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EE 422G Notes: Chapter 9 Instructor: Zhang

Chapter 9 Analysis and Design of Digital Filter

9-1 Introduction

What designs have we done in this course? What do we ean !" #i$ters here?

What do we ean !" #i$ters design?

Given speci#ications %re&uireents' ()H(z)

*et+s see how we can ip$eent a digita$ #i$ter %processor' i# itsH(z)is given?

9-2 ,tructures o# igita$ .rocessors

1/ irect-0or ea$iation

==

=

=

=

+==

m

j

j

r

i

i

m

j

j

j

r

i

i

i

jTnTykiTnTxLnTy

zk

zL

zX

zYzH

13

1

3

'%'%'%

1'%

'%'%

he #unction is rea$ied5

What+s the issue here?

Count how an" eor"

e$eents we need5

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Can we reduce this nu!er?

I# we can6 what is the concern?

'%

1'%

3

1

3

2

1

1

1

1

'%

zH

m

j

jjzH

r

i

iim

j

ij

r

i

ii

zk

zL

zk

zL

zH

==

=

=

+

=

+

=

'%'%'%'%'%'% 21 zXzHzHzXzHzY ==

enote '%'%'%'%'%'% 12 zVzHzYzXzHzV ==

Ip$eentH2(z) and thenH1(z) ?

Wh" H2 is ip$eented?

%1'

'%'%'%'% 11 zVzkzVzkzXzV m

m = %2'

'%1

1'%

'%'%'1%

1

11

zXzk

zV

zXzVzkzk

m

j

jj

mm

=

+=

=+++

H2 is rea$ied5

Can "ou te$$ wh"H1 is rea$ied?

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What can we see #ro this rea$iation? ,igna$s at jA and jB : a$wa"s the sae

irect 0or II ea$iation

E7ap$e 81

821

'2/31%

/3:/38/31'%

++

=z

zzzzH

,o$ution:821

821

33;/312/3/31

/3/38/31'%

+++

+=

zzz

zzzzH

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Iportant: %' %1' : 1@A/ %2' Coe##icients in : in the #eed!acB channe$

2/ Cascade ea$iation

0actorie '1'%1'%1%/3:/38/31 8

8

2

2

1

1

821 =+ zazazazzz

'%

1

1

8

'%

1

1

2

'%

1

1

1

81

821

'8%'2%'1%

2/31

1

2/31

1

2/31

1

'2/31%

9/3:/38/31

'%

zHzHzH

z

za

z

za

z

za

z

zzz

zH

+

+

+

=+

+

=

Genera$ 0or

pp$" irect II #or each5

8/ .ara$$e$ ea$iation %,ip$e .o$es'

polescongugae

complex!erealize

"

l ll

ll

polesreal

!erealize

"

k k

k

mr

#

i

i

iz$z$

ze%

z%BzAzH

=

=

>

=

+

+=

21

11C1

1

11

I#

3 '1'%1%

1

1

1'%

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E7ap$e 9-1

';

11'%

2

11%

'1%'%

11

81

=

zz

zzH

cascade and para$$e$ rea$iation5

,o$ution:

%1'Cascade:

'1%

';

11%

1

'2

11%

1'% 1

1

1

1

1

= z

z

z

z

zzH

%2'.ara$$e$

';

1'%

2

1%

'1%

';

11'%

2

11%

'1%'%8

11

81

=

=

zzz

z

zz

zzH

In order to aBe deg%nu'Ddeg%den'

.age 9-

.artia$-0raction

E7pansion #ors

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;>12>1

';

1'%

2

1%

'1%'%

2

2

8

+

++=

=

z

"

z

%

z

B

z

A

zzz

z

z

zH

1

';

1'%

2

1%

'1%

';>1'%2>1%

'1%'%$i

8

3

2

82

3=

=

==

zz zzz

zzzzHA

8

4

';>1%

'1%$i

'%'2>1%$i

2

8

2>12>1=

== zz

z

z

zHz%

zz

8

848

'2>1%

'1%$i

'%';>1%$i

2

8

;>1;>1=

== zz

z

z

zHz"

zz

z &an"thing other than ' 1*+ 1B & ,112 0or e7ap$e6z(2

112

';

