winer filter

8/9/2019 Winer Filter

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Week 3 ELE 774 - Adaptive Signal Processing 1

WIENER FILTERS


2/42

ELE 774 - Adaptive Signal 2Week

!o"ple#-val$ed stationar% &at least '(s(s() stoc*astic processes(

Linear discrete-ti"e +ilter, w , w ., w 2, ((( &IIR or FIR &in*erentl% sta/le))

%&n) is t*e esti"ate o+ t*e desired response d&n) e&n) is t*e esti"ation error , i(e(, di++erence /'( t*e +ilter o$tp$t and t*e

desired response

Linear 0pti"$" Filtering1 State"ent


3/42


4/42


Linear 0pti"$" Filtering1 State"ent

0pti"i8ation &"ini"i8ation) criterion1 .( E#pectation o+ t*e a/sol$te val$e,

2( E#pectation &"ean) s9$are val$e,

( E#pectation o+ *ig*er po'ers o+ t*e a/sol$te val$e

o+ t*e esti"ation error(

:ini"i8ation o+ t*e :ean S9$are val$e o+ t*e Error &:SE) is "at*e"aticall%

tracta/le(

Pro/le" /eco"es1

3esign a linear discrete-ti"e +ilter '*ose o$tp$t %&n) provides an

esti"ate o+ a desired response d&n), given a set o+ inp$t sa"ples $&),$&.), $&2) (((, s$c* t*at t*e "ean-s9$are val$e o+ t*e esti"ation error

e&n), de+ined as t*e di++erence /et'een t*e desired response d&n) and

t*e act$al response, is "ini"i8ed(


5/42

ELE 774 - Adaptive Signal ;Week

Principle o+ 0rt*ogonalit%

Filter o$tp$t is t*e convol$tion o+ t*e +ilter IR and t*e inp$t


6/42

ELE 774 - Adaptive Signal $adratic F$nc( = !onve# F$nc(

:ini"$" is attained '*en

&radient '(r(t( opti"i8ation varia/le

' is 8ero()


7/42


3erivative in co"ple# varia/les

Let

t*en derivation '(r(t( 'k is

6ence

or

??? @1 real, '*% ???


8/42

ELE 774 - Adaptive Signal Week


Partial derivative o+ @ is

Bsing and

6ence


9/42

ELE 774 - Adaptive Signal CWeek


Since , or

T*e necessar% and s$++icient condition +or t*e cost +$nction @ to

attain its "ini"$" val$e is, +or t*e corresponding val$e o+ t*eesti"ation error eo&n) to /e ort*ogonal to eac* inp$t sa"ple t*at

enters into t*e esti"ation o+ t*e desired response at ti"e n(

Error at t*e "ini"$" is $ncorrelated 'it* t*e +ilter inp$t?

A good /asis +or testing '*et*er t*e linear +ilter is operating in its

opti"$" condition(


10/42

ELE 774 - Adaptive Signal .Week


!orollar%1

I+ t*e +ilter is operating in opti"$" conditions &in t*e :SE sense)

W*en t*e +ilter operates in its opti"$" condition, t*e esti"ate o+ t*e

desired response de+ined /% t*e +ilter o$tp$t %o&n) and t*e

corresponding esti"ation error eo&n) are ort*ogonal to eac* ot*er(


11/42

ELE 774 - Adaptive Signal ..Week

:ini"$" :ean-S9$are Error

Let t*e esti"ate o+ t*e desired response t*at is opti"i8ed in t*e:SE sense, depending on t*e inp$ts '*ic* span t*e space

i(e( & ) /e

T*en t*e error in opti"al conditions is

or

Also let t*e "ini"$" :SE /e &D)

6W1 tr% to derive t*is

relation +ro" t*e corollar%(


12/42

ELE 774 - Adaptive Signal .2Week

:ini"$" :ean-S9$are Error

Nor"ali8ed :SE1 Let

:eaning

I+ is 8ero, t*e opti"$" +ilter operates per+ectl%, in t*e sense t*at

t*ere is co"plete agree"ent /'( d (n) and ( &0pti"$" case)

I+ is $nit%, t*ere is no agree"ent '*atsoever /'( d (n) and

&Worst case)


13/42


Wiener-6op+ E9$ations

We *ave &principle o+ ort*ogonalit%)

Rearranging

'*ere

Wiener-6op+

E9$ations&set o+

in+inite e9n(s)


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Wiener-6op+ E9$ations

Sol$tion Linear Transversal &FIR) Filter case

: si"$ltaneo$s e9$ations


15/42

ELE 774 - Adaptive Signal .;Week

Wiener-6op+ E9$ations &:atri# For")

Let

T*en

and


16/42

ELE 774 - Adaptive Signal .


