2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems

23/4/21 1Zhongguo Liu_Biomedical Engineering_Shandong U

niv.

Biomedical Signal processingChapter 5 Transform Analysis

of Linear Time-Invariant Systems

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

2

Chapter 5 Transform Analysis of Linear Time-

Invariant Systems5.0 Introduction5.1 Frequency Response of LTI Systems5.2 System Functions For Systems

Characterized by Linear Constant-coefficient Difference equation

5.3 Frequency Response for Rational System Functions

5.4 Relationship Between Magnitude and Phase5.5 All-Pass System5.6 Minimum-Phase Systems5.7 Linear Systems with Generalized Linear

Phase

3

5.0 IntroductionAn LTI system can be characterized in

time domain by impulse responseOutput of the LTI system:

h n

k

knhkxnhnxny

zXzHzY

in Z-domain by system function

in frequency-domain by Frequency response

jwjwjw eXeHeY

With Fourier Transform and Z-transform, an LTI system can be characterized H z

jwH e

4

5.1 Frequency Response of LTI Systems

jwjwjw eXeHeY

jwjwjw eXeHeY

Phase response (phase shift) jweH

jwjwjw eXeHeY

jweHFrequency response

jweHMagnitude response (gain)

distortions

change on useful signal

syst

em

Usefulinput signal

+deleteriou

ssignal

5

5.1.1 Ideal Frequency-Selective Filters

Ideal lowpass filter

1,

0,cjw

lpc

w wH e

w w

jweH

0 cwcw 22

1

sin,

c

lp

w n

nnh n

Noncausal, not computationally realizable

no phase distortion

6


Ideal highpass filter

0,

1,cjw

hpc

w wH e

w w

sin,c

hp

w nh n n n

n

0 cwcw 22

jweH

1

1 jwlpH e

7


Ideal bandpass filter

1 21,

0,

c cjwbp

w w wH e

others

01cw

1cw

jweH

1

2cw2cw

8


Ideal bandstop filter

1 20,

1,

c cjwbs

w w wH e

others

0

jweH

1

1cw1cw

2cw2cw

9

5.1.2 Phase Distortion and Delay

did nnnh

dnnxny

djwnjwid eeH

1jwid eH wwneH d

jwid ,

The frequency response

The impulse response

To understand the effect of the phase and the group delay of a linear system, first consider the ideal delay system:

10

Group Delay(群延迟， grd )

For ideal delay system

argjw jwdw H e Hd eg

wr

d

d d

dwn n

dw

0arg jwdIf H e wn dthen w n

The group delay represents a convenient measure of the linearity of the phase.

arg arg djwnjwd dw H e e

dw dw

Given a narrowband input x[n]=s[n]cos(w0n) for a system with frequency response H(ejw), it is assumed that X(ejw) is nonzero only around w =w0

11

Group Delay(群延迟， grd )

00 00[ ] cos( )d

jd

wy n H e s n wn wn n

0arg ,jwdI wf H e n dnthen w

it can be shown (see Problem 5.57) that the response y[n] to x[n] is

the time delay of the envelope s[n] is .

dn

Group Delay

12

25.0w85.0w

5.0w

Example 5.1 Effect of Attenuation and Group Delay

Three consecutive narrowband pulsesis applied to a filter

连贯的

13

Ex. 5.1 Effect of Attenuation and Group Delay

Filter frequency response

Group Delay

magnitude

25.0w

85.0w5.0w

14

25.0w85.0w 5.0w

Group Delay

50

Group Delay

200

15

5.1.2 Phase Distortion and Delay

Ideal lowpass filter with linear phase delay

,

0,

djwncjw

lpc

e w wH e

w w

nnnn

nnwnh

d

dclp ,

sin

delay distortion is a rather mild form of phase distortion, its effect is to shift the sequence in time. we accept linear phase response rather than zero phase response.

The impulse response (delayed by time nd )

16

5.2 System Functions For LTI Systems Characterized by Linear Constant-coefficient Difference

equation

Linear Constant-coefficient Difference equation

M

kk

N

kk knxbknya

00

M

k

kk

N

k

kk zXzbzYza

00

zXzbzYzaM

k

kk

N

k

kk

00

17

5.2 LTI System Characterized by Linear Constant-coefficient

Difference equation

M

kk

N

kk knxbknya

00

M

k

kk

N

k

kk zXzbzYza

00

If a system is not LTI, then the following Z-transform cannot be derived. (see P37, example 2.16, for x[n]=kδ[n], y[n]=an+1c + Kanu[n], for all n, x[n] have z-transform K, y[n] have no z-transform.)

18

11 0 :k kz c zec roz

1

0 0 1

10

0 1

1

1

MMk

kkk kN N

kk k

k k

c zb zY z b

H zX z aa z d z

11 0 :k kz d pod lez

For an LTI system:

its poles and zeros:

5.2 System Functions For Systems Characterized by Linear Constant-coefficient

Difference equation

19

Ex. 5.2 find difference equation for second-order System function

21

1 1

1

1 31 1

2 4

zH z

z z

zXzzzYzz 2121 218

3

4

11

21228

31

4

1 nxnxnxnynyny

1 2

1 2

1 21 3

14 8

Y zz z

X zz z

Solution:

20

5.2.1 Stability and Causality

The difference equation does not uniquely specify the impulse response of a linear time-invariant system.

Each possible choice for the ROC of the system function will lead to a different impulse response, but they will all correspond to the same difference equation.

21

Causality

For a causal system the impulse response must be right-sided sequence.

nh

The region of convergence (ROC) of must be outside the outermost pole.

zH

0, 0h n for n

22

Stability

n

nh

1

zforznhn

n

ROC of includes the unit circle

zH

For a stable system The impulse response must be absolutely summable, i.e.,

23

Ex. 5.3 Determine the ROC, Stability and causality for LTI

system:

nxnynyny 212

5

1121 21

21

1

1

25

1

1

zzzz

zH

poles: 1/2, 2; zeros(two) ： 0

Solution:

24

Example 5.3 Determining the ROC

2 : ,z causal not stable

12 :

2,

z

stable not causal

stablenotcausalnotz ,:2

1

1)

2)

3)

25

Causal and Stable system

Causal: ROC must be outside the outermost pole

Stable: ROC includes the unit circle

Causal and stable: all the poles of the system function are inside the unit circle. ROC is outside the outermost pole, and includes the unit circle.

