Beamforming design for multi-user two-way relaying with MIMO amplify and forward relays

Beamforming design for multi-user two-way relaying with MIMO amplify and forward relays
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  BEAMFORMINGDESIGNFORMULTI-USERTWO-WAYRELAYINGWITHMIMOAMPLIFYANDFORWARDRELAYS  Jianshu Zhang, Florian Roemer, and Martin Haardt  Ilmenau University of Technology, Communications Research LaboratoryP. O. Box 10 05 65, D-98684 Ilmenau, Germany,  {  jianshu.zhang,  orian.roemer,martin.haardt } ABSTRACT Relays represent a promising approach to extend the cell coverage,combat the strong shadowing effects as well as guarantee the QoSin dense networks. Among the numerous existing relaying tech-niques, two-way relaying uses the radio resource in a particular ef-  cient manner. Moreover, amplify and forward (AF) relays causeless delays and require lower hardware complexity. Therefore, weconsider multi-user two-way relaying with MIMO AF relays wherea base station (BS) and multiple users (UT) exchange messages viathe relay in this paper. We propose three sub-optimal algorithms forcomputing the transmit and receive beamforming matrices at the BSas well as the ampli  cation matrix at the relay. Simulations showthat block diagonalization (BD) combined with the algebraic norm-maximizing (ANOMAX) transmit strategy provides the best balancebetween complexity and performance, the zero-forcing dirty papercoding (ZFDPC) based design can perform well when the system isheavily loaded, and the channel inversion (CI) based design yieldsthe lowest complexity. All three algorithms outperform a recentlyproposed technique from the literature.  Index Terms —  two-way relaying, amplify and forward (AF),MIMO, block diagonalization (BD), algebraic norm-maximizing(ANOMAX), zero-forcing dirty paper coding (ZFDPC) 1. INTRODUCTION Recently, relays have received an increase interest due to their poten-tial abilities of reducing the deployment cost, enhancing the networkcapacity, mitigating shadowing effects, and so on. When placed atthe cell edge, relays can boost the coverage. In such applications, itis likely that each relay has to support multiple users. This motivatesthedevelopment ofmulti-userMIMOrelayingtechniques, wheretherelay forwards data to and from multiple users. Prior work on multi-user relay channels focuses on one-way relaying [1], [2]. However,it is known that the two-way relaying technique can compensate thespectral ef   ciency loss in one-way relaying due to the half-duplexconstraint of the relay and therefore uses the radio resources in a par-ticular ef   cient manner [3]. To our knowledge, only a few referencesdeal with multi-user two-way relaying, which include beamformingwith an AF relay [4], beamforming with a decode and forward (DF)relay [5], and relaying protocols with repeaters [6]. Therefore, weconsider the beamforming design for multi-user two-way relayingin our work. Moreover, we prefer the AF relays which retransmitan ampli  ed version of their received signal since these cause less This work has been performed in the framework of the European re-search project SAPHYRE, which is partly funded by the European Unionunder its FP7 ICT Objective 1.1 - The Network of the Future. BSUT 1 UT K  R 1 M  B 1  M  R Fig.1 . Multi-user two-way relaying with a MIMO amplify and for-ward relay.transmission delays and require lower hardware complexity than DFrelays.Finding the sum-rate optimal strategy involves a non-tractableoptimization problem. To avoid this complex problem, we intro-duce three sub-optimal algorithms for computing the transmit andreceive beamforming