Turbo-Like Beamforming Based on Tabu

Transcription

1 Tubo-Like Beamfoming Based on Tabu 1 Seach lgoihm fo Millimee-Wave Massive MIMO Sysems axiv: v1 [cs.it] 16 Jul 2015 Xinyu Gao, Suden Membe, IEEE, Linglong Dai, Senio Membe, IEEE, Chau Yuen, Senio Membe, IEEE, and Zhaocheng Wang, Senio Membe, IEEE bsac Fo millimee-wave (mmwave) massive MIMO sysems, he codebook-based analog beamfoming (including ansmi pecoding and eceive combining) is usually used o compensae he sevee aenuaion of mmwave signals. Howeve, convenional beamfoming schemes involve complicaed seach among pe-defined codebooks o find ou he opimal pai of analog pecode and analog combine. To solve his poblem, by exploing he idea of ubo equalize ogehe wih abu seach (TS) algoihm, we popose a Tubo-like beamfoming scheme based on TS, which is called Tubo-TS beamfoming in his pape, o achieve he nea-opimal pefomance wih low complexiy. Specifically, he poposed Tubo-TS beamfoming scheme is composed of he following wo key componens: 1) Based on he ieaive infomaion exchange beween he base saion and he use, we design a Tubo-like join seach scheme o find ou he nea-opimal pai of analog pecode and analog combine; 2) Inspied by he idea of TS algoihm developed in aificial inelligence, we popose a TS-based pecoding/combining scheme o inelligenly seach he bes pecode/combine in each ieaion of Tubo-like join seach wih low complexiy. nalysis shows ha he poposed Tubo-TS beamfoming can consideably educe he seaching complexiy, and simulaion esuls veify ha i can achieve he nea-opimal pefomance. Index Tems Beamfoming, millimee-wave, massive MIMO, abu seach, ubo equalize. X. Gao, L. Dai, and Z. Wang ae wih he Tsinghua Naional Laboaoy fo Infomaion Science and Technology (TNLis), Depamen of Eleconic Engineeing, Beijing , China ( gxy @sina.com, {daill, zcwang}@singhua.edu.cn). C. Yuen is wih he SUTD-MIT Inenaional Design Cene, Singapoe Univesiy of Technology and Design, 20 Dove Dive, Singapoe , Singapoe ( yuenchau@sud.edu.sg). This wok was suppoed by Naional Key Basic Reseach Pogam of China (Gan No. 2013CB329203), Naional High Technology Reseach and Developmen Pogam of China (Gan No ), and Naional Naue Science Foundaion of China (Gan Nos and ).

2 2 I. INTRODUCTION The inegaion of millimee-wave (mmwave) and massive muliple-inpu muliple-oupu (MIMO) is egaded as a pomising echnique fo fuue 5G wieless communicaion sysems [1], since i can povide odes of magniude incease boh in he available bandwidh and he specal efficiency [2]. On one hand, he vey sho wavelengh associaed wih mmwave enables a lage anenna aay o be easily insalled in a small physical dimension [3]. On he ohe hand, he lage anenna aay in massive MIMO can povide a sufficien anenna gain o compensae he sevee aenuaion of mmwave signals due o pah loss, oxygen absopion, and ainfall effec [1], as he beamfoming (including ansmi pecoding and eceive combining) echnique can concenae he signal in a naow beam. MmWave massive MIMO sysems usually pefom beamfoming in he analog domain, whee he ansmied signals o eceived signals ae only conolled by he analog phase shife (PS) newok wih low hadwae cos [1]. Compaed wih adiional digial beamfoming, analog beamfoming can decease he equied numbe of expensive adio fequency () chains a boh he base saion (BS) and uses, which is cucial o educe he enegy consumpion and hadwae complexiy of mmwave massive MIMO sysems [4]. Exising dominan analog beamfoming schemes can be geneally divided ino wo caegoies, i.e., he non-codebook beamfoming and he codebook-based beamfoming. Fo he noncodebook beamfoming, hee ae aleady some excellen schemes. In [5] [7], a low-complexiy analog beamfoming, whee wo PSs ae employed fo each eny of he beamfoming maix, is poposed o achieve he opimal pefomance of fully digial beamfoming. Howeve, hese mehods equie he pefec channel sae infomaion (CSI) o be acquied by he BS, which is vey challenging in pacice, especially when he numbe of chains is limied [1]. By conas, he codebook-based beamfoming can obain he opimal pai of analog pecode and analog combine by seaching he pe-defined codebook wihou knowing he exac channel. The mos inuiive and opimal scheme is full seach (FS) beamfoming [8]. Howeve, is complexiy inceases exponenially wih he numbe of chains and quanified bis of he angles of aival and depaue (o/ods). To educe he seaching complexiy of codebook-based beamfoming, some low-complexiy schemes, such as he ones adoped by sandads IEEE c [9] and IEEE ad [10], have aleady been poposed. Fuhemoe, a muli-level codebook ogehe wih a ping-pong seaching scheme is also poposed in [11]. These schemes can educe he seaching complexiy wihou obvious pefomance loss. Howeve, hey usually involve a lage numbe of ieaions o exchange he infomaion beween he use and he BS, leading o a high ovehead fo pacical sysems. To educe boh he seaching complexiy and he ovehead of codebook-based beamfoming, in his

