MUSIC-BASED Approaches for Hybrid Millimeter-Wave Channel Estimation

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1 016 8h Inenaional Symposium on elecommunicaions (IS'016) MUSIC-BASED Appoaches fo ybid Millimee-Wae Channel Esimaion MShahsi Dasgahian Elecical Engineeing Depamen edowsi Uniesiy of Mashhad Mashhad, Ian Khoshbin Ghomash Elecical Engineeing Depamen edowsi Uniesiy of Mashhad Mashhad, Ian Absac Millimee wae communicaion (mmwc) oe E- b-fequencies beween Gz- is a pomising olunee fo 5G indoo/oudoo communicaion sysems wih an ula high daa aes hans o he lage bwidh o subdue he channel popagaion chaaceisics in his fequency b, high dimensional anenna aays need o be deployed a boh he base saion (BS) mobile ses (MS) Due o lage numbe of anennas a ansceie, mmw sysems aoid o employ ADC o R chain in each banch of MIMO sysem because of powe consains hus Such sysems leeage o he hybid analog/digial pecoding/combining achiecue fo downlin single use deploymen insead of sic digial pocess Lage aay a boh BS MS inside of analog beamfoming equies deeloped channel esimaion his pape popose a new algoihm o esimae he mmw channel by exploiing he spase naue of he channel finding he subspace of eceied signal ecos based on MUSIC By combining he muliple measuemen eco (MMV) concep MISIC, a modified appoach ies o ecoe suppo of an unnown channel maic accuaely een unde he defecie- an condiion Simulaion esuls indicae MUSIC-based appoaches offe lowe esimaion eo highe sum aes compaed wih conenional MMV soluions Keywods Millimee wae MIMO sysems; spase channel esimaion; suppo; muliple measuemen ecos (MMV); hybid achiecue I IRODUCIO hans o he emendous incease in affic, dem fo ula highe daa aes in cellula newos is ineiable by shifing he opeaing fequency of cellula sysems fom he conenional micowae specum o pomising E-b millimee wae specum (30 o 300Gz) fo indoo een oudoo applicaions[19],[17] One of he challenges in oudoo ambience is seee pah loss shadowing phenomena due o oxygen absobion, humidiy fades, eflecie oudoo maeials in millimee-wae b ounaely, adanes in R cicuis combined wih ey small waelenghs of mmw signals (beween 1 o 10mm) maes i possible o pac a miniauized lage numbe of anennas ino ansceies heeby poiding high beamfoming gains ha can compensae pah loss een ou-of-cell inefeence[14] Moeoe, adapie beamfoming may mae sysems less ulneable o unfaoable shadowing effecs In conas o conenional lowe fequency sysems, millimee wae (mmw) sysems, aoid o dedicae one complee R chain one high-esoluion ADC o DAC o each banch of anenna due o a high cos powe consumpion of such componens o his eason, low complexiy sub-opimal analog beamfoming based on beam aining is poposed o be used insead of fully baseb soluions o suppo only single seam MIMO communicaion [16],[4] o suppo capabiliy of muliplexing seeal daa seams o achiee moe accuae beamfoming gain, a hybid achiecue has been poposed in [6],[1] whee he pocessing is diided acoss he analog sage wih numbe of R chains much lowe han he numbe of anennas baseb sage Baseb o digial sage is fo coecion of limiaion of analog R secion In efeences [6] [1], he spase naue of he poo scaeing mmw channel is exploied o deelop low-complexiy hybid beamfoming ybid beamfomes designing poblem has been