Lesson 5. Chapter 7. Wiener Filters. Bengt Mandersson. r k s r x LTH. September Prediction Error Filter PEF (second order) from chapter 4

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1 Optimal Sigal oceig Leo 5 Capte 7 Wiee Filte I ti capte we will ue te model ow below. Te igal ito te eceie i ( ( iga. Nomally, ti igal i ditubed by additie wite oie (. Te ifomatio i i (. Alo, we ofte ued te appoac tat te ifomatio igal i modeled a wite oie w( filteed i a filte g(. Capte 7. Wiee Filte w( g ( G ( ( oie ( ( ( ( etimated output deied output d( eo e( LT Septembe We will imie te output eo e(, wic we decibe a te diffeece betwee te deied output d( ad te etimated output. ˆ imie Ee [ ( ] E[( d ( d ( ] Begt Madeo Depatmet of Electical ad Ifomatio Tecology, Lud Uieity Lud Uieity Applicatio. Filteig (: Smootig: edictio: Equaliatio: Te deied igal i ( ad we will detee te optimum filte fo oie eductio. Like filteig but we allow a eta delay i te output igal (pecially image poceig. Te output i a pedictio of futue alue of (. Oe tep pedicto. pedict et alue (. Te deied igal i w( ad we will detee te optimum filte fo witeig te output pectum (iee filteig, decoolutio. Ote applicatio: Eco cacellatio. Noie cacellatio. ule apig edictio Eo Filte EF (ecod ode fom capte 4 Optimum Filte (poce te igal ( w( δ ( Model of te igal (. Iput: wite oie w ( o impule δ ( A ( ( ( k A e( A ( a( a( ( w( ( ( g ( δ ( k k We aume ucoelated oie (. ( d$( d( e( e( ( ˆ( l l a ( ( ( ˆ( wit a ( ( d ( could be: (, filteig oiy igal ( (, mootig (allow delay (, pedict futue alue δ (, iee filteig, decoolutio We ca ewite te figue w( δ ( - A d ( dˆ( ˆ( e( ( ˆ( d ( ( ( [ a, a ] e( d$( $( deied igal ( ökad etimated igal eo igal ( caual filte: eay, ueful (cap. 7. ( ocaual IIR filte: eay, le ueful (cap ( caual IIR filte moe difficult, ueful (cap We aume tat coelatio fuctio (k, (k ad d (k ae kow o could be etimated

2 Deiatio of te optimal olutio (Wiee filte. Real-alued adom igal. We tat wit ˆ Ee [ ( ] E[( d ( d ( ] wit l ( ( l (i geeal ocaual filte l ad e ( d ( d ( l ( ( l Set te deiatie of wit epect to (k equal to eo fo all k. ( Ee Ee k k ( e Ee k wic gie [ ( ] [ ( ( ] [ ( ( ( ] Ee [( ( k] (Te otogoality piciple Replace e( ad te E[( d( ( ( ( k] l Deiatio of te imum eo Witig [ ( ] [ ( ( ] { ( [ ( ( ( ]} l Ee Eee Ee d l l Eed {( ( l ( Ee {( ( } l e( ad gie data otogoal Ti gie te imum eo ad Eed [ ( ( ] E{[ d( l ( ( ] d( } l ( ( l ( ( ( l ( l d d d l l d( ( ( l ad we got te Wiee-opf equatio l l ( ( k Te Wiee filte wa deie fom adom igal. Fo a deteitic appoac we ae to ue te defiitio of autocoelatio ad co coelatio Te, imie ( ( k d( ( k ˆ e( ( d ( d ( ε Te Wiee-opf equatio will be te ame. Wiee filte (pp , table 7. page 339 Te Wiee-opf equatio ae ow p l o i mati fom l ( ( k k,,..., p ( ( ( ( p ( ( ( ( ( ( p ( ( ( ( ( ( p 3 ( (. ( p ( p ( p 3 ( p ( ( p 443 R R Now, we will look at te tee type of filte ( Wiee filte (i te tetbook deoted W( Nocaual IIR filte Caual Wiee filte (at te ed of ti capte Te olutio i ad te imum eo wic alo ca be witte R p d l ( ( ( p T d ( ( ( d ( opt d ( l T R 79 8

