PSO Selection of Surviving Nodes in QRM Detection for MIMO Systems

Transcription

1 PSO Selecton of Survvng Nodes n Detecton for MIMO Systems Hu Zhao, Hang Long, Wenbo Wang Wreless Sgnal Processng & Network Lab Bejng Unversty of Posts and Telecommuncatons, Bejng, PRhna Emal: hzhao@bupteducn, wbwang@bupteducn Abstract-Ths paper proposes a promsng detecton that apples PSO (partcle swarm optmzaton) as the selecton of survvng paths n detecton for the coded MIMO system, named as -PSO Because only a few chldren nodes s metrc are needed to calculate n the PSO search process, -PSO can cut down the complexty further, whle keepng the same performance as due to the quas-random movement of the partcles Smulaton results demonstrated that the advantage of -PSO n complexty s obvous especally when antenna number and modulaton level are hgh The requred real multplcaton number of -PSO s only 04% of that of when transmt antenna number s 0 and 6QAM s employed I INTRODUTION Multple-nput multple-output (MIMO) systems can potentally perform remarkably n terms of spectral effcency and acheve very hgh capactes The actual performance acheved wth a certan antenna confguraton s relatve wth the detecton algorthm at the recever The optmal detecton rule s the maxmum lkelhood (ML) crteron But ts complexty grows exponentally wth the number of transmt antennas and the modulaton level The suboptmal lnear algorthms use ZF crteron or the MMSE crteron wth the low complexty, whereas they can t explot the full dversty gan and underperform ML detecton very much In [-], an approach that apples QR decomposton assocated wth the M-algorthm to ML detecton (hereafter ) was proposed to remarkably reduce the computatonal cost Ownng to the applcaton of QR decomposton, the algorthm can be expressed as the search process n an nversed tree wth a heght of transmt antenna number In the orgnal algorthm, the node metrc s calculated for all symbol canddates at each layer of the tree Then, M nodes wth the smallest metrc are survved to generate the next layer, whereas all nodes are survved n the ML detecton Therefore, has already a small performance loss comparng ML detecton, but cuts down a lot of complexty The adaptve selecton of survvng paths [3] was proposed to further decrease the computaton cost by only calculatng metrc for M canddates However, the cost of adaptve selecton s the debasement of the relablty of the fnal survvng nodes at the last layer leadng to a worse performance than In ths paper, we propose the dea of partcle swarm optmzaton (PSO) to select the survvng nodes at each layer, whch also needs the few operatons of metrc calculaton, whle keepng same performance as PSO s an evoluton-based search and optmzaton technque [4-5], whch fnds optmal solutons of complex search spaces through nteracton of ndvduals n a populaton of partcles As a search strategy, the bnary and real-valued versons of PSO also were exploted for sgnal processng problems, such as multuser detecton and blnd extracton of sources n [6] When there are enormous canddates, the heurstc search of PSO can rapdly fnd the objecton wth a lttle cost We just utlze PSO to fnd the best survvng nodes wth the smallest metrc at each layer of the tree The rest of the paper s organzed as follows Secton II descrbes the tree structure used n algorthm In Secton III the PSO selecton algorthm for the survvng paths s proposed and llustrated n detal Then, the smulaton results n Secton IV show the effectveness of the proposed algorthm on reducng complexty, comparng wth Fnally, conclusons are drawn n Secton VI II QR DEOMPOSITION AND TREE EXPRESSION We assume a frequency flat MIMO channel, so the sgnal model s y = Hx+ n () where y s the receved Nr sgnal vector, H s the Nr uncorrelated channel matrx ( Nr) and t s elements are ndependent zero mean complex Gaussan random varables wth unt varance, x s a vector vector conssts of symbols, each chosen from an alphabet of sze Mc (Mc s modulaton level) And n s a complex whte Gaussan nose vector wth zero mean and covarance matrx σ I Nr Frst we compute QR decomposton of H and obtan an upper trangular matrx R and a untary matrx Q wth H=QR Then y s left-multpled by Q H (the conjugated transposton of Q): Supported by Panasonc Research & Development (hna) o Ltd X/06/$ IEEE Ths full text paper was peer revewed at the drecton of IEEE ommuncatons Socety subject matter experts for publcaton n the IEEE GLOBEOM 006 proceedngs