inesigaed fo ohe achiecues in [9] based on subspace esimaion ahe han esimaion of he whole channel by uilizing he concep of he ecipociy of he channel in DD MIMO sysems In his pape, we conside a hybid beamfoming model fo downlin single-use mmw sysems We assume o hae an nown sensing maix in seeal imes of aining mode as well as o now he geomey of he aays in souce desinaion hus, we uilize he muliple measuemen ecos (MMV) model of spase fo millimee channel popose diffeen appoaches fo soling he channel esimaion poblem he main conibuion of he pape is deeloping he MUSIC based mehods ahe han he exising simulaneous algoihms fo soling he join spase channel ecoey We use he following noaions houghou his pape he bold uppe-case lees denoes maices, bold lowe-cases epesen ecos uhemoe, A is obenius nom, * wheeas A, A, A A ae is anspose, conjugae anspose (emiian), conjugae Mooe-Penose pseudoinese, accodingly A B is he Konece poduc of A B [], is used o designae expecaion A is a submaix of A composed of columns indicaed by se R ( Ψ ) is abbeiaed o ange of he maix Ψ An n n uniay maix is epesened by I n /16/$ IEEE 66

2 igue1 ybid Model of millimee wae channel II SYSEM MODEL Assume a single use downlin Millimee-Wae MIMO sysem wih ansmi anennas a he base saion (BS) eceie anennas a he Mobile saion (MS) wheeas each side is equipped by f f R chains s sepaae daa seams is consideed o send ino spase channel In ou model, he numbe of componen of he ansmie aay is moe han he eceie also he numbe of R chains is saisfied o s f min(, ) igue 1 depics a hybid single use MIMO mmw ansceie wih spaial muliplexing gain phase shife as an analog beamfome he downlin signal a he eceie side afe fileing on baseb is gien by, y = G G +G n = C P+ C n (1) whee C is he complex spase channel assumed o f be slowly bloc-fading, C is he analog o R f pecode, C 1 G he baseb pecode, C is ansmi signal eco wih coaiance maix [ ]= 1 (P / I C n is he addiie Gaussian noise a ) s s s he eceie wih [ nn ]= σ n I Similaly, C f f C G ae he R b baseb combines, especiely maix C is defined as C G In ig1, he bloc of phase shifes as an analog pecode/combine can be chosen fom pedefined codeboos o as om maices wih aiable phase consan ampliude hus, a possible alue se fo μh phase shife νh R chain in maix is j [ ] 1 ϑ e μν, = () μν, whee ϑ μν, as a M-bis quanized angle is chosen fom unifom disibuion in ange of [0, π ) he oal powe consain is compelled by nomalizing G such ha G = s Based on paameic physical model of channel wih L scaees assumpion ha each scaee conibues a single popagaion pah beween he BS MS, he nonlinea channel in spaial angles (bu linea in he pah gains) can be indicaed as whee ([ 1,,, ]) C L β = diag β β β L L is he L dimensional popagaion pah gain diagonal maix wih independenly idenically disibued complex Gaussian diagonal enies wih zeo mean aiance 1/L he ( ) C L V θ ( ) C L V ϕ in (3) epesen aay esponse maices a he BS MS, especiely Such maices ae gien by V ( θ) = ( θ ), ( θ ),, ( θ ) / (4) [ L ] [ ] 1 V( ϕ) = ( ϕ1), ( ϕ),, ( ϕl) / (5) he ϕ i θ i denoe he Angle of Depaue (AoDs) Angle of Aial (AoAs) of ih independen pah fom L oal pah By assuming of an unifom linea aays (ULA) model, ( ) θ l ( ) ϕ l can be defined as π π j d sin( θl ) j ( 1) d sin( θl ) ( θl) 