Precoding: Difference between revisions

Revision as of 16:37, 1 January 2013

Revision as of 15:08, 2 April 2013

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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-layerstream 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-layerstream 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> Thus, inIn order to maximize the throughput in multiple receive antenna systems, multi-layerstream beamforming is generally required.

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 [[Spacespace-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 cointer-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 in the downlink of cellular networks, known as network MIMO or coordinated multipoint (CoMP), is a generalized form of multi-user MIMO that can be analyzed by the same mathematical techniques.<ref name=fnt2013>E. Björnson and E. Jorswieck, [http://kth.diva-portal.org/smash/get/diva2:608533/FULLTEXT01 Optimal Resource Allocation in Coordinated Multi-Cell Systems], Foundations and Trends in Communications and Information Theory, vol. 9, no. 2-3, pp. 113-381, 2013.</ref>

==Precoding for Point-to-Point MIMO Systems ==

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==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> but linear precoding approaches usually achieve reasonable performance with much lower complexity. Linear precoding strategies include MMSEmaximum precodingratio andtransmission the(MRT)<ref name=lo>T. simplifiedLo, [http://dx.doi.org/10.1109/26.795811 Maximum ratio transmission], IEEE Transactions on Communications, vol. 47, no. 10, pp. 1458–1461, 1999.</ref>, [[Zero-forcing precoding|zero-forcing]] (ZF) precoding, and transmit Wiener precoding<ref name=joham>M. Joham, W. Utschick, and J. Nossek, [http://dx.doi.org/10.1109/TSP.2005.850331 Linear transmit processing in MIMO communications systems], IEEE Transactions on Signal Processing, vol. 53, no. 8, pp. 2700–2712, 2005.</ref> There are also precoding strategies tailored for low-rate [[feedback]] of [[channel state information]], for example random beamforming.<ref name=sharif>M. Sharif and B. Hassibi, [http://iss.bu.edu/sharif/mimobc-final.pdf On the Capacity of MIMO Broadcast Channels With Partial Side Information], IEEE Transactions on Information Theory, vol. 51, no. 2, pp. 506-522, 2005.</ref> Nonlinear precoding is designed based on the concept of [[dirty paper coding]] (DPC), which shows that any known interference at the transmitter can be subtracted without the penalty of radio resources if the optimal precoding scheme can be applied on the transmit signal.

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 (or some other metric of achievable rates under linear precoding). Optimal
The optimal linear precoding isdoes knownnot ashave MMSEany precoding<refclosed-form name=gesbert/>expression, andbut isit simpletakes tothe characterizeform of a weighted MMSE precoding for single-antenna receivers;.<ref thename=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 meanscan be interpreted as finding the optimal balance between achieving strong signal gain and limiting cointer-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 weighted MMSE precoding is often difficult, leading to approximate approaches thatwhere 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>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 cointer-user interference is negligible compared to the noise. ZF precoding aims at nulling the cointer-user interference, at the expense of losing some signal gain. ZF precoding can achieve close to the systemsum capacity when the number of users is large or the system is interference-limited (i.e., the noise is weak compared to the interference). IfA receiversbalance havebetween multipleMRT antennas,and thenZF is obtained by the so-called regularized zero-forcing precoding<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> has(also theknown correspondingas propertiessignal-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 filter<ref name=joham/>) All of these heuristic approaches can also be applied to receivers that have multiple antennas.<ref name=peel/><ref name=sadek/><ref name=joham/>

Note that the optimal linear precoding can be computed using monotonic optimization techniques<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.

===Linear precoding with limited channel state information===