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Non-Regenerative Multi-Antenna Multi-GroupMulti-Way RelayingAditya Umbu Tana Amah1* and Anja Klein2

Abstract

We consider non-regenerative multi-group multi-way (MGMW) relaying. A half-duplex non-regenerative multi-antenna relay station (RS) assists multiple communication groups. In each group, multiple half-duplex nodesexchange messages. In our proposal, the required number of communication phases is equal to the maximumnumber of nodes among the groups. In the first phase, all nodes transmit simultaneously to the RS. Assumingperfect channel state information is available at the RS, in the following broadcast (BC) phases the RS appliestransceive beamforming to its received signal and transmits simultaneously to all nodes. We propose three BCstrategies for the BC phases: unicasting, multicasting and hybrid uni/multicasting. For the multicasting strategy,network coding is applied to maintain the same number of communication phases as for the other strategies. Weaddress transceive beamforming maximising the sum rate of non-regenerative MGMW relaying. Due to the highcomplexity of finding the optimum transceive beamforming maximising the sum rate, we design generalised lowcomplexity transceive beamforming algorithms for all BC strategies: matched filter, zero forcing, minimisation ofmean square error and BC-strategy-aware transceive beamforming. It is shown that the sum rate performance ofnon-regenerative MGMW relaying depends both on the chosen BC strategies and the applied transceivebeamforming at the RS.

Keywords: Multi-way relaying, Non-regenerative, Multi-antenna, Analog network coding, Transceive beamforming

IntroductionTwo-way relaying is a spectrally efficient protocol toestablish bidirectional communication between two half-duplex nodes via a half-duplex relay station (RS) [1-3].It was shown in [1,3] that two-way relaying outperformsthe traditional one-way relaying due to its smaller num-ber of communication resources. In two-way relaying,two communication phases are needed. The first phaseis the multiple access (MAC) phase where the two com-municating nodes send their data streams simulta-neously to the RS. The second phase is the broadcast(BC) phase where the RS sends the processed signalssimultaneously to both nodes. Consequently, the nodesneed to cancel their self-interference.Regarding the signal processing at the RS, it can be

either regenerative, cf. [1,2] or non-regenerative, cf.[1,3]. A regenerative RS regenerates (decodes and re-

encodes) the data streams of all nodes while a non-regenerative RS performs linear signal processing to thereceived signals and transmits the output to the nodes.The use of multiple antennas can improve the spectral

efficiency and/or the reliability of communication net-works [4,5]. For two-way relaying, a multi-antenna RSthat serves one bidirectional pair was considered in[6-8] for a regenerative RS and in [3,9-11] for a non-regenerative RS. For the non-regenerative case, while[3,9] assume multi-antenna nodes, [10,11] assume singleantenna nodes. Their works consider optimal transceivebeamforming at the RS maximising the sum rate as wellas linear transceive beamforming based on Zero Forcing(ZF) [3,9-11], Minimisation of Mean Square Error(MMSE) [3,9,10], Maximisation of Signal to Noise Ratio(MSNR) [3] and Matched Filter (MF) criteria [10,11].Multi-user two-way relaying, where an RS serves mul-

tiple bidirectional pairs, is treated in [12-14] for a regen-erative RS and in [15,16] for a non-regenerative RS. In[12], all bidirectional pairs are separated using CodeDivision Multiple Access. Every two nodes in a

* Correspondence: [email protected] School of Computational Engineering and CommunicationsEngineering Lab, Technische Universität Darmstadt, Darmstadt 64283GermanyFull list of author information is available at the end of the article

Amah and Klein EURASIP Journal on Wireless Communications and Networking 2011, 2011:29http://jwcn.eurasipjournals.com/content/2011/1/29

© 2011 Amah and Klein; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

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bidirectional pair have their own code which is differentfrom the other pairs’ codes. In contrast to [12], havingmultiple antennas at the RS and assuming perfect detec-tion in MAC phase, in [13,14] the separation of thepairs in the BC phase is done spatially using transmitbeamforming employed at the RS. In [15], ZF andMMSE transceive beamforming for multi-user non-regenerative two-way relaying is designed to separatethe nodes. In [16], block-diagonalisation-singular-value-decomposition (BD-SVD) transceive beamforming isdesigned for separating the two-way pairs by extendingBD-SVD transmit beamforming proposed in [17].In applications such as video conference and multi-

player gaming, multiple nodes exchange messages. In[18], the multi-way relay channel is considered, wherean RS assists multiple communication groups. In eachgroup, each member node exchanges messages withother member nodes, but not with other nodes from theother groups. A full-duplex communication is assumedand time division is used to separate the multi-waygroups. The full duplex assumption, however, is not yetpractical and half-duplex nodes and relays are more ofpractical importance [1,19]. Therefore, communicationprotocols for multi-way relaying for half-duplex nodesand a half-duplex RS are needed.Multi-way relaying protocols for one communication

group, where a half-duplex multi-antenna RS assists Nhalf-duplex nodes to exchange messages, is proposed bythe authors of this paper in [20] for a non-regenerativeRS and in [21] for a regenerative RS. The required num-ber of communication phases is only N, consisting ofone MAC phase and N - 1 BC phases. In [20,21], ineach BC phase, the RS sends N data streams to N nodessimultaneously. Thus, each node receives an intendeddata stream from a specific node, while seeing otherdata streams as interference. Nevertheless, the interfer-ence can be canceled by performing successive interfer-ence cancellation or by applying linear transceivebeamforming, e.g., ZF, that nullifies the interference[20]. In [20], it was also shown that instead of transmit-ting N different data streams per BC phase, the RS cantransmit one data stream simultaneously to all nodes ineach BC phase. This data stream is a superposition oftwo different data streams. Consequently, each node hasto perform self- and known-interference cancellation.Since in each BC phase the RS transmits only onesuperposed data stream to all nodes, there is no inter-stream interference and, thus, the performance isimproved.In this paper, we consider non-regenerative multi-

group multi-way (MGMW) relaying. A half-duplexmulti-antenna RS assists L multi-way groups where eachgroup consists of half-duplex single antenna nodes. Weconsider non-regenerative relaying where the RS

performs transceive beamforming. Non-regenerative,compared to regenerative, has three advantages: nodecoding error propagation, no delay due to decodingand deinterleaving, and transparency to the modulationand coding schemes that are used at the nodes [3].In each l-th multi-way group, l ∈ L, L = {1, · · · , L},

there are Nl ≥ 2 nodes that exchange messages. In ourproposal, the required number P of communicationphases is equal to maxl Nl. Our work is a generalisa-tion of some of the above mentioned publications. If L= 1 and N1 = 2, we have a non-regenerative two-wayrelaying as in [3,10,22]. If L > 1 and Nl = 2, ∀l, l ∈ L,we have a non-regenerative multi-user two-way relay-ing as in [15,16], and if L = 1 and N1 ≥ 2, we have anon-regenerative single-group multi-way relaying as in[20].We propose three BC strategies for the BC phases,

namely, unicasting, multicasting and hybrid uni/multi-casting. The proposed strategies are designed in such away that the number of communication phases remainsP = maxl Nl. We derive the sum rate expression fornon-regenerative MGMW relaying with the proposedBC strategies for asymmetric and symmetric traffic. Inasymmetric traffic all nodes in each group may commu-nicate with different rate, while in symmetric traffic allnodes in each group communicate with the same rate.We address the sum rate maximisation which requires

optimum transceive beamforming. Due to the high com-plexity of finding the optimum transceive beamformingmaximising the sum rate, we design generalised lowcomplexity transceive beamforming algorithms for allproposed BC strategies, namely, ZF, MMSE, MF andBC-strategy-aware (BCSA) transceive beamforming.BCSA transceive beamforming is designed by suppres-sing unwanted signals using either block diagonalisation(BD) [17] or regularised BD proposed in [23].This paper is organised as follows. Section II explains

the proposed broadcast strategies and the system modelof non-regenerative MGMW relaying. Section IIIexplains the sum rate expression. The transceive beam-forming algorithms are explained in Section IV. Thesimulation results are given in Section V. Finally, Sec-tion VI provides the conclusion.

NotationsBoldface lower and upper case letters denote vectorsand matrices, respectively, while normal letters denotescalar values. The superscripts (·)T, (·)* and (·)H stand formatrix or vector transpose, complex conjugate, andcomplex conjugate transpose, respectively. The opera-tors modN (x), E{X} and tr{X} denote the modulo N ofx, the expectation and the trace of X, respectively, andCN (0, σ 2) denotes the circularly symmetric zero-meancomplex normal distribution with variance s2.


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Broadcast Strategies And System ModelWe consider L multi-way communication groups. It isassumed that there are no direct links among the nodesand the MGMW communication can only be performedwith the assistance of a half-duplex multi-antenna RSwith M antenna elements. In the lth group, l ∈ L,L = {1, · · · , L}, there are Nl nodes which exchangemessages through an RS. For simplicity of notations, weconsider the same number of nodes in all groups, i.e., Nl= Nmw, ∀l ∈ L. However, the extension to the case ofdifferent numbers of nodes in the groups is straightfor-ward. The total number N of nodes in the network isN =

∑l∈LNl = LNmw.

