Linear Phase Finite Impulse Response

8/10/2019 Linear Phase Finite Impulse Response

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1. OBJECTIVES

Design the Linear Phase Finite Impulse Response (FIR) Filter and a lowpass filter andbandpass filter through Window Design Technique of FIR

2. COMPONENTS

De!stop P" #atlab $ % with &ignal Processing Toolbo'

3. THEORIES

3.1. LINEAR PHASE FIR FILTER

mong all the ob ious ad antages that digital filters offer* the FIR filter can guaranteelinear phase characteristics There are man+ commerciall+ a ailable software pac!agesfor filter design ,owe er* without basic theoretical !nowledge of the FIR filter* it will bedifficult to use them

Filter coefficient !


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Filter tr"ct"re causal FIR filter whose impulse response is s+mmetrical is guaranteed to ha e alinear phase response (- en s+mmetr+ . /dd s+mmetr+)

Frequenc+ Response of an FIR Filter

To full+ designand implement a filter fi e steps are required0

(1) Filter specification

(2) "oefficient calculation(%) &tructure selection(3) &imulation (optional)($) Implementation

There are se eral different methods a ailable* the most popular are0

Window method

Frequenc+ samplingPar!s4#c"lellan

We will 5ust consider the window method

Linear phase is a one t+pe of a filter Filter need to modif+ a signal6s magnitude4spectrum when preser ing the signal6s time4domain wa eform as much as possible Thislinear phase filter can be di ided into four t+pe of FIR0

s+mmetric sequence of odd length s+mmetric sequence of e en length


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anti4s+mmetric sequence of odd length anti4s+mmetric sequence of e en length

There are four possible situation0 filter length e en or odd* and impulse response iseither s+mmetric or antis+mmetric 0

FI#$RE 2.%

3.2. FIR I AN& FIR II T'PE

The s+mmetric coefficients shown that the frequenc+ responses are of the

following form0

FIR I (# is e en* sequence is s+mmetric and of odd length)

,owe er* this s+stem has linear phase (the quantit+ inside the parenthesis is a realquantit+) and the phase dela+ is #72 samples


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8 FIR II (# is odd* the sequence is s+mmetric and of e en length)

9otethat this of the form,(:) ; e4 5


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,owe er* this s+stem has linear phase (the quantit+ inside the parenthesis is a realquantit+) and the phase dela+ is #72 samples

8 FIR II (# is odd* the sequence is s+mmetric and of e en length)

9otethat this of the form,(:) ; e4 5


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,(:) ; e45


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). RES$LTS

P*rt A

Properties of Linear4Phase Finite Impulse Response (FIR) Filters

T+,e-1 FIR Filter

# TL H script0

function [Hr,w,a,L] = Hr_Type1(h !" #o$pute% &$p'itu e re%pon%e Hr(w of a Type)1 L* + - fi'ter " )))))))))))))))))))))))))))))))))))))))))))))))))))))))))))" [Hr,w,a,L] = Hr_Type1(h" Hr = &$p'itu e -e%pon%e

" w = 5.. fre/uencie% 0etween [. pi] o er which Hr i% co$pute" a = Type)1 L* fi'ter coefficient%" L = r er of Hr " h = Type)1 L* fi'ter i$pu'%e re%pon%e

= 'en th(h ! L = ( )1 2!a = [h(L 1 2 h(L8)181 ]! " 19(L 1 row ector n = [.818L]! " (L 1 91 co'u$n ector w = [.8185..]: pi 5..!Hr = co%(w n a:!

;;h = [)4 1 )1 )2 5 6 5 )2 )1 1 )4]

h =

)4 1 )1 )2 5 6 5 )2 )1 1 )4

;; = 'en th(h ! n = .8 )1

n =

0 1 2 3 4 5 6 7 8 9 10

;;[Hr,w,a,L] = Hr_Type1(h !;;a,L

a =

6 10 -4 -2 2 -8

L =

5

;;a$a9 = $a9(a 1! a$in = $in(a )1!;;%u0p'ot(2,2,1 ! %te$(n,h ! a9i%([)1 2 L 1 a$in a$a9];;9'a0e'( :n: ! y'a0e'( :h(n : ! tit'e(: $pu'%e -e%pon%e:;;%u0p'ot(2,2,3 ! %te$(.8L,a ! a9i%([)1 2 L 1 a$in a$a9];;9'a0e'( :n: ! y'a0e'( :a(n : ! tit'e(:a(n coefficient%:;;%u0p'ot(2,2,2 ! p'ot(w pi,Hr ! ri


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> M = 21; alpha = (M-1) 2; n = 0!M-1

n = "#l$%ns 1 &h'#$ h 16

0 1 2 3 4 5 6 7 8 9 10 1112 13 14 15

"#l$%ns 17 &h'#$ h 21

16 17 18 19 20

>> hd = (c#s(p *(n-alpha))). (n-alpha); hd(alpha+1)=0;>> , ha% = (ha%% n (M)) ; h = hd .* , ha%; / ' , L = ' p 3(h);

