12015-9-171Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 7...

123/4/19 1Zhongguo Liu_Biomedical Engineering_Shandong U

niv.

Biomedical Signal processingChapter 7 Filter Design

Techniques

Zhongguo Liu

Biomedical Engineering

School of Control Science and Engineering, Shandong University

2

Chapter 7 Filter Design Techniques

7.0 Introduction7.1 Design of Discrete-Time IIR Filters

From Continuous-Time Filters7.2 Design of FIR Filters by Windowing7.3 Examples of FIR Filters Design by

the Kaiser Window Method7.4 Optimum Approximations of FIR

Filters7.5 Examples of FIR Equiripple

Approximation7.6 Comments on IIR and FIR Discrete-

Time Filters

3

Filter Design Techniques

7.0 Introduction

4

7.0 Introduction

Frequency-selective filters pass only

certain frequencies

Any discrete-time system that

modifies certain frequencies is called

a filter.

We concetrate on design of causal

Frequency-selective filters

5

Stages of Filter Design

The specification of the desired properties of the system.

The approximation of the specifications using a causal discrete-time system.

The realization of the system.Our focus is on second stepSpecifications are typically given in

the frequency domain.

6

Frequency-Selective Filters

Ideal lowpass filter

ww

wweH

c

cjwlp ,0

,1

0cw

cw 22

jweH

1

nn

nwnh c

lp ,sin

7


Ideal highpass filter

0,

1,cjw

hpc

w wH e

w w

0cw

cw 22

jweH

1

sin,c

hp

w nh n n n

n

8


Ideal bandpass filter

others

wwweH ccjw

bp,0

,121

01c

w1c

w

jweH

1

2cw

2cw

9


Ideal bandstop filter

others

wwweH ccjw

bs,1

,021

01c

w1c

w

jweH

1

2cw

2cw

10

If input is bandlimited and sampling frequency is high enough to avoid aliasing, then overall system behave as a continuous-time system:

T

TeHjH

Tj

eff

,0

,

Linear time-invariant discrete-time system

,effjw w

H H j wT

e

continuous-time specifications are converted to discrete time specifications by:

Tw

11

Example 7.1 Determining Specifications for a Discrete-Time

FilterSpecifications of the continuous-time filter:1. passband2. stopband

20002001.0101.01 forjH eff

30002001.0 forjH eff

sT 410

max

12 2

22 5000

fT T

T

TeHjH

Tj

eff

,0

,

12


FilterSpecifications of the continuous-time filter:1. passband2. stopband

20002001.0101.01 forjH eff

30002001.0 forjH eff

sT 410

max

12 2

22 5000

fT T

1 0.01

2 0.001

2 (2000)p

2 (3000)s

13

sT 410T


FilterSpecifications of the discrete-time filter in

1 0.01 2 0.001

2 (2000)p 2 (3000)s

0.4p 0.6s

14

Filter Design Constraints

Designing IIR filters is to find the approximation by a rational function of z.

The poles of the system function must lie inside the unit circle(stability, causality).

Designing FIR filters is to find the polynomial approximation.

FIR filters are often required to be linear-phase.

15

Filter Design Techniques

7.1 Design of Discrete-Time IIR

Filters From Continuous-Time

Filters

16

7.1 Design of Discrete-Time IIR

Filters From Continuous-Time Filters

The traditional approach to the

design of discrete-time IIR filters

involves the transformation of a

continuous-time filter into a

discrete filter meeting prescribed

specification.

17

Three Reasons

1. The art of continuous-time IIR filter design is highly advanced, and since useful results can be achieved, it is advantageous to use the design procedures already developed for continuous-time filters.

18

Three Reasons

2. Many useful continuous-time IIR design method have relatively simple closed form design formulas. Therefore, discrete-time IIR filter design methods based on such standard continuous-time design formulas are rather simple to carry out.

19

Three Reasons

3. The standard approximation

methods that work well for

continuous-time IIR filters do not

lead to simple closed-form

design formulas when these

methods are applied directly to

the discrete-time IIR case.

20

Steps of DT filter design by transforming a prototype

continuous-time filter

The specifications for the continuous-time filter are obtained by a transformation of the specifications for the desired discrete-time filter.

Find the system function of the continuous-time filter.

Transform the continuous-time filter to derive the system function of the discrete-time filter.