Assuming that the RS already knows which nodesbelong to which communication group, the RS makesthe indexing of all nodes according to their group mem-bership. Nodes in group one are indexed within the set{0,..., N1 - 1}, nodes in group two are indexed within theset {N1,..., (N1 + N2) - 1}, and so on. In general, it canbe given as follows. The lth group consists of nodes Sil,il ∈ Il, where Il is the set of node indices given byIl = {al, · · · , bl}, with al = (l - 1)Nmw, bl = lNmw - 1.Each node only exchanges messages with the othernodes in its group and each node belongs only to onemulti-way group, i.e., Il ∩ Ik = ∅, ∀l ≠ k andI = ⋃Ll=1 Il = {0, · · ·,N − 1}.A. Broadcast StrategiesIn this subsection, the broadcast strategies for non-regenerative MGMW relaying are described. The num-ber P of communication phases to perform MGMWcommunication is given by the maximum number ofnodes among all groups, i.e., P = maxl Nl = Nmw. In thefirst phase, the MAC phase, all nodes transmit simulta-neously to the RS. In the following P - 1 BC phases, theRS transmits to the nodes. Let p, p ∈ P,P = {2 , . . . ,P}, denote the index of the BC phase. Inpth phase, in group l, receiving node rl ∈ Il is intendedto receive the data stream of transmitting nodetl ∈ Il\{rl}.1) Unicasting StrategyUsing unicasting strategy, in each BC phase, the RStransmits different data streams to different nodes. Eachdata stream is intended only for one receiving node.Consequently, in each BC phase each node sees theother data streams transmitted by the RS to the othernodes as interference. The data stream transmitted fromthe RS to each particular node is changed in each BCphase, such that within P - 1 BC phases, each nodereceives the data streams from all other nodes in itsgroup.The relationship of the parameters p, rl and tl is given

by

tl = al + modNl(rl + p − al − 1). (1)Using such strategy, assuming each node knows its

index and all other nodes’ indices in its group, there isno signalling required in the network. The proposedunicasting strategy is a generalization of the work in[3,10] for L = 1 and N1 = 2, in [15] for L > 1 and Nl =2, ∀l, and in [20] for L = 1 and N1 ≥ 2 with multiplexingtransmission. Figure 1a shows an example of the uni-casting strategy for MGMW relaying with L = 2 com-munication groups and N1 = N2 = 3 nodes.2) Hybrid Uni/Multicasting StrategyFor each served group, one data stream is transmitted toone node exclusively (unicast transmission) and onedata stream is transmitted to the other Nl - 1 nodes(multicast transmission). In each BC phase, the uni-casted data stream is fixed and is transmitted to a differ-ent node in the group. Consequently, the multicasteddata stream has to be changed in each BC phase toensure that each node in each group receives all datastreams of the other nodes in its group within P phases.Compared to the unicasting strategy, intra-group inter-ference in each BC phase is reduced since only two datastreams are transmitted simultaneously.The procedure can be described as follows. For each

group l, the RS chooses one data stream out of Nl datastreams. This data stream will be unicasted to differentnodes in different BC phases. Therefore, in each BCphase, to ensure that each node receives the Nl - 1 datastreams from the other Nl - 1 nodes, the multicasteddata stream is the transmitted data stream of the nodewho will receive the unicasted data stream. In the fol-lowing, we derive the mathematical formulation of theprocedure for hybrid uni/multicasting.In the lth group, given the index tlu ∈ Il of the trans-

mit node whose data stream is unicasted by the RS andthe index tlm ∈ IlM, IlM = Il\{tlu}, of the transmit nodewhose data stream is multicasted, the relationshipbetween rl, tl, and p is defined by

tl ={tlu , if rl = tlmtlm, otherwise

(2)

where

tlm ={(p + al) − 1, for (p + al) ≥ tlu + 2(p + al) − 2, for (p + al) ≤ tlu + 1

. (3)

The relationships in (2) and (3) are defined afterchoosing the data stream to be unicasted for group l, tlu,which remains the same in all Nl - 1 BC phases. In thepth phase, node rl = tlm, whose data stream is multi-casted by the RS, receives the unicasted data streamfrom tlu. The other nodes, rl, rl ∈ Il\{tlm }, receive thedata stream from node tlm which is multicasted by the


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RS to these Nl - 1 nodes. The multicasted data stream ischanged in every BC phase as defined in Equation (3).Using hybrid uni/multicasting strategy, the nodes need

to know which data stream is unicasted and which datastream is multicasted by the RS in the pth phase. How-ever, given (2) and (3), and by choosing the unicasteddata stream from the node with the lowest index, thatis, tlu = al, there is no signalling effort needed. The RS isthen multicasting the data streams in the P - 1 BCphases starting from the lowest index in the setIl\{tlu = al}. In case of one-pair two-way relaying and

multi-user two-way relaying, the hybrid uni/multicastingstrategy is the same as the unicasting strategy. The pro-posed hybrid uni/multicasting strategy is a generalisationof the work in [3,10] for L = 1 and N1 = 2, and in [15]for L > 1 and Nl = 2, ∀l. Figure 1b shows an example ofthe proposed hybrid uni/multicasting strategy forMGMW relaying when L = 2 and N1 = N2 = 3 nodes.3) Multicasting StrategyUsing multicasting strategy, the RS transmits only onedata stream for each served group in each BC phase.The RS transmits x̂vlwl, i.e., the superposition of the data

Figure 1 Multi-group multi-way relaying with L = 2, N1 = N2 = 3: (a) unicasting; (b) hybrid uni/multicasting; (c) multicasting.


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streams of nodes Svl, vl ∈ Il, and Swl, wl ∈ Il\{vl}, in thelth group, to all Nl nodes in group l. Prior to detection,each node has to cancel the self- and known-interfer-ence from each of the received data streams using theavailable side information. The side information can beits own transmitted data stream or a data stream whichhas been decoded in one of the previous BC phases.The general rule for selecting the two data streams foreach group in each BC phase is that we have to ensurethat the data stream of each node in each group isselected at least once.The relationship between rl, tl, and p can be written as

tl ={vl, for rl = wl,wl, otherwise.

(4)

Given the general rule, we may have several options todefine the superposed data stream. However, each optionwill lead to different signaling requirement, since the RShas to inform the nodes about the indices vl and wl ineach BC phase. In this work, we are interested in anoption that does not need any signalling. Therefore, weextend the proposal in [20] to the case of MGMW relay-ing. We always choose vl = al, and, consequently, wl ischanged in each BC phase and is selected successivelybased on the relationship defined by wl = vl + p - 1.Using such relationships, node Srl = al always per-

forms self-interference cancellation, i.e., xvlwl − xvl=al, toobtain all Nl - 1 data streams from other nodes tl = wl,∀wl ∈ Il\{al}. Regarding the other nodes rl ∈ Il\{al},they have to be able to decode xal and, afterwards, usexal to perform known-interference cancellation. Eachnode rl ∈ Il\{al} has to wait until its own data stream issuperposed with xvl=al, that is in the pth phase whichleads to rl = (p + al) -1. In this corresponding pthphase, node rl = (p + al) - 1 performs self-interferencecancellation, i.e., xvlwl − xwl=(p+al)−1 to obtain xvl=al. After-wards, using xvl=al, it performs known-interference can-cellation xvlwl − xvl=al to obtain the other data streamfrom the other nodes tl = wl, ∀wl ∈ Il\{al, rl} received inthe other BC phases. The proposed multicasting strategyis a generalisation of the work in [1,22,24] for L = 1 andN1 = 2, in [16] for L > 1 and Nl = 2, ∀l, and in [20] forL = 1 and N1 ≥ 2 with analog network coding transmis-sion. Figure 1c shows an example of MGMW relayingusing the multicasting strategy when L = 2 and N1 = N2= 3 nodes.In this subsection, we have mathematically formulated

the BC strategies. With the assumption of time-invariantchannels within P phases, for unicasting strategy, anyother relationship that one may derive will lead to thesame performance. The relationships given in SectionII-A1 has an advantage that it does not require any sig-naling in the network. For hybrid uni/multicasting, one

may also find other relationship than the relationshipsgiven in (2) and (3). However, with the time-invariantchannels assumption and given the same tlu, the sameperformance will be obtained. In Section II-A2, wechoose tlu = al such that there will be no signaling in thenetwork. This is sub-optimum and one may improvethe performance by exhaustively searching the best tlu,∀l. This will lead to a higher computational complexityand also requires signalling in the network since the RSneeds to inform all nodes in group l about the chosentlu. Regarding multicasting strategy, in this work we pro-pose vl = al and wl = p + al -1 which does not requireany signaling in the network. This is sub-optimum andone may improve the performance by exhaustivelysearching vl and wl which optimises the performancewhile fulfilling the general rule for multicasting strategyas described in Section II-A3. However, the computa-tional complexity will be higher and there are signalingneeded since the RS needs to inform the nodes in groupl about vl and wl.Note that all the relationships of parameters which are

described in this subsection can also be directly appliedfor the case when the numbers of nodes are not equalin all groups. If the numbers of nodes are not equal inall groups, in each pth phase, the RS serves only thegroups with Nl ≥ p.

