Antenna combining for multi-antenna multi-relay channels.

Full text

Page 1

EUROPEAN TRANSACTIONS ON TELECOMMUNICATIONS
Eur. Trans. Telecomms. 2007; 18:617–626
Published online 30 August 2007 in Wiley InterScience
(www.interscience.wiley.com) DOI: 10.1002/ett.1231
Antenna combining for multi-antenna multi-relay channels†
Yijia Fan1∗, Abdulkareem Adinoyi2, John S. Thompson1and Halim Yanikomeroglu2
1Institute for Digital Communications, University of Edinburgh, Edinburgh, EH9 3JL, UK
2Broadband Communications and Wireless Systems (BCWS) centre, Department of Systems and Computer Engineering,
Carleton University, Ottawa, K1S 5B6, Canada
SUMMARY
In this paper we analyse the performance of multiple relay channels when multiple antennas are deployed
only at relays. We apply two antenna diversity techniques at relays, namely maximum ratio combining
(MRC) on receive and transmit beamforming (TB). We show that for both decode-and-forward and amplify-
and-forwardrelayingprotocols,withKrelaysthenetworkcanbedecomposedintoKdiversitychannelseach
with a different channel gain, and that the signals can be effectively combined at the destination. We assume
that the total number of antennas at all relays is fixed at N. With a reasonable power constraint at the relays,
the network capacity will be lower bounded by that of N relay channels each with single antenna, and upper
bounded by that of single relay channel with N antennas. Copyright © 2007 John Wiley & Sons, Ltd.
1. INTRODUCTION
It is widely believed that ad hoc networking [1] or multi-
hop cellular networks [2] are important new concepts for
future generation wireless systems [3], where either mobile
or fixed nodes (often referred to as relays) are used to help
forward the information to the desired user. One advantage
of these structures is that it is possible to unite multiple
relays in the network as a ‘virtual antenna array’ to forward
theinformationcooperatively,whileappropriatecombining
at the destination realises diversity gain. The diversity
achieved in this way is often named as user cooperation
diversity or cooperative diversity [4], as it mimics the
performance advantages of multiple-input multiple-output
(MIMO) systems [5] in exploiting the spatial diversity
of the relay channels. The performance limits of space–
time codes, which can exploit cooperative diversity, are
discussed in References [6–8] for single-antenna relay
networks. For multiple-antenna relay channels where every
terminal in the network can be deployed with multiple
antennas, studies are mainly concentrated on spatial
multiplexing systems References [9–11].
*Correspondence to: Yijia Fan, Institute for Digital Communications, University of Edinburgh, Edinburgh, EH9 3JL, UK. E-mail: y.fan@ed.ac.uk
†A previous edition of this paper has been presented in the 12th European Wireless Conference (EW 2006), Athens, Greece.
In this paper we exploit the spatial diversity of the
relay channels in a different way from space–time coding
based approach. We apply two kinds of antenna combining
techniques at the relay, namely maximum ratio combining
(MRC) [12] for reception and transmit beamforming (TB)
[13] for transmission. Those techniques were often used
in point-to-point single-input multiple-output (SIMO) or
multiple-input single-output (MISO) wireless links, where
either the transmitter or receiver is equipped with multiple
antennas. It has been shown that MRC (TB) is able
to achieve information theoretic upper bound of SIMO
(MISO) systems [14]. In a relay context, we move the
multiple antennas to the relays, while the source and
the destination are only equipped with a single-antenna.
Compared with the single-antenna relay network where
everynodeisequippedwithoneantenna,oursystemmodel
allows certain wired cooperation between the antennas at
the relays, which is clearly a advantage. However, it will be
shownthatourschemescanalsoworkeffectivelyforsingle-
antenna networks. More specifically, we will show that by
applying the antenna combining techniques in the relay
network, a network with K relays can be decomposed into
Copyright © 2007 John Wiley & Sons, Ltd.
Accepted 1 December 2006

