2200 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 6, JUNE 2015

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1 00 IEEE TRANSACTIONS ON COMMUNICATIONS VOL. 63 NO. 6 JUNE 015 Two-Stae Beamformer Desin for Massive MIMO Downlink By ace Quotient Formulation Donun Kim Student Member IEEE Gilwon Lee Student Member IEEE and Younchul Sun Senior Member IEEE Abstract In this paper the problem of outer beamformer desin based only on channel statistic information is considered for two-stae beamformin for multi-user massive MIMO downlink and the problem is approached based on sinal-to-leakae-plusnoise ratio SLNR. To eliminate the dependence on the instantaneous channel state information a lower bound on the averae SLNR is derived by assumin zero-forcin ZF inner beamformin and an outer beamformer desin method that maximizes the lower bound on the averae SLNR is proposed. It is shown that the proposed SLNR-based outer beamformer desin problem reduces to a trace quotient problem TQP which is often encountered in the field of machine learnin. An iterative alorithm is presented to obtain an optimal solution to the proposed TQP. The proposed method has the capability of optimally controllin the weihtin factor between the sinal power to the desired user and the interference leakae power to undesired users accordin to different channel statistics. Numerical results show that the proposed outer beamformer desin method yields sinificant performance ain over existin methods. Index Terms Massive MIMO systems two-stae beamformin sinal-to-leakae-plus-noise ratio SLNR trace quotient problem TQP adaptive weihtin factor. I. INTRODUCTION TE multiple-input multiple-output MIMO technoloy has prevailed in wireless communications for more than a decade. The technoloy has been adopted in many wireless standards since it improves the spectral efficiency and reliability of wireless communication without requirin additional bandwidth. Recently the MIMO technoloy based on larescale antenna arrays at base stations so-called massive MIMO is considered to further improve the system performance for upcomin 5G wireless systems and viorous research is bein performed on this topic. Massive MIMO can support hih data rates and enery efficiency and simplify receiver processin based on the asymptotic orthoonality amon user channels Manuscript received November 5 014; revised March and April ; accepted April Date of publication May 5 015; date of current version June This research was supported in part by Basic Science Research Proram throuh the National Research Foundation of Korea NRF funded by the Ministry of Education 013R1A1AA This work was also supported by The Cross-Ministry Gia KOREA Proect of The Ministry of Science ICT and Future Plannin Korea [GK14N0100 5G mobile communication system development based on mmwave]. The associate editor coordinatin the review of this paper and approvin it for publication was M. Matthaiou. The authors are with Department of Electrical Enineerin Korea Advanced Institute of Science and Technoloy KAIST Daeeon Korea d.kim@kaist.ac.kr; wlee@kaist.ac.kr; ysun@ee.kaist.ac.kr. Diital Obect Identifier /TCOMM based on lare antenna arrays [1] []. owever realizin the benefits of massive MIMO in practical systems faces several challenes especially in widely-used frequency division duplexin FDD scenarios. In contrast to current small-scale MIMO systems downlink channel estimation is a difficult problem for FDD massive MIMO systems since the number of available trainin symbols required for downlink channel estimation is limited by the channel coherence time and the number of channel parameters to estimate is very lare [3] [6]. Furthermore channel state information CSI feedback overhead for downlink user schedulin for massive FDD multi-user MIMO can be overwhelmin without some smart structure on massive MIMO systems. To overcome these difficulties associated with massive MIMO two-stae beamformin for massive MIMO under the name of Joint Spatial Division and Multiplexin JSDM has been studied in [7] [11]. The two-stae beamformin idea is basically a divide-and-conquer approach and the key ideas of the two-stae beamformin stratey are 1 to partition the user population supported by the servin base station into multiple roups each with approximately the same channel covariance matrix this can be viewed as virtual sectorization and to decompose the MIMO beamformer at the base station into two steps: an outer beamformer and an inner beamformer as shown in Fi. 1. The outer beamformer faces the antenna array and rouhly distinuishes different roups by bolsterin in-roup transmit power and suppressin inter-roup interference and the inner beamformer views the product of the actual channel and the outer beamformer as an effective channel separates the users within a roup and provides spatial multiplexin amon in-roup users [7]. ere maor complexity reduction results from the approach that the outer beamformer is properly desined based only on channel statistic information not on CSI. In this case the actually required CSI for the inner beamformer adoptin typical zero-forcin ZF or reularized ZF RZF beamformin is sinificantly reduced since it only requires the CSI of the effective channel with sinificantly reduced dimensions. Several researchers followed the aforementioned framework for two-stae beamformin for massive MIMO. They adopted linear beamformin such as ZF for the inner beamformer and tackled the problem of outer beamformer desin based on channel statistic information [7] [10] [11]. In [7] Adhikary et al. proposed a simple block diaonalization BD alorithm for the outer beamformer desin which obtains the outer beamformer by proectin the dominant eienvectors of the desired roup channel covariance matrix onto the null IEEE. Personal use is permitted but republication/redistribution requires IEEE permission. See for more information.

