Robust Adatve Volterra Flter uder Maxmum Corretroy Crtera Imulsve Evromets Weyua Wag Haqua Zhao Badog Che Abstract: As a robust adatato crtero the maxmum corretroy crtero (MCC) has gaed creased atteto due to ts successful alcato the adatato esecally olear ad o-gaussa stuatos. I ths aer the secod-order Volterra (SOV) flter based o MCC s derved whch s called MCC-SOV. It combes the advatages of the SOV flter ad MCC. However smlar to the covetoal adatve algorthm the roosed MCC- SOV flter has a tradeoff betwee covergece rate ad steady-state error. I order to solve ths roblem we reset a combato of the MCC-SOV flter by combg Volterra kerels whch s called CK-MCC-SOV. I addto a weght trasfer method s aled to mrove the erformace of the roosed flter terms of the covergece rate ad the steady-state error. Fally smulatos are carred out to demostrate the advatages of the roosed flters. Keywords: Adatve flterg Covex combato Maxmum corretroy crtera Secod-order Volterra flter Imulsve ose Itroducto As we all kow the cocet of lear system detfcato lays a mortat role statstcal sgal rocessg [8]. However sce may real alcatos requre comlex olear models modelg by meas of the lear flter may result oor erformace. I such case the olear system detfcato techques have become a hot toc over the last decades. To accurately model ad characterze the olear system lots of olear aroaches have bee roosed such as Haqua Zhao () hqzhao@home.swjtu.edu.c Weyua Wag weyuawag@my.swjtu.edu.c Badog Che chebd@mal.xjtu.edu.c School of Electrcal Egeerg Southwest Jaotog Uversty Chegdu Cha. School of Electroc ad Iformato Egeerg X a Jaotog Uversty X a Cha
olyomal exaso [34] eural etworks [4] Kalma flter [ 30] ad so o. Sce eural etworks have outstadg erformace the aroxmato of olear fucto they have bee wdely used olear system alcatos. Furthermore dfferet tyes of olear flters based o eural etworks have bee roosed ad aled to lots of aers. To obta better erformace for dscrete-tme Markova jum fuzzy eural etworks wth tme delays authors reseted the mxed H-Ifty ad Passve Flterg [4]. Besdes to address the sem-markova jum system authors reseted a ew method of sldg mode cotrol [5]. Comared wth methods of eural etworks ad Kalma flter olyomal exaso method has lower comutatoal comlexty. As oe of the most famous methods of olyomal exaso algorthm the Volterra flter ca coe wth a geeral class of olear systems [34 35] because most olear systems ca be aroxmately modeled by olyomals. Moreover sce the arameters of Volterra system are learly assocated wth the outut detfcato of ths model s a lear estmato roblem. Because the Volterra flter eeds a large umber of coeffcets for accurately modelg olear systems oly the secod-order Volterra (SOV) ad thrd-order Volterra (TOV) flters ca be used for the mlemetato. Smlar to the covetoal adatve flterg algorthms the adatve SOV flter usually uses the mmum mea square error (MMSE) crtero as the adatato crtero [4 7]. The MMSE crtero has a otmal soluto uder Gaussa ose evromet. However whe the outut sgals are corruted by the mulsve ose ths crtero would ot be robust. To address ths ssue the maxmum corretroy crtero (MCC) has bee roosed [5 6]. Corretroy whch has already bee used may areas such as adatve flterg classfcato ad robust regresso s a ew kd of measure to estmate the smlarty betwee two radom varables [6 7 8 9 0 8]. Comared wth MMSE crtero the MCC ca rovde the strog atjammg caablty to mulsve terfereces though t may suffer from erformace degradg uder the Gaussa