IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 12, DECEMBER 2008 1 Weighted Sum-Rate Maximization using Weighted MMSE for MIMO-BC Beamforming Design Søren Skovgaard Christensen, Rajiv Agarwal, Elisabeth de Carvalho, and John M. Ciof fi Abstract—This paper studies linear transmit filter design for Weighted Sum-Rate (WSR) maximization in the Multiple Input Multiple Output Broadcast Channel (MIMO-BC). The problem of finding the optimal transmit filter is non-convex and intractable to solve using low complexity methods. Motivated by recent results highlighting the relationship between mutual information and Minimum Mean Square Error (MMSE), this paper establishes a relationship between weighted sum-rate and weighted MMSE in the MIMO-BC. The relationship is used to propose two low complexity algorithms for finding a local weighted sum-rate optimum based on alternating optimization. Numerical results studying sum-rate show that the proposed algorithms achieve high performance with few iterations. Index Terms—MIMO systems, transceiver design, smart an- tennas, antennas and propagation. I. I NTRODUCTION M IMO systems have great potential to achieve high throughput in wireless systems [1]. In cellular systems, multiple antennas can easily be deployed at the base station to enhance the system capacity. When Channel State Information (CSI) is available at the transmitter, the base station can transmit to multiple users simultaneously to achieve a linear increase of system throughput in the number of transmit anten- nas. This can be done using linear or non-linear transmission techniques. For the Multiple Input Multiple Output Broadcast Channel (MIMO-BC), non-linear techniques have been shown to outperform linear techniques and achieve channel capacity. The capacity-achieving downlink strategy is non-linear and uses Dirty Paper Coding (DPC) [2]. However, practical tech- niques to implement DPC [3], [4], [5], are in preliminary states of development and are difficult to implement in practice because of their high computational burden. This makes linear downlink transmission techniques (also called beamforming) an attractive alternative because of their simplicity [6], [7], [8], [9]. Transmit beamforming design entails finding the linear transmit filter, through which the data intended for the different users is passed before transmission on the channel. This paper focuses on transmit beamforming design to maximize Weighted Sum-Rate (WSR) subject to a transmit- power constraint, which is a non-convex and non-trivial prob- lem. WSR is useful for prioritizing different users and thus finds different practical applications. For instance the weights Manuscript received August 1, 2007; revised March 16, 2008 and October 27, 2008; accepted November 4, 2008. The associate editor coordinating the review of this paper and approving it for publication was K. Wong. S. S. Christensen is with Nokia Denmark, Modem Algorithm Design, Copenhagen, Denmark (e-mail: skovgaard@ieee.org). R. Agarwal and J. M. Cioffi are with Stanford University (e-mail: {rajivag, cioffi}@stanford.edu). E. de Carvalho is with the Department of Electronic Systems, Aalborg University, Aalborg, Denmark (e-mail: edc@es.aau.dk). Digital Object Identifier 10.1109/T-WC.2008.070851 can be chosen according to the state of the packet queues corresponding to a max-stability service [10] or, by using equal weights, to maximize sum-rate corresponding to a best effort service. A recent paper [11] studies the same problem and pro- poses an iterative algorithm based on uplink-downlink Mean Square Error (MSE)-duality. From a given starting point, the algorithm converges to a local WSR-optimum. The principle in the algorithm is to iterate between the downlink system and a virtual uplink system in order to update filter struc- tures, in addition to solving a Geometric Program (GP) for optimizing the transmit power distribution. In another recent paper [12], the authors attempt to solve the WSR problem using concepts from [8], however their algorithm is a 4-step iterative algorithm, two of which require solving a GP, which again is iterative and a Second-Order Cone Program (SOCP) respectively. This paper takes a different approach to solving the WSR- problem which leads to an iterative algorithm that is guaran- teed to converge to a local WSR-optimum. In the same line as recent results highlighting the relationship between informa- tion theoretic quantities (mutual information) and MMSE in single user Multiple Input Multiple Output (MIMO) channels [13], [14], we have established a relationship between WSR and Weighted sum-Minimum Mean Square Error (WMMSE) in the MIMO-BC. By comparing the gradients of resp. WSR and WMMSE cost functions we are able to show a simple relationship between the Karush-Kuhn-Tucker (KKT) condi- tions of the two problems. Essentially we show that the WSR- problem can be solved as a WMMSE-problem with optimized MSE-weights. Using the derived correspondence we propose an itera- tive algorithm for WSR-optimization in the MIMO-BC. The algorithm iterates between WMMSE transmit filter design, MMSE receive filter computation using well-known closed- form expressions and weighting matrix update. Each of the three steps is solved by evaluating closed-form expressions, and the proposed algorithm is less complex than state-of-the- art methods [11],[12] requiring multiple-level iterations. Two versions of the algorithm are given. In the first one, the weight matrix is computed based on the correspondence between WSR and WMMSE. In the second one, the weight matrix is additionally constrained to be diagonal which is shown to lead to a WSR-optimum with decorrelated streams at each user. Numerical results comparing convergence rates and sum- rate performance to other recently proposed algorithms are presented. Notation: m ij denotes the (i, j )th entry of the matrix M. M T /M H /Tr (M) denotes transpose/conjugate transpose/trace of a vector/matrix M. The dimension of a matrix M is denoted 1536-1276/08$25.00 c  2008 IEEE