On Capacity-Maximizing Angular Densities of Multipath in MIMO Channels

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1 On Capaci-Maimizing Angula Densiies of Mulipah in MIMO Channels Geog Levin and Sege Loka School of Infomaion Technolog and Engineeing Univesi of Oawa 6 Louis Paseu, Oawa, Onaio, Canada, KN 6N5 Tel: (63) e 98, Fa: (63) {glevin, sloka}@sieuoawaca Absac This pape povides a paial answe o he quesion: "wha is he bes angula densi of mulipah in MIMO channels?" using he size-asmpoic heo of Toepliz maices fo unifom -D and 3-D anenna aas A Konecke-pe appoimaion of he aa coelaion sucue is poposed and used o find he angula densiies ha compleel eliminae coelaion beween an elemens of anenna aas and hus maimize he asmpoic MIMO capaci fo a boad class of fading disibuions A half-wavelengh spacing, he bes angula densi is shown o be non-unifom, which implies ha he popula Clake s (Jake s) model does no epesen he bes case scenaio The asmpoic esuls ae validaed via Mone-Calo simulaions, and a numbe of pacical guidelines fo anenna design and opimal oienaion ae povided I INTRODUCTION Thee ae wo majo popagaion-elaed facos ha affec he capaci of MIMO fading channels: (i) anenna aa configuaion (eg geome, elemen spacing), and (ii) he angula densi of mulipah A numbe of capaci opimizaion poblems, which ake ino accoun hese facos, have been fomulaed and solved The impac of angula densi on spaial coelaion and channel capaci has been discussed in [] The effec of fading coelaion on he mean/insananeous capaci and cu-off ae has been sudied in [] The unifom angula densi wihin a seco has been invesigaed and he equiemens on anenna elemen spacing have been fomulaed in [3] Vaious models of non-unifom angula densiies (eg Gaussian, Laplacian, Cosine ec) have been poposed and validaed unde vaious popagaion condiions [4] The quesion adiionall asked in mos of he pio woks, in one fom o anohe, is wha is he effec of such-and-such angula densi on he channel capaci (coelaion, divesi, ec)? The onl ecepion is [5], whee an opposie quesion has been posed: "wha is he bes angula densi of mulipah in MIMO channels?" While being an iniguing quesion fom a heoeical viewpoin, i is also pacicallelevan Specificall, he capaci-maimizing angula densi found in [5] fo a boad class of MIMO channels wih unifom linea aas (ULA) povided a benchmak fo pacicall eising densiies o compae wih, as well as led o some pacical guidelines fo anenna design when he angula densi is Gaussian, as adoped b IEEE 80n Bes angula densi povides an eample of a scenaio whee he anenna aa and is elecomagneic envionmen ae mached in infomaion-heoeical sense (capaci-wise) In he pesen pape, he poblem of he bes angula densi is solved fo a boad class of MIMO channels wih unifom - D and 3-D anenna aas While he elaionship beween popagaion channel chaaceisics, anenna aa design and he channel capaci ae ael amenable o a closed-fom analsis, we go aound his difficul b using an asmpoic appoimaion of he channel capaci (when he numbe of anennas is lage) This allows us o obain he bes angula densi in a closed-fom fo a geneal class of fading channels (no onl Raleigh/Rice), and also an insigh as o wh i is of ha paicula fom The size-asmpoic appoach has alead been successfull used in a numbe of woks, and has been poven o be easonabl accuae fo channels wih a modeae numbe of anennas Fo eample, asmpoic ouage capaci disibuion of coelaed MIMO Raleigh-fading channels has been obained in [6] The impac of coelaion on he asmpoic capaci of unia-independen-unia (UIU) channels has been sudied in [7] B leing he numbe of ansmi anennas go o infini and using ciculan maices heo, an ineesing effec of capaci sauaion in cicula anenna aas of a fied apeue has been shown in [8] fo he unifoml disibued mulipah