21

Constraints of Transformation

to preserve the essential properties of the frequency response, the imaginary axis of the s-plane is mapped onto the unit circle of the z-plane. jwezjs

planes planez Im Im

Re Re

22

Constraints of Transformation

In order to preserve the property of stability, If the continuous system has poles only in the let half of the s-plane, then the discrete-time filter must have poles only inside the unit circle.

planes Im

Re

planez Im

Re

23

7.1.1 Filter Design by Impulse Invariance

The impulse response of discrete-time system is defined by sampling the impulse response of a continuous-time system.

dcd nThTnh

dc TjHif ,0

w

T

wjHeHthen

dc

jw ,

wforTw d

k ddc

jw kT

jT

wjHeH

2Relationship of frequencies

24

,, dT

relation between frequencies

S plane Z plane

-

3 / dT

j

/ dT

/ dT

k ddc

jw kT

jT

wjHeH


dc TjHif ,0

w

T

wjHeHthen

dc

jw ,

25

Aliasing in the Impulse Invariance

k ddc

jw kT

jT

wjHeH

2

dc TjHif ,0

,jwc

d

wthen H e H j

T

w

26

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

27

Time 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

28

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)()(

29

( ) ( ) ( ) ( ) ( )n n

x n x t t nT x nT t nT

For discrete-time signal ，

( ) ( ) st

n

x nT t nT e dt

( ) ( )snT sT

n

x nT e X e

[ ( )] ( ) stx n x n e dt

L

sTz e令：( ) ( )n

n

x n z X z

z-transform

of discrete-

time signal

the Laplace transformLaplace transform

30

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

n

nznxzX )()(

so ：Tr e

T

relation betweens zand

Laplace transformLaplace transform continuous time signal

z-transformz-transform discrete-time signalrelation

[ ( )] ( ) ( )snT sT

n

x n x nT e X e

LsTz elet ：

31

2 sT f f

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 / sT

j

/ sT

/ sT

1|j jrz re e

32

2 2s sT f f f

z plane

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

22s

s

f

T

sT

3

sT

3

sT

33

If input is bandlimited and fs>2fmax , :

T

TeHjH

Tj

eff

,0

,

discrete-time filter design by impulse invariance

wforTw d

k ddc

jw kT

jT

wjHeH

2

dc TjHif ,0

,jwc

d

wthen H e H j w

T

dcd nThTnh

34

,, dT

relation between frequencies

S plane Z plane

-

3 / dT

j

/ dT

/ dT

k ddc

jw kT

jT

wjHeH


dc TjHif ,0

w

T

wjHeHthen

dc

jw ,

35

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


2s

k

S j kT

1 1 2*

2 2s c c sk

X j X j S j X j k dT

1c s

k

X j kT

Review

36

proof of

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

n

nTtts

2s

k

S j kT

Review

2s

k

S j kT

sjk tk

n

a e

s(t) 为冲击串序列，周期为 T ，可展开傅立叶级数

1sjk t

n

eT

2 ( )sjk t Fse k

37

periodic sampling

s c c cn n


j Tnc

k

x nT e

( ) j ts c

n

X j x nT t nT e dt

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

cn

X e x nT e

( ) ( ) ( )j j Ts TX j X e X e

1( )j T

c sk

X e X j kT

1 2( ) c

k

j kX X j

T T Te

2s T

( ) 0,j Tif X eT

1( ) c

jthen X X jT T

e

1s c s

k

X j X j kT

38

discrete-time filter design by impulse invariance

k ddc

jw kT

jT

wjHeH

2

dc TjHif ,0

,jwc

d

wthen H e H j w

T

dcd nThTnh

1 2( ) c

k

j kX X j

T T Te

1

( ) cjX X j

T Te

1 2( ) c

k

j kH H j

T T Te

1( ) c

jH H jT T

e

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

39



Obtain the specifications for continuous-time filter by transforming the specifications for the desired discrete-time filter.



40

Transformation from discrete to continuous

In the impulse invariance design procedure, the transformation is

Assuming the aliasing involved in the transformation is neglected, the relationship of transformation is

w

T

wjHeH

dc

jw ,

dTw

41






42

Continuous-time IIR filters

Butterworth filtersChebyshev Type I filtersChebyshev Type II filtersElliptic filters