3 3 pape, we popose a Tubo-like beamfoming scheme based on abu seach algoihm [12] (called as Tubo- TS beamfoming) wih nea-opimal 1 pefomance fo mm-wave massive MIMO sysems. Specifically, he poposed Tubo-TS beamfoming scheme is composed of he following wo key componens: 1) Based on he ieaive infomaion exchange beween he BS and he use, we design a Tubo-like join seach scheme o find ou he nea-opimal pai of analog pecode and analog combine; 2) Inspied by TS algoihm in aificial inelligence, we develop a TS-based pecoding/combining o inelligenly seach he bes pecode/combine in each ieaion of Tubo-like join seach wih low complexiy. Fuhemoe, he conibuions of he poposed TS-based pecoding/combing can be summaized in he following hee aspecs: 1) Povide he appopiae definiions of neighbohood, cos, and sopping cieion involved in TS-based pecoding/combing; 2) Take he exac soluion insead of he convenional move as abu o guaanee a wide seaching ange; 3) Popose a esa mehod by selecing seveal diffeen iniial soluions unifomly disibued in he codebooks o fuhe impove he pefomance. I is shown ha he poposed Tubo-TS beamfoming can consideably educe he seaching complexiy. We veify hough simulaions ha Tubo-TS beamfoming can appoach he pefomance of FS beamfoming [8]. The es of his pape is oganized as follows. Secion II biefly inoduces he sysem model of mmwave massive MIMO. Secion III specifies he poposed Tubo-TS beamfoming. The simulaion esuls of achievable ae ae shown in Secion IV. Finally, conclusions ae dawn in Secion V. Noaion: Lowe-case and uppe-case boldface lees denoe vecos and maices, especively; ( ) T, ( ) H, ( ) 1, and de( ) denoe he anspose, conjugae anspose, invesion, and deeminan of a maix, especively; E( ) denoes he expecaion; Finally, I N is he N N ideniy maix. II. SYSTEM MODEL We conside he mmwave massive MIMO sysem wih beamfoming as shown in Fig. 1, whee he BS employs N anennas and N wih N anennas and N N N = N chains o simulaneously ansmi N s daa seams o a use chains. To fully achieve he spaial muliplexing gain, we usually have = N s [13]. TheN s independen ansmied daa seams in he baseband fisly pass hough chain o be conveed ino analog signals. fe ha, he oupu signals will be pecoded by an N N analog pecode P as x = P s befoe ansmission, whee s is he N s 1 ansmied signal veco subjec o he nomalized powe E ( ss H) = 1 N s I Ns. Noe ha he analog pecode P is usually ealized by a PS newok wih low hadwae complexiy [1], which equies ha all elemens of 1 Noe ha nea-opimal means achieving he pefomance close o ha of he opimal FS beamfoming.

