Low-Complexity Factor Graph Receivers for Spectrally Efficient MIMO-IDMA

Transcription

1 Low-Compexty Factor Graph Recevers for Spectray Effcent MIMO-IDMA Cemens Nova, Franz Hawatsch, and Gerad Matz Insttute of Communcatons and Rado-Frequency Engneerng, Venna Unversty of Technoogy Gusshausstrasse 25/389, A-1040 Venna, Austra e-ma: {cemens.nova, franz.hawatsch, Abstract Intereave-dvson mutpe access IDMA has recenty been ntroduced as an attractve aternatve to CDMA. IDMA empoys user-specfc ntereavers combned wth ow-rate channe codng for user separaton. In ths paper, we consder a MIMO-IDMA system wth ncreased spectra effcency due to the use of hgher-order symbo consteatons. Based on a factor graph framewor and the sum-product agorthm, we deveop an teratve turbo mutuser recever. Gaussan approxmatons for certan messages propagated through the factor graph ead to a compexty that scaes ony neary wth the number of users. To further reduce compexty, we ntroduce a seectve message update scheme. Numerca smuatons demonstrate the performance of the proposed recever agorthms. I. INTRODUCTION Code-dvson mutpe access CDMA s wdey used n mutuser communcatons due to ts many attractve propertes [1], [2]. Recenty, ntereave-dvson mutpe access IDMA has been proposed as an aternatve to CDMA [3]. Wth IDMA, user separaton s obtaned va user-specfc ntereavers combned wth ow-rate channe codng. Le CDMA, IDMA offers dversty aganst fadng and aows a mtgaton of nter-ce nterference [3]. However, IDMA has some mportant advantages over CDMA: t aows the use of mutuser detectors that are sgnfcanty ess compex than those requred for CDMA; t can outperform coded CDMA when teratve turbo recevers are used [3]; and t can be ntegrated nto a mutpe-nput mutpe-output MIMO system more easy than CDMA [4]. Most IDMA systems proposed so far use BPSK moduaton to avod excessve recever compexty. An excepton s [5], where resuts for a SISO-IDMA system wth QPSK are provded. In ths paper, we consder a MIMO-IDMA system that empoys hgher-order moduaton for ncreased spectra effcency. We use a factor graph framewor [6], [7] to deveop a correspondng teratve recever. Straghtforward appcaton of the sum-product agorthm [6] woud resut n a compexty that s exponenta n the number of users. Based on a Gaussan approxmaton for certan messages propagated through the factor graph, we derve an effcent mutuser detector whose compexty s ony near n the number of users. To further reduce compexty, we propose a varant of the sum-product agorthm wth seectve message updates. Ths resuts n a Ths wor was supported by the STREP project MASCOT IST wthn the Sxth Framewor of the European Commson. b m Fg. 1. channe encoder π m c m x m [n] N n=1 Boc dagram of the MIMO-IDMA transmtter for the mth user. mutuser detector that does not update messages correspondng to reabe bts whose og-ehood rato LLR magntude s above a threshod. Wth ths recever, compexty can easy be traded aganst performance by adjustng the LLR threshod. Ths paper s organzed as foows. In Secton II, we descrbe the MIMO-IDMA system. The factor graph and the messages on whch the teratve recever s based are derved n Secton III. In Secton IV, we deveop the ow-compexty mutuser detector and the seectve message update. Fnay, smuaton resuts demonstratng the performance of the proposed recever agorthms are presented n Secton V. II. MIMO-IDMA SYSTEM We consder an upn mutpe-access scenaro wth M users, each of whch empoys transmt antennas for spata mutpexng [8]. The base staton has M R receve antennas. Assumng fat fadng, and consderng the equvaent compex baseband after symbo-rate sampng, the ength-m R receve vector at symbo tme n s gven by M r[n] = H m [n] x m [n] +w[n] 1 = m=1 M m=1 =1 h m [n] x m [n] +w[n], n =1,...,N. Here, x m [n] = x m 1 [n] x m [n] T s the transmt vector of the mth user, H m [n] = h m 1 [n] h m [n] s the M R MIMO channe matrx from the mth user to the base staton, w[n] = w 1 [n] w MR [n] T N0,σ 2 I s..d. compex Gaussan nose, and N s the number of symbos per boc. A MIMO-IDMA transmtter s shown n Fg. 1; t extends the BPSK-based MIMO-IDMA transmtter of [4] to hgherorder moduaton aphabets. The ength-k sequence of nformaton bts of the mth user, b m = b m 1 b m T, K /08/$ IEEE

