Precoding
Encyclopedia
Precoding is a generalization of beamforming
to support multi-layer transmission in multi-antenna
wireless communications. In conventional single-layer 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-layer beamforming cannot simultaneously maximize the signal level at all of the receive antennas. Thus, in order to maximize the throughput in multiple receive antenna systems, multi-layer beamforming is 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) 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 co-user interference in multi-user systems.
) 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.
(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.
, 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, but linear precoding approaches usually achieve reasonable performance with much lower complexity. Linear precoding strategies include MMSE precoding and the simplified zero-forcing
(ZF) precoding. There are also precoding strategies tailored for low-rate feedback
of channel state information
, for example random beamforming. 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.
Finding the optimal MMSE precoding is often difficult, leading to approximate approaches that concentrate on either the numerator or denominator of the mentioned ratio; that is, maximum ratio transmission (MRT) and zero-forcing
(ZF) precoding. MRT only maximizes the signal gain at the intended user. MRT is close-to-optimal in noise-limited systems, where the co-user interference is negligible compared to the noise. ZF precoding aims at nulling the co-user interference, at the expense of losing some signal gain. ZF precoding can achieve close to the system capacity when the number of users is large or the system is interference-limited (i.e., the noise is weak compared to the interference). If receivers have multiple antennas, then regularized zero-forcing precoding has the corresponding properties.
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
(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.
Other precoding strategies have been developed for the case with very low channel feedback rates. Random beamforming (or opportunistic beamforming) 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 weights 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 spatially correlated
environments, the long-term channel statistics can be combined with low-rate feedback to perform SDMA 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.
is required everywhere to achieve the weighted sum rate. This category includes Costa precoding, Tomlinson-Harashima precoding and the vector perturbation technique.
, slowly-fading
channel model for point-to-point (single-user) MIMO communication is described in the page on MIMO communication.
transmit antennas and 
single-antenna users. The channel to user 
is described by the 
vector 
of channel coefficients and its 
th element describes the channel response between the 
th transmit antenna and the receive antenna. The input-output relationship can be described as
where
is the 
transmitted vector signal, 
is the received signal, and 
is the zero-mean unit-variance noise.
Under linear precoding, the transmitted vector signal is
, where 
is the (normalized) data symbol and 
is the 
linear precoding vector. The signal-to-noise-and-interference ratio (SINR) at user 
becomes
and the corresponding communication rate is
bits per channel use. The transmission is limited by a power constraint, for example 
.
A common performance measure in multi-user systems is the weighted sum rate
for some positive weights
that represent the user priority. The weighted sum rate is maximized by MMSE precoding that selects
for some positive coefficients
(related to the user weights) that satisfy 
and 
is the optimal power allocation.
The suboptimal MRT approach removes the channel inversion and only selects
while the suboptimal ZF precoding
makes sure that
for all i ≠ k and thus the interference can be removed in the SINR expression:
receive antennas. The input-output relationship can be described as
where
is the transmitted symbol for user 
, 
is the transmit power for this symbol, 
and 
are the 
vector of received signals and noise respectively, 
is the 
vector of channel coefficients. If the base station uses linear receive filters to combine the received signals on the 
antennas, the SINR for the data stream from user 
becomes
where
is the unit-norm receive filter for this user. Observe that the only difference is that the indices are switched in the interference term. Remarkably, the optimal receive filters are the same as the MMSE beamforming vectors, up to a scaling factor:
Observe that the coefficients
that was used in the MMSE beamforming are exactly the optimal power coefficients in the uplink (that maximize the weighted sum rate). This important relationship between downlink precoding and uplink receive filtering is known as the uplink-downlink duality. As the downlink precoding problem usually is more difficult to solve, it often useful to first solve the corresponding uplink problem.
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
In this case, the beamforming vectors are distorted as
, where 
is the optimal vector and 
is the error vector caused by inaccurate channel state information. The received signal can be rewritten as
where
is the residual interference 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.
Beamforming
Beamforming is a signal processing technique used in sensor arrays for directional signal transmission or reception. This is achieved by combining elements in the array in a way where signals at particular angles experience constructive interference and while others experience destructive...
to support multi-layer transmission in multi-antenna
MIMO
In radio, multiple-input and multiple-output, or MIMO , is the use of multiple antennas at both the transmitter and receiver to improve communication performance. It is one of several forms of smart antenna technology...
wireless communications. In conventional single-layer 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-layer beamforming cannot simultaneously maximize the signal level at all of the receive antennas. Thus, in order to maximize the throughput in multiple receive antenna systems, multi-layer beamforming is 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
In radio, multi-user MIMO is a set of advanced MIMO, multiple-input and multiple-output , technologies that exploit the availability of multiple independent radio terminals in order to enhance the communication capabilities of each individual terminal...
, the data streams are intended for different users (known as SDMA
Space-division multiple access
Space-Division Multiple Access is a channel access method based on creating parallel spatial pipes next to higher capacity pipes through spatial multiplexing and/or diversity, by which it is able to offer superior performance in radio multiple access communication systems...
