Optimal and Suboptimal Beamforming forMultiOperator TwoWay Relaying with a MIMOAmplifyandForward 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, D98694 Ilmenau, Germany Merckstrr. 25, D64283 Darmstadt, Germany
Abstract
—In this work, we consider optimal and suboptimalbeamforming designs in a multioperator twoway relaying network with a multipleinputmultipleoutput (MIMO) amplifyandforward (AF) relay. Such a network is interference limitedand thus, an interference nulling strategy is reasonable. We ﬁrstderive the necessary condition for interference nulling and introduce a closedform 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 signaltointerferenceplusnoise ratio (SINR) constraints at each user.The second is the SINR balancing technique with a relay transmitpower constraint. These two problems are nonconvex. However,we show that both problems can be efﬁciently solved using convexapproximation techniques. The simulation results verify the suboptimality 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
—twoway relaying; MIMO; semideﬁnite programming, secondorder 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, twoway 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 projectionbased separation of multipleoperators (ProBaSeMO) scheme (block diagonalization (BD)with algebraic normmaximizing (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. Multioperator twoway 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 nonconvex. 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 twoway AF relaying protocol consists of two phases:In the ﬁrst phase, all the UTs transmit their data simultaneously 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 symbols with zero mean and variance
P
(
ℓ
)
k
,
∀
k,ℓ
. The zeromean 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
selfinterference
+
¯
k
=1
,
2¯
ℓ
=
ℓ
h
(
ℓ
)
T
k
Gh
(¯
ℓ
)¯
k
x
(¯
ℓ
)¯
k
interoperator 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 selfinterference 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 Hermitian 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 interoperator interference completely, the following 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 lowcomplexity suboptimalrelay transmit strategy which fulﬁlls the constraint in (10). In[1] it is proposed for the twooperator 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, theinteroperator interference needs to be canceled. It is observedfrom our scenario that the interoperator 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 interoperator 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 interferencenulling 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’ concatenated 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 nonconvex. 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 semideﬁnite,
∀
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 thesemideﬁnite programming (SDP) approach [3] or the iterativesecondorder 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 semideﬁniteness, and the rank of a matrix,respectively. Dropping the rankone constraint, problem (18)can be approximated by the following convex SDP problemwhich can be solved efﬁciently by the interiorpoint 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 rankone, 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 rankone. Although a rankone 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 rankone 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 rankone 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 nonconvex 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 nonconvexfeasible 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 nonconvex. Following the idea of SDR in theprevious section, we introduce
X
=
gg
H
and drop the nonconvex rankone 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 quasiconvex 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 approximation using SDP and randomization technique [3] while“lower bound” is obtained from (19). “iSOCP” is the iterativeSOCP technique and “BiSDR” stands for SDP with rankoneextraction 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 balancing 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 amultioperator twoway 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 nonconvex. Thus, to solve theoptimization problems, we have applied convex approximationtechniques. We have introduced a suboptimal 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 Objective 1.1  The Network of the Future. This collaborativework is also supported by German Research Foundation orDeutsche Forschungsgemeinschaft (DFG) under contract no.HA 2239/21 and GE 1881/41. It is also partially supported bythe European Research Council (ERC) Advanced InvestigatorGrants Program under Grant 227477ROSE.R
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