Filtros FIR - Ejemplos de Diseño

8/17/2019 Filtros FIR - Ejemplos de Diseño

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2.4 Examples

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MikroElektronika

This chapter discusses various FIR flter desin methods. It also provides exampleso! all t"pes o! flters as #ell as o! all methodes descri$ed in the previous chapters.

The !our standard t"pes o! flters are used here%

lo#&pass flter'

hih&pass flter'

$and&pass flter' and

$and&stop flter.

The desin method used here is kno#n as the #indo# method.

The FIR flter desin process can $e split into several steps as descri$ed in (hapter2.2.4 entitled )esinin FIR flters usin #indo# !unctions. These are%

)efnin flter specifcations'

*peci!"in a #indo# !unction accordin to the flter specifcations'

(omputin the flter order accordin to the flter specifcations and specifed #indo#

!unction'

(omputin the coe+cients o! the #indo#'

(omputin the ideal flter coe+cients accordin to the flter order'

(omputin the FIR flter coe+cients accordin to the o$tained #indo# !unction and

ideal flter coe+cients' and

I! the resultin flter has too #ide or too narro# transition reion, it is necessar" to

chane the flter order. The specifed flter order is increased or decreased accordin

to needs, and steps 4, - and are repeated a!ter that as man" times as needed.

)ependin on the #indo# !unction in use, some steps #ill $e skipped. I! the flter

order is kno#n, step / is skipped. I! the #indo# !unction to use is predetermined,

step 2 is skipped.

In ever" iven example, the FIR flter desin process #ill $e descri$ed throuh thesesteps in order to make it easier !or "ou to note similarities and di0erencies $et#een

various desin methodes, #indo# !unctions and desin o! various t"pes o! flters as

#ell.

2.4.1 Filter desin usin Rectanular #indo#

2.4.1.1 Example 1

*tep 1%

T"pe o! flter lo#&pass flter

Filter specifcations%

Filter order N31

http://learn.mikroe.com/ebooks/digitalfilterdesign/chapter/window-functions/http://learn.mikroe.com/ebooks/digitalfilterdesign/chapter/finite-word-length-effects/http://learn.mikroe.com/ebooks/digitalfilterdesign/chapter/finite-word-length-effects/http://learn.mikroe.com/ebooks/digitalfilterdesign/chapter/window-functions/


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*amplin !re5uenc" !s32678

Pass$and cut&o0 !re5uenc" !c32.-678

*tep 2%

Method flter desin usin rectanular #indo#*tep /%

Filter order is predetermined, N31'

9 total num$er o! flter coe+cients is larer $" one, i.e. N:1311' and

(oe+cients have indices $et#een and 1.

*tep 4%

9ll coe+cients o! the rectanular #indo# have the same value e5ual to 1.

#;n< 3 1 ' = n =1

*tep -%

The ideal lo#&pass flter coe+cients >ideal flter impulse response? are expressed as%

#here M is the index o! middle coe+cient.

Normali8ed cut&o0 !re5uenc" @c can $e calculated usin the !ollo#in expression%

The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the

values o! M and @c #ith expression !or the impulse response coe+cients o! the

ideal lo#&pass flter%


3/112

The middle element is !ound via the !ollo#in expression

*tep %

The desined FIR flter coe+cients are o$tained via the !ollo#in expression%

The FIR flter coe+cients h;n< rounded to diits are%

*tep A%

The flter order is predetermined.

There is no need to additionall" chane it.

Filter reali8ation%

Fiure 2&4&1 illustrates the direct reali8ation o! desined FIR flter, #hereas Fiure 2&

4&2 illustrates the optimi8ed reali8ation o! desined FIR flter, #hich is $ased on the

!act that all FIR flter coe+cients are, !or the sake o! linear phase characteristic,

s"mmetric a$out their middle element.


4/112

Fiure 2&4&1. FIR flter direct reali8ation

Fiure 2&4&2. Bptimi8ed reali8ation structure o! FIR flter

2.4.1.2 Example 2


5/112

*tep 1%

T"pe o! flter hih&pass flter


Filter order N3C


Pass$and cut&o0 !re5uenc" !c3-678

*tep 2%

Method flter desin usin rectanular #indo#

*tep /%

Filter order is predetermined, N3C'

9 total num$er o! flter coe+cients is larer $" 1, i.e. N:13D'

(oe+cients have indices $et#een and C.*tep 4%

9ll coe+cients o! the rectanular #indo# have the same value e5ual to 1.

#;n< 3 1 ' = n = C

*tep -%

The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed

as%


Normali8ed cut&o0 !re5uenc" @c can $e calculated usin the !ollo#in expression%





6/112

*tep %

The desined FIR flter coe+cients are !ound via expression%


*tep A%



Filter reali8ation%Fiure 2&4&/ illustrates the direct reali8ation o! desined FIR flter, #hereas fure 2&

4&4 illustrates the optimi8ed reali8ation o! desined FIR flter #hich is $ased on the




7/112

Fiure 2&4&/. FIR flter direct reali8ation

Fiure 2&4&4. FIR flter optimi8ed reali8ation structure

2.4.1./ Example /

*tep 1%


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T"pe o! flter $and&pass flter


Filter order N314


Pass$and cut&o0 !re5uenc" !c13/678, !c23-.-678

*tep 2%


*tep /%

Filter order is predetermined, N314

9 total num$er o! flter coe+cients is larer $" 1, i.e. N:131-.


