Hybrid higher-order statistics learning in multiuser detection

Depósi o de In es igación de la Uni e sidad de Se illa h ps://idus.us.es/ This is an Accep ed Manusc ip o an a icle published by Camb idge Uni e si y P ess Applied Ca alysis A: Gene al 897 (2010), a ailable a : h ps://doi.o g/10.1016/j.apca a.2010.05.027 Copy igh 2010. En idUS Licencia C ea i e Commons CC BY-NC-ND 1 Hyb id Highe O de S a is ics Lea ning in Mul iuse De ec ion An onio J. Caama˜ no Membe , IEEE,, Ra ael Boloix-To osa, Ja ie Ramos Membe , IEEE,, Juan J. Mu illo-Fuen es Membe , IEEE, Abs ac —In his wo k we explo e he signi icance o second and highe o de s a is ics lea ning in communica ions sys ems. The inal goal in sp ead spec um communica ion sys ems is o e- cei e a signal o in e es comple ely ee om in e e ence caused by o he concu en signals. To achie e his end we exploi he s uc u e o he in e e ence, designing Second O de S a is ics (SOS) de ec o s, such as he Minimum Squa e E o (MMSE) (known o be op imum linea de ec o s in noisy en i onmen s) in conjuc ion wi h Highe O de S a is ics (HOS) echniques, such as he Blind Sou ce Sepa a ion (BSS) and he Independen Componen Analysis (ICA) (known o exhibi such ema kable p ope ies in lea ning such as scale in a iance and supe e iciency). Such Hyb id Highe O de S a is ics (HyHOS) app oach enables us o alle ia e BSS/ICA algo i hms o one o hei main p oblems, ha is, hei sensi i iness o high le els o noise as we bene i om hei speed o lea ning and independence o he ini ial se ings o he p oblem. We success ully applied he esul s o his app oach o he design o mul iuse de ec o s in Code Di ision Mul iple Access (CDMA) channels. Index Te ms—A ay Signal P ocessing, blind sou ce sepa a- ion, highe o de s a is ics, independen componen analysis, unsupe ised lea ning, CDMA. I. INTRODUCTION INTERFERENCE limi a ion due o he simul aneous access o mul iple use s sys ems has been he s imulus o he de elopmen o a powe ul amily o Signal P ocessing ech- niques, namely Mul iuse De ec ion (MUD). This echniques ha e been ex ensi ely applied o Di ec Sequence Code Di- ision Mul iple Access (DS-CDMA) sys ems. Thus, mos o las gene a ion digi al communica ion sys ems such as GPS, wi eless 802.11b, UMTS, e c, may ake ad an age o any imp o emen on his opic. In he blind case he algo i hms should cope no only wi h noise and he nea - a p oblem bu wi h no aining sequence a ailable. In his sense, he minimum mean squa e e o (MMSE) c i e ia [1] p o ides a good blind linea solu ion o he p oblem. Due o i s compu a ional complexi y, some o he al e na i e algo i hms ha e been p oposed [2], [3], [4]. On he o he hand, blind sou ce sepa a ion (BSS) and independen componen analysis (ICA) ha e la ely been main ields o esea ch due o hei g ea numbe o applica ions A.J. Caama˜ no and Ja ie Ramos a e wi h he DTSC, EPS- Telecomunicaciones, Uni e sidad Ca los III de Mad id. Bu a que, 15, 28911 Legan´ es, Mad id, Spain. Tl: +34 916248736. Fax: +34 916248749. E-mail: [email p o ec ed] J.J. Mu illo-Fuen es and R. Boloix-Tolosa a e wi h ATSC. Escuela Supe io de Ingenie os. Uni e sidad de Se illa. Paseo de los Descub imien os sn. 41092 Se illa. Spain. Tl: +34 954487333. Fax: +34 954487372. E-mail: [email p o ec ed]. in di e en a eas. ICA and BSS ha e been usually applied (see [5] and e e ences he ein) o communica ions signals (mainly in a ay p ocessing), biomedical signals such as ECG o EEG, moni o ing, as an al e na i e o p incipal componen analysis in image o inancial da a p ocessing, in moni o ing, e c... Besides, hey ha e been la ely applied o o he p oblems such as sp ead-spec um based communica ions [6], [7], digi al wa e ma king o images [8], audio spec um basis unc ions compu a ion [9], image classi ica ion, encoding o comp es- sion [10],... Se e al me hods ha e been p oposed as solu ions o he BSS/ICA p oblem. We b ing ou he e wo main app oaches. On he one hand, we ha e echniques based on he