- Open Access
A dynamic clustering algorithm for downlink CoMP systems with multiple antenna UEs
© Baracca et al.; licensee Springer. 2014
- Received: 13 March 2014
- Accepted: 11 July 2014
- Published: 8 August 2014
Coordinated multi-point (CoMP) schemes have been widely studied in the recent years to tackle inter-cell interference. In practice, latency and throughput constraints on the backhaul allow the organization of only small clusters of base stations (BSs) where joint processing (JP) can be implemented. In this work, we focus on downlink CoMP-JP with multiple antenna user equipments (UEs) where the additional degrees of freedom are used to suppress the residual interference by using an interference rejection combiner (IRC) and allow a multi-stream transmission. The main contribution of this paper is the development of a novel dynamic BS clustering algorithm with corresponding UE scheduling. In particular, we first define a set of candidate BS clusters depending on long-term channel conditions. Then, in each time block, we develop a resource allocation scheme where: (a) for each candidate BS cluster, with corresponding scheduled UEs, a weighted sum rate is estimated and then (b) we select the set of non-overlapping BS clusters that maximizes the downlink system weighted sum rate. Numerical results show that much higher rates are achieved when UEs are equipped with multiple antennas and dynamic BS clustering is used.
- Resource allocation and interference management
- MIMO systems
- Cellular technology
Coordination among base stations (BSs) has been widely studied in the recent years to tackle inter-cell interference which strongly limits the rates achieved in cellular systems, in particular by the user equipments (UEs) at the cell-edge . Supported by the first results promising huge gains with respect to the baseline non-cooperative system , a lot of attention has been paid to the topic both in the academia [3, 4] and in the industry [5, 6]. These techniques, known in the industry as coordinated multi-point (CoMP), are classified into (a) coordinated scheduling/beamforming (CS/CB), which requires channel state information (CSI) but no data sharing among the BSs, and (b) joint processing (JP), which requires both CSI and data sharing among the BSs. This paper focuses on downlink CoMP-JP, where BSs jointly serve the scheduled UEs by sharing the data to be sent. Although CoMP-JP is a very promising technique, many issues make its implementation still challenging. First, CSI at the transmitter may be unreliable because of noise on channel estimation in time division duplex (TDD) systems and limited bandwidth available for feedback in frequency division duplex systems. Then, sharing UE data among all the BSs is generally limited by throughput and delay constraints in the backhaul infrastructure. A possible approach to deal with backhaul throughput constraints relies on partial sharing of UE data among the BSs, i.e., a BS serving a certain UE may have only a partial knowledge of the data to be sent toward that UE [7, 8]. Although the promising results achieved under idealistic assumptions, partial UE data sharing has not found application in real systems mainly due to its complexity. To deal with limited throughput backhaul, most of the works in the literature focus on the simpler clustering approach where the BSs are organized in clusters and joint processing is applied within each cluster by sharing the whole data to be sent among all the BSs of the cluster. However, even if intra-cluster interference is mitigated by using CoMP schemes within each cluster, UEs at the cluster border suffer strong inter-cluster interference (ICI). Many clustering schemes have been developed in the literature to deal with ICI. In , static clustering with block diagonalization is considered and precoders are designed in each cluster by nullifying the interference towards UEs of neighboring clusters close to the border. A more flexible solution is obtained with dynamic clustering[10, 11] where the set of clusters changes over time by adapting to the network conditions. In particular, in , a greedy algorithm is developed where, for each cluster, the first BS is selected randomly to guarantee fairness, while the remaining BSs are selected by maximizing the cluster sum rate. In  instead, based on the long-term channel conditions, it is defined a set of candidate clusterings and then in each time block is selected the most suitable one. In , selected clusters maximize the increase of the achievable UE rate, whereas in [12, 13], they minimize the interference power. In , a BS negotiation algorithm is used for cluster formation within a given cluster size. In , active clusters are selected by minimizing an overall cost function which depends on the UE average received power. A framework for feedback and backhaul overhead reduction is developed in  where each UE feeds back CSI only to a subset of BSs, and UEs associated to the same subset are grouped together. In , a greedy UE scheduling algorithm with overlapping clusters is proposed where precoders are designed by considering the layered virtual signal to interference plus noise ratio (SINR) criterion . In , an iterative algorithm is proposed to jointly optimize beamforming and clustering in heterogeneous networks.
