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{{Other uses|Pre-code (disambiguation)}}
'''Precoding''' is a generalization of [[beamforming]] to support multi-layer transmission in [[MIMO|multi-antenna]] wireless communications. In conventional single-layer beamforming, the same signal is emitted from each of the transmit antennas with appropriate weighting such that the signal power is maximized at the receiver output. When the receiver has multiple antennas, single-layer beamforming cannot simultaneously maximize the signal level at all of the receive antennas.<ref>G.J. Foschini and M.J. Gans, [http://dx.doi.org/10.1023/A:1008889222784 On limits of wireless communications in a fading environment when using multiple antennas], Wireless Personal Communications, vol. 6, no. 3, pp. 311–335, 1998.</ref>
In point-to-point systems, precoding means that multiple data streams are emitted from the transmit antennas with independent and appropriate weightings such that the link throughput is maximized at the receiver output. In [[multi-user MIMO]], the data streams are intended for different users (known as [[Space-division multiple access|SDMA]]) and some measure of the total throughput (e.g., the sum performance) is maximized. In point-to-point systems, some of the benefits of precoding can be realized without requiring [[channel state information]] at the transmitter, while such information is essential to handle the co-user interference in multi-user systems.<ref name=gesbert>D. Gesbert, M. Kountouris, R.W. Heath Jr., C.-B. Chae, and T. Sälzer, [http://dx.doi.org/10.1109/MSP.2007.904815 Shifting the MIMO Paradigm], IEEE Signal Processing Magazine, vol. 24, no. 5, pp. 36-46, 2007.</ref>
==Precoding for Point-to-Point MIMO Systems ==
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===Statistical channel state information===
If the receiver knows the channel matrix and the transmitter has statistical information, eigenbeamforming is known to achieve the MIMO channel capacity.<ref name=dlove>D. Love, R. Heath, V. Lau, D. Gesbert, B. Rao and M. Andrews, [http://www.eurecom.fr/~gesbert/papers/JSAC_limitedfeedback_tutorial.pdf An overview of limited feedback in wireless communication systems], IEEE Journal on Selected Areas Communications, vol. 26, no. 8, pp. 1341-1365, 2008.</ref>
===Full channel state information===
If the channel matrix is completely known, [[singular value decomposition]] (SVD) precoding is known to achieve the MIMO channel capacity.<ref>E. Telatar, [http://mars.bell-labs.com/papers/proof/proof.pdf Capacity of multiantenna Gaussian channels], European Transactions on Telecommunications, vol. 10, no. 6, pp. 585-595, 1999.</ref>
==Precoding for Multi-user MIMO Systems==
In [[multi-user MIMO]], a multi-antenna transmitter communicates simultaneously with multiple receivers (each having one or multiple antennas). This is known as [[space-division multiple access]] (SDMA). From an implementation perspective, precoding algorithms for SDMA systems can be sub-divided into linear and nonlinear precoding types. The capacity achieving algorithms are nonlinear,<ref name=weingarten>H. Weingarten, Y. Steinberg, and S. Shamai, [http://www.stanford.edu/class/ee360/suppRead/read1/WeingartenSteinbergShamai2006.pdf The capacity region of the Gaussian multiple-input multiple-output broadcast channel], IEEE Transactions on Information Theory, vol. 52, no. 9, pp. 3936–3964, 2006.</ref>
While performance maximization has a clear interpretation in point-to-point MIMO, a multi-user system cannot simultaneously maximize the performance for all users. Thus, it is common to maximize the weighted sum capacity, where the weights correspond to user priorities. In addition, there might be more users than data streams, requiring a [[scheduling algorithm]] to decide which users to serve at a given time instant.
