8/17/2019 Filtros FIR - Ejemplos de Diseño
1/112
2.4 Examples
Previous Next
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 $et#een
various desin methodes, #indo# !unctions and desin o! various t"pes o! flters as
#ell.
2.4.1 Filter desin usin Rectanular #indo#
2.4.1.1 Example 1
*tep 1%
T"pe o! flter lo#&pass flter
Filter specifcations%
Filter order N31
http://learn.mikroe.com/ebooks/digitalfilterdesign/chapter/window-functions/http://learn.mikroe.com/ebooks/digitalfilterdesign/chapter/finite-word-length-effects/http://learn.mikroe.com/ebooks/digitalfilterdesign/chapter/finite-word-length-effects/http://learn.mikroe.com/ebooks/digitalfilterdesign/chapter/window-functions/
8/17/2019 Filtros FIR - Ejemplos de Diseño
2/112
*amplin !re5uenc" !s32678
Pass$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%
8/17/2019 Filtros FIR - Ejemplos de Diseño
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.
8/17/2019 Filtros FIR - Ejemplos de Diseño
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
8/17/2019 Filtros FIR - Ejemplos de Diseño
5/112
*tep 1%
T"pe o! flter hih&pass flter
Filter specifcations%
Filter order N3C
*amplin !re5uenc" !s32678
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%
#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%
8/17/2019 Filtros FIR - Ejemplos de Diseño
6/112
*tep %
The desined FIR flter coe+cients are !ound via 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&/ illustrates the direct reali8ation o! desined FIR flter, #hereas fure 2&
4&4 illustrates the optimi8ed reali8ation o! desined FIR flter #hich is $ased on the
!act that all FIR flter coe+cients are, !or the sake o! linear phase characteristic,
s"mmetric a$out their middle element.
8/17/2019 Filtros FIR - Ejemplos de Diseño
7/112
Fiure 2&4&/. FIR flter direct reali8ation
Fiure 2&4&4. FIR flter optimi8ed reali8ation structure
2.4.1./ Example /
*tep 1%
8/17/2019 Filtros FIR - Ejemplos de Diseño
8/112
T"pe o! flter $and&pass flter
Filter specifcations%
Filter order N314
*amplin !re5uenc" !s32678
Pass$and cut&o0 !re5uenc" !c13/678, !c23-.-678
*tep 2%
Method flter desin usin rectanular #indo#
*tep /%
Filter order is predetermined, N314
9 total num$er o! flter coe+cients is larer $" 1, i.e. N:131-.
(oe+cients have indices $et#een and 14.
*tep 4%9ll coe+cients o! the rectanular #indo# have the same value e5ual to 1.
#;n< 3 1 ' = n =14
*tep -%
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed
as%
#here M is the index o! middle coe+cient.
Normali8ed cut&o0 !re5uencies @c1 and @c2 can $e !ound usin the !ollo#in
expressions%
8/17/2019 Filtros FIR - Ejemplos de Diseño
9/112
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
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.
There is no need to additionall" chane it.
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.
8/17/2019 Filtros FIR - Ejemplos de Diseño
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%
8/17/2019 Filtros FIR - Ejemplos de Diseño
11/112
T"pe o! flter $and&stop flter
Filter specifcations%
Filter order N314
*amplin !re5uenc" !s32678
*top$and cut&o0 !re5uenc" !c13/678, !c23-.-678
*tep 2%
Method flter desin usin rectanular #indo#
*tep /%
Filter order is predetermined, N314'
9 total num$er o! flter coe+cients is larer $" 1, i.e. N:131-' and
(oe+cients have indices $et#een and 14.
*tep 4%9ll coe+cients o! the rectanular #indo# have the same value e5ual to 1.
#;n< 3 1 ' = n = 14
*tep -%
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed
as%
#here M is the index o! middle coe+cient.
Normali8ed cut&o0 !re5uencies @c1 and @c2 can $e !ound usin expressions%
8/17/2019 Filtros FIR - Ejemplos de Diseño
12/112
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M and @c1 and @c2 #ith expression !or the impulse response coe+cients
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?, $ut have the
opposite sin.
*tep %
The desined FIR flter coe+cients are !ound via expression%
The FIR flter coe+cients h;n
8/17/2019 Filtros FIR - Ejemplos de Diseño
13/112
Fiure 2&4&A. FIR flter direct reali8ation
Fiure 2&4&C. FIR flter optimi8ed reali8ation structure
8/17/2019 Filtros FIR - Ejemplos de Diseño
14/112
2.4.2 Filter desin usin artlett #indo#
2.4.2.1 Example 1
*tep 1%
T"pe o! flter lo#&pass flterFilter specifcations%
Filter order N!3D
*amplin !re5uenc" !s32678
Pass$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#%
#here M is the index o! middle coe+cient.
8/17/2019 Filtros FIR - Ejemplos de Diseño
15/112
*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 $e calculated usin expression%
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M and @c #ith expression !or the impulse response coe+cients o! the
ideal lo#&pass flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n =D
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&D illustrates the direct reali8ation o! desined FIR flter, #hereas fure 2&
4&1 illustrates optimi8ed reali8ation o! desined FIR flter #hich is $ased on the !act
that all FIR flter coe+cients are, !or the sake o! linear phase characteristic,
s"mmetric a$out their middle element.
8/17/2019 Filtros FIR - Ejemplos de Diseño
16/112
Fiure 2&4&D. FIR flter direct reali8ation
Fiure 2&4&1. FIR flter optimi8ed reali8ation structure
2.4.2.2 Example 2
*tep 1%
8/17/2019 Filtros FIR - Ejemplos de Diseño
17/112
T"pe o! flter hih&pass flter
Filter specifcations%
Filter order N!3C
*amplin !re5uenc" !s32678
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 -%
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed
as%
#here M is the index o! middle coe+cient.