However, most of the works on dynamic clustering ([10–13, 16, 17]) assume that UEs are equipped with only one antenna, although the Long Term Evolution (LTE) Advanced standard developed by the 3rd Generation Partnership Project (3GPP) considers that UEs may be equipped with up to eight antennas . Although this number seems a bit optimistic for current mobile devices, the technological innovation may allow in the near-future manufacturing smartphones or tablets with numerous antennas and hence much more attention should be paid to the study of CoMP schemes with multiple antenna UEs [21, 22]. Therefore, in this work, we consider downlink CoMP-JP with a constraint on the maximum cluster size and propose a novel dynamic BS clustering and UE scheduling algorithm by explicitly considering that UEs are equipped with multiple antennas. In our proposal, UEs exploit these additional degrees of freedom by implementing interference rejection combiner (IRC)  to partially suppress ICI and being served by means of a multi-stream transmission. Moreover, differently from many works on dynamic clustering where UE selection is not considered and a simple round robin scheduler is implemented ([10, 12–14, 16, 19]), here, we assume UE scheduling as a part of the optimization. In our approach, we first define a set of candidate BS clusters depending on long-term properties of the channels. Then, in each time block, the proposed algorithm follows a two-step procedure: (a) a weighted sum rate is estimated for each candidate cluster by performing UE selection, precoding design, power allocation, and transmission rank selection and then (b) the central unit (CU) coordinating all the BSs schedules the set of non-overlapping candidate clusters that maximizes the system weighted sum rate under the assumption of perfect successive interference cancellation (SIC) with IRC at each UE.
For a performance comparison, we use the effective achievable rate at UEs, by assuming that CSI is perfectly known at the receiver. In particular, we evaluate the achievable rate of the proposed solution in a LTE-TDD scenario and compare it against a baseline single-cell processing (SCP) scheme and two static clustering schemes, where clusters do not dynamically adapt to the network conditions. Numerical results show that the achievable rates strongly increase with the number of UE antennas. Moreover, as with CoMP part of the interference is managed at the transmit side, multi-stream transmission is more effective with the proposed scheme than with SCP. However, as most of the gain is due to the interference suppression capability of the IRC, the relative gain achieved by the proposed scheme with respect to SCP decreases by increasing the number of UE antennas. Finally, a further decrease of this gain is observed when imperfect CSI is considered at BSs.
Notation. We use (·) T to denote transpose and (·) H conjugate transpose. 0N×M denotes the matrix of size N×M with all zero entries, I N the identity matrix of size N, tr(X) the trace of matrix X, det(X) the determinant of matrix X, vec(X) the vectorization of X, ∥X∥ the Frobenius norm of X, [X]n,m the entry on row n and column m of X, [X]·,m the m th column of X, and diag(x) the diagonal matrix with the entries of vector x on the diagonal. Expectation is denoted by .
We consider a TDD system where a set of BSs , each equipped with M antennas, is serving a set of UEs , each equipped with N antennas, with K>J M. As the overall number of transmitting antennas is not sufficient to serve all the UEs at the same time, UE scheduling is part of the optimization problem. We assume a block fading channel model and denote with Hk,j(t), t=0,1,…,T−1, the multiple-input multiple-output (MIMO) channel matrix of size N×M between BS j and UE k in block t. We consider that the entries of matrix Hk,j(t) are identically distributed zero-mean complex Gaussian random variables, i.e., , for n=0,1,…,N−1 and m=0,1,…,M−1, where represents the large scale fading between BS j and UE k, which depends on path loss and shadowing. We assume that the statistical description of the channels does not change for all the T blocks, whereas fast fading realizations are independent among different blocks. Then, we denote with the covariance matrix of the channel matrix Hk,j(t). We indicate with L E the number of resource elements, i.e., time slots, forming a block. Note that the block fading model considered in this work can be adapted to represent a more realistic channel which changes continuously both in time and in frequency by suitably selecting the number of resource elements in each block. In fact, by denoting with W C and T C the coherence bandwidth and time of the channel, respectively, we have L E =W C T C .