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===Linear precoding with full channel state information===
This suboptimal approach cannot achieve the weighted sum capacity, but it can still maximize the weighted sum performance. Optimal linear precoding is known as MMSE precoding<ref name=gesbert/> and is simple to characterize for single-antenna receivers; 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 means finding the optimal balance between achieving strong signal gain and limiting co-user interference.<ref name=bjornson>E. Björnson, R. Zakhour, D. Gesbert, B. Ottersten, [http://www.ee.kth.se/php/modules/publications/reports/2010/IR-EE-SB_2010_005.pdf Cooperative Multicell Precoding: Rate Region Characterization and Distributed Strategies with Instantaneous and Statistical CSI], IEEE Transactions on Signal Processing, vol. 58, no. 8, pp. 4298-4310, 2010.</ref>
Finding the optimal MMSE precoding is often difficult, leading to approximate approaches that concentrate on either the numerator or denominator of the mentioned ratio; that is, '''maximum ratio transmission (MRT)'''<ref name=lo>T. Lo, [http://dx.doi.org/10.1109/26.795811 Maximum ratio transmission], IEEE Transactions on Communications, vol. 47, no. 10, pp. 1458–1461, 1999.</ref> 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 co-user interference is negligible compared to the noise. ZF precoding aims at nulling the co-user interference, at the expense of losing some signal gain. ZF precoding can achieve close to the system capacity when the number of users is large or the system is interference-limited (i.e., the noise is weak compared to the interference). If receivers have multiple antennas, then regularized zero-forcing precoding<ref>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> has the corresponding properties.
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In practice, the [[channel state information]] is limited at the transmitter due to estimation errors and quantization. Inaccurate channel knowledge may result in significant loss of system throughput, as the interference between the multiplexed streams cannot be completely controlled. In closed-loop systems, the feedback capabilities decide which precoding strategies that are feasible. Each receiver can either feedback a quantized version of its complete channel knowledge or focus on certain critical performance indicators (e.g., the channel gain).
If the complete channel knowledge is fed back with good accuracy, then one can use strategies designed for having full channel knowledge with minor performance degradation. Zero-forcing precoding may even achieve the full multiplexing gain, but only provided that the accuracy of the channel feedback increases linearly with [[signal-to-noise ratio]] (in dB).<ref name=jindal/>
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 weights 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 SDMA 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>
===DPC or DPC-like nonlinear precoding===
[[Dirty paper coding (DPC)|Dirty paper coding]] is a coding technique that pre-cancels known interference without power penalty. Only the transmitter needs to know this interference, but full [[channel state information]] is required everywhere to achieve the weighted sum rate.<ref name=weingarten/>
==Mathematical Description==
===Description of Point-to-Point MIMO===
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for some positive weights <math>a_k</math> that represent the user priority. The weighted sum rate is maximized by MMSE precoding that selects
:<math>\mathbf{w}^{\textrm{MMSE}}_k = \sqrt{p_k} \frac{( \mathbf{I} + \sum_{i \neq k} q_i \mathbf{h}_i \mathbf{h}_i^H )^{-1} \mathbf{h}_k}{\|( \mathbf{I} + \sum_{i \neq k} q_i \mathbf{h}_i \mathbf{h}_i^H )^{-1} \mathbf{h}_k\|} </math>
for some positive coefficients <math>q_1,\ldots,q_K</math> (related to the user weights) that satisfy <math>\sum_{i=1}^K q_i = P</math> and <math>p_i</math> is the optimal power allocation.<ref name=bjornson />
The suboptimal MRT approach removes the channel inversion and only selects
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:<math>\mathbf{v}_k = \frac{( \mathbf{I} + \sum_{i \neq k} q_i \mathbf{h}_i \mathbf{h}_i^H )^{-1} \mathbf{h}_k}{\|( \mathbf{I} + \sum_{i \neq k} q_i \mathbf{h}_i \mathbf{h}_i^H )^{-1} \mathbf{h}_k\|} </math>
Observe that the coefficients <math>q_1,\ldots,q_K</math> that was used in the MMSE beamforming are exactly the optimal power coefficients in the uplink (that maximize the weighted sum rate). This important relationship between downlink precoding and uplink receive filtering is known as the uplink-downlink duality.<ref>M. Schubert and H. Boche, [http://dx.doi.org/10.1109/TVT.2003.819629 Solution of the multiuser downlink beamforming problem with individual SINR constraints], IEEE Transactions on Vehicular Technology, vol. 53, no. 1, pp. 18-28, 2004.</ref><ref>A. Wiesel, Y.C. Eldar, S. Shamai, [http://dx.doi.org/10.1109/TSP.2005.861073 Linear precoding via conic optimization for fixed MIMO receivers], IEEE Transactions on Signal Processing, vol. 54, no. 1, pp. 161-176, 2006.</ref>
==== Limited feedback precoding ====
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