728

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 2, FEBRUARY 2011

Distributed Multicell Beamforming With Limited Intercell Coordination Yongming Huang, Member, IEEE, Gan Zheng, Member, IEEE, Mats Bengtsson, Senior Member, IEEE, Kai-Kit Wong, Senior Member, IEEE, Luxi Yang, Member, IEEE, and Björn Ottersten, Fellow, IEEE

Abstract—This paper studies distributed optimization schemes for multicell joint beamforming and power allocation in time-division-duplex (TDD) multicell downlink systems where only limited-capacity intercell information exchange is permitted. With an aim to maximize the worst-user signal-to-interference-and-noise ratio (SINR), we devise a hierarchical iterative algorithm to optimize downlink beamforming and intercell power allocation jointly in a distributed manner. The proposed scheme is proved to converge to the global optimum. For fast convergence and to reduce the burden of intercell parameter exchange, we further propose to exploit previous iterations adaptively. Results illustrate that the proposed scheme can achieve near-optimal performance even with a few iterations, hence providing a good tradeoff between performance and backhaul consumption. The performance under quantized parameter exchange is also examined. Index Terms—Limited capacity backhauling, multicell beamforming, uplink–downlink duality.

Manuscript received January 07, 2010; revised May 09, 2010, August 16, 2010; accepted October 03, 2010. Date of publication October 25, 2010; date of current version January 12, 2011. This work was supported in part by the NBRPC/973 under Grant 2007CB310603, the National Science and Technology Major Project of China under Grants 2009ZX03003-004, 2011ZX03003-001 and 2011ZX03003-003, the EPSRC under Grants EP/D058716/1 and EP/E022308/1, the British Council Research Exchange Programme, the NSFC under Grants 60902012 and 61071113, the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) and ERC grant agreement 228044, and the Ph.D. Programs Foundation of the Ministry of Education of China under Grants 20090092120013 and 20100092110010. Part of this work was conducted when Y. Huang was visiting ACCESS Linnaeus Center, KTH Signal Processing Lab, Royal Institute of Technology, Stockholm, Sweden. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Amir Leshem. Y. Huang and L. Yang are with the School of Information Science and Engineering, Southeast University, Nanjing 210096, China and also with the Key Laboratory of Underwater Acoustic Signal Processing of Ministry of Education, Southeast University, Nanjing 210096, China (e-mail: huangym, lxyang@seu. edu.cn). G. Zheng is with the Interdisciplinary Centre for Security, Reliability and Trust (SnT), University of Luxembourg, Coudenhove-Kalergi, Luxembourg (e-mail: [email protected]). M. Bengtsson is with the ACCESS Linnaeus Center, KTH Signal Processing Laboratory, Royal Institute of Technology, SE-100 44 Stockholm, Sweden (e-mail: [email protected]). K.-K Wong is with the Department of Electrical and Electronic Engineering, University College London, London, WC1E 7JE, U.K. (e-mail: k.wong@ee. ucl.ac.uk). B. Ottersten is with the ACCESS Linnaeus Center, KTH Signal Processing Laboratory, Royal Institute of Technology, SE-100 44 Stockholm, Sweden, and also with securityandtrust.lu, University of Luxembourg, Coudenhove-Kalergi, Luxembourg (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2010.2089621

I. INTRODUCTION ROVIDING coverage and supporting a large number of users with limited spectrum available is a challenge. In cellular systems, this is achieved by frequency reuse with a single-cell processing policy, meaning that users in a cell are served by the base station (BS) located at the center of the cell only and a BS is not supposed to enhance the channel links not belonging to its cell. Intercell interference (ICI) is handled by careful frequency planning and usually treated as noise. Nonetheless, it has become apparent that this single-cell approach has a major problem at the cell edges where users not only suffer from high signal attenuation, but also severe ICI and exhibit much poorer performance than the interior users. To tackle this, multicell processing based on cooperation between BSs has emerged as a promising solution in recent years. The idea is that neighboring BSs can form a cooperating cluster by exchanging some information between each other, to improve the system performance. Early works in this area largely focused on the uplink [1], [2], where several BSs jointly decode the signals from the users served by these cells. The problem of downlink multicell processing, however, is much more challenging, especially with multiple antennas at each BS. To fully exploit the potential of multicell processing, the cooperating BSs should perform joint or coherent transmission to multiple users, where each user is simultaneously served by a cluster of BSs and therefore both data and channel state information (CSI) are required to be shared between the BSs. In this case the BS cluster essentially becomes a single-cell multiuser multi-antenna system, and existing downlink beamforming techniques, e.g., [3]–[11], can be directly applied to obtain a significant capacity gain, e.g., [12]–[14]. Nevertheless, the optimal joint transmission requires full phase coherence between the signals from different BSs, which is hard to realize due to the differences in propagation delay. Moreover, the transmission needs to be implemented in a centralized manner which will cause a huge backhauling burden due to data sharing between the BSs. A distributed solution with limited intercell coordination, if possible, is therefore much more desirable. More recently, the achievable rates of downlink multicell processing using distributed schemes [15], [16] and with limited backhauling capacity [17] were studied. Several distributed multicell processing methods were also devised in [18]–[23]. In particular, given that data sharing is permitted, [19] addressed the design of distributed transmit beamforming by recasting the downlink beamforming problem into a linear minimum mean-square-error (LMMSE) estimation problem. However, the required message passing between the BSs is too high and

