Decentralized Power Control for Random Access with Iterative Multi-User Detection

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1 Dcntralzd Powr Control for Random Accss wth Itratv Mult-Usr Dtcton Chongbn Xu, Png Wang, Sammy Chan, and L Png Dpartmnt of Elctronc Engnrng, Cty Unvrsty of Hong ong, Hong ong SAR Emal: xcb5@mals.tsnghua.du.cn, pngwang@ctyu.du.h, schan@ctyu.du.h, lpng@ctyu.du.h Abstract Ths papr s concrnd wth th applcaton of tratv mult-usr dtcton (MUD n ALOHA-typ random accss systms. W focus on a random-powr transmsson schm, n whch th transmsson powr lvl of ach usr s randomly slctd accordng to a crtan dstrbuton f. W am at dsgnng f to maxmz th systm throughput gvn an avrag powr constrant for ach usr. W adopt a suboptmal schm, n whch only typ- collsons (collsons nvolvng two pacts ar rsolvd. W frst prov that, th support (.., th smallst closd st whos complmnt has zro probablty of th optmal f n ths cas s a dscrt st for systms wth crtan fasbl rgons (th st of powr profls that can support rlabl communcatons. Th rlatd dscrt st can b asly obtand accordng to th boundars of th fasbl rgon. W thn apply ths fndng to th dsgn of f. Numrcal rsults show that wth tratv MUD, th proposd schm can achv notcabl prformanc mprovmnt compard wth th convntonal ALOHA and offr flxbl tradoff btwn th systm throughput and powr consumpton. I. INTRODUCTION In a convntonal random accss schm such as ALOHA, pacts nvolvd n a collson ar usually assumd unrcovrabl. Ths s, howvr, a pssmstc assumpton. In many stuatons, t s possbl to rcovr som or all pacts from a collson []-[6]. Ths phnomnon s capturd by th multpl pacts rcpton (MPR modl []. Most xtng wor on MPR modl s focusd on sngl-usr dtcton (SUD [3]-[6]. In ths cas, randomzd powr allocaton has bn studd to mprov th MPR capablty of th systm, n whch ach usr randomly slcts a transmsson powr lvl accordng to a crtan dstrbuton dnotd by f [5]-[6]. It s conjcturd n [5] that th optmal powr lvl dstrbuton may b of a dscrt natur. Howvr, no rgorous proof s avalabl so far. Itratv dtcton tchnqus [7]-[] hav th potntal to provd furthr prformanc mprovmnt. It has bn shown that propr powr allocaton s crucal for cntralzd systms wth tratv mult-usr dtcton (MUD []. In ths papr, w study a dcntralzd ALOHA-typ random accss schm wth tratv MUD. W adopt th random-powr transmsson schm mntond abov and am at mprovng th systm throughput by dsgnng th dstrbuton f gvn an avrag powr constrant for ach usr. To rduc complxty, w adopt a suboptmal schm, n whch only typ- collsons (collsons nvolvng two pacts ar rsolvd. W frst prov that th support (.., th smallst closd st whos complmnt has zro probablty of th optmal f n ths cas s a dscrt st for systms wth crtan fasbl rgons (th st of powr profls that support rlabl communcatons. W thn apply ths fndng to th dsgn of f. Numrcal rsults dmonstrat that wth tratv MUD, th proposd schm can achv notcabl prformanc mprovmnt ovr th convntonal ALOHA and offr flxbl tradoff btwn systm throughput and powr consumpton. Th proposd schm provds an attractv opton for hgh throughput transmsson n nvronmnts wthout cntralzd control (such as a cogntv rado systm whr svral scondary usrs opportunstcally accss th channl whn th prmary usr s dl. II. PRELIMINARIES A. Itratv Mult-Usr Dtcton To facltat our dscussons blow, w frst brfly rvw th basc prncpl of tratv MUD. Ta a -usr ntrlav dvson multpl-accss (IDMA systm ovr an addtv wht Gaussan nos (AWGN channl for xampl. Th rcvd sgnal r s gvn by r = x + n. ( = In (, x s th transmttd sgnal of usr ( =,,, wth unt powr, th corrspondng transmsson powr, and n th complx AWGN sampl wth man and varanc N. ˆd dˆ π π / / Fg. Illustraton of a codd IDMA systm wth tratv MUD. At th rcvr, th adoptd tratv MUD conssts of two man moduls, as shown n Fg. []-[3]. Th frst modul, lmntary sgnal stmator (ESE, s usd to stmat th transmttd sgnals {x } usng th rcvd sgnal r and th a pror nformaton of {x } basd on th lnar mnmum man squar rror (LMMSE prncpl. Th scond modul, dcodr (DEC, s usd for a postror probablty (APP dcodng basd on th outputs of th ESE. Th outputs of th DECs provd rfnd stmaton of {x } wth whch th a pror nformaton of {x } s updatd. Thn th ESE prforms LMMSE dtcton agan. Ths procss oprats tratvly untl convrgnc. B. Random-Powr Transmsson In ths papr, w focus on random accss systms and ma th followng assumptons. ( Th transmsson rat of ach usr s fxd at R f t has a pact to transmt.

