Turbo-Like Beamforming Based on Tabu Search Algorithm for Millimeter-Wave Massive MIMO Systems

Transcription

1 IEEE TRNSCTIONS ON VEHICULR TECHNOLOGY, VOL. 65, NO. 7, JULY Turbo-Like Beamforming Based on Tabu Search lgorihm for Millimeer-Wave Massive MIMO Sysems Xinyu Gao, Linglong Dai, Chau Yuen, and Zhaocheng Wang bsrac For millimeer-wave mmwave massive muliple-inpu muliple-oupu MIMO sysems, codebook-based analog beamforming including ransmi precoding and receive combining is usually used o compensae he severe aenuaion of mmwave signals. However, convenional beamforming schemes involve complicaed search among predefined codebooks o find ou he opimal pair of analog precoder and analog combiner. To solve his problem, by exploring he idea of urbo equalizer ogeher wih he abu search TS algorihm, we propose a Turbo-like beamforming scheme based on TS, which is called Turbo-TS beamforming in his paper, o achieve near-opimal performance wih low complexiy. Specifically, he proposed Turbo-TS beamforming scheme is composed of he following wo key componens: 1 Based on he ieraive informaion exchange beween he base saion BS and he user, we design a Turbo-like join search scheme o find ou he near-opimal pair of analog precoder and analog combiner; and 2 inspired by he idea of he TS algorihm developed in arificial inelligence, we propose a TS-based precoding/combining scheme o inelligenly search he bes precoder/combiner in each ieraion of Turbo-like join search wih low complexiy. nalysis shows ha he proposed Turbo-TS beamforming can considerably reduce he searching complexiy, and simulaion resuls verify ha i can achieve near-opimal performance. Index Terms Beamforming, massive muliple-inpu muliple-oupu MIMO, millimeer wave mmwave, abu search TS, urbo equalizer. I. INTRODUCTION The inegraion of millimeer-wave mmwave and massive muliple-inpu muliple-oupu MIMO is regarded as a promising echnique for fuure fifh-generaion 5G wireless communicaion sysems [1] since i can provide an orders of magniude increase boh in he available bandwidh and he specral efficiency [2]. On one hand, he very shor wavelengh associaed wih mmwave enables a large anenna array o be easily insalled in a small physical dimension [3]. On he oher hand, he large anenna array in massive MIMO can provide a sufficien anenna gain o compensae he severe aenuaion of mmwave signals due o pah loss, oxygen absorpion, and rainfall effec [1] as he beamforming including ransmi precoding and receive combining echnique can concenrae he signal in a narrow beam. Manuscrip received March 14, 2015; revised June 1, 2015; acceped July 23, Dae of publicaion July 28, 2015; dae of curren version July 14, This work was suppored in par by he Inernaional Science and Technology Cooperaion Program of China under Gran 2015DFG12760; by Singapore s gency for Science, Technology, and Research STE Science and Engineering Research Council Projec under Gran ; by he Naional Naural Science Foundaion of China under Gran and Gran ; by Beijing Naural Science Foundaion under Gran ; and by he Foundaion of Shenzhen Governmen. The review of his paper was coordinaed by Prof. D. B. da Cosa. X. Gao, L. Dai, and Z. Wang are wih he Tsinghua Naional Laboraory for Informaion Science and Technology TNLis, Deparmen of Elecronic Engineering, Tsinghua Universiy, Beijing , China gxy @ sina.com; daill@singhua.edu.cn; zcwang@singhua.edu.cn. C. Yuen is wih he SUTD-MIT Inernaional Design Cenre, Singapore Universiy of Technology and Design, Singapore yuenchau@ sud.edu.sg. Color versions of one or more of he figures in his paper are available online a hp://ieeexplore.ieee.org. Digial Objec Idenifier /TVT MmWave massive MIMO sysems usually perform beamforming in he analog domain, where he ransmied signals or received signals are only conrolled by he analog phase shifer PS nework wih low hardware cos [1]. Compared wih radiional digial beamforming, analog beamforming can decrease he required number of expensive radio-frequency chains a boh he base saion BS and users, which is crucial o reduce he energy consumpion and hardware complexiy of mmwave massive MIMO sysems [4]. Exising dominan analog beamforming schemes can be generally divided ino wo caegories, i.e., he non-codebook beamforming and he codebookbased beamforming. For he non-codebook beamforming, here are already some excellen schemes. In [5] [7], a low-complexiy analog beamforming, where wo PSs are employed for each enry of he beamforming marix, is proposed o achieve he opimal performance of fully digial beamforming. However, hese mehods require he perfec channel sae informaion o be acquired by he BS, which is very challenging in pracice, paricularly when he number of chains is limied [1]. By conras, he codebook-based beamforming can obain he opimal pair of analog precoder and analog combiner by searching he predefined codebook wihou knowing he exac channel. The mos inuiive and opimal scheme is full-search FS beamforming [8]. However, is complexiy exponenially increases wih he number of chains and quanified bis of he angle of arrival o and he angle of deparure od. To reduce he searching complexiy of codebook-based beamforming, some low-complexiy schemes, such as he schemes adoped by he IEEE c [9] and IEEE ad [10] sandards, have already been proposed. Furhermore, a mulilevel codebook, ogeher wih a ping-pong searching scheme, is also proposed in [11]. These schemes can reduce he searching complexiy wihou obvious performance loss. However, hey usually involve a large number of ieraions o exchange he informaion beween he user and he BS, leading o a high overhead for pracical sysems. To reduce boh he searching complexiy and he overhead of codebook-based beamforming, in his paper, we propose a Turbo-like beamforming scheme based on he abu search TS algorihm [12] called as Turbo-TS beamforming wih near-opimal 1 performance for mmwave massive MIMO sysems. Specifically, he proposed Turbo- TS beamforming scheme is composed of he following wo key componens. 1 Based on he ieraive informaion exchange beween he BS and he user, we design a Turbo-like join search scheme o find ou he near-opimal pair of analog precoder and analog combiner. 2 Inspired by he TS algorihm in arificial inelligence, we develop a TS-based precoding/combining o inelligenly search he bes precoder/combiner in each ieraion of Turbo-like join search wih low complexiy. Furhermore, he conribuions of he proposed TS-based precoding/ combining can be summarized in he following hree aspecs. 1 Provide he appropriae definiions of neighborhood, cos, and sopping crierion involved in TS-based precoding/combining. 2 Take he exac soluion insead of he convenional move as abu o guaranee a wider searching range. 3 Propose a resar mehod by selecing several differen iniial soluions uniformly disribued in he codebooks o furher improve he performance. 1 Noe ha near-opimal means achieving he performance close o ha of he opimal FS beamforming IEEE. Personal use is permied, bu republicaion/redisribuion requires IEEE permission. See hp:// for more informaion.

2 5732 IEEE TRNSCTIONS ON VEHICULR TECHNOLOGY, VOL. 65, NO. 7, JULY 2016 Fig. 1. rchiecure of he mmwave massive MIMO sysem wih beamforming. I is shown ha he proposed Turbo-TS beamforming can considerably reduce he searching complexiy. We verify hrough simulaions ha Turbo-TS beamforming can approach he performance of FS beamforming [8]. The remainder of his paper is organized as follows. Secion II briefly inroduces he sysem model of mmwave massive MIMO. Secion III specifies he proposed Turbo-TS beamforming. The simulaion resuls of achievable rae are shown in Secion IV. Finally, conclusions are drawn in Secion V. Noaion: Lowercase and uppercase boldface leers denoe vecors and marices, respecively; T, H, 1,andde denoe he ranspose, conjugae ranspose, inversion, and deerminan of a marix, respecively; and E denoes he expecaion. Finally, I N is he N N ideniy marix. II. SYSTEM MODEL We consider he mmwave massive MIMO sysem wih beamforming, as shown in Fig. 1, where he BS employs N anennas and N chains o simulaneously ransmi N s daa sreams o a user wih N r anennas and Nr chains. To fully achieve he spaial muliplexing gain, we usually have N = Nr = N s [13]. The N s independen ransmied daa sreams in he baseband firs pass hrough N chain o be convered ino analog signals. fer his, he oupu signals will be precoded by an N N analog precoder P as x = P s before ransmission, where s is he N s 1 ransmied signal vecor subjec o he normalized power Ess H =1/N s I Ns. Noe ha he analog precoder P is usually realized by a PS nework wih low hardware complexiy [1], which requires ha all elemens of P should saisfy p i,j 2 = 1/N. Under he narrow-band blockfading