1'%

2

1%

'1%$i

''%%$i

'%$i'%$i

'%$i'%$i

8

3

3

33

33

=

=

=

+=

+=

zz

z

$z

$B

B$z

zzH$

BzAzzH

Bz

AzH

z

z

zz

zz

1E

;

8

848

;>1E

8>848

;

1

9>;2>8

8>4

21

44

1

4E

4

;

1E

2

84

1'%

2

8

=

=

=

=

=

=

=

=

z

"

z

%

z

A

z

zH

112F1E

;

8

848

9

;4

4E

4G2 =++=B

E7ap$e 9-2: ,"ste having a cop$e7 conHugate po$e pair at jaez =

rans#er #unction

22122

2

11

2

'%cos21

1

'%cos2

'1'%1%1

''%%'%

+

+=

+=

=

=

zazaazaz

z

zaezaeaezaezzzH

jjjj

rjrjrj

eaeaeH

422

2

'%cos21

1'%

+=

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.age 9-;

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9/8 iscrete-ie Integration

ethod o# iscrete-tie s"ste esign: ppro7iate continuous-tie s"ste

Integrator =

$xy3'%'% a sip$e s"ste

s"ste input

Jutput

iscrete-tie appro7iation o# this s"ste: discrete-tie Integrator

1/ ectangu$ar Integration

+=+==

$xy$x$x$xy3

3

3'%'%'%'%'%'% 333

change o#y#ro 'to : '6% 3 y

& nT '& nT, T

'6%'%

'%'%'%

nTTnTyTnTy

$xTnTynTynT

TnT

+=

+=

Tsa$$ enough () '%'% TnTxx

'%

'%

'%'%

TnTTx

$TnTx

$TnTx$x

nT

TnT

nT

TnT

nT

TnT

=

=

'%'%'% TnTTxTnTynTy +=

discrete-tie integrator: rectangu$ar integrator

1

1

11

1'%

'%'%

'%'%'%

==

+=

z

Tz

zX

zYzH

zTXzzYzzY

.age 9-13

ConstantF'6G% nTTnT

Constanty(nT): ,"ste output

x(nT): ,"ste input 6 to !e

integrated

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2/ rapeoida$ Integration

''%'%%

2

2

'%'%'%

nTxTnTxT

$nTxTnTx

$xnT

TnT

nT

TnT

+=

+=

'%2

'%2

'%'% TnTxT

nTxT

TnTynTy ++=

or

egraorsime$icree

.o/e.een$i00erence

TnTxnTxT

TnTTxTnTynTy

int

'F%'%G

2

'%'%'%

++=

1

1

11

1

1

2'%

'%2'%

2'%'%

+

=

++=

z

zTzH

zXzT

zXT

zYzzY

8/ 0re&uenc" Characteristics

8/1 ectangu$ar Integrator

1

1

1'%

=

z

TzzHr

0re&uenc" esponse2>sin21

'%2>

2>2>

2>

Tj

Te

ee

Te

e

TeeH

Tj

TjTj

Tj

Tj

TjTj

r

=

=

=

.age 9-11

Constants

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Jrrj

TeeH

rjrj

r

sin2

'% 2

=

= =

=

rTr

T

s

s

2>

12

p$itude esponse

'3%sin2

13

sin2'% = rr

r

TrAr

.hase esponse

2

13

2

'% == rrjer rjr

8/2 rapeoida$ Integrator

1

1

1

1

2'%

+=

z

zTzH

0re&uenc" esponse

rj

rT

rj

rT

ee

eeT

e

eTeH

rjrj

rjrj

rj

rjrj

sin

cos

2sin2

cos2

2

21

1

2'%

2

22

==

+

=+

=

p$itude:213

sin2cos'% = r

rrTrA

.hase:

>=>=

3cos

3sin

2

13

2'%

r

rrr

8/8 Kersus Idea$ Integrator

Idea$ %continuous-tie ' Integrator

2'%

2

1'%

2

1'%

221

'%

==

=

===

rr0

rA

r0j

rH

r00j

jH

s

s

s

when T&1 second %i##erent p$ots and re$ationships wi$$ resu$t i# Tis di##erent/'

.age 9-12

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*ow 0re&uenc" ange '12%1

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eterine a digita$ e&uiva$ent/

,o$ution

%1' =$ocB iagra o# the origina$ s"ste

%2' n e&uiva$ent

%8' rans#er 0unction erivation

1

1

1

1

1

1

1

1

'2%'2%

'1%

'%

'%'%

'%1

1

2'%

1

1

21

''%'%%1

1

2'%

+++

==

+

=

+

+

+

=

zTT

zT

z1

zYzH

zXz

zTzY

z

zT

zYzXz

zTzY

esign: 9-4 0ind E&uiva$ence o# a given ana$og #i$ter %II':

Inc$uding ethods in ie oain and 0re&uenc" oain/

9- No ana$og protot"pe6 #ro the desired #re&uenc" response: 0I

9- Coputer-ided esign

.age 9-14

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9.4 Infinite Impulse Response (IIR) Filter Design%GivenH(s) H$(z)'

9-4A Synthesis in the ime-Domain! In"ariant Design

1/ Ipu$se L Invariant esign

%1' esign .rincip$e

%2' I$$ustration o# esign Mechanis %Not Genera$ Case'

ssue:%1' Given ana$og #i$ter %rans#er 0unction'

= +

=m

i i

ia

ss

2sH

1

'% %a specia$ case'

%2' ,ap$ing .eriod T %sap$e !a()to generate !a(nT)'

erivation:

%1' Ipu$se esponse o# ana$og #i$ter

=

==m

i

s

iaaieksHL!

1

1''%%'%

%2' !a(nT): sap$ed ipu$se response o# ana$og #i$ter

=

=

==m

i

nTsi

m

i

nTsia

ii ekeknT!11

'%'%

%8' -trans#or o# !a(nT)

.age 9-1

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,ap$ed ipu$se response o# ana$og #i$ter

=

=

=

=

= =

=

===

==

m

iTs

i

m

iTsi

m

i n

nTsi

n

m

i

nnTs

i

n

n

aa

zek

zekzek

zekznT!nT!1

ii

i

i

11

11

1 3

3 13

111'%

'%'%''%%

%4' Ipu$se-Invariant esign .rincip$e''%%''%%'%'% nT!1nT!1nT!nT! aa

igita$ #i$ter is so designed that its ipu$se response !(nT)

e&ua$s the sap$ed ipu$se response o# the ana$og #i$ter !a(nT)

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%' esign E7ap$e sTs

sHa 21

1'% =

+=

,o$ution: 16161 11 === s2m

12121

1 1

2

1

12

1'%

=

=

=

=

zezeze

2TzH

m

iTs

i

i

2/ Genera$ ie L Invariant ,"nthesis%1' esign .rincip$e

%2' erivatio

n

Given: Ha(s) trans#er #unction o# ana$og #i$ter

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Xa(s) *apa$ce trans#or o# input signa$ o# ana$og 0i$ter

T sap$ing period

0ind H(z) -trans#er #unction o# digita$ #i$ter

%1' esponse o# ana$og #i$terxa()

'F%'%G'% 1 sXsHLy aaa=

%2'ya(nT) sap$ed signa$ o# ana$og #i$ter output

nTaaa sXsHLnTy

=

= F'%'%GG'% 1

%8' -trans#or o#ya(nT)NF'%'%GOG'F%G

1

nTaaa sXsHL1nTy1

=

=

%4' ie L invariant esign .rincip$e'F%G''%%'%'% nTy1nTy1nTynTy aa =

igita$ #i$ter is so designed

that its output e&ua$s the sap$ed

output o# the ana$og #i$ter

Incorporate the sca$ing :

