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Performance Analysis for Multi-Way Relaying in Rician
FadingChannels
Citation for published version:Xue, J, Sellathurai, M,
Ratnarajah, T & Ding, Z 2015, 'Performance Analysis for
Multi-Way Relaying inRician Fading Channels', IEEE Transactions on
Communications, vol. 63, no. 11, pp.
4050-4062.https://doi.org/10.1109/TCOMM.2015.2477085
Digital Object Identifier (DOI):10.1109/TCOMM.2015.2477085
Link:Link to publication record in Heriot-Watt Research
Portal
Document Version:Peer reviewed version
Published In:IEEE Transactions on Communications
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Download date: 02. Apr. 2021
https://doi.org/10.1109/TCOMM.2015.2477085https://doi.org/10.1109/TCOMM.2015.2477085https://researchportal.hw.ac.uk/en/publications/d9ea48fb-2283-4e9b-b5e4-fd25b99d15a7
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IEEE TRANSACTION ON COMMUNICATIONS 1
Performance Analysis for Multi-Way Relaying inRician Fading
Channels
J. Xue‡, M. Sellathurai†, T. Ratnarajah‡ and Z. Ding§
‡ Institute for Digital Communications, School of Engineering,
The University of Edinburgh, UK.† Heriot-Watt University,
Edinburgh, UK
§ School of Computing and Communications, Lancaster University,
UK.
Abstract—In this paper, the multi-way relaying scenario
isconsidered with M users who want to exchange their
informationwith each other with the help of N relays (N ≫ M )
amongthem. There are no direct transmission channels between anytwo
users. Particularly all users transmit their signals to allrelays
in the first time slot and M − 1 relays are selectedlater to
broadcast their mixture signals during the followingM − 1 time
slots to all users. Compared to the transmissionwith the help of
single relay, the multi-way relaying scenarioreduces the transmit
time significantly from 2M to M timeslots. Random and
semiorthogonal relays selections are applied.Rician fading channels
are considered between the users andrelays, and analytical
expressions for the outage probabilityand ergodic sum rate for the
proposed relaying protocol aredeveloped by first characterizing the
statistical property of theeffective channel gain based on random
relays selection. Also,the approximation of ergodic sum rate at
high signal-to-noiseratio (SNR) regime is derived. In addition, the
diversity order ofthe system is investigated for both random and
semiorthogonalrelay selections. Meanwhile, it is shown that when
the relays arerandomly separated into L groups of M−1 relays, the
group withmaximum average channel gain can achieve the diversity
order Lwhich will increase when more relays considered in the
scheme.Furthermore, when semiorthogonal selection (SS) algorithm
isapplied to select the relays with semiorthogonal channels, it
isshown that the system will guarantee that all the users can
decodethe others information successfully. Moreover, the maximum
ofchannel gain after semiorthogonal relays selection is
investigatedby using extreme value theory, and tight lower and
upper boundsare derived. Simulation results demonstrate that the
derivedexpressions are accurate.
Index Terms—Cooperative communication, extreme value the-ory,
multi-way relaying, semiorthogonal relays selection.
I. INTRODUCTION
Cooperative communications [1] has triggered enormousresearch
interest in understanding the performance of differentmulti-way
relay channels (MWRCs). The MWRC can beviewed as an extension of
the two-way relay channel (TWRC)[2]–[4] where two users exchange
their information via arelay. Such a channel was first introduced
in [5], where theachievable information rates were developed. In
multi-way
The work of J. Xue and T. Ratnarajah was supported, in part, by
the UKEngineering and Physical Sciences Research Council (EPSRC)
grant fundedby the UK government (No. EP/I037156/2). The work of M.
Sellathurai wassupported by the EPSRC project EP/M014126/1 on Large
Scale Antenna Sys-tems Made Practical: Advanced Signal Processing
for Compact Deployments[LSAS-SP]. The work of Z. Ding was supported
by the UK EPSRC undergrant number EP/L025272/1.
relay scenario, several users try to exchange their informa-tion
with each other with the help of relays, where directlinks between
the source nodes whether exist or are notconsidered either due to
large scale path loss or shadowingeffects. Similar to two-way
relaying, self interference can beremoved by exploring the priori
information at the sourcenodes [6]–[9]. Various relaying protocols,
such as amplify-and-forward (AF), decode-and-forward (DF) or
compress-and-forward (CF), were considered and the achievable
symmetricrate of all users were studied in [5]. In [8], the
authorsinvestigated the capacity of binary multi-way relay
systems.Considering Nakagami-m fading, the performance of multi-way
relay direct-sequence code-division multiple-access (MR-DS-CDMA)
systems was analyzed in [10]. The capacityregion of MWRC with
functional-decode-forward (FDF) wasstudied in [11]. The outage
performance of compute-and-forward (CPF) multi-way relay system was
investigated in[12]. Recently, the authors studied secure
performance of twoway relaying scenario with one pair of source
nodes, one relayand one eavesdropper in [13].
Meanwhile, different feasible types of multi-flow
relayingstrategies, network codings and cooperation schemes
wereanalyzed in [14]–[17]. Multi-way relay communications
werestudied for a group of single-antenna users with
regenerativerelaying strategies in [7]. By using stochastic
geometry andpercolation theory, the authors analyzed the
connectivity ofcooperative ad hoc network with selfishness in [18].
Usingnon-coherent fast frequency hopping (FFH) techniques,
infor-mation exchange among a group of users was studied in
[9]where information exchange could be accomplished withinonly two
time slots regardless of the number of users. Theauthors in [19],
studied a matching framework for coopera-tive networks with
multiple source and destination pairs. In[20], authors studied the
capacity gap for different relayingtechniques and showed that FDF
results in a capacity gap lessthan 12(M−1) bit.
The physical layer network coding (PLNC) [14] was studiedin
[21], [22] and references therein which allow several trans-mitters
to transmit signals simultaneously to the same receiverto improve
overall performance. In [22], PLNC was investi-gated for multi-user
relay channel with multiple source nodes,single relay and single
destination, and a novel decoder wasdesigned that offered the
maximum possible diversity order oftwo. In [23], a novel
cooperation protocol based on complex-field wireless network coding
was developed in a network with
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IEEE TRANSACTION ON COMMUNICATIONS 2
multiple sources and one destination. Meanwhile, relay
selec-tion strategy is inseparable from the cooperative network
cod-ing problem [24], because the relay selection can indeed
bene-fit the network performance from many aspects. In
[25]–[27],relay selection schemes were studied in ad hoc network.