Page 2

618
Y. FAN ET AL.
Kdiversitychannelseachwithadifferentchannelgain,and
thesignalsfromallKbranchescanbeeffectivelycombined
at the destination. We derive the capacity bounds for this
signalcombiningtechniquesandouranalysisresultscanbe
applied to both ergodic capacity and outage capacity [15]
performance.Ourinvestigationisbasedonboththedecode-
and-forward (digital) relaying mode, where the relays de-
code,re-encodeandre-transmitthesignals,andtheamplify-
and-forward(analogue)mode,wheretherelaysamplifyand
forward the signals without decoding the message.
Therestofthispaperisorganisedasfollows.InSection2,
the basic system model and assumptions are introduced.
Section 3 introduces the antenna combining techniques.
The capacity performance analysis are made in Section 4.
Section 5 presents and discusses simulation results and
finally, conclusions are drawn in Section 6.
2. SYSTEM MODEL
Weconsideratwo-hopnetworkmodelwithonesource,one
destinationandKrelays.Forsimplerpresentationweignore
thedirectlinkbetweenthesourceanddestination.However,
the extension of all the results to include the direct link is
straightforward. We assume that the source and destination
aredeployedwithsingleantennas,whilerelaykisdeployed
with mkantennas; the total number of antennas at all relays
is fixed to N. This can be expressed as:
K
?
k=1
mk= N
(1)
We restrict our discussion to the case where the channels
are frequency-flat fading. The data transmission is over two
timesslotsusingtwohops.Inthefirsttransmissiontimeslot,
the source broadcasts the signal to all the relay terminals.
The input/output relation for the source to the kth relay is
given by
rk=√ηhks + nk
(2)
where rkis the mk× 1 receive signal vector, and η denotes
thetransmitpoweratthesource.Thescalarsistheunitmean
powertransmitsignalandnkisthemk× 1complexcircular
additivewhiteGaussiannoisevectoratrelaykwithidentity
covariance matrix Imk. The vector hkis the mk× 1 channel
transfer vector from the source to the kth relay. The entries
of hk are independent and identically distributed (i.i.d.)
random variables. In the second hop, each relay processes
itsreceivedsignalsandre-transmitsthemtothedestination.
The signal received at the destination can be written as:
y =
K
?
i=1
gkdk+ nd
(3)
where the 1 × mkvector gkis the channel vector from the
kth relay to the destination, of which each entry is an i.i.d.
randomvariable.Thescalarndisthecomplexadditivewhite
Gaussian noise at the destination with unit variance. The
mk× 1 vector dkis the transmit signal vector at relay k,
which should meet the total transmit power constraint:
?
where ? · ?Fdenotes the Frobenius norm and E[·] denotes
theexpectation.Thispowerconstraintmeansthatthepower
is allocated at each relay in proportion to its number of
antennas. For presentation simplicity we assume here that
the total power at all relays is fixed to be η, i.e. the same
as at the source. However, all the conclusions in the paper
also hold when the total power at all relays is fixed to an
arbitrary constant. We assume a coherent relay channel
configuration context where the kth relay can obtain full
knowledgeofbothbackwardchannelvectorhkandforward
channelvectorgk.Forfaircomparison,wealsoassumethat
for each channel realisation, all the backward and forward
channel coefficients for all N antennas remain the same
regardless of the number of relays K. Figure 1 shows the
system model.
E
?dk?2
F
?
?ηmk
N
(4)
Source
Relay 1
Relay 2
RelayK
1
2
3
N
Destination
Figure 1.
destination are each deployed with one antenna. A total of N
antennas are deployed at K relays. For each channel realisation,
bothbackwardandforwardchannelcoefficientsforallNantennas
remain the same regardless of the number of relays K.
System model for a two-hop network: source and
Copyright © 2007 John Wiley & Sons, Ltd.
Eur. Trans. Telecomms. 2007; 18:617–626
DOI: 10.1002/ett

