2\pienar Filters

!. L i 'rc.aa- o p+ i rnrrrv., {i kto,f

tl(tiDo, tJ r, a)rr' ...

(.) Pv,t ci/e aft orr\ogon<1.'$ .

r,^1*f nrt') 'orat " 'l^'

gr"l= F" a{ utn-L1 (1.) l"t.6 a-1, LL(^, eL tr. -*C9

e(!^)e oltal-'Srnr= d.h)- E toiL>e

5, Eflet"11'1 = eteOdLif$rte *1.'"1 1^;4 = a.L * j $&

Vd-t = EI * i. el- eaE '" EbLus5 trr'

wr = E[#ef.) *flo", -i ffi&r2 f,eol frtUsw @) gec4)

<r-ll:ll_- d4tl-

;) ('l a)

"o&

z -_tt1n-L;

" - atcn-L1

e(^) a woY

ttb-&) (")

(r)

t'o,,, -. . c4)

e€td--Jl|= d 4(n11

a(tq)_TFL =n'.do-e4un, \4J = -zE[a6n41exc")JLef Etuu^-LlehlJ =e + oLla.'. 67o,(^;,,.-.-rtr ep};u"-f solrfir.. i" len4o) * 1go=[a,oo,,,orr ,.-.JT

dt- c.rrc*7or.r5 "", ,1 +ut n":-:""i<A .-r"t de^+<4 al ei4

^"^,* E[ctt,'--l-1e*r/J=o G)-t[ls is ."Ilcl gri.;?\e -* ot-tloyono,r.'\-.

lJ""' vls *nslltrE t y,1 e*@l = t [fi af uu-*1 et'tJ

=f". e[awbet^:7o0[L4-l

^f qto : f tceo, ur,r, r^1.: -"J7., (Jq Aa.^<- et") . €--Cu) a.J $(r

(7)

,i a,r

(4s

(r"J

=o

.tf.,. e t7"alehlhq,4 ** frot")

"9.,fl..i. .f dt-)

t^re dea"te

'LLn; ryue(z) f,(ia f ru,l,p. MS6

bz.= 7 r"ilt e [uu4 fi,fi =-ie e-4r" 9or,a) *o €o(n). He,,q- g,(h)

a.-) Q.ata 1 ,i t4 4rts..7l'rJ-. prrvt.

!.o1a"1= (ath) (8J

rpa.aQ U6 .rU ,l^p*ala*a ,.g *o {l'"€ tr,,

€o(n) = /r^1 - So(u,1 = A,r,) -f,P|&,\€Y dt"r = f,(nldr") + €o cvr)

OndZ tb r'..ea.1 9q+\ t<_

o-4L= o-4'+ efleotntll

tr}. 5r-,,,. = Ell ea"tlz)

65a ax 4o r^a.1,6<- T^,,., a,

&e,""rd Svn\Fwu e'Sa,*Lt e*cr\ = 6[utn-tq[ dfq -F-r.^

LJ? lwva- ;!,F* r,, e [uu4 ut^-;1j = efu"-g

tel f(,i-1") = ElQ1"-a1a%-;l;

f (h) e E [wn-at Xcat]-dott vz l,o; f (,;-./',) = f (4) t ( A,o ,t, --. )

t o E tncr.+Lg( e.pr.ai'i^ S

valx< on yolt c,Ve< rl++ 4""<-V

= 6Xz+3:ro.,,i^

= %i_ q3

-<€s (Z-.J-3 = 1*

Wieuer ^H"ff

izo'd.is t't caJ,le{

_dq'2.

ct)

{o\

ut-r11]

ahl(o,t.t. at>uJi^ )

L4-Z

co)

e) Ftt< +:ltzr .^se qa*v t>v.'i foatt)

5(!PfPse a\tc.e ute- Pt +"f: , LJo, u),,---lZ too;fkt-F\ =fe+), l_o,l

i,,--a

Lrf, ll\)>Ctt$) u(^-t) ---ab-u+r1 Jtp=2fwntfl^1y

f = efttt^laht) =a^Je A.t r{-

R,&=p.-u- N)eqor

L ftu)

r(u^t) 1r(u^ilr@ Jf(0 "

r tto) f(t\= I ft,] ^tto)

ls^, n,fr,

=> D, = itfse tn-*{o '.,

Dr.l-t r

_ - .lr{1

'lf'ren (lD le.naes

- 7tu-,tlr

(tL)

tt-t I'l'1 +*F-L._nualJ;n4:4) ftt,

(?) Erro" p*frw,axe e^rPt_-- {* np- (+k*,(c-.rt etnl= 6l1y,y- H""t ,tcni.)