Theauthors studied novel contention-free and contention-based
re-lay selection algorithms for multiple source-destination
systemin [28] where the best relay selection is based on the
channelgains. Considering the multiple relays system with
multiplesources and one destination, the authors in [29]
proposedoptimal and sub-optimal relay selection schemes based onthe
sum capacity maximization criterion. In [30], a relayselection
algorithm, called RSTRA (Relay Selection algorithmcombined
Throughput and Resource allocation), is proposedfor IEEE 802.16m
network in order to maximize the networkthroughput. However,
proposing more effective and practicalmulti-way relaying protocols
is still a hot topic requiring moreinvestigations.
Motivated by the previous works, a new transmission strat-egy of
multi-way relaying protocol has been proposed andinvestigated in
this paper which can reduce the transmissiontime slots and increase
the diversity order. We consider amultiple relaying scenario with
multiple sources and relays,where sources exchange information with
each other withthe help of the selected relays. According to the
coopera-tive transmission strategy proposed in this paper, the
timeconsumption will be reduced significantly. Also, the
diversityorder is equal to the number of randomly separated
relaygroups which will increase with the total number of relaysin
this scheme. By utilizing the statistical property of Ricianfading
channels, we first find the density function of effectivechannel
gains, from which the performance of the proposedmulti-way relaying
protocol can be analyzed by using twoinformation theoretic
criteria, outage probabilities and ergodicsum rates, respectively.
Analytical results are also providedto demonstrate the superior
performance of the proposedprotocol. To guarantee that all the
users can decode theothers messages, semiorthogonal selection
method [31] isapplied in our scenario. The advantages of
semiorthogonalselection can be seen by our simulation results
comparedto random selection. Moreover, the maximum channel
gainafter semiorthogonal relays selection is studied by
extremevalue theory when the number of total relays goes to
in-finity [32]–[34]. The maximum channel gain is bounded bylogN
+log logN +O(log log logN), where N is the numberof all relays. In
addition, the simulation results are shown tomatch the developed
analytical results, which demonstrate theaccuracy of the analytical
results.
Throughout the paper, following notations are adopted.Matrices
and vectors are denoted by bold uppercase and boldlowercase
letters. In denotes the n × n identity matrix and[A]i,j is the (i,
j)th element of matrix A. (·)† denotes theconjugate transpose of a
matrix or vector. CN (µ, σ2) denotesthe circularly symmetric
complex Gaussian distribution withmean µ and variance σ2. tr(·) and
det(·) denote the trace anddeterminant of a matrix, respectively.
E[·] and log(·) denotethe expectation operation and natural
logarithm.
The rest of the paper is organized as follows. The system
model is introduced in Section II. Section III presents the
keyanalytical results of cooperative transmission. The
semiorthog-onal relay selection is introduced in Section IV. In
SectionV, the properties of maximum channel gain are
investigated.Numerical results are discussed in Section VI.
Finally, SectionVII provides the conclusion of this paper.
II. SYSTEM MODEL
Assuming there are M single antenna users, they plan toexchange
their information with each other with help fromrelays, because
there are no direct transmission channelsbetween any two users. In
order to compare the performance ofmulti-way relaying scenario with
the single relay transmissionscheme, at first, a benchmark scheme
without cooperation(i.e., single relay transmission) will be
described. Then, thecooperative scheme with AF strategy will be
proposed andanalyzed where M users communicate with each other viaM
− 1 relays.
Firstly, considering the multi-user transmission scenariowith
help of single relay (only one relay1 in the system), eachuser
needs two time slots to transmit his own informationto all the
other users. In first time slot, one user transmitshis signal to
the relay and the relay broadcast this signal toall the other users
in the second time slot. Following thisstrategy, 2M time slots are
needed for M users sharing theirown information. Secondly, we
consider a multi-way relayingscenario in which M users transmit
their own signal in the firsttime slot simultaneously and N relays
listen, where N ≫Mand PLNC scheme is used at all relays2. As shown
in Fig.1, The proposed cooperative transmission strategy consistsof
two phases. During the first phase, all sources broadcasttheir
messages, where all the relays listen. During the secondphase, (M −
1) relays are collaborating with the sources bybroadcasting their
observations during the (M −1) time slots.Assuming there are M
users, the total time consumption isreduced to M time slots,
compared to 2M time slots oftransmission with single relay. All the
relays use amplify-and-forward (AF) strategy to transmit their
received mixtures.Meanwhile, we assume that all nodes are equipped
with singleantenna and the full channel state information (CSI) is
knownby all the nodes.
III. COOPERATIVE TRANSMISSION
Assuming there are M users, they need 2M time slots toexchange
their messages with help from single relay describedabove. In our
cooperative transmission, the transmission timeconsumption is
reduced to just M time slots. Furthermore, theproperties of the
cooperative transmission will be investigated.
1In this paper, the single relay system has been considered only
in twoplaces. One is in here, where we consider the single relay
system to comparethe time consumption of transmission. Another one
is in the section ofnumerical results where we compare the
performance of ergodic rate in Fig.5. In all the other places,
“single relay” means one (arbitrary) relay from theselected
relays.
2we assume that all the relays use physical layer network coding
whichallows all the users to transmit their signals simultaneously
to the relays inthe same time slot without interweaving with each
other’s signal. However,this topic is beyond the scope of this
paper and more details can be found in[14], [21], [22] and
references therein.
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IEEE TRANSACTION ON COMMUNICATIONS 3
S1
S2
S3 S4
Si
SM
R1
Rk
RM-1
hi,k
Fig. 1. System Model with multi-way relays.
A. Outage Probability
In the cooperative transmission protocol, M users transmittheir
own signal in the first time slot and each relay receivethe
superposition of M signals. Hence, received signal at thenth relay
is given by
yRn =M∑i=1
hi,nxi + µn, n = 1, . . . , N. (1)
where µn ∼ CN (0, 1) denotes the background noise of thenth
relay and hi,n ∼ CN
([Θ]i,n, ε
2)
denotes the channelcoefficient between the ith user and the nth
relay. Each relaynormalizes the received signal and forwards the
mixture whichcan be written as
rn =√Q
yRn√E {|yRn |2}
= η
(M∑i=1
hi,nxi + µn
), (2)
where
η =
√Q√
E{|yRn |2}=
√Q√∑M
i=1 E {|hi,nxi|2}+ 1
denotes the scaling factor of each relay which is used to
ensureE{|rn|2} = Q.
During the next M−1 time slots, the selected M−1 relaysare
invited respectively to transmit their received mixtures.
