Sparse Beamforming Design for Network MIMO System with Per-Base-Station Backhaul Constraints

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1 Sparse Beamforming Design for Networ MIMO System with Per-Base-Station Bachau Constraints Binbin Dai and Wei Yu Department of Eectrica and Computer Engineering University of Toronto, Toronto, Ontario M5S 3G4, Canada Emais: {bdai, Abstract This paper considers the joint beamforming and custering design probem in a downin networ mutipe-input mutipe-output (MIMO) setup, where the base-stations (BSs) are connected to a centra processor with rate-imited bachau ins. We formuate the probem as that of devising a sparse beamforming vector across the BSs for each user, where the nonzero beamforming entries correspond to that user s serving BSs. Differing from the previous wors, this paper expicity formuates the per-bs bachau constraints in the networ utiity maximization framewor. In contrast to the traditiona utiity maximization probem with transmit power constraint ony, the additiona bachau constraints resut in a discrete 0- norm formuation, which maes the probem more chaenging. Motivated by the compressive sensing iterature, we propose to iterativey approximate the per-bs bachau constraints using a reweighted 1-norm technique and reformuate the bachau constraints as weighted per-bs power constraints. This aows us to sove the weighted sum rate maximization probem through a generaized weighted minimum mean square error (WMMSE) approach. To reduce the computationa compexity of the proposed agorithm within each iteration, we propose two additiona techniques, iterative in remova and iterative user poo shrining, which dynamicay decrease the potentia BS custer size and user scheduing poo. Numerica resuts show that the proposed agorithm can significanty improve the system throughput as compared to the naive BS custering strategy based on the channe strength. I. INTRODUCTION Networ mutipe-input mutipe-output (MIMO) is a promising technique for improving the system performance of future wireess ceuar networs. In practice, networ MIMO system can be reaized by connecting the base-stations (BSs) to a centra processor (CP) with high-speed bachau ins. The idea impementation of networ MIMO system, where a the BSs fuy cooperate and each BS acquires the channe state information (CSI) of the entire networ, requires very high bachau capacities and is not practica. With finite-rate bachau ins, the CP can share each user s message with a imited set of BSs (caed BS custer in this paper), which then cooperativey serve the user through joint beamforming. To determine the optima BS custering and joint beamforming strategies for the downin networ MIMO system with imited bachau is not an easy tas. First, as each user wants to be served by the strongest BSs whie each BS has a imit on the number of users it can support, the optima BS custering design invoves a tradeoff between the user rates and the bachau capacities. The naive custer seection scheme purey based on the channe strength may ead to severey imbaanced traffic oad, especiay in heterogeneous networs where the macro-bss usuay have much higher transmit power than the pico-bss. Second, even with fixed BS custering, finding the optima joint beamforming is aso nontrivia since different custers for different users may overap. In this paper, we propose to jointy address the above two issues by devising a sparse beamforming strategy across the BSs for each user, where the nonzero beamforming entries correspond to that user s serving BSs. In contrast to [1] [4] which formuate the probem as radio resources minimization under fixed user rate constraints, this paper focuses on networ utiity maximization under given radio resources, namey transmit power and bachau capacities. We propose an efficient agorithm to seect the BS custer for each user dynamicay and to design the corresponding beamforming vector according to the given bachau imits. Utiity maximization for the networ MIMO system considered in existing iterature mosty taes the imited bachau capacities into account impicity by either fixing the BS custers [5] [7] or adding the bachau as a penaized term into the objective function [8] [10]. Specificay, [5] considers sum rate maximization under fixed and non-overapping custering scheme whie [6] and [7] maximize a more genera utiity function under potentiay overapping but predetermined BS custers. Dynamic custering design is considered in [8] by penaizing the objective function with an 2 -norm approximation of the custer size. Aternativey, [9] and [10] choose the bachau rate as the penaized term but sove the probem heuristicay. For fixed custering scheme assumed in [5] [7], the bachau consumption is ony nown afterwards by evauating the rates of user messages deivered in each bachau in. For dynamic custering designs considered in [8] [10], one has to optimay choose the price associated with each penaized term to ensure that the overa bachau stays within the budget, which is not easy. In contrast to a the above existing wors, the main contribution of this paper is that we expicity account for the per-bs bachau constraints in the overa networ utiity maximization framewor. Differing from the utiity maximization probems in conventiona wireess networs with ony power constraints, the additiona bachau constraints in networ MIMO system mae the probem more chaenging. The bachau consumption at a particuar BS is a function

2 Fig. 1: Networ MIMO with per-bs bachau constraints. of not ony the (continuous) user rates but aso the (discrete) number of associated users. Motivated by the compressive sensing iterature [11], this paper casts the BS custering probem as an 0 -norm optimization probem and proposes to iterativey approximate the mixed continuous and discrete per-bs bachau rate through an 1 -norm reweighting technique, which essentiay reformuates the per-bs bachau constraints into weighted per- BS power constraints. This nove approximation aows us to appy a generaized weighted minimum mean square error (WMMSE) approach [7] to sove the utiity maximization probem. The resuting agorithm efficienty determines the optima BS custering and beamforming strategies jointy for the networ MIMO system with imited bachau. In order to further improve the efficiency of the proposed agorithm, we propose two additiona techniques, iterative in remova and iterative user poo shrining, which iterativey excude those BSs with negigibe transmit power out of the candidate custer set for each user, and excude those users with negigibe rates out of the user scheduing poo, respectivey. Simuation resuts show that the proposed agorithm can fuy utiize the imited bachau resource and significanty improve the system throughput as compared to the conventiona user-centric custering scheme based on channe strength. II. SYSTEM MODEL AND PROBLEM FORMULATION Consider a downin ceuar networ with L BSs and K users, where each BS has M transmit antennas whie each user has N receive antennas. Each BS is connected to a CP via a bachau in with capacity imit C, =1, 2,,L,as depicted in Fig. 1. With inear transmit beamforming scheme, the received signa at user, denoted as y C N 1, = 1, 2,,K, can be modeed as y = H w s + j H w j s j + n, (1) where H C N Mt and w C Mt 1 =[w 1, w2,, wl ] denote the CSI matrix and beamforming vector respectivey from a the M t = LM transmit antennas 1 to user. Suppose BS is not part of user s serving custer, then the corresponding beamforming entries w CM 1 are set to 0. In this paper, we consider the case where each user has ony a singe data stream for simpicity and assume that user s message s C is independent and identicay distributed according to CN(0, 1). Here, n C N 1 is the received noise at user and modeed as n CN(0,σ 2 I). We assume that the CP has access to a the users data and has goba CSI for designing the optima sparse beamforming vector w. Once w is determined, the CP transmits user s message, aong with the beamforming coefficients, to those BSs corresponding to the nonzero entries in w through the bachau ins. In this paper, we assume that the channes are sow varying and ony consider the bachau consumption due to the user data sharing and ignore the bachau required for sharing CSI and deivering beamforming coefficients. Under this assumption, the per-bs bachau constraint can be cast in the foowing weighted 0 -norm format: w 2 R C, (2) where R is the achievabe rate for user defined as R = (3) og 1+w H H H 1 H w j wj H H H + σ 2 I H w j Intuitivey, the bachau consumption at the th BS is the accumuated data rates of the users served by BS. Here, w 2 characterizes whether or not BS serves user : { w 2 0, if w = 2 2 =0. (4) 1, otherwise This paper considers networ utiity maximization as the objective. Among the famiy of utiity functions, weighted sum rate (WSR) has been widey appied to networ contro and optimization probems. In this paper, we aso adopt the WSR utiity but point out that the proposed scheme can be readiy extend to any utiity function that hods an equivaence reationship with the WMMSE minimization probem (see [12] for a sufficient condition on the utiity functions hoding such an equivaence). With per-bs power and per-bs bachau constraints, the WSR maximization probem can be formuated as: maximize {w } α R w 2 2 P, w 2 R C, (5a) (5b) (5c) 1 To simpify the notations, we assume that a the L BSs can potentiay serve each schedued user. In practice, ony the strongest few BSs around each user need to be considered as the candidate serving BSs.

3 where α denotes the priority weight associated with user, P and C represent the transmit power budget and bachau capacity imit for BS, respectivey. III. PROPOSED ALGORITHM Conventiona WSR maximization probem is a we-nown nonconvex probem, for which finding the goba optimaity is aready quite chaenging even without the additiona bachau constraint. This paper therefore focuses on soving for the oca optimum soution of the probem (5) ony. Our main contribution is a new way of deaing with the discrete 0 -norm constraint (5c). In compressive sensing iterature, nonconvex 0 -norm objective is often approximated by the convex reweighted 1 -norm [11]. In this paper, we extend this idea to the 0 -norm in the constraint and approximate (5c) as βr w 2 2 C (6) where β is a constant weight associated with BS and user and is updated iterativey according to β 1 = w,, (7) τ with some sma constant reguarization factor τ > 0 and w 2 2 from the previous iteration. Even with the above approximation, the optimization probem (5) with the bachau constraint (5c) repaced by (6) is sti difficut to dea with due to the fact that the rate R in the constraint is unnown. To address this difficuty, we propose to sove the probem (5) iterativey with fixed rate ˆR in (6) and update ˆR by the achievabe rate R from the previous iteration. Under fixed β and ˆR, probem (5) now reduces to maximize {w } α R w 2 2 P, β ˆR w 2 2 C, (8a) (8b) (8c) where the approximated bachau constraint (8c) can be interpreted as a weighted per-bs power constraint bearing a resembance to the traditiona per-bs power constraint (8b). Athough the approximated probem (8) is sti nonconvex, we can reformuate it as an equivaent WMMSE minimization probem in order to reach a oca optimum soution. The equivaence between WSR maximization and WMMSE minimization is first estabished in [13] for MIMO broadcast channe and ater generaized to MIMO interfering channe in [12] and MIMO interfering channe with partia cooperation in [7]. It is not difficut to see that the generaized WMMSE equivaence estabished in [12] aso extends to the probem (8) with the newy introduced weighted per-bs power constraint (8c). We expicity state the equivaence as foows: Proposition 3.1 ( [12]): The WSR maximization probem (8) has the same optima soution with the foowing WMMSE minimization probem: minimize {ρ,u,w } α (ρ e og ρ ) (9) w 2 2 P, β ˆR w 2 2 C, where ρ denotes the MSE weight for user and e is the corresponding MSE defined as e = E [ u H y s 2 ] 2 (10) = u H H w j wj H H H + σ 2 I u 2Re{u H H w } +1 j under receiver u C N 1. The advantage of soving the WSR maximization probem (8) through its equivaent WMMSE minimization probem (9) is that (9) is convex with respect to each of the individua optimization variabes. This crucia observation aows the probem (9) to be soved efficienty through the boc coordinate descent method by iterating between ρ, u and w : The optima MSE weight ρ under fixed w and u is ρ = e 1,. The optima receiver u under fixed w and ρ is the MMSE receiver: u = 1 H w j wj H H H + σ 2 I H w,. (11) j The optimization probem to find the optima transmit beamformer w under fixed u and ρ is a quadraticay constrained quadratic programming (QCQP) probem, which can be soved using standard convex optimization sovers such as CVX [14]. A straightforward but computationay intensive way of appying the above WMMSE agorithm to sove the origina probem (5) woud invove two oops: an inner oop to sove the approximated WSR maximization probem (8) with fixed weight β and rate ˆR, and an outer oop to update β ˆR and. Instead, we propose to combine these two oops into a singe oop and update the weight β and rate ˆR inside the WMMSE agorithm, as summarized in Agorithm 1. Agorithm 1 has the same compexity order as the conventiona WMMSE agorithm since it ony introduces two additiona steps 4 and 5 in each iteration in updating β and ˆR, which are both cosed-form functions of the transmit beamformers. The main computationa compexity of Agorithm 1 comes from the optima transmit beamformer design in Step 3, which is a QCQP probem as mentioned before, but can aso be equivaenty reformuated as a second order cone programming (SOCP) probem as we do in the simuation part

4 Agorithm 1 Sparse Beamforming Design with Expicit Per- BS Bachau Constraints Initiaization: β, ˆR, w,, ; Repeat: 1) Fix w,, compute the MMSE receiver u and the corresponding MSE e according to (11) and (10); 2) Update the MSE weight ρ = e 1,. 3) Find the optima transmit beamformer w under fixed u and ρ,. 4) Compute the achievabe rate R according to (3), ; 5) Update ˆR = R and β according to (7),,. Unti convergence of this paper. The compexity of soving SOCP using interiorpoint method is approximatey O((KLM) 3 ) [15]. To improve the efficiency of Agorithm 1 in each iteration, in what foows, we further propose two techniques, iterative in remova and iterative user poo shrining. The former aims at reducing the number of potentia transmit antennas LM serving each user whie the atter is intended to decrease the tota number of users K to be considered in each iteration. A. Iterative Lin Remova Simiar to what we observed in [2], the transmit power from some of the candidate serving BSs woud drop down rapidy cose to zero as the iterations go on. By taing advantage of this, we propose to iterativey remove the th BS from the th user s candidate custer once the transmit power from BS to user, i.e. w 2 2, is beow a certain threshod, say 100dBm/Hz. This reduces the dimension of the potentia transmit beamformer for each user and reduces the compexity of soving SOCP in Step 3 of Agorithm 1. B. Iterative User Poo Shrining The WMMSE agorithm does user scheduing impicity. We observe from simuations that, it is beneficia for Agorithm 1 to consider a arge poo of users in the iterative process. However, to consider a the users in the entire networ a the time woud incur significant computationa burden. Instead, we propose to chec the achievabe user rate R in Step 4 iterativey and ignore those users with negigibe rates (beow some threshod, say 0.01 bps/hz) during the next iteration. Our simuations show that, after around 10 iterations, more than haf of the tota users can be taen out of the consideration with negigibe performance oss to the overa agorithm. This significanty reduces the tota number of variabes to be optimized during the foowing iterations. IV. SIMULATION RESULTS In this section, numerica simuations are conducted to show the effectiveness of the proposed agorithm. We consider a 7-ce wrapped around two-tier heterogeneous networ with the simuation parameters isted in Tabe I. Each ce forms a reguar hexagon with a singe macro-bs ocated in the center and 3 pico-bss equay separated within the ce as TABLE I: Simuation Parameters. Ceuar Hexagona Layout 7-ce wrapped-around Channe bandwidth 10 MHz Distance between ces 0.8 m Num. of (macro-bss, pico-bss, users)/ce (1, 3, 30) Num. of antennas/(macro-bs, pico-bs, user) (4, 2, 2) Max. Tx power for (macro-bs, pico-bs) (43, 30) dbm Antenna gain 15 dbi Bacground noise 169 dbm/hz Path oss from macro-bs to user og 10 (d) Path oss from pico-bs to user og 10 (d) Log-norma shadowing 8 db Rayeigh sma scae fading 0 db Reweighting function parameter τ = Macro BS Pico BS Mobie User User 3 in Ce Fig. 2: 7-ce wrapped around two-tier heterogeneous networ. iustrated in Fig. 2. To simpify the discussion, we set a the macro-bss to have equa bachau constraints and iewise for the pico-bss. The bachau constraints are denoted as (C macro,c pico ) respectivey. The proposed agorithm is simuated under the power constraints isted in Tabe I with various sets of (C macro,c pico ) bachau constraints. As indicated previousy, instead of considering a the L BSs in the entire networ as the candidates serving each user, in simuations we ony consider the strongest L c (L c L) BSs 2 around each user as its candidate custer. To iustrate how the sparse beamforming vector is formed in the proposed agorithm, we pot in Fig. 3 the power evoutions of the strongest 8 BSs for the third user in the second ce as an exampe, without removing any BS out of the candidate custer using the proposed iterative in remova technique. As we can see, after around 20 iterations ony the first and third strongest BSs maintain at a reasonabe transmit power eve. They eventuay form the custer for serving user 3 in ce 2. With the proposed iterative in remova technique and by setting the threshod to be 100dBm/Hz, the improved version of Agorithm 1 can narrow down the candidate BSs to ony the strongest 4 BSs after the 5th iteration, and the (1st, 3rd, 4th) strongest BSs after the 8th iteration, and finay the (1st, 3rd) strongest BSs after the 17th iteration. 2 The strength of a BS seen by a user is evauated by the maximum transmit power from that BS compensated with the path-oss to the user.

5 Power (dbm/hz) st Strongest BS 160 2nd Strongest BS 3rd Strongest BS 180 4th Strongest BS 5th Strongest BS 6th Strongest BS 200 7th Strongest BS 8th Strongest BS Iteration Fig. 3: Power evoutions of the strongest 8 BSs for user 3 in ce 2, (C macro,c pico ) = (245, 70)Mbps, α =1,, L c =8. Cumuative Distribution Proposed Agorithm (268, 73)Mbps Strongest 1 BS (268, 73)Mbps Proposed Agorithm (690, 107)Mbps Strongest 2 BSs (690, 107)Mbps Proposed Agorithm (1195, 172)Mbps Strongest 3 BSs (1197, 173)Mbps Proposed Agorithm (1828, 295)Mbps Strongest 4 BSs (1832, 297)Mbps User Rate (Mbps) Fig. 4: User rates comparison with L c =8and α updated according to proportiona fairness criterion. In Fig. 4, we compare the cumuative distributions of the ong-term average user rates between the proposed agorithm and the user-centric custering scheme where each user is served by the strongest S BSs (S =1, 2, 3, 4). We first run the baseine Strongest S BSs schemes then evauate the corresponding bachau requirement for each BS afterwards. For the proposed agorithm, the expicit bachau constraints (C macro,c pico ) are set to be the average bachau over the macro-bss and the pico-bss respectivey from the baseine. Each curve in Fig. 4 is obtained by iterativey simuating the corresponding scheme with fixed user priority weights α s and updating the weights according to the proportiona fairness criterion. As we can see, by optimizing with expicit bachau constraints, the proposed agorithm achieves significant performance gain. For instance, around 35% improvement is obtained for the 50th-percentie user as compared with the baseine where each user is served by the strongest 3 BSs. Note that since the baseine agorithm connects each user to an equa number of neighboring BSs, it inevitaby favors high-rate users. In contrast, the proposed agorithm can seect custers for each user adaptivey, and in particuar can choose a arger custer for the ow-rate users, thus achieving a significant overa gain from a networ utiity perspective. V. CONCLUSION This paper proposes a nove 1 -norm reweighting technique for soving the utiity maximization probem for a networ MIMO system with imited per-bs bachau capacities. The proposed agorithm dynamicay seects the BS custer and designs the corresponding transmit beamformer for each user in each time sot according to the given bachau constraints. To further reduce the computationa compexity of the proposed agorithm, we propose to iterativey remove those BSs with negigibe transmit power out of the candidate custer for each user, and remove those users with negigibe rates out of the user scheduing poo. Simuation resuts show that by expicity accounting for the bachau constraints into the utiity maximization framewor, the proposed agorithm can fuy utiize the bachau resource and significanty improve the system throughput as compared to the conventiona usercentric custer scheme purey based on the channe strength. REFERENCES [1] J. Zhao, T. Que, and Z. Lei, Coordinated mutipoint transmission with imited bachau data transfer, IEEE Trans. Wireess Commun., vo. 12, no. 6, pp , [2] B. Dai and W. Yu, Sparse beamforming for imited-bachau networ MIMO system via reweighted power minimization, in Proc. IEEE Goba Teecommunications Conf. (Gobecom), [3] F. Zhuang and V. Lau, Bachau imited asymmetric cooperation for MIMO ceuar networs via semidefinite reaxation, IEEE Trans. Signa Process., vo. 62, no. 3, pp , Feb [4] S. Luo, R. Zhang, and T. J. Lim, Downin and upin energy minimization through user association and beamforming in coud RAN, [Onine]. Avaiabe: [5] J. Zhang, R. Chen, J. Andrews, A. 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