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 thea multiple antenna transmitter can null the multiuser interference signals in [[wireless|wirelessa communications]]. Regularized '''zeromulti-forcinguser precoding'''MIMO iswireless enhancedcommunication processingsystem. to considerWhen the impactchannel onstate ainformation backgroundis [[noise]]perfectly andknown unknownat userthe [[Interference (communication)|interference]]transmitter,<ref>{{cite journal|author1=B.then C.the B.zero-forcing Peelprecoder |author2=B.is M.given Hochwaldby |author3=A. L. Swindlehurstthe [[Moore–Penrose inverse|name-listMoore-style=ampPenrose |title=A vectorpseudo-perturbationinverse]] techniqueof for near-capacity multiantenna multiuser communication - Part I:the channel inversion and regularization|journal=IEEE Trans. Commun.|pages=195–202|volume=53|date=Jan 2005|doi=10.1109/TCOMM.2004.840638|citeseerx=10.1.1.3.923 }}</ref> where the background noise and the unknown user interference can be emphasized in the result of (known) interference signal nullingmatrix.

==Mathematical description==

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 better use of the antenna arrays, we 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 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 system which comprises an <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

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

==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|author=N. Jindal|title=MIMO Broadcast Channels with Finite Rate Feedback|journal=IEEE Transactions on Information Theory|pages=5045–5059|volume=52|issue=11|date=Nov 2006|doi=10.1109/TIT.2006.883550|arxiv=cs/0603065}}</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 onewith canzero rewritemean theand receivedvariance signal<math>\mathbf{E}(|s_i|^2) as= 1</math>.

==Mathematical description==

In a multiple antenna downlink system which comprises an <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, one 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> is orthogonal to all the user channel vectors <math>\mathbf{h}_j</math> when <math>i \neq j</math>. That is,

=== Quantify the feedback amount ===

:<math>\mathbf{w}_i \perp \mathbf{h}_j \quad \mathrm{if} \quad i \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.

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_i} 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.,

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It follows from the above equation that the feedback 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. One, hence, sees 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|author=N. Jindal|title=MIMO Broadcast Channels with Finite Rate Feedback|journal=IEEE Transactions on Information Theory|pages=5045–5059|volume=52|issue=11|date=Nov 2006|doi=10.1109/TIT.2006.883550|arxiv=cs/0603065}}</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==