Thedetails of relay selection will be described in next
section.Hence, during the (M − 1) time slots, received signal at
theith user is given by
y(R)k,i = hk,irk + zk,i
= hk,iη
(M∑i=1
hi,kxi + µk
)+ zk,i, k = 1, . . . ,M − 1,
(3)
where hk,i ∼ CN([Θ]k,i, ε
2)
denotes the channel coefficientbetween the kth relay and the ith
user. zk,i ∼ CN (0, 1)
denotes noise imposed on the ith user at the time of
receivingsignal from the kth relay. Eliminating his own signal of
theith user, the received signal at the ith user is given by
ŷ(R)k,i = hk,iη
M∑j=1,j ̸=i
hj,kxj + µk
+ zk,i. (4)After (M − 1) time slots, the observation at the ith
user isexpressed as, y1,i...yM−1,i
= ηh1,i · · · 0... . . . ...
0 · · · ηhM−1,i
×
h1,1 · · · hM(j ̸=i),1... . . . ...h1,M−1 · · · hM(j
̸=i),M−1
x1...xM(j ̸=i)
+
z1,i + ηµ1h1,i...zM−1,i + ηµM−1hM−1,i
(5)which is written as
ŷ(R)i = DiGis + wi, (6)
where
ŷ(R)i =
y1,i...yM−1,i
, Di = h1,iη · · · 0... . . . ...
0 · · · hM−1,iη
,Gi =
h1,1 · · · hM(j ̸=i),1... . . . ...h1,M−1 · · · hM(j
̸=i),M−1
, s = x1...xM(j ̸=i)
and
wi =
z1,i + ηµ1h1,i...zM−1,i + ηµM−1hM−1,i
. (7)The principle of zero-forcing (ZF) detection is
considered
in this system, because a cooperative network can
outperformnon-cooperative ones at moderate or high SNR, which
moti-vates the use of ZF detection. In particular, at moderate or
highSNR, ZF can achieve performance similar to MMSE, but theuse of
ZF can facilitate performance evaluation significantly[35].
Applying the principle of zero-forcing (ZF) detection,we have(
G†iGi)−1
G†iD−1i y
(R)i = s +
(G†iGi
)−1G†iD
−1i wi
= s + w̃. (8)
Therefore, after M − 1 time slots, the effective channel
gain
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IEEE TRANSACTION ON COMMUNICATIONS 4
at the ith user due to the jth user’s signal is given by
γ(R)i,j =
P
E{
w̃w̃†}
=P
E{(
G†iGi)−1
G†iD−1i wiw
†i
(D−1i
)† Gi (G†iGi)−1}=
P(G†iGi
)−1G†iD
−1i E
{wiw†i
}(D−1i
)† Gi (G†iGi)−1=
P(G†iGi
)−1G†iD
−1i Φ
(D−1i
)† Gi (G†iGi)−1 , (9)where P is the transmit power at each relay
and
Φ = E{
wiw†i}
= E
1 + η
2|h1,i|2 · · · 0...
. . ....
0 · · · 1 + η2|hM−1,i|2
.
Denoting Di =
h1,iη · · · 0... . . . ...0 · · · hM−1,iη
and D−1i =1
h1,iη· · · 0
.... . .
...0 · · · 1hM−1,iη
.The D−1i Φ
(D−1i
)†can be derived as (10) at the top of next
page.Under the assumptions that no relay is scheduled twice
and
that the used relays have good enough outgoing channels
withunity channel gain3, i.e., 1|hk,i|2 = 1, k = 1, . . . ,M − 1,
wehave
D−1i Φ(D−1i
)†=
(1 +
1
η2
)IM−1. (11)
Therefore, to obtain the tractable analytical expression forthe
PDF of γ(R)i,j , we construct an auxiliary signal model
asfollows:
ŷ(R)i = s + q̃ (12)
which has the new noise covariance matrix as
Q̃ =(
G†iGi)−1
G†i
(1 +
1
η2
)IM−1Gi
(G†iGi
)−1=
(1 +
1
η2
)(G†iGi
)−1G†iGi
(G†iGi
)−1=
(1 +
1
η2
)[(G†iGi
)−1]jj
. (13)
3The “unity gain” is assumed so that the used relay selection
strategy inour paper can ensure that the channel gains of outgoing
channels are equal oreven larger than one (by assuming there are
large number of relays, ideallythe number of relays can go to
infinity), to simplify the analysis. Meanwhile,we focus on the
lower bound of the performance achieved by the
proposedprotocol.
Therefore, after M−1 time slots, the effective channel gainat
the ith user can be written as
γ(R)i,j =
P(1 + 1η2
)[(G†iGi
)−1]jj
=ρ[(
G†iGi)−1]
jj
, (14)
where ρ = P(1+ 1
η2
) .Proposition 1. The effective channel gains γ(R)i,j =
ρ[(G†i Gi)
−1]jj
, j = 1, . . . ,M, j ̸= i follow noncentral Chi-
squared distribution and the probability density
function(p.d.f.) can be expressed as
fγ(R)i,j
(γ) =1
2ρε2
(γ
ρ[Θ]2i,j
)− 14e−
[Θ]2i,j+γρ
2ε2 I− 12
([Θ]i,jε2
√γ
ρ
)(15)
where Ia(x) is the modified Bessel function of the first
kind.
Proof: See Appendix A.This proposition is the basis of following
analysis in this
paper and was derived by the random matrix theory ofnoncentral
Wishart matrix.
Proposition 2. The cumulative distribution function (c.d.f.)
ofeffective channel gains, γ(R)i,j , is given by
F(γ(R)i,j ≤ x
)= 1−Q 1
2
([Θ]i,jε
,
√x/ρ
ε
)(16)
where Qβ(a, b) is the generalized Marcum Q-function.
Proof: See Appendix B.By using Proposition 2, the following
proposition can be
derived.
Proposition 3. The outage probability of γ(R)i,j with
thresholdγth is given by
Pout
(γ(R)i,j ≤ γth
)= 1−Q 1
2
([Θ]i,jε
,
√γth/ρ
ε
). (17)
Proof: This can be derived easily from Proposition 2.
B. Ergodic Achievable Rate
The ergodic achievable rate at the ith user due the jth
user’ssignal is given by
Rzf−relayi,j = E{log2
(1 + γ
(R)i,j
)}. (18)
The following proposition presents the analytical expressionof
the ergodic sum rate of the ith user considering all theselected
relays.
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IEEE TRANSACTION ON COMMUNICATIONS 5
D−1i Φ(D−1i
)†=
1
h1,iη· · · 0
.... . .
...0 · · · 1hM−1,iη
1 + η
2|h1,i|2 · · · 0...
. . ....
0 · · · 1 + η2|hM−1,i|2
1
η†h†1,i· · · 0
.... . .
...0 · · · 1
η†h†M−1,i
=
1+η2|h1,i|2
h1,iη· · · 0
.... . .