P

1053-587X/$26.00 © 2010 IEEE

HUANG et al.: DISTRIBUTED MULTICELL BEAMFORMING WITH LIMITED INTERCELL COORDINATION

global convergence is not guaranteed. Later, [20] proposed a distributed design in time-division-duplex (TDD) systems using only local CSI (i.e., the channel from the BS to its users plus the crosstalk to other users) and demonstrated that performance close to the Pareto bound can be obtained. However, the main issue with [19], [20] is that both require data sharing between the BSs, which prohibits their use when the backhaul capacity is limited. Motivated by this, [21]–[24] looked into coordinated beamforming schemes without data sharing, where BSs do not cooperate at the signal level, but design their transmit precoders (or beamformers) and power levels jointly. In [21], Lagrangian duality theory was adopted to coordinate the BS beamforming with limited intercell information exchange but the solution was only applicable to the sum-BS power minimization problem with a signal-to-interference and noise ratio (SINR) constraint. Similar ideas will be used in this paper to design a distributed solution for the rate-fair optimization problem under a sum-BS power constraint and per-BS power constraints. Alternatively, it is also possible to devise distributed algorithms for coordinated precoding based on the signal-to-generated-interference plus noise ratio (SGINR) [22] and the signal-to-caused-interference ratio (SCIR) [23], where each BS only exploits its local CSI. However, this strategy is only asymptotically optimal in terms of the sum-rate for the scenarios with two BSs, when the signal-to-noise ratio (SNR) tends to infinity. In this paper, our aim is to find distributed algorithms for coordinated BS beamforming and power allocation in multicell downlink systems. In particular, we study the rate-fair optimization problem to maximize the minimum cell rate with a sum-BS power constraint and limited intercell information exchange. By exploiting the downlink-uplink SINR duality, we recast the problem into a more desirable equivalent form and propose to combine a bisection search with a distributed power allocation in order to provide an efficient distributed solution. We also extend the proposed solution to address the optimization problem subject to per-BS power constraints. Our main contributions are as follows. 1) We propose an iterative algorithm that provides a distributed solution for the rate-fair optimization, where only a few positive scalars are shared between cells. We prove that the proposed scheme always converges to the global optimum and results show that near-optimal performance can be achieved with only a few iterations. 2) To expedite the convergence and reduce the intercell parameter exchange, we propose to adaptively choose the initial point in each inner iteration using the previous iteration results. The effectiveness of this method is then analyzed and the impact on the amount of intercell parameter exchange is also investigated by computer simulations. 3) Finally, we examine the use of quantized parameter exchange between BSs and in particular, we show by simulations that the proposed approach can be efficiently implemented using a simple uniform linear quantization method. Moreover, we prove that, with an appropriate choice of initial conditions, the successive power parameters are monotonically increasing, implying that a more efficient differ-

729

ential quantization can be used to further reduce the burden of parameter exchange between BSs. The rest of this paper is organized as follows. In Section II, we describe the system model for coordinated BS beamforming and power allocation in the downlink. Section III proposes a hierarchical iterative scheme to realize distributed implementation of joint multicell beamforming1 and power allocation. In Section IV, we analyze the proposed algorithm regarding the intercell information exchange, global convergence and convergence acceleration. Simulation results are given in Section V, and we conclude the paper in Section VI. II. SYSTEM MODEL Consider a multicell downlink system composed of a cluster BSs, each with transmit antennas. The cluster perof forms coordinated beamforming and power control for serving the users in the cells covered by these BSs in the same frequency band in flat fading channels. The served user in each cell of the cluster will experience significant interference from all the other cells.2 As a consequence, a coordinated transmission design is needed. For convenience, it is first assumed that each BS is serving only one single-antenna user terminal. A generalization to cope with multiple users in each BS is addressed in Section III-C. Let denote the information symbol from BS to its served user , where denotes the (also indexed by ), with and denote, expectation operator. Also, we let respectively, the unit-norm beam vector (so that ) and the transmit power at BS . The received signal at mobile user can be expressed as [22], [23] (1)

denotes Hermitian transpose, where the superscript denotes the channel vector from BS to user , and denotes complex additive white Gaussian noise (AWGN) with . zero mean and variance of , or simply Note that in the above model, the assumption of perfect symbol-to-symbol synchronization is not necessary if SINR is used as the performance metric for the system design, due to the fact that the SINR criterion only involves the power of the interference. The above system model implies that each BS independently transmits the signal to its own user, instead of a joint transmission. Thus, data (or ) exchange between the BSs in the cluster and the beamis avoided. However, the power allocation 1In this paper, we use the terms “coordinated beamforming” and “multicell beamforming” interchangeably and both refer to the techniques where BSs in different cells cooperate to perform beamforming. 2For example, the cell cluster can be dynamically established according to the feedback from users. Specifically, each user terminal first determines how many dominant interfering cells it is experiencing by measuring the channel strength from the neighboring cells, and reports the information to its associated BS. Then, by a simple exchange between the cells, a cell cluster in which each user terminal sees significant interference from the member cells of the cluster can be formed to perform coordinated transmission. It is understandable that the users in the cell cluster are usually located at the cell edges, and coordination between the BSs in such a cell cluster has a great potential for improving the overall performance of the cluster.

730

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 2, FEBRUARY 2011

forming vectors are to be jointly optimized such that ICI is suppressed and the overall system performance can be improved. This joint optimization may suggest a centralized approach, which requires exchange of CSI between the BSs. Alternatively, our focus here is on distributed implementation of joint coordinated beamforming and power allocation, where exchange of only a few parameters is required. The availability of local CSI at the BS is crucial for the design and can be well justified for a TDD system. In TDD, the down) can be directly estilink CSI corresponding to BS (i.e., mated at the BS by exploiting the downlink–uplink reciprocity. On the other hand, it is reasonable to assume that each BS can estimate the crosstalk channels to other users in the same cluster. This can be explained by recognizing that usually a cell cluster is dynamically set up only if a user terminal inside the cluster sees significant interference from other cells in the same cluster. In other words, the channel between the BS and the users in other cells and that between the BS and its desired user in the same cluster are of similar strengths.3 For these reasons, we assume , which will be that each BS knows the CSI referred to as the local CSI and exploited in the subsequent distributed multicell system design.