4 P should saisfy p i,j 2 = 1 N. Unde he naowband block-fading massive MIMO channel [13], he N 1 eceived signal veco a he use can be pesened as = ρhp s+n, (1) whee ρ is he ansmied powe, H C N N denoes he channel maix which will be discussed in deail lae in his secion, and n = [n 1,,n N ] T is he addiive whie Gaussian noise (WGN) veco, whose enies follow he independen and idenical disibuion (i.i.d.) CN(0,σ 2 I N ). as he use side, an N N analog combine C is employed o pocess he eceived signal veco y = C H = ρc H HP s+c H n, (2) whee he elemens of C have he simila consains as ha of P, i.e., 2 = 1 Due o he limied numbe of significan scaes and seious anenna coelaion of mmwave communicaion [14], in his pape we adop he widely used geomeic Saleh-Valenzuela channel model [13], whee he channel maix H can be pesened as N N L H = α l f (φ l L )fh l=1 c i,j N. ( φ l ), (3) whee L is he numbe of significan scaes, and we usually have L min(n,n ) fo mmwave communicaion sysems due o he spase naue of scaes, α l C is he gain of he lh pah including he pah loss, φ l and φ l ae he azimuh of ods/os of he lh pah, especively. Finally, f ( φ l ) and f (φ l ) ae he anenna aay esponse vecos which depend on he anenna aay sucue a he BS and he use. When he widely used unifom linea aays (ULs) ae consideed, we have [13] f ( φ l ) = 1 N [1,e jkdsin(φ l ),,e j(n 1)kdsin(φ l )] T, (4) f (φ l ) = 1 N [1,e jkdsin(φ l ),,e j(n 1)kdsin(φ l )] T, (5) whee k = 2π λ, λ denoes he wavelengh of he signal, and d is he anenna spacing. 4 III. NER-OPTIML TURBO-TS BEMFORMING WITH LOW COMPLEXITY In his secion, we fis give a bief inoducion of he codebook-based beamfoming, which is widely used in mmwave massive MIMO sysems. fe ha, a low-complexiy nea-opimal Tubo- TS beamfoming scheme is poposed, which consiss of Tubo-like join seach scheme and TS-based pecoding/combining. Finally, he complexiy analysis is povided o show he advanage of he poposed Tubo-TS beamfoming scheme.

5 5. Codebook-based beamfoming ccoding o he special chaaceisic of mmwave channel, he beamseeing codebook [8] is widely used. Specifically, le F and W denoe he beamseeing codebooks fo he analog pecode and analog combine, especively. If we use B (B ) bis o quanify he od (o), F (W) will consis of all he possible analog pecode (combine) maices P (C ), which can be pesened as [8] [ ) ) )] P = f ( φ 1,f ( φ 2,,f ( φ N, (6) [ ) ) )] C = f ( φ 1,f ( φ 2,,f ( φ N, (7) whee he quanified od φ i fo i = 1,,N { } whee n 1, 2 B. Similaly, he quanified o φ j possible candidaes, i.e., φ { j = 2πn whee n 1, 2 B 2 B of W ae 2 B N and 2 B N a he BS has 2 B possible candidaes, i.e., φ i = 2πn 2 B fo j = 1,,N a he use has 2 B }. Thus, he cadinaliies F of F and W, especively. Then, by joinly seaching F and W, he opimal pai of analog pecode and analog combine can be seleced by maximizing he achievable ae as [13] ( R = max log 2 I Ns + ρ ) R 1 n C H P F,C W N HP P H H H C = max log 2(ϕ(P,C )), (8) s P F,C W whee R n = σ 2 C H C pesens he covaiance maix of noise afe combining, and ϕ(p,c )= I N s + ρ R 1 n N CH HP P H HH C (9) s is defined as he cos funcion. We can obseve ha o obain he opimal pai of analog pecode and analog combine, we need o exhausively seach he codebooks F and W. When N B = N = 2, = 6, he oally equied imes of seach is , which is almos impossible in pacice. In his pape, we popose a Tubo-TS beamfoming o educe he seaching complexiy. The poposed Tubo-TS beamfoming is composed of wo key componens, i.e., Tubo-like join seach scheme and TS-based pecoding/combining, which will be descibed in deail in he following Secion III-B and Secion III-C, especively. B. Tubo-like join seach scheme Based on he idea of he infomaion ineacion in he well-known ubo equalize, we popose a Tubolike join seach scheme o find ou he nea-opimal pai of analog pecode and analog combine, which is shown in Fig. 2. Le P op,k and C op,k denoe he nea-opimal analog pecode and analog combine obained in he kh ieaion, especively, whee k = 1,2,,K, and K is he pe-defined maximum numbe of ieaions. Fisly, he BS selecs an iniial pecodep op,0, which can be an abiay candidae

