Chapter 4Filter Design Techniques

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Chapter 4 Filter Design Techniques

• 7.0 Introduction

• 7.1 Design of Discrete-Time IIR Filters FromContinuous-Time Filters

• 7.2 Design of FIR Filters by Windowing

• 7.3 Examples of FIR Filters Design by the KaiserWindow Method

• 7.4 Optimum Approximations of FIR Filters

• 7.5 Examples of FIR Equiripple Approximation

• 7.6 Comments on IIR and FIR Discrete-TimeFilters

1



Filter Design Techniques

7.0 Introduction

2



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

3



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 step

• Specifications are typically given in the frequency

domain.

4



Frequency-Selective Filters

• Ideal lowpass filter

ww

wwe H

c

c jw

lp,0

,1

5

0cw

cw 2 2

jwe H

1

nn

nwnh clp ,sin




• Ideal highpass filter

0,

1,

c jw

hp

c

w w H e

w w

0

6

cwcw 2 2

jwe H

1

sin

,c

hp

w n

h n n nn




•

Ideal bandpass filter

others

wwwe H cc jw

bp,0

,121

0

7

1cw1cw

jwe H

1

2cw2cw




•

Ideal bandstop filter

others

wwwe H cc jw

bs,1

,021

0

8

1cw1cw

jwe H

1

2cw2cw



• If input is bandlimited and sampling frequency is

high enough to avoid aliasing, then overall systembehave as a continuous-time system:

T

T e H j H

T j

eff

,0

,

9

Linear time-invariant discrete-time system

,eff

jw w

H H j wT e

continuous-time specifications are converted to discretetime specifications by:

T w



Example 7.1 Determining Specifications for a

Discrete-Time Filter

•

Specifications of the continuous-time filter:• 1. passband

• 2. stopband

20002001.0101.01 for j H eff

30002001.0 for j H eff

10

sT 410

max12 22

2 5000

f T T

T

T e H j H

T j

eff

,0

,



Example 7.1 Determining Specifications for a

Discrete-Time Filter

•

Specifications of the continuous-time filter:• 1. passband

• 2. stopband

20002001.0101.01 for j H eff

30002001.0 for j H eff

11

sT 410

max12 22

2 5000

f T T

1 0.01

2 0.001

2 (2000) p

2 (3000)s



Example 7.1 Determining Specifications for a Discrete-

Time FiltersT 410

T

12

Specifications of the

discrete-time filter in

1 0.01

20.001

2 (2000) p 2 (3000)s

0.4 p 0.6s



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.

13



Filter Design Techniques

7.1 Design of Discrete-Time IIR Filters

From Continuous-Time Filters

14



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.

15



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.

16



Three Reasons

2. Many useful continuous-time IIR designmethod have relatively simple closed form

design formulas. Therefore, discrete-time IIR

filter design methods based on such standardcontinuous-time design formulas are rather

simple to carry out.

17



Three Reasons

3. The standard approximation methods that

work well for continuous-time IIR filters do

not lead to simple closed-form designformulas when these methods are applied

directly to the discrete-time IIR case.

18



Steps of DT filter design by transforming a prototype

continuous-time filter

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

•

Find the system function of the continuous-timefilter.

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

19



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. jwe z js

20

planes plane z Im Im

Re Re



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 unitcircle.

21

planes Im

Re

plane z Im

Re



7.1.1 Filter Design by Impulse

Invariance•

The impulse response of discrete-time system isdefined by sampling the impulse response of a

continuous-time system.

d cd nT hT nh

22

d c T j H if ,0

w

T w j H e H then

d

c

jw ,

w for T w d

k d d

c

jwk

T j

T

w j H e H

2Relationship of frequencies



23

,, d T

relation between frequencies

S plane Z plane

-

3 / d

T

j

/ d T

/ d

T

k d d

c jw k

T j

T w j H e H 2Relationship of

frequencies

d c T j H if ,0

w

T

w j H e H then

d

c

jw ,



Aliasing in the Impulse Invariance

24

k d d

c

jwk

T

j

T

w j H e H

2

d c T j H if ,0

, jw

c

d

wthen H e H j

T

w



periodic sampling

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

25

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

n

nT t t s

s c c c

n n

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

Review



26

Time domain： )(t x

Complex frequency

domain：

dt et xs X st )()(

Laplace transform

js

f 2

s pl ane

j

0

Relation between Laplace Transformand Z-transform

Review



27

dt et x j X t j)()(

Fourier Transform

frequency domain ：

s- pl ane

j

0

Fourier Transform is the Laplace transform when s

have the value only in imaginary axis, s=jΩ

js Since

So 0 s j

dt et xs X st )()(



28

( ) ( ) ( ) ( ) ( )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

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

L

sT z e： ( ) ( )