1te rr^.r', s?M wl+< -( r+t <rrqr , or it sE

5 = E tet'ltt,i3

= p.1ld,t lLl- S "f 6[at"4t4rtr)*^t

"a.fuT"-tta".tJ*H# 'li.i etuu-7tf t,.-)))

e i4'|= -f,- "+''

afpca: -,9J"^ ftaPra."qh9: .

(,') y '! a >nd-order p,^*rin * -r, n bo,-,1 _:htdQa+r)-dihe*r,io^4 t*f"t- -,4 U v*,t^hlsg.

fd,) T na4c5 ,'7s 2nliau.w-,. !a,;M-t.z ,d'i 16'4\'= /otk)iw

tii'l ! at a- -(nu**n ^f tD la aL4-3

alLe* f.1y :e

&u'',"'u't)nahti firn,

, t"Q,,

E'(ti) = qt- ^*P- f,^ + t^lHR u)

is n ini *',5r"4 +" o^'- pH E,p btfo,, N=.Jo=rTte4/€,ftre,

Ifu.r) = f^,- t (p-t^rr;xA gr^,-r^ra)

r^-/\o.rr 5-,, = 6n:- ptf 4-lp(iV) hlyt"*t;2,', R + n= &AgH ,, a.

T: ]-,^+ (ar-oo;'an aH(c->',o"1

:Jrt,L + VHA V

= 5,^,.,). * $.,,trlrrr*t'

= cor'-f H rq't * (r-i'f )* R (*dP)(ul- {rP)H ( f* - ft'gS >.o (r ,s p,s,,+ire- att'^tu) ,

gint<-

y(a)

4."t")

dtu,) = 4H4.(',) +tZ.)tb^(") = [(l".,("1 |t,,.,(h-t)

{lerrt-ot=ont+a,HR".a

&€(^i.:' 5-"L = 6i1Nq ha{^r_ I,.;6,11=

( tl)

a,J V =&Cg-o;; =[v, --.Uaf

+,""

dt"1

r't t*r,,1-7 h,^Jh k )

(r b)

+b " ndorj

A'P

C,9

3.

b,l^4d,4 lt i< 'b

erjo^"al"e of Rqt"4t i^ ( c) is call el c-a.n o 4 t' caJ

Line,.'y rvAtetg;\ -"tto!fufor. ti Acsl.A sl6n-l dtnl ,', 6o,'nofu) lma t*(,,) ++,,"fLrYfot" * o.:',:- tnTw 4..4) (5

1tevt?ra.te4 lmr, 4tb(4,

A lln*r m "olol "*J dbl c oa/:*fqc a-ad.i'r.,'v\4_. noise l,,,)

' -'u- (n-,..*r)]T (-, is

, Fn =eltl+u>alt^lpHR"-rp

trrro-* LaH R^a- doh p,.r)c^:"=R-lp at^A Ra"l )

L4-+

v(")

(l\ea< ure "<4

Fer f,iv+n n, $vez &ol"es 4 P '(r) Ur,/rrfttteA vrto&.I t MLta.t

TLq tinepr. $t+ar tN ca,^,. n qf, €+ *. w'&l 4o " 1l'o *n-nlnn4

[4s E M4 r,:, lhJ vafia-q ll,r=,o1 :

5p.; (.o) = qn" 4 ai Rr^^A.

o,*), da at et*zs a'* M i's tx rv'e^8d '

tiD ot:Jrc,,4l {:uA mold 1 l7=rvt

A+ r'r=-,0 a6 l,Jien".r $ w"t ;, &d* '-,tfol€J J-." rt''r<1rus,;n'

^nlJ at Q)e =4- a-d R= Oq, *) T,.;',grll=e-r"

(iit Or,'fl+ei *"lt) ' u>^=d^a. t Ji aner Sl [+*r +a,gl .\tn- tuv.rt- +la^ ,^if^.f r s n{.{.&.4nl+ Wiew",. Jol,^.f,,j..,., ts tn. O" "[XJtorr+7"vlruf, , l"L<^)=|,ff' f , s",- Llx-*,rn) r! (u-r',)xl

te*tt llr\il< \ ?-, Yi s"^1f cs rl,^^ertarc,tg 1rre../.g*o n-rd-f ve*q- |la,l,v,) (*"U y^?r\i )

E{&yle, A "".lel $+5 y4a,..,(ilyt r" A- =tq.o 4, ao) t , N vt6raei.., r^-eel a4 ,rllt /.3 ,