26

5.2.2 Inverse Systems

1,iG z H z H z 1

iH zH z

Time domain: nnhnhng i

Not all systems have an inverse. Ideal LPF hasn’t

For a LTI system , the inverse system

which cascaded with satisfies:

H z iH z zH

jwjw

i eHeH

1Frequency response

x[n] y [n] h n ih n( )H z ( )iH z

27

5.2.2 Inverse Systems

N

kk

M

kk

zd

zc

a

bzH

1

1

1

1

0

0

1

1

N

kk

M

kk

i

zc

zd

b

azH

1

1

1

1

0

0

1

1

ROC of and ROC of must overlap, for convolution theorem to hold:

zH zH i

zHzH i

1systems with

rational system functions:

ih n h n n 1,iH z H z

28

Ex. 5.4 analyse Inverse System for First-Order

System

1

1

1 0.5:

1 0.0

9.9

zH z R zOC

z

1

1

1 0.9:

1 0.0.5

5i

zH z ROC

zz

15.09.05.0 1 nununh nni

:i stable and cH z ausal

Solution:

29

Ex. 5.5 find Inverse for System with a Zero in the

ROC

1

1

0.5:

1 0.0.9

9

zH z R zOC

z

1

1

1

1

21

8.12

5.0

9.01

z

z

z

zzH i

1

1: 2 2 1 1.8 22n n

iROC h n uz n u n

1

2: 2 2 1.82 2 1n n

iROC h n u n u nz

20.9Solution:

: ,i nos tta caH blez usal

1)

2)

: ,i noc tau stH salz able

30

Minimum-phase Systems

A LTI system is stable and causal and also has a stable and causal inverse if and only if both the poles and the zeros of are inside the unit circle

zH

Such systems are referred as minimum-phase systems

1

9.0:9.01

5.011

1

zROCz

zzH

5.0:5.01

9.011

1

zROCz

zzH i

31

5.2.3 Impulse Response for Rational System Functions

For a LTI system

N

k k

kNM

r

rr zd

AzBzH

11

0 1

N

k

nkk

NM

rr nudArnBnh

10

can be infinite impulse response (IIR)

or finite impulse response (FIR)

If causal,

0

0

, M N

Mk

kkN

kk

k

b zH z

a z

32

FIR System

0

, 0

0,

M

kn

k

b n Mh n n k

otherb

wise

0

Mk

kk

H z b z

0

M

kk

y n b x n k

( )

( )

Y Z

X Z

0

[ ]M

k

nb k k

33

Ex.5.6 A First-Order IIR System

nxnayny 1

1

1, :

1H z ROC a

azz

1,a

nuanh n

Determine System function, condition of stability, h[n] for stable, causal System.

Solution:∵ it is causal

condition of stability:

34

Ex.5.7 A Simple FIR System

otherwise

Mnanh

n

,0

0,

1

11

0 1

1

az

zazazH

MMM

n

nn

2 1: , 0,1, ,j k Mkz ae kzeros M

7M

Determine System function, zero-pole plot, stability, difference equation For h[n]:

Solution:

system is stable: kpole p a

, 0z

35

Example 5.7 A Simple FIR System

Difference equation

0

Mk

k

y n a x n k

11 1My n ay n x n a x n M

0

( )

( )

Mn n

n

aY

HZ

z zX Z

1 1

1

1

1

M Ma z

az

36

5.3 Frequency Response for Rational System

FunctionsIf a stable LTI system has a

rational system function

N

k

jwkk

M

k

jwkk

jw

ea

ebeH

0

0

Its frequency response is

N

k

kk

M

k

kk

za

zbzH

0

0

37

5.3 Frequency Response for Rational System

Functions

N

k

jwk

M

k

jwk

jw

ed

ec

a

beH

1

1

0

0

1

1

N

k

jwk

M

k

jwk

jw

ed

ec

a

beH

1

1

0

0

1

1

2jw jw jwH e H e H e

22

0 1

0 *

1

1 1

1 1

Mjw

k kjw

j

jw kN

jwk k

w

k

c e c eb

H ea

d e d e

magnitude-squared function:

38

Log magnitude

010 10 10

1 10

20log 20log 1 20log 1M N

jw jwk k

k k

bc e d e

a

1020log jwGain in d H eB

1020log jwAtt H eenuatio Gain in dn in dB B

1020log jwH e log magnitude in dB

N

k

jwk

M

k

jwk

jw

ed

ec

a

beH

1

1

0

0

1

1

39

Output: Log magnitude, phase

jwjwjw eXeHeY

jwjwjw eXeHeY 101010 log20log20log20

jwjwjw eXeHeY

jwjwjw eXeHeY

40

N

k

jwk

M

k

jwk

jw edeca

beH

110

0 11

1 1

arg 1 arg 1N M

jw jw jwk k

k k

gd d

H e d e c edw

rddw

0 1

0

1

1

1

Mjw

kjw k

Njw

kk

c eb

H ea

d e

Phase Response for a rational system function

group delay:

41

Phase Response

1 1

a arg 11rg jwN M

jwk

j

kk

k

wd dgrd H e

dwe c

dd

we

22

22

{ }cos

1 2 cos 1 2 { }

jwk k

jwk k

d Re d er r w

r r w d Re d e

( )11 1j jw jk

jw we red ree

sinarg 1 arctan

1 cosjw

k

r wd e

r w

1 cos sinr w jr w

sinarctan

1 cos

r wd

dw r w

2

1arctan '

1x

x

42

Phase Response

1 1

arg 1 arg 1N M

jw jwjk

k k

wk

dd e

dgrd H e

ddwe c

w

2

2

{ }

1 2 { }

jwk k

jwk k

d Re d e

d Re d e

sinarctan

1 cos

r wd

dw r w

2 2

2 21 1

{ } { }

1 2 { } 1 2 { }

jw jwN Mk k k k

jw jwk kk k k k

d Re d e c Re c e

d Re d e c Re c e

arg 1 jwk

dd e

dw

43

Principal Value（主值）Principal Value of the phase of jweH

jwHARG e

2

where r w is a positive or negative integer

jw jwH e H eA r wRG

wredARG

ecARGa

bARGeHARG

N

k

jwk

M

k

jwk

jw

21

1

1

10

0

0 1

0

1

1

1

Mjw

kjw k

Njw

kk

c eb

H ea

d e

441 2 3 4 5 6 7

2

jw jw

jw

arg

AR

H e H e

H e r wG

we refer to ARG [H(ejw)] as the "wrapped" phase,

continuous (unwrapped) phase curve is denotedas arg [H (ejw)]