n

n

x n z X z

z-transform

of discrete-

time signal

the Laplace transform



29

( )sT j T T j T j z e e e e re

n

n zn x z X )()(

so：

T r e

T

relation between

s zand

Laplace transform continuous time signal

z-transform discrete-time signalrelation

[ ( )] ( ) ( )snT sT

n

x n x nT e X e

L

sT z elet：

( )sT j T T j T j



30

2s

T f f

DTFT :

Discrete Time

Fourier Transform

( )sT j T T j T j z e e e e re

( ) ( )

j j n

n X e x n e

S plane Z plane

-

3 / s

T

j

/ s

T

/ s

T

1| j j

r z re e



31

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

:

s f f f

j

s pl ane

0

22

s

s

f T

sT

3

sT

3

sT

di i fil d i b i l i i



• If input is bandlimited and f s>2f max , :

d cd nT hT nh

T T e H j H

T j

eff

,0

,

32

discrete-time filter design by impulse invariance

w for T w d

k d d

c

jwk

T

j

T

w j H e H

2

d c T j H if ,0

, jw

cd

wthen H e H j w

T

l i b f i



33

,, d T

relation between frequencies

S plane Z plane

-

3 / d

T

j

/ d T

/ d

T

k d d

c jw k

T j

T w j H e H 2Relationship of

frequencies

d c T j H if ,0

w

T

w j H e H then

d

c

jw ,

i



periodic sampling

[ ] ( ) | ( )c t nT c

x n x t x nT

1 1 2*

2 2s c c s

k

X j X j S j X j k d T

34

T：sample period; fs=1/T:sample rate

Ωs=2π/T:sample rate

n

nT t t s

s c c c

n n


2

s

k S j k T

1

c s

k X j k T

Review

2

R i



proof of

35

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

n nT t t s

2s

k

S j k T

Review

2s

k

S j k T

s jk t

k n a e

s(t)为冲击串序列，周期为T，可展开傅立叶级数

1s jk t

neT

2 ( )s jk t F

se k

2



periodic sampling

36

s c c c

n n


j Tn

c

k

x nT e

( ) j t

s c

n

X j x nT t nT e dt

[ ] ( ) | ( )c t nT c

x n x t x nT ( ) j j n

cn

X e x nT e

( ) ( ) ( )

j j T

s T X j X e X e

1( ) j T c s

k

X e X j k T

1 2

( ) c

k

j k X X j

T T T

e

2s

T

( ) 0, j T i f X eT

1( ) c

jthen X X j

T T

e

1

s c s

k

X j X j k T

di t ti filt d i b i l i i



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

d cd nT hT nh

37

discrete-time filter design by impulse invariance

k d d

c

jwk

T

j

T

w j H e H

2

d c T j H if ,0

, jw

cd

wthen H e H j w

T

1 2( ) c

k

j k X X j

T T T e

1( ) c

j X X j

T T e

1 2

( ) c

k

j k

H H jT T T e

1

( ) c

j

H H jT T e

[ ] ( )ch n h nT





• Obtain the specifications for continuous-time

filter by transforming the specifications for the

desired discrete-time filter.• Find the system function of the continuous-time

filter.

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

38



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

39

wT

w

j H e H d

c

jw

,

d T w







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.

40



Continuous-time IIR filters

• Butterworth filters

• Chebyshev Type I filters

•Chebyshev Type II filters

• Elliptic filters

41







desired discrete-time filter.

• Find the system function of the continuous-time

filter.