'F%G'F%G

'%'%

nTy13nTy1 aT3zXzH

=

-trans#er #unction o# digita$ #i$ter

%' esign E&uation

NF''%'%%OG'%

'% 1nTaa

sXsHL1zX

3zH

==

specia$ caseX%z'&1 Xa%s'& 1 %ipu$se'

() NF''%%OG'% 1

nTa sHL31zH

=

=

%' esign procedure

: 0ind '%'F%'%G1 ysXsHL aaa = %output o# ana$og #i$ter'

=: 0ind nTaa ynTy == '%'%

C: 0ind ''%% nTy1 a : ''%%'% nTy31zH a=

E7ap$e 9-'2'%1%

'4%E/3'%++ += ss

ssHa

0ind digita$ #i$ter

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2

1

1

E/1

'2'%1%

'4%E/3'%

1'%

+

+=

++

+=

=

ssss

ssH

sX

a

a

aaa eesXsHLy 21 E/1'F%'%G'% ==

%2' 0ind nTaa ynTy == '%'%nTnT

a eenTy '%'%E/1'%2 =

%8' 0ind ''%% nTy1 a

121 1

1

1

E/1''%%

=

zezenTy1

TTa

%4' 0ind -trans#er #unction o# the digita$ #i$ter

=

=

121 1

1

1

E/1

''%%'%

zeze3

nTy31zH

TT

a

use G (

121 11

E/1'%

=

ze

T

ze

TzH

TT

%' Ip$eentation

Characteristics

%1' 0re&uenc" esponse e&uations: ana$og and digita$

na$og :'2'%1%

'4%E/3'%

jj

jjHa ++

+=

igita$ : TjTTjTTj

ee

T

ee

TeH

=

211

E/1'%

%2' dc response coparison % 3= '

.age 9-19

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na$og: 121

4E/3'3% =

=aH

igita$: TTj

e

T

e

THeH

2

3

11

E/1'1%'%

==

Kar"ing with %shou$d !e'

:3T Te T 1 6 Te T 212

1'21%1'1%1

E/1'1% =

T

T

T

TH

:3T #or e7ap$e '23%8141/323

2=== s0sT

834/3=Te 6 E88E/32 = Te 34E/1'1% =H good enough

%8''% jH

a versusI'%I TjeH

:on$T sec8141/3

23

2==

Psing nora$ied #re&uenc" ss00r >> ==

'%

222

E4

1E/3'%

4cos982/32cosE12E/8492E/2

2cos39;8/32E4;;/3

13'%

42

2

2

rH

rT

r00jH

rr

reH

a

sa

rj

===++

+=

+

=

.age 9-23

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%4' H versus aH

%' Gain adHustent when 3T3T () #re&uenc" response ine&ua$it"

adHust G () '%'% jHeH aTj = at a specia$

#or e7ap$e 3=I# 3 & T & 3/8142 () 34E/1I'% 3==

TjeH

I# se$ecting 3 & T*1/34 () 1'% 3==TjeH

8/ ,tep L invariance s"nthesis

ssXa1

'% = 111

'% = zzX

NI'FF%

1GOG'1%'%

NI'FF%'%GOG'%

'%

11

1

nTa

nTaa

sHs

L1z3zH

sXsHL1zX

3zH

=

=

=

=

E7ap$e 9-'2'%1%

'4%E/3'%

++

+=

ss

ssHa / 0ind its step-invariant e&uiva$ent/

,o$ution o# esign

2

E/3

1

E/11

'2'%1%

'4%E/3'%

1

++

+=

+++

=ssssss

ssH

s a

nTnT

nTanTaa eesHs

LynTy 21 E/3E/11'F%1

G'%'% =

= +===

1211

1

1

1

E/3

1

E/1

1

1'%1%

'F%G'1%'%

+

=

=

zezezz3

nTy1z3zH

TT

a

.age 9-21

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Coparison with ipu$se-invariant e&uiva$ent/