卷绕的

解卷绕的

45

5.3.1 Frequency Response of a Single Zero or Pole

1 1 1jw jwk kc e or d e

1 : , :j jw r radiusre e wher anglee

2

2

1 1 1

1 2 cos

j jw j jw j jwre e re e re e

r r w

wrrere jwj cos21log101log20 21010

1. formular method 2. Geometrical method

magnitude-squared function:

log magnitude in dB

46

5.3.1 Frequency Response of a Single

Zero or Pole

wr

wrereARG jwj

cos1

sintan1 1

2 2

22

cos cos1

1 2 cos 1

j jw

j jw

r r w r r wgrd re e

r r w re e

1 1 cos sinj jwre e r w jr w

group delay:

Phase Response :

47

Log Magnitude response for a single zero with r=0.9

0

2

wrrere jwj cos21log101log20 21010

1 j jwre e

48

for a single zero with r=0.9

0

2

Group Delay

Phase response

1 sintan

1 cos

r w

r w

2

2

cos1

1 2 cosj jw r r w

grd re er r w

0.470.45 0.470.461 j jwre e

0

49

frequency response is associated with vector diagrams in the complex plane and

pole-zero plots

2. Geometrical method

50

frequency response is associated with vector diagrams in the complex plane and

pole-zero plots

2. Geometrical method

51

Magnitude response for a single zero with

5.0r

7.0r

9.0r

1r

52

Phase response for a single zero with

5.0r

7.0r9.0r

1r

53

Group Delay for a single zero with

5.0r

7.0r

9.0r

54

55

for a single zero outside the unit circle, with

56

Magnitude response for a single zero outside the unit circle, with

9.01r

25.1r

0.2r

57

Phase response for a single zero outside the unit circle,

with

9.01r

25.1r

0.2r

58

Group Delay for a single zero outside the unit circle, with

9.01r

25.1r

0.2r

59

5.3.2 Examples with Multiple Poles and Zeros

（ self study）

60

5.4 Relationship Between Magnitude and Phase

In general, knowledge about the magnitude provides no information about the phase, and vice versa.

If the magnitude of the frequency response and the number of poles and zeros are known, then there are only a finite number of choices for the associated phase.

For frequency response of LTI system

61


Under a constraint referred to as minimum phase, the frequency-response magnitude specifies the phase uniquely, and the frequency-response phase specifies the magnitude to within a scale factor.

If the number of poles and zeros and the phase are known, then, to within a scale factor, there are only a finite number of choices for the magnitude.

62


2* * *1

jw

jw jw jw

z eH e H e H e H z H z

N

kk

M

kk

zd

zc

a

bzH

1

1

1

1

0

0

1

1

*

* 0 1*

*0

1

11

1

M

kkN

kk

zb

ad z

cH

z

1 *2

* * 0 1

1 *0

1

1 11

1 1

M

k kkM

k kk

c z cb

zC z H z H z

azd z d

kc *1 kc

kd*1 kd

共轭倒数对conjugate reciprocal pairs

* *1 1 ( ) 1jw

jw jw

z e

jwz e e e z

magnitude-squared system

function

63


If is causal and stable, then all its poles are inside the unit circle ，the poles of H (z) can be identified from the poles of C(z).

zH

The poles and zeros of occur in conjugate reciprocal pairs, with one element of each pair associated with and one element of each pair associated with

C z

* *1H z H z

* *1C z H z H z

but its zeros are not uniquely identified by C(z)

pz

*pz

1 pz

*1 pz

2z21 z

*11 z

1z

*1z

11 z

64

Example 5.11

1

1 1 4 1

1

4

1

1 0.8 1 0.

2 1 5

8

0.j j

zH z

e z e z

z

1

2 1 1

1

4 4

1 21

1 0.8 1 0.8j j

zH z

e z e

z

z

* *2 2 2 1C z H z H z

Two systems with

have same magnitude-squared system function

* *1 1 1

1

4 1

1

4 1 4 4

1

1 1

1 0.8 1 0.

2 1 0

8 1 0.8 1 0

2 1

.8

0 5. .5j j j j

C z H z H z

z z

e z e

z

z e e z

z

z

11 12 1 0.5 1 22 1 0.5 1 2zzz zz z

65

Example

1H z

1 1

2 4 1 4 1

1 1 2

1 0.8 1 0.8j j

z zH z

e z e z

2H z

* * * *1 1 2 21/ 1/C z H z H z H z H z

1 1

1 4 1 4 1

2 1 1 0.5

1 0.8 1 0.8j j

z zH z

e z e z

66

Ex. 5.12 given

1 *

1 11

z aH z H z

az

1 *

11

z a

az

1

* 1 *

1

1

z a az

a z z a

* * * *1 11 1C H z H z H z H zz

apH z * *1apH z 1

Determine zeros and poles of stable,causal system H(z), if coefficients are real.

number of poles, zeros is known(3)

H(z): conjugate pole,zero pairs

* *1apH z apH z1

* *1C z H z H z

67

Ex. 5.12 given

Determine zeros and poles of stable,causal system H(z), if coefficients are real.

number of poles, zeros is known(3)

H(z): conjugate pole,zero pairs

Solution:

H(z) poles: p1, p2, p3;H(z) Zeros: (z1, z2, z3),

or (z1, z2, z6), or (z4, z5, z3), or (z4, z5, z6),

* *1C z H z H z

68

5.5 All-Pass(全通 ) SystemA stable system function of the form

1 *

1,

1ap

z aH z

az

z a*1z a , Pole:

Zero:

*1 az

az

apH z * *1apH z 1

jw

jwjw

jw

jwjw

ap ae

eae

ae

aeeH

1

1

1

**

11

1 *

jw

jwjwjw

ap ae

eaeeH

frequency-response

magnitude is unity

,1 c la ausa

,1 non la causa*1 az

az

unit circle

69

General Form of All-Pass System

An all-pass system is always stable, since when frequency response characteristics (such as allpass) are discussed, it is naturally assumed that the Fourier transform exists, thus stability is implied.

all-pass system: A system for which the frequency-response magnitude is a constant.