B. Generalised System ModelIn this subsection, we explain the generalised systemmodel for non-regenerative MGMW relaying which isvalid for all BC strategies. In order to have non-regen-erative MGMW relaying with a specific BC strategy, therelationship of p, rl, and tl as described in the previoussubsection has to be set accordingly.The overall channel matrix from the nodes to the RS

is given by H = [h0, ...,hN−1] ∈ CM×N, withi ∈ I , i ∈ I , the channel vector between node Si and theRS. The channel coefficient hi , m,m ∈ M,M = {1, . . . , M}, follows CN (0, σ 2x ). The vectorx ∈ CN×1 is equal to (x0,..., xN-1)T, with xi the transmitsignal of node Si that follows CN (0, σ 2x ). The AWGNnoise vector at the RS is denoted aszRS = (zRS1, ..., zRSM)T ∈ CM×1 with zRSm followingCN (0, σ 2zRS). In this work, we assume that all nodestransmit with fixed and equal transmit power.In the first phase, all nodes transmit simultaneously to

the RS and the received signal at the RS is given by

yRS = Hx + zRS. (5)

Assuming reciprocal and time-invariant channels in Pphases, the downlink channel from the RS to the nodesis simply the transpose of the uplink channel H. In thepth phase, the RS performs transceive beamforming,


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denoted by matrix Gp, to the received signals and trans-mits to the nodes. Therefore, Gp has to be designed toensure that the MGMW relaying is performed accordingto the chosen BC strategy. It is assumed that there is atransmit power constraint at the RS. The received signalvector of all nodes in the pth phase can be written as

ypnodes = HTGp(Hx + zRS) + z

pnodes, (6)

where zpnodes = (zp0, · · ·, zpN−1)T, with zprl the noise at

receiving node rl which follows CN (0, σ 2znode). Accord-ingly, the received signal at node Srl, rl ∈ Il, whilereceiving the data stream from node Stl, tl ∈ Il\{rl}, inthe pth phase is given by

yprl,tl = hTrlG

phtl xtl︸︷︷︸useful signal

+N−1∑j = 0j �= tl

hTrlGphjxj

︸︷︷︸interference signals

+ hTrlGpzRS︸︷︷︸

RS’s propagated noise

+ zprl︸︷︷︸node rl ’s noise

.

(7)

Sum Rate ExpressionIn this section, we derive the sum rate expression ofnon-regenerative MGMW relaying. We start by definingthe signal to interference and noise ratio (SINR) for theBC strategies. The achievable sum rate of MGMWrelaying for both asymmetric and symmetric traffic areexplained afterwards. The achievable sum rate is thesum of the rates received at all nodes. Asymmetric traf-fic refers to the situation where we allow all nodes inthe group to transmit with different rates. Each nodetransmits with a rate that ensures that in the followingNl - 1 consecutive BC phases, all Nl - 1 nodes in itsgroup can decode its data stream correctly. Symmetrictraffic is when all nodes in group l have to transmitsimultaneously with the same rate that is defined by thelowest rate among all possible link combinations ofreceive and transmit node (rl, tl) in group l.

A. Signal to Interference and Noise RatioIt is assumed that xi, ∀i, zRSm, ∀m, and zi, ∀i, are all sta-tistically independent. Therefore, given the received sig-nal in (7), the SINR for the link between receiving nodeSrl and transmitting node Stl is given by

γprl,tl =

srlIrl + ZRSrl + Zrl

, (8)

with the useful signal power at node Srl

Srl = E{|hTrlGphtl xtl |2} = |hTrlGphtl |2σ 2x , (9)the RS’s propagated noise power which appear at node

Srl

ZRSrl = E{|hTrlG

pzRS|2} = |hTrlGp|2σ 2zRS (10)and the node Srl ’s noise power

Zrl = E{|zrl |2} = σ 2znode . (11)The interference power at receiving node Srl, is given

by

Irl = Isgrl + Iogrl , (12)

with Isgrl the same group interference power and Iogrlthe other group interference power. While Isgrl depends

on the applied BC strategy, Iogrl does not depend on theBC strategy and it is given by

Iogrl =∑d�∈Il

E{|hTrlGphdxd|2} =∑d�∈Il

|hTrlGphd|2σ 2x . (13)

At each receiving node Srl, Isgrl includes the interfer-ence power caused by its own data stream and otherdata streams that have been decoded in the previous BCphases. These a priori known data streams can be can-celed by each receiving node prior to detection by per-forming self- and known-interference cancellation. Ifself- and known-interference cancellation is performed,the remaining interference power which is not canceledby the receiving node, Inot - cancrl, is given by

Inot - cancrl = Isgrl − Icancrl (14)with Icancrl the interference power caused by the data

streams which are a priori known by the receiving nodeSrl and is canceled. With interference cancellation, theinterference power in (12) can be rewritten as

Irl = Inot - cancrl + Iogrl . (15)

In the following, we explain Isgrl and Inot - cancrl for eachBC strategy.1) UnicastingThe same group interference power is given by

Iusgrl=

bl∑j = alj �= tl

E{|hTrlGphjxj|2} =bl∑

j = alj �= tl

|hTrlGphj|2σ 2x . (16)

In every pth phase, node Srl may perform interferencecancellation. It subtracts the a priori known self-interfer-ence as well as the a priori known same group other-stream interference from the previous BC phases. Oncethe nodes have decoded other nodes’ data streams inthe previous BC phases, they may use it to performknown-interference cancellation in a similar fashion toself-interference cancellation. Using interference


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cancellation, Inot - cancrl for unicasting strategy is given by

Iunot - cancrl=

bl∑j = al

j �= {rl, tl}j �∈ Brl

|hTrlGphj|2σ 2x(17)

with Brl the set of the nodes’ indices whose datastreams have been decoded by receiving node rl in theprevious BC phases.2) Hybrid uni/multicastingThe same group interference power can be decoupled intotwo parts. The first part is the interference caused by theunicasted or the multicasted data stream, denoted by Iu/mrl.The second part is the interference caused by other datastreams which can only appear at the receiving node rl ifthe transceive beamforming applied at the RS cannot fullysuppress it. The same group interference power is given by

Iu/msgrl = Iu/mrl +bl∑

j = alj �= {tlu , tlm}

|hTrlGphj|2σ 2x , (18)

with tlu the index of the transmitting node whose datastream is unicasted by the RS, tlm the index of the trans-mitting node whose data stream is multicasted by theRS, and

Iu/mrl =

⎧⎪⎨⎪⎩E{|hTrlGphrl xrl |2} = ||hTrlGphrl |2σ 2x , if rl = tlm,

E{|hTrlGphtlu xtlu |2} = |hTrlGphtlu |2σ 2x , otherwise,(19)

the interference at the nodes which only can be eitherfrom the unicasted data stream (at Nl - 1 nodes whichare intended to receive the multicasted data stream) orfrom the multicasted data stream (at the node whichreceives the unicasted data stream). Similar to the uni-casting strategy, interference cancellation at the nodescan also be applied. For hybrid uni/multicasting trans-mission, Inot - cancrl is defined by

Iu/mnot - cancrl = Iul +bl∑

j = alj �= {rl, tlm }j �∈ Bl

|hTrlGphj|2σ 2x ,(20)

with Bl the sets of nodes’ indices whose data streamshave been multicasted by the RS in the previous BCphases and

Iul ={ |hTrlGphtlu |2σ 2x ,

0,if rl �= tlu and rl �∈ Blotherwise.

(21)

3) MulticastingThe same group interference power can be decoupledinto two parts. The first part is the inherent interferencewithin the superposed data stream which can only beeither self- or known-interference, denoted by Is|k. Thesecond part is the interference caused by other datastreams which can only appear at the receiving node rlif the transceive beamforming applied at the RS cannotfully suppress it. The same group interference power isgiven by

Imsgrl= Is|krl +

bl∑g = al

j �= {vl,wl}

|hTrlGphj|2σ 2x , (22)

with {vl, wl} the indices of the two nodes in group lwhose data streams are superposed by the RS in the pthphase.Is|krl is the self- or known-interference power, which

can only be either self-interference power at nodes rl =wl and rl = vl given by

Is|krl = Isrl = E{|hTrlG

phrl xrl |2} = |hTrlGphrl |2σ 2x , (23)or known-interference power at nodes rl ≠ wl ≠ vl

given by

Is|krl = Ikrl = E{|hTrlG

phvw̃l xvw̃l |2} = |hTrlGphvw̃l |2σ 2x , (24)with vw̃l the index of the known-interference which

can only be either wl or vl. As explained in Section II-A3, Is|krl can be cancelled and, thus, Is|krl = 0. Moreover,once the nodes have decoded other nodes’ data streamsfrom the previous BC phases, they may use them toreduce the amount of interference in the second sum-mand in (22). For the multicasting strategy, Inot - cancrl isdefined by

Imnot - cancrl=

bl∑j = al

j �= {vl,wl}j �∈ Brl

|hTrlGphj|2σ 2x(25)

with Brl the set of the nodes’ indices whose datastreams have been decoded by receiving node rl in theprevious BC phases.