2 s encoded nto a ength-l sequence of code bts, wth rate R = K/L < 1. The code s a concatenaton of a termnated convoutona code and a ow-rate repetton code. The code bt sequence s passed through a user-specfc ntereaver π m, yedng the bt sequence c m = c m 1 c m T L = C m b m. Here, the one-to-one functon C m denotes the combned effect of channe codng and ntereavng. Dfferent users empoy dentca codes but dfferent ntereavers. The repetton code together wth the user-specfc ntereaver repaces the spreadng empoyed n CDMA systems. The compex transmt symbo x m [n] on the th antenna of the mth user at tme n s obtaned by mappng a group of B successve ntereaved bts c m [n] = m T, [ c n,+1 n,+b cm wth n, = n 1MT + 1 ] B, to a symbo from an aphabet S of sze S =2 B. Ths w be denoted as x m [n] = c m [n] S wth the one-to-one symbo mappng. We w refer to c m [n] as the symbo abe assocated to x m [n]. The transmt symbo vector of the mth user at tme n w be smary wrtten as x m [n] = c m [n] where c m [n] = c mt 1 [n] c mt [n] T.Note that the number N of symbo vectors per user and the number L of code bts are reated as N = L/ B. III. FACTOR GRAPH AND MESSAGES We next derve a factor graph for an teratve MIMO- IDMA recever. Ths w be used n Secton IV as a bass for deveopng a ow-compexty MIMO mutuser detector. A. Dervaton of the Factor Graph The proposed MIMO-IDMA recever s based on the optma maxmum a posteror bt detector [1], [9] ˆbm = arg max pb m b m {0,1} r. 2 Here, b m s the th nformaton bt of the mth user, r = r T [1] r T [N] T s the receved vector sequence cf. 1, and pb m r denotes the condtona probabty of b m gven r. In what foows, et b = b 1T b MT T and c = c 1T c MT T denote the vectors contanng a nformaton bts and code bts, respectvey; furthermore, et X = X[1] X[N] wth X[n] = x 1 [n] x M [n] be the NM matrx of a transmt vectors x m [n], n =1,...,N, m =1,...,M. Note that there s a one-toone correspondence between b, c, and X. To compute pb m r n 2, we frst wrte t as a margna of pb r and appy Bayes rue assumng aprorequay ey nformaton bt sequences b: pb m r = pb r pr b, 3 b m b m where denotes summaton wth respect to a components of b except b m b m, pr b s the condtona probabty densty functon of r gven b, and denotes equaty up to factors rreevant to the maxmzaton n 2. Wth pr b = X,c pr, X, c b = X,c pr XpX cpc b, we can wrte 3 as pb m r b m pr X px c pc b, 4 where from now on denotes summaton wth respect b m to a unnown varabes except b m n the present case X, c, and a components of b except b m. Note that pr X corresponds to the channe cf. 1, px c descrbes the moduator symbo mappngs x m [n] = c m [n], and pc b represents the channe encoder and ntereaver oneto-one correspondences c m = C m b m. There s pc b =1 f c m = C m b m for a m and pc b =0otherwse. Usng the ndcator functon I{ }, whch equas 1 f ts argument s true and 0 otherwse, we thus have pc b = M I { c m = C m b m }. 5 m=1 We note that the code constrant I { c m = C m b m } can be expressed n a more detaed manner by usng the code structure n partcuar, a tres/state representaton for the convoutona code [6], [7], [9]. A smar reasonng yeds M px c = I { x m [n] =c m [n] } = n=1 m=1 M n=1 m=1 =1 I { x m [n] =c m [n] }. 6 Fnay, because the receve vectors r[n] are condtonay ndependent gven the transmt vectors x m [n] cf. 1, pr X = pr[n] X[n]. 