IN synchronous Code Division Multiple Access (CDMA)

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1 1 Optmal Groupng Algorthm for a Group Decson Feedbac Detector n Synchronous CDMA Communcatons J. Luo, K. Pattpat, P. Wllett, G. Levchu Abstract he Group Decson Feedbac (GDF) detector s studed n ths paper. he computatonal complexty of a GDF detector s exponentaln the largest sze of the groups. Gven the maxmum group sze, a groupng algorthm s proposed. It s shown that the proposed groupng algorthm maxmzes the Asymptotc Symmetrc Energy (ASE) of the multuser detecton system. Furthermore, based on a set of lower bounds on Asymptotc Group SymmetrcEnergy (AGSE) of the GDF detector, t s shown that the proposed groupng algorthm, n fact, maxmzes the AGSE lower bound for every group of users. ogether wth a fast computatonal method based on branch-and-bound, the theoretcal analyss of the groupng algorthm enables the o²ne estmaton of the computatonal cost and the performance of GDF detector. Smulaton results on both small and large sze problems are presented to verfy the theoretcal conclusons. All the results n ths paper can be appled to the Decson Feedbac (DF) detector by smply settng the maxmum group sze to 1. Keywords Multuser detecton, decson feedbac, optmzaton methods, code dvson multple access. I. Introducton IN synchronous Code Dvson Multple Access (CDMA) communcaton systems, the near-far problem caused by the nteruser nterference has been wdely studed. Wth the addtve whte Gaussan nose assumpton and when the source sgnal s bnary- or nteger-valued, the conventonal detector does not produce relable decsons for the CDMA channel [?]. he computaton of theoptmal detecton, however, s generally NP-hard and thus s exponental n the number of users [], unless the sgnature wave form correlaton matrx has a specal structure [10] [9]. Several new algorthms have been proposed to provde relable solutons wth relatvely low computatonal cost. Among the sub-optmal algorthm groups, the decson-drven detecton methods, ncludng decson feedbac (DF) [5] [11], group detecton [6], and multstage detecton [3] [4], are popular. Although the DF method s smple and performs well, there are stuatons when a margnal ncrease n computaton can provde sgn cant mprovement n performance [1]. he man drawbac of DF s that detectons are made for one user at a tme; the decson on the strong user s obtaned by treatng the wea users as nose. However, when user chp sequences are correlated, ths nose becomes he Authors are wth the Electrcal and Computer Engneerng Department, Unversty of Connectcut, Storrs, C0669, USA. E- mal:rshna@engr.uconn.edu 1 hs wor was supported by the O±ce of Naval Research under contract N , N , and by NUWC under contract N based, and thus s naturally harmful to the userwse detecton. he dea of sequental group detecton was rst ntroduced by Varanas n [6] and can be vewed as the Group Decson Feedbac (GDF) detector. GDF detector dvdes users nto several groups. he users wth relatvely hgh correlatons are assgned to the same group, and the correlaton between users n d erent groups are relatvely low. Smlar to DF detector, GDF detector maes decsons sequentally based on successve cancelaton. However, nstead of mang decsons on sngle user at a tme, GDF detector maes decsons groupwse,.e., the decsons on users n the same group (the correlated users) are made smultaneously. he computatonal expense for a GDF detector s approxmately exponental n the largest group sze, and ths s expected to be small f the largest group sze s small. In [6], the szes of the groups are desgn parameters. However, n practce, gven a user sgnal set, t s not easy for one to nd the correlated users and assgn them to groups. Snce the largest group sze s closely related