*tep 4%9ll coe+cients o! the rectanular #indo# have the same value e5ual to 1.

#;n< 3 1 ' = n =14

*tep -%


as%


Normali8ed cut&o0 !re5uencies @c1 and @c2 can $e !ound usin the !ollo#in

expressions%


9/112


values o! M and @c1 and @c2 #ith expression !or the impulse response coe+cients

o! the ideal $and&pass flter%

*tep A%

Filter order is predetermined.


Filter reali8ation%

Fiure 2&4&- illustrates the direct reali8ation o! desined FIR flter, #hereas fure 2&

4& illustrates optimi8ed reali8ation o! desined FIR flter #hich is $ased on the !act

that all FIR flter coe+cients are, !or the sake o! linear phase characteristic,s"mmetric a$out their middle element.


10/112

Fiure 2&4&-. FIR flter direct reali8ation

Fiure 2&4&. FIR flter optimi8ed reali8ation structure

2.4.1.4 Example 4

*tep 1%


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T"pe o! flter $and&stop flter


Filter order N314


*top$and cut&o0 !re5uenc" !c13/678, !c23-.-678

*tep 2%


*tep /%

Filter order is predetermined, N314'

9 total num$er o! flter coe+cients is larer $" 1, i.e. N:131-' and


*tep 4%9ll coe+cients o! the rectanular #indo# have the same value e5ual to 1.

#;n< 3 1 ' = n = 14

*tep -%


as%


Normali8ed cut&o0 !re5uencies @c1 and @c2 can $e !ound usin expressions%


12/112



o! the ideal $and&stop flter%

Note that, exceptin the middle element, all coe+cients are the same as in the

previous example >$and&pass flter #ith the same cut&o0 !re5uencies?, $ut have the

opposite sin.

*tep %


The FIR flter coe+cients h;n


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Fiure 2&4&A. FIR flter direct reali8ation

Fiure 2&4&C. FIR flter optimi8ed reali8ation structure


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2.4.2 Filter desin usin artlett #indo#

2.4.2.1 Example 1

*tep 1%

T"pe o! flter lo#&pass flterFilter specifcations%

Filter order N!3D


Pass$and cut&o0 !re5uenc" !c32.-678

*tep 2%

Method flter desin usin arlett #indo#

*tep /%

Filter order is predetermined, N!3D'

9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:131' and

(oe+cients have indices $et#een and C.

*tep 4%

The coe+cients o! artlett #indo# are expressed as%

*tep -%

The ideal lo#&pass flter coe+cients >ideal flter impulse response? are iven in the

expression $elo#%



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*ince the value o! M is not an inteer, the middle element representin a center o!coe+cients s"mmetr" doesnt exist.

Normali8ed cut&o0 !re5uenc" @c can $e calculated usin expression%




*tep %


h;n< 3 #;n< G hd;n< ' = n =D


*tep A%



Filter reali8ation%

Fiure 2&4&D illustrates the direct reali8ation o! desined FIR flter, #hereas fure 2&

4&1 illustrates optimi8ed reali8ation o! desined FIR flter #hich is $ased on the !act

that all FIR flter coe+cients are, !or the sake o! linear phase characteristic,



16/112

Fiure 2&4&D. FIR flter direct reali8ation


2.4.2.2 Example 2

*tep 1%


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Filter order N!3C


Pass$and cut&o0 !re5uenc" !c3-678

*tep 2%

Method flter desin usin artlett #indo#

*tep /%

Filter order is predetermined, N!3C'

9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:13D'

(oe+cients have indices $et#een and C.

*tep 4% The artlett #indo# !unction coe+cients are !ound via expression%

*tep -%


as%



18/112

Normali8ed cut&o0 !re5uenc" @c ma" $e calculated via the !ollo#in expression%



ideal hih&pass flter%

*tep %


h;n< 3 #;n< G hd;n< ' = n = C


*tep A%

Filter order is predetermined.


Filter reali8ation%

Fiure 2&4&11 illustrates the direct reali8ation o! desined FIR flter, #hereas fure

2&4&12 illustrates optimi8ed reali8ation o! desined FIR flter #hich is $ased on the




19/112



2.4.2./ Example /

*tep 1%


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Filter order N!314'

*amplin !re5uenc" !s32678' and

Pass$and cut&o0 !re5uencies !c13/678, !c23-.-678.

*tep 2%


*tep /%

Filter order is predetermined, N!314'

9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:131-' and


*tep 4% The alett #indo# coe+cients are !ound via expression%

*tep -%


as%



21/112

Normali8ed cut&o0 !re5uencies @c1 and @c2 can $e calculated usin expressions%



o! the ideal $and&pass flter%

*tep %


h;n< 3 #;n< G hd;n< ' = n = 14


*tep A%



Filter reali8ation%

Fiure 2&4&1/ illustrates the direct reali8ation o! desined FIR flter, #hereas fure2&4&14 illustrates optimi8ed reali8ation structure o! desined FIR flter #hich is

$ased on the !act that all FIR flter coe+cients are, !or the sake o! linear phase

characteristic, s"mmetric a$out their middle element.


22/112

Fiure 2&4&1/. FIR flter direct reali8ation



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2.4.2.4 Example 4

*tep 1%



Filter order N!314'


*top$and cut&o0 !re5uencies !c13/678, !c23-.-678.

*tep 2%


*tep /%


9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:131-' and(oe+cients have indices $et#een and 14.