The emphasis of his pape is on infomaion-heoeic ahe han elecomagneic analsis of unifom -D and 3-D anenna aas, which also dicaes he assumpions we make In paicula, we follow a simplified model of isoopic anenna aa elemens, which is widel used in he anenna aa heo and infomaion-heoeic lieaue [9] The main conibuions ae: We show ha when he numbe of anennas is lage, he insananeous and mean capaci of a boad class of MIMO channels (no necessail Raleigh-fading) wih an abia coelaion sucue (no necessail unia-independenunia UIU) does no depend on a paicula channel disibuion, bu onl on he coelaion beween anennas This genealizes/eends he known esuls obained ealie fo Raleigh and UIU channels We popose an appoimaion of mulidimensional aa coelaion sucue as a Konecke poduc of ULA coelaion maices The appoimaion advanages ae: (i) i

2 is easonabl accuae, (ii) i allows appling he well developed heo of Toepliz maices o analsis of mulidimensional anenna aas, and (iii) i is scalable (a -D anenna coelaion mai is appoimaed b he Konecke poduc of wo ULA coelaion maices, ec) Using he Konecke-pe appoimaion and Szego Theoem [0], we find he mulipah angula densi ha maimizes capaci of a boad class of MIMO channels wih unifom -D and 3-D anenna aas The capaci-maimizing angula densi is shown o be non-unifom, ie he popula Clake s (Jake s) model [] does no alwas epesen he bes-case mulipah scenaio II ASYMPTOTIC CAPACITY Conside an equivalen base-band discee model of a MIMO channel wih n T and n R anennas = + w, () whee and ae ansmi and eceive vecos especivel, is he channel mai whose elemens ( ) km, k = n, h m = n, epesen he comple channel gains fom m h ansmi o k eceive anennas, and w is he AWGN noise veco We adop he following assumpions: (i) he channel sae infomaion (CSI) is available a he R end bu no a he T end, (ii) CN ( 0, PT / ni), whee means idenicall disibued, I is ideni mai, and P T is he oal ansmied powe, which does no depends on n (his achieves he egodic capaci of he iid Raleigh fading channel [], and is a easonable ansmission saeg wih no T CSI in geneal), (iii) w CN ( 0, N0I), whee N 0 is he noise vaiance in each eceive anennas, (iv) he channel is fequenc fla and quasi-saic (slow block fading) The insananeous capaci C (ie he capaci of a given channel ealizaion) and mean capaci C pe R anenna of he MIMO channel in naual unis [ na ] ae [] C = n ln de[ I + γ / n ], C = E( C), () 0 whee opeao E denoes epecaion and γ 0 is he SNR pe R anenna Wihou loss of geneali is nomalized so ha E{ } = n n, whee is Fobenius nom The ansmi and eceive coelaion maices ae defined as R { = n E } and R { = n E } especivel Due o he adoped nomalizaion { R } = n and { R } = n, whee sands fo ace The following heoem gives asmpoic insananeous and mean capaci of he MIMO channel when n and n go o infini Theoem : Le be a comple cicula smmeic andom mai (no necessail Gaussian, iid o wih a sepaable coelaion sucue as in [6], [7]), and he following condiions ae saisfied as n, n : n n (i) n κ ( ) 0 4 nm nk lk lm k, m= n, l=, (3) whee κ 4 ( nm nk lk lm ) is he fouh ode cumulan of cicula smmeic andom vaiables In pacice his assumpion coesponds o capaci-appoaching codes (ii) / / n n kl kl n Q G R k, l= k, l= { } + { } 0, (4) whee Q { kl = E hkh l } and G { kl = E gkg l }, h k and g k ae he k h columns of and especivel As n, he capaci pe R anenna conveges as p, (5) ( a) C C ; ( b) C C p denoes convegence whee C = n ln de[ I + γ0r ], and in pobabili Poof: Main seps o pove (a) ae given in [5] A poof of (b) follows fom (a) using Jensen inequali 3 Theoem allows spliing he effec of coelaion a he ansmi and eceive ends and indicaes ha in asmpoic appoimaion he channel capaci does no depend on a paicula disibuion of, o R, bu onl on R Since he condiion (4) does no hold