17/42


S$/stit$te =

Re'riting

Error-Per+or"ance S$r+ace


18/42


Error-Per+or"ance S$r+ace

>$adratic +$nction o+ t*e +ilter coe++icients = conve# +$nction, t*en

or

Wiener-6op+

E9$ations


19/42

ELE 774 - Adaptive Signal .CWeek

:ini"$" val$e o+ :ean-S9$are Error

We calc$lated t*at

T*e esti"ate o+ t*e desired response is

6ence its variance is

T*en

At wo(

& J min

is independent o+ w)


20/42


!anonical For" o+ t*e Error-Per+or"ance S$r+ace

Re'rite t*e cost +$nction in "atri# +or"

Ne#t, e#press @&w) as a per+ect s9$are in w

T*en, /% s$/stit$ting

In ot*er 'ords,


21/42

ELE 774 - Adaptive Signal 2.Week


0/servations1 @&w) is 9$adratic in w,

:ini"$" is attained at wGwo,

@"in is /o$nded /elo', and is al'a%s a positive 9$antit%,

@"inH =


22/42



Trans+or"ations "a% signi+icantl% si"pli+% t*e anal%sis,

Bse Eigendeco"position +or R

T*en

Let

S$/stit$ting /ack into @

T*e trans+or"ed vector v is called as t*e principal a#es o+ t*e s$r+ace(

a vector

!anonical +or"


23/42



'.

'2wo

@&wo)G@"in

@&w)Gc c$rve

v.&.)

v2&2)

@"in

@&v)Gc c$rve

>

Trans+or"ation


24/42


:$ltiple Linear Regressor :odel

Wiener Filter tries to "atc* t*e +ilter coe++icients to t*e "odel o+ t*e

desired response, d&n)(

3esired response can /e generated /%

.( a linear "odel, a

2( 'it* nois% o/serva/le data, d&n)

( noise is additive and '*ite(

:odel order is m, i(e(

W*at s*o$ld t*e lengt* o+ t*e Wiener +ilter /e to ac*ive "in( :SE


25/42

ELE 774 - Adaptive Signal 2;Week

:$ltiple Linear Regressor :odel

T*e variance o+ t*e desired response is

J$t 'e kno' t*at

'*ere wo is t*e +ilter opti"i8ed '(r(t( :SE &Wiener +ilter) o+ lengt* :(

.( Bnder+itted "odel1 :K"

Per+or"ance i"proves 9$adraticall% 'it* increasing :( Worst case1 :G,

2( !riticall% +itted "odel1 :G"

woGa, RGR",


26/42

ELE 774 - Adaptive Signal 2


27/42


E#a"ple

Let t*e "odel lengt* o+ t*e desired response d&n) /e , t*e a$tocorrelation "atri# o+ t*e inp$t $&n) /e &+or conse9( sa"ples)

T*e cross-correlation o+ t*e inp$t and t*e &o/serva/le) desiredresponse /e

T*e variance o+ t*e o/serva/le data &desired response) /e

T*e variance o+ t*e additive '*ite noise /e

We do not kno' t*e val$es


28/42


E#a"ple >$estion1

a) Find @"in +or a &Wiener) +ilter lengt* o+ :G.,2,,4

/) 3ra' t*e error-per+or"ance &cost) s$r+ace +or :G2

c) !o"p$te t*e canonical +or" o+ t*e error-per+or"ance s$r+ace(

Sol$tion1

a) 'e kno' t*at and t*en


29/42

ELE 774 - Adaptive Signal 2CWeek

E#a"ple Sol$tion, /)


30/42


E#a"ple

Sol$tion, c) 'e kno' t*at '*ere +or :G2

T*en

v.&.)