...0 · · · 1+η
2|hM−1,i|2hM−1,iη
1
η†h†1,i· · · 0
.... . .
...0 · · · 1
η†h†M−1,i
=
1+η2|h1,i|2η2|h1,i|2 · · · 0
.... . .
...0 · · · 1+η
2|hM−1,i|2η2|hM−1,i|2
=
1 +1η2 ×
1|h1,i|2 · · · 0
.... . .
...0 · · · 1 + 1η2 ×
1|hM−1,i|2
. (10)
Proposition 4. The ergodic sum rate of the ith user is
givenby
C =M − 1ln 2
e−[Θ]2i,j
2ε2
∞∑k=0
2−kε−2k[Θ]2ki,j
k!Γ(k + 12
)×G1,33,2
[2ρε2
∣∣∣∣ 12 − k, 1, 11, 0], (19)
where Gm,np,q
[x
∣∣∣∣ a1, · · · , apb1, · · · , bq]
is the Meijer’s G-function [36,
Eq. (9.3)] and Γ(x) is the gamma function [36, Eq. (8.31)].
Proof: We know that
C =E
M−1∑j=1
log2
(1 + γ
(R)i
)=(M − 1)E
{log2
(1 + γ
(R)i
)}=(M − 1)
ln 2
∫ ∞0
G1,22,2
[γ(R)i
∣∣∣∣ 1, 11, 0]fγ(R)i,j
(γ(R)i
)dγ
(R)i .
(20)
Using [36, Eq. (8.445)], the modied Bessel functionI− 12
([Θ]i,jε2
√γρ
)can be expressed as
I− 12
([Θ]i,jε2
√γ
ρ
)=
∞∑k=0
1
k!Γ(k + 12 )
([Θ]i,j2ε2
√γ
ρ
)2k− 12.
Therefore, we derive Proposition 4 with the help of [36,
Eq.(7.813.1)].
To gain better insight into the ergodic sum rate performanceand
reduce the computation complexity, we investigate theergodic sum
rate at the high SNR regime in the followingproposition.
Proposition 5. At high SNR regime, the ergodic sum rate can
be approximated by
Chigh−SNR =(M − 1)ln 2
∞∑k=0
2−kε−2k[Θ]2ki,je−
[Θ]2i,j
2ε21
k!
×[ψ
(k +
1
2
)+ ln(2ρε2)
], (21)
where ψ(x) is the Euler psi function [36, Eq. (8.36)].
Proof: At high SNR regime,
Chigh−SNR ≈ (M − 1)ln 2
∫ ∞0
ln(γ(R)i
)fγ(R)i,j
(γ(R)i )dγ
(R)i ,
(22)
with the help of [36, Eq. (8.445), Eq. (4.352.1)], Proposition5
can be derived after some algebraic manipulations.
C. Diversity Order
Considering the number of relays to be large enough inthis
scenario, we can randomly separate the relays into Ldifferent
groups, L =
⌊N
M−1
⌋where ⌊x⌋ denotes the largest
integer which is smaller than x, and there are M − 1 re-lays in
each group. Based on this, L groups of relays areindependent of
each other. We denote the average channelgain in each group as
{γ(R)i,1 , γ
(R)i,2 , . . . , γ
(R)i,L
}where γ(R)i,n
denote the average channel gains (which can be seen as
thechannel gain of the channel between an arbitrary groupedrelay
and the ith user) in group n. Considering the orderstatistics and
assuming γmin = min
{γ(R)i,1 , γ
(R)i,2 , . . . , γ
(R)i,L
}and γmax = max
{γ(R)i,1 , γ
(R)i,2 , . . . , γ
(R)i,L
}, the p.d.f. of γmin
and γmax are given by
fγ(R)i,j
(γmin) = Lfγ(R)i,j(γ)[1− F
γ(R)i,j
(γ)]L−1
;
fγ(R)i,j
(γmax) = Lfγ(R)i,j(γ)[Fγ(R)i,j
(γ)]L−1
. (23)
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IEEE TRANSACTION ON COMMUNICATIONS 6
Proposition 6. Considering random separation, the
outageprobability of the relay group with maximum channel gainγmax
is given by
Pout(γmax ≤ γth) =
[1−Q 1
2
([Θ]i,jε
,
√γth/ρ
ε
)]L.
(24)
Proof: This can be derived by using eq. (23).Considering the
definition of marcum Q-function and the
basic integration property of boundary, there exist real
num-bers t ≤ T , so that
t
(√γth/ρ
ε
)L≤
[1−Q 1
2
([Θ]i,jε
,
√γth/ρ
ε
)]L
≤ T
(√γth/ρ
ε
)Lwhere it shows the diversity order is L. The algebraic
manip-ulations and proof are omitted here. It is worth to notice
thatthe diversity order will increase if more relays are involvedin
the scheme. Moreover, the diversity order based on therandom
selection is a lower bound of the diversity order ofusing
semiorthogonal selection.
IV. SEMIORTHOGONAL RELAY SELECTION
In this section, we consider how to select M − 1 relaysto
construct the full rank channel matrix HS . Semiorthogonalselection
(SS) is applied which can select the relay with thebest channel
gain and all the selected relays orthogonal toeach other as much as
possible. Full rank channel matrix willguarantee all the users can
decode the messages of others suc-cessfully and can potentially
provide some fairness among themultiple source nodes. Because of
these, the semiorthogonalselection algorithm [31] is applied in the
form of pseudo-code.
Algorithm 1 Semiorthogonal Relays Selection1: procedure
SEMIORTHOGONAL RELAYS SELECTION2: Initial: R = ∅, H = h1, . . . ,hN
, where R is the set of
selected relays, ∅ is the empty set, Sβ is the set of index
ofsubchannel in the βth selection and hi is the subchannelvector
from each relay to all users;
3: Calculation: g1 = h1, gi = hi −∑i−1j=1
g†jhi
∥gj∥2gj ,where the component of hi orthogonal to the
subspacewhich is spanned by vectors {g1,g2, . . . ,gi−1};
4: Select the βth relay: k = argmaxi∈Sβ ∥gi∥, R← R∪k, HS(:, β) =
hk is the βth column of HS , H(:, k) = 0,gβ = gk;
5: If size of set R is less than M − 1, improvethe set of index
Sβ+1 for next selection by Sβ+1 ={λ ∈ Sβ , λ ̸= k,
|h†λgβ |∥hλ∥∥gβ∥ < α
}, β ← β + 1, where
α = 0.4 [31]. If Sβ+1 ̸= ∅, go to Step 3);6: Else Quit
After repeating step two to four for M − 1 times, we selectM−1
relays from N relays and construct the full rank channel
matrix HS . The reason is that gi, i = 1, 2, . . . ,M − 1
areorthogonal vectors created by step 3) in the pseudo-code. Instep
5), we improve the selection index set Sβ by droppingoff the
subchannels that are not semiorthogonal to one ofthe g1, . . .