We assume that the BSs are coordinated by a CU, and the backhaul links have zero latency and are error-free. Each block is organized in three phases: (a) in the first phase, all the UEs send pilot sequences to allow channel estimation at BSs, (b) in the second phase, BS clustering, UE scheduling, beamforming design, transmission rank selection, and power optimization are performed by the CU, and finally, (c) in the third phase, the BSs perform data transmission toward the set of scheduled UEs.
List of symbols, time block t has been omitted for simplicity
Set of BSs, each equipped with M antennas
Set of UEs, each equipped with N antennas
Set of integers, each identifying a candidate BS clusters
Set of BSs forming candidate BS cluster c,
Set of scheduled UEs
Set of UEs corresponding to BSs in (see (12))
Set of scheduled UEs corresponding to BSs in (see (16a))
Channel between BS j and UE k
Estimated channel between BS j and UE k
Estimated channel between BSs in and UE k
Precoding matrix used by BS j to serve UE k
Precoding matrix used by BSs in to serve UE k
Transmission rank allocated to UE k
Power vector allocated to UE k
Estimated weighted sum rate achieved by BSs in
Estimated rate achieved by UE k
Effective rate achieved by UE k
2.1 First phase: uplink pilot transmission
2.2 Second phase: resource allocation at the CU
After uplink pilot transmission, each BS j forwards the channel estimates , , to the CU, which, in turn, organizes BSs in clusters and schedules in each time block t a subset of UEs.
Here, denotes the set of integers, each identifying a candidate BS cluster, while is the c-th candidate BS cluster and the corresponding set of scheduled UEs. For complexity reasons, in this work, we consider non-overlapping clusters: hence, in each time block t, the set of BSs is partitioned into non-overlapping clusters and no UE can be served in the same time block by two different BS clusters. Although a solution with overlapping clusters would provide higher rates, it would be much more challenging in terms of computational complexity, in particular when the number of BSs J managed by the CU is high.
In this work, we make the following assumptions regarding power allocation and beamforming design.
Equal power is allocated to the streams sent toward the UEs scheduled within the same cluster, i.e., Pk,l(t)=P(c)(t), , l=0,1,…,l k (t)−1, where P(c)(t) can be analytically computed from (5) as(6)
Beamformers are designed by using the multi-user eigenmode transmission (MET) scheme , where the precoding matrix used to serve UE k is optimized with the aim of nullifying the interference toward the eigenmodes selected for the co-scheduled UEs . In detail, let be the singular value decomposition (SVD) of matrix , where the eigenvalues in are arranged so that the ones selected for transmission toward UE k appear in the leftmost columns. By defining matrix(7)
precoding matrix used to serve UE k satisfies constraints(8)
Note that MET has been proven to outperform in a MIMO broadcast channel other linear precoding schemes such as block diagonlization , whereas the assumption of equal power allocation among the scheduled streams reduces the computational complexity and is asymptotically optimal at high SNR.
The main contribution of this work, described in Section 3, is a practical algorithm for dynamic BS clustering and corresponding UE scheduling developed in the considered setup with multiple antenna BSs, with equal power allocation and MET, that transmit toward multiple antenna UEs. Moreover, we recall that at UEs, the multiple receive antennas are used to perform IRC with SIC (, Ch. 10); in fact, while the rank l k (t) allocated to UE k is given by the number of columns of the precoder, the remaining degrees of freedom at UE are used to partially suppress the residual ICI. Note that IRC both minimizes the mean square error and maximizes the SINR at the detection point .
