Errorless Codes for Over-loaded CDMA with Active User Detection

Abstrac t — In t his paper w e introduc e a new class of codes f or over-loaded sy nchronous w ireless CDMA syst ems which increases the nu mber of use rs for a fix ed number of chi ps wi thout i ntr oduci ng any error s . In addit ion t he se c odes sup port active user det ection. We derive an u pper bound o n the number of users w it h a fixed spre ading factor. Also w e propose a n ML decoder fo r a subclas s of these co des that is computati onall y impl e menta ble. Alt houg h for our si mulat ions w e consi de r a scenario t hat is w orse than what occ urs in practi ce, si mulation results i ndicate that this c oding/ decoding scheme is rob ust against ad ditive nois e. As an ex ample, for 64 chips and 88 users we pr opos e a c odi ng/de co ding sche me t hat can o btai n an arbitra ry small probability of error which is computationally feasible an d can detect act ive users. Further more, we prove t hat for this to be possible t he number of us ers cannot be beyond 2 30. I. INTRODUCTION In Direct-Sequence Code Division Multiple Access (DS- CDMA) due to pract ical c ondition s it i s desirabl e to use binary anti podal si gnatu res (s preading codes) in conjun ction with BPSK modulation. In these sys tems we can obtain errorless transm issi on using orth ogonal codes (e.g. Hadam ard codes) under the as sump tion o f no ise less cha nne l. T his is o nly tr ue if the num be r of use rs does not exc eed th e sprea ding fac tor (un der-l oaded or ful ly-loade d CDMA) . When the number of use rs excee ds th e sprea ding fa ctor, s uch orthog onal c odes do not exi st. Als o, usin g Pseudo-N oise (PN) spreadin g si gnatures creates interf erence that cannot be completely removed and results in error s in the Multi-User Detectio n (MUD) rece iver [1-3]. There are papers tha t discu ss doubl e orth ogonal c odes for increasing capacity [4-5]. T hese codes are non-binary complex codes (equivalent t o  phases for MC-OFDM) and thus are not a fa ir com parison to bin ary codes. In bandwidth limited channels, over-loaded CDMA is required. Most of the research in the o ver-loaded case is focus ed on code de sign and Mu lti-A ccess Int erfe rence cance llat ion for de creas ing the proba bili ty of error. Exam ples of thes e type of re searc h are pseu do ran dom spreadi ng (PN) codes [6- 7]; OC DMA/OC DMA (O/O) c odes [8- 10], Mul tipl e- OCDMA (MO) codes [11], and Binary Welch Bound Equality (BWBE) codes [12-14]. None of the sign atures and decodin g schemes introduced above guarantee errorless communication in an i dea l (no isel ess) sync hro nous chan nel i n the over -lo ade d case. All of these codes were designed with the idea of minimizing the criterio n of To tal Squared Cor relation (T SC) in mind . Acco rd ing to [15] minimi zin g T SC maxi mize s channel capacity when the input distributio n is Gaussian. Ho wever, whe n the inp ut alp hab et is  these codes do not necessarily maximize the ch annel capacity and thus may not result in a low er probability of error. In [16] we presen ted a new class of codes named CO W. Contrary to the aforementioned codes these codes can achieve error less transmission in the ideal ( noiseles s) channel. Altho ugh t he CO W co des c an achi eve hi gh o ver- loa ding factors 1 , we need to know the active users for proper decodin g. In ma ny ra ndo m acc ess c ommu nicat io n syste ms, identification of the active users will help inc rease the system capacity as n oted in [17]. For exam ple in ad-h oc networks as observ ed in [18] , “Opt imal tran smis sion st rategi es requi re the identification a nd localizati on of active