,gβ−1 by the condition4
|h†λgβ |∥hλ∥∥gβ∥ = cos θ < α,
where θ is the angle between vectors hλ and gβ . When HSis full
rank, all the users can decode the other M − 1 users’information by
solving the M−1 equations of received mixedsignals.
In the following, we analyze the computational complexityof the
relays selection above. The computational complexityin each step
can be given by
• It takes 12M(i − 1) real multiplications and (10M −3)(i− 1) +
2M real additions to compute gi in step 3).Assuming Ni is the size
of Sβ , The total flop count inthis step is (22M − 3)(i− 1)Ni +
2MNi.
• In step 4), it takes Ni(2M + 1) real multiplications
andNi(2M−1) real additions to compute all ∥gi∥ for i ∈ Sβ .In
addition, it takes Ni − 1 real comparisons to select arelay. The
total flop count is (4NiM) + (Ni − 1) in thisstep.
• In step 5), during the ith relay selection, it takes (Ni
−1)(8M + 4) real multiplications, (Ni − 1)(8M − 3) realadditions
and Ni−1 real comparisons to compute Sβ+1.Thus the flop count is
(Ni − 1)(16M + 1) in this step.
Since the exact closed-form expression of Ni is unknown,
theexact flop count of the relays selection could be calculatedby
simulation. However, it should be noted that Ni ≤ N andN ≥ M in our
system. In this way, the upper bound of theflop count of the relays
selection can be given by
ε ≤M−1∑i=1
[(22M − 3)(i− 1)N + (N − 1)(16M + 1)
+6NM +N − 1]
=1
2
(4 + 28M − 32M2 − 10N + 13MN − 25M2N
+22M3N)
=O(M3N
), (25)
where ε denotes the flop count of the relays selection.According
to the Lemma 2 in [31], the average channel
gain between the βth selected relay and the users, γβ , is
lowerbounded by
γβ >∥gβ∥2
1 + (M−2)4α2
1−(M−2)α2. (26)
Considering the lower bound in-eq. (26) and the channel gainof
eq. (14), the semiorthogonal selection always chooses theith relay
which has ∥gmax∥ and γmax first. Analyzing thestatistical
characters of γmax can help us to understand theperformance of the
system, because it will determine theproperties of the whole system
when the number of relays
4It should be noticed that α changes from 0.2 to 0.4 when the
total numberof relays changes from 100000 to 100. It means we
should relax the conditionof α when the searching space is just
hundreds or less relays. α = 0.4 hasbeen chosen in this paper
according to the system assumption and the resultshave shown that
this condition can be satisfied in this system.
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IEEE TRANSACTION ON COMMUNICATIONS 7
is large enough. In next section, extreme value theory will
beapplied to get deep insight of the properties of γmax based
onsemiorthogonal relays selection.
V. MAXIMUM CHANNEL GAIN ANALYSIS
In the following, the asymptotic behavior of the distributionof
the maximum channel gain γmax of the best relay isinvestigated.
Extreme value theory [34], [38]–[40] is used toevaluate the upper
and lower bounds of γmax. First, it is provedthat the p.d.f. of
γmax converges to Gumbel distribution as asufficient condition of
using extreme value theory. Second,the unique root x∗ for the
equation 1 − Fγi,j (x∗) = 1N isderived. Finally, the value of γmax
can be bounded by theunique solution of x∗. Meanwhile, the bounds
of ergodic rateis derived based on the bounds of γmax.
Generally speaking, extreme value theory is used to dealwith
extreme values, such as maxima or minima of
asymptoticdistributions. Assuming γi,j , j = 1, . . . , N are N
i.i.d randomvariables of the effective channel gains from the ith
user to thejth relay (equally as the channel from the relay to the
user).Different to the previous works the addressed variable is not
aChi-square variable, but the non-central Chi-square variable.
By extreme value theory [39], [40], if there exist constantsa ∈
R, b > 0, and some non-degenerate distribution functionG(x) such
that the distribution of γmax−ab converges to G(x),then G(x)
converges to one of the three standard extremevalue distributions:
Frechet, Webull, and Gumbel distributions,where γmax = max{γi,1, .
. . , γi,N}.
There are only three possible non-degenerate limiting
dis-tributions for maxima, which can be expressed as
• G(x) = e−e−x
;• G(x) = e−x
−αu(x), α > 0;
• G(x) =
{e−(−x)
α
, α > 0, x ≤ 0;1, x ≥ 0.
where u(x) is the step function.The distribution of γi,j , F
(x), determines the exact lim-
iting distribution. A distribution function F (x) belongs tothe
domain of attraction of the limiting distribution, if
thatdistribution function F (x) results in one limiting
distributionfor extreme.
Lemma 1. (Gnedenko, 1947) Assume x1, x2, . . . , xn are
i.i.d.random variables with distribution function F (x). Defineψ(x)
= sup{x : F (x) < 1}. Let there be a real numberx1 such that,
for all x1 ≤ x ≤ ψ(x), f(x) = F ′(x) andF ′′(x) exist and f(x) ̸=
0. If
limx→ψ(x)
d
dx
[1− F (x)f(x)
]= 0, (27)
then there exist constants a and b > 0 such that
γmax−abuniformly converges in distribution to a normalized
Gumbelrandom variable as n→∞. The normalizing constants a andb are
determined by
a = F−1(1− 1
N
),
b = F−1(1− 1
Ne
)− F−1
(1− 1
N
). (28)
where F−1(x) = inf{y : F (y) ≥ x}.
For a random variable X with the normalized Gumbeldistribution,
whose distribution function is given by
G(x) = e−e−x, −∞ < x 0 when
ρ < 12ε2 ⇔P(
1+ 1η2
) < 12ε2 .It turns out that the class of distribution
functions for our
scenario in this paper is the type of normalized
Gumbeldistribution as N → ∞. Therefore, we further look
intosufficient conditions on the distribution of γi.j , such that
thedistribution of maximum is Gumbel distribution.
Given the existence of limit of the growth function, we alsoneed
to find x∗ which is the unique root for the equation1− Fγi,j (x∗) =
1N and it will be used to bound the value ofγi,j [39]. It should be
noticed that x∗ is unique because thec.d.f. Fγi,j (x) is continuous
and strictly increasing for x ≥ 0.
