Dsp-unit 5.1 Analog Filters

7/23/2019 Dsp-unit 5.1 Analog Filters

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UNIT-V

IIR DIGITAL FILTERS



Jan 6, 2016 2

Contents

• Introduction– Filtering of signals– Classification of filters

• Analog and digital• Based on frequenc res!onse

• Practical analog filter specifications– LPF. HPF, PF and !F.

• "nalog filters appro#i$ation– utter%ort& and C&e'(s&e)– Finding *rder and nor$ali+ed !ta'le filter– esign e#a$ples

• "nalog to analog transfor$ations

• esign of II- igital filters fro$ analog filters– I$pulse in)ariance, step in)ariance and 'ilinear

transfor$ations– esign e#a$ples

• "nalog to digital transfor$ations



Introduction

Filtering of signals



Jan 6, 2016

T"e DTFT is re#e#$ered again%

∑∞

−∞=

−=n

jwn jw en xe X ][)( ∫ −

=π

π

π dwee X n x jwn jw )(

2

1][

•#/n is e#pressed as a su$$ation of sinusoids %it&scaled a$plitude.

•sing a s(ste$ %it& a freuenc( selecti)e to t&eseinputs, t&en it is possi'le to pass so$e freuenciesand attenuate t&e ot&ers.

•Suc" a sste# is called a Filter.

&"at is #eant $ a filter'



Jan 6, 2016 3

4&e function of a filter is to re$o)eun%anted parts of t&e signal, suc&

as rando$ noise, orto e#tract useful parts of t&e signal,

suc& as t&e co$ponents l(ing %it&ina certain freuenc( range.

Filtered signal

Unfiltered signal

or raw signal

&"at is #eant $ a filter'



Jan 6, 2016 6

E(a#!le)*

C&oose freuenc( response of a s(ste$ suc& t&at

≤≤

≤=

π ω ω

ω ω

c

c jw

for

for e H

0

1)(

If

"l$ost 5 0

&ic& indicating t&e LPF effect of t&e L4I s(ste$

Adawy

π ω ω ω ω ω <<<<+= 2121 0)cos()cos(][ c for n Bn An x

( ) ( ))(cos)()(cos)(][2211

21 ω θ ω ω θ ω ω ω +++= ne H Bne H An y

j j

( ))(cos)(][ 11

1ω θ ω ω

+= ne H An y j



Jan 6, 2016 7

h[n]=[ a b a]

y[n]=h[0]x[n]+h[1]x[n-1]+&/2#/n82]=ax[n]+bx[n-1]+ax[n-2]

y[n]=h[n]*x[n]

ω ω ω ω ω 22]2[]1[]0[)( j j j j j aebeaehehhe H −−−− ++=++=ω ω

ω ω

ω ω j j

j j

j j beeee

abeea −−−

−− +

+=++=

22)1( 2

ω

ω j

eba

−

+= )cos2(bae H j += ω

ω cos2)( !"#= -"

$esign a %& digital filter that 'asses

the 0.rad9sec( and sto's the 0)1

radse fre,eny)

0)1.0cos(2)( 1.0 =+= bae H j 1)4.0cos(2)( 4.0 =+= bae H j

!ol)ing for t&e t%o euations gi)es a5 86.761:3,

'51;.36;;3

E(a#!le)+



Jan 6, 2016 <

(/n5 86.761:3 = #/n>#/n82 ?>1;.36;;3 #/n81

If #/n5@cos=0.1n?>cos=0.n?Au=n?

x1+x2

x1

.t't of

the filter /ransient

.t't is

alost

e,al to x2

(

the high

fre,eny



,lassification of filters

Analog and digital



Jan 6, 2016 10

,lassification of filters asanalog or digital

Analog filters Digital filters"n analog filter processesanalog inputs and generates

analog outputs)

"nalog filters are constructedfro$ passi)e or acti)eelectronic co$ponents suc& asresistors, capacitors andopa$ps to produce t&e reuired

filtering effect)

"n "nalog filter is descri'ed '(a differential euation)

" digital filter processesand generates digital data)

" digital filter consists ofele$ents liBe adder,$ultiplier and dela(ele$ent

igital filter is descri'ed '(difference euation)



Jan 6, 2016 11

,lassification of filters asanalog or digital

Analog filters Digital filters4&e freuenc( response of ananalog filter can 'e $odified '(c&anging t&e co$ponents.

!uc& filter circuits are %idel(used in suc& applications asnoise reduction, )ideo signalen&ance$ent, grap&iceuali+ers in &i8fi s(ste$s, and

$an( ot&er areas)

4&e freuenc( response ofdigital filter can 'e c&anged '(c&anging t&e filter coefficients

" digital filter uses a digitalprocessor to perfor$ nu$ericalcalculations on sa$pled )aluesof t&e signal)

4&e processor $a( 'e ageneral8purpose co$putersuc& as a PC, or a speciali+ed!P =igital !ignal Processor?c&ip)



Jan 6, 2016 12



,lassification of filters

Based on Frequencres!onse



Jan 6, 2016 1

• " analog filter is a net%orB used to s&ape t&e freuenc(spectru$ of an electrical signal.

• 4&ese net%orBs are essential parts of co$$unication andcontrol s(ste$s.

• Filters are classified as lo% pass, &ig& pass, 'and pass and'and reect, a$plitude euali+ers and dela( euali+ers.