nod es in the neigh borhood of th e transm itter”. In th is paper w e present a new set of ov er- loaded code s which guarantee er rorless trans mission in noiseless cha nnels and a re al so rob ust aga inst a d ditive noise . In a ddi tion t hey a re capable of detecting active users. We call this new class of codes, Codes for Over-loaded Wireless C DMA with Detection of Active users (COWDA). These codes are w ell s uited for synchr o nous Co de Divis io n Mult iple xin g (CDM) in broadcast ing and dow nlin k wirel ess a pplicat ions . Du e to thei r active user detection capability they can also be used in spatial mult iplexi ng appli cati ons and ad- hoc ne tworks. Furth ermore, these c ode s can r esul t in ba ndwid th sa vin g b y the use o f lo wer chip ra tes. In addit ion, we w ill propos e a dec odin g alg orithm at the receiver that is ML and computationally feasible. As an example, for a signature length of  these new codes can achi eve ov er-loa ding f actors of about  t hat can be practically decoded in time. Furthermore, we have proven t he existence of codes w ith an overload fact or of alm ost  . We will also obt ain an upper boun d on th e over-loadi ng factor, where it cann ot be bey ond % 260. In se ction II nec ess ary and su ffi cient c ondit ions f or errorless transmission with activ e user detection in over- loaded C DMA sys tems are discu ssed. A lso m ethods of const ruct ing la rge C OWD A co des with a hig h per centa ge of over- loading fa ctors wil l be pres ent ed. In secti on III an upper bound on th e number of use rs is obt aine d for a giv en 1 The perc enta ge of the num ber of us ers divi ded by the nu mbe r of chips m inus 1. Pedram Pad, Mahdi S oltanolkotab i, Saeed Hadik hanlou, Arash Enay ati and Farokh Ma rvasti Advanced Communications Research Institute (ACRI), Department of Electrical Engineering , Sharif University of Technology Te hran, Iran Emai l: {pedram _pad, m sol tan, s_h adikh anlou }@ee.sh arif .edu, aras henay ati@gm ail.c om and marva sti @shar if.ed u Errorless Co des for Ov er -loaded C DMA with Active User D etection spreadi ng f actor. The decod in g algor ithm is presen ted i n section IV. Si mulation result s app ear in section V. II. C ODES FOR O VER - LO AD ED W IR E LE S S CDM A WITH D ETECTIO N OF A CTI VE USERS (C OWDA) For dev elopin g C OWDA codes , w e will first presen t an intui tive geo metri c inter pre tat ion a nd then d eve lop the co des mathe mat icall y. At a gi ve n time the multi -use r bi nar y data ca n be represented by an  -dimens ional v ector (with  denoting active user data and  denoti ng n on-act ive us ers); t hese vectors can be interpreted as a set of discrete points on the vertices and ins ide an  -dimens ional hy per-cu be        . The data pertaining to the active u sers are multiplied b y their respective  -chi p long bi nary antipoda l signa ture s and fi nall y the ir summa tio n is tra nsmi tted . T hus, the transmitted  -tu ple v ector can be vi ewed as the res ult of the multiplication of an   matrix (with columns be ing the signa ture s o f di ffere nt use rs) by t he inp ut  -dimensi onal vector. Alterna tively, this ca n be viewed as a linear mapping of th e poin ts on the v ertic es and i nsi de the hy per-c ube ont o poi nts in a n  -dim ension al spac e     . As long as, the resul tin g poi nts in t he  -di mensional space are distin ct, the mapping is on e-to- one and theref ore, w e can uni quely decode each received  -tuple vector at the receiver. However, if the points in the  -dimensio nal space a re not distinct, the mapping is not one -to-one and thus not invertible. T his results in irremovable interference. Cons