(6) -fu Cr?rela.*-,".;^ y\,\,..anV Ot tlvtI /'[ o.S o, L -o.of T

4a: I u.r t.L D.s- o.f /I s.,t o.9 t-L o.s I

Lnog o.d_ ,.J r.!, J

(t -r|,c wogt -c€rr(!j-riu. ya-.*rr bdl^)eoq. *ltt .1 a...\ /C"1 , ',

f>to'St'l-lt, -0'@!'6 ' '6'tuol ' - o'olo6 tra

flote +1.* tt! ta4+ ol"*r.* 4 f rs o^l*l,et"d s^'1" 1L"+ a3 =o

t:I^ ml[?l " U:J -) &^6n'r' ' r'!*a' "r^-a4_5

[ae:l,i', r ,

Lo 7+,ve4-,,\odz-

\*-*-5tl M* r

eo Dtsplo.X t6.

6Jfi ir'(:t90 9derr4|r.

5otr*to^,lo . dt") = 4n 4t,") +An LLbl) + ALL$'L) + nh)

t"te,r,e- our- rc aaw,-i, emLtrl=t1161 )

zc :i{'^r.,.^r,Jl .. AaL^HF-FHr,l+ l,lHR\^)

= 6t'-. P"r^: * u.rH R. ta1

-".1q86- 1[osz72 -a.co5t) LI,] *ro. w,r

Ij i i:ll l::_l= e -]6$ :1-o$4o do1 q8l 6r r^lr + 1.1 [roi+r"ri)

F,:rr Tl)-L I luoi o,g\6, , wt--'-o.1gct

-Il^.r-r )n i.= A*"- ptr *-lf

(c1 -rt^-q yanr\a.,tra-- 'f t\ obsov.la.tle- d"c.**

O) -ft* Va,1a\a.*r. .( nC") I 6-"t = o. 1ol5

qt= o.1+a1

T,",; t 9J,?,i,+

a h),,e,^er 4rl+-r + Va{S'g

errur p*ftrr**e- /*F a( o'' rl.licacr ftlaT

ib aa,.ont.^l .&r,^ + k> ttc? s",(n-'.

al" ru>4 see Lt"[

= 6-.tL- PH,^) o = 6d"t- nr$ Rrllo

o\. T'',n1 = c-*t- pH r;o = o.Qu$- [o.Sz-lr-

s o.rs1]Fvo'^. R' f 4.1- o's l , L,{ 9c{- -!r -- t,6 , \,p= c,l

L o'9 L'n' J

J\ls,vq)= !rr.,,; +1,, qla o.gi,L

Hebe ly:l = ofl;/

I o.&*o 7-o.*rt][-_"r$s\

)o

L4-&

€ ip.-'

o

og

o

o

Fr-

oot

J O (n Z=' E) -a

F-l -i:i-oitsoc H I -;::-a t1. )irl :-'u X ^';-_ c-; F -iE = ! \,o:i--?;-< ; ^ :

J.?th..f^

4 o 1A =-o tl .r' * l39c!-

:-3

aE E i i'-l ,

tr:.: -iia E i'" a,aw; -ll\;E lrlS aY'!n={9(!o, c :: ^:'aof, (* Xo tr-._q - a -oa n. - (

-;\<!or:

Y -. -l- A .',n +4 \on f"

t

l.J

E

E

3

Ee

I

!!

z$

i.-._

o

q:

gag

z-E?E

='- E

Ri

; E ?E: :-.a-o a O, *-(, 1

.i\L? Ea ?r.q= .-

I

tl

ll

llI

tt

l

l

lLt

I

LI

Section 2.7 Example 113

\4.41

\t.61

E.7i

342

0.00 6.;x

FIGURE 2.7 !{or lerlormlrc. miJac. ol the $,.lap r;:rs,enal tule. dcs.rtbed ia tleermplc ol Se.tion 2.7.

Figure 2.7 shows a tbee-dimensjonal compurer plor ofthe mean square elror,rk%,2_.r ). r. 5.t-r'r.:ob. -0ar J..Figure 2.8 shon's contour plois oI the tap weilht rrr re^us 2r. for valling \'alues

of the mean-sqrarc error "/. We sce that the locrs of 1,rl,elsus 1lh Ior a fjjed Jis ill thcfo1'i11 oJ an eLlipse. The elliptical locus shrjnls in size a,c ./ alpl oachcs the rntuimun vat LLe

J-i.. For "I - 1.j., thc locus reduccs ro a pojnr \vi!h coordinaies zd.o - 1:r_Sj60 xnd,,,1 : -0.7E5i;tlar is,1he opti urr tap-weiglrt vecLor is

": [-';X'iL]The mirrjmrnn mean-sq!arc ellor is,ii accordance wirtr Eq (2.49),

r",i,, = 0.e4t6 1osrr, -0..r,.l"1[ 11111]= 0.1519.