*tep 4%

The coe+cients o! artlett #indo# are !ound via expression%

*tep -%


as%



24/112



values o! M, @c1 and @c2 #ith expression !or the impulse response coe+cients o!

the ideal $and&stop flter%

Note that, exceptin the middle element, all the coe+cients are the same as in the


opposite sin.

*tep %


h;n< 3 #;n< G hd;n< ' = n = 14


*tep A%



Filter reali8ation%

Fiure 2&4&1- illustrates the direct reali8ation o! desined FIR flter, #hereas fure

2&4&1 illustrates optimi8ed reali8ation o! desined FIR flter #hich is $ased on the




25/112

Fiure 2&4&1-. FIR flter direct reali8ation



26/112

It is determined on purpose that FIR flters, explained in examples / and 4, have the

same order. The similarit" $et#een the coe+cients o! $and&pass and $and&stop FIR

flters is o$vious. 9ll coe+cients o! the $and&stop FIR flter have the same a$solute

values as the correspondin coe+cients o! the $and&pass FIR flter. The onl"

di0erence is that the" are o! the opposite sin. The middle element o! the $and&stop

flter is defned as%

$$s 3 1 $$p

#here%

$$s is the middle coe+cient o! the $and&stop flter' and

$$p is the middle coe+cient o! the $and&pass flter.

ecause o! such similarit", it is eas" to convert a $and&pass FIR flter into a $and&

stop FIR flter havin the same cut&o0 !re5uencies, samplin !re5uenc" and flter

order.

esides, lo#&pass and hih&pass FIR flters are interrelated in the same #a", #hich

can $e seen in examples descri$in 7ann #indo#.

2.4./ Filter desin usin 7ann #indo#

2.4./.1 Example 1

*tep 1%



Filter order N!31'*amplin !re5uenc" !s32678' and

Pass$and cut&o0 !re5uenc" !c32.-678.

*tep 2%

Method flter desin usin 7ann #indo#

*tep /%




*tep 4%

The 7ann #indo# !unction coe+cients are !ound via expression%


27/112

*tep -%





values o! M and @c #ith expression !or the impulse reaponse coe+cients o! the


*tep %


h;n< 3 #;n< G hd;n< ' = n = 1



28/112

*tep A%



Filter reali8ation%

Fiure 2&4&1A illustrates the direct reali8ation o! desined FIR flter, #hereas fure

2&4&1C illustrates optimi8ed reali8ation structure o! desined FIR flter #hich is



Fiure 2&4&1A. FIR flter direct reali8ation


29/112

Fiure 2&4&1C. FIR flter optimi8ed reali8ation structure

2.4./.2 Example 2

*tep 1%

Filter t"pe hih&pass flter Filter specifcations%

Filter order N!31'


Pass$and cut&o0 !re5uenc" !c32.-678.

*tep 2%


*tep /%




*tep 4%


*tep -%

The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressedas%


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*tep %


h;n< 3 #;n< G hd;n< ' = n = 1


*tep A%



Filter reali8ation%

Fiure 2&4&1D illustrates the direct reali8ation o! desined FIR flter, #hereas fure2&4&2 illustrates optimi8ed reali8ation structure o! desined FIR flter #hich is


31/112



Fiure 2&4&1D. FIR flter direct reali8ation


2.4././ Example /


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*tep 1%



Filter order N!314'


Pass$and cut&o0 !re5uenc" !c13/678, !c23-.-678.

*tep 2%


*tep /%

Filter order is predtermined, N!314'

9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:131-' and

(oe+cients have indices $et#een and 14.*tep 4%


*tep -%


as%




33/112



the ideal $and&pass flter%

*tep %


h;n< 3 #;n< G hd;n< ' = n = 14


*tep A%



Filter reali8ation%


2&4&22 illustrates optimi8ed reali8ation structure o! desined FIR flter #hich is

$ased on the !act that all FIR flter coe+cients are, !or the sake o! linear phasecharacteristic, s"mmetric a$out their middle element.


34/112




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2.4././ Example /

*tep 1%



Filter order N!314'


Pass$and cut&o0 !re5uenc" !c13/678, !c23-.-678.

*tep 2%


*tep /%


9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:131-'(oe+cients have indices $et#een and 14.

*tep 4%


*tep -%


as%




36/112




Note that, exceptin the middle element, all coe+cients are the same as in the


opposite sin.

*tep %


h;n


37/112




38/112

It is specifed on purpose that FIR flters, explained in examples 1 and 2, have the

same order. The similarit" $et#een lo#&pass and hih&pass FIR flter coe+cients is

o$vious. 9ll coe+cients o! the lo#&pass FIR flter have the same a$solute values as

the correspondin coe+cients o! the hih&pass FIR flter. The onl" di0erence is that

the" are o! the opposite sin. The middle element is defned as%

$lp 3 1 $hp

#here%

$lp is the middle coe+cient o! a lo#&pass flter' and

$hp is the middle coe+cient o! a hih&pass flter.

ecause o! such similarit", it is eas" to convert a lo#&pass FIR flter into a hih&pass

FIR flter havin the same cut&o0 !re5uencies, sampilin !re5uenc" and flter order.

2.4.4 Filter desin usin artlett&7annin #indo#

2.4.4.1 Example 1

*tep 1%



Filter order N!3D'

*amplin !re5uenc" !s322-78' and

Pass$and cut&o0 !re5uenc" !c34678.

*tep 2%

Method flter desin usin artlett&7annin #indo#

*tep /%

Filter order is predetermined, N!3D'


(oe+cients have indices $et#een and D.