43






44

Transformation from continuous to discrete

N

k k

k

ss

AsH

1

0,0

0,1

t

teAth

N

k

tsk

c

k

: k ds Tkpole s s z e

two requirements for transformation

N

k

nTskd

N

k

nTskdcd nueATnueATthTnh dkdk

11

N

kTskd

ze

ATzH

dk1

11

45

Example 7.2 Impulse Invariance with a Butterworth Filter

Specifications for the discrete-time filter:

weH

weH

jw

jw

3.0,17783.0

2.00,189125.0

dd TwTlet 1

Assume the effect of aliasing is negligible

3.0,17783.0

2.00,189125.0

jH

jH

c

c

46


0.89125 1, 0 0.2

0.17783, 0.3

c

c

H j

H j

0.2 0.89125

0.3 0.17783

c

c

H j

H j

0.30.2

2 20.2 1

10.89125

N

c

2 2

0.3 11

0.17783

N

c

2

2

1

1Nc

cH j

j j

2

2

11

N

cc

j jH j

47


2

2

1

1Nc

cH j

j j

2 20.2 1

1 1.258930.89125

N

c

7032.0,6 cN

2 20.3 1

1 31.622040.17783

N

c

20.2

0.25893

N

c

20.3

30.62204

N

c

23

118.263782

N

70470.0,8858.5 cN

0.2 0.89125

0.3 0.17783

c

c

H j

H j

48

Example 7.2 Impulse Invariance with a Butterworth

Filter

2

2

1

1Nc

cH j

j j

2

2

1

1Nc c c

cH s H s H s

s j

0,1, , 2 1k N 2 2 11 21 ,k

j N k NNc cs j e

6, 0.7032cN

0.182 0.679,j

0.497 0.497,j

0.679 0.182j

Plole pairs: cH s

49


4945.03585.14945.09945.04945.03640.0

12093.0222

ssssss

sH c

22

2 2 2

1

1

N

N N N

c

cc c c

cH s H s H s

ss j

0.182 0.679,j 0.497 0.497,j 0.679 0.182j Plole pairs: cH s

0.12093

0.182 0.679 0.182 0.679 0.497 0.497cH ss j s j s j

1

0.497 0.497 0.679 0.182 0.679 0.182s j s j s j

60.7032Nc

50


1 1

1 2 1 2

1

1 2

0.2871 0.4466 2.1428 1.1455

1 1.2971 0.6949 1 1.0691 0.3699

1.8557 0.6303

1 0.9972 0.2570

z z

z z z z

z

z z

0.12093

0.182 0.679 0.182 0.679 0.497 0.497cH ss j s j s j

1

0.497 0.497 0.679 0.182 0.679 0.182s j s j s j

1

Nk

k k

A

s s

1 1

1 11 1

N Nd k k

k kk d ks T s

T A AH z

e z e z

1dT

51

Basic for Impulse Invariance

To chose an impulse response for the discrete-time filter that is similar in some sense to the impulse response of the continuous-time filter.

If the continuous-time filter is bandlimited, then the discrete-time filter frequency response will closely approximate the continuous-time frequency response.

The relationship between continuous-time and discrete-time frequency is linear; consequently, except for aliasing, the shape of the frequency response is preserved.

52

7.1.2 Bilinear Transformation

Bilinear transformation can avoid the problem of aliasing.

Bilinear transformation maps onto

w

1

1

2 1

1dc

zH z H

T z

1

1

1

12

z

z

Ts

d

Bilinear transformation:

cH s

1

1

1

12

z

z

Ts

d

53


1

1

1

12

z

z

Ts

d

sT

sTz

d

d

21

21

js 221

221

dd

dd

TjT

TjTz

anyforz 10

anyforz 10

1 12 (1 ) 1dT s z z

11 2 ] 1 2d dT s z T s

54


jsaxisj

21

21

d

d

Tj

Tjz

1z 1 2

1 2d

d

jw j T

j Te

planes Im

Re

planez Im

Re

221

221

dd

dd

TjT

TjTz

anyforz 10

anyforz 10

js

55


1

1

1

12

z

z

Ts

d

2 1

1d

jw

jwjT

ee

2tan 2

d

j wT

2 2 sin 2

2cos 2d

j w

T w

2tan2

wTd

2tan2 1dTw

/2 /2 /2

/2 /2 /22 ( )

( )d

jw jw jw

jw jw jwT

e e ee e e

56

2tan

2

2tan

2

:

s

ds

p

dp

T

T

prewarp

2

tan2)()(

dT

cj jHeH

relation between frequency response of Hc(s), H(z)

57

Comments on the Bilinear Transformation

It avoids the problem of aliasing encountered with the use of impulse invariance.

It is nonlinear compression of frequency axis.

S plane Z plane

-

3 / dT

j

/ dT

/ dT

2tan2

wTd

2tan2 1dTw

58

Comments on the Bilinear Transformation

The design of discrete-time filters using bilinear transformation is useful only when this compression can be tolerated or compensated for, as the case of filters that approximate ideal piecewise-constant magnitude-response characteristics.