massive MIMO channel [13], N r 1 received signal vecor r a he user can be presened as r = ρhp s + n 1 where ρ is he ransmied power; H C Nr N denoes he channel marix, which will be discussed in deail laer in his secion; and n =[n 1,,n Nr ] T is he addiive whie Gaussian noise vecor, whose enries follow he independen and idenical disribuion i.i.d. CN0,σ 2 I Nr [14]. he user side, an N r Nr analog combiner C is employed o process he received signal vecor r as y = C H r = ρc H HP s + C H n 2 where he elemens of C have he similar consrains as hose of P, i.e., c i,j 2 = 1/N r. Due o he limied number of significan scaers and serious anenna correlaion of mmwave communicaion [15], [16], in his paper, we adop he widely used geomeric Saleh Valenzuela channel model [13], where he channel marix H can be presened as N N r L H = α l f r φ r l L f H φ l 3 l=1 where L is he number of significan scaers, and we usually have L minn,n r for mmwave communicaion sysems due o he sparse naure of scaers; α l C is he gain of he lh pah including he pah loss; and φ l and φr l are he azimuh of ods/os of he lh pah, respecively. Finally, f φ l and f rφ r l are he anenna array response vecors, which depend on he anenna array srucure a he BS and he user. When he widely used uniform linear arrays ULs are considered, we have [13] f φ 1 [ l = 1,e jkd sinφ l,...,e jn 1kd sinφ l ] T 4 N f r φ r l = 1 [ 1,e jkd sinφr l,...,e jn r 1kd sinφ r l ] T 5 Nr where k =2π/λ, λ denoes he wavelengh of he signal, and d is he anenna spacing. III. NER-OPTIML TURBO-TBU SERCH BEMFORMING WITH LOW COMPLEXITY Here, we firs give a brief inroducion of he codebook-based beamforming, which is widely used in mmwave massive MIMO sysems. fer his, a low-complexiy near-opimal Turbo-TS beamforming scheme is proposed, which consiss of Turbo-like join search scheme and TS-based precoding/combining. Finally, he complexiy analysis is provided o show he advanage of he proposed Turbo-TS beamforming scheme.. Codebook-Based Beamforming ccording o he special characerisic of a mmwave channel, he beamseering codebook [8] is widely used. Specifically, le F and W denoe he beamseering codebooks for he analog precoder and he analog combiner, respecively. If we use B Br bis o quanify he od o, F W will consis of all he possible analog precoder combiner marices P C, which can be presened as [8] [ ] P = f φ 1, f φ 2,, f φn 6 [ C = f r φr 1, fr φr 2,, fr φr N r where he quanified od φ i for i =1,...,N possible candidaes, i.e., φ i = 2πn/2B ] 7 a he BS has 2 B where n {1,...,2 B }.

3 IEEE TRNSCTIONS ON VEHICULR TECHNOLOGY, VOL. 65, NO. 7, JULY of analog precoder and analog combiner, which is expeced o achieve near-opimal performance, as will be verified laer in Secion IV. Noe ha, in each ieraion, searching he bes analog precoder combiner afer a poenial analog combiner precoder has been seleced from he codebook W F can be realized by he proposed TS-based analog precoding/combining wih low complexiy, which will be described in deail nex. C. TS-Based Precoding/Combining Fig. 2. Proposed Turbo-like join search scheme. Similarly, he quanified o φ r j for j = 1,...,N r a he user has 2 B r possible candidaes, i.e., φr j = 2πn/2 B r where n {1,...,2 B r }. Thus, he cardinaliies F of F and W of W are 2 B N and 2 B r N r, respecively. Then, by joinly searching F and W, he opimal pair of analog precoder and analog combiner can be seleced by maximizing he achievable rae as [13] R = max P F,C W log 2 I Ns + ρ N s R 1 n CH HP P H HH C = max log 2 ϕp, C 8 P F,C W where R n = σ 2 C H C presens he covariance marix of noise afer combining, and ϕp, 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 observe ha, o obain he opimal pair of analog precoder and analog combiner, we need o exhausively search he codebooks F and W.WhenNr = N = 2 and B = Br = 6, he oally required ime of search is , which is almos impossible in pracice. In his paper, we propose Turbo-TS beamforming o reduce he searching complexiy. The proposed Turbo-TS beamforming is composed of wo key componens, i.e., Turbo-like join search scheme and TS-based precoding/ combining, which will be described in deail in Secion III-B and C, respecively. B. Turbo-Like Join Search Scheme Based on he idea of he informaion ineracion in he well-known urbo equalizer, we propose a Turbo-like join search scheme o find ou he near-opimal pair