Proposition 8. The maximum value of channel gains, γi,j , j =1,
. . . , N which are i.i.d. random variables satisfies
P
(logN − log logN +O(log log logN)
≤ max1≤j≤N
γi,j ≤ logN + log logN +O(log log logN))
> 1−O(
1
logN
). (32)
Proof: See Appendix D.The proposition derives lower and upper
bound of the
maximum value of the channel gain after semiorthogonalrelays
selection. It is obvious that the performance of thesystem is a
monotone increasing function depending on thenumber of total
relays. The bounds, derived in Proposition8, are significant for
analyzing the properties of the system.Such extreme value results
can be used to bound the outage
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IEEE TRANSACTION ON COMMUNICATIONS 8
probability and ergodic rate, such as the rate based on
themaximum channel gain is bounded by
P
(log(1 + logN − log logN) ≤ C
(max
1≤j≤Nγi,j
)≤ log(1 + logN + log logN)
)> 1−O (− log logN) .
(33)
VI. NUMERICAL RESULTS
In this section, we provide the analytical results derivedin the
previous sections which are verified by Monte Carlosimulations.
Note that in all simulations, unless otherwisespecified, we assume
that K = 10, [Θ]i,j =
√KK+1 and
ε =√
1K+1 .
Fig. 2 shows the outage probability of a random relay in
themulti-way relaying system for different values of factor K
withthreshold γth = 5dB based on Proposition 3. It is seen thatthe
analytical results are in perfect agreement with the MonteCarlo
simulation results, confirming the correctness of theanalytical
expressions. The outage probability decreases withincreasing K but
changes slowly when K is small. It is be-cause the stronger
line-of-sight will improve the performanceof the system when
considering the single input single output(SISO) scenario. When
[Θ]i,j =
√KK+1 ≃ 1 as K →∞, the
outage probability converges to 1−Q 12
(1ε ,
√γth/ρ
ε
), which
is a lower bound when K tends to infinity. Moreover, the slopeof
the outage curves declines when K decreases in the lowSNR regime
which also fits our expectation when consideringRician fading
channels. However, all the curves will have thesame slope at high
SNR regime, which means the LOS factorK does not affect the slopes
of the outage curves when SNRis large.
Fig. 3 depicts the ergodic sum rate of multi-way relayingsystem
with different numbers of users based on Proposi-tion 4 and
high-SNR approximation of ergodic sum rate inProposition 5. The
ergodic sum rate increases when moreusers are in this system, but
the decoding becomes morecomplex. However, the effect of M on
ergodic sum ratereduces when M increases. The ergodic sum rate
increasessharply when both of SNR and M are large. In addition,
thehigh SNR approximation works quite well when SNR is
large,especially the computation complexity is reduced
significantlythat provides significant computational advantage.
Moreover,the slopes of the ergodic sum rate curves can be derived
bythe high SNR approximations.
Fig. 4 shows the performance comparison between relaysselection
based on semiorthogonal, random relays selectionmethods and
exhaustive search5. The special case is presentedwhen M = 3 and γth
= 10dB. One of the straightforwardstrategies for maximizing the sum
rate is to carry out ex-haustive search, whereas our semiorthogonal
approach yieldsless computational complexity. However, it shows
that the
5Here, we define the outage probability of the whole system as:
the systemis in outage if and only if the maximum channel gain (the
relay with the bestchannel) is in outage.
exhaustive search performs better when SNR increases6, butthere
is no obvious advantage at low SNR regime comparedto semiorthogonal
selection. Meanwhile, the performance isimproved by the
semiorthogonal relays selection method com-pared to random
selection which is because the semiorthogonalmethod selects the
relays with the channels which betweenthe relays and users are as
orthogonal as possible. This resultconfirms that the users can
decode their messages correctly andimprove the outage probability
performance when semiorthog-onal selection is applied. It is
obvious that the semiorthogonalselection method will be more
effective when there are morecandidate relays to choose from.
Meanwhile, Fig. 5 presentsthe comparison of ergodic rate between
general single relaysystem (only one relay in the system) and the
selected multi-relay scheme during the same time slots.
It is worth to notice that Fig. 2 and Fig. 4 are based
ondifferent scenarios. Fig. 2 presents the outage probability
ofaverage channel gain for unbiased randomly selected relays.It
means that we consider the outage probability of single av-erage
channel gain of the system. Also, the outage probabilityshown in
Fig. 2 is independent of relay selection which meansit has the same
properties as single relay system. On the otherhand, Fig. 4 is the
outage probability when the whole group ofselected relays are
considered, according to the semiorthogonalrelays selection, which
means the curves shown in Fig. 4 arethe outage probability of the
whole system.
Considering the multiple relay scenario as in Fig. 4, the
out-age probability is presented for different values of parameterK
with M = 5, N = 10 and threshold γth = 10dB in Fig. 6. Itis shown
that the outage probability increases with K, but theeffect of K
reduces when K is large. When the whole systemwith multiple sources
and relays is considered, the increasingK will degrade the
performance of the system. In this case, itis same as the multiple
input multiple output (MIMO) systemwhere the Rician factor K
represents the ratio between thedeterministic (specular) and the
random (scattered) energies.The performance will decrease with K,
because the increasein K emphasizes the deterministic part of the
channels but thedeterministic channels are of rank 1.
The upper and lower bound of the maximum channel gainγi,j , j =
1, 2, . . . , N based on the formula (32) are presentedin Fig. 7.
The difference between the lower bound and upperbound is less than
3 when N = 200, which means the twobounds which have been derived
by extreme value theory aretight. Also, from the two bounds, it can
be noticed that themaxj γi,j , j = 1, 2, . . . , N increases
quickly when N is lessthan 60, but will converge when N goes to
infinity. From thesimulation result, It shows the maximum channel
gain close tothe upper bound when N is more, and converge to the
lowerbound when N is large.
Fig. 8 presents the upper and lower bounds of ergodicrate based
on the bounds of maxj γi,j , j = 1, 2, . . . , N withdifferent N .
It should be noticed that the curves in Fig. 8are the rates of the
single maximum channel gain calculatedusing lower and upper bounds
provided in formula (32). It is
6The exhaustive search achieves the optimal and largest
diversity gain. Weneed to point out that our relay protocol is not
diversity optimal, i.e., theremight be a loss of diversity
gain.
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IEEE TRANSACTION ON COMMUNICATIONS 9
expected that the bounds of ergodic sum rate of the system
isequal to M − 1 times of the values in the figure, becausethe
channel gain of each selected relay will be close tomaxj γi,j , j =
1, 2, . . . , N when N is large enough. Thedifference between the
upper and lower bounds is less than0.6. Moreover, the slopes of the
bounds converge to zero whenN increases.
