Variable Forgetting Factor Recursive Total Least Squares Algorithm for FIR Adaptive filtering

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Lucas Derek Andrews
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1 01 Inernanal Cnference n Elecrncs Engneerng and Infrmacs (ICEEI 01) IPCSI vl 49 (01) (01) IACSI Press Sngapre DOI: /IPCSI01V4931 Varable Frgeng Facr Recursve al Leas Squares Algrhm fr FIR Adapve flerng Sang Mk Jung 1 and PGyen Park 1 Deparmen f Elecrcal Engneerng Phang Unversy f Scence and echnlgy Phang Krea Dvsn f I Cnvergence Engneerng Phang Unversy f Scence and echnlgy Phang Krea Absrac hs paper prpses a varable frgeng facr recursve al leas squares (VFF-RLS) algrhm recursvely cmpue he al leas squares slun fr adapve fne mpulse respnse (FIR) flerng he frgeng facr f he VFF-RLS algrhm s updaed by mnmzng he cs funcn (Raylegh quen In smulans he VFF-RLS algrhm prvded bh fas rackng capably and small mean square devan Based n mprved precsn esmae he FIR f an unknwn sysem and adapably change n he sysem he VFF-RLS algrhm can be appled exensvely n adapve sgnal prcessng areas Keywrds: Adapve flerng parameer esman fne mpulse respnse Raylegh quen recursve leas squares 1 Inrducn he al leas squares (LS) mehd [1] can prvde unbased esmaes f sysem parameers even f bh npu and upu are crruped by whe nse LS slun can be baned by cmpung he sngular value decmpsn (SVD) f an augmened daa marx r s cvarance marx and s he generalzed egenvecr asscaed wh he smalles egenvalue f he marx [1] [] Hwever he number f 3 mulplcan perans f he SVD fr an N N marx s generally ON ( ) s real-me sgnal prcessng applcans usng LS are lmed effcenly cmpue he slun several mehds have been suded Sme f hese sudes have deermned by cnsderng he Raylegh quen as a cs funcn and hese mehds have been used develp effcen recursve LS algrhms [3] [4] A new recursve al leas squares (N-RLS) algrhm [4] was prpsed fr adapve FIR flerng; hs algrhm uses fas cmpuan f he fas gan vecr (FGV) and adapan mnmzan f he Raylegh quen n [3] Hwever he N-RLS algrhm has a cnsan frgeng facr (e uses he prevus esman resul wh a fxed ra represen nex esman) s s n suable fr rackng me-varyng parameers because s cnvergence s slw when he frgeng facr s clse ne whereas he mean square devan (MSD) s large when he frgeng facr s small herefre yeld precse esman f he FIR f an unknwn sysem n me-varyng envrnmens he N-RLS algrhm mus be mdfed hs paper prpses he varable frgeng facr recursve al leas squares (VFF-RLS) algrhm whch adaps he N-RLS algrhm use a varyng he frgeng facr he cnrl f he frgeng facr s based n he mnmzan f he cs funcn (Raylegh quen In smulans he VFF-RLS algrhm had bh faser rackng capably and smaller MSD han he N-RLS algrhms Varable Frgeng Facr Recursve al Leas Squares Algrhm PGyen Park el: ; fax: E-mal address: llus@psechackr 170