6 6 in F, o ansmi a aining sequence o he use. Then he use can seach he bes analog combine C op,1. fe ha, he use uses C op,1 o ansmi a aining sequence o he BS, and in eun he BS can seach he bes analog pecode P op,1. We epea such ieaion fo K imes in a simila way as he ubo equalize, and oupu P op,k and C op,k as he final pai of analog pecode and analog combine, which is expeced o achieve he nea-opimal pefomance as will be veified lae in Secion IV. Noe ha in each ieaion, seaching he bes analog pecode (combine) afe a poenial analog combine (pecode) has been seleced fom he codebook W (F) can be ealized by he poposed TS-based analog pecoding/combinging wih low complexiy, which will be descibed in deail in he nex subsecion. C. TS-based pecoding/combining In his subsecion, we fis focus on he pocess of seaching he bes analog pecode P afe a poenial analog combine C has been seleced. The pocess of seaching he bes analog combine C afe a ceain analog pecode P has been seleced can be deived in he simila way. The basic idea of he poposed TS-based analog pecoding can be descibed as follows. TS-based analog pecoding sas fom an iniial soluion, i.e., an analog pecode maix seleced fom he codebook F, and defines a neighbohood aound i (seveal analog pecode maices fom F based on a neighboing cieion). fe ha, i selecs he mos appopiae soluion among he neighbohood as he saing poin fo he nex ieaion, even if i is no he global opimum. Duing he seach in he neighbohood, TS aemps o escape fom he local opimum by uilizing he concep of abu, whose definiion can be changed accoding o diffeen cieions (e.g., convegence speed, complexiy, ec). This pocess will be coninued unil a ceain sopping cieion is saisfied, and finally he bes soluion among all ieaions will be declaed as he final soluion. Nex, five impoan aspecs of he poposed TS-based pecoding, including neighbohood definiion, cos compuaion, abu, sopping cieion, and TS algoihm, will be explained in deail as follows. 1) Neighbohood definiion: Noe ha he mh column of analog pecode P can be pesened by an { } ( ) index q m 1,2,,2 B, which coesponds o he veco f 2πqm as defined in (4) and (6). Then 2 B an analog pecode is defined as a neighbo of P if: i) i has only one column ha is diffeen fom he coesponding column in P ; ii) he index diffeence beween he wo coesponding columns equals one. Fo example, when N = 2 and B = 3, fo a possible analog pecode P = [ ( f 3π ) ( 7π )] 4,f 4, anohe pecode [ ( f 2π ) ( 7π )] 4,f 4 is a neighbo of P. Le P (i) denoe he saing poin in he ih ieaion of he poposed TS-based analog pecoding, and ( ) { } V P (i) = V (i) 1,V(i) 2, V(i) V pesens he neighbohood of P (i), whee V is he cadinaliy of V.

7 7 ccoding o he neighbohood definiion above, i is obvious ha V = 2N. We hen define ha he ( ) uh neighbo in V P (i) is diffeen fom P (i) in he u 2 h column, and he index of he coesponding column is q u/2 +( 1) mod (u,2), whee q u/2 is he index of his column. To avoid oveflow of he definiion above, we se 2 B ( ) 1+( 1) mod (u,2) = max 1+( 1) mod (u,2),1, (10) ( +( 1) mod (u,2) = min 2 B +( 1) mod (u,2),2 B ). (11) Fo example, he neighbohood of one analog pecodep (i) =[ ( f 3π ) ( 7π )] (i) 4,f 4 isv 1 =[ ( f 2π ) ( 7π )] 4,f 4, V (i) 2 =[ ( f 4π ) ( 7π )] (i) 4,f 4, V 3 =[ ( f 3π ) ( 6π )] (i) 4,f 4, and V 4 =[ ( f 3π ) ( 8π )] 4,f 4. 2) Cos compuaion: We define he value of he cos funcion ϕ(p,c ) in (9) as he eliabiliy meic of a possible soluion, i.e., a soluion P leading o a lage value of ϕ(p,c ) is a bee soluion. Fuhe, accoding o he neighbohood definiion, we can obseve ha once we obain he cos of P, we do no need o ecompue (9) o obain he cos of is neighbohood hough infomaion exchange beween he BS and he use. This is due o he fac ha he neighbo V u of P only has he u 2 h column ha is diffeen fom he coesponding one in P, hen he updaed effecive channel maix C H HV u in (9) also has he u 2 h column ha is diffeen fom he coesponding one in he oiginal effecive channel maix C H HP, whee such diffeence can be easily calculaed since P and V u ae known. Moe impoanly, his special popey indicaes ha fo he poposed TS-based analog pecoding, we can only esimae he effecive channel maix C H HP of size N N hough ime-domain and/o fequency-domain aining sequence [15], whose dimension is much lowe han he oiginal dimension N N of he exac channel maix H. 3) Tabu: In he convenional TS algoihm [12], he abu is usually defined as he move, which can be egaded as he diecion fom one soluion o anohe one fo he analog pecoding poblem. The move can be denoed by (a,b), whee a = 1,,N denoes ha he ah column of he oiginal soluion is diffeen fom ha of he cuen soluion, b { 1, 1} means he changed index of his paicula column fom he oiginal soluion o he cuen soluion. Conside he example above, he move (diecion) fom [ ( f 3π ) ( 7π )] [ ( 4,f 4 o 2π ) ( 7π )] f 4,f 4 can be wien as (1, 1). Regading he move as abu can save soage of he abu lis, since i only equies a abu lis of size 2N 1, whose elemen akes he value fom {0,1} o indicae whehe a move is abu o no (i.e., 1 is abu, and 0 is unconsained). Howeve, as shown in Fig. 3 (a), his mehod may lead o he unexpeced fac ha one soluion will be seached wice, and he cos funcion of he same neighbohood will be compued again. To solve his poblem, we popose o ake he exac soluion as abu. Specifically, le p = 1,2,,2 B N pesen

