Zero-forcing precoding: Difference between revisions

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Zero-forcing (or null-steering) precoding is a method of spatial signal processing by which a multiple antenna transmitter can null the multiuser interference in a multi-user MIMO wireless communication system.<ref>{{Cite conference |first1=Taesang |last1=Yoo |first2=Andrea J. |last2=Goldsmith |date=2005 |title=Optimality of zero-forcing beamforming with multiuser diversity |book-title=IEEE International Conference on Communications, 2005 |publisher=IEEE |volume=1 |pages=542–546 |doi=10.1109/ICC.2005.1494410 |isbn=978-0-7803-8938-0 |___location=Seoul, Korea (South)}}</ref> When the channel state information is perfectly known at the transmitter, the zero-forcing precoder is given by the pseudo-inverse of the channel matrix. Zero-forcing has been used in [[LTE (telecommunication)|LTE]] mobile networks.<ref>{{Cite journal |last1=Aslan |first1=Yanki |last2=Roederer |first2=Antoine |last3=Fonseca |first3=Nelson |last4=Angeletti |first4=Piero |last5=Yarovoy |first5=Alexander |date=Oct 2021 |title=Orthogonal Versus Zero-Forced Beamforming in Multibeam Antenna Systems: Review and Challenges for Future Wireless Networks |journal=IEEE Journal of Microwaves |volume=1 |issue=4 |pages=879–901 |doi=10.1109/JMW.2021.3109244 |doi-access=free |issn=2692-8388}}</ref>

Zero-forcing (or Null-Steering) precoding is a spatial signal processing by which the multiple antenna transmitter can null multiuser interference signals in [[wireless|wireless communications]]. Regularized zero-forcing precoding is enhanced processing to consider the impact on a background [[noise]] and unknown user [[Interference (communication)|interference]]<ref>{{cite journal|author=B. C. B. Peel, B. M. Hochwald, and A. L. Swindlehurst|title=A vector-perturbation technique for near-capacity multiantenna multiuser communication - Part I: channel inversion and regularization|journal=IEEE Trans. Commun.|pages=195–202|volume=53|date=Jan. 2005|doi=10.1109/TCOMM.2004.840638}}</ref>, where the background noise and the unknown user interference can be emphasized in the result of (known) interference signal nulling.

==Mathematical Descriptiondescription==

In particular, '''Null-Steering''' is a method of [[beamforming]] for [[narrowband]] [[signal processing|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.

In a multiple antenna downlink systemssystem which comprises a <math>N_t</math> transmit antenna access point (AP)points and <math>K</math> single receive antenna users, such that <math>K \leq N_t</math>, the received signal of user <math>k</math> is described as

:<math>y_k = \mathbf{h}_k^T \mathbf{x} + n_k, \quad k=1,2, \ldots, K</math>

==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<ref name="Jindal_ZF">{{cite journal|author=N. Jindal|title=MIMO Broadcast Channels with Finite Rate Feedback|journal=IEEE Trans. Information Theory.|pages=5045–5059|volume=52|issue=11|date=Nov. 2006|doi=10.1109/TIT.2006.883550}}</ref>. 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.

where <math>\mathbf{x} = \sum_{i=1}^K s_i \sqrt{P_i} s_i \mathbf{w}_i</math> is the <math>N_t \times 1</math> vector of transmitted symbols, <math>n_k</math> is the noise signal, <math>\mathbf{h}_k</math> is the <math>N_t \times 1</math> channel vector and <math>\mathbf{w}_i</math> is thesome <math>N_t \times 1</math> linear precoding vector. FromHere <math>(\cdot)^T</math> is the factmatrix thattranspose, each<math>\sqrt{P_i}</math> Beamis generatedthe bysquare ZF-precodingroot isof orthogonaltransmit topower, alland the<math>s_i</math> otheris userthe channelmessage vectors,signal wewith canzero rewritemean theand receivedvariance signal<math>\mathbf{E}(|s_i|^2) as= 1</math>.

==Mathematical Description==

In multiple antenna downlink systems which comprises a <math>N_t</math> transmit antenna access point (AP) and <math>K</math> single receive antenna users, the received signal of user <math>k</math> is described as

The above signal model can be more compactly re-written as

:<math>y_k = \mathbf{h}_k^T\mathbf{x}+n_k, \quad k=1,2, \ldots, K</math>

:<math>y_k \mathbf{y} = \mathbf{hH}_k^T \sum_mathbf{i=1W}^K s_i P_i \mathbf{wD}_i+n_k = \mathbf{hs}_k^T s_k P_k+ \mathbf{wn}_k +n_k, \quad k=1,2, \ldots,. K</math>

where <math>\mathbf{x} = \sum_{i=1}^K s_i P_i \mathbf{w}_i</math> is the <math>N_t \times 1</math> vector of transmitted symbols, <math>n_k</math> is the noise signal, <math>\mathbf{h}_k</math> is the <math>N_t \times 1</math> channel vector and <math>\mathbf{w}_i</math> is the <math>N_t \times 1</math> 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

:<math>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</math>

where

For comparison purpose, we describe the received signal model for multiple antenna uplink systems. In the uplink system with a <math>N_r</math> receiver antenna AP and <math>K</math> K single transmit antenna user, the received signal at the AP is described as

