A Charrelation Matrix-Based Blind Adaptive Detector for DS-CDMA Systems

Transcription

1 Sensos 05, 5, ; do:0.3390/s50805 Atcle OPEN ACCESS sensos ISSN A Chaelaton Matx-Based Blnd Adaptve Detecto fo DS-CDMA Systems Zhongqang Luo * and Ldong Zhu Natonal ey Laboatoy of Scence and echnology on Communcatons, Unvesty of Electonc Scence and echnology of Chna (UESC), Chengdu 673, Chna; E-Mal: zld@uestc.edu.cn * Autho to whom coespondence should be addessed; E-Mal: luozhongqang@6.com; el.: Academc Edto: Vttoo M. N. Passao Receved: July 05 / Accepted: 0 August 05 / Publshed: 4 August 05 Abstact: In ths pape, a blnd adaptve detecto s poposed fo blnd sepaaton of use sgnals and blnd estmaton of speadng sequences n DS-CDMA systems. he blnd sepaaton scheme explots a chaelaton matx fo smple computaton and effectve extacton of nfomaton fom obsevaton sgnal samples. he system model of DS-CDMA sgnals s modeled as a blnd sepaaton famewok. he unknown use nfomaton and speadng sequence of DS-CDMA systems can be estmated only fom the sampled obsevaton sgnals. heoetcal analyss and smulaton esults show that the mpoved pefomance of the poposed algothm n compason wth the exstng conventonal algothms used n DS-CDMA systems. Especally, the poposed scheme s sutable fo when the numbe of obsevaton samples s less and the sgnal to nose ato (SNR) s low. eywods: blnd souce sepaaton; DS-CDMA; chaelaton matx; jont dagonalzaton. Intoducton he poblem of blnd sepaaton n DS-CDMA systems has attacted extensve attenton fo the past few yeas [ 8]. Reseach woks elated to blnd sepaaton fo DS-CDMA systems ae of patcula nteest n felds such as ant-jammng n mltay communcatons (MILCOM) and satellte communcatons (SACOM). hee ae thee man poblems whch need to be solved, ncludng blnd use sepaaton [ 6], blnd speadng/chp sequence estmaton [7,8] and blnd ntefeence suppesson [9 8]. he blnd

2 Sensos 05, sepaaton poblems ae denomnated blnd to ndcate the lack of nfomaton concenng the souce sgnals and mxng matx. he mxng matx s equvalent to the effect of the channel matx n DS-CDMA systems. he only po nfomaton utlzed s the often soundly justfed assumpton of statstcal ndependence between the souce sgnals. hs technque s called as ndependent component analyss (ICA), whch has mpotant applcatons fo blnd sepaaton n weless communcatons [,9 4]. So fa, some classcal ICA algothms have been poposed to solve blnd sepaaton poblems n DS-CDMA systems. A second ode blnd dentfcaton (SOBI) algothm s used to sepaate the desed sgnal and ntefeence sgnal n spead spectum communcaton systems [9]. Fast Independent Component Analyss (FastICA) [ 4] and Jont Appoxmatve Dagonalzaton of Egen-matces (JADE) [,0] ae used to sepaate multuse sgnals to esst multple access ntefeence (MAI) o sepaate the useful sgnal and jamme sgnals fo ntefeence mtgaton n DS-CDMA systems [0 6]. JADE and FastICA ae classcal blnd sepaaton algothms. When evaluatng these two algothms fo obustness, sepaaton accuacy and elablty, t s found that the JADE algothm s obust and elable, and the FastICA algothm s not stable and sometmes fals. In geneal, the exstng blnd sepaaton technques utlze the second-ode statstcs (SOS) and hghe-ode statstcs (HOS) of the obsevatons fo souce sepaaton []. Fo example, SOBI explots second ode moment nfomaton, and FastICA and JADE make use of the fou ode moments/cumulants nfomaton to sepaate the souce sgnal fom mxed sgnals. Classcal HOS ae poweful tools n the context of multvaate statstcal analyss, often entalng valuable statstcal nfomaton beyond SOS. Howeve, longe obsevaton ntevals mght be equed n ode to fully ealze the advantages of HOS ove SOS. HOS s bette than SOS at the expense of nceased computatonal and notatonal complexty and compomsed statstcal stablty [,5]. In ths pape, we consde a new genec tool, whch offes the stuctual smplcty and contollable statstcal stablty of SOS on one hand, yet etans HOS-qualty nfomaton on the othe hand. As s well known, the cumulants ae elated to hghe-ode devatves of the second chaactestc functon at the ogn. Howeve, the new statstcal tools ae elated to lowe-ode (fst and second) devatves of the second chaactestc functon away fom the ogn, at locatons called pocessng-pont, and temed chamean and chaelaton [5]. he use of chaelaton matces s extemely useful fo extactng statstcal nfomaton n ode to establsh the optmzed cost functon fo the poblem of blnd sepaaton wok. Due to the stuctual smplcty and ample statstcal nfomaton, we consde usng the new statstcs nstead of the HOS statstcs used n blnd sepaaton poblems fo DS-CDMA systems. As fa as we know thee s lttle lteatue epotng on the use of chaelaton matx-based blnd sepaaton n DS-CDMA systems [6]. heefoe, the man contbuton of ths pape s to extend blnd sepaaton usng chaelaton matx and mplement blnd use sepaaton and blnd speadng/chp sequence estmaton. hen the new tools statstc based on off-ogn Hessans of the second chaactestc functon wll be dscussed and analyzed. We consde a synchonous DS-CDMA systems model. Smulatons have been caed out to obseve Intefeence to Sgnal Rato (ISR) [4] as a functon of sample numbe of obsevatons and vaaton n bt eo ate (BER) as a functon of SNR. A pefomance compason of DS-CDMA systems s executed usng two assessment ctea to vefy the advantages of the poposed method.