8 8 he index of a candidae of he analog pecode (soluion) ou of F wih 2 B N possible candidaes. Paiculaly, p can be calculaed by each column index q m (1 q m N ) 2 of his analog pecode as Fo example, when B p = = 3 and N N m=1 ( (q m 1) 2 B ) N m +1. (12) = 2, if an analog pecode has he column indexes {2,7}, hen he index of his analog pecode in F is p = 15 accoding o (12). In his way, ou mehod can efficienly avoid one soluion being seached wice, and heefoe a wide seaching ange can be achieved as shown in Fig. 3 (b). Noe ha he only cos of ou mehod is he inceased soage size of he abu lis fom 2N o 2 B N. 4) Sopping cieion: We define flag as a paamee o indicae how long (in ems of numbe of ieaions) he global opimal soluion has no been updaed. Tha means in he cuen ieaion, if a subopimal soluion is seleced as he saing poin fo he nex ieaion, we have flag = flag+1, ohewise, if he global opimal soluion is seleced, we se flag = 0. Based on his mechanism, TSbased analog pecoding will be eminaed when eihe of he following wo condiions is saisfied: i) The oal numbe of ieaions eaches he pe-defined maximum numbe of ieaions max ie; ii) The numbe of ieaions fo he global opimal soluion no being updaed eaches he pe-defined maximum value max len, i.e., flag = max len. Noe ha we usually se max len < max ie, which means if TSbased analog pecoding has aleady found he opimal soluion a he beginning, all he saing poins in following ieaions will be subopimal, so we don need o wai max ie ieaions. Theefoe, he aveage seaching complexiy can be educed fuhe. 5) Tabu seach algoihm: Le G (i) denoe he analog pecode achieving he maximum cos funcion (9) ha has been found unil he ih ieaion. TS-based analog pecoding sas wih he iniial soluion P (0). Noe ha in ode o impove he pefomance of TS-based analog pecoding, we can selec M diffeen iniial soluions unifomly disibued in F o sa TS-based analog pecoding M imes, hen, he bes one ou of M obained soluions will be declaed as he final analog pecode. Fo each iniial soluion, we se G (0) = P (0), flag = 0. Besides, all he elemens of he abu lis ae se as zeo. Consideing he ih ieaion, TS-based analog pecoding execues as follows: Sep 1: Compue he cos funcion (9) of he 2N neighbos of P (i) given he effecive channel 2 I is woh poining ou ha o fully achieve he spaial muliplexing gain, he column index q m should be diffeen fo diffeen chains, i.e., q 1 q 2 q N. ll he possible pecode/combine maices ha do no obey his consain will be declaed as abu o avoid being seached.