) and some measure of the total throughput (e.g., the sum performance) is maximized. In point-to-point systems, some of the benefits of precoding can be realized without requiring channel state information
Channel state information
In wireless communications, channel state information refers to 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 example, scattering, fading, and power decay with...
at the transmitter, while such information is essential to handle the co-user interference in multi-user systems.
Precoding for Point-to-Point MIMO Systems
In point-to-point multiple-input multiple-output (MIMOMIMO
In radio, multiple-input and multiple-output, or MIMO , is the use of multiple antennas at both the transmitter and receiver to improve communication performance. It is one of several forms of smart antenna technology...
) systems, a transmitter equipped with multiple antennas communicates with a receiver that has multiple antennas. Most classic precoding results assume narrowband
Narrowband
In radio, narrowband describes a channel in which the bandwidth of the message does not significantly exceed the channel's coherence bandwidth. It is a common misconception that narrowband refers to a channel which occupies only a "small" amount of space on the radio spectrum.The opposite of...
, slowly fading
Fading
In wireless communications, fading is deviation of the attenuation that a carrier-modulated telecommunication signal experiences over certain propagation media. The fading may vary with time, geographical position and/or radio frequency, and is often modelled as a random process. A fading channel...
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
Channel capacity
In electrical engineering, computer science and information theory, channel capacity is the tightest upper bound on the amount of information that can be reliably transmitted over a communications channel...
, depends on the channel state information
Channel state information
In wireless communications, channel state information refers to 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 example, scattering, fading, and power decay with...
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. In this approach, the transmitter emits multiple streams in eigendirections of the channel statistics. As the actual channel realization is unknown at transmitter, interference will appear between the streams.Full channel state information
If the channel matrix is completely known, singular value decompositionSingular value decomposition
In linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....
(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 MIMOMulti-user MIMO
In radio, multi-user MIMO is a set of advanced MIMO, multiple-input and multiple-output , technologies that exploit the availability of multiple independent radio terminals in order to enhance the communication capabilities of each individual terminal...
, 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
Space-Division Multiple Access is a channel access method based on creating parallel spatial pipes next to higher capacity pipes through spatial multiplexing and/or diversity, by which it is able to offer superior performance in radio multiple access communication systems...
(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, but linear precoding approaches usually achieve reasonable performance with much lower complexity. Linear precoding strategies include MMSE precoding and the simplified zero-forcing
Zero-forcing precoding
Zero-forcing precoding is a spatial signal processing by which the multiple antenna transmitter can null multiuser interference signals in wireless communications...
(ZF) precoding. There are also precoding strategies tailored for low-rate feedback
Feedback
Feedback describes the situation when output from an event or phenomenon in the past will influence an occurrence or occurrences of the same Feedback describes the situation when output from (or information about the result of) an event or phenomenon in the past will influence an occurrence or...
of channel state information
Channel state information
In wireless communications, channel state information refers to 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 example, scattering, fading, and power decay with...
, for example random beamforming. Nonlinear precoding is designed based on the concept of dirty paper coding
Dirty paper coding
In telecommunications, dirty paper coding is a technique for efficient transmission of digital data through a channel subjected to some interference known to the transmitter...
(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.
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. Optimal linear precoding is known as MMSE precoding and is simple to characterize 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 means finding the optimal balance between achieving strong signal gain and limiting co-user interference.Finding the optimal MMSE precoding is often difficult, leading to approximate approaches that concentrate on either the numerator or denominator of the mentioned ratio; that is, maximum ratio transmission (MRT) and zero-forcing
Zero-forcing precoding
Zero-forcing precoding is a spatial signal processing by which the multiple antenna transmitter can null multiuser interference signals in wireless communications...
(ZF) precoding. MRT only maximizes the signal gain at the intended user. MRT is close-to-optimal in noise-limited systems, where the co-user interference is negligible compared to the noise. ZF precoding aims at nulling the co-user interference, at the expense of losing some signal gain. ZF precoding can achieve close to the system capacity when the number of users is large or the system is interference-limited (i.e., the noise is weak compared to the interference). If receivers have multiple antennas, then regularized zero-forcing precoding has the corresponding properties.
Linear precoding with limited channel state information
In practice, the channel state informationChannel state information
In wireless communications, channel state information refers to 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 example, scattering, fading, and power decay with...
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
Signal-to-noise ratio is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. It is defined as the ratio of signal power to the noise power. A ratio higher than 1:1 indicates more signal than noise...
(in dB). Quantization and feedback of channel state information is based on vector quantization
Vector quantization
Vector quantization is a classical quantization technique from signal processing which allows the modeling of probability density functions by the distribution of prototype vectors. It was originally used for data compression. It works by dividing a large set of points into groups having...
, and codebooks based on Grassmannian line packing have shown good performance.
