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The z-Transform

主講人：虞台文

Content

Introduction z-Transform Zeros and Poles Region of Convergence Important z-Transform Pairs Inverse z-Transform z-Transform Theorems and Properties System Function

The z-Transform

Introduction

Why z-Transform?

A generalization of Fourier transform Why generalize it?

– FT does not converge on all sequence– Notation good for analysis– Bring the power of complex variable theory deal with

the discrete-time signals and systems

The z-Transform

z-Transform

Definition The z-transform of sequence x(n) is defined by

n

nznxzX )()(

Let z = ej.

( ) ( )j j n

n

X e x n e

Fourier Transform

z-Plane

Re

Im

z = ej

n

nznxzX )()(

( ) ( )j j n

n

X e x n e

Fourier Transform is to evaluate z-transform on a unit circle.

Fourier Transform is to evaluate z-transform on a unit circle.

z-Plane

Re

Im

X(z)

Re

Im

z = ej

Periodic Property of FT

Re

Im

X(z)

X(ej)

Can you say why Fourier Transform is a periodic function with period 2?

Can you say why Fourier Transform is a periodic function with period 2?

The z-Transform

Zeros and Poles

Definition

Give a sequence, the set of values of z for which the z-transform converges, i.e., |X(z)|<, is called the region of convergence.

n

n

n

n znxznxzX |||)(|)(|)(|

ROC is centered on origin and consists of a set of rings.

ROC is centered on origin and consists of a set of rings.

Example: Region of Convergence

Re

Im

n

n

n

n znxznxzX |||)(|)(|)(|

ROC is an annual ring centered on the origin.

ROC is an annual ring centered on the origin.

xx RzR ||

r

}|{ xx

j RrRrezROC

Stable Systems

Re

Im

1

A stable system requires that its Fourier transform is uniformly convergent.

Fact: Fourier transform is to evaluate z-transform on a unit circle.

A stable system requires the ROC of z-transform to include the unit circle.

Example: A right sided Sequence

)()( nuanx n )()( nuanx n

1 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8

n

x(n)

. . .


)()( nuanx n )()( nuanx n

n

n

n znuazX

)()(

0n

nn za

0

1)(n

naz

For convergence of X(z), we require that

0

1 ||n

az 1|| 1 az

|||| az

az

z

azazzX

n

n

10

1

1

1)()(

|||| az

aa

Example: A right sided Sequence ROC for x(n)=anu(n)

|||| ,)( azaz

zzX

|||| ,)( az

az

zzX

Re

Im

1aa

Re

Im

1

Which one is stable?Which one is stable?

Example: A left sided Sequence

)1()( nuanx n )1()( nuanx n

1 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8n

x(n)

. . .


)1()( nuanx n )1()( nuanx n

n

n

n znuazX

)1()(

For convergence of X(z), we require that

0

1 ||n

za 1|| 1 za

|||| az

az

z

zazazX

n

n

10

1

1

11)(1)(

|||| az

n

n

n za

1

n

n

n za

1

n

n

n za

0

1

aa

Example: A left sided Sequence ROC for x(n)=anu( n1)

|||| ,)( azaz

zzX

|||| ,)( az

az

zzX

Re

Im

1aa

Re

Im

1

Which one is stable?Which one is stable?

The z-Transform

Region of Convergence

Represent z-transform as a Rational Function

)(

)()(

zQ

zPzX where P(z) and Q(z) are

polynomials in z.

Zeros: The values of z’s such that X(z) = 0

Poles: The values of z’s such that X(z) =


)()( nuanx n |||| ,)( azaz

zzX

Re

Im

a

ROC is bounded by the pole and is the exterior of a circle.


)1()( nuanx n|||| ,)( az

az

zzX

Re

Im

a

ROC is bounded by the pole and is the interior of a circle.

Example: Sum of Two Right Sided Sequences

)()()()()( 31

21 nununx nn

31

21

)(

z

z

z

zzX

Re

Im

1/2

))((

)(2

31

21

121

zz

zz

1/3

1/12

ROC is bounded by poles and is the exterior of a circle.

ROC does not include any pole.

Example: A Two Sided Sequence

)1()()()()( 21

31 nununx nn

21

31

)(

z

z

z

zzX

Re

Im

1/2

))((

)(2

21

31

121

zz

zz

1/3

1/12

ROC is bounded by poles and is a ring.


Example: A Finite Sequence

10 ,)( Nnanx n

nN

n

nN

n

n zazazX )()( 11

0

1

0

Re

ImROC: 0 < z <


1

1

1

)(1

az

az N

az

az

z

NN

N

1

1

N-1 poles

N-1 zeros

Always StableAlways Stable

Properties of ROC

A ring or disk in the z-plane centered at the origin. The Fourier Transform of x(n) is converge absolutely iff the ROC includ

es the unit circle. The ROC cannot include any poles Finite Duration Sequences: The ROC is the entire z-plane except possibl

y z=0 or z=. Right sided sequences: The ROC extends outward from the outermost fi

nite pole in X(z) to z=. Left sided sequences: The ROC extends inward from the innermost nonz

ero pole in X(z) to z=0.

More on Rational z-Transform

Re

Im

a b c

Consider the rational z-transform with the pole pattern:

Find the possible ROC’s

Find the possible ROC’s


Re

Im

a b c


Case 1: A right sided Sequence.


Re

Im

a b c


Case 2: A left sided Sequence.


