Reduced Cluster Search ML Decoding for QO-STBC Systems

Transcription

1 Reduced Cluster Search ML Decodng for QO-STBC Systems Isaque Suzuk, Taufk Abrão Dept. of Electrcal Engneerng State Unversty of Londrna , PR, Brazl Bruno A. Angélco, Fernando Craco Paul Jean E. Jeszensky Dept. of Telecomm. & Control Engneerng Escola Poltécnca, Unversty of São Paulo São Paulo, , SP, Brazl {angelco, fcraco, Fernando Casadevall Dept. of Sgnal Theory & Communcaton Unverstat Poltècnca de Catalunya Barcelona, Span Abstract Snce the maxmum lkelhood (ML) decodng results too complex when the modulaton order and the number of receve antennas ncrease, an effcent reduced complexty ML-based decodng scheme appled to a multple-nput-multpleoutput (MIMO) antenna systems wth quas-orthogonal spacetme block code (QO-STBC) s proposed, and named reduced cluster search ML decodng (RCS-ML). Its performance and complexty aspects are compared to the conventonal ML decodng approach. Hgh-order modulaton ndexes and short low densty party check codes (LDPC) are consdered. Numercal results have ndcated no degradaton n the performance and an ncreasng reducton n the complexty of RCS-ML decodng wth respect to the conventonal ML when the modulaton order ncreases. Index Terms MIMO system, QO-STBC, ML decodng, cluster search, LDPC. I. INTRODUCTION The last fve years have been domnated by hgh demands on vdeo and audo data wth relablty real-tme applcatons. Multple-nput-multple-output (MIMO) schemes, assocated wth space-tme block codes (STBC), for nstance, Alamout rate STBC [] (R STBC), represent a sutable soluton and are frequently ncorporated by many standards lke WMAX. Furthermore, hgher throughput wth acceptable performance complexty trade-off can be acheved through the ncluson of a bt-mapped coded modulaton (BMCM) structure. BMCM n conjuncton wth parallel short low densty party check codes (LDPC), quas-orthogonal STBC (QO-STBC) [2], and teratve soft parallel nterference cancellaton (PIC) detector, s dscussed n [3]. The am s to acheve low complexty schemes, hgh throughput wth good performance and to generate low processng delay n the overall processng of detecton and decodng. Good STBC desgns must take nto account jontly performance crtera codng gan, dversty gan, multplexng gan, and the decoder complexty. In ths scenaro, the so called fast-decodable SBTC MIMO schemes have been consdered recently wth great nterest. Prevous works on fast-decodable low-complexty SBTC MIMO systems ncludes [4] [9]. In [7], a famly of full-rate, fulldversty 2 2 codes, whose detecton complexty grows only quadratcally wth the sze of the sgnal constellaton have been proposed. Thus, the optmum decoder complexty reduced by a factor of 256 for the 6-QAM sgnal constellaton (and by 4, 096 for the 64-QAM modulaton). A dfferent approach to acheve low-complexty near-maxmum lkelhood QO-STBC decodng based on teratve nterference cancellaton (IICIS) was proposed n [6]. In order to fulfll the requrements of system capacty, low power consumpton and low termnal sze, development of LDPC decoders has been focused by many recent papers. However, the LDPC were ntroduced snce 962 by Gallager [0], showng the possblty of achevng capacty by codng a message usng long codes, resultng n a good trade-off between complexty and performance. At that tme there was no way to mplement LDPC codes and they remaned forgotten untl 999, when Mackay brought them back to scene []. Ths paper proposes a low-complexty and effcent decodng algorthm for QO-STBCs schemes, based on reduced ML cluster search. The RCS-ML decoder performs smlarly to the conventonal ML n terms of bt error rate (BER), but wth lower complexty that s more evdent for hgher order modulatons. Performance results were obtaned wth and wthout LDPC codes. The remanng sectons are organzed as follows: Secton II descrbes the QO-STBC MIMO system wth short parallel LDPC nner codes. The proposed reduced cluster search ML decoder n the context of QO-STBC codes s dscussed n Secton III. Numercal results and complexty analyss for the proposed QO-STBC decoder wth an arbtrary number of receve antennas are analysed n Secton IV. Man conclusons are presented n Secton VI. II. QO-STBC SYSTEM MODEL AMIMOsystemwth =4transmt antennas and n R receve antennas s consdered, wth 4 symbols transmtted smultaneously. Addtonally, n R ndependent flat fadng subchannels, M-QAM modulaton, Rate QO-STBC scheme of [2], and short LDPC (optonally) are employed, Fg.. A. LDPC Encodng In order to acheve hgh throughput (heren, up to 4 bts per symbol perod was consdered, dependng on the modulaton In many of them, complexty scales lnearly wth code length.