3 Sensos 05, he est of ths pape s oganzed as follows: Secton gves a descpton of the DS-CDMA sgnal model and descbes the elatonshp between the DS-CDMA sgnal model and basc blnd sepaaton one. he chaelaton matx s llustated and deved, and the blnd use sepaaton and blnd speadng/chp sequence estmaton ae dscussed n Secton 3. he smulaton esults and dscussons and concludng emaks ae gven n Sectons 4 and 5, espectvely.. System Model In ths secton a concete dscete-tme model of a DS-CDMA system s constucted fo the dscusson of the poblem fomulaton. he tansmtte and eceve stuctues of DS-CDMA systems nvolve smultaneous uses. We consde the DS-CDMA model as a synchonous one, whch s a base band model wth fadng channel [,6]. Assumng that the system has uses, the sgnal s sent by use k as follows: M - () = ( - ) x t å b m c t m () k k k m= 0 whch contans the nfomaton of M symbols b km. hs symbol b km denotes the m th symbol of the k th use. ck () s the k th use s bnay chp sequence,.e., the speadng code, suppoted by é0, êë ), whee s the symbol duaton. he sgnal passed though channel whch s assumed to be fxed dung one symbol duaton: M- åå k k k () m= 0 k= () = ( - ) + () t Ab m c t m n t whee A s the channel gan facto of the kth use; M s the numbe of symbols pe use; s the k numbe of uses; nt () denotes the addtve whte Gaussan nose wth zeo mean and vaable vaance; he chp sequence length (pocessng gan) s C =, whee s chp duaton. Snce c c the chp sequence k c t s now contnuous by defnton, t ncludes not only the bnay chps but also a chp wavefom pt. () Moe pecsely: C - () = () ( - ) k k c = 0 c, c t å c p t (3) whee pt () s suppoted by é0, ù êë c úû only. hs pape assumes a ectangula wavefom fo each use. Contnuous-to-dscete tme conveson of the above model can be ealzed by a chp-matched flte, whch s a smple ntegate-and-dump devce. Usng chp-ate samplng, whch means ntegatng ove a chp-duaton : c c () = ( -( -) c ) () ò p t t dt (4) - c c he sampled data, whch have been obtaned by chp-matched flteng usng a pocessng wndow sze of one symbol, can be wtten as: k

4 Sensos 05, ( m) = å Ab k k( m) + k ( m) c n (5) k= he chp sequence vecto c has a sze of C, and the nose vecto n has a sze of C. Wth k m a smple manpulaton, we can get a compact epesentaton fo the data [,6]: ( m) = Ab ( m) + + A b ( m) + ( m C C ) ( C ) = é A,, A ù êë c c úû b( m) + n ( m) C ( C ) = Gb( m) + n( m) c c n whee c = éc, c,, c ù k ê ë k k Ck ú û, = én, n,, n ù m ê ë C ú û denotes tanpose. Futhemoe, we can obtan: (6) n, and m éb m, b m,, b m ù b. () = ê ë ú û X= GB+ N (7) whee X é(, ), ( M ) ù, B éb(),, b( M ) ù, and N én(, ), n( M ) ù = ê ë ú = û ê ë ú = û ê ë ú. hs data model û Equaton (7) s same as the blnd souce sepaaton one [,6]. In next secton, we wll analyze the blnd detecto elated to the model Equatons (6) and (7). 3. Chaelaton Matx Based Blnd Detecto fo DS-CDMA System he chaelaton matx can be consdeed as the off-ogn Hessans of the second chaactestc functon. he theoy of chaelaton matx s bult on the deal of genealzed cumulants, whch ae defned as the aylo sees coeffcents of the second chaactestc functon same pespecfed pont n the doman of second chaactestc functon. he pespecfed pont s pocessng pont, whch s away fom the zeo pont. If ths pont s chosen as the ogn, then the genealzed cumulants educe to the tadtonal cumulants [5]. As an appealng altenatve, t s also possble to eman at the moe comfotable second-ode dffeentaton, but to move away fom the ogn. hese second ode devatves mantan the convenent fom of matces. he poposed blnd adaptve detecto fo DS-CDMA system s shown n the Fgue. Next, the chaelaton matx fo the blnd sepaaton algothm wll be analyzed. b b b ˆb c c c n ˆb ˆ b Fgue. Blnd adaptve detecto based DS-CDMA Systems

