Linear Transceive Filters for Relay Stations with
Multiple Antennas in the Two-Way Relay Channel
Timo Unger, Anja Klein
Institute of Telecommunications, Communications Engineering Lab
Technische Universität Darmstadt
Merckstrasse 25, 64283 Darmstadt, Germany
{t.unger, a.klein}@nt.tu-darmstadt.de
Abstract— This paper considers the two-hop relaying case
where two nodes in a wireless network can communicate with
each other via an intermediate relay station (RS). Assuming
that a RS can only receive and transmit on orthogonal channel
resources, the required resources in amplify-and-forward (AF) relaying are doubled compared to a direct communication between
two nodes, i.e., although two-hop relaying aims at an increase
in spectral efficiency, there exists also a conceptual degradation
of the spectral efficiency by the factor 1/2. Two-way relaying
is a promising protocol in order to compensate for that factor
1/2. In this paper, the two-way relaying protocol is extended
to the case of nodes and RSs with multiple antennas where
both nodes have the same number of antennas and the number
of antennas at the RS is twice as much. This multiple input
multiple output (MIMO) two-way relaying protocol only requires
channel state information (CSI) at the RS which reduces the CSI
signalling overhead in the two-way relay network compared to
previous work. In MIMO two-way relaying, both nodes transmit
simultaneously to the RS which can separate the two signals by
receive processing. The RS rearranges these separated signals in
order to provide each node with the desired signal and finally
applies transmit processing before retransmission. This approach
makes receive processing unnecessary at the two communicating
nodes while the factor 1/2 in the spectral efficiency is still
compensated.
I. I NTRODUCTION
Recently, relaying and in particular cooperative relaying
gain much attention in the wireless communications research
community [1]. This paper considers the two-hop relaying
case where two nodes can communicate with each other via
an intermediate relay station (RS) assuming that a direct
communication between the two nodes is not possible, e.g.,
due to shadowing or limited transmit power. In general, the RS
receives a signal from one node on a first hop and retransmits
a signal addressed to the other node on a second hop. There
are two prominent concepts for this two-hop relaying case:
amplify-and-forward (AF) and decode-and-forward (DF) [2].
In AF, the receive signal at the RS is only amplified and
retransmitted by the RS. In DF, the receive signal is decoded,
re-encoded, and retransmitted by the RS. In the following,
only AF relaying is considered which has a lower complexity.
However, the results of this work can be easily extended to
DF relaying.
Assuming that a RS can only receive and transmit on
orthogonal channel resources, the required resources in AF
relaying are doubled compared to a direct communication
between two nodes, i.e., although two-hop relaying aims at an
increase in spectral efficiency there exists also a conceptual
degradation of the spectral efficiency by the factor 1/2.
There exist several protocols which aim at compensating
this conceptual drawback of two-hop relaying. Common to
all these protocols is that it is not possible to improve the
spectral efficiency of a single two-hop connection between
source and destination, but the overall spectral efficiency of
different two-hop connections. In [3], the authors consider
several two-hop connections with multiple RSs. These RSs
are divided into two groups. While the first group of RSs
receives signals, the second group of RSs transmits signals
on the same channel resource and vice versa. This protocol
significantly increases the spectral efficiency of the network.
Two other protocols are introduced in [4][5]. The first
protocol is named two-path relaying and is similar to
the protocol of [3]. For this paper, the second protocol,
which is named two-way relaying, is of particular interest.
The principle of this cooperative protocol is based on the
framework of network coding [6] in which data packets
from different sources in a multi-node computer network are
jointly encoded at intermediate network nodes, thus saving
network resources. For two-way relaying, two nodes transmit
simultaneously on a first channel resource to a RS which
receives a superposition of both signals. On the second
channel resource, the RS retransmits this superposition. Due
to the broadcast nature of the wireless channel, both nodes
receive that superposition and may detect the desired signal
from the other node by subtracting their own known signal. In
[4], it is shown that the spectral efficiency of two-hop relaying
is significantly increased by the two-way relaying protocol.
However, in order to design an adequate receive filter both
nodes of the two-way relay network require channel state
information (CSI) about their own link to the RS as well as
CSI about the link from the other node to the RS. Exchanging
this CSI requires a high signalling effort. Furthermore, the
investigations in [4][5] are restricted to the case of nodes and
RSs with single antennas.
In this paper, the two-way relaying approach is extended to
nodes and RSs with multiple antennas leading to a multiple
input multiple output (MIMO) two-way relaying protocol.
