Signal and Systems Prof. H. Sametice.sharif.edu/courses/98-99/1/ce242-2/resources/root... · 2020. 9. 7. · • Unilateral z-Transforms. The z-Transform Book Chapter10: Section 1

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. Causality

b. Stability

•Geometric Evaluation of z-Transforms and DT Frequency Responses

• First- and Second-Order Systems

• System Function Algebra and Block Diagrams

• Unilateral z-Transforms

The z-Transform

Book Chapter10: Section 1

Computer Engineering Department, Signal and Systems 2

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

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


Example #1


This

form for

PFE and

inverse z-

transform

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

outside a circle

This form to find

pole 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


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

z

za

za

za

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


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


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)

nxZznxzXnx

n

n

)(

n

nj rnxrez ||at which ROC

( )jX e


Some Intuition on the Relation

between ZT and LT


The (Bilateral) z-Transform

Can think of z-transform as DT

version of Laplace transform with

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

TenTx nsT

n nx

T

)()(lim

0

n

nsT

TenxT )(lim

0

}{)( nxzznxzXnx

n

n

sTez

Let t=nT


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


Properties of the ROCs of Z-Transforms


(2) The ROC does not contain 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)


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


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


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.



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

d

nz

erreXnx njnj

2

)(2

1

dzzj

ddjredzrez jj 11

dzzzX

jnx 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

1

6

53

)(

z

B

z

A

zz

z

zz

zz

zX

nxnxnx

zz

zX

21

11

][

3

11

2

4

11

1)(



ROC III:

ROC II:

ROC I:

nunx

nunx

z

n

n

3

12

4

1

signal sidedright - 3

1

2

1

13

12

4

1

signal sided two- 3

1

4

1

2

1

nunx

nunx

z

n

n


13

12

14

1

signal sidedleft - 4

1

2

1

nunx

nunx

z

n

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)

azaz

azazaz

zX

,i.e.,1for convergent

)(11

1)(

1

211

1

nuanx n

azeiza

zazaa

zazaza

zaza

azzX

.,.,1for convergent

))(1(

1

1

1

1 )(

1

33221

2111

1

1

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:

),(0

0 zXznnxn

dz

zdXznnx

)(

n

n

n

n

znnxdz

zdX

znxzX

1)(

)(

n

nznnxdz

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 of

H(z) is the exterior of a circle including z = ∞

1

)(

Nn

nznhzH

. include doesbut , circle a outside ROC

at ][ rerm then the,0 If 1

11

not

zzNhNN

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

N

N

N

N

M

M

M

M

if ,at poles No

)(01

1

1

01

1

1

)(

)()(

zD

zNzH

)(degree)(degree zDzN


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 𝐻(𝑒𝑗ω) (DTFT of h[n]) exists.

n

h n


Geometric Evaluation of a Rational

z-Transform

Example 1

Example 2

Example 3

Book Chapter10:


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

poleorder -first A - 1

)(2az

zX

)()(X ,)(

1)( 12

1

2 zXzzX

zX

)(

)()(

1

1

j

P

j

i

R

i

z

zMzX

j

P

j

i

R

i

z

zMzX

1

1)(

R

i

P

j

ji zzMzX

1 1

)()()(

Geometric Evaluation of DT

Frequency Responses

Book Chapter10:


First-Order System one real pole

1 ,

, 1

1)(

1

anuanh

azaz

z

azzH

n

221

22

1

2

1 )( ,1

)( ,)(

jjj eHeHeH

Second-Order System

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

Book Chapter10:


0 ,10 ,)cos2(1

1

))(()(

221

21

2

rzrzrzzzz

zzH

nun

rnhreeree

eH n

jjjj

j

sin

1sin ,

))((

1)(

Clearly, |H| peaks near ω = ±θ

Demo: DT pole-zero diagrams, frequency response,

vector diagrams, and impulse- & step-responses

Book Chapter10:


DT LTI Systems Described by LCCDEs

Book Chapter10:


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 knxbknya

00

N

k

M

k

k

k

k

k zXzbzYza

0 0

)()(

N

k

k

k

M

k

k

k

za

zbzH

zXzHzY

0

0)(

)()()(

—Rational

System Function Algebra and Block Diagrams

Book Chapter10:


Feedback System

(causal systems) negative feedback

configuration

Example #1:)()(1

)(

)(

)()(

21

1

zHzH

zH

zX

zYzH

1

4

11

1)(

z

zH

nxnyny

14

1

𝑧−1 D

Delay

Example 2

Book Chapter10:


— Cascade of

two systems 1

1

411

41

1

211

1

1

21)(

z

zz

zzH

Unilateral z-Transform

Book Chapter10:


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

Properties of Unilateral z-Transform

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

Book Chapter10:


• 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

zmxzx

m

m

0

11

Use of UZTs in Solving Difference Equations

with Initial Conditions

Book Chapter10:


UZT of Difference Equation

ZIR — Output purely due to the initial conditions,

ZSR — Output purely due to the input.

][]1[2 nxnyny

11

,]1[

z

nunxy

11

]}1[{

)(12)(z

ny

zYzz

UZ

Y

ZSR

zz

ZIR

zz

)11)(121(121

2)(

Y

Example (continued)

Book Chapter10:


β = 0 ⇒ System is initially at rest:

ZSR

α = 0 ⇒ Get response to initial conditions

ZIR

121

1)()(

)(

11

)(

121

1)()()(

zzHz

z

z

z

zzzz

H

XH

XHY

121

2)(

zzY

][)2(2][ nuny n

Documents

Signal and Systems Prof. H. Sametice.sharif.edu/courses/98-99/1/ce242-2/resources/root... · 2020. 9. 7. · • Unilateral z-Transforms. The z-Transform Book Chapter10: Section 1