5 Sensos 05, Chaelaton Matx Based Blnd Souce Sepaaton akng nto account the blnd souce sepaaton mxtue model lnkng wth DS-CDMA system fom above dscusson, the system model s ewtten as: = +, =,, m Gb m n m m M (8) C fom the pespectve of blnd sepaaton famwok, the stochastc veto ( m) Î epesents b m Î obsevaton sgnals, and denotes eal feld. he components of the stochastc vecto coespond to the unobseved souce sgnals. C n m Î denotes addtve Gaussan whte nose. he unknown mxng matx chaactezes the way the souces ae combned n the obsevaton. he goal C of blnd sepaaton conssts of estmatng the mxng matx G Î fom the obsevatons and ecoveng the souce sgnals, on the assumpton that the souce sgnal ae non-gaussan and statstcally ndependent. Fom the sngle antenna/senso DS-CDMA ecepton pont of vew, n ode to have a standad blnd sepaaton model avalable n the eceve, the numbe of uses can be at most C,.e., C. hat s to say, the mxng matx G s full column ank. Equaton (8) s an ovedetemned o detemned BSS model fo C n mxng matx G. In the pocess of blnd sepaaton, the whtenng opeaton s always mplemented to convet a model Equaton (8) as: ( m) = Q( m) = Gb ( m) + n ( m) whee Q s a whtenng matx whch s deved n followng Secton 3.3. Afte whtenng opeaton, we can obtan that ( m) Î, G Î and (9) n Î. Next, the pncple of blnd sepaaton based on chaelaton matces s llustated. Let u Î denote an abtay (detemnstc) vecto, called pocessng pont. he genealzed chaactestc functon and the genealzed second chaactestc functon of the obesevaton vecto ae defned espectvely as: E exp( ( m) ) f u é ù ê u ë úû (0) j ( u) log( f ( u )) () E éù êú ëû denotes the expectaton opeaton. Next we dscuss the way to estmate the mxng matx m by ts epesentaton and based on the chaelaton matx of the obsevatons. Replacng negelectng the nose contbuton gets: ( u) = E éexp( u Gb( m ù )) = f ( G u b ) f ëê úû () Futhemoe, usng the ndependence popety of souce vecto, the genealzed second chaactestc functon can be ewtten as: ( u) = ( G u) =å b b ( g u ) j j j (3) =

6 Sensos 05, whee g s column vecto of G. As a consequence, devng the chaelaton matx Y ( u) obtaned by calculatng the second devatve of j ( u) chaelaton matx ( u) wth: Y It s wothwhle to note that Y b whch s wth espect to u, we can obtan the followng (the detals ae llustated n Appendx A): b Y u = GY G u G (4) ( u) é j ù é j ù Y u = ê u ú = u ë u û (5) u ê u ú ë û Gu s a dagonal matx (the detals ae llustated n Appendx B). In a detemned blnd sepaaton model, souce sgnals ae usually sepaated by multplyng the obsevatons wth the pseudo-nvese/nvese of mxng matx estmate. he estmaton of the mxng matx can be caed out by appoxmate jont dagonalzaton (AJD) of a sees of chaelaton matces. By choosng a set of pocessng ponts { u, u,, u L}, we can constuct a sees of chaelaton matces that obey the tansfomaton Equaton (4). hen the estmaton poblem of the mxng matx can be descbed as the followng jont dagonalzaton [6,7]: mn L G b = å Y u -GY G u G (6) whee denotes the squaed Fobenus nom. In BSS, the dagonal matces Y ( F b ) statstcal o stuctual popetes of souces. he taget matces Y ( u) smla matces petanng to the obseved mxtues. he dagonalzaton of the matces Y b ( ) F Gu contan some usually denotes estmates of Gu, whch s often attbuted to the statstcal ndependence of the souces, seves as the key to u. he optmzaton of Equaton (6) s a dentfablty of mxng matx G fom the matces well-known jont dagonalzaton poblem. A numbe of jont dagonalzaton methods have been epoted, such as n the lteatues [6,7]. Among these, a state-of-the-at algothm called weght exhaustve dagonalzaton usng Gauss teaton (WEDGE) s used to mnmze the cteon n Equaton (6). he WEDGE appoach wll not be llustated n ths pape, as detals ae povded n [6]. 3.. Estmaton Chaelaton Matx ( u) Y Y In pactcal applcatons the exact chaelaton matx ( u) of obsevatons s estmated by samplng of a andom vaable. he genealzed chaactestc functon of obsevaton vecto s estmated as: m= Y M f ( u) = å exp( u ( m )) (7) M Smlaly, the fst and second devatves of the genealzed chaactestc functon s as follows:

7 Sensos 05, ( u) f M G ( u) = å exp( u ( m) ) ( m ) (8) u M ( u) m= m= f M X ( u) = exp( ( m) ) ( m) ( m ) å u (9) u u M Based on the pevous analyss, the chaelaton matx Y ( u) chaactestc functon ( u) j can be easly gven by: ( u) f ( u) X u G u G u Y ( u) = - f 3.3. Blnd Use Sepaaton and Blnd Speadng/Chp Sequence Estmaton of the second genealzed akng nto account the poblem of blnd use sepaaton n a DS-CDMA system, the whtenng pocess wll be caed out fst n ode to smplfy the blnd sepaaton poblem and suppess nose. Usng the model Equaton (7), the whtenng pocessng s executed as follows: he autocoelaton of the obsevatons s descbed as: é ù é ù R = E êxx ú = GE êbb úg + s I ë û ë û = USU + s I = U( S + s I) U = ULU whee E é ê ù ú = L = dag ë û l, l,, l = dag s + s, s + s,, s + s, s,, s. C s, =,,, ae egenvalues of the sgnal subspace, whch contans the egenvalues of R n descendng ode. I denotes an dentty matx of sutable sze. s s nose vaance. he numbe of actve uses s known n DS-CDMA systems. he nose vaance can be estmated as: BB S, s l l + (0) ()» + + C - C () he coespondng egenvalues of sgnal subspace ae s = l - s, =,,,. Futhemoe: the matx U B s the sze of C C ( C - ) matx N é s ù l + I 0 B úé ù é ù úê UB R = êu U ë B N úû s úê 0 I êu ë N ú ê û ( C- ) ( C-) ú ë û (3), whch contans the othonomal sgnal egenvectos, and U contans the nose egenvecto. l = dag ( s,, s ) B mplement a whtenng opeaton, and X can be compessed as X :. It s convenent to X - = QX = l U GB + N B B = GB + N (4)

8 Sensos 05, he model Equaton (4) s a whtened BSS model, Q - = B U B l s whtenng matx same as n Equaton (9), and G s an othogonal matx. heefoe, the poblem of Equaton (6) s a conveted nto a othogonal jont dagonalzaton to seek a mxng matx. Assume that the sepaaton matx s W obtaned afte the optmzaton of Equaton (6). hen the goal of blnd use sepaaton s ealzed by: Bˆ = sgn ( WX ) (5) Ideally, thee exsts the elatonshp of WG = I. In fact, thee exsts an nheent ambguty poblem n blnd souce sepaaton. heefoe, V = WG s not an dentty matx but athe a genealzed pemutaton matx. he ambguty poblem ncludes ampltude and ode ambguty. he ampltude ambguty wll be elmnated f we assume the covaance of B satsfes E é êbb ù ú = I. Afte the ë û whtenng opeaton, ths condton can be obtaned easly. he ode ambguty can be solved based on the followng pncple: he ecoveed souce sgnals can be gven by: B ˆ = WGB = VB (6) whee V s a genealzed pemutaton matx, o global matx, whee each column (o ow) contans only one non-zeo element whose absolute value s. It s wothwhle to note that: E é ˆ ù = E é ù êë BB úû V êë BB úû (7) he global matx V can be estmated by fndng the coss-coelaton matx between the vectos of sepaated sgnal ˆB and souce sgnal B. If an estmate of V s acqued, the souce sgnal wth pope ode can be obtaned as: B= V Bˆ (8) In summay, we use the followng steps whch ae shown n able to ovecome the ode ambguty, whee we assume each use tansmts the shot length M of the plot symbols. able. Ode ambguty elmnated. (a) Nomalze the plot symbols so that: ( M ) M p p å b( ) b ( ) = I = (b) Estmate V va the tme aveage as: Mp ˆ V = M å b p () b () = (c) Fo each column of G, nomalzed the ampltude of the element whch has the maxmum absolute-value to be one, and set all of othe elements to be zeo. Denote the nomalzed global matx as V ; (d) Restoe the ode of the outputs of blnd sepaaton by B= V B. ˆ Next, we wll dscuss the blnd speadng sequence estmate usng BSS. Accodng to Equaton (7), when the nose N s not consdeed except smulaton expements, we can ave at: p X = GB (9)