7 n=1 Here, pr[n] X[n] s compex Gaussan wth mean M m=1 Hm [n] x m [n] = M MT m=1 =1 hm [n] x m [n] and covarance matrx σ 2 I. Insertng the expressons 5 7 nto 4, we obtan the overa factorzaton pb m r b m M pr[n] X[n] I { c m = C m b m } n=1 m =1 =1 I { x m [n] =c m [n] }, whch can be represented by the factor graph [6], [7], [9] shown n Fg. 2. There are factor nodes for the channe, symbo mapper constrants, and code constrants, and varabe nodes for the transmt symbos, code bts, and nformaton bts. B. Sum-Product Agorthm and Messages For a factor graph wthout cyces, margnas e 4 and the assocated bt decsons 2 can be determned exacty and effcenty usng the sum-product agorthm [6]. For a factor

3 b 1 1 b 1 2 b 1 K I { c 1 = C 1 b 1 } π 1 I { c M = C M b M } 1 n,1+1 1 µ 4 c n,1+1 c 1 1 [1] c1 [1] MT cm 1 [1] c M [1] MT c 1 1 [N] c1 c M 1 [N] c M x 1 1 [1] x1 [1] MT 1 µ 1 x 1 [1] xm 1 [1] x M [1] MT x 1 1 [N] x1 1 µ 2 x 1 [1] p r[1] X[1] p r[n] X[N] 1 [N] x M [N] MT xm Fg. 2. Factor graph for a MIMO-IDMA system wth convoutona encodng and hgher-order moduaton. graph wth cyces as n Fg. 2, the sum-product agorthm can st be used but t generay becomes teratve, yeds ony approxmate resuts, and requres sutabe message schedung. In what foows, we cacuate the messages to be propagated aong the edges of our factor graph accordng to the update rues of the sum-product agorthm [6]. Because the code bt varabe nodes c m n,+j and the transmt symbo varabe nodes [n] are connected to ony two neghborng factor nodes, x m they just pass the messages from one neghborng factor node to the other. Thus, we ony need to consder the message updates for the factor nodes. For the code factor nodes I { c m = C m b m } n Fg. 2, the sum-product agorthm amounts to the BCJR agorthm for soft-decodng the convoutona code [6], [10], whe the repetton code s soft-decoded by summng the aprorllrs of successve bts after ntereavng. The LLRs produced by the overa soft channe decoder are the sum of extrnsc LLRs and pror LLRs [2]. The extrnsc LLRs, denoted by ξ m, correspond to messages beefs [6], [7], [9] m = expξm c m 1 + expξ m, cm {0, 1} 8 that eave the code factor node I { c m = C m b m } and are propagated to the code varabe nodes c m [n] and further to the moduator factor node. Agan usng the sum-product agorthm, the message m µ 1 x [n] passed from the moduator factor node to the varabe node x m [n] and further to the channe factor node pr[n] X[n] s obtaned from the messages m as m µ 1 x [n] = I { x m [n]=c m [n] } m [n] c m [n] = µ 3 1 x m [n], 9 where denotes summaton over a the 2 B symbo c m [n] m abes and [n] = B j=1 µ m 3 c n,+j. m The message µ 2 x [n] passed from the channe factor node pr[n] X[n] to the varabe node x m [n] and further to the moduator factor node s obtaned as µ 2 x m [n] = x m [n] pr[n] X[n],m,m m µ 1 x [n], 10 where denotes summaton wth respect to a entres x m [n] of X[n] except x m [n]. Fnay, the message µ 4 c m passed from the moduator factor node to the code varabe node c m [n] and further to the code factor node I { c m = C m b m } s obtaned as m µ 4 c n,+j = I { x m [n]=c m [n] } m µ 2 x [n] c m n,+j m n,+j j j = m µ 2 c [n] m n,+j. c m j j n,+j 11 Combnng 9 and 10 and nsertng the resut nto 11 yeds a message update that taes the code bt beefs m from the channe decoder and yeds refned code bt beefs µ 4 c m. Hence, these message updates taen together can be thought of as a soft-n/soft-out MIMO mutuser detector. Snce 9 and 11 nvove ony one antenna of one user, the overa compexty of the sum-product agorthm s domnated m by 10. Indeed, the compexty of cacuatng µ 2 x [n] s exponenta n the number of transmt antennas and n the number of users M because the sum n 10 nvoves S MTM 1 terms. For exampe, S MTM for four users wth two transmt antennas and 16-QAM moduaton.