"

"

%!e "#

0

LPF

Classification of filters According to

frequency response

0 "1

"

BPF%!e

"

#

"2"1

"

0

BSF%!e "#

"2

"

"

HPF%!e "#

0



ractical analog filters!ecifications



Jan 6, 2016 16

Practical Analog Filter specifications

3andsto'filter

3and'assfilter

4ow'ass

filter %igh'ass

filter

Ωp ΩpΩs

Ω2

Ωs

ΩUΩLΩ1 ΩU

Ω2Ω1ΩL

0 0

0 0

Ω

Ω

Ω

Ω



Jan 6, 2016 17

ractical analog Lo. !ass filters!ecifications



Jan 6, 2016 1<

Practical analog low pass filterspecifications

fre,eny

5 a g n i t - d e ! d

3 #

A min=αs

Ω p

Ωs

&assband/ransition

band6to'band

0

0

A max

=α p

0

Ω

ti l l L filt



Jan 6, 2016 1:

• 4&e 'asic function a of L* P"!! filter is to pass L*freuencies %it& )er( little loss and to attenuate &ig&

freuencies.• It is reuired to pass signals fro$ C up to pass 'and

edge freuenc( Dp %it& at $ost "$a#=E!?d ofattenuation.

• 4&e freuencies a'o)e stop 'and edge freuenc( Ds

are reuired to &a)e atleast "$in=Es?d of attenuation.

• 4&e 'and of freuencies fro$ 0 to Dp is called t&e pass'and.

• 4&e 'and of freuencies fro$ Ds to infinit( is called t&estop 'and.

• 4&e freuenc( 'and fro$ Dp to Ds is referred to astransition 'and.

ractical analog Lo. !ass filters!ecifications



Jan 6, 2016 20

ractical analog "ig" !ass filter s!ecifications

• 4&e 'asic function a of HIH P"!! filter is to passHIH freuencies %it& )er( little loss and to

attenuate lo% freuencies.• It is reuired to pass signals fro$ pass 'and edge

freuenc( Dp up to infinit( %it& at $ost "$a#=Ep? dof attenuation.

• 4&e freuencies 'elo% stop 'and edge freuenc( Ds are reuired to &a)e atleast "$in=Es?d ofattenuation.

• 4&e 'and of freuencies fro$ Dp to infinit( is calledt&e pass 'and.

• 4&e 'and of freuencies fro$ +ero to Ds is called t&estop 'and.

• 4&e freuenc( 'and fro$ Ds to Dp is referred to astransition 'and.



Jan 6, 2016 21

Practical analog band pass filterspecifications

ΩLΩ1 Ω2ΩU

Ω

A max=α p

A min

=αs A

min=α

s

0

5

a g n i t - d e ! d 3 #

&assband

6to'band 6to'band



Jan 6, 2016 22

ractical analog Band !ass filter s!ecifications

• 4&e 'asic function a of "G P"!! filter is to pass ILfreuencies %it& )er( little loss and to attenuate lo% and &ig&

freuencies.• It is reuired to pass signals fro$ lo%er pass 'and edge

freuenc( DL to upper pass 'and edge freuenc( Du %it& at$ost "$a#=Ep? d of attenuation.

• 4&e freuencies 'elo% lo%er stop 'and edge freuenc( D1 anda'o)e upper stop 'and edge freuenc( D2 are reuired to&a)e atleast "$in=Es?d of attenuation.

• 4&e freuenc( 'and fro$ DL to D is called t&e pass 'and.

• 4&e 'and of freuencies fro$ 0 to D1 and D2 to infinit( are

called t&e stop 'ands.• 4&e 'and of freuencies fro$ D1 to DL and D to D2 are

referred to as transition 'ands.



Jan 6, 2016 2;

ractical analog Band sto! filter s!ecifications

• 4&e 'asic function a of "G !4*P filter is to attenuateIL freuencies and to pass lo% and &ig& freuencies %it&

)er( little loss .• It is reuired to attenuate signals fro$ lo%er stop 'and edge

freuenc( D1 to upper stop 'and edge freuenc( D2 %it& atleast "$in=Es?d of attenuation.

• 4&e freuencies 'elo% lo%er pass 'and edge freuenc( DL anda'o)e upper pass 'and edge freuenc( D are reuired to&a)e at $ost "$a#=Ep? d of attenuation.

• 4&e freuenc( 'and fro$ D1 to D2 is called t&e stop 'and.

• 4&e 'and of freuencies fro$ 0 to DL and D to infinit( are

called t&e pass 'ands.• 4&e 'and of freuencies fro$ DL to D1 and D2 to D are

referred to as transition 'ands.



Jan 6, 2016 2

Design of digital filters fro# analog filters

• 4&e $ost co$$on tec&niues used for designing

II- digital filters Bno%n as indirect $et&od, in)ol)esfirst designing an analog protot(pe filter and t&entransfor$ing t&e protot(pe to a digital filter.

• For t&e gi)en specifications of a digital filter, t&e

deri)ation of t&e digital filter transfer functionreuires t&ree steps1. ap t&e desired digital filter transfer function into

eui)alent analog filter.

2. eri)e t&e analog transfer function for t&e analog

protot(pe.;. 4ransfor$ t&e transfer function fro$ t&e analog protot(pe

into an eui)alent digital transfer function.