matrices at the BS as well as the ampli  ca-tion matrix at the relay. They are based on conventional channelinversion (CI), BD [7] combined with ANOMAX (BD ANOMAX)and ZFDPC (OWR ZFDPC). We also compare our algorithms withthe algorithm in reference [4]. It turns out that BD ANOMAX pro-vides the best balance between complexity and performance, OWRZFDPC can still perform well for large loaded systems, while CIyields the lowest complexity. Notation:  Uppercase and lower case bold letters denote matri-ces and vectors, respectively. The expectation, trace of a matrix,transpose, Hermitian transpose, and Moore-Penrose pseudo inverseare denoted by E {·} , Tr {·} ,  {·} T ,  {·} H , and  {·} + , respectively. The m − by − m identity matrix is  I  m . The Euclidean norm of a vectorand the Frobenius norm of a matrix is denoted by   ·   and   ·  F ,respectively. The Kronecker product is denoted by  ⊗  and blkdiag {·} is a block diagonal matrix containing several matrices. The rank of a matrix is denoted by rank {·}  and vec {·}  stacks the columns of amatrix into a vector. 2. SYSTEMMODEL The scenario under investigation is shown in Figure 1. Due to thepoor quality of the direct channel between the BS and the UTs, theycan only communicate with each other with the help of the relay.Assume that we have K   single antenna UTs. The BS is equippedwith M  B  antennas and the relay has M  R  antennas. For notationalsimplicity, in the rest of our work we assume that M  B  =  K  . Thechannelis  atfading. Thechannelbetweenthe k thuserandtherelay 2824978-1-4577-0539-7/11/$26.00 ©2011 IEEEICASSP 2011  is denoted by h k  ∈ C M  R . The channel between the base station andthe relay is full rank and denoted by H  B  ∈ C M  R × M  B .The two-way AF relaying protocol consists of two transmissionphases: in the   rst phase all the users and the base station transmittheir data simultaneously to the relay. Let the BS transmit the datasymbol vector  d B  = [ d B , 1 ,...,d B ,K ] T ∈  C K using the transmitbeamforming matrix  F  B  ∈  C M  B × K . The data symbols in  d B  areindependently distributed with zero mean and unit variance. Let usfurther assume that  d B ,k  is the symbol transmitted from the BS tothe  k th UT and the relay knows the order of the data streams fromthe BS. The total power at the base station is denoted by  P  B . Thetransmit power constraint can be written as E { F  B d B  2 }  =  Tr { F  B F  HB  } ≤  P  B .  (1)Then, the received signal vector at the relay is given by r  = K  k =1 h k  ·  d k  + H  B F  B d B  + n R  ∈ C M  R ,  (2)where  d k  is the transmitted scalar from the  k th user to the BS and n R  ∈  C M  R is the zero mean circularly symmetric complex Gaus-sian (ZMCSCG) noise with E { n R n HR }  =  σ 2R I  M  R . Moreover, weassume that each user has identical transmit power  P  U   and the trans-mit power constraint is equivalent to  E {| d k | 2 } ≤  P  U .In the second phase, the relay ampli  es the received signal andthen forwards it to all the UTs as well as the BS. The signal trans-mitted by the relay can be expressed as ¯ r  =  γ  0  · G · r .  (3)where  G  ∈  C M  R × M  R is the relay ampli  cation matrix and  γ  0  ∈ R + is chosen such that the transmit power constraint at the relay isful  lled, i.e., E { Tr { ¯ r ¯ r H }}  =  Tr { γ  20  · G { P  U H  U H  HU + P  B H  B F  B F  HB H  HB  +  σ 2R I  M  R } G H }  =  P  R ,  (4)where H  U  = [ h 1 ,..., h K ]  ∈ C M  R × K is the concatenated channelmatrix of all UTs.For notational simplicity, we assume that the reciprocity as-sumption between the   rst- and second- phase channels is valid.This assumption is ful  lled in a TDD system if identical RF chainsare applied. Then the received signal vector at the BS can be ex-pressed as y B  =  W  B ( H  TB ¯ r + n B )=  γ  0 W  B H  TB GH  U d U      useful signal + γ  0 W  B H  TB GH  B F  B d B      self-interference + γ  0 W  B H  TB Gn R  + W  B n B      effective noise ∈ C M  R (5)where  d U  = [ d 1 ,...,d K ] T ∈  C K is the concatenated data vec-tor of all the UTs and  n B  ∈  C M  B is the ZMCSCG noise withE { n B n HB }  =  σ 2B I  M  B . The receive beamforming matrix is denotedby  W  B  ∈  C K × M  B . It can be seen from (5) that the BS only expe-riences the self-interference caused by its own transmitted signal. If the BS has perfect channel knowledge, the self-interference can besubtracted.On the other hand, the received scalar  y k  at the  k th UT can bewritten as y k  =  h T k  ¯ r +  n k  =  γ  0 h T k GH  B f  B ,k d B ,k      useful signal + γ  0 h T k Gh k d k      self-interference + K  m =1 m  = k γ  0 h T k GH  B f  B ,m d B ,m      interference from other streams to other UTs + K  j =1 j  = k γ  0 h T k Gh j d j          interference from other UTs + γ  0 h T k Gn R  +  n k      effective noise (6)where  f  B ,k  is the  k th column of   F  B  and  n k  is ZMCSCG noise ateach UT with identical variance  σ 2U . As can be seen from (6), unlikethe BS, each UT experiences self-interference, interference causedby other UTs, and the interference caused by the signal which istransmitted from the BS but intended for another UT.The overall sum rate of the system could be written as R sum  =  R U  +  R B  (7)where  R B  and  R U  are the achievable data rate at the BS and the cu-mulated achievable data rate at all UTs, respectively. The optimiza-tion problem to   nd the relay ampli  cation matrix structure whichmaximizes (7) subject to transmit power constrains in (1) and (4)is non-convex. To avoid a non-tractable optimization problem, weresort to sub-optimal algorithms instead.In [4], a linear beamforming is proposed such that G  =  γ  1 ( H  TU ) − 1 H  − 1U F  B  =  γ  2 H  − 1B  H  U W  B  =  H  TU ( H  TB ) − 1 (8)where γ  1  and γ  2  are the normalizing coef   cients satisfying the trans-mit power constraint at the relay and the BS, respectively.However, it can be seen that the inverses of  H  U  and H  B  do notalways exist. Hence, this method can hardly be utilized since (8)requires that  M  R  =  M  B  =  K  . Our algorithms in Sections 3, 4, and5 are applicable for a broader range of antenna con  gurations. Wespecify the corresponding dimensionality constraints below.Moreover, to have a common framework for the proposed sub-optimal solutions, we decompose G into G  =  G T G S G R  ∈ C M  R × M  R (9) 3. CHANNELINVERSIONBASEDDESIGN In this section, we introduce a straightforward beamforming de-sign based on channel inversion. Using this method, orthogonalchannels are created between the BS and the UTs for interferencefree communication. This algorithm can ef   ciently eliminate theself-interference as well as the co-channel interference. However,the well-known disadvantage of it is the enhancement of the noisepower.Let us de  ne  H   = [ H  B  H  U ]  ∈  C M  R × ( K + M  B ) . Then thechannel inversion receive beamforming is then given by G R  =  H  + = ( H  H H  ) − 1 H  H (10) 2825  and the transmit beamforming is given by G T  = G TR .In this case, the matrix  G S  is chosen to be a block matrix of the form  G S  =  Π 2  ⊗ I  K  ∈  C 2 · K × 2 · K , where  Π 2  = [ 0 11 0 ]  is theexchange matrix which ensures that the BS and the UTs will notreceive their own transmitted signals. Furthermore, for simplicity,we choose  F  B  =  W  B  =    P  B M  B I  M  B . As can be seen from (10),this channel inversion method requires that  M  R  ≥  2 K  . Moreover,compared to the other algorithms proposed in the following sections,the complexity for calculating the Moore-Penrose pseudo inverse ismuch lower. 