linear phase finite impulse response
TRANSCRIPT
-
8/10/2019 Linear Phase Finite Impulse Response
1/30
1
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 !
-
8/10/2019 Linear Phase Finite Impulse Response
2/30
2
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)($) Implementation
There are se eral different methods a ailable* the most popular are0
Window method
Frequenc+ samplingPar!s4#c"lellan
We will 5ust consider the window method
Linear phase is a one t+pe of a filter Filter need to modif+ a signal6s magnitude4spectrum when preser ing the signal6s time4domain wa eform as much as possible Thislinear phase filter can be di ided into four t+pe of FIR0
s+mmetric sequence of odd length s+mmetric sequence of e en length
-
8/10/2019 Linear Phase Finite Impulse Response
3/30
3
anti4s+mmetric sequence of odd length anti4s+mmetric sequence of e en length
There are four possible situation0 filter length e en or odd* and impulse response iseither s+mmetric or antis+mmetric 0
FI#$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
-
8/10/2019 Linear Phase Finite Impulse Response
4/30
4
8 FIR II (# is odd* the sequence is s+mmetric and of e en length)
9otethat this of the form,(:) ; e4 5
-
8/10/2019 Linear Phase Finite Impulse Response
5/30
5
,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
-
8/10/2019 Linear Phase Finite Impulse Response
6/30
6
,(:) ; e45
-
8/10/2019 Linear Phase Finite Impulse Response
7/30
7
). 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
-
8/10/2019 Linear Phase Finite Impulse Response
8/30
> 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 $n &s ); a s(/01 0 1 );
-
8/10/2019 Linear Phase Finite Impulse Response
20/30
2.
/utput &imulation0
0 2 4 6 8 10 12 14 16 18 20
-1
-0.5
0
0.5
1
Ideal Impulse Response
n
h d ( n )
0 2 4 6 8 10 12 14 16 18 20
-1
-0.5
0
0.5
1
Actual Impulse R esponse
n
h ( n )
-
8/10/2019 Linear Phase Finite Impulse Response
21/30
21
. &ISC$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$$an which corre%pon in the + - e/uation
are entere into at'a0 to 0e proce%%e A There are inc'u in %o$e con%tant% an aria0'e%A
Be'ow are con%tant% which 0een u%e in *art 1 e$on%tration%8
GL i% or er of HrA GL i% written a%, L = ( )1 2A
Gw i% fre/uencie% 0etween [., pi] o er which Hr i% co$pute A Gw i% w = [.8185..] G pi 5..A
Gn written a% n = [.818L]
Type-1 FIR Filters
+or type Hr, e/uation u%e i%, Hr = co% (w n a , where, a = [h(L 1 2 h(L8)181 ]A
-
8/10/2019 Linear Phase Finite Impulse Response
22/30
22
Type- FIR Filters
+or type Hr, e/uation u%e i%, Hr = co% (w n 0 , where, 0 = 2 [h(L8)181 ]A
Type-! FIR Filters
+or type Hr, e/uation u%e i%, Hr = %in (w n c , where, c = [ 2 h (L 18 )1 8 1 ]A
Type-4 FIR Filters
+or type Hr, e/uation u%e i%, Hr = %in (w n , where, = 2 [h(L8 )181 ]A
6.1. O"TP"T #RAP$S
+ro$ raph enerate 0y at'a0, there are no re%triction% on Hr(w either w=. or w=piA Fy%te$
po'e% %how% that there are three (3 po'e% on the ri ht of rea' part, one 9 p'ane whi'e the other
two %y$$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$e with
Type)2 an Type)3A
.2. PART
+or the %econ part of the 'a0, we were a%Ee to con%truct a i ita' 0an pa%% + - fi'ter u%in the
-
8/10/2019 Linear Phase Finite Impulse Response
23/30
23
fo''owin %pecification%!
Be'ow i% the i$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$$in win ow, e9cept it
contain% a %econ har$onic ter$ i en a% fo''ow%!
dB A
dBR
dBR
dB A
ss
p p
p p
ss
60 8.0 :edgestopbandupper
1 65.0 :edgepassbandupper
1 35.0 :edgepassbandlower
60 2.0 :edgestopbandlower
2
2
1
1
==
==
==
==
s p 111 = s p 222 = 1 2
== 21
-
8/10/2019 Linear Phase Finite Impulse Response
24/30
24
( )
+
=
otherwise;0
10;1
4cos08.0
12
cos5.042.0 MnM
nM
nn
-
8/10/2019 Linear Phase Finite Impulse Response
25/30
25
Ie a'%o nee the i ea' 0an pa%% fi'ter i$pu'%e re%pon%e function, A Therefore, the
&TL&B routine d al lp(,c M) i% %ufficient to eter$ine the i$pu'%e re%pon%e of an i ea'
0an pa%% fi'terA Ie a'%o inc'u e the $o ifie function of the fre/uency) o$ain p'ot% or :' < % ,
which return% the $a nitu e re%pon%e in a0%o'ute a% we'' a% in re'ati e B %ca'e, the pha%e re%pon%e an
the roup e'ay re%pon%eA
Ba%e on the re/uire$ent% i en, u%in the co$$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$$ary ta0'e i en a0o e u%in the e9act a'ue% ,
= cei'(11 pi tr_wi th 1
( )nhd
p s
4
M
1.8 M
8
M
6.1 M
8
M
6.2 M
8
M
6.6 M
12 M
11 M
-
8/10/2019 Linear Phase Finite Impulse Response
26/30
26
which return% the fo''owin re%u't% fro$ the &TL&B output co$$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
-
8/10/2019 Linear Phase Finite Impulse Response
27/30
27
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
-
8/10/2019 Linear Phase Finite Impulse Response
28/30
2