Page 3

MULTI-ANTENNA MULTI-RELAY CHANNELS
619
3. ANTENNA COMBINING TECHNIQUES IN
RELAY CHANNELS
In this section we apply MRC and TB techniques to the
system model described in Section 2. We discuss both
decode-and-forward and amplify-and-forward relaying
modes below.
3.1. Decode-and-forward relaying
We assume that each relay performs MRC of the received
signals, by multiplying the received signal vector by the
vectorhH
kdenotesthecomplexconjugate
transposeofhk.Thesignalattheoutputoftherelayreceiver
is given by
k/?hk?F,wherehH
˜ rk= s
?
?
?
?η
mk
?
i=1
??hi,k
??2+
mk
?
?
i=1
h∗
i,kni,k
mk
?
i=1
??hi,k
??2
(5)
wherehi,kdenotestheithantennaatrelayk,andni,kdenotes
the noise factor for ith receiver input branch. We denote
h∗
(SNR) at the output of the receiver can be written as:
i,kthecomplex-conjugateofhi,k.Thesignal-to-noiseratio
ρmk
k
= η
mk
?
i=1
??hi,k
??2
(6)
After the relays decode the signals, each relay then
performs TB of the decoded waveform. If we denote the
transmitted signals as tkwith unit variance, the transmitted
signal vector dkfor relay k can be written as
dk=
?ηmk
N
gH
k
?gk?F
tk
(7)
The destination receiver simply detects the combined
signals from all K relays. If we adjust the transmission
data rate so that the signals are correctly decoded at all
the relays (i.e. tk= s), the output signal at the destination
can be written as:
?
N
i=1
y = s
K
?
k=1
?
?
?ηmk
mk
?
??gi,k
??2+ nd= s
K
?
k=1
˜ gk+ nd
(8)
It can be seen from Equation (8) that by applying
antenna diversity schemes at relays, the networks can be
decomposedtoKdiversitychannelseachwithchannelgain
˜ gk.TheoutputSNRatthedestinationreceivercantherefore
be written as:
ρmk
d
=


K
?
k=1
?
?
?
?ηmk
N
mk
?
i=1
??gi,k
??2


2
(9)
Whenalltherelaysaredeployedwithasingleantenna,there
is no traditional MRC gain at the relays and the destination.
However, the destination still observes a set of equal gain
combined [16] amplitude signals from all relays.†Since we
assume that the backward and forward channel coefficients
for each antenna are kept the same for different values of K
and mi, the output SNR at the destination can be rewritten
as:
ρ1
d=η
N
?K
k=1
?
mi
?
i=1
??gi,k
??
?2
(10)
when all the antennas are deployed in one relay (i.e. K = 1
and m1= N), the network can be separated into a SIMO
(source-relay) and a MISO (relay-destination) channel.
Using MRC and TB can achieve the information theoretic
upper bound of each link. The SNR for this case can be
rewritten as:
ρN
d= η
K
?
k=1
mi
?
i=1
??gi,k
??2
(11)
3.2. Amplify-and-forward relaying
For amplify-and-forward relaying, after each relay receiver
performingMRCofthesignalvector,itamplifiesthesignal
(5) by a factor that can meet the power constraint (4). The
amplifying factor can be computed as:
γk=
ηmk
??hi,k
N
η
mk
?
i=1
??2+ 1
(12)
The transmitted signal tkat each relay can be expressed
by tk= γk˜ rk. Note that unlike decode-and-forward mode,
it now becomes a combination of the source signal and the
noise at the relay. The relay then applies TB to tkto form
†Unlike Reference [16], the equal gain combining weights for the relay
channels are applied at the transmitter(s) instead of the receiver.
Copyright © 2007 John Wiley & Sons, Ltd.
Eur. Trans. Telecomms. 2007; 18:617–626
DOI: 10.1002/ett

