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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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===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.
===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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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, [
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, [https://dx.doi.org/10.1109/TSP.2011.2179538 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.</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, [https://dx.doi.org/10.1109/ICASSP.2012.6288607 Weighted sum rate optimization for downlink multiuser MIMO systems with per antenna power constraint:Downlink-uplink duality approach] IEEE International Conference On Acoustics, Speech and Signal Processing (ICASSP), Kyoto, Japan, 25 – 30 Mar. 2012, pp. 3245 – 3248.</ref> and <ref>T. E. Bogale and L. Vandendorpe, [https://dx.doi.org/10.1109/TSP.2013.2272554 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.</ref> 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.▼
▲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.
▲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, [http://dx.doi.org/10.1109/TIT.2003.817466 Grassmannian Beamforming for Multiple-Input Multiple-Output Wireless Systems], IEEE Transactions on Information Theory, vol. 49, no. 10, pp. 2735-2747, 2003.</ref>
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, [
▲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 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>
▲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://www.ee.kth.se/php/modules/publications/reports/2009/IR-EE-SB_2009_010.pdf Exploiting Quantized Channel Norm Feedback through Conditional Statistics in Arbitrarily Correlated MIMO Systems], IEEE Transactions on Signal Processing, vol. 57, no. 10, pp. 4027-4041, 2009</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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{{Main|MIMO}}
The standard [[narrowband]], [[fading|slowly
===Description of Multi-user MIMO===
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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 ====
The precoding strategies described above was based on having perfect [[channel state information]] at the transmitter. However, in real systems, receivers can only feed back quantized information that is described by a limited number of bits. If the same precoding strategies are applied, but now based on inaccurate channel information, additional interference appears. This is an example on limited feedback precoding.
The received signal in multi-user MIMO with limited feedback precoding is mathematically described as
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