4 IV. LOW-COMPLEXITY RECEIVER We w next derve a mutuser detector whose compexty s ony near n the number of users. A. Gaussan Approxmaton m To smpfy 10, we approxmate the beefs µ 1 x [n] by Gaussan dstrbutons wth the same means and varances m as those of µ 1 x [n],.e., m µ 1 x [n] exp xm [n] m m [n] 2. [n] σ m2 Usng 9, the means and varances are obtaned as m m [n] = x m [n] µ 3 1 x m [n], σ m2 [n] = x m [n] x m x m [n] [n] m m [n] 2 µ 3 1 x m [n]. m Repacng n 10 µ 1 x [n] m wth µ 1 x [n] and the correspondng summaton wth an ntegraton aows us to derve m the foowng cosed-form approxmaton to µ 2 x [n] : m µ 2 x [n] exp C m wth mean vector r[n] h m [n] 1 m r[n] h [n]x m m m [n] = and covarance matrx C m [n] =σ 2 I + h m,m,m σ m 2,m,m =1 m=1 [n]x m [n] m m [n] H [n] m m [n], 12 [n] mm [n] = C r [n] σ m2 [n] h m Here, M C r [n] =σ 2 I + [n] h m [n] hm [n] H hm [n] σ m2 [n] h mh [n]. 13 [n] h mh [n] s the current estmate of the covarance matrx of r[n]. Hence, the exponentay compex computaton of 10 s repaced wth the computaton of 12. Accordng to 13, the M covarance matrces C m [n] are ran-one updates of C r [n]. Thus, they can be effcenty nverted va Woodbury s dentty [11]. The overa compexty of computng m µ 2 x [n] can be shown to scae neary wth the number of users and cubcay wth the number of transmt antennas. B. Seectve Message Updates To further reduce computatona compexty, we propose a seectve message update scheme that avods the computaton of updated beefs for code bts wth hgh reabty. The code bt reabtes are measured va the posteror LLRs ξ m = og µ 3c m =0µ 4 c m m =0 =1µ 4 c m =1. If ξ m exceeds a prescrbed threshod, the correspondng message µ 4 c m s not updated.e., the vaue from the prevous teraton s reused. The dea s that, as the sum-product teratons progress, the code bt reabtes mprove and hence fewer and fewer message updates have to be performed. We note that such a seectve message update can be vewed as a specfc schedung [7] of the sum-product agorthm that adapts dynamcay to the current bt reabtes. The choce of the threshod affects both the number of message updates to be done and the performance of the sumproduct agorthm convergence behavor and fna bt error rate. Snce the LLRs generay ncrease wth the SNR, the threshod has to be adapted to the SNR. The mpact of the LLR threshod on the performance and compexty of the recever w be studed expermentay n Secton V. C. Overa Recever Structure The sum-product agorthm deveoped above can be nterpreted as an teratve turbo recever structure. The dotted boxes n the ower part of Fg. 2 correspond to a soft-n/soft-out MIMO mutuser detector that exchanges bt reabty nformaton wth M parae snge-user soft-n/soft-out channe decoders the dotted boxes n the upper part of Fg. 2. The proposed recever uses parae message schedung [7],.e., the extrnsc LLRs ξ m for a users are smutaneousy updated by the channe decoders, converted to beefs m va 8, and then used by the mutuser detector to cacuate refned messages µ 4 c m for a users concurrenty. When the sum-product agorthm s termnated after a predefned number of teratons, the sgns of the a posteror LLRs of the nformaton bts computed by the channe decoder m provde the fna bt decsons ˆb approxmatng 2. V. SIMULATION RESULTS We next ustrate the performance of the proposed MIMO- IDMA recever. A. Smuaton Setup We smuated a MIMO-IDMA system wth M =2 users, =2 antennas per user, and M R =2 base staton antennas. The number of nformaton bts was K = 512. A termnated rate-1/2 convoutona code code poynoma [23 35] 8 seray concatenated wth a rate-1/2 repetton code was used; thus, the overa code rate was R =1/4. The ntereavers were randomy generated for each data boc. The ntereaved code bts were mapped to 16-QAM symbos, yedng a tota number of N = K/R B = 256 transmt vectors per user. The sum rate of ths system s 4 bts per channe use. The channe matrces H m [n] of sze 2 2 were generated ndependenty for each n fast fadng channe, wth eements that were..d. Gaussan wth zero mean and unt varance.