to the overall computatonal cost, n ths paper, we consder the largest group sze as the only desgn parameter. A groupng and orderng algorthm s proposed to nd the optmal sze and users for each group. heoretcal results are gven to show the optmalty n terms of the Asymptotc Symmetrc Energy (ASE). ogether wth a fast computatonal method mod ed from [1], the proposed GDF detecton method provdes an e±cent way to mprove the DF detecton wth margnal ncrease n computatonal cost. Smulaton results on small and large sze problems are presented to verfy the theoretcal conclusons. he rest of the paper s organzed as follows. In secton II, we revew the problem model and the theoretcal results on the performance measure gven n [6]. In secton III, gven the largest group sze, a groupng and orderng algorthm s proposed to maxmze the ASE of the system. Proof of optmalty s gven n the appendx. A fast computatonal method s proposed for the GDF detector and a theoretcal upper bound on computatonal cost s derved. Smulaton results on a small example as well as on a system of 100 users are presented n secton IV. Conclusons are provded n secton V. II. Problem Formulaton and Performance Measure of GDF Detector A dscrete-tme equvalent model for the matched- lter outputs at the recever of a CDMA channel s gven by the

2 K -length vector [] y Hb + n (1) where bf 1; +1g K denotes the vector of bts transmtted by thek actve users. Here H s a nonnegatve de nte sgnature waveform correlaton matrx, n s a real-valued zero-mean Gaussan random vector wth a covarance matrx H. It has been shown that ths model holds for both baseband [] and passband [11] channels wth addtve Gaussan nose. When all the user sgnals are equally probable, the optmal soluton of (1) s the output of a Maxmum Lelhood (ML) detector [] Á ML : ^b arg mn b Hb y b () bf 1;+1g K whch s generally NP-hard and exponentally complex to mplement. he sequental group detecton based on the dea of successve cancellaton was rst ntroduced by Varanas n [6]. Suppose users are parttoned nto an ordered set of P groups, G 0 ;:::;G. he number of users n group G s denoted by G, and naturally P 0 G K. he decson on group fg 0 g s made by " ^b G0 arg mn mn b Hb y b (3) b G0 f 1;+1g G 0 b ¹G0 where b G0 denotes the part of vector b that corresponds to users n group G 0, and ¹G 0 denotes the complement of G 0,.e., the unon of G 1 ;:::;G. he decsons of (3) are then used to subtract the multple-access nterference due to users n G 0 from the remanng decson statstcs y ¹G 0. he detector for the next group G 1 s desgned under the assumpton that the multple-access nterference cancelaton s perfect. hs process of nterference cancelaton and group detecton s carred out sequentally for users n groups G ;:::;G, wth the group detector for group G tang advantage of the decsons made by group detectors for G 0 ;:::;G 1. Denote the channel model for the user expurgated channel that only has users n groups G ;:::;G by y () H () b () + n () (4) he decsons on group G can be represented as ^b G arg mn 4mn ³b () 3 () H () b () y () b 5 b () f 1;+1g G G b () (5) In mult-user detecton, the Asymptotc Symmetrc Energy (ASE) s an mportant performance measure. De ne the probablty that not all users are detected correctly as P (;Á), then the ASE for the detector Á [11] s gven by 9 < P (;Á) (Á) sup e 0; lm <1 :!0 Q ; ³p e (6) where s the addtve nose varance (see (1)), and Q(x) R 1 p 1 x ¼ e x dx. he ASE for the optmal detector Á ML s gven by (Á ML ) d mn mn ef 1;0;1g K nf0g e He (7) K where d mn s nown as the mnmum dstance of matrx H [] and \n" s the set subtracton. Smlarly, we can de ne the Asymptotc Group Symmetrc Energy (AGSE) for each user group. For a group detector, de ne the probablty that not all users n group fgg are detected