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Optimal and suboptimal beamforming for multi-operator two-way relaying with a MIMO amplify-and-forward relay

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Optimal and suboptimal beamforming for multi-operator two-way relaying with a MIMO amplify-and-forward relay
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  Optimal and Suboptimal Beamforming forMulti-Operator Two-Way Relaying with a MIMOAmplify-and-Forward Relay  Jianshu Zhang 1  , Nils Bornhorst  2  , Florian Roemer  1  , Martin Haardt  1  , Marius Pesavento 21 Communications Research Laboratory  2 Communication Systems GroupIlmenau University of Technology Technische Universit¨at DarmstadtP. O. Box 100565, D-98694 Ilmenau, Germany Merckstrr. 25, D-64283 Darmstadt, Germany  Abstract —In this work, we consider optimal and suboptimalbeamforming designs in a multi-operator two-way relaying net-work with a multiple-input-multiple-output (MIMO) amplify-and-forward (AF) relay. Such a network is interference limitedand thus, an interference nulling strategy is reasonable. We firstderive the necessary condition for interference nulling and intro-duce a closed-form algebraic solution, i.e., the projection basedseparation of multiple operators (ProBaSeMO). This solution canbe adjusted to satisfy various system design criteria. However,it is only suboptimal. For the design of optimal relay transmitstrategies, we study two QoS design criteria for the network. Thefirst is to minimize the relay transmit power subject to a signal-to-interference-plus-noise ratio (SINR) constraints at each user.The second is the SINR balancing technique with a relay transmitpower constraint. These two problems are non-convex. However,we show that both problems can be efficiently solved using convexapproximation techniques. The simulation results verify the sub-optimality of the ProBaSeMO method when compared to theoptimal designs. However, the ProBaSeMO technique approachesoptimality as the number of relay antennas increases and enjoysa significantly reduced computational complexity.  Index Terms —two-way relaying; MIMO; semidefinite pro-gramming, second-order cone programming. I. I NTRODUCTION Relaying is a means of reducing the deployment cost,enhancing the network capacity and mitigating shadowingeffects. Among various relaying techniques, two-way relayinguses the spectrum in an efficient manner and it can beused to facilitate the resource sharing in wireless networks.For instance, we have studied a resource sharing scenarioin [1] where the spectrum as well as a MIMO AF relay(infrastructure) is shared between multiple operators to enablethe communication between users of different operators. Toaccomplish this form of spectrum and infrastructure sharing,we have proposed the projection-based separation of multipleoperators (ProBaSeMO) scheme (block diagonalization (BD)with algebraic norm-maximizing (ANOMAX) and regularizedBD (RBD) with ANOMAX in [1]). It has been shown that theProBaSeMO method can achieve a significant sharing gain interms of sum rate in the case of two operators.In this work, we consider a similar resource sharing scenarioand deal with the optimal design of relay transmit strategies.We first derive the interference nulling condition. Further- UT (1)1  UT (1)2 UT ( L )1  UT ( L )2 R 1  M  R Fig. 1. Multi-operator two-way relaying system model. Each UT has a singleantenna and the relay has M  R  antennas. more, we show that our ProBaSeMO algorithm [1] fulfillsthis condition and can be extended to the  L  operator case.Afterwards, we investigate two different optimal beamformingdesign criteria. The first