Precoding is a generalization of beamforming to support multi-stream (or multi-layer) transmission in multi-antenna wireless communications.
[1] 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.
[4] In this approach, the transmitter emits multiple streams in eigendirections of the channel covariance matrix.
[5] 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.
[8] There are also precoding strategies tailored for low-rate feedback of channel state information, for example random beamforming.
This can be viewed as a multi-objective optimization problem where each objective corresponds to maximization of the capacity of one of the users.
Thus, precoding can be interpreted as finding the optimal balance between achieving strong signal gain and limiting inter-user interference.
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-forcing[12] (also known as signal-to-leakage-and-interference ratio (SLNR) beamforming[13] and transmit Wiener filtering[8]) All of these heuristic approaches can also be applied to receivers that have multiple antennas.
On the other hand, duality approach also considered in [15] and [16] to get sub-optimal solution for weighted sum rate optimization.
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.
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).
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.
In spatially correlated environments, the long-term channel statistics can be combined with low-rate feedback to perform multi-user precoding.
In order to achieve multiuser diversity and apply zero-forcing precoding, the CSI of all users are required at the base station.
Only the transmitter needs to know this interference, but full channel state information is required everywhere to achieve the weighted sum capacity.
for all i ≠ k and thus the interference can be removed in the SINR expression: 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
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: Observe that the coefficients
This important relationship between downlink precoding and uplink receive filtering is known as the uplink-downlink duality.
The precoding strategies described above was based on having perfect channel state information 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.
The received signal in multi-user MIMO with limited feedback precoding is mathematically described as In this case, the beamforming vectors are distorted as