VII. CONCLUSIONS
In this paper, multi-way relaying scenario was studied
withmultiple sources and relays. The new scenario presented inthis
paper reduces the transmit time significantly compared tothe
traditional single relay transmission. In order to reduce
thetransmit time, M − 1 relays were selected to help M usersto
exchange their information. For random relays selection,the
analytical expression of outage probability and ergodicsum rate
were derived based on the statistical property ofthe average
channel gain. Meanwhile, the approximation ofergodic sum rate was
investigated at high SNR regime to gainbetter insight into this
system and simplify the calculation.Based on our network coding
scheme, the multi-way relayingscenario has achieved diversity order
of L which increaseswith the total number of relays and is a lower
bound of thediversity order based on semiorthogonal selection.
Moreover,the semiorthogonal relays selection method was applied
toselect the relays to guarantee that all the users can
decodeothers’ information and improve the properties of the
system.In addition, the performance of random and semiorthogonal
re-lays selection methods were compared through outage
proba-bility. Furthermore, the maximum channel gain was studied
byextreme value theory and tight upper and lower bounds
werederived. Especially, the maximum channel gain is boundedby logN
+ log logN + O(log log logN), where N is thetotal number of relays.
The simulation and analytical resultsshow that the multi-way
relaying protocol not only reducesthe transmission time, but also
improves system properties.
APPENDIX APROOF OF PROPOSITION 1
Suppose G̃i is the matrix Gi without jth column gj , wehave
γ(R)i,j =
ρ[(G†iGi
)−1]jj
= ρdet(G†iGi
)det(G̃i
†G̃i
) , (34)using the property of the block matrices determinant, we
have
γ(R)i,j = ρ
[g†jgj − g
†jG̃i
(G̃i
†G̃i
)−1G̃i
†gj
]= ρg†j [IM−1 −PM−1]gj , (35)
where
PM−1 = G̃i
(G̃i
†G̃i
)−1G̃i
†. (36)
We note that matrix (IM−1 − PM−1) is a Hermitianmatrix,
perpendicular to matrix G̃i
†and independent of gj .
Considering Gi ∼ CN(Θ, ε2I
), g†j [IM−1 −PM−1]gj is
distributed as noncentral Wishart distribution W1(1,
ε2I,Ω),where Ω = Θ†Θ is the noncentral parameter, i.e.,
α = g†j [IM−1 −PM−1]gj
is a noncentral Chi-squared variable distributed as
f(α) =1
2ε2
(α
[Θ]2i,j
)− 14e−
[Θ]2i,j+α
2ε2 I− 12
([Θ]i,jε2√α
)(37)
applying the change of variable, γ(R)i,j = ρα, we derive
thep.d.f. of effective channel gains shown in the Proposition
1.
APPENDIX BPROOF OF PROPOSITION 2
By the definition of c.d.f., we have
F(γ(R)i,j ≤ x
)=
∫ x0
fγ(R)i,j
(γ)dγ(R)i,j . (38)
Assuming γ(R)i,j = ε2ρy2, we have
F(γ(R)i,j ≤ x
)=
∫ √x/ρε
0
y
(εy
[Θ]i,j
)− 12e−
[Θ]2i,j
ε2+y2
2
× I− 12
([Θ]i,jε
y
)dy. (39)
With the help of [37, Eq. (2.3-37)], we derive Proposition
2directly.
APPENDIX CPROOF OF PROPOSITION 7
Using L’Hospital’s rule, we have
limx→∞
g(x) = limx→∞
1− Fγi,j (x)fγi,j (x)
= limx→∞
(1− Fγi,j (x))′
f ′γi,j (x)
= limx→∞
−fγi,j (x)
f ′γi,j (x)
= limx→∞
−1
2ρε2
(x
ρ[Θ]2i,j
)− 14e−
[Θ]2i,j+xρ
2ε2 I− 12
([Θ]i,jε2
√xρ
)(
12ρε2
(x
ρ[Θ]2i,j
)− 14e−
[Θ]2i,j
+ xρ
2ε2 I− 12
([Θ]i,jε2
√xρ
))′ .(40)
Using the following identity
I− 12
([Θ]i,jε2
√x
ρ
)=
√2ε2
π[Θ]i,j
√ρ
xcosh
[Θ]i,j√
xρ
ε2
,(41)
we have
limx→∞
g(x) = limx→∞
−(−34x−1 − 1
2ρε2+ex + e−x
ex − e−x
)=
1− 2ρε2
2ρε2. (42)
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IEEE TRANSACTION ON COMMUNICATIONS 10
APPENDIX DPROOF OF PROPOSITION 8
The following Lemma has been used to proof the Theorem,
Lemma 2. (Uzgoren, 1956) Let x1, x2, . . . , xn be a sequenceof
i.i.d. positive random variables with continuous and
strictlypositive p.d.f. f(x) for x > 0 and c.d.f. of F (x).
Also, assumethat g(x) be the growth function. Then if
limx→∞
g(x) = c > 0, (43)
then,
log{− logFn(x∗ + ug(x∗))}
=− u− u2g′(x∗)
2!− · · · − u
mg(m)(x∗)
m!
+O
(e−u+O(u
2g′(x∗))
n
)(44)
where x∗ is defined before.
Considering the scenario in this paper, such a unique rootcan be
found by solving the equation
1
N= 1− Fγi,j (x∗).
(45)
After submitting Fγi,j (x∗) in this equation, we have
1
N= Q 1
2
([Θ]i,jε
,
√x/ρ
ε
)(46)
when x is large enough, we can approximate the equation as[41],
[42]
1
N=
(√x/ρ
[Θ]i,j
)1/2−1/2Q
(√x/ρ
ε− [Θ]i,j
ε
),
= Q
(√x/ρ
ε− [Θ]i,j
ε
). (47)
For solving this equation, a pure exponential approximationis
used which given by [43]
Q
(√x/ρ
ε− [Θ]i,j
ε
)=
1
12exp
−(√
x/ρ
ε −[Θ]i,jε
)22
+
1
4exp
−23
(√x/ρ
ε− [Θ]i,j
ε
)2+O( 1x
). (48)
Using this approximation, the equation (47) can be approx-imate
as
1
N= Q
(√x∗/ρ
ε− [Θ]i,j
ε
)
≈ exp(−x∗) +O(
1
x∗
), when x∗ →∞. (49)
Compared to the results in [39], the unique solution x∗ toour
above equation is given by
x∗ = logN +O(log log logN). (50)
It is obvious that g′(x∗) = O( 1x∗ ). Therefore, the
maximumvalue of channel gains, γi,j , j = 1, . . . , N which are
i.i.d.random variables satisfies Proposition 8.