equently, we look for codes that map the point s in the  -di mensional space onto d istinct poi nts in t he  -dime nsion al s pace. Most of th e over- loaded codes dis cuss ed in th e lite rature do n ot poss ess th is property and thus their MUD scheme cannot be perfect. We call the class of codes with the above men tion ed property Codes for Over-loaded Wireless CDMA with Detection of Active Users (COWDA). Now we will develop a systematic method for the gener atio n of s uch c ode s. Lemma 1 Denote the  -dimensi onal input ve ctors    with the set  . The neces sary and suff icient condition for the multiplica tion of a    -mat rix  w ith elements of  to be a one- to- one m apping is tha t           , where  is the null space of  . The proof is tr ivi al. Corollar y 1 If  is a COWDA matrix then: a- New COWDA matrices can be generated by multiplying each r ow or col umn of  by  . b- New COWDA matrices can be ge nerated by permutations of colum ns or row s of  . c- By adding an arbi trary bin ary an tipodal row t o  , we obtai n another COWDA matrix. The proof is tr ivi al. Note 1 To ver ify tha t a    -matrix is COWDA using Le mma 1 it is sufficient to check   vecto rs. I gnor ing the zero -vecto r and considering the fact that half of the vectors are t he negat ive o f the o ther half, we ne ed t o sea rch o nly amo ngst        vectors, which is a very huge num ber. Now we suggest a method that can decrease this number dramatically. Partitio n  as  " # where is an  invertible matrix (accordin g t o Coroll ary 1 t his ca n be don e in most cases). S uppose there exis ts a vector $%         s uch that $   . Then there exists vectors $ & and $ ' of si ze  and    respectivel y, with entries in      such t hat $ &  "$ '   . Conse quently , we only need to s earch among st   ()     likely vectors $ ' in      () and c heck t hat  ( & "$ ' belong s to      ) . For e xample , to check t hat t he  & *'' matrix of Table. 1 is COWDA we only need to search among   *     vectors. Theore m 1 Assume that  is an  COWDA matrix and + is a n invertible ,,    -matri x, then +-  is a ,!  , COWDA matrix, w here - denotes the Kronecker matr ix product . Proof : Obvious ly , +-  is a    -matrix. S uppose that  +-  . where . has entr ie s in    . Multip lyi ng bo th si des o f th is eq uatio n by + (& /0 )  , we have  0 1 -  .  The above set of equations can be decoupled into , diff erent equations of the form . 2  where . 2 denotes the 3  4    5 th to the  4  th entries of . wit h  646, . How ever, w e know that  is COWDA, thus accordin g t o Lemm a 1, . 2 equals the ze ro vector for 64 6 , . Hence, . is t he zer o-ve cto r and agai n acco rd ing to Lem ma 1 +-  is C OWDA. 7 In the follo wi ng t heor em we will p ro ve the exis tenc e o f COWDA c odes wi th a h igher perce nta ge of the ov er-loadi ng factor. Theore m 2 Assume  is an   COWDA matrix and 8 9 is a   Hadam ard matrix. We can add :   ;<= > ' ? column s t o 8 9 - to obta in another COWDA matr ix. This t heorem is pr oved in the Appendi x. Note 2 @ A B as A B . This observation is a direct res ult of Theorem 2 since @ is of order CDEF   . This im plies that as the chip rate increases the nu mber o f use rs gr o w much fa ste r. G H H H H H H H H H H H H H I                                                                 J K K K K K K K K K K K K K L Table. 1.  &*'' where + denotes  and  denotes  . Notice t ha t the first  columns o f the abo ve matrix is a    Hadamard matrix. Exampl e 1 In the f irst st ep, apply ing Theorem 2 on a  Hadamard matrix, w e first get a    COWDA matrix (  & *&M ). By computer search we also found a    COWD A matrix (  &*'' which is shown i n T able . 