9-4# Design in the Fre$uen%y Domain --- he #ilinear &-transform

1/ Motivation %pro!$e in ie oain esign'

'%

'%

0X

x

+++++=

=

='F%G'%G'%

'%'%

'%

11 ssn

nsns

s

00X%00X%0X%

n00X%0X

nTx

Introduced !" sap$ing6 undesired5

x%' !and$iited % '3'2>% == s00X

.age 9-22

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'%'% 0X0X s=

#or2

s0

0 1 = / We have no pro!$e to #ind a digita$ e&uiva$ent

.age 9-24

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'%S

'%1

1

a

Tj

$ HeH without a$iasing5

*et+s use '%L as a nu!er %#or e7ap$e 3/2' representing an" $ow

#re&uenc"6

hen6 !ecause '2/3%S '% =LaH is a good appro7iation o# '2/3%

'% =LaH 6

= '% '2/3% '% Tj$

LeH '2/3% '% =LaH shou$d !e a good appro7iation/

digita$ #i$ter can thus !e designed #or an ana$og" #i$ter '%aH which is not

!and$ited5

wo ,tep esign .rocedure:

Given: ana$og" #i$ter '%sHa %1' 0ind an !and$iited ana$og" appro7iation % '%

S1sHa ' #or '%sHa

%2' esign a digita$ e&uiva$ent ''%% zH$ #or the !and$iited #i$ter '%S

1sHa /

=ecause o# the re$ationship !etween % '%S

1aH ' #or '%aH 6

'%zH$ is a$so digita$ e&uiva$ent o# '%sHa /

he over$apping %a$iasing' pro!$e is avoided5

he designed digita$ #i$ter can appro7iate '%aH %#or 1 and taBe the

sae va$ue' at $ow #re&uenc"/

8/ a7is to 1a7is %sp$ane tos1p$ane' trans#oration

e&uireent :2

1s

== % s is given sap$ing #re&uenc"/'

.roposed trans#oration :

2tan22

1tan 11 T

%%s

=

=

E##ect o# C:

We want the trans#oration ap

r= %#or e7ap$e6 sra$r >133= ' to r =1 '%C

()2

cot2

tan T

%T

% rrr

r

==

i/e/ when the sap$ing period Tis given6 %is the on$" paraeter

which deterines what wi$$ !e apped into 1a7is with the

sae va$ue/

E7ap$e: 22

2

2'%

cc

ca

sssH

++

=

cc

ca

jjH

2'%'%

22

2

+

= not !and$iited

.age 9-2

Karia!$e in

doain

Karia!$e in 1

doain

Constant

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I# we want to ap 1332 = to 13321 = 233=r

2

233cot233

T%

=

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%rans#er rep$ace Tse 1 !"z #unction o#

digita$ #i$ter'

Tuestion: Can we direct$" o!tainH$(z)#roHa(s)? Res5 %=ut how?'

=i$inear -trans#or

.reparation : %1'jxjx

jxjx

ee

eej

x

xx

+

==cos

sintan

%2'2

tan 1T

%

=

2>

2>2>

2>2>

2>2>

+

=

+

=

+

=

ep$ace ze Ts =1 #or digita$ #i$ter

+

=

1

1

1

1

z

z%s direct trans#oration #rostoz%!"passs1'

E7ap$e 22

2

2'%

cc

ca

sssH

++

=

igita$ 0i$ter

2122212

212

2

1

1

21

212

2

'1%'1%2'1%

'1%

1

12

'1%

'1%'%

+++

+

=

++

+

+

=

zzz%

z

z

z

z

z%

zH

cc

c

cc

c

$

C: on$" undeterined paraeter in the digita$ #i$ter/

o deterine C: %1' '% sT

%2' r %re$ated to the #re&uenc" range o# interest'