9 9 maix C H HP(i). Le V 1 = ag max 1 u 2N ϕ(v u,c ). (13) Calculae he index p 1 of V 1 in F accoding o (12). Then, V 1 will be seleced as he saing poin fo he nex ieaion when eihe of he following wo condiions is saisfied: If V 1 canno be seleced, we find he second bes soluion as ϕ ( V 1,C ) > ϕ ( G (i),c ), (14) V 2 = ag max ( p 1) = 0. (15) 1 u 2N Vu V 1 ϕ(v u,c ). (16) Then we decide whehe V 2 can be seleced by checking (14) and (15). This pocedue will be coninued unil one soluion V is seleced as he saing poin fo he nex ieaion. Noe ha if hee is no soluion saisfying (14) and (15), all he coesponding elemens of he abu lis will be se o zeo, and he same pocedue above will be epeaed. Sep 2: fe a soluion has been seleced as he saing poin, i.e., P (i+1) = V, we se ( ) (p ) = 0, G (i+1) = P (i+1), if ϕ P (i+1),c > ϕ ( G (i) ),C, ( ) (p ) = 1, G (i+1) = G (i), if ϕ P (i+1),c ϕ ( G (i) ) (17),C. TS-based analog pecoding will be eminaed in Sep 2 and oupu G (i+1) as he final soluion if he sopping cieion is saisfied. Ohewise i will go back o Sep 1 and epea he pocedue above unil i saisfies he sopping cieion. I is woh poining ou ha seaching he nea-opimal analog combine C afe a ceain analog pecode P has been seleced can be also solved by simila pocedue descibed above, whee he definiions such as neighbohood should be changed accodingly o seach he nea-opimal analog combine C. D. Complexiy analysis In his subsecion, we povide he complexiy compaison beween he poposed Tubo-TS beamfoming and he convenional FS beamfoming. I is woh poining ou ha alhough he poposed Tubo- TS beamfoming equies some exa infomaion exchange beween he BS and he UE (K imes of ieaions) as discussed in Secion III-B, he coesponding ovehead is ivial compaed wih he seaching complexiy, since K is usually small (e.g., K = 4 as will be veified by simulaion esuls). Theefoe,

10 10 in his secion we evaluae he complexiy as he oal numbe of soluions need o be seached. I is obvious ha he seaching complexiy of FS beamfoming C FS is C FS = N 2 B N 2 B. (18) By conas, he seaching complexiy of he poposed Tubo-TS beamfoming C TS is N C TS = ( 2N max ie+2n max ie ) MK. (19) Compaing (18) and (19), we can obseve ha he complexiy of Tubo-TS beamfoming is linea wih and N, and i is independen of B and B, which indicaes ha Tubo-TS beamfoming enjoys a much lowe complexiy han FS beamfoming. Table I shows he compaison of he seaching complexiy beween Tubo-TS beamfoming and FS beamfoming when he numbes of chains a he BS and he use ae N = N = 2, whee hee cases ae consideed: 1) Fo B = 4, we se max ie = 500 and max len = 100, and unifomly selec M = 1 diffeen iniial soluions o iniiae he TS-based pecoding/combining; 2) Fo B and M = 2; 3) Fo B = 5, we se max ie = 1000, max len = 200, = 6, we se max ie = 3000, max len = 600, and M = 5. Besides, fo all hese cases above, we se he oal numbe of ieaions K = 4 fo he Tubo-like join seach scheme. Fom Table I, we can obseve ha he poposed Tubo-TS beamfoming scheme has much lowe seaching complexiy han he convenional FS beamfoming, e.g., when B =B = 6, he seaching complexiy of Tubo-TS beamfoming is only 2.1% of ha of FS beamfoming. IV. SIMULTION RESULTS We evaluae he pefomance of he poposed Tubo-TS beamfoming in ems of he achievable ae. Hee we also povide he pefomance of he ecenly poposed beam seeing scheme [16] wih coninuous angles as he benchmak fo compaison, since i can be egaded as he uppe bound of he poposed Tubo-TS beamfoming wih quanified o/ods. The sysem paamees fo simulaion ae descibed as follows: The caie fequency is se as 28GHz; We geneae he channel maix accoding o he channel model [13] descibed in Secion II; The os/ods ae assumed o follow he unifom disibuion wihin [0,π]; The complex gain α l of he lh pah follows α l CN (0,1), and he oal numbe of scaeing popagaion pahs is se as L = 3; Boh he ansmi and eceive anenna aays ae ULs wih anenna spacingd = λ/2. Thee cases of quanified bis pe os/ods, i.e., B and B = 4, B = 5, = 6 ae evaluaed; SNR is defined as ρ σ 2 ; ddiionally, he paamees used fo he poposed TS-based pecoding/combing ae he same as hose in Secion III-D.

