Advances in MIMO Techniques for Mobile Communications—A Survey <s> Successive Opt ization <s> Downlink beamforming in a multi-user MIMO channel can provide significant gain in system throughput by allowing space division multiple access (SDMA). The exact solution for the sum capacity of such channels does not exist in closed form, but requires an expensive iterative algorithm. By imposing certain constraints on the capacity equation, a sub-optimal closed-form solution can be obtained. The paper presents two such solutions. The first, referred to as "block-diagonalization" arises from forcing all inter-user interference to zero. The second - "successive optimization" - is an alternative method of minimizing transmit power for an arbitrary information rate per user, and allows solutions which, under certain circumstances, are superior to the block-diagonalization approach. Both algorithms have sub-optimal performance, but they lead to simpler transmitter and receiver structures, and allow a tradeoff between performance and complexity. <s> BIB001 </s> Advances in MIMO Techniques for Mobile Communications—A Survey <s> Successive Opt ization <s> The use of space-division multiple access (SDMA) in the downlink of a multiuser multiple-input, multiple-output (MIMO) wireless communications network can provide a substantial gain in system throughput. The challenge in such multiuser systems is designing transmit vectors while considering the co-channel interference of other users. Typical optimization problems of interest include the capacity problem - maximizing the sum information rate subject to a power constraint-or the power control problem-minimizing transmitted power such that a certain quality-of-service metric for each user is met. Neither of these problems possess closed-form solutions for the general multiuser MIMO channel, but the imposition of certain constraints can lead to closed-form solutions. This paper presents two such constrained solutions. The first, referred to as "block-diagonalization," is a generalization of channel inversion when there are multiple antennas at each receiver. It is easily adapted to optimize for either maximum transmission rate or minimum power and approaches the optimal solution at high SNR. The second, known as "successive optimization," is an alternative method for solving the power minimization problem one user at a time, and it yields superior results in some (e.g., low SNR) situations. Both of these algorithms are limited to cases where the transmitter has more antennas than all receive antennas combined. In order to accommodate more general scenarios, we also propose a framework for coordinated transmitter-receiver processing that generalizes the two algorithms to cases involving more receive than transmit antennas. While the proposed algorithms are suboptimal, they lead to simpler transmitter and receiver structures and allow for a reasonable tradeoff between performance and complexity. <s> BIB002 </s> Advances in MIMO Techniques for Mobile Communications—A Survey <s> Successive Opt ization <s> Multiple-input multiple-output (MIMO) communication techniques have been an important area of focus for next-generation wireless systems because of their potential for high capacity, increased diversity, and interference suppression. For applications such as wireless LANs and cellular telephony, MIMO systems will likely be deployed in environments where a single base must communicate with many users simultaneously. As a result, the study of multi-user MIMO systems has emerged recently as an important research topic. Such systems have the potential to combine the high capacity achievable with MIMO processing with the benefits of space-division multiple access. In this article we review several algorithms that have been proposed with this goal in mind. We describe two classes of solutions. The first uses a signal processing approach with various types of transmitter beamforming. The second uses "dirty paper" coding to overcome the interference a user sees from signals intended for other users. We conclude by describing future areas of research in multi-user MIMO communications. <s> BIB003
Successive optimization (SO) BIB001 BIB002  ] is a successive precoding algorithm which addresses the power control problem in BD where capacity loss occurs due to the nulling of overlapping subspaces of different users. First, an optimum ordering of the users is determined like in case of V-BLAST detection. The precoding matrix for each user is then designed in a successive manner so that it lies in the null space of the channel matrices of the previous users only. In other w rice o interfere with users 1, , 1 i " . Ho with the successive users is allowed. The combined channel matrix for the previous 1 i users can be written as and its SVD is given by Dirty paper coding is a ed by adding interalogous to "wri irty paper" where the writing is the desired signal and the dirt represents the interference. Since the transmitte knows where the "dirt" or interference is, writing on di paper is the same as paper BIB003 . ion to SU-MIMO s tem Z coding is also applicable to mission. In a MU-MIMO downlink system, CSI feedback from the users is available at the BS and it can figure out the interference produced at a particular user by the signals meant for other users. Therefore, dirty paper coding can be applied to each user's signal at the BS so that the known interference from other users is avoided. Various dirty paper coding techniques for the MU-MIMO downlink are discussed in BIB003 . A well-known approach is to use QR decom x, given by H LQ , where L is a lower triangular matrix and Q is a unitary matrix. H Q is then used for transmit precoding which results in the effective channel L. Therefore, the first user does not see any interference from the other users and no further processing of its signal is required at the BS. However, each of the subsequent users sees interference from the preceding users and dirty paper coding is applied to eliminate this known interference. Another technique called vector precoding jointly precodes the users' signals rather than applying dirty paper coding to the us als individually. The vector precoding technique is shown in Figure 51 . The desired signal vector d is offset by a vector l of integer values and this operation is followed by channel inversion, resulting in the transmitted signal x, given by ers' sign where the vector l is chosen to minimize the power of x, i.e.  The signal received at the k-th user is expressed as  Regularized vector precoding is a modification of vector precoding which uses regularized (MMSE) hannel invers transmitte c ion in place of simple channel inversion. The d signal vector is then given by here the vector l is chosen to minimize the norm of x and / T K P D . Dirty paper coding techniques based on vector precoding approach the sum capacity of the MU-MIMO downlink channel, which is defined as the maximum system throughput achieved by maximizing the sum of the information rates of all the us Figure 52 ers BIB003 . shows the performance comparison of vari- ous channel inversion and dirty paper coding techniques for uncoded MU-MIMO downlink transmission with 10 T M BS transmit antennas and singleantenna UEs, using QPSK modulation. Th vector preues clearly outperform the others at high r, regularized channel inversion performs even better than regularized vector precoding in the low SNR region. A possible reason for the performance loss of regularized vector precoding at low SNR is the use of a finite cubical lattice in this algorithm BIB003 . Use of different lattice strategies may result in improved performance.