Average SIR of the desired user ([7]) β=0.99. β= Average NSE of the desired user (BADD) β=0.95. µ= β=0.9

Transcription

1 A Blnd Adaptve Decorrelatng Detector for CDMA Systems Sennur Ulukus Roy D. Yates Department of lectrcal and Computer ngneerng Wreless Informaton Networks Laboratory (WINLAB) Rutgers Unversty, PO Box 909, Pscataway NJ Abstract: Te decorrelatng detector s known to elmnate te mult access nterference gven tat te sgnature sequences of te users are lnearly ndependent, at te cost of enancng te Gaussan recever nose. In ts paper, we present and study te convergence of a blnd adaptve decorrelatng detector wc s based on te observaton of te avalable statstcs. Te algortm s teratve and dstrbuted. Only two parameters are needed to be known for te constructon of te recever lter of a user: te user's sgnature sequence and te varance of te wte Gaussan recever nose. Te computatonal complexty of te proposed algortm per teraton s lnear n te number of users. 1 Introducton Code Dvson Multple Access (CDMA) systems suer from te near-far eect, because of non-ortogonalty of te users' sgnature sequences. Multuser detecton [1] can be used to overcome te near-far problem. Multuser detectors explot te specal structure of te multple access nterference to effectvely demodulate te non-ortogonal sgnals of te users. It was sown n [2] tat te optmal multuser detector as a computatonal complexty wc ncreases exponentally wt te number of actve users. Several suboptmum detectors, ncludng te decorrelatng detector [3], ave been proposed to aceve a performance comparable to tat of optmum detector wle keepng te complexty low. Te decorrelatng detector [3] wc as lnear (n number of users) computatonal complexty was sown to elmnate te mult access nterference totally f te sgnature sequences of te users are lnearly ndependent at te cost of enancng te addtve wte Gaussan nose (AWGN). Te decorrelatng detector of [3] s centralzed and non-teratve. Te constructon of te decorrelatng detector lter for a certan user requres te knowledge of te sgnature sequences of all nterferng users as well as te sgnature sequence of te user of nterest. In addton to tat, te constructon requres nverson of te N N cross correlaton matrx, were N s te number of actve users. Blnd adaptve algortms are desrable to overcome te need for te knowledge about te parameters of te nterferng users and teratve algortms are needed to avod te matrx nverson wc may ave a Supported by NSF Grant NCR large computatonal complexty. A blnd adaptve algortm based on te mnmzaton of te output energy was gven n [4]. Ts algortm was sown to converge to te MMS multuser detector [5]. In [6] an adaptve multuser detector wc converges to te decorrelatng detector s proposed. Ts detector stll needs te sgnature sequences of all te users and transmsson of tranng sequences. In [7] blnd algortms based on sgnal subspace trackng are nvestgated and two algortms wc converge to te decorrelatng and MMS multuser detectors are proposed. Te blnd adaptve decorrelatng detector proposed n [7] needs te nformaton about te varance of te AWGN and te number of users bot of wc can be estmated agan usng subspace trackng tecnques. Te computatonal complexty of te algortm of [7] s O(GN) per teraton, were G s te processng gan. In ts paper we present a blnd adaptve multuser detector wc uses observables tat are readly avalable at te recever and wc converges to a decorrelatng detector. Te detector s constructed va a dstrbuted teratve algortm wc updates te recever lter coecents of a desred user by usng te prevous output of te lter under constructon. Snce te lter output s random due to te randomness of te mult access nterference and exstence of ambent Gaussan cannel nose, te algortm evolves stocastcally. We prove te convergence of te lter coecents to a decorrelatng detector n te mean squared error (MS) sense. We develop lower and upper bounds for te MS of te lter coecents at teraton n + 1 n terms of te MS of te lter coecents at teraton n, and nvestgate te condtons under wc te MS sequence converges to zero as number of teratons grows to nnty. Te computatonal complexty of te proposed algortm s O(G) per teraton. Te proposed algortm requres te knowledge of only two parameters for te constructon of te lter coecents for a desred user: te desred user's sgnature sequence and te varance of te AWGN. Te varance of te AWGN s a xed quantty (not tme varyng) and can be estmated easly, peraps before te nformaton transfer starts wen only te samples of AWGN can be observed wtout any nterference at te output of an arbtrary (nonzero) recever lter. In suc a case, a relable estmate of te varance of AWGN can be obtaned by tme-averagng te squares of te output of te recever lter.

