Precoding
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Precoding is a generalization of
beamforming Beamforming or spatial filtering is a signal processing technique used in sensor arrays for directional signal transmission or reception. This is achieved by combining elements in an antenna array in such a way that signals at particular angles e ...
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. 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 Multi-user MIMO (MU-MIMO) is a set of multiple-input and multiple-output (MIMO) technologies for multipath wireless communication, in which multiple users or terminals, each radioing over one or more antennas, communicate with one another. In cont ...
, the data streams are intended for different users (known as SDMA) and some measure of the total
throughput Network throughput (or just throughput, when in context) refers to the rate of message delivery over a communication channel, such as Ethernet or packet radio, in a communication network. The data that these messages contain may be delivered ov ...
(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 In wireless communications, channel state information (CSI) is the known channel properties of a communication link. This information describes how a signal propagates from the transmitter to the receiver and represents the combined effect of, for ...
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 in Simple Words

Precoding is a technique that exploits transmit diversity by weighting the information stream, i.e. the transmitter sends the coded information to the receiver to achieve pre-knowledge of the channel. The receiver is a simple detector, such as a matched filter, and does not have to know the channel state information. This technique will reduce the corrupted effect of the communication channel. For example you are sending the information s and it will pass through the channel, h, and add Gaussian noise, n. The received signal at the receiver front-end will be r = sh + n; The receiver will have to know the information about h and n. It will suppress the effect of n by increasing SNR, but what about h? It needs information about the channel, h, and this will increase the complexity. The receiver (mobile units) has to be simple for many reasons like cost or size of mobile unit. So, the transmitter (the base station) will do the hard work and predict the channel. Let us call the predicted channel h_ and for a system with precoder the information will be coded: . The received signal will be r = \left(\frac\right) s + n. If your prediction is perfect, h_ = h and r = s + n and it turns out to be the detection problem in Gaussian channels which is simple. To prevent a potential misunderstanding here, precoding does not cancel out the impact of the channel, but it aligns the vector containing the transmit symbols (i.e. transmit vector) with the eigenvector(s) of the channel. In simple terms, it transforms the transmit symbols' vector in such a way that the vector reaches the receiver in the strongest form that is possible in the given channel. Why do they call it "coding"? It is a preprocessing technique that performs transmit diversity and it is similar to equalization, but the main difference is that you have to optimize the precoder with a decoder. Channel equalization aims to minimize channel errors, but the precoder aims to minimize the error in the receiver output.