A. Duality-Enabled Problem Formulation By viewing the cluster of BSs as a virtual BS (VBS), our system model in Section II can be seen as a multiuser system user terminals, a single VBS with a precoder consisting of , where the imposed block-diagonal structure of is used to avoid the data sharing between the actual BSs. This permits us to turn the downlink optimization problem into a virtual uplink problem using uplink-downlink duality [6], [24]. Let denote the virtual uplink transmit power as the virtual receive beamforming vector, vector. Viewing the virtual uplink SINR of user can be written as (4) It is known from [6] and [7] that (2) and (4) have the same achievable SINR region for the same given beam set and sumpower constraint. This means that despite the additionally imposed diagonal structure of , the uplink–downlink SINR duality can still be used and the downlink problem can be equivalently addressed by solving the virtual uplink problem given by

III. COORDINATED BEAMFORMING AND POWER ALLOCATION Several different design criteria, such as sum-rate, are often used to design the beamformers. Here, however, we study the maximization of the minimum of the user’s SINRs in the cluster. This provides user fairness in the optimization and is particularly important for cell-edge users. Let and denote respectively, the downlink transmit power vector and the multicell precoding matrix, where the superscript denotes transposition. Note that the block-diis due to the exclusion of data sharing agonal structure of between BSs. The downlink SINR for each user is calculated as

(5) has been obtained, the optimal Once the solution is easily calculated based on the downlink power vector uplink–downlink duality theory. In addition, it is important to note that the achievable max-min SINR, denoted by (6) is upper bounded by (7)

(2)

where problem with respect to

and

. Then, the optimization can be written as (3)

denotes the 1-norm of the input vector, and where denotes the sum-power limit of the BS cluster. Here, we shall mainly consider a sum-power constraint, and leave the discussion regarding the more challenging problem with per-BS power constraints in the Appendix. The direct optimization of the above problem is difficult especially with the extra constraints of distributed implementation and limited intercell coordination. To proceed, we find it useful to reformulate this problem into an equivalent form by using the uplink-downlink duality. 3To do so, a coordinated training phase for the cell cluster is required such that each BS can estimate the crosstalk channels together with the channel for it own user in predefined time slots, in which the training signals from different users are designed to be orthogonal in order to avoid mutual interference.

denotes the maximum eigenvalue of . where in the deThis result can be easily seen from (4) by setting nominator to zeros and will become useful for the development of a distributed multicell processing solution in Section III-B. For the virtual uplink problem (5), it appears that the joint can now be achieved in a distributed optimization of fashion at each individual BS. Two issues however remain. One issue is that the optimization of still couples the BSs through their power levels, making a distributed optimization difficult to achieve. Another issue is that the distributed implementation is also subject to constraints on the intercell information exchange, i.e., a given upper bound on the parameter exchange between the BSs. To address the first issue, we shall propose an iterative algorithm that is able to converge to the optimum as the BSs continue to exchange power allocation parameters. As a practical solution to the second issue, in the proposed algorithm, the signaling overhead for exchanging parameters between the BSs can be flexibly reduced to a desired level by fixing the number of iterations. To proceed, we rewrite our problem into a form related to the problem of sum-power minimization, so that the distributed

HUANG et al.: DISTRIBUTED MULTICELL BEAMFORMING WITH LIMITED INTERCELL COORDINATION

fixed-point iteration algorithm in [8]–[10] can be applied. More specifically, we first define a virtual SINR target for each user. To fulfill this virtual SINR constraint with minimum sumpower, the optimization problem can be expressed as

731

Otherwise, the result of the previous inner iteration loop is “infeasible” and

(11) (8) If the target, , is feasible, it is well known that the solution will always lead to an SINR balancing strategy. That is, using the solution, all the users will obtain an SINR of . Based on this observation, it is easy to understand that the optimization of the problem (5) is equivalent to finding a maximum virtual SINR target at which the minimum required sum-power denote the minimum sum-power is exactly equal to . Let required to satisfy a target , i.e., . Problem (5) can then be reexpressed as (9)

If , then the outer iteration is said to have is a carefully chosen converged and is stopped, where threshold (typically with value very close to zero). , from the • Inner iteration—For a given target SINR, outer iteration loop, each BS carries out the following iterations with respect to the virtual uplink, in which the power vector and the beamforming vector at the th inner iterate are, respectively, denoted by and , where the index for the outer iteration is omitted denotes the th inner iteration. and the superscript The initialization for each inner iteration will be discussed , at the end of this section. For a given power vector each BS first computes

B. The Proposed Two-Layer Iteration Based on the formulation (9), we propose a hierarchial iteraand , which is comtive method to distributively optimize posed of an outer iteration and an inner iteration, where only a limited number of parameter exchanges between the BSs is required. In the outer iteration, a bisection search is used to find the maximum achievable SINR target. For each particular virtual target, the optimal power allocation and beamforming, i.e., the solution of (8), will be obtained in the inner iteration using a generalized fixed-point method, to determine whether the target is feasible or not. We shall show in Section IV that this hierarchical iterative scheme can achieve the solution of (9) or (5) with rapid convergence and hence the intercell coordination is mild. The original problem (3) can then be solved by calculating the downlink solution from the virtual uplink solution.4 The proposed hierarchical iterative algorithm for solving (9) includes three main parts: outer iteration, inner iteration and the conversion between uplink and downlink, as follows. to denote the • Outer iteration—Using the superscript denote the teniteration number in the outer loop, let tative SINR target for a given sum-BS power. From (7), . This SINR target will be used as a constraint in the inner iteration to be described later. In the outer iteration, a bisection search is performed to update the lower and upper bounds, denoted, respectively, by and . For each iteration, if the result of the previous inner iteration loop is “feasible,” then update

(12)

where denotes the normalization factor. Then, each BS updates the power value based on and by