2 nput data stream output data stream Mux m : Demux :m B B m B B m BP Decoder It BP BP Decoder FEC LDPC FEC C {c,..., c p,..., c N } C m Q I Q I Mappng M QAM LDPC m 4 M =2 m symb. L 0, L, L 0 m, L m, Demappng M QAM, λ,...,λm Bt Metrc Calculaton ŝ() s() QO STBC Encoder (RCS ML) s QO-STBC Reduced Cluster Search ML Decoder mn f,3 (s,s 3 ) & mn f 2,4 (s 2,s 4 ) Fgure. QO-STBC wth short LDPC MIMO system and reduced cluster search ML decodng approach (RCS-ML). =4and n R antennas. ndex and codng rate), assocated to sutable performance and smplcty of decodng at the recever, the bt-mapped coded modulaton (BMCM) scheme s used, jontly wth m short parallel LDPC codng [2]. The adopted modulaton s hghorder squared-qam modulaton and m =2, 4, 6 or 8. Intally, as shown n Fg., the nput data stream s demultplexed nto m data substreams {B } m = wth block lengths K. To keep the decodng complexty low, each substream B s encoded usng a LDPC code, obtanng m codewords C,.e., {C } m =, of length N; thepth encoded bt n C s denoted c p. Observe that there s an nherent flexblty n the BMCM structure: substreams can be encoded usng LDPC codes wth dfferent values of K, as long as all m codes produce codewords of the same length N. Hence, the overall rate of the m LDPC codes s gven by m K R ldpc = mn. () Assumng dentcal block lengths, K = K, the overall rate of the m LDPC(N,K) codes s smplfed to: R ldpc = K N. All the m LDPC encodng processes are smply done n parallel. The th LDPC encodng process s gven by C = B G, where G s the K N generator matrx of the th LDPC code component. B. R QO-STBC The pth bts from all m LDPC codewords smultaneously select the pth 2 m -ary constellaton pont s S, where S s a set of all vald symbols belongng to the adopted constellaton at the transmtter. The topology of Fg. employs rate quas-orthogonal space-tme block code (R QO-STBC) proposed n [3], descrbed by the code matrx s s 2 s 3 s 4 A = s 2 s s 4 s 3 s 3 s 4 s s. (2) 2 s 4 s 3 s 2 s s nt r r nr n R Channel MIMO h j,k Four new constellaton ponts (s,s 2,s 3,s 4 ) are transmtted usng =4and L =4tme slots, such that #symbols transmtted R stbc = = (Rate ), (3) #tme slots used,l and the overall throughput for a system usng a M-ary constellaton s defned as Θ=R stbc R ldpc log 2 M [bts per symbol perod]. (4) C. Recever Let x() be the th modulated symbol wth durato s, and s j (t) the transmtted symbol by the jth transmt antenna at tme t. Each transmtted symbol goes through the wreless channel to arrve at each of n R receve antennas. Denotng the path gan from transmt antenna j to receve antenna k at each symbol nterval by h kj (t), the baseband dscrete-tme sgnal receved at the kth antenna s gven by r k (t) = j= h kj (t)s j (t)+n k (t), t =,..., L, (5) where h kj (t), k {, 2,...,n R }, j {, 2,..., } are assumed to be..d. complex Gaussan random varables (fadng ampltudes are Raylegh dstrbuted) wth zero mean and E[(h I kj )2 ] = E[(h Q kj )2 ] = 2, where hi kj and hq kj are the real and magnary parts of h kj (t). The complex addtve whte Gaussan nose (AWGN) at the kth receve antenna, {n k },k =,...,n R, has zero mean and varance E[n 2 