:<math>\mathbf{y}</math> =is \sum_{i=1}^{the <math>K} s_i \mathbf{h}_itimes + \mathbf{n}1</math> received signal vector,

where :<math>s_i</math>\mathbf{H} is= the transmitted signal of user <math>i</math>, <math>[\mathbf{nh}</math> is the <math>N_r_1, \times 1</math> noise vectorldots, <math>\mathbf{h}_k_K]</math> is the <math>N_rN_t \times 1K</math> channel vector.matrix,

:<math>\mathbf{W} = [\mathbf{w}_1, \ldots, \mathbf{w}_K]</math> is the <math>N_t \times K</math> precoding matrix,

:<math>\mathbf{D} = \mathrm{diag}(\sqrt{P_1}, \ldots, \sqrt{P_K})</math> is a <math>K \times K</math> diagonal power matrix, and

:<math>\mathbf{s} = [s_1, \ldots, s_K]^T</math> is the <math>K \times 1</math> transmit signal.

A ''zero-forcing precoder'' is defined as a precoder where <math>\mathbf{w}_i</math> intended for user <math>i</math> is orthogonal to every channel vector <math>\mathbf{h}_j</math> associated with users <math>j</math> where <math>j \neq i</math>. That is,

==== 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.,,

:<math>\Deltamathbf{w}_i R =\perp R_\mathbf{ZFh}_j -\quad R_\mathrm{FBif} \leqquad log_2i g\neq j.</math>.

Thus the interference caused by the signal meant for one user is effectively nullified for rest of the users via zero-forcing precoder.

Jindal showed that the required feedback bits of an [[Spatial Correlation|spatially uncorrelated]] channel should be scaled according to SNR of the downlink channel, which is given by<ref name=jindal />:

From the fact that each beam generated by zero-forcing precoder is orthogonal to all the other user channel vectors, one can rewrite the received signal as

:<math>y_k = \mathbf{h}_k^T \sum_{i=1}^K \sqrt{P_i} s_i \mathbf{w}_i + n_k = \mathbf{h}_k^T \mathbf{w}_k \sqrt{P_k} s_k + n_k, \quad k=1,2, \ldots, K</math>

The orthogonality condition can be expressed in matrix form as

:<math>\mathbf{H}^T \mathbf{W} = \mathbf{Q}</math>

where <math>\mathbf{Q}</math> is some <math>K \times K</math> diagonal matrix. Typically, <math>\mathbf{Q}</math> is selected to be an identity matrix. This makes <math>\mathbf{W}</math> the right [[Moore–Penrose inverse|Moore-Penrose pseudo-inverse]] of <math>\mathbf{H}^T</math> given by

:<math>\mathbf{W} = \left( \mathbf{H}^T \right)^+ = \mathbf{H} (\mathbf{H}^T \mathbf{H})^{-1}</math>

Given this zero-forcing precoder design, the received signal at each user is decoupled from each other as

:<math>y_k = \sqrt{P_k} s_k + n_k, \quad k=1,2, \ldots, K.</math>

==== 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.,,

:<math>\Delta R = R_{ZF} - R_{FB} \leq \log_2 g</math> .

Jindal showed that the required feedback bits of ana [[Spatial Correlation|spatially uncorrelated]] channel should be scaled according to SNR of the downlink channel, which is given by:<ref name=jindal"Jindal_ZF" />:

:<math> B = (M-1) \log_2 \rho_{b,m} - (M-1) \log_2 (g-1) </math>

where ''M'', is the number of transmit antennas and <math>\rho_{b,m}</math> is the SNR of the downlink channel.

To feedbackfeed back ''B'' bits though the uplink channel, the throughput performance of the uplink channel should be larger than or equal to 'B'

:<math> b_{FB} \log_2(1+\rho_{FB}) \geq B </math>

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

:<math> 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})} </math>.

Note that differently from the feedback bits case, the required feedback resource is a function of both downlink and uplink chanelchannel conditions. It is reasonable to include the uplink channel status in the calcurationcalculation 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. ConsidedrConsider a case when SNR of the downlink and uplink are proportion such that <math>\rho_{b,m} / \rho_{FB}) = C_{up,dn}</math> is constant and both SNRs are sufficiently high. Then, the feedback resource will be only proportionproportional to the number of transmit antennas

:<math> 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</math>.

It follows from the above equation that the feebackfeedback resource (<math>b_{FB}</math>) is not necessary to scale according to SNR of the downlink channel, which is almost contradict to the case of the feedback bits. WeOne, hence, seesees that the whole systematic analysis can reverse the facts resulted from each reductioned situation.

==Performance==

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.<ref name="Jindal_ZF">{{cite journal |authorfirst=N.Nihar |last=Jindal |title=MIMO Broadcast Channels with Finite Rate Feedback |journal=IEEE Trans.Transactions on Information Theory. |pages=5045–5059 |volume=52|issue=11 |date=Nov. 2006 |doi=10.1109/TIT.2006.883550 |arxiv=cs/0603065|s2cid=265096041 }}</ref>. 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.

==See also==