B. Sum Rate for Asymmetric TrafficGiven the SINR as in (8), the information rate at receiv-ing node rl when it receives from transmitting node tl inthe pth phase is given by

Rrl,tl = log2(1 + γprl,tl). (26)


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Since in MGMW relaying there is only one MACphase, the transmitting node tl has to ensure that itsdata stream can be decoded correctly by all Nl - 1intended receiving nodes. Consequently, we have

Rtl = minrl∈Il\{tl}(Rrl ,tl), (27)

which is the minimum rate among all receiving nodesrl in group l when they receive the data stream from acertain transmitting node tl. The achievable sum rate ofnon-regenerative MGMW relaying is given by

SRasym =1P

L∑l=1

⎛⎝(Nl − 1) ∑tl∈Il

Rtl

⎞⎠ . (28)The factor Nl - 1 is since in group l there are Nl - 1

nodes that receive the same data stream from a certaintransmitting node tl. The scaling factor

1P is due to P

channel uses for MGMW relaying.One important note regarding (27) is that by taking

the minimum, we ensure each node Si transmits xi withthe rate that can be decoded correctly by all othernodes in its group. Thus, knowing xi, all other nodes inthe group can use it to perform known-interference can-cellation in a similar fashion to their self-interferencecancellation.

C. Sum Rate for Symmetric TrafficIn certain scenarios, there may be a requirement to havea symmetric traffic between all nodes in group l. Allnodes communicate with the same data rate defined bythe minimum of Rtl,∀tl, tl ∈ Il. The achievable sum ratefor symmetric traffic for all BC strategies is given by

SRsymm =1P

L∑l=1

(Nl − 1)Nl(mintl∈Il

Rtl

). (29)

Transceive BeamformingIn this section, first, we formulate the optimisation pro-blem of finding the optimum transceive beamformingmaximising the sum rate. Afterwards, we explain thedesign of generalised low complexity transceive beam-forming algorithms for all BC strategies. It is assumedthat perfect channel state information is available at theRS whose number of antennas is higher than or equalto the total number of the nodes, i.e., M ≥ N.

A. Sum Rate MaximisationThe optimisation problem of finding the optimumtransceive beamforming maximising the sum rate ofnon-regenerative MGMW relaying for asymmetric traf-fic can be written as

maxGp

∑i

∑f (i,p)

Rf (i,p),i

s.t. tr{Gp(HRxHH + RzRS )GpH} = ERS,(30)

with Rx = E{|| xxH||22}, RzRS = E{|| zRSzHRS ||22}, and f(i, p)the receiving node index, which is a function of trans-mitting index i and BC phase index p, and depends onthe applied BC strategy.In this work, we assume that the transmit powers at

the nodes are fixed and equal. In order to improve thesum rate, one could have the transmit powers at thenodes as variables to be optimised subject to a powerconstraint at each node. However, since there is onlyone MAC phase, one has to find the optimum transmitpower at each node and, simulateneously, the transceivebeamforming for all BC phases, i.e., Gp, ∀p, p ∈ P. Thisjoint optimisation problem would further increase thecomputational effort.The optimisation problem in (30) is non-convex and it

requires high computational complexity to find the glo-bal optimum solution. Thus, in the following, we pro-pose generalised low complexity transceive beamformingalgorithms for all proposed BC strategies.As mentioned in Section II-B and as seen in (30), the

transceive beamforming Gp depends on the BC strategyapplied at the RS. In order to design generalised trans-ceive beamforming for all BC strategies and to make theproblem more tractable, we decouple Gp into transmitbeamforming GpT, BC-strategy-defining permutationmatrix Πp and receive beamforming GpR, such that

Gp = GpT�pGpR.

In the following, we explain specially designed trans-ceive beamforming for MGMW relaying. First, weexplain the generalised linear transceive beamformingbased on three different optimisation criteria, namely,MF, ZF, and MMSE. Afterwards, we explain the general-ised BC-strategy-aware (BCSA) transceive beamforming.

B. Linear Transceive BeamformingIn this subsection, we explain the design of three lowcomplexity generalised linear transceive beamformingalgorithms, namely, MF, ZF, and MMSE. Since we haveonly one MAC phase, the receive beamforming is com-puted only once, i.e., GpR = GR, ∀p ∈ P. The BC-strategy-defining permutation matrix Πp defines the transmissionfrom the RS according to the BC strategies. Table 1shows Πp for the example in Figure 1 for MF, ZF, andMMSE for all BC strategies. One important note is that,even though the derivation for the MF, ZF, and MMSEgeneralised transceive beamforming appears to be simi-lar with the three-step transceive beamforming for two-way relaying in [9]; however, our generalised transceivebeamforming algorithms have a different approach and


Page 8 of 19

are based on different motivations. In [9], the downlink(from the RS to the nodes) channel matrix is a per-muted matrix of the uplink (from the nodes to the RS)channel matrix. Therefore, Πp in [9] is a diagonal matrixwith weighting factors in each of its diagonal elements.Such approach as in [9] is only suitable for unicastingstrategy. Hence, our generalised transceive beamformingis a generalisation of the three-step transceive beam-forming for two-way relaying in [9].1) Matched FilterGiven the received signal at RS as in (5), the output ofthe receive filtering is given by

X̂RS = GRyRS = GR(Hx + zRS). (31)

The MF optimisation problem for receive beamform-ing can be written as

GRMF = arg maxGR

|E{xHx̂RS}|2E{||x||22}E{||GRzRS||22}

. (32)

The objective function in (32) can be written as

|E{xHx̂RS}|2E{||x||22}E{||GRzRS||22}

=|tr(GRHRx)|2

tr(Rx)tr(GRRzRSGHR )

. (33)

By taking the derivative of (33) with respect to GR andsetting it equal to zero, we have [see, e.g, [25,26]]

GRMF = RxHHR−1zRS . (34)

The received signal at the nodes in (6) can now berewritten as

ypnodes = HTGpT

∏px̂RS + zpnodes = H

TGpTx̃pRS + z

pnodes, (35)

with x̃pRS = �px̂RS the transmitted signals from the RS

in the pth phase. The MF optimization problem fortransmit beamforming can be written as

GpTMF = argmaxGpT

|E{x̃pHRS ypnodes}|2E{||x̃pRS||22}E{||zpnodes||22}

s.t. E{||GpTx̃pRS||2} = ERS.(36)

Let Rpx̃RS = E{||x̃pRSx̃

pH

RS||22} andRpznodes = E{||z

pnodesz

pH

nodes||22}. Using the same steps as in[26] by deriving the Lagrangian function and solving theKarush-Kuhn-Tucker (KKT) conditions, we have

GpTMF = βpMFH

∗, (37)

where βpMF ∈ R+ is needed to fulfill the power con-straint and given by

βpMF =

√√√√ ERStr

(H∗Rpx̃RSH

T) . (38)

2) Zero ForcingGiven (31), the ZF optimisation problem for receivebeamforming can be written as

GRZF = argminGR

E{||x − x̂RS||22}

s.t. x̂RS = x|zRS=0.(39)

where x̂RS = x|zRS=0 is the ZF constraint which impliesthat GRH = IM. Due to the ZF constraint, the objectivefunction in (39) can be written as

E{||x − x̂RS||22} = tr(GRRzRSG

HR

). (40)

Using the same steps as in [26] by deriving theLagrangian function and solving the KKT conditions, wehave [see, e.g., [25]]

GRZF =(HHR−1zRSH

)−1HHR−1zRS . (41)Given (35), the ZF optimisation problem for transmit

beamforming can be written as

GpTZF = argminGpT

E{||x̃pRS − ypnodes||22}

s.t. E{||GpTx̃pRS||2} = ERS,ynodes = x̃RS|znodes=0.

(42)

where ynodes = x̃RS|znodes=0 is the ZF constraint whichimplies that HTGpT = IN. Due to the ZF constraint, the

Table 1 BC-strategy-defining permutation matrices of L =2, N1 = N2 = 3

Π2 Π3

Unicasting ⎛⎜⎜⎜⎜⎜⎜⎝

0 1 0 0 0 00 0 1 0 0 01 0 0 0 0 00 0 0 0 1 00 0 0 0 0 10 0 0 1 0 0

⎞⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎝

0 0 1 0 0 01 0 0 0 0 00 1 0 0 0 00 0 0 0 0 10 0 0 1 0 00 0 0 0 1 0

⎞⎟⎟⎟⎟⎟⎟⎠Uni/multicasting ⎛⎜⎜⎜⎜⎜⎜⎝

0 1 0 0 0 01 0 0 0 0 00 1 0 0 0 00 0 0 0 1 00 0 0 1 0 00 0 0 0 1 0

⎞⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎝

0 0 1 0 0 00 0 1 0 0 01 0 0 0 0 00 0 0 0 0 10 0 0 0 0 10 0 0 1 0 0

⎞⎟⎟⎟⎟⎟⎟⎠Multicasting ⎛⎜⎜⎜⎜⎜⎜⎝

1 1 0 0 0 01 1 0 0 0 01 1 0 0 0 00 0 0 1 1 00 0 0 1 1 00 0 0 1 1 0

⎞⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎝

1 0 1 0 0 01 0 1 0 0 01 0 1 0 0 00 0 0 1 0 10 0 0 1 0 10 0 0 1 0 1

⎞⎟⎟⎟⎟⎟⎟⎠


Page 9 of 19

objective function in (42) can be written as

E{||x̃pRS − ypnodes||22} = tr(Rpznodes

). (43)

Using the same steps as in [26] by deriving theLagrangian function and solving the KKT conditions, wehave