2 H H H Q y = Q Hx+Q n = Rx+n r r r x, 0 r r, x = + n 0 0 r x Due to the upper trangular matrx R, the set of all possble symbol vectors can therefore be represented by a tree structure, n whch each node denotes a canddate symbol and Mc branches are generated from each node, as shown n Fg e 4 e 3 e E = e + e + e Fgure Illustraton of tree search (=4, Mc=) 4 3 Each node from the root node stands for a sequence of symbols xˆ, ˆ ˆ N x,, t N x wth a metrc t E, where = N t,(t s xˆ that N t s detected frstly, leadng to the descendng order) Every possble symbol vector ˆx corresponds to a node at the last layer of the tree, and has a total metrc, whch denotes the estmaton error after the detecton: = y-hxˆ = Rx-ρ ˆ Nr rxˆ ( ˆ ρ rx j j ) ρ = j= + = + E = + H T where ρ = Q y = ( ρ, ρ,, ρnr ) Snce the second tem n (3) s same for all the nodes at the same layer, the metrc E can be rewrtten as: E = r xˆ ( ρ r xˆ ) kk k k kj j k= j= k+ r ˆ ˆ, x ( ρ r, jx j) E+ j=+ = e + E + () (3) = + (4) where e s the metrc of the path from the above layer to ths layer, just as the gray path llustrated n Fg The objecton of the detecton wth hard decson s to fnd a node at the last layer wth the smallest E But for the coded MIMO system, the decoder needs soft nformaton as the nput So t s necessary to survve several nodes at the last layer to calculate the log lkelhood rato (LLR) of a posteror probablty (APP) of each bt of the soluton vector III PSO SELETION OF SURVIVING NODES The orgnal detecton must calculate the metrc of all the M Mc canddates and choose M among them to survve However, t s the operaton of metrc calculaton accordng to (4) that expends the prmary computaton complexty Therefore, we utlze the heurstc PSO algorthm to select the M survvng nodes at each layer, whch only requres a few operatons of metrc calculaton A Sortng the chldren nodes Before PSO search, the chldren nodes of each survved parent node stll need to be sorted Here, we propose a smple method whch needn t calculate the node s metrc out Frstly, solve the equaton e = 0 wth the known x, x+ to get the undetected soluton on -th dmenson, denoted as the black pont n Fg(a) Here, there are 6 chldren nodes (the hollow crcle) for 6QAM Then each chldren pont can be denoted by (r,), where r/ s ths pont s ndex after sortng the projected dstance on real/magnary axs between ths pont and And these dstances are stored and wll be used n the followng metrc calculaton In ths process, for 6QAM, there are only 8 operatons of the subtracton between real numbers totally (4 for real part, and 4 for magnary part) Aforehand, the whole constellaton plane s dvded as 64 grds And a correspondng order of each chldren pont can be obtaned ntutvely accordng to ther rough Eucld dstance to the gray grd contanng n ascendng order Here t needn t the exactly calculaton to get the order For example n Fg(a), accordng to the poston of gray grd, the pont (,) must be the nearest to But the second may be the pont (,) or (,) We can regulate that the pont wth a large r s preferental n the sortng Hence, wherever the gray grd locates, the mappng relatonshp between the coordnates (r,) of each chldren pont and ts order-ndex s changeless, stored as a look-up lst wth sze of Mc as Fg(b) Then we can look up the order-ndex of each chldren pont (the number n the dashed crcle n Fg(b)) by ths lst Now, our problem s how to choose M canddates ponts from the M Mc chldren nodes as shown n Fg X/06/$ IEEE Ths full text paper was peer revewed at the drecton of IEEE ommuncatons Socety subject matter experts for publcaton n the IEEE GLOBEOM 006 proceedngs

3 Fgure The method of sortng the chldren nodes Intalze partcles Whle maxmum teratons s not attaned For each partcle alculate the metrc hoose the partcle wth the smallest metrc among all the partcles and let p g equal to ts poston For other partcles except the partcle at the best poston alculate partcle velocty accordng (5) Update partcle poston accordng (6) (a),,3,,,3, Fgure 3 Sorted chldren nodes B PSO selecton In PSO selecton algorthm, the search regon s just the ndex of parent nodes, e ~M Each partcle denotes a canddate wth a metrc and has a velocty whch drects ts flyng The randomly ntalzed partcles fly through the search space to search for the parent nodes, whch chldren node has the smallest metrc after updatng generatons Due to the quas-random movement of the partcles, all the parent nodes almost can be accessed In every teraton, each partcle s updated by the hstorcal best poston The best poston p g that s tracked by the partcle swarm optmzer s the ndex of parent node whch chldren node has the smallest metrc, obtaned so far by any partcle n the populaton After fndng the best poston, ths vsted chldren nodes wll be replaced by ts brother nodes The partcle at p g doesn t evolve, but explore n ts chldren nodes and ts metrc s updated by the next chldren node s metrc Other partcles update ther velocty and poston wth followng formulas v = w v + r ( p x ) (5) p p g p x = x + v (6) p p p where r s a random number between (0,), drawn from a unform dstrbuton; w s nerta weght, when