A Robust Total Least Mean Square Algorithm For Nonlinear Adaptive Filter

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1 A Robust otal east Mea Square Algorthm For Nolear Adaptve Flter Ruxua We School of Electroc ad Iformato Egeerg X'a Jaotog Uversty X'a 70049, P.R. Cha Chogzhao Ha, azhe u School of Electroc ad Iformato Egeerg X'a Jaotog Uversty X'a 70049, P.R. Cha czha@xjtu.edu.c Abstract - he robust olear adaptve flterg problem based o olterra model s researched whe the put ad output observato data are both corrupted by ose ths paper. O the bass of mmzg olterra total mea square error (MSE), a robust total least mea square adaptve flterg algorthm for olear olterra flter s proposed. he performace aalyss demostrates that the robust performace of the preseted algorthm s cer tha other exsted algorthms. Ad smulato results have also show the promet advatages of the preseted algorthm, t ca't oly permt to use larger learg factor, but ts covergece precso s remarkably hgher tha other algorthms uder hgher ose evromets. Keywords: Nolear flter, olterra seres, total least mea squares, robust adaptve flter. Itroducto It s well kow that olear adaptve flter has play a mportat role trackg fuso system[, commucato system[, ad cotrol system etc. Because olterra seres ca completely descrbe the put ad output trasfer characterstc of a large type of olear system, the olterra adaptve flter has bee wdely studed recet years. Usually, the put ad output data sampled from a practcal system are all corrupted by ose. Hece, t s ecessary to fd a effcet robust adaptve flter algorthm, ad ts covergece performace should be good. he traoal olterra adaptve flter algorthm s bult up based o mmzg the mea squares error [5. Because there s a uderlyg assumpto that the ose s mposed oly ad totally the output observato data, the performace of ths kd of algorthm becomes worse whe t s appled to a practcal flter problem. Although MCA algorthm for the total least squares (S) soluto ca provde better flter effects for ths problem [4, but ts robust performace wll become worse for larger learg factor or uder hgher ose evromets. O the bass of the dea of mmum total mea square error of olear olterra flter, by modfyg the gradet of olterra total mea squares error (MSE), a robust total least mea square (RMS) adaptve flterg algorthm for olear olterra flter s preseted the paper. he algorthm ca effcetly decrease the bad effect of the ose, ad more mportatly, t permts to use a larger learg factor, meawhle, t stll keeps fe covergece performace, that s very magetc for the practcal applcatos. he paper s orgazed as follows: the secto s devoted for ordary olterra adaptve flter algorthm. he robust olterra total least mea squares (MS) algorthm s derved secto 3, meawhle, ts performace s aalyzed. he smulato results are show secto 4. Fally, the secto 5 s the cocluso. olterra adaptve flter algorthm. olterra MS adaptve flter For SISO tme-varat casual olear system, ts output y( ca be wrtte as the olterra seres tme doma as followg [6: where y ( t ) y ( t ), () y (! h ( τ,!, τ) u( t τ) dτ, N, ad N s the atural umber set. y ( s the output of the thorder olterra seres, ad h τ, τ! τ ) s the th-order (

