VERY large multiple-input multiple-output (MIMO), or. Beam Division Multiple Access Transmission for Massive MIMO Communications

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1 1 Beam Dvson Multple Access Transmsson for Massve MIMO Communcatons Chen Sun, Student Member, IEEE, Xq Gao, Fellow, IEEE, Sh Jn, Member, IEEE, Mchal Matthaou, Senor Member, IEEE, Zh Dng, Fellow, IEEE, Chengshan Xao, Fellow, IEEE Abstract We study multcarrer multuser multple-nput multple-output MU-MIMO systems, n whch the base staton BS employs an asymptotcally large number of antennas. We analyze a fully correlated channel matrx and provde a beam doman channel model, where the channel gans are ndependent of sub-carrers. For ths model, we frst derve a closed-form upper bound on the achevable ergodc sum-rate, based on whch, we develop asymptotcally necessary and suffcent condtons for optmal downln transmsson, that requre only statstcal channel state nformaton at the transmtter. Furthermore, we propose a beam dvson multple access BDMA transmsson scheme that smultaneously serves multple users va dfferent beams. By selectng users wthn non-overlappng beams, the MU- MIMO channels can be equvalently decomposed nto multple sngle-user MIMO channels; ths scheme sgnfcantly reduces the overhead of channel estmaton, as well as, the processng complexty at transcevers. For BDMA transmsson, we wor out an optmal plot desgn crteron to mnmze the mean square error MSE, and provde optmal plot sequences by utlzng the Zadoff-Chu sequences. Smulatons demonstrate the near-optmal performance of BDMA transmsson and the advantages of the proposed plot sequences. Index Terms Beam dvson multple access BDMA, massve MIMO, statstcal channel state nformaton CSI, sum-rate upper bound. I. INTRODUCTION VERY large multple-nput multple-output MIMO, or massve MIMO, employs a large number of antennas Manuscrpt receved October 21, 2014; revsed January 12, 2015; accepted Aprl 13, Ths wor was supported by Natonal Natural Scence Foundaton of Chna under Grants , , and , the Chna gh-tech 863 Plan under Grants 2015AA and 2014AA01A704, Natonal Scence and Technology Major Project of Chna under Grant 2014ZX , and the Program for Jangsu Innovaton Team. The wor of S. Jn was supported by the Natonal Natural Scence Foundaton of Chna under Grant and the Internatonal Scence & Technology Cooperaton Program of Chna under Grant 2014DFT The wor of Z. Dng was supported by Natonal Scence Foundaton under Grants ECCS , CNS , and CNS The wor of C. Xao was supported n part by US Natonal Scence Foundaton under grants ECCS and ECCS Ths wor was presented n part at the IEEE Internatonal Conference on Communcatons ICC, London, U.K., C. Sun, X. Gao, and S. Jn are wth the Natonal Moble Communcatons Research Laboratory, Southeast Unversty, Nanjng, , Chna. X. Gao s the correspondng author emal: {sunchen, xqgao, jnsh}@seu.edu.cn. M. Matthaou s wth the School of Electroncs, Electrcal Engneerng and Computer Scence, Queen s Unversty Belfast, BT3 9DT, Belfast, U.K. and wth the Department of Sgnals and Systems, Chalmers Unversty of Technology, , Gothenburg, Sweden emal: m.matthaou@qub.ac.u. Z. Dng s wth the Department of Electrcal and Computer Engneerng, Unversty of Calforna, Davs, CA 95616, USA emal: zdng@ucdavs.ca. C. Xao s wth the Department of Electrcal and Computer Engneerng, Mssour Unversty of Scence and Technology, Rolla, MO 65409, USA emal: xaoc@mst.edu. at base-statons BS to smultaneously and jontly serve multple.e., dozens moble users. Thans to ts potental to sgnfcantly mprove spectral effcency and power effcency, massve MIMO has spurred much recent research nterest [1, 2. The semnal paper by Marzetta [1 consdered noncooperatve massve MIMO systems wth unlmted numbers of antennas at the BS and sngle-antenna users. Assumng that propagaton channels are ndependently and dentcally dstrbuted, the propagaton beam vectors of dfferent users become asymptotcally orthogonal. ence, the matched flter MF recever/beamformng BF precoder becomes optmal, and the effect of nose and ntra-cell nterference dmnshes. Furthermore, the transmt power can be made arbtrarly low wthout performance loss. By explotng the upln-donwln recprocty n tme-dvson duplex TDD systems, the requred channel state nformaton CSI for downln at the BS can be acqured va upln tranng. Nevertheless, because of plot sequence reuse among cells, nter-cell nterference perssts for multcell systems [3, 4. Such problem s nown as plot contamnaton. In the practcal case of BS equpped wth a large, but fnte number of antennas, plot contamnaton has a more severe mpact on system performance relatve to the effect of addtve whte nose [5. The performance of MF/BF and more complex lnear recevers/precoders has been analyzed n [6 10. For fnte number of antennas, mnmum meansquare error/regularzed zero-forcng MMSE/RZF schemes can provde better performance than MF/BF schemes [10. owever, the complexty of MMSE/RZF s substantally hgher for massve MIMO systems, because of the requred matrx nverse computaton. Moreover, n the above wors [6 10, the BS requres nstantaneous CSI of all users. CSI plays a central role n massve MIMO systems. TDD systems can explot channel recprocty by estmatng downln channels from upln tranng. Yet, tranng overhead scales lnearly wth the total number of user antennas. When the number of users s large or when each user has multple antennas, the overhead becomes prohbtvely hgh. In addton, as user moblty ncreases, channel coherence tme becomes relatvely short. As a result, t becomes much more dffcult to obtan accurate nstantaneous CSI at transmtter CSIT for medum or hgh-moblty user applcatons. For frequencydvson duplex FDD systems wthout channel recprocty, the overhead of downln tranng wth orthogonal plots, as well as, the number of feedbac bts ncrease lnearly wth the number of BS antennas [11; thus, the sheer large number of antennas maes t mpractcal to obtan nstantaneous CSI at c 2015 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See

2 2 BS through panstang downln tranng and feedbac for CSIT. For ths reason, the bggest body of massve MIMO lterature assumes TDD systems. owever, n practce, there are system mperfectons that lmt the performance gans of TDD massve MIMO systems, such as the calbraton errors n upln/downln RF chans [12. In addton, FDD domnates current wreless cellular systems. In ths lght, Jont Spatal Dvson and Multplexng JSDM proposed n [13 15, offers strong potental to acheve massve MIMO ln gans for FDD systems. The ey dea les n parttonng the users nto groups wth approxmately smlar covarance matrces and splttng beamformng nto a pre-beamformng and a MU- MIMO precoder. After the pre-beamformng stage, the tranng dmensons are sgnfcantly reduced [13. The wor n [13 showed that under some condtons, n whch the dfferent user groups have non-overlappng supports of ther angles of arrval and angular spreads, JSDM acheves optmalty n sum-rate. To meet these condtons, the authors studed user selecton and opportunstc beamformng n [14. Notce that, opportunstc beamformng yelds sgnfcant gan only n the regme of fxed number of BS antennas and large number of users. In the regme of large number of BS antennas or lmted number of users, opportunstc beamformng wth user selecton fals to provde any gan. Because nstantaneous CSIT acquston represents a sgnfcant bottlenec, we advocate the use of statstcal CSIT nstead of the less robust nstantaneous CSI. The motvaton stems from the fact that statstcal CSI vares over much larger tme scales than nstantaneous channel parameters. Most mportantly, statstcal CSI acquston requres much less overhead than that of nstantaneous CSI, even when users are equpped wth multple antennas. Regardng the nstantaneous CSIT acquston, t s typcally assumed that users are equpped wth a sngle antenna, such that multplexng gans are acheved due to multple users. owever, when the number of users wthn a cell s not very large, users equpped wth multple antennas wll brng larger throughput gans. Moreover, multple antenna users are consdered n wreless networ standards such as LTE/LTE-Advanced LTE-A [16, 17. Although massve MIMO has been well nvestgated n the lterature [1 10, to the best of our nowledge, there has been lttle lterature nvolvng a complete transmsson scheme sutable for massve FDD-based topologes wth only statstcal CSIT. In addton, current standards face some challenges ncludng the CSIT acquston and the complexty of the transcevers [2. Motvated by the above dscusson, n ths paper, we nvestgate massve MIMO systems wth FDD operaton by consderng multple antennas at each user, whle only statstcal CSIT s avalable at the BS. Assumng orthogonal frequency dvson multplexng OFDM, we focus on a wdeband channel model. As the number of antennas at the BS grows asymptotcally large, we consder a fully correlated channel matrx, from whch we derve a beam doman channel model. For ths channel model, we frst deduce a closedform upper bound on the achevable ergodc sum-rate of the downln transmsson, whch depends only on the transmt covarance matrces and nput sgnal covarance matrces. The upper bound s rather tght n the low and mddle sgnalto-nose rato SNR regmes. Based on ths upper bound, we analyze the optmal downln transmsson strategy and derve asymptotcally necessary and suffcent condtons for the egenvectors and egenvalues of the optmal nput covarance matrx. Although maxmzng an upper bound does not guarantee maxmum sum-rate, nterestngly, our results are n perfect lne wth some specal case results [13, 18, 19. Inspred by these condtons, we propose a complete beam dvson multple access BDMA transmsson scheme approachng optmal transmsson. Our dea s to decompose massve MU- MIMO channels nto small dmensonal sngle-user MIMO SU-MIMO channels va user schedulng. The BDMA transmsson taes place n the beam doman and conssts of the followng steps: 1 acqurng channel couplng matrces CCMs, 2 schedulng users and beams, 3 transmttng plot and data n the upln and downln. In the thrd phase, we wor out the optmal plot condtons of a least square LS channel estmator and provde optmal plot sequences va the Zadoff-Chu ZC sequences to mnmze the mean square error MSE. Our numercal smulatons demonstrate the near-optmal performance of BDMA transmsson and that the proposed plot sequences can substantally mprove the MSE performance of channel estmaton. To further valdate the effectveness of BDMA transmsson, we evaluate the bt error rate BER of dfferent modulaton schemes. Smulatons llustrate that for QPSK and 16QAM modulatons, an dentcal plot sequence shared among users brngs lttle performance loss. owever, for 64QAM modulaton, the BER of utlzng an dentcal plot sequence gets hgher than In contrast, the BER performance of utlzng the proposed plot sequences approaches that of deal channel estmaton, thereby sufferng neglgble loss. We use the followng notaton throughout the paper: Upper lower bold-face letters denote matrces column vectors; I denotes the dentty matrx whle ts subscrpt, f needed, represents ts dmensonalty. The superscrpts, T, stand for the conjugate-transpose, transpose, and conjugate of a matrx, respectvely. We use E{ } to denote ensemble expectaton and tr, det to represent matrx trace and determnant operatons, respectvely, x 2 to denote the Eucldean norm of vector x, and B to denote the cardnalty of a set B. The vec operator stacs the columns of a matrx nto a tall vector. The symbols and denote the adamard product and Kronecer product of two matrces, respectvely. We use bold talc symbols to represent the channel matrces and sgnal matrces n the tme doman and standard bold symbols to represent those n the frequency doman. The nequalty A 0 means that A s ermtan postve semdefnte, and A 0 means that A s ermtan postve defnte; S n denotes symmetrc n n matrces. The operator δ represents the Kronecer delta functon. We use [A mn to denote the m, nth element of matrx A; [A B denotes the sub-matrx of A by eepng ts rows ndexed by the nteger set B, and [A B denotes the sub-matrx of A obtaned by eepng the columns ndexed by B. The operators eg and In stand for the sorted egenvalues n decreasng order and the nerta of a matrx, respectvely c 2015 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See

3 3 II. CANNEL MODEL AND ITS SPATIAL CARACTERISTICS We consder a sngle-cell FDD system consstng of one BS equpped wth M antennas. Moreover, K users, each wth N antennas, are randomly and unformly dstrbuted throughout the cell. Under the bloc fadng channel assumpton, the MIMO channel matrx stays constant for the coherence bloc contanng plot and data transmssons, and changes from bloc to bloc accordng to some ergodc process. We consder far-feld scatterng characterstcs 1, that s, the scatterers are far away from the BS and the angular spread AS around the BS s relatvely small. Throughout ths paper, we consder a physcal channel model, whch descrbes the propagaton between transmt array and receve array. Suppose that there are P physcal paths between the BS and users, and the pth path of the th user has an attenuaton of a p,, an angle of φ p, wth the transmt antenna array, and an angle of θ p, wth the receve antenna array. Then, the N M physcal MIMO channel matrx n the downln assocated wth the pth path of the th user s gven by [20 d p, = a p, e j2πd p,/λ c e r θ p, e t φ p, 1 where the superscrpt d means downln, d p, s the physcal dstance between transmt antenna 1 and receve antenna 1 along path p, and λ c s the carrer wavelength. Moreover, e r θ C N 1 whch satsfes e r θ 2 = 1 s the user antenna array response vector correspondng to the angle of arrval AoA θ, and e t φ C M 1 whch satsfes e t φ 2 = 1 s the BS antenna array response vector correspondng to the angle of departure AoD φ. For a wdeband channel, after OFDM operaton, the channel frequency response matrx n the lth sub-carrer of the th user s gven by P d,l = a p, e j2πd p,/λ c e r θ p, e t φ p, e j2πlτ p, 2 p=1 where τ p, s the propagaton delay assocated wth the pth path. We suppose that the upln physcal channel has the same small scale parameters θ p,, φ p,, τ p, as the downln channel except for the carrer frequency and the ntal phases [21. ence, we can calculate the upln channel matrx smlarly. ere, we analyze the downln channel propertes as an example, whle the upln analyss admts smlar manpulatons and s, thus, omtted. We assume that users are qute far away from the BS such that the phases 2πd p, /λ c are unformly dstrbuted on [0, 2π and mutually ndependent due to uncorrelated scatterng. Based on these assumptons, we have } {a p, e j2πd p,/λ c E E = 0 3 } {a p, e j2πd p,/λ c a p, e j2πd p, /λ c = β p, δp p, 4 1 Ths assumpton reles on the fact that the BS antennas are located at the top of a tall buldng such that there s no sgnfcant scatterng around the BS. where β p, = E { a p, 2} s the channel gan of path p. We also assume that the array response vectors correspondng to dstnct AoDs are asymptotcally orthogonal wth nfnte number of antennas at the BS 2 [22,.e., lm M e t φe t η = δφ η. 5 For each user, when the samplng of θ n, satsfes the condton 3 : e r θ n, e r θ n, = δn n 6 we can rewrte the channel matrx n 2 as d,l = N n=1 m=1 M [ d,l nm e rθ n, e t φ m = U d,l V 7 where V = [e t φ 1, e t φ 2,..., e t φ M C M M, and U = [e r θ 1,, e r θ 2,,..., e r θ N, C N N s a untary matrx. Note that, as the number of BS antennas M tends to nfnty, V becomes an asymptotcally untary matrx. We call d,l as the beam doman channel matrx wth one drecton of AoD called a beam. Followng [23, we have the followng samplng approxmaton: [ d,l a p, e j2πdp,/λc e j2πlτp, 8 nm p S r,n St,m where S r,n s the set of all paths whose receve angles are nearest to the angle θ n,, and S t,m s the set of paths whose transmt angle s exactly equal to the angle φ m. When the BS antenna number M tends to nfnty, the beam doman channel gans, whch are gven by [ d,l 2 nm 2 P a p, e r θ n, e r θ p, e t φ p, e t φ m e j2πlτ p, p=1 { a p, e r θ n, e r θ p, 2, φ p, φ m as M 0, φ p, φ m 9 become ndependent of sub-carrers. From 8 and 9, we can see that n the beam doman, dfferent elements of the channel matrx represent the sgnals of dfferent transmt and receve angles. Dfferent from the summaton of all path sgnals n the physcal doman, the beam doman channel can separate the paths of dfferent angles by dfferent beams. Moreover, the channel covarance matrces at the BS and the th user are respectvely gven by R t,,l V R t, V, as M 10 2 Massve antenna arrays can provde hgh resoluton of the sgnal angles. Therefore, arrays wth an nfnte number of antennas can dstngush sgnals from dfferent angles even wth nfntesmal spatal separaton [2. 3 The angle θ n, s the samplng of receve sgnals. When the response vectors are orthogonal, the user can separate these orthogonal drecton sgnals perfectly. For unform lnear antenna arrays, unform samplng of snθ n,,.e., snθ n, = n/n, s a classcal choce for spatal angles [ c 2015 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See

4 4 and R r,,l = U Rr, U 11 where Rt, = dag p S β t,1 p,,..., p S β t,m p, and R r, =dag β p Sr,1 p,,..., p, β are the transmt p Sr,N and receve covarance matrces n the beam doman, respectvely. Although 10 s an asymptotc result, for suffcently large M, the transmt channel covarance matrx can henceforth be well approxmated by R t,,l V R t, V. 12 We can see that the channel covarance matrces are ndependent of the sub-carrers, whch s a common result n all uncorrelated scatterng channels [15, 24. ere, we analyze the asymptotcally channel spatal characterstcs for general antenna array topologes wth nfnte large number of BS antennas. Usng the method n [25, we can construct the u- ntary matrx V by lettng φ m = ϑm/m, m = 0, 1,..., M, where ϑ s a strctly ncreasng contnuous functon over the support [0, 1. Consderng the specal case of unform lnear arrays ULAs wth half wavelength antenna spacng, by lettng φ m = ϑm/m = arcsn 2m M, V becomes a u- ntary dscrete Fourer transform DFT matrx F M C M M [F M m = e j2πm/m / M [13. Wthout loss of generalty, we assume that U = I n the remanng of ths paper, whch means that the user antennas are uncorrelated. We can now stac the columns of the channel matrx d,l from 7 nto a tall vector to get vec d,l = N n=1 m=1 M [ d,l nm e t φ m e r θ n, 13 and calculate the full correlaton matrx of d,l as R, = N M n=1 m=1 { [ [ } E d,l d,l nm nm e t φ m e r θ n, e t φ m e r θ n,. 14 It s nterestng to note that the full correlaton matrx s dentcal to that of the jont correlaton channel model [26. Smlarly to [26, we can defne the egenmode CCM as Ω d,l = E { d,l d,l } 15 whose elements specfy the mean amount of energy that s coupled from the mth egenvector of the BS to the nth egenvector of users. We fnd that the egenmode CCM Ω d,l s lewse ndependent of sub-carrers. Thus, we can omt the subscrpt l. As mentoned before, n ths paper, we consder that the BS has access only to the CCMs. In ths case, the optmal transmt strateges are the same for all sub-carrers. Ths observaton wll sgnfcantly decrease the overhead of CSIT acquston. Moreover, accordng to [27, 28, the upln and downln statstcal CSI are usually recprocal n both TDD and FDD systems. Thus, we can denote the upln and downln CCMs as Ω d,l = Ω u,l T = Ω 16 whose elements can be approxmated by [Ω nm p S r,n St,m β p,. Note that the superscrpt u denotes u- pln. III. DOWNLINK OPTIMAL TRANSMISSION WIT SUM-RATE UPPER BOUND We now analyze the downln transmsson, where the BS wth M antennas serves K users wth N antennas. Suppose that the BS has access only to the CCMs of each user va a channel soundng process, whlst under the assumpton of bloc fadng, each user has nowledge of the nstantaneous C- SI, as well as, the aggregate nter-user nterference covarance matrx usng channel estmaton. 4 We wll provde a channel estmaton method n Secton IV-C. Under these assumptons, we frst analyze the sum-rate of downln transmsson and derve an upper bound on the achevable sum-rate. Based on the achevable sum-rate upper bound, we propose the asymptotcally necessary and suffcent condtons to maxmze the upper bound. A. Sum-Rate Upper Bound The receved sgnal at the th user n the lth OFDM subcarrer can be rewrtten n the form y d,l = d,lx d,l + n,l 17 E n,x {n,l n,l } = I +d,l, Qd,l where x d,l C M 1 s the sgnal ntended for the th user, the { covarance matrx of whch s denoted by } Q d,l = E x d,l x d,l, n,l = K, d,l xd,l + n,l s the aggregate nstantaneous nterference-plus-nose, K, d,l xd,l represents nter-user nterference, n,l s the N 1 complex Gaussan nose vector wth n,l CN 0, I N. Suppose that the th user has nowledge of the nstantaneous channel matrx d,l and of the aggregate nstantaneous nterference-plus-nose covarance matrx K,l =.,l d ere, we assume that the BS employs lnear precodng. When the recever treats the aggregate nterference-plus-nose as Gaussan nose wth covarance K,l [29, we can derve the followng achevable ergodc rate [30: { R,l = E log det { I + K = E log det,l d,lq d,l K,l + d,lq d,l d,l } d,l } E {log det K,l }. 18 By substtutng the expresson of K,l nto R,l n 18, t yelds { K R,l = E log det Q d d,l,l d,l + I log det Q d,l d,l d,l + I We provde a method to estmate the aggregate nstantaneous nterferenceplus-nose covarance matrx n Secton IV-C c 2015 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See

5 5 We can calculate the achevable ergodc sum-rate of K users as R sum,l = R,l. 20 =1 The achevable sum-rate s dfferent from some exstng results [8, 29. In [8, the authors assume that the recever nows only the channel dstrbuton and not the nstantaneous channel. ere, we assume that the recever has access to the nstantaneous channel matrx, as well as, the nstantaneous aggregate nterference-plus-nose covarance matrx nstead of the statstcal nterference-plus-nose covarance matrx [29. In general, t s very dffcult to derve a sum-rate n closedform. Therefore, t needs Monte-Carlo smulatons to calculate the achevable ergodc sum-rate, whch s computatonally cumbersome. Moreover, the sum-rate gven by 20 s a nonconcave functon on Q d,l, whch appears n both the subtracton terms smultaneously. Thus, t s challengng to determne the optmal nput covarance matrx Q d,l maxmzng the sumrate. In order to derve a closed-form upper bound, we frst derve the followng lemmas as prelmnary results. Lemma 1: If symmetrc postve semdefnte M M matrces A, B, C, D satsfy A B and C D, then the sorted egenvalues satsfy egac egbd. Proof: See Appendx A. Lemma 2: Let A, B be symmetrc postve semdefnte M M matrces, satsfyng A B 0. Defne a matrx functon as fx = log deti + AX log deti + BX. 21 Then, fx s matrx concave on X 0. Proof: See Appendx B. Wth the help of the above lemmas, we can derve a closedform upper bound on the sum-rate, whch s shown by the followng theorem: Theorem 1: The sum-rate R sum,l n 20 s upper bounded by K R sum,l log det R t,,l + I =1 log det, Q d,l Q d,l R t,,l + I = R ub,l. 22 Proof: Usng Lemma 2, the sum-rate s a matrx concave of,l d d,l. Owng to the propertes of the concave functon, we have the upper bound of 22. Remar 1: In the above proof, Jensen s nequalty [31, 32 s appled to obtan the upper bound n 22. It should be noted that applyng Jensen s nequalty to two subtracton terms smultaneously wll not, n general, return an upper bound to the orgnal problem. Nevertheless, we prove that when the condton n Lemma 2 s satsfed, the matrx functon n 21 s strctly matrx concave, resultng n a strct theoretcal upper bound nstead of smply an approxmaton. In [33, t was the frst tme to reveal an upper bound of two subtracton terms, where the constrants A, B 0 were requred. ere, we relax the constrants of [33 and derve a more general result. Remar 2: Unfortunately, there s no closed-form expresson for 19 under the channel model we consder n ths paper. In Theorem 1, we derve a closed-form upper bound on the achevable rate of MU-MIMO broadcast transmsson. The closed-form upper bound depends only on the nput covarance matrces Q d,l and the transmt correlaton matrces R t,,l for = 1, 2,..., K. Thus, t s a convenent means to evaluate the system performance, nstead of tedous Monte- Carlo smulaton for every channel realzaton. Moreover, t sgnfcantly reduces the complexty of the transmt sgnal desgn. Most mportantly, we can apply 22 to maxmze the upper bound R ub,l wth respect to Q d,l. B. Optmal Transmsson Based on the Upper Bound Based on the closed-form upper bound n Theorem 1, we can now wor out the optmal downln transmsson. Our man objectve s to fnd the optmal nput covarance matrx Q d,l maxmzng the sum-rate upper bound. Ths can be wrtten as a maxmzaton problem: max R ub,l 23 Q d 1,l,Qd 2,l,...,Qd K,l K subject to tr P =1 Q d,l where P s the total power constrant. 5 Let us decompose the nput covarance matrx as Q d,l = Φ,lΛ,l Φ,l dentfyng the egenvectors of Q d,l wth the columns of the untary matrx Φ,l and ts egenvalues wth the dagonal entres of Λ,l. Both the egenvectors and the egenvalues have an mmedate engneerng meanng: the former ndcate the drectons on whch sgnals are beng transmtted whle the latter ndcate the transmt powers allocated onto each drecton. Next, we wll focus on the egenvectors and the egenvalues of the nput covarance matrx Q d,l. In the general MU-MIMO case, the sum-rate upper bound s not a concave functon on Q d,l. Furthermore, the egenvectors of channel covarance matrces are dstnct for dfferent users. Therefore, although the optmal nput covarance matrx for SU-MIMO was determned n [18, t s very dffcult to deal wth the MU-MIMO case. Fortunately, as the number of BS antennas ncreases, the egenvector matrces of channel covarance matrces become dentcal. Thus, we can start our analyss of Q d,l, and frst derve the followng lemmas as prelmnares. Lemma 3: For arbtrary M M matrx Q 0, and M M dagonal matrces A B 0, we have: A I + AQA A B I + BQB B. 24 Proof: See Appendx C. Lemma 4: Let A, B = 1, 2,..., K be symmetrc postve semdefnte M M matrces, and B be dagonal matrces. We can construct A = K =1 A, dagonal matrx 5 Massve MIMO systems wor n the low and mddle SNR regmes, snce the total transmt power at the BS should be lmted. Therefore, we consder a total power constrant and not ndvdual constrants, under whch the transmt power wll ncrease as more users communcate wth the BS smultaneously. On top of ths, ndvdual power constrants render the analyss rather challengng c 2015 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See

6 6 Ã = dag{a}, and dagonal matrces Ã, whose dagonal elements are [Ã { } = arg max [Bj [Ã = j { } arg max [Bj j Then, the followng nequalty holds log det I + AB log det I + A A B log det I + ÃB log det I + Ã Ã B. 26 Proof: See Appendx D. Recallng the above lemmas, we can derve the asymptotcally necessary condtons of the optmal nput covarance, whch s shown by the followng theorem: Theorem 2: The optmal nput covarance desgn maxmzng the upper bound gven by 22, satsfes the followng asymptotcally condtons: the egenvectors of the optmal nput covarance matrx Q d,l are gven by the columns of V,.e., Φ,l V 27 and the egenvalues of the optmal nput covarance satsfy [Λ,l mm = 0, for arg max {[ R } t, mm. 28 Proof: From 10, the upper bound n 22 can be rewrtten as K R ub,l log det V V R t, + I =1 log det V, Q d,l Q d,l V R t, + I. 29 Wth the help of Lemma 4, the upper bound R ub,l K wth dagonal matrces V Qd,l V and K V, Qd,l V s larger than that wth non-dagonal matrces. Therefore, the egenvector matrces of Q d,l are equal to V. Moreover, from Lemma 4, the egenvalues of optmal nput covarance Q d,l satsfy [Λ,l mm = 0, for arg max {[ R } t, mm. Remar 3: From 27, we can see that the egenvectors of the optmal nput covarance matrx Φ,l equal those of the transmt channel covarance matrx V, thereby beng asymptotcally ndependent of users and sub-carrers. Thus, f we transform the sgnals nto the beam doman, the condton 27 wll be satsfed. Ths mples that beam doman transmsson s optmal. Moreover, we see from 28 that on each beam t s optmal to transmt sgnals to the user wth the strongest beam gans. The optmal transmtted beams of dfferent users should be non-overlappng. Although we consder the optmal transmsson wth the sum-rate upper bound, the results are consstent wth some exstng results n the specal case. For example, n the SU-MIMO case, when the transmtter nows only the channel dstrbuton, the egenvectors of the capactyachevng nput covarance are exactly the same as that of the transmt channel covarance matrx [18, 31. Turnng now our attenton to the beam doman, the receved sgnal of th user n the lth sub-carrer can be rewrtten as ỹ d,l = d,l xd,l +, d,l xd,l + ñ,l 30 where x d,l = V x d,l and ỹd,l = yd,l are the transmt and receve sgnals n the beam doman respectvely, ñ,l s the Gaussan nose n the beam doman. We defne the ndces of the non-zero elements of Λ,l as the beam set: { } B = b 1, b 2,..., b B 31 where b j represents the jth beam ndex of user, and B = B represents the total number of beams. Selectng the nonzero elements n Λ,l and x d,l, we have Λ,l = [Λ,l B B R B B and s d,l = [ xd,l B C B 1. Thus, we can further smplfy the sgnal model n 30 as ỹ d,l = [ d,l B s d,l +, [ d,l B s d,l + ñ,l. 32 The sum-rate upper bound can be expressed as K R ub,l log det Λ,l R t, + I =1 log det, Λ,l Rt, + I. 33 Utlzng the fact that Λ,l and R t, are dagonal matrces, we can further smplfy the upper bound n 33 as R ub,l =1 log det Λ,l [ R t, B B + I B. 34 We fnd from 34 that beam doman transmsson s optmal. Moreover, dfferent from 22, we can see from 34 that the sum-rate upper bound s matrx concave on Λ,l. Thus, t s easy for us to perform power allocaton. avng analyzed the necessary condtons of the optmal transmsson wth the upper bound, we can now focus on the power allocaton, whch s the suffcent condton of the optmal transmsson. In ths case, the maxmzaton problem becomes max Λ,l s.t. =1 =1 log det Λ,l [ R t, B B + I B tr Λ,l P 35 Λ,l The maxmzaton s over all power allocaton polces Λ,l based on the transmt channel covarance matrces. Thus, we c 2015 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See

7 7 Fg. 1: Bloc dagram of a BDMA transmsson scheme. have the soluton [34 [ Λ,l = max 0, 1 ν 1 [ [ R t, B B 37 where ν s the Lagrange multpler wth the condton 0, 1 ν 1 [ [ R = P. 38 t, B =1 B We can use a well-nown water-fllng algorthm [35 to evaluate the power allocaton polcy. IV. BEAM DIVISION MULTIPLE ACCESS TRANSMISSION From Theorem 2, beam doman transmsson s optmal from the pont of maxmzng the sum-rate upper bound. Moreover, n the beam doman, t s better for one beam to contan only one user sgnal. For far-feld scatterng condtons, where the AS at the BS s relatvely small, nonvanshng channel coeffcents are concentrated wthn some beams correspondng to the AoD. Inspred by these propertes, we propose a BDMA transmsson scheme for MU-MIMO n the beam doman wth CCMs avalable at the BS. Fgure 1 llustrates the overall communcaton scheme, whch conssts of the followng ey components: 1 The BS acqures CCMs from all users n ts own cell. CCMs acquston conssts of two steps: Frst n the upln, dfferent users transmt soundng sgnals on separate sub-carrers. Secondly, the BS estmates the nstantaneous CSI to calculate CCMs gven by 15. Snce the statstcal CSI vares much slower than the nstantaneous CSI, we do not need to estmate the whole channel realzatons and, therefore, the overhead of statstcal CSI s much less than that of nstantaneous CSI. 2 The BS schedules users to be suffcently separated n beam doman. Based on CCMs, the user scheduler selects users for maxmzng the sum-rate upper bound wth the constrants that dfferent user beams are nonoverlappng. After user schedulng, beams at the BS are dvded nto dfferent sets, leadng to decomposng the massve MU-MIMO ln nto small dmensonal SU- MIMO lns. ere, we consder the beam set selecton wth equal power allocaton nstead of performng power allocaton. 3 Upln and downln transmssons consst of plot tranng and data transmsson. In the tranng step, snce users are separated by dfferent beam sets at the BS, upln and downln plot sequences do not need to be orthogonal. Accordng to the number of selected users, optmal plot sequences are desgned to mnmze the channel estmaton error. The recevers n the upln and downln estmate the reduced dmensonal channel matrces, as well as, the aggregate nterference covarance matrx for data detecton. Wth channel nformaton, the recevers utlze the teratve soft-nput and soft-output SISO detecton and SISO decodng. The BDMA scheme s sutable for FDD systems wth only statstcal CSI at the BS. Note, however, that the proposed approach also apples to TDD systems, when nstantaneous CSI at the BS s not avalable, e.g., for cases nvolvng hgh user moblty. Next, we wll elaborate on the above problems and provde practcal schemes n detal. A. Channel Couplng Matrces Acquston In ths subsecton, we provde a method to acqure CCMs of all users n the cell. Our method s based on the fact that CCMs are ndependent of sub-carrers, and on the statstcal recprocty of upln and downln for both FDD and TDD systems [27, 28. Consder a wdeband system, and after OFDM operaton, there are L sub-carrers. Let L = {1, 2,..., L} denote the OFDM sub-carrer set. Dfferent antennas transmt soundng sgnals n the dfferent sub-carrers. The sub-carrer set of the nth antenna of the th user s gven by: L n, = {l l = 1N + n + 1NK, = 1, 2,..., N S }, where s the sample ndex, and N S s the total sample number of each antenna. ere, we assume that L N S NK, and the remanng L N S NK sub-carrers are not used. At the start of each perod, every antenna transmts a soundng sgnal n ts correspondng sub-carrer set. The BS estmates the channel based on the relatonshp: ỹ u n,,l = h u n,,l x u l + ñ l 39 where x u l represents the soundng sgnal transmtted on the lth sub-carrer, h u n,,l CM 1 denotes the beam doman channel between the nth antenna of the th user and all the beams at the BS. Because antennas are dstnct n the frequency doman, x u l can be any determnstc sgnal nown by the BS and users, wth power constrant E { x u l xu l } = P s. The LS estmaton of h u n,,l s ĥ u n,,l = 1 x u ỹn,,l. u 40 l We reconstruct the channel estmator of user as follows: Ĥ u,l = [ĥu 1,,l, ĥu 2,,l+1,..., N,,l+N ĥu C M N. 41 Then, the CCM for user can be obtaned by ˆΩ T = 1 N Ĥu,l Ĥu,l. 42 S l L 1, c 2015 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See

8 8 B. User Schedulng We consder the beam set selecton wth equal power allocaton, nstead of performng power allocaton. The problem can be formulated as [U, B 1, B 2,..., B K = arg max R ub,l 43 P s.t. Λd,l = K B I B B Bj =, j 44 where U s the selected user set, B s the beam set for user, whch s an empty set f the user s not selected. Ths problem s a combnatoral problem, and we can utlze exhaustve search or other heurstc search methods. ere, usng the optmal condtons n Theorem 2, we present an algorthm wth a low complexty. Let r,m be the mth beam gans of the th user defned as r,m = N [ ˆΩ T n=1 nm. 45 The strongest user ndex on the mth beam and the correspondng value of beam gan can be respectvely expressed as and b m = arg max r,m 46 r m = max r,m. 47 Then, the maxmzaton problem gven by 43 can be reformulated as [U, B 1, B 2,..., B K = arg max log 1 + PBS r m U m B 48 s.t. B {m b m = } B Bj =, j where B S = U B s the total number of selected beams. Wth 48, we present an algorthm wth reduced complexty to select users and beams as follows. Algorthm 1: User Schedulng Algorthm. Step 1: Calculate the beam gans r,m and select the strongest user on each beam b m accordng to 45 and 46. Step 2: Intalze = 1, R sum = 0, B =, = 1, 2,..., K, and the remanng beam set B r = B. Step 3: For teraton, select the strongest beam n the remanng beam set m = arg max r m. Set B b m B m = r B bm {m }, U = U {b m }, and calculate R accordng to 48. Step 4: If R > R sum and < M, set R sum = R, B bm = B b m, B r = B r \ {m }, U = U, = + 1, and go to Step 3. Else, go to Step 5. Step 5: U s the selected user set and B s the selected beam set for user. Algorthm 1 provdes a method to search the selected user set and the beam sets for each user. The complexty of Algorthm 1 s only OK, whch s lnear wth the number of users n the cell. In the smulaton, we wll compare Algorthm 1 wth a genetc algorthm [36 dealng wth the problem max R sum,l ; the smulaton results valdate the near optmal performance of ths user schedulng algorthm. C. Plot Desgn and Channel Estmaton After user schedulng, K S users are selected to communcate wth the BS employng the same tme and frequency resources. In the upln dedcated tranng phase, all users transmt plot sequences whch allow the BS to estmate the CSI, whle n the downln, the BS transmts plot sequences to the th user through the beam set B, the number of whch s much less than that of the BS antennas. ere, we analyze the downln tranng sequences as an example. Let x p,l be the plot sequence transmtted by the B beam set to the th user on the lth sub-carrer. The receved sgnal at the th user n the lth sub-carrer can be wrtten as ỹ d,l = [ d,l B x p,l + KS, [ d,l B x p,l + ñ,l. 49 Let Ỹ [ỹ d =,1 d, ỹd,2,..., ỹd,l C N L, and then 49 can be wrtten n a matrx form as K S Ỹ d = [ d Xp B + Ñ 50 where [ d B = [[ d,1 B, [ d,2 B,..., [ d,l B C N BL, Ñ d [ñ = d,1, ñd,2,..., ñd,l C N L, and x p,1 0 0 X p 0 x p =, CBL L x p,l The channel parameters [ d B are determned by the tme doman CSI: [ d B = [ d [ B IB P 0 BP B L P LFL I B 52 where [ d B = [[ 1, d B, [ 2, d B,..., [ P, d B C N B P, the number of whch s usually much less than that of [ d B. Then, the receved sgnal of the th user n 50 can be rewrtten as K S Ỹ d = [ d [ B IB P 0 B P B L P LFL I Xp B + Ñ. 53 Let T = [ I BP 0 B P B L P LFL I Xp B, R p ab = T at b and the LS channel estmaton of [ d B s obtaned as [Ĥd B = Ỹd T Rp c 2015 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See

9 9 Then, we can calculate the MSE of the th user as MSE = Ntr Rp K S + tr, where K = E K R p Rp 2 Rp 55 { [ d B [ d B }. To mnmze the MSE n 55 regardless of K, the optmzaton problem can be expressed as R p = arg mn R p { } max MSE. 56 K Through the smlar procedure as that n [37, we have the followng result. Theorem 3: The MSE of the th user s channel estmate regardless of K s lower-bounded by max MSE NB P + B P K L 57 where the condton for equalty s Rp = I B P and B P p a=1 [ R ba 2 = B P L, for arbtrary 1 b B P. From Theorem 3, the optmal plot sequence set should have the perfect auto-correlaton property and meet the Welch bound. Among well-nown perfect sequences, the class of ZC sequences s the largest and nown to be Welch bound equalty sgnal sets [37. Moreover, to satsfy the Toepltz structure of the tme doman sgnal, let X = T F L, the B p + b, lth p = 0, 1,..., P, b = 1, 2,..., B, l = 1, 2,..., L element of whch can be expressed as [ X Bp+b,l = [ a L r 58 l p+bp L where a L r s the length-n ZC sequence wth the root of r and L denotes the modulo-l operaton. Then, we can calculate p the plot sequence matrx X as X p = 1 [ F X L I B F L. 59 L 0 BL P L Accordng to 52, we can calculate the estmate [Ĥd,l B wth [Ĥd B. In prncple, once an estmate [Ĥd,l B s avalable, we could estmate K d,l va ˆK d,l,1 = ỹ,l d [Ĥd,l B x p,l ỹ,l d [Ĥd,l B x,l p. 60 owever, the sample covarance estmaton wll be poor n general. As the nterference s a movng-average process, we wll use a conceptually smple but suboptmal approach to estmate the receve covarance matrx [38, and have the mproved estmaton ˆK d,l,2= P l= P P + 1 l P + 1 L 1 ˆK d L,l,1e j2πll/l e j2πll/l. l=0 61 Sum Rate n b/s/z Power allocaton Schedulng wth upper bound Schedulng wth genetc algorthm Upper bound SNR n db Fg. 2: Comparson of user schedulng wth upper bound and genetc algorthm. Achevable Net Sum-Rate n bts/s/z WMMSE, 8 antennas WMMSE, 1 antenna BDMA, 8 antennas SNR n db Fg. 3: Comparson of BDMA scheme wth WMMSE algorthm gven nstantaneous CSI. P +1 l P +1 The weghtng factor nsde the sum maes the estmaton ˆK d,l,2 based. Nevertheless, ths weghtng factor s essental because t guarantees that ˆK d,l,2 s postve defnte. V. NUMERICAL RESULTS In ths secton, we gve some examples to llustrate the performance advantages of the proposed BDMA transmsson. Our smulatons are based on the 3GPP spatal channel model SCM wth a suburban-macro scenaro [39, whch consders a MIMO system wth M = 128 antennas at the BS and N = 8 antennas at each user. We consder a ULA topology at the BS wth 0.5λ antenna spacng. Assume that there are L = 2048 sub-carrers and N S = 12 samples to acqure the CCMs. We do not consder shadow fadng or path loss n ths paper, and only test the non-lne-of-sght NLOS condton. There are K = 20 users n the cell that are unformly dstrbuted randomly. Fgure 2 presents the near optmal performance of user schedulng usng Algorthm 1 n 22. Frstly, we evaluate Algorthm 1 wth power allocaton usng 37, 6 and see that 6 We perform power allocaton for all users. If the user s power s found to be zero, then the user s not selected c 2015 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See

10 10 MSE The proposed plot scheme The same plot scheme SNR n db Fg. 4: The MSE performance comparson of dfferent plot desgns. BER QPSK 16QAM 64QAM Proposed plot Identcal plot 10 5 Ideal CSI SNRn db Fg. 5: BER of BDMA transmsson wth dfferent modulatons. power allocaton wth 37 can brng lttle sum-rate gan. Moreover, we compare the closed-form ergodc sum-rate upper bound 22 wth Monte-Carlo smulated results. The upper bound s rather tght n the low and mddle SNR regmes. Furthermore, we show the performance of user schedulng wth a genetc algorthm [36, whch s a heurstc search to maxmze the sum-rate R sum,l. From Fgure 2, we observe that at the low and moderate SNR, the performance of Algorthm 1 s almost the same as that of the genetc algorthm. Although, maxmzng the upper bound drectly mght result n a slght performance loss at hgh SNR, Algorthm 1 reduces the complexty of user schedulng to OK, whch s lnear wth the user number and much less than that of the genetc algorthm. Fgure 3 compares the achevable net sum-rate of the BDMA scheme aganst that of the WMMSE algorthm gven nstantaneous CSI [8. To obtan the CSI at the BS for the WMMSE algorthm, we assume TDD operaton. For our scheme, we assume that there are 5 users n the cell, each equpped wth 8 antennas. For the WMMSE algorthm, we consder two confguratons: the frst s the same as ours, and the second s 40 users n the cell, each equpped wth 1 antenna. We assume bloc fadng channels wth a frame contanng 7 OFDM symbols. Wth the above settngs, n our scheme, the overhead of the proposed plot scheme n Secton IV-C s only 1/7. owever, for the WMMSE algorthm, the requred plot segment length s 3 [1, and the overhead s 3/7. eren, we do not consder channel estmaton errors, and compare the achevable net sum-rates of these two schemes, whch s R net,l = 1 ηr sum,l, where η s the overhead. Wthout CSI at the users sde, the achevable sum-rate of the WMMSE algorthm can be calculated followng [8. Our smulaton results show that by tang nto consderaton the overhead, the performance of the BDMA scheme s slghtly better than that of the WMMSE algorthm under the second confguraton. Most mportantly, under the frst confguraton, the achevable net sum-rate of the WMMSE algorthm becomes substantally worse. Ths s because the users cannot jontly process the receved sgnals due to the lac of nstantaneous CSI. To further llustrate the performance of BDMA transmsson scheme, we smulate the MSE and BER of ln transmsson of selected 5 users by user schedulng. In Fgure 4, we compare the MSE performance of the proposed plot sgnals 59 wth a scheme usng an dentcal plot sequence across users. We can clearly see that the proposed plot sgnals can provde substantally better MSE performance than the dentcal plot sgnal scheme; n fact, the performance gap ncreases as the SNR ncreases, whch confrms that the proposed plot sequences can provde better performance even f the plot sequence length s not a prme number. We now present the coded BER performances of dfferent modulatons by utlzng the MMSE teratve recever proposed n [40. We employ the Turbo encoder and decoder n our smulatons, for codng rate 0.5. Fgure 5 presents the coded BER curves wth dfferent plot sequences under QPSK, 16QAM and 64QAM modulatons. For the purpose of comparng the error performance, we also assume that the recevers have deal CSI. For QPSK modulaton, both the BERs of utlzng the proposed plot and the dentcal plot are almost the same to the BER of the deal CSI case. For 16QAM modulaton, the BER of the proposed plot approaches the BER of the deal CSI scheme agan. Also, at the targeted BER of 10 3, the proposed plot acheves about 1 db SNR gans compared to the dentcal plot. For 64QAM modulaton, the BER of the proposed plot s stll close to the deal CSI case. owever, the BER of the dentcal plot reaches an error floor at a hgh BER value due to nter-user nterference. Ths ndcates that the dentcal plot cannot wor effectvely for 64QAM modulaton. VI. CONCLUSION Ths wor nvestgated downln massve MU-MIMO transmsson wth multple antennas at each user, where only statstcal CSI s avalable at the BS. Based on the full correlaton profle, we provded a beam doman channel model, whch reveals the frequency-flat propertes of channel gans n the beam doman. For ths channel model, we derved a sum-rate upper bound on the downln sum-rate. Based on the upper bound, we developed asymptotcally necessary and suffcent condtons for the optmal downln transmsson wth statstcal CSIT. The optmal transmsson s n the c 2015 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See

11 11 beam doman, and the optmal transmtted beams of dfferent users should be non-overlappng. These condtons lead to the proposed BDMA transmsson whch provded the foundaton of a novel framewor for the practcal mplementaton for FDD massve MIMO systems. By selectng users wth non-overlappng beams, we decomposed MU-MIMO channels nto multple sngle-user MIMO channel lns. Owng to the reduced-dmensonal MIMO lns, the decomposton reduces sgnfcantly the overhead of channel estmaton and the processng complexty at the transmtters and recevers. For BDMA transmsson, we derved the optmal plot desgn crteron mnmzng the MSE of an LS channel estmator, and provded optmal plot sequence sets by utlzng ZC sequences. The defnton doman of t s gven by {t Z+tV 0}, whch can be rewrtten as I + tz /2 VZ /2 0. Let {α } = egz /2 VZ /2 = egvz, and t satsfes 1+α t 0. Moreover, usng the matrx nverse lemma, we have Z + Ã Ã = Z ZÃ I + ÃZÃ ÃZ 65 whch yelds {λ } = egvã I + ÃZÃ Ã Smlarly, we have = egvz Z Z + A Z. 66 {ξ } = egvz Z Z + B Z. 67 APPENDIX A PROOF OF LEMMA 1 Owng to C D, B 0, we have B 1/2 CB 1/2 B 1/2 DB 1/2, ths means that the sorted egenvalues satsfy egc 1/2 BC 1/2 egd 1/2 BD 1/2. Therefore, there exsts a untary matrx U, satsfyng UC 1/2 BC 1/2 U D 1/2 BD 1/2 [41. In addton, due to A B, we have UC 1/2 AC 1/2 U UC 1/2 BC 1/2 U. Thus, we have egac = eguc 1/2 AC 1/2 U egd 1/2 BD 1/2 = egbd. 62 APPENDIX B PROOF OF LEMMA 2 As A 0, B 0, we can decompose the matrx A and B as A = Ã Ã, B = B B. We can verfy concavty by consderng an arbtrary lne, gven by X = Z + tv, where Z, V S n. We defne gt = fz + tv, where fx was defned n 21, and restrct gt to the nterval of values of t for whch Z + tv 0. We have gt = logdeti+ãz + tvã logdeti+ BZ+tV B = log deti + I + ÃZÃ /2 tãvã I + ÃZÃ /2 + log deti + ÃZÃ log deti + BZ B log deti + I + BZ B /2 t BV B I + BZ B /2 N N = log1 + tλ log1 + tξ + log deti + ÃZÃ log deti + BZ B 63 where, λ are the sorted egenvalues of I + ÃZÃ /2ÃVÃ I + ÃZÃ /2 and ξ are the sorted egenvalues of I + BZ B /2 BV B I + BZ B /2. Therefore, we have N g ξ 2 t = 1 + tξ 2 λ tλ 2 N ξ λ ξ + λ + 2tξ λ = 1 + tξ tλ And they have the followng relatonshp Z Z Z Z + A Z Z Z Z + B Z We call µ, ν, ζ the nerta of the matrx V, and wrte InV = µ, ν, ζ, where V has µ egenvalues wth postve real part, ν wth negatve real part, and ζ purely magnary ones. Let InVZ = µ 1, ν 1, ζ 1 69 InVZ Z Z + A Z = µ 2, ν 2, ζ 2 70 InVZ Z Z + B Z = µ 3, ν 3, ζ From [42, we have µ 1 µ,µ 2 µ, µ 3 µ 72 ν 1 ν,ν 2 ν, ν 3 ν. 73 Let the th sorted egenvalues of V be γ. When γ > 0, f α < 0, then for arbtrary j >, α j < 0 holds, whch s n contradcton wth ν 1 ν. Therefore, α 0 holds. Smlarly, when γ > 0, we have λ 0 and ξ 0. When γ < 0, f α > 0, for arbtrary j <, α j > 0 holds, whch s n contradcton wth µ 1 µ. So, α 0 holds. Smlarly, we have λ 0 and ξ 0. When γ = 0, Owng to µ 1 µ, ν 1 ν, therefore α = 0, λ = ξ = 0 hold. For sorted egenvalues, there only exst two cases: α, λ, ξ 0 and α, λ, ξ 0. Meanwhle, {α 2 } = egvz VZ 74 {λ 2 } = eg VZ Z Z + A Z VZ Z Z + A Z 75 {ξ 2 } = eg VZ Z Z + B Z VZ Z Z + B Z. 76 Invong Lemma 1, we have {α 2 } {λ2 } {ξ2 }. When {α } {λ } {ξ } 0 or {α } {λ } {ξ } 0, due to 1 + tα 0, we have ξ λ ξ + λ + 2tξ λ Snce g t 0, we conclude that fx s concave c 2015 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See

12 12 APPENDIX C PROOF OF LEMMA 3 When B = 0, 24 holds obvously. When B 0, B can be expressed as [ B1 0 B = P P where B 1 0 s a dagonal matrx contanng the non-zero elements of B, and P s the correspondng permutaton matrx. Smlarly, A can be expressed as [ A1 0 A = P P 79 0 A 2 where A 1 s full ran, A 1 B 1 and A 2 0. The substtuton of 78 and 79 nto 24 yelds [ [ [ [ A1 0 A1 0 A1 0 A1 0 I + Q 0 A 2 0 A 2 0 A 2 0 A 2 [ [ [ [ B1 0 B1 0 B1 0 B1 0 I + Q where Q = P QP. Next, we wll prove that 80 holds. Let [ Q11 Q Q = Q 21 Q 22 Invong the bloc matrx nverse formula, we can rewrte the rght-hand sde of 80 as [ [ [ [ B1 0 B1 0 B1 0 B1 0 I + Q [ B 2 = 1 + Q and the left-hand sde of 80 as [ [ [ [ A1 0 A1 0 A1 0 A1 0 I + Q 0 A 2 0 A 2 0 A 2 0 A [ [ 2 [ A1 0 I + A1 Q = 11 A 1 A 1 Q 12 A 2 A A 2 A 2 Q 21 A 1 I + A 2 Q 22 A 2 0 A 2 For smplcty of notaton, let [ I + A1 Q 11 A 1 A 1 Q 12 A 2 A 2 Q 21 A 1 I + A 2 Q 22 A 2 = M = [ M11 M 12 M 21 M From the bloc matrx nverse formula, we have [ [ M M = 11 0 M + 11 M 12 J [ M 0 0 I 21 M 11 I 85 where J represents the Schur complement of bloc M 11 of matrx M [43, whch s gven by Let M = J = M 22 M 21 M 11 M [ M 11 M 12 I J [ M 21 M 11 I. 87 Therefore, the left-hand sde of 80 can be further expressed as [ [ [ A1 0 M 11 0 A1 0 + M 0 A A 2 [ = A 2 [ [ 1 + Q 11 0 A1 0 A1 0 + M A 2 0 A 2 Owng to M 0, then we have J 0 and M 0. In addton, because A Q 11 B 2, 1 + Q holds. APPENDIX D PROOF OF LEMMA 4 Defne a dagonal matrx B wth the dagonal elements gven by [B = max {[B }. Due to Ã, Ã, and B beng dagonal matrces, and à à = 0, for, we have I + ÃB à B = I + ÃB à B I + ÃB j. 89 j Then, we can calculate the rght-hand sde of 26 as log det I + ÃB log det I + à à B = log det I + ÃB log det I + ÃB à B I + ÃB j j = log det I + ÃB log det I + ÃB + log det I + ÃB = log det I + ÃB. 90 In addton, we have log det I + AB log det I + A A B = log det I + B 1/2 AB1/2 log det I + B 1/2 AB1/2 B 1/2 A B 1/2 B = log det I A 1/2 B1/2 I + B 1/2 AB1/2 1/2 A1/2 log det I A 1/2 B1/2 I + B 1/2 AB 1/2 B 1/2 A 1/2. 91 The last nequalty holds from Lemma 3. We can further calculate the left-hand of 26 as log det I + AB log det I + A A B c 2015 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See

13 log det I B 1/2 I + B 1/2 AB 1/2 B 1/2 A = log det I B 1/2 I + B 1/2 AB 1/2 B 1/2 A log det I B 1/2 I + B 1/2 AB 1/2 B 1/2 A. 92 The last nequalty holds due to log det I A B log det I A+log det I B, for arbtrary symmetrc postve semdefnte matrces A and B satsfyng I A B 0. Usng matrx nverse lemma, we have I B 1/2 I + B 1/2 AB 1/2 B 1/2 A = I + B 1/2 AB 1/2. 93 Thus, log det I + AB log det I + A A B log det I + AB log det I + ÃB. 94 The last nequalty holds due to the adamard nequalty. ACKNOWLEDGMENT We would le to than the edtor and the anonymous revewers for ther helpful comments and suggestons. REFERENCES [1 T. L. Marzetta, Noncooperatve cellular wreless wth unlmted numbers of base staton antennas, IEEE Trans. Wreless Commun., vol. 9, no. 5, pp , Nov [2 F. Ruse, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta, O. Edfors, and F. Tufvesson, Scalng up MIMO: Opportuntes and challenges wth very large arrays, IEEE Sgnal Process. Mag., vol. 30, no. 1, pp , Jan [3. Q. Ngo, E. G. 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14 Ths artcle has been accepted for publcaton n a future ssue of ths journal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI 14 [36 T. Whte, Global Optmzaton Algorthm: Theory and Applcaton. Abrufdatum, [37 J. W. Kang, Y. Whang,. Y. Lee, and K. S. Km, Optmal plot sequence desgn for mult-cell MIMO-OFDM systems, IEEE Trans. Wreless Commun., vol. 10, no. 10, pp , Oct [38 E. G. Larsson, Sem-structured nterference suppresson for orthogonal frequency dvson multplexng, n Proc. IEEE ISSPIT, Dec [39 3rd Generaton Partnershp Project. TR V10.0.0, Spatal channel model for multple nput multple output MIMO smulatons, [40 X. Wang and. Poor, Iteratve turbo soft nterference cancellaton and decodng for coded CDMA, IEEE Trans. Commun., vol. 47, no. 7, pp , Aug [41 G. A. Seber, A Matrx andboo for Statstcans. John Wley & Sons, [42 A. Ostrows and. Schneder, Some theorems on the nerta of general matrces, Journal of Mathematcal analyss and applcatons, vol. 4, no. 1, pp , Feb [43 F. Zhang, The Schur Complement and ts Applcatons. Sprnger, Sh Jn S 06-M 07 receved the B.S. degree n communcatons engneerng from Guln Unversty of Electronc Technology, Guln, Chna, n 1996, the M.S. degree from Nanjng Unversty of Posts and Telecommuncatons, Nanjng, Chna, n 2003, and the Ph.D. degree n communcatons and nformaton systems from the Southeast Unversty, Nanjng, n From June 2007 to October 2009, he was a Research Fellow wth the Adastral Par Research Campus, Unversty College London, London, U.K. e s currently wth the faculty of the Natonal Moble Communcatons Research Laboratory, Southeast Unversty. s research nterests nclude space tme wreless communcatons, random matrx theory, and nformaton theory. e serves as an Assocate Edtor for the IEEE Transactons on Wreless Communcatons, and IEEE Communcatons Letters, and IET Communcatons. Dr. Jn and hs co-authors have been awarded the 2011 IEEE Communcatons Socety Stephen O. Rce Prze Paper Award n the feld of communcaton theory and a 2010 Young Author Best Paper Award by the IEEE Sgnal Processng Socety. Chen Sun S 13 receved the B.E. degree n electrcal engneerng from Southeast Unversty, Nanjng, Chna, n Currently he s worng towards the Ph.D. degree n the Natonal Moble Communcatons Research Laboratory, Southeast Unversty, Nanjng, Chna. s research nterests nclude communcatons, and nformaton theory, wth emphass on massve MIMO communcatons. Xq Gao S 92 AM 96 M 02 SM 07 F 15 receved the Ph.D. degree n electrcal engneerng from Southeast Unversty, Nanjng, Chna, n e joned the Department of Rado Engneerng, Southeast Unversty, n Aprl Snce May 2001, he has been a professor of nformaton systems and communcatons. From September 1999 to August 2000, he was a vstng scholar at Massachusetts Insttute of Technology, Cambrdge, MA, USA, and Boston Unversty, Boston, MA. From August 2007 to July 2008, he vsted the Darmstadt Unversty of Technology, Darmstadt, Germany, as a umboldt scholar. s current research nterests nclude broadband multcarrer communcatons, MIMO wreless communcatons, channel estmaton and turbo equalzaton, and multrate sgnal processng for wreless communcatons. From 2007 to 2012, he served as an Edtor for the IEEE T RANSACTIONS ON W IRELESS C OMMUNICATIONS. From 2009 to 2013, he served as an Edtor for the IEEE T RANSACTIONS ON S IGNAL P ROCESSING. e now serves as an Edtor for the IEEE T RANSACTIONS ON C OMMUNICATIONS. Dr. Gao receved the Scence and Technology Awards of the State Educaton Mnstry of Chna n 1998, 2006 and 2009, the Natonal Technologcal Inventon Award of Chna n 2011, and the 2011 IEEE Communcatons Socety Stephen O. Rce Prze Paper Award n the feld of communcaton theory. Mchal Matthaou S 05 M 08 SM 13 was born n Thessalon, Greece n e obtaned the Dploma degree 5 years n Electrcal and Computer Engneerng from the Arstotle Unversty of Thessalon, Greece n e then receved the M.Sc. wth dstncton n Communcaton Systems and Sgnal Processng from the Unversty of Brstol, U.K. and Ph.D. degrees from the Unversty of Ednburgh, U.K. n 2005 and 2008, respectvely. From September 2008 through May 2010, he was wth the Insttute for Crcut Theory and Sgnal Processng, Munch Unversty of Technology TUM, Germany worng as a Postdoctoral Research Assocate. e s currently a Senor Lecturer at Queen s Unversty Belfast, U.K. and also holds an adjunct Assstant Professor poston at Chalmers Unversty of Technology, Sweden. s research nterests span sgnal processng for wreless communcatons, massve MIMO, hardwareconstraned communcatons, and performance analyss of fadng channels. Dr. Matthaou was the recpent of the 2011 IEEE ComSoc Best Young Researcher Award for the Europe, Mddle East and Afrca Regon and a corecpent of the 2006 IEEE Communcatons Chapter Project Prze for the best M.Sc. dssertaton n the area of communcatons. e was co-recpent of the Best Paper Award at the 2014 IEEE Internatonal Conference on Communcatons ICC and was an Exemplary Revewer for IEEE C OMMUNICATIONS L ETTERS for In 2014, he receved the Research Fund for Internatonal Young Scentsts from the Natonal Natural Scence Foundaton of Chna. e has been a member of Techncal Program Commttees for several IEEE conferences such as ICC, GLOBECOM, VTC etc. e currently serves as an Assocate Edtor for the IEEE T RANSACTIONS ON C OMMUNICATIONS, IEEE C OMMUNICATIONS L ETTERS and was the Lead Guest Edtor of the specal ssue on Large-scale multple antenna wreless systems of the IEEE J OURNAL ON S ELECTED A REAS IN C OMMUNICATIONS. e s an assocate member of the IEEE Sgnal Processng Socety SPCOM and SAM techncal commttees c 2015 IEEE. 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15 Ths artcle has been accepted for publcaton n a future ssue of ths journal, but has not been fully edted. Content may change pror to fnal publcaton. Ctaton nformaton: DOI 15 Zh Dng S 88-M 90-SM 95-F 03 s Professor of Electrcal and Computer Engneerng at the Unversty of Calforna, Davs. e receved hs Ph.D. degree n Electrcal Engneerng from Cornell Unversty n From 1990 to 2000, he was a faculty member of Auburn Unversty and later, Unversty of Iowa. Prof. Dng has held vstng postons n Australan Natonal Unversty, ong Kong Unversty of Scence and Technology, NASA Lews Research Center and USAF Wrght Laboratory. Prof. Dng has actve collaboraton wth researchers from several countres ncludng Australa, Chna, Japan, Canada, Tawan, Korea, Sngapore, and ong Kong. Dr. Dng s a Fellow of IEEE and has been an actve volunteer, servng on techncal programs of several worshops and conferences. e was assocate edtor for IEEE Transactons on Sgnal Processng from , , and assocate edtor of IEEE Sgnal Processng Letters e was a member of techncal commttee on Statstcal Sgnal and Array Processng and member of Techncal Commttee on Sgnal Processng for Communcatons Dr. Dng was the Techncal Program Char of the 2006 IEEE Globecom. e s also an IEEE Dstngushed Lecturer Crcuts and Systems Socety, , Communcatons Socety, e served on as IEEE Transactons on Wreless Communcatons Steerng Commttee Member and ts Char Dr. Dng receved the 2012 IEEE Wreless Communcaton Recognton Award from the IEEE Communcatons Socety and s a coauthor of the text: Modern Dgtal and Analog Communcaton Systems, 4th edton, Oxford Unversty Press, Chengshan Xao M 99-SM 02-F 10 earned a Bachelor of Scence degree n electronc engneerng from the Unversty of Electronc Scence and Technology of Chna n 1987, a Master of Scence degree n electronc engneerng from Tsnghua Unversty n 1989, and a Ph.D. n electrcal engneerng from the Unversty of Sydney n From 1989 to 1993, he was a faculty member wth the Department of Electronc Engneerng, Tsnghua Unversty, Bejng, Chna. From 1997 to 1999, he was a Senor Member of Scentfc Staff wth Nortel Networs, Ottawa, ON, Canada. From 1999 to 2000, he was a Faculty Member wth the Unversty of Alberta, Edmonton, AB, Canada. From 2000 to 2007, he was wth the Unversty of Mssour, Columba, where he was an Assstant Professor and then an Assocate Professor. e s currently a Professor wth the Department of Electrcal and Computer Engneerng, Mssour Unversty of Scence and Technology, Rolla formerly Unversty of Mssour, Rolla. s research nterests nclude wreless communcatons, sgnal processng, and underwater acoustc communcatons. e s the holder of three U.S. patents. s nvented algorthms have been mplemented nto Nortel s base staton rados after successful techncal feld trals and networ ntegraton. Dr. Xao s the Drector of Conference Publcaton of IEEE Communcatons Socety ComSoc, an Elected Member of IEEE ComSoc Board of Governors, a Member of IEEE ComSoc Fellow Evaluaton Commttee, and a Dstngushed Lecturer of the IEEE Vehcular Technology Socety. Prevously, he served as an Edtor, an Area Edtor and the Edtor-n-Chef of the IEEE Transactons on Wreless Communcatons; an Assocate Edtor of the IEEE Transactons on Vehcular Technology, the IEEE Transactons on Crcuts and Systems-I, and the nternatonal journal of Multdmensonal Systems and Sgnal Processng. e was the Techncal Program Char of the 2010 IEEE Internatonal Conference on Communcatons ICC, the Lead Co-char of the 2008 IEEE ICC Wreless Communcatons Symposum, and a Phy/MAC Program Co-char of the 2007 IEEE Wreless Communcatons and Networng Conference. e served as the foundng Char of the IEEE Techncal Commttee on Wreless Communcatons and the Vce-Char of the IEEE Techncal Commttee on Personal Communcatons. e s a recpent of the 2014 umboldt Research Award from the Alexander von umboldt Foundaton, and he also receved the 2014 Joseph LoCcero Award from IEEE Communcatons Socety c 2015 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See