cancella ion o es ima ion equa ions by using he s eepes descen algo i hm [11], [12], [13], [7]. These me hods usually ha e he maximum likelihood (ML) app oach as s a ing poin and a e no usually obus as he p obabili y densi y unc ions (p.d. .) a e gi en a p io i h ough he sepa a ion [14], also sco e [12], unc ions. He e obus ness s ands o he me hods o be a ailable o mix u es o any s a is ical dis ibu ions. On he o he hand, we ha e con as unc ions. Con as a e cos unc ions whose minimiza ion yields he solu ion o he BSS/ICA. These me h- ods usually ace he BSS/ICA p oblem s a ing om lowe o highe o de momen s analysis. In his sense he mix u es a e i s educed o ze o mean signals. Then a whi ening p ocess is ca ied ou o achie e deco ela ion. Finally, highe o de momen s a e used o compu e a uni a y ans o ma ion o diagonalize he associa ed highe o de enso o he whi ened ou pu s. I a mos one ma ginal ou h-o de cumulan is null, his uni a y ma ix may be compu ed by diagonalizing he ou h o de enso . This enso may be diagonalized by ei he canceling he whole se o cumulan s ou o he diagonal (minimiza ion o he mu ual in o ma ion, MI), o by he maximiza ion o he diagonal en ies (minimiza ion o he ma ginal en opy, ME [15]) as he SICA me hod used in his pape . Some blind sou ce sepa a ion (BSS) algo i hms ha e been p oposed as ully blind MUD’s, in he sense ha sp eading codes a e unknown [16], [17], [7], [18]. Howe e his blind echniques a e no usually noise obus . We p opose in his pape o in oduce a simila echnique o BSS such as inde- penden componen analysis (ICA) [19], [5] in o he s uc u e o he MMSE MUD o exploi i s p o en noise obus ness. We will compa e he pe o mance o a block ICA algo i hm such as SICA [19] o an adap i e me hod o compu e he ma ix such as M-EASI algo i hm [16]. The pape is o ganized as ollows. Sec ion II summa izes 2 he ma ix model, main assump ions and de ini ions o a DS- CDMA communica ion sys em. The MMSE de ec o will be also add essed. Then, BSS/ICA echniques a e in oduced in Sec ion III, whe e we will ocus on he SICA algo i hm. Sec ion IV is de o ed o ela e bo h o hese app oaches BSS/ICA and he MMSE o p opose a new HOS based solu ion, he so called BH MUD de ec o . These a e e e ed o h oughou his pape as HyHOS I and II espec i ely. Sec ion V includes expe imen s o compa e MMSE wi h he Hyb id High O de S a is ics de ec o s de eloped in his wo k and o show noise esis ance en supe e icien lea ning. Sec ion VI is de o ed o conclusions. II. COMMUNICATION SYSTEM MODEL:THE MMSE Conside a CDMA ansmi e [1].The ansmi ed baseband signal, assuming ha he p opaga ion channel o each use is memo yless, can be ep esen ed as x( ) = L X l=1 n X i=1 χisici(l−τi)g( −lTc−τi),0≤ ≤Ts(1) whe e Lis he leng h o he sp eading code, nis he numbe o ac i e use s, χnis a scaling ac o speci ied by he powe con ol loop, snis he symbol ansmi ed by use n,g( ) ep esen s he ansmi e pulse shaping il e , Tcis he chip pe iod, and τnis he delay o each use . The sp eading code o use nis deno ed by cn. No e ha indi idual chips in cn a e deno ed by he elemen s in [cn(1),...,cn(L)]Twi h each cn(·)∈ ±1. Equa ion 1 usually is exp essed as: x( ) = N X n=1 χnsncn( −τn)(2) whe e cn( )is he no malized signa u e wa e o m assigned o he n- h use . I he ansmi ed pulse-shape il e s a e Nyquis , we assume he synch onous case and he ecei e uses a chip- a e sampled ma ched il e ollowed by a se ial o pa allel con e e , he ma ix o m o he sys em equa ion is as ollows: x =A s +n (3) whe e x is a L×1 ec o , A is a L×nma ix ep e- sen ing he channel esponse and n is a noise ec o . The memo yless easies solu ion is a synch onized ma ched il e (MF). Howe e , i he di e ence be ween powe s o use s is high enough and use ’s codes a e no o hogonal, we ha e he nea - a p oblem and o he well-known SOS based de ec o s such as he MMSE ha e much be e a pe o mance. A. Linea Mul iuse De ec ion A linea mul iuse de ec o Cgi es an es ima