correctly as P G (;Á), and correspondngly we have 9 < P G (;Á) G (Á) sup e 0; lm ³ pe <1 () :!0 Q ; as the AGSE for group fg g. Although an exact performance analyss of GDF detector s ntractable [6], one can obtan upper and lower bounds for the AGSE of all groups. Inhthe above descrpton of the GDF detector, de- ne J () H () 1, and denote J () G G to be the sub-matrx of J () that only contans the columns and rows correspondng to users n G. De ne d G ;mn to be the mnmum dstance of matrx J () G G 1, ³.e., d G ;mn ³ mn e J () ef 1;0;1g G nf0g G G G 1 e (9) hen the AGSE for group G can be bounded by mn(d G 0;mn;:::;d G ;mn) G (Á)d G ;mn (10) A smlar result can be found n [6]. he upper bound n (10) s reached when all decsons on the users n group G 1 through group G 1 are correct. III. Optmal Groupng and Detecton Order for GDF Detector It s nown that the performance of the decson-drven mult-user detector s sgn cantly a ected by the order of the users [?]. Snce the overall computaton for GDF detector s exponental n the maxmum group sze, whch s de ned by G max max(g 0 ;:::;G ), n ths secton, we develop a groupng and orderng algorthm that maxmzes the ASE of the GDF detector gven G max as a desgn parameter. Denote the Cholesy decomposton of H by L L H, where L s a lower trangular matrx. Multply both sdes of (1) by (L 1 ) to obtan the whte nose model [5] (L 1 ) y Lb + (L 1 ) n (11) De ne ~y (L 1 ) y, ~n (L 1 ) n, partton the matrces and the vectors accordng to G 0 and ¹G 0 to obtan ~yg0 LG0 G 0 0 bg0 ~ng0 + (1) ~y ¹G 0 L ¹G 0 ¹G 0 b ¹G 0 ~n ¹G 0 L ¹G 0G 0

3 3 Snce L ¹ G0 ¹ G0 s a full ran matrx by assumpton, the decson on group G 0 n (3) can be wrtten as ^b G0 arg mn LG0 G 0 b G0 ~y G0 b G0 f 1;+1g G 0 (13) herefore, the AGSE of group G 0 s determned by the mnmum dstance of matrx L G 0G 0 L G0G0. Snce H L L, we have L G 0G 0 LG 0 G 0 (H 1 1 h 1 )G 0 G 0 J (0) G 0 G 0 G0 (Á GDFD ) d G 0 ;mn (14) A smlar result can be obtaned for group G. In the descrpton of GDFD n secton II, f we let H () L () L (), ³ then L () G L () G J () G G 1. Snce H () s the southeast sub-dagonal matrx of H, L () s the south-east subdagonal matrx of L and L () G L G. Hence, ³ L G G L G G J () G G 1 (15) he above result shows that d G;mn s determned by the dagonal bloc-matrx L G of L. Now, gven all the decsons on group G 0 to group G 1 are correct, denote the probablty that not all the users n group ³ G are detected dg ;mn correctly by P e (G G 0 ;:::;G 1 )¼ Q. he probablty that not all the K users are detected correctly can be represented as P (;Á)¼1 Y 0 1 Q µ dg;mn herefore, the ASE of GDF detector s gven by (16) (Á GDFD ) mn(d G 0;mn;:::;d G ;mn) (17) SnceG max s gven as a desgn parameter, the problem s then to nd an optmal partton and detecton order that maxmzes mn(d G 0;mn;:::;d G ;mn ). Notce that d erent GDF detectors may have the sameg max but d erent numbers of groups snce P s not a desgn parameter. Groupng and Order Algorthm : Fnd the optmal groupng and detecton order va the followng steps. Step 1: Partton the K users nto two groups fg 0 g and f ¹G 0 g wth G 0 G max. Among these parttons (fg 0 g and G 0 are not xed), select the one that maxmzes d G0;mn (whch s the mnmum dstance of matrx h 1). J (0) G 0G 0 Step : Partton the remanng K G 0 users nto two groups G 1 and G ¹ 1 wth G 1 G max. Among these parttons, select the one that maxmzes d G1 ;mn. Step 3: Contnue ths process untl all the users are assgned to groups. Example 1 : he algorthm s llustrated by the followng 4-user example. Suppose theh matrx s gven by H 4 4:30 1:00 0:60 0:30 1:00 3:00 1:70 0:50 0:60 1:70 :0 0:70 0:30 0:50 0:70 1: (1) Assume that the desred maxmum group sze sg max. In step 1 of the algorthm, the possble choces for group G 0 and the resultng d G 