8/17/2019 Filtros FIR - Ejemplos de Diseño
18/112
Normali8ed cut&o0 !re5uenc" @c ma" $e calculated via the !ollo#in expression%
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M and @c #ith expression !or the impulse response coe+cients o! the
ideal hih&pass flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = C
The FIR flter coe+cients h;n< rounded to diits are%
*tep A%
Filter order is predetermined.
There is no need to additionall" chane it.
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 $ased on the
!act that all FIR flter coe+cients are, !or the sake o! linear phase characteristic,
s"mmetric a$out their middle element.
8/17/2019 Filtros FIR - Ejemplos de Diseño
19/112
Fiure 2&4&11. FIR flter direct reali8ation
Fiure 2&4&12. FIR flter optimi8ed reali8ation structure
2.4.2./ Example /
*tep 1%
8/17/2019 Filtros FIR - Ejemplos de Diseño
20/112
T"pe o! flter $and&pass flter
Filter specifcations%
Filter order N!314'
*amplin !re5uenc" !s32678' and
Pass$and cut&o0 !re5uencies !c13/678, !c23-.-678.
*tep 2%
Method flter desin usin artlett #indo#
*tep /%
Filter order is predetermined, 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% The alett #indo# coe+cients are !ound via expression%
*tep -%
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed
as%
#here M is the index o! middle coe+cient.
8/17/2019 Filtros FIR - Ejemplos de Diseño
21/112
Normali8ed cut&o0 !re5uencies @c1 and @c2 can $e calculated usin expressions%
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M and @c1 and @c2 #ith expression !or the impulse response coe+cients
o! the ideal $and&pass flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 14
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 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.
8/17/2019 Filtros FIR - Ejemplos de Diseño
22/112
Fiure 2&4&1/. FIR flter direct reali8ation
Fiure 2&4&14. FIR flter optimi8ed reali8ation structure
8/17/2019 Filtros FIR - Ejemplos de Diseño
23/112
2.4.2.4 Example 4
*tep 1%
T"pe o! flter $and&stop flter
Filter specifcations%
Filter order N!314'
*amplin !re5uenc" !s32678' and
*top$and cut&o0 !re5uencies !c13/678, !c23-.-678.
*tep 2%
Method flter desin usin artlett #indo#
*tep /%
Filter order is predetermined, 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%
The coe+cients o! artlett #indo# are !ound via expression%
*tep -%
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed
as%
#here M is the index o! middle coe+cient.
8/17/2019 Filtros FIR - Ejemplos de Diseño
24/112
Normali8ed cut&o0 !re5uencies @c1 and @c2 can $e calculated usin expressions%
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M, @c1 and @c2 #ith expression !or the impulse response coe+cients o!
the ideal $and&stop flter%
Note that, exceptin the middle element, all the coe+cients are the same as in the
previous example >$and&pass flter #ith the same cut&o0 !re5uencies?, $ut have the
opposite sin.
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 14
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 fure
2&4&1 illustrates optimi8ed reali8ation o! desined FIR flter #hich is $ased on the
!act that all FIR flter coe+cients are, !or the sake o! linear phase characteristic,
s"mmetric a$out their middle element.
8/17/2019 Filtros FIR - Ejemplos de Diseño
25/112
Fiure 2&4&1-. FIR flter direct reali8ation
Fiure 2&4&1. FIR flter optimi8ed reali8ation structure
8/17/2019 Filtros FIR - Ejemplos de Diseño
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%
T"pe o! flter lo#&pass flter
Filter specifcations%
Filter order N!31'*amplin !re5uenc" !s32678' and
Pass$and cut&o0 !re5uenc" !c32.-678.
*tep 2%
Method flter desin usin 7ann #indo#
*tep /%
Filter order is predetermined, N!31'
9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:1311' and
(oe+cients have indices $et#een and 1.
*tep 4%
The 7ann #indo# !unction coe+cients are !ound via expression%
8/17/2019 Filtros FIR - Ejemplos de Diseño
27/112
*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 $e calculated usin expression%
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M and @c #ith expression !or the impulse reaponse coe+cients o! the
ideal lo#&pass flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 1
The FIR flter coe+cients h;n< rounded to diits are%
8/17/2019 Filtros FIR - Ejemplos de Diseño
28/112
*tep A%
The flter order is predetermined.
There is no need to additionall" chane it.
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
$ased on the !act that all FIR flter coe+cients are, !or the sake o! linear phase
characteristic, s"mmetric a$out their middle element.
Fiure 2&4&1A. FIR flter direct reali8ation
8/17/2019 Filtros FIR - Ejemplos de Diseño
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'
*amplin !re5uenc" !s32678' and
Pass$and cut&o0 !re5uenc" !c32.-678.
*tep 2%
Method flter desin usin 7ann #indo#
*tep /%
Filter order is predetermined, N!31'
9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:1311' and
(oe+cients have indices $et#een and 1.
*tep 4%
The 7ann #indo# !unction coe+cients are !ound via expression%
*tep -%
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressedas%
8/17/2019 Filtros FIR - Ejemplos de Diseño
30/112
#here M is the index o! middle coe+cient.
