Weighted Sum Rate of Multi-Cell MIMO Downlink Channels in the Large System Limit Sung-Hyun Moon ∗ , Hoon Huh † , Young-Tae Kim ∗ , Giuseppe Caire † , Inkyu Lee ∗ ∗ Korea University, Seoul, Korea, Email: {shmoon, reftm, inkyu}@korea.ac.kr † University of Southern California, Los Angeles, CA, USA, Email: {hhuh, caire}@usc.edu Abstract—The optimization of the weighted ergodic sum rate is considered for the downlink of a cellular networks with multiple cells and multi-antenna base stations. We focus on the large system limit where the number of base station antennas and the number of users per cell go to infinity with a fixed ratio. We consider two extreme cases of full inter-cell cooper- ation (network MIMO) and no inter-cell cooperation (single- cell multiuser MIMO). Using the large random matrix theory and Lagrangian optimization, we obtain a numerical algorithm that exactly computes the maximum weighted sum rate in this asymptotic regime. Numerical results are presented for a simple case of two interfering cells in a linear arrangement. I. I NTRODUCTION Multiuser multiple-input multiple-output (MU-MIMO) tech- nology holds the potential to drastically improve the spec- tral efficiency of the next generation cellular systems by exploiting spatial multiplexing without requiring cumbersome multi-antenna user terminals. Many research efforts have been focused on the single cell setting with the full information theoretic understanding of the underlying MIMO Gaussian broadcast channel (see [1]–[3] and references therein). In order to appreciate the full potential of such technology in a realistic wireless cellular setting, the following important aspects need to be considered: 1) a multi-cell coverage with realistic pathloss model and users’ spatial distribution; 2) the type of inter-cell cooperation (e.g., see [4]); 3) multiuser scheduling, taking into account fairness issues; 4) the type of downlink precoding and signal processing employed and the type of channel state information available at the transmitter (CSIT). While taking into account all the above aspects is very complicated and results in a model (MIMO Gaussian broadcast and interference channel with imperfect CSIT) that is not even fully understood from an information theoretic viewpoint, it makes sense to tackle a subset of these aspects at a time, seeking a clean closed-form or at least semi-analytic expressions towards a better understanding of the possible capacity gains and the tradeoffs involved to achieve them. This work represents a step forward in the above direction, where we consider multiple cells, pathloss and users’ spatial distribution, and limit ourselves to two extreme cases of inter- cell cooperation: full multi-cell joint processing (akin to the so-called “Wyner model” [5]) and no inter-cell cooperation where inter-cell interference (ICI) is treated as noise. Fairness is a very important aspect in cellular networks, since users may be in very different conditions (near or far from the base The work of S.-H. Moon, Y.-T. Kim and I. Lee was supported in part by Seoul R&BD Program (ST090852) and in part by Seoul R&BD Program (WR080951). The work of G. Caire was supported in part by NSF Grant CIF-0917343. station (BS), suffering from interference from adjacent BSs, etc.). The sum capacity is, in the sense of fairness, generally not a good measure for the system performance, since with a sufficiently large number of users it is typically obtained by scheduling only the users close to a BS (“center” users). While this maximizes the sum rate, it results in an unacceptably poor quality of service for the users in unfavorable locations (“edge” users). Since the ergodic achievable rate region R is obtained as the convex hull of all achievable rates, it is easy to see that R is a convex and bounded region. It follows that operating at any desirable point on the region boundary can be defined as a weighted sum rate maximization problem of R ⋆ = arg max R∈R  k∈users W k R k (1) where k runs over all the users in the system and {W k } are suitable non-negative weights. In this paper, we propose a method to compute (1) under the assumptions said above, in the large system limit where the number of antennas per BS and the number of users per cell become large with a fixed ratio. In order to achieve this goal, we combine the results of [6] on the asymptotic sum capacity for the MIMO multiple access channel and the well-known uplink-downlink duality [2], [7] with the input covariance matrix optimization technique of [8]. II. SYSTEM MODEL We consider a multi-cell MIMO downlink system where M BSs with γN antennas each communicate to KN single antenna users. KN users are divided into K co-located “user groups” of equal size N and γ indicates the ratio of the number of BS antennas to the number of users per group. Co- located users are characterized by (approximately) the same set of distances from the BSs. However, since (typically) a wavelength is much smaller than the distances between BSs and users 1 , we assume that users (also co-located users) are separated by a sufficiently large number of wavelength such that they undergo i.i.d. small-scale fading. Notice that this model accounts for an arbitrary placement of BSs’ and users’ locations. Specific examples will be given in Section IV. The received signal vector y k =[y k,1 ··· y k,N ] T ∈ C N for the k-th user group is given by y k = M  m=1 α m,k H H m,k x m + n k (2) 1 A wavelength here is referred to the carrier frequency f 0 . For example, for f 0 =2 GHz the wavelength is λ = 15 cm. On the other hands, distances between users and BS may range from 10 to 10 3 m. 978-1-4244-6404-3/10/$26.00 ©2010 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2010 proceedings