•

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

42



Transformation from continuous to discrete

43

N

k k

k

ss

As H

1

0,0

0,1

t

t e At h

N

k

t sk

c

k

: k d s T

k pole s s z e

two requirements for transformation

N

k

nT s

k d

N

k

nT sk d cd nue AT nue AT t hT nh d k d k

11

N

k T s

k d

ze

AT z H

d k 1 11



Example 7.2 Impulse Invariance with a

Butterworth Filter

•

Specifications for the discrete-time filter:

44

we H

we H

jw

jw

3.0,17783.0

2.00,189125.0

d d T wT let 1

Assume the effect of aliasing is negligible

3.0,17783.0

2.00,189125.0

j H

j H

c

c




Butterworth Filter

45

0.89125 1, 0 0.2

0.17783, 0.3c

c

H j

H j

0.2 0.8912

0.3 0.1778c

c

H j

H j

0.3 0.2

2 20.2 1

1

0.89125

N

c

2 2

0.3 11

0.17783

N

c

2

2

1

1 N c c

H j j j

2

2

1

1

N

cc j j H j




Butterworth Filter

46

2

21

1N c

c

H j j j

22

0.2 11 1.258930.89125

N

c

7032.0,6 c N

2

20.3 11 31.62204

0.17783

N

c

2

0.2 0.25893

N

c

2

0.3 30.62204

N

c

2

3 118.263782

N

70470.0,8858.5 c N

0.2 0.89125

0.3 0.17783

c

c

H j

H j

l l h




Butterworth Filter

2

2

1

1 N c

c H j j j

47

2

2

1

1 N c c c

c H s H s H s

s j

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

k

j N k N N

c cs j e

6, 0.7032c N

0.182 0.679, j 0.497 0.497, j

0.679 0.182 j

Plole pairs: c H s




Butterworth Filter

48

4945.03585.14945.09945.04945.03640.0

12093.0222

ssssss

s H c

22

2 2 2

1

1

N

N N N

c

cc c c

c H s H s H s ss j

0.182 0.679, j

0.497 0.497, j

0.679 0.182 j

Plole pairs: c H s

0.12093

0.182 0.679 0.182 0.679 0.497 0.497c

H ss j s j s j

1

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

60.7032 N

c




Butterworth Filter

1d T

49

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 0.497c H s s j s j s j

1

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

1

N

k

k k

As s

1 1

1 11 1

N N d k k

k k k d k s T s

T A A H z

e z e z



Basic for Impulse Invariance• To chose an impulse response for the discrete-

time filter that is similar in some sense to theimpulse response of the continuous-time filter.

• If the continuous-time filter is bandlimited, thenthe discrete-time filter frequency response will

closely approximate the continuous-timefrequency response.

• The relationship between continuous-time anddiscrete-time frequency is linear; consequently,

except for aliasing, the shape of the frequencyresponse is preserved.

50



7.1.2 Bilinear Transformation

51

Bilinear transformation can avoid the

problem of aliasing.

Bilinear transformation maps

onto

w

1

1

2 1

1d

c z

H z H T z

1

1

1

12

z

z

T s

d

Bilinear transformation:

c H s

1

1

1

12

z

z

T s

d




1

1

1

12

z

z

T sd

52

sT

sT

z d

d

21

21

js 221

221

d d

d d

T jT

T jT z

any for z 10

any for z 10

1 12 (1 ) 1d T s z z

11 2 ] 1 2d d T s z T s




jsaxis j

53

2121

d

d

T jT j z

1 z 1 21 2

d

d

jw j T j T

e

planes Im

Re

plane z Im

Re

221

221

d d

d d

T jT

T jT

z

any for z 10

any for z 10

js




1

1

1

12

z

z

T

sd

54

2 1

1d

jw

jw j

T

e

e

2

tan 2d

j wT

2 2 sin 2

2cos 2d

j w

T w

2tan

2w

T d

2tan2 1

d T w

/2 /2 /2

/2 /2 /2

2 ( )

( )d

jw jw jw

jw jw jwT

e e e

e e e

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



55

2tan

2

2tan

2

:

s

d s

p

d p

T

T

prewarp

2tan

2)()(

d T

c j j H e H

q y p c( ), ( )

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.

56

S plane Z plane

-

3 / d

T

j

/ d

T

/ d

T

2tan2 wT d

2tan21

d T w





• 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-responsecharacteristics.

57

0cw

cw

2 2

jwe H

1

Bilinear Transformation ofs

e



Bilinear Transformation of

58

1

1

1

12

z

z

T

sd

2tan2

w

T d

e j

e

2tan 2

d

wT

d T



Comparisons of Impulse Invariance and Bilinear

Transformation

• The use of bilinear transformation is restricted tothe design of approximations to filters with

piecewise-constant frequency magnitude

characteristics, such as highpass, lowpass andbandpass filters.