Normali8ed cut&o0 !re5uenc" @c can $e calculated usin expression%
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M and @c #ith expression !or the impulse response coe+cients o! the
ideal hih&pass flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 1
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&1D illustrates the direct reali8ation o! desined FIR flter, #hereas fure2&4&2 illustrates optimi8ed reali8ation structure o! desined FIR flter #hich is
8/17/2019 Filtros FIR - Ejemplos de Diseño
31/112
$ased on the !act that all FIR flter coe+cients are, !or the sake o! linear phase
characteristic, s"mmetric a$out their middle element.
Fiure 2&4&1D. FIR flter direct reali8ation
Fiure 2&4&2. FIR flter optimi8ed reali8ation structure
2.4././ Example /
8/17/2019 Filtros FIR - Ejemplos de Diseño
32/112
*tep 1%
T"pe o! flter $and&pass flter
Filter specifcations%
Filter order N!314'
*amplin !re5uenc" !s32678' and
Pass$and cut&o0 !re5uenc" !c13/678, !c23-.-678.
*tep 2%
Method flter desin usin 7ann #indo#
*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%
The 7ann #indo# !unction coe+cients are !ound via expression%
*tep -%
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed
as%
#here M is the index o! middle coe+cient.
Normali8ed cut&o0 !re5uencies @c1 and @c2 can $e calculated usin expressions%
8/17/2019 Filtros FIR - Ejemplos de Diseño
33/112
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M, @c1 and @c2 #ith expression !or the impulse response coe+cients o!
the ideal $and&pass flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 14
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&21 illustrates the direct reali8ation o! desined FIR flter, #hereas fure
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.
8/17/2019 Filtros FIR - Ejemplos de Diseño
34/112
Fiure 2&4&21. FIR flter direct reali8ation
Fiure 2&4&22. FIR flter optimi8ed reali8ation structure
8/17/2019 Filtros FIR - Ejemplos de Diseño
35/112
2.4././ Example /
*tep 1%
T"pe o! flter $and&stop flter
Filter specifcations%
Filter order N!314'
*amplin !re5uenc" !s32678' and
Pass$and cut&o0 !re5uenc" !c13/678, !c23-.-678.
*tep 2%
Method flter desin usin 7ann #indo#
*tep /%
Filter order is predetermined, N!314'
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%
The 7ann #indo# !unction coe+cients are !ound via expression%
*tep -%
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed
as%
#here M is the index o! middle coe+cient.
Normali8ed cut&o0 !re5uencies @c1 and @c2 can $e calculated usin expressions%
8/17/2019 Filtros FIR - Ejemplos de Diseño
36/112
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M, @c1 and @c2 #ith expression !or the impulse response coe+cients o!
the ideal $and&stop flter%
Note that, exceptin the middle element, all coe+cients are the same as in the
previous example >$and&pass flter #ith the same cut&o0 !re5uencies?, $ut have the
opposite sin.
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n
8/17/2019 Filtros FIR - Ejemplos de Diseño
37/112
Fiure 2&4&2/. FIR flter direct reali8ation
Fiure 2&4&24. FIR flter optimi8ed reali8ation structure
8/17/2019 Filtros FIR - Ejemplos de Diseño
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
$hp is the middle coe+cient o! a hih&pass flter.
ecause o! such similarit", it is eas" to convert a lo#&pass FIR flter into a hih&pass
FIR flter havin the same cut&o0 !re5uencies, sampilin !re5uenc" and flter order.
2.4.4 Filter desin usin artlett&7annin #indo#
2.4.4.1 Example 1
*tep 1%
T"pe o! flter lo#&pass flter
Filter specifcations%
Filter order N!3D'
*amplin !re5uenc" !s322-78' and
Pass$and cut&o0 !re5uenc" !c34678.
*tep 2%
Method flter desin usin artlett&7annin #indo#
*tep /%
Filter order is predetermined, N!3D'
9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:131' and
(oe+cients have indices $et#een and D.
*tep 4%
The artlett&7annin #indo# !unction coe+cients are !ound via expression%
*tep -%
The ideal lo#&pass flter coe+cients >ideal flter impulse response? are expressed as%
8/17/2019 Filtros FIR - Ejemplos de Diseño
39/112
#here M is the index o! middle coe+cient.
Normali8ed cut&o0 !re5uenc" @c can $e calculated usin expression%
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M and @c #ith expression !or the impulse response coe+cients o! the
ideal lo#&pass flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = D
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&2- illustrates the direct reali8ation o! desined FIR flter, #hereas fure
2&4&2 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.
8/17/2019 Filtros FIR - Ejemplos de Diseño
40/112
Fiure 2&4&2-. FIR flter direct reali8ation
Fiure 2&4&2. FIR flter optimi8ed reali8ation structure
8/17/2019 Filtros FIR - Ejemplos de Diseño
41/112
2.4.4.2 Example 2
*tep 1%
T"pe o! flter hih&pass flter
Filter specifcations%
Filter order N!31'
*amplin !re5uenc" !s322-78' and
Pass$and cut&o0 !re5uenc" !c34678.
*tep 2%
Method flter desin usin artlett&7annin #indo#
*tep /%
Filter order is predetermined, N!31'
9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:1311' and(oe+cients have indices $et#een and 1.
*tep 4%
The artlett&7annin #indo# !unction coe+cients are !ound via expression%
*tep -%
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed
as%
#here M is the index o! middle coe+cient.
8/17/2019 Filtros FIR - Ejemplos de Diseño
42/112
8/17/2019 Filtros FIR - Ejemplos de Diseño
43/112
Fiure 2&4&2A. FIR flter direct reali8ation
Fiure 2&4&2C. FIR flter optimi8ed reali8ation structure
8/17/2019 Filtros FIR - Ejemplos de Diseño
44/112
2.4.4./ Example /
*tep 1%
T"pe o! flter $and&pass flter
Filter specifcations%
Filter order N!312'
*amplin !re5uenc" !s322-78' and
Pass$and cut&o0 !re5uenc" !c132678, !c23-678.
