Suboptimal MIMO Detector based on Viterbi Algorithm

Transcription

1 Proceedings of he 7h WSEAS Inernaional Conference on ulimedia Sysems & Signal Processing, Hangzhou, China, April 5-7, Subopimal IO Deecor based on Vierbi Algorihm Jin Lee and Sin-Chong Park Sysem Inegraion Technology Insiue Informaion and Communicaions Universiy 9, unji-ro, Yuseong-gu, Daejeon Republic of Korea {mygenie, scpark}@icu.ac.kr Absrac: - Subopimal deecors of muliple-inpu muliple-oupu (IO) have been sudied because he implemenaion of he opimum deecor, he maximum-likelihood (L) deecor, has so far been considered infeasible for high-rae sysem. Sphere decoder (SD) using deph-firs ree searching and K-bes algorihm are used for near opimum deecor. SD has he non-deerminisic compuaional hroughpu and K-bes requires he soring uni whose complexiy is significanly high when a large K is used ogeher wih high modulaion consellaion. In his paper, we propose a subopimal IO deecor employing Vierbi algorihm insead of ree searching. This deecor can keep he compuaional hroughpu consan and reduce he complexiy because he soring is no required. In he simulaion, we analyze he advanage and he drawback of he proposed deecor in he environmen of IEEE 80.n sysem. Key-Words: - IO sysem, Vierbi deecion, Sphere decoder. Inroducion A large number of works on he physical-layer sudy of IO echniques have been done in he pas decade. The opimal IO deecion, maximum likelihood (L) decoder, is infeasible by is complexiy when a large number of anennas are used ogeher wih higher modulaion consellaion []. The efficien algorihms for reducing complexiy of L have been proposed such as linear deecors, sphere decoder (SD) and K-bes algorihm. Linear deecors are based on principles of minimum mean-square error (SE) and zero-forcing (ZF). Alhough hey can reduce he complexiy dramaically, heir performances are significanly degraded. Cerain echniques such as sof inerference cancellaion can improve he performance. To achieve he near opimal performance, SD based on non-exhausive ree searching has been researched. In general, subopimal deecors using ree searching are divided ino wo ypes, deph-firs SD and breadh-firs using -algorihm, also known as K-bes []. Deph-firs SD can obain beer performance han K-bes deecion wih small K. In spie of his penaly, K-bes deecion has he advanage in erms of he fixed compuaion hroughpu and laency while hose of deph-firs SD vary randomly. For he sof-oupu deecor, when ieraive deecion or sof decoder is used, deph-firs SD is advanced o lis sphere decoder (LSD), and i degrades he compuaional hroughpu severely []. The hroughpu of K-bes deecion is no decreased even hough i provides he sof-oupu. In K-bes deecion, he nodes a a deph mus be sored according o heir merics and he complexiy of soring is significanly high when a large K is used ogeher wih high modulaion consellaion. In his paper, we propose he Vierbi algorihm based subopimal deecor which convers ree searching o rellis searching. The rellis searching in Vierbi algorihm is runcaed version of breadh-firs ree searching, hence he performance is degraded. However, he proposed deecor overcomes he weak poin of deph-firs SD because i can guaranee he consan hroughpu, and i can ease he pressure of soring in K-bes algorihm. This paper consiss of he following conens. In Secion, he fundamenals of deph-firs SD, K-bes algorihm and sof-oupu generaion are summarized. In Secion, he proposed Vierbi algorihm based IO deecor is described. The error performance in he environmen of IEEE 80.n sysem is simulaed in Secion 4. We conclude he paper in Secion 5. Fundamenals of IO Deecors. Sysem odel