2 olterra tme-doma kerel or the th-order mpulse respose fucto. I practcal flter problem, the dscrete trucated form of olterra seres s used as followg: y ( k ) N N y k ( ) M h ( m, m,!, m ) m 0 M! u( k m ), () m 0 where M s a memory legth of the olterra kerel, ad N s the maxmum order of the trucated olterra seres. he olterra MS adaptve flter s just ole to adjust the kerel coeffcets of olterra flter by a recursve algorthm to mmze the mea square error of the flter. Cosder that the put ad output observato data are both corrupted by ose, ad wrte the put ad output at k as ( ) ( ) ( k u k u k + ) ad d( d( + o (, where ( ad o ( are the adve ose. Defe the put observato vector of olterra flter at k as: [ x ( x ( ( k X ( k )! ), (3) xn where x ( (,!, N) s the th order put observato vector of the olterra flter, for example: [ u u u! u u u! uu u u x ( m m! M (4) where u ( k m ). Note that, as usual, the olterra u m + kerel s assumed to be symmetrc wthout loss of geeralty[7. Correspodg defe the th order kerel vector, for example: h [ h 0,0)! h (0, M ) h (,) h ( M, M ) (! Ad the defe the olterra kerel vector as: [ h h h N (5) H!. (6) he, the olterra MS adaptve flter algorthm ca be descrbed as: [5 H ( k + ) H ( + µε( X (, (7) where ε ( d( y( s error, µ s the learg factor whch cotrols the stablty ad rate of covergece of the adaptve algorthm. he olterra MS adaptve flter algorthm has bee extesvely appled the olear adaptve flter. However, there s a uderlyg assumpto that the put sgal s kow exactly, practcal stuato t sometmes may be mpossble to avod oses whe the put sgal s measured. Due to the presece of oses both put ad output of the aalyzed system, the olterra MS algorthm ca oly obta the suboptmal solutos.. olterra S adaptve flter I order to mprove the performace of the adaptve flter for ths kd of practcal problem, some adaptve methods based o optmal total least squares (S) techque has bee proposed. By usg Mor Compoets Aalyss (MCA), a modfed at-hebba learg rule for S soluto ca be obtaed [4. Smlar to olterra MS method, by descrbg the olterra flter as a pseudo-lear oe, the olterra MCA adaptve flter algorthm ca be bult as follow: [ ε( ) k + ) µε( k (8) where W [ H, s the olterra augmeted kerel vector, Z ( k ) [ X (, d( s called as the olterra augmeted observato vector, ad ε ( Z (. µ s the learg factor whch cotrols the stablty ad rate of covergece of the adaptve algorthm. However, the flterg performace of the olterra MCA algorthm wll become worse whe a larger learg factor s used or sgal-ose-rate (SNR) s lower. So a more effectve adaptve flter algorthm wll be proposed based o mmzg the olterra total mea squared error. 3 Robust olterra total least mea squares (MS) algorthm 3. he dervato of robust MS algorthm Defe the total error at k as ε ( e(. he W ( refer to the Wdrow s dea [8, the olterra total mea squared error (MSE) ca be obtaed by takg the expected value of e (. ε ( MSE E{ e ( } E{ } W (

3 W ( Z ( W ( R E{ }, (9) W ( W ( where R E{ Z ( }. Now the adaptve flter s just to adjust the kerel coeffcets of olterra flter to mmze the MSE. It s called olterra total least mea squares (MS) flter. Accordg to (9), t ca be kow that the soluto of MS problem s the ormalzed egevector assocated wth the smallest egevalue of the put autocorrelato matrx R. Now usg the method of steepest descet solves the MS problem. o develop the adaptve algorthm, a estmated gradet should be obtaed by usg e ( as the estmated value of E { e ( } ad dfferetatg t. Ad the modfyg ths estmated gradet as : E e ε ( { } [ W ( ε( k ) ( [ W (. (0) So the robust MS algorthm s a mplemetato of steepest descet usg above modfed estmated gradet: µε( [ W ( ε( k ) k + ) [ W ( () 3. he performace aalyss of the proposed algorthm Now aalyze the robust performace of the preseted algorthm by comparg wth olterra MCA algorthm. Assume that the ose ( ad o ( are both depedet statoary adve whte ose, the deote X ( k ) ( ( X + d( d( + o (. () For the sake of smplcty, assume the olterra augmeted kerel vector at k has coverged to true value W, so Z ( 0. Ad the: ε ( Z ( + Z ( Z (. (3) So the expected value of the weght gap ca be obtaed from (), () ad (3) as: Z ( W ( k + ) k + ) µ E. (4) W ( Whle for the olterra MCA algorthm (8), t s: { Z ( )} { W ( Z ( } k + ) µ E k + µ E. (5) Comparg (4) wth (5), t s clear that the robust performace of the preseted algorthm s cer tha the olterra MCA adaptve flter algorthm. 3.3 he covergece of robust MS algorthm Assume Z ( s statoary, ad Z ( s ot correlated wth W (, ad followg the reasog of Xu [4, () ca be approxmated by the followg dfferetal equato: [ R W ( + W ( R t ) [ W ( d, (6) where R E{ Z ( }, R E{ Z ( k )}. he asymptotc property of (6) approxmates that of (), ad the asymptotc property of (6) ca be sured by the followg theorem. heorem: et R s a sempostve defte matrx, λ + ad c are respectvely t s smallest egevalue ad + correspodg ormalzed egevector wth the o-zero last compoet. If W ( 0) c + 0, the lm α + ( c, + t where α +( t ) s a scalar fucto. I.e. W ( teds to be the drecto of c asymptotcally as + t. Proof: Deote + egevalues of R by λ, λ,!, λ +, where λ s the smallest egevalue, ad deote a set of + correspodg ormalzed egevectors by c, c,!, c +. So R ad W ( ca be wrtte as: + R λ c c, + α ( c. Ad the t ca be obtaed:

4 d + + dα ( c [( λ + R t ) ) α ( c [ λ + R t ) 4. (7) dα ( α (. (8) 4 α Because of α ( W ( c ad W 0) c+ 0 + ( 0( t 0). Defe (, so y( 0.64u( + u( k ) + 0.9u ( + u ( k ) Defe the learg error of the olterra kerel vector as: Error( 0log H ˆ ( H, () where [ H s the true olterra kerel vector of the aalyzed olear system, H ˆ ( s estmated olterra kerel vector. Assume that the SNR (sgal-ose rate) of the put sgal s equal to the SNR of the output sgal. he adve ose s depedet zero-mea whte ose. Ad α ( t ) η ( t ). α ( ) + t SNR E{ y( k ) } 0 log( ), () E{ ( k ) } o Ad the the followg dfferetal equato ca be obtaed: dη ( α + ( [ dα ( α ( [ dα α ( + + ( λ + λ η ( t ). (9) W ( t ) Because λ s the smallest egevalue, the dfferetal + system descrbed by (9) wll asymptotcally coverge. herefore, lm η ( 0 (,!, ). Because of α + ( 0 t for t 0, the lmα ( 0 (,!, ). t So our cocluso ca be obtaed: α. (0) + lm W ( lm ( ) + ( ) + t c α t c t t hs completes the proof of the theorem. 4 Smulato I the smulatos, three algorthms are used for olear adaptve flter,.e. the traoal olterra MS adaptve flter algorthm [5, the olterra MCA adaptve flter algorthm [4 ad the preseted robust olterra total least mea squares (RMS) adaptve flter algorthm. her covergece performaces are compared uder dfferet extet ose evromet ad by usg dfferet learg factor. he olear system s gve as followg: where y ( s the output sgal vector, o ( s the terferece of the output. Fg ad Fg show respectvely the learg curves for SNR0dB ad SNR0dB wth µ hese two learg curves dcate that the preseted RMS olear adaptve flter algorthm ca keep well covergece performace whe a larger learg factor s used or SNR s lower, whle the olterra MS ad MCA algorthms ca ot. Moreover, Fg 3 ad Fg 4 have show respectvely the covergece curves of h ( ) ad h (, ) at SNR0dB, ad these covergece curves at SNR0dB are gve Fg 5 ad Fg 6. Because the olterra MS ad MCA algorthms dverge severely, ther correspodg curves are ot plotted. Fg 7 ad Fg 8 show respectvely the covergece curves of h ( ) ad h (, ) for SNR0dB ad µ hs set of curves llustrates the covergece performace of these algorthms, whe a smaller learg factor has bee used, ad SNR s hgher. he smulato results have show that the preseted robust olterra total least mea squares algorthm for olear adaptve flter has excellet robust covergece performace, ad t permts to use a larger learg factor, that s very fe for the practcal applcatos. 5 Cocluso O the bass of mmzg MSE, the paper proposes a robust adaptve flter algorthm for olear olterra flter wth the corrupted put ad output sgal. he olterra seres s used to descrbe ths kd olear system. he aalyss ad smulato results have show that ths algorthm s more effcet to a practcal olear flter.

5 Fg learg curve atµ 0. 05, 0dB Fg learg curve atµ 0. 05, 0dB Fg 3 h ( ) covergece curve atµ 0. 05, 0dB Fg 4 h(, ) covergece curve atµ 0. 05, 0dB Fg 5 h( ) covergece curve atµ 0. 05, 0dB Fg 6 h (, ) covergece curve atµ 0. 05, 0dB Fg 7 h( ) covergece curve atµ , 0dB Fg 8 h(, ) covergece curve atµ , 0dB

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