2015-7-31zhongguo liu_biomedical engineering_shandong univ. discrete-time signal processing chapter...

23/4/19 1Zhongguo Liu_Biomedical Engineering_Shandong U

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Discrete-Time Signal processingChapter 3 the Z-transform

Zhongguo Liu

Biomedical Engineering

School of Control Science and Engineering, Shandong University

山东省精品课程山东省精品课程《《生物医学信号处理生物医学信号处理 (( 双语双语 )) 》》http://course.sdu.edu.cn/bdsp.htmlhttp://course.sdu.edu.cn/bdsp.html

23/4/192Zhongguo Liu_Biomedical Engineering_Shandong U

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Chapter 3 The z-Transform

3.0 Introduction3.1 z-Transform3.2 Properties of the Region of

Convergence for the z-transform3.3 The inverse z-Transform3.4 z-Transform Properties


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3.0 Introduction

Fourier transform plays a key role in analyzing and representing discrete-time signals and systems, but does not converge for all signals.

Continuous systems: Laplace transform is a generalization of the Fourier transform.

Discrete systems : z-transform, generalization of DTFT, converges for a broader class of signals.


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3.0 Introduction

Motivation of z-transform:The Fourier transform does not

converge for all sequences and it is useful to have a generalization of the Fourier transform.

In analytical problems the z-Transform notation is more convenient than the Fourier transform notation.


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3.1 z-Transform

jw

n

jwnX e x n e

[ ]n

nX z x n x nz

Z

0n

nznxzX

one-sided, unilateral z-transform

z-Transform: two-sided, bilateral z-transform

[ ]x n X zZ

If , z-transform is Fourier transform.

jwez


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Express the complex variable z in polar form as

Relationship between z-transform and Fourier transform

jwrez

n

jwnnjw ernxreX

The Fourier transform of the product of and the exponential sequence

nxnr

1, jwIf r X Z X e


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Complex z planecircleunitw

8

periodic sampling

[ ] ( ) | ( )c t nT cx n x t x nT

T ： sample period; fs=1/T:sample rateΩs=2π/T:sample rate

n

nTtts

s c c cn n

x t x t s t x t t nT x nT t nT

Review

9

ContinuousTime domain ： )(tx

Complex frequency domain ：

dtetxsX st)()(

Laplace transformLaplace transform

js

f2

s pl ane

j

0

Relation between Laplace Relation between Laplace Transform and Z-transformTransform and Z-transform

Review

10

dtetxjX tj)()(

Fourier Transform

frequency domain ：

s-pl ane

j

0

Fourier Transform is the Laplace transform Laplace transform

when s have the value only in imaginary axis, s=jΩ

js Since

So 0 s j

dtetxsX st)()(

Laplace Transform and Fourier Laplace Transform and Fourier transformtransform

11

For sampling signal ，

( ) stc

n

x nT t nT e dt

snTc

n

x nT e

[ ] sts sx t x t e dt

L

j TsTz e e 令[ ] ( )n

n

x n z X z

z-transform

of discrete-

time signal

the Laplace transformLaplace transform

s c cn n

x t x t t nT x nT t nT

( )STX es j T

( )jX e [ ]x n

jre

12

( )sT j T T j T jz e e e e re

( ) [ ] n

n

X z x n z

so ：Tr e

T

relation betweens zand

Laplace transformLaplace transform continuous time signal

z-transformz-transform discrete-time signalrelation

[ ] ( ) ( )snT sTs

n

x t x nT e X e

LsTz elet ：

13

,T

DTFT : Discrete Time Fourier Transform

( )sT j T T j T jz e e e e re

( ) ( )j j n

n

X e x n e

S plane Z plane

-

3 / T

j

/ T

/ T

1|j jrz re e

0, 1r

1r

1r 0

0, 1r ( ) [ ] n

n

X z x n z

T

Go

14

2 2s sT f f f

zplane

Re[ ]z

Im[ ]z

0

r

0 2 / 2

0 2

2

s

s s

s s

f

f

: 0

0 2

2 4

:sf f f

j

s pl ane

0

22

s

s

f

T

sT

3

sT

3

sT


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Region of convergence (ROC)

For any given sequence, the set of values of z for which the z-transform converges is called the Region Of Convergence (ROC).

n

n

X z x n z

jw n

n

X re x n r

0

n

n

X z x n z

jwrez

Absolute Summability

n

jwnnjw ernxreX

the ROC consists of all values of z such that the inequality in the above holds

z1

if some value of z, say, z =z1, is in the ROC, then all values of z on the circle defined by |z|=|z1| will also be in the ROC.if ROC includes unit circle, then Fourier transform and all its

derivatives with respect to w must be continuous functions of w.