k]=n 0 = E s γ = m0 SNR E s 0 R stbc R ldpc, (6) where E s s the average energy of the transmtted symbols, gven a constellaton format, SNR s the sgnal-to-nose rato per receve antenna n decbels (db), and γ s the average SNR per receve antenna [2]. The receved sgnals from all receve antennas can be rearranged n a vectoral form, such that r(t) =H(t)s(t)+n(t), t =,..., L, (7) where, n each tme slot t =,..., 4 of the adopted QO-STBC scheme, r(t) =[r (t) r 2 (t)... r nr (t)] T s the receved sgnal vector, s(t) =[s (t) s 2 (t)... s nt (t)] T s the transmtted symbol vector, H(t) s the n R channel matrx wth channel nt,nr coeffcents {h kj } k,j= between the jth transmtted antenna and kth receve antenna, and n(t) =[n (t) n 2 (t)... n nr (t)] T s the sampled nose vector. The channel matrx coeffcents are assumed to be perfectly known at the recever, but completely unknown at the transmtter. D. ML Decodng The maxmum lkelhood decson metrc s obtaned by mnmzng the two sum terms [3] (ŝ, ŝ 2, ŝ 3, ŝ 4 )= (8) arg mn f,4(s,s 4 ), arg mn f 2,3(s 2,s 3 ), s,s 4 S s 2,s 3 S

3 n R n T =4 f,4 (s,s 4 )= h jk 2 ( s 2 + s 4 2) +2R{( h k rk () h 2k r k(2) h 3k r k(3) h 4k rk (4))s k= j= + ( h 4k rk () + h 3k r k(2) + h 2k r k(3) h k rk (4)) s 4 +(h k h 4k h 2k h 3k h 2k h 3k + h k h 4k)s s 4 }] (9) n R n T =4 f 2,3 (s 2,s 3 )= h jk 2 ( s s 3 2) +2R{( h 2k rk () + h k r k(2) h 4k r k(3) + h 3k rk (4))s 2 k= j= +( h 3k rk () h 4k r k(2) + h k r k(3) h 2k rk (4))s 3 +(h 2k h 3k h k h 4k h k h 4k + h 2k h 3k)s 2 s 3 }] (0) where the cost functon to be ndependently mnmzed are gven by (9) and (0), at the top of the page. So, for MIMO system wth small constellaton sze, t s computatonally vable to evaluate, ndependently, all possble values for the pars (s,s 4 ) and (s 2,s 3 ), usng the two cost functons (9) and (0), obtanng drectly the ML estmates. However, once the computaton complexty ncreases exponentally wth m, t s computatonally neffcent to evaluate all par combnatons when the dmenson of the constellaton s hgh, for nstance M 6. Secton III descrbes an alternatve low complexty procedure to compute (9) and (0), sutable for hgh squared-order modulaton MIMO QO-STBC wth =4and n R. E. Bt Metrc and Belef Propagaton Decoders The last block n the decodng process s performed by low complexty m short parallel LDPC decoders. Heren, the LDPC decoders use belef propagaton (BP) decodng algorthm, wth a maxmum number of teratons It BP. As BP decoders requre soft symbol estmates L 0 m,, L m,, and admttng dentcal nose varance n all receve antennas E[n 2 k ]=σ2,k=,...,, the bt metrcs are calculated as: L 0 m, =, m =,...,m, () +e λm wth L m, = L0 m,, and ( λ m = 2σ 2 mn ŝ κs 2 s S m,(0) mn p S m,() ) ŝ κp 2, (2) where κ takes nto account path loss and shadowng effects, and the channel coeffcents assocated to s n (9) or (0) as s the set of all constellaton ponts wth a zero at -th poston and S m,() s the set of all constellaton ponts wth one at -th poston. In fact, the soft estmates symbols can be nterpreted as well; S m,(0) ŝ = κs + η, (3) where η s the sum of recever nose plus nterference terms n(9)or(0). The soft symbol