.age 9-2

=i$inear Ltrans#oration

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E7ap$e 9- 22

2

2'%

cc

ca

sssH

++

=

c : !reaB #re&uenc"

aBe cr = Consider 'E332%E33 == cc Hz0

sec'333E/31

%2333 ===s

s0

THz0

C deterined ()H$%z' deterined

221

11E8/31

292;98/3E;E;:/3292;98/3'%

+++=

z

zzzH$

'% 2 rj

$ eH

6 '%2 rj

$ eH

6 '% 2 rj

$ eH

o copare the #re&uenc" response with the origina$ ana$og #i$terHa:

';%'2%'2%'%4

ca

00

saaa r0jHr0jH0jHjH

cs

=

===

% rep$aces!" '%; sHinr0j ac '

II aH 6 aH

oo $ow0s () poor accurac" in0c/

.age 9-2;

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9-4C #ilinear &-ransform #andpass Filter

1/ Construction Mechanis

%1'0ro an ana$og $ow-pass #i$terHa(s)

to ana$og !andpass #i$ter '%22

/

ca

ssH

+

i/e/6 rep$aces!"/

c

s

s

22 +

to #or a !andpass #i$ter

0or e7ap$e1

1'%

+=

ssHa $ow-pass

1

122

++

/

c

s

s

!and-pass

Wh"? Jrigina$ $ow-pass'% jHa

*ow ()

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1

1

1

'1%

'1%

1

1

1

1

1

2

21

21222

+

+

++

++

/

c

/

c

z

z

z

z%

s

s

2/ =i$inear -trans#or e&uation

na$og *ow-pass =andpass %ana$og'

s/

c

s

s

22

+

'1%

'1%'1%

1

1

1

1

2

212212

1

1

2

2

1

1

++

=

+

+

+

z%

zz%

z

z

%

z

z%

/

c

/

c

2

21

22

22

22

2

212212

1

21

'1%

'1%'1%

+

+

+

=

++

z

zz%

%

%

%

z%

zz%

c

c

/

c

/

c

2

21

1

1

+

z

zBzA

with

22

22

22

2

c

c

/

c

%

%

%

%A

+

=

+=

#

8/

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Iportant paraeters o# !andpass

%1' center #re&uenc" c

%2' u upper critica$ #re&uenc"

%8' l $ow critica$ #re&uenc"

,e$ection o# c #or !andpass: luc =2

esign o# % /c% 66 '

We want2

tan T

% uu

= 6 $so want2

tan T

% ll

=

one paraeter C () ipossi!$e

so$ution2

tan2

tan22 TT

% luc

!andwidth2

tan2

tan T%T% lu/

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2'cos%

2'cos%

2'cos%

'cos%

tantan1

tantan1

T

T

lu

luyx

yx

yx

yx

+

+

=+

2

21

1

1

+

z

zBz

As

E7ap$e : '1%'1%

1

1

1

11221

2

2

21

++

=

+

+

+ zzBzA

z

z

zBzA

s

/ In the nora$ied #re&uenc"

e#erence #re&uenc": sap$ing #re&uenc" '% ss0

s

u

s

uu

0

0r ==

s

l

s

l

l

0

0r ==

()u

su

u r0

0T

==2

12

2

6 ll rT

=2

()

+=

=

'%cos

'%cos2

'%cot

lu

lu

lu

rr

rrB

rrA

s () 2

21

1

1

+

z

zBzA

E7ap$e 9-9 *owpass1

1'%

+=

ssHa

rans#er #unction o# !andpass digita$ #i$ter

1

1

1

1'%

2

21

+

+

z

zBzA

zH$

and =? eterined !" design re&uireents/

=

=

Hz0

Hz0

l

uE33

1333sap$ing #re&uenc" 0s& 333Hz

==

==

1/3>

2/3>

sll

suu

00r

00r

.age 9-82

In $ow-pass

In $ow-pass

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28:3:;3/1

'1/32/3%cot

'1/32/3%cot2

3:;8E/8'1/32/3%cot

=+=

==

B

A

21

2

3;8E/2;34221/83;8E/4

1'%

+

=

zz

zzH

'% 2 rjeH 6 I'%I 2 rjeH 6 '% 2 rjeH

.age 9-88

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9-' Design of Finite-Duration Impulse Response (FIR) Digital Filter

irect esign o# igita$ 0i$ters with no ana$og protot"pe/

Can we a$so do this #or II? Res5

Psing coputer progra in ne7t section/9-'A A fe $uestions

1/

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/ CanH( r ) !e e7pressed in 0ourier ,eries ? Res5