w s large, the search trends to explore n a wde regon, otherwse, to explot n the vcnty Partcles veloctes are clamped to the maxmum velocty And the evolved poston needs to be rounded to nearest nteger for the partcle s poston s the ndex of the parent node The second tem on the rght-hand sde of the equaton (5) just expresses the nteracton of ndvduals n a populaton of partcles Each partcle studes from each other and fles toward the optmal drecton That s the key dea of PSO algorthm The pseudo code of the procedure s as follows: (b) (c) (d),,,4, p g,,,3,,,,,,,,3,,,,,,4,,,,3 Fgure 4 An example of PSO searchng,4 For vvdly explanng how PSO search performs, we gve an example, resortng to Fg4 For the sake of smplfcaton, there are only two partcles and M=4, e there should be 4 nodes to be survved fnally The subplots on the left of Fg4 show the flyng procedure of two partcles and those on the rght are the lst of survvors Here,,p,q means the q-th chldren node of the p-th parent node at -th layer Step(a) Two partcles (the whte poles) are chosen from M parent nodes, and ther chldren nodes are sorted at the same tme (not shown) The gray poles are ther frst chldren nodes respectvely and the heght sgnfes the sze of the metrc Due to the smaller metrc, s pcked as the best node (p, g ) and ts pole s smeared wth shadow Now that the rght lst s empty, both of them are coped to the lst Step(b) explots n the local so ts second chldren nodes, replaces the survved frst chldren nodes At the same tme, the X/06/$ IEEE Ths full text paper was peer revewed at the drecton of IEEE ommuncatons Socety subject matter experts for publcaton n the IEEE GLOBEOM 006 proceedngs

4 other partcle evolves to the new poston from,3 to, accordng the formulas (5-6) Now, becomes p, g The lst sn t full so that these two chldren nodes are all saved n the lst Step(c) The p g partcle moves from the frst chldren nodes of to the second and the other partcle fles to Now, the,,4 two partcles at the new poston are checked whether they can enter nto the lst Only the heght of s lower than that of the,4 hghest node, so,3, s deleted from the lst and,3, jons,4, Step(d) After enough teratons (these wll be a maxmum lmt), the M survved chldren nodes n the lst are sorted accordng to ther metrc and renamed as {,,,,, M } They become the new parent nodes There are some rules to accelerate the search: ) If the frst chldren node of a partcle s excluded from the lst due to the entrance of the nodes wth smaller metrc, then ths parent node and ts all chldren nodes are deleted from the search space ) If no chldren node of all the current partcles s survved, the partcles are deleted and the current p g partcle fly to the parent node accessed rarely Sequentally, PSO search at the next layer begns Up to the last layer, the fnal M survved paths (symbol vector) are chosen and used to generate the soft nformaton For the far comparson, the calculaton of APP employed n our algorthm s same to the method as [] Because PSO s employed only as the method of survvng nodes n detecton, the proposed detecton s abbrevated as -PSO n the followng IV SIMULATION RESULTS A The system structure The system structure n the smulaton s shown n Fg5 At the transmtter, bnary nformaton data bts are frst seral-to-parallel-converted nto data streams Then the nformaton data sequence s encoded by Turbo encoder and modulated After passng the flat-fadng channels, the transmtted streams are receved by Nr receve antennas At the recever, the receved sgnals are detected by or -PSO nto coded soft nformaton sequences Then they are decoded by Max-Log-MAP wth eght teratons and parallel-to seral-converted to recover the transmtted bnary data In the smulaton, the antenna number and the modulaton are all varous but the codng rate s changeless as Rc=3/4 In the followng fgures, tr s the maxmum teraton number of PSO search The two detecton wll survve the same number M = Mc nodes n smulaton We assume the number of partcles s p n = Mc The parameter n (5) s determned experentally, set asw =08 S/P : P/S :Nr Modulator Encoder Modulator Encoder Decoder Decoder or -PSO Fgure 5 The system model n smulatons B Performance comparson BER 00 E-3 E-4 -PSO 4x4 6QAM M=6 tr= SNR(dB) Fgure 6 Performance comparson between and -PSO Due to the search wth strategy, PSO can guarantee that the metrc of M survvors are smallest n all the canddates Therefore, -PSO has the almost same performance as, as shown n Fg6 On the other hand, the metrc of all of the chldren nodes at each layer wll be calculated to fnd the M survvors wth the smallest metrc n So t s not necessary to prune at the last layer, because the prunng at the last layer can t reduce the complexty but lose the vald nformaton Fnally, has M Mc nodes to generate the