0;mn are shown n able I. he User(s) ,1 d G 0 ;mn User(s) 0, 0,3 1, 1,3,3 d G 0 ;mn ABLE I Dfferent choces of group G 0 and the correspondng d G 0;mn best choce for group G 0 s fuser 0g. hen, for the user expurgated channel, we have " 3:00 1:70 0:50 H (1) 1:70 :0 0:70 (19) 0:50 0:70 1:90 he possble choces for group G 1 and the resultng d G1;mn are shown n able II. We can see that the best choce for User(s) 1 3 1, 1,3,3 d G 1;mn ABLE II Dfferent choces of group G1 and the correspondng d G1 ;mn group G 1 s fuser 1, user g. Naturally fuser 3g wll be the last group. he resultng ASE for ths parttonng and orderng s 1:7. Note that the above example has 4 users andg max. One may thn that parttonng users nto groups wth users n each group s a good choce. However, snce user 0 s a strong user, t has to be detected rst. And snce user 1 and user are serously correlated, they have to be assgned to the same group. If, for example, we assgn two groups as fuser0;user3g and fuser1;userg. As a punshment of detectng the wea user (user 3) rst, we get 1:6 < 1:7. Proposton 1 : he proposed groupng and orderng algorthm maxmzes the ASE n (17). See Appendx for the proof. he proposed groupng and orderng algorthm s also optmal n the followng sense. Proposton : he proposed groupng and orderng algorthm maxmzes the performance lower bound n (10) for every group. In other words, suppose G s the groupng and orderng result obtaned from the proposed algorthm, and G s one of the groups n G. Further suppose there s another group and detecton sequence ^G wth ^G l beng one of the groups n ^G, and ^G l G. hen the followng result holds, mn(d G 1 ;mn;:::;d G ;mn) mn(d ^G 1 ;mn ;:::;d G l ;mn) (0)

4 4 See Appendx for the proof. In addton to the above propostons, we can derve a fast computatonal method for GDF detector, whch s a mod ed verson of the method proposed n [1]. We propose the followng steps for the group detecton. Computatonal Method for GDF Detector: Suppose the GDF detector has P groups, G 0,..., G 1) Intalze ~y (1) (L 1 ) y, L (1) L. Let 1; ) Form the whte nose system model for the userexpurgated channel, and partton the vectors and matrces accordng to group G and ts complement G ¹ as " ~y () G ~y () " L () G G 0 L () G L () ¹ G " b () G b () + " ~n () G ~n () (1) Fnd the decson on group G by LGG ^b G arg mn b G ~y () b G f 1;+1g G G 3) Compute ~y (+1) by () Fg. 1. Performance of varous methods (4 users, Monte- Carlo runs. Á D s the conventonal decorrelator; Á D DF s the decorrelaton-based decson feedbac detector; Á GDFD s the group decson feedbac detector wth G max ; and Á ML s the maxmum lelhood detector.) Let ~y (+1) ~y () L () G ^bg (3) L (+1) L () ¹ G (4) 4) Let + 1. If < P, go to step ; otherwse, stop the computaton. he computatonal cost for step 1 s K(K+1) multplcatons and K(K 1) addtons. Assume the computatonal cost for step can be bounded by \ " M(G ) ; \ + "S(G ) (5) where \ " denotes the number of multplcatons and \+" denotes the number of addtons. In step 3, snce b can only tae nown dscrete values, Lb can be precomputed and stored. hus, only G P +1 G addtons are needed. herefore, the overall computatonal cost s bounded by s assumed to be 3. Fgure shows one of the smulaton results. he respectve computatonal costs for the three detectors are Á D \ " \ + " 9900 Á D DFD \ " 5050 \ + " 9900 Á GDFD \ " 530 \ + " 1000 (7) Ben tng from the optmal groupng and the branch-andbound-based computatonal method, GDFD shows a sgn cant mpreovement on the performance whle the computatonal cost s even less than that of the conventonal decorrelator. Due to the NP-hard nature of the optmal ML detector, the results on optmal detector could not be computed. \ " K(K + 1) \ + " K(K 1) X [M(G )] + 0 X + 4S(G ) +G 0 X +1 3 G 5 (6) IV. Smulaton