Jan 6, 2016 23

Ad/antages of digital filters

1. nliBe analog filters, t&e digital filters perfor$ance

is not influenced '( co$ponent aging, te$peratureand po%er suppl( )ariations.

2. " digital filter is &ig&l( i$$une to noise andposses considera'le para$eter sta'ilit(.

;. igital filters afford a %ide )ariet( of s&apes for t&ea$plitude and p&ase responses.

. 4&ere are no pro'le$s of input or outputi$pedance $atc&ing %it& digital filters.

3. igital filters can 'e operated o)er a %ide range offreuencies.



Jan 6, 2016 26

Ad/antages of digital filters

6. 4&e coefficients of digital filter can 'e progra$$ed

and altered an( ti$e to o'tain t&e desiredc&aracteristics.

7. ultiple filtering is possi'le onl( in digital filters.

Disad/antage of digital filters

1. 4&e uanti+ation error arises due to finite %ord

lengt& in t&e representation of signals andpara$eters.



Analog filtera!!ro(i#ations



Jan 6, 2016 2<

Analog filter a!!ro(i#ations 4&e rational function lo% pass appro#i$ations %&ic& %e descri'e

in t&is &a)e t&e general for$.

)1(1

1

)(1

1)()( 2

)(

)(2

2

22

→+=Ω+==Ω=ΩΩ j D

j N IN

OUT

j K V

V j H S H agnitude function

&ere H=!? is t&e desired $agnitude function and =!? is t&erational function in !.

•4&e function =!? is c&osen suc& t&at

•its $agnitude is s$all in pass 'and to $aBe t&e $agnitude of H=!? close to GI4K.•Its $agnitude is large in t&e stop 'and to $aBe t&e

$agnitude of H=!? close to -*.In particular =!? $a( 'e c&osen to 'e a pol(no$ial of t&e

for$ )2(...)()( 2210 →++++== N

N N S aS aS aaS P S K

&ere t&e coefficients of t&e nt& order pol(no$ial Pn=!? are

c&osen so t&at t&e corresponding loss function satisfies t&egi)en filter reuire$ents.



Jan 6, 2016 2:

Lo. !ass filter a!!ro(i#ation In particular =!? $a( 'e c&osen to 'e a pol(no$ial of t&e for$

IN OUT V

V V V j K j K asei

OUT

IN =⇒=Ω+=⇒→Ω→Ω 1)(10)(,0. 22

"s e#pected in t&e pass 'and of Lo% pass filter =near to C? noloss of signal t&e signal. ut practicall( t&ere %ill 'e so$e loss.

In t&e pass 'and

In t&e stop 'and

0)(1)(,.2

2

=⇒∞=Ω+=⇒∞→Ω∞→Ω OUT V

V V j K j K asei

OUT

IN

"s e#pected in t&e stop 'and of Lo% pass filter =&ig& freuencies?no pass of signal t&e signal. ut practicall( t&ere %ill 'e so$ePass of signal



Jan 6, 2016 ;0

Lo. !ass filter a!!ro(i#ation 4&ere are four t(pes of pol(no$ials %&ic& satisf( t&ese conditions.

Butter.ort" filter a!!ro(i#ation and

,"e$s"e/ filter a!!ro(i#ation

4&e( are1. utter%ort& Pol(no$ial =a#i$all( flat appro#i$ation?1. C&e'(s&e) pol(no$ial2. lliptic pol(no$ials

;. essel Pol(no$ials

e are going to stud(



Jan 6, 2016 ;1

IIR Filter ypes

3tterworth 7hebyshe8 /y'e I

9lli'ti 3essel

7hebyshe8 /y'e II



Jan 6, 2016 ;2

Ω

( ) ( ) ( ) )4()()( →=== Ω

Ω

Ω

Ω N N N

S

S

N C P P or S P S K ε ε

C Ω

&ere

4&e corresponding $agnitude function is

( ))5(,...3,2,1

1

1)(

2→=

+==Ω

ΩΩ

N for V

V j H

N IN

OUT

C

N is the order of the filter

is the operating frequenc and

is the cutoff frequenc

( ) ( ) P C P C

N N N

ΩΩ

ΩΩ

ΩΩ

ΩΩ =⇒= !1

ε ε )"(!1 →Ω=Ω C

N

P ε or

P Ω is the pass #and edge frequenc

Butter.ort" filter A!!ro(i#ation01a(i#all flat a!!ro(i#ation2

ε is a constant



Jan 6, 2016 ;;

( ) 122 <<Ω

Ω N

P

ε

!o

"t C $eans near D 50

$his e%pression sho&s that the first 2N'1 deriaties are ero at*+0.


( )[ ] ( ) ( ) ( ) ...11""

1"144

122

2122 2

1

++−+=+ ΩΩ

ΩΩ

ΩΩ

ΩΩ N N N N

P P P P ε ε ε ε

-ince (-) &as chosen to #e an nth order polno/ial this

is the /a%i/u/ nu/#er of deriaties that can #e /ade ero.

$hus the slope is as flat as possi#le at .

or this reason the #utter &orth appro%i/ation is also no&n as

the a%i/all flat ppro%i/ation.