0cw

cw 22

jweH

1

59

Bilinear Transformation of

1

1

1

12

z

z

Ts

d

2tan2

wTd

se

je

2tan 2

d

wT

dT

60

Comparisons of Impulse Invariance and Bilinear

Transformation

The use of bilinear transformation is restricted to the design of approximations to filters with piecewise-constant frequency magnitude characteristics, such as highpass, lowpass and bandpass filters.

Impulse invariance can also design lowpass filters. However, it cannot be used to design highpass filters because they are not bandlimited.

61

Comparisons of Impulse Invariance and Bilinear

Transformation

Bilinear transformation cannot design filter whose magnitude response isn’t piecewise constant, such as differentiator. However, Impulse invariance can design an bandlimited differentiator.

62

Butterworth Filter, Chebyshev Approximation, Elliptic Approximation

7.1.3 Example of Bilinear Transformation

weH

weH

jw

jw

6.0,001.0

4.0,01.199.0

63

0.0160.01

Example 7.3 Bilinear Transformation of a Butterworth Filter

0.89125 1, 0 0.2

0.17783, 0.3

jw

jw

H e w

H e w

2 0.20.89125 1, 0 tan

2

2 0.30.7783, tan

2

cd

cd

H jT

H jT

, 1dFor convenience we choose T

,17783.015.0tan2

,89125.01.0tan2

jH

jH

c

c

2tan2

wTd

64


,17783.015.0tan2

,89125.01.0tan2

jH

jH

c

c

N

c

cjj

jH 2

2

1

1

2 2

2 2

2 tan 0.1 11

0.89125

3tan 0.15 11

0.17783

N

c

N

c

305.5N

766.0

,6

c

N

0.0160.01

65

Locations of Poles

0,1, , 2 1k N 2 2 11 21 ,k

j N k NNc cs j e

0.1998 0.7401,j

0.5418 0.5418,j

0.7401 0.1998j

Plole pairs: cH s

2

2

1

1Nc

cH j

j j

2

2

1

1Nc c c

cH s H s H s

s j

6, 0.766cN

66


5871.04802.15871.00836.15871.03996.0

20238.0222

ssssss

sH c

21

2121

61

2155.09904.01

1

3583.00106.117051.02686.11

10007378.0

zz

zzzz

zzH

22

2 2 2

1

1

N

N N N

c

cc c c

cH s H s H s

ss j

0.1998 0.7401,j 0.5418 0.5418,j 0.7401 0.1998j Plole pairs: cH s

1

1

1

12

z

z

Ts

d

67

Ex. 7.3 frequency response of discrete-time filter

68

Example 7.4 Butterworth Approximation (Hw)

weH

weH

jw

jw

6.0,001.0

4.0,01.199.014Norder

69

Example 7.4 frequency response

70

Chebyshev filters

C Chebyshev filter (type I)

)/(1

1|)(| 22

2

cN

cV

jH

)coscos()( 1 xNxVN

c

1

1

Chebyshev polynomial

Chebyshev filter (type II)

122

2

)]/([1

1|)(|

cN

cV

jH

1

c

71

Example 7.5 Chebyshev Type I , II Approximation

weH

weH

jw

jw

6.0,001.0

4.0,01.199.08Norder

Type I Type II

72

Example 7.5 frequency response of Chebyshev

Type I Type II

73

E

elliptic filters

Elliptic filter

)(1

1|)(|

222

Nc U

jH

sp

1

11

2Jacobian elliptic function

74

Example 7.6 Elliptic Approximation

weH

weH

jw

jw

6.0,001.0

4.0,01.199.06Norder

75

Example 7.6 frequency response of Elliptic

76

*Comparison of Butterworth, Chebyshev, elliptic filters: Example

-Given specification

0.4| | .011 |)(| 99.0 jeH ||0.6 0.001 |)(| jeH

6.0 ,4.0 001.0 ,01.0 s21 p

)( s

-Order

Butterworth Filter : N=14. ( max flat)

Chebyshev Filter : N=8. ( Cheby 1, Cheby 2)

Elliptic Filter : N=6 ( equiripple)

B

C

E

77

-Pole-zero plot (analog)

-Pole-zero plot (digital)

B C1 C2 E

B C1 C2 E

(14) (8)

78

-Magnitude -Group delay

B

C1

C2

E

B

C1

C2

E

4.0 6.0 4.0 6.0

5

20

79

7.2 Design of FIR Filters by Windowing

FIR filters are designed based on directly approximating the desired frequency response of the discrete-time system.

Most techniques for approximating the magnitude response of an FIR system assume a linear phase constraint.