It passes all frequency components of its input with a constant gain A (is not restricted to be unity in this text).

70

General Form of All-Pass System

1 * 11

1 1 * 11 11 1 1

tan''

cr MMk kk

apk kk

k

k

ap

ap

k

k

z e z ez dH z A

d z e z e zwhere A is a pd real pol

ositive cons ts are the of H zs are thee complex pol o zs

se f H

e

71

5.5 All-Pass System

( )

( )

(1 )

(1 )

jw j w

j w

e re

re

sin2arctan

1 cosjw

ap

r wH e w

r w

j

jwj

jjw

jw

jwjw

ap reaifere

ree

ae

aeeH

11

*

(1 cos sin )

1 cos sin

jwe r w jr w

r w jr w

phase response

72

5.5 All-Pass System group delay of a causal all-pass system is positive

1

jw j

j jw

e re

re e

sin2arctan

1 cos

r ww

r w

2

2

cos1 2

1 1 2 cos

jw j

j jw

r r we regrd

re e r r w

2 2

2

1 2 cos 2 2 cos

1 2 cos

r r w r r w

r r w

2

2

1

1 2 cos

r

r r w

2

2

1

1 j jw

r

re e

0

1then r

,if stable causal

73

nonpositivity of the unwrapped Phase of All-Pass Systems

0

0arg argwjw j

ap ap apjH e grd H e d H e

0arg arg 0jap AeH 0jw

apgrd H e

0arg 0jwap fH o we r

1 * 11

1 1 * 11 11 1 1

cr MMk kk

apk kk k k

z e z ez dH z A

d z e z e z

0for w

*

*

0

1 1

1 11

1 1 1

crk kk

k k k

MMj

apk k

e ed

d e eH e A A

74

Example 5.13 analyse First-Order All-Pass System

: 0.9, 0.9, 0rpole z

: 0.9, 0.9,rpole z

11

1 *

jw

jwjwjw

ap ae

eaeeH

Log magnitude 1

jw jjw

ap j jw

e reH e

re e

76

Example 5.13 First-Order All-Pass System: "wrapped"

phase

: 0.9, 0.9,pole z r

: 0.9, 0.9, 0pole z r

sin2arctan

1 cos

r ww

r w

1

jw jjw

ap j jw

e reH e

re e

77

Example 5.13 First-Order All-Pass System: group

delay

,9.0

,9.0:

r

zpole

0,9.0

,9.0:

r

zpole

2

2

1

1 j jw

r

re e

2

2

1

1 2 cos

r

r r w

78

Second-Order All-Pass System

with poles at and

.

( )

(

( )

(1 )1 )

jw jjw

ap

j

j jw

w j

j jw

e re

re e

e re

re eH e

1

1

sin( )( )2 2 tan

(1 )(1 ) 1 cos

sin 2 tan

1 cos

jw j jw j

j jw j jw

r we re e rew

re e re e r w

r w

r w

jz re jz re

79

Ex. 5.13 Second-Order All-Pass System :

4: 0.9 jpole z e Magnitude

Phase"wrapped"

Group delay 4

2

4

80

Fig. 5.21 fourth order all-pass system

81

Frequency response of Fig. 5.21

Magnitude

Phase

Group delay

"wrapped"

24

4

82

Application of All-Pass Systems

Used as compensators for phase or group delay (Chapter 7)

Be useful in the theory of minimum-phase systems (Section 5.6)

Be useful in transforming frequency-selective lowpass filters into other frequency-selective forms and in obtaining variable-cutoff frequency-selective filters (chapter 7)

83

11 0 :k kz c zec roz

1

0 0 1

10

0 1

1

1

MMk

kkk kN N

kk k

k k

c zb zY z b

H zX z aa z d z

11 0 :k kz d pod lez

5.6 Minimum-Phase Systems

For an LTI system:

its poles and zeros:

Its inverse:

1invH z

H z

:kz d zero:kz c pole

84


For a stable and causal LTI system, all the poles must be inside the unit circle.

If its inverse system is also stable and causal, all the zeros must be inside the unit circle.

1p

3z

1z

2z 2p

3pUnit

Circle

1p

3z1z

2z

2p

3p

Unit Circle

85


Minimum-phase system: all the poles and zeros of an LTI system are inside unit circle, so the system and its inverse is stable and causal.

1p

3z

1z

2z 2p

3pUnit

Circle

86

5.6.1 Minimum-Phase and All-Pass Decomposition

Any rational system function can be expressed as

zHzHzH apminSuppose has one zero outside the

unit circle at , , and the remaining poles and zeros are inside the unit circle.

zH*1z c 1c

1

1min

*

1

z

cz

cH z

11 *zH z H z c

stable, causal in the text, but it applies more generally.

reflect the zero to conjugate reciprocal locations inside the unit circle:

1

*

1

1

11

1cz

z

z

cH z

c

87

Example 5.14 Minimum-Phase/All-Pass Decomposition

11

11 31

12

z

zH z

circleunittheinsidezpole2

1:

: 3 :z outside the unit czero ircle1

3z

1

1

1

11

3 ,

3

apH zz

z

1

min1

11

31

13

2

H zz

z

reflect this zero to conjugate reciprocal locations inside the unit circle:

(1)

1 min apH z H z H z

88

Example 5.14 Minimum-Phase/ All-Pass Decomposition

4

21

1 4 13 31

2

3

21

1

1 j je z

z

zz

eH

circleunittheinsidezpole3

1:

43: :

2j outside the uzer nito z c re i cle

42

3jz e reflect two zeros to conjugate

reciprocal locations inside the unit circle:

(2)

89

Example 5.14 Minimum-Phase/All-Pass Decomposition

4 1

4 1

1 1

1 4

41

1

44 2 29 3

21

32

1

21

321

31

3

3

34 1

j jj j

jj

e z

e z

ee z

e z

z

z

ez

4 1

21

413 31

2

3

21

11

j jz ze

z

eH z

min apH z H z

90

5.6.2 Frequency-Response Compensation

When a signal has been distorted by an LTI system with an undesirable frequency response, perfect compensation:

nsnsc zHzH

dc

1If poles and zeros of Hd(z)

are inside the unit circle :

assume that the distorting system is stable and causal and require the compensating system to be stable and causal,

91


zH d

zH c

then perfect compensation is possible only if is a minimum-phase system.

dH z

92


min

1

dc H

Hz

z

minmin

1d pc pd a

daH z H z HH z

H zG zzz H

min ,apdd H zH z H z

If isn’t minimum-phase, its inverse then isn’t stable, so we decompose

zH d

zH d

G z

cs n s n

stable

93

Example 5.15 Compensation of an FIR System

0.6 1 0.6 1

0.8 1 0.8 11 1.25 1 1.