Butter.ort" filter A!!ro(i#ation



Jan 6, 2016 ;


( ))6(

1

1)(

22

2

2 →+

==ΩΩΩ N

IN

OUT

P

V

V j H

ε

)(. /a% P P Aisoss!hea! ei α Ω=Ω

( ) N P ΩΩε 10log20

as

4&e loss in d is gi)en fro$ eu i.e

t pass #and edge frequenc

( )[ ] )(1log10)(22

10 →+=Ω ΩΩ dB A

N

P ε

t high frequencies the loss as/ptoticall approaches

#ecause

( )[ ] ( )[ ] ( ) N N N

P P P ΩΩ

ΩΩ

ΩΩ ==+ ε ε ε 10

210

2210 log20log101log10

$hese loss is seen to increase &ith the order N. t high frequencies

the slope is "N d7!8ctae. $herefore the stop #and loss increases

&ith the order N.

[ ]⇒+=Ω= 2

10

1log10)( ε α P P

A )9(1101.0 →−= P α

ε

:t is the para/eter related to pass #and

Butter.ort" filter A!!ro(i#ation



Jan 6, 2016 ;3


( ) ( ) ( )110

110

110

110110

1.0

1.0

1.0

1.02221.0

−−

=⇒−−

=⇒=− ΩΩ

ΩΩ

ΩΩ

P

S

P

S

P

S

P

S

P

S S N N N

α

α

α

α

α ε

)(. /in S S Aisoss!hea! ei α Ω=Ω

( )[ ] N

S S P

S A 2210 1log10)( ΩΩ+=Ω= ε α

t stop #and edge frequenc

-ince this e%pression nor/all does not result in an integer alue

&e therefore round off N to the ne%t higher integer to satisf the/ini/u/ required specifications.

( ) ( ) ( ) P

S

P

S

P

S

P

S

P

S

N N ΩΩ

ΩΩΩ

Ω

=−−

=⇒−−

=10

10

10

1.0

1.0

10

1.0

1.0

1010log

log

log

110

110log

110

110

loglog ε

λ

α

α

α

α

( ) )11(

log

log

10

10

→

≥ΩΩ P

S

N ε

λ

)10(110 1.0 →−= S α

λ ;here is a <ara/eter related to stop #and



Jan 6, 2016 ;6

)12(1

1)(

2

2 →Ω+

=Ω N

j H

Sta$le nor#ali3ed Butter.ort"filter design

$he /agnitude function of the 7utter&orth lo& pass filter is gien #

$he /agnitude squared function of a nor/alied 7utter&orth lo& passfilter &ith *c+1 rad!sec is gien #

( )[ ])11(

1

1)(

21

2→

+=Ω

ΩΩ N

C

j H

$he function is /onotonicall decreasing. $he /a%i/u/ response isero at *+0. $he response approaches ideal characteristics as the

order N increases.



Jan 6, 2016 ;7

( ))13(

1

1)()()(

2

2 →+

=Ω−Ω=Ω N

jS

j H j H j H


t *+*c the cure passes through 0.606

&hich corresponds to

3d7 point.

No& let us derie the transfer function of a sta#le filter.

or this purpose su#stitute *+-!= in equ.12 then &e hae

)14()1(1

1)()(

2 →

−+=−

N N S S H S H



Jan 6, 2016 ;<

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

N=1

N=2

N=200

N=100




Jan 6, 2016 ;:

)1"(1 22 →=⇒== N " j

"

" j N eS eS π

π

)15(0)(1 2 →=−+ N S

)16(1)12(

)12(2 →=⇒=−= −

− N

" j

"

" j N eS eS π

π


$he a#oe relation tells us that this function has poles in the >?< as &ell as inthe @?< #ecause of the presence of t&o factors ?(-) and ?('-).

$hese roots &e can get # equating the deno/inator to ero i.e

$he solution of the a#oe equation is

:f ?(-) has poles in the >?< then ?('-) has the corresponding poles in the

@?<.

or N odd it reduces to

or N een it reduces to

[ ])1(2,...,2,1

122 →== −+

N " for eS N N " j

"

π



Jan 6, 2016 0

)19()(

1)( →

−=∏ j

jS S S H


$he - do/ain /agnitude function is therefore gien #

$hese 2N roots are located on the unit circle and are equall spaced atA!N radians interals

;here - = are the left half plane poles.r

Re

ImB'<lane

<oles of 7utter&orth filter are located

on the circle in the -'plane and areequall spaced at A!N radians interals

Sta$le nor#ali3ed Butter.ort" filter design



Jan 6, 2016 1

[ ] ( ) ( ) 1sincos33

2

−=+=== π π π

π

jeeS j j

[ ][ ][ ])"".05.0()1()"".05.0(

1

))()((

1)(

321 jS S jS S S S S S S S H

−−−−−+−−=

−−−=

[ ] ( ) ( ) "".05.0sincos32

32

13

2

j jeS j +−=+== π π

π



-olutionC N+3 for third order and +1 to 2N + 1 to ".

r

Re

ImB'<lane

E(a#!le)4 Find t&e utter%ort& appro#i$ation

function for t&e ;rd order Gor$ali+ed lo% Pass Filter

[ ] ( ) ( ) "".05.0sincos3

43

43

34

j jeS j −−=+== π π

π

[ ] [ ] [ ]π π π

31

322

212

2++−+

=== " "

N N " j j j

" eeeS

[ ][ ] )1)(65.025.0(

1

1)"".0()5.0(

1)(

222 ++++=

+−+=

S S S S jS S H

)1)(1(

1)(

2 +++=

S S S S H




Jan 6, 2016 2

[ ] ( ) ( ) 326.09239.0sincos6

6

26

j jeS j

+−=+== π π π

))()()((

1)(

4321 S S S S S S S S S H

−−−−=

[ ] ( ) ( ) 9239.0326.0sincos

5

51

5

j jeS j +−=+== π π

π



-olutionC N+4 for fourth order and +1 to 2N + 1 to .