For the proposed MIMO two-way relaying protocol, both
nodes have the same number of antennas and the number of
antennas at the RS is twice as much. For MIMO systems,
the choice between transmitter and receiver oriented filters is
well-known [7]. Since the RS in the two-way relay channel
is a transmitter as well as a receiver, a transceive filter can be
applied at the RS if CSI is available. Firstly, the RS separates
the signals from both simultaneously transmitting nodes by
receive processing. Secondly, the RS rearranges the separated
signals in order to provide each node with the desired signal.
Thirdly, the RS also applies transmit processing before
retransmitting to the nodes. Thus, receive processing and in
particular subtracting the own signal become unnecessary
at the two communicating nodes, i.e., the effort at the two
communicating nodes is decreased and the CSI signalling
effort in the network can be reduced while maintaining the
increased spectral efficiency of the two-way relay channel. In
this paper, the MIMO two-way relaying protocol is verified
for transceive filters which are based on the Zero Forcing
(ZF) and the Minimum Mean Square Error (MMSE) criteria.
The paper is organized as follows: In Section II, the
system model of MIMO two-way relaying is introduced.
Section III gives the ZF and MMSE transceive filters at the
RS. The performance of the filters is analysed by means of
simulations in Section IV. In Section V, some characteristics
of MIMO two-way relaying are discussed and Section VI
concludes this work.
II. S YSTEM M ODEL
In the following, the communication between two nodes
S1 and S2 is considered which cannot exchange information directly, e.g., due to shadowing conditions, but via an
intermediate RS. S1 and S2 are equipped with M1 and M2
antennas, respectively. For the introduced MIMO two-way
relaying protocol
M1 = M2 = M
(1)
is required while it is assumed that the RS is equipped with
MRS = M1 + M2 = 2M
(2)
antennas.
Data vector x1 = [x11 , . . . , x1M ]T of data symbols x1l , l =
1, . . . , M, shall be transmitted from S1 to S2, and data vector
T
x2 = [x21 , . . . , x2M ] of data symbols x2l , l = 1, . . . , M,
T
shall be transmitted from S2 to S1, where [·] denotes the
transpose. For simplicity, but without loss of generality, the
wireless channel is assumed to be flat fading. Hence, the
channel between Sk, k = 1, 2, and the RS may be described
by the channel matrix


(k)
(k)
h11
...
h1M


..
..
..
,
(3)
H(k) = 
.
.
.


(k)
hMRS 1
(k)
...
(k)
hMRS M
where hmn , m = 1, . . . , MRS and n = 1, . . . , M , are complex
fading coefficients. In Fig. 1, the described relay network is
Fig. 1. Relay network for M1 = 1 antenna at S1, M2 = 1 antenna at S2,
and MRS = 2 antennas at the RS
depicted for the case of M = 1 and MRS = 2. In MIMO
two-way relaying, the data vectors x1 and x2 are exchanged
during two orthogonal time slots. During the first time slot,
S1 and S2 transmit
to the RS. The overall
T simultaneously
T T
is
defined
with covariance matrix
data vector
x
=
x
,
x
1 2
Rx = E xxH where E {·} denotes the expectation. Since
spatial filtering shall only be applied at the RS, only a scalar
transmit filter Q = qI at S1 and S2 is required in order to
fulfill the transmit energy contraint at nodes S1 and S2 which
is given by
(4)
E k qx k22 ≤ ET ,
where I is an identity matrix, k · k2 is the Euclidian norm
of a vector, and ET is the maximum transmit energy shared
between
(1) S1
and S2. Defining the overall channel matrix H =
H , H(2) , the receive vector yRS at the RS is given by
yRS = qHx + nRS ,
(5)
where nRS is an additive white
noise vector with
Gaussian
covariance matrix RnRS = E nRS nH
RS .
In the following, a linear transceive filter G at the RS shall be
designed. This linear transceive filter G is a combination of a
linear receive filter GR and a linear transmit filter GT where
both filters can be determined independently.
The receive vector yRS is multiplied with the linear receive
filter GR resulting in the RS estimation vector
T
x̂RS = x̂TRS,1 , x̂TRS,2 = GR yRS = qGR Hx + GR nRS (6)
with the estimate x̂RS,1 for x1 and the estimate x̂RS,2 for x2 ,
respectively. After the second time slot, S1 should receive an
estimate of data vector x2 and S2 should receive an estimate
of data vector x1 . Therefore, before applying the transmit filter
GT , the RS estimation vector x̂RS is multiplied with the RS
mapping matrix
0M×M
IM
(7)
Gmap =
IM
0M×M
where 0M×M is a null matrix with M rows and M columns.