ion o he o iginal ansmi ed signals as y( ) = Cx( ) = CAs( ) + Cn( )(4) The cen alized de ec o minimizing he mean-squa e-e o (MSE) o each use k, MSEk=E|yk( )−sk( )|2, is he CMMSE CMMSE =R−1 H∗=H∗HW ∗ W H∗(5) whe e ( ) = H∗x( )and W is he deco ela ing ma ix o ec o . He e, as in he ollowing, ∗deno es anspose- conjuga e. In [17] he au ho s de ine a whi ening– o a ion de ec o (WR) as he CW R ma ix ha minimizes he MSE sum o MSEk,k= 1, ..., n. A posible s uc u e o his ecei e was gi en as ollows CW R =JQW (6) whe e Wis a m×mwhi ene , Qam×m o a ion ma ix, and J = [I0] is n×m. In he con ex o he BSS, he p oblem consis o compu ing he ma ix WB ha minimizes s a is ical dependence a he ou pu . O he p e ious BSS app oaches o ully blind MUD’s use a SVD decomposi ion. Howe e , his in ol es a highe compu- a ional complexi y. The na u al o ela i e g adien (NG) has been also used as a s eepes descen algo i hm o adap i e app oach in bo h BSS and MUD [20]. We ace in his pape he de elopmen o a HOS based de ec o . Le ’s i s in oduce he HOS-BSS/ICA ools we a e going o use. III. BSS AND SICA A. Main assup ions and model Blind sepa a ion o sou ces (BSS) in ol e he ask o ob- aining a non-obse able se o signals, he so-called sou ces, om ano he se o obse able signals ega ded as mix u es. He e, he adjec i e ”blind” s ands o he ac ha nei he he o iginal sou ces no he mix u e i sel a e known. Usually, and in he con ex o his pape , he assump ion o spa ial s a is ical independence is he key o achie e sepa a ion. In his sense, he ela ed [21] and mo e gene al echnique o independen componen analysis (ICA) compu es he p ojec ion o a se o componen s (mix u es in BSS) ha minimizes he s a is ical dependence o he oupu s [15]. In i s simples o m, he BSS educes o he ollowing ma ix o m. The m=nmix u es sampled a ime ,x , a e ins an aneous linea combina ions o he nsou ces s , i.e., x( ) = As( )(7) Usually, and in he con ex o his pape , s a is ical indepen- dence o he inpu s is assumed. Thus, i x is a s a iona y e godic andom sequence and he mixing ma ix Ais non- singula , i is possible o es ima e a sepa a ion ma ix B o ob ain he sou ces as y=Bx =BAs. y =Bx =BAs =Cs = 1,2,... (8) In Blind Sou ce Sepa a ion (BSS) we compu e ma ix Bso ha Cis ideally he iden i y ma ix. Howe e , he o iginal scaling and a angemen canno be es ima ed om he inde- pendence assump ion. In his sense, Cis a non mixing ma ix i i has one and only one non-ze o en y in each column and each ow. The ela ed [21] and mo e gene al p oblem o independen componen analysis (ICA) [15] consis s o ob aining he change o basis B ha p ojec s componen s x in o a se y as s a is ically independen as possible (sou ces). As ma ix Bcan be decomposed in o he p oduc o a 3 whi ening Wand a uni a y Vma ix, his sphe ing s age gi es us signals zi( ): y =Bx =V W As =V z = 1,2,... (9) In addi ion o he sou ce independence hypo hesis, he e a e o he mino conside a ions. Fi s , al hough he ins an a- neous mix u e model conside ed he e applies o many o he applica ions e e ed a he In oduc ion, i may be ex ended o he con olu i e case [22], [23] when needed. Besides, we may conside he e he case we e he numbe o mix u es m is g ea e o equal o he numbe o sou ces n, i.e. m≥n. In he case m>n, a subspace app oach such as a singula alue decomposi ion may be used o p ojec he mix u e space on o he signal subspace, educing he e ec o weak noises. Fo noisy mix u es see [24], [25], [26]. We hus assume he noiseless case wi h m=n. In addi ion, a necessa y and su icien condi ion o he wa e o m-p ese ing sou ce es ima ion o be easible is ha no mo e han one gaussian dis ibu ed sou ce be p esen in he mix u e [24], [15], [27]. This model in (7) may be iden i ied wi h he na owband m- senso linea -a ay applica ion, he synch onous-CDMA case wi h sp eading ac o L=m, o he gene al ins an aneous BSS/ICA model. In synch onous CDMA communica ions, ma ix Amay be decomposed in o A=Hχ, whe e χis a diagonal