B'<lane

E(a#!le)5 Find t&e utter%ort& appro#i$ation function for t&e t& order Gor$ali+ed lo% Pass Filter

[ ]

( ) ( ) 326.09239.0sincos 9

9

3

9

j jeS j

−−=+== π π

π

[ ] [ ]4

322

122

+−+

== "

N N " j j

" eeS π π

)1466".1)(16"53".0(

1)(

22 ++++=

S S S S S H

[ ] ( ) ( ) 9239.0326.0sincos

11

114

11

j jeS j −−=+== π π

π

[ ][ ]2222 )326.0()9239.0()9239.0()326.0(

1)(

jS jS S H

−+−+=

[ ][ ][ ][ ])9239.0326.0()326.09239.0()326.09239.0()9239.0326.0(

1)(

jS jS jS jS S H

−−−−−−+−−+−−=

r

Re

Im

Li t f tt t& l i l



Jan 6, 2016 ;

List of utter%ort& pol(no$ials

G eno$inator of H=!?

1 =!>1?

2 =!2>M2!>1?

; =!>1? =!2>!>1?

=!2>0.763;7!>1? =!2>1.<776!>1?

3 =!>1? =!2>0.61<0;!>1? =!2>1.61<0;!>1?

6 =!2>1.:;1<33!>1?=!2>M2!>1? =!2>0.3176!>1?

7 =!>1? =!2>1.<01:!>1?=!2>1.27!>1? =!2>0.3!>1?

,"e$ "e filte A o i tio



Jan 6, 2016

oreo)er t&e attenuation pro)ided in t&e stop 'and is less t&ant&at attaina'le using so$e ot&er pol(no$ial t(pes, suc& asC&e'(s&e) pol(no$ial.

4&e $ain feature of t&e utter%ort& appro#i$ation is t&at t&e loss

is $a#i$all( flat at t&e origin.

4&ere are t%o t(pes of C&e'(s&e) filters .i.e. 4(pe8I and 4(pe8II

,"e$s"e/ filter A!!ro(i#ation

4(pe8I are all8pole filters t&at e#&i'its euiripple 'e&a)ior in t&e

Pass 'and and a $onotonic c&aracteristics in t&e stop 'and.

4&us t&e appro#i$ation to a flat pass 'and is )er( good at t&eorigin 'ut it gets progressi)el( poorer as freuenc( approac&espass 'and edge.

4(pe8II contains 'ot& poles and +eros and e#&i'its a $onotonic'e&a)ior in t&e pass 'and and an euiripple 'e&a)ior in t&estop 'and.




Jan 6, 2016 3


4KP8I

," $ " filt A i ti



Jan 6, 20166


[ ] [ ]

)(2)coscos(2

)cos(cos)coscos(2

cos)1(coscos)1(cos)()(

1

11

1111

ΩΩ=ΩΩ=

ΩΩ=Ω−+Ω+=Ω+Ω

−

−−

−−−+

N

N N

C N

N

N N C C

)2(1)coshcosh(

1)coscos()(

1

1

→

>ΩΩ≤ΩΩ

=Ω −

−

s!o#band N

#assband N C

N

&ere

4&e $agnitude suare response of Gt& order 4(pe8I filter can 'ee#pressed as

( ) )1(,...3,2,1

1

1)(

22

22

→=+

==ΩΩΩ

N for C V

V j H

P N IN

OUT

ε

is t&e Gt& order C&e'(s&e) pol(no$ial defined as

It can 'e e#pressed '( recursi)e for$ula fro$

ε is a filter para$eter related to t&e ripple in t&e pass 'and.

( )Ω N C

)3()()(2)(11

→Ω−ΩΩ=Ω −+ N N N C C C as

and




Jan 6, 20167


)()(2)(11

Ω−ΩΩ=Ω −+ N N N C C C

1)(0 =ΩC and

e Bno% t&at

Ω=Ω)(1

C

4&en fro$

12)()(2)( 2

012 −Ω=Ω−ΩΩ=Ω C C C

( ) Ω−Ω=Ω−−ΩΩ=Ω−ΩΩ=Ω 34122)()(2)(32

123 C C C

( ) ( ) 112342)()(2)( 2423

234

+Ω−Ω=−Ω−Ω−ΩΩ=Ω−ΩΩ=Ω C C C

( ) ( ) Ω+Ω−Ω=Ω−Ω−+Ω−ΩΩ=Ω−ΩΩ=Ω 5201"3412)()(2)(45324

345 C C C




Jan 6, 2016<





Jan 6, 2016:


Ω−Ω−−

=Ω e$en N for C

odd N for C

C N

N

N )(

)(

)(

oscillates %it& eual ripple 'et%een

C&e'(s&e) Pol(no$ial &as t&e follo%ing properties

for

1.