estmates L 0 m, and L m, are used by BP decoders to compute the log-lkelhood ratos (LLRs) passed from varable nodes to check nodes. LLRs are used nstead of probabltes because of ther hgher numercal stablty [4]. Hence, BP decoders [2] n Fg. consst of an teratve algorthm that passes messages (LLR values) between varable nodes and check nodes. Computaton of (2) s performed recursvely untl the BP teraton process fnshes. If the matrx of party checks s satsfed or the algorthm reaches the maxmum number of teratons, a hard decson s performed, where postve values of LLRs are consdered bt one and negatve values are bt zero. After all, each LDPC decoder outputs ˆB m. III. REDUCED CLUSTER SEARCH ML DECODING (RCS-ML) The dea of cluster search came up from observng the cost functon behavor, Eqs. (9) and (0), for all possble symbol combnatons. As can be seen from the llustratve example n Fg. 2, there s a pattern that repeats tself. Ths fgure was generated consderng 6-QAM modulaton, resultng n clusters. f others possble cluster pars st cluster s Local mnmum poston Global mnmum ndex s poston ndex 4 Fgure 2. Typcal f 4 values, consderng (s,s 4) mappng symbol pars n a 6-QAM modulaton as descrbed n Fg. 4.a) and 4.b); crcles ndcate search values nsde the frst cluster, and the flled crcle s the local mnmum. The postons of local mnma wthn the other clusters have the same pattern, regardng the frst cluster, as represented by square markers. The same behavor s observed wth f 23 values, consderng (s 2,s 3). Ths pattern can be descrbed ntutvely as follows: gven a QAM constellaton and a receved symbol corrupted by nose, frstly, from constellaton mappng n Fg. 3, t could be consdered the symbols belongng to one column (or row);

4 these columns (rows) dentfyng cluster regons. For 6-QAM, four symbols of frst column (.e., frst cluster, consttuted by symbols ndex, 2, 3, and 4) are consdered. In terms of Eucldean dstance from receved symbol to the constellaton column (row) symbols, two cases can be dstngushed: a) receved symbol s closer to the symbol located at the extremtes of the selected constellaton column (row); n the Fg. 3, ths stuaton s dentfed by hypothetcal receved symbols and, located on cluster-regons I and III, respectvely. b) receved symbol s located closer to one or two (f halfway stuaton) nternal symbols of the same column (row) n the constellaton; n ths example, the receved symbol. In case of hypothess a) occurs, and f the, 2, 3 and 4-th symbol constellaton dstances computaton order s assumed, a monotonc decreasng (receved symbols nto Regon I) or ncreasng (Regon III) pattern for cost functons f,4 or f 2,3 values s observed. On the other hand, n case of hypothess b) take place, a parabolc pattern wth upwards concavty appears on the f,4 or f 2,3 evaluatons. Besdes, ths behavor repeats f we consder other columns (or clusters) of the constellaton, as can be nferred from Fg. 2. Ths fgure explans the case of cost functon be employed to evaluate the dstance of a par of symbols, s and s 4 (or alternatvely s 2 and s 3 ), nstead of only one symbol evaluaton n eq. (9), say s, holdng s 4 fxed. Hence, the