when he T end is full coelaed [5], which coesponds o small anenna spacing, we conjecue ha (4) is saisfied if he anenna spacing eceeds a ceain minimum Noe ha C is he uppe bound on he mean capaci of MIMO channels wih a finie numbe of anennas ( C C ) [3], ie following Theoem, his bound is asmpoicall igh Condiions (3) and (4) ae elaboaed in deail in [5] I can be shown ha as special cases Theoem includes a numbe of popula channel models such as iid and coelaed Raleigh/Rice, UIU channels fo which he esuls in (5) ae known, see eg [6], [7] In paicula, if is iid comple cicula smmeic Gaussian, i is saighfowad o show ha Theoem holds if lim n, n n / n = 0, ie n has o incease much fase han n Theoem will be used lae on o find he bes mulipah angula densi in ems of channel capaci oweve, in ode o poceed owad his goal, an appoimaion of he anenna aa coelaion sucue is poposed in he ne secion III KRONECKER-TYPE APPROXIMATION Conside a unifom ecangula aa (URA) ling on he XY plane wih n and n anennas along and coodinaes especivel, so ha n = n n Assume ha he coelaion beween he anenna elemens along coodinae does no depend on and is given b mai R, and he coelaion along coodinae does no depend on and is given b mai R The following Konecke-pe appoimaion of he URA coelaion mai is poposed: R = R R, (6) whee denoes he Konecke poduc Following (6), zeo coelaion beween wo anennas A and A locaed in a ow (along o coodinae), enfoces zeo coelaion beween A and all he anenna elemens in he column conaining A, R is he k, m elemen of eg if ( R ) k, m = 0, whee ( ), We show lae a case when his condiion implies ha n has o incease much fase han n 3 In geneal convegence in he mean implies convegence in pobabili, bu no vise vesa [] k m

3 R, = + Fom (6), R is he Konecke poduc of wo ULA coelaion maices R and R, which ae Toepliz 4 Theefoe, even hough R ma no be Toepliz, is appoimaion (6) has a Toepliz sucue To assess he accuac of he Konecke-pe appoimaion R, hen he whole block ( ), = 0 i j i = ( k ) n +,, kn, j ( m ) n,, mn we use he scala measue of coelaion n R [/ n,] ha accouns fo oal coelaion beween muliple anennas and simulaneousl affecs he divesi measue and ouage capaci disibuion of MIMO channels when a numbe of anennas is lage [3], [4] n R = / n coesponds o a case whee he anennas ae compleel uncoelaed, ie R = I, fo full coelaed anennas n R =, ie R has a single non-zeo eigenvalue equal n The highe he n R, he lowe he ouage capaci a ouage pobabiliies < 05 [4] Fom (6) n R = n R n R, (7) ie unde he Konecke-pe appoimaion he measue of coelaion of he R anenna is he poduc of he coesponding measues of R and R Measue of Coelaion n -/ d=d =d [wavelengh] Kon Appo Mone-Calo Fig Measue of coelaion of 33 URA vs anenna spacing Unifoml disibued mulipah Conside a URA in a unifoml disibued 3-D mulipah I is saighfowad o show ha in his case he Toepliz mai-geneaing vecos 6 of R and R ae he same and given b R k = J0 ( π kd ), (8) whee J0 ( ) is zeo-ode Bessel funcion of he fis kind, and d, eihe d o d, is he anenna spacing in wavelenghs along o coodinaes especivel Fig shows he measue of coelaion of a 33 URA vs anenna spacing d = d = d The solid line is calculaed analicall using (6) 4 Toepliz coelaion mai phsicall coesponds o unifom anenna aa geome, when coelaion depends on he spacing beween elemens onl, bu no on hei posiions 5 The measue applies o a se of nomalized coelaion maices such ha R = n 6 The geneaing veco of an n n Toepliz mai R is defined as Rk m = ( R ) km, k, m = n, whee ( R ) km is an elemen of R [0] 5 and (8) The doed line is obained via Mone-Calo simulaion, when he mulipah is unifoml disibued I can be seen ha he main lobe ( d < 05 ) is well appoimaed b (6), while hee is small discepanc on he ail ( d 05 ) A d 075, he measue of coelaion