of analog precoder and analog combiner, which is shown in Fig. 2. Le P op,k and C op,k denoe he nearopimal analog precoder and analog combiner obained in he kh ieraion, respecively, where k = 1, 2,...,K,andK is he predefined maximum number of ieraions. Firs, he BS selecs an iniial precoder, which can be an arbirary candidae in F, o ransmi a raining sequence o he user. Then, he user can search he bes analog combiner C op,1. fer his, he user uses C op,1 o ransmi a raining sequence o he BS, and in reurn, he BS can search he bes analog precoder P op,1. We repea such ieraion for K imes in a similar way as he urbo equalizer and oupu P op,k and C op,k as he final pair P op,0 Here, we firs focus on he process of searching he bes analog precoder P afer a poenial analog combiner C has been seleced. The process of searching he bes analog combiner C afer a cerain analog precoder P has been seleced can be derived in he similar way. The basic idea of he proposed TS-based analog precoding can be described as follows. TS-based analog precoding sars from an iniial soluion, i.e., an analog precoder marix seleced from he codebook F, and defines a neighborhood around i several analog precoder marices from F based on a neighboring crierion. fer his, i selecs he mos appropriae soluion among he neighborhood as he saring poin for he nex ieraion, even if i is no he global opimum. During he search in he neighborhood, TS aemps o escape from he local opimum by uilizing he concep of abu, whose definiion can be changed according o differen crieria e.g., convergence speed and complexiy. This process will be coninued unil a cerain sopping crierion is saisfied, and finally, he bes soluion among all ieraions will be declared as he final soluion. Nex, five imporan aspecs of he proposed TS-based precoding, including neighborhood definiion, cos compuaion, abu, sopping crierion, and TS algorihm, will be explained in deail as follows. 1 Neighborhood Definiion: Noe ha he mh column of analog precoder P can be presened by an index q m {1, 2,...,2 B }, which corresponds o he vecor f 2πq m /2 B as defined in 4 and 6. Then, an analog precoder is defined as a neighbor of P if i i has only one column ha is differen from he corresponding column in P and ii if he index difference beween he wo corresponding columns is equal o one. For example, when N = 2andB = 3, for a possible analog precoder P =[f 3π/4, f 7π/4], anoher precoder [f 2π/4, f 7π/4] is a neighbor of P. Le P i denoe he saring poin in he ih ieraion of he proposed TS-based analog precoding, and VP i ={Vi 1, Vi 2,...,Vi V } presens he neighborhood of P i,where V is he cardinaliy of V. ccording o his neighborhood definiion, i is obvious ha V = 2N. We hen define ha he uh neighbor in VP i is differen from P i in he u/2 h column, and he index of he corresponding column is q u/2 + 1 mod u,2,whereq u/2 is he index of his column. To avoid overflow of his definiion, we se mod u,2 =max mod u,2, B + 1 mod u,2 =min 2 B + 1 mod u,2, 2 B. 11 For example, he neighborhood of one analog precoder P i = [f 3π/4, f 7π/4] is V i 1 =[f 2π/4, f 7π/4], V i 2 =[f 4π/4, f 7π/4], V i 3 =[f 3π/4, f 6π/4], and V i 4 =[f 3π/4, f 8π/4]. 2 Cos Compuaion: We define he value of he cos funcion ϕp, C in 9 as he reliabiliy meric of a possible soluion, i.e., a soluion P leading o a larger value of ϕp, C is a beer soluion. Furhermore, according o he neighborhood definiion, we can observe ha, once we obain he cos of P, we do no

4 5734 IEEE TRNSCTIONS ON VEHICULR TECHNOLOGY, VOL. 65, NO. 7, JULY 2016 TBLE I COMPLEXITY COMPRISON Fig. 3. Illusraion of how he soluion abu can avoid one soluion being searched wice. a Convenional move abu. b Proposed soluion abu. need o recompue 9 o obain he cos of is neighborhood hrough informaion exchange beween he BS and he user. This is due o he fac ha he neighbor V u of P only has he u/2 h column ha is differen from he corresponding column in P, and hen he updaed effecive channel marix C H HV u in 9 also has he u/2 h column ha is differen from he corresponding column in he original effecive channel marix C H HP, where such difference can be easily calculaed since P and V u are known. More imporanly, his special propery indicaes ha, for he proposed TS-based analog precoding, we can only esimae he effecive channel marix C H HP of size Nr N hrough ime-domain and/or frequency-domain