• Impulse invariance can also design lowpass filters.

However, it cannot be used to design highpassfilters because they are not bandlimited.

59



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.

60



• Butterworth Filter,

•

Chebyshev Approximation,• Elliptic Approximation

61

7.1.3 Example of Bilinear

Transformation

we H

we H

jw

jw

6.0,001.0

4.0,01.199.0

xamp e . near



pTransformation of a Butterworth

Filter

0.89125 1, 0 0.2

0.17783, 0.3

jw

jw

H e w

H e w

620.0160.01

2 0.20.89125 1, 0 tan

2

2 0.30.7783, tan

2

c

d

c

d

H j

T

H jT

, 1d For convenience we choose T ,17783.015.0tan2

,89125.01.0tan2

j H

j H

c

c

2tan2 wT d



Example 7.3 Bilinear Transformation of a Butterworth Filter

,17783.015.0tan2

,89125.01.0tan2

j H

j H

c

c

63

N

c

c j j

j H 2

2

11

2 2

2 2

2tan 0.1 1

1 0.89125

3tan 0.15 11

0.17783

N

c

N

c

305.5 N

766.0

,6

c

N

0.0160.01



Locations of Poles

64

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

k

j N k N N

c cs j e

0.1998 0.7401, j

0.5418 0.5418, j

0.7401 0.1998 j

Plole pairs: c H s

2

2

1

1 N c

c

H j j j

2

2

1

1

N c c c

c

H s H s H s

s j

6, 0.766c N

Examp e 7.3 Bi inear



pTransformation of a Butterworth

Filter

5871.04802.15871.00836.15871.03996.0

20238.0222

ssssss

s H c

65

21

2121

61

2155.09904.01

1

3583.00106.117051.02686.11

10007378.0

z z

z z z z

z

z H

22

2 2 2

1

1

N

N N N

c

cc c cc

H s H s H s ss j

0.1998 0.7401, j 0.5418 0.5418, j 0.7401 0.1998 j

Plole pairs: c H s

1

1

1

12

z

z

T s

d

. . time filter



time filter

66



Example 7.4 Butterworth Approximation (Hw)

67

we H

we H jw

jw

6.0,001.0

4.0,01.199.014 N order

Example 7.4 frequency response



p q y p

68

Chebyshev filters



69

y

C Chebyshev filter (type I)

) / (1

1|)(|22

2

c N

cV

j H

)coscos()( 1 x N xV N

c

1

1

Chebyshev polynomial

Chebyshev filter (type II)

122

2

)] / ([1

1|)(|

c N

cV

j H

1

c

E l 7 5 Ch b h T I II A i i



Example 7.5 Chebyshev Type I , II Approximation

70

we H we H

jw

jw

6.0,001.04.0,01.199.0 8 N order

Type I Type II

.Cheb she



Chebyshev

71

Type I Type II

elliptic filters



72

E

p f

Elliptic filter

)(1

1|)(|

22

2

N

c

U

j H

s p

1

11

2 Jacobian elliptic function



Example 7.6 Elliptic Approximation

73

we H

we H

jw

jw

6.0,001.0

4.0,01.199.0

6 N order

.Elliptic



Elliptic

74

*Comparison of Butterworth, Chebyshev, elliptic filters: Example



75

p f , y , p f p

-Given specification

0.4|| .011 |)(| 99.0 je H

||0.6 0.001 |)(| j

e H

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

-Pole-zero plot (analog)



76

p ( g)

-Pole-zero plot (digital)

B C1 C2 E

B C1 C2 E

(14) (8)



77

-Magnitude -Group delay

B

C1

C2

E

B

C1

C2

E

4.0 6.0 4.0 6.0

5

20



7.2 Design of FIR Filters by Windowing

•FIR filters are designed based on directlyapproximating the desired frequency response

of the discrete-time system.

•

Most techniques for approximating themagnitude response of an FIR system assume a

linear phase constraint.