ose evromet. I [9] authors used corretroy as a cost fucto lear adatve flters. I [36] authors aled the MCC to kerel adatve flter whch was robust agast the o-gaussa measuremet ose. Nevertheless the aforemetoed algorthms have the tradeoff betwee covergece rate ad steady-state error o the choce of the fxed ste sze. To overcome the roblem above the covex combato of adatve flters has bee roosed [ ]. Due to combg the good erformace of dfferet adatve flters the covex combato method ca offer comlemetary caabltes. Sce ths method has such vrtue t has bee aled several felds of sgal rocessg cludg bld equalzato sgal characterzato ad so o [7 5 6 3]. Referece [5] troduced a robust adatve algorthm by covexly combg two MCC-based adatve algorthms wth dfferet ste szes whch outerforms each of the dvdual flters. I [6] authors roosed a ovel affe rojecto sg algorthm (APSA) by adatvely combg two APSA flters to rovde good trackg erformace o-gaussa ose evromet. I ths aer a ew SOV flter based o MCC whch s called MCC-SOV flter s reseted. It s robust agast the mulsve ose. We vestgate the erformace of roosed MCC-SOV flter comarso wth the covetoal SOV flter uder the
mulsve ose crcumstace. Comared wth the algorthms [5] ad [4] whch are also robustess agast the o-gaussa ose ad ca oly obta good erformace secfc systems the roosed MCC-SOV flter ca be aled geeral olear system. Ufortuately there s a heret tradeoff betwee covergece rate ad steady-state error MCC-SOV flter. To overcome ths drawback a ovel algorthm derved by covexly combg two MCC-SOV flters s roosed where the scheme of the combato s obtaed by combg kerels (CK). It s called CK-MCC-SOV flter. The roosed algorthm rovdes sgfcatly comutatoal savgs comared wth the combato of Volterra flters (CVF) [3]. To rovde good robustess agast the o-gaussa mulsve terferece the mxg factor s obtaed by maxmzg the corretroy. Addtoally order to further mrove the erformace of combatos of kerel we med a weght trasfer method roosed [3] ad aly t to reseted algorthm. The rest of the aer s orgazed as follows. The bref revew of corretroy s gve Secto. I Secto 3 we reset MCC-SOV flter. The covex combato of the kerel of the SOV flter s rovded Secto 4. The smulato results are gve Secto 5. Fally the cocluso s draw Secto 6. Bref Revew of Corretroy The cocet of corretroy was reseted the lear adatve flter to deal wth o-gaussa ose esecally mulsve ose. The corretroy s a method to estmate the smlarty betwee two arbtrary radom varables X [ x... x ] T N ad Y y y N [... ] T. The corretroy s usually defed by [9] where E[ ] V ( X Y) E ( X Y) ( x y) df ( x y) () deotes the exectato oerator XY s a symmetrc ostve defte kerel FXY ( x y ) deotes the jot dstrbuto fucto ad s the kerel wdth. I ths aer we use a ormalzed Gaussa kerel to exress corretroy [6] e ( xy ) ex () where e x y ad reresets the kerel sze of corretroy. I ractce stuatos oly a fte umber of samles ( x y ) of the varables X ad Y are avalable. Therefore t s more commo to use the samle estmator for the exectato oerator N V( X Y) ( x y). (3) N Substtutg () to () ad alyg Taylor seres exaso yeld
