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{{Other uses|Pre-code (disambiguation)}}
'''Precoding''' is a generalization of [[beamforming]] to support multi-stream (or multi-layer) transmission in [[MIMO|multi-antenna]] wireless communications. In conventional single-stream beamforming, the same signal is emitted from each of the transmit antennas with appropriate weighting (phase and gain) such that the signal power is maximized at the receiver output. When the receiver has multiple antennas, single-stream beamforming cannot simultaneously maximize the signal level at all of the receive antennas.<ref>G.J. Foschini and M.J. Gans, [
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 or max-min fairness) 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 inter-user interference in multi-user systems.<ref name=gesbert>D. Gesbert, M. Kountouris, R.W. Heath Jr., C.-B. Chae, and T. Sälzer, [
==Precoding for Point-to-Point MIMO Systems ==
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===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] {{webarchive|url=https://web.archive.org/web/20110707223008/http://mars.bell-labs.com/papers/proof/proof.pdf |date=2011-07-07 }}, European Transactions on Telecommunications, vol. 10, no. 6, pp. 585-595, 1999.</ref> In this approach, the channel matrix is diagonalized by taking an SVD and removing the two unitary matrices through pre- and post-multiplication at the transmitter and receiver, respectively. Then, one data stream per singular value can be transmitted (with appropriate power loading) without creating any interference whatsoever.
==Precoding for Multi-user MIMO Systems==
In [[multi-user MIMO]], a multi-antenna transmitter communicates simultaneously with multiple
While performance maximization has a clear interpretation in point-to-point MIMO, a multi-user system cannot simultaneously maximize the performance for all users. This can be viewed as a [[multi-objective optimization]] problem where each objective corresponds to maximization of the capacity of one of the users.<ref name=fnt2013/> The usual way to simplify this problem is to select a system utility function; for example, the weighted sum capacity where the weights correspond to the system's subjective user priorities. Furthermore, 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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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://kth.diva-portal.org/smash/get/diva2:373150/FULLTEXT01 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 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, [
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, [
Note that the optimal linear precoding can be computed using monotonic optimization algorithms,<ref>W. Utschick and J. Brehmer, [
===Linear precoding with limited channel state information===
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
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/> Quantization and feedback of channel state information is based on [[vector quantization]], and codebooks based on Grassmannian line packing have shown good performance.<ref name=dlove2>D.J. Love, R.W. Heath, and T. Strohmer, [
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 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, [
In multiuser MIMO systems where the number of users are higher than the number of transmit antennas, a multiuser diversity can be achieved by performing user scheduling before applying zero-forcing beamforming. Multiuser diversity is a form of selection diversity among users, the base station can schedule its transmission to those users with favorable channel fading conditions to improve the system throughput. In order to achieve multiuser diversity and apply zero-forcing precoding, the CSI of all users are required at the base station. However, the amount of overall feedback information increases with the number of users. Therefore, it is important to perform a user selection at the receiver to determine the users which feed back their quantized CSI to the transmitter based on a pre-defined threshold. <ref> B. Özbek, D. Le Ruyet, [https://www.springer.com/us/book/9781461477402 Feedback strategies for wireless communication], Springer-Verlag New York, U.S.A, December 2014. </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 capacity.<ref name=weingarten/> This category includes Costa precoding,<ref>M. Costa, [
==Mathematical Description==
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:<math>\mathbf{x} = \sum_{i=1}^K \mathbf{w}_i s_i,</math>
where <math>s_i</math> is the (normalized) data symbol and <math>\mathbf{w}_i</math> is the <math>N \times 1</math> linear precoding vector. The [[Signal-to-noise plus interference|signal-to-interference-and-noise]] ratio (SINR) at user <math>k</math> becomes
:<math>\textrm{SINR}_k = \frac{|\mathbf{h}_k^H\mathbf{w}_k|^2}{
where <math>\sigma_k^2</math> is the noise variance for channel to user <math>k</math> and the corresponding achievable information rate is <math>\log_2(1+\textrm{SINR}_k)</math> bits per channel use. The transmission is limited by power constraints. This can, for example, be a total power constraint <math>\sum_{i=1}^K \|\mathbf{w}_i\|^2 \leq P</math> where <math>P</math> is the power limit.
A common performance metric in multi-user systems is the weighted sum rate
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:<math>\mathbf{w}^{\mathrm{MRT}}_k = \sqrt{p_k} \frac{\mathbf{h}_k}{\|\mathbf{h}_k\|}, </math>
while the suboptimal [[zero-forcing precoding|ZF precoding]] makes sure that <math>\mathbf{h}_i^H\mathbf{w}^{\mathrm{ZF}}_k=0</math> for all i ≠ k and thus the interference can be removed in the SINR expression:
:<math>\textrm{SINR}^{\mathrm{ZF}}_k = \frac{|
====Uplink-downlink duality====
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:<math>\mathbf{y} = \sum_{k=1}^{K} \mathbf{h}_k \sqrt{q_k} s_k + \mathbf{n}</math>
where <math>s_k</math> is the transmitted symbol for user <math>k</math>, <math>q_k</math> is the transmit power for this symbol, <math>\mathbf{y}</math> and <math>\mathbf{n}</math> are the <math>N \times 1</math> vector of received signals and noise respectively, <math>\mathbf{h}_k</math> is the <math>N \times 1</math> vector of channel coefficients. If the base station uses linear receive filters to combine the received signals on the <math>N</math> antennas, the SINR for the data stream from user <math>k</math> becomes
:<math>\textrm{SINR}^{\mathrm{uplink}}_k = \frac{q_k|\mathbf{h}_k^H\mathbf{v}_k|^2}{
where <math>\mathbf{v}_k</math> is the unit-norm receive filter for this user. Compared with the downlink case, the only difference in the SINR expressions is that the indices are switched in the interference term. Remarkably, the optimal receive filters are the same as the weighted MMSE precoding vectors, up to a scaling factor:
:<math>\mathbf{v}^{\textrm{MMSE}}_k = \frac{(\sigma_k^2 \mathbf{I} + \sum_{i \neq k} q_i \mathbf{h}_i \mathbf{h}_i^H )^{-1} \mathbf{h}_k}{\|(\sigma_k^2 \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 weighted MMSE precoding are not exactly the optimal power coefficients in the uplink (that maximize the weighted sum rate) except under certain conditions. This important relationship between downlink precoding and uplink receive filtering is known as the uplink-downlink duality.<ref>M. Schubert and H. Boche, [
==== Limited feedback precoding ====
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