*tep 2%
Method flter desin usin artlett&7annin #indo#
*tep /%
Filter order is predetermined, N!312'
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%
The artlett&7annin #indo# !unction coe+cients are !ound via expression%
*tep -%
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed
as%
#here M is the index o! middle coe+cient.
8/17/2019 Filtros FIR - Ejemplos de Diseño
45/112
Normali8ed cut&o0 !re5uencies @c1 and @c2 can $e calculated usin expressions%
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M, @c1 and @c2 #ith expression !or the impulse response coe+cients o!
the ideal $and&pass flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 12
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&2D illustrates the direct reali8ation o! desined FIR flter, #hereas fure
2&4&/ 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.
8/17/2019 Filtros FIR - Ejemplos de Diseño
46/112
Fiure 2&4&2D. FIR flter direct reali8ation
8/17/2019 Filtros FIR - Ejemplos de Diseño
47/112
Fiure 2&4&/. FIR flter optimi8ed reali8ation structure
2.4.4.4 Example 4
*tep 1%
T"pe o! flter $and&stop flterFilter specifcations%
Filter order N!312'
*amplin !re5uenc" !s322-78' and
Pass$and cut&o0 !re5uencies !c132678, !c23678.
*tep 2%
Method flter desin usin artlett&7annin #indo#
*tep /%
Filter order is predetermined, N!312'
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%
The artlett&7annin #indo# !unction coe+cients are !ound via expression%
*tep -%
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed
as%
#here M is the index o! middle coe+cient.
8/17/2019 Filtros FIR - Ejemplos de Diseño
48/112
Normali8ed cut&o0 !re5uencies @c1 and @c2 can $e calculated usin expressions%
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M, @c1 and @c2 #ith expression !or the impulse reaponse coe+cients o!
the ideal $and&stop flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 12
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 fure
2&4&/2 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.
8/17/2019 Filtros FIR - Ejemplos de Diseño
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%
8/17/2019 Filtros FIR - Ejemplos de Diseño
50/112
T"pe o! flter lo#&pass flter
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
8/17/2019 Filtros FIR - Ejemplos de Diseño
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 $e hal! o!
that.
Filter order is N!31'
9 total num$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 coe+cients are !ound via expression%
*tep -%
The ideal lo#&pass flter coe+cients >ideal flter impulse response? are expressed as%
8/17/2019 Filtros FIR - Ejemplos de Diseño
52/112
#here M is the index o! middle coe+cient.
Normali8ed cut&o0 !re5uenc" @c can $e calculated usin expression%
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M and @c #ith expression !or the impulse response coe+cients o! the
ideal lo#&pass flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 1
The FIR flter coe+cients h;n< rounded to diits are%
*tep A%
9nal"se in the !re5uenc" domain is per!ormed usin the Filter )esiner Tool
proram.
8/17/2019 Filtros FIR - Ejemplos de Diseño
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%
8/17/2019 Filtros FIR - Ejemplos de Diseño
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 $" t#o. The #hole process o! desinin flter is
repeated !rom the step /.
*tep /%
Filter order is N!312'
9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:131/' and(oe+cients have indices $et#een and 12.
*tep 4%
The 7ammin #indo# !unction coe+cients are !ound via expression%
8/17/2019 Filtros FIR - Ejemplos de Diseño
55/112
*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 $e calculated usin expression%
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M and @c #ith expression !or the impulse response coe+cients o! the
ideal lo#&pass flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 12
The FIR flter coe+cients h;n< rounded to diits are%
*tep A%
8/17/2019 Filtros FIR - Ejemplos de Diseño
56/112
9nal"se in the !re5uenc" domain is per!ormed usin the Filter )esiner Tool
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
2&4&/ 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.
8/17/2019 Filtros FIR - Ejemplos de Diseño
57/112
Fiure 2&4&/-. FIR flter direct reali8ation
Fiure 2&4&/. FIR flter optimi8ed reali8ation structure
8/17/2019 Filtros FIR - Ejemplos de Diseño
58/112
2.4.-.2 Example 2
*tep 1%
T"pe o! flter hih&pass flter
Filte specifcations%
Filter order N!31'
*amplin !re5uenc" !s322-78' and
Pass$and cut&o0 !re5uenc" !c34678.
*tep 2%
Method flter desin usin 7ammin #indo#
*tep /%
Filter order is predetermined, N!31'
9 total num$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 coe+cients are !ound via expression%
The 7ammin #indo# !unction is one o! rare standard #indo#s #here #;< O is in
e0ect.
*tep -%
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed
as%
#here M is the index o! middle coe+cient.
8/17/2019 Filtros FIR - Ejemplos de Diseño
59/112
Normali8ed cut&o0 !re5uenc" @c can $e calculated usin expression%
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M and @c #ith expression !or the impulse response coe+cients o! the
ideal hih&pass flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 1
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&/A illustrates the direct reali8ation o! desined FIR flter, #hereas fure
2&4&/C 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.
8/17/2019 Filtros FIR - Ejemplos de Diseño
60/112
Fiure 2&4&/A. FIR flter direct reali8ation
Fiure 2&4&/C. FIR flter optimi8ed reali8ation structure
8/17/2019 Filtros FIR - Ejemplos de Diseño
61/112
2.4.-./ Example /
*tep 1%
T"pe o! flter $and&pass flter
Filte specifcations%
Filter order N!312'
*amplin !re5uenc" !s31678'
Pass$and cut&o0 !re5uenc" !c132678, !c23-678.