0

n

n

X z x n z

r zConvergence of the z-transform for a

given sequence depends only on .



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sin,lp

cw nh n

n 0[ ] cos ,x n w n

Neither of them is absolutely summable, neither of them multiplied by r-n (-∞<n<∞) would be absolutely summable for any value of r. Thus, neither them has a z-transform that converges absolutely.

-∞<n<∞


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The Fourier transforms are not continuous, infinitely differentiable functions, so they cannot result from evaluating a z-transform on the unit circle. it is not strictly correct to think of the Fourier transform as being the z-transform evaluated on the unit circle.

0cos w n 0 02 2k

w w k w w k

ww,

ww,eH

c

cjwlp 0

1 sin

lpcw n

h nn


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Zero and pole

zQ

zPzX

0zXZero: The value of z for which

The z-transform is most useful when the infinite sum can be expressed in closed form, usually a ratio of polynomials in z (or z-1).

zXPole: The value of z for which


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Example 3.1: Right-sided exponential

sequence

1

0 0

nn n

n n

X z a z az

az:polez:zero 0

nuanx n

Determine the z-transform, including the ROC in z-plane and a sketch of the pole-zero-plot, for sequence:

Solution:

ROC:1 1az or z a

1

1

1

z

az z a


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: zeros : poles

Gray region: ROC

nuanx n

zX z

z a

for z a


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Ex. 3.2 Left-sided exponential sequence

1

1n n n n

n n

X z a u n z a z

1 nuanx n

,z a

1

1 1

nn n

n n

a z a z

Determine the z-transform, including the ROC, pole-zero-plot, for sequence:

Solution:

ROC:

1

11

a z z

a z z a

: 0 :zero z pole z a


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zX z

z a

for z a

1 nuanx n


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Ex. 3.3 Sum of two exponential sequences

1 1

2 3

n n

x n u n u n

1 1

2 3

n nn

n

X z u n u n z

1 1

0

1 1

2 3

n n

n n

z z

1 1

2 3

n nn n

n n

u n z u n z


Solution:


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Example 3.3: Sum of two exponential sequences

1 1 1:

2 3 2z and z ROC z

1 1

0

1 1

2 3

n n

n n

X z z z

1 1

12

121 1

1 12 3

z z

z z

1 1

1 11 1

1 12 3

z z

ROC:


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1

11

12

z

1

11

13

z

1 1

12

121 1

1 12 3

z z

z z

1 1

2 3

n n

x n u n u n


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Example 3.4: Sum of two exponential

2

1

21

1

1

2

1

1

z,

znu

Zn

3

1

31

1

1

3

1

1

z,

znu

Zn

2

1

31

1

1

21

1

1

3

1

2

1

11

z,

zznunu

Znn

1 1

2 3

n n

x n u n u n

nuanx n

zX z

z a

for z a

Solution:

ROC:


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1

11

12

z

1

11

13

z

1 1

12

121 1

1 12 3

z z

z z

1 1

2 3

n n

x n u n u n


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Example 3.5: Two-sided exponential sequence

12

1

3

1

nununx

nn

3

1

31

1

1

3

1

1

z,

znu

Zn

2

1

21

1

11

2

1

1

z,

znu

Zn

2

1

3

1

21

131

1

121

2

21

1

1

31

1

1

1111

z:ROCzz

zz

zzzX

zX z

z a

for z a

1 nuanx n

Solution:


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ROC, pole-zero-plot

12

1

3

1

nununx

nn

2

1

3

1

21

131

1

121

2

21

1

1

31

1

1

1111

z:ROCzz

zz

zzzX


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Finite-length sequence

5 nnnx

01 5 z:ROCzzX

2

1

N

n N

nX z x n z

Example :


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Example 3.6: Finite-length sequence

otherwise,

Nn,anx

n

0

10

1 1

1

0 0

N N nn n

n n

X z a z az

: 0ROC z 1

1 N N

N

z a

z z a

1

1

1

1

Naz

az


Solution:


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: 0ROC z 1

1 N N

N

z aX z

z z a

N=16, a is real

pole-zero-plot


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z-transform pairs

zall:ROC,n 1

11

11

z:ROC,z

nu

11

11

1

z:ROC,

znu

,

: 0 0 0

mn m

ROC all z except if m or if m

z


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z-transform pairs

az:ROC,az

nuan

11

1

az:ROC,az

nuan

11

11

az:ROC,az

aznunan

21

1

1

az:ROC,az

aznunan

21

1

11


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z-transform pairs

1

21

121

0

10

0

z:ROC,zzwcos

zwcosnunwcos

1

21 210

10

0

z:ROC,zzwcos

zwsinnunwsin

00 01

cos2

jw n jw nw n u n u ne e

1 10 0

1 1 1

2 1 1jw jw

e z e z


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z-transform pairs

00 01

cos2

n n njw n jw nr w n u n r r u ne e

1 10 0

1 1 1

2 1 1jw jw

re z re z

rz:ROC,

zrzwcosr

zwcosrnunwcosr n

2210

10

0 21

1

rz:ROC,

zrzwcosr

zwsinrnunwsinr n

2210

10

0 21


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z-transform pairs

0

1

1

0

101

z:ROC

,az

za

otherwise,

Nn,a NNn

3.2 Properties of the ROC for the

z-transform

LR rzr0

zFor a given x[n], ROC

is dependent only on .

Property 1: The ROC is a ring or disk in the z-plane centered at the origin.

1n nR Lr u n r a u n

1 1

1 1, :

1 1 R LR L

ROC r z rr z r z


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3.2 Properties of the ROC for the z-transform

Property 2: The Fourier transform of converges absolutely if the ROC of the z-transform of includes the unit circle.

nx

nx

The z-transform reduces to the Fourier transform when ie.1z jwez

0n

nznxzX jw

n

jwnX e x n e


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z-transform

Property 3: The ROC cannot contain any poles.

is infinite at a pole and therefore does not converge.

zX

0

n

n

P zX z x n z

Q z


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z-transform

Property 4: If is a finite-duration sequence, i.e., a sequence that is zero except in a finite interval :

nx

1 2N n N

then the ROC is the entire z-plane, except possible or 0z z

2

1

Nn n

n N

X z a z


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Property 5: If is a right-sided sequence, i.e., a sequence that is zero for , the ROC extends outward from the outermost finite pole in to (may including)

nx 1Nn

zX z

1

,N

n

k kk

x n A d

1,n NProof:

1N1N

. . Ni e r d1 1 ,N Nif r d d d


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Property 6: If is a left-sided sequence, i.e., a sequence that is zero for , the ROC extends inward from the innermost nonzero pole in to .

nx

2Nn

zX 0z

2 ,n N 1

,N

n

k kk

x n A d

Proof:

2n N2n N

n n

1r d1 , , ,Nr d r d ,kr d


z-transformProperty 5: If is a right-sided

sequence, i.e., a sequence that is zero for , the ROC extends outward from the outermost finite pole in to (possibly including)

nx

1Nn

zX z

1

Nn

k kk

x n A d

1n NProof:

1

1 0, ., 0N

n

k kk

NA d u n ef i ni


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Property 5: right-sided sequence

1

Nn

k kk

x n A d

0

n nk k

n

A d z

1

0,N

n

k kk

A d u n n

1,

1k

kk

Az r d

d z

1 , , Nr d r d Nr d

1 1N Nif d d d

n

k kA d u nfor the z-transform:

ROCFor other terms:


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z-transformProperty 6: If is a left-sided

sequence, i.e., a sequence that is zero for , the ROC extends inward from the innermost nonzero pole in to

nx

2Nn

zX 0z

2n N 1

Nn

k kk

x n A d

Proof:

21

01 , , . 0N

n

k kk

A d u n if N ie n


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Property 6: left-sided sequence

1

Nn

k kk

x n A d

1

n nk k

n

A d z

0n

1,

1k

kk

Az r d

d z

1 , , Nr d r d 1r d

1

1N

n

k kk

A d u n

1n

k kA d u n

for

the z-transform:

For other terms:

ROC

23/4/1949


z-transformProperty 7: A two-sided sequence

is an infinite-duration sequence that is neither right-sided nor left-sided.