11 11 fis, we povide he achievable ae pefomance of Tubo-TS beamfoming agains diffeen paamees o explain why we choose hese values as lised in Secion III-D. Fig. 4 shows a example when N N = 16 64, N = N = N s = 2, B = 6, and SNR = 0 db. We can obseve ha when max ie = 3000 (Fig. 4 (a)), max len = 600 (Fig. 4 (b)), M = 5 (Fig. 4 (c)), and K = 4 (Fig. 4 (d)), he poposed Tubo-TS beamfoming can achieve moe han 90% of he ae of FS beamfoming, which veifies he aionaliy of ou selecion. Fig. 5 shows he achievable ae compaison beween he convenional FS beamfoming and he poposed Tubo-TS beamfoming fo an N N = mmwave massive MIMO sysem wih N = N = N s = 2. We can obseve ha Tubo-TS beamfoming can appoach he achievable ae of FS beamfoming wihou obvious pefomance loss. Fo example, when B = 4 and SNR = 0 db, he ae achieved by Tubo-TS beamfoming is 7 bi/s/hz, which is quie close o 7.2 bi/s/hz achieved by FS beamfoming. When he numbe of quanified bis pe os/ods inceases, boh Tubo- TS beamfoming and FS beamfoming can achieve bee pefomance close o he beam seeing scheme wih coninuous os/ods [16]. Meanwhile, Tubo-TS beamfoming can sill guaanee he saisfying pefomance quie close o FS beamfoming. Consideing he consideably educed seaching complexiy of Tubo-TS beamfoming, we can conclude ha he poposed Tubo-TS beamfoming achieves a much bee ade-off beween pefomance and complexiy. Fig. 6 shows he achievable ae compaison fo an N N = mmwave massive MIMO sysem, whee he numbe of chains is sill se as N = N = N s = 2. Fom Fig. 6, we can obseve simila ends as hose fom Fig. 5. Moe impoanly, compaing Fig. 5 and Fig. 6, we can find ha he pefomance of he poposed Tubo-TS beamfoming can be impoved by inceasing he numbe of low-cos anennas insead of inceasing he numbe of expensive chains. Fo example, when N N = 16 64, B = 6, and SNR = 0 db, Tubo-TS beamfoming can achieve he ae of 10.1 bi/s/hz, while when N N = , he achievable ae can be inceased o 14 bi/s/hz wihou inceasing he numbe of chains. V. CONCLUSIONS In his pape, we popose a Tubo-TS beamfoming scheme, which consiss of wo key componens: 1) a Tubo-like join seach scheme elying on he ieaive infomaion exchange beween he BS and he use; 2) a TS-based pecoding/combining uilizing he idea of local seach o find he bes pecode/combine in each ieaion of Tubo-like join seach wih low complexiy. nalysis has shown ha he complexiy of he poposed scheme is linea wih N and N, and i is independen of B and B, which