2 KIM et al.: TWO-STAGE BEAMFORMER DESIGN FOR MASSIVE MIMO DOWNLINK 01 Fi. 1. Multi-roup massive MIMO downlink with two-stae beamformin [9]. space of the dominant eienspace of all other roup channel covariance matrices. In [10] Chen and Lau considered the outer beamformer desin criterion of minimizin the total interroup interference power minus the weihted total desired roup sinal power. In this case for a iven weihtin value between the total inter-roup interference power and the total in-roup sinal power the outer beamformer is iven by a set of dominant eienvectors of the weihted difference between the total undesired roup channel covariance matrix sum and the desired roup channel covariance matrix. In [11] Liu and Lau considered the outer beamformer desin from a fairness perspective. In this work they desined the outer beamformer by choosin a set of columns from a discrete Fourier transform DFT matrix to maximize the minimum averae rate amon all the users. In this paper we also consider the outer beamformer desin based only on channel statistic information for the aforementioned two-stae beamformin framework for massive MIMO. As already shown in the previous works computation of the sinal-to-interference-plus-noise-ratio SINR for each receiver is difficult in this downlink scenario with interferin roups. To circumvent this difficulty as our desin criterion we adopt the averae sinal-to-leakae-plus-noise ratio SLNR criterion [1] which is shown to be Pareto-optimal in the achievable rate reion in certain interference channel cases [13] [14] and propose an averae SLNR-based outer beamformer desin framework in sinle cell massive MIMO systems. 1 The sinal power to the desired receiver and the leakae power to other undesired receivers by the transmitter required for the SLNR method cannot be computed by considerin only the outer beamformer. Instead both the outer beamformer and the inner beamformer should ointly be considered to derive the two quantities. Thus to simplify analysis we assume a ZF beamformer with equal power allocation for the inner beamformer althouh simulation is performed for both ZF and reularized ZF RZF inner beamformers. Even with this assumption of ZF for the inner beamformer the derivation of averae SLNR is not straihtforward due to the oint nature. Thus exploitin the fact that ZF is used for the inner beamformer we derive a 1 The multi-cell scenario considered in [10] can be cast into this sinle-cell multi-roup settin simply by considerin each base station in the multi-cell case as one roup in the sinle-cell multi-roup case. lower bound on the averae SLNR that is a function of only channel statistics and the outer beamformer and our desin criterion is to maximize this lower bound on the averae SLNR under the constraint that the outer beamformer matrix has orthonormal columns. Then we cast this constrained optimization problem as a trace quotient problem TQP which is often encountered in the field of pattern reconition computer vision and machine learnin [15] [17]. To obtain an optimal solution to the formulated TQP we modify the alorithm in [17] to fit into the considered case and show the optimality and converence of the modified alorithm based on existin results [15] [17]. Numerical results show that the proposed outer beamformer desin approach yields sinificant sum rate ain over the existin alorithms in [7] and [10]. Notation and Oranization: We will make use of standard notational conventions. Vectors and matrices are written in boldface with matrices in capitals. All vectors are column vectors. For a matrix X X X T X [X] i and X indicate the complex conuate transpose conuate transpose i -th element and trace of X respectively. I n stands for the identity matrix of size n the subscript is omitted when unnecessary. For vector x x represents the -norm of x.dia x 1 x x n means a diaonal matrix with diaonal entries x 1 x x n. The notation x CNμ means that x is complex circularly-symmetric Gaussian distributed with mean vector μ and covariance matrix. E{ } denotes the expectation. ι 1 and C is the set of complex numbers. The remainder of this paper is oranized as follows. The system model is described in Section II. In Section III a lower bound on the averae SLNR is derived and the outer beamformer desin problem is formulated as a TQP. An iterative alorithm for the TQP is presented and its optimality and converence are shown. The performance of the proposed alorithm is investiated in Section IV followed by the conclusion in Section V. II. SYSTEM MODEL We consider a sinle-cell massive MIMO downlink system in which a base station with a uniform linear array ULA of M transmit antennas serves K sinle-antenna users. We assume that K users in the cell are partitioned into G roups such that K = G =1 K where K is the number of users in roup and K users in roup have the same M M channel covariance matrix R as in [7] [9] [11]. We assume a typical spatial correlation channel model [4] [18]. That is the channel of user k in roup is iven by h k = R 1/ h i.i.d 1 where h i.i.d CM 1 i.i.d. CN0 I M. We assume that the outer beamformer implements virtual sectorization i.e. the outer The outer beamformer matrix s havin orthonormal columns is very desirable for effective downlink channel estimation [3] and downlink user schedulin [8] [9] purposes under the two-stae beamformin framework for massive MIMO.

3 0 IEEE TRANSACTIONS ON COMMUNICATIONS VOL. 63 NO. 6 JUNE 015 beamformer divides the overall azimuthal anle typically 10 derees in conventional cellular networks into multiple virtual sectors with rouhly 3 derees for azimuthal coverae for each virtual sector. A channel model considerin such a situation is the one-rin scatterin model [19] [0] which captures the base station s elevation and local scatterin around the users. Under this model the channel covariance matrix for each sector coverin azimuthal anle centered at θ can be precomputed as [19] [0] [ R ]kl = 1 θ+ θ e ιπk l d λc sin ω dω where λ c is the carrier wavelenth d is the antenna spacin θ is the anle of the center of the subsector and is the anle spread AS. From here on we will assume that the channel covariance matrices R = 1 G are iven to the base station. As in [7] [11] we consider two-stae beamformin for downlink transmission with outer beamformers {V = 1 G} for roup separation or virtual sectorization and an inner beamformer W for user separation within roup for each = 1 G as shown in Fi. 1. Denote the overall K M channel matrix as = [ 1 G] where =[h 1 h h K ] is the K M channel matrix for the users in roup. Then the sinal vector received by all the users in the cell is iven by y = VWs + n 3 where the overall outer beamformer matrix V is partitioned into G submatrices as V =[V 1 V V G ] with V C M M satisfyin 4 V V = I; 4 the overall inner beamformer W has a block diaonal structure as W = diaw 1 W W G with the inner beamformer W =[w 1 w w K ] C M S for roup ; s =[s 1 s s ] CN0 I S is the data vector with s CN0 I S ; and n =[n 1 n n G ] CN0σ I K is the noise vector. Thus M is the dimension of the effective MIMO channel seen by the inner beamformer W and S is the number of data streams for roup. We assume that the base station has an averae transmit power constraint as E[VWss W V ]=VWE[ss ]W V = VWW V = WW P T. Combinin V we can rewrite the data model 3 as y = Ws + n 5 3 Each sector cannot be completely separated because the number of transmit antennas is finite and there always exists some overlap amon virtual subsectors even in the case we desin the main coverae anle of of each sector to be small. 4 The orthoonality constraint is desirable for random beamformin type user schedulin [8] or ReDOS-PBR user schedulin [9] for two-stae beamformin based massive MIMO. where 1 V 1 1 V 1 V G V 1 V V G V = G V 1 G V G V G and V C K M is the effective MIMO channel seen by the inner beamformer W for roup. We assume that the CSI of the effective MIMO channel = 1 G is available to the transmitter please see [3] and the inner beamformer W C M S M S for each = 1 G is desined as a zero-forcin beamformer with equal power w k = 1for each user based on the effective CSI i.e. W = V V V 1 P 7 where P = diap 1 p S and p k is the transmit power scalin factor for user k in roup for satisfyin w k = 1. The received sinal vector for the users in roup is iven by y = V W s + = V W s + n 8 where s =[s 1 s s S ] T C S 1 and n =[n 1 n n K ] T C K 1 are the data and noise vectors for roup respectively. The received sinal of user k in roup is iven by y k = h V w k s k + k =k h V w k s k K + h V w s + n k 9 = =1 where the second and third terms in the riht-hand side RS of 9 are the intra-roup and inter-roup interference respectively. With the assumed ZF inner beamformin the intra-roup interference is completely eliminated and the sinal-tointerference-plus-noise ratio SINR at user k in roup is iven by SINR k = K = =1 h V w k. 10 h V w + σ Usin the SINR as the optimization criterion one could try to desin the outer beamformer {V = 1 G} to maximize a relevant measure such as the sum rate. owever this criterion enerally leads to a challenin nonconvex optimization problem since each user s SINR is ointly dependent on {V = 1 G} in a nonconvex manner [1] [13]. To circumvent the difficulty we here adopt the SLNR approach proposed in [1] and [13].