ma ix wi h he ampli udes o each use and His a ma ix whose columns a e he sp eading codes. No ice ha BSS/ICA applied o CDMA would lead o a de ec o whe e nei he a aining sequence no he sp eading codes hemsel es a e needed. Howe e , noise ec o in (3) is no included in he BSS/ICA model in (7). Thus, i noise is high enough, he pe o mance o he BSS/ICA model de e io a es. Adap i e solu ions o his p oblem a e usually based on maximum likelihood and he na u al (o ela i e) g adien [11], [28], [29]. On he con a y, mos o he o -line solu ions o ICA [15], [30] minimize one c i e ion, con as unc ion, o cancella ion o mul iple c i e ia. Fo ins an aneous linea mix u es o wo sou ces, di ec me hods, which consis o di ec ly es ima ing he mixing ma ix om he mix u es, a e also possible. Di ec me hods sugges ed in [31], [32], [33] sol e polynomial equa ions based on cumulan s up o ou h- o de . The solu ions o he equa ions p o ide he en ies o he mixing ma ix. Howe e , me hods based on he op imiza ion o c i e ia usually ha e a be e pe o mance. In his sense we conside he e he con as unc ion based on he minimiza ion o he ma ginal en opies. B. Minimum ma ginal en ophy based con as A con as unc ion φ(·)maps Y, he se o andom ec o s y(mul i a ia e p.d. .’s), on R. I has a minimum when he en ies o ya e s a is ically independen , i.e, i yhas inde- penden componen φ(y)≤φ(Ay)∀Anon-singula . Thus, he minimiza ion o a con as unc ion yields he solu ion o he BSS/ICA p oblem. I he numbe o mix u es m > 2, hese con as s a e complex unc ions o minimize. The ‘Jacobi op- imiza ion’ [15] consis o sol ing he n-dimensional p oblem by decomposing i in a se o 2-dimensional op imiza ions. The o a ion ma ix Vis decomposed in o g=m(m−1)/2Gi ens o a ions. Thus, minimiza ion o he con as s o each angle θh,h= 1, ..., g, d i es he sys em o he solu ion. We will i s ocus on a 2-dimenisonal con as , he SICA. Unde he whi eness cons ain , i.e. E[yyT] = I, i yields he ollowing o hogonal (deno ed by φ2) con as [21], φME 2(y)c = n X i=1 H[yi](10) whe e he c i e ia o minimize is he ma ginal en opy H[y]. He e, as in he ollowing, c =means equali y up o an addi i e cons an c. Besides, we will deno e by µy ijkl =E[yiy∗ jyky∗ l] and µy ij =E[yiy∗ j] he ou h o de and second-o de momen s. I we app oxima e he possible dis ibu ions o sby and Edge- wo h expansion, ew i e he esul in e ms o second-o de and ou h-o de cumulan s, Cy ijkl =µy ijkl −µy ijµy kl −µy ikµy jl − µy ilµy jk and hen minimize i o all possible dis ibu ions [15], i ollows ha φME 2(y)≈1 48φME 24 (y) = −1 48 X i (Cy iiii)2(11) whe e Cy iiii =µy iiii −3µ2 ii is he au ocumulan o ou pu yi, symbol ∗deno ing complex conjuga ion. Thus, he minimiza- ion o he ou h-o de cumulan based ME con as yields he diagonaliza ion o he associa ed enso when a mos one ma ginal ou h-o de cumulan is null. in he ollowing, we will ace he eal case. The ex ension o complex- alued sou ces is somehow immedia e. By using he complex no a ion and unde he whi ening cons ain , he independen compo- nen s yp( )and yq( )in he wo-dimensional app oach yield y∗( ) = ( )ejρpq ( )= ( )ej(θpq +βpq ( )) =ejθpq z∗( )(12) whe e j=√−1. The whi enned mix u e ec o z∗( ) = zp( ) + jzq( )a e a simple o a ion o he no malized (uni a iance) sou ces ¯s∗( ) = ¯sp( ) + j¯sq( ) = ( )ejα. No ice ha a he solu ion ρ( ) = θ+β( ) = α( ) + kπ/2,k= 0,1,2, ... The associa ed es ima ion o he o a ion angle θ in he op imiza ion o 2-dimensional con as s may be easily exp essed as a closed unc ion o he ollowing complex- alued linea combina ions (cen oids) o he s a is ics o he ou pu s [34] ξγ=E[ 4( )ej4βpq( )](13) ξη=E2[ 4( )ej2βpq( )](14) γ=κ −8(15) Based on he so called ‘weigh ed es ima o s’ (WE) o WAML [35], [36] a gene al 2-dimensional es ima o yields ˆ θGW E(ωγ, ωξ) = 1 4∠(ωξωγξγ+ (1 −ωξ)ξη)(16) 0< ωξ<1, ωγ=±1, γ This es ima es yields EML es ima o in [34] ˆ θEML = ˆ θGW E(sign(γ),1) o he [37], MK [38], [30], SKSE and SKDE [39], ML [40] es ima es ˆ θGW E(±1,1),ˆ θMaSSF OC = ˆ θGW E(γ, 1/2) in [39], o he ˆ θAML =ˆ θGW E(γ, 1/3) [35]. The discussion on he op imum wξ[36] is open as i is di icul 4 o compu e he s a is ics o hese es ima es and analyze hem o all possible p.d. ’s. In his pape , we p opose o use θSICA =ˆ θGW E(γ, 3/7) (17) as i may be p o ed ha he SICA app oach is as obus as he φME 24 (y), i.e, i p o ides a solu ion o he ICA p oblem wi h he only condi ion o no mo e han one sou ce wi h null ou h o de s a is ics in he mix u e. The minimiza ion o φSICA(θ)is immedia e as he solu ion is minus ou imes he phase o he esul ing unc ion, a sinusoid. This may be easily implemen ed by a look-up able so ha we minimize he numbe o ope a ions needed. C. n-dimensional case The 2-dimensional case may be easily ex ended o ndimen- sions by using he ’Jacobi Op imiza ion’ [15]. We ha e he e ew i en he algo i hm using φSICA(θ). Such an algo i hm can be summa ized as ollows. Algo i hm 1: n-dimensional SICA using Jacobi Op imiza- ion. 1) Whi enning. Compu e a whi ening ma ix Wand he ou pu ec o z=W x. Se c= 1 and y=z. 2) Sweep c. Fo all g=n(n−1)/2pai s, i.e., o 1≤p < q≤n, do (a) Compu e he Gi ens angle θpq θSICA(y)in (17) wi h [zp, zq]T= [yp, yq]T. (b) i θpq > θmin, do o a e he pai (yp, yq)by θpq acco ding o (12). 3) End? I he numbe o sweeps csa is ies c≥K= 1 + √no no angle θpq has been upda ed, s op. O he wise go o s ep 2 o ano he sweep wi h c=c+ 1. D. The Na u al G adien The s eepes descen me hod upda es Cacco ding o he di ec ion o he g adien ∇Lo a loss unc ion L(C). The na u al [23] o ela i e g adien [5] p oposes o use e ∇L(C) = e ∇E[l(C)] = ∇L(C)CTC. The s ochas ic e sion uses he in an aneous alue e ∇l(C). The lea ning law yields C←− C−λ∇l(C)CTC(18) In he BSS p oblem i is usually assumed s a is ical indepen- dence a he ou pu s y( ). Maximum likelihood (ML) is an ex ended echnique o de i e a loss unc ion o his c i e ia [23], [5]. I can be shown ha by o cing he es ima ing unc ion K(y) = ∇l(C)·CT=ϕ(y)yT−I o cancel, we achie e independence a he ou pu . The lea ning law yields C←− C−λK(y)C(19) whe e in he ML app oach ϕi(yi) = −q0 i(yi)/qi(yi), being qi(·) he p obabili y densi y unc ion o sou ce bi. Howe e , as sou ce dis ibu ions a e supposed unknown, each au ho in oduces his own ac i a ion unc ion ϕi(yi). A amily o hem may be ound, o sou ces wi h nega i e o posi i e ku oses, in [41]. By no malizing (19) he lea ning law may be w i en as ollows C←− C−λϕ(y)yT−I 1 + λ|ϕT(y)y|C(20) In [16] he au ho s p oposed a NG based algo i hm o sep- a a e signals in digi al communica ion, he M-EASI (Median- Equi a ian Adap i e Sepa a ion ia Independence). This me- hod assumes ze o-mean, symme ic, ‘ci cula ly dis ibu ed’ signals and in oduces he sign unc ion o educe he bias in oduced by he noise. I es ima ing unc ion yields K(y) = ysgn(y)∗−I 1 + λsgn(y)∗y+1 α ϕsgn(y)∗−sgn(y)ϕ∗ 1 + λ|y∗ϕ|(21) whe e sgn(y) = sgn(<(y)) + jsgn(=(y)),y∈C. Wi h his me hod we imp o e he s abili y o he algo i hm [29], p o ide he me hod wi h phase eco e ing p ope ies and make he me hod mo e obus agains noise. IV. HYBRID HIGH ORDER STATISTICS MULTIUSER DETECTORS A. Na u al G adien Based HyHOS We could use he na u al g adien o compu e a de ec o Cby imposing some c i e ia such as he MSE. Bu we may di ec ly exploi he MMSE es uc u e in 5 o he cen alized (i.e., base s a ion, all codes a ailable) and non-cen alized case (i.e., use equipmen , only use code a ailable). Fi s , we p opose he WWH, cen alized de ec o by com- pu ing W in CWWH =H∗HW ∗ W H∗(22) by using (20) wi h y= =W H∗xand ac i a ion unc ions ϕ( ) = . The lea ning law yields he deco ela ing algo i hm W ←− W +λI−(W )(W )T 1 + λ|(W )T(W )|W (23) Now, we p opose a non-cen alized MUD, he HWW de ec- o . The non-cen alized MMSE-MUD was gi en in (5) as CHW W =H∗W∗ xWx(24) I we ede ine y=Wxxand ew i e he na u al g adien blind sou ce sepa a o in (18) as we did in (23), i ollows ha Wx←− Wx+λI−(Wxx)(Wxx)T 1 + λ|(Wxx)T(Wxx)|Wx(25) Following he same s uc u e as be o e, we ha e cen alized and non-cen alized BSS based de ec o algo i hms. In he case o he BSS cen alized MUD, he BH de ec o , we can p oceed as ollows I is posible o subs i u e he ma ix p oduc H∗HW ∗ W in (22) by a sepa a ing ma ix B. The new de ec o yields CBH =BH∗=QW H∗ (26) whe e Bis compu ed as a blind sou ce sepa a o , ha is, a whi ening- o a o . No ice ha in he cen alized case he dimensional educ ion is ca ied ou by he ma ched il e H∗, 5 hus J=Iin (6). Ma ix in B(31) is compu ed by using he M-EASI algo i hm B←− B−λKB(y)·B(27) whe e KB(y) = ysgn(y)∗−I 1 + λsgn(y)∗y+1 α ϕsgn(y)∗−sgn(y)ϕ∗ 1 + λ|y∗ϕ|, (28) y=B =BH∗x, and ϕ(y)may be chosen as desc ibed in [41]. Thus, he MMSE s uc u e is u he enhanced by he p ope ies o he M-EASI algo i hm. On he o he hand, i is no possible o de ine an a chi ec u e o he non-cen alized MUD as in (31). Suppose he candida e now o be he de ec o CHB =H∗B=H∗QW x(29) He e, ma ix CB=Bis al eady a solu ion, as independence is imposed a he ou pu s. Thus CHB is no a de ec o as H6=I. Compu ing ma ix CBis basically a blind sepa a ion p oblem and i is ou o he scope o his pape . No ice ha his de ec o is ully blind as i does no use he sp eading codes [16], [7], [17], bu a he cons uc s hem. Now, we include a discussion on some heo e ical aspec s o he me hods abo e. We ocus on he nea - a p oblem, noise, con e gence and complexi y. The main poin in using he na u al g adien in MUD is ha i is equi a ian [5], i.e., he con e gence has a uni o m pe o mance. Le ’s de ine D=CS and igh mul iply (19) by H o ob ain D←− D+λK(Db)D(30) Ma ix S=AH is only p esen a he ini ial alue D0= C0AH. Thus, he con e gence does no depend on ma ix AH. I can be concluded ha he me hods p oposed in his pape a e nea - a esis ance, as con e gence is independen o he use ’s ampli udes. Algo i hms in BSS usually do no cope wi h he noisy case whene e he numbe o sou ces and mix u es a e he same m=n. By using he M-EASI algo i hm we comba he e ec o noise in he sepa a ion p ocess o digi al communica ion. On he o he hand i m>na signal subspace p ojec ion allows noise educ ion. P e ious BSS app oaches o ully blind MUD’s use a SVD decomposi ion, a MPLL,[17] ... Howe e , his in ol es a highe compu a ional complexi y. As he sp eading codes a e usually a ailable (a leas a he base s a ion), i is s aigh o wa d o in oduce hem as a subspace algo i hm a a null complexi y cos [4]. Besides, he s uc u e o he MMSE de ec o has been used in (22) and (24) o cope wi h noise. Ano he impo an cha ac e is ic o na u al g adien BSS echniques is ha o supe e iciency [42]. In his sense, p o- ided E[ϕ(y)] = 0 (an usual case), he co a iance be ween wo ou pu s dec eases o he o de o 1/ 2in ba ch es ima ion and o he o de o λ2in on-line lea ning. Fu he mo e, λ= 1/ gi es, asymp o ically, he bes pe o mance, which is he same as he op imal ba ch es ima o . Thus, wi h λ= 1/ we achie e an on-line algo i hm wi h ba ch ea u es a e e y . On he o he hand, i adap i e ea u es a e needed, we may use λ < 1 o achie e an ou pu co a iance dec easing as λ2. On he ques ion o complexi y, he exac MMSE solu- ion a e e y ime ein ol es compu ing he eigen alues o he au oco ela ion ma ix. Thus, he compu a ional esou ces needed a e signi ican . The me hods p oposed in his pape allow compu ing, as desc ibed in he las sec ion de o ed o supe e iciency, he ba ch solu ion as an on-line algo i hm a a low compu a ional bu den. B. SICA Based HyHOS In he sec ion abo e, i is p oposed o subs i u e he ma ix p oduc H∗HW ∗ W in (5) by a ma ix B ha makes he ou pu s yias s a is ically independen as possible. The new de ec o yields CBH =BH∗=QW H∗ (31) whe e Bis a whi ening– o a o compu ed by using he SICA algo i hm [19]. No ice ha in he cen alized case he dimen- sional educ ion is ca ied ou by he ma ched il e H∗, hus J=Iin (6). In [19] au ho s p o e ha unde whi eness cons ains some ou h-o de con as s may