GpTZF = βpZFH

∗(HTH∗)−1, (44)

where βpZF ∈ R+ is needed to fulfill the power con-straint and given by

βpMF =

√√√√ ERStr

((H∗HT)Rpx̃RS

) . (45)

3) Minimisation of Mean Square ErrorGiven (31), the MMSE optimisation problem for receivebeamforming can be written as

GRMMSE = argminGR

E{||x − x̂RS||22}. (46)

The objective function in (46) can be written as

E{||x − x̂RS||22} = tr(Rx − 2R(GRHRx) +GRHRxHHGHR +GRRzRSGHR

). (47)

By taking the derivative of (47) with respect to GR andsetting it equal to zero, we have [see, e.g., [25,26]]

GRMMSE = RXHH(HRXHH + RzRS

)−1. (48)

Given (35), the MMSE optimisation problem fortransmit beamforming can be written as

{GpTMMSE , βpMMSE} = argminE

GpT,βp||x̃pRS −

1βp

ypnodes||22}

s.t. E{||GpTx̃pRS||2} = ERS.(49)

where 1/bp is introduced to modify the mean squareerror as in [27,28]. Using the same steps as in [28] byderiving the Lagrangian function and solving the KKTconditions, we have

GpTMMSE = βpMMSE

⎛⎝H∗HT + tr(Rpznodes

)ERS

IM

⎞⎠−1H∗, (50)where βpMMSE ∈ R+ is needed to fulfill the power con-

straint and given by

βpMMSE =

√√√√√√√√ERS

tr

⎛⎜⎝⎛⎝H∗HT + tr

(Rpznodes

)ERS

IM

⎞⎠−2H∗Rpx̃RSHT⎞⎟⎠.

(51)

Finally, for MF, ZF, and MMSE the transceive beam-forming is given by

Gp = βpa1gorithmGpTalgorithm�

pGpRalgorithm , (52)

where the subscript (·)algorithm refers to either MF, ZF,or MMSE.

C. Broadcast-Strategy-Aware Transceive BeamformingIn the following, we explain the design of BCSA trans-ceive beamforming. Based on the chosen BC strategy,the RS separates the data streams which are going to betransmitted in the BC phase and transmits to the corre-sponding node or nodes. For unicasting strategy, the RSseparates all data streams and transmits each datastream to each corresponding receiving node. For hybriduni/multicasting, for each group, the RS separates theunicasted data stream from the other data streams andtransmits it to the corresponding node whose datastream is multicasted. The RS also separates the multi-casted data stream from the other data streams andtransmits it to the remaining nodes in the correspond-ing group. For multicasting strategy, the RS separatesthe superposition of two data streams from the othersand transmits the superposed data stream to all nodesin the group.In order to compute the transceive beamforming, we

first compute the equivalent channels for receive beam-forming and transmit beamforming. The equivalentchannels are needed to ensure that there will be nointerstream interference received at the unintendedreceiving node or nodes. In order to find the equivalentchannel, BD as proposed in [17] can be applied. Severalworks have considered BD for separation of datastreams, e.g., [16,20,29,30]. In this work, we also con-sider regularised BD (RBD) as proposed in [23]. RBDavoids the drawbacks of BD which has a quite poor per-formance if the subspaces of the users channel matricesoverlap significantly [23].Equivalent channelWithout loss of generality, in the following we omit theBC phase index p. Let HTin ∈ Cηin×M andH̃

Tun ∈ C(N−ηin )

×M denote the channel matrix of theintended nodes and the channel matrix of the otherunintended nodes, respectively, with ηin the number ofintended nodes. Both channel matrices are parts of the

overall channel matrix, i.e., HT = HTin ∪ H̃Tun. Since the

steps of computing the equivalent channel for receivebeamforming and transmit beamforming are similar, wegenerally explain the methods for finding the equivalent

channel using HTin and H̃Tun. In order to relate them with

the receive beamforming and transmit beamforming for


Page 10 of 19

BC strategies, we have to set HTin and H̃Tun

accordingly.

Table 2 shows the corresponding HTin and H̃Tun

for all BC

strategies. Given the singular value decomposition(SVD) of the unintended nodes’ channels as

H̃Tun = Ũun�̃un

[Ṽ(1)un , Ṽ

(0)un

]︸︷︷︸

Ṽun

,(53)

we compute the equivalent channel for the intendednodes Heqin . The equivalent channel is given by

Heqin = HTinFnull, (54)

where Fnull is the null-space matrix which can becomputed either using BD or RBD.

Using BD, Fnull = Ṽ(0)un ∈ CM×(N−r̃un ) with r̃un denoting

the rank of matrix H̃Tun. The BD approach can be used

directly for receive and transmit beamforming, since itonly deals with the channels without considering thenoise. Using RBD, however, the equivalent channels forreceive and transmit beamforming need to be computeddifferently. RBD for transmit beamforming has beenderived in [23] and in this work, we provide the deriva-tion of RBD for receive beamforming in the appendix.

Using RBD, Fnull = Ṽun(�̃

Tun�̃un + κIM

)∈ CM×M with

κ =Nσ 2nodeERS

for transmit beamforming [23] and κ =σ 2RS

σ 2xfor receive beamforming, see appendix.Having Heqin , we can now compute the receive beam-

forming and transmit beamforming. In the following,

when computing the receive beamforming, Heqin and ηin

relate to HTin and H̃Tun

as defined in Table 2 for receive

beamforming, while when computing transmit beam-

forming, Heqin and ηin relate to HTin and H̃

Tun

as defined in

Table 2 for transmit beamforming.In this work, we consider signal processing algo-

rithms which do not deal with interference since Heqin isfree from unwanted data streams, namely, MF, SVD,and semidefinite relaxation (SDR) of maximising theminimum SNR. MF and SVD for single-pair two-wayrelaying have been investigated in [3,10]. The BD-MFand BD-SDR have been designed in [20] for single-group multi-way relaying for multicasting strategy. BD-SVD has been designed in [16] only for multi-usertwo-way relaying with multicasting strategy. In thiswork, BCSA transceive beamforming is designed fornon-regenerative MGMW relaying for all proposed BCstrategies. Due to the requirement to make generalisedBCSA transceive beamforming also suitable for non-regenerative MGMW relaying with multicasting strat-egy, it has a slight difference to [3,10,16]. Using BCSAfor multicasting strategy, for each group l, the RS hasto transmit one data stream, which is a superpositionof two data streams, to Nl nodes in the group whereNl can be any number higher than two. For that rea-son, in the design of receive beamforming using MFand SVD, we have to do a superposition of two datastreams. This makes the proposed BCSA not a directgeneralisation of [3,10,16]. However, for cases of one-pair two-way relaying and multi-user two-way relaying,if the superposition is not performed, BCSA is a gener-alisation of [3,10,16].

Table 2 The corresponding HTin and H̃Tun

for all BC strategies

Receive beamforming Transmit beamforming

∀tl : ∀rl :UC HTin = H

Ttl ∈ C1×M HTin = HTrl ∈ C1×M

H̃Tun = H

TI\{tl} ∈ C(N−1)×M H̃

Tun = H

TI\{rl} ∈ C(N−1)×M

For tlu : For rl = tlm :HTin = H

Ttlu

∈ C1×M HTin = HTrl ∈ C1×MH̃

Tun = H

TI\{tlu } ∈ C(N−1)×M H̃

Tun = H

TI\{rl=tlm } ∈ C(N−1)×M

U/MC

For tlm : ∀rl, rl ∈ Il\tlm :HTin = H

Ttlm

∈ C1×M HTin = HTIl\{rl=tlm } ∈ C(Nl−1)×M

H̃Tun = H

TI\{tlm } ∈ C

(N−1)×M H̃Tun = HTI\{tlm } ∈ C(N−(Nl−1))×M

∀l : ∀l :MC HTin = H

Tvlwl ∈ C2×M HTin = HTIl ∈ CNl×M

H̃Tun = H

TI\{vlwl} ∈ C(N−2)×M H̃

Tun = H

TI\Il ∈ C(N−Nl)×M

UC, unicasting; U/MC, hybrid uni/multicasting; MC, multicastingtlu : index of transmitting node whose data stream is unicasted, tlu ∈ Iltlm : index of transmitting node whose data stream is multicasted, tlm ∈ Il\{tlu}


Page 11 of 19

Matched filter The receive beamforming vector isgiven by

m = �in 1TηinHeq∗in F

Tnull,︸︷︷︸

m̃

(55)

where 1ηin a vector of ones of length ηin. 1ηin super-poses (adds) two-data streams from two nodes in eachgroup for multicasting strategy. �in = mean(|HTinm̃T|)can be seen as receive power loading where the modulusoperator | · | is assumed to be applied element-wise andthe mean function returns the mean of a vector.The transmit beamforming vector is given by

mDL = FnullHeqHin

1ηin︸︷︷︸m̃DL

�inDL, (56)

with �inDL = mean(|HTinm̃DL|) the transmit powerloading. 1ηin is a vector of ones with size ηin × 1. Formulticasting strategy, 1ηin replicates the superposed datastream ηin times.Singular value decomposition Let the SVD of the

equivalent channel be given by

Heqin = Ueqin

�eqin[Veq(1)in ,V

eq(0)in

], (57)

The receive beamforming vector is given by

m = �in 1TηinVeq(1)