80

Window MethodAn ideal desired frequency response

n

jwnd

jwd enheH

dweeHnh jwnjw

dd 2

1

Many idealized systems are defined by piecewise-constant frequency response with discontinuities at the boundaries. As a result, these systems have impulse responses that are noncausal and infinitely long.

ww

wweH

c

cjwlp ,0

,1

sin clp

w nh n

n

0cw

cw

jweH1

81

Window Method

otherwise

Mnnhnh d

,0

0,

nwnhnh d

otherwise

Mnnw

,0

0,1

deWeHeH wjjwd

jw

2

1

The most straightforward approach to obtaining a causal FIR approximation is to truncate the ideal impulse response.

82

Windowing in Frequency Domain

Windowed frequency response

deWeH21

eH jjd

j

The windowed version is smeared version of desired response

83

Window Method

1

2j wjw j jw

d dH e H e W e d H e

If nnw 1

kn

jwnjw kwenweW 22

0cw

cw

jweH1

0 5 10510 1515

2

2

24 4 6

jwW e

84

Choice of Window is as short as possible in duration.

This minimizes computation in the implementation of the filter.

nw

approximates an impulse. jweW

0

Mjw jwn jwn

n n

W e w n e e

1

2sin 1 21

1 sin 2

jw MjwM

jw

w Mee

e w

otherwise

Mnnw

,0

0,1

1M

2

1M

2

1M

jwW e

85

Window Method

jwd

wjjwd

jw eHdeWeHeH

21

then would look like , except where changes very abruptly.

jweH jwd eH

jwd eH

nw jweW0w

If is chosen so that is concentrated in a narrow band of frequencies around

0cwcw

jwdH e

11M

2

1M

2

1M

jwW e

86

Rectangular Window for the rectangular window has a

generalized linear phase. jweW

As M increases, the width of the “main lobe” decreases. 4 1

mw M

While the width of each lobe decreases with M, the peak amplitudes of the main lobe and the side lobes grow such that the area under each lobe is a constant.

2sin 1 2

sin 2jw jwM

w MW e e

w

1M

2

1M

2

1M

1

M

M

1

M

M

87

Rectangular Window will oscillate at the

discontinuity.

deWeH wjjwd

The oscillations occur more rapidly, but do not decrease in magnitude as M increases.

The Gibbs phenomenon can be moderated through the use of a less abrupt truncation of the Fourier series.

88

Rectangular WindowBy tapering the window smoothly to zero

at each end, the height of the side lobes can be diminished.

The expense is a wider main lobe and thus a wider transition at the discontinuity.

89

7.2 Design of FIR Filters by Windowing Method

To design an ilowpass FIR Filters

ww

wweH

c

cjwlp ,0

,1

sin clp

w nh n

n

0cw

cw

jweH1

Review

dh n h n w n 1, 0

0,

n Mw n

otherwise

deWeHeH wjjwd

jw

2

1

1M

2

1M

2

1M

jwW e

sin 2

2cw n M

n M

02M0

2M0

M

M

2M0 M

90

7.2.1 Properties of Commonly Used Windows

Rectangular

otherwise

Mnnw

,0

0,1

otherwise

MnMMn

MnMn

nw

,0

2,22

20,2

Bartlett (triangular)

91


Hanning

otherwise

MnMnnw

,0

0,2cos5.05.0

otherwise

MnMnnw

,0

0,2cos46.054.0

Hamming

92


Blackman

otherwise

MnMn

Mnnw

,00,4cos08.0

2cos5.042.0

93


94

Frequency Spectrum of Windows

(a) Rectangular, (b) Bartlett, (c) Hanning, (d) Hamming, (e) Blackman , (M=50)

(a)-(e) attenuation of sidelobe increases, width of mainlobe increases.

95


biggest ， high oscillations at discontinuity

smallest ， the sharpest transition

Table 7.1

96

7.2.2 Incorporation of Generalized Linear Phase

In designing FIR filters, it is desirable to obtain causal systems with a generalized linear phase response.

otherwise

MnnMwnw

,0

0,

The above five windows are all symmetric about the point ,i.e.,2M

97


Their Fourier transforms are of the form

2Mjwjwe

jw eeWeW woffunctionevenandrealaiseW jw

e

causalnwnhnh d :

d d dif h M n h n h n h n w n

2Mjwjwe

jw eeAeH

:h M n h n generalized linear phase

2MM

98


phaselineardgeneralizenhnMh

nwnhnhnhnMhif ddd

:

2Mjwjwo

jw eejAeH

2MM

99

Frequency Domain Representation

nhnMhif dd 2Mjwjwe

jwd eeHeH

1

2jw

d

j wjH e H W de e

1

2 e e

j wjw jwewhere A H W de e e

221

2 e e

j w j w Mj j MH W de e e e

2Mjwjwe

jw eeWeW w n w M n

2jw jwMeA e e

dh n h n w n

100

Example 7.7 Linear-Phase Lowpass Filter

The desired frequency response is

ww

wweeH

c

cjwM

jwlp ,0

,2

lph M n

nw

Mn

Mnwnh c

2

2sin

21

2

sin 2

2

c

clp

c

w

w

jwM jwnh n dw

w n Mfor n

n M

e e

2M

wweH

wweH

sjw

pjw 01

101

magnitude frequency response

pw

sw

1 0jwp p

jws s

H e w w

H e w w

s pw w w

1020logp p

0.1 0.15jwH e w

1 0.05 0 0.25jwH e w

0.1s pw w w 1020log 0.05 26p dB

1020logs s

20s dB

102


smallest ， the sharpest transition

biggest ， high oscillations at discontinuity

103

7.2.3 The Kaiser Window Filter Design Method

1 22

0

0

1, 0

0,

I nn Mw n

I

otherwise

2,where M

0 :I u zero order modified Bessel function of the first kind

2

01

21

!r

ru

I ur

: 1,length M :parametershape:two parameters

Trade side-lobe amplitude for main-lobe width

104

Figure 7.24

As increases, attenuation of sidelobe increases, width of mainlobe increases.

As M increases, attenuation of sidelobe is preserved, width of mainlobe decreases.

M=20

(a) Window shape, M=20, (b) Frequency spectrum, M=20, (c) beta=6

=6

105

Table 7.1

Transition width is a little less than mainlobe width

106

Increasing M wile holding constant causes the main lobe to decrease in width, but does not affect the amplitude of the side lobe.

ComparisonIf the window is tapered more, the side lobe

of the Fourier transform become smaller, but the main lobe become wider.

M=20

=6

M=20

107

pw

sw

Filter Design by Kaiser Window

wweH

wweH

sjw

pjw 01

ps www 10log20A

108

Filter Design by Kaiser Window

ps www 10log20A

21,0.0

5021,2107886.0215842.0

50,7.81102.04.0

A

AAA

AA

2285.2

8

w

AM

1 22

0

0

1, 0

0,

I nn Mw n

I

otherwise

M=20

109

Example 7.8 Kaiser Window Design of a Lowpass Filter

weH

weH

jw

jw

6.0,001.0

4.0,01.199.0

ps www 10log20A

1 22

0

0

1sin

, 0c

I nw n

n Mn I

0,h n

otherwise

5.182 Mwhere

21,0.0

5021,2107886.0215842.0

50,7.81102.04.0

A

AAA

AA

8

2.285

AM

w

110


weH

weH

jw

jw

6.0,001.0

4.0,01.199.0

001.0,min,001.0,01.0

,6.0,4.0

:1

2121

sp ww

step

0.52

s pc

w wcutoff frequency w

2 :step

3:step100.2 20log 60

0.5653 37

s pw w w A

M

111


0.5653

5.182 Mwhere 2

01

21

!r

ru

I ur

1 22

0

0

1sin

, 0c

I nw n

n Mn I

0,h n

otherwise

21,0.0

5021,2107886.0215842.0

50,7.81102.04.0

A

AAA

AA

837

2.285

AM

w

10

3:

0.2 20log 60s p

step

w w w A

112

1 22

0

0

1sin

, 0c

I nw n

h n n Mn I

Ex. 7.8 Kaiser Window Design of a Lowpass Filter

113

7.3 Examples of FIR Filters Design by the Kaiser Window

MethodThe ideal highpass filter with

generalized linear phase

wwe

wweH

cMjw

cjwhp ,

,02

jwlp

Mjwjwhp eHeeH 2

nMn

Mnw

Mn

Mnnh c

hp ,2

2sin

2

2sin

hph n h n w n

114

Example 7.9 Kaiser Window Design of a Highpass Filter

Specifications:

wweH

wweH

pjw

sjw

,11

,

11

2

021.0,5.0,35.0 211 ps wwwhere

24,6.2 M

By Kaiser window method

115

Example 7.9 Kaiser Window Design of a Highpass Filter

Specifications:

wweH

wweH

pjw

sjw

,11

,

11

2

021.0,5.0,35.0 211 ps wwwhere

24,6.2 M

By Kaiser window method

116

7.3.2 Discrete-Time Differentiator

wejweH Mjwjwdiff ,2

nMn

Mn

Mn

Mnnhdiff ,

2

2sin

2

2cos2

nwnhnh diff

phaselineardgeneralizeIVtypeorIIItypenMhnh :

117

Example 7.10 Kaiser Window Design of a Differentiator

Since kaiser’s formulas were developed for frequency responses with simple magnitude discontinuities, it is not straightforward to apply them to differentiators.