be app oxima ed by a sinusoid. Thus, he minimiza ion o he con as educes o compu ing i s phase. The s a ing poin is he Minimum En opy (ME) ICA con as gi en by Comon in [15] so ha we use he ’Jacobi op imiza ion’ o cope wi h highe dimensions. This me hod, called SICA (Sinusoidal ICA,) has a good pe o mance along wi h a low compu a ional cos , ou pe o ming he ME by Comon and he JADE me hods [43]. Applying SICA o ou p oblem, assuming inpu s iha e been deco ela ed, his algo i hm compu es y=B minimiz- ing he ollowing unc ion, deno ed as con as , φME [y]≈φ4[y] = 1 48 X ijkl6=iiii (Cumy ijkl)2(32) whe e Cumy ijkl is he ou h o de cumulan o ou pu s yi, yj, ykand yl. Thus, his ICA algo i hm may be seen as a ou h o de deco ela ion, and we go u he han he second o de based MMSE o ind a new solu ion o he p oblem. The algo i hm SICA is a block me hod ha compu es ma ix B om a whole se o obse a ions, simila o he MMSE me hod. As he sp eading codes a e usually a ailable (a leas a he base s a ion), i is s aigh o wa d o in oduce hem as a subspace algo i hm a a null complexi y cos [3]. Besides, he s uc u e o he MMSE de ec o has been used o cope wi h noise. We will compa e he pe o mance o his algo i hm o ha o he M-EASI algo i hm, an adap i e me hod o compu e ma ix B ha y o minimize a simila ou h o de based con as by using he na u al g adien [16]. V. EXPERIMENTAL RESULTS The es en i onmen used he e is a synch onous CDMA sys em in which he use s he e a e sp ead using Gold se- quences wi h sp eading ac o 31. We will p esen he e wo 6 main esul s: i s , he ypical e alua ion o he pe o mance in a digi al communica ions sys em, i.e., Bi E o Ra es om 10−3 o 10−6and he equi alen SNR ≥8dBs, a Low Noise En i onmen (LNE); second, an e alua ion in a High Noise En i onmen (HNE) whe e we es he noise esis ance capabili ies o he HyHOS algo i hm wi h Bi E o Ra es as low as 2×10−1≤BER <5×10−1and equi alen Signal o Noise Ra io o −15 ≤SNR ≤0dBs. As he BSS p oblem is posed in he absence o noise, one o he main issues is o es he pe o mance o he esul ing algo i hms in noisy en i onmen s. The compa ison o he a ious algo i hms p esen ed he e mus be done wi h equal con e gence speed (i.e. lea ning speed) on one hand, and equal inal a iance on he o he , o ai h ully cha ac e ize hei pe o mance. The me hods desc ibed in [44], [45] and [46] a e o be used o analyze he inal a iance and he con e gence speed o each algo i hm. Those me hods a e use ul and s aigh o wa d in he case o linea algo i hms such as he MMSE bu , in he case o he nonlinea unc ionals o he HyHOS algo i hms, his is a much di icul ask and, as such, me i s a sepa a e analysis which is cu en ly unde way. In he p esen wo k, and as a means o compa e he algo i hms on an equal oo ing, he aining leng h o each o he de ec o s is 2000 samples, he con e gence speed (i.e. s ep-size) and he inal a iance a e ixed h oghou he expe imen s. The numbe o Mon e Ca lo simula ions a e aged is 100 in he HNE and 500 in he LNE, which amoun s o a con idence ma gin o a leas 95% o any BER es ima ion p esen ed in his wo k. A. Low Noise En i onmen In Low Noise En i onmen we p esen , o cla i y, he con e gence pe o mance o he HyHOS I and he MMSE algo i hms bo h wi h 15 and 20 use s. We see a clea di e ence in he con e gence beha iou o he HyHOS I and he MMSE algo i hm. Tha o he HyHOS II algo i hms closely esembles ha o he HyHOS I. The a e age dec ease in SINR o he HyHOS I algo i hm whi an inc ease o 5use s is hal o he dec ease su e ed by he MMSE. In Figu e 1 we can see ha 2000 samples is mo e han enough aining leng h o he HyHOS I algo ihm whils o he MMSE, con e gence canno be achie ed in a highly occupied channel. In he e alua ion o he Bi E o Ra e (see Figu e 2), his speed o con e gence o he di e en algo i hms is e lec ed in he pe o mace o bo h algo i hms, HyHOS I and II and ha o he MMSE. In his Low Noise En i onmen (LNE), he sough ea u es in mul iuse de ec o is ha o a con olled deg