2.−

=e$en N for

odd N for C

N N 2)1(

0)0(

;. N a for C N 1)1( =

.−

=−e$en N for

odd N for C

N 1

1)1(

3. )(Ω N

C

1±

.1≤Ω

6.)(Ω

N C is $onotonicall( increasing forFor all G 101)(0 ≤Ω≤≤Ω≤ for C

N and 11)( >Ω>Ω for C N

7. .1 N a for >Ω

<. )er( coefficient is an integer and t&e one associated %it& N Ωis .2 1− N




Jan 6, 201630

7hebyshe8 /y'e-I





Jan 6, 201631


)(. /in S S Aisoss!hea! ei α Ω=Ω

)(. /a% P P Aisoss!hea! ei α Ω=Ω

)5(1101.0 →−= P α

ε

as

4&e loss in d is gi)en fro$ eu.1 i.e

"t pass 'and edge freuenc(

( )[ ] )4(1log10)( 22

10 →+=Ω ΩΩ dBC A P N ε

( )[ ] [ ]⇒+=+=Ω= 2

10

22

10 1log1011log10)( ε ε α

N P P C A

( ) P N IN

OUT

C V

V j H

ΩΩ+

==Ω22

2

2

1

1)(

ε

"t stop 'and edge freuenc(

( )[ ] ( )[ ])(coshcosh1log101log10)( 122

10

22

10 P

S

P

S N C A N S S Ω

Ω−ΩΩ +=+=Ω= ε ε α

( )110

110cosh)(cosh)(coshcosh

1101.0

1.0

1112

2

1.0

−

−=⇒=

− −ΩΩ−

ΩΩ−

P

S

P

S

P

S

S

N N α

α α

ε

( ) )"(

cosh

cosh

)(cosh

110

110cosh

1

1

1

1.0

1.01

→

=−−

≥ΩΩ−

−

ΩΩ−

−

P

S

P

S

P

S

N ε

λ

α

α

;here )6(1101.0 →−= S α

λ

,"e$s"e/ A!!ro(i#ation !ro/ides 60N-*2dB #ore



Jan 6, 201632

attenuation t"an Butter.ort" for t"e sa#e order

( )[ ] ( )[ ] ( ) 1

1010

1

10102log20log202log20log20)( −

ΩΩ

ΩΩ−

ΩΩ +===Ω N N N N

N P

S

P

S

P

S C A ε ε ε

P Ω>>Ω

( ) ( ) III C N N

N P P

S

P

S →≅Ω>>Ω ΩΩ−

ΩΩ 12

4&e attenuation in !4*P "G of utter%ort& filter forin d is gi)en '(

( )[ ] ( )[ ] ( )[ ] II C C C A P

S

P

S

P

S

N N N →=≈+=Ω Ω

ΩΩΩ

ΩΩ

ε ε ε 10

22

10

22

10log20log101log10)(

ut for

dB N N

)1("2log20 1

10 −=−

!o a'o)e euation8II reduces to

P Ω>>Ω

( )[ ] ( )[ ] ( )[ ] I A N N N

P

S

P

S

P

S

→=≈+=Ω ΩΩΩΩΩΩ ε ε ε 10

22

10

22

10 log20log101log10)(

4&e attenuation in !4*P "G of C&e'(s&e) filter forin d is gi)en '(

Co$paring a'o)e eu.I and IN it is seen t&at t&e C&e'(s&e)appro#i$ation pro)ides $ore attenuationt&an a utter%ort& of t&e sa$e order.

( ) ( ) IV N N A N N

P

S

P

S →−+=−+=Ω ΩΩ

ΩΩ )1("log202log)1(20log20)(

101010 ε ε

Sta$le nor#ali3ed ,"e$s"e/ filter



Jan 6, 20163;

)(1

1)()()(

22

2

Ω+=Ω−Ω=Ω

N C j H j H j H

ε

1=Ω P

( )[ ] ( ))(sinhsinhsin 111122 ε

π σ −−±= N N

"

"

Sta$le nor#ali3ed ,"e$s"e/ filterdesign

$o find the poles of the he#she appro%i/ation transfer function

$ae the deno/inator of the equ(1) su#stitute to get

nor/alied function and equate it to ero. .i.e.

:t can #e proed that the roots of a#oe equation are

;here

ε ε

j

N N C C ±=Ω⇒=Ω+ )(0)(1 22

( )[ ] ( ))(sinhcoshcos 111122 ε π −−=Ω N N

" "

N " for jS " " " 2,...,2,1=Ω±=σ

further

( ) ( ) 1

)(sinhcosh)(sinhsinh

2

111

2

111=

Ω+

−−

ε ε

σ

N

"

N

"

Sta$le nor#ali3ed ,"e$s"e/ filter



Jan 6, 20163

+=

= e$en N for

odd N for S H

S

2

0

1

11

)(

ε

∏ −=

j j

S S

H S H

)()( 0

design


$hese 2N roots are located on the ellipse in the s'plane spaced atA!N radians interals

;here - = are the left half plane poles and

?0 is the order dependent constantr

R

e

Im

:t can #e found fro/

( ))(sinhsinh 111ε

− N

B'<lane( ))(sinhcosh 111

ε

− N

Sta$le nor#ali3ed ,"e$s"e/ filter design



Jan 6, 201633

-olutionC Dien *p+200 rad!s, *s+"00 rad!s, Ep+0.5d7, Es+20d7.