2 2 Te Decorrelatng Detector Trougout ts paper, a syncronous CDMA system wt BPSK modulaton s assumed to smplfy te analyss. We wll use G dmensonal vector s to denote te pre-assgned unque sgnature sequence of user. Let us dene an N G dmensonal matrx S wt ts ( j)t element beng (s ) j, te jt component of s. Terefore, te rows of S (equvalently te columns of S > ) are te sgnature sequences of te users. For future use, we dene a subspace L n G dmensonal vector space to be te subspace spanned by te sgnature sequences of te users,.e., L = spanfs 1 ::: s N g = column space of S > (1) We consder te decorrelatng detector for te rst user wtoutlossofgeneralty and represent ts G dmensonal recever lter by c. Ten c sould satsfy te followng condton. Sc = e 1 (2) were e 1 s te rst unt vector n N dmensonal space,.e., e 1 = [1 0 0 ::: 0] >, and s a non-negatve real number. It can be easly sown tat te bt error rate performance of te decorrelatng detector s nsenstve to scalng of te lter coecents as long as te lter elmnates te mult access nterference totally. Te reason for ts s tat te scalar factor multples bot te receved power level of te desred user and te AWGN. We rst note tat (2) as more tan one soluton, because t as N equaltes and G unknowns, and typcally G N. Te unque decorrelatng detector for te rst user, ~c, sgven [3] as ~c = S > (SS > ) ;1 e 1 (3) Let us denote any soluton of (2) as c. Ten nsertng (2) nto (3) we obtan (assumng =1) ~c = S > (SS > ) ;1 Sc = P c (4) Note tat P = S > (SS > ) ;1 S s te projecton matrx wc projects any vector onto te column space of S >. Also note tat te column space of S > s te subspace spanned by s 1 ::: s N, te subspace wc was denoted as L. Terefore, altoug (2) as more tan one soluton, all of te solutons ave te same projecton onto L and ts projecton s equal to te unque decorrelatng detector of Lupas and Verdu [3]. Let be an N N dmensonal dagonal matrx wt receved power of user, p,astst dagonal element. Multplyng bot sdes of (2) rst wt, ten wt S > yelds S > Sc = p 1 s 1 (5) We observe tat altoug (5) as G equatons n G unknowns t does not ave a unque soluton for c, snce (G ; N) egenvalues of S > S are equal to zero. Te soluton spaces of (2) and (5) are related as stated n te followng Remark. Remark 1 If s 1 ::: s N are lnearly ndependent, ten all solutons of (2) and (5) concde. Let us dene A = S > S. At ts pont we coose = 1=p 1 and devse te followng determnstc gradent descent algortm: c(n +1)=c(n) ; (Ac(n) ; s 1 ) (6) Note tat te teratve algortm gven n (6) converges to te soluton of (5) and equvalently, by Remark 1, to te soluton of (2). Te fact tat te soluton of (2) s not unque s noted earler. Note tat, for any c, PAc= S > (SS > ) ;1 SS > Sc = S > Sc = Ac (7) Ts means tat for any c(n), Ac(n) les entrely n L. Usng ts result and (6) we obtan c(n +1)=Pc(n +1)+c(n) ; Pc(n) (8) By nducton (8) yelds c(n) =Pc(n)+c(0) ; Pc(0) (9) Terefore, f te teratve algortm (6) s started n te subspace L, mplyng c(0) = Pc(0), ten from (9) we wll ave: c(n) =Pc(n) for all n. In ts case c(n) wll always stay n L and wll converge to te scaled verson of te unque decorrelatng detector soluton of [3] c = 1 p 1 ~c as n goes to nnty. Note tat te algortm converges to te scaled verson of ~c, nstead of convergng to ~c because s not cosen to be 1. Te restrcton tat te teratons sould be started n L for algortm (6) to converge to a decorrelatng detector s farly mld. Selectons c(0) = 0, c(0) = s 1 or any lnear combnaton of te sgnature sequences mply c(0) 2L, satsfyng te convergence condton of (6). Te sgnature sequences of all of te users mustbeknown for te algortm gven n (6). Also, te algortm of (6) s an o-lne algortm wc must be run before te real nformaton transmssons of te users start. After runnng te algortm for some tme, te lter coecents would converge to a decorrelatng detector lter and ten te communcaton can be started. In ts paper, our am s to develop a blnd adaptve algortm wc would converge to a decorrelatng detector soluton n a stocastc sense by usng real random measurements wle te users are actve and transmttng bts. To ts end, we wll propose an algortm wc can be vewed as te stocastc verson of te determnstc algortm gven n (6). 