( X Y) V ( X Y) E ex ( X Y) E 0!. (4) Obvously corretroy ca be see as a measure of quatfyg how dfferet radom varable X from radom varable Y robablty cotrolled by the kerel wdth. Also corretroy s a geeralzed correlato fucto for two radom rocesses whch cludes more tha two order momets of the radom varable. Utlzg the corretroy as a cost fucto s effectve uder mulsve ose evromet. I the secfc case of adatve flterg alcato the weght udate equato ca be obtaed by maxmzg the corretroy betwee the outut of the flter ad observed sgal. We call ths crtero the maxmum corretroy crtero. Sce corretroy s sestve to outlers the MCC based adatve flterg algorthm ca offer a mactful mechasm to relef the ll effects of the large outlers resultg from mulsve ose. E X Y 3 Proosed MCC-SOV flter 3. Volterra flter algorthm v ( ) z ( ) x ( ) d ( ) Nolear System (Plat) e ( ) Model (SOV flter) y ( ) Adatve algorthm Fg. Nolear system detfcato model based o olyomal flter Sce the Volterra seres model has may advatages t s the most dffusely used model for olear systems. I artcular ths model s wdely used the olear adatve flter because the covetoal coceto or crtera of lear adatve flters ca be easly exteded to sut ths model. The structure of olear system detfcato based o adatve olyomal flter s show Fg where x() z() e() d() ad v() rereset the ut sgal lat outut error sgal desred sgal ad the mulsve ose resectvely. I ths aer we maly study the trucated SOV flter whose outut ca be rereseted as
N y h ( m ) x( m ) m 0 w N N h ( m m ) x( m ) x( m ). (5) m 0 m m T u where h( m ) ad h ( m m ) are frst ad secod-order kerels resectvely. The ta weght vector The ut vector w u s gve by The ta weght vector w [ h (0) h ()... h ( N ) h (00) s exressed as... h ( N N )] T u [ x... x( N ) x... x ( N )] T. (7) w (6) of SOV flter ca be udated by the least mea squares (LMS) algorthm whch the MMSE crtero s used as the cost fucto T where e d w u. J E e (8) MMSE Usg the method of stochastc gradet descet yelds [8] e w( ) w w. (9) Arragg (9) leads to the LMS-SOV flter as follows w( ) w e u (0) where s the ste sze. It s brght to have dfferet covergece factors for the frst- ad secod-order kerels of the Volterra flter []. I ths case the weght udate equato of the LMS- SOV flter are gve by w( ) w e ul () w( ) w e uq where w [ h (0) h ()... h ( N )] s the lear ta weght vector w [ h (00)... h ( N N )] T s the quadratc ta weght vector u l [ x... x( N )] s the ut vector for frst-order kerel the ut u ( ) [ ( )... ( )] T s the ut vector for secod-order kerel ad x x N q e d w ul w u q. ad lear ad quadratc kerels resectvely. () are the ste szes for
3. Proosed MCC-SOV flter However o-gaussa ose esecally mulsve ose the error sgal varace may be fte. Therefore the LMS algorthm s mroer. From [9] we ca get that the MCC s more robust agast mulsve ose tha the MMSE crtera. Thus we modfy the cost fucto as follows e ( ) J MCC d( ) y( ) ex. N N N N () Rearragg () yelds d( ) w ( ) ul w( ) uq( ) J MCC ex. N N (3) Smlar to MMSE crtero usg a teratve gradet ascet aroach to search the otmal soluto we have w( ) w w J MCC (4) w( ) w w J MCC (5) where ad are ostve costat. Substtutg (3) to (4) ad (5) yelds w( ) w N N e () ex (6) w e () ex w ( ) w. (7) N N w The gradets (6) ad (7) ca be derved as follows e () ex e () e () ex w w e () e ( ) ex e ( ) w e () d( ) w( ) ul( ) w( ) uq( ) ex e ( ) w e () ex e ( ) ul ( ) (8)
e () ex w e () ex w e () e ( ) ex e ( ) w e () d( ) w( ) ul( ) w( ) uq( ) e () ex e ( ) w e () ex e ( ) uq ( ). (9) Table. Summary of the roosed MCC-SOV flter Parameters settg: Italzato: w (0) w (0) For =.. do y w u w u ) l ) e d y e ( ) 3) w( ) w eex u ( ) l q e ( ) 4) w( ) w eex u ( ) q Ed for Cosequetly we ca get the MCC-SOV flter e () w( ) w ex e( ) ( ) 3 u l (0) N N e () w( ) w ex e( ) ( ) 3 u q. () N N Usg the curret value (.e. N =) to aroxmate the sum term (0) ad () yelds e ( ) w( ) w eex u ( ) l ()