2 Proceedings of he 7h WSEAS Inernaional Conference on ulimedia Sysems & Signal Processing, Hangzhou, China, April 5-7, We consider a spaial muliplexing IO sysem wih N ransmi and N r receive anennas. The ransmier sends N spaial sreams. Assume ha he ransmied symbol is aken from a Gray-labeled -QA consellaion ( = q ). A once, he ransmier maps one qn coded bi vecor x ono a N symbol vecor s. The ransmission of each vecor s over IO channels can be modeled as y = Hs + n, where y is a N r vecor of received signals. H is a Nr N IO channel marix which is i.i.d. zero-mean uni variance complex Gaussian marix, perfecly known o he receiver. n is a vecor of independen zero-mean complex Gaussian noise enries wih variance N /. 0 a-priori choice of he radius [4]. If r is chosen oo small, no soluion is found, and he search mus be resared wih a larger radius. If i is chosen oo large, many candidae vecor symbols lie wihin he sphere, and he deecion effor is high. A echnique known as radius reducion allows us o avoid his problem if he deph-firs SD is used in conjuncion wih Schnorr-Euchner (SE) enumeraion [5]. However, he deecion effor varies randomly according o he received SNR and channel marix, so he compuaional hroughpu is non-deerminisic. In pracice, he maximum deecion effor mus be limied o deph-firs SD because he effor o find he soluion someimes even corresponds o an exhausive search.. Deph-firs Sphere Decoder Deph-firs SD algorihm can be divided ino wo pars, he firs par is he PED calculaion and he oher par is he ree raversal. The following inequaliy expresses he sphere consrain []: y Hs <r. () The lef par of inequaliy () can be decomposed wih PEDs as N N N i = yi rijsj = Ti y Rs s, () i= j= i i= where H ZF y = Q y = Rs, Q and R are he resuls of QR decomposiion of H, and r ij is he elemen of he upper riangular marix R. y i and s j are he elemen of y i and s, respecively. i T s is called PED. An efficien deph-firs ree searching is operaed wih PED calculaions of whole children nodes of a moher node. Fig. shows he simple example of ree searching and ree raversal of deph-firs SD when N = and = 4. The number in each circle (node) is he accumulaed PED of he node. In his paper, ree level i is presened in descending order. A doed recangle means he nodes which are calculaed a he same ime. Dark gray nodes are pruned nodes because he accumulaed PEDs of heir moher nodes already exceed he radius r. A considerable problem of deph-firs SD is he fac ha he searching complexiy criically depends on he Fig.. Example of deph-firs SD. K-bes Algorihm The K-bes algorihm raverses he ree in breadh-firs manner. The deecor selecs only K nodes a each level of he ree which have he minimum accumulaed PEDs and compues he PEDs of all heir children. Among hese children, i selecs he K nodes wih he smalles PEDs as he paren nodes o be visied a he nex level. This algorihm is designed such ha he deecion effor is consan for each ransmied vecor. K-bes have beer parallelism han deph-firs SD. I is clear ha K deermines he rade-off among silicon area, hroughpu and he error performance of he sysem. Compuaional complexiy for PED grows wih increasing K and he soring complexiy o selec K nodes also increases wih larger K and. K-bes deecion wih small K has he worse error performance han deph-firs SD..4 Sof-oupu IO Deecor Sof-oupu IO deecor provides a poseriori probabiliy (APP) informaion for each coded bi.

