = |z|, just like the ROC in s-plane only depends on Re(s)

nxZznxzXnxn

n

)(

n

nj rnxrez ||at which ROC

( )jX e


8

Some Intuition on the Relation between ZT and LT


The (Bilateral) z-Transform

Can think of z-transform as DTversion of Laplace transform with

( ) ( ) ( ) { ( )}stx t X s x t e dt L x t

TenTx nsT

n nxT

)()(lim

0

n

nsT

TenxT )(lim

0

}{)( nxzznxzXnxn

n

sTez

Let t=nT


9

More intuition on ZT-LT, s-plane - z-plane relationship

LHP in s-plane, Re(s) < 0 |⇒ z| = | esT| < 1, inside the |z| = 1 circle.

Special case, Re(s) = -∞ ⇔|z| = 0. RHP in s-plane, Re(s) > 0 |⇒ z| = | esT| > 1, outside the |z| = 1 circle.

Special case, Re(s) = +∞ ⇔|z| = ∞. A vertical line in s-plane, Re(s) = constant⇔| esT| = constant, a circle in z-plane.


zesT )( plane-sin axis jsj plan-zin circleunit a1 Tjez


10

Properties of the ROCs of Z-Transforms


(2) The ROC does notcontain any poles (same as in LT).

(1) The ROC of X(z) consists of a ring in the z-plane centered about the origin (equivalent to a vertical strip in the s-plane)


11

More ROC Properties


(3) If x[n] is of finite duration, then the ROC is the entire z-plane, except possibly at z = 0 and/or z = ∞.

Why?

Examples:

CT counterpart

2

1

)(N

Nn

nznxzX

stzn all ROC 1)( all ROC 1

seeTtzzn sT )( 0 ROC 1 1

seeTtzzn sT )( ROC 1


12

ROC Properties Continued


(4) If x[n] is a right-sided sequence, and if |z| = ro is in the ROC, then all finite values of z for which |z| > ro are also in the ROC.

1

1

0

1

nfaster tha converges

Nn

n

n

Nn

rnx

rnx


13

Side by Side


What types of signals do the following ROC correspond to?

right-sided left-sided two-sided

(5) If x[n] is a left-sided sequence, and if |z| = ro is in the ROC, then all finite values of z for which 0 < |z| < ro are also in the ROC.

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


14Computer Engineering Department, Signal and Systems

Example #1

0 ,|| bbnx n

1 nubnubnx nn

b

zzb

nub

bzbz

nub

n

n

1 ,

1

11

,1

1

From：

11

1



Example #1 continued

Clearly, ROC does not exist if b > 1 ⇒ No z-transform for b|n|.

bzb

zbbzzX

1 ,

1

1

1

1)(

111



Inverse z-Transforms

for fixed r:

ROC },][{)()( jnj rezrnxreXzX

dereXreXrnx njjjn

2

1 )(2

1)}({F

dnz

erreXnx njnj

2)(

2

1

dzzj

ddjredzrez jj 11

dzzzXj

nx n 1)(2

1



Example #2

Partial Fraction Expansion Algebra: A = 1, B = 2

Note, particular to z-transforms:

1) When finding poles and zeros, express X(z) as a function of z.

2) When doing inverse z-transform using PFE, express X(z) as a function of z-1.

1111

12

3

11

4

11

3

11

4

11

6

53

3

1

4

16

53

)(

z

B

z

A

zz

z

zz

zzzX

nxnxnx

zzzX

21

11

][

3

11

2

4

11

1)(



ROC III:

ROC II:

ROC I:

nunx

nunx

z

n

n

3

12

4

1

singnal sided-right- 3

1

2

1

13

12

4

1

singnal sided-two- 3

1

4

1

2

1

nunx

nunx

z

n

n

1

2

1 -left-sided singnal

4

11

4

12 1

3

n

n

z

x n u n

x n u n



4

1

3

23 )( 43

nx

x

x

x

z-zzzX

Inversion by Identifying Coefficients in the Power Series

Example #3:

nznx oft coefficien -

n

nznxzX )(

3

-1

2

0 for all other n’s— A finite-duration DT sequence



Example #4:

(a)

(b)

zzaz

azazaz

zX

,i.e.,1for convergent

)(11

1)(

1

2111

nuanx n

azeiza

zazaa

zazaza

zaza

azzX

.,.,1for convergent

))(1(

1

1

1

1 )(

1

33221

2111

11

1

1 nuanx n



Properties of z-Transforms

(1) Time Shifting

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

no> 0 ⇒ ROC z ≠ 0 (maybe)no< 0 ⇒ ROC z ≠ ∞ (maybe)

(2) z-Domain Differentiation same ROC

Derivation:

),(00 zXznnx n

dz

zdXznnx

)(

n

n

n

n

znnxdz

zdX

znxzX

1)(

)(

n

nznnxdx

zdXz

)(



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 be bigger if there is pole/zero cancellation. e.g.