e ( ) w( ) w eex u ( ) q (3) where ad 3 are the ste szes for the frst- ad 3 secod-order kerels resectvely. The summary of the roosed MCC-SOV flter s show Table. 4 Proosed CK-MCC-SOV flter 4. Proosed CK-MCC-SOV flter Accordg to [ 8 33] there s a heret tradeoff betwee covergece rate ad steady-state errors all adatve flters. I other words the large ste sze would lead to large steady-state error ad fast covergece rate. Ad vce versa. Thereby the roosed MCC-SOV flter has the drawback of coflctg requremet betwee fast covergece rate ad small steady-state error though t s more robust agast mulsve ose tha the covetoal algorthms. I order to overcome ths roblem we focus o the covex combato of two MCC-SOV flters whch the combato of two kerels wth dfferet ste szes s used to relace the corresodg kerel. The block dagram of the roosed combato of the SOV flters s llustrated Fg.. v ( ) z ( ) x ( ) d ( ) Nolear System + + e ( ) e () Lear kerel y w e Lear kerel w( ) e y + e ( ) y + y ( ) Quadratc kerel w Quadratc kerel w e y y + e ( ) y Fg.. Block dagram of the roosed combato of MCC Volterra flters scheme
For the sake of keeg the heret roertes of all comoet flters dvdual kerels should be deedetly adated accordg to ther ow crtera. I the sequel two lear kerels of the roosed scheme are adated through the followg recurso e w( ) w e ex u ( ) l e w' ( ) w' e ex u ( ) l (4) where the ad are the ste szes of lear kerel. The recurso of the quadratc kerels Fg. are gve by e w ( ) w e ex ( ) uq e w' ( ) w' e ex ( ) uq (5) where are the ste szes of the quadratc kerel. (4) ad (5) ad e deotes the ovel quadratc error sgals where = ad =. I order to make the dvdual kerels be adated deedetly the quadratc error sgals are defed as. e ( ) ad e ( ) are the outut error sgals of the comoet flters e d y y ' (6) ' where y deotes the combato of two costtuet kerels ad y s the outut of kerel. The relatosh betwee y ( ) ad y ( ) s gve by where ( ) y y y (7) y y y s the mxg arameter the rage of 0 to. Ths arameter cotrols the combato of the two flters at each terato ad comes from a sgmodal actvato fucto gve by sgm a (8) a e where a ( ) s the mxg arameter. I order to make the roosed CK-MCC-SOV flter robust agast mulsve ose a ew stochastc ostve gradet method based o the corretroy maxmzato crtera s utlzed to derve the udate equato for a as follows
a ( ) a a e ex a e a a ex e a e e a ex e ( ) ( ) a a e a a e ( ) ( ) ( ) ( ) ex ( e e ) where / s a ste-sze arameter. a a Although the udate equato (9) s robust agast the mulsve ose t s dffcult to choose the arorate ste sze [4]. To overcome ths dffculty a ormalzed rule s aled to adat the mxg arameter whch ca make selecto of ste sze smle ad lead to the more stable erformace of the flter. The the mxg arameter udate equato (9) ca be modfed as follows e ex a a( ) a r a (30) where r s gve by a e a e e ex e ( ) ( ) r s the estmate of where s the forgettg factor. e e a. A recursve way for comutg r (9) r r ( ) ( ) e e (3) The erformace of the roosed scheme by usg the way of the trasfer of the coeffcet [3] we adot ad modfy ths method as follows. ad are defed as wdow legth. The f mod N0 0 ad N 0 N 0 a ( ) a we use the followg equato to calculate w ' ( ) w' ( ) w ( ). (3)