monotonc (de)creasng or parabolc patterns become 3D surfaces. I) s quadrature symbol mappng ndex at each 2 m symbols for symbol pars (s,s 4 ) (alternatvely (s 2,s 3 )), and all pars nsde ths pattern consttute a cluster. Fnally, the ML search s performed over the generated set of local mnma n order to fnd the global mnmum of (9) (or alternatvely (0)). Fgure 2 shows f,4 values, obtaned va cluster search procedure, wth global mnmum occurrence at (s =2, s 4 =6) ndexes, square bold marker. In order to mplement the RCS-ML QO-STBC decodng, three steps are carred out consderng each of 2 m clusters as ndcated n Algorthm. Algorthm RCS-ML QO-STBC Input: r, H Output: ŝ, ŝ 2, ŝ 3, ŝ 4 step : Perform a ML search decodng nsde of the frst cluster over (s,s 4 ) and (s 2,s 3 ) symbol pars, Eqs. (9) and (0); Symbols of M-QAM constellaton are hypothetcally mappng as s,s 2,s 3,s 4 {, 2,, 2 m }. step 2: Record the two pars (š, š 4 ) and (š 2, š 3 ) that locally mnmze Eqs. (9) and (0), respectvely. step 3: Generate sets S clst = {š + k 2 m k =0,..., 2 m }, =,...,4; Perform decodng through all possble cluster pars (s,s 4 ) and (s 2,s 3 ), where s S clst : (ŝ, ŝ 4 ) = arg mn f,4 (s,s 4 ), and s S clst ; s 4 S 4 clst (ŝ 2, ŝ 3 ) = arg mn s 2 S 2 clst ; s 3 S 3 clst f 2,3 (s 2,s 3 ) II) III) s * s selected column n-fase Fgure 3. 6-QAM Eucldean dstance evaluaton by clusters. Two stuaton can be dentfed: external and nternal receved symbol s, =,...,4, locaton regards the selected constellaton column symbols. In summary, the basc dea s to perform a ML search only nsde one cluster (for nstance, n Fgure 2 the frst cluster was chosen) n order to fnd a local mnmum, and to generate a sub-set of symbol pars from the other cluster (other 5 clusters n the 6-QAM example) wth the same relatve poston regardng the local mnmum prmarly found n the frst cluster, resultng n a set of local mnma (square markers n Fgure 2). The pattern observed n Fgure 2 repeats tself In order to llustrate the steps of the Algorthm RCS-ML, consder the clustered 6-QAM QO-STBC decodng sketched n Fgure 4. Snce the clusters pattern s smlar for (s,s 4 ) and (s 2,s 3 ) pars, only the cluster search for the (s,s 4 ) par s descrbed. Intally (step ), the frst column of symbols n Fgure 4.a s chosen as a cluster to be evaluated; ths set generates all symbol pars n the whte shadng cluster of Fgure 4.d, used n computaton of f,4, and s called S clst = {, 2, 3, 4}. The dark (red) block nto whte shadng cluster of Fgure 4.d, (s =2, s 4 =4), ndcates the selected par ndexes that yelds the mnmum value of f,4 nsde step 2 (local mnmum). { The } cluster-sets for (s,s 4 ) symbol par, labelled S clstm, S clstm 4, m =,...,m, are obtaned by selectng (š )-th and (š 4 )-th row of constellaton mappng n Fgure 4.a and 4.b, respectvely. The resultng symbol par set (s,s 4 ) calculated nsde step 3 s shown n Fgure 4.c. After all, the ML search s performed over all pars generated by the symbol set of Fgure 4.c,.e., n ths hypothetcal example, S clst = {2, 6, 0, 4} and S4 clst = {4, 8, 2, 6}. The estmated symbol par (ŝ, ŝ 4 ) wll be that whch produces the mnmum f,4 value.