Precoding for Point-to-Point MIMO Systems

In point-to-point multiple-input multiple-output (
MIMO In radio, multiple-input and multiple-output, or MIMO (), is a method for multiplying the capacity of a radio link using multiple transmission and receiving antennas to exploit multipath propagation. MIMO has become an essential element of wir ...
) systems, a transmitter equipped with multiple antennas communicates with a receiver that has multiple antennas. Most classic precoding results assume
narrowband Narrowband signals are signals that occupy a narrow range of frequencies or that have a small fractional bandwidth. In the audio spectrum, narrowband sounds are sounds that occupy a narrow range of frequencies. In telephony, narrowband is usua ...
, 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 In telecommunications, orthogonal frequency-division multiplexing (OFDM) is a type of digital transmission and a method of encoding digital data on multiple carrier frequencies. OFDM has developed into a popular scheme for wideband digital commun ...
. The precoding strategy that maximizes the throughput, called channel capacity, depends on the
channel state information In wireless communications, channel state information (CSI) is the known channel properties of a communication link. This information describes how a signal propagates from the transmitter to the receiver and represents the combined effect of, for ...
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 In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is related ...
(SVD) precoding is known to achieve the MIMO channel capacity. 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 Multi-user MIMO (MU-MIMO) is a set of multiple-input and multiple-output (MIMO) technologies for multipath wireless communication, in which multiple users or terminals, each radioing over one or more antennas, communicate with one another. In cont ...
, a multi-antenna transmitter communicates simultaneously with multiple receivers (each having one or multiple antennas). This is known as
space-division multiple access Space-division multiple access (SDMA) is a channel access method based on creating parallel spatial pipes (focused signal beams) using advanced antenna technology next to higher capacity pipes through spatial multiplexing and/or diversity, by ...
(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 Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handled ...
of
channel state information In wireless communications, channel state information (CSI) is the known channel properties of a communication link. This information describes how a signal propagates from the transmitter to the receiver and represents the combined effect of, for ...
, 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 In telecommunications, dirty paper coding (DPC) or Costa precoding is a technique for efficient transmission of digital data through a channel subjected to some interference known to the transmitter. The technique consists of precoding the data ...
(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 Multi-objective optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, multiattribute optimization or Pareto optimization) is an area of multiple criteria decision making that is concerned with ...
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 In computing, scheduling is the action of assigning ''resources'' to perform ''tasks''. The ''resources'' may be processors, network links or expansion cards. The ''tasks'' may be threads, processes or data flows. The scheduling activity is c ...
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. 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 and to get sub-optimal solution for weighted sum rate optimization. Note that the optimal linear precoding can be computed using monotonic optimization algorithms, 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 In wireless communications, channel state information (CSI) is the known channel properties of a communication link. This information describes how a signal propagates from the transmitter to the receiver and represents the combined effect of, for ...
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 that 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 Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to the noise power, often expressed in deci ...
(in dB). Quantization and feedback of channel state information is based on
vector quantization Vector quantization (VQ) is a classical quantization technique from signal processing that allows the modeling of probability density functions by the distribution of prototype vectors. It was originally used for data compression. It works by di ...
, 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.
Other precoding strategies have been developed for the case with very low channel feedback rates. Random beamforming (or opportunistic beamformingP. Viswanath, D. N. C. Tse, Member, and R. Laroia
Opportunistic Beamforming Using Dumb Antennas
IEEE Transactions on Information Theory, vol. 48, no. 6, pp. 1277–1294, 2002.
) 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 spatially correlated environments, the long-term channel statistics can be combined with low-rate feedback to perform multi-user precoding. 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. 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.


DPC or DPC-like nonlinear precoding

Dirty paper coding In telecommunications, dirty paper coding (DPC) or Costa precoding is a technique for efficient transmission of digital data through a channel subjected to some interference known to the transmitter. The technique consists of precoding the data ...
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 In wireless communications, channel state information (CSI) is the known channel properties of a communication link. This information describes how a signal propagates from the transmitter to the receiver and represents the combined effect of, for ...
is required everywhere to achieve the weighted sum capacity. This category includes Costa precoding, Tomlinson-Harashima precoding and the vector perturbation technique.


Mathematical Description


Description of Point-to-Point MIMO

The standard
narrowband Narrowband signals are signals that occupy a narrow range of frequencies or that have a small fractional bandwidth. In the audio spectrum, narrowband sounds are sounds that occupy a narrow range of frequencies. In telephony, narrowband is usua ...
, slowly fading channel model for point-to-point (single-user) MIMO communication is described in the page on
MIMO In radio, multiple-input and multiple-output, or MIMO (), is a method for multiplying the capacity of a radio link using multiple transmission and receiving antennas to exploit multipath propagation. MIMO has become an essential element of wir ...
communication.