Ste Italzato Table. Summary of the roosed CK-MCC-SOV flter Algorthm w (0) 0 m N 0 Calculate 3 4 5 m y w' (0) 0 a (0) N 0 ad e w (0) 0 a 0 w' (0) 0 usg (6) ad (7) Udate fast flter accordg to (4) ad (5). Let. m m Calculate a m m ad ( ) usg (30) ad (8). If a ( ) a let a ( ) a ( ) 0. Else f a ( ) let a ( ) ( ).If also a m 0 let a w' ( ) w ( ) ad m N0 If the codtos ste 4 are ot met udate usg (4). w ' ( ) 6 If a ( ) a let a ( ) a ( ) 0. Else f a ( ) a let a ( ) a ( ). 7 If also m 0 let w' ( ) w( ) ad m N. 8 0 If the codtos ste 7 are ot met udate w ' ( ) usg (5). 9 Let ad retur to ste. Smlar to the lear kerel the quadratc kerel ca be udated as follows If mod N0 0 ad a ( ) a the followg equato s aled to comute w ' ( ) w' ( ) w ( ). (33) Comared wth the stadard combato method the addtoal oeratos of ths method are the mod N0 ad mod oeratos. However whe equatos (3) ad (33) are used to comute w ' ( ) ad w ' ( ) resectvely the comlexty of ths method s actually lower tha that of the orgal combato method. The summary of the roosed CK-MCC-SOV flter s exhbted Table. N 0 4. Covexly costraed mxtures of the roosed algorthm Accordg to the method [9 0-] the Taylor seres exaso of the error sgal e s exressed as e e ( ) e a a o a ( ) ( ) e a a (34)
where o a ( ( )) deotes the hgher order terms of the Taylor seres exaso. Equato (30) mles that e y y ( ) a ad a e ( ) a ex ( ) e y y ( ) r (36) Substtutg (35) ad (36) to (34) ad omttg the hgher order momet yeld a e ( ) e ( ) e ex ( ) y y (37) r I order to make the error sgal close to zero whe teds to fty the error sgal e satsfes e a e ( ) e ( ) e y y ex (38) r The we obta a e ( ) y y ex ( ) r (39) Accordgly the rages of the mxg arameters roosed algorthm are exressed as r 0 a ( ) e y y ex a (35) (40) 5 Smulatos To evaluate the effectveess of two roosed algorthms smulato exermets are carred out olear system detfcato uder mulsve ose evromet. The smulato results show Fgs. 3-7 are obtaed by averagg over 00 deedet trals. To assess the estmato erformace we use the followg ormalzed kerel error (NKE) [9] w Ψ NKE Ψ (4) w Ψ F NKE Ψ F where Ψ s the lear kerel of system Ψ s the quadratc kerel of system s the Eucldea -orm ad F s the Frobeus orm. The mea square devato
(MSD) s also used as a measure of the erformace whch s defed as MSD 0 log w w. s the otmal weght vector. 0 o w o 5. Exermet I ths exermet the erformace of the MCC-SOV flter s comared wth that of LMS-SOV [7] ad LMP-SOV [3] flters. Ths exermet s o the olear system detfcato where the olear system has the followg form [3] y x 0.8 x( ).9 x( 3) 0.95 x (4). x x( ) 0.63 x( ) x( 3) The ut sgals are obtaed by a zero-mea whte Gaussa rocess wth ut varace. The outut sgal-to-ose rato (SNR) s set to 30 db. The mulsve terferece s geerated as c A [7] where s a Beroull c ( ) rocess wth the robablty desty fucto descrbed by c c P (wth ( ) 0 r terferece) ad A ( ) T o P r ( ) Pr ad beg the robablty of the occurrece of the mulsve s a zero mea whte Gaussa ose wth varace A 00 E ( ) u w. I ths exermet Pr s set to 0.0. The ste szes of the MCC-SOV ad LMP-SOV flters are chose such that the tal covergece rates of the MCC-SOV ad LMP-SOV flters are aroxmately same. The ste sze of the LMP-SOV flter s same as that of MCC-SOV flter. The learg curves of ormalzed lear kerel error ad ormalzed quadratc kerel error are show Fg. 3. As we ca see both MCC-SOV ad LMP-SOV flters are robust agast the mulsve ose whle LMS-SOV flter has large fluctuatos. Sce the egevalue sread of the correlato matrx for quadratc terms s larger tha that of lear terms the estmato error for the lear kerel s less tha that of the quadratc kerel both methods whch ca be see Fg. 3. It s also foud that the roosed MCC- SOV flter has the less estmato error tha LMP-SOV flter.