293.2)3(cosh

)44.2(cosh

)(cosh

110

110cosh

)(cosh

110

110cosh

1

1

200"001

05.0

2

1

1

1.0

1.0

1

==−−

=−−

≥−

−

−

−

ΩΩ−

−

P

S

P

S

N α

α

-o the required order is N+3, for third order +1 to 2N + 1 to ".

( )[ ] ( ))(sinhsinhsin 111122 ε

π σ −−±= N N "

" ;here

N " for jS " " " 2,...,2,1=Ω±=σ

( )[ ] ( ))(sinhcoshcos 111122 ε

π −−=Ω N N "

"

E(a#!le)7 Find t&e C&e'(s&e) appro#i$ation function order

for t&e filter reuire$ents

Dp5200 rad9s, Ds5600 rad9s, Ep50.3d, Es520d.




Jan 6, 201636

95.99910011.0 2

1

12

==⇒=+⇒=+

λ λ λ

2=Ω P

121 221

11

2 =⇒=+⇒=+ ε ε ε

8rder to #e selected is N+3 and +1 to 2N + 1 to ".

-olutionC :n general specifications are gien as

( ) ( ) 2"9.2)(cosh

cosh)(cosh

cosh)(cosh

110

110cosh

241

1

95.91

1

1

1

1.0

1.0

1

===−

−

≥ −

−

ΩΩ−

−

ΩΩ−

−

P

S

P

S

P

S

N ε

λ α

α

E(a#!le)6 o$tain an analog ,"e$s"e/ filter transfer function

t"at satisfies t"e constraints

201)(2

1 ≤Ω≤≤Ω≤ for j H 41.0)( ≥Ω≤Ω for j H

# for j H Ω≤Ω≤≤Ω≤

+ 01)(

21

1

ε s for j H Ω≥Ω≤Ω

+ 21

1)(λ

4=ΩS




Jan 6, 201637

-olution cont.dC

E(a#!le)6 o$tain an analog ,"e$s"e/ filter transfer function

t"at satisfies t"e constraints201)(

2

1 ≤Ω≤≤Ω≤ for j H 41.0)( ≥Ω≤Ω for j H

( )[ ] ( ))(sinhsinhsin 111122 ε

π σ −−±= N N

"

" ;here

N " for jS " " " 2,...,2,1=Ω±=σ

( )[ ] ( ))(sinhcoshcos 111122 ε

π −−=Ω N N "

"

29369.0)(sinh)(sinh111

31111 == −−

ε N

( ) ( ) 29.029369.0sinh)(sinhsinh 111 === −ε N

A

( ) ( ) 043.129369.0cosh)(sinhcosh 111 === −ε N B




Jan 6, 2016

3<

( )[ ] 149.0)29.0(sin31

21 ±=±= π σ ( )[ ] 903.0)043.1(cos 31

21 ==Ω π

( )[ ] B"

" 3

12

2cos

−

=Ω π

( )[ ] 0)043.1(cos 33

22 ==Ω π

E(a#!le)6 cont)d

( )[ ] A"

" 3

12

2

sin −±= π σ

( )[ ] 29.0)29.0(sin33

22 ±=±= π σ

( )[ ] 903.0)043.1(cos 35

23 −==Ω π ( )[ ] 149.0)29.0(sin 3

523 ±=±= π σ

903.0149.0111 j jS +±=Ω+=σ

29.0222

±=Ω+= jS σ

903.0149.0333 j jS −±=Ω+=σ

>eft half plane <oles are gien #




Jan 6, 2016

3:

]3.0)(29.0))[(29.0(

25.0

)3.029.0)(29.0(

25.0

)()(2

2

222

2 +++=+++== =Ω

= s s s s

S

sS S S S S H s H C

[ ][ ] )3.029.0)(29.0(29.0)903.0()149.0()(

2

0

22

0

+++=

+−+=

S S S

H

S jS

H S H

Odd N for S H S

1)(0=

=Fsing

E(a#!le)6 cont)d

)354.359".0)(59".0(

2)(

2 +++=

s s s s H

Nor/alied $ransfer function is

[ ] [ ] [ ])903.0149.0()29.0()903.0149.0())()(()( 0

321

0

jS S jS H

S S S S S S H S H

−−−−−+−−=

−−−=

)3.029.0)(29.0(

25.0)(25.0)3.0)(29.0(

20 +++=⇒==

S S S S H H

enor/alied $ransfer function &ith is21!2! !1 ==Ω=Ω N

P C ε

Design of Analog Butter.ort" L8& ASS



Jan 6, 2016

60

filter

1. Fro$ t&e gi)en specifications find t&e order of

t&e filter G.

2. -ound off *rder G to t&e ne#t &ig&er integer.

;. Find t&e Gor$ali+ed 4ransfer function H=!?.

. Calculate t&e )alue of cut off freuenc( Dc.

3. Find t&e e8nor$ali+ed transfer function H=s? '(replacing ! %it& s9Dc.

Design of Analog ,"e$s"e/ L8&ASS filter



Jan 6, 2016

61

Design of Analog ,"e$s"e/ L8&ASS filter

1. Fro$ t&e gi)en specifications find t&e order oft&e filter G.

2. -ound off *rder G to t&e ne#t &ig&er integer.