is ve close o he minimum n / = / 3, which coesponds o an uncoelaed URA The following coolla is based on Theoem and uilizes he Konecke-pe appoimaion of URA coelaion sucue Coolla : Conside a MIMO channel ha saisfies he condiions of Theoem and whose coelaion mai R = R R has a Toepliz sucue Le R be nondegeneae and squae-summable 7, ie 0 < R k = k <, whee R k is he geneaing veco of R As n, n and n go o infini, he channel capaci pe R anenna conveges as π π ( ) ln[ 0 ( ) ( )], (9) C π + γ λ u λ v dudv π π whee λ and λ ( v), u, v ( π; π ] ae he specums of R and R especivel 8 Poof: Due o Szego Theoem [0], using he fac ha he eigenvalues of R ae given b he poduc of he coesponding eigenvalues of R and R Maimum Asmpoic Capaci: I is saighfowad o show using adamad inequali ha 0 de( R ) fo an n n coelaion mai R nomalized such ha ( R ) = n Theefoe, he asmpoic capaci in (5) is maimal when R = I, ie he channel is uncoelaed a he R end Using Konecke appoimaion (6), his implies ha boh R = I, R I wih coesponding specums = λ = λ =, u ( π; π ] (0) Fom (9) and (0), he channel maimal capaci pe R anenna is C ma = ln( + γ 0 ), ie in he asmpoic appoimaion, each addiional R anenna ma incease he oal capaci b he amoun no beond he capaci of an AWGN channel The mulipah angula densi ha achieves he maimum C ma fo a boad class of MIMO channels wih unifom -D and 3-D anenna aas is deived in he following secions IV BEST ANGULAR DENSITY FOR URA Conside a MIMO channel in a 3-D mulipah envionmen Le R(, ) denoe he spaial coelaion beween wo anennas a spacing + The mulipah wave-numbe specum and he coelaion funcion ae elaed b he Fouie Tansfom [9] j( k + k ) f ( k, k ) ( ) R(, ) e dd = π, () 7 If R is non-degeneae and absoluel summable, i also saisfies he condiion of Coolla, since R ( ) n n R = n 8 jnu The specum of a mai R is given b λ = R n ke, whee R = k is / he geneaing veco of R and j = ( ) [0]

4 whee k and k ae and componens of he waveveco k f ( k, k ) is ofen efeed as he join pobabili densi funcion (PDF) of k and k due o he following popeies [9]: (i) f ( k, k ) is eal and non-negaive assuming ha R(, ) is emiian fo boh and, (ii) unde nomalizaion R (0, 0) =, f ( k, k ) dk dk = Conside now a eceiving URA of isoopic anenna elemens, when he coelaion beween he elemens is given b a Toepliz mai R = R R (as suggesed b appoimaion (6)) In his case R, n = R( d n,0), R, n = R(0, d n), n = 0,,, () whee R, n and R, n ae he Toepliz mai-geneaing vecos of R and R especivel I is saighfowad o show ha due o (6) R(, ) can be hus facoized as R(, ) = R(,0) R(0, ), so ha using () f ( k, k ) = f ( k ) f ( k ), (3) ie unde he Konecke appoimaion, k and k ae independen Fom he geome of he poblem, he link beween k, k and he angles of aival of mulipah componens is ψ = kd = πd sin θcos φ, ψ = k d = πd sin θsin φ, whee θ [0; π ] and φ [0; π ) ae he elevaion and azimuh angles, ψ, ψ epesen he phase diffeence beween wo adjacen anennas along and coodinaes especivel Fom () R, n and R, n ae he samples of he coninuous funcion R(, ), hence he elaionship beween he specums λ, λ of R, R and he specum f ( k, k ) of R(, ) is given b he sampling heoem, ( ) λ = π f u πn ψ ( ) λ = π f u πn ψ, u ( π; π ], (4) whee fψ = f ( u / d ) / d, fψ = f ( u / d ) / d ae he PDF's of ψ and ψ especivel Theefoe, unde he condiions of Theoem and following Coolla, one obains he condiion o achieve he maimal capaci C ma b subsiuing (0) in (4): C C if ma π f ( u π n) = ψ π f ( u π n) = ψ fom which we obain he following:, u ( π; π ] (5) Theoem : Conside a MIMO channel equipped wih a URA a he R end Unde he condiions of Theoem, as boh n and n go o infini, he following holds: (i) if eihe d o d < /, hee ae no such fψ, fψ ha C Cma, ie he uppe bound is no achievable (ii) if boh d, d /, hee is a class