4 KIM et al.: TWO-STAGE BEAMFORMER DESIGN FOR MASSIVE MIMO DOWNLINK 03 The SLNR approach considers the ratio between the sinal power to the desired receiver and the power of the total interference to undesired receivers caused by the desired sinal plus noise not the power of the total interference received at the desired receiver. The rationale for this approach is that it is reasonable for the transmitter to maximize the sinal power to the desired receiver for a iven allowed level of interference to undesired receivers in multi-user interference channels. The SLNR method is shown to be Pareto-optimal in the rate reion in certain MIMO interference channel scenarios [13] [14]. Under the assumption of ZF inner beamformin with equal power allocation the SLNR for user k in roup is iven by SLNR k = K = =1 h V w k h. 11 V w k + σ Note that the key difference between 10 and 11 is that the SINR of user at the receiver side is a oint function of {V = 1 G} whereas the SLNR of user at the transmitter side is a function of only V not of {V = }. Note also that the SLNR of user is a function of the channel {h k h = = 1 K } and the inner beamformer w k in addition to V. III. OUTER BEAMFORMER DESIGN CRITERION AND OPTIMIZATION Recall that the main advantae of the two-stae beamformin results from the fact that the outer beamformer is desined without knowin the CSI of the actual channel {h k k = 1 K = 1 G} [7] [11]. Thus the outer beamformer should be desined based only on the channel covariance matrices {R = 1 G} and this leads to usin the averae SLNR as our desin criterion. ence we formulate the outer beamformer desin problem as follows: P 1 : V = ar max V V =I K E[SLNR k ] 1 for each = 1 G. Althouh Problem P 1 is conceptually simple solvin the optimization problem is not straihtforward. The first difficulty is the derivation of the averae SLNR since the random quantities i.e. the channel vectors are both in the numerator and the denominator as seen in 11 and a closed-form expression of the averae SLNR is not available. To circumvent this difficulty we first derive a lower bound on the averae SLNR and maximize this lower bound on the averae SLNR under the constraint V V = I. For a iven outer beamformer V and a iven inner beamformer W the SLNR of user averaed over channel realizations is lower bounded as follows: E { } SLNR k h V w k = E 13 K = =1 h V w k + σ a h V w k E 14 K = =1 h V w k + σ h b V w k = E 15 = K =1 h V + σ { h } c = E V w k E = K = =1 1 h { d h } 1 E V w k { K E = { h ] E V w k e = { K = =1 V E h h { h } E V w k =1 h V + σ } V + σ } V + σ V R V ere a follows from the sub-multiplicativity of norm AB A B ; b follows from the equal power allocation w k =1; c follows from the independence between the desired-roup channels and other roup channels; d results from Jensen s inequality since the function f x = 1 x is convex for x 0; and e follows from R K R + σ I. 18 = M The next difficulty in derivin a lower bound on the averae SLNR is that the ZF inner beamformer w k is based on the CSI of the effective channel V. owever at the time of desinin V this effective channel is not determined. This difficulty is properly circumvented by exploitin the property of ZF inner beamformin and a lower bound on the averae sinal power appearin in the numerator of the RS of 17 is iven in the followin theorem. Theorem 1: When the outer beamformer V is iven and the inner-beamformer W =[w 1 w w K ] is a ZF beamformer with equal power allocation E[ h V w k ] is lowerbounded as { h } E V w k V R V K 1λ 19

5 04 IEEE TRANSACTIONS ON COMMUNICATIONS VOL. 63 NO. 6 JUNE 015 where λ is the larest eienvalue of the channel covariance matrix R of roup. Proof: See Appendix. Applyin Theorem 1 to 17 we obtain a lower bound on the averae SLNR iven by [ h ] E [ E ] V w k SLNR k V R V V R V K 1λ V R V V R 1V = 0 V R V where R 1 is defined as R 1 R K 1 λ I. 1 M Note that the derived lower bound on the averae SLNR depends only on the roup index not on the user index k. This makes sense since we assume that each user in the same roup has the same channel statistics. Finally the proposed outer beamformer desin based only on the channel statistic information {R = 1 G} is formulated as V R 1V P : V = ar max V V =I V R V for each = 1 G where R 1 and R are iven by 1 and 18 respectively. Note that in Problem the outer beamformer desin is performed for each roup separately. This is an advantae of the proposed desin method; complicated oint optimization is not required. Note that in Problem P R 1 is ermitian and R is positive definite due to the added identity matrix in R in 18. Problem P maximizes the quotient or ratio of two traces under the constraint V V = I and is known as a trace quotient problem TQP or trace ratio problem TRP which is often encountered in linear discriminant analysis LDA for feature extraction and dimension reduction [15] [16]. Several research works have been performed to understand the theoretical properties of TQP and to develop numerical alorithms for TQP [15] [17]. If we relax the orthoonality constraint V V = I for the outer beamformer then Problem P reduces to the followin simpler optimization problem: V P 3 : V = ar max R 1V 3 V R V for each = 1 G. It was shown in [1] that the optimal V of Problem P 3 is the M eneralized eienvectors correspondin to the M larest eneralized eienvalues of the matrix pencil R 1 λr. There exist many available fast alorithms to obtain the eneralized eienvectors of the positive definite matrix pencil R 1 λr [1] []. Note that