*tep 4%

The artlett&7annin #indo# !unction coe+cients are !ound via expression%

*tep -%



39/112






*tep %


h;n< 3 #;n< G hd;n< ' = n = D


*tep A%



Filter reali8ation%

Fiure 2&4&2- illustrates the direct reali8ation o! desined FIR flter, #hereas fure




40/112

Fiure 2&4&2-. FIR flter direct reali8ation



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2.4.4.2 Example 2

*tep 1%



Filter order N!31'



*tep 2%


*tep /%


9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:1311' and(oe+cients have indices $et#een and 1.

*tep 4%


*tep -%


as%



42/112


43/112

Fiure 2&4&2A. FIR flter direct reali8ation

Fiure 2&4&2C. FIR flter optimi8ed reali8ation structure


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2.4.4./ Example /

*tep 1%



Filter order N!312'


Pass$and cut&o0 !re5uenc" !c132678, !c23-678.

*tep 2%


*tep /%


9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:131/' and(oe+cients have indices $et#een and 12.

*tep 4%


*tep -%


as%



45/112





*tep %


h;n< 3 #;n< G hd;n< ' = n = 12


*tep A%



Filter reali8ation%

Fiure 2&4&2D illustrates the direct reali8ation o! desined FIR flter, #hereas fure

2&4&/ illustrates optimi8ed reali8ation structure o! desined FIR flter #hich is




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Fiure 2&4&2D. FIR flter direct reali8ation


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Fiure 2&4&/. FIR flter optimi8ed reali8ation structure

2.4.4.4 Example 4

*tep 1%

T"pe o! flter $and&stop flterFilter specifcations%

Filter order N!312'


Pass$and cut&o0 !re5uencies !c132678, !c23678.

*tep 2%


*tep /%


9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:131/' and


*tep 4%


*tep -%


as%



48/112



values o! M, @c1 and @c2 #ith expression !or the impulse reaponse coe+cients o!


*tep %


h;n< 3 #;n< G hd;n< ' = n = 12


*tep A%



Filter reali8ation%

Fiure 2&4&/1 illustrates the direct reali8ation o! desined FIR flter, #hereas fure

2&4&/2 illustrates optimi8ed reali8ation structure o! desined FIR flter #hich is




49/112

Fiure 2&4&/1. FIR flter direct reali8ation

Fiure 2&4&/2. FIR flter optimi8ed reali8ation structure

2.4.- Filter desin usin 7ammin #indo#

2.4.-.1 Example 1

*tep 1%


50/112


Filte specifcations%

*amplin !re5uenc" !s322-78'

Pass$and cut&o0 !re5uenc" !c13/678'

*top$and cut&o0 !re5uenc" !c23678' and

Minimum stop$and attenuation 4d.

*tep 2%

Method flter desin usin 7ammin #indo#

*tep /%

For the frst iteration, the flter order can $e determined !rom the ta$le 2&4&1 $elo#.

HIN)BH

FN(TIB

N

NBRM9JIKE)

JENLT7

BF T7E

M9IN

JBE FBR

N32

TR9N*ITI

BN

RELIBN

FBR

N32

MINIMM

*TBP9N)

9TTEN9TI

BN BF

HIN)BH

FN(TIBN

MINIMM

*TBP9N)

9TTEN9TI

BN BF

)E*ILNE)

FIJTER

Rectanu

lar.1 .41 1/ d 21 d

Trianula

r

>artlett?

.2 .11 2 d 2 d

7ann .21 .12 /1 d 44 d

artlett&

7annin.21 .1/ / d /D d

7ammin

.2/ .14 41 d -/ d

ohman ./1 .2 4 d -1 d

lackma

n./2 .2 -C d A- d

lackma

n&7arris.4/ ./2 D1 d 1D d

Ta$le 2&4&1. (omparison o! #indo# !unctions


51/112

sin the specifcations !or the transition reion o! the re5uired flter, it is possi$le

to compute cut&o0 !re5uencies%

The re5uired transition reion o! the flter is%

The transition reion o! the flter to $e desined is approximatel" t#ice that o! the

flter iven in the ta$le a$ove. For the frst iteration, the flter order can $e hal! o!

that.

Filter order is N!31'



*tep 4% The 7ammin #indo# !unction coe+cients are !ound via expression%

*tep -%



52/112






*tep %


h;n< 3 #;n< G hd;n< ' = n = 1


*tep A%

9nal"se in the !re5uenc" domain is per!ormed usin the Filter )esiner Tool

proram.


53/112

Fiure 2&4&//. Fre5uenc" characteristic o! the resultin flter

Fiure 2&4&// illustrates the !re5uenc" characteristic o! the resultin flter. It is

o$tained in the Filter )esiner Tool proram. 9s seen, the resultin flter doesnt

satis!" the re5uired specifcations. The attenuation at the !re5uenc" o! 678

amounts to /2.Dd onl", #hich is not su+cient. It is necessar" to increase the flter

order.

9nother #a" is to compute the attenuation at the !re5uenc" o! 678. *tartin !rom

the impulse response, the frst thin that should $e done is the K&trans!orm. It is

explained, alon #ith Fourier trans!ormation, in chapter 2&2&2.

It is eas" to o$tain the Fourier trans!ormation via the K&trans!orm%


54/112

9ccordin to the anal"se per!ormed usin Filter )esiner Tool, it is confrmed that

the flter order has to $e incremented.