soft nformaton, whle -PSO has only M chosen nodes Hence, there s a lttle dfference between them and s just better a lttle omplexty Analyss As to complexty, we lst the number of real multplcatons per symbol vector for two detecton algorthms n Table Here, for, the man operaton s the multplcatons to calculate the node s metrc accordng to the defnton (4) As to -PSO, some addtonal outlay s pad for partcle s evoluton Accordng to the formulas n Table, ther complexty are all the lnear functon of the pont number of modulaton constellaton and the quadratc functon of transmt antenna number It s noted that the complexty of detecton algorthm s relatve wth, but not wth Nr The latter only nfluences the performance of detecton The sum of several tems n Table I for two detectons respectvely s plotted n Fg7 We can see that the advantage of -PSO s obvous especally when antenna number s large Ths advantage attrbutes to two aspects One reason s that the number of the nodes whch need metrc calculaton s cut down greatly The other s because the metrc n Nr X/06/$ IEEE Ths full text paper was peer revewed at the drecton of IEEE ommuncatons Socety subject matter experts for publcaton n the IEEE GLOBEOM 006 proceedngs

5 PSO are calculated by the square sum of the stored projected dstance on real/magnary axs b between the canddate nodes and Ths method only requres a few multplcatons and some addtons TABLE I NUMBER OF REAL MULTIPLIATIONS FOR TWO DETETIONS Method Operaton Number of multplcatons - PSO Number of Real Multplcatons alculate the node s metrc at each layers alculate the node s metrc at -th layer Fnd the undetected soluton and calculate the projected dstance Intalze the partcles 6x0 4 x0 4 80x0 3 40x0 3 Evolve Mc= tr=5 -PSO Mc M[4+ 4 ( )] = Mc 4 = = = M[+ M + 4( )] p n [ ( p n ) tr ] (transmtter antenna number) Fgure 7 omplexty comparson of two detectons at dfferent antenna number omplexty Rato of -PSO to % 0 QPSK tr={5,5,40} for Mc={,4,6} 6QAM 64QAM (transmtter antenna number) Fgure 8 omplexty rato of -PSO to at dfferent modulaton level For exhbtng the nfluence of modulaton level on the complexty, Fg8 shows the requred real multplcaton number rato of -PSO to Wth the ncrease of, ths rato s ncreasng slghtly and gradually tends to a stable value For QPSK, 6QAM and 64QAM, ths stable rato s respectvely 30%, 4% and 04% It s demonstrated that -PSO s preferred for a system wth large antenna number and hgh modulaton level Ths s because the heurstc search algorthm s very ft to the search space wth a large amount of canddate ponts V ONLUSION Addressng the problem of computaton complexty of MIMO s detecton algorthm, ths paper proposes a promsng detecton for coded MIMO system that apples PSO as the selecton of survvng paths n detecton PSO s an evoluton-based search and optmzaton technque Due to the quas-random movement of the partcles, the survved nodes are guaranteed to be the best nodes wth the smallest metrc So the proposed -PSO has the same performance as the orgnal At the same tme, the reducton of the nodes number requrng metrc calculaton and a proposed smple metrc calculaton method make -PSO very effectve on complexty Through the statstc of the requred real multplcaton used n the detecton, the complexty between -PSO and s compared under the condton of dfferent transmt antenna number and modulaton level Smulaton results show that the advantage of -PSO n complexty s obvous especally when antenna number and modulaton level are hgh The requred real multplcaton number of -PSO may approach 04% of that of when transmt antenna number s 0 and 64QAM s employed REFERENES [] Kyeong Jn Km, Jang Yue, Jont hannel Estmaton and Data Detecton Algorthms for MIMO-OFDM Systems, Proc 36th Aslomar onference on Sgnals, Systems and omputers, pp , Nov00 [] Hroyuk Kawa etc, Lkelhood functon for -MLD sutable for soft-decson turbo decodng and ts performance for OFDM MIMO multplexng n multpath fadng channel, IEIE Transommun, vole88-b, No, pp47-56, Jan,005 [3] Kench Hguch, Hroyuk Kawa etc Adaptve selecton of survvng symbol replca canddates based on maxmum relablty n -MLD for OFDM MIMO multplexng, IEEE Globecom, pp , 004 [4] Eberhart R, and Kennedy J, A new optmzer usng partcle swarm theory IEEE Proc Sxth Intl Symposum on Mcro Machne and Human Scence (Nagoya, Japan),pp 39-43, 995 [5] Maurce lerc, Kennedy J, The partcle swarm exploson stablty and convergence n a multdmensonal complex space, IEEE Transactons on evolutonary computaton, Vol6, No, pp: 58-73, 00 [6] Yng Zhao, Junl Zheng, Partcle swarm optmzaton algorthm n sgnal detecton and blnd extracton, IEEE ISPAN, pp37-4, X/06/$ IEEE Ths full text paper was peer revewed at the drecton of IEEE ommuncatons Socety subject matter experts for publcaton n the IEEE GLOBEOM 006 proceedngs