*tep 2%
Method flter desin usin 7ammin #indo#
*tep /%
Filter order is predetermined, N!312'
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%
The 7ammin #indo# !unction coe+cients are !ound via expression%
*tep -%
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed
as%
#here M is the index o! middle coe+cient.
8/17/2019 Filtros FIR - Ejemplos de Diseño
62/112
Normali8ed cut&o0 !re5uencies @c1 and @c2 can $e calculated usin expressions%
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M, @c1 and @c2 #ith expression !or the impulse response coe+cients o!
the ideal $and&pass flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 12
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&/D illustrates the direct reali8ation o! desined FIR flter, #hereas fure
2&4&4 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 elements.
8/17/2019 Filtros FIR - Ejemplos de Diseño
63/112
Fiure 2&4&/D. FIR flter direct reali8ation
Fiure 2&4&4. FIR flter optimi8ed reali8ation structure
8/17/2019 Filtros FIR - Ejemplos de Diseño
64/112
2.4.-.4 Example 4
*tep 1%
T"pe o! flter $and&stop flter
Filte specifcations%
Filter order N!312'
*amplin !re5uenc" !s3178' and
Pass$and cut&o0 !re5uenc" !c132678, !c23678.
*tep 2%
Method flter desin usin 7ammin #indo#
*tep /%
Filter order is predetermined, N!312'
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%
The artlett&7annin #indo# !unction coe+cients are !ound via expression%
*tep -%
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed
as%
#here M is the index o! middle coe+cient.
8/17/2019 Filtros FIR - Ejemplos de Diseño
65/112
Normali8ed cut&o0 !re5uencies @c1 and @c2 can $e calculated usin expressions%
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M, @c1 and @c2 #ith expression !or the impulse response coe+cients o!
the ideal $and&stop flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 12
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&41 illustrates the direct reali8ation o! desined FIR flter, #hereas fure
2&4&42 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.
8/17/2019 Filtros FIR - Ejemplos de Diseño
66/112
Fiure 2&4&41. FIR flter direct reali8ation
Fiure 2&4&42. FIR flter optimi8ed reali8ation structure
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
8/17/2019 Filtros FIR - Ejemplos de Diseño
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%
T"pe o! flter lo#&pass flter
Filte specifcations%
Filter order N!31'
*amplin !re5uenc" !s32678' and
Pass$and cut&o0 !re5uenc" !c3-678.
*tep 2%
Method flter desin usin ohman #indo#
*tep /%
Filter order is N!31'
9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:1311' and
(oe+cients have indices $et#een and 1.
*tep 4%
The ohman #indo# !unction coe+cients are !ound via expression%
8/17/2019 Filtros FIR - Ejemplos de Diseño
68/112
*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 ma" $e computed usin expression%
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M and @c #ith expression !or the impulse response coe+cients o! the
ideal lo#&pass flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 1
The FIR flter coe+cients h;n< rounded to diits are%
*tep A%
8/17/2019 Filtros FIR - Ejemplos de Diseño
69/112
The flter order is predetermined.
There is no need to additionall" chane it.
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$ased on the !act that all FIR flter coe+cients are, !or the sake o! linear phase
characteristic, s"mmetric a$out their middle element.
Fiure 2&4&4/. FIR flter direct reali8ation
8/17/2019 Filtros FIR - Ejemplos de Diseño
70/112
Fiure 2&4&44. FIR flter optimi8ed reali8ation structure
2.4..2 Example 2
*tep 1%
T"pe o! flter hih&pass flter
Filte specifcations%
*amplin !re5uenc" !s322-78'
Pass$and cut&o0 !re5uenc" !c131.-678'
*top$and cut&o0 !re5uenc" !c234678' and
Minimum stop$and attenuation /-d.*tep 2%
Method flter desin usin ohman #indo#
*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 $elo#%
HIN)BH
FN(TIB
N
NBRM9JI
KE)
JENLT7
BF T7E
M9IN
JBE FBR
N32
TR9N*ITI
BN
RELIBN
FBR
N32
MINIMM
*TBP9N)
9TTEN9TI
BN BF
HIN)BH
FN(TIBN
MINIMM
*TBP9N)
9TTEN9TI
BN BF
)E*ILNE)
FIJTER
Rectanu
lar.1 .41 1/ d 21 d
Trianula .2 .11 2 d 2 d
8/17/2019 Filtros FIR - Ejemplos de Diseño
71/112
r
>artlett?
7ann .21 .12 /1 d 44 d
artlett&
7annin.21 .1/ / d /D d
7ammin
.2/ .14 41 d -/ d
ohman ./1 .2 4 d -1 d
lackma
n./2 .2 -C d A- d
lackma
n&7arris.4/ ./2 D1 d 1D d
Ta$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 $" 2 !or the purpose o!
defnin initial value.
8/17/2019 Filtros FIR - Ejemplos de Diseño
72/112
Filter order is N!31C'
9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:131D' and
(oe+cients have indices $et#een and 1C.
*tep 4%
The coe+cients o! ohman #indo# are !ound via expression%
*tep -%
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed
as%
#here M is the index o! middle coe+cient.
Normali8ed cut&o0 !re5uenc" @c is e5ual to pass$and cut&o0 !re5uenc"%
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 hih&pass flter%
8/17/2019 Filtros FIR - Ejemplos de Diseño
73/112
8/17/2019 Filtros FIR - Ejemplos de Diseño
74/112
*tep /%
Filter order is N!31'
9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:131A' and
(oe+cients have indices $et#een and 1.