1,e,,e / (3) = (6) π π j d sin( ϕl ) j ( 1) d sin( ϕl ) ( ϕl) = 1,e,,e / (7) whee is he signal waelengh d is he ine-anenna disance se o a boh he BS MS I is assumed ( ) θ l ( ) ϕ l ay slowly hey can be well esimaed a boh sides III L = βl( θl) ( ϕl) L l = 1 = V( θ) βv ( ϕ) L PROBLEM ORMULAIO In his secion, we ae he adanages of he spase naue of he mmw channel fomulize channel esimaion as a compessie sensing poblem Conay o [16],[1] whee aining analogue ecos ae obained fom a muli-leel hieachically, hee hey ae found as om ecos wih 67

3 flucuaing phase by using one R chain in aining sep As a esul, he eceied signal can be wien as υ = f,f, + f,n (8) wheeυ ae eceied ansmied symbol, f, f, ae aining analog beamfome a he MS BS n is addiie eceied eco noise a he h insan o epesening he spase chaaceisics of he channel, we can apply lemma ec ( ABC) = ( C A) ec ( B) o (8) fom [5] hus we can ewie (8) as υ ( ) ( ) = f, f, ec + f,n (9) We can assume ha BS sends equal symbols in sepaae M ime slo wih diffeen pecode ecos also MS eceied in M diffeen combines Wih such an assumpion, MS sacs he M measuemens in a eco as y = Θh + ζ (10) whee y = [ υ,, υ ], = ( ),,( ) Θ 1 M f f f f,1,1,m,m, h =ec( ) ζ = f,1n1,, f,mn M Unde he consideaion of iual model of he channel, we can chaaceize physical channels by join spaial beams in fixed iual ansmi eceie diecions deemined by esoluion of he aays [15] Using his linea model of channel, use channel can be modelled as m n m=1 n=1 = ( ˆ θ ˆ ˆ 1), ( θ),, ( θ ) = (m,n) ( θ ) ( ϕ ) = U U (11) U is an aay esponse maix Simila o (3) exceping ha insead of spaial fequencies π d sin( θl ), l = 1,, L, we subsiue he iual spaial fequencies π, = 1,,, Similaly, U = ˆ ˆ ˆ ( ϕ1), ( ϕ),, ( ϕ ) is an aay esponse maix wih iual spaial fequencies πi, i = 1,,, hans o hese spaial iual diecions, he maices U U ae full-an D maices heefoe, is uniaily equialen o capues all of channel infomaion epesens he iual complex channel maix C is no geneally diagonal o iual angles whee hee is no scaeing, he coesponding enies ae appoximaely zeo oe ha, he iual epesenaion does no disinguish beween scaees ha ae wihin he spaial esoluion By ecoising he channel maix in equaion (11), we hae h = ec( ) = ( U U) ec( ) (1) = ( U U ) h = Wh (13) whee C W is defined as a complex dicionay 1 maix of he channel h epesens a spase C eco wih L non zeo enies as L << Replacing (13) in he saced measuemen eco in (10) assuming = 1, we can wie, y = ΘWh + ζ (14) = Ψh + ζ (15) M whee Ψ C is sensing maix wih he consain of M < Equaion (15) can be seen as a single measuemen eco (SMV) compessie sensing poblem due o L-leel spase eco h Some of impoan geedy echniques such as ohogonal Maching Pusui (OMP) is deiaions, ad Ieaie hesholding (I) is exensions hae been offeed o esole he SMV spase poblems [3] When he SR is ey low, which is ypical case a mmw sysems, we need o enhance he numbe of measuemens compaable o he dimension of unnown spase eco o peen lage sacing of he measuemens, exploiing Muliple Measuemen Veco (MMV) o join spasiy is poposed Rahe han ecoeing he K unnown eco, one aemps o simulaneously ecoe all ecos by finding he ow suppo of he unnown fom he maix fomulaion as Y = Ψ + Ε (16) Whee, =[,1,,,K] C K h h, [ 1,, ] C M E = ζ ζ K K hus Y=[ y1,, y