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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/>
Also for multiuser MIMO system setup, another approach has been used to reformulate the weighted sum rate optimization problem to a weighted sum MSE problem with additional optimization MSE weights for each symbol in <ref>T. E. Bogale and L. Vandendorpe, “Weighted sum rate optimization for
downlink multiuser MIMO coordinated base station systems: Centralized
and distributed algorithms,” IEEE Trans. Signal Process., vol. 60, no. 4,
pp. 1876 – 1889, Dec. 2011. [http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=6104172&queryText%3DWeighted+sum+rate+optimization+for+downlink+multiuser+MIMO+coordinated+base+station+systems%3A+Centralized+and+distributed+algorithms] </ref>. However, still this work is not able to solve this problem optimally (i.e., its solution is suboptimal). On the other hand, duality approach also considered in <ref>T. E. Bogale and L. Vandendorpe, “Weighted sum rate optimization for downlink multiuser MIMO systems with per antenna power constraint:
Downlink-uplink duality approach,” in IEEE International Conference
On Acuostics, Speech and Signal Processing (ICASSP), Kyoto, Japan, 25
– 30 Mar. 2012, pp. 3245 – 3248. [http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=6288607&queryText%3DWeighted+sum+rate+optimization+for+downlink+multiuser+MIMO+systems+with+per+antenna+power+constraint%3A+Downlink-uplink+duality+approach]</ref> and <ref>T. E. Bogale and L. Vandendorpe, “Linear transceiver design for downlink
multiuser MIMO systems: Downlink-interference duality approach,”
IEEE Trans. Sig. Process., vol. 61, no. 19, pp. 4686 – 4700, Oct. 2013. [http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=6556946&queryText%3DLinear+transceiver+design+for+downlink+multiuser+MIMO+systems%3A+Downlink-interference+duality+approach]
</ref> also to get sub-optimal solution for weighted sum rate optimization.
Note that the optimal linear precoding can be computed using monotonic optimization algorithms,<ref>W. Utschick and J. Brehmer, [http://dx.doi.org/10.1109/TSP.2011.2182343 Monotonic optimization framework for coordinated beamforming in multicell networks], IEEE Transactions on Signal Processing, vol. 60, no. 4, pp. 1899–1909, 2012.</ref><ref>E. Björnson, G. Zheng, M. Bengtsson, and B. Ottersten, [http://arxiv.org/pdf/1104.5240v4 Robust monotonic optimization framework for multicell MISO systems], IEEE Transactions on Signal Processing, vol. 60, no. 5, pp. 2508–2523, 2012.</ref> but the computational complexity scales exponentially fast with the number of users. These algorithms are therefore only useful for benchmarking in small systems.
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