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Badly/inappropriately written section with accuracy issues, taken almost verbatim from a discussion on a web forum. Tag: section blanking |
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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-
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 [[
==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>
==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.
===Linear precoding with full channel state information===
This suboptimal approach cannot achieve the weighted sum
The optimal linear precoding Finding the optimal weighted MMSE precoding is
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, [https://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, [https://arxiv.org/abs/1104.5240 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.
===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/>
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.
In [[spatial correlation|spatially correlated]] environments, the long-term channel statistics can be combined with low-rate feedback to perform
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
==Mathematical Description==
===Description of Point-to-Point MIMO===
{{Main|MIMO}}
The standard [[narrowband]], [[fading|slowly
===Description of Multi-user MIMO===
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where <math>\mathbf{x}</math> is the <math>N \times 1</math> transmitted vector signal, <math>y_k</math> is the received signal, and <math>n_k</math> is the zero-mean unit-variance noise.
Under linear precoding, the transmitted vector signal is
:<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- :<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
A common performance
:<math>\underset{\{\mathbf{w}_k\}:\sum_i \|\mathbf{w}_i\|^2 \leq P}{\mathrm{maximize}} \sum_{k=1}^K a_k \log_2(1+\textrm{SINR}_k)</math>
for some positive weights <math>a_k</math> that represent the user priority. The weighted sum rate is maximized by weighted MMSE precoding that selects
:<math>\mathbf{w}^{\textrm{W-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
:<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====
For comparison purposes, it is instructive to compare the downlink results with the corresponding uplink MIMO channel where the same single-antenna users transmit to the same base station, having <math>N</math> receive antennas. The input-output relationship can be described as
:<math>\mathbf{y} = \sum_{k=1}^{K} \mathbf{h}_k \sqrt{
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.
:<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
==== Limited feedback precoding ====
The precoding strategies described above was based on
The received signal in multi-user MIMO with limited feedback precoding is mathematically described as
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:<math>y_k = \mathbf{h}_k^H \sum_{i=1}^K \mathbf{w}_i s_i + \mathbf{h}_k^H \sum_{i=1}^K \mathbf{e}_i s_i+ n_k, \quad k=1,2, \ldots, K</math>
where <math>\mathbf{h}_k^H \sum_{i
==See also==
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