1). Acc ording t o Theorem 1,  & *'' leads to a    COW DA matri x by t he Kro necke r pr oduct 8 N - &*'' (where 8 N is a  Hadamard matrix ); this im plies that we can h ave errorless decoding f or  users with onl y  chip s; which ha s an ov er-l oading fact or of abou t  (we w ill introduce a suitable decoder for this code in section IV). However, reuse of Theore m 2 f or  &*'' leads to a   O COWDA matrix (  * NP> ). Thi s im plies an over- loading fac tor of  . III. U PPER BOUND FOR THE OVER - LOADI NG FA CT OR The fol lowin g the orem provi des an upper- boun d for th e over- loading fa ctor of a C OWDA matrix. Theore m 3 If  Q 2R # is a COWDA matrix with  columns (users ) and  ro ws (chips), t hen 6   ST U  ,     1V( ;<= W U  ,   X whe re U   ,  T Y   ZY  , Z [ (1 ' \ ]V ^ _ Proof : Suppose ` $ , wh ere  is a COWD A matrix and $ is a random variable from the set    with unifo rm distribution. Since t he code matri x (  denotes a on e-to-one transformation between $ and ` , t hen the vecto rs ` and $ have t he sa me a mount of in for matio n. T hus, a  `  a  $   ! ; < = '  where a denotes the entr opy function i n bits. Since the entrie s of $ are from the set   and the entries of the code matrix are  , the entries of ` are integers between b and  . The probabi lit y th at an entry in ` takes th e value , is equal to, c  d 2 ,   U   ,    where U   , is the num ber of s olutions of the equat ion, e & e '  f e  , wit h e 2 belongi ng to th e set   . U  , can be calculated from the f o llowing formula, U   ,  T Y   ZY   , Z [ (1 ' \ ]V^ each e ntry i n th e above summati on is t he num ber of such solutions with   ’ s (there are 3  ] 5 s uch e ntri es) a nd , entries of  ’s (there are 3 (] ]g1 5 such entri es). Kno wing t he probabil ity of t he s ingle en tries of th e vec tor ` ( c  d 2 ,  ), we can calculat e an upper bo und f or a  `  with the assumption of independence betw een the entries of ` . Thu s, a `  6  S T U  ,    1V ( ;<= ' U  ,   X The refo re, !  ;<= ' 6  S T U  ,     1V ( ;<= ' U  ,    X which impl ies the u pper bound in th e t heorem . 7 Fig.1. Uppe r bou nd on the num ber of user s vs . the n umber of chips The upper bou nd stated in Theorem 3 is shown in Fig. 1. This figure shows that we cann ot have errorless communication with 64 chi ps an d beyon d 230 use rs, wh ich im plies an over- loading factor lower than %26 0. IV. D ECOD I NG ALGOR ITHM In thi s sect ion we wil l pre sent a dec oding a lgori th m for the propose d clas s of codes . This decodi ng sc heme has th e added advan tage th at it works with an unknown num ber of active users. At the receiver a combination of the signatures of different users embedded in AWGN is received. This vector can be mode led as ` $  h where  is the code matrix and h denotes the noise vector whic h has a Ga ussia n di strib uti on, with zero mea n, and aut o- covariance mat rix i ' 0 (where 0 de notes the ide ntit y ma tri x). To impl ement ML decodi ng for ea ch us er, it mus t minim ize j`  $ k j ' , where the user ’s e ntry i n $ k is  an d the rest of the entries (o f $ k ) belong to t he s et   . The reason behind this f act, is that each activ e user is not aw are of the status of other users; it o nly knows that its own sig nature contributes to the received vector by a  occurren ce and not  . Conseq ue ntly the user has to choo se be tween  (& input vectors $ k (w ith  denoting t he num ber of users). The computatio nal complexit y of this i mplementat ion of the ML decoder is tre m endously high . In the foll owing we will pre sen t a decoding m ethod w ith much l ower co m plexity. We w ill also derive condi tions un der whi ch this decoder is M L. This is accom