**Orignal source from Wikipedia: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 ^{[1]}, 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 order to maximize the throughput performance of a multiple receive antenna system. In precoding, the multiple streams of the signals are emitted from the transmit antennas with independent and appropriate weighting per each antenna such that the link throughtput is maximized at the receiver output.

## Single-user MIMO PrecodingEdit

*Main article: Single-user MIMO Precoding*

The capacity of the single-user MIMO system can be achieve by unitary precoding at the transmitter. In particular, static random unitary matrix and adaptive SVD precoding must be used to achieve the open-loop and closed-loop MIMO capacities, respectively. The precoding vectors for a single user MIMO system are generally determined based on the downlink channel of one corresponding user, while the precoding vectors for a multiuser MIMO system are generally determined based on the downlink channels of a set of candidate users.

### Static random unitary precodingEdit

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

### Adaptive SVD precodingEdit

The SVD based precoding approach is known to achieve the (real) channel capacity of MIMO systems at the expense of closed-loop signaling burden^{[2]}.

## Multi-user MIMO PrecodingEdit

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 PrecodingEdit

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.

According to precoding forms the following category is given by:

- Strict Unitary Precoding
- Semi-Unitary Precoding
- Rotated Unitary Precoding

According to CQI forms, the following category is given by

- SU-MIMO Rank 1 CQI
- Hybrid CQI or Dynamic CQI covering both SU-MIMO Rank 1 CQI and Unitary full rank CQI
- Unitary full rank CQI

### Zero-forcing precoding Edit

This category includes zero-forcing and regularized zero-forcing precoding^{[3]}. 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^{[4]}. 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 or DPC-like precodingEdit

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 ^{[5]}, Tomlinson-Harashima precoding^{[6]}^{[7]} and the vector perturbation technique^{[8]}.

## Mathematical DescriptionEdit

### Description for Single-user MIMOEdit

We consider a Precoded MIMO system with $ N_t $ transmit antennas and $ N_r $ receive antennas. The received signal 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 MIMOEdit

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 Edit

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 transmit 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 which results in decreasing the uplink resource.

#### Qantify the feedback amount Edit

Quantify the amount of the feedback resource required to maintain at least a given throughput performance gap between zeroforcing with perfect feedback and with limited feedback, i.e.,,

- $ \Delta R = R_{ZF} - R_{FB} \leq log_2 b $.

Jindal showed that the required feedback bits as the required resource should be scaled acording to SNR of the downlink channel, which is given by^{[4]}:

- $ B = (M-1) \log_2 \rho_{b,m} - (M-1) \log_2 (b-1) $

where *M*, is the number of transmit antennas and $ \rho_{b,m} $ is SNR of the downlink channel.

To feedback *B* bits though uplink channel, the throughput performance of the uplink channel should be larger than or equal to *B *

- $ b_{FB} \log_2(1+\rho_{FB}) \geq B $

where $ b = \Omega_{FB} T_{FB} $ is the feedback resource consisted by multiplying the feedback frequency resource and the frequency temporal resource subsequently and $ \rho_{FB} $ is SNR of the feedback channel. Then, the required feedback resource to satisfy $ \Delta R \leq \log_2 g $ is

- $ b_{FB} \geq \frac{B}{\log_2(1+\rho_{FB})} = \frac{(M-1) \log_2 \rho_{b,m} - (M-1) \log_2 (g-1)}{\log_2(1+\rho_{FB})} $.

Note that differently from the feedback bits case, the required feedback resource is function of both downlink and uplink chanel conditions. It is resonable to incldue the uplink channel status in the calcuration of the feedback resource since the uplink channel status determines the capacity, i.e., bits/second per unit frequency band (Hz), of the feedback link. Considedr a case when SNR of the downlink and uplink are proportion such that $ \rho_{b,m} / \rho_{FB}) = C_{up,dn} $ is constant and both SNRs are sufficiently high. Then, the feedback resource will be only proportion to the number of transmit antennas

- $ b_{FB,min}^* = \lim_{\rho_{FB} \to \infty } \frac{(M-1) \log_2 \rho_{b,m} - (M-1) \log_2 (g-1)}{\log_2(1+\rho_{FB})} = M - 1 $.

It follows the above equation that the feeback resource ($ b_{FB} $) is not necessary to scale acording to SNR of the downlink channel, which is almost contradict to the case of the feedback bits. We, hence, see that the whole systematic analysis can reverse the facts resulted from each reductioned situation.

### Performance AnalysisEdit

We introduce one example of the performance analysis when precoding is used for multiple antenna communication systems. The analysis use beta distribution to obtain performance analytically. In multiple antenna systems, the distribution of two vector production is used for the performance estimation in frequent. Assume that **s** and **v** are vectors the (*M*-1)-dimensional nullspace of **h** with isotropic i.i.d. where **s**, **v** and **h** are in *C*^{M} and the elements of **h** are i.i.d complex Gaussian random values. Then, the production of **s** and **v** with absolute of the result |**s**^{H}**v**| is beta(1,*M*-2) distributed.

## ReferencesEdit

- ↑ Gerard J. Foschini and Michael. J. Gans (January 1998). "On limits of wireless communications in a fading environment when using multiple antennas". Wireless Personal Communications 6 (3): 311–335. doi:10.1023/A:1008889222784.
- ↑ E. Telatar (June 1995.). "Capacity of multiantenna gaussian channels". AT&T Bell Laboratories, Tech. Memo.
- ↑ B. C. B. Peel, B. M. Hochwald, and A. L. Swindlehurst (Jan. 2005). "A vector-perturbation technique for near-capacity multiantenna multiuser communication - Part I: channel inversion and regularization". IEEE Trans. Commun. 53: 195–202. doi:10.1109/TCOMM.2004.840638.
- ↑
^{4.0}^{4.1}N. Jindal (Nov. 2006). "MIMO Broadcast Channels with Finite Rate Feedback". IEEE Trans. Information Theory. 52: 5045–5059. doi:10.1109/TIT.2006.883550. - ↑ M. Costa (January 2007). "Writing on dirty paper". IEEE Trans. Information Theory 29: 439–441. doi:10.1109/TIT.1983.1056659.
- ↑ M. Tomlinson (Mar. 1971). "New automatic equalizer employing modulo arithmetic". Electron. Lett. 7: 138–139. doi:10.1049/el:19710089.
- ↑ H. Harashima and H. Miyakawa (Aug. 1972). "Matched-transmission technique for channels with intersymbol interference". IEEE Trans. Commun. COM-20: 774–780.
- ↑ B. M. Hochwald, C. B. Peel, and A. L. Swindlehurst (March 2005.). "A vector-perturbation technique for near-capacity multiantenna multiuser communication - Part II: Perturbation". IEEE Trans. Commun. 53: 537–544. doi:10.1109/TCOMM.2004.841997.

## External linksEdit

- C. B. Chae, D. Mazzarese, N. Jindal, and R. W. Heath, Jr., Coordinated Beamforming with Limited Feedback in the MIMO Broadcast Channel, IEEE Jour. on Selected Topics in Comm., 2008.