3 A Blnd Adaptve Decorrelatng Detector (BADD) Te receved base band sgnal at te nput of te recever lters after cp-matced lterng followed by cp rate samplng s NX p r = p b s + n (10) =1 were b s te nformaton bt (1 equprobably) transmtted by user and n s a Gaussan random vector wt zero mean

3 and [nn > ]= 2 I. Note tat, rr > = NX =1 p s s > + 2 I = A + 2 I (11) Terefore, usng (11), te determnstc teraton of (6) can be wrtten n an exact form as c(n +1)=c(n) ; ; rr > ; 2 I c(n) ; s 1 (12) Note tat rr > ; 2 I s an unbased estmate for A matrx. In order to obtan a practcal algortm we replace te exact expresson for A n (12) wt ts unbased estmator bysmply removng te expectaton n (12). Tus we obtan, c(n +1)=c(n) ; ; rr > ; 2 I c(n) ; s 1 (13) Before analyzng te convergence of (13), we state t n terms of avalable observatons, and lst te parameters needed at eac teraton. Let y(n) be te output of te recever lter of te desred user at tme n. Ten, y(n) =r > (n)c(n), were r(n) s te sampled receved sgnal before te recever lters and c(n) s te current lter of te desred user. Tus, te mplementaton orented verson of te algortm (13) s c(n +1)= ; 1 ; 2 c(n) ; (y(n)r(n) ; s 1 ) (14) Snce r(n) and y(n) are readly avalable at te nput and output of te recever lter tat s under constructon, only two system parameters are needed to be known n order to run te algortm: te sgnature sequence of te desred user s 1, and te varance of te AWGN, 2. Te varance of AWGN s a xed quantty wc can be easly estmated before te communcaton starts as dscussed n Secton 1. 4 Convergence of te BADD In ts Secton we wll nvestgate te convergence of te blnd adaptve decorrelatng detector proposed n te prevous secton. Let us dene te zero mean random vector (n) as (n) = ; rr > ; A ; 2 I c(n) (15) Notng tat Ac = s 1,we can wrte te stocastc teratons (13) and (14) as c(n +1)=c(n) ; [A (c(n) ; c )+(n)] (16) Subtractng c from bot sdes of (16) and denng (n) = c(n) ; c yelds (n +1)=(n) ; (A(n)+(n)) (17) Note tat norm of (n) s a measure of te dstance of te current recever lter from c, te convergence pont. Squarng of bot sdes of (17) yelds k(n +1)k 2 = k(n)k 2 ; 2(n) > A(n)+2 2 (n) > A(n) ;2(n) > (n)+ 2 (n) > A 2 (n)+ 2 k(n)k 2 (18) By takng te condtonal expectaton of bot sdes, condtoned on (n)=, and observng tat [(n) j (n)=]=0, we obtan k(n+1)k 2 j (n)= = kk 2 ; 2 > A + 2 > A k(n)k 2 j (n) = (19) We wll be usng te results of te followng Lemmas to develop bounds for te rgt and sde of (19). Lemma 1 If P = 0 ten > A = 0 and > A 2 = 0. If P6= 0 ten tere exst 0 <k 0 k 1 < 1, suctat k 0 kk 2 > A k 1 kk 2 (20) k 2 0 kk 2 > A 2 k 2 1 kk 2 (21) Lemma 2 Tere exst 0 <c 0 c 1 < 1, suc tat 0 k(n)k 2 j (n) = c 0 + c 1 kk 2 (22) Te outlne of te proof of Lemma 1 s as follows. Snce A s a postve semdente matrx wt rank N, (G ; N) egenvalues are equal to zero and remanng N egenvalues are postve. Snce for any x, Ax 2 L by (7), egenvectors of A are eter completely n L wt postve egenvalues, or completely out of L (meanng tat ter projectons onto L are zero) wt zero egenvalues. In ts case Lemma 1 s a smple result of Rayleg quotent [8]. Te proof of Lemma 2 wc can be found n [9] s more lengty and s omtted ere. In wat follows, we wll denote te condtonng on (n) =, P = 0 and P 6= 0 by, P and P, respectvely. For example [k(n +1)k 2 j (n) = P = 0] wll be denoted by [k(n +1)k 2 j P]. If P=0, followng lower and upper bounds can be developed for te rgt and sde of (19) usng Lemmas 1 and 2, k(n +1)k 2 j P k(n +1)k 2 j P And smlarly f P6= 0, k(n+1)k 2 j P ; 1+ 2 c 1 kk 2 + c 0 2 kk 2 (23) ; 1 ; 2k 0 +(k c 1 ) 2 kk 2 +c 0 2 k(n+1)k 2 j P ; 1 ; 2k 1 + k0 2 2 kk 2 (24) An mportant observaton toward te convergence proof s tat te projecton of c(n) (equvalently te projecton of (n)) on L would be non-zero almost surely (a.s.) [10]. Ts means tat for any n te probabltyofteevent[p(n) =0] s zero. Smlar to te determnstc result n (9), t can be sown usng nducton on (17) tat (see also [11, qn. (16)]), (n) =P(n)+(0) ; P(0) a.s. (25) Tus, f (0) 2Lten (0) = P(0) and (25) mples (n) = P(n) a.s. An upper bound for [k(n +1)k 2 j ] can be