1

2>1

2

2

'%

'%

$rerHX

eXrH

rjn

n

n

rjn

n

What does this ean? Ever" desired #re&uenc" responseH%r' o# digita$

#i$ter can !e e7pressed into 0ourier ,eries 5 0urther6 the coe##icients o#

the 0ourier series can !e ca$cu$ated usingH%r'5

9-# Design prin%iple

=

=n

rjn

neXrH2'%

enote

==

==

=

2>1

2>1

2

2

'%'%

'%'%'%

$rerHXnT!

enT!rHXnT!

rjn

n$

n

rjn$

n$

Consider a #i$ter with trans#er #unction

=

n

n$ znT! '%

What+s its #re&uenc" response ?

'%'%''%% 22 rHenT!enT!n

rnj$

n

nrj$ ==

=

=

given speci#ication o# digita$ #i$ter+s #re&uenc" response5

9- C Design *ro%edure

%1' GivenH%r'

%2' 0ind H(r)s Fourier series

=

=n

rnj

$ enT!rH2'%'%

.age 9-8

3 : sap$ing #re&uenc"? No5 3T : period o#x

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where =2>1

2>1

2'%'% $rerHnT!

rjn

$

%8' esigned #i$ter+s trans#er #unction

=

n

n$ znT! '%

What+s !$%nT' ? Ipu$se response5E7ap$e 9-13: '2cos1%

2

1'% rrH +=

,o$ution :

%1' Given '%rH : done

%2' 0ind '%rH +s 0ourier series

=

=+=n

rjn

$ enT!rrH 2'%'2cos1%

2

1'%

where += 2>1

2>1

2'2cos1%2

1'% $rernT! rjn$

n & 3

2

12cos

2

1

2

1'2cos1%

2

1'3%

2>1

2>1

2>1

2>1

2>1

2>1=+=+= r$r$r$rr!$ 3n

+

+=

++=

2>1

2>1

'1%22>1

2>1

'1%2

2>1

2>1

222

2>1

2>1

2

4

1

4

1

22

1

2

1'%

$re$re

$reee

$renT!

rnjrnj

nrjrjrj

rjn

$

1633'%

4

1

4

1'%

4

1

4

1'%

2>1

2>1

2>1

2>1

=

==

==

nnT!

$rT!

$rT!

$

$

$

===

++==

=

2II63'%64

1'%6

2

1'3%

4

1

2

1

4

1'%'% 222

nnT!T!!

eeenT!rH

$$$

rjrj

n

rnj

$

%8' igita$ 0i$ter

111

41

21

41'%'3%'%'%

= ++=++= zzzT!!zT!znT! $$$n

n$

9. D *ra%ti%al Issues ! Infinite num+er of terms and non-%ausal

%1'

=

=n

n$nc znT!zH '%'%

.age 9-8

2M@1 ters

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runcation () ''%'%%'%'%

=

=

==n

n

$r

#

#n

n

$nc znT!n.znT!zH

ectangu$ar window #unction

>

= #n

#n

n.rII3

II1'%

runcationwindow

E##ect o# runcation %windowing':

ie oain: Mu$tip$ication % !and .'

0re&uenc" oain: Convo$ution

r

r#ee.

