In signal processing, precoding is a generalization of beamforming to support multi-stream (or multi-layer) transmission in 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.G.J. Foschini and M.J. Gans, On limits of wireless communications in a fading environment when using multiple antennas, Wireless Personal Communications, vol. 6, no. 3, pp. 311–335, 1998. In order to maximize the throughput in multiple receive antenna systems, multi-stream transmission 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 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.D. Gesbert, M. Kountouris, R.W. Heath Jr., C.-B. Chae, and T. Sälzer, Shifting the MIMO Paradigm, IEEE Signal Processing Magazine, vol. 24, no. 5, pp. 36-46, 2007. 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.E. Björnson and E. Jorswieck, 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.
Precoding for Point-to-Point MIMO Systems In point-to-point multiple-input multiple-output (MIMO) systems, a transmitter equipped with multiple antennas communicates with a receiver that has multiple antennas. Most classic precoding results assume narrowband, slowly fading channels, meaning that the channel for a certain period of time can be described by a single channel matrix which does not change faster. In practice, such channels can be achieved, for example, through OFDM. The precoding strategy that maximizes the throughput, called channel capacity, depends on the channel state information available in the system.
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.D. Love, R. Heath, V. Lau, D. Gesbert, B. Rao and M. Andrews, An overview of limited feedback in wireless communication systems, IEEE Journal on Selected Areas Communications, vol. 26, no. 8, pp. 1341–1365, 2008. In this approach, the transmitter emits multiple streams in eigendirections of the channel covariance matrix.
Full channel state information If the channel matrix is completely known, singular value decomposition (SVD) precoding is known to achieve the MIMO channel capacity.E. Telatar, Capacity of multiantenna Gaussian channels , European Transactions on Telecommunications, vol. 10, no. 6, pp. 585-595, 1999. 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 user's receiver (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,H. Weingarten, Y. Steinberg, and S. Shamai, 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. but linear precoding approaches usually achieve reasonable performance with much lower complexity. Linear precoding strategies include maximum ratio transmission (MRT),T. Lo, Maximum ratio transmission, IEEE Transactions on Communications, vol. 47, no. 10, pp. 1458–1461, 1999. zero-forcing (ZF) precoding,M. Joham, W. Utschick, and J. Nossek, Linear transmit processing in MIMO communications systems, IEEE Transactions on Signal Processing, vol. 53, no. 8, pp. 2700–2712, 2005. and transmit Wiener precoding. There are also precoding strategies tailored for low-rate feedback of channel state information, for example random beamforming.M. Sharif and B. Hassibi, On the Capacity of MIMO Broadcast Channels With Partial Side Information, IEEE Transactions on Information Theory, vol. 51, no. 2, pp. 506-522, 2005. 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. This can be viewed as a multi-objective optimization problem where each objective corresponds to maximization of the capacity of one of the users. 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.
Linear precoding with full channel state information
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. 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.E. Björnson, R. Zakhour, D. Gesbert, B. Ottersten, 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.
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) and zero-forcing (ZF)N. Jindal, MIMO Broadcast Channels with Finite Rate Feedback, IEEE Transactions on Information Theory, vol. 52, no. 11, pp. 5045–5059, 2006. precoding. MRT only maximizes the signal gain at the intended user. MRT is close-to-optimal in noise-limited systems, where the inter-user interference is negligible compared to the noise. ZF precoding aims at nulling the inter-user interference, at the expense of losing some signal gain. ZF precoding can achieve a performance close to the sum capacity when the number of users is large or the system is interference-limited (i.e., the noise is weak compared to the interference). A balance between MRT and ZF is obtained by the so-called regularized zero-forcingB. C. B. Peel, B. M. Hochwald, and A. L. Swindlehurst, 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. (also known as signal-to-leakage-and-interference ratio (SLNR) beamformingM. Sadek, A. Tarighat, and A. Sayed, A leakage-based precoding scheme for downlink multi-user MIMO channels, IEEE Transactions on Wireless Communications, vol. 6, no. 5, pp. 1711–1721, 2007. and transmit Wiener filtering) All of these heuristic approaches can also be applied to receivers that have multiple antennas.
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.T. E. Bogale and L. Vandendorpe, 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. 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 T. E. Bogale and L. Vandendorpe, 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. and T. E. Bogale and L. Vandendorpe, 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. to get sub-optimal solution for weighted sum rate optimization.
Note that the optimal linear precoding can be computed using monotonic optimization algorithms,W. Utschick and J. Brehmer, Monotonic optimization framework for coordinated beamforming in multicell networks, IEEE Transactions on Signal Processing, vol. 60, no. 4, pp. 1899–1909, 2012.E. Björnson, G. Zheng, M. Bengtsson, and B. Ottersten, Robust monotonic optimization framework for multicell MISO systems, IEEE Transactions on Signal Processing, vol. 60, no. 5, pp. 2508–2523, 2012. 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 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). Quantization and feedback of channel state information is based on vector quantization, and codebooks based on Grassmannian line packing have shown good performance.D.J. Love, R.W. Heath, and T. Strohmer, Grassmannian Beamforming for Multiple-Input Multiple-Output Wireless Systems, IEEE Transactions on Information Theory, vol. 49, no. 10, pp. 2735–2747, 2003.