*tep 4%
The coe+cients o! ohman #indo# are !ound via expression%
*tep -%
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressedas%
#here M is the index o! middle coe+cient.
Normali8ed cut&o0 !re5uenc" @c is e5ual to the pass$and cut&o0 !re5uenc"%
8/17/2019 Filtros FIR - Ejemplos de Diseño
75/112
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 hih&pass flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 1
The FIR flter coe+cients h;n< rounded to diits are%
*tep A%
9nal"se in the !re5uenc" domain is per!ormed usin the Filter )esiner Tool
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.
8/17/2019 Filtros FIR - Ejemplos de Diseño
76/112
*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
(oe+cients have indices $et#een and 14.
*tep 4%
The coe+cients o! ohman #indo# are !ound via expression%
*tep -%
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed
as%
#here M is the index o! middle coe+cient.
Normali8ed cut&o0 !re5uenc" @c is e5ual to pass$and cut&o0 !re5enc"%
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 reaponse coe+cients o! the
ideal hih&pass flter%
8/17/2019 Filtros FIR - Ejemplos de Diseño
77/112
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 14
The FIR flter coe+cients h;n< rounded to diits are%
*tep A%
9nal"se in the !re5uenc" domain is per!ormed usin the Filter )esiner Tool
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%
8/17/2019 Filtros FIR - Ejemplos de Diseño
78/112
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
$ased on the !act that all FIR flter coe+cients are, !or the sake o! linear phase
characteristic, s"mmetric a$out their middle element.
Fiure 2&4&4C. FIR flter direct reali8ation
8/17/2019 Filtros FIR - Ejemplos de Diseño
79/112
Fiure 2&4&4D. FIR flter optimi8ed reali8ation structure
2.4../ Example /
*tep 1%
T"pe o! flter $and&pass flter
Filte specifcations%
Filter order N!312'
*amplin !re5uenc" !s344178' and
Pass$and cut&o0 !re5uenc" !c134678, !c231-2-78.
*tep 2%
Method flter desin usin ohman #indo#
*tep /%
Filter order is predetermined, N!312'
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%
The coe+cients o! ohman #indo# are !ound via expression%
*tep -%
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed
as%
8/17/2019 Filtros FIR - Ejemplos de Diseño
80/112
#here M is the index o! middle coe+cient.
Normali8ed cut&o0 !re5uencies @c1 and @c2 can $e calculated usin expressions%
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
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
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.
8/17/2019 Filtros FIR - Ejemplos de Diseño
81/112
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
$ased on the !act that all FIR flter coe+cients are, !or the sake o! linear phase
characteristic, s"mmetric a$out their middle element.
Fiure 2&4&-. FIR flter direct reali8ation
8/17/2019 Filtros FIR - Ejemplos de Diseño
82/112
Fiure 2&4&-1. FIR flter optimi8ed reali8ation structure
2.4..4 Example 4
*tep 1%
T"pe o! flter $and&stop flter
Filte specifcations%
Filter order N!312'
*amplin !re5uenc" !s3178' and
Pass$and cut&o0 !re5uenc" !c132678, !c23678.
*tep 2%
Method flter desin usin ohman #indo#
*tep /%
Filter order is predetermined, N!312'
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%
The coe+cients o! ohman #indo# are !ound via expression%
*tep -%
8/17/2019 Filtros FIR - Ejemplos de Diseño
83/112
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed
as%
#here M is the index o! middle coe+cient.
Normali8ed cut&o0 !re5uencies @c1 and @c2 can $e calculated usin expressions%
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M, @c1 and @c2 #ith expression !or the impulse response coe+cients o!
the ideal $and&stop flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 12
The FIR flter coe+cients h;n< rounded to diits are%
8/17/2019 Filtros FIR - Ejemplos de Diseño
84/112
*tep A%
The flter order is predetermined.
There is no need to additionall" chane it.
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
$ased on the !act that all FIR flter coe+cients are, !or the sake o! linear phase
characteristic, s"mmetric a$out their middle element.
Fiure 2&4&-2. FIR flter direct reali8ation
8/17/2019 Filtros FIR - Ejemplos de Diseño
85/112
Fiure 2&4&-/. FIR flter optimi8ed reali8ation structure
2.4.A Filter desin usin lackman #indo#
2.4.A.1 Example 1*tep 1%
T"pe o! flter lo#&pass flter
Filte specifcations%
Filter order N!312'
*amplin !re5uenc" !s3441678'
Pass$and cut&o0 !re5uenc" !c31-678' and
9ttenuation o! d at 78 d.
*tep 2%
Method Filter desin usin lackman #indo#
*tep /%
Filter order is predetermined, N!312'
9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:131/'
(oe+cients have indices $et#een and 12.
*tep 4%
The coe+cients o! lackman #indo# are !ound via expression%
*tep -%
8/17/2019 Filtros FIR - Ejemplos de Diseño
86/112
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 $e computed usin expression%
The values o! coe+cients are o$tained >rounded to six diits? $" com$inin the
values o! M and @c #ith expression !or the impulse response coe+cients o! the
ideal lo#&pass flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 12
The FIR flter coe+cients h;n< rounded to diits are%
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%
8/17/2019 Filtros FIR - Ejemplos de Diseño
87/112
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%
The flter order is predetermined.
There is no need to additionall" chane it.
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
$ased on the !act that all FIR flter coe+cients are, !or the sake o! linear phase
characteristic, s"mmetric a$out their middle element.