Note that the RS mapping matrix Gmap is an essential part of
the introduced MIMO two-way relaying protocol since Gmap
ensures that the RS transmits the estimate x̂RS,2 in the direction
of S1 and the estimate x̂RS,1 in the direction of S2. Finally,
Fig. 2.
System model with linear transceive filter G at the RS
the transmit filter GT can be applied yielding the RS transmit
vector
xRS = GyRS = qGHx + GnRS
(8)
of the second time slot with the overall transceive filter
G = GT Gmap GR .
(9)
The RS transmit vector has to fulfill the transmit energy
contraint at the RS
E k xRS k22 ≤ ERS ,
(10)
where ERS is the maximum transmit energy at the RS. Since
spatial filtering shall only be applied at the RS, only a scalar
receive filter P = pI at S1 and S2 is assumed. Note that
the channel matrix from the RS to nodes S1 and S2 is the
transpose HT of channel matrix H assuming that the channel
is constant during two consecutive time slots. In the following,
the estimate for data vector x2 at S1 is termed x̂1 and the
estimate for data vector x1 at S2 is termed
T x̂2 . Then, the
after the scalar
overall estimated data vector x̂ = x̂T1 , x̂T2
receive filter is given by
x̂ = p qHT GHx + HT GnRS + nR ,
(11)
where it is assumed that nR is an additive white
Gaussian
noise vector with covariance matrix RnR = E nR nH
R . The
overall system model is depicted in Fig. 2.
III. L INEAR T RANSCEIVE F ILTERS
One major drawback of the proposed two-way relaying
protocol of [4] is that both nodes S1 and S2 require CSI
about their own link to the RS as well as CSI about the link
from the other node to the RS. Exchanging this CSI requires
a high signalling effort. Compared to this signalling effort, it
is relatively easy to obtain CSI at the RS, e.g., by estimating
the channel at the RS in a time division duplex (TDD) system.
In [8], the impact of knowing CSI at the RS is analysed for
AF relaying. Applying CSI, the scalar amplification factor at
the RS can be adapted to the current channel state which
significantly improves the performance compared to a constant
amplification factor at the RS for unknown CSI. If CSI is
already available at the RS, it is also intuitive to use this CSI
for receive and transmit processing at the RS which motivates
a transceive filter at the RS. Firstly, the RS separates the
signals from nodes S1 and S2 by receive processing. Secondly,
the RS rearranges the separated signals in order to provide
each node with the desired signal. Thirdly, the RS applies
transmit processing before the simultaneous retransmission to
S1 and S2. By introducing this transceive filter at the RS, the
signal processing effort is completely moved from nodes S1
and S2 to the RS and no further CSI is required at nodes S1
and S2 which significantly reduces the signalling overhead for
CSI in the network.
In a linear system, the transceive filter G of (9) is a multiplication of a receive filter GR , the RS mapping matrix Gmap ,
and a transmit filter GT where GR and GT can be determined
independently. In the following, two different linear MIMO
transceive filters G at the RS are derived which are based on
the ZF and MMSE criteria. The filters are based on the results
for transmit and receive filters in [7][9].
A. Zero Forcing Transceive Filter
1) Zero Forcing Receive Filter: For the ZF criterion, the
receive filter GR,ZF at the RS has to be designed such that the
mean squared error of the estimate vector x̂RS for data vector
x is minimized. With the ZF constraint and the transmit power
contraint of (4), the ZF optimization may be formulated as
(12a)
{GR,ZF , qZF } = arg minE k x̂RS − x k22
{GR ,q}
subject to:
x̂RS = x for nRS = 0MRS ×1
E k qx k22 ≤ ET .
(12b)
(12c)
As shown in [9], the ZF optimization expression in (12a) is
not convex. In this case, the Karush-Kuhn-Tucker conditions
[10] are only required to determine the global minimum of
(12a) under the constraints (12b) and (12c) which leads to the
ZF receive filter
−1 H −1
1
HH R−1
(13)
H RnRS
GR,ZF =
nRS H
qZF
with the scalar transmit filter
s
ET
qZF =
(14)
tr {Rx }
equal for both nodes S1 and S2, and tr {·} denotes the sum
of the main diagonal elements of a matrix.