9 Sensos 05, he autocoelaton of X s denoted as (SVD),.e.,: we assume that the expanded space of column vecto of the column vecto of Î R, then X = R X s executed sngula value decomposton R UDU X (30) C U denotes the matx composed of man egenvectos, then the B C U Î belongs to the same space as the expanded space of B G. Note that thee exst a lnea tansfomaton elatonshp between If the lnea tansfomaton s assumed as A, we can establsh the blnd sepaaton model: = U and G B U AG B (3) he matx A denotes the mxng matx n the BSS model, G s the souce matx, and the matx U s the obsevaton matx. heefoe, the blnd sepaaton can be used to estmate the matx G. B Assume that Y denotes the sepaated sgnal. Afte the sepaaton s executed, the had decson s caed out fo Y, namely: Gˆ = sgn Y (3) hen we can estmate the speadng sequence fom Equaton (3) as Gˆ = é,, ù ê ë c c ú. he steps û C of the poposed method can be outlned as shown n able. able. Blnd Adaptve Detecto fo DS-CDMA system. (a) he whtenng pepocessng of the obsevaton usng Equatons () and (4); (b) Intalze pocessng pont u, = L fom the ange of [-, ]; (c) Estmate chaelaton matces Y ( u ) of the whtened sgnal usng Equaton (0); (d) Cay out WEDGE jont dagonalzaton fo optmzng poblem Equaton (6); (e) Seek sepaaton matx and estmate uses ognal sgnal of DS-CDMA usng Equaton (5), ode ambguty elmnated by Equatons (7) and (8); (f) Based on the model (7), to estmate speadng sequence of DS-CDMA usng Equatons (30) (3) Pefomance Analyss In ths subsecton, we evaluate the blnd sepaaton pefomance of the poposed algothm compaed wth the conventonal scheme (see detals n the smulaton analyss) and the HOS-based JADE algothm. Due to the fact the chaelaton matx ncopoates HOS chaactestcs, whch can suppess the Gaussan nose [5]. heefoe, the poposed blnd scheme can mpove the system pefomance compaed to the conventonal scheme n the nose envonment. Moeove, the poposed algothm explots a hybd statstcs (SOS and HOS) manne to extact statstcal nfomaton, whch can acque moe pefect estmated nfomaton compaed to the JADE wth HOS-based manne when the length of sgnal samples s not enough. In addton, we know that the pncple of JADE s jont dagonalzaton of estmated fouth-ode cumulant egenmatces []. he pncple of the poposed method s jont

10 Sensos 05, 5 06 dagonalzaton of estmated chaelaton matces. Fom the pespectve of computatonal complexty, the cumulant matx needs fou devatves of the second chaactestc functon compaed to the two devatves n chaelaton matx. Fnally, the computatonal complexty of the jont dagonalzaton method s also consdeed fo both these blnd sepaaton algothms. he computatonal complexty of the 4 jont dagonalzaton method of the JADE and the poposed method s 3 O M [] and O M [8] espectvely. hus we can know the new method s computed smply and extacts pefect nfomaton fom samples of obsevaton sgnals. Fom the pevous analyss, we can evaluate the mpoved blnd sepaaton pefomance that can be obtaned. In ode to cay out moe pefomance assessments, the pefomance n tems of ISR s dsplayed n mathematcal analyss. he ISR s computed fom the estmated mxng matx by post-multplyng ts nvese by the tue mxng matx and aveagng the mnmum to maxmum powe ato n each ow of the esults. Moe specfcally, f we defned V = WG as the esultng oveall contamnaton matx, ISR E é ù j êv ë j ú s the esdual mean contamnatng powe of souce j n the econstucton of souce. û hus, n the vcnty of a non-mxng condton G = I, t s easy to obseve that ISR» E é ù j ê W ë j ú. û Unde the small-eos assumpton and sub-gaussan souce sgnal consdeed (most communcaton sgnals ae sub-gaussan sgnals), the covaance n the estmaton of the elements of G, and hence the ISR can be pedcted. he ISR pefomance of JADE can be shown [8] to gven by: ISR JADE j k + k + k» M 4 j j ( k + k j ) Lkewse, takng nto account the dagonal stuctue (at G = I), we can obtan appoxmatvely the ISR pefomance of the poposed method as follows: ISR Poposed j» M k j k + k + k k j j whee, the k and k ae statstcal moment paametes about unknown th and j th souce. Fo j convenence, wthout loss of genealty, the souces come fom the unfom dstbuton, assumng k = k = k. We defne the ato of two ISR to evaluate the pefomance. We can obtan: j (33) (34) 6 4 Poposed JADE k h = ISR ISR = = j j k + 4k + k + k + k 4 (35) Based on the above analyss, we fnd that ISR Poposed j < ISR and the ISR pefomance s nvesely popotonal to the numbe of samples M. ISR denotes that the ISR s smalle when the V matx s close to the genealzed pemutaton matx, and the demxng pefomance of the algothm s bette. We can know that the poposed method outpefoms JADE, and the ISR mpovement becomes moe ponounced as the numbe of sample nceases. In the next secton we wll gve the smulaton esults and demonstate the pefomance of the DS-CDMA system aded by the poposed method to vefy the analyzed case. JADE j