2 1 Sysem mdel M 1 Cnsder a sysem denfcan mdel he unknwn sysem has a FIR vecr h R In hs sysem bh npu and upu are crruped by whe Gaussan nse he adapve fler s used esmae he unknwn sysem he desred upu s represened as d( x ( h n ( (1) where x( [ x( x( 1) x( M 1)] denes he nse-free npu vecr and n () s bservan nse whch s whe Gaussan wh zer mean and varance () he nsy npu vecr f he adapve FIR fler s gven by x ( x( n ( () where n ( [ n ( n ( 1) n ( M 1)] and () s whe Gaussan nse wh zer mean and varance () hen an aucrrelan marx f x () s defned as n [ ( ) ( )] R E x x R I (3) where R E[ x( x ( ] s he aucrrelan marx f he nse-free npu vecr he augmened npu vecr s defned as x( ) [ x ( ) ( )] where b E[ x ( d( )] and c E[ d( d( )] ( M1) 1 d R and s aucrrelan marx can be represened as R b R E[ x( x ( ] b c Raylegh quen and N-RLS algrhm One mehd slve he LS prblem s fnd he vecr whch mnmzes he fllwng Raylegh quen [3] [ w 1] R[ w 1] J( w) (6) [ w 1] D[ w 1] M 1 where wr s a wegh f he adapve fler and ag(1 1 ) / hen he LS slun s gven by arg mn J( w ) w LS D d R w MM (4) s a weghng marx wh Schasc quanes can be replaced by me-averaged values wh a frgeng facr (0 1) fr suffcenly large such as R ( R( 1) x ( x ( (7) b( b( 1) x ( d( ) (8) c( c( 1) d( d( ) (9) In he N-RLS algrhm [4] he wegh vecr f he adapve fler s updaed by where () can be deermned by mn J( w ( ) () w( w( 1) ( x ( (10) Usng equan (10) mn J( w ( ) can be rewren as mn J( w( 1 ) ( x ( ) () fnd mn J( w( 1 ) ( x ( ) le he graden f J( w( 1 ) ( x ( ) wh respec () be equal () () zer hen J ( w( 1 ) () x () ) 0 ( We ban () by slvng equan (11) 3 Prpsed varable frgeng facr recursve al leas squares algrhm (11) 171

3 adjus n he N-RLS algrhm he seepes descen mehd was appled mnmze he Raylegh quen J( w) wh respec as fllws where 1 s a unng parameer Jw ( ) ( ( 1) 1 Usng equan (7) (9) we can shw ha equan (6) s equvalen (1) ( ) ( 1) ( ) w R x x ( ) w ( c( 1) d( d( J( w) w( w() and s dervave s easly cmpued as J ( w) w( R ( 1) w( c( 1) w( w( Fnally we ban he updae equan fr he frgeng facr w( R ( 1) w( c( 1) ( ( 1) 1 w( w( Hwever frgeng facr shuld be zer because pas nfrman dsurbs precse esman when unknwn sysem s changed suddenly When he FIR f he sysem h s changed e () s cnsderably ncreased If esman f h s gven by w () ( h w ( ) hen e () can be represened as e( = d( x ( w( x( h n ( ( x( n ( ) w( n n ( h (13) (14) (15) (16) decde wheher sysem has changed r n we need w assumpns: E x( x ( ; E ( x( h ) Frm hese assumpns E e () can be represened as where Usng (18) n (17) yelds Frm ndependency f x() and n () Subsung (0) n (19) E e ( = h h E x( x ( (17) E ( x( h) = h E x( x ( (18) h x x (19) E e ( E ( ( E x x E x x M M E x x ( ) ( ) ( ) ( ) (1 ) ( ) ( ) If we use a suffcenly large value f (0) E e ( h E ( ( 1 M x x (1) (1) can be apprxmaed as e h x w x ( ) ( ) ( ) ( ) herefre we can decde ha he sysem s n changed f e () sasfes () Frm (15) and () s nferred ha he mehd f varable frgeng facr based n selecve updae s gven by 0 f e( w( x ( () = w( R ( 1) w( c( 1) ( 1) 1 (3) herwse w() w() () 17