>> s$bpl#&(2 2 1); s& %(n hd); & &l ( d al %p$ls R sp#ns )>> a s(/-1 M -1.2 1.2 ); lab l( n ); lab l( hd(n) )>> s$bpl#&(2 2 2); s& %(n , ha%);& &l ( a%% n nd#, )>> a s(/-1 M 0 1.2 ); lab l( n ); lab l( ,(n) )>> s$bpl#&(2 2 3); s& %(n h);& &l ( Ac&$al %p$ls R sp#ns )>> a s(/-1 M -1.2 1.2 ); lab l( n ); lab l( h(n) )>> s$bpl#&(2 2 4);pl#&(, p ' p ); & &l ( A%pl &$d R sp#ns ); ' d;>> lab l( :' $ nc n p $n &s ); lab l( sl#p n p $n &s ); a s(/01 0 1 );


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2.

/utput &imulation0

0 2 4 6 8 10 12 14 16 18 20

-1

-0.5

0

0.5

1

Ideal Impulse Response

n

h d ( n )

0 2 4 6 8 10 12 14 16 18 20

-1

-0.5

0

0.5

1

Actual Impulse R esponse

n

h ( n )


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. &ISC$SSIONS

.1. PART 1 / TYPE 1 - TYPE 4 LINEAR FIR FILTER)

6.1.1. MATLAB COMMANDS

n or er to et the wante output, %o$e at'a0 co$$an which corre%pon in the + - e/uation

are entere into at'a0 to 0e proce%%e A There are inc'u in %o$e con%tant% an aria0'e%A

Be'ow are con%tant% which 0een u%e in *art 1 e$on%tration%8

GL i% or er of HrA GL i% written a%, L = ( )1 2A

Gw i% fre/uencie% 0etween [., pi] o er which Hr i% co$pute A Gw i% w = [.8185..] G pi 5..A

Gn written a% n = [.818L]

Type-1 FIR Filters

+or type Hr, e/uation u%e i%, Hr = co% (w n a , where, a = [h(L 1 2 h(L8)181 ]A


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Type- FIR Filters

+or type Hr, e/uation u%e i%, Hr = co% (w n 0 , where, 0 = 2 [h(L8)181 ]A

Type-! FIR Filters

+or type Hr, e/uation u%e i%, Hr = %in (w n c , where, c = [ 2 h (L 18 )1 8 1 ]A

Type-4 FIR Filters

+or type Hr, e/uation u%e i%, Hr = %in (w n , where, = 2 [h(L8 )181 ]A

6.1. O"TP"T #RAP$S

+ro$ raph enerate 0y at'a0, there are no re%triction% on Hr(w either w=. or w=piA Fy%te$

po'e% %how% that there are three (3 po'e% on the ri ht of rea' part, one 9 p'ane whi'e the other

two %y$$etrica''y near y p'aneA n the ne ati e %i e of rea' part, there are two $ore po'e%

on the 9)p'aneA Fince po'e% pre%ent on the po%iti e %i e of rea' part, %y%te$ i% not %ta0'eA Ie

cannot ana'y?e ?ero% 0ecau%e error in co in A t %how% that we were $i%%in 'i0rary to

enerate ?ero co in A

+ro$ the p'ot%, Hr(w i% ?ero at w = piA *o'e% coor ination i% %a$e a% + - +i'ter% type)1 e9cept

that on the ne ati e %i e of rea' part, in%tea of the po'e% p'ace on 9)p'ane, po'e% f'oatin up to

.A5 an to ).A5A Fy%te$ a'%o un%ta0'eA

+ro$ the p'ot%, Hr(w = . at w = . an w = piA *o'e% coor ination are %a$e with + - +i'ter%Type)3A

+ro$ p'ot%, it can 0e o0%er e that Hr(w i% ?ero at w = .A The po'e% pattern are %a$e with

Type)2 an Type)3A

.2. PART

+or the %econ part of the 'a0, we were a%Ee to con%truct a i ita' 0an pa%% + - fi'ter u%in the


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fo''owin %pecification%!

Be'ow i% the i$a e of the i ita' 0an pa%% fi'ter that $u%t 0e pro uce A Jotice that the hi h'i hte

one% are the %e'ecte 0an pa%% fi'ter 0a%e on the re/uire$ent% i en a0o eA

To o thi%, we nee to i entify which

win ow e%i n to 0e u%e A +ir%t, we nee to $ea%ure the 0an wi th of the 0an %A The two tran%ition

0an %, an $u%t 0e the %a$e in the win ow e%i n (there i% no in epen ent contro' o er an A +or

, we can u%e the B'acE$an win owA

B'acE$an win ow e%i n u%e% the %a$e function of Hann win ow an Ha$$in win ow, e9cept it

contain% a %econ har$onic ter$ i en a% fo''ow%!