3 Nocaual IIR Wiee filte (pp , table 7. Te Wiee-opf equatio ae ee l l ( ( k all k Filteig igal fo oie eductio Te igal i ditubed by additie eo mea wite oie ( ( ( ee we ae a complete coolutio ad it ca be oled uig -tafom o Fouie tafom ( ; j ω ( e e ( j ω ( e Te imum eo i d( ( ( l w( g ( G ( Deied igal i ow (. Te E[ d( ( k] E[ ( ( ( k ( k] ( oie ( ( ( ( Etimated deied d(( eo e( We ca ue te aeal elatio ad alo wite ti i te fequecy domai. Te, (ee popetie of te Fouie tafom, ee page 356, Table 7. π π π [ ( e ( e ( e ] dω d 8 8 Caual -filte fo oie eductio Te -filte equatio ae p l Now, tey will be p o ad l l ( ( k k,,..., p (( l ( k ( k ( R R ( R R opt Te pectum we fid fom te Fouie Tafom ( e Fouie{ ( } opt Nocaual IIR-filte fo oie eductio Fo o-caual IIR filte, we ae ( ; ( e e ( j ω ( e I te filteig poblem te powe pecta ae wic gie te Wiee filte ; ( e e ( ; ( e ( e We ee tat fo fequecie wit low oie, ( e 83 84

4 edictio I a oe-tep pedicto, te deied igal i (. Noie cacellatio (page 349 A igal i ditubed by additie oie (. w( g ( G ( oie ( ( ( ( ( etimated deied d(( eo e( Ty to meaue te oie ( fom te ouce ad etimate te oie ( added to te igal. Te ubtact te oie ( fom te igal. Deied igal i ow (. Te ( k E[ d(( k] E[( (( k] ( k Ti gie te Wiee-opf equatio Sigal ouce Noie ouce ( ( ( ( ( ( ( ( Wiee filte Etimate of ( p l l ( ( k ( k k,,..., p Decoolutio (equaliig, iee filteig Deied igal ee i w( (o allow delay, w(-. (ee alo poblem 4.9 Ti mea tat g( ( δ ( Caual IIR Wiee filte (page Deiatio of te caual filte i moe difficult. Te Wiee olutio i l ( ( k w( g ( G ( ( oie ( ( ( ( etimated deied d(w(- eo e( We diide te olutio ito two tep. Step Step w( ( ε( F( /F( ( G( ( I tep, we witeig te iput igal (. Fom capte 3, we ae pectal factoiatio If te coe F( aiace equal to. te igal ε( will be wite wit 87 88

5 IIR, caual filte Step I tep we kow ae (Wiee-opf equatio l g( ε ( k wit ε δ IIR, caual filte 3 Vi ae to detee (. Te wee E{ d( ε( k} E{ d( ( l l f ( f ( Z f ( ( k } ( k { F( } Te Te optimal filte (te caual filte, k i te wit te -tafom g u( k G [ ( ] Te otatio [ ] mea te caual pat of te agumet. ( ( F( To fid G( we take te caual pat G ( Combiig tep ad tep gie fially F G ( ( ( 89 9 Relatio betwee caual ad o caual IIR Wiee filte No caual IIR Wiee filte ( ( ( Caual IIR Wiee filte We ca ee bot filte a a cacade two filte tee te fit i a witeig filte. Te imum eo i a befoe d l ( ( ( Eample 7.3. (page 36 Gie: (.8 ( w( ( ( ( k 3.8 { w } Tak: Etimate ( wit a b ocaual IIR c caual IIR. d etimate (-N/ wit a of legt N Solutio: (ee te tetbook page fo detai w( g ( G ( ( oie ( 3 [ ], δ (, w, Fomula table gie.36 ; (.8 (.8 (,.36 (.5.6 (.8 (.8 (.8 ( ( etimated d deied d(( (.5 (.8 9 9