Re

Im

a b c


Case 3: A two sided Sequence.


Re

Im

a b c


Case 4: Another two sided Sequence.

The z-Transform

Important

z-Transform Pairs

Z-Transform Pairs

Sequence z-Transform ROC

)(n 1 All z

)( mn mz All z except 0 (if m>0)or (if m<0)

)(nu 11

1 z

1|| z

)1( nu 11

1 z

1|| z

)(nuan 11

1 az

|||| az

)1( nuan 11

1 az

|||| az

Z-Transform Pairs

Sequence z-Transform ROC

)(][cos 0 nun 210

10

]cos2[1

][cos1

zz

z1|| z

)(][sin 0 nun 210

10

]cos2[1

][sin

zz

z1|| z

)(]cos[ 0 nunr n 2210

10

]cos2[1

]cos[1

zrzr

zrrz ||

)(]sin[ 0 nunr n 2210

10

]cos2[1

]sin[

zrzr

zrrz ||

otherwise0

10 Nnan

11

1

az

za NN

0|| z

The z-Transform

Inverse z-Transform

The z-Transform

z-Transform Theorems and Properties

Linearity

xRzzXnx ),()]([Z

yRzzYny ),()]([Z

yx RRzzbYzaXnbynax ),()()]()([Z

Overlay of the above two

ROC’s

Shift

xRzzXnx ),()]([Z

xn RzzXznnx )()]([ 0

0Z

Multiplication by an Exponential Sequence

xx- RzRzXnx || ),()]([Z

xn RazzaXnxa || )()]([ 1Z

Differentiation of X(z)

xRzzXnx ),()]([Z

xRzdz

zdXznnx

)()]([Z

Conjugation

xRzzXnx ),()]([Z

xRzzXnx *)(*)](*[Z

Reversal

xRzzXnx ),()]([Z

xRzzXnx /1 )()]([ 1 Z

Real and Imaginary Parts

xRzzXnx ),()]([Z

xRzzXzXnxe *)](*)([)]([ 21R

xj RzzXzXnx *)](*)([)]([ 21Im

Initial Value Theorem

0for ,0)( nnx

)(lim)0( zXxz

Convolution of Sequences

xRzzXnx ),()]([Z

yRzzYny ),()]([Z

yx RRzzYzXnynx )()()](*)([Z

Convolution of Sequences

k

knykxnynx )()()(*)(

n

n

k

zknykxnynx )()()](*)([Z

k

n

n

zknykx )()(

k

n

n

k znyzkx )()(

)()( zYzX

The z-Transform

System Function

Shift-Invariant System

h(n)h(n)

x(n) y(n)=x(n)*h(n)

X(z) Y(z)=X(z)H(z)H(z)

Shift-Invariant System

H(z)H(z)X(z) Y(z)

)(

)()(

zX

zYzH

)(

)()(

zX

zYzH

Nth-Order Difference Equation

M

rr

N

kk rnxbknya

00

)()(

M

rr

N

kk rnxbknya

00

)()(

M

r

rr

N

k

kk zbzXzazY

00

)()(

N

k

kk

M

r

rr zazbzH

00)(

N

k

kk

M

r

rr zazbzH

00)(

Representation in Factored Form

N

kr

M

rr

zd

zcAzH

1

1

1

1

)1(

)1()(

N

kr

M

rr

zd

zcAzH

1

1

1

1

)1(

)1()(

Contributes poles at 0 and zeros at cr

Contributes zeros at 0 and poles at dr

Stable and Causal Systems

N

kr

M

rr

zd

zcAzH

1

1

1

1

)1(

)1()(

N

kr

M

rr

zd

zcAzH

1

1

1

1

)1(

)1()( Re

ImCausal Systems : ROC extends outward from the outermost pole.

Stable and Causal Systems

N

kr

M

rr

zd

zcAzH

1

1

1

1

)1(

)1()(

N

kr

M

rr

zd

zcAzH

1

1

1

1

)1(

)1()( Re

ImStable Systems : ROC includes the unit circle.

1

Example

Consider the causal system characterized by

)()1()( nxnayny

11

1)(

azzH 11

1)(

azzH

Re

Im

1

a

)()( nuanh n

Determination of Frequency Response from pole-zero pattern

A LTI system is completely characterized by its pole-zero pattern.

))(()(

21

1

pzpz

zzzH

))(()(

21

1

pzpz

zzzH

Example:

))(()(

21

1

00

0

0

pepe

zeeH jj

jj

))(()(

21

1

00

0

0

pepe

zeeH jj

jj

0je

Re

Im

z1

p1

p2



))(()(

21

1

pzpz

zzzH

))(()(

21

1

pzpz

zzzH

Example:

))(()(

21

1

00

0

0

pepe

zeeH jj

jj

))(()(

21

1

00

0

0

pepe

zeeH jj

jj

0je

Re

Im

z1

p1

p2

|H(ej)|=?|H(ej)|=? H(ej)=?H(ej)=?



Example:

0je

Re

Im

z1

p1

p2

|H(ej)|=?|H(ej)|=? H(ej)=?H(ej)=?

|H(ej)| =| |

| | | | 1

2

3

H(ej) = 1(2+ 3 )

Example

11

1)(

azzH 11

1)(

azzH

Re

Im

a

0 2 4 6 8-10

0

10

20

0 2 4 6 8-2

-1

0

1

2d

B

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