2.3 Third phase: downlink data transmission
Hence, f k (1) is the anchor BS for UE k.
which rapidly increases with J. However, as most of the interference at each UE comes from the closest BSs, we can limit the number of candidate BS clusters. Hence, we assume that set identifies all and only the sets whose size is not bigger than JMAX. As an example, selecting JMAX=3, set identifies three candidate clusters for each UE k: (i) the first cluster includes only its anchor BS, , (ii) the second cluster is composed of the two closest BSs, , and (iii) the third cluster is composed of the three closest BSs, . Note that, as different UEs often have the same candidate clusters, the cardinality of set turns out to be much lower than K JMAX. The considered assumption yields an important saving in terms of computational complexity by strongly limiting the number of candidate clusters with respect to (11): this complexity saving is evaluated in Section 4 for a typical LTE scenario.
We highlight that (12) allows BSs in cluster to serve all the UEs in its coverage area, even UEs close to the border. Although a different choice could be taken for instance by forcing clusters to serve only the UEs far away from the border, it has been shown in  that this alternative choice provides worse performance than (12) when a huge network is considered and fairness among the UEs is taken into account.
For each candidate BS cluster , we estimate the weighted sum rate by selecting a suitable subset of UEs , designing precoders, selecting transmission ranks, and allocating powers.
After computing the weighted sum rate for all the candidate BS clusters in , the CU schedules a set of non-overlapping BS clusters, where each BS belongs to at most one cluster.
Moreover, based on (12), we observe that UE k can be selected only by candidate clusters that include its anchor BS f k (1). Hence, if we enforce a non-overlapping solution, each UE is never scheduled by two different non-overlapping clusters in the same block. However, we highlight that the proposed dynamic solution allows the flexibility of scheduling a given UE in different clusters across successive blocks.
3.1 Cluster weighted sum rate estimation
Note that (13) represents the average ICI power at the UE k when all the BSs outside cluster c are transmitting at full power (, (2)), and for each candidate cluster allows the computation of independently of the other candidates.
where scaling factor α k in (16a) represents the quality of service (QoS) for UE k which depends on the employed scheduler.
Maximization (16) is a well-studied multi-user MIMO problem  involving (a) UE selection, (b) transmission rank selection, (c) precoding design, and (d) power allocation.
We solve problem (16) by enforcing the assumptions of equal power allocation among the streams sent within cluster and MET introduced in Section 2.2. Moreover, the eigenmodes (and accordingly the set of scheduled UEs and the transmission rank allocated to each UE ) are selected by using a greedy iterative algorithm which, at each iteration, includes the eigenmode which maximizes the weighted sum rate among the ones not scheduled in the previous iterations. The algorithm starts with no UE scheduled and stops when no increase in the weighted sum rate is observed. Cluster , among the possible eigenmodes, selects a maximum of eigenmodes, due to the limited number of BS antennas. Note that the considered method flexibly adapts to the channel conditions by allowing the allocation of (a) different ranks to different UEs in the same block and (b) different ranks to the same UE across successive blocks.
3.2 Clustering optimization
Note that (18) differs from the optimization carried out in  where the objective function simply depends on the received power measured by the UEs.
where is a matrix of size N×M whose entries are independent and identically distributed zero-mean complex Gaussian random variables with as statistical power.
Results are obtained by simulating 100 UE drops and T=200 block channel realizations for each UE drop. We assume that proportional fair scheduling  is implemented to provide fairness among UEs, i.e., , with , t=0,1,…,T−1, where γ=0.1 is the forgetting factor and we initialize . However, to allow the scheduler to reach a steady state, only the last T/2 channels of each UE drop are considered for system performance evaluation.
We compare the developed scheme based on dynamic clustering (DC) against the three static schemes SCP, ISC, and SC, introduced in Section 2. Moreover, we assume that UE scheduling, beamforming design, transmission rank selection, and power allocation are performed, as described in Sections 2.2 and 3.1, also for the static schemes: in particular, UEs are served by using MET  with equal power allocation among the eigenmodes and a greedy UE selection is performed within each BS cluster.