8/17/2019 Filtros FIR - Ejemplos de Diseño
88/112
Fiure 2&4&-4. FIR flter direct reali8ation
Fiure 2&4&--. FIR flter optimi8ed reali8ation structure
8/17/2019 Filtros FIR - Ejemplos de Diseño
89/112
2.4.A.2 Example 2
*tep 1%
T"pe o! flter hih&pass flter
Filte specifcations%
Filter order N!312'
*amplin !re5uenc" !s322-78'
Pass$and cut&o0 !re5uenc" !c34678'
Prevention o! possi$le flterin overSo#s.
*tep 2%
Method flter desin usin lackman #indo#
*tep /%
Filter order is N!312'
9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:131/'
(oe+cients have indices $et#een and 12.
*tep 4%
The coe+cients o! lackman #indo# !unction are !ound via%
*tep -%
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed
as%
#here M is the index o! middle coe+cient.
8/17/2019 Filtros FIR - Ejemplos de Diseño
90/112
Normali8ed cut&o0 !re5uenc" @c can $e computed usin expression%
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M and @c #ith expression !or the impulse response coe+cients o! the
ideal hih&pass flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 12
The FIR flter coe+cients h;n< rounded to diits are%
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%
8/17/2019 Filtros FIR - Ejemplos de Diseño
91/112
*tep A%
The flter order is predetermined.
There is no need to additionall" chane it.
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
$ased on the !act that all FIR flter coe+cients are, !or the sake o! linear phase
characteristic, s"mmetric a$out their middle element.
Fiure 2&4&-. FIR flter direct reali8ation
8/17/2019 Filtros FIR - Ejemplos de Diseño
92/112
Fiure 2&4&-A. FIR flter optimi8ed reali8ation structure
2.4.A./ Example /
*tep 1%
T"pe o! flter $and&pass flter
Filte specifcations%
Filter order N!312'
*amplin !re5uenc" !s344178'
Pass$and cut&o0 !re5uenc" !c134678, !c231-2-78'
Prevention o! possi$le flterin overSo#.
*tep 2%Method flter desin usin lackman #indo#
*tep /%
Filter order is predetermined, N!312'
9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:131/'
(oe+cients have indices $et#een and 12.
*tep 4%
The coe+cients o! lackman #indo# !unction are !ound via expression%
*tep -%
8/17/2019 Filtros FIR - Ejemplos de Diseño
93/112
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed
as%
#here M is the index o! middle coe+cient.
Normali8ed cut&o0 !re5uencies @c1 and @c2 can $e computed usin expressions%
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M, @c1 and @c2 #ith expression !or the impulse response coe+cients o!
the ideal $and&pass flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 12
The FIR flter coe+cients h;n< rounded to diits are%
8/17/2019 Filtros FIR - Ejemplos de Diseño
94/112
In order to prevent flterin overSo#s, 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? $" 1.12-. 9!ter this, their
values are%
*tep A%
The flter order is predetermined.
There is no need to additionall" chane it.
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
$ased on the !act that all FIR flter coe+cients are, !or the sake o! linear phase
characteristic, s"mmetric a$out their middle element.
8/17/2019 Filtros FIR - Ejemplos de Diseño
95/112
Fiure 2&4&-C. FIR flter direct reali8ation
Fiure 2&4&-D. FIR flter optimi8ed reali8ation structure
8/17/2019 Filtros FIR - Ejemplos de Diseño
96/112
2.4.A.4 Example 4
*tep 1%
T"pe o! flter $and&stop flter
Filte specifcations%
Filter order N!312'
*amplin !re5uenc" !s3178'
Pass$and cut&o0 !re5uencies !c132678, !c23678'
Prevention o! possi$le flterin overSo#s.
*tep 2%
Method flter desin usin lackman #indo#
*tep /%
Filter order is predetermined, N!312'
9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:131/'
(oe+cients have indices $et#een and 12.
*tep 4%
The coe+cients o! lackman #indo# are !ound via expression%
*tep -%
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed
as%
#here M is the index o! middle coe+cient.
8/17/2019 Filtros FIR - Ejemplos de Diseño
97/112
Normali8ed cut&o0 !re5uencies @c1 and @c2 can $e computed usin expressions%
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin thevalues o! M, @c1 and @c2 #ith expression !or the impulse response coe+cients o!
the ideal $and&stop flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 12
The FIR flter coe+cients h;n< rounded to diits are%
In order to prevent flterin overSo#s, the !ollo#in condition must $e met%
The sum o! a$solute values o! the resultin FIR flter coe+cients is%
8/17/2019 Filtros FIR - Ejemplos de Diseño
98/112
The o$tained coe+cients must $e scaled >divided? $" .D1A. 9!ter this, their
values are%
*tep A%
The flter order is predetermined.
There is no need to additionall" chane it.
Filter reali8ation%
Fiure 2&4& illustrates the direct reali8ation o! desined FIR flter, #hereas fure
2&4&1 illustrates optimi8ed reali8ation structure o! desined FIR flter #hich is$ased on the !act that all FIR flter coe+cients are, !or the sake o! linear phase
characteristic, s"mmetric a$out their middle element.
Fiure 2&4&. FIR flter direct reali8ation
8/17/2019 Filtros FIR - Ejemplos de Diseño
99/112
Fiure 2&4&1. FIR flter optimi8ed reali8ation structure
2.4.C Filter desin usin lackman&7arris #indo#
2.4.C.1 Example 1
*tep 1%
T"pe o! flter lo#&pass flter
Filte specifcations%
Filter order N!312'
*amplin !re5uenc" !s3441678'
Pass$and cut&o0 !re5uenc" !c31-678'
9ttenuation o! d at 78.