plished in tw o major st eps. In t he fir st st ep, we sho w t hat i f t he cod es have been gen erated a ccording to Theorem 1, the decoding probl em can be reduc ed to a set of decoding problems with smaller code matrices. Consider a COWD A code matrix  ]l!]1 + ] ] - m l1 gener ated by the K rone cke r p roduc t of a n inver tibl e matr ix + wi th a sma ller COWD A mat rix m (according t o T heorem 1). The received vect or is ` $  h  +-m  $ h Multip lyi ng bo th si des b y  + (& -0 l  we h ave  + (& -0 l  `  + (& -0 l  3  +-m  $h 5   0 ] -m  $  + (& -0 l  h This implies that t he first ; elements o f  + (& -0 l  ` depends only on the first , elements of $ p lus the new noise term  + (& -0 l  h , the se cond ; elem ents of  + (& -0 l  ` depends only on the se cond , elements of $ pl us the ne w noise term  + ( & -0 l  h , and so o n. Henc e, for extr acting t he first , bits of $ it’s sufficient to consider o nly the first ; elements of  + ( & -0 l  ` , for extracting the secon d , bits of $ it’s sufficient to c onsider onl y the second ; elements of  + ( & -0 l  ` , and so on. T hus, we have d ivi ded the pr ob lem of decodi ng a CDMA system w ith   ; chips and  , users to d eco ding  CDM A s yst em s wi th ; chips and , users . Thi s result s i n a hu ge re ducti on o f co mputa tio nal c ost s. If t he matr ix + is Hadamard, the ML decoder of the big ger sys tem becomes equivalent to the ML decod ers of the smaller systems (because n  + (& -0 l is a uni tar y matr ix a nd do es not chan ge the auto-covariance m atrix of noise). In th e second major step, we will further reduce the complexity of the smaller s ystems. Con sider m " # where is an ; ; i nverti ble matr ix and " is an ; ,;  matrix. The reason th at can be considered invertible is that the a ssum pt ion of m being f ull r ank is no t ver y rest rict ing an d due to C orollary 1, c olum ns of m can be pe rmuted. Us ing this partitioning `m $  h  " # o $ & $ ' ph  $ & " $ ' h where $ & and $ ' are ;  and ,  ;   vectors, respect ive ly. Mult iply ing both sides by (& , we arrive at the equation: ! (& `$ &  (& "$ '  (& h Thus the stated minimizatio n prob lem can be simplified to qrs t k u t k v j (& `  $ k &  (& "$ k ' j ' For extracting the data of th e 4 th user, the best estimati on of $ & is as follows 1- If 46 ; w 3$ k & 5 R  x4= Y3 (& ` (& "$ k ' 5 R Zy  4 3$ k & 5 R  x  ? , and the r esult ant matrix, ‚ , is a COW DA matrix. W e wish to p rove that o ne can a dd a nothe r co lumn to ‚ to obtain a COWDA m atrix w ith   ,   co lumns. Assume that ‚  m ˆ - # , w here -  © & ˆ ® ˆ © 1 # , and © 2 is a    vector, for 4   f  , . Le t $%      Ng1 , $  $ & • $ ' • # • , where $ & is a    ve ctor and $ ' is a ,  vector. Hence, m$ & ‚ $ - $ ' . By s teps 2 and 4, and the fact that $ ' has  1 different possibilities, we have ˆ      N)  ¯       N) ˆ  •ˆ  m°      N     -$ '      N)  ¯ t v  -$ '      N) ˆ 6 1   W)  )(&  ')g&  6 1  W)g& wh er e   ‚$ ˆ $%      Ng1  . Now, if  1  W )g&  ˆ     N ) ˆ  N ) , then we ca n ad d anoth er column to m atrix D by ap pl ying st ep 1 . T hus, we can add at least :   ;<= >  ? vectors to m and obtain a bigger COWDA matrix. !7 R EFERENCES [1] S. V erdu , Mu ltiu ser Det ect ion , C ambrid ge Un ivers it y Press , N ew Yor k, NY, USA, 1998. [2] A. Kapur an d M.K. V aranasi , “Multiuser de tection for ov er-loaded CDMA sy stems,” IEEE T ransact ions on I nformati on Theo ry , vol. 49, no. 7 , pp. 1728–174 2, Ju l. 200 3. [3] S. Mosha vi, “Multi -User detection for DS-CDMA communi cations,” IEEE C ommuni cat ions Mag azi ne , vol. 34, no. 10 , pp. 124–136, Oct. 1996. [4] B. Natarj an, C .R. 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