5 Table I MIMO SYSTEM, RCS-ML DECODING AND CHANNEL PARAMETERS. Parameter Adopted Values QO-STBC MIMO System #Txantennas =4 # Rx antennas n R =or 4 Modulaton format squared M-QAM: M =4, 6, 64, 256 QO-STBC code Rate, R stbc =[3] Rx SNR per antenna Throughput SNR [ 0; 42] db Θ =.0, 2.0, 3.0, or 4.0 [bts/symb. perod] LDPC codes Number and sze m short LDPC [2], [5] Rate LDPC(204, 02), R ldpc = 2 Belef Prop. Decoder It BP 20 teratons Raylegh Channel sub-channel fadng flat-raylegh channel type quas-statc (slow), L =4 Channel state nfo. perfectly known at Rx RCS-ML Decodng cluster sze 2 m 2 m use of clusterng approach, but yet performng the ML testng overall clusterng par-canddates. Fgure 4. Clusters mappng employed n RCS-ML procedure, consderng 6-QAM: hypothetcal cluster search for a) symbol s ; b) symbol s 4 ;c) symbol par (s,s 4 ), and d) all clusters mappng for symbol par (s,s 4 ). 0 =4 n R =; R; Unocoded IV. NUMERICAL RESULTS 0 2 The man system and channel parameters used n Monte- Carlo smulatons are summarzed able I. In all numercal results shown here, QO-STBC and M QAM modulaton were adopted; n the cases of source codng, short LDPC(204,02) was adopted. For smplcty, t s assumed perfect knowledge of channel state nformaton (CSI) at the recever sde. In accordance wth the most common channel model n the lterature [], [3], heren a quas-statc tme-varyng channel model s adopted: the fadng coeffcents reman fxed durng each QO-STBC block of L =4tme slots (the quas-statc fadng condton s satsfed: L T s < (Δt) c, the channel coherence tme), and vary ndependently from one block to the next. Fg. 5 descrbes the performance of ML and RCS-ML decodng, both n the absence of LDPC codng, for the case of QO-STBC MIMO system wth =4 n R =antennas. Besdes, Fg. 6 ndcates the behavor of both decoders for the case of short LDPC(204,02) QO-STBC MIMO system wth =4 n R =antennas employng the ML aganst RCS- ML decodng. Fnally, Fg. 7 compares the performance of both decoders but consderng n R =4receve antennas under the same short LDPC and QO-STBC codng. It s worth notng that the dfference n system performances wth RCS-ML and conventonal ML decodng s undstngushed for all SNR and constellaton sze condtons. Those results ndcate that even for undetermned system condton ( >n R ) the RCS-ML algorthm acheves the same BER performances of the ML wth exhaustve search. Ths s due to the adopton of cluster search strategy, allowng a reducton n space search by the BER x 4QAM ML 4x 4QAM RCS 4x 6QAM ML 4x 6QAM RCS 4x 64QAM ML 4x 64QAM RCS 4x 256QAM ML 4x 256QAM RCS SNR db Fgure 5. ML aganst RCS-ML decodng performance comparson for the QO-STBC MIMO system wth dfferent QAM constellaton sze, =4 n R =antennas. V. COMPLEXITY ANALYSIS FOR THE RCS-ML DECODING In order to evaluate the complexty of the proposed algorthm, real multplcatons and sums are consdered. Analyzng (9) and (0) and consderng each complex multplcaton as four real multplcatons and each complex sum as two real sums, there are 90 real multplcatons and 27 real sums for each f 4 or f 23 evaluaton. As the proposed algorthm needs only 2 m ML evaluaton for step and 2 m for step 2, so, the overall evaluaton number s only 2 m+. Table II compares the complexty of RCS-ML QO-STBC and ML QO-STBC decodng schemes. Those complextes can be compared usng the perceptual complexty reducton factor, expressed by CR = C RCS C ML 00 = 2 m 00 [%]. (4) where the second equally holds for all analyzed constellatons and number of receved antennas. From Table II one can conclude an ncreasng reducton n the computatonal complexty ndex CR when the modulaton