K] C M K When pedominan nonzeo enies of h, ae shaed in he same locaions, MMV algoihms can lead o compuaional pomoion [8] In his pape, we assume ha is L-ow-spase, ie, ha is owsuppo, which is defined as supp( ) : = { i :,(i) 0}, has cadinaliy a mos L One of he famous geedy algoihms fo soling he MMV poblem is Simulaneously Ohogonal Maching Pusui (SOMP) [18] owee, SOMP is a an blind appoach amely, i does no allow fo pefec ecoey in he full an case wih small numbe of measuemens Anohe geedy blind an algoihms such as Simulaneous Ieaie ad hesholding (SI), Simulaneous ad hesholding Pusui (SP) ae poposed in [] In conas o high compuaional complexiy of an-blind mehods, MUSIC (Muliple Signal Classificaion) appoach, whee is a anawae, poides guaaneed ecoey in he full ow an cases wih he mild complexiy IV MUSIC-BASED MEODS One of he simples appoaches eoles he an of obseaions is MUSIC, a popula algoihm in aay signal pocessing which is idenical o an awae hesholding echniques in he full an case [1] owee, one of he main disadanages of he MUSIC echnique is is ending o failue unde he condiion of an(,( )) < L o compensae his limiaion, one can use a geedy selecion algoihm o find s = L aoms of dicionay (o equialenly suppos) hen apply MUSIC o an 68

4 augmened daa maix o ecognize he es suppos Moiaed by hese facs, we inesigae an awae algoihms o impoe esimaion of unnown channel maix A mmw channel esimaion by MUSIC Inspied by using of he MUSIC fo join spase ecoey in [7], signal subspace is needed o esimae by eigenalue decomposiion (EVD) of he appoximaed coaiance maix YY / K o singula alue decomposiion (SVD) of obseaion maix Y o disinguish beween he signal noise subspace, i is bee o uncae he numbe of eigenalues o spasiy leel of he channel, L he esuled subspace maix is consiued of L eigenecos popoional o L dominan eigenalues Bu MUSIC is ending o failue when Ψ,( ) is ill-condiioned in he pesence of he noise, o when,( ) does no hae full ow an If he condiion numbe, defined as he aio of he lages o he smalles singula alue, is lage enough hen he maix is said o be illcondiioned Also wheeas K, he numbe of snapshos, is smalle han he spasiy leel L, hen no moe han K ows can be linealy independen, he nonzeo ows of unnown maix uns ino an defecie Coelaion beween souces o muli-pah popagaion is anohe eason ha caused o an defecie maix heefoe, in such condiions we choose lage eigenalues, whee is he an of Ψ, insead of L lage eigenalues whee < L Bu an is usually unnown in eceie o esimae an, diffeence beween (i)h (i+1)h descended soed eigenalue of M eigenalues of YY /K is calculaed oe ha calculaion is begun o (M- 1)h Mh eigenalues If he esul of diiding of subaced alue o he lages eigenalue is less han a pedefined heshold hen an is deceased one uni an finding pocedue will be pesis unil M-1 compuaions Afe he idenificaion of he an, signal subspace esimaion is pefomed Gien he signal subspace S ˆ, MUSIC fo soling he join spase ecoey poblem acceps each columns of Ψ if ha column is posed on he esimaed subspace coninues unil L columns is seleced MUSIC mehod is summaized in algoihm 1 o find he unnown iual channel maix In algoihm 1, funcion Eigen efes o eigenalue decomposiion euns bac a diagonal maix included eigenalues Algoihm 1 MUSIC 1 EVs =so(eigen( YY /K)) degeneaely; if K<L 3 =whole of numbe of EVs lage han zeo; 4 else 5 ρ = an esimaion fom M EVs as explained in secion IVA; 6 = min ( ρ, L ); 7 consuc maix U ŝ fom eigeneco popoional o lages eigenalues; 8 P ŝ = UU s ˆ s ˆ ; 9 = { }; 10 fo j=1: 11 κ = PˆΨ / Ψ j s j j 1 end fo 13 = { selec L lages elemens of eco κ }; 14 ˆ = Ψ Y ;,( ) ( ) B SUBSPACE AUGMEAIO MEODS he ange of he maix Ψ is nohing moe han he space spanned by se of all possible linea combinaions columns of Ψ is denoed by R ( Ψ ) Pacically speaing, Signal subspace, S ˆ, is esimaed fom he finiely many snapsho maix Y = Ψ + Ε exacly compued fom YY /K when K Mahemaically speaing, signal subspace is defined by Δ S = R( Ψ ) = R( Ψ,( ) ) (17) When,( ) is a full ow an maix, he ange of signal subspace of Y = Ψ + Ε, ie R ( Ψ,( ) ), is coincided wih R ( Ψ ) If we assume ha,( ) is a full ow an maix SR is high, hen R( Ψ ) = R( Y )amely, only he columns of Ψ ha seleced by elemens of, lie wihin R ( Ψ,( ) ) Consequenly, one can find he alue of elemens of suppo by pojecing he columns of Ψ ino ohogonal subspace of ( Ψ,( ))[11] R owee, in pacical issue of esimaing he an, due o limiaions in SR, finie numbe of K o uncaion of eigenalues, esimaion of signal subspace is inaccuae hus, equaion (19) is no saisfied we hae o minimize Pŝ Ψ o equialenly maximize Pŝ Ψ in algoihm 1 Also, in he an defecie case, ie, an(,( )) = < L, in spie of exac an esimaion, i may happen ha some of columns of sensing maix as an elemen of he suppo no belong o he signal subspace S hus MUSIC maes a misae o selec a ue componen of suppo se o pohibi he wong esimaing of he suppo, one can esimae a subspace spanned by L columns of Ψ as an augmenaion subspace by conenional MMV algoihms lie SOMP in a pobabilisic way aach his subspace o he -dimensional subspace obained by MUSIC mehod deeminisically he deails will be discussed in nex secion 1) Subspace Augmenaion MUSIC-MMV Suppose ha has L nonzeo ows wihin suppo {1,, } also is an defecie, ie < L Le be an abiay subse of wih L elemens If we esimae -dimensional subspace fom R ( Ψ,( ) ) hen we can wie, R( Ψ ) = Sˆ + R( Ψ ) (18) 69

5 whee S ˆ is esimaed subspace of R ( Ψ,( ) ) which obains by applying he EVD oe augmened signal subspace YY / he goal is o find an heoy 1: Suppose ha S is an augmened signal subspace wihin R ( Ψ ) If we hae equaion (18) as S = Sˆ + R( Ψ ) whee S ˆ is esimaed subspace, hen pojecion maix on augmened signal subspace aains as P = P + ( P Ψ )( P Ψ ) (19) S Sˆ Sˆ Sˆ he implemenaion on (19) seems impacical We now ha he pojecion ono R ( Ψ ) is achieed by = = PR ( Ψ ) ΨΨ UU 1 1 whee 1 = [ ] [ 1 1 ] U is aained by SVD on Ψ U U Σ V V [13] A pope subsiuion o consuc he pojecion ono Sˆ + R( Ψ ) is exploiing he ohonomal basis fom [ Uŝ Ψ ] by SVD o QR decomposiions owee, SVD doubles he compuaion ime ahe han QR facoizaion, bu poides moe eliable consisen an deeminaion Poposiion1: he selecion ule of augmened suppo by SA-MUSIC-MMV is aained as κ = U Ψ Ψ whee γ = {1: }\ / j aug γ( j) γ( j ) obseed maix is noisy Algoihm Subspace Augmenaion MUSIC-MMV (SA- MUSIC) 1 = { }; Coninue by sep 1 o 8 fom algoihm1, exploi Uŝ ; 3 γ = {1 : } \ 4 fo j=1 : lengh( γ ) 5 η ag max P UˆΨ = R( Ψ ) s :, [ ]\ 6 = η ; 7 end fo 8 U =[ U ˆ; Ψ ] ;find augmenaion