78



Window Method• An ideal desired frequency response

79

n

jwn

d

jw

d enhe H

dwee H nh jwn jw

d d 2

1

Many idealized systems are defined by

piecewise-constant frequency response withdiscontinuities at the boundaries. As a result,these systems have impulse responses that

are noncausal and infinitely long.

ww

wwe H

c

c jwlp

,0

,1

sinc

lp

w nh n

n

0cw

cw

jw

e H 1



Window Method

80

otherwise

M nnhnh

d

,0

0,

nwnhnh d

otherwise

M n

nw ,0

0,1

d eW e H e H w j jw

d

jw

2

1

The most straightforward approach toobtaining a causal FIR approximation is totruncate the ideal impulse response.

Windowing in Frequency Domain



• Windowed frequency response

81

deWeH21eH j j

d j

The windowed version is smeared version

of desired response

Window Method



82

1

2

j w jw j jw

d d H e H e W e d H e

If nnw 1

k n

jwn jwk wenweW 22

0cw

cw

jwe H

1

0 5 10510 1515

2

2

2 4 4 6

jwW e

h f d



Choice of Window• is as short as possible in duration. This

minimizes computation in the implementation of the filter.

83

nw

approximates an impulse. jweW

0

M jw jwn jwn

n n

W e w n e e

1

2sin 1 21

1 sin 2

jw M

jwM

jw

w M ee

e w

otherwise

M nnw

,0

0,1

1 M

2

1 M

2

1 M

jwW e

d h d



Window Method

84

jw

d

w j jw

d

jw e H d eW e H e H

21

then would look like , exceptwhere changes very abruptly.

jwe H jw

d e H

jw

d e H

nw jweW

0w

If is chosen so that is concentrated

in a narrow band of frequencies around

0cwcw

jw

d H e

1

1 M

2

1 M

2

1 M

jwW e

R l Wi d



Rectangular Window• for the rectangular window has a

generalized linear phase.

85

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 andthe side lobes grow such that the area undereach lobe is a constant.

2sin 1 2

sin 2

jw jwM w M

W e ew

1 M

2

1 M

2

1 M

1

M

M

1

M

M

R l Wi d



Rectangular Window

86

will oscillate at

the discontinuity.

d eW e H w j jw

d

The oscillations occur more rapidly, butdo not decrease in magnitude as Mincreases.

The Gibbs phenomenon can bemoderated through the use of a lessabrupt truncation of the Fourier series.

R l Wi d



Rectangular Window• By tapering the window smoothly to zero at

each end, the height of the side lobes can bediminished.

• The expense is a wider main lobe and thus a

wider transition at the discontinuity.

87

.Method



Method• To design an ilowpass FIR Filters

88

wwwwe H

c

c jw

lp,0

,1

sinc

lp

w nh n

n

0cw

cw

jwe H

1

Review

d h n h n w n

1, 0

0,

n M w n

otherwise

d eW e H e H w j jw

d

jw

2

1

1 M

2

1 M

2

1 M

jwW e

sin 2

2

cw n M

n M

02 M 0

2 M 0

M

M

2 M 0 M



7.2.1 Properties of Commonly Used Windows• Rectangular

89

otherwise

M nnw

,0

0,1

otherwise

M n M M n M n M n

nw

,0

2,2220,2

Bartlett (triangular)

7 2 1 P i f C l U d Wi d



7.2.1 Properties of Commonly Used Windows

•

Hanning

90

otherwise

M n M nnw

,0

0,2cos5.05.0

otherwise

M n M nnw

,0

0,2cos46.054.0

Hamming




• Blackman

91

otherwise

M n M n

M n

nw

,0

0,4cos08.0

2cos5.042.0




92



93

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.

7 2 1 P i f C l U d Wi d




94

biggest，high oscillations

at discontinuity

smallest，

the sharpest transition

Table 7.1

7 2 2 I i f G li d Li Ph



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.

95

otherwise M nn M wnw

,00,

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

7 2 2 I ti f G li d Li Ph




• Their Fourier transforms are of the form

96

2 M jw jw

e

jweeW eW

wof functionevenand realaiseW jw

e

causalnwnhnh d :

d d d if h M n h n h n h n w n

2 M jw jw

e

jwee Ae H

:h M n h n generalized linear phase

2 M

M




phaselinear d generalizenhn M h

nwnhnhnhn M hif d d d

:

97

2 M jw jwo jw ee jAe H

2 M

M

Frequency Domain Representation



Frequency Domain Representation

98

nhn M hif d d 2 M jw jw

e

jw

d ee H e H

1

2

jw

d

j w j H e H W d e e

1

2e e

j w jw jwewhere A H W d e e e

221

2e e

j w j w M j j M H W d e e e e

2 M jw jw

e

jweeW eW

w n w M n

2 jw jwM e A e e

d h n h n w n

Example 7 7 Linear Phase Lowpass Filter



Example 7.7 Linear-Phase Lowpass Filter

• The desired frequency response is

99

ww

wwee H

c

c jwM

jw

lp,0

,2

lph M n

nw

Mn

M nwnh c

2

2sin

21

2

sin 2

2

c

clp

c

w

w

jwM jwn

h n dw

w n M for n

n M

e e

2 M

wwe H

wwe H

s

jw

p

jw01

magnitude frequency response



magnitude frequency response

100

pw

sw

1 0 jw

p p

jw

s s

H e w w

H e w w

s pw w w

1020log p p

0.1 0.15 jw

H e w

1 0.05 0 0.25 jw H e w

0.1s p

w w w 10

20log 0.05 26 p

dB

1020logs s

20s dB

7 2 1 Properties of Commonly Used Windows




101

smallest，the sharpest

transitionbiggest，high oscillations

at discontinuity

7 2 3 The Kaiser Window Filter Design Method



7.2.3 The Kaiser Window Filter Design Method

102

1 22

0

0

1, 0

0,

I nn M w n

I

otherwise

2,where M

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

2

0

1

21

!r

r u I u

r

: 1,length M : parameter shape

:two parameters

Trade side-lobe amplitude for main-lobe width



103Figure 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

Table 7.1



104

Transition width is a little less than

mainlobe width

Comparison



Comparison• If the window is tapered more, the side lobe of the

Fourier transform become smaller, but the mainlobe become wider.

105

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

M=20

=6

M=20

Filter Design by Kaiser Window




106

pw

sw

wwe H

wwe H

s

jw

p

jw

01

ps www

10log20 A

Filter Design by Kaiser Window1 2




1 22

0

0

1

, 0

0,

I n

n M w n I

otherwise

107

ps www 10log20 A

21,0.0

5021,2107886.0215842.0

50,7.81102.04.0

A

A A A

A A

2285.2

8

w

A M M=20

Example 7.8 Kaiser Window Design of a Lowpass

Filt



Filter

we H

we H

jw

jw

6.0,001.0

4.0,01.199.0

108

ps

www 10log20 A

1 22

0

0

1

sin , 0c

I n

w n n M n I

0,

h notherwise

5.182 M where

21,0.0

5021,2107886.0215842.0

50,7.81102.04.0

A

A A A

A A

8

2.285

A M

w




p g p

Filter

we H

we H

jw

jw

6.0,001.0

4.0,01.199.0

109

001.0,min,001.0,01.0,6.0,4.0

:1

2121

s p ww

step

0.52

s p

c

w wcutoff frequency w

2 :step

3:step100.2 20log 60

0.5653 37

s pw w w A

M




Filter

2

0

1

21

!r

r u

I ur

110

0.5653

5.182 M where

1 22

0

0

1

sin , 0c

I n

w n n M n I

0,

h notherwise

21,0.0

5021,2107886.0215842.0

50,7.81102.04.0

A

A A A

A A

837

2.285

A M

w

10

3:

0.2 20log 60s p

step

w w w A

. .Filter

1 2

2

0 1sin

I nw n



111

0

0

sin, 0

cw nh n n M

n I

7.3 Examples of FIR Filters Design by the Kaiser



Window Method• The ideal highpass filter with generalized linear

phase

112

wwe

wwe H

c

M jw

c jw

hp

,

,02

jw

lp

M jw jw

hp e H ee H 2

n M n

M nw M n

M nnh chp ,

22sin

22sin

hph n h n w n

Example 7.9 Kaiser Window Design of a Highpass



Filter• Specifications:

113

wwe H

wwe H

p

jw

s

jw

,11

,

11

2

021.0,5.0,35.0 211 ps wwwhere

24,6.2 M

By Kaiser window method

Example 7.9 Kaiser Window Design of a Highpass



Filter• Specifications:

114

wwe H

wwe H

p

jw

s

jw

,11

,

11

2

021.0,5.0,35.0 211 ps wwwhere

24,6.2 M

By Kaiser window method

7.3.2 Discrete-Time Differentiator



7.3.2 Discrete Time Differentiator

115

we jwe H M jw jw

diff ,2

n

M n

M n

M n

M nnhdiff ,

2

2sin

2

2cos2

nwnhnh diff

phaselinear d generalize IV typeor III typen M hnh :

Example 7.10 Kaiser Window Design of a



Differentiator

• Since kaiser’s formulas were developed forfrequency responses with simple magnitude

discontinuities, it is not straightforward to

apply them to differentiators.