T

inFTnull,︸︷︷︸

m̃

(58)

with �in = mean(|HTinm̃T|) the receive power loading.The transmit beamforming vector is given by

mDL = FnullVeq(1)in

1ηin︸︷︷︸m̃DL

�inDL, (59)

with �inDL = mean(|HTinm̃DL|) the transmit powerloading.Semidefinite RelaxationSince in MGMW relaying all member nodes in eachgroup exchange messages, we are also interested in afair beamforming algorithm which aims at balancing theSNRs at the RS as well as at the receiving nodes in eachgroup.The SNR balancing problem for receive beamforming

can be written as

msdr = argmaxm

miniin∈Iin

∣∣∣∣∣mheqiin

σ 2zRS

∣∣∣∣∣2

,

s.t. ||m||22 ≤ 1(60)

with Iin the set of intended nodes with cardinalityequal to ηin. iin is the index of a member node in Iin andheqiin ∈ H

eqin . The receive beamforming is given by

m = �in mTsdrF

TNull

,︸︷︷︸m̃sdr

(61)

with �in = mean(|HTinm̃Tsdr|) the receive power loading.Equation (60) is a non-convex quadratically con-

strained quadratic program. A similar optimisation isalso considered in [31]. It is proved to be NP-hard in[31]. Nonetheless, it can be approximately solved usingSDR techniques [31,32]. We will not go further into thisrelaxation and the interested reader may find moredetailed derivation in [20,31,32].The SNR balancing problem for transmit beamform-

ing can be written as

msdr DL = argmaxm

miniin∈Il

∣∣∣∣∣mheqiin

σ 2znode

∣∣∣∣∣2

s.t. ||mDL||22 ≤ 1,(62)

with heqiin ∈ Heqin . The transmit beamforming is given by

mDL = FNu11msdr DL︸︷︷︸m̃sdrDL

�inDL, (63)

with �inDL = mean(|HTinm̃sdrDL |) the transmit powerloading. Note that to compute the transmit beamform-ing with SDR, we assume that the information of thenoise power at the nodes is available at the RS. Similarto (60), (62) can be approximately solved with semidefi-nite relaxation techniques using a solver such asSEDUMI [33].In the following, we use again the BC phase index p to

describe the BCSA transceive beamforming. For unicast-ing strategy, the receive beamforming matrix is given by

GpR = [mp1, . . . , m

pN], (64)

where mptl, ∀tl ∈ I , is the receive beamforming as in(55), (58) or (61) given the equivalent channel of nodetl. The transmit beamforming matrix is given by

GpT =[mpDL1 , . . . , m

pDL1

], (65)

where mpDLrl

, ∀rl ∈ I , is the transmit beamforming asin (56), (59), or (63) given the equivalent channel ofnode rl.For hybrid uni/multicasting strategy, the receive beam-

forming matrix is given by


Page 12 of 19

GpR =[mpt1u ,m

pt1m

, . . . , mptLu , mptLm

], (66)

where m mptlu, ∀l ∈ L, and mptlm, ∀l ∈ L are the receive

beamforming as in (55), (58), or (61) given the equiva-lent channels of nodes tlu and tlm, respectively. The trans-mit beamforming matrix is given by

GpT =[mpDLr1=t1m

,mpDLI1\{t1m }, . . . ,mpDLrL=tLm

,mpDLIL\{tLm }

], (67)

where mpDLrl=tlm

, ∀l ∈ L, and mpDLIl\{tlm }, ∀l ∈ L, are thetransmit beamforming as in (56), (59), or (63) given theequivalent channel of node rl = tlm and the equivalentchannel of all other nodes in group l, ∀rl ∈ Il\{tlm},respectively.For multicasting strategy, the receive beamforming

matrix is given by

GpR = [mpl , . . . , m

pL], (68)

where mpl , ∀l ∈ L, is the receive beamforming as in(55), (58), or (61) given the equivalent channels of twonodes vl and wl whose data streams are superposed. Thetransmit beamforming matrix is given by

GpT =[mpDL1 , . . . , m

pDLL

], (69)

where mpDLl, ∀l ∈ L, is the transmit beamforming as in(56), (59), or (63) given the equivalent channels of allnodes in group l.Finally, BCSA transceive beamforming is given by

Gp = βpGpT�pGpR. (70)

where bp is needed in order to satisfy the transmitpower constraint at the RS, with

βp =

√√√√ ERStr

{GpT�pG

pR(σ 2x HH

H + σ 2RSI)GpH

R �pHGp

H

T

} . (71)Note that Πp is not the same for all BC strategies. For

unicasting strategy, Πp is the same as for MF, ZF, andMMSE, where an example for L = 2, N1 = N1 = 3 isgiven in Table 1. For hybrid uni/multicasting strategy,Πp = I2L and for multicasting strategy, Π

p = IL.

Simulation ResultsIn this section, the sum rate performance is analysedbased on simulation results. We set σ 2zRS = σ

2znode = 1,

σ 2x = 1, and ERS = 1. The channel coefficients are i.i.d.CN (0, σ 2x ), i.e., Rayleigh fading. Hence, the SNR valueis given by

σ 2xσ 2znode

|hi,m|2 = σ2x

σ 2zRS|hi,m|2.

We consider three scenarios. The first two scenariosare the well-known scenarios, namely, one-pair two-wayrelaying and multi-user two-way relaying. We considerboth scenarios to show that the proposed BC strategiesand the generalised transceive beamforming designed inthis work are valid for both well-known scenarios. Thethird scenario is the two-group multi-way case, whereeach group consists of three nodes.

A. First Scenario: L = 1 and N1 = 2In one-pair two-way relaying, unicasting and hybrid uni/multicasting are the same. Figure 2 shows the sum rateperformance of one-pair two-way relaying with MF, ZF,and MMSE transceive beamforming for both asym-metric traffic and symmetric traffic. Regarding sym-metric traffic, to reduce the number of lines in thefigure, we only plot the result for MF transceive beam-forming. The approximation of maximum sum rate isalso provided for two cases, i.e., with optimised andwith fixed transmit power at the nodes. Both optimumtransceive beamforming solutions maximising the sumrate were computed using fmincon from MATLAB toprovide performance bounds for two-way relaying. Weuse the value of MMSE transceive beamforming as theinitial value. In general, using MF, ZF and MMSE trans-ceive beamforming, unicasting and hybrid uni/multicast-ing outperform multicasting strategy. A directsuperposition of the output of receive beamforming forthe multicasting strategy doubles the amount of the RS’sfiltered noise. Moreover, the RS transmit power is dis-tributed within the superposed data stream and after theself-interference cancellation, each node only receiveshalf of the power. Since each node performs self-inter-ference cancellation, no interference appears at thenodes, and thus, for all BC strategies MF outperformsMMSE and ZF. At low SNR, MMSE converges to MFand in high SNR, ZF converges to MMSE. In this work,we assume fixed transmit power at all nodes and theperformance of unicasting and hybrid uni/multicastingwith MF is close to the approximation of maximumsum rate with fixed transmit power. If the nodes canoptimise their transmit power, the sum rate is improvedwith a penalty of having higher computational complex-ity. It can also be seen that asymmetric traffic leads to ahigher rate compared to symmetric traffic since the ratefor symmetric traffic is defined by the weakest linkamong all available links. Therefore, in the following, weonly consider asymmetric traffic.Figure 3 shows the sum rate performance of two-way

relaying with BCSA transceive beamforming. For multi-casting strategy, since there is no separation neededboth for receive beamforming and transmit beamform-ing, BD and RBD are the same. For unicasting and


Page 13 of 19

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

10

SNR in dB

Ave

rag

e S

um

Rat

e (b

/s/H

z)

Approx. Max Sum Rate with optimised transmit powerApprox. Max Sum Rate with fixed transmit powerUC=U/MC: MF asymmetricUC=U/MC: MF symmetricMC: MF asymmetricMC: MF symmetricUC=U/MC: MMSE asymmetricMC: MMSE asymmetricUC=U/MC: ZF asymmetricMC: MF asymmetric

Figure 2 Sum rate performance of first scenario with MF, ZF, and MMSE; UC, unicasting; U/MC, hybrid uni/multicasting; MC,multicasting.

0 5 10 15 20 25 300

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/s/H

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Approx. Max Sum Rate with optimised transmit power

Approx. Max Sum Rate with fixed transmit power

MC: RBD−SDR=BD−SDR

MC: RBD−MF=BD−MF

UC=U/MC: RBD−MF=RBD−SVD=RBD−SDR

MC: RBD−SVD=BD−SVD

UC=U/MC: BD−MF=BD−SVD=BD−SDR

Figure 3 Sum rate performance of first scenario with BCSA; UC, unicasting; U/MC, hybrid uni/multicasting; MC, multicasting.


Page 14 of 19

hybrid uni/multicasting strategies, BD-MF, BD-SVD, andBD-SDR perform the same and they have similar perfor-mance to multicasting strategy with SVD. Different tomulticasting strategy, for unicasting and hybrid uni/mul-ticasting, since there is a stream separation both inreceive beamforming and transmit beamforming, RBDimproves the performance in low SNR region. In highSNR region, BD converges to RBD. For unicasting andhybrid uni/multicasting strategies, since the equivalentchannels (which are free from interference) always cor-respond only to one intended node for both receivebeamforming and transmit beamforming, MF, SVD, andSDR will always have the same performance. It can beseen that multicasting strategy with SDR performs best.Hence, having a suitable transceive beamforming, onecan exploit the benefit of beamforming-based physicallayer network coding for non-regenerative single groupmulti-way relaying as proposed in [20].

