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This suboptimal approach cannot achieve the weighted sum rate, but it can still maximize the weighted sum performance (or some other metric of achievable rates under linear precoding).
The optimal linear precoding does not have any closed-form expression, but it takes the form of a weighted MMSE precoding for single-antenna receivers.<ref name=fnt2013/> The precoding weights for a given user are selected to maximize a ratio between the signal gain at this user and the interference generated at other users (with some weights) plus noise. Thus, precoding can be interpreted as finding the optimal balance between achieving strong signal gain and limiting inter-user interference.<ref name=bjornson>E. Björnson, R. Zakhour, D. Gesbert, B. Ottersten, [http://
Finding the optimal weighted MMSE precoding is difficult, leading to approximate approaches where the weights are selected heuristically. A common approach is to concentrate on either the numerator or the denominator of the mentioned ratio; that is, maximum ratio transmission (MRT)<ref name=lo/> and [[zero-forcing precoding|zero-forcing]] (ZF)<ref name=jindal>N. Jindal, [http://dx.doi.org/10.1109/TIT.2006.883550 MIMO Broadcast Channels with Finite Rate Feedback], IEEE Transactions on Information Theory, vol. 52, no. 11, pp. 5045–5059, 2006.</ref> precoding. MRT only maximizes the signal gain at the intended user. MRT is close-to-optimal in noise-limited systems, where the inter-user interference is negligible compared to the noise. ZF precoding aims at nulling the inter-user interference, at the expense of losing some signal gain. ZF precoding can achieve performance close to the sum capacity when the number of users is large or the system is interference-limited (i.e., the noise is weak compared to the interference). A balance between MRT and ZF is obtained by the so-called regularized zero-forcing<ref name=peel>B. C. B. Peel, B. M. Hochwald, and A. L. Swindlehurst, [http://dx.doi.org/10.1109/TCOMM.2004.840638 A vector-perturbation technique for near-capacity multiantenna multi-user communication - Part I: channel inversion and regularization], IEEE Transactions on Communications, vol. 53, no. 1, pp. 195–202, 2005.</ref> (also known as signal-to-leakage-and-interference ratio (SLNR) beamforming<ref name=sadek>M. Sadek, A. Tarighat, and A. Sayed, [http://dx.doi.org/10.1109/TWC.2007.360373 A leakage-based precoding scheme for downlink multi-user MIMO channels], IEEE Transactions on Wireless Communications, vol. 6, no. 5, pp. 1711–1721, 2007.</ref> and transmit Wiener filtering<ref name=joham/>) All of these heuristic approaches can also be applied to receivers that have multiple antennas.<ref name=joham/><ref name=peel/><ref name=sadek/>
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Other precoding strategies have been developed for the case with very low channel feedback rates. Random beamforming<ref name=sharif/> (or opportunistic beamforming<ref name= viswanath>P. Viswanath, D. N. C. Tse, Member, and R. Laroia, [http://www.eecs.berkeley.edu/~dtse/oppbf_it.pdf Opportunistic Beamforming Using Dumb Antennas], IEEE Transactions on Information Theory, vol. 48, no. 6, pp. 1277-1294, 2002.</ref>) was proposed as a simple way of achieving good performance that scales like the sum capacity when the number of receivers is large. In this suboptimal strategy, a set of beamforming directions are selected randomly and users feed back a few bits to tell the transmitter which beam that gives the best performance and what rate they can support using it. When the number of users is large, it is likely that each random beamforming weight will provide good performance for some user.
In [[spatial correlation|spatially correlated]] environments, the long-term channel statistics can be combined with low-rate feedback to perform multi-user precoding.<ref>D. Hammarwall, M. Bengtsson, and B. Ottersten, [http://dx.doi.org/10.1109/TSP.2008.920484 Utilizing the spatial information provided by channel norm feedback in SDMA systems], IEEE Transactions on Signal Processing, vol. 56, no. 7, pp. 3278–3293, 2008</ref> As spatially correlated statistics contain much directional information, it is only necessary for users to feed back their current channel gain to achieve reasonable channel knowledge. As the beamforming weights are selected from the statistics, and not randomly, this approach outperforms random beamforming under strong spatial correlation.<ref>E. Björnson, D. Hammarwall, B. Ottersten, [http://
===DPC or DPC-like nonlinear precoding===
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