For performance evaluation, we assume perfect CSI at the UE side, which employs IRC with SIC, and perfect detection, i.e., there is no error propagation. Moreover, in Section 4.6, we further provide some numerical results that validate the assumption of perfect CSI at the UE employed in most of the CoMP literature.
In (25), the overhead due to the UE pilot transmission is taken into account in the scaling factor before the logarithm.
The proposed schemes are compared in terms of:
UE rate, defined as(26)
Average cell rate, defined as(27)
Number of candidate clusters with J MAX = 3: comparison between DC and exhaustive search
4.1 Effect of multiple antennas at UEs
Distribution (%) of l k with N = 4
l k =1
l k =2
l k =3
l k =4
Average cell rate and fifth percentile of the UE rate with N = 4 and l (MAX) = 1,4
Fifth percentile of [bit/s/Hz]
4.2 Effect of antenna correlation
4.3 Effect of UE selection
4.4 Effect of cluster size
4.5 Effect of imperfect CSI at BSs
Note that above expressions are only used to determine the block size L E such that the channel can be modeled as uncorrelated between adjacent blocks. Indeed, if fd or increase, L E is reduced and this lowers the rate of each UE as given by (25). Due to the problem of obtaining a reliable CSI at BSs in a high mobility scenario, in the following, we consider fd=5 Hz, which at 2.5 GHz carrier frequency roughly corresponds to a mobile velocity of 2 km/h (, Ch. 21). In this section, we also assume N=4, l(MAX)=1, uncorrelated antennas and JMAX=3 with DC.
4.6 Effect of imperfect CSI at UEs
Average cell rate and fifth percentile of the UE rate for the ETU channel with imperfect CSI at UEs
Fifth percentile of [bit/s/Hz]
In this paper, we have considered a downlink CoMP-JP system and, by assuming a maximum cluster size, we have developed a dynamic BS clustering algorithm where the clusters change over time adapting to the channel conditions. We consider that UEs are equipped with multiple antennas that implement IRC and are served by a multi-stream transmission. The proposed algorithm first defines a set of candidate BS clusters depending on the large scale channel fading. Then, a two-step procedure is applied following a fast fading time scale: (a) first, a weighted sum rate is estimated within each candidate BS cluster by performing UE selection, precoding, power and transmission rank selection, and then (b) the CU schedules the set of non-overlapping BS clusters that maximizes the estimated system weighted sum rate. Numerical results show that much higher effective rates can be achieved when UEs are equipped with multiple antennas. In fact, by reducing the level of interference suffered by UEs, the proposed approach exploits more the multi-stream transmission than SCP. However, as most of the gain is due to the IRC, the gain achieved by the proposed approach decreases with respect to SCP by increasing the number of UE antennas. Finally, when channel estimation is considered at BSs, the gain promised in the perfect CSI scenario may be achieved only in part; in fact, a better estimate requires a longer training sequence and this lowers the system rate.
Part of this work has been performed in the framework of the FP7 project ICT-317669 METIS, which is partly funded by the European Union. The authors would like to acknowledge the contributions of their colleagues in METIS, although the views expressed are those of the authors and do not necessarily represent the project. Part of this work has been presented at the International Symposium on Wireless Communication Systems (ISWCS) 2012, Paris (France), and at the International Conference on Signal Processing, Computing and Control (ISPCC) 2013, Shimla (India). This work was carried out when Federico Boccardi was with Bell Labs, Alcatel-Lucent.