Suppose 4.210 M

118

Group Delay

Phase:

Group Delay:

25

22

ww

M

samplesM

52

119

Group Delay

Phase:

Group Delay:

Noninteger delay

22

5

22

ww

M

samplesM

2

5

2

120

7.4 Optimum Approximations of FIR Filters

Goal: Design a ‘best’ filter for a given MIn designing a causal type I linear phase

FIR filter, it is convenient first to consider the design of a zero phase filter.

Then insert a delay sufficient to make it causal.

nhnh ee

121


nhnh ee

2, MLenheAL

Ln

jwne

jwe

functionperiodicevenrealwnnhheAL

nee

jwe ,,:cos20

1

.2 samplesMLbyitdelaying

bynhfromobtainedbecansystemcausalA e

nMhMnhnh e 2

2Mjwjwe

jw eeAeH

122


Designing a filter to meet these specifications is to find the (L+1) impulse response values

Lnnhe 0,

In Packs-McClellan algorithm,

is fixed, and is variable. 21,,, andwwL sp

21 or

Packs-McClellan algorithm is the dominant method for optimum design of FIR filters.

123


wnwTwn n coscoscoscoscos 1 1coscos0coscos0cos 1

0 wwTw

wwwTw coscoscos1coscos1cos 11

1cos2cos2cos 22 wwTw

wTwTw

wTwn

nn

n

coscoscos2

coscos

21

wwwww

wwww

cos3cos4cos1cos2cos2

cos2coscos23cos32

124


L

k

kk

wx

L

k

kk

L

nee

jwe

xaxPwhere

xPwawnnhheA

0

cos01

coscos20

functionweightingtheiswWwhere

eAeHwWwE

functionerrorionapproximatanDefinejw

ejw

d

125


ww

wweH

s

pjwd ,0

0,1

ww

wwKwW

s

p

,1

0,1

1

2

126

Minimax criterion

Within the frequency interval of the passband and stopband, we seek a frequency response that minimizes the maximum weighted approximation error of

jwe eA

jwe

jwd eAeHwWwE

wE

FwLnnhe maxmin

0:

127

Other criterions

00:1 min dwwEH

Lnnhe

0

2

0:2 min dwwEH

Lnnhe

wEH

FwLnnhe maxmin0:

128

Let denote the closet subset consisting of the disjoint union of closed subsets of the real axis x.

Alternation Theorem

pF

is an r th-order polynomial.

r

k

kk xaxP

0

denotes a given desired function of x that is continuous on

xDP

pF is a positive function, continuous on pF xWP

The weighted error is xPxDxWxE PPP

xEE PFx P

max

The maximum error is defined as

129

Alternation Theorem

A necessary and sufficient condition that be the unique rth-order polynomial that minimizes is that exhibit at least (r+2) alternations; i.e., there must exist at least (r+2) values in such that

xPE xEP

ix PF

221 rxxx

ExExE iPiP 1

1,,2,1 ri

and such that

for

130

Example 7.11 Alternation Theorem and Polynomials

Each of these polynomials is of fifth order.

The closed subsets of the real axis x referred to in the theorem are the regions

11.01.01 xandx

1xWP

131

7.4.1 Optimal Type I Lowpass Filters

For Type I lowpass filter

The desired lowpass frequency response

Weighting function

L

k

kk wawP

0

coscos

wwww

wwwwwD

ss

ppp coscos1,0

01coscos,1cos

wwww

wwwwKwW

ss

ppp

coscos1,1

01coscos,1

cos

132


The weighted approximation error is

The closed subset is

or

wPwDwWwE PPP coscoscoscos

wwandww sp0

sp wwandww cos11coscos

xEP

133


The alternation theorem states that a set of coefficients will correspond to the filter representing the unique best approximation to the ideal lowpass filter with the ratio fixed at K and with passband and stopband edge and if and only if exhibits at least (L+2) alternations on , i.e., if and only if alternately equals plus and minus its maximum value at least (L+2) times.

Such approximations are called equiripple approximations.

ka

21

pwsw )(coswEP

PF

)(coswEP

134


The alternation theorem states that the optimum filter must have a minimum of (L+2) alternations, but does not exclude the possibility of more than (L+2) alternations.