4. BDCOMBINEDWITHANOMAX For simplicity, we again choose  F  B  =  W  B  =    P  B M  B I  M  B  in thissection. Let us further  x the order of the users such that the k th usercommunicates with the BS only via the  k th antenna at the BS. Thenthis system can be regarded as multiple pairs of single-antenna userswhich communicate with each other with the help of the relay. Thisscheme is mathematically analogous to multi-pair two-way relayingin [8] or multi-operator two-way AF relaying in [9].Although all proposed schemes in the above papers can be ap-plied, we recommend BD combined with ANOMAX since accord-ingtoourworkin[9]forthemulti-operatortwo-wayrelayingcase(2UTs per operator) it provides the best performance. Here we brie  yextend this BD ANOMAX method to the multi-user two-way relay-ing case.First, partition G T , G S , and G R  as G T  =  G (1)T  , ...,  G ( K )T   ∈ C M  R × KM  R G S  =  blkdiag  G (1)S  , ...,  G ( K )S   ∈ C KM  R × KM  R G R  =  G (1) T R  , ...,  G ( K ) T R  T ∈ C KM  R × M  R (11)where G ( k )T  , G ( k )S  , and G ( k )R  ∈ C M  R × M  R .BD ANOMAX consists of two steps. In the  rst step, the systemis converted into  K   parallel independent sub-systems via the BDdesign of  G R  and G T . Then, in the second step, for each single-pairtwo-way relaying sub-system, we use the ANOMAX algorithm tocalculate G ( k )S  .Let us de  ne the combined channel matrix  ˜ H  ( k ) for all UTsexcept for the  k th UT as ˜ H  ( k ) =  H  (1) ...  H  ( k − 1) H  ( k +1) ...  H  ( K )  ,  (12)where H  ( k ) = [ h B ,k  h k ]  and h B ,k  is the  k th column of  H  B .Let  ˜ L ( k ) =  rank {  ˜ H  ( k ) }  and calculate the singular value de-composition (SVD) ˜ H  ( k ) = [ ˜ U  ( k )s  ˜ U  ( k )n  ] ˜Σ ( k ) ˜ V   ( k ) H .  (13)where  ˜ U  ( k )n  contains the last  ( M  R  −  ˜ L ( k ) )  left singular vectors.Thus,  ˜ U  ( k )n  forms an orthogonal basis for the null space of   ˜ H  ( k ) .Therefore, we choose  G ( k )R  =  ˜ U  ( k )n  ˜ U  ( k ) H n  ∈  C M  R × M  R which isa projection matrix that projects any matrix into the null space of  ˜ H  ( k ) . Due to the channel reciprocity, we can simply set  G ( k )T  = G ( k ) T R  .Next, we de  ne the matrix K  ( k ) β  =  β  (( G ( k )R  h k ) ⊗ ( G ( k ) T T  h B ,k )) , (1 − β  )(( G ( k )R  h B ,k ) ⊗ ( G ( k ) T T  h k )))  .  (14)which is needed to calculate the ANOMAX solution of   G ( k )S  [10].The parameter  β   ∈  [0 , 1]  is a weighting factor.ThenwecomputetheSVDof  K  ( k ) β  as K  ( k ) β  = U  ( k ) β  Σ ( k ) β  V   ( k ) H β  .Let the   rst column of   U  ( k ) β  , i.e., the dominant left singular vectorof  K  ( k ) β  be denoted by u ( k ) β, 1 . According to the ANOMAX concept,the matrix G ( k )S  is then obtained via G ( k )S  =  unvec M  R × M  R { u ( k ) ∗ β, 1  } .  (15)where the operator unvec M  R × M  R {·}  inverts the vec {·}  operationby forming a  M  R -by- M  R  matrix  G ( k )S  . In this paper we use equalweighting and therefore  β   is set to 0.5. This algorithm has the di-mensionality constraint that  M  R  >  (2 K   − 2) . 5. ZFDPCBASEDDESIGN The multi-antenna BS has the ability of jointly encoding its transmit-ted data streams or jointly decoding of its received data streams. Tofurther make use of this capability, we introduce the ZFDPC basedbeamforming design.Let us partition G R  = [ G TB  G TU ] T and assume that G T  = G TR .Moreover, let  L U  =  rank ( H  U )  and de  ne the SVD of  H  U  as H  U  = [ U  U , s  U  U , n ] Σ U V   HU  ∈ C M  R × K .  (16)where U  U , n  contains the last  ¯ L U  =  M  R  − L U  left singular vectors.Thus, with the same reasoning as in Section 4, we choose  G B  = U  U , n U  HU , n  ∈ C M  R × M  R .Furthermore, let us de  ne  G S  =  Π 2  ⊗ I  M  R  ∈  C 2 · M  R × 2 · M  R and  0 K × K  to be the  K  -by- K   matrix with all zero elements. Thenthe concatenated received signal at the BS and all UTs can be writtenas   y B y U   =   γ  0 W  B H  TB GH  B F  B  W  B H  T B G TB G U H  U H  T U  G TU G B H  B F  B  0 K × K      H  eff  ·   d B d U   +  ˜ n ∈ C ( M  B + K ) .  (17)In equation (17), the   rst  M  B  rows represent the received signalat the BS ( y B ). We further assume that the BS has perfect channelknowledge, and thus, the self-interference term which corresspondsto the upper left block of  H  eff   can be subtracted from y B . Then, thesystem is further decomposed into two-sub systems where the upperright part is equivalent to the uplink of a one-way relay broadcastchannel and the lower left part is equivalent to the downlink of aone-way relay multiple access channel. In the next step, we showhow to design G U , F  B , and W  B  using ZFDPC.ZFDPC is a sub-optimal beamforming solution which has beenused in several multi-user MIMO relaying references ([1], [2], [5]).Thus, we will also modify the ZFDPC design for our scenario.First, we apply the QR decomposition and the SVD to the chan-nel matrices H  TU  and G B H  B  respectively, H  TU  = M  U Q U  ∈ C K × M  R ,  (18)where M  U  is a lower triangular matrix and Q U  is a unitary matrix.The singular value decomposition of  G B H  B  is denoted by G B H  B  = U  B Σ B V   HB  ∈ C M  R × M  B .  (19)Then the linear processing matrix G U  can be expressed as: G U  = U  ∗ B Q ∗ U  ∈ C M  R × M  R .  (20) 2826   5051015202530350510152025SNR [dB]    S  u  m   r  a   t  e   [   B   i   t  s   /  s   /   H  z   ]   BD ANOMAXCIOWR ZFDPCToh09 Fig. 2 . Sum rate comparison for  M  R  = 8  and  K   = 2 .Moreover, theprecodingmatrix F  B  ischosenas F  B  =   P  B M  B V   B and the decoding matrix  W  B  is constructed as  W  B  =  F  TB  ∈ C M  B × M  B .Inserting  G U ,  F  B  and  W  B  into (17), the upper right matrix in H  eff   is converted into an upper-triangular matrix while the lower leftpart of it is converted into a lower-triangular matrix. Thus, for eachUT, the interference can be canceled by applying a successive in-terference cancellation (SIC) receiver with perfect knowledge of theinterfering signals. Assuming that the BS has also perfect knowl-edge of the interference signals, it can also utilize a SIC receiver todecode each data stream. Unfortunately, the ZFDPC design has alsoa dimensionality constraint, which means  M  R  ≥  2 K  . Furthermore,since this is a non-linear algorithm, it has the highest computationalcomplexity among the three proposed algorithms. 6. SUM RATE NUMERICAL RESULTS In this section, the performance of the proposed algorithms is eval-uated via Monte-Carlo simulations. The simulated MIMO   at fad-ing channels h k  and H  B  are spatially uncorrelated Rayleigh fadingchannels. The SNRs at all nodes are de  ned as SNR  = 1 /σ 2B  =1 /σ 2R  = 1 /σ 2U . All the simulation results are obtained by averag-ing over 1000 channel realizations. “CI”, “BD ANOMAX”, “OWRZFDPC”, and “Toh09” denote the algorithms in Section 3, 4, 5 and[4], respectively. Note that the curves labeled “Toh09” in our resultsare obtained by using the