UNIT-4 Final Ppt

7/25/2019 UNIT-4 Final Ppt

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Unit-4

IIR & FIR Digital Filters


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Filter Filteris a frequency selective network.

Filtersgenerally do not add frequency components to a signal

It oost or attenuate selected frequency regions!eneral types of filters are"

#ractical c$aracteristics


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%ypes of Filters

%$ere are two types of filters.

nalog Filters

Digital Filters.

nalog Filter uses passive elements suc$ as resistors' inductors an

d (apacitors. %$ey descri)ed )y t$e Differential equations.

Digital Filter *inear time invariant Discrete time system. Descri)ed

)y t$e difference equations.

+," IIR'FIR


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(oncept of nalog *#F

Here input signal is a 5v DC signal. Noise is a high frequency

component. So it is suppressed here. DC Component is a low

frequency component so it is passed here.


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(oncept of Digital *#F


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Implementation of Digital filters

(an )e implemented in software like c or assem)ly language.

Usually suc$ a languages are compiled and an e,ecuta)le code for

processors is prepared.

Digital filters are also implemented )y a dedicated $ardware contai

ns flip flops' counters' s$ift registers' *U.

ut digital filters wit$ dedicated $ardware can perform one type of

filtering only $ence not possi)le to modify t$em.


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(omparison )etween nalog & Digital Filters

Analog Filter Digital FilterAnalog lters wors on analog signals !t operates on the digital samples of

the signals!t is dened "y linear di#erential equations $hese inds of lters are dened

using linear di#erence equations%hile implementing the analog lters in

hardware or software simulation& electrical

components lie resistors& capacitors andinductors are used.

%hile implementing the digital lters

in hardware or software(for

simulation)& we need adders&su"tractors& delays& etc which are

classied under digital logic

components.$he frequency response is modied "y

changing the components.

$he frequency response is modied

"y changing the lter coe*cients.+aplace transform is used for the analysis of

analog lter.

, transforms are used for the anaysis

of digital lters-or sta"ility and causality& the poles should

lie on the left half of splane.

!n order to "e sta"le and causal& the

poles of the transfer function should

lie inside the unit circle in /plane.
http://amitbiswal.blogspot.com/2012/02/reason-why-transformers-are-rated-in.htmlhttp://amitbiswal.blogspot.com/2011/08/most-popular-open-source-softwares-list.htmlhttp://amitbiswal.blogspot.com/2011/08/most-popular-open-source-softwares-list.htmlhttp://amitbiswal.blogspot.com/2012/02/reason-why-transformers-are-rated-in.html


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dvantages of Digital Filters. Digital filter performance is not influenced )y component ageing' temperat

ure & power supply variations.

digital filter is $ig$ly immune to noise & possess considera)le parameter

sta)ility.

Digital filters afford a wide variety of s$apes for amplitude & p$ase respons

es.

/o pro)lems of i0p 'o0p impedance matc$ing.

1perated over a wide range of frequencies.

%$e coefficients of digital filters can )e c$anged at any time to o)tain desir

ed response.

2ultiple filtering is possi)le only in digital filters.


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Disadvantages of Digital Filters.

%$ere are few disadvantages also. 3uantiation error occurs due to finite word lengt$ in t$e represe

ntation of signals and parameters.

Digital filters also suffer from andwidt$ pro)lems.


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Infinite Impulse Response5IIR6 Digital Filter

In IIR Digital Filter' present o0p samples depends upon present i0p' past i0p also o

n past o0p.

IIR filter is a recursive filter.

/t$ order Difference equation is given )y

*et a7 8-7 and )9 8 a9 87 wit$ k87 wit$ remaining coefficients as eros' t$e a)ove eq

uation )ecomes

y5n68y5n-76:,5n6


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pply ;-%ransform we get

Its inverse ;-%ransform is $5n68u5n6

It indicates t$at t$e impulse response of IIR filter is $aving infi

nite duration.

%$e %.F of IIR filter is

Design of IIR filter for given specifications is to find t$e filter co

efficients


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Design of Digital filter from nalog Filter


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nalog *#F into Digital *#F


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& alog Filter Desig

%$e most general form of a alog tra sfer fu ctio is

. ($e)ys$ev filter &ppro,imatio


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utterwort$ filter

%$e utterwort$ filteris a type of signal processing filter designed

to $ave as flat aAfreque cy respo seas possi)le i t$eApass)a d.

