Zero-forcing precoding
Encyclopedia
Zero-forcing 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.
(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.
transmit antenna access point (AP) and 
single receive antenna users, the received signal of user 
is described as
where
is the 
vector of transmitted symbols, 
is the noise signal, 
is the 
channel vector and 
is the 
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
For comparison purpose, we describe the received signal model for multiple antenna uplink systems. In the uplink system with a
receiver antenna AP and 
K single transmit antenna user, the received signal at the AP is described as
where
is the transmitted signal of user 
, 
is the 
noise vector, 
is the 
channel vector.
.
Jindal showed that the required feedback bits of an spatially uncorrelated
channel should be scaled according to SNR of the downlink channel, which is given by:
where M, is the number of transmit antennas and
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'
where
is the feedback resource consisted by multiplying the feedback frequency resource and the frequency temporal resource subsequently and 
is SNR of the feedback channel. Then, the required feedback resource to satisfy 
is
.
Note that differently from the feedback bits case, the required feedback resource is function of both downlink and uplink chanel conditions. It is reasonable to include 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
is constant and both SNRs are sufficiently high. Then, the feedback resource will be only proportion to the number of transmit antennas
.
It follows the above equation that the feeback resource (
) is not necessary to scale according 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.
Wireless
Wireless telecommunications is the transfer of information between two or more points that are not physically connected. Distances can be short, such as a few meters for television remote control, or as far as thousands or even millions of kilometers for deep-space radio communications...
. Regularized zero-forcing precoding is enhanced processing to consider the impact on a background noise
Noise
In common use, the word noise means any unwanted sound. In both analog and digital electronics, noise is random unwanted perturbation to a wanted signal; it is called noise as a generalisation of the acoustic noise heard when listening to a weak radio transmission with significant electrical noise...
and unknown user interference
Interference (communication)
In communications and electronics, especially in telecommunications, interference is anything which alters, modifies, or disrupts a signal as it travels along a channel between a source and a receiver. The term typically refers to the addition of unwanted signals to a useful signal...
, 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
Beamforming
Beamforming is a signal processing technique used in sensor arrays for directional signal transmission or reception. This is achieved by combining elements in the array in a way where signals at particular angles experience constructive interference and while others experience destructive...
for narrowband
Narrowband
In radio, narrowband describes a channel in which the bandwidth of the message does not significantly exceed the channel's coherence bandwidth. It is a common misconception that narrowband refers to a channel which occupies only a "small" amount of space on the radio spectrum.The opposite of...
signals
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
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 informationChannel state information
In wireless communications, channel state information refers to known channel properties of a communication link. This information describes how a signal propagates from the transmitter to the receiver and represents the combined effect of, for example, scattering, fading, and power decay with...
(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
Channel state information
In wireless communications, channel state information refers to known channel properties of a communication link. This information describes how a signal propagates from the transmitter to the receiver and represents the combined effect of, for example, scattering, fading, and power decay with...
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 transmit antenna access point (AP) and single receive antenna users, the received signal of user is described aswhere
is the vector of transmitted symbols, is the noise signal, is the channel vector and is the 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 asFor comparison purpose, we describe the received signal model for multiple antenna uplink systems. In the uplink system with a
receiver antenna AP and K single transmit antenna user, the received signal at the AP is described as where
is the transmitted signal of user , is the noise vector, is the channel vector.Quantify the feedback amount
Quantify the amount of the feedback resource required to maintain at least a given throughput performance gap between zero-forcing with perfect feedback and with limited feedback, i.e.,,.Jindal showed that the required feedback bits of an spatially uncorrelated
Spatial Correlation
Theoretically, the performance of wireless communication systems can be improved by having multiple antennas at the transmitter and the receiver. The idea is that if the propagation channels between each pair of transmit and receive antennas are statistically independent and identically...
channel should be scaled according to SNR of the downlink channel, which is given by:
where M, is the number of transmit antennas and
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'
where
is the feedback resource consisted by multiplying the feedback frequency resource and the frequency temporal resource subsequently and is SNR of the feedback channel. Then, the required feedback resource to satisfy is.Note that differently from the feedback bits case, the required feedback resource is function of both downlink and uplink chanel conditions. It is reasonable to include 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
is constant and both SNRs are sufficiently high. Then, the feedback resource will be only proportion to the number of transmit antennas.It follows the above equation that the feeback resource (
) is not necessary to scale according 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.External links
- Schelkunoff Polynomial Method (Null-Steering) www.antenna-theory.com