raining sequence [17], [18], whose dimension is much lower han he original dimension N r N of he channel marix H. 3 Tabu: In he convenional TS algorihm [12], he abu is usually defined as he move, which can be regarded as he direcion from one soluion o anoher soluion for he analog precoding problem. The move can be denoed by a, b, wherea = 1,...,N denoes ha he ah column of he original soluion is differen from ha of he curren soluion, and b { 1, 1} means he changed index of his paricular column from he original soluion o he curren soluion. Considering his example, he move direcion from [f 3π/4, f 7π/4] o [f 2π/4, f 7π/4] can be wrien as 1, 1. Regarding he move as abu can save sorage of he abu lis since i only requires a abu lis of size 2N 1, whose elemen akes he value from {0, 1} o indicae wheher a move is abu or no i.e., 1 is abu, and 0 is unconsrained. However, as shown in Fig. 3a, his mehod may lead o he unexpeced fac ha one soluion will be searched wice, and he cos funcion of he same neighborhood will be compued again. To solve his problem, we propose o ake he exac N soluion as abu. Specifically, le p = 1, 2,...,2 B presen he index of a candidae of he analog precoder soluion ou of F wih 2 B N possible candidaes. In paricular, p can be calculaed by 2 of his analog precoder as each column index q m 1 q m N p = N m=1 q m 1 2 B N m For example, when B = 3andN = 2, if an analog precoder has he column indexes {2, 7}, hen he index of his analog precoder in F is p = 15 according o 12. This way, our mehod can efficienly avoid one soluion being searched wice, and herefore, a wider searching range can be achieved, as shown in Fig. 3b. Noe ha he only 2 I is worh poining ou ha o fully achieve he spaial muliplexing gain, he column index q m should be differen for differen chains, i.e., q 1 q 2 q N. ll he possible precoder/combiner marices ha do no obey his consrain will be declared as abu o avoid being searched. cos of our mehod is he increased sorage size of he abu lis from o 2 B N. 2N 4 Sopping Crierion: We define flag as a parameer o indicae how long in erms of he number of ieraions he global opimal soluion has no been updaed. This means ha, in he curren ieraion, if a subopimal soluion is seleced as he saring poin for he nex ieraion, we have flag = flag + 1; oherwise, if he global opimal soluion is seleced, we se flag = 0. Based on his mechanism, TS-based analog precoding will be erminaed when eiher of he following wo condiions is saisfied: i The oal number of ieraions reaches he predefined maximum number of ieraions max_ier, or ii he number of ieraions for he global opimal soluion no being updaed reaches he predefined maximum value max_len, i.e., flag = max_len. Noe ha we usually se max_len < max_ier, which means ha, if TS-based analog precoding has already found he opimal soluion a he beginning, all he saring poins in following ieraions will be subopimal; hus, we do no need o wai max_ier ieraions. Therefore, he average searching complexiy can be reduced furher. 5 TS lgorihm: Le G i denoe he analog precoder achieving he maximum cos funcion 9 ha has been found unil he ih ieraion. TS-based analog precoding sars wih he iniial soluion P 0.Noe ha, o improve he performance of TS-based analog precoding, we can selec M differen iniial soluions uniformly disribued in F o sar TS-based analog precoding M imes; hen, he bes one ou of M obained soluions will be declared as he final analog precoder. For each iniial soluion, we se G 0 = P 0, flag = 0. In addiion, all he elemens of he abu lis are se as zero. Considering he ih ieraion, TS-based analog precoding execues as follows. Sep 1: Compue he cos funcion 9 of he 2N P i given he effecive channel marix CH HPi.Le V 1 =arg max 1 u 2N neighbors of ϕv u, C. 13 Calculae he index p 1 of V 1 in F according o 12. Then, V 1 will be seleced as he saring poin for he nex ieraion when eiher of he following wo condiions is saisfied: ϕv 1, C >ϕ G i, C 14 p 1 =0. 15 If V 1 canno be seleced, we find he second bes soluion as V 2 =arg max ϕv u, C. 16 Vu V 1 1 u 2N Then, we decide wheher V 2 can be seleced by checking 14 and 15. This procedure will be coninued unil one soluion V is seleced as he saring poin for he nex ieraion. Noe ha, if here is no soluion saisfying 14 and 15, all he corresponding elemens of he abu lis will be se o zero, and he same procedure will be repeaed.