(13) If

or , where is defined similarly as , then the inner iteration is , return the result stopped. In particular, if , as “infeasible”; otherwise, if return the result as “feasible.” • Conversion from the virtual uplink to the downlink—After we solve the problem (5), we have and the maximum achievable target . Then, the optimal downlink , can be easily determined power allocation vector, using the uplink-downlink SINR duality. In particular, to do so, we first define

(10) (14) 4The problem subject to per-BS power constraints can be solved in a similar way. However, our analysis shows that the solution to this problem requires more intercell backhaul communication and therefore it is more difficult to implement in practice. For details, please refer to the Appendix.

and the matrix

with its

th element given by for for

(15)

732

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 2, FEBRUARY 2011

Then, the solution to the downlink problem can be obtained by solving the following linear system of equations [24, (19)–(21)], given by (16) where denotes the all one vector. Though it will be shown in Section IV that an arbitrary choice satisfying the sum-power constraint already guarantees of convergence, the intermediate result from the previous outer iteration can be used to provide a more suitable initial power vector, so that the convergence rate of the overall iteration can be improved. The idea is to use the power vector of the most recent feasible SINR target as the initial power vector for the next SINR target search. Specifically, we denote the final power vectors correSINR target searches as sponding to the first . As the inner iteration for each SINR target search may return the result of “feasible” or “infeasible,” to distinguish between them we use the set to denote the collection of the successful “feasible” to searches, and use the set denote the corresponding power vectors. Then, for the next th SINR target search with , we start its inner as the iteration using the most recent power vector in is empty, then the all zero vector initial point. If the set will be used as the initial power vector. As will be shown in Section IV-C, this strategy can speed up the convergence of the iterative algorithm, and therefore reduce the parameter exchange burden required between the BSs. C. Generalizations Here, we discuss some possible generalizations of using the proposed algorithm in the multicell environments. So far, our discussion assumed that only one user is served for each cell. All the results are extended immediately to scenarios with multiple user per cell, if each such cell, notation-wise, is duplicated into multiple cells, each serving one user and viewed as a virtual micro-cell (VmC). Obviously, there are no communication constraints between these notational copies of the same cell, thus centralized design for these VmCs of the same cell might be used instead of distributed design. However, it does not help to achieve performance improvement over our generalized distributed solution, as our scheme already achieves the performance of the optimal centralized linear beamforming design by recasting the max-min rate optimization problem into a virtual uplink problem. The SINR balancing approach described above will result in equal SINR for all users. Another generalization is to handle different SINR requirements among the users, in order to accommodate different QoS classes or to increase the system throughput at the expense of reduced fairness between users. This can be accomplished by extending our proposed scheme with a weighted fairness metric. Formally, that is, (17)

denotes the weighting factor for cell . The weights have the effects to steer the optimization in such a way that a larger weighting factor for a user will give rise to a higher rate for that user. Problem (17) above can readily be solved by the proposed iterative algorithm by replacing the SINR of each user with a weighted SINR.

where

IV. ANALYSIS A. Distributed Implementation and Parameter Exchange In (12), the beamforming optimization at each iteration can be separately implemented at each BS, say , with local CSI . Also, from (13), the power allocation in the virtual uplink for user can be iteratively updated in a distributed th inner iteration fashion. The only issue is that in the of the th outer iteration, the power vector needs to be known at the BSs, which will require parameter exchange between the BSs. That is to say, BS needs to broadcast the value to all other BSs in the cluster. In practice, the exchanged parameters will be quantized before transmission and a differential quantization of the power parameters is possible to reduce the required number of bits for a given accuracy, by recognizing that the power update in the proposed algorithm is element-wise (see Propomonotonically increasing, i.e., sition 4 for details). For the computation of the optimal downlink power vector, , it is seen from (16) that each BS can compute its optimal power if both and are available. This can be achieved by exreal-valued nonzero elements in and bechanging the tween the BSs. This is arguably practically more viable than requiring to exchange full CSI between the BSs, especially when the number of cooperating BSs and/or the number of antennas is large. To show the complexity saving of the proposed method in terms of required intercell coordination, we compare it with the scheme where each BS aims to directly optimize its downlink beamforming and transmit power based (as compared to the local CSI on full CSI for BS ). Let be the number of quantized bits for a real-valued scalar. Then, the sharing of full CSI among BSs requires about bits for intercell coordination, where the factor of 2 arises from the complex nature of the channel coefficients. In contrast, the parameter exchange in the proposed scheme takes bits, is the total number of iterations taken to meet the where stopping criterion. Thus, the ratio between the numbers of bits , required by the two methods, can be used to quantify the complexity saving of the proposed algorithm. Results in Section V will reveal that is very small, especially for large . For the case where each BS is serving multiple users, there will not be any increase of intercell information exchange when using the proposed algorithm. B. Global Convergence Here, the convergence property of the inner iteration and its infeasibility judgement will be first investigated. Then, the ef-

HUANG et al.: DISTRIBUTED MULTICELL BEAMFORMING WITH LIMITED INTERCELL COORDINATION

ficiency of the outer iteration will be analyzed. By combining these results, the global convergence of the overall iterative algorithm is established. 1) Convergence of the Inner Iteration: In the following, the convergence of the inner iteration is studied for a given target . From the well-known fixed-point iteration theory [10], the power control interference function resulting from the iterative methods in (12) and (13) can be written as

733

monotonically increasing property also implies that for any , an arbitrary must be given feasible target feasible. With the continuity and strict monotonicity of the function , it can be concluded that the bisection approach used in the outer iteration can guarantee to effectively find the solution , or to converge to the globally optimal point. of C. Convergence Acceleration