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5 6 7 8 9 10 11 12 13 14 15 1610
−4
10−3
10−2
10−1
100
SNR (dB)
Out
age
Pro
babi
lity
Outage Probability
Simulation
K=∞, 20, 10, 5 and 2
Fig. 2. Outage probability of average channel gain based on
random selectionfor multi-way relaying system.
0 2 4 6 8 10 12 14 16 18 200
5
10
15
20
25
SNR (dB)
Erg
odic
Sum
Rat
e
Ergodic Sum Rate
Simulation
Approximation (High SNR)
M=5, 4, 3 and 2
Fig. 3. Ergodic sum rate of multi-way relaying system based on
randomselection.
Jiang Xue received the B.S. degree in Informationand Computing
Science from the Xian JiaotongUniversity, Xian, China, in 2005, the
M.S. degreesin Applied Mathematics from Lanzhou University,China
and Uppsala University, Sweden, in 2008 and2009, respectively. Dr.
J. Xue reveived the Ph.D.degree in Electrical and Electronic
Engineering fromECIT, the Queen’s University of Belfast, U.K.,
in2012. He is currently a Research Fellow with theUniversity of
Edinburgh, UK. His main interestlies in the performance analysis of
general multi-
ple antenna systems, Stochastic geometry, cooperative
communications, andcognitive radio.
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IEEE TRANSACTION ON COMMUNICATIONS 12
0 5 10 15 20 25
10−1
100
SNR (dB)
Out
age
prob
abili
ty
SUS N=3SUS N=5SUS N=10SUS N=15SUS N=25SUS N=50SUS N=100SUS
N=200Random Selection M=N=3Exhaustive Search M=N=3
Fig. 4. Outage probability based on different relays
selections.
0 1 2 3 4 5 6 7 8 9 102
4
6
8
10
12
14
16
18
20
22
SNR (dB)
Erg
od
ic S
um
Ra
te
Single Relay Sytem, M=3SS M=3, N=4SS M=3, N=5SS M=3, N=6SS M=3,
N=10
Fig. 5. Ergodic rate comparison between single relay system and
multi-relayscheme.
Mathini Sellathurai is presently a Reader with theHeriot-Watt
University, Edinburgh, U.K and leadingresearch in signal processing
for intelligent systemsand wireless communications. Her research
includesadaptive, cognitive and statistical signal
processingtechniques in a range of applications including Radarand
RF networks, Network Coding, Cognitive Radio,MIMO signal
processing, satellite communicationsand ESPAR antenna
communications. She has beenactive in the area of signal processing
research forthe past 15 years and has a strong international
track record in multiple-input, multiple-output (MIMO) signal
processing withapplications in radar and wireless communications
research. Dr. Sellathuraihas 5 years of industrial research
experience. She held positions with Bell-Laboratories, New Jersey,
USA, as a visiting researcher (2000); and with theCanadian
(Government) Communications Research Centre, Ottawa Canada asa
Senior Research Scientist (2001-2004). Since 2004 August, she has
beenwith academia. She also holds an honorary Adjunct/Associate
Professorshipat McMaster University, Ontario, Canada, and an
Associate Editorship for theIEEE Transactions on Signal Processing
between 2009 -2013 and presently
0 5 10 15 20 25
100
SNR (dB)
Outa
ge p
robabili
ty b
ase
d o
n S
S
K=5K=10K=15K=20
Fig. 6. Outage probability based on different value of parameter
K forsemiorthogonal selection.
20 40 60 80 100 120 140 160 180 200
−4
−2
0
2
4
6
8
10
12
Relays number N
max
γi,j
Lower boundUpper BoundSimulation
Fig. 7. Lower and upper bounds of channel gain maxj γi,j .
serving as an IEEE SPCOM Technical Committee member. She has
publishedover 150 peer reviewed papers in leading international
journals and IEEEconferences; given invited talks and written
several book chapters as well as aresearch monograph titled
“Space-Time Layered Processing” as a lead author.The significance
of her accomplishments is recognized through internationalawards,
including an IEEE Communication Society Fred W. Ellersick Best
Pa-per Award in 2005, Industry Canada Public Service Awards for
contributionsin science and technology in 2005 and awards for
contributions to technologyTransfer to industries in 2004. Dr.
Sellathurai was the recipient of the NaturalSciences and
Engineering Research Council of Canadas doctoral award forher Ph.D.
dissertation.
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IEEE TRANSACTION ON COMMUNICATIONS 13
20 40 60 80 100 120 140 160 180 2000
0.5
1
1.5
2
2.5
3
Relays number N
Erg
odic
sum
rate
base
d o
n m
ax j
γi,j
Lower boundUpper bound
Fig. 8. Lower and upper bounds of ergodic rate based on the
maximumchannel gain maxj γi,j .
Tharmalingam Ratnarajah (A’96-M’05-SM’05) iscurrently with the
Institute for Digital Communi-cations, University of Edinburgh,
Edinburgh, UK,as a Professor in Digital Communications and Sig-nal
Processing. His research interests include signalprocessing and
information theoretic aspects of 5Gwireless networks, full-duplex
radio, mmWave com-munications, random matrices theory,
interferencealignment, statistical and array signal processing
andquantum information theory. He has published over260
publications in these areas and holds four U.S.
patents. He is currently the coordinator of the FP7 projects
HARP (3.2Me)in the area of highly distributed MIMO and ADEL (3.7Me)
in the area oflicensed shared access. Previously, he was the
coordinator of FP7 Future andEmerging Technologies project CROWN
(2.3Me) in the area of cognitiveradio networks and HIATUS (2.7Me)
in the area of interference alignment.Dr Ratnarajah is a Fellow of
Higher Education Academy (FHEA), U.K., andan associate editor of
the IEEE Transactions on Signal Processing.
Zhiguo Ding (S’03-M’05) received his B.Eng inElectrical
Engineering from the Beijing Universityof Posts and
Telecommunications in 2000, and thePh.D degree in Electrical
Engineering from ImperialCollege London in 2005. From Jul. 2005 to
Aug.2014, he was working in Queen’s University Belfast,Imperial
College and Newcastle University. SinceSept. 2014, he has been with
Lancaster Universityas a Chair Professor.
Dr Ding’ research interests are 5G networks, gametheory,
cooperative and energy harvesting networks
and statistical signal processing. He is serving as an Editor
for IEEETransactions on Communications, IEEE Transactions on
Vehicular Networks,IEEE Wireless Communication Letters, IEEE
Communication Letters, andJournal of Wireless Communications and
Mobile Computing. He received thebest paper award in IET Comm.
Conf. on Wireless, Mobile and Computing,2009, IEEE Communication
Letter Exemplary Reviewer 2012, and the EUMarie Curie Fellowship
2012-2014.