Results Example 1 - contnued : In the prevous 4-user example, (Á GDFD ) 1:7. he ASE for optmal DDFD and the ML detector can be obtaned from [11] as (Á DDFD ) 1:69 and (Á ML ) 1:. he smulaton results are shown n Fgure 1, whch are consstent wth the theoretcal analyss. Example : Suppose we have 100 users. he sgnature sequences for each user are bnary and of length 115. hey are generated randomly. he maxmum group sze Fg.. Performance of varous methods (100 users, Monte- Carlo runs. Á D s the conventonal decorrelator; Á D DF s the decorrelaton-based decson feedbac detector; Á GDFD s the group decson feedbac detector wthg max 3)

5 5 V. Concluson An optmal groupng and orderng algorthm for Group Decson Feedbac Detector s proposed. ogether wth a fast computatonal method based on the dea of branch and bound, the proposed algorthm provdes a systematc way of mprovng the Decson Feedbac Detector, especally when correlaton exsts among the users. Smulaton results show that GDF detector wth the optmal groupng and orderng algorthm provdes a sgn cant mprovement over DF detector, whle the ncrease n computatonal cost s margnal and even negatve n some cases. he proposed method can be easly extended to nte-alphabet sgnals nstead of bnary ones. Appendx I. Pre-proved Lemmas Before provng the propostons n ths paper, we present the followng three lemmas that wll be used n the proof. Lemma 1: Suppose H L L s parttoned on two arbtrary dagonal elements as H11 H 1 H 1 H L11 0 L11 0 L 1 L L 1 L () For any permutaton matrx P of the same sze as H, f I 0 H11 H 1 I 0 0 P H 1 H 0 P ~L11 0 ~L11 0 ~L 1 L ~ ~L 1 L ~ (9) then the followng results hold. ~L 11 L 11 ; ~ L ~ L PL L P (30) he proof s qute straght forward and s therefore gnored n ths paper. Lemma : Suppose H s a m m symmetrc and postve de nte matrx. Suppose H L L s the Cholesy decomposton. Partton H and L on the last (south-east) dagonal component as H11 h 1 L11 0 L11 0 h 1 h l 1 l l 1 l (31) Now \move up" the last \user" to the rst, denote the acton and the new Cholesy decomposton matrx by 0 1 H11 h 1 0 I I 0 h 1 h 1 0 ~l 11 0 ~l 11 0 ~ l1 L ~ ~ l1 L ~ (3) hen matrx ~ L ~ L L 11 L 11 s non-negatve de nte. Pro of : Substtutng (31) nto (3) yelds ~L ~ L L 11 L 11 l 1 l 1 0 (33) Lemma 3: Suppose L and ~ L are two lower trangular matrces of sze m m, assume that L L L ~ L 0. ~ Partton L on an arbtrary dagonal component, and partton ~L accordngly as L11 0 L ; ~ ~L11 0 L L ~L 1 L ~ (34) We have L 1 L 11 L 11 ~ L 11 ~ L 11 0 ; L L ~ L ~ L 0 (35) Proof : Snce L L ~ L ~ L 0, we can nd a lower trangular matrx C whch sats es Accordng to (34), partton C as C11 0 C L L ~ L I + C C ~L (36) C 1 C Substtute (34)(37) nto (36) to obtan L L ~ L I + C C ~L (37) L 11 L 11 ~ L 11 I + C 11 C 11 ~L11 +4 (3) where 4 s a symmetrc non-negatve de nte matrx. he proof s complete. Note that n Lemma 3, we can contnue parttonng the sub-dagonal bloc matrces, and apply Lemma 3 teratvely to get a result smlar to (35) for an arbtrary partton. II. Proof of Proposton 1 Denote the optmal group and detecton sequence determned by the proposed algorthm as G, whch has groups G 0 ;:::;G. Denote the group decson feedbac detector usng detecton sequence G by Á G GDFD. he dea of the proof can be summarzed as follows. Suppose there s another group and detecton sequence G (), whch has groups G () 0 ;:::;G() P () 1. Wthout loss of generalty, assume (0 < ) G () G (he superscrpt () means that the rst groups n G () are dentcal to the rst groups n G). Now construct a new group and detecton sequence. he groups of are de ned by G () G 0 < G G () 1 n G > (39) o smplfy the notaton, n the above constructon, f NULL, we stll eep group and de ne d (+1) G ;mn 1. Evdently, G(+1) has one more group than G (). he followng result holds for.