5 IEEE TRNSCTIONS ON VEHICULR TECHNOLOGY, VOL. 65, NO. 7, JULY Fig. 4. chievable rae of Turbo-TS beamforming agains differen parameers. a max_ier.b max_len.c M.d K. Sep 2: fer a soluion has been seleced as he saring poin, i.e., = V,wese p =0, G i+1 = P i+1, if ϕ P i+1, C >ϕ G i, C, 17 p =1, G i+1 = G i, if ϕ P i+1, C ϕ G i, C. P i+1 TS-based analog precoding will be erminaed in Sep 2 and oupu G i+1 as he final soluion if he sopping crierion is saisfied. Oherwise, i will go back o Sep 1, and repea he procedure unil i saisfies he sopping crierion. I is worh poining ou ha searching he near-opimal analog combiner C afer a cerain analog precoder P has been seleced can be also solved by he similar procedure previously described, where he definiions such as neighborhood should be changed accordingly o search he near-opimal analog combiner C. D. Complexiy nalysis Here, we provide he complexiy comparison beween he proposed Turbo-TS beamforming and he convenional FS beamforming. I is worh poining ou ha, alhough he proposed Turbo-TS beamforming requires some exra informaion exchange beween he BS and he UE K imes of ieraions as discussed in Secion III-B, he corresponding overhead is rivial compared wih he searching complexiy since K is usually small e.g., K =4as will be verified by simulaion resuls. Therefore, here, we evaluae he complexiy as he oal number of soluions needed o be searched. I is obvious ha he searching complexiy of FS beamforming C FS is C FS = N 2 B N r 2 B r. 18 By conras, he searching complexiy of he proposed Turbo-TS beamforming C TS is C TS = 2N max_ier + 2N r max_ier MK. 19 Comparing 18 and 19, we can observe ha he complexiy of Turbo-TS beamforming is linear wih N and Nr, and i is independen of B and Br, which indicaes ha Turbo-TS beamforming enjoys much lower complexiy han FS beamforming. Table I shows he comparison of he searching complexiy beween Turbo-TS beamforming and FS beamforming when he numbers of chains

6 5736 IEEE TRNSCTIONS ON VEHICULR TECHNOLOGY, VOL. 65, NO. 7, JULY 2016 a he BS and he user are N considered. = N r = 2, where hree cases are 1 For B = Br = 4, we se max_ier = 500 and max_len = 100 and uniformly selec M = 1 differen iniial soluions o iniiae he TS-based precoding/combining. 2 For B 200, and M = 2. 3 For B and M = 5. = B r = 5, we se max_ier = 1000, max_len = =B r =6, we se max_ier=3000, max_len = 600, In addiion, for all hese cases, we se he oal number of ieraions K = 4 for he Turbo-like join search scheme. In Table I, we can observe ha he proposed Turbo-TS beamforming scheme has much lower searching complexiy han he convenional FS beamforming, e.g., when B = Br =6, he searching complexiy of Turbo-TS beamforming is only 2.1% of ha of FS beamforming. IV. SIMULTION RESULTS We evaluae he performance of he proposed Turbo-TS beamforming in erms of he achievable rae. Here, we also provide he performance of he recenly proposed beam seering scheme [19] wih coninuous angles as he benchmark for comparison since i can be regarded as he upper bound of he proposed Turbo-TS beamforming wih quanified os/ods. The sysem parameers for he simulaion are described as follows: The carrier frequency is se as 28 GHz; We generae he channel marix according o he channel model [13] described in Secion II. The os/ods are assumed o follow he uniform disribuion wihin [0,π]. The complex gain α l of he lh pah follows α l CN0, 1, and he oal number of scaering propagaion pahs is se as L = 3. Boh he ransmi and receive anenna arrays are ULs wih anenna spacing d = λ/2. Three cases of quanified bis per o/od, i.e., B = Br = 4, B = Br = 5, and B = Br = 6, are evaluaed. SNR is defined as ρ/σ 2. ddiionally, he parameers used for he proposed TS-based precoding/combining are he same as hose in Secion III-D. firs, we provide he achievable rae performance of Turbo-TS beamforming agains differen parameers o explain why we choose hese values as lised in Secion III-D. Fig. 4 shows an example when N r N = 16 64, N r = N = N s = 2, B = B r = 6, and SNR = 0 db. We can observe ha, when max_ier = 3000 [see Fig. 4a], max_len = 600 [see Fig. 4b], M = 5 [see Fig. 4c], and K = 4 [see Fig. 4d], he proposed Turbo-TS beamforming can achieve more han 90% of he rae of FS beamforming, which verifies he raionaliy of our selecion. Fig. 5 shows he achievable rae comparison beween he convenional FS beamforming and he proposed Turbo-TS beamforming for an N r N = mmwave massive MIMO sysem wih Nr = N = N s = 2. We can