• Suppose

116

4.210 M

Group Delay



Group Delay

• Phase:

• Group Delay:

117

25

22 ww M

samples M

5

2

Group Delay



Group Delay

• Phase:

• Group Delay:

• Noninteger delay

118

225

22 ww M

samples M

2

5

2

7 4 Optimum Approximations of FIR Filters



7.4 Optimum Approximations of FIR Filters

• Goal: Design a ‘best’ filter for a given M

• In 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.

119

nhnh ee





nhnh ee

120

2, M Lenhe A L

Ln

jwn

e

jw

e

function periodicevenrealwnnhhe A

L

n

ee

jw

e ,,:cos201

.2 samples M Lbyit delaying

bynh fromobtained becansystemcausal A e

n M h M nhnh e 2

2 M jw jw

e

jwee Ae H





• Designing a filter to meet these specifications is to

find the (L+1) impulse response values

121

Lnnhe 0,

In Packs-McClellan algorithm,is fixed, and is variable.

21,,, and ww L s p

21 or

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





wnwT wn n coscoscoscoscos 1

122

1coscos0coscos0cos 1

0 wwT w

wwwT w coscoscos1coscos1cos 1

1

1cos2cos2cos 22 wwT w

wT wT w

wT wn

nn

n

coscoscos2

coscos

21

wwwww

wwww

cos3cos4cos1cos2cos2

cos2coscos23cos

32





L

k

k

k

w x

L

k

k k

L

n

ee jw

e

xa xPwhere

xPwawnnhhe A

0

cos01

coscos20

123

functionweightingtheiswW where

e Ae H wW w E

functionerror ionapproximat an Define

jw

e

jw

d





124

ww

wwe H

s

p jw

d ,0

0,1

ww

wwK wW

s

p

,1

0,

1

1

2

Minimax criterion



• Within the frequency interval of the passband

and stopband, we seek a frequency responsethat minimizes the maximum weightedapproximation error of

125

jw

e e A

jw

e

jw

d e Ae H wW w E

w E

F w Lnnhe maxmin

0:

Other criterions



00:1 min dww E H

Lnnhe

126

0

2

0:2 min dww E H Lnnhe

w E H

F w Lnnhe

maxmin0:

Alternation Theorem• Let denote the closet subset consisting of theF



• Let denote the closet subset consisting of the

disjoint union of closed subsets of the real axis x.

127

pF

is an r th-order polynomial.

r

k

k

k xa xP0

denotes a given desired function of x that is continuous on

x DP

pF

is a positive function, continuous on pF xW P

The weighted error is xP x D xW x E PPP

x E E PF x P

max

The maximum error is defined as

Alternation Theorem



• A necessary and sufficient condition that be the

unique r th-order polynomial that minimizes isthat exhibit at least (r+2) alternations; i.e.,

there must exist at least (r+2) values in such

that

128

xP

E x E P

i x PF

221 r x x x

E x E x E iPiP 1

1,,2,1 r i

and such that

for

Example 7.11 Alternation Theorem and

Polynomials



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

129

11.01.01 xand x

1 xW P

7.4.1 Optimal Type I Lowpass Filters



p yp p

• For Type I lowpass filter

• The desired lowpass frequency response

•

Weighting function

130

L

k

k

k wawP0

coscos

wwww

wwwww D

ss

p p

pcoscos1,0

01coscos,1cos

wwww

wwwwK wW

ss

p p p

coscos1,1

01coscos,1

cos




• The weighted approximation error is

• The closed subset is

or

131

wPw DwW w E PPP coscoscoscos

wwand ww s p0

s p wwand ww cos11coscos

x E P




p yp p

•The alternation theorem states that a set of coefficients will correspond to the filterrepresenting the unique best approximationto the ideal lowpass filter with the ratio

fixed at K and with passband and stopbandedge and if and only if exhibits atleast (L+2) alternations on , i.e., if and onlyif alternately equals plus and minus its

maximum value at least (L+2) times.• Such approximations are called equiripple

approximations.