*tep 2%
Method flter desin usn lackman&7arris #indo#
*tep /%
Filter order is predetermined, N!312'
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%
The coe+cients o! lackman&7arris #indo# are !ound via expression%
*tep -%
8/17/2019 Filtros FIR - Ejemplos de Diseño
100/112
The ideal hih&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 ma" $e calculated usin expression%
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M and @c #ith expression !or the impulse response coe+cients o! the
ideal lo#&pass flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 12
The FIR flter coe+cients h;n< rounded to diits are%
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%
8/17/2019 Filtros FIR - Ejemplos de Diseño
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%
The flter order is predetermined.
There is no need to additionall" chane it.
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
$ased on the !act that all FIR flter coe+cients are, !or the sake o! linear phase
characteristic, s"mmetric a$out their middle element.
8/17/2019 Filtros FIR - Ejemplos de Diseño
102/112
Fiure 2&4&2. FIR flter
Fiure 2&4&/. Bptimi8ed FIR flter desin
2.4.C.2 Example 2
*tep 1%
8/17/2019 Filtros FIR - Ejemplos de Diseño
103/112
T"pe o! flter hih&pass flter
Filte specifcations%
Filter order N!312'
*amplin !re5uenc" !s322-78'
Pass$and cut&o0 !re5uenc" !c34678'
Prevention o! flterin overSo#s.
*tep 2%
Method flter desin usin lackman&7arris #indo#
*tep /%
Filter order is N!312'
9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:131/' and
(oe+cients have indices $et#een and 12.*tep 4%
The coe+cients o! lackman&7arris #indo# are !ound via%
*tep -%
The ideal hih&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 $e computed usin expression%
8/17/2019 Filtros FIR - Ejemplos de Diseño
104/112
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M and @c #ith the expression !or the impulse response coe+cients o! the
ideal hih&pass flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 12
The FIR flter coe+cients h;n< rounded to diits are%
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./DAAD1. 9!ter this, theirvalues are%
*tep A%
The flter order is predetermined.
There is no need to additionall" chane it.
Filter reali8ation%
8/17/2019 Filtros FIR - Ejemplos de Diseño
105/112
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
$ased on the !act that all FIR flter coe+cients are, !or the sake o! linear phase
characteristic, s"mmetric a$out their middle element.
Fiure 2&4&4. FIR flter direct reali8ation
8/17/2019 Filtros FIR - Ejemplos de Diseño
106/112
Fiure 2&4&-. Bptimi8ed FIR flter desin
2.4.C./ Example /
*tep 1%
T"pe o! flter $and&pass flterFilter specifcation%
Filter order N!312'
*amplin !re5uenc" !s344178'
Pass$and cut&o0 !re5uencies !c134678, !c231-2-78' and
Prevention o! possi$le flterin overSo#s.
*tep 2%
Method flter desin usin lackman&7arris #indo#
*tep /%
Filter order is predetermined, N!312'
9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:131/' and
(oe+cients have indices $et#een and 12.
*tep 4%
The coe+cients o! lackman&7arris #indo# are !ound via expression%
*tep -%
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed
as%
#here M is the index o! middle coe+cient.
8/17/2019 Filtros FIR - Ejemplos de Diseño
107/112
Normali8ed cut&o0 !re5uencies @c1 and @c2 can $e computed usin expressions%
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M, @c1 and @c2 #ith expression !or the impulse response coe+cients o!
the ideal $and&pass flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 12
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 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
$ased on the !act that all FIR flter coe+cients are, !or the sake o! linear phasecharacteristic, s"mmetric a$out their middle element.
8/17/2019 Filtros FIR - Ejemplos de Diseño
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%
8/17/2019 Filtros FIR - Ejemplos de Diseño
109/112
T"pe o! flter $and&stop flter
Filter specifcation%
Filter order N!32'
*amplin !re5uenc" !s3178'
Pass$and cut&o0 !re5uenc" !c132678, !c23678' and
Prevention o! possi$le flterin overSo#s.
*tep 2%
Method flter desin usin lackman&7arris #indo#
*tep /%
Filter order is predetermined, N!32'
9 total num$er o! flter coe+cients is larer $" 1, i.e. N3N!:1321' and
(oe+cients have indices $et#een and 2.*tep 4%
The coe+cients o! lackman&7arris #indo# are !ound via expression%
*tep -%
The ideal hih&pass flter coe+cients >ideal flter impulse response? are expressed
as%
#here M is the index o! middle coe+cient.
Normali8ed cut&o0 !re5uencies @c1 and @c2 can $e computed usin expressions%
8/17/2019 Filtros FIR - Ejemplos de Diseño
110/112
The values o! coe+cients >rounded to six diits? are o$tained $" com$inin the
values o! M, @c1 and @c2 #ith expression !or the impulse response coe+cients o!
the ideal $and&stop flter%
*tep %
The desined FIR flter coe+cients are !ound via expression%
h;n< 3 #;n< G hd;n< ' = n = 2
The FIR flter coe+cients h;n< rounded to diits are%
In order to prevent flterin overSo#s, 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? $" 1.122/2. 9!ter this, their
values are%
8/17/2019 Filtros FIR - Ejemplos de Diseño
111/112
*tep A%
The flter order is predetermined.
There is no need to additionall" chane it.
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
$ased on the !act that all FIR flter coe+cients are, !or the sake o! linear phase
characteristic, s"mmetric a$out their middle element.
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.
8/17/2019 Filtros FIR - Ejemplos de Diseño
112/112