2 ( Each usr has no nowldg of th transmsson stat of th othrs. W adopt th random-powr transmsson schm mntond n Scton I. W randomly draw th transmsson powr of ach usr accordng to a crtan dstrbuton f. Our objctv s to dsgn f so as to maxmz th systm throughput. C. Suboptmal Soluton For a random accss systm wth usrs, w rfr to a collson nvolvng ( pacts as a typ- collson. Whn s larg, th optmzaton of th dstrbuton f s vry complcatd. To smplfy th problm, w adopt a suboptmal schm as follows. Whn collsons occur at th rcvr, pacts nvolvd n typ- collsons ar dcodd whl thos nvolvd n othr typs of collsons ar smply dscardd. Whn th systm arrval rat λ s low, typ- collsons domnat th ovrall prformanc. Rsolvng typ- collsons can provd notcabl prformanc mprovmnt. W wll solv th problm through two stps: w frst fnd th support of th optmal f n Scton III and thn dsgn f basd on ths support n Scton IV. A. Fasbl Rgon powr of usr ( E, E III. -USER CASE powr of usr Fg.. An xampl of th fasbl rgon for a -usr systm. A Lt us start wth two usrs. (W wll consdr mor usrs n Scton IV. Sttng = n (, w hav r = = S x + n. ( In th systm n (, dffrnt powr pars (, may lad to dffrnt bt rror rat (BER prformanc. W say that a pact s rlably transmttd f ts BER s no hghr than -5 aftr tratv dtcton. Thn w rfr to a fasbl rgon, dnotd by S, as th st of th powr pars that can support rlabl communcatons for both usrs,.., (, BER (, 5 S =, BER (, 5 { } whr BER (, s th achvabl BER of usr ( = or gvn th powr par (,. Gvn a forward rror corrcton (FEC cod, th fasbl rgon of th systm n ( can b obtand by smulatng th BER prformanc of th systm ovr all possbl powr pars. Fg. shows th fasbl rgon (s [4] for a smlar xampl of th systm n ( wth a codword-lngth 4 (3, 6 rgular LDPC cod [5]-[6] followd by PS and tratv LMMSE rcvr. It can b sn that th fasbl rgon s boundd by four curvs, = φ(, and = φ( whr th functon φ( s vsualzd n Fg.. Ths boundary curvs wll b usful n th dscussons n Sctons III-C and III-D. Intutvly, th shap of th fasbl rgon n Fg. can b xpland as follows. Notc th ndntd ara boundd by = φ( and = φ( n Fg. (mard by A. Abov or blow th boundary, tratv dtcton can convrg n two stuatons. Frst, th sgnals from two usrs hav a suffcnt powr dffrnc, so th sgnal from on usr s rcovrd frst and ts ntrfrnc to th othr usr s gradually canclld. Thn th sgnal of th lattr can also b rcovrd. Scond, th sgnals from both usrs ar suffcntly hgh and so both sgnals can b rcovrd gradually at th sam spd, whch happns nar th qual powr ln abov th star marr n Fg.. B. Comparson of Two Dstrbuton Pars Lt and b randomly drawn from two probablty dnsty functons (PDFs f and f, rspctvly. Th avrag systm sum powr s gvn by E ( f, f = f ( d + f ( d. (3 sum For th systm n (, w say that a powr par (, succds f t falls n th fasbl rgon S. Th jont succss probablty of a dstrbuton par (f, f s thn dfnd as P ( f, f f ( f ( d d. (4 = S JS (, Not that P JS (f, f n (4 s th probablty of th vnt that th transmssons for both usrs ar succssful. Whn th systm throughput s concrnd, th succss probablty of ach ndvdual usr should also b consdrd. W wll rturn to ths ssu n Scton III-E. Dfnton blow s a crtron to compar two PDF pars (f *, f * and (f, f. Dfnton : W say that (f *, f * s bttr than or quvalnt to (f, f, dnotd by (f *, f * b.. (f, f, f th nqualts * * * * Esum ( f, f Esum ( f, f and PJS ( f, f PJS ( f, f hold smultanously. Furthrmor, w say that (f *, f * s bttr than (f, f, dnotd by (f *, f * b. (f, f, f at last on nqualty