It is also referred to as a ma,imally flat mag itude filter.

It was first descri)ed in 7B?9 )y t$e ritis$ engineer and p$ysicist =t

ep$e utterwort$in $is paper entitled C1 t$e %$eory of Filter

&mplifiersC


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& alog *ow pass5*#F6 utterwort$ filt

er

*owpass utterwort$ filters are all-pole filters c$aracteried )y

t$e magnitude response given )y

or

...........

576

N

p

N

c

jH2

2

2

2

1

1

1

1|)(|

+

=

+

=

N

c

jH2

1

1|)(|

+

=

passbandallowablespecifyingparameter

frequencyPassbandp

frequencycutofforfrequencydB

frequency

filtertheoforderN

c

=

===

==

3

,....3,2,1


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& alog *ow pass5*#F6 utterwo rt$ f i l ter

%$is mag itude respo se is mo oto ically decreasi g fu ctio

w$ere ma,imum respo se is unity at 89 as s$own in )elow


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%$e response )ecomes ideal as t$e order / is increases.

%$e 2agnitude response equation576 )ecomes

#ut in a)ove equation 5>6 we get

int3707.0;

)(;

1)(;

podBaiswhichthroughpassediscurvethe

rapidlydecreasesjH

jHfor

c

c

c

=

>

=


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( )( )

......(3)1

1

2

2

Ns

jH+

=

( ) )4.(..........01 2 =+ Ns

0quating denominator equal to /ero in equation(1) we get

roots

Nkes

areequationofrootsthenow

es

becomesequationthenoddisNfor

Nkj

k

kjN

2,...3,2,1;

)4(

1

)4(

2/2

22

==

==


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Nkes

areequationthisofrootsthenow

es

becomesequationthenevenisNfor

Nkj

k

kjN

2,...3,2,1;

1

)4(

2/)12(

)12(2

==

==

!f N21 we get the following roots


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3oles on left half of splane gives sta"ility. 3oles which are

left in splane are


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%$erefore /8? rd order utterwort$ *owpass filte %ransfer functio

n at

is given )y

$his is the denominator polynomial of $ransfer function H(s)

sec/1radc=


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%$e poles w$ic$ are present only in left $alf of s-plane can )e calc

ulated using.

%$e poles given )y a)ove equation5E6 are /ormalied poles )ecau

se t$ey are calculated at

=o unnormalied poles are given )y %$e transfer function of suc$ a u ormali4ed utterwort$ filter

can )e o)tained )y su)stituting

NkN

k

where

es

k

j

kk

,...3,2,1;2

)12(

2

)5.........(

=

+=

=

sec/1radc=

kck ss .| =

c

ss

=


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1rder of t$e utterwort$ filter

5/6

ss

pp

frequencystopbandatnattenuatiostopbandthebelet

frequencypassbandatnattenuatiopassbandthebelet

.max

.max


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1rder of t$e utterwort$ filter5/6Consider )1.........(

1

1|)(|

2

2

2

N

p

jH

+

=

+

=N

p

jH

getweeqnofsidesbothonarithmtake

2

2

2

1

1log10|)(|log10

)1(log

)1log(10)1log(10|)(|log20

2

2

N

p

jH

+=

)2)........(1log(10|)(|log20

2

2

N

p

jH

+=

.|)(|log20 nattenuatiocalledisjHhere


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1rder of t$e utterwort$ filter5/6

)3...(..........)110(

110

10)1(

1.0)1log(

)1log(10

)1log(10

)1log(10|)(|log20

)2(

2/11.0

1.02

1.02

2

2

2

2

2

2

=

==+

=+

=+

=

+

=

+=

=

p

p

p

p

p

p

N

p

p

p

N

p

p

pp

jH

becomeseqnwhen


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110

110110

110

10)1(

1.0)1log(

)1log(10

)1log(10

)1log(10|)(|log20

)2(

1.0

1.0

2

1.02

1.0

2

2

1.0

2

2

2

2

2

2

2

2

2

2

=

=

=

=

+

=

+

=

+

=

+

=

+=

=

p

ss

s

s

N

p

s

N

p

s

N

p

s

s

N

p

s

s

N

p

s

s

N

p

s

s

N

p

s

ss

jH

becomeseqnwhen

N


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=

=

=

=

p

s

p

s

p

s

N

p

s

N

p

s

p

s

p

s

p

s

p

s

p

s

N

aswrittenbecaneqnthereforeegerresultsnotdoesequationThis

filtertheofordertheisthisN

N

eqntosidesbothonarithmtake

log

110

110log

)5(int

)5...(....................

