Signals & Systems - Sharifce.sharif.edu/courses/92-93/1/ce242-1/resources/root/... · 2020. 9. 7. · Signals & Systems Chapter 10: The Z-Transform Adapted from: Lecture notes from

Signals & Systems

Chapter 10: The Z-Transform

Adapted from: Lecture notes from MIT, Binghamton University

Dr. Hamid R. Rabiee

Fall 2013

Lecture 15 (Chapter 10)Lecture 16 (Chapter 10)

Introduction to the z-Transform

Properties of the ROC of the z-Transform

Inverse z-Transform

Properties of the z-Transform

System Functions of DT LTI Systems

o Causality

o Stability

Geometric Evaluation of z-Transforms and DT Frequency Responses

First- and Second-Order Systems

System Function Algebra and Block Diagrams

Unilateral z-Transforms

Outline

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


The z-Transform

Motivation: Analogous to Laplace Transform in CT

We now do not

restrict ourselves

just to z = ejω

The (Bilateral) z-Transform

[ ] ( ) ny n H z z

convergesit assuming ][)(

n

nznhzH

[ ] ( ) [ ] { [ ]}Z n

n

x n X z x n z Z x n

3

[ ] x nEigen function for DT LTI

( )H z

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


The ROC and the Relation Between zT and DTFT

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

only depends on Re(s)

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

jrez ||, zr

n

njn

n

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

[ ] nx n r F

n

nj rnxrez |][|at which ROC •



Example #1 sided-right- nuanx n



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: sided-left-]1[ nuanxn



Example #2:

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

sided-left-]1[ nuanx n

1

1

1 )(

n

nn

n

nn

za

znuazX

0

11

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

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

only depends on Re(s)

• Last time:

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

o Rational transforms correspond to signals that are

linear combinations of DT exponentials

( )Z nn

x n X z x n z Z x 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

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

TenTx nsT

n nx

T

)()(lim0

n

nsT

TenxT )(lim

0

}{)( nxzznxzXnxn

n

sTez

Let t=nT



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

zesT

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

• 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.



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



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



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.

Side by Side



Example #1 0 ,|| bbnx n



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 )(21

dzzj

ddjredzrez jj 11

dzzzX

jnx n 1)(

2

1



Example #2

?

3

1

4

16

53

)(

2

zz

zz

zX



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

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

13

12

4

1

singnal sided-left- 4

1

2

1

nunx

nunx

z

n

n

Example #2 (Continued)



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 )(



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

1

1

1

1 )(

1

1

1

zaza

azzX



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

(2) z-Domain Differentiation

?,0 nnx

?nnx



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 zXznnxn

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

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 possiblyincluding 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

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

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

nnh



Geometric Evaluation of a Rational z-Transform

Example #1:

Example #2:

Example #3:

zeroorder -firstA - )(1 azzX

poleorder -first A - 1

)(2az

zX

)(

)()(

1

1

j

P

j

i

R

i

z

zMzX



Geometric Evaluation of a Rational z-Transform

Example #1:

Example #2:

Example #3:

All same as

in s-plane

zeroorder -firstA - )(1 azzX

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

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*)

0 ,10 ,)cos2(1

1

))(()(

221

21

2

rzrzrzzzz

zzH

nun

rnhreeree

eH njjjj

j

sin

1sin ,

))((

1)(

Clearly, |H| peaks near ω = ±θ



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 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




Feedback System

(causal systems)

negative feedback

configuration

Example #1:)()(1

)(

)(

)()(

21

1

zHzH

zH

zX

zYzH




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:

1

41

1

1

21)(

z

zzH



Example #2:

— Cascade of

two systems 1

1

411

41

1

211

1

1

21)(

z

zz

zzH



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



Properties of Unilateral z-Transform

• Convolution property (for x1[n


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[

z

nunxy

11

]}1[{

)(12)(z

ny

zYzz

UZ

Y

ZSR

zz

ZIR

zz

)11)(121(121

2)(

Y



Example (continued)

β = 0⇒System is initially at rest:

ZSR

α = 0⇒ Get response to initial conditions

ZI

R

121

1)()(

)(

11

)(

121

1)()()(

zzHz

z

z

z

zzzz

H

XH

XHY

121

2)(

zzY

][)2(2][ nuny n


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Signals & Systems - Sharifce.sharif.edu/courses/92-93/1/ce242-1/resources/root/... · 2020. 9. 7. · Signals & Systems Chapter 10: The Z-Transform Adapted from: Lecture notes from