Precoding

Precoding is generalized beamforming to support multi-layer transmission in MIMO radio systems. Conventional beamforming considers linear single-layer precoding so that 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 , the single-layer beamforming cannot simultaneously maximize the signal level at all of the receive antenna and so precoding is used for multi-layer beamforming. In precoding, the multiple streams of the signals are emitted from the transmit antennas with independent and appropriate weighting such that the link throughtput is maximized at the receiver output.

Precoding for Single-user MIMO

In Single-user MIMO, identity matrix precoding and SVD precoding can be used to achieve the open-loop and closed-loop MIMO capacities, respectively.

Random unitary precoding

Random unitary precoding including identity transformation matrix can achieve the open-loop MIMO capacity where no signaling burden in the reverse link is required.

Optimal unitary precoding (SVD precoding)

SVD precoding has been proven to achieve the (real) channel capacity of MIMO systems at the cost of closed-loop signaling burden.

Precoding for Multi-user MIMO

In the implementation prospective, precoding algorithms for multi-user MIMO can be sub-divided into linear and nonlinear precoding algorithms. Linear precoding can achieve reasonable performance while the complexity is lower than nonlinear approaches. Linear precoding includes unitary precoding and zero-forcing (ZF) precoding. Nonlinear precoding can achieve near optimal capacity at the expense of complexity. 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.

Unitary based precoding

This category includes unitary and semi-unitary precoding both of which are simple extension of (matched filter) SVD precoding in single-user MIMO with the addition of the SDMA-based user scheduling technique. The SDMA-based opportunistic user scheduling technique pairs near orthogonal users to avoid intra-group interferences at the minimal cost of the feedback signaling burden, which results in high performance advantage relative to the single user MIMO. For example, it can increase diversity order to almost the number of transmitter antennas times even with simple linear decoding at the receiver.

ZF based precoding (or pre-DPC precoding)

This category includes zero-forcing and regularized zero-forcing precoding. If the transmitter knows the downlink channel status information almost perfectly, ZF-based precoding can achieve close to the system capacity when the number of users is large. With limited channel status information at the transmitter, ZF-precoding requires the feedback overhead increasement with respect to signal-to-noise-ratio (SNR) to achieve the full multiplexing gain. Hence, inaccurate channel state information at the transmitter may result in the significant loss of the system throughput because of the residual interference among transmit streams.

DPC concept based precoding

Dirty paper coding is a coding technique that pre-cancels known interference without power penalty once the transmitter is assumed to know the interference signal regardless of channels state information knowledge at the receiver. This category includes Costa precoding , Tomlinson-Harashima precoding and the vector perturbation technique.

Mathematical Description

Description for Single-user MIMO

In a Precoded MIMO system with $N_{t}$ transmitter antennas and $N_{r}$ receiver antennas, the input-output relationship can be described as

$\mathbf {y} =\mathbf {HWs} +\mathbf {n}$

where $\mathbf {s} =[s_{1},s_{2},\ldots ,s_{N_{s}}]^{T}$ is the $N_{s}\times 1$ vector of transmitted symbols, $\mathbf {y,n}$ are the $N_{r}\times 1$ vectors of received symbols and noise respectively, $\mathbf {H}$ is the $N_{r}\times N_{t}$ matrix of channel coefficients and $\mathbf {W}$ is the $N_{t}\times N_{s}$ linear precoding matrix. The column dimension $N_{s}$ of $\mathbf {W}$ can be selected smaller than $N_{t}$ which is useful if the system requires $N_{s}\leq N_{t}$ streams.

Description for Multi-user MIMO

In a Precoded MIMO BC system with $N_{t}$ transmitter antennas at AP and a receiver antenna for each user $k$, the input-output relationship can be described as

$y_{k}=\mathbf {h} _{k}^{T}\mathbf {x} +n_{k},\quad k=1,2,\ldots ,K$

where $\mathbf {x} =\sum _{i=1}^{K}s_{i}P_{i}\mathbf {w} _{i}$ is the $N_{t}\times 1$ vector of transmitted symbols, $y_{k}$ and $n_{k}$ are the received symbol and noise respectively, $\mathbf {h} _{k}$ is the $N_{t}\times 1$ vector of channel coefficients and $\mathbf {w} _{i}$ is the $N_{t}\times 1$ linear precoding vector.

For the comparison purpose, we describe the mathematical description of MIMO MAC. In a MIMO MAC system with $N_{r}$ receiver antennas at AP and a transmit antenna for each user $k$ where $k=1,2,\ldots ,K$, the input-output relationship can be described as

$\mathbf {y} =\sum _{i=1}^{K}s_{i}\mathbf {h} _{i}+\mathbf {n}$

where $s_{i}$ is the transmitted symbol for user $i$, $\mathbf {y}$ and $\mathbf {n}$ are the $N_{r}\times 1$ vector of received symbols and noise respectively, $\mathbf {h} _{k}$ is the $N_{r}\times 1$ vector of channel coefficients.

Description for Multi-user MIMO with limited feedback precoding

To achieve the capacity of a multi-user MIMO channel, the accurate channel state information is necessary at the transmitter. However, in real systems, receivers feedback the partial channel state information to the transmitter in order to efficiently use the uplink feedback channel resource, which is the Multi-user MIMO system with limited feedback precoding.

The received signal in MIMO BC with limited feedback precoding is mathematically described as

$y_{k}=\mathbf {h} _{k}^{T}\sum _{i=1}^{K}s_{i}P_{i}{\hat {\mathbf {w} }}_{i}+n_{k},\quad k=1,2,\ldots ,K$

Since the trasmit vector for limited feedback precoding is ${\hat {\mathbf {w} }}_{i}=\mathbf {w} _{i}+\mathbf {e} _{i}$ where $\mathbf {e} _{i}$ is the error vector caused by the limited feedback such as quantization, the received signal can be rewritten as

$y_{k}=\mathbf {h} _{k}^{T}\sum _{i=1}^{K}s_{i}P_{i}\mathbf {w} _{i}+\mathbf {h} _{k}^{T}\sum _{i=1}^{K}s_{i}P_{i}\mathbf {e} _{i}+n_{k},\quad k=1,2,\ldots ,K$

where $\mathbf {h} _{k}^{T}\sum _{i=1}^{K}s_{i}P_{i}\mathbf {e} _{i}$ is the residual interference according to the limited feedback precoding. To reduce this interference, we should use the higher accuracy channel information feedback the amound of which decreases the uplink resource propotianally.

See also

• 802.11n
• MIMO
• Space–time code
• Space–time trellis code
• Spatial multiplexing
• Dirty paper coding (DPC)