(18) If we define the interference function vector as , then it can be easily seen by rewriting (18) into (19)

that it satisfies the conditions of positivity, monotonicity and scalability. Together with the results in [10], we have the following proposition. Proposition 1: If the target, , is feasible, then the iterations with any initial power vector, , will converge to a unique fixed point, , which can exactly meet . the target and has the minimum sum-power, Proof: See [10]. That is to say, if the target is feasible, the iterative algorithm will produce a unique optimal , i.e., the fixed point. Note that , since the the optimization in (18) determines both and optimization of is implicitly included in the definition of . In Section IV-C, we will show how to determine, in a distributed way, whether the SINR target is feasible or not. 2) Convergence of the Outer Iteration: Thus far, we have shown that for each target SINR , the inner iteration loop can always return the feasible fixed point or an indication of infeasibility. It has been seen previously that the outer iteration loop can be viewed as a search for the numerical solution of via a bisection approach. To show the convergence of the outer iteration, we first develop the following Proposition is continuous and strictly monoto show that the function tonically increasing. Proposition 2: For any feasible SINR target , the minimum is continuous and strictly monotonically insum-power . creasing with respect to , for the range of Proof: The continuity of follows directly from the continuity of the interference function. The monotonicity can be verified by contradiction. Assume that for some feasible , we have . Their solutions are denoted as and , respectively. Observe that for all , we can always such that . The find some scalar assumption yields . is not optimal for and contradicts This means that always holds for the assumption. Therefore, , i.e., is strictly monotonically increasing. This

Since the convergence rate directly determines the amount of intercell parameter exchange, it is of great importance to accelerate the convergence of the proposed algorithm (and hence reducing ). To achieve this, an efficient infeasibility detection are the keys, for the inner iteration and a good choice of which lead to the following two propositions. Proposition 3: In the inner iteration, the SINR target is infeasible for a given power constraint if the iterative algo, starting with a zero vector, results rithm for all , for some . in Proof: It has been shown in [10, Lemma 2] that for a feasible problem, the iterative algorithm starting from a zero vector will produce an element-wise monotonically increasing , i.e., and . sequence Based on this observation, if we generalize the definition of the by adding to represent the convergence to a point case where the SINR target cannot be met even at infinite power, for some , then the then it can be seen that if convergence point must have the property of , i.e., this target cannot be supported with the power . The above proposition implies that starting with a zero vector, the infeasibility of each inner iteration can be easily detected. Additionally, further analysis will show that we can adaptively choose the initial power vector in each inner iteration by exploiting the results from already completed inner iteration loops, by which the convergence rate can be improved. Before presenting this result, the following proposition is useful. denote the fixed point corresponding Proposition 4: Let is used as the initial to a given feasible SINR target . If as the target SINR value in the inner iteration, with value, then the iteration will produce a monotonically increasing sequence of power vectors that converges to the fixed point corresponding to . denote the intermediate point after Proof: Let iterations in the inner iteration loop corresponding to target . Since is the fixed point for , we have

(20) yields . Based on this, the As a result, following inequalities can be easily derived using the property of the standard fixed-point iteration algorithm [10]

(21)

734

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 2, FEBRUARY 2011

The uniqueness of together with the convergence guaranteed by any arbitrary initial point of the fixed-point iterative algorithm implies that the nondecreasing sequence must converge to the new unique fixed point , which completes the proof. Using this proposition, we see that if the inner iteration loop starts with the largest point in , i.e., corresponding to the fixed point corresponding to the most recent feasible SINR target search, it will not ruin the infeasibility detection based on Proposition 3, and will have faster convergence than using the all zero vector as the initial point. This is easy to understand by is much more closer to recognizing that the largest point in than the all zero vector. the fixed point corresponding to V. SIMULATION RESULTS Here, the performance of the proposed iterative scheme is investigated via computer simulations. In the simulations, a cell cluster with an inter-BS distance of 1 km and 10 MHz channel bandwidth was considered. It is also assumed that all the users are located in the edges such that each user will receive significant ICI in the same cluster. We consider configurations of 2-BS and 3-BS clusters, in which each user has a distance of 400 m from its associated BS and a distance of 600 m from other BSs in the cluster. The channel vector between , is generated based on the any BS and user , denoted as scenario of urban macro with non-line-of-sight (NLoS) so that , where denotes the small-scale fading and is assumed to be zero-mean Gaussian distributed with the covari, and is the large-scale ance matrix of , or path loss, given by (22) in which is a scaling factor, is the path loss exponent (typ), is the distance between user and BS , and ically denotes the lognormal shadowing. In particular, we choose , which in decibels gives , where denotes the shadow fading in decibels and follows the distribution (0, 8 dB). As for the sum-power constraint of the cell cluster, we assume that over the entire bandwidth, each BS has an average power of and for each channel realization the BSs in the same cluster can . Each user also cooperate to consume a total power of in has a fixed receive noise figure of 9 dB, and we shall use the figures to indicate the transmit SNR. In the simulation results, the optimal performance of the proposed algorithm is obtained after convergence defined by the . Note that in stopping condition with practice, the performance with a given amount of exchanged bits between the BSs is of great interest. For this reason, we also provide the performance of the proposed scheme with a fixed number of outer iterations. Specifically, if no feasible SINR target can be found after a fixed number of outer iterations, the achieved rate will be set to be zero. For comparison, we also provide the results for the distributed multicell beamforming scheme using the SGINR criterion of [22] and [23], in which . each BS will transmit at power

Fig. 1. The average rate of the worst cell in the proposed multicell distributed . beamforming system with