11 Sensos 05, Smulatons and Dscussons o demonstate the effectveness of ou poposed method, smulaton expements ae used to llustate the ISR pefomance of blnd sepaaton by the chaelaton matx and the Bt Eo Rate (BER) pefomance of ths method executed n DS-CDMA systems. he esults ae shown n Fgues 6. he ISR and BER pefomance ndex ae utlzed to show the advantage of pefomance of the poposed appoach Aveage ISR [db] JADE Poposed method Numbe of samples M Fgue. ISR pefomance compason of JADE and the poposed method Use=4 BER Use=3 0-3 JADE Poposed Method JADE Poposed Mehod SNR/dB Fgue 3. BER pefomance compason between JADE and the poposed method fo use sepaaton n DS-CDMA systems.

12 Sensos 05, DEC MF MMSE PIC SIC JADE Poposed method BER 0. Conventonal Scheme 0.05 Blnd Sepaaton Scheme SNR/dB Fgue 4. BER pefomance compason among the poposed blnd sepaaton scheme and conventonal scheme. Use Use Use Use4 0 Ogn Chp Code Estmate Chp Code Chp Sequence Numbe Fgue 5. Pefomance of estmate the chp code fo fou uses n a DS-CDMA system JADE JADE Poposed Method Poposed Method Estmaton Accuacy Use=3 Use= SNR/dB Fgue 6. Estmaton accuacy of the chp code fo dffeent uses n a DS-CDMA system wth the JADE method and the poposed method.

13 Sensos 05, o begn wth, we demonstate the pefomance of the ISR, whch can be computed n numecal analyss as follows [9]: whee V v j ISR num ì é ù é ùü vj vj = ï í ï ý = j= max v j= = ê max v j j ú ê j ú ïî ë û ë û ïþ åå åå (36) = = WG s the unmxng-mxng global matx, and - W = G s the estmated unmxng matx. he smulaton paametes ae that the numbe of souces s 4, the mxng matx s andomly geneated, and 0 smulatons ae mplemented, and the channel fadng gans of 4 uses ae set as, 0.8, 0. and 0.05, whch ae same as n the followng smulaton condtons. he othe paametes ae maked n Fgue. It s eadly seen that the poposed method outpefoms JADE fom the Fgue as pevously exposed. Next the Bt Eo Rate (BER) of use sepaaton pefomance n DS-CDMA systems by ths method s gven. he smulaton paametes n DS-CDMA systems ae that the numbe of uses s fou, the length of speadng code s 3, and Gold Sequence s consdeed as speadng code, the length of samples s set as shot wth 000 bts, 0 smulatons ae executed, and the modulaton mode s BPS. Fo compason, the pefomance of anothe BSS algothm, JADE (usng the same data) s ndcated as well. Fom the smulaton esults shown n Fgues and 3, we can conclude that the new appoach has bette pefomance than the JADE algothms appled n DS-CDMA systems when the numbe of samples s shot and the SNR of eceve sgnal s low. We also know fom Fgue 3 that the pefomance becomes wose wth the nceasng numbe of uses. Next, we compaed the new method wth the conventonal scheme and blnd scheme used n multuse detecton. he conventonal schemes nclude Decoelaton (DEC), Matched Flte (MF) Mnmum Mean-Squae-Eo (MMSE), Paallel Intefeence Cancellaton (PIC) and Successve Intefeence Cancellaton (SIC). he blnd schemes ae JADE and the new method. All schemes wee tested usng Gold code of length C = 3. he numbe of uses s = 4. he numbe of samples s 0,000 bts, the modulaton mode s BPS, and 0 smulatons ae caed out. he dffeent paametes ae maked n Fgue 4. Accodng to Fgue 4, we can know that blnd scheme s supeo to the conventonal scheme and poposed method s bette than the JADE at lowe SNR. In the end, the blnd chp sequence estmaton s nvestgated fo DS-CDMA systems. he smulaton paametes ae that the Walsh code of length 64 s tested, and 0 smulatons ae mplemented. he dffeent paametes ae maked n Fgue 5 (SNR = 8 db) and Fgue 6, espectvely. Fgues 5 and 6 show the chp sequence can be estmated completely at low SNR. Fom Fgue 6, we can know that the poposed method has bette pefomance compaed to JADE. In summay, we can acque the chp sequence wth hgh accuacy n the case of low SNR. 5. Conclusons In ths pape, we nvestgate the chaelaton matx (the genealzed covaance matx) n DS-CDMA systems fo blnd use sepaaton and blnd chp/speadng sequence estmaton. he unknown mxng matx s estmated by jont dagonalzaton of the chaelaton matx of the