abov s rplacd by strct nqualty. It can b vrfd that th rlaton b.. s transtv,.., f (f **, f ** b.. (f *, f * and (f *, f * b.. (f, f, thn (f **, f ** b.. (f, f. If w furthr hav (f **, f ** b. (f *, f * or (f *, f * b. (f, f, thn (f **, f ** b. (f, f. Exampl : Consdr f ( = δ ( E and f ( = δ ( E, whr E and E s mard n Fg., and δ ( s th Drac dlta functon. Ths transmsson schm ffctvly rducs to a dtrmnstc on wth powr par (E, E mard n Fg.. Th rlatd sum powr E +E s th mnmum nformaton thortc valu that guarants jont succss probablty. Thrfor, (f (, f ( b.. (f, f for all (f, f wth jont succss probablty. ( ( Not that n Exampl, f f and so a cntralzd mchansm s ssntal to proprly assgn f ( and f ( btwn th two usrs. For random accss systms consdrd n ths papr, thr s no cntralzd control. Ths mans that th dstrbuton functons adopt by th two usrs should b th sam,.., f = f = f. Dfnton : W say that a dstrbuton par (f, f s dcntralzd f f = f = f.

3 Exampl : Consdr f ( = f ( = f ( =.5δ( +.5δ ( E, whr E s gvn n Exampl. Each usr slcts powr lvl or E wth qual probablts. Ths producs four powr pars,.., (,, (, E, (E,, and (E, E, ach wth probablty.5. Th succss vnts ar rlatd to (E, for usr, and (, E for usr. Exampl can b vwd as th convntonal ALOHA (S Scton IV. Its prformanc can b mprovd by optmzng th rlatd probablty profl. Such a schm s gnrally suboptmal snc th powr dstrbuton s optmzd ovr two lvls and E only. Th focus of ths papr s to drv mprovd powr dstrbutons wth a mor gnral form. Dfnton 3: W say that f * s optmal f thr s no f such that (f, f b. (f *, f *. C. Support wth rspct to Jont Succss Probablty As shown n Fg., a typcal fasbl rgon s boundd by four curvs, = φ(, and = φ(. Assumng that th boundary functon φ( s monotoncally ncrasng (th non-monotoncally ncrasng cas wll b dscussd n Scton III-D, w can dfn a dscrt st E = {E } as, = ; E = (5 φ ( E, >. Mor spcfcally, E s th mnmum powr lvl that guarants succssful dcodng of on usr whn th ntrfrnc powr from th othr usr s E -. Partcularly, E s th mnmum powr for th succssful transmsson of on usr whn th othr usr s n slnc. Not that E s dpndnt on th adoptd codng schm of ach usr. So s th whol dscrt st {E } consquntly. Th dtald powr lvls n ths dscrt st wll b dffrnt onc th codng schm changs. Fg. 3 shows an xampl. Not that for th fasbl rgon n Fg. 3, th boundary functon φ( s monotoncally ncrasng, but t s not th cas for that of th fasbl rgon n Fg.. W wll rturn to ths ssu n Scton III-D. E 5 E 4 E 3 E E ( E, E 3 pont = ( E3, E E E 3 Fg. 3. Illustraton of th powr lvls dfnd n (5. E E4 E5 In Fg. 3, th two boundary functons = φ( and = φ( ntrsct wth ach othr at a crtan pont. W dnot ths pont by = (E, E wth E bng th soluton to quaton = φ(. Th transmsson of on usr wth powr E s always succssful, rgardlss of th powr lvl of th othr. Ths ndcats that rducng a powr lvl > E to E dos not affct P JS. Thus w hav th followng. Rmar : For a systm wth monotoncally ncrasng boundary functon φ(, th support of th optmal f s confnd wthn [, E ], whr E s th soluton to quaton = φ(. It can b vrfd that th powr lvls E = {E } dfnd n (5 has th followng proprty: E [, E, and lm E. (6 Hnc E [, E. Nxt w furthr prov that th support of th optmal f n th rang [, E s confnd wthn E for th systms wth monotoncally ncrasng boundary functon φ(. For a dstrbuton f, w dfn a nw dstrbuton f [] constructd as follows. [ ] δ ( E, ( τ < E f = (7 f ( E. whrτ = f ( d, =,,,. It can b vrfd E < E+ that f [] = f. Rmar : (f [+], f [] b.. (f [], f [],. (8 Proof: From (7, a sampl of f [+] can b quvalntly obtand through th followng stps: Stp : Draw a powr valu accordng to f [] ; Stp : If E < < E +, rduc to E ; othrws, p unchangd. [3] f ( E 5 E 4 E 3 E E ( E, E Fg. 4. Illustraton of Rmar. A A A E E E3 E4 E5 f f A [3] [4] ( ( Clarly, w hav E sum (f [+], f [] E sum (f [], f [] from Stp. In what follows, w wll show that th powr chang n Stp dos not dcras P JS. Lt us focus on th mpact of powr chang from (, to (E, whn E < < E +. Ths can b dscussd cas by cas blow. In Fg. 4, wht crcls {A } rprsnt powr pars drawn from (f [], f [] whl blac crcls {A } rprsnt thos aftr th powr chang n Stp, whch ar also sampls drawn from (f [+], f []. From Fg. 4, t can b sn that ths powr chang lads to th followng thr possblts. a A fals whl A succds. Such vnts lad to hghr P JS ; b A succds whl A fals. Such vnts cannot happn as f [] ( =, (E -, E ; c In all th othr stuatons, both powr pars fal or succd smultanously and so P JS rmans unchangd. Hnc, P JS (f [+], f [] P JS (f [], f []. Ths lads to (8. Followng a smlar dscusson as that n Rmar, w can furthr show (f [+], f [+] b.. (f [+], f []. From th transtv proprty of th rlaton b.., w hav th followng.

4 Rmar 3: (f [+], f [+] b.. (f [], f []. (9 Basd on Rmars and 3, w can obtan th thorm blow. Thorm : Th support of th optmal powr dstrbuton f for a -usr systm wth monotoncally ncrasng boundary functon φ( s a dscrt subst of {E, E,, E,, E }. Proof: W only nd to show that th optmal dstrbuton f tas zro valu n (E, E +,. Ths can b sn by rducton to absurdty as follows. Assum that th optmal powr dstrbuton f tas non-zro valus n ntrvals (E, E + for som ntgrs. Dnot by * th mnmum ntgr for such ntrvals. W can always gnrat a nw dstrbuton f * wth rducd powr by mrgng th probablts of powr lvls n [E *, E *+ to that of E *. Accordng to Rmar 3, (f *, f * b.. (f, f. Furthrmor, E sum (f *, f * < E sum (f, f, and so (f *, f * b. (f, f. Hnc f s not optmal. Th dscrt natur of th optmal dstrbuton support shown n Thorm gratly smplfs th dstrbuton dsgn problm, as wll b dtald n Scton IV. D. Tratmnt for Non-monotoncally Boundary Functon powr of usr nnr bound powr of usr Fg. 5. Approxmaton of tratv MUD fasbl rgon. nnr bound Th dscusson n Scton III-C s basd on th assumpton of a monotoncally ncrasng boundary functon φ(. Ths may not b th cas for a practcally codd systm such as th on n Fg.. Th boundary functon n Fg. s mostly ncrasng, xcpt for a small scton, as shown wth th nlargd vw n Fg. 5. To ovrcom ths problm, w ntroduc an nnr bound and an outr bound, as shown n Fg. 5. Th boundary functon bcoms monotoncally ncrasng f t s rplacd partally by th nnr or. Thn w can apply Thorm. Followng (5, w obtan th dscrt powr lvls for th nnr bound as [E, E,, E 5 ] = [,.43, 3., 4.8, 6.6, 6.8], (a and thos for th as [E, E,, E 6 ] = [,.43, 3., 4.8, 6.6, 6.8, 6.88]. (b Latr w wll show that th prformanc curvs obtand from th two bounds almost ovrlap, ndcatng that thy provd good stmats of th tru prformanc. E. Throughput Optmzaton Th dscussons abov ar basd on th jont succss probablty P JS dfnd n (4, whch s wth rspct to th rlabl transmssons of both usrs. Whn th systm throughput s concrnd, th rlabl transmsson of only on usr should also b consdrd. It can b vrfd that Rmar 3 stll holds n ths cas. So dos Thorm. W conjctur that th support of th optmal powr dstrbuton for systms wth mor than two usrs s stll a dscrt st. Howvr, th rlatd analyss s vry complcatd du to th