4 developed as k(n+1)k 2 j = k(n+1)k 2 j P ProbfP = 0g + k(n+1)k 2 j P ProbfP6= 0g = k(n+1)k 2 j P ; 1 ; 2k 0 +(k c 1 ) 2 kk 2 +c 0 2 (26) were we used ProbfP = 0g = 0 and ProbfP 6= 0g =1. By smlar arguments a lower bound can be found to be k(n +1)k 2 j ; 1 ; 2k 1 + k kk 2 (27) Takng te expectaton of bot sdes of te nequaltes n (26) and (27) wt respect to (n) and lettng b n = [k(n)k 2 ], we obtan te followng bounds for b n, te mean squared error (MS) of te lter coecents at teraton n from c, b n+1 ; 1 ; 2k 1 + k b n (28) b n+1 ; 1 ; 2k 0 +(k c 1 ) 2 b n + c 0 2 (29) Denng, 0 =1;2k 0 +(k 2 1+c 1 ) 2 and 1 =1;2k 1 +k 2 0 2, we can rewrte quatons (28) and (29) as 1 b n b n+1 0 b n + c 0 2 (30) We observe from (30) tat te nonnegatve sequence b n s sandwced between te two sequences generated accordng to b 0 n+1 = 0 b 0 n+c 0 2 and b 00 n+1 = 1 b 00 n. Tese two sequences converge to nte numbers f and only f s cosen suc tat j 0 j< 1andj 1 j< 1. Note tat bot 0 and 1 are equal to1at =0. We also note tat bot 0 and 1 are locally decreasng as ncreases, snce d 0 d =0 = ;2k 0 < 0 and d 1 d =0 = ;2k 1 < 0 (31) Ts means tat we can always coose small enoug so tat j 0 j < 1 and j 1 j < 1, n wc case te sequences b 0 n and b 00 n converge and te lmtng MS,.e., lm n!1 b n, as nte lower and upper bounds. From te sandwc teorem, 0 lm n!1 b n c ; 0 (32) Te upper bound can be evaluated as! 0as, lm!0 c 0 2 c 0 = lm 1 ; 0!0 2k 0 ; (k1 2 + c =0 (33) 1) Terefore, f te step sze () s cosen extremely small ten te MS of te lter coecents from te decorrelatng lter coecents goes to zero as te number of teratons grows to nnty. But note tat as! 0, te numbers 0 and 1 go to 1 n wc case te convergence rate goes to zero. Tus, we observe te trade o between te lmtng value of te MS and te convergence rate. If a large value s cosen as te step sze ten te convergence rate s faster but te lmtng MS s larger and f a small value s cosen as te step sze te lmtng MS s smaller but te convergence rate s slower too. Hence, a tme dependent step sze sequence wc takes large values at te begnnng and smaller values at te end may be preferable. An teraton ndex (n) dependent step sze sequence can be used to accompls ts. Replacng te xed step sze n (14) wt te tme varyng step sze sequence a n we obtan te new algortm to be: c(n +1)= ; 1 ; 2 a n c(n) ; an (y(n)r(n) ; s 1 ) (34) Te convergence of (34) s stated n te followng Lemma. Te arguments of te proof wc can be found n [9] follow tose made n [12]. Lemma 3 If a n satses 1X n a n = 1 and 1X n a 2 n < 1 ten te stocastc teraton gven n (34) converges to c n te MS sense,.e., lm n!0 [kc(n) ; c k 2 ]=0. Note tat a n = a=(n + n 0 ), for any a>0 and n 0 > 0 satsfy te condtons of Lemma 3. 5 Smulaton Results In te smulatons we consder a system wt N = 6 users usng randomly generated sgnature sequences wt a processng gan of G =31. Powers of te users are assgned so tat te nal sgnal to nterference rato (SIR) of all users wll be 20 db. Tus, te power of te t user s found by p = [; ;1 ] were ; s te cross correlaton matrx wc s equal to SS >. Te algortm s run for 100 tmes and averaged results are plotted n te gures. In Fgure 1, we plot te averaged normalzed squared error (NS) of te lter coecents of te desred (rst) user versus teraton ndex n, were NS at teraton n s dened as NS(n) =kc(n) ; c