Page 4

620
Y. FAN ET AL.
the transmit signal vector dk, which now can be expressed
as:
dk=
gH
k
?gk?F
γk˜ rk
(13)
The destination receiver receives the sum of the signals
from all K relays and performs data detection. The output
signal at the destination (3), after some modification, can
be written as:
y = s
K
?
k=1
?
?
?
?
?
?η
h∗
mk
?
i=1
??gi,k
??2
mk
?
i=1
??hi,k
??2
ηmk
??hi,k
N
η
mk
?
i=1
??2+ 1
+
K
?
k=1
mk
?
?
i=1
i,kni,k
mk
?
i=1
??hi,k
??2
?
?
?
?
?
?
mk
?
i=1
??gi,k
??
??2
ηmk
??hi,k
N
η
mk
?
i=1
??2+ 1
?
?
nr
+nd
(14)
where we denote nras the equivalent noise generated from
the relays. It can be seen from Equation (14) that compared
with decode-and-forward mode, the signals can also be
coherently combined at the destination, with a channel gain
whichtakesintoaccountthesourcetorelaychannelgainsat
thecostofadditionalnoisenr.Furthermore,wecanobserve
from nrthat the noise generated at different relays is not
coherently combined at the destination, though the signal
can.Beamformingattherelaysworksonlyforthesignalbut
not for the noise. Therefore, while the signal is enhanced
by the beamforming, the noise generated at the relays is
not. This implies that besides the equal gain combining
gain, beamforming the signal from the different relays can
offer an additional coherent combining gain for reducing
theimpactofthenoisegeneratedatdifferentrelaysspecially
fortheamplify-and-forwardmode.Specifically,theSNRat
the destination can be written as:
ρmk
a
=



K ?
k=1
?
?
?
?η
K ?
mk
?
i=1
??gi,k
mk
?
??2mk
??2
?
i=1
??hi,k
ηmk
mk
?
??2
ηmk
N
η
mk
?
i=1|hi,k|2+1



2
k=1
i=1
??gi,k
N
η
i=1|hi,k|2+1
+ 1
(15)
where we can clearly see an additional coherent combining
gain of the signal power over the noise power generated at
the relays.
Similar to the analysis for decode-and-forward mode,
when all the relays are deployed with single antenna, the
SNR at the destination can be rewritten as:
ρ1
a=
η
?K ?
k=1
K ?
mk
?
mk
?
i=1
??gi,k
??gi,k
????hi,k
??2
??
?
η
N
η
N
η|hi,k|2+1
?2
k=1
i=1
η|hi,k|2+1+ 1
(16)
It can be seen that no maximal ratio combining gain
can be obtained in this case. However, the additional
coherent combining gain and equal gain combining gain
aremaximisedasthenumberofrelaysismaximised.When
all the antennas are deployed in one relay, the SNR can be
rewritten as:
ρN
a=
η
K ?
k=1
mk
?
i=1
??gi,k
??2
η
K ?
k=1
mk
?
i=1
??hi,k
K ?
??2
η
η
K ?
k=1
mk
?
i=1|hi,k|2+1
??2+ 1
η
K ?
k=1
mk
?
i=1|hi,k|2+1
k=1
mk
?
i=1
??gi,k
(17)
In this case there is no equal gain combining gain or
additional coherent combining gain, as all the antennas
belongstoonerelay.However,themaximalratiocombining
gain can be obtained due to the full cooperation of the
antennasattherelay.Ithasbeenshownthatinthisscenario,
using MRC and TB can achieve the information theoretic
upper bound of the single relay channel [17] if the direct
link is ignored.
4. CAPACITY BOUNDS
4.1. Decode-and-forward relaying
The network capacity for decode-and-forward relaying for
each channel realisation can be written as:
Cmk
D= min
?
C1,m1
r
,C2,m2
r
,...,CK,mk
r
,Cmk
d
?
(18)
where Ck,mk
capacity for the source to the kth relay channel, and
Cmk
d
r
= 0.5log2(1 + ρmk
= 0.5log2(1 + ρmk
k) denoting the Shannon
d) denoting the Shannon capacity
Copyright © 2007 John Wiley & Sons, Ltd.
Eur. Trans. Telecomms. 2007; 18:617–626
DOI: 10.1002/ett