1 0.9 1

5

0.9

2j

j jd

j

H z e z e

e z e

z

z

for causal

unstable, is d

inverz

seH

zeros: outside the unit circle,

Solution:

min ,apdd H z HH z zdecomposition is needed:

min

1c

d

H zH z

94

frequency response of

Magnitude

Phase

Group delay

zH d

"wrapped"

95

Example 5.15 Compensation of an FIR System

18.018.0

16.016.0

25.1125.11

9.019.01

zeze

zezezHjj

jjd

1 0.8 1 0.8

0.8 1 0.8 11 0

0.8 0.

. 0

8

8 1 .8

j j

ap j j

zH

e

e z e

z ez

z

min ,apdd H z HH z z

2 1 0.8 1 0.81.25 0.8 0.8j jz e z e min ofCompe nsation , causal, stable: dinverse H z

min

1c

d

H zH z

0.6 1 0.

0.8 1 0.8 1

6min

2 11 0.9 1 0

1 0.8 1 0.

1.25

8

.9j j

j

d

j

H z e z e z

e z e z

96

Frequency response of

Magnitude

Phase

Group delay

mindH z

"wrapped"

Minimum Phase-Lag

97

Frequency response of apH z

Magnitude

Phase

Group delay

"wrapped"Maxmum Phase-Lag

98

5.6.3 Properties of Minimum-Phase Systems

1. Minimum Phase-Lag Property

jwap

jwjw eHeHeH min

minarg arg argjw jw jwapH e H e H e

arg 0jwapH e for all w

minarg argjw jwH e H e For all systems that have a given magnitude response , minimum-phase system has the Minimum Phase-Lag.

minjwH e

99

0 0j

n

H e h n

1. Minimum Phase-Lag Property

to make the interpretation of Minimum Phase-Lag systems more precise, it is necessary to impose the additional constraint that be positive at

jwH e

0.w

h n jwH eIts system function with same poles and

zeros, is also a minimum-phase system, according to its defination, but the phase is altered by π.

since

100


2. Minimum Group-Delay Property

jwap

jwjw eHeHeH min

jwap

jwjw eHgrdeHgrdeHgrd min

0jwapgrd H e for all w

minjw jwgrd H e grd H e

For all systems that have a given magnitude response , minimum-phase system has the Minimum Group Delay.

minjwH e

101


3. Minimum Energy-Delay PropertyFor any causal, stable, LTI systems

min ,jw jwH e H e min0 0h h

2

02

min

2

2

min0

1

21

2

n

w

jw

n

jH e dw

H e h nw

n

d

h

HW 5.65

If then

1 *

min 1lim ,

1z

z cH z H

zz

c

min0 0 ch h

102

Minimum Energy-Delay Property

For any causal LTI systems, define the partial energy of the impulse response

n

m

mhnE0

2

2

mi

2

n00

n

m

n

m

h h mm

HW 5.66

For all systems that have a given magnitude response , minimum-phase system has the Minimum Energy-Delay.

minjwH e

103

Four systems, all having the same frequency-response magnitude.

Zeros are at all combinations of the complex conjugate zero pairs and and their reciprocals.

0.60.9 je 0.80.9 je

Fig. 5.30

minimum-phase maximum-

phase

104

Minimum-Phase System and Maximum-Phase System

A maximum-phase system is the opposite of a minimum-phase system. A causal and stable LTI system is a maximum-phase system if its inverse is causal and unstable. (From Wikipedia)

Maximum-Phase System: poles are all in the unit circle, zeros are all outside the unit circle. It’s causal and stable.

(noncausal)Maximum-Phase System: anti-causal, stable System whose System function has all its poles and zeros outside the unit circle. (problem5.63).

Maximum energy-delay systems are also often called maximum-phase systems.

105

Sequences corresponding to the pole-zero plots of Fig. 5.30

minimum-phase sequence ha[n]

maximum-phase sequence hb[n].

106

Fig.5.32 Partial energies for the four sequences of Fig. 5.30. (Note that Ea[n] is for the minimum-phase sequence ha[n] and Eb[n] is for the maximum-phase sequence hb[n].

the maximum energy delay occurs for the system that has all its zeros outside the unit circle. Maximum energy-delay systems are also often called maximum-phase systems.

107

5.7 Linear Systems with Generalized Linear Phase

In designing filters, it’s desired to have nearly constant magnitude response and zero phase in passband.

For causal systems, zero phase is not attainable, and some phase distortion must be allowed.

108

5.7 Linear Systems with Generalized Linear Phase

The effect of linear phase (constant group delay) with integer slope is a simple time shift.

A nonlinear phase, on the other hand, can have a major effect on the shape of a signal, even when the frequency-response magnitude is constant

109

5.7.1 System with Linear Phase

, :jw jwid reaH e w le

1jwid eH weH jw

id

jwid eHgrd

sin,

if is in t eger

id

nh n n

n

n

k kn

knkx

n

nnxny

sinsin

110

5.7.1 System with Linear Phase

Specially, if : integerdn

did nnnh

dd nnxnnnxny

nn

nnhid ,

sin

111

Interpretation of

ch t t T

Tjc ejH

( ) ,jw jwH e e w


sinC

n

nt T Tx n

nt x

t T T

c cy t x t T

c ny n x T T

sin

Ck

kt T Tx k

kt x

t T T

comes from sampling a continuous-time signal, if is not an integer *cx t t T

112

Interpretation of

is not an integer


comes from sampling a continuous-time signal, if

sinC

k

kt T Tx k

kt x

t T T

c ny n x T T

sin

k

kxk

n

n

k

sin nx n

n

2T

113

General frequency response with linear

phase

,jw jw jwH e H e e w

For nonconstant magnitude response

multiplication in frequency domain

114

Linear-phase ideal lowpass filter

ww

wweeH

c

cjw

jwlp ,0

,

ww

wweH

c

cjwlp ,0

,1 jwe

n

nwnh c

sin

n

nwnh c

lp

sin

nx nw ny

sinid

nh n

n

The corresponding impulse response is

*

time domain

115

Ex. 5.16 symmetry of impulse response of Ideal Lowpass with Linear Phase in three cases:

α is integer; 2α is integer; 2α is not integerdn

sin

c dlp

d

nnw n

hn n

sin 22

2c d d

lp dd d

w n n nh n n

n n n

sin

clp

w nh n

n

0.4 , 5c dw n

sin c d

lpd

w n nh n

n n

Solution: (1)

:

2lp lp

even symmetric abouth n h n

lph n 2 lph n lph n

116

:

2

lp

lp lp

h n is even

symmetric about

h n h n

, .50.4 4 cw

n

nwnh c

lp

sin(2)

Ex. 5.16 symmetry of impulse response of Ideal Lowpass with Linear

Phase

is an integer2

4.5

117

0. 3, 44 .cw

n

nwnh c

lp

sin(3) is not an integer2

4.3


Phase

118

2

n

nwnh c

lp

sin

ww

wweeH

c

cjw

jwlp ,0

,


Phase

, constant

( )j

jw j j

w

w jw and are

A e is a real b

H

i

e A e

pol

e

possibly funct war ion of

119

5.7.2 Generalized Linear phase

jwjw eHdw

deHgrd arg

1 2 22 111 2

sin 1 21

sin 2jw M M

M M

w M M

we

For moving average system (Ex.2.20, Page 45)

1 2

21

sin ( 1) 21,

si,

n0

2

jwMM

w M

wM M Me

1 2 0,if M M M if negtive, it’s not, strictly speaking,a linear-phase system, since π is added to the phase. It’s the form:

arg , 0jwH e w w

it is referred to Generalized Linear phase system

120

cos sin

wjjw jw

jw jw

H e A e e

A e w jA e w

cos sinjwn

n nn

h n wn j hh n wne n

tan( )

sinsin

cc ososn

n

hw

n wn

h n wn

w

w

cos sin sin cos 0

n n

h n wn w h n wn w

If a system with h[n] has Linear phase

sin 0 n

for allh n w n w

, ,condition o h nn

121

j wjw jwH e A e e

sin 0n

for alln w n wh

This equation is a necessary condition

on h[n], for the system to have constant group delay.

and

It is not a sufficient condition, however, and, owing to its implicit nature,it does not tell us how to find a linear-phase system.

If a system with h[n] has Linear phase

122

One set of condition: even symmetry

jw jw jw jH e A e e sin 0

n

h n w n for all w

2 : integerM 2h n h n

0 or sin 0

n

h n w n

:jwA e even function of w

0 M/2=α

sati

sf

y

Shown inType I,II FIR0 M/2=α M=5

M

M even

M odd sin i2 s nnw w n

sin sin 2 02n n n nwhwh

123

Another set of condition : odd symmetry

2 : integerM 2h n h n

jw jw jw jH e A e e sin 0

n

h n w n for all w

2 3 2or cos 0

n

h n w n

woffunctionoddeA jw :

0

M

M/2

sati

sf

y

Shown inType III, IV FIR0 M/2

M=3

M even

M odd

cos cos 2 02n n n nwhwh

124

5.7.3 Causal Generalized Linear-Phase Systems

0

sin 0n

h n w n for all w

0 0causal h n n

22 or n nh n hh hn

0 2, nh n M

0

sin 0M

n

h n w n for all w

125


Causal FIR systems have generalized linear phase if they have impulse response length

and satisfy 1M

0 or

integer:2 M h M n h n

2 3 2or integer:2 M

h M n h n

jwjw jw jA e eH e

126


If

, 0

0,

n Mh n

otherw

h M

ise

n

, ,jwe real even periodwhere A e is a function of w

2jwj jw wMeA e eH e

then

It’s sufficient condition, not necessary condition

127


If

, 0

0,

n Mh n

otherw

M

s

h

i e

n

, ,jwo real odd periodwhere A e is a function of w

then

2 2 2jw jwM jw j jwMo o

jwH A e e A ej ee

It’s sufficient condition, not necessary condition

128


The above two FIR conditions are sufficient to guarantee a causal system with generalized linear phase.

Clements (1989) showed that causal IIR can also have Fourier transforms with generalized linear phase.

The corresponding system function, however, are not rational, and thus, the systems cannot be implemented with difference equations.

M is even M is odd

h[Mn] = h[n]

h[Mn] = h[n]

5.7.3 Causal FIR Linear-Phase Systems

satisfies: symmetric or Antisymmetric impulse response

h[Mn] = ±h[n] for n = 0,1,…,M

Type I

Type III

Type II

Type VI

symmetric

Antisymmetric

0 M/2

M=3

0

M=6M/2

0 M/2 M=6 0 M/2 M=5

130

Type I FIR Linear-Phase Systems

Symmetric impulse response

2M k

2

k 1

2 2 2( ) ( )

2 2 2

M M M Mjw k w jkj wh h hM M Mk ke e e

2M k

MnnMhnh 0,

0

Mjw jwn

n

H e h n e

2

k 1

2 2

2 2

M MM jwk jjw wk jwM Mh k he e ee

0 M/2 M=10K=0 1 2 M/212M/2

h[n] 2

Mh k 2Mh k

M: even integer,

M/2 : integer.

131


2

k 1

2 2

2 2

MM Mjwk jjw wk jwM Mh k he eee

0 2

2 1, , , 22 2

where a h M

a k h M k k M

2

2 2

0

cosM

jwM jw jwM

kea k wk e A e e

0

Mjw jwn

n

H e h n e

k 1

22

2 22 cos cos( 0)

M MjwM Mh k hwk w e

, ,real even period

132



MnnMhnh 0,

0

Mjw jwn

n

H e h n e

0 2

2 1, , , 22 2

where a h M

a k h M k k M

2

2 2

0

cosM

jwM jw jwMe

k

wka k e A e e

0 M/2 M=10K=01 2 M/212M/2

, ,real even period

M: even integer,

M/2 : integer.

