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. 73 74 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. 7.3. ( caual IIR filte moe difficult, ueful (cap. 7.3. We aume tat coelatio fuctio (k, (k ad d (k ae kow o could be etimated. 75 76
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 {( ( } 443 44443 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 77 78 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. 337-339, 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 ( 4444444444444444443 443 ( 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
Nocaual IIR Wiee filte (pp. 353-356, 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
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 85 86 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 358-36 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
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 {...8.8 w.36 3.8.8.8.8 } Tak: Etimate ( wit a b ocaual IIR c caual IIR. d etimate (-N/ wit a of legt N Solutio: (ee te tetbook page 36-364 fo detai w( g ( G ( ( oie ( 3 [.8.8.8 ], 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. 5 caualiir (.375.5 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.395.3.375.375.3.3.375 93 94 ome eecie Te ame eample a aboe but etimate (, pedictio oe tep Gie: w( g ( G ( (.8 ( w( ( ( ( k 3.8 {...8.8 ( oie ( ( ( ( etimated w.36 3.8.8.8.8 } 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 {...8.8 Te caual IIR-filte epoe wa w( caualiir g ( G ( ( oie ( ( (.375.5 u( ( ( etimated dˆ ( ˆ( w.36 3.8.8.8.8 } deied d(( Ti coepod to te diffeece eq. ˆ(.8 ˆ(.375 ( (.8ˆ( 3 Kalma gai ( fia [ ] 95 96
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