B. Second Scenario: L = 2 and N1 = N2 = 2Figure 4 shows the sum rate performance of multi-usertwo-way relaying with MF, ZF, and MMSE transmitbeamforming. In this scenario, unicasting and hybriduni/multicasting are the same and they outperform mul-ticasting strategy. The reason is the same as in the caseof one-pair two-way relaying. Moreover, the directsuperposition of the output of receive beamforming for

multicasting strategy not only increases the amount ofthe RS’s filtered noise but also increases the unwantedinterference at the receiving nodes. For all strategies,MMSE performs best and in high SNR region, ZF con-verges to MMSE, while in low SNR region, MF con-verges to MMSE. Different to the case of one-pair two-way relaying, in multi-user two-way relaying MF per-forms worse since it does not cancel the interferencefrom other pairs which appears at each node. The trans-ceive beamforming maximising the sum rate was com-puted using fmincon from MATLAB to provide abound for multi-user two-way relaying. We use thevalue of MMSE tranceive beamforming as initial value.It can be clearly seen that if the transmit power at thenodes can be optimised, the sum rate can be improvedat the expense of computational complexity.Figure 5 shows the sum rate performance of multi-

user two-way relaying with BCSA transceive beamform-ing. In general, RBD outperforms BD in low SNR regionand BD converge to RBD in high SNR. Only for multi-casting strategy, BD-SVD outperforms RBD-SVD for allSNR values and it has similar performance as unicastingand hybrid uni/multicasting with BD-MF, BD-SVD, andBD-SDR. The gain of RBD compared to BD is obtainedmost for unicasting and hybrid uni/multicasting strate-gies, while for multicasting strategy (with MF and SDR),the gain is small. In medium to high SNR region,

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Approx. Max Sum Rate with optimised transmit powerApprox. Max Sum Rate with fixed transmit powerUC=U/MC: MMSEMC: MMSEUC=U/MC: ZFMC:ZFUC=U/MC: MFMC: ZF

Figure 4 Sum rate performance of second scenario with MF, ZF, and MMSE; UC, unicasting; U/MC, hybrid uni/multicasting; MC,multicasting.


Page 15 of 19

multicasting strategy with RBD-SDR or BD-SDR per-forms best. While for both unicasting and hybrid uni/multicasting strategies, the performance of MF, SVD,and SDR is the same, for multicasting strategy SDRalways performs best followed by MF and SVD.

C. Third Scenario: L = 2 and N1 = N2 = 3Figure 6 shows the sum rate performance of two-groupthree-way relaying using MF, ZF, and MMSE. In generalhybrid uni/multicasting performs best followed by uni-casting and multicasting strategies. While hybrid uni/

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Approx. Max Sum Rate with optimised transmit powerApprox. Max Sum Rate with fixed transmit powerMC: RBD−SDRMC: BD−SDRUC=U/MC : RBD−MF=RBD−SVD=RBD−SDRUC=U/MC: BD−MF=BD−SVD=BD−SDRMC: RBD−MFMC: BD−MFMC: BD−SVDMC: RBD−SVD

Figure 5 Sum rate performance of second scenario with BCSA; UC, unicasting; U/MC, hybrid uni/multicasting; MC, multicasting.

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U/MC: MMSEUC: MMSEMC: MMSEU/MC: ZFUC: ZFMC: ZFU/MC: MFUC: MFMC: MF

Figure 6 Sum rate performance of third scenario with MF, ZF, and MMSE; UC, unicasting; U/MC, hybrid uni/multicasting; MC,multicasting.


Page 16 of 19

multicasting strategy with MMSE slightly ouperformsunicasting strategy with MMSE, both strategies havesimilar ZF performance. With MMSE, we find the tradeoff between the noise enhancement and the interferencesuppression. Since, hybrid uni/multicasting has smallernumber of transmit data streams from the RS, it per-forms better than unicasting strategy both for MMSEand MF. ZF perfectly cancels the interference, and, thus,both unicasting and hybrid uni/multicasting performsimilar. In general, for all strategies, ZF converges toMMSE in high SNR region and in low SNR region, MFconverges to MMSE. It can be seen that multicastingstrategy is outperformed by other strategies since it suf-fers from the increase of RS’s filtered noise and thereduced received power at the nodes. This shows thatanalog network coding for non-regenerative MGMWrelaying obtained by directly adding the output ofreceive beamforming (using MF, ZF, and MMSE receivebeamforming) is not an efficient strategy and, thus,appropriate transceive beamforming is required.Figs. 7 and 8 show the sum rate performance of two-

group three-way relaying using BCSA transceive beam-forming with BD and RBD, respectively. Comparingboth the figures, in general, RBD outperforms BD, andthey converge in high SNR. Only when using SVD, forboth hybrid uni/multicasting and multicasting strategies,RBD-SVD performs worse than BD-SVD and BD-SVDdoes not converge to RBD-SVD in high SNR region. In

medium to high SNR, multicasting strategy outperformsother strategies when using SDR and MF. However, ifSVD is applied, unicasting strategy performs best. Forunicasting strategy, MF, SVD and SDR have similarperformance.Comparing Figures 6, 7, and 8, one can clearly see

that BCSA transceive beamforming improves the sumrate performance compared to MF, ZF, and MMSEtransceive beamforming, especially for multicastingstrategy. For the multicasting strategy, only RBD-SVDperforms worse than ZF and MMSE. For the hybriduni/multicasting strategy, BD-MF performs similar toZF, RBD-MF performs similar to MMSE and both BD-SDR and RBD-SDR outperform MF, ZF, and MMSE.For the unicasting strategy, BD-MF, BD-SVD, and BD-SDR perform similar to ZF, while RBD-MF, RBD-SVD,and RBD-SDR perform similar to MMSE. The highestsum rate (especially in high SNR) is obtained by multi-casting strategy with BCSA BD-SDR and RBD-SDR.Therefore, provided a suitable transceive beamformingis used which can exploit analog network coding, thesum rate of non-regenerative MGMW relaying can beimproved.

ConclusionIn this paper, we consider non-regenerative MGMWrelaying. A multi-antenna RS assists L communicationgroups, where Nl nodes in each group communicate to

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U/MC: BD−SDRUC: BD−MF=BD−SVD=BD− SDRMC: BD−MFU/MC: BD−MFMC: BD−SVDU/MC: BD−SVDMC: BD−SDR

Figure 7 Sum rate performance of third scenario with BCSA BD; UC, unicasting; U/MC, hybrid uni/multicasting; MC, multicasting.


Page 17 of 19

each other but not with other nodes in other groups.The number P of communication phases is equal tomaxl Nl. Three BC strategies are proposed, namely,unicasting, hybrid uni/multicasting, and multicasting.We derive the sum rate expression for non-regenera-tive MGMW relaying for asymmetric and symmetrictraffic. We address the optimum transceive beamform-ing maximising the sum rate of non-regenerativeMGMW relaying. We design generalised low complex-ity sub-optimum transceive beamforming for all BCstrategies, namely, MF, ZF, MMSE, and BCSA trans-ceive beamforming. It is shown that the performanceof non-regenerative MGMW relaying depends on theBC strategy and the applied transceive beamforming.While multicasting strategy using BCSA SDR providesbetter sum rate performance compared to other strate-gies, however, if either MF, ZF, or MMSE are applied,multicasting strategy is outperformed by the otherstrategies.

AppendixRBD for Receive BeamformingFor receive beamforming, the RS has to ensure that theinterference from other users to the intended user i canbe minimised while taking into consideration theappearance of noise at the RS. The matrix FNull isdesigned to achieve the aim and, by rewriting the opti-misation problem for transmit beamforming in [23]

Equation (9) we have the optimisation problem forreceive beamforming,

FNu11i,∀i = arg minFi,∀i

E

{N∑i=1

||FiH̃iun ||2 +||FizRS||2

β−1

},

s.t. βE{ ||xixHi ||} = Pnodes,(72)

where b is a scaling factor needed to fulfill the nodes’transmit power constraint. In this work, we assume thatall nodes transmit with fixed and equal unit power and,thus, the constraint can be written as

β =Pnodes

E{||xixHi ||}=

1σ 2x

. (73)

The objective function in (72), f(Fi), can be written as

f (Fi) =N∑i=1

(tr

(FiH̃iun H̃

HiunFHi

)+

σ 2RSIMσ 2x

tr(FHi Fi

)).(74)

Let the SVD of H̃iun be given by

H̃iun = Ũiun ̃iun Ṽiun , (75)

(74) can be rewritten as

f (Fi) =N∑i=1

(tr

(FiŨiun

(̃iun ̃

Tiun

+σ 2RSIMσ 2x

)Ũ

HiunFHi

)). (76)

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MC: RBD−SDRU/MC: RBD−SDRUC: RBD−MF=RBD−SVD=RBD−SDRMC: RBD−MFU/MC: RBD−MFMC: RBD−SVDU/MC: RBD−SVD

Figure 8 Sum rate performance of third scenario with BCSA RBD; UC, unicasting; U/MC, hybrid uni/multicasting; MC, multicasting.