dB A

dBR

dBR

dB A

ss

p p

p p

ss

60 8.0 :edgestopbandupper

1 65.0 :edgepassbandupper

1 35.0 :edgepassbandlower

60 2.0 :edgestopbandlower

2

2

1

1

==

==

==

==

s p 111 = s p 222 = 1 2

== 21


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( )

+

=

otherwise;0

10;1

4cos08.0

12

cos5.042.0 MnM

nM

nn


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Ie a'%o nee the i ea' 0an pa%% fi'ter i$pu'%e re%pon%e function, A Therefore, the

&TL&B routine d al lp(,c M) i% %ufficient to eter$ine the i$pu'%e re%pon%e of an i ea'

0an pa%% fi'terA Ie a'%o inc'u e the $o ifie function of the fre/uency) o$ain p'ot% or :' < % ,

which return% the $a nitu e re%pon%e in a0%o'ute a% we'' a% in re'ati e B %ca'e, the pha%e re%pon%e an

the roup e'ay re%pon%eA

Ba%e on the re/uire$ent% i en, u%in the co$$an win ow of &TL&B, the fo''owin %trin of

co e are entere %o that it ec'are the po%ition or the e9act 0an wi th of the 0an pa%% fi'ter8

w%1 = .A2 pi! wp1 = .A35 pi! wp2 = .A65 pi! w%2 = .A< pi! &% = 6.!

Je9t, the tran%ition wi th i% ca'cu'ate 0y fin in the $ini$u$ ifference 0etween an !

tr_wi th = $in((wp1)w%1 ,(w%2)wp2 !

Fu$$ary of co$$on'y u%e win ow function characteri%tic%A

WindowNa e

!ransition Width" #in. $topband%ttenuation%ppro&i ate '&act (alues

)ectangular 21 d*

*artlett 25 d*

+anning 44 d*

+a ing 53 d**lac, an -4 d*

Je9t, fin the a'ue of fi'ter 'en th, M for the tran%itiona' wi th, (thi% i% eci e fro$ the

%u$$ary ta0'e i en a0o e u%in the e9act a'ue% ,

= cei'(11 pi tr_wi th 1

( )nhd

p s

4

M

1.8 M

8

M

6.1 M

8

M

6.2 M

8

M

6.6 M

12 M

11 M


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which return% the fo''owin re%u't% fro$ the &TL&B output co$$an !

M =

75

Je9t, entere the center fre/uency of the 0an pa%% fi'ter i en a% fo''ow%!

n = [.818 )1]!

wc1 = (w%1 wp1 2! wc2 = (wp2 w%2 2!

Then, we entere the i ea' 0an pa%% fi'ter functionA Fince there are two ifferent center fre/uency, the

a'ue of hd i% ca'cu'ate a% fo''ow%!

h = i ea'_'p(wc2, ) i ea'_'p(wc1, !

The a'ue% of hd wi'' %tore the attache a'ue of nu$0er of pu'%e%, n i en a0o eA

Je9t, run the B'acE$an win ow a% blac %an(M) an $u'tip'ie with the i en hd , an %tore in

h A

w_0'a = (0'acE$an( :!

h = h A w_0'a!

The function of :' < % i% u%e in thi% ti$e with the %tore a'ue of the pre iou% h A

[ 0,$a ,pha, r ,w] = fre/?_$(h,[1] !

i% i en 0y the efau't for$u'a a% !

e'ta_w =2 pi 1...!

100

2


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Then, fina''y we o0taine the actua' 0an pa%% ripp'e, R p an $ini$u$ %top0an attenuation, A sA The%e

are i en 0y the for$u'a a% fo''ow%!

-p = )$in( 0(wp1 e'ta_w 1818wp2 e'ta_w

&% = )roun ($a9( 0(w%2 e'ta_w 18185.1

which co$e% the re%u't% a% fo''ow%!

Rp =

0.0030

As =

75

Then, to con%truct the fo''owin output %i$u'ation, the fo''owin %trin co e of &TL&B co$$an

i% inputte a% fo''ow%!

%u0p'ot(2,2,1 ! %te$(n,h ! tit'e(: ea' $pu'%e -e%pon%e:

a9i%([. )1 ).A4 .A5] ! 9'a0e'( :n: ! y'a0e'( :h (n :

%u0p'ot(2,2,2 ! %te$(n,w_0'a ! tit'e( :B'acE$an Iin ow:

a9i%([. )1 . 1A1] ! 9'a0e'( :n: ! y'a0e'( :w(n :

%u0p'ot(2,2,3 ! %te$(n,h ! tit'e( :&ctua' $pu'%e -e%pon%e:

a9i%([. )1 ).A4 .A5] ! 9'a0e'( :n: ! y'a0e'( :h(n :

%u0p'ot(2,2,4 ! p'ot(w pi, 0 ! a9i%([. 1 )15. 1.] !

( )

( )1 01

log20

0 011log20

1

210

1

110

>>>+=

>+=

s

p

A

R


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Linear Phase Finite Impulse Response

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