ada ion o Bi E o Ra e wi h an inc ease in he numbe o use s. Bo h he SICA based HyHOS algo i hm (HyHOS I) and he Na u al G adien based HyHOS (HyHOS II) show a pe o mance close o ha o he MMSE (i no be e in he HyHOS I case) wi h a low occupa ion o he channel (5use s) bu he di e ence in beha iou a ises in a highly occupied channel (25 use s) , whe e he BER o he MMSE deg ades om 10−6a app oxima ely 14 dB o SNR o 10−5, while bo h HyHOS algo i hms keep he BER o app oxima ely 10−6in he same condi ions. 0 500 1000 1500 2000 Samples 8 9 10 11 12 13 14 15 SINR (dB) HyHOS (15 use s) HyHOS (20 use s) MMSE (15 use s) MMSE (20 use s) Fig. 1. Signal o In e p e ence plus Noise Ra e, wi h SNR o he Use O In e es (UOI) = 15 dB, o he HyHOS de ec o and he MMSE de ec o , bo h wi h 15 and 20 use s. The powe s o he in e e ing use s is dis ibu ed homogeneously be ween 0and 30 dB abo e ha o he UOI. 8 10 12 14 SNR (dB) 1×10-6 1×10-4 1×10-2 BER HyHOS I (5 use s) HyHOS I (25 use s) HyHOS II (5 use s) MMSE (5 Use s) MMSE (25 use s) Fig. 2. Bi E o Ra e in Low Noise En i onmen (LNE), i.e. SNR o he UOI ≥8dB, o he HyHOS de ec o and he MMSE de ec o , bo h wi h 5and 25 use s. The powe s o he in e e ing use s is dis ibu ed homogeneously be ween 0and 30 dB abo e ha o he UOI. B. High Noise En i onmen In he case o High Noise En i onmen , o HNE, he con e gence speed (see Figu e 3) o bo h HyHOS algo i hms is comple ely simila bu o highe a iance in he HyHOS I algo i hm. Bo h in a low o high occupa ion channel, he con e gence is ai ly simila . The MMSE shows a lack o con e gence in a high-noise, low-in e e ence channel, qui expec ed o he MMSE because he in e e ence noise s uc- u e is comple ely masked by gaussian noise, a oiding his algo i hm o “lock” on o he signal o he Use o In e es . Qui e he con a y is he case o he HyHOS algo i hms. Thei in insic nonlinea p ope ies show he e he limi a ions o linea unc ionals in linea de ec o s. The Bi E o Ra e o he HyHOS algo i hms show he noise esis ance p ope o he MMSE wi hou showing i s demeano s. The con e gence o a Ma ched Fil e solu ion in he asymp o ic limi o σ→ ∞ is shown expe imen ally in Figu e 4. 7 0 500 1000 1500 2000 Samples -26 -24 -22 -20 -18 -16 -14 SINR (dB) HyHOS I (5 use s) HyHOS I (25 use s) HyHOS II (5 use s) HyHOS II (25 use s) MMSE (5 use s) MMSE (25 use s) Ma ched Fil e Fig. 3. Signal o In e p e ence plus Noise Ra e, wi h SNR o he UOI =−15 dB, o he HyHOS de ec o and he MMSE de ec o , bo h wi h 5and 25 use s. The powe s o he in e e ing use s is dis ibu ed homogeneously be ween 0 and 30 dB abo e ha o he UOI. -15 -10 -5 0 SNR (dB) 2.0×10 -1 2.5×10 -1 3.1×10 -1 4.0×10 -1 5.0×10 -1 BER HyHOS I (5 use s) HyHOS I (25 use s) HyHOS II (5 use s) HyHOS II (25 use s) MMSE (5 use s) MMSE (25 use s) Ma ched Fil e Fig. 4. Bi E o Ra e in High Noise En i onmen (HNE), i.e. SNR o he UOI ≤0dB, o he HyHOS de ec o and he MMSE de ec o , bo h wi h 5 and 25 use s. The powe s o he in e e ing use s is dis ibu ed homogeneously be ween 0and 30 dB abo e ha o he UOI. VI. CONCLUSION In his pape he au ho s p oposed bo h ICA and NG as alid echniques o exploi he ma ix es uc u e o he pa ame e s in ol ed in he MUD p oblem, an applica ion o he mo e gene al na owband m-senso linea -a ay case. He e, we p opose a no el Hyb id MUD by in oducing an ICA ma ix in o he s uc u e o he blind cen alized MMSE MUD. This way we pallia e one o he main p oblems o he BSS/ICA algo i hms, hei sensi i eness o noise. The esul s included he e show a good nea - a esis an pe o mance in synch onous CDMA. ACKNOWLEDGMENTS w The au ho s would like o acknowledge D . Za zoso and D . Ma ´ ınez-Ram´ on o hei aluable commen s. REFERENCES [1] S. Ve d´ u, Mul iuse De ec ion. Camb idge Uni e si y P ess, 1998. [2] B. Yang, “P oyec ion app oxima ion subspace acking,” IEEE T ansac- ions on Signal P ocessing, ol. 43, pp. 95–107, Janua y 1995. [3] X. Wang and H. V. 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