If is a two-sided sequence, the ROC will consist of a ring in the z-plane, bounded on the interior and exterior by a pole and not containing any poles.

x n


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z-transformProperty 8: ROC must be a connected region.

for finite-duration sequence

0 z possibleROC:0zz

for right-sided sequence

Rr z possibleROC: zfor left-sided sequence

0 Lz r possibleROC: 0z

R Lr z r for two-sided sequence


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Example: Different possibilities of the ROC define different

sequences

A system with three poles


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(b) ROC to a right-sided sequence

Different possibilities of the ROC.

(c) ROC to a left-handed sequence


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(e) ROC to another two-sided sequence

Unit-circleincluded

(d) ROC to a two-sided

sequence.


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LTI system Stability, Causality, and ROC

A z-transform does not uniquely determine a sequence without specifying the ROC

It’s convenient to specify the ROC implicitly through time-domain property of a sequence

stable system(h[n] is absolutely summable and therefore has a Fourier transform): ROC include unit-circle.

causal system (h[n]=0,for n<0) : right sided

Consider a LTI system with impulse response h[n]. The z-transform of h[n] is called the system function H (z) of the LTI system.


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Ex. 3.7 Stability, Causality, and the ROC

Consider a LTI system with impulse response h[n]. The z-transform of h[n] i.e. the system function H (z) has the pole-zero plot shown in Figure. Determine the ROC, if the system is:

(1) stable system: (ROC include unit-circle)

(2) causal system: (right sided sequence)


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ROC: , the impulse response is two-sided, system is non-causal. stable.


22

1 z

Solution: (1) stable system (ROC include unit-

circle),


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ROC: ,the impulse response is right-sided. system is causal but unstable.


2z

21

2

A system is causal and stable if all the poles are inside the unit circle.

(2) causal system: (right sided sequence)


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ROC: , the impulse response is left-sided, system is non-causal, unstable since the ROC does not include unit circle.

2

1z



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3.3 The Inverse Z-TransformFormal inverse z-transform is based

on a Cauchy integral theorem.11

( )2

nn c

x X z z dz c ROCj

围线 c ： X(z) 的环状收敛域内环绕原点的一条逆时针的闭合单围线。

（留数 residue法）

1[ ( ) ]Res n

iiz zX z z

Zi 是 X(z)zn-1 在围线 C 内的极点。

Re[ ]z

Im[ ]j z

0Rr Lr

C


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3.3 The Inverse Z-Transform

Less formal ways are sufficient and preferable in finding the inverse z-transform. :Inspection methodPartial fraction expansionPower series expansion


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3.3 The inverse z-Transform3.3.1 Inspection Method

1

1,

1

Zna u n

azz a

2

1

21

1

1

1

z,

zzX

nunxn

2

1


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3.3 The inverse z-Transform

3.3.1 Inspection Method

1

11 ,

1

Zna u n

azz a

2

1

21

1

1

1

z,z

zX

12

1

nunx

n


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3.3 The inverse z-Transform3.3.2 Partial Fraction Expansion

1

0 0 0 0

10

0 0 0

1

1

MM Mk N M k

kk kk k kN N N

k M N kk k k

k k k

c zb z z b zb

X zaa z z a z d z

11

1

1

1

Nk

k k

k kkz d

A

d z

w

if M

here A d z X z

N


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Example 3.8Second-Order z-Transform

1 1

1,

1 11 1

22

4

1X z

z zz

1

2

1

1

21

141

1 z

A

z

AzX

14

11

4

1

11

z

zXzA

22

11

2

1

12

z

zXzA


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Example 3.8Second-Order z-Transform

2

1

21

1

2

41

1

1

11

z,

zzzX

nununxnn

4

1

2

12


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NMifza

zbzX N

k

kk

M

k

kk

0

0

N

k k

kNM

r

rr zd

AzBzX

11

0 1Br is obtained by long division

Inverse Z-Transform by Partial Fraction Expansion


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Inverse Z-Transform by Partial Fraction Expansion

s

1mm1

i

mN

ik,1k1

k

kNM

0r

rr

zd1

Czd1

AzBzX

Br is obtained by long division

kdz

1kk zXzd1A

1

111

!i

s m

s m s m

w d

m

s

i

i

dd

C X ws dm d

ww

X zif M>N, and has a pole of order s at d=di


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Example 3.9:Inverse by Partial Fractions