- Marsch P, Fettweis G: Coordinated multi-point in mobile communications. Cambridge University Press, Cambridge, England; 2011.View ArticleGoogle Scholar
- Karakayali MK, Foschini GJ, Valenzuela RA: Network coordination for spectrally efficient communications in cellular systems. IEEE Wireless Commun Mag 2006, 13(4):56-61. 10.1109/MWC.2006.1678166View ArticleGoogle Scholar
- Gesbert D, Hanly S, Huang H, Shamai S, Simeone O, Yu W: Multi-cell MIMO cooperative networks: a new look at interference. IEEE J. Sel. Areas Commun 2010, 28(9):1380-1408.View ArticleGoogle Scholar
- Björnson E, Jorswieck E: Optimal resource allocation in coordinated multi-cell systems. Foundations and Trends in Communications and Information Theory 2012, 9(2-3):113-381.View ArticleMATHGoogle Scholar
- 3GPP TR 36.819 v11.1.0: Coordinated multi-point operation for LTE physical layer aspects (Release 11). 2011.Google Scholar
- Irmer R, Droste H, Marsch P, Grieger M, Fettweis G, Brueck S, Mayer H-P, Thiele L, Jungnickel V: Coordinated multipoint: concepts, performance, and field trial results. IEEE Commun. Mag 2011, 49(2):102-111.View ArticleGoogle Scholar
- Zakhour R, Gesbert D: Optimized data sharing in multicell MIMO with finite backhaul capacity. IEEE Trans. Signal Process 2011, 59(12):6102-6111.MathSciNetView ArticleGoogle Scholar
- Baracca P, Tomasin S, Benvenuto N: Constellation quantization in constrained backhaul downlink network MIMO. IEEE Trans. Commun 2012, 60(3):830-839.View ArticleGoogle Scholar
- Zhang J, Chen R, Andrews JG, Ghosh A, Heath RW: Networked MIMO with clustered linear precoding. IEEE Trans. Wireless Commun 2009, 8(4):1910-1921.View ArticleGoogle Scholar
- Papadogiannis A, Gesbert D, Hardouin E: A dynamic clustering approach in wireless networks with multi-cell cooperative processing. In Proc. IEEE International Conference on Communications (ICC). Beijing, China; 2008.Google Scholar
- Boccardi F, Huang H, Alexiou A: Network MIMO with reduced backhaul requirements by MAC coordination. In Proc. IEEE Conference on Signals, Systems and Computers (Asilomar). Pacific Grove, CA; 2008.Google Scholar
- Moon J-M, Cho D-H: Inter-cluster interference management based on cell-clustering in network MIMO systems. In Proc. IEEE Vehicular Technology Conference (VTC Spring). Budapest, Hungary; 2011.Google Scholar
- Liu J, Wang D: An improved dynamic clustering algorithm for multi-user distributed antenna system. In Proc. IEEE International Conference on Wireless Communications & Signal Processing (WCSP). Nanjing, China; 2009.Google Scholar
- Zhou S, Gong J, Niu Z, Jia Y, Yang P: A decentralized framework for dynamic downlink base station cooperation. In Proc. IEEE Global Communications Conference (GLOBECOM). Honolulu, HI; 2009.Google Scholar
- Weber R, Garavaglia A, Schulist M, Brueck S, Dekorsy A: Self-organizing adaptive clustering for cooperative multipoint transmission. In Proc. IEEE Vehicular Technology Conference (VTC Spring). Budapest, Hungary; 2011.Google Scholar
- Papadogiannis A, Bang HJ, Gesbert D, Hardouin E: Efficient selective feedback design for multicell cooperative networks. IEEE Trans. Veh. Technol 2011, 60(1):196-205.View ArticleGoogle Scholar
- Gong J, Zhou S, Niu Z, Geng L, Zheng M: Joint scheduling and dynamic clustering in downlink cellular networks. In Proc. IEEE Global Communications Conference (GLOBECOM). Houston, TX; 2011.Google Scholar
- Zakhour R, Gesbert D: Distributed multicell-MISO precoding using the layered virtual SINR framework. IEEE Trans. Wireless Commun 2010, 9(8):2444-2448.MathSciNetView ArticleGoogle Scholar
- Hong M, Sun R, Baligh H, Luo Z-Q: Joint base station clustering and beamformer design for partial coordinated transmission in heterogeneous networks. IEEE J. Sel. Areas Commun 2013, 31(2):226-240.View ArticleGoogle Scholar