a (legt 5 i te figue R ; R d lot of te impule epoe fo te aiou filte fo filte legt5 ad N b Nocaual IIR ocaualiir.36 /.6 (.5 (.5 ( ocaualiir (.3.5 c Caual IIR (ee page 363.375.

6 a (legt 5 i te figue R ; R d lot of te impule epoe fo te aiou filte fo filte legt5 ad N b Nocaual IIR ocaualiir.36 /.6 (.5 (.5 ( ocaualiir (.3.5 c Caual IIR (ee page caualiir ( u( d Etimate (- wit N/, N filte legt Te te igt ide i [ (... ( ( ( ( ]' ad agai R ; delayd delayed R Te fial eo fo tee filte ae d ( ( ( legt 5 :.3755 delayed o caual IIR Caual IIR legt : delayed o caual IIR Caual IIR ome eecie Te ame eample a aboe but etimate (, pedictio oe tep Gie: w( g ( G ( (.8 ( w( ( ( ( k 3.8 { ( oie ( ( ( ( etimated w } Tak: Etimate ( (pedict oe tep wit a b o caual IIR c caual IIR. deied d(( Solutio: Now ( k, Te, modify te olutio aboe to te pedictio poblem ad plot te coepodig impule epoe i Matlab. Recuie olutio, Kalma filte I Wiee filteig, te aume tat we kow te coelatio fuctio of te iput o tat we ca etimate it befoe we deig ou filte. Te, we mut ae tatioay (WSS. Aote appoac i to ue oly iput data fom te pat, i.e. old alue of ( ad (. Ti lead u to te Kalma filte, ectio 7.4. Eample 7.3. Gie: (.8 ( w( ( ( ( k 3.8 { Te caual IIR-filte epoe wa w( caualiir g ( G ( ( oie ( ( ( u( ( ( etimated dˆ ( ˆ( w } deied d(( Ti coepod to te diffeece eq. ˆ(.8 ˆ(.375 ( (.8ˆ( 3 Kalma gai ( fia [ ] 95 96

7 Ti lead to te Kalma fomulatio [( (.8ˆ( ] ˆ(.8 ˆ( K( wic i a iteatie fomulatio. Te cotat K( (Kalma gai i oled fo,,... ad ue oly te eo up to time ide. Deiatio of te Kalma gai ue tate pace fomulatio ad ca be foud i ectio 7.4. Fo a WSS poce, te olutio goe to te Wiee olutio fo goe to ifiity. Adaptie filteig. Capte 9 o te coue Adaptie Sigal oceig. We wat to imie te eo Ee [ ( ] E[( d ( ] Iteatie olutio We ca ole ti iteatiely uig te update equatio ( k δ μ' δ μ' E{ e( ( k} tee μ' i te tep ie. Adaptie olutio (Leat Mea Squae, LMS Ue te appoimatio E{ e( ( k} e( ( k wic gie ( k ' e( ( k μ μ '? ow to coe tep ie Doe te algoitm coege? ow fat? 97 98

Lesson 5. Chapter 7. Wiener Filters. Bengt Mandersson. r x k We assume uncorrelated noise v(n). LTH. September 2010

Lesson 5. Chapter 7. Wiener Filters. Bengt Mandersson. r x k We assume uncorrelated noise v(n). LTH. September 2010 Optimal Sigal Poceig Leo 5 Chapte 7 Wiee Filte I thi chapte we will ue the model how below. The igal ito the eceive i ( ( iga. Nomally, thi igal i ditubed by additive white oie v(. The ifomatio i i (.