k 2 = kc k 2. Varous curves n Fgure 1 correspond to te blnd adaptve decorrelatng detector algortms wt xed step sze for derent step sze values and tat wt a tme dependent step sze sequence of te form of a n = 1=(n + n 0 ) wt n 0 = 5. We observe tat f te step sze s too large, ten te algortm does not converge see ncreasng NS curve for =0:1 nfgure 1. We also observe tat for large step sze values te convergence rate s fast but te lmtng NS s large too compare te curves correspondng to =0:01 and =0:001 n Fgure 1. Averaged (over 100 runs) SIR of te desred (rst) user s plotted n Fgure 2 for te same system. At teraton n, te SIR of te desred user s calculated as p 1 (s > 1 SIR(n) = P c(n))2 p j6=1 j(s > j c(n))2 + 2 (c > (n)c(n)) (35) We observe tat te convergence propertes of te SIRs to te target SIR (wc s 20 db n ts experment) s smlar

5 to te propertes of te convergence of te NS to zero. Te blnd adaptve decorrelatng detector of [7] s also mplemented for te same system and te SIR of te desred user s plotted as a functon of teraton ndex n Fgure 3 for derent values of te forgettng factor β=0.999 Average SIR of te desred user ([7]) β=0.99 β= Average NS of te desred user (BADD) µ=0.1 Average SIR(n) 10 1 β=0.95 µ= β=0.9 Average NS(n) 10 1 µ=0.001 teraton ndex (n) µ=0.01 Fgure 3: Averaged SIR of te desred user (algortm of [7]). a n =1/(n+n 0 ) 10 2 teraton ndex (n) Fgure 1: Averaged normalzed squared error (NS) of te desred user. Average SIR(n) Average SIR of te desred user (BADD) a n =1/(n+n 0 ) µ=0.001 µ=0.01 µ= µ=0.1 teraton ndex (n) Fgure 2: Averaged SIR of te desred user. 6 Concluson In ts paper we presented an teratve and dstrbuted adaptve decorrelatng detector algortm wc s based on te observaton of te avalable statstcs, and studed ts convergence. Te update equaton of te algortm needs te observaton of te cp sampled nput sgnal before te recever lter and te output of te lter wt te current lter coecents. For te mplementaton of te algortm to construct te decorrelatng recever lter of a user only two parameters are needed to be known: te user's sgnature sequence and te varance of te addtve wte Gaussan recever nose. We studed te convergence of te proposed algortm bot for a xed step sze sequence and for a tme varyng step sze sequence. For te rst case we developed te condtons of avng lower and upper bounds on te MS and sowed tat as te step sze goes to zero te algortm converges n te MS. For te second case we drectly proved te convergence n te MS. References [1] S. Verdu. Multuser detecton. Advances n Statstcal Sgnal Processng, 2:369{409, [2] S. Verdu. Mnmum probablty of error for asyncronous gaussan multple-access cannels. I Trans. on Inform. Te., 32:85{96, January [3] R. Lupas and S. Verdu. Lnear multuser detectors for syncronous code-dvson multple-access cannels. I Trans. on Inform. Te., 35(1):123{136, January [4] M. Hong, U. Madow, and S. Verdu. Blnd adaptve multuser detecton. I Trans. on Inform. Te., 41(4):944{ 960, July [5] U. Madow and M. L. Hong. MMS nterference suppresson for drect-sequence spread-spectrum CDMA. I Trans. on Comm., 42(12):3178{3188, December [6] D. S. Cen and S. Roy. An adaptve multuser recever for CDMA systems. I JSAC, 12(5):808{816, June [7] X. Wang and H. V. Poor. Blnd adaptve nterference suppresson for CDMA communcatons based on egenspace trackng. In CISS-97, pages 468{473, Marc [8] G. Strang. Lnear Algebra and Its Applcatons. Saunders College Publsng, rd edton. [9] S. Ulukus and R. D. Yates. Convergence of a blnd adaptve decorrelatng detector. To be submtted. [10] W.F. Stout. Almost Sure Convergence. Academc Press, [11] L. Gyor. Adaptve lnear procedures under general condtons. I Trans. on Inform. Te., IT-30(2):262{267, Marc [12] D. J. Sakrson. Stocastc approxmaton : A recursve metod for solvng regresson problems. Advances n Comm. Systems 2, pages 51{106, A. V. Balakrsnan, d.