2) Zero Forcing Transmit Filter: The ZF transmit filter
GT,ZF at the RS has to be designed such that the mean squared
error of the estimate vector x̂ for transmit vector Gmap x̂RS is
minimized. With the ZF constraint and the RS transmit power
contraint of (10), the ZF optimization may be formulated as
(15a)
{GT,ZF, pZF } = arg minE k x̂ − Gmap x̂RS k22
{GT ,p}
subject to:
x̂ = x̂RS for nR = 0MRS ×1
E k xRS k22 ≤ ERS .
(15b)
(15c)
According to [9], the ZF transmit filter GZF for (15) is given
by
−1
1 ∗
(16)
H HT H∗
GT,ZF =
pZF
with the scalar receive filter
v n
o
u
u tr (HT H∗ )−1 Rx
t
pZF =
.
(17)
ERS
Since the derived receive and transmit filters GR,ZF and GT,ZF
require the same channel coefficients there also exists a high
potential for saving processing effort at the RS.
B. Mimimum Mean Square Error Transceive Filter
ET/N1=0dB
−1
10
E /N =5dB
T
T
2
subject to: E k qx k2 ≤ ET
T
equal for both nodes S1 and S2.
2) Mimimum Mean Square Error Transmit Filter: The
MMSE transmit filter GT,MMSE at the RS has to be designed
such that the mean squared error of the estimate vector x̂ for
transmit vector Gmap x̂RS is minimized. With the RS transmit
power contraint of (10), the MMSE optimization may be
formulated as
{GT,MMSE, pMMSE } = arg minE k x̂ − Gmap x̂RS k22
{GT ,p}
(21a)
subject to: E k xRS k22 ≤ ERS .
(21b)
According to [9], the MMSE transmit filter GMMSE for (21)
is given by
−1
1
tr {RnR }
∗ T
GT,MMSE =
H H +
I
H∗
(22)
pMMSE
ERS
with the scalar receive filter
v (
)
u
−2
u
tr{RnR }
u tr
∗
T
∗
T
H Rx H
H H + ERS I
u
t
(23)
pMMSE =
ERS
equal for both nodes S1 and S2.
Similar to the ZF filter, there exists a high potential for saving
processing effort at the RS since transmit and receiver filters
require the same channel coefficients.
IV. S IMULATION R ESULTS
In this section, the bit error rate (BER) performance of
the ZF and MMSE transceive filters is evaluated by means
of simulations. It is assumed that nodes S1 and S2 are each
equipped with M = 1 antenna and the RS is equipped with
MRS = 2 antennas according to the requirement in Eq. (2).
1
ET/N1=20dB
ET/N1=25dB
(18b)
As shown in [9], the MMSE optimization expression in (18a)
is not convex. In this case, the Karush-Kuhn-Tucker conditions
[10] are only required to determine the global minimum of
(18a) under the constraint (18b) leading to
−1
2
GR,MMSE = qMMSE Rx HH qMMSE
HRx HH + RnRS
(19)
with the scalar transmit filter
s
ET
qMMSE =
(20)
tr {Rx }
1
E /N =15dB
−2
10
{GR ,q}
1
E /N =10dB
BER
1) Mimimum Mean Square Error Receive Filter: For the
MMSE criterion, the receive filter Gr.MMSE at the RS has to
be designed such that the mean squared error of the estimate
vector x̂RS for data vector x is minimized. Additionally, the
transmit power contraint of (4) at S1 and S2 has to be met.
The constrained MMSE optimization may be formulated as
(18a)
{GR,MMSE , qMMSE } = arg minE k x̂RS − x k22
−3
10
0
5
10
15
20
25
E /N in dB for RS−S2 link
RS
30
35
2
Fig. 3. BER performance at S2 for the ZF transceive filter for different fixed
values of ET /N1 on the first hop
The data symbols of S1 and S2 are BPSK modulated. The
average transmit energy ERS at the RS is equal to the average
transmit energy at the node S1 and S2, i.e., ERS = ET . The
channel coefficients are spatially white and their amplitude is
Rayleigh distributed. Because of the symmetry in the network,
it is sufficient to consider only the BER performance at S2
which receives data from S1. For that purpose, the signal-tonoise ratio (SNR) ET /N1 of the first hop from S1 to the RS
is fixed at a certain value while the BER is depicted depending
on the SNR ERS /N2 of the second hop from the RS to S2
where N1 and N2 are the constant noise power density spectra
of the first and second hop, respectively.
