BEAMFORMINGDESIGNFORMULTIUSERTWOWAYRELAYINGWITHMIMOAMPLIFYANDFORWARDRELAYS
Jianshu Zhang, Florian Roemer, and Martin Haardt
Ilmenau University of Technology, Communications Research LaboratoryP. O. Box 10 05 65, D98684 Ilmenau, Germany, http://tuilmenau.de/crlemail:
{
jianshu.zhang,
orian.roemer,martin.haardt
}
@tuilmenau.de
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 techniques, twoway 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 multiuser twoway relaying with MIMO AF relays wherea base station (BS) and multiple users (UT) exchange messages viathe relay in this paper. We propose three suboptimal 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 normmaximizing (ANOMAX) transmit strategy provides the best balancebetween complexity and performance, the zeroforcing 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
—
twoway relaying, amplify and forward (AF),MIMO, block diagonalization (BD), algebraic normmaximizing(ANOMAX), zeroforcing dirty paper coding (ZFDPC)
1. INTRODUCTION
Recently, relays have received an increase interest due to their potential 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 ofmultiuserMIMOrelayingtechniques, wheretherelay forwards data to and from multiple users. Prior work on multiuser relay channels focuses on oneway relaying [1], [2]. However,it is known that the twoway relaying technique can compensate thespectral ef
ciency loss in oneway relaying due to the halfduplexconstraint of the relay and therefore uses the radio resources in a particular ef
cient manner [3]. To our knowledge, only a few referencesdeal with multiuser twoway 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 multiuser twoway 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 research 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
. Multiuser twoway relaying with a MIMO amplify and forward relay.transmission delays and require lower hardware complexity than DFrelays.Finding the sumrate optimal strategy involves a nontractableoptimization problem. To avoid this complex problem, we introduce three suboptimal algorithms for computing the transmit andreceive beamforming matrices at the BS as well as the ampli
cation 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 provides 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 matrices and vectors, respectively. The expectation, trace of a matrix,transpose, Hermitian transpose, and MoorePenrose 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
28249781457705397/11/$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 twoway 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 Gaussian (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 transmit 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 transmitted 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 assumption 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 expressed 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
selfinterference
+
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 vector 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 experiences the selfinterference caused by its own transmitted signal. If the BS has perfect channel knowledge, the selfinterference 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
selfinterference
+
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 selfinterference, 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 cumulated achievable data rate at all UTs, respectively. The optimization problem to
nd the relay ampli
cation matrix structure whichmaximizes (7) subject to transmit power constrains in (1) and (4)is nonconvex. To avoid a nontractable optimization problem, weresort to suboptimal 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 transmit 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 suboptimal 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 design 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 theselfinterference as well as the cochannel interference. However,the wellknown 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 MoorePenrose 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 singleantenna userswhich communicate with each other with the help of the relay. Thisscheme is mathematically analogous to multipair twoway relayingin [8] or multioperator twoway AF relaying in [9].Although all proposed schemes in the above papers can be applied, we recommend BD combined with ANOMAX since accordingtoourworkin[9]forthemultioperatortwowayrelayingcase(2UTs per operator) it provides the best performance. Here we brie
yextend this BD ANOMAX method to the multiuser twoway relaying 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 subsystems via the BDdesign of
G
R
and
G
T
. Then, in the second step, for each singlepairtwoway relaying subsystem, 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 decomposition (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 dimensionality constraint that
M
R
>
(2
K
−
2)
.
5. ZFDPCBASEDDESIGN
The multiantenna BS has the ability of jointly encoding its transmitted 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 selfinterference term which corresspondsto the upper left block of
H
eﬀ
can be subtracted from
y
B
. Then, thesystem is further decomposed into twosub systems where the upperright part is equivalent to the uplink of a oneway relay broadcastchannel and the lower left part is equivalent to the downlink of aoneway relay multiple access channel. In the next step, we showhow to design
G
U
,
F
B
, and
W
B
using ZFDPC.ZFDPC is a suboptimal beamforming solution which has beenused in several multiuser 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 channel 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 uppertriangular matrix while the lower leftpart of it is converted into a lowertriangular matrix. Thus, for eachUT, the interference can be canceled by applying a successive interference cancellation (SIC) receiver with perfect knowledge of theinterfering signals. Assuming that the BS has also perfect knowledge 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 nonlinear 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 evaluated via MonteCarlo simulations. The simulated MIMO
at fading 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 averaging 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 pseudoinverse 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 multiplexing 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 multiuser twoway 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 performance, “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].
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