Zero-forcing Beamforming (ZF-BF) is a spatial signal processing in multiple antenna wireless devices. For downlink, the ZF-BF algorithm allows a transmitter to send data to desired users together with nulling out the directions to undesired users and for uplink, ZF-BF receives from the desired users together with nulling out the directions from the interference users.

The concept of interference users in the receive mode is information theoretically dual to undesired users in the transmit mode.

## Literature ReviewEdit

This category summarizes techniques of zero-forcing and regularized zero-forcing precoding^{[1]}. If the transmitter knows the downlink channel status information perfectly, ZF-based precoding can achieve close to the optimal capacity especially when the number of users is sufficient. With limited channel status information at the transmitter, ZF-BF requires the amount of feedback overhead proportional to the average signal-to-noise-ratio (SNR) to achieve the full multiplexing gain^{[2]}. Hence, inaccurate channel state information at the transmitter may suffer the significant performance loss of the system throughput because of the interference among transmit streams is remained.

## System Model for Multi-user BeamformingEdit

### Downlink and uplink signal modelsEdit

In the downlink multi-user beamforming system with $ N_t $ transmitter antennas at an access point and a receiver antenna for each user $ k $, the received signals can be written 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, $ n_k $ is the scalar value of the noise symbol, $ \mathbf{h}_k $ is the $ N_t \times 1 $ vector of downlink channel coefficients and $ \mathbf{w}_i $ is the $ N_t \times 1 $ linear precoding vector.

In the uplink multi-user beamforming system with $ N_r $ receiver antennas at AP and a transmit antenna for each user $ k $ where $ k=1,2, \ldots, K $, the received signal can be written 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{n} $ is the $ N_r \times 1 $ vector of noise symbols, and $ \mathbf{h}_k $ is the $ N_r \times 1 $ vector of uplink channel coefficients.

### Beamforming vectorsEdit

ZFBF lets each beamforming vector $ \mathbf{w}_i^o \in \mathcal{C}^{M \times 1} $ for arbitrary user $ i $ be orthogonal to other users' accurate channel state information vectors, i.e., $ \{ \mathbf{h}_j \},~j \neq i $.

The beamforming vector obtained using the perfect CSIT is denoted as $ \mathbf{w}_m^o $.

If the perfect channel state information at the transmitter (CSIT) is assumed, the beamforming vectors are chosen to be the normalized rows of the inverse of the channel matrix $ [\mathbf{h}_{1}, \mathbf{h}_{2}, \cdots, \mathbf{h}_{M}] $. Notice that if the channel state information is not perfect, the beamforming vectors can not be orthogonal to the real channel vectors.

In order to consider more general cases, we discuss the beamforming vector obtained using the limited feedback channel information as $ \mathbf{w}_m $.

## Throughput AnalysisEdit

The SNR of transmit stream $ k $ in the zero-forcing beam forming systems is given by

- $ \gamma_k = \frac{1}{[(\mathbf{I} + \rho \mathbf{H}_k^H \mathbf{H}_k)^{-1}]_{k,k}} - 1 $

Assuming $ \mathbf{H}_k $ is a $ M\times N $ Wishart matrix with $ M < N $, the distribution of $ \gamma_k $ is given by

- $ F_{\gamma_k}(z) = 1 - \frac{e^{-\frac{z}{\rho}}}{(1+z)^{M-1}}L(z) $

where

- $ L(z) = \sum_{n=1}^N \frac{ \sum_{i=0}^{N-n} ( M - 1, i) z^i }{(n-1)!} \left( \frac{z}{\rho} \right)^{n-1} $

## ReferencesEdit

- ↑ 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.
- ↑ N. Jindal (Nov. 2006). "MIMO Broadcast Channels with Finite Rate Feedback". IEEE Trans. Information Theory. 52: 5045–5059. doi:10.1109/TIT.2006.883550.