1,1

21

1

1

21

23

1

21

11

21

21

21

zzz

z

zz

zzzX


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1 2

1 20 1

1 2 1

1 23 1 1 11 12 2 2

A Az zX z B

zz z z

2

15

23

1212

3

2

1

1

12

1212

z

zz

zzzz

11

1

121

1

512

zz

zzX


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11 2

111 1

1 52 2

11 111 122

A AzX z

zzz z

9

2

11

121

1

51

2

1

1

11

1

1

z

zzz

zA

81

121

1

51

1

1

11

1

2

z

zzz

zA


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11

9 82 , 1

1 112

X z zzz

nZ 22 nuz

nZ

2

1

21

1

1

1

nuz

Z 11

1

nununnxn

82

192


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For a LTI system with impulse response h[n], if it is causal, what do we know about h[n]? Is h[n] one-sided or two-sided sequence? Left-sided or right-sided?

nh nx ny

Then what do we know about the ROC of the system function H (z)?

If the poles of H (z) are all in the unit circle, is the system stable?

Review


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For H (z) with the poles as shown in figure , 1 1 1

1

1 1 1H z

az bz cz

Unit-circle

included

can we uniquely determine h[n] ?

is the system stable ?

If ROC of H(z) is as shown in figure, can we uniquely determine h[n] ?

Review


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For H (z) with the poles as shown in figure , 1 1 1

1

1 1 1H z

az bz cz

If the system is causal (h[n]=0,for n<0,right-sided ), What’s the ROC like?

If ROC is as shown in figure, is h[n] one-sided or two-sided? Is the system causal or stable?

Review


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3.3 The Inverse Z-Transform

Inspection methodPartial fraction expansion

Power series expansion zall:ROC,n 1

az:ROC,az

nuan

11

11

az:ROC,az

nuan

11

1

Review


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Partial Fraction Expansion

s

1mm1

i

mN

ik,1k1

k

kNM

0r

rr

zd1

Czd1

AzBzX

Br is obtained by long division

kdz

1kk zXzd1A

1

111

!i

s m

s m s m

w d

m

s

i

i

dd

C X ws dm d

ww

X zif M>N, and has a pole of order s at d=di

Review


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3.3 The inverse z-Transform

3.3.3 Power Series Expansion

2 1 1 22 1 0 1 2

nn

n

X z x n z

x z x z x x z x z


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Example 3.10:Finite-Length Sequence

121112

2

11

2

111

2

11

zzzzzzzzX

otherwise

n

n

n

n

nx

,0

1,2

10,1

1,2

12,1

12

11

2

12 nnnnnx


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Ex. 3.11: Inverse Transform by power series expansion

1log 1 ,X z az z a

11 1

nn a

u nn

1

2 3 41 1 12 3 4

1

( 1)log 1

n n

n

xx x x x x

n

1x

1

11( 1

1)

,n n n

n

z zXn

aa z

11 , 1

( )

0, 1

nn a

nx n n

n


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Example 3.12: Power Series Expansion by Long Division

azaz

zX

,1

11

221

22

221

1

1

11

1

11

zaaz

za

zaaz

az

az

az

2211

11

1zaaz

az

nuanx n

Z

na u n


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Example 3.13: Power Series Expansion for a Left-sided Sequence

azaz

zX

,1

11

1 2 2

1

1

1

1 2 2

2 2

1 1

1

a z a z

az

a z

a z

a z a z

a z

1 2 21

1

1a z a z

az

1 nuanx n

1Z

na u n


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3.4 z-Transform Properties

xZ RROCzXnx ,

1

,11 xZ RROCzXnx

2

,22 xZ RROCzXnx

1 2

1 2 1 2

( more than)

Z

x xcon

ax n bx n aX z b

tai

X z

ROC R Rns may

3.4.1 Linearity


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Example of Linearity

1

1, :

1Zna u n ROC z

aa

z

1, :

1

N NZn a z

a u n N ROC za

az

1 1

1

1 1

N NZ n na z

a u n a u n Naz az

n nx n a u n a u n N

0z 11 1 2 2 1

1

1 1

1

NNaz az a z a z

az

11 2 2 11 NNaz a z a z


niv.

3.4.2 Time Shifting

00

nZx n n z X z

nx0n

0n nx

is positive, is shifted right

is negative, is shifted left

0x

except for the possible addition

or deletion of z or zROC R

0n is an integer


niv.

Time Shifting: Proof

0nnxnyif

n

nznnxzY 0 0nnmLet

0m n

m

Y z x m z

0 0n nm

m

z x m z z X z


niv.