the obtained eneralized eienvectors from Problem P 3 do not satisfy the orthoonality constraint V V = I in eneral. The optimal solution V of Problem P 3 can be decomposed by thin sinular value decomposition SVD as V = D 4 where is an M M matrix satisfyin = I; D is an M M diaonal matrix; and is an M M unitary matrix. By settin V = 5 we can obtain an outer beamformer satisfyin the constraint V V = I. Althouh this approach does not necessarily yield an optimal solution to Problem P this solution is useful for the proposed iterative alorithm in the next section to reduce the computational complexity. Specifically the solution 5 can be used as an initialization point for the iterative alorithm. The performance of this initialization and random initialization will be evaluated in Section IV. A. Optimal Solution to TQP In this section we briefly discuss the optimal solution to the TQP P. In Theorem.1 of [16] the optimal solution V to the TQP P is known as the eienvectors correspondin to the M larest eienvalues of E ρv R 1 ρv R 6 when the value ρv V R 1V V is maximized. In [16] R V and [3] it is also shown that M ={V C M M E ρv V = V M V and V V = I M } is the set of all Karush-Kuhn-Tucker KKT points of TQP P where M V = V E ρv V.This implies that any KKT point V of the TQP P is an orthonormal eienbasis of the matrix E ρv with the sum of the correspondin eienvalues zero i.e. M V = 0. It is easy to see the fact that M V = 0 from 6. Note that 6 is a nonlinear eienvalue problem since the weihtin factor ρv itself is a function of the desin variable V. It is known that there is no closed-form solution to the TQP [15] [17]. In [16] however it is shown that any iterative method that monotonically converes to a KKT point in M achieves a lobal maximizer V for the TQP by repeatedly applyin the update stratey for ρv specified in Section IV of [16]. The update stratey is that in every iteration step ρv is computed as ρv n where V n is determined as the eienvectors correspondin to the M larest eienvalues of R 1 ρv n 1 R. The proof of this is based on the radient and essian of the cost function and optimization theory [16]. Althouh the statement was proved in the case that R 1 and R are real symmetric and real positivedefinite symmetric respectively in [16] the statement can easily be extended to the case that R 1 and R are ermitian and positive-definite respectively. Interested readers are referred to [16]. B. The Outer Beamformer Desin Alorithm and Its Converence In [17] an iterative alorithm was proposed under the condition that R 1 and R are both real symmetric positive

6 KIM et al.: TWO-STAGE BEAMFORMER DESIGN FOR MASSIVE MIMO DOWNLINK 05 semidefinite. owever in our case R 1 is ermitian and R is positive-definite. Thus to apply the iterative alorithm in [17] to our case we need sliht modification of the alorithm from the real matrix case to the complex matrix case. Basically the presented alorithm iteratively optimizes the obective function subect to the orthoonality constraint with ρv update specified in [16]. Denote the obective function of Problem P at the n-th iteration by V n 1 R1 V n 1 ρ n = V n 1 R V n 1 7 where V n 1 is the outer beamformer at the n 1-th iteration. The outer beamformer V n is updated by solvin the followin trace difference problem: V n = ar max V R 1 ρ n R V 8 V V =I where ρ n is iven by 7. The outer beamformer V n of the trace difference problem in the n-th iteration is updated by the M eienvectors correspondin to the M larest eienvalues of the ermitian matrix R 1 ρ n R for iven ρ n. This procedure is iterated until the iteration converes. The proposed alorithm for Problem P is summarized in Alorithm 1. Note that in the proposed alorithm the initialization of V 0 is done based on 5. This initialization sinificantly reduces the required number of iterations compared to random initialization. The monotonic increase of the obective function ρ n by Alorithm 1 is uaranteed by the followin theorem: Alorithm 1 Outer BeamformerDesinby TQP Based on [17] Require: The channel covariance matrices {R =1 G} the noise variance σ and the stoppin tolerance ɛ>0. Construct R 1 and R by 1 and 18. Set ρ 1 = and initialize V 0 by 5. for n = 1 do Compute the trace ratio ρ n from V n 1 ρ n = V n 1 V n 1 : R1 V n 1 R V n 1 Solve the trace difference problem V n = ar max V V =I That is update V n =[v n R 1 ρ n R v n. 9 V R 1 ρ n R V vn vn M ] by solvin k = νn k v n k 31 where v n k is the eienvector correspondin to the k-th larest eienvalue ν n k of R 1 ρ n R. if ρ n ρ n 1 <ɛ Break the loop. end if end for Theorem : Alorithm 1 monotonically increases the trace quotient ρ n i.e. ρ n+1 ρ n for all n. Proof: See Appendix. Furthermore the boundedness of the obective function ρ n for all n can also be shown as stated in the followin theorem: Theorem 3: Let ψ 1 ψ ψ M and ϕ 1 ϕ ϕ M > 0 be the sorted eienvalues of R 1 and R 0 respectively. Then the obective function of the TQP is lowerbounded and upper-bounded as follows: M i ψ M k 0 and ψ k 0 : M M ψ M k M ϕ k M ρv ii ψ M k < 0 and ψ k 0 : M ψ M k M ϕ M k M M ρv iii ψ M k < 0 and ψ k < 0 : M ψ M k M ϕ M k M ρv M ψ k M ϕ M k M ψ k M ϕ M k M ψ k M. ϕ k Proof: See Appendix. Theorems and 3 toether imply that the proposed outer beamformer desin alorithm monotonically converes by the monotone converence theorem. The converence of the proposed alorithm actually uarantees its lobal optimality. Let us first assume that the proposed alorithm converes in the n-th iteration i.e. ρ n+1 = ρ n. Then the same trace difference problem 30 in the n + 1-th iteration as that in the n-th iteration will be solved by eienvalue decomposition. Thus it is easy to see that the obtained solution V from the proposed alorithm is a KKT point in M. This implies the local optimality of the proposed alorithm. In Theorem 1.1. of [16] it was shown that any local optimum is also a lobal optimum for the TQP P.In addition to this Alorithm 1 adopts the ρv update stratey specified in [16]. As the optimality result in [16] explained in Section III-A Alorithm 1 yields a lobal optimizer for the TQP P. C. Discussion Now consider the difference between the proposed outer beamformer desin method and the alorithm in [10]. In Section IV it will be seen that both alorithms perform better

7 06 IEEE TRANSACTIONS ON COMMUNICATIONS VOL. 63 NO. 6 JUNE 015 than the outer beamformer desin method in [7]. The alorithm in [10] basically minimizes R wr 3 = where w is an arbitrary chosen constant weihtin factor. Problem 3 is equivalent to maximizin R λ R 33 = where λ = 1/w. Note that the first difference is that R and = R are respectively used in [10] instead of R 1 and R used in the proposed alorithm. Thus in the proposed alorithm there is sliht chane in the sinal power by includin the impact of ZF inner beamformin and the inclusion of the thermal noise in the leakae part. owever the maor difference of the proposed approach from the existin method in [10] is the formulation of the trace ratio in Problem P. As already mentioned in Section III-A it was shown by Zhan et al. that the optimal solution of TQP P is the M dominant eienvectors of R 1 ρv R 34 with ρv = a lower bound of E{SLNR k } achievin its maximum value [16]. Thus the proposed alorithm updates the weihtin factor ρv for different channel statistics by solvin a nonlinear eienvalue problem 5 note that the weihtin factor itself is a function of the desin variable V whereas the existin method solves a linear eienvalue problem by simply fixin the weihtin factor. The procedure of updatin the weihtin factor of the proposed outer beamformer desin approach can yield sinificant performance ain over the existin method as seen in Section IV. Now consider the difference between Problem P and Problem P 3. The optimal solution to Problem P 3 is iven by the set of M dominant eneralized eienvectors of R 1 R satisfyin [1] R 1 ξ i = λ i R ξ i i = 1 M 35 where λ 1 λ M. Since R 1 is ermitian and R is positive-definite {ξ i } are R -orthoonal [4] i.e. ξ R ξ i = 0 i =. Thus V =[ξ 1 ξ M ] does not satisfy the orthoonality constraint V V = I unless R = ci for some constant c. owever the matrix in 34 is ermitian and thus it is diaonalizable by unitary similarity i.e. it has orthonormal eienvectors [5]. ence it yields an outer beamformer satisfyin the orthoonality constraint. Note that 33 involves a linear ermitian eien-system composed of a weihted matrix difference with a fixed weihtin factor and Problem P 3 involves a linear 5 This procedure is clearly seen in Alorithm 1. eneralized eien-system 35 with a matrix pencil whereas the proposed problem P involves a nonlinear ermitian eiensystem 34 aain composed of a weihted matrix difference but with a weihtin factor dependin on the desin variable. IV. NUMERICAL RESULTS In this section we provide some numerical results to evaluate the performance of the proposed outer beamformer desin method in Section III. Throuhout the simulation we considered a massive multiple-input sinle-output MISO downlink system in which a base station is equipped with a ULA of M = 18 antenna elements and each of K users has a sinle receive antenna. The K users were rouped into four roups G = 4 and the base station supported five users simultaneously for each roup i.e. K = 5 for each = 1 3 and 4. M and S were set as M = M/G and S = K for = 1 3 and 4. The channel covariance matrix for each roup is specified by with the center anle θ and the AS parameter. The channel vector for each user was independently enerated accordin to the model 1. The stoppin tolerance of the proposed outer beamformer desin alorithm was set as ɛ = The noise power is set as σ = 1. From here on all db power values are relative to σ = 1. Fi. a shows the sum rate performance of the proposed outer beamformer desin alorithm and two existin alorithms in [7] and [10]. Throuhout the simulation the weihtin factor for the alorithm in [10] was set as w = 1. We also considered reularized ZF RZF [6] inner beamformin with the same outer beamformer desined by the proposed alorithm under the assumption of ZF inner beamformin. The reularization factor was set as α = K/P T which is approximately optimized based on [6] [7]. The parameters for four virtual sectors coverin a conventional cell sector were θ = and 45 with = π/13 i.e. = 7.7. Note that still inter-roup interference leakae comes in even thouh <30 for the channel covariance model when the number of transmit antennas is finite. The power azimuth spectrum PAS for this settin is shown in Fi. b. The curves in Fi. a is the averae sum rate over 000 independent channel realizations accordin to the model 1 and. It is seen that the proposed alorithm has sinificant ain over the other two alorithms. It is also seen that in the low SNR reion the proposed outer beamformer combined with RZF inner beamformin has performance ain over the proposed outer beamformer combined with ZF inner beamformin as expected. Althouh it is not shown due to space limitation similar relative performance is observed for different channel models such as truncated Gaussian and Laplacian power azimuth spectra. Fi. 3 shows the sum of the total sinal power to each desired roup and the sum of the total leakae power to undesired roups correspondin to each roup under the same settin as in Fi.. The shown curves are averae values over 000 channel realizations. Now the cause of the performance ain of the proposed alorithm over the existin alorithms is clear. It achieves almost the same sinal power to the desired roup with reduced leakae power to undesired roups compared to the two other alorithms by solvin a nonlinear eienvalue

8 KIM et al.: TWO-STAGE BEAMFORMER DESIGN FOR MASSIVE MIMO DOWNLINK 07 Fi.. M = 18 G = 4 M = 3 K = S = 5 = π/13 and θ = and 45 : a averae sum rate versus transmit power and b PAS of the model. Note that the db transmit power in the x-axis in Fi. a is not the received SNR because of the channel ain and the beamformin ain. Fi. 3. a Total desired roups power versus transmit power b total interference leakae power versus transmit power c total desired roups power in a db scale versus transmit power and d total interference leakae power in a db scale versus transmit power M = 18 G = 4 M = 3 K = S = 5 = π/13 and θ = and 45. problem with adaptive weihtin resultin from the trace quotient formulation for the sinal power and the leakae power. In the case of the alorithm [10] the performance can chane with respect to the weihtin factor between the desired-roup sinal power and the inter-roup interference power. owever the optimal weihtin factor dependin on the channel statistics is not known aprioriin the case of a constant weihtin factor and the weiht factor is not adaptively optimized accordin to the iven channel statistic information in [10]. Thus we evaluated the sum rate performance for different channel covariance

9 08 IEEE TRANSACTIONS ON COMMUNICATIONS VOL. 63 NO. 6 JUNE 015 Fi. 4. Averae sum rate over various alorithms versus anle spread M = 18 G = 4 M = 3 K = S = 5 P T = 10 db and θ = and 45. Moreover the alorithm in [7] is based on the sequential proection from Group 1 to Group 4. Thus the effective channel power for each roup is deraded at every proection step and this is clearly shown in Fi. 5. Finally we tested the converence of the proposed alorithm. Fi. 6a shows the obective value ρ n of the TQP for each roup by the proposed alorithm as a function of iteration time. It is seen that the proposed alorithm monotonically converes as expected from Theorem and furthermore the proposed alorithm converes after only a few iterations. Next we tested the converence of the proposed alorithm with different initial points. Fi. 6b shows the obective value ρ n of the TQP for Group 1 for different initial points. It is seen that the alorithm with the proposed initial point based on 5 converes much faster than the other two random initial points. Since the proposed alorithm requires eienvalue decomposition EVD for each iteration step the number of required iterations directly translates into computational complexity. Thus the initialization based on 5 reduces computational complexity sinificantly compared to random initialization althouh the alorithm with a random initial point also converes eventually. Fi. 5. Averae SLNR versus each roup for various alorithms M = 18 G = 4 M = 3 K = S = 5 = π/13 P T = 15 db and θ = and 45. matrix setup to see the impact of a fixed weihtin factor. The channel covariance matrix is a function of the center anle and anular spread accordin to the model. We fixed the center anles of four virtual subsectors as θ = and 45 but varied AS. Fi. 4 shows the averae sum rate performance with respect to when the transmit power is 10 db. Note the transmit power 10 db point in Fi.. It is seen that at a certain AS the fixed weihtin factor w = 1 yields ood performance but deteriorated performance for different AS s. Thus the optimization of the weihtin factor dependin on different channel statistics is necessary for ood performance for arbitrary channel statistics. Fi. 5 shows the averae SLNR performance for each roup under the same settin as in Fi.. The shown values are averae values over 000 channel realizations. It is seen that the proposed alorithm outperforms the other two alorithms. The alorithm in [7] has relatively poor performance compared to the other alorithms since it only considers the subspace spanned by the dominant eienvectors of each channel covariance matrix. The channel component enerated by the nondominant eienvectors can result in nonneliible interference. V. C ONCLUSION In this paper we have considered the outer beamformer desin problem based only on channel statistic information for two-stae beamformin for massive MIMO downlink and have proposed an outer beamformer desin method which maximizes a lower bound on the averae SLNR. We have shown that the proposed SLNR-based outer beamformer desin problem reduces to a TQP which is often encountered in the field of machine learnin and have presented an iterative alorithm to obtain an optimal solution to the proposed TQP. Numerical results show that the proposed outer beamformer desin method yields sinificant performance ain over existin methods. The proposed method can easily be adapted to the multi-cell scenario in which each base station serves one roup of users. APPENDIX A PROOF OF TEOREM 1 Usin similar techniques to those used in the proof of Lemma 1 in [8] and the proof of Theorem 1 in [11] we first prove several lemmas necessary for proof of Theorem 1. Lemma 1: For ZF inner beamformin we have h V w k = h k V Ũ k Ũ V h 36 where the column space of Ũ k is the null space of the composite effective channel matrix k except the channel of user i.e. k [V h 1 V h 1 V h +1 V h K ]. Proof: k C K 1 M K M is decomposed by sinular value decomposition SVD as [ U ] k = ϒ k D k k Ũ 37

10 KIM et al.: TWO-STAGE BEAMFORMER DESIGN FOR MASSIVE MIMO DOWNLINK 09 Fi. 6. a The obective value of the TQP for each roup versus iteration number and b the obective value of the TQP of Group 1 for various initial points versus iteration number M = 18 G = 4 M = 3 K = S = 5 = π/13 and θ = and 45. where ϒ k is a K 1 K 1 unitary matrix D k is a M M diaonal matrix U k is a M K 1 submatrix whose column space is the row space of k and Ũ k is a M M K + 1 submatrix whose column space is the null space of k. The ZF inner beamformer w k with equal power can be expressed by the proection of the effective channel V h of user to Ũ k as [8] w k = By 38 h V w k is expressed as h V w k = h k V w k w V h Ũ k Ũ V h Ũ Ũ V h. 38 = h V Ũ k Ũ V h h V Ũ k Ũ V h Ũ Ũ V h = h V Ũ k Ũ V h. This concludes the proof. Lemma : The conditional expectation of h V w k for iven k is iven by { h } E V w k k = k 39 where k = Ũ V R V Ũ k 40 and Ũ k is defined in 37 in Lemma 1. Proof: By Lemma 1 we have h V w k =h V Ũ k Ũ V h. For iven k Ũ k is also iven because Ũ k deterministically depends on k accordin to 37. Define ξ k Ũ V h. Then ξ k k is a zero-mean complex Gaussian random vector with the covariance matrix k = Ũ V R V Ũ k i.e. ξ k k CN0 k since h k CN0 R from 1. Let the eienvalue decomposition of k be k = U k k U where k = dia λ k 1 λ k N and N = M K + 1. Then ξ k can be expressed as ξ k k = 1/ η = U k 1/ η where η CN0 I M K +1. Thus h V w k can be rewritten as h V w k = h k V Ũ k Ũ V h = ξ ξ k = ξ k ξ = U k 1/ ηη / U. 41 Usin 41 we now take the conditional expectation of h V w k for iven k : { h } E V w k k { = E U k 1/ ηη / U } k = U k 1/ E { ηη } / k U = U k 1/ / U N = λ k i = k. 4 i=1 Lemma 3: [5] [Corollary ] Let M be any J J ermitian matrix and let be a iven inteer such that 1 J. Then we have and λ 1 + λ + +λ = λ J +1 + λ J + + +λ J = min U MU 43 U U=I max U MU 44 U U=I

11 10 IEEE TRANSACTIONS ON COMMUNICATIONS VOL. 63 NO. 6 JUNE 015 where λ k is the k-th smallest eienvalue of M and equality holds if the columns of the J matrix U are chosen to be the orthonormal eienvectors associated with the correspondin eienvalues of M. Lemma 4: For the matrix k defined in Lemma k is lower bounded as k V R V K 1λ 45 where V is the outer beamformer for roup satisfyin V V = I R is the channel covariance matrix of roup and λ is the larest eienvalue of R. Proof: Applyin the Rayleih quotient technique one can easily show that the maximum eienvalue of the matrix V R V is less than or equal to λ where λ is the larest eienvalue of R since V has orthonormal columns. That is λ max V R V max x V R V x x x = max x V R V x x V V x = max x R x x x = λ. Define λ 1 V R V and let 0 χ 1 χ χ M be the eienvalues of. Then the maximum eienvalue χ M of is less than or equal to one from the above. Note from 40 that λ 1 k = Ũ Ũ k where Ũ Ũ k = I. Applyin Lemma 3to λ 1 k with M = wehave 1 λ k a b N χ k 46 M χ k K 1χ M 47 M χ k K 1 48 = K 1 49 where a follows from the relationship N = M K + 1 and the fact that χ M is the maximum eienvalue of and b follows from the fact that χ M 1. Multiplyin both sides of 49 by λ we obtain the desired result. Proof of Theorem 1: Finally we prove Theorem 1 by usin the above lemmas. By Lemma and Lemma 4 the conditional expectation of h V w k for iven k is lowerbounded as { h } E V w k k = k 50 V R V K 1λ 51 where 50 is by Lemma and 51 is by Lemma 4. Note that the lower bound 51 is independent of k and is a constant. By takin expectation over k { on both sides of 51. The left-hand side LS becomes E h V w k } by the law of iterated expectation and the RS does not chane since the RS is a constant. ence we have { h } E V w k V R V K 1λ. 5 APPENDIX B PROOF OF TEOREM As in the proof of Lemma 1 in [17] we first define f n V V R 1 ρ n R V. Then we have f n V n 1 = 0 from 7. The step 30 of Alorithm 1 maximizes f n V over the set {V : V V = I} which includes V n 1 and V n is the maximizer of f n V. ence we have f n V n f n V n 1 = Based on the positive-definiteness of R f n V n = R1 ρ n R V n 0 can be rewritten as V n R1 V n ρ n+1 = ρ n. 54 R V n V n This concludes the proof. V n APPENDIX C PROOF OF TEOREM 3 By Lemma 3 the numerator and denominator of the obective function of TQP are lower-bouned and upper-bounded as M ψ M k V R 1V M M ψ k and ϕ M k V R V M ϕ k. 55 R 1 is ermitian and R is positive-definite. ence we have M ϕ M k > 0 and M ϕ k > 0. We consider three cases: i M ψ M k 0 and M ψ k 0 ii M ψ M k <0 and M ψ k 0 and iii M ψ M k < 0 and M ψ k < 0. By applyin 55 to each case we have the desired result. REFERENCES [1] T. L. Marzetta Noncooperative cellular wireless with unlimited numbers of base station antennas IEEE ans. Wireless Commun. vol. 9 no. 11 pp Nov [] F. Rusek et al. Scalin up MIMO: Opportunities and challenes with very lare arrays IEEE Sinal Process. Ma. vol. 30 no. 1 pp Jan [3] S. Noh M. D. Zoltowski Y. Sun and D. J. Love Pilot beam pattern desin for channel estimation in massive MIMO systems IEEE J. Sel. Topics Sinal Process. vol. 8 no. 5 pp Oct [4] J. Choi D. J. Love and P. Bidiare Downlink trainin techniques for FDD massive MIMO systems: Open-loop and closed-loop trainin with

12 KIM et al.: TWO-STAGE BEAMFORMER DESIGN FOR MASSIVE MIMO DOWNLINK 11 memory IEEE J. Sel. Topics Sinal Process. vol. 8 no. 5 pp Oct [5] J. So D. Kim Y. Lee and Y. Sun Pilot sinal desin for massive MIMO systems: A received sinal-to-noise-ratio-based approach IEEE Sinal Process. Lett. vol. no. 5 pp May 015. [6] S. Noh M. Zoltowski and D. Love ainin sequence desin for feedback assisted hybrid beamformin in massive MIMO systems IEEE ans. Commun. submitted for publication 015. [Online]. Available: [7] A. Adhikary J. Nam J. Ahn and G. Caire Joint spatial division and multiplexin The lare-scale array reime IEEE ans. Inf. Theory vol. 59 no. 10 pp Oct [8] J. Nam A. Adhikary J. Ahn and G. Caire Joint spatial division and multiplexin: Opportunistic beamformin user roupin and simplified downlink schedulin IEEE J. Sel. Topics Sinal Process. vol. 8 no. 5 pp Oct [9] G. Lee and Y. Sun A new approach to user schedulin in massive multiuser MIMO broadcast channels IEEE ans. Inf. Theory submitted for publication. [Online]. Available: [10] J. Chen and V. Lau Two-tier precodin for FDD multi-cell massive MIMO time-varyin interference networks IEEE J. Sel. Areas Commun. vol. 3 no. 6 pp Jun [11] A. Liu and V. Lau Phase only RF precodin for massive MIMO systems with limited RF chains IEEE ans. Sinal Process. vol. 6 no. 17 pp Sep [1] M. Sadek A. Tarihat and A.. Sayed A leakae-based precodin scheme for downlink multi-user MIMO channels IEEE ans. Wireless Commun. vol. 6 no. 5 pp May 007. [13] R. Zakhour and D. Gesbert Coordination on the MISO interference channel usin the virtual SINR framework in Proc. Int. ITG Workshop Smart Antennas Berlin Germany Feb. 009 pp [14] J. Park G. Lee Y. Sun and M. Yukawa Coordinated beamformin with relaxed zero forcin: The sequential orthoonal proection combinin method and rate control IEEE ans. Sinal Process. vol. 61 no. 1 pp Jun [15]. Shen K. Diepold and K. üper A eometric revisit to the trace quotient problem in Proc. Int. Symp. MTNS 010 pp [16] L. Zhan W. Yan and L. Liao A note on the trace quotient problem Optim. Lett. vol. 8 no. 5 pp Jun [17]. Wan S. Yan D. Xu X. Tan and T. uan ace ratio vs. ratio trace for dimensionality reduction in Proc. IEEE Int. Conf. CVPR007 pp [18] J.. Kotecha and A. M. Sayeed ansmit sinal desin for optimal estimation of correlated MIMO channels IEEE ans. Sinal Process. vol. 5 no. pp Feb [19] A. Abdi and M. Kaveh A space-time correlation model for multielement antenna systems in mobile fadin channels IEEE J. Sel. Areas Commun. vol. 0 no. 3 pp Apr. 00. [0] M. Zhan P. J. Smith and M. Shafi An extended one-rin MIMO channel model IEEE ans. Wireless Commun.vol.6no.8pp Au [1] G.. Golub and C.F. V. LoanMatrix Computations nd ed. Baltimore MD USA: The Johns opkins Univ. Press [] R.-C. Li Rayleih quotient based numerical methods for eienvalue problems The Fourth Gene Golub SIAM Summer School Fudan University Shanhai China Jul [3] L. Zhan L. Liao and M. N Fast alorithms for the eneralized Foley-Sammon discriminant analysis SIAM J. Matrix Anal. Appl. vol. 31 no. 4 pp [4] B. N. Parlett The Symmetric Eienvalue Problem. Philadelphia PA USA: SIAM [5] R. A. orn and C. R. Johnson Matrix Analysis. Cambride U.K.: Cambride Univ. Press [6] C. B. Peel B. M. ochwald and A. L. Swindlehurst A vectorperturbation technique for near-capacity multiantenna multiuser communication Part I: Channel inversion and reularization IEEE ans. Commun. vol. 53 no. 1 pp Jan [7] M. Joham K. Kusume M.. Gzara and W. Utschick ansmit Wiener filter for the downlink of TDD DS-CDMA systems in Proc. IEEE 7th Symp. Spread-Spectrum Technol. Appl. Praue Czech Republic Sep. 00 pp [8] P. Li D. Paul R. Narasimhan and J. Cioffi On the distribution of SINR for the MMSE MIMO receiver and performance analysis IEEE ans. Inf. Theory vol. 5 no. 1 pp Jan Donun Kim S 10 received the B.S. deree in electronics and communications enineerin from anyan University Seoul Korea in 010. e is currently workin toward the Ph.D. deree in the Department of Electrical Enineerin KAIST. is research interests include convex optimization matrix theory and sinal processin for next eneration wireless communications. Gilwon Lee S 10 received the B.S. and M.S. derees in electrical enineerin from KAIST Daeeon South Korea in 010 and 01 respectively. e is currently with the wireless information systems research roup KAIST workin towards the Ph.D. deree. is research interests are in the desin and analysis of lare-scale MIMO systems and sinal processin for next wireless communications. Younchul Sun S 9 M 93 SM 09 received the B.S. and M.S. derees from Seoul National University Seoul Korea in electronics enineerin in 1993 and 1995 respectively. After workin at LG Electronics Ltd. Seoul Korea from 1995 to 000 he oined the Ph.D. proram and received the Ph.D. deree in electrical and computer enineerin from Cornell University Ithaca NY USA in 005. From 005 to 007 he was a Senior Enineer in the Corporate R&D Center of Qualcomm Inc. San Dieo CA USA and participated in desin of WCDMA base station modem. Since 007 he has been on the faculty of the Department of Electrical Enineerin in KAIST Daeeon Korea. is research interests include sinal processin for communications statistical sinal processin asymptotic statistics and information eometry. Dr. Sun is currently a member of UNESCO/Netexplo University Advisory Board Sinal and Information Processin Theory and Methods SIPTM Technical Committee of Asia-Pacific Sinal and Information Processin Association APSIPA Vice-Chair of IEEE ComSoc Asia-Pacific Board ISC and was an Associate Editor of the IEEE SIGNAL PROCESSING LETTERS from 01 to 014.