#

#n

rnjrj

r

sin

'12%sin'% 22

+===

%#ter runcation: he desired #re&uenc"Hr rr.H #re&uenc" response o# truncated #i$ter '

he e##ect wi$$ !e seen in e7ap$es5

%2' Causa$ 0i$ters:

=

+

=

==#

#n

n#

$

#

#n

n

$

#

c znT!znT!zzH '%'%'%'%

k & n4# =

=#

k

k$c z#TkT!zH

2

3

'%'%

e#ine '% #kT!L $k = = =#

k

kkc zLzH

2

3'%

e$ationship: #ncc zzHzH = '%'%

0re&uenc" esponse

=

==

#rrr

rArAeeHeH

ncc

ncc#rjrj

nc

rj

c

2'%'%

'%'%'%'% 222

esign E7ap$es

+=

#n

#n#

n

n.!

II3

IIcos4:/3E4/3'%

E7ap$e 9-11 esign a digita$ di##erentiator

,tep1 : ssign '%rH

.age 9-8

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'%rH shou$d !e the #re&uenc" response o# the ana$og di##erentiator

H(s) & s

() esiredr0jjrH s

r00

0

s

2I'% 2 ==

=

=

,tep2 : Ca$cu$ate !$%nT'

[ ] [ ]

[ ]njnj

0n

n

0

eenj

0ee

n

0

enjn

0e

njn

0

enj

ren

0

r$eren

0

rr$en

0

$rnrejn

0

$rer0j$rerH

$rer0j$rerHnT!

ss

njnjsnjnjs

njsnjs

r

r

nrjnrjs

nrjr

r

nrjs

nrjs

nrjs

nrj

s

nrj

nrj

s

nrj

$

sin22

cos22

22

2

1

2

1

2

1

2

1

2

1

'%

2

'2%'%

'2%'%'%

2

2

2>1

2>1

22

2>1

2>1

22>1

2>1

2

2>1

2>1

2

2>1

2>1

2

2>1

2>1

22>1

2>1

2

2>1

2>1

22>1

2>1

2

=

+=

=

=

=

=

=

=

==

=

=

=

=

=

=

=

22

2

FsinGsin

3'1%sin2

2'1%

'%

n

nn0n

n

0

n

0

nn

0n

nj

j0

n

0

nT!sss

nssns

$

.age 9-8;

nrj

/

a

/

a

/

a

e5ru

5$uu5u$5

26

==

=

3

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33sin

$i2

>'%

>'cos%$i

2

cos

$i2

FsinG

$i

2

3

3

3

23

===

=

=

nn0

$nn$

$nn$0

n

n0

$n

n$

nn$n

$

0

n

s

n

s

n

s

ns

i/e/6

=

=

33

1'1%'%

n

nn

0

nT!

ns

$

,tep 8: Construct nc #i$ter with haing window %M('

'%'%

'%'1%'%'1%'%

1

1

zHzzH

zn.n

0zn.

n

0zH

c

n

n

n

!

nsn

n

n

!

ns

=

=

=

=

=

+=

Example 9- 12:

Desired low- pass

FIR digital filter

characteristic

.age 9-89

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'%

'%>'8/3%sin$i'3%

33===

n

n$n$

n$n$T!

nn$

=

=

38/3sin

'%

8/3'3%

n

n

nnT!

!

$

$

NC filter with 17 weights window: =

=;

;

'%'%'%

n

n

!$6% zn.nT!zH ,

'%; zHzH 6%%=

.age 9-43

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Example 9-13 (90ophase shifter)

1

3

23

2>1

2

+=++=

=

+=

n= 0 => 3

1

sin$i

cos1$i'3%

33

==

=

n

n

n!

nn

$

=

=

3F'1%1G1

'%

3'3%

nn

nT!

!

n$

$

.age 9-41

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=

==

e5ennn

o$$nn

633

2

=>

=

==

e5enn

o$$nnnT!$

3

2

'%

Filter: 9'%'%'%

===

#zn.n!zHn

n

!$6%

'% zHzH 6%%=

.age 9-42

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Fig. 9-32 Amplitude response of digital 90 degree phase shifter

9. Computer-Aided Design of Digital Filters

9- Coand "u$ewa$B #or II

E7ap$e 9-14

9-= Coand ree #or 0I

E7ap$e 9-1

Chapter 9

EE422-9

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Transcript of EE422-9