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Linear Precoding and Equalization for Network MIMO with Partial Cooperation Saeed Kaviani, Student Member, IEEE, Osvaldo Simeone, Senior Member, IEEE, Witold A

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Linear Precoding and Equalization for Network MIMO with Partial Cooperation Saeed Kaviani, Student Member, IEEE, Osvaldo Simeone, Senior Member, IEEE, Witold A IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 5, JUNE 2012 2083 Linear Precoding and Equalization for Network MIMO With Partial Cooperation Saeed Kaviani, Student Member, IEEE, Osvaldo Simeone, Senior Member, IEEE, Witold A. Krzymien,´ Senior Member, IEEE, and Shlomo Shamai (Shitz), Fellow, IEEE Abstract—A cellular multiple-input–multiple-output (MIMO) network-level interference management appears to be of fun- downlink system is studied, in which each base station (BS) trans- damental importance to overcome this limitation and harness mits to some of the users so that each user receives its intended the gains of MIMO technology. Confirming this point, multicell signal from a subset of the BSs. This scenario is referred to as network MIMO with partial cooperation since only a subset cooperation, which is also known as network MIMO, has been of the BSs is able to coordinate their transmission toward any shown to significantly improve the system performance [3]. user. The focus of this paper is on the optimization of linear Network MIMO involves cooperative transmission by multi- beamforming strategies at the BSs and at the users for network ple base stations (BSs) to each user. Depending on the extent of MIMO with partial cooperation. Individual power constraints at multicell cooperation, network MIMO reduces to a number of the BSs are enforced, along with constraints on the number of streams per user. It is first shown that the system is equivalent to scenarios, ranging from a MIMO broadcast channel (BC) [4] in a MIMO interference channel with generalized linear constraints case of full cooperation among all BSs to a MIMO interference (MIMO-IFC-GC). The problems of maximizing the sum rate (SR) channel [5], [6] in case no cooperation of the BSs is allowed. and minimizing the weighted sum mean square error (WSMSE) In general, network MIMO allows cooperation only among a of the data estimates are nonconvex, and suboptimal solutions cluster of BSs for transmission to a certain user [7], [8] (see with reasonable complexity need to be devised. Based on this, suboptimal techniques that aim at maximizing the SR for the also references in [3]). MIMO-IFC-GC are reviewed from recent literature and extended In this paper, we consider a MIMO interference channel with to the MIMO-IFC-GC where necessary. Novel designs that aim at partial cooperation of the BSs. It is noted that all BSs cooperat- minimizing the WSMSE are then proposed. Extensive numerical ing for transmission to a certain user have to be informed about simulations are provided to compare the performance of the con- the message (i.e., the bit string) intended for the user. This can sidered schemes for realistic cellular systems. be realized using the backhaul links among the BSs and the Index Terms—Cooperative communication, linear precoding, central switching unit. We focus on the sum-rate maximization multicell processing, network MIMO, partial cooperation. (SRM) and the minimization of weighted sum-MSE (WSMSE) under per-BS power constraints and constraints on the number I. INTRODUCTION of streams per user. Moreover, although nonlinear processing NTERFERENCE is known to be a major obstacle in techniques such as vector precoding [9], [10] may generally be I realizing the spectral efficiency increase promised by useful, we focus on more practical linear processing techniques. multiple-antenna techniques in wireless systems. Indeed, Both the SRM and WSMSE minimization (WSMMSE) prob- multiple-input–multiple-output (MIMO) capacity gains are lems are nonconvex [11], and thus suboptimal design strategies severely degraded and limited in cellular environments due of reasonable complexity are called for. to the deleterious effect of interference [1], [2]. Therefore, The contributions of this paper are as follows. 1) It is first shown in Section II-A that network MIMO with Manuscript received June 24, 2011; revised November 14, 2011; accepted partial BS cooperation, i.e., with partial message knowl- January 5, 2012. Date of publication February 11, 2012; date of current version edge, is equivalent to a MIMO interference channel, in June 12, 2012. This paper was presented in part at the IEEE Global Commu- nication Conference, Houston, TX, December 2011. The work of S. Kaviani which each transmitter knows the message of only one and W. A. Krzymien´ was supported in part by TRLabs, by the Rohit Sharma user under generalized linear constraints, which we refer Professorship, and by the Natural Sciences and Engineering Research Council to as MIMO-IFC-GC. of Canada. The work of O. Simeone was supported by the U.S. National Science Foundation under Grant CCF-0914899. The work of S. Shamai was supported 2) We review the available suboptimal techniques that have by the Israel Science Foundation. The review of this paper was coordinated by been proposed for the SRM problem [12]–[14] and ex- Prof. H.