6 6 Proposton 3: If s constructed accordng to the above de nton, then (1) (0 < ), d ;mn d G () ;mn. () d ;mn d. G () ;mn (3) ( < P () ), d ;mn d G () ;mn. 1 Proof : (1) For any <, the decson for group G () s made by treatng the sgnal correspondng to G () +1,..., G() P () 1 as nose and mnmzng the probablty of error n ML sense. herefore, any swappng of users wthn groups of ndex larger than wll not a ect the performance of G (). hs result can be formally proved by usng Lemma 1. () Snce G () ( < ), ths result can be drectly obtaned from the de nton of the optmal groupng and orderng algorthm. (3) he proof for ths part s relatvely trcy. In fact, the constructon of from G () can be dvded nto three stages. De ne the users n group G as K 0,..., K G 1. For the convenence of dscusson, we rst consder user K 0. Stage 1 Suppose, n G (), user K 0 belongs to group G () ( ). De ne the the acton \tae out user K 0 from group G () ", whch converts G () to, as, G () < fuserk 0 g G () fuserk 0 g + 1 G () 1 > + 1 (40) Stage Now n, we have fuserk 0 g. De ne the acton \move up user K 0 to follow group 1 ", whch converts to G (S), as follows, G (S) G (S) G (S) G (S) < fuserk 0 g 1 < > (41) Contnue performng the above two stages on all users K 0,..., K G 1. Denote the resultng group and detecton sequence as G (S3). Denote the number of groups n G (S3) by P (S3). Stage 3 In G (S3), combne groups fk G 1g,..., fk 0 g, whch converts G (S3) to, as, G (S3) < fuserk 0 ;:::;K G 1g G (S3) G +1 > (4) In the rst stage, wthout loss of generalty, suppose user K 0 s the rst user n group G (). he \tae out" acton does not change the order of the users, thus the Cholesy decomposton matrx L remans unchanged. hs shows that L (S1) G s the south-east dagonal sub-bloc of +1 G(S1) +1 L () G G(). herefore, d +1 ;mn d G () ;mn (43) In the second stage, snce the \mnmum dstance" of a sub-bloc s the performance measure for the correspondng user group gven all the user groups wth smaller ndces are correctly detected, puttng more users nto the detected user lst wll result n a better performance and a larger \mnmum dstance". In fact, from Lemma and Lemma 3, for any groups G (S) L G (S) herefore, G (S) 1 L (S) G G (S) L, <, we have, L 1 G(S1) (S1) 1 G 1 G(S1) 1 0 (44) d G (S) ;mn d (45) 1 ;mn Hence, n, for any >, d ;mn d G () ;mn, 1 whch proves part (3) of proposton 3. Based on proposton 3, suppose (Á GDFD ) d ;mn. hen, we have (Á GDFD) (Á G () GDFD) (46) By teratvely usng the above constructon procedure n the proof of Proposton 1, we wll nally get G (P) G and (Á G (P) GDFD) (Á G () GDFD) (47) whch completes the proof. III. Proof of Proposton In the above proof for proposton 1, let G () ^G. Construct usng the same procedure. Note that G () l ^G l G, and G \ G NULL. herefore, n, we have l+1 G. Suppose mn(d 0 ;mn ;:::;d ;mn) d. From l+1 ;mn Proposton 3, ² If <, we have d ;mn mn(d ^G 0 ;mn ;:::;d^g ). l ;mn ² If, we have d ;mn mn(d ^G ;:::;d^g ). 0;mn l;mn ² If >, we have d ;mn mn(d ^G ;:::;d^g ). 0;mn l;mn Hence, d G () ;mn d G () ;mn d G () 1 ;mn mn(d 0 ;mn ;:::;d l+1 ;mn) mn(d^g ;:::;d^g ) 0;mn l;mn (4) By teratvely usng the constructon procedure, we wll nally get G (P) G whch sats es mn(d G 0;mn ;:::;d G ;mn ) mn(d^g0;mn ;:::;d^gl;mn ) (49) Hence the proof s complete.

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