The flter order is incremented $" t#o. The #hole process o! desinin flter is

repeated !rom the step /.

*tep /%



*tep 4%

The 7ammin #indo# !unction coe+cients are !ound via expression%


55/112

*tep -%







*tep %


h;n< 3 #;n< G hd;n< ' = n = 12


*tep A%


56/112


proram.

Fiure 2&4&/4. Fre5uenc" characteristic o! the resultin flter

Fiure 2&4&/4 illustrates the !re5uenc" characteristic o! the resultin flter. 9s seen,the resultin flter doesnt satis!" the iven specifcations. The attenuation at the

!re5uenc" o! 678 amounts to 4-.2d onl", #hich is not su+cient. It is necessar"

to chane the flter order.

Filter reali8ation%

Fiure 2&4&/- illustrates the direct reali8ation o! desined FIR flter, #hereas fure





57/112

Fiure 2&4&/-. FIR flter direct reali8ation

Fiure 2&4&/. FIR flter optimi8ed reali8ation structure


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2.4.-.2 Example 2

*tep 1%



Filter order N!31'



*tep 2%


*tep /%


9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:1311' and(oe+cients have indices $et#een and 1.

*tep 4%


The 7ammin #indo# !unction is one o! rare standard #indo#s #here #;< O is in

e0ect.

*tep -%


as%



59/112





*tep %


h;n< 3 #;n< G hd;n< ' = n = 1


*tep A%



Filter reali8ation%

Fiure 2&4&/A illustrates the direct reali8ation o! desined FIR flter, #hereas fure

2&4&/C illustrates optimi8ed reali8ation structure o! desined FIR flter #hich is




60/112

Fiure 2&4&/A. FIR flter direct reali8ation

Fiure 2&4&/C. FIR flter optimi8ed reali8ation structure


61/112

2.4.-./ Example /

*tep 1%



Filter order N!312'

*amplin !re5uenc" !s31678'

Pass$and cut&o0 !re5uenc" !c132678, !c23-678.

*tep 2%


*tep /%



*tep 4%


*tep -%


as%



62/112





*tep %


h;n< 3 #;n< G hd;n< ' = n = 12


*tep A%



Filter reali8ation%

Fiure 2&4&/D illustrates the direct reali8ation o! desined FIR flter, #hereas fure



characteristic, s"mmetric a$out their middle elements.


63/112

Fiure 2&4&/D. FIR flter direct reali8ation



64/112

2.4.-.4 Example 4

*tep 1%



Filter order N!312'


Pass$and cut&o0 !re5uenc" !c132678, !c23678.

*tep 2%


*tep /%



*tep 4%


*tep -%


as%



65/112





*tep %


h;n< 3 #;n< G hd;n< ' = n = 12


*tep A%



Filter reali8ation%






66/112



The frst example >lo#&pass flter desined usin 7ammin #indo#? explains thealorithm used to compute the needed flter order #hen it is unkno#n. The flter


67/112

order can also $e !ound usin 6aiser #indo#, a!ter #hich the num$er o! iterations,

i.e. correction steps is reduced.

The !orth example explains the #a" o! desinin a $and&stop flter. 9s can $e seen,

the impulse response o! the resultin flter contains lare num$er o! 8ero values,

#hich results in reducin the num$er o! multiplication operations in desin process.

These 8eros appear in impulse response $ecause o! the stop$and #idth #hich

amounts to .- 3 2.

I! it is possi$le to speci!" the samplin !re5uenc" !rom a certain !re5uenc" rane,

"ou should tend to speci!" the value representin a multiple o! the pass$and #idth.

The num$er o! 8eros contained in an impulse response is larer in this case,

#hereas the num$er o! multiplications, other#ise the most demandin operation in

flterin process, is less.

In the iven example, onl" - multiplication operations are per!ormed in direct

reali8ation o! a t#el!th&order FIR flter, i.e. / multiplication operations in optimi8ed

reali8ation structure.

2.4. Filter desin usin ohman #indo#

2.4..1 Example 1

*tep 1%



Filter order N!31'


Pass$and cut&o0 !re5uenc" !c3-678.

*tep 2%

Method flter desin usin ohman #indo#

*tep /%




*tep 4%

The ohman #indo# !unction coe+cients are !ound via expression%


68/112

*tep -%



Normali8ed cut&o0 !re5uenc" @c ma" $e computed usin expression%




*tep %


h;n< 3 #;n< G hd;n< ' = n = 1


*tep A%


69/112



Filter reali8ation%

Fiure 2&4&4/ illustrates the direct reali8ation o! desined FIR flter, #hereas fure

2&4&44 illustrates optimi8ed reali8ation structure o! desined FIR flter #hich is$ased on the !act that all FIR flter coe+cients are, !or the sake o! linear phase




70/112


2.4..2 Example 2

*tep 1%




Pass$and cut&o0 !re5uenc" !c131.-678'

*top$and cut&o0 !re5uenc" !c234678' and

Minimum stop$and attenuation /-d.*tep 2%


*tep /%

The needed flter order is determined via iteration.

It is necessar" to speci!" the initial value o! flter order that is to $e chaned as

man" times as needed. This value is specifed accordin to the data contained in

the ta$le 2&4&2 $elo#%

HIN)BH

FN(TIB

N

NBRM9JI

KE)

JENLT7

BF T7E

M9IN

JBE FBR

N32

TR9N*ITI

BN

RELIBN

FBR

N32

MINIMM

*TBP9N)

9TTEN9TI

BN BF

HIN)BH

FN(TIBN

MINIMM

*TBP9N)

9TTEN9TI

BN BF

)E*ILNE)

FIJTER

Rectanu

lar.1 .41 1/ d 21 d

Trianula .2 .11 2 d 2 d


71/112

r

>artlett?