14 Sensos 05, obsevatons. heoetcal analyss and smulaton esults show that the poposed blnd sepaaton usng chaelaton matx pefoms bette than the conventonal scheme n low SNR. Especally, the poposed blnd sepaaton method has supeo pefomance than that of the exstng classcal JADE algothm-based HOS when the numbe of samplngs s shot and the SNR of the eceved sgnal s low. Futhemoe, we can acque the chp sequence n the case of low SNR and hgh accuacy asssted blnd sepaaton based on the chaelaton matx, so the poposed method has stong ablty antntefeence, whch s pomsng n applcatons fo ant-jammng n mltay communcatons and satellte communcatons. Acknowledgments hs wok s fully suppoted by a gant fom the Natonal Hgh echnology Reseach and Development Pogam of Chna (863 Pogam) (No. 0AA0A50), Natonal Natual Scence Foundaton of Chna (No ), Scence and echnology Suppot Pogam of Schuan Povnce (No. 04GZX0004). Autho Contbutons Zhongqang Luo and Ldong Zhu conceved and desgned the expements; Zhongqang Luo pefomed the expements and analyzed the data; Zhongqang Luo and Ldong Zhu wote the pape. Conflcts of Inteest he authos declae no conflct of nteest. Appendx A. Chaelaton Matx Devaton Devaton of the chaelaton matx Equaton (4) s llustated n ths Appendx. Fst, the dffeentaton of fomula Equaton (3) wth espect to u gves: ( u ) j j = å = å = = ( gu ) jb ( gu ) = å g = ( gu ) j gu gu gu b b u u u (A) Second, the dffeentaton of Equaton (A) wth espect to j ( u) j ( gu) = å = ( ) ( ) j ( gu) = å = ( gu ) ( gu ) u affods: ( gu) b g u u gu gu b gg u (A) In a smple opeaton, we can deve a moe compact fom of the coe equaton:

15 Sensos 05, B. Dagonalzaton Vefcaton b Y = Y u G G u G (A3) Let f b ( u ) denote the genealzed chaactestc functon of souce sgnal b ( m) statstcal ndependence of elements of b ( m ) = é b ( m ),, b ( m ) ù ê ë ú û, we get: = ( u ) ( u b b 3) b ( u) whee f ( u ) = E éexp ( ub ( m ù )),, b ê = ú. Defnng j ( u) = logf b b ë û chaactestc functon of souce sgnal b ( m). Hence, we obtan:. Due to the f u f f f b (B) j = j ( u ) + j ( u b b ) + + jb ( u) Consequently, the chaelaton matx Y b ( u ) can be easly obtaned: é j ù j ê u é ú ù b u u Refeences u s called second genealzed b u (B) Y = = b u êë u b úû u ê u ú ë û æ jb ( u) jb ( u) jb ( u) ö = dag,,, ç u u u çè ø. Yu, X.; Hu, D.; Xu, J. Blnd Souce Sepaaton: heoy and Applcatons; John Wley & Sons, Sngapoe Pte. Ltd.: Sngapoe, 04.. Rstanem,.; Joutsensalo, J. Advanced ICA-based eceves fo block fadng DS-CDMA channels. Sgnal Pocess. 00, 8, Albataneh, Z.; Salem, F. Blnd multuse detecton DS-CDMA algothm on the fast elatve Newton algothm. In Poceedngs of IEEE Weless telecommuncaton symposum, Washngton DC, USA, 9 0 Apl 04, pp Huovnen,.; Rstanem,. Independent component analyss usng successve ntefeence cancellaton fo ovesatuated data. Eu. ans. elecommun. 006, 7, Albataneh, Z.; Salem, F. Robust Blnd Multuse Detecton Algothm Usng Fouth-Ode Cumulant Matces. Ccuts Syst. Sgnal Pocess. 05, 34, Luo, Z.Q.; Zhu, L.D.; L, C.J. Explotng Chaelaton Matx to Impove Blnd Sepaaton Pefomance n DS-CDMA Systems. In Poceedngs of the 9th Intenatonal Confeence on Communcatons and Netwokng n Chna, Maomng, Chna, 4 6 August 04; pp Lu, F.B.; Huang, Z..; Jang, W.L. Blnd estmaton of speadng sequence of CDMA sgnals based on FastICA and pefomance analyss. J. Commun. 0, 3, (B3)