- Boccardi F, Clerckx B, Ghosh A, Hardouin E, Jöngren G, Kusume K, Onggosanusi E, Tang Y: Multiple-antenna techniques in LTE-advanced. IEEE Commun. Mag 2012, 50(3):114-121.View ArticleGoogle Scholar
- Hwang I, Chae C-B, Lee J, Heath RW: Multicell cooperative systems with multiple receive antennas. IEEE Wireless Commun. Mag 2013, 20(1):50-58.View ArticleGoogle Scholar
- Clerckx B, Lee H, Hong Y-J, Kim G: A practical cooperative multicell MIMO-OFDMA network based on rank coordination. IEEE Trans. Wireless Commun 2013, 12(4):1481-1491.View ArticleGoogle Scholar
- Winters J: Optimum combining in digital mobile radio with cochannel interference. IEEE J. Sel. Areas Commun 1984, 2(4):528-539.View ArticleGoogle Scholar
- Kay S: Fundamentals of statistical signal processing, volume I: estimation theory. Prentice Hall, Upper Saddle River, New Jersey; 1993.MATHGoogle Scholar
- Boccardi F, Huang H: A near-optimum technique using linear precoding for the MIMO broadcast channel. In Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). Honolulu, HI; 2007.Google Scholar
- Tse D, Viswanath P: Fundamentals of wireless communication. Cambridge University Press, Cambridge, England; 2005.View ArticleMATHGoogle Scholar
- Baracca P, Boccardi F, Braun V: A dynamic joint clustering scheduling algorithm for downlink CoMP systems with limited CSI. In Proc. IEEE International Symposium on Wireless Communication Systems (ISWCS). Paris, France; 2012.Google Scholar
- Huh H, Tulino AM, Caire G: Network MIMO with linear zero-forcing beamforming: large system analysis, impact of channel estimation, and reduced-complexity scheduling. IEEE Trans. Inf. Theory 2012, 58(5):2911-2934.MathSciNetView ArticleGoogle Scholar
- Spencer QH, Peel CB, Swindlehurst AL, Haardt M: An introduction to the multi-user MIMO downlink. IEEE Commun. Mag 2004, 42(10):60-67. 10.1109/MCOM.2004.1341262View ArticleGoogle Scholar
- Hoffman K, Padberg M: Set covering, packing and partitioning problems. Springer Encyclopedia of Optimization 2001, 2348-2352.View ArticleGoogle Scholar
- Sesia S, Toufik I, Baker M: LTE: The UMTS Long Term Evolution. John Wiley & Sons, Hoboken, New Jersey; 2009.View ArticleGoogle Scholar
- Hytönen T: Optimal wrap-around network simulation. Helsinki University of Technology, Report A432 2001.Google Scholar
- Kermoal JP, Schumacher L, Pedersen KI, Mogensen PE, Frederiksen F: A stochastic MIMO radio channel model with experimental validation. IEEE J. Sel. Areas Commun 2002, 20(6):1211-1226. 10.1109/JSAC.2002.801223View ArticleGoogle Scholar
- Viswanath P, Tse D, Laroia R: Opportunistic beamforming using dumb antennas. IEEE Trans. Inf. Theory 2002, 48(6):1277-1294. 10.1109/TIT.2002.1003822MathSciNetView ArticleMATHGoogle Scholar
- Benvenuto N, Cherubini G: Algorithms for communications systems and their applications. John Wiley & Sons, Hoboken, New Jersey; 2002.View ArticleGoogle Scholar
- Rappaport T: Wireless communications: principles and practice. Prentice Hall, Upper Saddle River, New Jersey; 2002.MATHGoogle Scholar
- Marzetta T, Hochwald BM: Fast transfer of channel state information in wireless systems. IEEE Trans. Signal Process 2006, 54(4):1268-1278.View ArticleGoogle Scholar
- Gomadam KS, Papadopoulos HC, Sundberg C-EW: Techniques for multi-user MIMO with two-way training. In Proc. IEEE International Conference on Communications (ICC). Beijing, China; 2008.Google Scholar
- Hoydis J, Hosseini K, ten Brink S, Debbah M: Making smart use of excess antennas: massive MIMO, small cells, and TDD. Bell Labs Tech. J 2013, 18(2):5-21. 10.1002/bltj.21602View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.