Page 18 of 19

Let Fi = FaiFbi and let Fbi = ŨHiun, the optimisation

problem reduces to

argminFai,∀i

N∑i=1

(tr

((̃iun ̃

Tiun

+σ 2RSIMσ 2x

)Fa2i

))(77)

where Fai needs to be positive definite in order to finda nontrivial solution [23]. Using the results from [23,34],we have

Fai = (̃iun ̃Tiun

+ κIM)−1/2, (78)

with κ =σ 2RS

σ 2x.

AcknowledgementsThe work of Aditya U. T. Amah is supported by the ‘Excellence Initiative’ ofthe German Federal and State Governments and the Graduate School ofComputational Engineering, Technische Universität Darmstadt. The authorswould like to thank the anonymous reviewers whose review comments arevery helpful in improving this article.Some parts of this paper have been presented at the IEEE ICASSP 2010,Dallas, USA, and IEEE WCNC 2010, Sydney, Australia.

Author details1Graduate School of Computational Engineering and CommunicationsEngineering Lab, Technische Universität Darmstadt, Darmstadt 64283Germany 2Communications Engineering Lab, Technische UniversitätDarmstadt, Darmstadt 64283 Germany

Competing interestsThe authors declare that they have no competing interests.

Received: 23 December 2010 Accepted: 5 July 2011Published: 5 July 2011

References1. B Rankov, A Wittneben, Spectral efficient protocols for half-duplex relay

channels. IEEE Journal on Selected Areas in Communications, 25(2),379–389 (Feb. 2007)

2. T Oechtering, Spectrally Efficient Bidirectional Decode-and-Forward Relayingfor Wireless Networks. Ph.D dissertation, TU Berlin (2007)

3. T Unger, Multi-antenna two-hop relaying for bi-directional transmission inwireless communication systems. Ph.D dissertation, TU Darmstadt (2009)

4. E Biglieri, R Calderbank, A Constantinides, A Goldsmith, HV Poor, MIMOWireless Communication (Cambridge University Press, 2007)

5. D Tse, P Viswanath, Fundamentals of Wireless Communication (CambridgeUniversity Press, 2005)

6. I Hammerström, M Kuhn, C Esli, J Zhao, A Wittneben, G Bauch, MIMO two-way relaying with transmit CSI at the relay. in Proc IEEE Signal ProcessingAdvances in Wireless Communications, Helsinki, 1–5 (June 2007)

7. TJ Oechtering, RF Wyrembelski, H Boche, Multiantenna bidirectionalbroadcast channels - optimal transmit strategies. IEEE Transactions onSignal Processing, 57(5), 1948–1958 (May 2009)

8. TJ Oechtering, EA Jorswieck, RF Wyrembelski, H Boche, On the optimaltransmit strategy for the mimo bidirectional broadcast channel. IEEETransactions on Communications, 57(12), 3817–3826 (2009)

9. T Unger, A Klein, Duplex schemes in multiple-antenna two-hop relaying.EURASIP Journal on Advances in Signal Processing 2008. Article ID 128592(2008)

10. Y-C Liang, R Zhang, Optimal analogue relaying with multiantennas forphysical layer network coding, in Proc IEEE International Conference onCommunications, Beijing. 3893–3897 (2008)

11. R Zhang, Y-C Liang, CC Chai, S. Cui, Optimal beamforming for two-waymulti-antenna relay channel with analogue network coding. IEEE Journal onSelected Areas in Communications, 27(5), 699–712 (June 2009)

12. M Chen, A Yener, Multiuser two-way relaying for interference limitedsystems. in Proc IEEE International Conference on Communications, Beijing,3883–3887 (2008)

13. C Esli, A Wittneben, One- and two-way decode-and-forward relaying forwireless multiuser mimo networks. in Proc IEEE Global CommunicationsConference, New Orleans, 1–6 (2008)

14. AUT Amah, A Klein, YCB. Silva, A Fernekeß, Multi-group multicastbeamforming for multiuser two-way relaying. in Proc International ITGWorkshop on Smart Antennas, Berlin. (2009)

15. J Joung, AH Sayed, Multiuser two-way amplify-and-forward relay processingand power control methods for beamforming systems. IEEE Transactions onSignal Processing, 58(3), 1833–1846 (March 2010)

16. E Yilmaz, R Zakhour, D Gesbert, R Knopp, Multi-pair two-way relay channelwith multiple antenna relay station. in Proc IEEE International Conference onCommunications, Cape Town, (May 2010)

17. QH Spencer, AL Swindlehurst, M Haardt, Zero-forcing methods for downlinkspatial multiplexing in multiuser MIMO channels. IEEE Transaction on SignalProcessing, 52, 461–471 (Feb 2004). doi:10.1109/TSP.2003.821107

18. D Gündüz, A Yener, A Goldsmith, HV Poor, Multi-way relay channel. ProcIEEE International Symposium on Information Theory, Seoul. (2009)

19. S Simoens, Cooperative MIMO Communications, Information TheoreticalLimits and Practical Coding Strategies. Ph.D dissertation, TU Catalonia (UPC)(2009)

20. AUT Amah, A Klein, Beamforming-based physical layer network coding fornon-regenerative multi-way relaying. EURASIP Journal on WirelessCommunications and Networking, Special Issue on Physical Layer NetworkCoding for Wireless Cooperative Networks 2010. Article ID 521571 2010.

21. A transceive strategy for regenerative multi-antenna multi-way relaying,Proc. IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, Aruba. (Dec. 2009)

22. S Katti, S Gollakota, D Katabi, Embracing wireless interference: analognetwork coding. Proc ACM Special Interest Group on Data Communication,Kyoto, 397–408 (2007)

23. V Stankovic, M Haardt, Generalized design of multi-user mimo precodingmatrices. IEEE Transactions on Wireless Communications, 7, 953–961 (2008)

24. P Popovski, H Yomo, Wireless network coding by amplify and forward forbi-directional traffic flows. IEEE Communications Letters, 11(1), 16–18 (2007)

25. JA Nossek, M Joham, W Utschick, Transmit processing in mimo wirelesssystems. Proc IEEE International Symposium on Emerging Technologies,Shanghai (May 2004)

26. M Joham, Optimization of linear and non-linear transmit signal processing.Ph.D dissertation, TU München (2004)

27. M Joham, K Kusume, MhH Gzara, W Utschick, JA Nossek, Transmit wienerfilter for downlink of tddds-cdma systems. Proc IEEE InternationalSymposium on Spread-Spectrum Techniques and Applications (May 2002)

28. M Joham, W Utschick, JA Nossek, Linear transmit processing in mimocommunications systems. IEEE Transactions on Signal Processing, 53(8),2700–2712 (Aug. 2005)

29. YCB Silva, Adaptive beamforming and power allocation in multi-carriermulticast wireless network. Ph.D dissertation, TU Darmstadt (2008)

30. YCB Silva, A Klein, Linear transmit beamforming techniques for the multi-group multicast scenario. IEEE Transactions on Vehicular Technology, 58,4353–4367 (Oct. 2009)

31. ND Sidiropoulos, TN Davidson, ZQ Luo, Transmit beamforming for physical-layer multicasting. IEEE Transaction on Signal Processing. 54, 2239–2251(2006)

32. E Karipidis, ND Sidiropoulos, ZQ Luo, Quality of service and max-min fairtransmit beamforming to multiple cochannel multicast groups. IEEETransaction on Signal Processing. 56(3), 1268–1279 (2008)

33. JF Sturm, Using sedumi 1.02, a matlab toolbox for optimisation oversymmetric cones. Optimisation Methods and Software. 11-12, 625–653(June 1999)

34. A Scaglione, P Stoica, S Barbarosa, G Giannakis, H Sampath, Optimal designsfor space-time linear precoders and decoders. IEEE Transactions on SignalProcessing, 50(5), 1051–1064 (May 2002). doi:10.1109/78.995062

doi:10.1186/1687-1499-2011-29Cite this article as: Amah and Klein: Non-Regenerative Multi-AntennaMulti-Group Multi-Way Relaying. EURASIP Journal on WirelessCommunications and Networking 2011 2011:29.


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AbstractIntroductionNotations

Broadcast Strategies And System ModelA. Broadcast Strategies1) Unicasting Strategy2) Hybrid Uni/Multicasting Strategy3) Multicasting Strategy

B. Generalised System Model

Sum Rate ExpressionA. Signal to Interference and Noise Ratio1) Unicasting2) Hybrid uni/multicasting3) Multicasting

B. Sum Rate for Asymmetric TrafficC. Sum Rate for Symmetric Traffic

Transceive BeamformingA. Sum Rate MaximisationB. Linear Transceive Beamforming1) Matched Filter2) Zero Forcing3) Minimisation of Mean Square Error

C. Broadcast-Strategy-Aware Transceive BeamformingEquivalent channelSemidefinite Relaxation

Simulation ResultsA. First Scenario: L = 1 and N1 = 2B. Second Scenario: L = 2 and N1 = N2 = 2C. Third Scenario: L = 2 and N1 = N2 = 3

ConclusionAppendixRBD for Receive Beamforming

AcknowledgementsAuthor detailsCompeting interestsReferences

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