design criterion is to minimize therelay transmit power subject to a SINR constraint at eachuser terminal (UT). The second one is to maximize theminimum SINR of the UTs in the network subject to arelay transmit power constraint, i.e., SINR balancing. The twoproblems are non-convex. Nevertheless, we show that bothproblems can be solved efficiently using established convexapproximation techniques. The simulation results show thatthe ProBaSeMO algorithms can approach optimality when thenumber of antennas at the relay increases. Furthermore, ithas less computational complexity compared to the convexoptimization based techniques.II. S YSTEM  M ODEL We consider the system in Fig. 1. Several pairs of UTswhich belong to  L  operators communicate with each otherwith the help of one relay. The relay has  M  R  antennas whileeach UT has a single antenna. We assume that the channelis flat fading. The channel between the  k th user of the  ℓ thoperator and the relay is denoted by  h ( ℓ ) k  ∈ C M  R ( k  ∈ { 1 , 2 } for users,  ℓ  ∈ { 1 , . . . , L }  for operators).The two-way AF relaying protocol consists of two phases:In the first phase, all the UTs transmit their data simultane-ously to the relay. The received signal vector at the relay is  then r  = L  ℓ =12  k =1 h ( ℓ ) k  x ( ℓ ) k  + n R  ∈  C M  R ,  (1)where  x ( ℓ ) k  stands for mutually uncorrelated transmitted sym-bols with zero mean and variance  P  ( ℓ ) k  , ∀ k,ℓ . The zero-mean circularly symmetric complex Gaussian (ZMCSCG)noise vector at the relay is denoted by  n R  ∈  C M  R and E { n R n HR }  =  σ 2R I  M  R . In the second phase, the relay forwardsthe signal to all UTs simultaneously. The signal transmittedby the relay can be expressed as ¯ r  =  G · r .  (2)where  G  ∈  C M  R × M  R is the relay amplification matrix. Thereceived signal  y ( ℓ ) k  at the  k th UT of the  ℓ th operator can beexpressed as y ( ℓ ) k  =  h ( ℓ ) T k  ¯ r +  n ( ℓ ) k =  h ( ℓ ) T k  Gh ( ℓ )3 − k x ( ℓ )3 − k          desired signal + h ( ℓ ) T k  Gh ( ℓ ) k  x ( ℓ ) k              self-interference +  ¯ k =1 , 2¯ ℓ  = ℓ h ( ℓ ) T k  Gh (¯ ℓ )¯ k  x (¯ ℓ )¯ k      inter-operator interference + h ( ℓ ) T k  Gn R  +  n ( ℓ ) k      effective noise ,  (3)where  n ( ℓ ) k  denotes the ZMCSCG noise with variance  σ ( ℓ ) 2 k  .Taking into account that the self-interference can be subtractedfrom the received signal  y ( ℓ ) k  at the receiver, the SINR at the k th UT of the  ℓ th operator can thus be written by η ( ℓ ) k  = E {| h ( ℓ ) T k  Gh ( ℓ )3 − k x ( ℓ )3 − k | 2 } E {|  ¯ k, ¯ ℓ  = ℓ h ( ℓ ) T k  Gh (¯ ℓ )¯ k  x (¯ ℓ )¯ k  | 2 } + E { h ( ℓ ) T k  Gn R  2 } + σ ( ℓ ) 2 k (4)To derive the optimal  G , further algebraic manipulationsare required. The transmit power at the relay can be expandedas E { ¯ r  2 } =  E { Tr { Gr ( Gr ) H }} = Tr  G  k,ℓ P  ( ℓ ) k  h ( ℓ ) k  h ( ℓ ) H k  +  σ 2R I  M  R  G H  =  k,ℓ Tr  P  ( ℓ ) k  Gh ( ℓ ) k  h ( ℓ ) H k  G H  + Tr  σ 2R GG H  =  k,ℓ P  ( ℓ ) k  ( Gh ( ℓ ) k  ) H Gh ( ℓ ) k  +  σ 2R g H g =  g H Cg  (5)where   ·   denotes the Euclidean norm and  g  = vec { G } .Moreover,  C   is a positive definite Hermitian matrix which isdefined as C   =  k,ℓ P  ( ℓ ) k  (( h ( ℓ ) k  h ( ℓ ) H k  ) T ⊗ I  M  R ) +  σ 2R I  M  2R .  (6)The fact that  Tr { Γ ζ  }  = Tr { ζ  Γ }  and  vec { Γ Xζ  }  = ( ζ  T ⊗ Γ )vec { X  }  is used in the derivation. Following a similarprocedure, the SINR  η ( ℓ ) k  can be rewritten as η ( ℓ ) k  =  g H D ( ℓ ) k  gg H ( E  ( ℓ ) k  + F  ( ℓ ) k  ) g +  σ ( ℓ ) 2 k (7)where  D ( ℓ ) k  ,  E  ( ℓ ) k  , and  F  ( ℓ ) k  are defined as D ( ℓ ) k  =  P  ( ℓ ) k  ( h ( ℓ ) T 3 − k  ⊗ h ( ℓ ) T k  ) H ( h ( ℓ ) T 3 − k  ⊗ h ( ℓ ) T k  ) E  ( ℓ ) k  =  ¯ k =1 , 2˜ ℓ  = ℓ P  (˜ ℓ )¯ k  ( h (˜ ℓ ) T ¯ k  ⊗ h ( ℓ ) T k  ) H ( h (˜ ℓ ) T ¯ k  ⊗ h ( ℓ ) T k  ) F  ( ℓ ) k  =  σ 2R ( I  M  R  ⊗  ( h ( ℓ ) k  h ( ℓ ) H k  ) T ) .  (8)The matrices  D ( ℓ ) k  and  E  ( ℓ ) k  are positive semidefinite Her-mitian matrices while the matrices  F  ( ℓ ) k  are positive definiteHermitian matrices,  ∀ k,ℓ .Our goal is threefold. First, we extend the ProBaSeMOconcept to the case of   L  operators. The second goal is tofind the matrix  G  which minimizes the relay power and thethird is to study the SINR balancing technique.III. S UBOPTIMAL AND  O PTIMAL  B EAMFORMING  D ESIGN  A. Interference Nulling Condition and ProBaSeMO To null the inter-operator interference completely, the fol-lowing condition needs to be fulfilled, i.e., g H E  ( ℓ ) k  g  = 0 ,  ∀ k,ℓ.  (9)Equation (9) implies that a common null space has to exist forall E  ( ℓ ) k  . Recalling that E  ( ℓ ) k  are positive semidefinite matrices,it is clear that (9) is equivalent to g H (  k,ℓ, ∀ k,ℓ E  ( ℓ ) k  ) g  =  g H E  a g  = 0 .  (10)Using subspace analysis, it can be shown that the rank of   E  a is equal to  min( M  R , 2( L − 1)) . Thus, to null the interferencecompletely  M  R  >  2( L − 1)  antennas are required at the relay.The ProBaSeMO algorithm is a low-complexity suboptimalrelay transmit strategy which fulfills the constraint in (10). In[1] it is proposed for the two-operator case. In the followingwe extend the concept of ProBaSeMO to  L  operators.The main idea of the ProBaSeMO algorithm is to parallelizethe system design of each operator, i.e., decoupling differentoperators and then revAdesigning the transmit strategy forthe system of each operator separately. Towards this end, theinter-operator interference needs to be canceled. It is observedfrom our scenario that the inter-operator interference is createdin both the first and second transmission phases. To null allthe interference, we find it is useful to decompose the relayamplification matrix  G  into G  =  γ  0  · G 0  =  γ  0  · G T  · G S  · G R  ∈  C M  R × M  R (11)  where  G R  ∈  C LM  R × M  R and  G T  ∈  C M  R × LM  R are filtersdesigned to suppress the inter-operator interference during thefirst and second phase, respectively. The parameter  γ  0  ∈  R + is chosen such that the transmit power constraint at the relayis fulfilled. Moreover, G S  is block diagonal since it representsthe processing performed for each operator individually. Theoverall transmit and receive filter matrices  G T  and  G R  canalso be partitioned as G T  =  G (1)T  , ...,  G ( L )T  ,  G R  =  G (1) T R  , ...,  G ( L ) T R  T (12)To design G T  and G R , a typical interference-nulling routinesuch as the BD approach can be followed [1]. Now we brieflyintroduce how to derive  G ( ℓ )R  using BD. Let us define thecombined channel matrix  ˜ H  ( ℓ ) ∈  C M  R × 2( L − 1) for all UTsexcept for the UTs of the  ℓ th operator as ˜ H  ( ℓ ) =  H  (1) ...  H  ( ℓ − 1) H  ( ℓ +1) ...  H  ( L )  ,  (13)where  H  ( ℓ ) = [  h ( ℓ )1  h ( ℓ )2  ]  ∈  C M  R × 2 is the users’ con-catenated uplink channel matrix of the  ℓ th operator. Then thecolumns of the matrix  G ( ℓ )R  should lie in the left null spaceof   ˜ H  ( ℓ ) so that the signal of the  ℓ th operator will not causeinterference to all the other operators. Let  ˜ L ( ℓ ) = rank {  ˜ H  ( ℓ ) } and define the singular value decomposition (SVD) of   ˜ H  ( ℓ ) as ˜ H  ( ℓ ) = [  ˜ U  ( ℓ )s  ˜ U  ( ℓ )n  ] ˜Σ ( ℓ ) ˜ V   ( ℓ ) H ,  (14)where  ˜ U  ( ℓ )n  contains the last  ( M  R − ˜ L ( ℓ ) )  left singular vectors.Thus,  ˜ U  ( ℓ )n  forms an orthogonal basis for the left null spaceof   ˜ H  ( ℓ ) such that  ˜ U  ( ℓ ) H n  ˜ H  ( ℓ ) =  0 . Then a linear combinationof the rows of   ˜ U  ( ℓ ) H n  is the candidate for matrix  G ( ℓ )R  and wechoose G ( ℓ )R  =  ˜ U  ( ℓ )n  ˜ U  ( ℓ ) H n  ∈ C M  R × M  R .  (15)Due to the reciprocity of the channel, we have G ( ℓ )T  = G ( ℓ ) T R  .Moreover, we deploy the ANOMAX algorithm in [2] to design G S  in this paper.It is also clear that  M  R  >  2( L − 1) M  U  has to be fulfilledso that the left null space of   ˜ H  ( ℓ ) cannot be empty. Since theProBaSeMO algorithm is modularized it can be easily adaptedto various system design criteria, e.g., sum rate maximization,relay power minimization, SINR balancing, etc..  B. Relay Power Minimization In this part, we look for the optimal g  which minimizes thetransmit power at the relay subject to an SINR constraint ateach UT. The optimization problem is expressed as min g g H Cg s.t.  g H D ( ℓ ) k  gg H ( E ( ℓ ) k  + F  ( ℓ ) k  ) g + σ ( ℓ )2 k ≥  γ  ( ℓ ) k  ,  ∀ k,ℓ.  (16)Problem (16) is mathematically similar to the beamformingproblems in [3] and [4] which are in general non-convex. Itcan be further expanded as the following equivalent problem min g g H Cg s.t.  g H B ( ℓ ) k  g ≥  γ  ( ℓ ) k  σ ( ℓ ) 2 k  ,  ∀ k,ℓ.  (17)where  B ( ℓ ) k  =  D ( ℓ ) k  − γ  ( ℓ ) k  ( E  ( ℓ ) k  + F  ( ℓ ) k  ) . Each constraint in(17) is a superlevel set of a quadratic function [5]. Such aset is convex if and only if the quadratic function is concave,i.e.,  B ( ℓ ) k  is negative semi-definite,  ∀ k,ℓ . It is clear that inthis case the feasible set is empty since  g H B ( ℓ ) k  g  ≤  0 ,  ∀ k,ℓ .Hence, problem (17) may not be solvable in polynomial time,but its approximate solution can be obtained by using either thesemi-definite programming (SDP) approach [3] or the iterativesecond-order cone programming (SOCP) approach [4]. In thesequel we will discuss the two approaches.In general, the SDP approach which uses semidefiniterelaxation technique (SDR) works as follows [3]. We introducea new variable  X   = gg H and rewrite problem (17) as min X Tr { CX  } s.t.  Tr { B ( ℓ ) k  X  } ≥  γ  ( ℓ ) k  σ ( ℓ ) 2 k  ,  ∀ k,ℓ X     0 ,  rank { X  }  = 1  (18)where  Tr {·} ,   , and  rank {·}  denote the trace of a matrix,the positive semi-definiteness, and the rank of a matrix,respectively. Dropping the rank-one constraint, problem (18)can be approximated by the following convex SDP problemwhich can be solved efficiently by the interior-point method[5]. min X Tr { CX  } s.t.  Tr { B ( ℓ ) k  X  } ≥  γ  ( ℓ ) k  σ ( ℓ ) 2 k  ,  ∀ k,ℓ X     0  (19)Obviously, problem (19) is a relaxed version of the srcinalproblem (16), i.e., the optimal value of (19) is a lower boundof problem (16). If the optimal solution X  opt  of (19) is rank-one, it is also optimal for the srcinal problem and the optimal g opt  is the principle component of  X  opt . Due to the relaxation, X  opt  is generally not rank-one. Although a rank-one solutionof (19) always exists if the number of constraints in (19) isless or equal to three [6], our problem has always more thanthree constraints, i.e, at least two operators and two UTs peroperator. Thus, we apply the randomization method in [3] toextract the rank-one approximation from  X  opt .Since the SDP solution is in general not optimal for ourproblem, it is worth applying an alternative approach whichis the iterative SOCP method [4]. In the traditional SOCPmethod, the rank-one property of the matrix D ( ℓ ) k  is exploitedand the constraints in (16) are rewritten as   P  ( ℓ ) k  | g H ( h ( ℓ ) T 3 − k  ⊗ h ( ℓ ) T k  ) H |   g H ( E  ( ℓ ) k  + F  ( ℓ ) k  ) g +  σ ( ℓ ) 2 k ≥   γ  ( ℓ ) k  ,  ∀ k,ℓ  (20)  If we introduce ˜ U  ( ℓ ) k  =   σ ( ℓ ) 2 k  0 T 0  ( E  ( ℓ ) k  + F  ( ℓ ) k  )  12 , ˜ g = [1 ,  g T ] T ,  ˜ h ( ℓ ) k  = [0 ,  ( h ( ℓ ) T 3 − k  ⊗ h ( ℓ ) T k  ) ∗ ] T ,  (21)(20) can be rewritten as | ˜ g H ˜ h ( ℓ ) k  | ≥   γ  ( ℓ ) k  /P  ( ℓ ) k   ˜ U  ( ℓ ) H k  ˜ g  ,  ∀ k,ℓ  (22)With the conservative approximation [4] | ˜ g H ˜ h ( ℓ ) k  | ≥  Re  ˜ g H ˜ h ( ℓ ) k   (23)where  Re {·}  denotes the real part, the non-convex part of theconstraint (22) can be strengthened as Re  ˜ g H ˜ h ( ℓ ) k   ≥   γ  ( ℓ ) k  /P  ( ℓ ) k   ˜ U  ( ℓ ) H k  ˜ g  ,  ∀ k,ℓ.  (24)Introducing the auxiliary variable  t  and the matrix ˜ V    =   0  0 T 0  C   12 ,  (25)problem (16) can be approximated by the following convexSOCP problem min t, ˜ g t s.t.   ˜ V   H ˜ g  ≤  t,  ˜ g 1  = 1Re  ˜ g H ˜ h ( ℓ ) k   ≥   γ  ( ℓ ) k  /P  ( ℓ ) k   ˜ U  ( ℓ ) H k  ˜ g  ,  ∀ k,ℓ.  (26)Since replacing (22) by (24) yields a restricted convexfeasible set which is a subset of the srcinal feasible set of the srcinal feasible set of problem (16), it guarantees that theoptimal solution of (26) is always feasible for (16). However,the drawback of this approach is that the solution of (16) maynot be optimal for (26) and it may turn the srcinal feasibleproblem into an infeasible one. Thus, the performance andfeasibility strongly depend on how accurately the non-convexfeasible set of problem (16) is approximated. To improve theconvex approximation, we apply the iterative SOCP approachwhich is proposed in [4]. C. SINR Balancing In this section, we study the SINR balancing problemsubject to a relay power constraint. The optimization problemcan be formulated as max g min ∀ k,ℓ η ( ℓ ) k s.t.  g H Cg ≤  P  R  (27)or equivalently as max g ,t t s.t.  g H Cg ≤  P  R , g H D ( ℓ ) k  gg H ( E ( ℓ ) k  + F  ( ℓ ) k  ) g + σ ( ℓ )2 k ≥  t,  ∀ k,ℓ  (28)where  P  R  is the maximum allowable relay transmit power.Problem (27) is non-convex. Following the idea of SDR in theprevious section, we introduce  X   =  gg H and drop the non-convex rank-one constraint. The problem is then reformulatedinto max X ,t t s.t.  Tr { CX  } ≤  P  R ,  X     0Tr { ( D ( ℓ ) k  −  t ( E  ( ℓ ) k  + F  ( ℓ ) k  )) X  } ≥  tσ ( ℓ ) 2 k  ,  ∀ k,ℓ  (29)Problem (29) is a quasi-convex problem similar as in [7].Hence, it can be solved using the same procedure as in [7], i.e.,using a simple bisection search algorithm in which a feasibilityproblem is solved at each step. Due to the relaxation, thesolution  X  opt  might not be feasible for the srcinal problem.The randomization techniques [3] can still be applied to obtainthe final  g .IV. S IMULATION  R ESULTS In this section we present simulation results only for L  = 2 . The simulated flat fading channels are spatiallyuncorrelated Rayleigh fading channels. The noise variancesat all nodes are the same, i.e.,  σ ( ℓ ) 2 k  =  σ 2R  =  σ 2 , ∀ k,ℓ . Allthe simulation results are obtained by averaging over 1000Monte Carlo runs. “ZF” is a channel inversion technique in[8]. “ProBaSeMO(BA)” is the algorithm derived from theframework of ProBaSeMO. “SDP” is the convex approxi-mation using SDP and randomization technique [3] while“lower bound” is obtained from (19). “iSOCP” is the iterativeSOCP technique and “BiSDR” stands for SDP with rank-oneextraction plus bisection search. −5 0 5 10 15−20−100102030SINR constraint [dB]    A  v  e  r  a  g  e  r  e   l  a  y  p  o  w  e  r   [   d   B   W   ]     ZF M   R =5ProBaSeMO(BA) M   R =5iSOCP M   R =5SDP M   R =5Lower bound M   R =5ZF M   R =8ProBaSeMO(BA) M   R =8iSOCP M   R =8SDP M   R =8Lower bound M   R =8 Fig. 2. Relay transmit power vs. SINR constraint, SNR = 15 dB Fig. 2 shows the relay transmit power vs. a common SINRconstraint with SNR = 15 dB, i.e., the transmit power of theUTs is 15 dB above the noise power level. It can be observedthat the difference of the ProBaSeMO solution to the lowerbound reduces for increasing  M  R . Moreover, the two convexapproximation techniques SDP and iterative SOCP mergewith the lower bound. This implies that both approximationtechniques are accurate enough for our problem.  −5 0 5 10 15−20−100102030SNR [dB]    A  v  e  r  a  g  e  m   i  n   i  m  u  m    S   I   N   R   [   d   B   ]     ZF M   R =4ProBaSeMO(BA) M   R =4BiSDR M   R =4ZF M   R =8ProBaSeMO(BA) M   R =8BiSDR M   R =8 Fig. 3. SINR balancing,  P  R  = 1  W Fig. 3 depicts the results corresponding to the SINR bal-ancing approach where this time, the maximized minimumSINR vs. SNR is shown. The total relay power  P  R  is fixedto unity and thus SNR  = 1 /σ 2 . Again the method basedon convex approximation yields the best results. However,the ProBaSeMO method, which yields competitive results,requires a significantly lower computational complexity.V. C ONCLUSION In this paper, we have studied the beamforming design in amulti-operator two-way relaying network with a MIMO AFrelay. Two system design criteria have been chosen. First,we have minimized the transmit power at the relay subjectto an SINR constraint per user. Second, we have discussedthe SINR balancing problem with a relay power constraint.Both problems are generally non-convex. Thus, to solve theoptimization problems, we have applied convex approximationtechniques. We have introduced a sub-optimal algorithm basedon the ProBaSeMO framework which can be applied in bothdesign criteria cases. Simulation results demonstrate that theProBaSeMO algorithm yields competitive results compared tothe convex approximation techniques especially when a largenumber of antennas are deployed at the relay. However, itrequires much less computational complexity.A CKNOWLEDGMENTS This work has been performed in the framework of the European research project SAPHYRE, which is partlyfunded by the European Union under its FP7 ICT Objec-tive 1.1 - The Network of the Future. This collaborativework is also supported by German Research Foundation orDeutsche Forschungsgemeinschaft (DFG) under contract no.HA 2239/2-1 and GE 1881/4-1. It is also partially supported bythe European Research Council (ERC) Advanced InvestigatorGrants Program under Grant 227477-ROSE.R EFERENCES[1] 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 Summit 2010 , Florence, Italy, June 2010.[2] F. Roemer and M. Haardt, “Algebraic Norm-Maximizing (ANOMAX)transmit strategy for Two-Way relaying with MIMO amplify and forwardrelays,”  IEEE Sig. Proc. Lett. , vol. 16, Oct. 2009.[3] Z.-Q. Luo, W.-K. Ma, A. M.-C. So, Y. Ye, and S. Zhang, “Semidefi-nite relaxation of quadratic optimization problems,”  Signal Processing Magazine , May 2010.[4] N. Bornhorst, M. Pesavento, and A. B. Gershman, “Distributed beam-forming for multi-group multicasting relay networks,”  IEEE Transactionson Signal Processing , vol. 60, pp. 221–232, Jan. 2012.[5] S. Boyd and L. Vandenberghe,  Convex Optimization , Cambridge, U.K.,2004.[6] Y. Huang and D. P. Palomar, “Rank-constrained separable semidefiniteprogramming with applications to optimal beamforming,”  IEEE Trans-actions on Signal Processing , vol. 58, pp. 664–678, Feb. 2010.[7] A. B. Gershman, N. D. Sidiropoulos, S. Shahbazpanahi, M. Bengtsson,and B. Ottersten, “Convex optimization based beamforming,”  SignalProcessing Magazine , May 2010.[8] J. Joung and A. H. 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