subspace aug s 9 U =oh( U ) ; find ohonomal basis fo he ange of aug aug Uaug by SVD o QR 10 γ = {1 : } \ 11 fo j=1 : lengh( γ ) 1 κ / j = UaugΨγ( j) Ψ γ( j ) 13 end fo 14 = { lages elemens of κ }; 15 ˆ = Ψ Y ;,( ) ( ) P R ( Ψ ) is pependicula complemenay pojecion on ange of Ψ Afe esimaion of iual channel hus exacing esimaion of mmw channel, hans o assumpion of he Gaussian signalling oe he lin in (), we ae able o achiee he esimaed specal efficiency as follow, ˆ P s ˆ 1 log ˆ ˆ R = + I s n eff eff R (0) s whee ˆ ˆ ˆ ˆ eff = C P is effecie esimaed channel wih Cˆ = ˆ ˆ G P ˆ = G ˆ ˆ as esimaed combine pecode maices especiely Also, ˆ ˆˆ Rn = σ ncc is he esimaed combined noise coaiance maix in he downlin Consideing he esimaed iual channel fom one of he poposed algoihms using (11), he esimaed mmw channel model can be wien ˆ = U ˆ U (1),( ) By applying he singula alue decomposiion on Ĥ in (1) choosing he fis columns of lef-hed uniay maix, ie, Û, choosing he fis s s columns of ighhed uniay maix, ie, ˆV, finally eplacing hem ahe han Ĉ ˆP especiely, we can design ˆ, G ˆ, ˆ G ˆ by soling he geneal spase opimizaion poblem as follow, ( ˆ, ˆ ) G agmin Wˆ ˆ G ˆ, s x x = x x x :, l :, l i ˆ x x = s x x x {[ ]:, x,cod} ˆ ˆ G C 1 i, l = 1,,, G ˆ () whee subscip x can be subsiued fo o, W ˆ x can be eplaced by Û o ˆV dependen on x Cx is a geneal codeboo included of quanized seeing ecos fo ansmie o eceie is chosen fom π Q ( ), = 0,1,,x,cod 1,Q = Q (3) j π cos( ϑ) jπ( 1)cos( ) x ϑ x ( ϑ) = 1,e,,e / x Q in (3) is numbe of bis fo conolling he phase shifes x,cod is numbe of seeing ecos exising in ansmie/eceie codeboo V SIMULAIO RESULS In his secion, we ealuae numeical esuls of poposed algoihms o esimae a ypical millimee-wae MIMO channel wih hybid pecoding sucue compae hei pefomance o conenional MMV SMV poblems BS MS ae equipped wih 64 3 miniaue anenna 10 6 R chains especiely We assume ha scaee numbe is 6 independen on indoo o oudoo enionmen x,f 70

6 igue omalized Minimum Mean Squae Eo of Channel Esimaion Each scaee as a cluse fuhe assumed o conibue a single popagaion pah beween he BS MS R phase shifes in analog pas of pecode combine ae able o be conolled wih 7 quanizaion bis he opeaional caie fequency of he sysem is 8Gz wih consideaion of bwidh of 100Mz he pah-loss exponen is assumed o be β loss = 35 he angles of aial depaue ae seleced omly wih a unifom disibuion fom ange of [0, π ] In ig(a), we ealuae he MSE behaiou of iual channel esimaion in he poposed algoihms fo he cases L = 6 as mulipah numbe o spasiy leel of channel se K = o 6 while numbe of aining beam ecos, ie M is held o consan o 0 he impac of K on he pefomance of he poposed algoihms especially on measuemen in an enionmen wih consan AoDs AoAs aiable channel gain, has incemenal gowh When K=1, he algoihm uns ino he SMV needs o incease he numbe of measuemens, ie, M, fo bee esuls owee, enlaging of he measuemen eco inceases he ime of soling he poblem exhausingly he pefomance of poposed SA-MUSIC (MMV is seleced hee o be SOMP) wih deploymens of M=0 deceasing an o 3 oupefomances han he ull-ran MUSIC SOMP he pefomance of MUSIC has moe impoemen when K, ie numbe of columns of In ig4(a), he specal efficiency is epesened by he poposed mmwae channel esimaion