6 0 0 2 =4 n R =; M QAM; R; QO STBC; LDPC(204,02) BP=20 VI. CONCLUSIONS In ths work, a reduced complexty ML decodng scheme based on cluster search, sutable for QO-STBC coded MIMO systems wth hgher-order modulaton ndexes, has been proposed. Numercal results for the RCS-ML have ndcated no degradaton n the performance n all analyzed cases. Thanks to reduced cluster search procedure, the RCS-ML acheves the ML performance wth an ncreasng reducton n the computatonal complexty when the modulaton order ncreases, beng 2.5% of ML decodng complexty for 6-QAM, and < % for 256- QAM. BER x 4QAM ML 4x 6QAM ML 4x 64QAM ML 4x 256QAM ML 4x 4QAM RCS 4x 6QAM RCS 4x 64QAM RCS 4x 256QAM RCS SNR [db] Fgure 6. ML aganst RCS-ML decodng performance comparson for the QO-STBC MIMO system wth short LDPC(204,02) and =4 n R = antennas. BER x4 4QAM ML 4x4 6QAM ML 4x4 64QAM ML 4x4 256QAM ML 4x4 4QAM RCS 4x4 6QAM RCS 4x4 64QAM RCS 4x4 256QAM RCS =4 n R =4; M QAM; R; QO STBC; LDPC(204,02) BP= SNR [db] Fgure 7. ML aganst RCS-ML decodng performance comparson for the QO-STBC MIMO system wth short LDPC(204,02) and =4 n R =4 antennas. Table II NUMBER OF REAL MULTIPLICATIONS/SUMS PER RECEIVE ANTENNA PER SYMBOL PAIR NECESSARY FOR QO-STBC DECODING, =4 n R =. REFERENCES [] S. Alamout, A smple transmt dversty technque for wreless communcatons, IEEE Journal on Selected Areas n Communcatons, vol. 6, no. 8, pp , October 998. [2] H. Jafarkhan, Space-Tme Codng: Theory and Practce. Cambrdge Unversty Press, [3] N. S. J. Pau, D. P. Taylor, and P. A. Martn, Robust hgh throughput space tme block codes usng parallel nterference cancellaton, IEEE Transactons on Wreless Communcatons, vol. 7, no. 5, pp , May [4] E. Bgler, Y. Hong, and E. Vterbo, On fast-decodable space-tme block codes, n IEEE Internatonal Zurch Semnar on Communcatons, March 2008, pp [5] C. Jang, H. Zhang, D. Yuan, and H.-H. Chen, A low complexty decodng scheme for quas-orthogonal space-tme block codng, n SAM th IEEE Sensor Array and Multchannel Sgnal Processng Workshop, July 2008, pp [6] J. Km and K. Cheun, An effcent decodng algorthm for qo-stbcs based on teratve nterference cancellaton, IEEE Communcatons Letters, vol. 2, no. 4, pp , Aprl [7] S. Sezgner and H. Sar, Full-rate full-dversty 2 x 2 space-tme codes of reduced decoder complexty, IEEE Communcatons Letters, vol., no. 2, pp , December [8] D. Cho, C. Chae, and T. Jung, Desgn of new mnmum decodng complexty quas-orthogonal space-tme block code for 8 transmt antennas, n IEEE Internatonal Symposum on Sgnal Processng and Informaton Technology, Dec. 2007, pp [9] C. Yuen, Y. L. Guan, and T. T. Tjhung;, Quas-orthogonal stbc wth mnmum decodng complexty: further results, n IEEE Wreless Communcatons and Networkng Conference, vol., March 2005, pp [0] R. Gallager, Low densty party check codes, IEEE Transactons on Informatoheory, vol. 8, no., pp. 2 28, January 962. [] D. Mackay and R. M. Neal, Near shannon lmt performance of low densty party check codes, Electronc Letters, vol. 33, no. 6, pp , March 997. [2] D. Mackay, Good error-correctng codes based on very sparse matrces, IEEE Transactons on Informatoheory, vol. 45, no. 2, pp , March 999. [3] H. Jafarkhan, A quas-orthogonal space-tme block code, IEEE Transactons on Communcatons, vol. 49, no., pp. 4, Jan [4] G. Lechner, Effcent decodng technques for ldpc codes, Master s thess, Venna Unversty of Technology, July [5] D. J. MacKay, Encyclopeda of sparse graph codes, Avalable at: http: // Decoder 4-QAM 6-QAM 64-QAM 256-QAM RCS-ML 720/ / / /3824 ML 440/ / / / CR RCS 50% 2.5% 3.25% 0.78% order ncreases, ndcatng that the RCS-ML becomes an attractve opton when M>6.