133

Ex. 5.17 determine H(ejw) of Type I FIR Linear-Phase Systems

2sin

25sin

1

1 254

0 w

we

e

eeeH jw

jw

jw

n

jwnjw

4 2

2 2

0

cos (1 2cos 2cos 2 )jwM j w

k

e a k wk e w w

Solution:

20 2 , 2 1,2, , 2 where a h M a k h M k k M

2

2

0

cosM

jw jwM

k

H e a ewk k

134

22

0

cos jw

k

a ekk w

Frequency response

Magnitude

Phase

Group delay

252sin

sin 2j we

w

w

"wrapped"

2

5

4

5

6

5

8

5

Type I

M: odd integer. integer plus one-half.

Type II FIR Linear-Phase Systems

Symmetric impulse response MnnMhnh 0,

2 :M

0

M/2

M=9K 1 212(M+1)/2 (M+1)/2

12

M k 12

M k 1

2Mh k 1

2Mh k

1

2

k

1 12

12( ) ( )1

21

2jw

M MMjw kjwk MMH e h kk he e

（）

1 2

k 1

21

221

12

M Mjw kj jw k wMh k e e e

（）（（））

1

2M

2M 1

2M


1

21 2 1

cos2

M

k

jwMb k w k e

1 2 1,2, , 22 1where b k h M k k M

2jw jwMeB e e

1 2

k 1

21 1

1 2 22

Mjw

Mjw k jw k jwMH e h k e e e

（）（（））

1 2

k 1

212 co

1s2

2

M MjwM kh k w e

（）

, ,real even period

M: odd integer. integer plus one-half.



MnnMhnh 0,

2

21

1

1s

2co

jw jwM

M

k

H e b k w ek

2 1 2 11 2, , 2,whe M Mre b k h k k

2M

2jw jwMeB e e

, ,real even period

0 M/2 M=5

138

Ex. 5.18 determine H(ejw) of Type II FIR Linear-Phase Systems

2sin

3sin

1

1 2565

0 w

we

e

eeeH jw

jw

jw

n

jwnjw

Solution:

139

Frequency response Type II

Magnitude

Phase

3

5 2

1

cos1

2jw

k

wb k ek

"wrapped"

5 2sin 3

sin 2 jww

ew

Group delay

140

Frequency response

Magnitude

Phase

Group delay

"wrapped"

Type I

22

0

cos jw

k

a ekk w

141

Type III FIR Linear-Phase Systems

Antisymmetric impulse response

, 0h n h M n n M

2

1

2

sinM

jw jwM

k

H e ewkc kj

2,,2,122 MkkMhkcwhere

M: even integer.

integer.2M

2 22jw jwM jw jwMo

joA e e Aj e e

0 M/2

M=4

, ,real odd period

142

Ex. 5.19 determine H(ejw) of Type III FIR Linear-Phase

Systems

jwwjjw ewjeeH sin21 2

Solution:

143

Ex. 5.19 Frequency response Type III

Magnitude

Group delay

Phase"wrapped"

2sinjw jwH e j w e

144

Type IV FIR Linear-Phase Systems


MnnMhnh 0,

2 22jw jwM jw jwMo

joA e e Aj e e

21,,2,1212 MkkMhkdwhere

1 2

2

1

1sin

2jw jwM

k

M

H e d w ekk j

M: odd integer. integer plus one-

half.2M 0 M/2

M=3

, ,real odd period

145

Ex. 5.20 determine H(ejw) of Type IV FIR Linear-Phase Systems

22sin21 wjjwjw ewjeeH

Solution:

146

Ex. 5.20 Frequency response Type IV

Magnitude

Phase"wrapped"

22sin 2 jwj ew

Group delay

M is even M is oddh[Mn] = h[n]

h[Mn] = h[n]


satisfies:

h[Mn] = ±h[n] for n = 0,1,…,M

Type I

Type III

Type II

Type VI

148

Type I ， I I FIR Linear-Phase Systems


MnnMhnh 0,

jwH e

2

2

0

cosM

jwM

k

a k wk e

0 M/2 M=10

1 2

2

1

1cos

2

Mjw jwM

k

H e b k w k e

M: odd integer. : integer plus one-half.

2M

0 M/2 M=7

M: even integer. integer.2M

, ,real even period

149

Type III , IV FIR Linear-Phase Systems


MnnMhnh 0,

2

2

1

sinM

jw jwM

k

H e c k wk ej

M: even integer.

integer.2M

0 M/2

M=4

1 2

2

1

1sin

2

Mjw jwM

k

jH e d k w k e

M: odd integer. :integer plus one-

half.2M 0 M/2

M=3

, ,real odd period

150

Locations of Zeros for FIR Linear-Phase Systems

For Type I and II, nhnMh

0

Mn

n

H z h n z

1 1

0

( )M

M k

k

Mz z zh k Hz

0

Mn

n

H z h M n z

0

( )M k

k M

h k z

M n k n M k

For Type III and IV,

1MH z z H z

151

Type I and II

If is a zero of , 0z zH

This implies that if is a zero of

, then is also a zero of

0jz re zH

1 10

jz r e zH

1 MH z z H z nhnMh

01

0 0MozH H zz then

oz z

The same result for Type III and IV

factor

10z z 1

0 0H z has factor H z

152

Type I and II

When is real and is a zero of

,

will also be a zero of

, so will .

h n 0z zH

0jz re zH

1 10

jz r e

0

Mn

n

H z h n z

1Mz H z

0jz re

0jz re 1 1

0jz r e

1 10

jz r e

real coefficient equation has conjugate complex roots pair:

So there are four possible complex zeros:

same result for Type III and IV

153

Type I and II ,Type III and IV

111111 1111 zerzerzrezre jjjj

When is real, each complex zero not on the unit circle will be part of a set of four conjugate reciprocal zeros of the form

h n

complex zeros on the unit circle

1 11 1j je z e z

10 0z z

1

0 0z z

Type IIType I

154

Type I and II ,Type III and IV

if a zero of is real, and not on the unit circle, the reciprocal is also a zero, and

have the factors of the form

H z

1 1 11 1rz r z

H z

Type IIType I

155

Type I and IIThe case of a zero at is important in

designing filter of some types of frequency responses (such as high-pass,low-pass filter).

1z

1 1 1M

H H

so z=-1 must be zero of Type II generalized linear-phase systems.