Description of Multi-user MIMO

Consider a downlink multi-user MIMO system where a base station with N transmit antennas and K single-antenna users. The channel to user k is described by the N \times 1 vector \mathbf_k of channel coefficients and its ith element describes the channel response between the ith transmit antenna and the receive antenna. The input-output relationship can be described as :y_k = \mathbf_k^H \mathbf+n_k, \quad k=1,2, \ldots, K where \mathbf is the N \times 1 transmitted vector signal, y_k is the received signal, and n_k is the zero-mean unit-variance noise. Under linear precoding, the transmitted vector signal is :\mathbf = \sum_^K \mathbf_i s_i, where s_i is the (normalized) data symbol and \mathbf_i is the N \times 1 linear precoding vector. The signal-to-interference-and-noise ratio (SINR) at user k becomes :\textrm_k = \frac where \sigma_k^2 is the noise variance for channel to user k and the corresponding achievable information rate is \log_2(1+\textrm_k) bits per channel use. The transmission is limited by power constraints. This can, for example, be a total power constraint \sum_^K \, \mathbf_i\, ^2 \leq P where P is the power limit. A common performance metric in multi-user systems is the weighted sum rate :\underset \sum_^K a_k \log_2(1+\textrm_k) for some positive weights a_k that represent the user priority. The weighted sum rate is maximized by weighted MMSE precoding that selects :\mathbf^_k = \sqrt \frac for some positive coefficients q_1,\ldots,q_K (related to the user weights) that satisfy \sum_^K q_i = P and p_i is the optimal power allocation. The suboptimal MRT approach removes the channel inversion and only selects :\mathbf^_k = \sqrt \frac, while the suboptimal ZF precoding makes sure that \mathbf_i^H\mathbf^_k=0 for all i ≠ k and thus the interference can be removed in the SINR expression: :\textrm^_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 N receive antennas. The input-output relationship can be described as :\mathbf = \sum_^ \mathbf_k \sqrt s_k + \mathbf where s_k is the transmitted symbol for user k, q_k is the transmit power for this symbol, \mathbf and \mathbf are the N \times 1 vector of received signals and noise respectively, \mathbf_k is the N \times 1 vector of channel coefficients. If the base station uses linear receive filters to combine the received signals on the N antennas, the SINR for the data stream from user k becomes :\textrm^_k = \frac where \mathbf_k 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: :\mathbf^_k = \frac Observe that the coefficients q_1,\ldots,q_K 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.A. Wiesel, Y.C. Eldar, S. Shamai
Linear precoding via conic optimization for fixed MIMO receivers
IEEE Transactions on Signal Processing, vol. 54, no. 1, pp. 161-176, 2006.
As the downlink precoding problem usually is more difficult to solve, it often useful to first solve the corresponding uplink problem.


Limited feedback precoding

The precoding strategies described above was based on having perfect
channel state information In wireless communications, channel state information (CSI) is the known channel properties of a communication link. This information describes how a signal propagates from the transmitter to the receiver and represents the combined effect of, for ...
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 :y_k = \mathbf_k^H \sum_^K \hat_i s_i +n_k, \quad k=1,2, \ldots, K. In this case, the beamforming vectors are distorted as \hat_i = \mathbf_i + \mathbf_i, where \mathbf_i is the optimal vector and \mathbf_i is the error vector caused by inaccurate channel state information. The received signal can be rewritten as :y_k = \mathbf_k^H \sum_^K \mathbf_i s_i + \mathbf_k^H \sum_^K \mathbf_i s_i+ n_k, \quad k=1,2, \ldots, K where \mathbf_k^H \sum_ \mathbf_i s_i is the additional interference at user k according to the limited feedback precoding. To reduce this interference, higher accuracy in the channel information feedback is required, which in turn reduces the throughput in the uplink.


See also

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802.11n IEEE 802.11n-2009 or 802.11n is a wireless-networking standard that uses multiple antennas to increase data rates. The Wi-Fi Alliance has also retroactively labelled the technology for the standard as Wi-Fi 4. It standardized support for multipl ...
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Channel state information In wireless communications, channel state information (CSI) is the known channel properties of a communication link. This information describes how a signal propagates from the transmitter to the receiver and represents the combined effect of, for ...
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Cooperative diversity Cooperative diversity is a cooperative multiple antenna technique for improving or maximising total network channel capacities for any given set of bandwidths which exploits user diversity by decoding the combined signal of the relayed signal and t ...
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Space–time code A space–time code (STC) is a method employed to improve the reliability of data transmission in wireless communication systems using multiple transmit antennas. STCs rely on transmitting multiple, redundant copies of a data stream to the ...
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Space–time trellis code Space–time trellis codes (STTCs) are a type of space–time code used in multiple-antenna wireless communications. This scheme transmits multiple, redundant copies of a generalised TCM signal distributed over time and a number of antennas ('s ...
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Spatial multiplexing Spatial multiplexing or space-division multiplexing (often abbreviated SM, SDM or SMX) is a multiplexing technique in MIMO wireless communication, fibre-optic communication and other communications technologies used to transmit independent chan ...
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Zero-forcing precoding Zero-forcing (or null-steering) precoding is a method of spatial signal processing by which a multiple antenna transmitter can null the multiuser interference in a multi-user MIMO wireless communication system. When the channel state information is ...


References

{{reflist Radio resource management