(2.6s)

(2.6e)

The pojnl reprcsenied j olntlv b-i il]ri op tlmur lap-rrejghL vector $, : l0.t-160, -0.7Sjj ll.,ri .:, ,in.r, 0-. qLr ,o, 0. oJ.,. I """ --.,"

F,,l O :i,-

_2

(2.aii)

r114 Chapier 2 wiener Fi teu

200

f 1.1 o.5 lR: [o.s r.r ]'$bL\i.gi.eno) rhc J.r nr ri' ,1 of rl-e ra.r {

0i0 1.00 r i0 2.00

2.8

Th,

fhc locuellipsc hl

lQ t"

LINEARLY

Thc csse

tior en<output.I

outputpicfrequesuch soll

Cot]]e tap i

For the !

FLGURE 2.8 C.nbur tlots olthc crroFp.fonnancc su.Iace lepicred inFig.2.7.

Computation of the Canonical Form of the Error-Performance Sudace

For eigenvalue analysis,we fomulatc lhc charactcislic equation ofthe two-by-two cor-relation nlatlix

Frr r o: I| 0., r.r ,l'

the saml'I1r

FN

that is.

(1 1 nf (o.s): - o.

lte two eigenvalues ofihe coirelatior matrir R are thereforewtth(

.--1 . L i'rec.-tg uo '-S,|"o,.JrcA 'y' r'n i nq""1r-.\y'e,rliariL U- $ l+.,V(erg {ttt-.r a. t\qet ^t*e > !

sG) = Sl dtlzw0 ^ J,^sv, Qt't

Fdr d\ 9,i*to ,A^) "-* a +^+'.o.-.- , atl tra\ =4-

s(^; = .it" fluf e-i"Je

Jh 9op.c- ^fphax;-.i, a" no.) +p n'"''ytrL:'* "'\:4^7e'1';';-- 4 Zr^, su.lole-* 1o a- l""c*r ac".alra-r;{

9' ,f {J."L -t , 4lutoeT , z ,\ co^slo,*p>o

ty\ol&, r'c- uv,".rt._

tseol.{6.---.-Tte o.+pr^* 4 ,U b". *.{oc,^-.e r

U*,= 4"(,) H arf .*Sr'o"

[u. ti vJ"*eA 1- 40)

nfr |}" clr1Q.,'.' lzn'oew'c4

rr,t.L g CfaC"llll:

s,f ' +-f t^rf, e-jLe" =uh;4

Uc2' Iagu.1a. FJ+l TIto's,6ict^a- l ftvr"" )nrn *t*)

*4" *";crJ''"JF'

( t?)

0tl

5. f,. crr"5= o +d o

* ff1ar1= &"rt + L1 Re[cr.->1 +tr -f.'-Ect*)]or {t^)= fa*> * R.e[,Sec^! , ,\=l'+itrz-

*#-- + # *fufn.nt,llo^r;;--L&-+

t1s.a ttt "Uo,*- yr,,"dtol,

r = il.ll t^rf ru; no-#-y *p"gl, Y,*tu:,"LuSJ 0u

V43 = t!-tt; ?.&'f) +I* e-i o"L -oor

a€ t,rrr,r'-L1"-;f .-le'A, (l=o,t,-- u)g o, =*$ scoo) , 5 (&") = f , dP" -- J@rof,,c=-Sot".t"j @"1

1,^ ",ndr--'to ddrrr"vr)-- )-,, rzei +!, c^-alrarit*

asll@i=7 6.1"^g *it\ (,"), "^^nU,-4 #rW {1.;ce.1 =fi + 1= - qffi;6

fl- p'l^irr.r.na., rrreo^-q,4u,oc4 V^|,,*c- -$+, --n"* "t{,b€4}4*

510") R{sts)

nr^ €slir.-i-e "{ A tpia*."- ap tu" <l7xd

&rr+ aleg fb ,{r\rc^*r"; ao'rr.ay"4 7 }o

(r,J

e9

I hrtn 's\ dt

,;7tVfre".

'rt^t-.t *

5rnn.^^ = 9-^t-.A.i.o a?

tte'g €'), Tn.i-:

tplp *"; t(n'-h-) = L)oH Bl,Jo

l3 l'

L+-&