4 he frgeng facr s updaed n he drecn f he Raylegh quen s seepes descen when h s mananed Furhermre frgeng facr value becme zer when h s changed he frgeng facr f he N-RLS algrhm s fxed a 0997 A fxed frgeng facr cann prvde lw esman errr when he sysem has arrved a he seady sae Hwever hs dsadvanage can be vercme by usng ur varable frgeng facr algrhm 4 Smulan n MALAB We used cmpuer smulan cnfrm ha he VFF-RLS algrhm prvdes mre accurae esman f h han her algrhms We cmpare f he VFF-RLS algrhm he N-RLS algrhm and he cnvennal recursve leas squares algrhm (RLS) [5] wh respec MSD he MSD value can be cmpued as MSD E h w( / h (4) he adapve fler and he unknwn channel were assumed have he same ap-lenghs ( M ) A FIR f unknwn sysem h was randmly generaed he npu sgnals were baned by generang a whe zer mean Gaussan randm sequence he sgnal nse ra (SNR) was calculaed as fr he npu sgnal and fr he bservan SNR 10lg 10 x ( n ( (5) SNR 10lg 10 ( x ( h ) n ( (6) Fg1 MSD curves f hree algrhms fr sysem denfcan smulan wh whe npu sequence ver 10 ndependen rals where SNR 10 db SNR 10 db M 16 (a) RLS and (b) N-RLS wh = 0997 (c) he prpsed VFF-RLS wh he h s changed a erans 5000 Fg Evlun n me f he frgeng facr () f he VFF-RLS 173

5 We cmpued MSD curves f he VFF-RLS algrhm cmpared wh he RLS and he N-RLS algrhm (Fg 1) MSD was larger when usng he RLS algrhm han when usng he her w algrhms because he sysem has a nsy npu sequence (LS prblem) he MSD f he VFF-RLS was less han ha f he N-RLS afer 1000 erans Alhugh he MSD curve f he N-RLS algrhm reached a he seady sae he MSD curve f he prpsed VFF-RLS algrhm decreased cnnuusly (Fg 1) An abrup change f he sysem s nrduced a erans 5000 In hs case N-RLS dverges fr a whle Hwever ur VFF-RLS prvdes gd rackng capably because he varable frgeng facr allws ur algrhm respnd sensvely changes n he sysem (Fg ) Frm hs resul ur VFF-RLS algrhm esmaed he FIR f he unknwn sysem h mre precsely han des he N-RLS algrhm whch uses a fxed frgeng facr fr bh f sanary and nnsanary envrnmens 3 Cnclusn hs sudy prpsed he VFF-RLS algrhm whch uses a varable frgeng facr slve he LS prblem fr adapve FIR flerng Our mehd f varyng he frgeng facr was descrbed and cmpared wh he exsng N-RLS algrhm In cmpuer smulan he prpsed VFF-RLS algrhm acheved beer esman f he FIR f he unknwn sysem han dd he N-RLS algrhm he adapve flerng algrhm has been used n numerus adapve sgnal prcessng applcans ncludng ech cancelan and sysem denfcan Because precse esman f he FIR f an unknwn sysem s mpran he prpsed VFF-RLS algrhm wll be useful n adapve sgnal prcessng applcans 4 Acknwledgemens hs research was suppred by he MKE(he Mnsry f Knwledge Ecnmy) Krea under he IRC(Infrman echnlgy Research Cener) suppr prgram (NIPA-01-H ) supervsed by he NIPA(Nanal I Indusry Prmn Agency) 5 References [1] G H Glub and C F Van Lan An analyss f he al leas squares prblem n SIAM Number Anal vl 17 pp Dec 1980 [] G H Glub and C F van Lan Marx Cmpuans Balmre MD: Jhns Hpkns Unv Press 1996 [3] C E Davla An effcen recursve al leas squares algrhm fr FIR adapve flerng IEEE rans Sgnal Prcessng vl 4 pp Feb 1994 [4] DZ Feng X-D Zhang D-X Chang and W X Zheng A Fas Recursve al Leas Squares Algrhm fr Adapve FIR Flerng IEEE rans Sgnal Prcessng vl 5 pp Oc 004 [5] S Haykn Adapve Fler hery Englewd Clffs NJ: Prence-Hall 1996 [6] S-H Leung and C F S Graden-based varable frgeng facr RLS algrhm n me-varyng envrnmens IEEE rans Sgnal Prcess vl 53 n 8 pp Aug 005 [7] C Palelgu J Benesy and S Cchna A Rbus Varable Frgeng Facr Recursve Leas-Squares Algrhm fr Sysem Idenfcan IEEE Sgnal Prcessng Leers vl 15 pp Oc

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