K=2

Figs. 1 and 2 show the average rate results achieved by the worst cell in the proposed distributed multicell beamforming scheme and the SGINR scheme. The results demonstrate that in , all the configurations the proposed scheme exhibits a significant advantage over the SGINR scheme, especially at high SNRs. Even with a fixed number of two outer iterations, our scheme still outperforms the SGINR scheme. It is also seen that the proposed scheme can obtain near-optimal performance with only four outer iterations. It is worth mentioning here that on the average only around two inner iterations are required to reach the stopping condition. As , the pera result, for instance in the case of centage of required exchanged bits in our scheme over direct CSI exchange is given by , showing a significant reduction in the need for backhaul signaling. Fig. 3 illustrates the rate achieved by the worst cell in the proposed scheme after a given number of outer iterations, relative to the optimal rate. The simulation results reveal that in all the given configurations, above 90% of the optimal rate can be obtained with 3 outer iterations, and almost 100% of the optimal rate can be attained with only six outer iterations. This suggests that our proposed scheme is able to achieve a good tradeoff between the performance and the amount of intercell parameter exchange. In order to evaluate the impact of quantized intercell parameter exchange on the performance, results are provided in Figs. 4 and 5 for the average rate of the proposed scheme employing a simple uniform linear quantization of the power parameter. The results show that the performance loss caused by a 5-bit parameter quantization is inappreciable, while a 3-bit parameter quantization only causes very little rate loss. The efficiency of quantization can be explained by the fact that the power param. This highlights eter is strictly included in the interval the advantage of the proposed scheme for casting the intercell CSI exchange into power parameter exchange, among the BSs without compromising the performance.

HUANG et al.: DISTRIBUTED MULTICELL BEAMFORMING WITH LIMITED INTERCELL COORDINATION

Fig. 2. The average rate of the worst cell in the proposed multicell distributed . beamforming system with

K=3

Fig. 3. The percentage of the optimal rate, achieved by the worst cell in the proposed multicell distributed beamforming system after a fixed number of outer 46 dBm. iterations, with P

=

Finally, the results in Figs. 6 and 7 quantify the required number of bits for BS coordination using the proposed scheme for varying number of transmit antennas at the BSs and transmit power. As can be seen, the required number of bits for the parameter exchange is insensitive to the size of the channels as well as the range of the power.

735

Fig. 4. The average rate of the worst cell in the proposed multicell distributed beamforming system using quantized intercell parameter exchange with M , and a maximum of four outer iterations. ;K

4

=

=2

Fig. 5. The average rate of the worst cell in the proposed multicell distributed beamforming system using quantized intercell parameter exchange with M ;K , and a maximum of four outer iterations.

4

=

=3

near-optimal rate performance with a small number of outer iterations, which means that our proposed scheme provides a good tradeoff between performance and intercell information exchange.

VI. CONCLUSION

APPENDIX COPING WITH PER-BS POWER CONSTRAINTS

In this paper, we proposed a distributed downlink solution of joint BS multicell beamforming and power allocation, in which the requirement of intercell information exchange is greatly reduced. It was proved that our proposed scheme can always converge to the global optimum. Furthermore, we proposed techniques to accelerate the convergence and reduce the backhaul burden. Simulation results showed that our scheme can achieve

The proposed iterative method can be extended to deal with individual BS power constraints. In particular, the resulting problem can still be cast into an equivalent form similar to (9) and can thus be solved by a bisection search with respect to a virtual SINR target . The only difference is that the power minimization problem is subject to additional per-BS power constraints. To tackle this, we adopt the idea in [3], originally

736

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 2, FEBRUARY 2011

, and is where a diagonal matrix of dual variables associated with the per-BS power constraints. This problem can be interpreted as a virtual uplink power minimization problem with an uncertain noise matrix . Similar to [3], the following subgradient based algorithm can be used to iteratively solve (23), in which for each we define the virtual uplink subproblem fixed noise matrix as

(25) Fig. 6. The required amount of intercell power exchange of the proposed iter46 dBm, and a maxative scheme for varying number of antennas with P imum of two outer iterations.

=

0) 1)

2)

3)

and initialize arbitrarily. Set , find the optimal virtual uplink power For a fixed , for the subproblem and beamformers, i.e., via an iterative method similar to (12) and (13). from the uplink Calculate the downlink power vector via a method similar to [3, Corollary 2] power vector or the method introduced in Section III-B. using the subgradient projection method Update with a preset step size as (26)

4) Fig. 7. The required amount of intercell power exchange of the proposed iter, and a maximum ative scheme for varying transmit power levels with M of two outer iterations.

=4

designed for multiuser downlink systems with per-antenna power constraints, to recast the problem into the form:

(23)

where is the Euclidean projection of the subon the constraint set gradient of . Set and return to Step 1) until convergence.

In contrast to the case of sum-BS power constraint, the above iterative solution tackling the per-BS power constraints involves an extra iteration loop dealing with the noise matrices. This gives rise to a multiplicative increase in the number of parameters exchanging between the BSs as the subgradient iteration proceeds. In addition, the subgradient method usually converges very slowly and typically takes more than 20 iterations to converge [3, Fig. 4]. As a result, the increased amount of information exchange can be unacceptable, or the performance will be severely compromised if only a small amount of intercell parameter exchange is allowed. ACKNOWLEDGMENT

where is the individual power constraint for BS . If can be found, then the virtual target is deemed feasible with the power constraints . It was shown in [3] that this downlink problem (23) is equivalent to the following dual uplink problem: diagonal and (24)

The authors would like to thank Dr. Y. Du from Huawei Technologies Corporation for his helpful discussion. The authors would also like to thank all the anonymous reviewers and the editor for their valuable comments that have helped to improve the quality of this paper. REFERENCES [1] J. D. Herdtner and E. K. P. Chong, “Analysis of a class of distributed asynchronous power control algorithms for cellular wireless systems,” IEEE J. Sel. Areas Commun., vol. 18, no. 3, pp. 436–446, Mar. 2000.

HUANG et al.: DISTRIBUTED MULTICELL BEAMFORMING WITH LIMITED INTERCELL COORDINATION

[2] R. Chen, J. G. Andrews, R. W. Heath Jr., and A. Ghosh, “Uplink power control in multi-cell spatial multiplexing wireless systems,” IEEE Trans. Wireless Commun., vol. 6, no. 7, pp. 2700–2711, Jul. 2007. [3] W. Yu and T. Lan, “Transmitter optimization for the multi-antenna downlink with per-antenna power constraints,” IEEE Trans. Signal Process., vol. 55, no. 6, pp. 2646–2660, Jun. 2007. [4] O. Somekh, O. Simeone, Y. Bar-Ness, A. M. Haimovich, and S. Shamai (Shitz), “Cooperative multicell zero-forcing beamforming in cellular downlink channels,” IEEE Trans. Inf. Theory, vol. 55, no. 7, pp. 3206–3219, Jul. 2009. [5] Y. Huang, L. Yang, and J. Liu, “A limited feedback SDMA for downlink of multiuser MIMO communication system,” EURASIP J. Advances Signal Process., vol. 2008, pp. 13–, Oct. 2008, ID. 947849. [6] M. Schubert and H. Boche, “Solution of the multiuser downlink beamforming problem with individual SINR constraints,” IEEE Trans. Veh. Technol., vol. 53, no. 1, pp. 18–28, Jan. 2004. [7] P. Viswanath and D. N. C. Tse, “Sum capacity of the vector Gaussian broadcast channel and uplink-downlink duality,” IEEE Trans. Inf. Theory, vol. 49, no. 8, pp. 1912–1921, Aug. 2003. [8] M. Schubert and H. Boche, “A generic approach to QoS-based transceiver optimization,” IEEE Trans. Commun., vol. 55, no. 8, pp. 1557–1566, Aug. 2007. [9] F. Rashid-Farrokhi, K. J. R. Liu, and L. Tassiulas, “Transmit beamforming and power control for cellular wireless systems,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1437–1450, Oct. 1998. [10] R. D. Yates, “A framework for uplink power control in cellular radio systems,” IEEE J. Sel. Areas Commun., vol. 13, no. 7, pp. 1341–1347, Sep. 1995. [11] A. Wiesel, Y. C. Eldar, and S. Shamai, “Linear precoding via conic optimization for fixed MIMO receivers,” IEEE Trans. Signal Process., vol. 54, no. 1, pp. 161–176, Jan. 2006. [12] S. Jing, D. Tse, J. Soriaga, J. Hou, J. Smee, and R. Padovani, “Multicell downlink capacity with coordinated processing,” EURASIP J. Wireless Commun. Net., 2008, ID. 586878. [13] O. Somekh, B. M. Zaidel, and S. Shamai (Shitz), “Sum rate characterization of joint multiple cell-site processing,” IEEE Trans. Inf. Theory, vol. 53, no. 12, pp. 4473–4497, Dec. 2007. [14] W. Choi and J. G. Andrews, “The capacity gain from intercell scheduling in multi-antenna systems,” IEEE Trans. Wireless Commun., vol. 7, no. 2, pp. 714–725, Feb. 2008. [15] E. Larsson and E. Jorswieck, “Competition versus cooperation on the MISO interference channel,” IEEE J. Sel. Areas Commun., vol. 26, pp. 1059–1069, 2008. [16] E. Jorswieck, E. Larsson, and D. Danev, “Complete characterization of the Pareto boundary for the MISO interference channel,” IEEE Trans. Signal Process., vol. 56, no. 10, pp. 5292–5296, Oct. 2008. [17] O. Simeone, O. Somekh, H. V. Poor, and S. Shamai (Shitz), “Downlink multicell processing with limited backhaul capacity,” EURASIP J. Adv. Signal Process. (Special Issue on Multiuser MIMO Transmission With Limited Feedback, Cooperation, and Coordination), pp. 10–, 2009, Article ID: 840814. [18] A. Papadogiannis, E. Hardouin, and D. Gesbert, “Decentralising multicell cooperative processing on the downlink: A novel robust framework,” EURASIP J. Wireless Commun. Netw. (Special Issue on Broadband Wireless Access), pp. 10–, 2009, Article ID: 890685. [19] B. L. Ng, J. Evans, S. Hanly, and D. Aktas, “Distributed downlink beamforming with cooperative base stations,” IEEE Trans. Inf. Theory, vol. 54, no. 12, pp. 5491–5499, Dec. 2008. [20] E. Björnson, R. Zakhour, D. Gesbert, and B. Ottersten, “Cooperative multicell precoding: Rate region characterization and distributed strategies with instantaneous and statistical CSI,” IEEE Trans. Signal Process. vol. 58, no. 8, pp. 4298–4310, Aug. 2010. [21] H. Dahrouj and W. Yu, “Coordinated beamforming for the multi-cell multi-antenna wireless system,” IEEE Trans. Wireless Commun., vol. 9, no. 5, pp. 1748–1759, May 2010. [22] B. O. Lee, H. W. Je, I. Sohn, O.-S. Shin, and K. B. Lee, “Interference-aware decentralized precoding for multicell MIMO TDD systems,” presented at the IEEE Global Commun., New Orleans, LA, Nov. 30–Dec. 4, 2008.

737

[23] N. Hassanpour, J. Smee, J. Hou, and J. Soriaga, “Distributed beamforming based on signal-to-caused-interference ratio,” in Proc. IEEE Int. Sym. Spread Spectrum Tech. Appl., Bologna, Italy, Aug. 25–28, 2008, pp. 405–410. [24] M. Bengtsson and B. Ottersten, “Optimal and suboptimal transmit beamforming,” in Handbook of Antennas in Wireless Commun., L. C. Godara, Ed. Boca Raton, FL: CRC Press, Aug. 2001. [25] “Spatial channel model for multiple input multiple output (MIMO) simulations,” 3GPP TR25.996 v.8.0.0, Jan. 2009, Release 8.

Yongming Huang (M’10) received the B.S. and M.S. degrees from Nanjing University, China, in 2000 and 2003, respectively, and the Ph.D. degree in electrical engineering from Southeast University, China, in 2007. After graduation, he held teaching/research positions at the School of Information Science and Engineering, Southeast University, China, where he is currently an Associate Professor. During 2008–2009, he was visiting the Signal Processing Laboratory, Electrical Engineering, Royal Institute of Technology (KTH), Stockholm, Sweden. His current research interests include MIMO communication systems, multiuser MIMO communications, cooperative communications, and satellite mobile communications.

Gan Zheng (S’05–M’09) received the B.Eng. and M.Eng. degrees from Tianjin University, China, in 2002 and 2004, respectively, both in electronic and information engineering, and the Ph.D. degree in electrical and electronic engineering from The University of Hong Kong, Hong Kong, in 2008. He worked as a Research Associate at the University College London, U.K., from December 2007 to September 2010. Since September 2010, he has been a Research Associate at the Interdisciplinary Centre for Security, Reliability and Trust (SnT), The University of Luxembourg. His research interests are in the general area of signal processing for wireless communications, with particular emphasis on the design, optimization and analysis for multiuser multiple-input multiple-output (MIMO) antenna systems, cooperative network, and cognitive radio.

Mats Bengtsson (M’00–SM’06) received the M.S. degree in computer science from Linköping University, Linköping, Sweden, in 1991 and the Tech.Lic. and Ph.D. degrees in electrical engineering from the Royal Institute of Technology (KTH), Stockholm, Sweden, in 1997 and 2000, respectively. From 1991 to 1995, he was with Ericsson Telecom AB Karlstad. He currently holds a position as Associate Professor in the Signal Processing Laboratory of the School of Electrical Engineering at KTH. His research interests include statistical signal processing and its applications to antenna-array processing and communications, radio resource management, and propagation channel modeling. Dr. Bengtsson served as Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING from 2007 to 2009. He is a member of the IEEE SPCOM Technical Committee.

738

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 2, FEBRUARY 2011

Kai-Kit Wong (SM’09) received the B.Eng., the M.Phil., and the Ph.D. degrees, all in electrical and electronic engineering, from the Hong Kong University of Science and Technology, Hong Kong, in 1996, 1998, and 2001, respectively. After graduation, he joined the Department of Electrical and Electronic Engineering, the University of Hong Kong, as a Research Assistant Professor. From July 2003 to December 2003, he visited the Wireless Communications Research Department of Lucent Technologies, Bell-Labs, Holmdel, NJ, where he was a Visiting Research Scholar studying optimization in broadcast MIMO channels. After that, he then joined the Smart Antennas Research Group of Stanford University, Stanford, CA, as a Visiting Assistant Professor conducting research on overloaded MIMO signal processing. From 2005 to August 2006, he was with the Department of Engineering, the University of Hull, U.K., as a Communications Lecturer. Since August 2006, he has been with the Department of Electronic and Electrical Engineering, University College London, U.K., where he is currently a Senior Lecturer. Dr. Wong received the IEEE Vehicular Technology Society Japan Chapter Award of the International IEEE Vehicular Technology—Spring 2000 and was also a corecipient of the First Prize Paper Award in the IEEE Signal Processing Society Postgraduate Forum Hong Kong Chapter in 2004. In 2002 and 2003, he received, respectively, the SY King Fellowships and the WS Leung Fellowships from the University of Hong Kong. Also, he was awarded the Competitive Earmarked Research Grant Merit and Incentive Awards in 2003–2004. He is also on the Editorial Board of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, the IEEE COMMUNICATIONS LETTERS, the IEEE SIGNAL PROCESSING LETTERS, and the IET Communications.

Luxi Yang (M’96) received the M.S. and Ph.D. degree in electrical engineering, from the Southeast University, Nanjing, China, in 1990 and 1993, respectively. Since 1993, he has been with the Department of Radio Engineering, Southeast University, where he is currently a Professor of Information Systems and Communications, and the Director of Digital Signal Processing Division. His current research interests include signal processing for wireless communications, MIMO communications, cooperative relaying

systems, and statistical signal processing. He is the author or coauthor of two published books and more than 100 journal papers, and holds ten patents. Prof. Yang received the first- and second-class prizes of Science and Technology Progress Awards of the State Education Ministry of China in 1998 and 2002. He is currently a member of Signal Processing Committee of Chinese Institute of Electronics.

Björn Ottersten (S’87–M’89–SM’99–F’04) was born in Stockholm, Sweden, in 1961. He received the M.S. degree in electrical engineering and applied physics from Linköping University, Linköping, Sweden, in 1986 and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, in 1989. He has held research positions at the Department of Electrical Engineering, Linköping University; the Information Systems Laboratory, Stanford University, Stanford, CA; and the Katholieke Universiteit Leuven, Leuven, Belgium. In 1991, he was appointed Professor of signal processing at the Royal Institute of Technology (KTH), Stockholm, where he was head of the Department for Signals, Sensors, and Systems from 1992 to 2004 and Dean of the School of Electrical Engineering from 2004 to 2008. Since 2009, he has been Director of securityandtrust.lu at the University of Luxembourg. During 1996–1997, he was Director of Research at ArrayComm Inc, San Jose, CA, a start-up company based on Ottersten’s patented technology. His research interests include wireless communications, stochastic signal processing, sensor array processing, and time-series analysis. Dr. Ottersten has coauthored papers that received an IEEE Signal Processing Society Best Paper Award in 1993, 2001, and 2006. He has served as Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING and on the Editorial Board of the IEEE Signal Processing Magazine. He is currently Editor-in-Chief of the EURASIP Signal Processing Journal and a member of the Editorial Board of the EURASIP Journal of Advances Signal Processing. Dr. Ottersten is a Fellow of EURASIP. He is a first recipient of the European Research Council advanced research grant.