hgh-dmnsonal fasbl rgon nvolvd and w hav no rgorous proof so far. F. Fadng Channls W hav lmtd our focus on Gaussan channls n th abov dscusson. For fadng channls, dffrnt tratmnts can b dvlopd basd on dffrnt assumptons. Th outcoms of ths papr can b drctly xtndd to th followng cass. If ach usr has prfct channl nowldg of both usrs but no nowldg about th transmsson stat of ach othr, th support of th optmal transmsson powr dstrbuton s stll dscrt. (Ths can b shown n a smlar way as Thorm ; If ach usr has prfct nowldg of ts own channl and no nowldg about th channl dstrbuton as wll as th transmsson stat of th othr, channl rvrs s th optmal stratgy. Th corrspondng optmal transmsson powr dstrbuton s th sam as that drvd n ths papr apart from a scalng factor. Howvr, f ach usr prfctly nows ts own channl and th statstcal channl dstrbuton of both usrs, thn th optmal transmsson powr dstrbuton may not b dscrt any mor. Ths ssu s mor complcatd and s undr nvstgaton. IV. -USER CASE W now consdr th dsgn of f n an ALOHA-typ random accss systm wth usrs basd on th support gvn n Thorm. For convnnc, w assum that th pacts of all usrs arrv ndpndntly, followng th Brnoull procss wth paramtr λ. W also assum that a pact s droppd f th transmsson fals,.., all pacts ar rgardd as nw arrvals. As mntond prvously, w assum that collsons othr than typ- ar un-rsolvabl. A. Throughput Optmzaton Dnot by p th probablty of transmsson powr tang valu E and by p th probablty of transmsson powr tang valu E. Th systm throughput s now gvn by T = T + T. (a In (a, T s th throughput rlatd to transmssons wthout collsons: T = C ( ( λ λ C p p = (b = ( p λ( λ + λ p whr C λ ( λ s th probablty of usrs among total usrs havng pacts to transmt and C ( p p s th probablty of only on usr among ths usrs transmttng wth non-zro powr. Also n (a, T s th throughout rlatd to typ- collsons: λ λ ( ( = ( ( = > T C C p p p (c = ( p p ( λ ( λ + λ p > whr ( p p s th probablty that both pacts > nvolvd n typ- collsons ar rcovrabl. Th throughput T n ( s n gnral non-convx wth rspct to {p }. Howvr, for a gvn p, maxmzng T s quvalnt to

5 mnmzng p, whch s convx wth rspct to {p } [7]. Hnc th throughput maxmzaton problm can b solvd through th followng two stps. Frst w solv th followng convx optmzaton problm for any gvn p. mnmz p (a subjct to p + p = (b E p + E p (c p, and p (d whr ē s th avrag powr constrant of ach usr. Thn th optmal p s obtand by full sarch. B. Numrcal Rsults Now w prsnt som numrcal rsults for th proposd schm. W consdr an IDMA systm adoptng a (3, 6 rgular LDPC cod as mntond n Scton II-A. Th rsultant fasbl rgon has bn gvn n Fgs. and 5. For rfrnc, th prformanc of th convntonal slottd ALOHA s also ncludd, whch can b rgardd as a spcal cas of th schms dscussd n ths papr whr th powr dstrbuton s optmzd only ovr two powr lvls E and E. (Ths mans p = and p = for and so p = p. For a -usr slottd ALOHA systm wth transmsson probablty p, th systm pact throughput s T = C ( λ λ C p ( p = (3 = p λ( λ p. It can b vrfd that th optmal transmsson probablty p = mn{,/(λ}, and th avrag powr consumpton s ē ALOHA (λ mn{, /(λ}. (4 pact throughput T = 3 db =.5 db = db = -.7 db arrval rat of ach usr λ Fg. 6. Throughput of th proposd schm n a -usr LDPC codd systm. nnr bound wthout MUD For a gvn pact arrval rat λ, w wrt th powr constrant of ach usr, ē(λ, as ( λ = ( λ ( λ (5 ALOHA whr ē ALOHA (λ n (4 s usd as a rfrnc and (λ s a constant. Fg. 6 compars th prformanc of systms wth and