of fψ, fψ such ha C Cma, ie he maimizing fψ, fψ ae no unique (iii) if d = d = /, C Cma fo ψ and ψ disibued unifoml, ie f ( ) ( ) ( ) u ψ = ψ = π The coesponding mulipah angula densi is "Sine" disibued f θ, φ ( θ, φ ) = sin( θ) 4π, θ [0; π ], φ [0; π ) (6) and he coesponding spaial coelaion funcion is R(, ) = sinc( ) sinc( ), (7) whee sinc( ) = sin( π) /( π ) Poof: Using (5) and Jacobian ansfomaion fom Caesian o spheical coodinaes Theoem suggess ha when d = d = /, he bes mulipah angula densi f θ, φ ( θ, φ ) is non-unifom in ems of θ, and cicula smmeic in ems of φ The lae is well eplained b he oaional smme of he URA wih an asmpoicall lage numbe of anennas Fuhemoe, since he maginal disibuion of φ is unifom, i follows ha in he d = d = / case he elevaion and azimuh angles ae independen As a paicula case of saemen (ii) of Theoem, i is saighfowad o show using (4), ha fo an coninuous mulipah angula densi wih no specula componens (no Diac s dela funcions in he densi), (5) alwas holds ue as d, d, and hence C Cma As a geneal emak, we noe ha when a given angula densi eliminaes coelaion beween an wo elemens of a URA of spacing +, i also does so fo spacing ( n) + ( m), whee n and m ae ineges Mean Capaci [Na/z/sec] C ma 5% γ 0 =0dB 5 Asmp via Kon Appo Mone-Calo d=d =d [wavelengh] Fig Mean capaci pe R anenna vs anenna spacing 44 T URA, R URA "Sine" disibued mulipah (6) Even hough Theoems and ae based on he sizeasmpoic assumpions and he Konecke-pe appoimaion, simulaions show ha hei esuls appl o he channels wih a finie (modeae) numbe of anennas, and heefoe pacicall elevan As an eample, Fig shows he mean capaci pe R anenna vs anenna spacing d = d = d The solid line is calculaed analicall using (5b) and (7) The doed line is obained via Mone-Calo simulaion of a MIMO Raleigh-fading channel wih 44 ansmiing and eceiving URA's when he mulipah a boh ends is disibued accoding o (6) I can be seen ha he mean and asmpoic capaciies well coincide, he discepanc does no eceed 5% of he capaci maimum even hough he simulaed MIMO channel has a modeae

5 numbe of anennas Moeove, he asmpoic heo pedics well he behavio of he mean capaci In boh asmpoic and finie cases he capaci maima occu a d = n /, n =,,, whee he channel becomes uncoelaed Theefoe, even hough (6) is obained unde he asmpoic assumpions, i is sill he bes angula densi in ems of he mean capaci when he numbe of anennas is finie Simulaneous analsis of Figs and shows ha he ae of coelaion deceasing o capaci inceasing wih anenna spacing depends ve much on he angula densi The ae is fase when he densi follows (6) as compaed o he unifom one Paiculal, in he fome case he channel becomes uncoelaed (achieves capaci maimum) a d = 05, while in he lae his happens a d 075 Thus he Clake s (Jake s) model [], whee f θ, φ ( θ, φ ) is assumed o be unifom, does no epesen he bes case scenaio when d = d = / V BEST ANGULAR DENSITY FOR UCA Conside a unifom cubic aa (UCA) of eceive anenna elemens in a 3-D mulipah envionmen As in he URA case, assume ha he coelaion beween he anenna elemens along coodinae does no depend on and z, and is given b mai R, he coelaion along coodinae does no depend on and z, and is given b mai R and he coelaion along z coodinae does no depend on and, and is given b mai R z Appling he same concep as fo he URA, he Konecke-pe appoimaion of he UCA coelaion mai is R = R R R z (8) Fom he geome of he poblem he link beween he componens of he wave-veco k : k, k, k z and he angles of aival of mulipah is ψ = kd = πd sin θcos φ, ψ = k d = πd sin θsin φ, and ψ z = kzd z = πd z cos θ, whee d, d and d z ae he anenna elemen spacing in wavelenghs along, and z coodinaes especivel Le fψ, fψ and fψ z be he maginal PDF's of ψ, ψ and ψ z, which can be poven o be independen unde appoimaion (8) Adding z coodinae and appling he same agumens as fo he URA we obain he following: Theoem 3: Conside a MIMO channel equipped wih a UCA a he R end Unde he condiions of Theoem and fo d = d = d z = /, C Cma as boh n and n go o infini, when ψ, ψ and ψ z ae unifoml disibued, ie f ( ) ( ) ( ) ( ) z u ψ = ψ = ψ = π The coesponding mulipah angula densi is non-unifom and given b (6) Poof: following he same agumens as fo Theoem The fac ha he bes mulipah angula densi (6) is nonunifom has ceain pacical implicaions Conside, fo eample, a 3-D envionmen, whee he mulipah is no isoopic, bu concenaed aound a hoizonal plane Theoems and 3 sugges ha egadless of an specifics of he ssem design, he URA and UCA should be mouned in paallel o ha plane in ode o incease he capaci We noe ha following he discussion in Secion II, his guideline holds fo a boad class of MIMO channels, no necessail Raleighfading VI CONCLUSION I is of paicula inees o seach popagaion scenaios, whee he mulipah disibuion fis he bes angula densi In such scenaios he anenna aa and he mulipah would be mached in a pobabilisic sense poviding maimal MIMO capaci In addiion, he bes f θ, φ ( θ, φ ) can be used as a heoeical benchmak indicaing, fo eample, how fa awa a given angula densi is fom he bes one, which, in un, can povide useful ips on opimal anenna spacing and aa oienaion fo pacicall eising popagaion envionmens REFERENCES [] I E Telaa, Capaci of Muli-Anenna Gaussian Channels, AT&T Bell Labs, Inenal Tech Memo, pp -8, June 995, (Euopean Tans Telecom, v0, no 6, pp , Dec 999) [] M T Ivlac, W Uschick, J A Nossek, "Fading Coelaions in Wieless MIMO Communicaion Ssems," IEEE Jounal on Seleced Aeas in Communicaions, vol, no5, pp 89-88, June 003 [3] S Loka, G Tsoulos, "Esimaing MIMO Ssem Pefomance Using he Coelaion Mai Appoach," IEEE Communicaions Lees, vol 6, no, pp 9-, Jan 00 [4] K Li, MA Ingam, A V Nguen, "Impac of Cluseing in Saisical Indoo Popagaion Models on Link Capaci," IEEE Tansacions on Communicaions, vol50, no4, pp5-53, Ap 00 [5] G Levin, S Loka, Wha is he Bes Angula Densi of Mulipah in MIMO Channels?, in Poc Canadian Wokshop on Infomaion Theo (CWIT009), Oawa, ON, Ma 009 [6] C Main, B Oesen, "Asmpoic Eigenvalue Disibuions and Capaci fo MIMO Channels unde Coelaed Fading," IEEE Tansacions on Wieless Communicaions, vol3, no4, pp , Jul 004 [7] A M Tulino, A Lozano, S Vedu, "Impac of Anenna Coelaion on he Capaci of Mulianenna Channels," IEEE Tansacions on Infomaion Theo, vol5, no7, pp , Jul 005 [8] T S Pollock, T D Abhaapala, R A Kenned, Anenna Sauaion Effecs on Dense Aa MIMO Capaci, in Poc ICASSP 003, IEEE Inenaional Confeence on Acousics, Speech, and Signal Pocessing, vol4, no 6-0, pp IV-36-4, 003 [9] L Van-Tees, Opimum Aa Pocessing: Pa IV of Deecion, Esimaion, and Modulaion Theo, John Wile & Sons, Inc, NY, 00 [0] R M Ga, Toepliz and Ciculan Maices: A Review, Foundaions and Tends in Commun and Infom Theo, vol, no 3, pp 55 39, 006 [] W C Jakes, Micowave Mobile Communicaions, Wille, NY, 974 [] T S Feguson, A Couse in Lage Sample Theo, Chapman & all/crc, s Ed Repin, 00 [3] M T Ivlac, J A Nossek, "Quanifing Divesi and Coelaion in Raleigh Fading MIMO Communicaion Ssems," Signal Pocessing and Infomaion Technolog, 003 ISSPIT 003 Poceedings of he 3d IEEE Inenaional Smposium on, pp 58-6, 4-7 Dec 003 [4] G Levin, S Loka, S, "On he Ouage Capaci Disibuion of Coelaed Kehole MIMO Channels", IEEE Tans on Infom Theo, vol54, no7, pp33-345, Jul 008

Low-complexity Algorithms for MIMO Multiplexing Systems

Low-complexity Algorithms for MIMO Multiplexing Systems Low-complexiy Algoihms fo MIMO Muliplexing Sysems Ouline Inoducion QRD-M M algoihm Algoihm I: : o educe he numbe of suviving pahs. Algoihm II: : o educe he numbe of candidaes fo each ansmied signal. :