16 Sensos 05, Qu, P.; Huang, Z.; Jang, W; Zhang, C. Blnd multuse speadng sequences estmaton algothm fo the dect-sequence code dvson multple access sgnals. IE Sgnal Pocess. 00, 4, Belouchan, A.; Amn, M.G. Jamme mtgaton n spead spectum communcatons usng blnd souce sepaaton. Sgnal Pocess. 000, 80, Rstanem,.; Raju,.; ahunen, J. Jamme Mtgaton n DS-CDMA Aay System Usng Independent Component Analyss. In Poceedngs of the IEEE Intenatonal Confeence on Communcatons (ICC), New Yok, NY, USA, 8 Apl May 00; pp Raju,.; Rstanem,.; ahunen, J.; Oja, E. Jamme Suppesson n DS-CDMA Aays Usng Independent Component Analyss. IEEE ans. Wel. Commun. 006, 5, Zhang, J.; Zhang, Hang; Cu, Z.; Guo, J. Blnd Jont Jammng Cancellaton and Mult-use Detecton fo Asynchonous DS-CDMA Systems. J. Sgnal Pocess. 03, 9, Huovnen,.; Shahed, A.; Valkama, M. Blnd dvesty ecepton and ntefeence cancellaton usng ICA. In Poceedngs of the 007 IEEE Intenatonal Confeence on Acoustcs, Speech and Sgnal Pocessng (ICASSP 07), Honolulu, HI, USA, 5 0 Apl 007; pp Huovnen,. Independent Component Analyss n DS-CDMA Multuse Detecton and Intefeence Cancellaton. Ph.D. hess, ampee Unvesty of echnology, ampee, Fnland, Zhang, J.; Zhang, H.; Cu, Z.F. Dual-antenna-based blnd jont hostle jammng cancellaton and mult-use detecton fo uplnk of asynchonous dect-sequence code-dvson multple access systems. IE Commun. 03, 7, Syananda, M.G.S.; Joutsensalo, J.; Hämälänen.. Intefeence Cancellaton Schemes fo Spead Spectum Systems wth Blnd Pncples. In Poceedngs of the 7th Intenatonal Confeence on Advanced Infomaton Netwokng and Applcatons, Bacelona, Span, 5 8 Mach 03; pp Syananda, M.G.S.; Joutsensalo, J.; Hämälänen,. Blnd souce sepaaton based ntefeence suppesson schemes fo OFDM and DS-CDMA. elecommun Syst. 05, 4, Ch-We, J.; Shuh-Jye, J. Blnd ICA detecton based on second-ode cone pogammng fo MC-CDMA systems. EURASIP J. Adv. Sgnal Pocess. 04, 5, Solé-Casals, J.; Valatte, F.-B. owads Sem-Automatc Atfact Rejecton fo the Impovement of Alzheme s Dsease Sceenng fom EEG Sgnals. Sensos 05, 5, Wang, X.; Huang, Z; Zhou, Y. Sem-Blnd Sgnal Extacton fo Communcaton Sgnals by Combnng Independent Component Analyss and Spatal Contants. Sensos 0,, Luo, Z.Q.; Zhu, L.D; L, C.J. Employng ICA fo Inte-Cae Intefeence Cancellaton and Symbol Recovey n OFDM Systems. In Poceedngs of the IEEE Global Communcatons Confeence, Austn, X, USA, 8 Decembe 04; pp Jménez-Henández, H. Backgound Subtacton Appoach Based on Independent Component Analyss. Sensos 00, 0, Qn, S.; Guo, J.; Zhu, C. Spase Component Analyss Usng me-fequency Repesentatons fo Opeatonal Modal Analyss. Sensos 05, 5, Yeedo, A. Blnd channel estmaton usng fst and second devatves of the Chaactestc Functon. IEEE Sgnal Pocess. Lett. 00, 9,

17 Sensos 05, Slapak, A.; Yeedo, A. Chaelaton and Cham: Genec Statstcs Incopoatng Hghe-Ode Infomaton. IEEE ans. Sgnal Pocess. 0, 60, chavský, P.; Yeedo, A. Fast Appoxmate Jont Dgonalzaton Incopoatng Weght Matces. IEEE ans. Sgnal Pocess. 009, 57, Gong, X.; Wang, X.; Ln Q. Genealzed Non-Othogonal Jont Dagonalzaton wth LU Decomposton and Successve Rotatons. IEEE ans. Sgnal Pocessng 05, 63, Smekhov, A. Asymptotcally Optmzed Blnd Souce Sepaaton Based on Fouth- and Second-Ode Statstcs. Maste s hess, el-avv Unvesty, el-avv, Isael, by the authos; lcensee MDPI, Basel, Swtzeland. hs atcle s an open access atcle dstbuted unde the tems and condtons of the Ceatve Commons Attbuton lcense (