algoihms when he desied numbe of pahs L equals 4 Algoihm SA-MUSIC MUSIC algoihms ae esed fo consan alues of BS MS anenna numbe wih diffeen R chains quanized bis in phase shifes, compaed wih he specal efficiency of he pefec channel VI COCLUSIO We hae exploed he abiliy of MUSIC based on subspace augmenaion fo Ran-defecie mmw-channel esimaion wih lage anenna aay umeical esuls show ha SA-MUSIC poides good specal efficiency nomalized MMSE REERECES [1] A Alhaeeb, O El Ayach, G Leus, R W eah, "Channel Esimaion ybid Pecoding fo Millimee Wae Cellula Sysems," Seleced opics in Signal Pocessing, IEEE Jounal of, ol 8, pp , 014 [] J D Blanchad, M Cema, D anle, J Yiong, "Geedy Algoihms fo Join Spase Recoey," Signal Pocessing, IEEE ansacions on, ol 6, pp , 014 igue3 Specal Efficiency of Esimaed mmw-channel [3] J D Blanchad, J anne, W Ke, "Conjugae Gadien Ieaie ad hesholding: Obseed oise Sabiliy fo Compessed Sensing," Signal Pocessing, IEEE ansacions on, ol 63, pp , 015 [4] J Bady, Behdad, A M Sayeed, "Beamspace MIMO fo Millimee-Wae Communicaions: Sysem Achiecue, Modeling, Analysis, Measuemens," Anennas Popagaion, IEEE ansacions on, ol 61, pp , 013 [5] C DMeye, "Maix Analysis Applied Linea Algeba," p 730, May 010 [6] O El Ayach, S Rajagopal, S Abu-Sua, P Zhouyue, R W eah, "Spaially Spase Pecoding in Millimee Wae MIMO Sysems," Wieless Communicaions, IEEE ansacions on, ol 13, pp , 014 [7] P eng, "Uniesal minimum-ae sampling specum-blind econsucion fo mulib signals," PhD disseaion, Decembe 1997 [8] S ouca, "Recoeing joinly spase ecos ia had hesholding pusui,," in Poc of SAMPA, 011,online [9] Ghauch, M Bengsson, K aejoon, M Soglund, "Subspace esimaion decomposiion fo hybid analog-digial millimee-wae MIMO sysems," in Signal Pocessing Adances in Wieless Communicaions (SPAWC), 015 IEEE 16h Inenaional Woshop on, 015, pp [10] R A a C R Johnson, "Maix Analysis," nd ed Cambidge Uniesiy Pess, 013 [11] K Jong Min, L O Kyun, Y Jong Chul, "Compessie MUSIC: Reisiing he Lin Beween Compessie Sensing Aay Signal Pocessing," Infomaion heoy, IEEE ansacions on, ol 58, pp , 01 [1] R a P V K S R Gibonal, "Aoms of all channels, unie! aeage case analysis of muli-channel spase ecoey using geedy algoihms," Jounal of ouie Analysis Applicaions, ol 14, pp , 008 [13] L Kiyung, Y Besle, M Junge, "Subspace Mehods fo Join Spase Recoey," Infomaion heoy, IEEE ansacions on, ol 58, pp , 01 [14] M L Riza Adeniz, Yuanpeng; Samimi, Mahew K; Sun, Shu; Rangan, Sundeep; Rappapo, heodoe S; Eip, Elza, "Millimee Wae Channel Modeling Cellula Capaciy Ealuaion," IEEE J Sel Aeas Comm, ol 3, pp , June 014 [15] A M Sayeed, "Deconsucing mulianenna fading channels," Signal Pocessing, IEEE ansacions on, ol 50, pp , 00 [16] Sooyoung, K aejoon, D J Loe, J V Kogmeie, A homas, A Ghosh, "Millimee Wae Beamfoming fo Wieless Bachaul Access in Small Cell ewos," Communicaions, IEEE ansacions on, ol 61, pp , 013 [17] R W J Rappapo, Daniels, J Mudoc, "Millimee wae wieless communicaions," Penice all, 014 [18] J A opp, "Algoihms fo simulaneous spase appoximaion Pa II: Conex elaxaion," Sig Poc, ol 86, pp , 006 [19] P Zhouyue Khan, "An inoducion o millimee-wae mobile boadb sysems," Communicaions Magazine, IEEE, ol 49, pp ,

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. :