7ann .21 .12 /1 d 44 d

artlett&

7annin.21 .1/ / d /D d

7ammin

.2/ .14 41 d -/ d

ohman ./1 .2 4 d -1 d

lackma

n./2 .2 -C d A- d

lackma

n&7arris.4/ ./2 D1 d 1D d

Ta$le 2&4&2. (omparison o! #indo# !unctions

9ccordin to the specifcations !or the transition reion o! re5uired flter, it is

possi$le to compute cut&o0 !re5uencies%

The re5uired transition reion is%

The transition reion o! the flter to $e desined is some#hat #ider than that o! the

flter iven in ta$le 2&4&2. For the frst iteration, durin flter desin process, the

flter order can $e lo#er.

nlike the lo#&pass FIR flter, the hih&pass FIR flter must $e o! even order. The

same applies to $and&pass and $and&stop flters. It means that flter order can $e

chaned in odd steps. The smallest chane is Q2. In this case, the flter order,

comparin to that !rom the ta$le >2?, can $e decreased $" 2 !or the purpose o!

defnin initial value.


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Filter order is N!31C'

9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:131D' and

(oe+cients have indices $et#een and 1C.

*tep 4%

The coe+cients o! ohman #indo# are !ound via expression%

*tep -%


as%


Normali8ed cut&o0 !re5uenc" @c is e5ual to pass$and cut&o0 !re5uenc"%





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*tep /%


9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:131A' and


*tep 4%


*tep -%

The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressedas%


Normali8ed cut&o0 !re5uenc" @c is e5ual to the pass$and cut&o0 !re5uenc"%


75/112




*tep %


h;n< 3 #;n< G hd;n< ' = n = 1


*tep A%


proram.

Fiure 2&4&4. Fre5uenc" characteristic o! the resultin flter

Fiure 2&4&4 illustrates the !re5uenc" characteristic o! the resultin flter. The

fure is o$tained in the Filter )esiner Tool proram. 9s seen, the resultin flter

satisfes the re5uired specifcations. The o$ective is to fnd the minimum flter

order. *ince the attenuation is close to the re5uired attenuation, the correct order is

pro$a$l" 1. 7o#ever, it is necessar" to check it.


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*ince the flter order must $e chaned $" an even num$er, the specifed value is &2.

The flter order is decreased $" 2, there!ore. The #hole process o! desinin flter is

repeated !rom the step / on.

*tep /%

Filter order is predetermined, N!314'9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:131-' and


*tep 4%


*tep -%


as%


Normali8ed cut&o0 !re5uenc" @c is e5ual to pass$and cut&o0 !re5enc"%


values o! M and @c #ith expression !or the impulse reaponse coe+cients o! the



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*tep %


h;n< 3 #;n< G hd;n< ' = n = 14


*tep A%


proram.

Fiure 2&4&4A. Fre5uenc" characteristic o! the resultin flter

Fiure 2&4&4A illustrates the !re5uenc" characteristic o! the resultin flter. The

fure is o$tained in the Filter )esiner Tool proram. 9s seen, the resultin flter

doesnt satis!" the re5uired specifcations. The attenuation at the !re5uenc" o!

1-678 amounts to 2.24d onl", #hich is not su+cient. The previous value

>N!31? represents the minimum FIR flter order that satisfes the iven

specifcations.

The flter order is N!31, #hereas impulse response o! the resultin flter is as

!ollo#s%


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Filter reali8ation%

Fiure 2&4&4C illustrates the direct reali8ation o! desined FIR flter, #hereas fure

2&4&4D illustrates optimi8ed reali8ation structure o! desined FIR flter #hich is



Fiure 2&4&4C. FIR flter direct reali8ation


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Fiure 2&4&4D. FIR flter optimi8ed reali8ation structure

2.4../ Example /

*tep 1%



Filter order N!312'


Pass$and cut&o0 !re5uenc" !c134678, !c231-2-78.

*tep 2%


*tep /%




*tep 4%


*tep -%


as%


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values o! M, @c1 and @c2 #ith expression !or the impulse response coe+cients o!the ideal $and&pass flter%

*tep %

The coe+cients o! desined FIR flter are !ound via expression%

h;n< 3 #;n< G hd;n< ' = n = 12


*tep A%

The flter order is predetermined. There is no need to additionall" chane it.


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Filter reali8ation%

Fiure 2&4&- illustrates the direct reali8ation o! desined FIR flter, #hereas fure

2&4&-1 illustrates optimi8ed reali8ation structure o! desined FIR flter #hich is





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Fiure 2&4&-1. FIR flter optimi8ed reali8ation structure

2.4..4 Example 4

*tep 1%



Filter order N!312'


Pass$and cut&o0 !re5uenc" !c132678, !c23678.

*tep 2%


*tep /%




*tep 4%


*tep -%


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as%






*tep %


h;n< 3 #;n< G hd;n< ' = n = 12



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*tep A%



Filter reali8ation%

Fiure 2&4&-2 illustrates the direct reali8ation o! desined FIR flter, #hereas fure

2&4&-/ illustrates optimi8ed reali8ation structure o! desined FIR flter #hich is



Fiure 2&4&-2. FIR flter direct reali8ation


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Fiure 2&4&-/. FIR flter optimi8ed reali8ation structure

2.4.A Filter desin usin lackman #indo#

2.4.A.1 Example 1*tep 1%



Filter order N!312'


Pass$and cut&o0 !re5uenc" !c31-678' and

9ttenuation o! d at 78 d.