Page 5

MULTI-ANTENNA MULTI-RELAY CHANNELS
621
for the relay to destination channels.‡The factor 0.5
denotes the half multiplexing factor compared with non-
relay channels.
We firstly analyse channel capacity for the relays to
destination link by bounding ρmk
the destination.
d, i.e. the output SNR at
Lemma 1.
For any mk, ρ1
d? ρmk
d
? ρN
d.
Proof.See Appendix A.
?
From Lemma 1, we can see that
C1
d? Cmk
d
? CN
d
(19)
where C1
channel when K = N, and CN
the relays to destination channel when K = 1. Now also
considering the capacity for the source to relay links and
extending the analysis to the whole network scenario, we
have the following theorem:
ddenotes the capacity for the relays to destination
ddenotes the capacity for
Theorem1.
asC1
DandforK = 1asCN
Proof.Considering the SNR ρmk
relay link, if we denote it as ρ1
K = 1, it can be shown that
?
Therefore, we have the following:
IfwedenotethenetworkcapacityforK = N
D,foranymk,C1
D? Cmk
D? CN
D.
k
for the source to the kth
nfor K = N and ρN
1for
min
ρ1
n
?
? min?ρmk
k
?? ρN
1
(20)
min
?
C1,1
r ,...,CN,1
r
?
? min
?
C1,m1
r
,...,CK,mk
r
?
? C1,N
r
(21)
Combining Equations (21) and (19), we thus complete the
proof.
?
From the above analysis we have shown that for the
antenna combining techniques discussed in the paper, the
network capacity will be lower bounded by that of N relay
channels each with a single antenna, and upper bounded by
that of a single relay channel with N antennas. This means
that even there are more relays, the increased ‘equal gain
combining’ gain at the destination cannot compensate for
the loss of MRC gain at the relay and the destination when
the numbers of antennas at each relay are reduced.
‡HeretheSNRforthedirectlinkshouldbeincludedinsidethelogfunction
when direct link is accounted for.
4.2. Amplify-and-forward relaying
The network capacity for amplify-and-forward relaying for
each channel realisation can be written as:
Cmk
A= 0.5log2
?1 + ρmk
a
?
(22)
The capacity analysis for the decode-and-forward mode
cannot be directly extended to the amplify-and-forward
mode, as they have different relaying mechanisms. In
fact, the analysis for amplify-and-forward relaying is more
difficult due to the following two reasons: (a) the source-
relay link and the relay-destination link in amplify-and-
forwardmodecannotbeconsideredseparately,asdecoding
is not performed at the relays; (b) the impact of the noise
component generated at the relays is complicated and play
a vital role in the capacity performance.
However, we can conjecture the same conclusion as of
the decode-and-forward mode by imagining an extreme
case,i.e.averyhightransmitpowerlevel.Theperformance
of amplify-and-forward mode in this scenario mimics that
of decode-and-forward mode, as the noise component
generated at the relays becomes negligible. In fact, the
same upper bound for the output SNR at the destination
for amplify-and-forward mode can be made as in Lemma
1 for the decode-and-forward mode. This is stated in the
following lemma:
Lemma 2.
For any mk, ρmk
a
? ρN
a.
Proof. See Appendix B.
?
Based on this lemma, we directly obtain the following
theorem for the amplify-and-forward relaying mode:
Theorem 2.
as CN
If we denote the network capacity for K = 1
A, for any mk, Cmk
A? CN
A.
Thecapacitylowerbound,however,isdifficulttoobtain.
The reason is that unlike decode-and-forward relaying, it is
much more complicated to compare ρmk
additional coherent combining gain among different relays
for amplify-and-forward relaying as stated in Section 3.
However,wecanstillgiveacomparisonbetweenthehighest
achievable capacity for both cases. In the following we
derive two tight (achievable) upper bounds for both ρmk
and ρ1
larger than that for ρ1
a.
a with ρ1
adue to the
a
a, and show that the upper bound for ρmk
a
is strictly
Copyright © 2007 John Wiley & Sons, Ltd.
Eur. Trans. Telecomms. 2007; 18:617–626
DOI: 10.1002/ett