observe ha Turbo-TS beamforming can approach he achievable rae of FS beamforming wihou obvious performance loss. For example, when B = Br = 4andSNR= 0 db, he rae achieved by Turbo-TS beamforming is 7 bis/s/hz, which is quie close o 7.2 bis/s/hz achieved by FS beamforming. When he number of quanified bis per o/od increases, boh Turbo-TS beamforming and FS beamforming can achieve beer performance close o he beam seering scheme wih coninuous os/ods [19]. Meanwhile, Turbo-TS beamforming can sill guaranee saisfying performance quie close o FS beamforming. Considering he considerably reduced searching complexiy of Turbo-TS beamforming, we can conclude ha he proposed Turbo-TS beamforming achieves a much beer radeoff beween performance and complexiy. Fig. 6 shows he achievable rae comparison for an N r N = mmwave massive MIMO sysem, where he number of Fig. 5. chievable rae comparison for an N r N = mmwave massive MIMO sysem wih Nr = N = N s = 2. Fig. 6. chievable rae comparison for an N r N = mmwave massive MIMO sysem wih Nr = N = N s = 2. chains is sill se as Nr = N = N s = 2. In Fig. 6, we can observe similar rends as hose in Fig. 5. More imporanly, comparing Figs. 5 and 6, we can find ha he performance of he proposed Turbo-TS beamforming can be improved by increasing he number of low-cos anennas insead of increasing he number of expensive chains. For example, when N r N = 16 64, B = B r = 6, and SNR = 0 db, Turbo-TS beamforming can achieve he rae of 10.1 bis/s/hz, whereas when N r N = , he achievable rae can be increased o 14 bis/s/hz wihou increasing he number of chains. V. C ONCLUSION In his paper, we have proposed a Turbo-TS beamforming scheme, which consiss of wo key componens: 1 a Turbo-like join search scheme relying on he ieraive informaion exchange beween he BS

7 IEEE TRNSCTIONS ON VEHICULR TECHNOLOGY, VOL. 65, NO. 7, JULY and he user and 2 a TS-based precoding/combining scheme uilizing he idea of local search o find he bes precoder/combiner in each ieraion of Turbo-like join search wih low complexiy. nalysis has shown ha he complexiy of he proposed scheme is linear wih N and Nr, and i is independen of B and Br, which can considerably reduce he complexiy of convenional schemes. Simulaion resuls have verified he near-opimal performance of he proposed Turbo-TS beamforming. Our furher work will focus on exending he proposed Turbo-TS beamforming o he muliuser scenario. REFERENCES [1] W. Roh e al., Millimeer-wave beamforming as an enabling echnology for 5G cellular communicaions: Theoreical feasibiliy and prooype resuls, IEEE Commun. Mag., vol. 52, no. 2, pp , Feb [2] T. L. Marzea, Noncooperaive cellular wireless wih unlimied numbers of base saion anennas, IEEE Trans. Wireless Commun., vol. 9, no. 11, pp , Nov [3] S. Han, C.-L. I, Z. Xu, and C. Rowell, Large-scale anenna sysems wih hybrid precoding analog and digial beamforming for millimeer wave 5G, IEEE Commun. Mag., vol. 53, no. 1, pp , Jan [4] T. E. Bogale and L. B. Le, Beamforming for muliuser massive MIMO sysems: Digial versus hybrid analog digial, in Proc. IEEE GLOBECOM, Dec. 2014, pp [5] X. Zhang,. F. Molisch, and S.-Y. Kung, Variable-phase-shif-based -baseband codesign for MIMO anenna selecion, IEEE Trans. Signal Process., vol. 53, no. 11, pp , Nov [6] E. Zhang and C. Huang, On achieving opimal rae of digial precoder by -baseband codesign for MIMO sysems, in Proc. IEEE VTC Fall, Sep. 2014, pp [7] T. E. Bogale, L. B. Le, and. Haghigha, Hybrid analog digial beamforming: How many chains and phase shifers do we need? arxiv preprin arxiv: , [8] T. Kim e al., Tens of Gbps suppor wih mmwave beamforming sysems for nex generaion communicaions, in Proc. IEEE GLOBECOM, Dec. 2013, pp [9] J. Wang e al., Beam codebook based beamforming proocol for muli- Gbps millimeer-wave WPN sysems, IEEE J. Sel. reas Commun., vol. 27, no. 8, pp , Oc [10] C. Cordeiro, D. khmeov, and M. Park, IEEE ad: Inroducion and performance evaluaion of he firs muli-gbps WiFi echnology, in Proc. CM In. Workshop mmwave Commun., 2010, pp [11] S. Hur e al., Millimeer wave beamforming for wireless backhaul and access in small cell neworks, IEEE Trans. Commun., vol. 61, no. 10, pp , Oc [12] F. Glover, Tabu search Par I, ORS J. Compu., vol. 1, no. 3, pp , [13] O. El yach, S. Rajagopal, S. bu-surra, Z. Pi, and R. Heah, Spaially sparse precoding in millimeer wave MIMO sysems, IEEE Trans. Wireless Commun., vol. 13, no. 3, pp , Mar [14] L. Dai e al., Low-complexiy sof-oupu signal deecion based on Gauss Seidel mehod for uplink muli-user large-scale MIMO sysems, IEEE Trans. Veh. Technol., vol. 64, no. 10, pp , Oc [15] Z. Pi and F. Khan, n inroducion o millimeer-wave mobile broadband sysems, IEEE Commun. Mag., vol. 49, no. 6, pp , Jun [16] L. Wei, R. Q. Hu, Y. Qian, and G. Wu, Key elemens o enable millimeer wave communicaions for 5G wireless sysems, IEEE Wireless Commun., vol. 21, no. 6, pp , Dec [17] L. Dai, Z. Wang, and Z. Yang, Specrally efficien ime frequency raining OFDM for mobile large-scale MIMO sysems, IEEE J. Sel. reas Commun., vol. 31, no. 2, pp , Feb [18] Z. Gao, L. Dai, and Z. Wang, Srucured compressive sensing based superimposed pilo design in downlink large-scale MIMO sysems, Elecron. Le., vol. 50, no. 12, pp , Jun [19] O. El yach, R. Heah, S. bu-surra, S. Rajagopal, and Z. Pi, The capaciy opimaliy of beam seering in large millimeer wave MIMO sysems, in Proc. SPWC Workshops, 2013, pp Bivariaeκ-μ Disribuion Mirko lbero Gomez Villavicencio, Rausley driano maral de Souza, Member, IEEE, Geordan Caldeira de Souza, and Michel Daoud Yacoub, Member, IEEE bsrac In his paper, a bivariae κ-μ model is presened. Exac expressions for he 1 join probabiliy densiy funcion, he 2 join cumulaive disribuion funcion, 3 join arbirary momens, and he 4 normalized envelope correlaion coefficien are derived. The join saisics are given in erms of heir respecive parameers κ 1,μ 1 and κ 2,μ 2,wih μ 1 = μ 2 = μ>0 and arbirary κ 1 > 0andκ 2 > 0. The parameer describing he correlaion beween κ-μ fading channels is hen wrien in erms of he physical insances known o affec i in a wireless medium, namely, Doppler shif, he separaion disance beween wo recepion poins, frequency, and delay spread. s an applicaion example, he ouage probabiliy of a dual-branch selecion-combining scheme is presened. The effec of correlaion in he various aspecs of sysem performance is hen invesigaed. The validiy of he analyical resuls is suppored by reducing hem o paricular cases, for which resuls are available in he lieraure, and by means of simulaion for he general cases. Index Terms κ-μ disribuion, bivariae disribuion, correlaion, fading channel, ouage probabiliy, selecion combining SC. I. INTRODUCTION The κ-μ fading model describes a fading scenario in which he radio channel exhibis clusers of mulipah and dominan componens in each cluser [1]. I is characerized by wo physical parameers, namely, κ > 0 and μ > 0. The parameer κ corresponds o he raio beween he oal power of he dominan componens and he oal power of he scaered waves. The parameer μ is relaed o he number of mulipah clusers. The κ-μ fading model conains, as special cases, 1 Nakagami-m, obained from i as κ 0 wih μ = m where m is he Nakagami parameer, and 2 Rice, obained from i wih μ = 1 and κ = k where k is he Rice parameer. Of course, semi-gaussian and Rayleigh are also special cases of i, because hey are special cases of Nakagami-m and Rice. The κ-μ fading scenario has been explored in a wide range of applicaions. The usefulness of he κ-μ channel has been recognized in pracical siuaions, and is saisics have been invesigaed in differen environmens. In [1] iself, he κ-μ disribuion was shown o yield he bes fi o daa colleced in field rials whenever dominan componens were presen, boh for indoor and oudoor environmens, for signals Manuscrip received December 10, 2014; revised June 19, 2015; acceped July 23, Dae of publicaion July 28, 2015; dae of curren version July 14, This work was suppored in par by Finep, wih resources from Funel, under Gran , hrough he Radiocommunicaion Reference Cener Cenro de Referência em Radiocomunicações CRR projec of he Naional Insiue of Telecommunicaions Insiuo Nacional de Telecomunicações Inael, Brazil. The review of his paper was coordinaed bydr.d.w.maolak. M.. G. Villavicencio and M. D. Yacoub are wih he Wireless Technology Laboraory WissTek, Deparmen of Communicaions DECOM, School of Elecrical and Compuaion Engineering FEEC, Sae Universiy of Campinas UNICMP, Campinas, Brazil mirkoalbero2@homail. com; michel@wissek.org. R... de Souza and G. C. de Souza are wih he Naional Insiue of Telecommunicaions Inael, Sana Ria do Sapucaí, Brazil rausley@inael.br; geordancal@yahoo.com.br. Color versions of one or more of he figures in his paper are available online a hp://ieeexplore.ieee.org. Digial Objec Idenifier /TVT IEEE. Personal use is permied, bu republicaion/redisribuion requires IEEE permission. See hp:// for more informaion.