Fg.3. Learg curve of MCC-SOV LMS-SOV ad LMP-SOV flters 5. Exermet I ths subsecto the erformace of the roosed CK-MCC-SOV flter s comared wth that of MCC-SOV flter. The ukow system ad the mulsve terferece are the same as those the frst exermet. The ut sgal s a zero-mea uform ose wth ut varat. The ste szes of slow flter s lear kerel ad quadratc kerel are set to 0.0 ad 0.0 resectvely. The ste szes of fast flter s lear kerel ad quadratc kerel are set to 0.5. ad Pr are set to 4 ad 0.0 resectvely. Besdes the ste sze of the mxture arameter s set to 0.. The arameter s set to 0.9. As show Fg. 4 the CK-MCC-SOV ad MCC-SOV flters are robust agast mulsve terferece. It s also see that the CK-MCC-SOV flter outerforms the MCC-SOV flter terms of the covergece rate ad the steady-state error. a
Fg.4. Learg curve of MCC-SOV ad CK-MCC-SOV flter Table 3. Covergece rates of MCC-SOV ad CK-MCC-SOV flter 34 Algorthm Covergece Rate/Iteratos (5.5dB for Lear kerel ad 0 db for Quadratc kerel) MCC (small ste sze) Lear Quadratc kerel kerel MCC (large ste sze) Lear Quadratc kerel kerel CK-MCC (o trasfer) Lear Quadratc kerel kerel Lear kerel CK-MCC Quadratc kerel 60 500 0 300 0 300 0 300 Covergece Rate/Iteratos (38dB for Lear kerel ad 30 db for Quadratc kerel) 900 500 -- -- 900 500 900 00 The covergece rates of the MCC-SOV ad CK-MCC-SOV flters are also lsted Table 3 where covergece rates are deoted by the terato umber frstly 3 The MCC-SOV ad CK-MCC-SOV flters are abbrevated as MCC ad CK-MCC resectvely. 4 The covergece rates are measured by the terato umber where the algorthms frstly reach the corresodg MSD
Algorthm reachg the corresodg MSD. To comare the steady-state MSD we calculate the steady-state MSD by averagg over more tha 500 stataeous MSD values the steady state for each algorthm. The results of steady-state MSD are gve Table 4. It s see from Table 3 ad Table 4 that CK-MCC-SOV flter wth trasfer has fastest covergece rate ad lowest MSD. I the other word the roosed CK-MCC-SOV flter does mrove the drawback of MCC-SOV flter. Table 4. Steady-state MSD of MCC-SOV ad CK-MCC-SOV flter MCC(small ste sze) Lear Quadratc kerel kerel MCC(large ste sze) Lear Quadratc kerel kerel CK-MCC (o trasfer) Lear kerel Quadratc kerel Lear kerel CK-MCC Quadratc kerel Steady-state MSD (db) -30.0-38.0-6.0 -. -30.0-38.0-30.03-38.0 The comutatoal tme of the MCC-SOV ad CK-MCC-SOV flters s show Table 5. It s see that the comutato tme s ot related to the value of ste sze. I addto the CK-MCC-SOV flter wth trasfer has lower comutato tme tha the CK-MCC-SOV flter wthout trasfer. Although CK-MCC-SOV flter has loger comutato tme tha the MCC-SOV flter t has the great better erformace tha MCC-SOV flter. Observg Table 5 aga we ca also get that the large ste sze would result large steady-state MSD ad fast covergece rate. O the cotrary the large ste sze would lead to small steady-state MSD ad slow covergece rate whch s corresodg to the cocluso at the begg of Secto 4. I a word there s a tradeoff betwee the comutato tme the covergece rate steady-state MSD ad the ste sze. Table 5. Comutatoal tme of the MCC-SOV ad CK-MCC-SOV flters for obtag Fg. 4 by usg MATLAB R04b o Itel(R) Core(TM)5 CPU 4460 at 3.0 GHz rocessor wth 8.00 GB RAM Algorthm MCC(small ste sze) MCC(large ste sze) CK-MCC (o trasfer) CK-MCC Comutato tme (s).45744.457455 3.7877.935537 Fg. 5 evaluates the erformace of the CK-MCC-SOV flter wth dfferet levels of mulsveess. Pr s set to 0.0ad 0.. It s foud that the CK-MCC-SOV flter algorthm s robust agast mulsveess varous levels of mulsveess. We also exame the erformace of the CK-MCC-SOV flter wth dfferet levels of SNR Fg.6. The results dcate that CK-MCC-SOV flter outerforms the MCC- SOV flter terms of the covergece rate ad the steady-state error varous levels