H(z) + ROC tells us everything about the system

nx nh nhnxny

zzY

zazzX

azaz

zH

all ROC 1)(

,)(

,1

)(

Function System The )(

n

nznhzH



CAUSALITY

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

including z = ∞:

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

1

)(Nn

nznhzH

. include doesbut , circle a outside ROC

at ][ rerm then the,0 If 111

not

zzNhN N

0Causal 1 N

No zm terms with m>0 =>z=∞ ROC



Causality for Systems with Rational System Functions

A DT LTI system with rational system function H(z) is causal

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

and (b) if we write H(z) as a ratio of polynomials

then

NM

azazaza

bzbzbzbzH

NN

NN

MM

MM

if ,at poles No

)(01

11

011

1

)(

)()(

zD

zNzH

)(degree)(degree zDzN


25

Stability

LTI System Stable⇔ ROC of H(z) includes the unit circle |z| = 1

A causal LTI system with rational system function is stable ⇔ all poles are inside the unit circle, i.e. have magnitudes < 1


⇒ Frequency Response (DTFT of h[n]) exists.

n

h n


Book Chapter10:

26

Geometric Evaluation of a Rational z-Transform

Example 1

Example 2

Example 3


1( ) -A first-order zeroX z z a

poleorder -first A - 1

)(2 azzX

)()(X ,)(

1)( 12

12 zXz

zXzX

)(

)()(

1

1

jPj

iRi

z

zMzX

j

Pj

iRi

z

zMzX

1

1)(

R

i

P

j

ji zzMzX1 1

)()()(

Book Chapter10:

27

Geometric Evaluation of DT Frequency Responses


First-Order System one real pole

1 ,

, 1

1)(

1

anuanh

azaz

z

azzH

n

22122

1

2

1 )( ,1

)( ,)(

jjj eHeHeH

Book Chapter10:

28

Second-Order System

Two poles that are a complex conjugate pair (z1= rejθ =z2*)


0 ,10 ,)cos2(1

1

))(()(

22121

2

rzrzrzzzz

zzH

nun

rnhreeree

eH njjjj

j

sin

1sin ,

))((

1)(

Clearly, |H| peaks near ω = ±θ

Book Chapter10:

29

Demo: DT pole-zero diagrams, frequency response, vector diagrams, and impulse- & step-responses


Book Chapter10:

30

DT LTI Systems Described by LCCDEs


Use the time-shift property

ROC: Depends on Boundary Conditions, left-, right-, or two-sided.

For Causal Systems ⇒ ROC is outside the outermost pole

M

k

k

N

k

k knxbknya00

N

k

M

k

kk

kk zXzbzYza

0 0

)()(

N

k

kk

M

k

kk

za

zbzH

zXzHzY

0

0)(

)()()(

—Rational

Book Chapter10:

31

System Function Algebra and Block Diagrams


Feedback System(causal systems)

negative feedback configuration

Example #1:

)()(1

)(

)(

)()(

21

1

zHzH

zH

zX

zYzH

1

4

11

1)(

zzH

nxnyny

14

1

D Delay

Book Chapter10:

32

Example 2


— Cascade of two systems

11

411

41

1

211

1

1

21)(

zzz

zzH

Book Chapter10:

33

Unilateral z-Transform


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

(2) UZT of x[n] = BZT of x[n]u[n]⇒ROC always outside a circle and includes z = ∞

(3) For causal LTI systems,

0

)(n

nznxz

)()( zXz

)()( zHz H

Book Chapter10:

34

Properties of Unilateral z-Transform

Many properties are analogous to properties of the BZT e.g.


• Convolution property (for x1[n<0] = x2[n<0] = 0)

• But there are important differences. For example, time-shift

Derivation: Initial

condition

)()(

2121 zzUZ

nxnx

)(1]1[][ 1 zzxznxny Y

100

111)(n

n

n

n

n

n znxxznxznyzY

z

zmxzxm

m

0

11

Book Chapter10:

35

Use of UZTs in Solving Difference Equations with Initial Conditions


UZT of Difference Equation

ZIR — Output purely due to the initial conditions,ZSR — Output purely due to the input.

][]1[2 nxnyny

11

,]1[

znunxy

11

]}1[{

)(12)(z

ny

zYzz

UZ

Y

ZSR

zzZIR

zz

)11)(121(121

2)(

Y

Book Chapter10:

36

Example (continued)


β = 0 ⇒ System is initially at rest:

ZSR

α = 0 ⇒ Get response to initial conditions

ZIR

121

1)()(

)(

11)(

121

1)()()(

zzHz

zz

zz

zzz

H

XH

XHY

121

2)(

zzY

][)2(2][ nuny n

Documents

Signal and Systems Prof. H. Sameti Chapter 10: Introduction to the z-Transform Properties of the ROC of the z-Transform Inverse z-Transform Examples Properties