v2&2)

@"in


31/42


Application !*annel E9$ali8ation

Trans"itted signal passes t*ro$g* t*e dispersive c*annel and a

corr$pted version &/ot* c*annel noise) o+ #&n) arrives at t*e receiver(

Pro/le"1 3esign a receiver +ilter so t*at 'e can o/tain a dela%ed

version o+ t*e trans"itted signal at its o$tp$t(

!riterion1 .( Mero Forcing &MF)

2( :ini"$" :ean S9$are Error &::SE)

Filter, w!*annel, h

3ela%, O

#&n) %&n)

#&n-O)

&n)8&n)

-


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::SE cost +$nction is1

Filter o$tp$t

Filter inp$t

!onvol$tion

!onvol$tion


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!o"/ine last t'o e9$ations

!o"pact +or" o+ t*e +ilter o$tp$t

3esired signal is #&n-O), or

!onvol$tion

Toeplit8 "atri# per+or"s convol$tion


34/42



Re'rite t*e ::SE cost +$nction

E#panding &data and noise are $ncorrelated E#&n)v&k)QG +or all n,k)

Re-e#pressing t*e e#pectations


35/42

ELE 774 - Adaptive Signal ;Week


>$adratic +$nction = gradient is 8ero at "ini"$"

T*e sol$tion is +o$nd as

And @"in is

@"in depends on t*e design para"eter O


36/42

ELE 774 - Adaptive Signal


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Application Linearl% !onstrained

:ini"$" - ariance Filter

Pro/le"1 2( We 'ant to design a /ea"+or"er '*ic* can resolve an

incident 'ave co"ing +ro" angle o &'it* a scaling +actor g),

'*ile at t*e sa"e ti"e s$ppress all ot*er 'aves co"ing +ro"

ot*er directions(


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Alt*o$g* t*ese pro/le"s are p*%sicall% di++erent, t*e% are"at*e"aticall% e9$ivalent(

T*e% can /e e#pressed as +ollo's1

S$ppress all co"ponents &+re9( or dir( ) o+ a signal '*ile

setting t*e gain o+ a certain co"ponent constant &o or o)

T*e% can /e +or"$lated as a constrained opti"i8ation pro/le"1

!ost +$nction1 variance o+ all co"ponents &to /e "ini"i8ed)

!onstraint &e9$alit%)1 t*e gain o+ a single co"ponent *as to /e g(

0/serve t*at t*ere is no desired response?(


39/42

ELE 774 - Adaptive Signal CWeek



:at*e"atical "odel1 Filter o$tp$t U Jea"+or"er o$tp$t

!onstraints1


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!ost +$nction1 o$tp$t po'er = 9$adratic = conve# !onstraint 1 linear

:et*od o+ Lagrange "$ltipliers can /e $tili8ed to solve t*e pro/le"(

Sol$tion1 Set t*e gradient o+ @ to 8ero

0pti"$" /ea"+or"er 'eig*ts are +o$nd +ro" t*e set o+ e9$ations

si"ilar to Wiener-6op+ e9$ations(

o$tp$t po'er constraint


41/42

ELE 774 - Adaptive Signal 4.Week



Re'rite t*e e9$ations in "atri# +or"1

6ence

6o' to +ind Bse t*e linear constraint1

to +ind

T*ere+ore t*e sol$tion /eco"es

For o,w

o is t*e linearl% !onstrained :ini"$"-ariance &L!:) /ea"+or"er

For o, wo is t*e linearl% !onstrained :ini"$"-ariance &L!:) +ilter


42/42

:ini"$"-ariance 3istortionless Response

Jea"+or"erVFilter

3istortionless = set gG., t*en

We can s*o' t*at &6W)

@"in represents an esti"ate o+ t*e variance o+ t*e signal i"pinging on t*eantenna arra% along t*e direction (

enerali8e t*e res$lt to an% direction &ang$lar +re9$enc% )1

"ini"$"-variance distortionless response &:3R) spectr$" An esti"ate o+ t*e po'er o+ t*e signal co"ing +ro" direction An esti"ate o+ t*e po'er o+ t*e signal co"ing +ro" +re9$enc%

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