5 no seectve update seectve update scheme C seectve update scheme B seectve update scheme A BER x2 MIMO IDMA, M=2 users snge user bound snge user bound wth optma detector E /N b 0 Fg. 3. Average BER versus SNR E b /N 0 for a 2 2 MIMO-IDMA system wth M =2 users, after 10 teratons of the ow-compexty recever wthout seectve message updates. The snge user bound and the snge user bound for the optma recever cf. 10 are shown for comparson. BER cumuatve number of message updates x 10 4 Fg. 4. BER of a 2 2 MIMO-IDMA system wth M =2 users, at an SNR of 11 db, versus number of message updates for the same system as n Fg. 3 and dfferent seectve message update schemes. The marers on the curves ndcate teraton cyces. B. Resuts We frst study the performance of the proposed owcompexty teratve recever agorthm wthout seectve message updates. Fg. 3 shows the bt error rate BER after 10 teratons, averaged over the two users, versus the sgnato-nose rato SNR E b /N 0. It s seen that the recever features the typca turbo-behavor, wth an SNR of more than 8 db requred for convergence, and a waterfa regon above that SNR. For SNR 10 db, our recever performs cose to the snge user bound. We aso show the BER of the optma recever cf. 10 for M =1 user. The proposed owcompexty recever snge user bound performs amost as we, whch justfes the approxmatons that ed to 12. Next, we consder the ow-compexty recever wth seectve message updates, at an SNR of E b /N 0 =11dB. We compare three dfferent schemes: schemes A and C use a constant LLR threshod of 5 and 30, respectvey, whe scheme B uses an LLR threshod that ncreases neary from 5 frst teraton to 30 10th teraton. Scheme B s motvated by the fact that the LLRs tend to ncrease wth the teratons. Fg. 4 shows the BER versus the compexty number of message updates for schemes A, B, C and for the recever wthout seectve message updates. It s seen that the seectve message update offers a very favorabe performance compexty tradeoff. Scheme A exhbts the qucest BER decrease, but saturates at a BER sghty above 10 4 and a compexty of about 9000 message updates. The ast teratons reduce the BER ony sghty but at the same tme requre ony very few message updates snce most posteror LLR magntudes are aready arger than 5. The behavor of scheme C ntay equas that observed wthout seectve message update. Eventuay, however, LLR threshodng sets n and the further BER decrease down to beow 10 5 s acheved wth sgnfcanty ess compexty than wthout seectve update. The behavor of scheme B s ntermedate between those of schemes A and C, wth quc nta BER decrease and saturaton at reasonaby ow BER. To acheve a target BER of 10 4 or better, the method wthout seectve update requres sx teratons wth amost message updates. Scheme B aso requres sx teratons but ony message updates, correspondng to computatona savngs of about 50%. VI. CONCLUSION Based on a factor graph framewor and the sum-product agorthm, we have deveoped a computatonay effcent MIMO-IDMA recever sutabe for hgher-order moduaton. A further reducton of compexty has been acheved by a seectve message update scheme that aows an easy compextyperformance tradeoff. The proposed system can be extended n varous ways. In partcuar, the smpe convoutona code can be repaced by more sophstcated codes such as LDPC codes [12], whch can be optmzed for a gven recever. An anaytca study of the seectve message update scheme s an nterestng topc for further research. REFERENCES [1] J. G. Proas, Dgta Communcatons. New Yor: McGraw-H, 3rd ed., [2] X. Wang and H. V. Poor, Iteratve turbo soft nterference canceaton and decodng for coded CDMA, IEEE Trans. Comm., vo. 47, pp , Juy [3] L. Png, L. Lu, K. Wu, and W. K. Leung, Intereave-dvson mutpeaccess, IEEE Trans. Wreess Comm., vo. 5, pp , Apr [4] C. Nova, F. Hawatsch, and G. 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