-F. Lu. tend them to the MIMO-IFC-GC scenario where neces- S. Kaviani and W. A. Krzymien´ are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6A 2G4, sary in Section V. Since these techniques are generally Canada, and also with TRLabs, Edmonton, AB T5K 2M5, Canada (e-mail: unable to enforce constraints on the number of streams, [email protected]; [email protected]). we also review and generalize techniques that are based O. Simeone is with the Center for Wireless Communications and Signal Processing Research, New Jersey Institute of Technology, Newark, NJ 07102 on the idea of interference alignment [5] and are able to USA (e-mail: [email protected]). impose such constraints. S. Shamai (Shitz) is with the Department of Electrical Engineer- 3) Then, we propose two novel suboptimal solutions for ing, Technion–Israel Institute of Technology, Haifa 32000, Israel (e-mail: [email protected]). the WSMMSE problem in Section VI under arbitrary Digital Object Identifier 10.1109/TVT.2012.2187710 constraints on the number of streams. It is noted that the 0018-9545/$31.00 © 2012 IEEE 2084 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 5, JUNE 2012 We now detail the signal model for the channel at hand, which is referred to as MIMO interference channel with partial T ∈ Cdk message sharing. Define as uk =[uk,1,...,uk,dk ] the dk × 1 complex vector representing the dk ≤ min(Mknt,nr) independent information streams intended for user k.Weas- sume that uk ∼CN(0, I). The data streams uk are known to all the BSs in the set Mk. In particular, if m ∈Mk,themth n ×d BS precodes vector uk via a matrix Bk,m ∈ C t k so that the n signal x˜m ∈ C t sent by the mth BS can be expressed as x˜m = Bk,mUk. (1) k∈Km Imposing a per-BS power constraint, the following constraint must be then satisfied: E 2 E H x˜m =tr x˜mx˜m H ≤ Fig. 1. Downlink model with partial BS cooperation or equivalently partial = tr Bk,mBk,m Pm message knowledge. k∈Km m = 1,...,M (2) WSMMSE problem without such constraints would be trivial, as it would result in zero minimum mean square where Pm is the power constraint of the mth BS. error (MMSE) and no stream transmitted. The proposed The signal received at the kth user can be written as solutions are based on a novel insight into the single-user M MMSE problem with multiple linear constraints, which is y = H x˜ + n˜ (3a) discussed in Section IV. k k,m m k m=1 4) Finally, extensive numerical simulations are provided in Section VII to compare the performance of the proposed = Hk,mBk,mUk + Hk,jBl,jUl + n˜k schemes in realistic cellular systems. m∈Mk l= k j∈Ml (3b) Linear MMSE precoding and equalization techniques pro- posed in this paper were discussed briefly in [15]. The detailed nr ×nt where Hk,m ∈ C is the channel matrix between the mth analysis and discussion (including the proofs to the lemmas) are BS and kth user, and nk is additive complex Gaussian noise included in this paper. Additionally, we have also reviewed and 1 n˜k ∼CN(0, I). We assume ideal channel state information extended available solutions to the SRM problem. Furthermore, (CSI) at all nodes. In (3b), we have distinguished between we have included discussions of the complexity and overhead the first term, which represents useful signal, the second term, of the proposed techniques and previously available (and/or which accounts for interference, and the noise. extended) solutions. Notation: We denote the positive definite matrices as A 0. [·]+ denotes max(·, 0). Capital bold letters represent ma- A. Equivalence With MIMO-IFC-GC trices, and small bold letters represent vectors. We denote We now show that the MIMO interference channel with · T the transpose operator with ( ) and conjugate (Hermitian) partial message sharing and per-BS power constraints previ- · H −(1/2) transpose with ( ) . A represents the inverse square of ously described is equivalent to a specific MIMO interference positive definite matrix A. channel with individual message knowledge and generalized linear constraints, which we refer to as MIMO-IFC-GC. Definition 1: MIMO-IFC-GC: The MIMO-IFC-GC consists II. SYSTEM MODEL AND PRELIMINARIES of K transmitters and K receiver, where the kth transmitter Consider the MIMO downlink system shown in Fig. 1, with has mt,k antennas and the kth receiver has mr,k antennas. The M BSs forming a set M and K users forming a set K. Each BS received signal at the kth receiver is is equipped with nt transmit antennas, and each mobile user employs nr receive antennas. The mth BS is provided with the yk = Hk,kxk + Hk,lxl + nk (4) messages of its assigned users set Km ⊆K.
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