log

110

110log

110

110loglog

110110loglog

)4(log

)4......(..........110

110

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0


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110,110

log

log

log

110

110log

1.01.0

1.0

1.0

==

ps

p

s

whereN

or

NisfilterhButterworttheoforderso

p

s

p

s

( )

=

==

A

bygivenisandparameteraisAratiotransitioncallediskwhere

k

AN

aswrittenbealsocanrder

s

p,

1log

log


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( ) ( )1 1

0.1 0.12 210 110 1p s

p sc

N N

= =

3rove that

3roof4

( )

( ) )1..(..........

110

110

110

11

11

1

1

1

1|)(|

2/11.0

2/11.0

1.02

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

cNp

N

c

p

N

c

p

N

c

p

N

p

N

c

N

p

N

c

N

p

N

c

N

p

N

c

p

p

p

getwecomparingbyjHconsider

=

=

==

=

=

=

+=

+

+

=

+

=


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( )

( )

( )writecanweandfrom

!onsider

s

p

s

p

p

s

p

s

p

p

s

p

s

sc

NN

cs

N

ps

NN

c

N

ps

N

p

s

)2()1(

)2........(..........110

110

110110

110

110

110110110

110110

110

110

2/11.0

2/1

1.0

1.02/11.0

2/1

1.0

1.0

2/1

1.0

1.02/1

1.0

2/1

1.0

1.0

1.0

1.02

=

=

=

= =

=

( ) ( )1 1

0.1 0.12 210 110 1p s

p sc

N N

= =


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FIR Filter Design ased on 2:7 w$ose D%F%

appro,imates t$e desired D%F% In s

ome sense.one commonly used appro,imation criterion is to mini

mie t$e integral-squared error.

)(eh j

d

{ }][nht)(eH

j

t


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FIR Filter Design ased on


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FIR Filter Design ased on 6t$e rectangular window $as an a)rupt transit

ion to ero.

=otherwise

"nn

$,0

0,1][


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FIR Filter Design ased on 6providing a smoot$ transition from t$e pass)and to t$e stop)and.


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FIR Filter Design ased on 6get

5?6determine t$e cutoff frequency )y setting"

5462 is estimated using 't$e value of t$e constant c is o)tain fromta)le given.

[ ] ][][ nwnnh hd =

2/)( spc +=

"

c


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(omputer-ided Design of Digital Filter

%wo specific design approac$es )ased in iterative potimiation tec$niques.

%$e aim is to determine iteratively t$e coefficients of t$e digital transfer function so t$at t$e difference )etween and

for all value of over closed su

)intervals of is minimied 'andusually t$e difference is specified as a weig$ted error function given )y"

)(ej

H

)(ej

&

0

)([ ])()()()( eee

jjj&H'

=

( t id d D i f Di it


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($e)ys$ev criterion

--to minimie t$e peak a)solute value of t$eweig$ted error

*east-p criterion

--to minimie t$e integral of pt$ power of t$e weig$ted error function

)(

)(max $=

)(

{ } =

=(

i

p

e&e' ijij

1

)()(



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Design of +quiripple *inear-#$ase FIR Filter

%$e frequency response of a linear-p$ase FIR filter is"

%$e weig$ted error function in t$is case involves t$e amplitude response and is given)y

)()( 2/

= HHeee

jjNj

=

)()()()( &H'



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%ype 7 linear-p$ase FIR filter

%$e amplitude response is "

It can )e rewrite using t$e notation

in t$e form


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%ype > linear-p$ase FIR filter


It can )e rewrite in t$e form"


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%ype ? linear-p$ase FIR filter