Other precoding strategies have been developed for the case with very low channel feedback rates. Random beamforming (or opportunistic beamforming) 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 weights 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 spatially correlated
Spatial Correlation
Theoretically, the performance of wireless communication systems can be improved by having multiple antennas at the transmitter and the receiver. The idea is that if the propagation channels between each pair of transmit and receive antennas are statistically independent and identically...
environments, the long-term channel statistics can be combined with low-rate feedback to perform SDMA 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.
DPC or DPC-like nonlinear precoding
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 informationChannel state information
In wireless communications, channel state information refers to 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 example, scattering, fading, and power decay with...
is required everywhere to achieve the weighted sum rate. This category includes Costa precoding, Tomlinson-Harashima precoding and the vector perturbation technique.
Description of Point-to-Point MIMO
The standard narrowbandNarrowband
In radio, narrowband describes a channel in which the bandwidth of the message does not significantly exceed the channel's coherence bandwidth. It is a common misconception that narrowband refers to a channel which occupies only a "small" amount of space on the radio spectrum.The opposite of...
, slowly-fading
Fading
In wireless communications, fading is deviation of the attenuation that a carrier-modulated telecommunication signal experiences over certain propagation media. The fading may vary with time, geographical position and/or radio frequency, and is often modelled as a random process. A fading channel...
channel model for point-to-point (single-user) MIMO communication is described in the page on MIMO communication.
Description of Multi-user MIMO
Consider a downlink multi-user MIMO system where a base station with transmit antennas and single-antenna users. The channel to user is described by the vector of channel coefficients and its th element describes the channel response between the th transmit antenna and the receive antenna. The input-output relationship can be described aswhere
is the transmitted vector signal, is the received signal, and is the zero-mean unit-variance noise.Under linear precoding, the transmitted vector signal is
, where is the (normalized) data symbol and is the linear precoding vector. The signal-to-noise-and-interference ratio (SINR) at user becomesand the corresponding communication rate is
bits per channel use. The transmission is limited by a power constraint, for example .A common performance measure in multi-user systems is the weighted sum rate
for some positive weights
that represent the user priority. The weighted sum rate is maximized by MMSE precoding that selectsfor some positive coefficients
(related to the user weights) that satisfy and is the optimal power allocation.The suboptimal MRT approach removes the channel inversion and only selects
while the suboptimal ZF precoding
Zero-forcing precoding
Zero-forcing precoding is a spatial signal processing by which the multiple antenna transmitter can null multiuser interference signals in wireless communications...
makes sure that
for all i ≠ k and thus the interference can be removed in the SINR expression:Uplink-downlink duality
For comparison purposes, it is instructive to compare with the uplink MIMO channel where the same single-antenna users transmit to the same base station, having receive antennas. The input-output relationship can be described aswhere
is the transmitted symbol for user , is the transmit power for this symbol, and are the vector of received signals and noise respectively, is the vector of channel coefficients. If the base station uses linear receive filters to combine the received signals on the antennas, the SINR for the data stream from user becomeswhere
is the unit-norm receive filter for this user. Observe that the only difference is that the indices are switched in the interference term. Remarkably, the optimal receive filters are the same as the MMSE beamforming vectors, up to a scaling factor:Observe that the coefficients
that was used in the MMSE beamforming are exactly the optimal power coefficients in the uplink (that maximize the weighted sum rate). This important relationship between downlink precoding and uplink receive filtering is known as the uplink-downlink duality. 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 have perfect channel state informationChannel state information
In wireless communications, channel state information refers to 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 example, scattering, fading, and power decay with...
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
In this case, the beamforming vectors are distorted as
, where is the optimal vector and is the error vector caused by inaccurate channel state information. The received signal can be rewritten aswhere
is the residual interference according to the limited feedback precoding. To reduce this interference, higher accuracy in the channel informationChannel state information
In wireless communications, channel state information refers to 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 example, scattering, fading, and power decay with...
feedback is required, which in turn reduces the throughput in the uplink.
See also
- 802.11n
- Channel state informationChannel state informationIn wireless communications, channel state information refers to 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 example, scattering, fading, and power decay with...
- Cooperative diversityCooperative diversityCooperative 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 the direct signal in wireless multihop networks...
- Space–time codeSpace–time codeA space–time code is a method employed to improve the reliability of data transmission in wireless communication systems using multiple transmit antennas...
- Space–time trellis codeSpace–time trellis codeSpace–time trellis codes are a type of space–time code used in multiple-antenna wireless communications. This scheme transmits multiple, redundant copies of a trellis code distributed over time and a number of antennas . These multiple, 'diverse' copies of the data are used by the receiver to...
- Spatial multiplexingSpatial multiplexingSpatial multiplexing is a transmission technique in MIMO wireless communication to transmit independent and separately encoded data signals, so-called streams, from each of the multiple transmit antennas...
- Zero-forcing precodingZero-forcing precodingZero-forcing precoding is a spatial signal processing by which the multiple antenna transmitter can null multiuser interference signals in wireless communications...