12 12 can consideably educe he complexiy of convenional schemes. Simulaion esuls have veified ha he nea-opimal pefomance of he poposed Tubo-TS beamfoming. Ou fuhe wok will focus on exending he poposed Tubo-TS beamfoming o he muli-use scenaio. REFERENCES [1] W. Roh, J.-Y. Seol, J. Pak, B. Lee, J. Lee, Y. Kim, J. Cho, K. Cheun, and F. yanfa, Millimee-wave beamfoming as an enabling echnology fo 5G cellula communicaions: Theoeical feasibiliy and pooype esuls, IEEE Commun. Mag., vol. 52, no. 2, pp , Feb [2] T. L. Mazea, Noncoopeaive cellula wieless wih unlimied numbes of base saion anennas, IEEE Tans. Wieless Commun., vol. 9, no. 11, pp , Nov [3] S. Han, C.-L. I, Z. Xu, and C. Rowell, Lage-scale anenna sysems wih hybid pecoding analog and digial beamfoming fo millimee wave 5G, IEEE Commun. Mag., vol. 53, no. 1, pp , Jan [4] T. E. Bogale and L. B. Le, Beamfoming fo muliuse massive MIMO sysems: Digial vesus hybid analog-digial, in Poc. IEEE Global Communicaions Confeence (GLOBECOM 14), Dec. 2014, pp [5] X. Zhang,. F. Molisch, and S.-Y. Kung, Vaiable-phase-shif-based -baseband codesign fo MIMO anenna selecion, IEEE Tans. Signal Pocess., vol. 53, no. 11, pp , Nov [6] E. Zhang and C. Huang, On achieving opimal ae of digial pecode by -baseband codesign fo MIMO sysems, in Poc. IEEE Vehicula Technology Confeence (VTC 14 Fall), Sep. 2014, pp [7] T. E. Bogale, L. B. Le, and. Haghigha, Hybid analog-digial beamfoming: How many chains and phase shifes do we need? axiv pepin axiv: , [8] T. Kim, J. Pak, J.-Y. Seol, S. Jeong, J. Cho, and W. Roh, Tens of Gbps suppo wih mmwave beamfoming sysems fo nex geneaion communicaions, in Poc. IEEE Global Communicaions Confeence (GLOBECOM 13), Dec. 2013, pp [9] J. Wang, Z. Lan, C.-W. Pyo, T. Baykas, C.-S. Sum, M.. Rahman, J. Gao, R. Funada, F. Kojima, H. Haada e al., Beam codebook based beamfoming poocol fo muli-gbps millimee-wave WPN sysems, IEEE J. Sel. eas Commun., vol. 27, no. 8, pp , Oc [10] C. Codeio, D. khmeov, and M. Pak, IEEE ad: inoducion and pefomance evaluaion of he fis muli-gbps WiFi echnology, in Poc CM In. Wokshop on mmwave commun., 2010, pp [11] S. Hu, T. Kim, D. Love, J. Kogmeie, T. Thomas, and. Ghosh, Millimee wave beamfoming fo wieless backhaul and access in small cell newoks, IEEE Tans. Commun., vol. 61, no. 10, pp , Oc [12] F. Glove, Tabu seach-pa I, ORS J. Compu., vol. 1, no. 3, pp , [13] O. El yach, S. Rajagopal, S. bu-sua, Z. Pi, and R. Heah, Spaially spase pecoding in millimee wave MIMO sysems, IEEE Tans. Wieless Commun., vol. 13, no. 3, pp , Ma [14] Z. Pi and F. Khan, n inoducion o millimee-wave mobile boadband sysems, IEEE Commun. Mag., vol. 49, no. 6, pp , Jun [15] L. Dai, Z. Wang, and Z. Yang, Specally efficien ime-fequency aining OFDM fo mobile lage-scale MIMO sysems, IEEE J. Sel. eas Commun., vol. 31, no. 2, pp , Feb [16] O. El yach, R. Heah, S. bu-sua, S. Rajagopal, and Z. Pi, The capaciy opimaliy of beam seeing in lage millimee wave MIMO sysems, in Poc. Signal Pocessing dvances in Wieless Communicaions (SPWC 13) Wokshops, 2013, pp

13 13 Tansmied Symbols Baseband Signal Pocessing Base Saion Chain nenna ay mmwave MIMO channel nenna ay Use Chain Baseband Signal Poessing Received Symbols Chain Chain N N Tansmi Pecode Receive Combine N N Fig. 1. chiecue of mmwave massive MIMO sysem wih beamfoming. BS Use op,0 P op,1 C TS-based analog combining TS-based analog pecoding op,1 P op,2 C TS-based analog combining TS-based analog pecoding P op, K 1 C op, K TS-based analog combining TS-based analog pecoding P op, K Fig. 2. Poposed Tubo-like join seach scheme.

14 14 Iniial soluion Candidae soluion Pohibied move Pohibied soluion (a) (b) Fig. 3. Illusaion of how he soluion abu can avoid one soluion being seached wice; (a) Convenional move abu; (b) Poposed soluion abu. TBLE I COMPLEXITY COMPRISON Convenional FS beamfoming [8] Poposed Tubo-TS beamfoming Complexiy aio (TS/FS) B B B = % = % = %

15 15 M K Fig. 4. chievable ae of Tubo-TS beamfoming agains diffeen paamees: (a) max ie; (b) max len; (c) M; (d) K.

16 FS beamfoming, B =B =4 [11] Tubo TS beamfoming, B =B =4 FS beamfoming, B =B =5 [11] chievable ae (bps/hz) Tubo TS beamfoming, B =B =5 FS beamfoming, B =B =6 [11] Tubo TS beamfoming, B =B =6 Beam seeing [20] SNR (db) Fig. 5. chievable ae compaison fo an N N = mmwave massive MIMO sysem wih N = N = N s = chievable ae (bps/hz) FS beamfoming, B =B =4 [11] Tubo TS beamfoming, B =B =4 FS beamfoming, B =B =5 [11] Tubo TS beamfoming, B =B =5 FS beamfoming, B =B =6 [11] Tubo TS beamfoming, B =B =6 Beam seeing [20] SNR (db) Fig. 6. chievable ae compaison fo ann N = mmwave massive MIMO sysem wihn = N = N s = 2.