wthout tratv MUD. At (λ = db, t can b sn that combnd wth tratv MUD, th proposd schm obtans a systm throughput mprovmnt of about 5% or a powr savng of about.7 db compard wth convntonal ALOHA whn λ. (.., λ. It s also sn that th curvs for th nnr and outr bounds almost concd. Ths justfs th approxmaton usd. Nxt, w chang th powr constrant n (5 and r-optmz f. Incrasng th powr constrant mpls a hghr succssful transmsson probablty and so an mprovd systm throughput as shown n Fg. 6. Ths mans that th proposd schm offrs flxbl tradoff btwn th systm throughput and powr consumpton wthout changng modulaton or codng format, whch may b attractv from mplmntaton pont of vw. V. CONCLUSIONS In ths papr, w dvlopd a dcntralzd powr allocaton schm for random accss systms wth tratv MUD. W provd that th support of th optmal powr dstrbuton f undr th monotoncty assumpton s dscrt. Basd on ths fndng, w dsgnd f. Numrcal rsults dmonstrat that sgnfcant prformanc gan can b obtand by th proposd schm. W hav lmtd our focus to ALOHA-typ random accss schms n ths papr. It s xpctd that th rsults can b xtndd to systms wth mor sophstcatd random accss protocols such as CSMA. ACNOWLEDGMENT Ths wor has bn prformd n th framwor of th ICT projcts ICT-733 WHERE and ICT WHERE whch ar partly fundd by th Europan Unon. REFERENCES [] J. J. Mtznr, On mprovng utlzaton n ALOHA ntwors, IEEE Trans. Commun., vol. 4, no. 4, pp , Apr [] S. Ghz, S. Vrdu, and S. Schwartz, Stablty proprts of slottd ALOHA wth multpact rcpton capablty, IEEE Trans. Autom. Control, vol. 33, no. 7, pp , Jul [3] L. Tong and V. Nawar, Sgnal procssng n random accss, IEEE Sgnal Procssng Mag., vol., no. 5, pp Sp. 4. [4] J. Luo and A. Ephrmds, Powr lvls and pact lngths n random multpl accss wth multpl-pact rcpton capablty, IEEE Trans. Inf. Thory, vol. 5, no. pp. 44-4, Fb. 6. [5] R. O. LaMar, A. rshna, and M. Zorz, On th randomzaton of transmttr powr lvls to ncras throughput n multpl accss rado systms, Wrlss Ntwors, vol. 4, no. 4, pp , Mar [6] Y. Lung, Man powr consumpton of artfcal powr captur n wrlss ntwors, IEEE Trans. Commun., vol 45, no. 8, pp , Aug [7] C. Brrou, A. Glavux, and P. Thtmajshma, Nar shannon lmt rrorcorrctng codng and dcodng: Turbo-cods, n Proc. Int. Conf. Communcatons, Gnva, Swtzrland, pp. 64-7, May 993. [8] S. Moshav, Mult-Usr Dtcton for DS-CDMA Communcatons, IEEE Commun. Mag., vol. 34, no., pp. 4-36, Oct [9] S. Vrd u, Multusr Dtcton, Cambrdg, U..: Cambrdg Unv. Prss, 998. [] D. Ts and P. Vswanath, Fundamntals of Wrlss Communcatons, Cambrdg: Cambrdg Unvrsty Prss, 5. [] L. Lu, J. Tong, and L Png, Analyss and Optmzaton of CDMA Systms wth Chp-Lvl Intrlavrs, IEEE J. Slct. Aras Commun. vol. 4, no., pp. 4-5, Jan. 6. [] X. Wang and H. V. Poor, Itratv (turbo soft ntrfrnc cancllaton and dcodng for codd CDMA, IEEE Trans. Commun., vol. 47, no. 7, pp. 46-6, Jul [3] L Png, L. Lu,. Wu, t al, Intrlav dvson multpl-accss, IEEE Trans. Wrlss Commun., vol. 5, no. 4, pp , Apr. 6. [4] A. Ydla, H. Pfstr, and. Narayanan, Unvrsalty for th nosy Slpan-Wolf problm va spatal couplng, n Proc. IEEE Int. Symp. Inform. Thory, St. Ptrsburg, Russa, pp , Jul.. [5] R. G. Gallagr, Low-Dnsty Party-Chc Cods. Cambrdg, MA: MIT Prss, 963. [6] S. Y. Chung, T. J. Rchardson, and R. Urban, Analyss of sum product dcodng of low-dnsty party-chc cods usng a Gaussan approxmaton, IEEE Trans. Inf. Thory, vol. 47, no., pp , Fb.. [7] S. Boyd and L. Vandnbrgh, Convx Optmzaton, Cambrdg: Cambrdg Unvrsty Prss, 4.