Precoding

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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.

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 at a given time instant can be described by a single channel matrix and is unaffected by previous transmissions. 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. 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 decomposition (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, 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 M. Sharif and B. Hassib. 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.

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 (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 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 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 P. 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.</ref>) 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.

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 information is required everywhere to achieve the weighted sum rate. This category includes Costa precoding and the vector perturbation technique.

References

Precoding

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