)sin(22)(

2/

1 nn

N

hH

N

n =

=

)cos()(sin

)sin(][)(

1

0

0

kkc

kkcH

"

k

"

k

=

=

=

=



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%ype 4 linear-p$ase FIR filter



))2

1(sin(]

2

1[2)(

2/)1(

1

+

= +

=

nnNhH

N

n

)cos(][)2

sin(

)2

1(sin][)(

2/)12(

0

2/)12(

1

kkd

kkdH

"

k

"

k

=

+

=

=

=

(omp ter ided Design of Digit


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%$e amplitude response for all four typesof linear-p$ase FIR filters can )e e,pressed in t$e form

%$en t$e we modify t$e form of t$e weig$t appro,imation function as"

)()()( A)H =

[ ]

=

=

)()()()()(

)()()()()(

w)&A)'

&A)'

(omputer ided Design of Digit


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Using t$e notions and

we can rewrite it as"

%$en we determine t$e coefficientsto minimie t$e peak a)solute value of t$e weig$ted appro,imation error over t$e specified frequency )ands

)()()( )'' =

)(/)()( )&& =

=

)()()()( &A'

][ka

$

(omputer ided Design of Digit


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lternation %$eorem

%$e amplitude function is t$e )est unique apro,imation of t$e desired amplitud

e response o)tained )y minimiing t$e peak a)solute valu

of if and only if t$ere e,ist at least

e,tremal angular frequencies''in a closed su)set R of t$e frequency ran

ge

)(A

)( 2+# 110 ,, + #

0 110 +


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Digital Filter Design Using 2atla)

IIR Digital Filter Design Using 2atla)

=teps"576determine t$e filter order / and t$e frequency scaling factor


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5>6determine t$e coefficients of t$e transfer function.

J)'aK8)utter5/'5/'Rs'


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FIR Digital Filter Design Using 2atla)

=teps576.estimate t$e filter order from t$egiven specification.

reme4ord'kaiserord

5>6determine t$e coefficient of t$e transfer function using t$e estimated order and t$e filter specification.

reme

Digital Filter Design Using 2atla


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FIR Digital Filter 1rder +stimation Using

2atla)

J/'fpts'mag'wtK8remeord5fedge'mval'dev6

J/'fpts'mag'wtK8remeord5fedge'mval'dev'F%6

For FIR filter design using t$e aiser window't$e window order s$ould )e estimatedusing kaiserord

J/'


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+quiripple *inear-p$ase FIR Design Using2atla)

--emplying t$e #arks-2c(lellan algorit$m.

)8reme5/'fpts'mag6

)8reme5/'fpts'mag'wt6

)8reme5/'fpts'mag'ftype6

)8reme5/'fpts'mag'wt'ftype6



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FIR equiripple lowpass filter of +,ample L.>L for /8>M

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-200

-150

-100

-50

0

50

\omega/pi\

Gain,dB



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!ain response of t$e FIR equiripple )andpass filter of +,ample L.>M.



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Filter Designfir7 is used to design conventional lowpass'$ig$pass' )andpass')andstop and multi)and FIR filter.

)8fir75/'


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e,ample of a conventional lowpass FIR filter

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-300

-250

-200

-150

-100

-50

0

50

\omega/pi\

Gain,dB



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Filter Design

fir>is employed to design FIR filters wit$ ar)itarily s$aped magnitude response.

)8fir>5/'f'm6

)8fir>5/'f'm'window6

)8fir>5/'f'm'npt6

)8fir>5/'f'm'npt'window6

)8fir>5/'f'm'npt'lap'window6



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+,amples of multilevel filter

--2agnitude response of t$e multilevel filter designed wit$ fir>

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

/pi

magnitude



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*east-squares +rror FIR Filter Design Using 2atla)

firlsto design any type of multi)and linea

r-p$ase FIR filter )ased on t$e least-squares met$od

)8firls5/'fpts'mag6

)8firls5/'fpts'mag'wt6

)8firls5/'fpts'mag'ftype6

)8firls5/'fpts'mag'wt'ftype6



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e,ample of t$e linear-p$ase FIR lowpass filter

--!ain response of t$e linear-p$ase FIR low

pass filter

-160

-140

-120

-100

-80

-60

-40

-20

0

20

gain,dB

UNIT-4 Final Ppt

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