1MH z z H z If M is even 1 1H H

If M is odd,

1 1H H 01 HType I can be HP filter

Type II cannot be HP filter

Both can be LP filter

z=1 is not zero for both,

156

Type I and II

Type IIType I

cannot be HP filtercan be HP,LP filtercan be LP filter

157

Type III and IV

nhnMh 1 zHzzH M


Type IVType III

158

Type III and IVThe case of 1z

11 HH 01 HFor both M is even and

odd,1z must be zero of Type III and IV generalized linear-phase systems.

1MH z z H z

Type IVType IIIcannot be LP filter

159

Type III and IVThe case of 1z

111 1 HH M

must be zero of Type III generalized linear-phase systems.

1z

1 zHzzH M

01 HIf M is even, 1 1H H

Type IVType III

If M is odd (Type IV) ,

1 1H H

cannot be HP filter

can be HP filter

160

Fig.5.41Typical plots of zeros for linear-phase systems

Type IV

(a) Type I

Type III

(b) Type II

z0=-1, 不能做 HP filter

z0=±1, 不能做 LP, HP filter

z0=1, 不能做 LP filter

能做 LP, HP filter

161

5.7.4 Relation of FIR Linear-Phase Systems to Minimum-

SystemsAll FIR linear-phase systems have zero of


mi axn mucH z H z HH z z

has all zeros inside the unit circle.

zHmin iM

has all zeros on the unit circle.

zHuc oM

has all zeros outside the unit circle

zHmax iM

iMzzHzH 1minmax

2r 1rSame

magnitude

162

Example 5.21 Decomposition of a Linear-Phase System

18.018.0

16.016.02min

8.018.01

9.019.0125.1

zeze

zezezHjj

jj

For Minimum-Phase System of Page 287, Eq. 5.109

2 0.6 1 0.6 1max

0.8 1 0.8 1

0.9 1 1.1111 1 1.1111

1 1.25 1 1.25

j j

j j

H z e z e z

e z e z

iMzzHzH 1minmax

min maxH z H z H z

Determine the frequecny response of Maximum-Phase System and the system cascaded by two.Solution:

163

Example 5.21 Decomposition of a Linear-

Phase System zHzHzH maxmin

jw

jwjwjw

eH

eHeHeH

min10

max10min1010

log40

log20log20log20

iMzzHzH 1minmax

max min mini ijM w jM wjw jw jwH e H e e H e e

max minjw jwH e H e

164

Example 5.21 Decomposition of a Linear-Phase System

min minjw jw jw

i iH e wM H e H e wM

4jwigrd H e M

min maxjw jw jwH e H e H e

iMzzHzH 1minmax

jwi

jw eHwMeH minmax

zHzHzH maxmin

max min mini ijM w jM wjw jw jwH e H e e H e e

min maxjw jw jwH e H e H e

165


Magnitude

Phase

Group delay

minH z

"wrapped"

2 0.6 1 0.6 1min

0.8 1 0.8 1

1.25 1 0.9 1 0.9

1 0.8 1 0.8

j j

j j

H z e z e z

e z e z

166


Magnitude

maxH z

Phase

Group delay

"wrapped"

1max min

iMH z H z z

167

Frequency response of H z

Magnitude

Phase

Group delay

"wrapped"

min maxH z H z H z

Review

168


a LTI bsystem is said to be minimum-phase if the system and its inverse are causal and stable.

Minimum-Phase System: all the zeros and poles are in the unit circle.

Properties:The Minimum Phase-Lag PropertyThe Minimum Group-Delay PropertyThe Minimum Energy-Delay Property

Review

169


A maximum-phase system is the opposite of a minimum-phase system. A causal and stable LTI system is a maximum-phase system if its inverse is causal and unstable. (From Wikipedia)

Maximum-Phase System: poles are all in the unit circle, zeros are all outside the unit circle. It’s causal and stable.

(noncausal)Maximum-Phase System: anti-causal, stable System whose System function has all its poles and zeros outside the unit circle. (problem5.63).

170

Rational System Function has an equal number of poles and

zeros

zeroczzc kk :01 1

1

0 0 1

10

0 1

1

1

MMk

kkk kN N

kk k

k k

L L

c zb zY z b

H zX z aa z

z zd z

poledzzd kk :01 1

0H z

H z

Review

z=∞, z=0 may be poles or zeros

171

1

0 0 1

10

0 1

1

1

MMk

kkk kN N

kk k

k k

L L

c zb zY z b

H zX z aa z

z zd z

If L=0 and M>N, then M−N extra poles at z = 0 are induced by the numerator.

If L=0 and M<N, then N−M zeros at z = 0 appear from the denominator.

if L<0 then H(z) has L zeros at z=∞, L poles at z=0 [+(M-N) ,↑for M>N, ↓for M<N].

If L>0, then H(z) has L poles at z=∞, L zeros at z = 0 [-(M-N) ,↓for M>N, ↑for M<N] .

Review

Rational System Function has an equal number of poles and zeros

M is even M is odd

h[Mn] = h[n]

h[Mn] = h[n]


satisfies: symmetric or Antisymmetric impulse response

h[Mn] = ±h[n] for n = 0,1,…,M

Type I

Type III

Type II

Type VI

symmetric

Antisymmetric

0 M/2

M=3

0

M=6M/2

0 M/2 M=6 0 M/2 M=5

Review

173



2M k

2

k 1

2 2 2( ) ( )

2 2 2

M M M Mjw k jw k jwM M Mh k h k he e e

2M k

MnnMhnh 0,

0

Mjw jwn

n

H e h n e

2

k 1

2 2

2 2

MM Mjwk jwkjw jwM Mh k he e e e

0 M/2 M=10K=0 1 2 M/212M/2

h[n] 2

Mh k 2Mh k

M: even integer,

M/2 : integer.

Review

174


2

k 1

2 2

2 2

MM Mjw jwjwk jwkM Mh k he e e e

2,,2,122

20

MkkMhka

Mhawhere

2

2 2

0

cosM

jwM jw jwMe

k

a k wk e A e e

0

Mjw jwn

n

H e h n e

2

k 1

22 cos cos( 0)2 2

M MjwM Mh k wk h w e

, ,real even period

Review

23/4/21175Zhongguo Liu_Biomedical Engineering_Shandong U

niv.

Chapter 5 HW5.3, 5.4, 5.32, 5.12, 5.15, 5.19, 5.22, 5.43, 5.65, 5.66,

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2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems