IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 12, DECEMBER 2010 6355 Optimization of MIMO Relays for Multipoint-to-Multipoint Communications: Nonrobust and Robust Designs Batu Krishna Chalise, Member, IEEE, and Luc Vandendorpe, Fellow, IEEE Abstract—In this paper, we propose algorithms to jointly opti- mize the multiple multiantenna relays which assist multipoint-to- multipoint communications in wireless networks. Assuming that the knowledge of the second order statistics of the channels such as covariance matrices are available, the multiple-input-multiple- output (MIMO) relays are designed using two different methods: 1) minimize the sum of the powers of the relays while fulfilling the signal-to-interference-plus-noise ratio (SINR) requirements for all destinations and 2) maximize the minimum of the SINRs of all des- tinations satisfying the transmit power constraint of each MIMO relay. Furthermore, considering the fact that the covariance ma- trices of the channels between the MIMO relays and destinations are subject to uncertainty due to feedback and quantization er- rors, the robust versions of the aforementioned methods based on worst-case concept are proposed. It is shown that the proposed nonrobust as well as robust designs are nonconvex optimization problems but they can be solved accurately and efficiently using the standard semidefinite relaxation and randomization techniques. Computer simulations verify the improved performance of the ro- bust designs over nonrobust methods. Index Terms—Channel state information, channel uncertainty and convex optimization, robust MIMO relays, worst-case perfor- mance optimization. I. INTRODUCTION T HE application of relays for cooperative communications has received a lot of interest in recent years [1], [2]. It is well known that the channel impairments such as shadowing, multipath fading, distance-dependent path losses and interfer- ence, often degrade the link-quality between the source and destination in a wireless network. If the link-quality degrades severely, relays can be employed between the source and desti- nation nodes for assisting the data transmission [1], [2]. In the literature, various types of cooperative communications such as amplify-and-forward (AF), decode-and-forward [2], coded-co- operation [3], and compress-and-forward [4] have been pre- Manuscript received December 30, 2009; accepted September 02, 2010. Date of publication September 20, 2010; date of current version November 17, 2010. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Phillipe Ciblat. B. K. Chalise was with the Universitè Catholique de Louvain, Belgium. He is now with the Center for Advanced Communications, Villanova University, Villanova, PA 19085 USA (e-mail: batu.chalise@villanova.edu). L. Vandendorpe is with the ICTEAM Instititute, Universitè Catholique de Louvain, B-1348 Louvain la Neuve, Belgium (e-mail: luc.vandendorpe@uclouvain.be). 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.2010.2077632 sented. In [5], the outage and ergodic capacities have been ana- lyzed for a three-node network where one of the nodes relays the messages of another node towards the third one. Among several cooperative communication schemes [2]–[5], the AF scheme is more attractive due to its simplicity since the relay simply for- wards the signal and does not decode it. Recently, space-time coding strategies have been developed for relay networks [6]. In [7], the authors study distributed beamforming for a cooper- ative network which consists of a transmitter, a receiver and an arbitrary number of relay nodes. This work has been extended in [8] to a multiple number of sources and destinations. The afore- mentioned works consider that the transmitter, receiver as well as the relays are all single-antenna nodes and do not take into account that the channel state information (CSI) (either instan- taneous or second order statistics) available in the network can be erroneous. The advantages of multiple-input-multiple output (MIMO) systems can be exploited in relay communications by accom- modating multiple antennas at the nodes [9], [10]. The optimal design of AF-MIMO relays has been proposed in [11] and [12] for a point-to-point communication. MIMO fixed relay that uses linear processing for enhancing multiuser transmission in the downlink of a cellular system has been investigated in [13] and [14]. The problem of designing the optimum AF-MIMO relay for multipoint-to-multipoint communication in wireless networks has been solved in [15] using semidefinite relaxation method. The latter work has been extended in [16] to the robust design where the MIMO relay has only the imperfect estimates of the source-relay and relay-destination channels. However, the works of [14]–[16] have been limited to the design of a single MIMO relay. The optimal relay processing for multiple AF-MIMO relays in a point-to-point communication scenario has been designed in [17] and [18] to minimize the mean-square-error (MSE) and satisfy the quality of service (QoS) requirements. In [19], the authors consider a multipoint-to multipoint scenario and jointly design the cooperative MIMO relays by minimizing both the noise received at each destination and interference caused by the sources not targeting this destination while preserving the re- ceived signal from each source at its targeted destination. How- ever, this approach requires a mechanism to control the ampli- tude and phase of the signals received at each destination which is difficult to implement in practice. It should be noted that the relay powers are important resources and must be optimally uti- lized [20]. In [20], it is assumed that the MIMO relays have in- stantaneous CSI, which is actually the perfect knowledge of all channels between sources and relays, and relays and destina- 1053-587X/$26.00 © 2010 IEEE