*tep 2%

Method Filter desin usin lackman #indo#

*tep /%


9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:131/'


*tep 4%

The coe+cients o! lackman #indo# are !ound via expression%

*tep -%


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Normali8ed cut&o0 !re5uenc" @c can $e computed usin expression%

The values o! coe+cients are o$tained >rounded to six diits? $" com$inin the



*tep %


h;n< 3 #;n< G hd;n< ' = n = 12


The resultin coe+cients must $e scaled in order to provide attenuation o! d at

78. In order to provide attenuation o! d, the !ollo#in condition must $e met%

The sum o! the previousl" o$tained coe+cients is%


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9s the sum is reater than one, it is necessar" to divide all coe+cients o! theimpulse response $" 1.2A4. 9!ter division, these coe+cients have the !ollo#in

values%

The sum o! scaled coe+cients is e5ual to 1, #hich means that attenuation at 78

!re5uenc" amounts to d. Note that these coe+cients cannot $e used in desinin

a FIR flter sa!e !rom flterin overSo#. In order to prevent a flterin overSo# !romoccurin it is necessar" to satis!" the condition $elo#%

The resultin flter doesnt meet this condition. Neative coe+cients in impulse

response make that $oth conditions cannot $e met. The sum o! apsolute values o!

coe+cients in the resultin flter is%

The sum o! coe+cients apsolute values $e!ore scalin amounts to 1./A1

>1./D/1.2A4?. 9!ter scalin, it is some#hat less, so it is less likel" that an

overSo# occurs. In such cases, possi$le flterin overSo#s are not danerous.

Namel", most processors containin hard#are multipliers >#hich is almost

necessar" !or flterin? have reisters #ith extended $and. In this case, it is !ar more

important to !aith!ull" transmit a direct sinal to a FIR flter output.

*tep A%



Filter reali8ation%

Fiure 2&4&-4 illustrates the direct reali8ation o! desined FIR flter, #hereas fure

2&4&-- illustrates optimi8ed reali8ation structure o! desined FIR flter #hich is




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Fiure 2&4&-4. FIR flter direct reali8ation

Fiure 2&4&--. FIR flter optimi8ed reali8ation structure


89/112

2.4.A.2 Example 2

*tep 1%



Filter order N!312'


Pass$and cut&o0 !re5uenc" !c34678'

Prevention o! possi$le flterin overSo#s.

*tep 2%

Method flter desin usin lackman #indo#

*tep /%




*tep 4%

The coe+cients o! lackman #indo# !unction are !ound via%

*tep -%


as%



90/112





*tep %


h;n< 3 #;n< G hd;n< ' = n = 12


In order to prevent flterin overSo#, the !ollo#in condition must $e met%

The sum o! a$solute values o! the resultin FIR flter coe+cients is%

The o$tained coe+cients must $e scaled >divided? $" 1./4CA. 9!ter that, their

values are%


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*tep A%



Filter reali8ation%

Fiure 2&4&- illustrates the direct reali8ation o! desined FIR flter, #hereas fure

2&4&-A illustrates optimi8ed reali8ation structure o! desined FIR flter #hich is





92/112

Fiure 2&4&-A. FIR flter optimi8ed reali8ation structure

2.4.A./ Example /

*tep 1%



Filter order N!312'


Pass$and cut&o0 !re5uenc" !c134678, !c231-2-78'

Prevention o! possi$le flterin overSo#.

*tep 2%Method flter desin usin lackman #indo#

*tep /%




*tep 4%

The coe+cients o! lackman #indo# !unction are !ound via expression%

*tep -%


93/112


as%


Normali8ed cut&o0 !re5uencies @c1 and @c2 can $e computed usin expressions%




*tep %


h;n< 3 #;n< G hd;n< ' = n = 12



94/112

In order to prevent flterin overSo#s, the !ollo#in condition must $e met%


The o$tained coe+cients must $e scaled >divided? $" 1.12-. 9!ter this, their

values are%

*tep A%



Filter reali8ation%

Fiure 2&4&-C illustrates the direct reali8ation o! desined FIR flter, #hereas fure

2&4&-D illustrates optimi8ed reali8ation structure o! desined FIR flter #hich is




95/112

Fiure 2&4&-C. FIR flter direct reali8ation

Fiure 2&4&-D. FIR flter optimi8ed reali8ation structure


96/112

2.4.A.4 Example 4

*tep 1%



Filter order N!312'


Pass$and cut&o0 !re5uencies !c132678, !c23678'


*tep 2%

Method flter desin usin lackman #indo#

*tep /%




*tep 4%

The coe+cients o! lackman #indo# are !ound via expression%

*tep -%


as%



97/112


The values o! coe+cients >rounded to six diits? are o$tained $" com$inin thevalues o! M, @c1 and @c2 #ith expression !or the impulse response coe+cients o!