In fact, for a lowpass filter, the maximum possible number of alternations is (L+3).

135


Because all of the filters satisfy the alternation theorem for L=7 and for the same value of , it follows that and/or must be different for each ,since the alternation theorem states that the optimum filter under the conditions of the theorem is unique.

21 K

pwsw

136

Property for type I lowpass filters from the alternation

theorem

The maximum possible number of alternations of the error is (L+3)

Alternations will always occur at andAll points with zero slop inside the

passband and all points with zero slop inside stopband will correspond to alternations; i.e., the filter will be equiripple, except possibly at and

pw sw

0w

w

137

7.4.2 Optimal Type II Lowpass Filters

For Type II causal FIR filter:The filter length (M+1) is even, ie, M is oddImpulse response is symmetricThe frequency response is

Mnnh 0

nhnMh

21,,2,1,212

2

1cos

2cos2

21

1

2

21

0

2

MnnMhnbwhere

nwnbe

nM

wnheeH

M

n

Mjw

M

n

Mjwjw

138


21

0

21

1

cos~

2cos2

1cos

M

n

M

n

wnnbwnwnb

wPweeH Mjwjw cos2cos2

21cos0

MLandwawPwhereL

k

kk

nMhnbnbnbafind k 212~

139


For Type II lowpass filter,

ww

wwwwDeH

s

pP

jwd

,0

0,2cos

1cos

ww

wwK

wwWwW

s

pP

,2cos

0,2cos

cos

140

7.4.3 The Park-McClellan Algorithm

From the alternation theorem, the optimum filter will satisfy the set of equation

2,,2,11 1 LieAeHwW ijwe

jwd

jwe eA

ii

jwd

jwd

jwd

L

LLLLL

L

L

wxwhere

eH

eH

eH

a

a

wWxxx

wWxxx

wWxxx

L

cos

11

11

11

2

2

1

1

0

2

2

22

22

22

222

11

211

141


Guessing a set of alternation frequencies

2

12

1

1

2

1 cos,1

1

L

kii

iiik

kL

k k

kk

L

k

jwdk

wxxx

bwhere

wW

b

eHb k

2,,2,1 Liforwi slpl wwww 1,and

142


2

1

1

1

1

1

1

1

cos,cos

Lkk

L

kii ik

k

kkL

kkk

k

L

kkk

jwe

xxbxx

d

wxxxxd

CxxdwPeA

143


For equiripple lowpass approximation

Filter length: (M+1)

ps wwwwherew

M

324.2

13log10 2110

144

7.5 Examples of FIR Equiripple Approximation

7.5.1 Lowpass Filter

weH

weH

jw

jw

6.0,001.0

4.0,01.199.0

26M

wweA

wweA

wW

wEwE

errorionapproximatunweighted

sjw

e

pjw

eA

,0

0,1

145

Comments

M=26, Type I filterThe minimum number of

alternations is (L+2)=(M/2+2)=157 alternations in passband and 8

alternations in stopbandThe maximum error in passband and

stopband are 0.0116 and 0.0016, which exceed the specifications.

146

7.5.1 Lowpass Filter

M=27, , Type II filter, zero at z=-1The maximum error in passband and

stopband are 0.0092 and 0.00092, which exceed the specifications.

The minimum number of alternations is (L+2)=(M-1)/2+2=15

7 alternations in passband and 8 alternations in stopband

w

147

Comparison

Kaiser window method require M=38 to meet or exceed the specifications.

Park-McClellan method require M=27Window method produce

approximately equal maximum error in passband and stopband.

Park-McClellan method can weight the error differently.

148

7.6 Comments on IIR and FIR Discrete-Time Filters

What type of system is best, IIR or FIR?

Why give so many different design methods?

Which method yields the best result?

149

7.6 Comments on IIR and FIR Discrete-Time Filters

Closed-Form

Formulas

Generalized Linear Phase

Order

IIR Yes No Low

FIR No Yes High

150


Their Fourier transforms are concentrated around

They have a simple functional form that allows them to be computed easily.

0w

The Fourier transform of the Bartlett window can be expressed as a product of Fourier transforms of rectangular windows.

The Fourier transforms of the other windows can be expressed as sums of frequency-shifted Fourier transforms of rectangular windows.(Problem7.34)

151

Homework

Simulate the frequency response (magnitude and phase) for Rectangular, Bartlett, Hanning, Hamming, and Blackman window with M=21 and M=51

152

23/4/19152Zhongguo Liu_Biomedical Engineering_Shandong U

niv.

Chapter 5 HW7.2, 7.4, 7.15,

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