;. Find t&e deno$inator of t&e Gor$ali+ed 4ransferfunction H=!?.

. Calculate t&e )alue of cut off freuenc( Dc and

find nu$erator constant H0 depending on t&e )alue

of G.

3. Find t&e e8nor$ali+ed transfer function H=s? '(replacing ! %it& s9Dc.

+=

= e$en N for

odd N for

S H S

20

1

1

1

)(

ε

Frequenc transfor#ations of analog filters



Jan 6, 2016

62

•4&e appro#i$ations descri'ed so far %ere directl( applica'le to lo%8pass filters.•4&ese appro#i$ations can 'e adapted to &ig& pass,

s($$etrical 'and pass and s($$etrical 'and reect filters fro$ anor$ali+ed lo% pass filter=Dc51 rad9sec?

4ranslate t&e gi)en reuire$ents to OIN"LG4 lo% passreuire$ents.

?<, 7< or 7@

require/ents

Gquialent

><

@equire/ents

$>< (-)

$ ?< (s)

$7<(s)

$7@ (s)

"ppro#i$ate t&e resulting lo% pass reuire$ent using t&especified appro#i$ation $et&od

Finall( translate t&e lo% pass appro#i$ation function to t&edesired HP, P or - appro#i$ation function

4aBe t&e gi)en filter reuire$ents

Frequenc transfor#ations of analog filters



Jan 6, 2016

6;

Practical Analog Filter specifications

Band stop

filter

Band pass

filter

Low pass

filter Hig! pass

filter

Ωp ΩpΩs

Ω2

Ωs

ΩUΩLΩ1 ΩU

Ω2Ω1ΩL

0 0

0 0

Ω

Ω

Ω

Ω

q g

Design of 9:B:BR filters of Butter.ort" :



Jan 6, 2016

6

,"e$s"e/ t!e Analog filter1. Find t&e eui)alent lo% pass reuire$ents.

.i.e.

2. Find t&e nor$ali+ed Lo% pass filter order and

4ransfer function for t&e gi)en appro#i$ation 4(pe.;. Find t&e reuired e8nor$ali+ed 4ransfer function'( replacing ! in H=!? %it& 'elo% transfor$ations

and # s # 1=Ωα α

12

12

Ω−ΩΩ−ΩΩ−ΩΩ−Ω

ΩΩ

=Ω=Ω=Ω

%

%

s

#

r

r

r

B&' for

BP' for

HP' for

% s

% s

% s % s

sC

S

S

S

S H s H B&' or B&' for

S H s H BP' for

S H s H HP' for

ΩΩ+

Ω−Ω

Ω−ΩΩΩ+

Ω

=

=

=

=

=

=

2

)(

)(

2

)()(

)()(

)()(

!ta'le "nalog filter design8 #a$ple.7



Jan 6, 2016

63

For t&e gi)en specifications Ep5;d, Es513d, Dp51000 rad9sand Ds5300 rad9s design a 'utter %ort& appro#i$ated Hig& passfilter.

srad srad dBdBS

P

r # s #!2!1153 500

1000 ===Ω=Ω== ΩΩ

α α

srad dB P # !10003 =Ω=α

( )( )

( )4".2

2log

533.5log

log

log

10

10

10

10

==

≥ΩΩ #

r

N ε

λ

srad dBS s !50015 =Ω=α -olutionC

1110110 3.01.0 =−=−= P α ε 533.5110110 5.11.0 =−=−= S α

λ

ui)alent lo% pass reuire$ents are

533.5! =ε λ

Sta$le analog filter design- E(a#!le);% ,ont)d



Jan 6, 2016

66

[ ] ( ) ( ) 1sincos33

2 −=+=== π π π

π

jeeS j j

[ ][ ][ ])"".05.0()1()"".05.0(

1

))()((

1)(

321 jS S jS S S S S S S S H

−−−−−+−−=

−−−=

[ ] ( ) ( ) "".05.0sincos32

32

13

2

j jeS j +−=+== π π

π

g g !

Gor$ali+ed 4ransfer function of eui)alent Lo%8pass filter is

-olutionC select N+3 and +1 to 2N + 1 to ".

[ ] ( ) ( ) "".05.0sincos3

43

43

34

j jeS j−−=+== π π π

[ ] [ ] [ ]π π π 31

322

212

2 ++−+ === " "

N N " j j j

" eeeS

[ ][ ] )1)(65.025.0(

1

1)"".0()5.0(

1)(

222 ++++=

+−+=

S S S S jS S H

)1)(1(

1)(

2 +++=

S S S S H

Sta$le analog filter design- E(a#!le);% ,ont)d



10001!1000!!1!1 ==Ω=Ω⇒Ω=Ω N

P C C

N

P ε ε

g g !

e8nor$ali+ed 4ransfer function of reuired Hig&8pass filter isi)en '(

-olutionC

++

+

=

110001000

11000

1)(

2

s s s

s H

s

S s

S S S S S H s H c

10002

)1)(1(

1)()(

=

Ω= +++

==

( )( )"323

3

101010)(

+++=

s s s

s s H

e8nor$ali+ed 4ransfer function of reuired Hig&8pass filter is

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Dsp-unit 5.1 Analog Filters

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