3 Proceedings of he 7h WSEAS Inernaional Conference on ulimedia Sysems & Signal Processing, Hangzhou, China, April 5-7, 007 The log-likelihood raio (LLR) of a poseriori is defined as Pr[ xk = y] LD( xk y) = ln = LA( xk) + LE( xk y), () Pr[ xk = 0 y] where k = 0,..., qn. LA( x k) is a priori log-likelihood raio and LE( xk y ) is an exrinsic log-likelihood raio. If we assume ha Pr[ xk = ] = Pr[ xk = 0], he exrinsic log-likelihood raio can be approximaed as [6] LE( xk y) min min N y Hs y Hs, (4) x Xk, x X k,0 where kb { k } Compuing ( ) 0 X, = x x = b, b= 0,. L x y requires exhausive search o E k find he minimum values for each x k because here qn are candidaes for each erm. In order o reduce he complexiy, he searching ses o selec he minimum values are changed from X kb, o L kb,, where L is he lis which includes a cerain number of candidaes which have he smalles disances [6]. Equaion (4) can be approximaed as LE( xk y) = min min N y Hs y Hs, (5) x Lk, x L k,0 0 Lkb, = x xk = b, x L, b= 0,. In deph-firs SD, A simple modificaion o he SD helps us o generae he lis, bu degrades he compuaional hroughpu severely []. This modificaion is called o lis sphere decoder (LSD). LSD does no decrease radius and add all searched poins o lis if he lis is no full, or if he lis is full, i compares he searched poin in he lis wih he larges radius and replaces his poin if he new poin has smaller radius. In K-bes deecion, we can generae he lis of K candidaes wihou any degradaion of hroughpu. where { } Proposed IO Deecion Algorihm In his paper, we propose he subopimal IO deecor using Vierbi algorihm insead of ree searching. In Fig., he original ree srucure and rellis srucure for Vierbi algorihm are showed. In he rellis srucure, here are N dephs and sages a each deph. Every sage has oupus oward every sage a he nex deph and inpus from every sage a he previous deph. oupus represen he pah merics which are accumulaed PEDs of all children nodes of he curren sage. (a) (b) Fig.. (a) Original ree srucure and (b) Trellis srucure From he equaion (), we re-formulae he hree parameers, he iniial values, branch meric and pah meric for he rellis searching. The iniial values of he firs deph are expressed as P N,,,,,,, i= yn r N Ns N i i = (6) where s ni, represens he symbol of he sage i on he deph n. The branch meric of he sage j a he nex deph n- from he sage i a he curren deph n can be wrien as N B = y r s r s ni,, j n n, m m n, n n, i m= n, i =,,, and n= N, N,, where s m is he symbol in he memorized pah hisory. In his paper, he deph n is presened in descending order. The branch meric corresponds o PED. (7)

4 Proceedings of he 7h WSEAS Inernaional Conference on ulimedia Sysems & Signal Processing, Hangzhou, China, April 5-7, 007 From he branch meric, he pah meric of he sage j a he nex deph n- can be wrien as Pn, j = min Pn, i + Bn, i, j (8) i Among pairs of previous pah meric and branch meric, every sage selecs he minimum pah meric and memorizes is pah. A he final deph, we can obain candidaes and heir disances. For hard-oupu deecion, he candidae wih he smalles disance a he final deph becomes he soluion. For sof-oupu deecion, candidaes make up he lis for generaing sof-oupu as in (5). The effor required o find he soluion of deph-firs SD varies randomly according o he realizaion of he channel and noise and someimes even corresponds o an exhausive search. However, he proposed deecor can keep he compuaion hroughpu consan and i can generae oupu a every cycle if compuaion is fully parallelized and pipelined. In erms of he complexiy, he proposed deecor is more efficien han K-bes because he soring is no required. If PED compuaion uni is implemened as described in [], which can compue PEDs of all children a one ime, he number of PED compuaion is K ( N ) + for K-bes and ( N ) + for he proposed deecor. The number of compare K K / for K-bes operaions is based on he soring nework in [8] and ( ) for he proposed deecor because only searching he minimum is required insead of he sor as shown in equaion (8). Table I compares he complexiy beween K-bes and he proposed deecor in erms of he number of comparisons for a level. The proposed deecor can reduce he complexiy, significanly. Fig. shows an example of he proposed IO deecion when four ransmi anenna and 4 QA are used. The hick lines represen he seleced pahs and he hick and doed lines represen he pah of he soluion for hard-oupu deecion. The number on each sage is he pah meric and he binary bis in he bracke are he bis of he survived symbols. The decision bis are he reverse order of he survived symbols. Table. Number of compare operaions for K-bes and he proposed deecor 4-QA 6-QA (K=6) 64-QA (K=64) (K=4) K-bes,5 8,84,5 Proposed Fig.. Example of he proposed IO deecion [ ] [ ] [ ] [ 0 0 0] Simulaion Resul In his secion, he error performance of he proposed IO deecion is compared wih deph-firs SD, K-Bes and ZF deecor. In his simulaion, he arge sysem is IEEE 80.n [7] which employs a spaial muliplexing, convoluional code and orhogonal frequency division muliplexing (OFD) wih up o 64-QA modulaion. Only 0 Hz bandwidh is considered in his simulaion. Ieraive deecion and decoding is no considered, bu sof Vierbi decoder for convoluional code is used. Frequency selecive IO channel wih hree delay aps is considered. I has 50 ns of he delay resoluion, exponenial power delay profile and no spaial correlaion. Since he block fading channel is assumed, he channel coefficiens of each ap are consan during he duraion of a packe ransmission and changed randomly beween wo consecuive packe ransmissions. The lengh of packe is 000 byes. Fig. 4 shows he error performance of he proposed deecor agains oher IO deecors. In his simulaion, he modulaion and coding scheme (CS) is 6-QA and convoluional code wih code rae of /. The number of ransmi anenna ( N ) is four. The size of he lis for sof-oupu deecor is 6 so ha K is also 6.