Example 3.14:Shifted Exponential

Sequence

4

1,

41

1

z

zzX

11

1

41

1

44

41

1

zz

zzX

14 4

4

n

x n n u n

1

1

41

1

1

zzzX

14

11

nunxn

1

4

nZ u n


niv.

3.4.3 Multiplication by an Exponential sequence

xZn RzROCzzXnxz 000 ,/

0 0 0,R L R Lxx z Rr z r z rif R is i z z rs

jwez 00

jwezif

AMeXnxe wwjFnjw :00


niv.

Example 3.15:Exponential Multiplication

1

1,

11Zu n z

z

0 0

0cos

1 1

2 2

n

n njw jw

x n r w n u n

re u n re u n

0

0 1

11 2 ,2 1

njw Zjwre u n z

rer

z


niv.

0 01 1

10

1 2 20

1 12 2

1 1

1 cos,

1 2 cos

jw jw

z r

X zre z re z

r w z

r w z r z

0 01 1

2 2

n njw jwx n re u n re u n

0

0 1

11 2 ,2 1

njw Zjw

re u nre z

z r


niv.

3.4.4 Differentiation of X(z)

,Z

x

dX znx n z

dzROC R

n

nznxzX

1n

n

dX zz z n x n z

dz

n

n

nx n z Z nx n


niv.

Example 3.16:Inverse of Non-Rational z-

Transform

azazzX ,1log 1

1

2

1

az

az

dz

zdX

1

1,

1 ( )Z dX z az

nx n z z adz za

11

nx a uan n n

azaznun

a Zn

n ,1log11 11

Look


niv.

azaz

aznuna Zn

,1

21

1

Example 3.17: Second-Order Pole

nuannunanx nn

azaz

nua Zn

,1

11

azaz

az

azazdz

dzzX

,1

,1

1

21

1

1


niv.

3.4.5 Conjugation of a complex Sequence

* * * ,Zxx n X z ROC R

n

nznxzX

n n

n n

x n z x n z

n

n

x n z X z

X z


niv.

3.4. 6 Time Reversal

* **

1 1,Z

xx n X ROC Rz

1 1: :1R L Lx R

xRr z r r z rR

if x n is real or we do not conjugate x n

x

Z

RROCzXnx 1,1


niv.

Example 3.18: Time-Reverse Exponential

Sequence

nuanx n

1

1X z

az

1

1,

1Zna u n z a

az

1 11

1 1,

1

a zz a

a z

x

Z

RROCzXnx 1,1


niv.

3.4. 7 Convolution of Sequences

21

,2121 xxZ RRcontainsROCzXzXnxnx

k

knxkxnxnxny 2121 *

1 2n n

n n k

Y z y n z x k x n k z

1 2n

k n

x k x n k z

1 2m k

k m

x k x m z z

1 2X z X z 2 1k

k

X z x k z


niv.

Ex. 3.19: Evaluating a Convolution Using the z-transform

1nx n a u n

2x n u n

1if a

Y z

1 1

1

1 1

1

az z

z

1 2*y n x n x n

1 1

1,

1Z X z z a

az

2 1

1, 1

1Z X z z

z

Solution:


niv.

Example 3.19: Evaluating a Convolution Using the z-

transform

1 1

1 1

1, 1

1 1

1 1, 1

1 1 1

Y z zaz z

az

a z az

nuanua

ny n 1

1

1


niv.

3.4. 8 Initial Value Theorem

00 nfornxif

zXxzlim0

0 1

0nn

n n

x nX z x n z x

z

1

0 0lim lim nz z n

x nX z x x

z


niv.


00 01

cos ( )2

jw n jw nw n e e

0 02 2k

w w k w w k

-∞<n<∞,

ww,

ww,eH

c

cjwlp 0

1 sinlp

cw nh n

n

does not converge

uniformly to the discontinuous

function .

jwn

n

c en

nwsin

jwlp eH

-∞<n<∞


niv.

Example

For nunx

0

jw jwn

n

X e e

does not absolutely converge

10

1

1

1 : 1

jw n jwnjw

n

X z X re r er e

absolutely converge if r ROC z

12

1 jwk

w ke

There’s no Z-Transform for 1,x n n


niv.23/4/19102Zhongguo Liu_Biomedical Engineering_Shandong U

niv.

Chapter 3 HW3.3, 3.4, 3.9, 3.16

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