Zero-forcing precoding

Zero-forcing (or Null-Steering) precoding is a spatial signal processing by which the multiple antenna transmitter can null multiuser interference signals in wireless communications. Regularized zero-forcing precoding is enhanced processing to consider the impact on a background noise and unknown user interference, where the background noise and the unknown user interference can be emphasized in the result of (known) interference signal nulling.

In particular, Null-Steering is a method of beamforming for narrowband signals where we want to have a simple way of compensating delays of receiving signals from a specific source at different elements of the antenna array. In general to make use of the antenna arrays, we better to sum and average the signals coming to different elements, but this is only possible when delays are equal. Otherwise we first need to compensate the delays and then to sum them up. To reach this goal, we may only add the weighted version of the signals with appropriate weight values. We do this in such a way that the frequency domain output of this weighted sum produces a zero result. This method is called null steering. The generated weights are of course related to each other and this relation is a function of delay and central working frequency of the source.

Performance of Zero-forcing Precoding

If the transmitter knows the downlink channel state information (CSI) perfectly, ZF-precoding can achieve almost the system capacity when the number of users is large. On the other hand, with limited channel state information at the transmitter (CSIT) the performance of ZF-precoding decreases depending on the accuracy of CSIT. ZF-precoding requires the significant feedback overhead with respect to signal-to-noise-ratio (SNR) so as to achieve the full multiplexing gain. Inaccurate CSIT results in the significant throughput loss because of residual multiuser interferences. Multiuser interferences remain since they can not be nulled with beams generated by imperfect CSIT.

Mathematical Description

In multiple antenna downlink systems which comprises a $N_{t}$ transmit antenna access point (AP) and $K$ single receive antenna users, the received signal of user $k$ is 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, $n_{k}$ is the noise signal, $\mathbf {h} _{k}$ is the $N_{t}\times 1$ channel vector and $\mathbf {w} _{i}$ is the $N_{t}\times 1$ linear precoding vector. From the fact that each Beam generated by ZF-precoding is orthogonal to all the other user channel vectors, we can rewrite the received signal as

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

For comparison purpose, we describe the received signal model for multiple antenna uplink systems. In the uplink system with a $N_{r}$ receiver antenna AP and $K$ K single transmit antenna user, the received signal at the AP is described as

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

where $s_{i}$ is the transmitted signal of user $i$, $\mathbf {n}$ is the $N_{r}\times 1$ noise vector, $\mathbf {h} _{k}$ is the $N_{r}\times 1$ channel vector.

See also

• Channel state information
• Precoding
• MIMO

External links

• Schelkunoff Polynomial Method (Null-Steering) www.antenna-theory.com