pseudo-inverse of  H  B  and H  U  in (8).As can be seen from Figure 2, “BD ANOMAX” provides thebest performance and is 8 dB away form “Toh09” in the high SNRregime. The “OWR ZFDPC” curve is as good as “CI” and is close to“BD ANOMAX”. However, it should be noted that “OWR ZFDPC”has the highest complexity. Moreover, all the curves have the sameslope at high SNRs which implies that they possess the same multi-plexing gain.Figure 3 show the system loading when M  R  = 20  and the SNRsat all nodes are 25 dB. It can be seen that due to the dimensionalityconstraint for “BD ANOMAX” and “CI”, there is an in  exion pointafter which increasing the number of UTs will decrease the systemsum rate. For “OWR ZFDPC”, although there seems to be also anin  exion point when the system is heavily loaded (at  K   = 9 ), thesum rate does not drop as quickly as in the case of the other twoalgorithms. 7. CONCLUSION In this paper, we consider multi-user two-way relaying with MIMOAF relays. We propose three different algorithms for computing the 24681010152025303540455055K = M  B    S  u  m   r  a   t  e   [   B   i   t  s   /  s   /   H  z   ]   BD ANOMAXCIOWR ZFDPCToh09 Fig. 3 . Sum rate comparison for  M  R  = 20  and  SNR = 25  dB.transmitandreceivebeamformingmatricesatthebasestationaswellas the linear ampli  cation matrix at the relay. Among those “BDANOMAX” provides the best balance between complexity and per-formance, “OWR ZFDPC” can still perform well in a heavily loadedsystem and “CI” yields the lowest complexity. All the algorithmsoutperform the recently proposed algorithm in [4]. 8. REFERENCES [1] Y. Yu and Y. Hua, “Power allocation for a MIMO relay system withmultiple-antenna users,”  IEEE Transactions on Signal Processing , vol.58, pp. 2823–2835, May 2010.[2] T. Tang, C-B. Chae, R. W. Heath, and S. Cho, “On achievable sumrates of a multiuser MIMO relay channel,” in  Proc. IEEE Int. Symp. on Information Theory , Seattle, USA, July 2006.[3] B. Rankov and A. Wittneben, “Spectral ef   cient protocols for half-duplex fading relay channels,”  IEEE Journal on Selected Areas inCommunications , vol. 25, pp. 379–389, Feb. 2007.[4] S. Toh and D. T. M. Slock, “A linear beamforming scheme for multi-user MIMO af two-phase two-way relaying,” in  IEEE InternationalSymposium on Personal, Indoor and Mobile Radio Communications(PIMRC) , Sept. 2009.[5] C. Esli and A. Wittneben, “Multiuser MIMO two-way relaying forcellular communications,” in  IEEE International Symposium on Per-sonal, Indoor and Mobile Radio Communications (PIMRC) , New Or-leans, LA, Sept. 2008.[6] L. Weng and R. D. Murch, “Muti-user MIMO relay system with self-interference cancellation,” in  IEEE Wireless Communications and Net-working Conference (WCNC) , Mar. 2007.[7] Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing meth-ods for downlink spatial multiplexing in multi-user MIMO channels,”  IEEE Transactions on Signal Processing , vol. 52, pp. 461–471, Feb.2004.[8] J. Joung and A. H. Sayed, “Multiuser two-way amplify-and-forwardrelayprocessingandpowercontrolmethodsforbeamformingsystems,”  IEEE Transactions on Signal Processing , vol. 58, pp. 1833–1846, Mar.2010.[9] F. Roemer, J. Zhang, M. Haardt, and E. Jorswieck, “Spectrum andinfrastructure sharing in wireless networks: A case study with Relay-Assisted communications,” in  Proc. Future Network and Mobile Sum-mit 2010 , Florence, Italy, June 2010.[10] F. Roemer and M. Haardt, “Algebraic Norm-Maximizing (ANOMAX)transmit strategy for Two-Way relaying with MIMO amplify and for-ward relays,”  IEEE Sig. Proc. Lett. , vol. 16, Oct. 2009. 2827
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