of SNR. I ths exermet we also comare the erformace of the CK-MCC-SOV flter wth that of CK-SOV flter [3]. The results are show Fg.7. As oe ca see the MCC-SOV flter s robust agast mulsve terferece whle the CK algorthm s dverget. Fg.5. Learg curve of MCC-SOV ad CK-MCC-SOV flters dfferet levels of mulsveess
Fg.6. Learg curve of MCC-SOV ad CK-MCC-SOV flters dfferet levels of SNR Fg.7. Learg curve of CK-LMS-SOV ad CK-MCC-SOV flters 6 Cocluso The maxmum corretroy crtero (MCC) s robust agast the mulsve ose. Motvated by ths dea ths aer resets the MCC-SOV ad CK-MCC-SOV flters
for olear system detfcato. Addtoally a weght trasfer aroach s aled to mrove the erformace of the roosed algorthm. Dfferet from the methods of flter desg reseted [3 5 4] the roosed algorthms ths aer are adatve flter algorthm where arameters of flters are tme-varat. I addto the roosed algorthms ca be aled a geeral olear system whle algorthms [5 4] oly address the sem-markova jum system or Markova jum system. Fally smulato exermets are carred out to demostrate the advatages of the roosed algorthms. Exermets results show that the MCC-SOV flter could obta the sueror erformace over the covetoal LMS-SOV ad LMP-SOV flters whe the measuremet ose s mulsve. It s also foud that the CK-MCC-SOV flter ca acheve hgh covergece rate wthout sacrfcg steady-state MSE erformace comarso wth the MCC-SOV flter. Ackowledgmets Ths work was artally suorted by Natoal Scece Foudato of the P.R. Cha (Grat: 657374 67340 ad 64330). Referece. J. Areas-García A. R. Fgueras-Vdal Adatve combato of ormalsed flters for robust system detfcato. Electro. Lett. 4(5) 874-875 (005). J. Areas-García A. R. Fgueras-Vdal A. H. Sayed Mea-square erformace of a covex combato of two adatve flters. IEEE Tras. Sgal Process. 54(3) 078-090 (006) 3. L. A. Azcueta-Ruz M. Zeller A. R. Fgueras-Vdal J. Areas-García W. Kellerma Adatve combato of Volterra kerels ad ts alcato to olear acoustc echo cacellato. IEEE Tras. Audo Seech Lag. Process. 9() 97-0 (0) 4. L. A. Azcueta-Ruz A. R. Fgueras-Vdal J. Areas-García A ormalzed adatato scheme for the covex combato of two adatve flters Proc. IEEE It. Cof. Acoustcs Seech Sgal Process. (008). 330-3304 5. R. J. Bessa V. Mrada J. Gama Etroy ad corretroy agast mmum square error offle ad ole three-day ahead wd ower forecastg. IEEE Tras Power Syst. 4(4) 657-666 (009) 6. B. Che X. Lu H. Zhao J. C. Prce Maxmum Corretroy Kalma Flter. Automatca 76 70-77 (07) 7. B. Che L. Xg H. Zhao N. Zheg J. C. Prce Geeralzed corretroy for robust adatve flterg. IEEE Tras. o Sgal Processg 64(3) 3376-3387 (06) 8. B. Che J. Wag H. Zhao N. Zheg J. C. Prce Covergece of a Fxed-Pot Algorthm uder Maxmum Corretroy Crtero. IEEE Sgal Processg Letters (0) 73-77 (05) 9. B. Che L. Xg J. Lag N. Zheg J. C. Prce Steady-state Mea-square Error Aalyss for Adatve Flterg uder the Maxmum Corretroy Crtero. IEEE Sgal Processg Letters (7) 880-884 (04) 0. B. Che J. C. Prce Maxmum corretroy estmato s a smoothed MAP estmato IEEE Sgal Processg Letters 9(8) 49-494 (0)