Fig. 3 gives the BER performance of the ZF transceive filter.
The performance depends significantly on the SNR of the first
hop which defines a minimum achievable BER. All curves
have a saturation region where an increase of the SNR on the
second hop does no longer improve the BER performance.
For small ET /N1 values on the first hop, this saturation
region is also achieved for smaller ERS /N2 values on the
second hop, e.g., for ET /N1 = 0dB the BER is constant
for ERS /N2 ≥ 20dB while for ET /N1 = 15dB the BER is
constant for ERS /N2 ≥ 30dB. However, for the high value of
ET /N1 = 25dB the BER performance for ERS /N2 ≤ 25dB
converges to the BER performance of the single transmit filter
introduced in [9]. In this case, the receive filter part at the
RS almost achieves an error-free BER performance and the
overall BER performance of the transceive filter only depends
on the transmit filter part of the transceive filter.
Fig. 4 gives the BER performance of the MMSE transceive
filter. Obviously, the general observations made for the ZF
transceive filter are also valid for the MMSE transceive filter.
Fig. 5 gives a comparison of the BER performance of both
transceive filters for the two extreme values of the SNR on
the first hop, i.e., for ET /N1 = 0dB and ET /N1 = 25dB.
As already known from transmit and receiver oriented MIMO
filters [7], the BER performance of the MMSE filter for low
E /N =0dB
−1
10
T
1
E /N =5dB
BER
T
1
E /N =10dB
T
1
−2
10
ET/N1=15dB
required. For BSs with M antennas, MIMO two-way relaying
can also be extended to a simultaneous communication with M
MTs if the RS is equipped with 2M antennas. Furthermore,
MIMO two-way relaying for nodes and RSs with multiple
antennas is transparent to the applied relaying concept, i.e., it
can be applied for AF relaying as well as for DF relaying.
These characteristics of MIMO two-way relaying make it very
promising for future wireless communications.
VI. C ONCLUSION
E /N =20dB
T
E /N =25dB
T
−3
10
0
1
5
10
15
20
25
E /N in dB for RS−S2 link
RS
30
1
35
2
Fig. 4. BER performance at S2 for the MMSE transceive filter for different
fixed values of ET /N1 on the first hop
ET/N1=0dB
−1
BER
10
−2
10
ET/N1=25dB
−3
10
0
MMSE transceive filter
ZF transceive filter
5
10
15
20
25
E /N in dB for RS−S2 link
RS
30
35
2
Fig. 5. Comparison of BER performance at S2 for the ZF and MMSE
transceive filter for two different fixed values of ET /N1 on the first hop
to moderate SNR is better than the BER performance of the
ZF filter. Only for high SNR, the ZF filter converges to the
MMSE filter. For the transceive filter in Fig. 5, the gain of
the MMSE filter is about 3dB in the considered SNR region.
The MMSE filter shows a better BER performance since it
considers the SNR while the ZF filter implicitly increases the
noise level.
V. C HARACTERISTICS
OF
T WO -WAY MIMO R ELAYING
The concept of MIMO two-way relaying can be easily
integrated into a conventional two-hop relay network without
any changes at base stations (BSs) and mobile terminals
(MTs). Only multiple antennas have to be installed at the RS
and no additional site is required. The typical degradation of
the spectral efficiency by factor 1/2 for two-hop relaying is
avoided since up- and downlink can be processed simultaneously. Transmit and receive processing can be restricted to the
RS and signalling of CSI between the BSs and the MTs is not
In this paper, the MIMO two-way relaying protocol has
been introduced which compensates for the degradation of
the spectral efficiency by the factor 1/2 in conventional AF
two-hop relay networks. Compared to previous work on the
two-way relay channel, MIMO two-way relaying only requires
CSI at the RS which significantly reduces the CSI signalling
overhead in the two-way relay network. In MIMO two-way
relaying, two communicating nodes have the same number of
antennas and the number of antennas at the RS is twice as
much. Both nodes transmit simultaneously to the RS which
can separate the two signals by receive processing. The RS
rearranges these separated signals in order to provide each
node with the desired signal, and applies transmit processing before retransmission. Thus, receive processing becomes
unnecessary at the two communicating nodes. In particular,
the linear transceive filters at the RS according to the ZF
and MSSE criteria are given in this paper where the MMSE
transceive filter outperforms the ZF transceive filter.
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