5 Proceedings of he 7h WSEAS Inernaional Conference on ulimedia Sysems & Signal Processing, Hangzhou, China, April 5-7, 007 Table. Summary of he comparison of he error performance Oupu Type SNR (db) Gain (db) Hard ZF Deph-firs K-bes Proposed Sof ZF.4. Deph-firs 7. - K-bes Proposed 9. 0 deecion. This is obvious because he proposed deecor can los he L soluion in he early deph. In pracice, he maximum deecion effor mus be limied in deph-firs SD because he effor required o find he soluion of deph-firs SD varies randomly and someimes even corresponds o an exhausive search. Fig. 5 shows he comparison of he error performance beween he proposed deecor and he limied deph-firs SD when CS is 7 in IEEE 80.n. The limied deph-firs SD has he real-ime consrain by early erminaion. The deecion effor, he number of visied nodes, of deph-firs SD wih he archiecure described in [] and [] does no exceed 6 for hard-oupu 50 for sof-oupu. The proposed deecor can have beer performance han he limied deph-firs SD. PER PER SNR(dB) Fig. 4. Error performance in IEEE 80.n sysem ZF (Hard) ZF (Sof) Deph-firs (Hard) Deph-firs (Sof) K-bes (Hard) K-bes (Sof) Proposed (Hard) Proposed (Sof) Deph-firs (Hard) Deph-firs (Sof) Proposed (Hard) Proposed (Sof) 4 Conclusion This paper presens a subopimal hard/sof-oupu IO deecion algorihm wih fixed hroughpu and low complexiy. The basic idea is o re-formulae he ree srucure o he rellis srucure similar as Vierbi algorihm. Since he rellis runcaes he ree, he rellis srucure of he proposed deecor causes performance degradaion. However, in erms of complexiy, he proposed deecor is more efficien han K-bes, and guaranees he consan hroughpu while he hroughpu of deph-firs SD varies randomly. In addiion, he error performance is beer han he consrained deph-firs SD for fixed hroughpu. Acknowledgemen This work was parly sponsored by ETRI SoC Indusry Promoion Cener, Human Resource Developmen Projec for IT SoC Archiec SNR(dB) Fig. 5. Comparison of he error performance wih limied deph-firs SD Table summarizes he comparison of he error performance a he packe error rae of 0 -. In his simulaion, he proposed IO deecor performs worse han deph-firs and K-bes abou 0.7 db in hard-oupu deecion and db in sof-oupu References: [] A.Burg, e al., VLSI Implemenaion of IO Deecion Using he Sphere Decoding Algorihm, IEEE Jour. Of Solid-Sae Circuis, vol 40, Issue 7, pp , July 005. [] Z. Guo and P. Nilsson, VLSI implemenaion issues of laice decoders for IO sysems, IEEE Inernaional Symposium on Circuis and Sysems, ay 004, pp. IV-477-IV480. [] Jin Lee and Sin-chong Park, A pipelined VLSI archiecure for a lis sphere decoder, IEEE Inernaional Symposium on Circuis and Sysems, ay 006, pp

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