132

k a

21

pwsw )(cos w E P

PF

)(cos w E P




• The alternation theorem states that theoptimum 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).

133




p yp p

•

Because all of the filters satisfy the alternationtheorem 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.

134

21 K

pw sw

Property for type I lowpass filters from the

alternation theorem



alternation theorem

•

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

• Alternations will always occur at and

•

All points with zero slop inside the passband andall points with zero slop inside stopband will

correspond to alternations; i.e., the filter will be

equiripple, except possibly at and

135

pw sw

0w

w

7.4.2 Optimal Type II Lowpass Filters



p yp p

• For Type II causal FIR filter:

• The filter length (M+1) is even, ie, M is odd

• Impulse response is symmetric

• The frequency response is

136

M nnh 0

nhn M h

21,,2,1,212

2

1cos

2cos2

21

1

2

21

0

2

M nn M hnbwhere

nwnbe

n M

wnhee H

M

n

M jw

M

n

M jw jw




21

0

21

1cos

~

2cos2

1

cos

M

n

M

nwnnbwnwnb

137

wPwee H M jw jw cos2cos2

21cos0

M Land wawPwhere L

k

k

k

n M hnbnbnba find k 212~




• For Type II lowpass filter,

138

ww

ww

ww De H s

p

P

jw

d

,0

0,

2cos

1

cos

ww

wwK

w

wW wW

s

pP

,2cos

0,2cos

cos

7.4.3 The Park-McClellan Algorithm



• From the alternation theorem, the optimum filter

will satisfy the set of equation

139

2,,2,111

Lie Ae H wW i jw

e

jw

d

jwe e A

ii

jw

d

jw

d

jw

d

L

L

L

L L L

L

L

w xwhere

e H

e H

e H

a

a

wW x x x

wW x x x

wW x x x

L

cos

11

11

11

2

2

1

1

0

2

2

2

2

22

2

2

2

22

1

1

2

11




•

Guessing a set of alternation frequencies

140

2

12

1

1

2

1 cos,1

1

L

k ii

ii

ik

k L

k k

k

k

L

k

jw

d k

w x x x

bwhere

wW

b

e H b k

2,,2,1 Li for wi sl pl wwww 1,and




141

2

1

1

1

1

1

1

1

cos,cos

Lk k

L

k ii ik

k

k k L

k

k k

k

L

k

k k jw

e

x xb x x

d

w x x

x xd

C x xd

wPe A




• For equiripple lowpass approximation

• Filter length: (M+1)

142

ps wwwwherew

M

324.2

13log10 2110

. xamp es o qu r pp eApproximation

7 5 1 L Fil



7.5.1 Lowpass Filter

we H

we H jw

jw

6.0,001.0

4.0,01.199.0

143

26 M

wwe Awwe A

wW w E w E

error ionapproximat unweighted

s

jw

e

p

jw

e A

,00,1

Comments



• M=26, Type I filter• The minimum number of alternations is

(L+2)=(M/2+2)=15

• 7 alternations in passband and 8 alternationsin stopband

• The maximum error in passband and

stopband are 0.0116 and 0.0016, whichexceed the specifications.

144

7.5.1 Lowpass Filter



• M=27, , Type II filter, zero at z=-1• The 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 alternationsin stopband

145

w

Comparison



• Kaiser window method require M=38 to meetor exceed the specifications.

• Park-McClellan method require M=27

• Window method produce approximatelyequal maximum error in passband and

stopband.

• Park-McClellan method can weight the errordifferently.

146

7.6 Comments on IIR and FIR Discrete-Time

Filters



Filters

•

What type of system is best, IIR or FIR?• Why give so many different design methods?

• Which method yields the best result?

147

7.6 Comments on IIR and FIR Discrete-Time

Filters



Filters

148

Closed-Form

Formulas

GeneralizedLinear Phase

Order

IIR Yes No Low

FIR No Yes High


h f d



• 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 Bartlettwindow can be expressed as a product of Fourier transforms of rectangular windows.

The Fourier transforms of the otherwindows can be expressed as sums of

Chapter 4Filter Design Techniques

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Transcript of Chapter 4Filter Design Techniques