*tep %


h;n< 3 #;n< G hd;n< ' = n = 12





98/112

The o$tained coe+cients must $e scaled >divided? $" .D1A. 9!ter this, their

values are%

*tep A%



Filter reali8ation%

Fiure 2&4& illustrates the direct reali8ation o! desined FIR flter, #hereas fure

2&4&1 illustrates optimi8ed reali8ation structure o! desined FIR flter #hich is$ased on the !act that all FIR flter coe+cients are, !or the sake o! linear phase


Fiure 2&4&. FIR flter direct reali8ation


99/112


2.4.C Filter desin usin lackman&7arris #indo#

2.4.C.1 Example 1

*tep 1%



Filter order N!312'


Pass$and cut&o0 !re5uenc" !c31-678'

9ttenuation o! d at 78.

*tep 2%

Method flter desin usn lackman&7arris #indo#

*tep /%




*tep 4%

The coe+cients o! lackman&7arris #indo# are !ound via expression%

*tep -%


100/112


as%


Normali8ed cut&o0 !re5uenc" @c ma" $e calculated usin expression%




*tep %


h;n< 3 #;n< G hd;n< ' = n = 12


The resultin coe+cients must $e scaled in order to provide attenuation o! d at

78. To provide d attenuation, the !ollo#in condition must $e met%

The sum o! the previousl" o$tained coe+cients is%


101/112

9s the sum is reater than one, it is necessar" to divide all the impulse responsecoe+cients $" .DAAD4A. 9!ter this, the values o! these coe+cients are%

The sum o! scaled coe+cients is e5ual to 1, #hich means that attenuation at 78

!re5uenc" amounts to d. Note that these coe+cients cannot $e used in desinin

a FIR flter sa!e !rom flterin overSo#. In order to prevent a flterin overSo# !romoccurin it is necessar" to satis!" the condition $elo#%

The resultin flter doesnt meet this condition. Neative coe+cients in impulse

response indicate that $oth conditions cannot $e met. The sum o! apsolute values

o! coe+cients in the resultin flter is%

The sum o! coe+cients apsolute values $e!ore scalin amounts to 1./A1

>1./D/1.2A4?. 9!ter scalin, the sum o! coe+cients apsolute values is

some#hat less, so it is less possi$le that an overSo# occurs. In such cases, possi$le

flterin overSo#s are not danerous. Namel", most processors containin hard#are

multipliers >#hich is almost necessar" !or flterin? have reisters #ith extended

$and. In this case, it is !ar more important to !aith!ull" transmit a direct sinal to aFIR flter output.

*tep A%



Filter reali8ation%






102/112

Fiure 2&4&2. FIR flter

Fiure 2&4&/. Bptimi8ed FIR flter desin

2.4.C.2 Example 2

*tep 1%


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Filter order N!312'


Pass$and cut&o0 !re5uenc" !c34678'

Prevention o! flterin overSo#s.

*tep 2%

Method flter desin usin lackman&7arris #indo#

*tep /%




The coe+cients o! lackman&7arris #indo# are !ound via%

*tep -%


as%




104/112


values o! M and @c #ith the expression !or the impulse response coe+cients o! the


*tep %


h;n< 3 #;n< G hd;n< ' = n = 12


In order to prevent flterin overSo#, the !ollo#in condition must $e met%


The o$tained coe+cients must $e scaled >divided? $" 1./DAAD1. 9!ter this, theirvalues are%

*tep A%



Filter reali8ation%


105/112


2&4&- illustrates optimi8ed reali8ation structure o! desined FIR flter #hich is





106/112

Fiure 2&4&-. Bptimi8ed FIR flter desin

2.4.C./ Example /

*tep 1%

T"pe o! flter $and&pass flterFilter specifcation%

Filter order N!312'


Pass$and cut&o0 !re5uencies !c134678, !c231-2-78' and


*tep 2%


*tep /%




*tep 4%


*tep -%


as%



107/112





*tep %


h;n< 3 #;n< G hd;n< ' = n = 12


*tep A%



Filter reai8ation%

Fiure 2&4& illustrates the direct reali8ation o! desined FIR flter, #hereas fure

2&4&A illustrates optimi8ed reali8ation structure o! desined FIR flter #hich is



108/112

Fiure 2&4&. FIR flter direct reali8ation

Fiure 2&4&A. Bptimi8ed FIR flter desin

2.4.C.4 Example 4

*tep 1%


109/112


Filter specifcation%

Filter order N!32'


Pass$and cut&o0 !re5uenc" !c132678, !c23678' and


*tep 2%


*tep /%





*tep -%


as%




110/112




*tep %


h;n< 3 #;n< G hd;n< ' = n = 2




The o$tained coe+cients must $e scaled >divided? $" 1.122/2. 9!ter this, their

values are%


111/112

*tep A%



Filter reali8ation%

Fiure 2&4&C illustrates the direct reali8ation o! desined FIR flter, #hereas fure

2&4&D illustrates optimi8ed reali8ation structure o! desined FIR flter #hich is



This FIR flter is an excellent example sho#in the importance o! the samplin!re5uenc". It is specifed to ive the pass$and amountin to .-. This causes most

impulse response coe+cients o! the resultin FIR flter to $e 8eros. It !urther makes

the flter reali8ation structure simpler. 9s !or optimi8ed FIR flter desin, there are

onl" 4 multiplications, even thouh the flter is o! 2th order. n!ortunatell", the

$u0er lenth cannot $e minimi8ed. It is fxed and corresponds to the flter order.

7o#ever, it is possi$le to a0ect desin complexit", #hether it is hard#are or

so!t#are implementation.


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Filtros FIR - Ejemplos de Diseño

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Transcript of Filtros FIR - Ejemplos de Diseño