. P. S. R. Dz Adatve flterg algorthms ad ractcal mlemetato 4th ed. (Srger New York 03). G. A. Ecke L. B. Whte Robust exteded Kalma flterg IEEE tas. Sgal rocess. 47 (9) 596-599 (999) 3. M. M. Foud M. G.Mostafa A.S.Mashat T. F. Gharb WSAWF: A weghted sldg wdow flterg algorthm for frequet weghted termets mg. It. J. Iov. Comut. If. Cotrol ( 4) 4-439 (05) 4. T. Koh E. J. Powers Secod-order Volterra flterg ad ts alcato to olear system detfcato. IEEE Tras. Audo Seech Lag. Process. 33(6) 445-455 (985) 5. F. L L. Wu P. Sh C. Lm State estmato ad sldg mode cotrol for sem- Markova jum systems wth msmatched ucertates. Automatcs. 5 386-393 (05) 6. W. Lu P. P. Pokharel J. C. Príce Corretroy: roertes ad alcatos o- Gaussa sgal rocessg. IEEE Tras. Sgal Process. 55() 586-598 (007) 7. L. Lu H. Zhao K. L ad B. Che A ovel ormalzed sg algorthm for system detfcato uder mulsve ose terferece Crcuts Systems ad Sgal Processg 35(9) 344 365 (06) 8. L. Lu H. Zhao Actve mulsve ose cotrol usg maxmum corretroy wth adatve kerel sze Mechacal Systems ad Sgal Processg 87 80 9 (07) 9. L. Lu H. Zhao B. Che Collaboratve adatve Volterra flters for olear system detfcato α-stable ose evromets. Joural of the Frakl Isttute 353 4500 455 (06) 0. D. P. Madc NNGD algorthm for eural adatve flters Electro. Lett. 39(6) 845-846 (000). D. P. Madc J. A. Chambers Toward the otmal learg rate for backroagato Neural Process. Lett. () (000). D. P. Madc A. I. Haa M. Razaz A ormalzed gradet descet algorthm for olear adatve flters usg a gradet adatve ste sze IEEE Sgal Process. Lett. 8() 95 97 (00) 3. V. H. Nascmeto R. C. De. Lamare A low-comlexty strategy for seedg u the covergece of covex combatos of adatve flters I Proc. IEEE It. Cof. Acoustcs Seech Sgal Process. (0). 3553-3556 4. P. Sh Y. Zhag M. Chadl R. K. Agarwal Mxed H-Ifty ad Passve Flterg for Dscrete Fuzzy Neural Networks Wth Stochastc Jums ad Tme Delays. IEEE Tras. Neural Netw. Lear. Syst. 7(4) 903-909 (06) 5. L. Sh Y. L Covex Combato of Adatve Flters uder the Maxmum Corretroy Crtero Imulsve Iterferece. IEEE Sgal Process. Lett. () 385-388 (04) 6. L. Sh Y. L X. Xe Combato of affe rojecto sg algorthms for robust adatve flterg o-gaussa mulsve terferece. Electro. Lett. 50(6) 466-467 (04) 7. M. Sayad F. Faech M. Najm A LMS adatve secod-order Volterra flter wth a zeroth-order term: steady-state erformace aalyss a tme-varyg evromet. IEEE Tras. Sgal Process. 47(3) 87-876 (999) 8. A. H. Sayed Fudametal of Adatve Flterg (Wley New Jersey 003) 9. A. Sgh J. C. Prce Usg corretroy as a cost fucto lear adatve flters IEEE Iteratoal Coferece o Acoustcs Seech ad Sgal Processg (009). 950-955 30. S. Wag G. Q L. Wag Hgh degree cubature H-fty flter for a class of olear dscrete-tme systems. It. J. Iov. Comut. If. Cotrol ( ) 67-640 (05)
3. B. Weg K. E. Barer Nolear system detfcato mulsve evromets. IEEE Tras. Sgal Process. 53(7) 588-594 (005) 3. Y. Yu H. Zhao Adatve Combato of Proortoate NSAF wth the Ta-Weghts Feedback for Acoustc Echo Cacellato Wreless Pers. Commu. 9 467 48 (07) 33. Y. Yu H. Zhao B. Che A ew ormalzed subbad adatve flter algorthm wth dvdual varable ste szes Crcuts Systems ad Sgal Processg 35 (4) 407-48 06 34. H. Zhao J. Zhag A ovel adatve olear flter-based eled feedforward secodorder Volterra archtecture IEEE Tras. Sgal Process.57() 37-46 (009) 35. H. Zhao X. Zeg Z. He T. L ad W. J Adatve exteded eled secod-order Volterra flter for olear actve ose cotroller IEEE Tras. Audo. Seech. Lag. Process. 0(4) 394-399 (0) 36. S. Zhao B. Che J. C. Prce Kerel adatve flterg wth maxmum corretroy crtero Proceedgs of Iteratoal Jot Coferece o Neural Networks (0). 0 07