[11]W.Su,“Performance analysis for a suboptimum ML receiver in decode-
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nel knowledge,”IEEE Trans.Signal Process.,vol.58,no.3,pp.1928–1934,Mar.2010.
Relay Selection for Two-Way Relaying With
Amplify-and-Forward Protocols
Lingyang Song,Member,IEEE
Abstract—In this paper,we propo a relay lection amplify-and-forward(RS-AF)protocol in general bidirectional relay networks with two sources and N relays.In the propod scheme,the two sourcesfirst simultaneously transmit to all the relays,and then,a single relay with a minimum sum symbol error rate(SER)will be lected to broadcast the received signals back to both sources.To fac
ilitate the lection process, we propo a simple suboptimal min–max criterion for relay lection, where a single relay that minimizes the maximum SER of two source nodes will be lected.Simulation results show that the propod min–max lection has almost the same performance as the optimal lection with lower complexity.We also prent a simple asymptotic SER expression and make a comparison with the conventional all-participate AF relay-ing scheme.The analytical results are verified through simulations.To improve the system performance,optimal power allocation(OPA)be-tween the sources and the relay is determined bad on the asymptotic SER.Simulation results indicate that the propod RS-AF scheme with OPA yields considerable performance improvement over an equal-power-allocation scheme,particularly with a large number of relay nodes.
Index Terms—Amplify-and-forward(AF)protocol,analog network coding,relay lection,two-way relaying.
I.I NTRODUCTION
Bidirectional relay communications have recently attracted con-siderable interest,and transmission schemes in bidirectional relay networks have been analyzed and compared[1].In[2]and[3], an amplify-
and-forward(AF)-protocol-bad network coding scheme was discusd.The transmission in this AF bidirectional relay network takes place in two time slots.Two source nodesfirst transmit at the same time through one or multiple relays.The relay receives a superimpod signal,then amplifies the received signal,and forwards it back to both source nodes.Analog network coding has been proved particularly uful in wireless networks,becau the wireless channels are ud as a natural implementation of network coding by summing the wireless signals over the air[1]–[5].
Manuscript received May4,2010;revid July27,2010,November4, 2010,and January20,2011;accepted February19,2011.Date of publication March7,2011;date of current version May16,2011.This work was supported in part by the National Nature Science Foundation of China under Grant 60972009and Grant61061130561,the National Science and Technology Major Project of China under Grant2009ZX03003-011,Grant2010ZX03003-003,and Grant2011ZX03005-002,and by the Shanghai Pujiang Program under Grant08PJ14057.The review of this paper was coordinated by Dr.X.Dong. The author is with the State Key Laboratory of Advanced Optical Communication Systems and Networks,School of Electrical Engineering and Computer Science,Peking University,Beijing100871,China(e-mail: lingyang.song@pku.edu).
Color versions of one or more of thefigures in this paper are available online at ieeexplore.ieee.
org.
Digital Object Identifier10.1109/TVT.2011.2123120
Recently,it has been shown that the performance of wireless relay networks can further be enhanced by properly lecting the relays for transmission[6],[7],[11]–[15],[19].In[6]–[8],relay lection meth-ods were reported for conventional one-way AF schemes to achieve full spatial diversity,and hybrid one-way relay lection schemes were discusd in[9]and[10].In[11],the authors propod the max–min sum rate lection algorithm for AF bidirectional networks bad on the outage probability.In[12],the authors analyzed the diversity orders of various relay lection schemes.In[13],two-way relay lection was introduced for differential modulation systems to improve system performance.In[14],the authors prented a max–min signal-to-noi ratio(SNR)-bad relay lection algorithm for two-way relay networks.In[15],opportunistic relay lection was propod,and in[19],the authors discusd the performance bound by maximizing the overall channel capacity to realize the best relay node lection.Conquently,it is beneficial to design an effective relay lection scheme for the coherent bidirectional transmission scheme with multiple relays and to achieve spatial diversity.
Power allocation(PA)in one-way relay systems has intensively been studied[16]–[18].Becau two-way relay system works quite differently and is more complex than one-way relay system,the PA algorithms developed for one-way relaying cannot readily be ud in two-way relay systems.Most work on PA for two-way , [11]and[20],was propod to maximize the sum rate of the ur pair. In[21],the authors consider PA with wireless network coding in a multiple-relay,multiple-ur networks using convex optimization.In [22],the authors prented PA strategies to maximize the sum rate and the diversity order,respectively.In[23],two PA algorithms were propod to maximize the upper bound of the average sum rate and to achieve the tradeoff of outage probability between two terminals, respectively.It is well known that the symbol error rate(SER)per-formance also plays an important role for many applications,but the optimal power allocation(OPA)optimization problem in a two-way relay system has not been investigated to minimize the system SER. Hence,it is extremely uful to study the OPA problem,minimizing the SER for bidirectional relay networks.
陶渊明的田园诗In this paper,we propo a relay lection amplify-and-forward (RS-AF)protocol for bidirectional relay networks using AF with two sources and N relays.In the propod scheme,two source nodesfirst transmit to all the relays at the same time.The signals received at the relay are a superposition of two transmitted symbols from both sources.Then,a single relay that minimizes the sum SER of two s
ource nodes is lected out of N relays to forward the network-coded signals back to both sources.However,the performance of the optimal relay lection amplify-and-forward(O-RS-AF)scheme is very difficult to analyze.
To facilitate the lection process,we introduce a suboptimal but low-complexity min–max-criterion-bad method,where a single re-lay that minimizes the maximum SER of two source nodes is lected. Bad on the min–max lection procedure,we prent a simple asymptotic SER expression and make comparisons with the conven-tional all-participate amplify-and-forward(AP-AF)relaying scheme. To improve the system performance,OPA between the sources and the relay is determined bad on the asymptotic SER.The performance of the propod RS-AF scheme is verified through simulations.
The rest of this paper is organized as follows.In Section II,we describe the system model.In Section III,we prent the propod RS-AF scheme.The performance is analyzed and compared with AP-AF in Section IV.In Section V,the OPA solution of RS-AF is given.Simulation results are provided in Section VI.In Section VII, we draw the main conclusions.
0018-9545/$26.00©2011IEEE
Fig.1.Block diagram of the propod RS-AF scheme and the AP-AF scheme. Notation:Boldfaced low
erca letters denote vectors,and(·)∗, (·)T,and(·)H reprent the conjugate,transpo,and conjugate trans-po,respectively.E is ud for expectation,Var reprents variance, and x 2=x H x.
II.S YSTEM M ODEL
We consider a general bidirectional relay network,which consists of two source nodes,denoted by S1and S2,and N relay nodes,denoted by R1,...,R N.We assume that all nodes are equipped with a single antenna.In the propod RS-AF scheme,as shown by the solid lines in Fig.1,each message exchange between two source nodes takes place in two phas.In thefirst pha,both source nodes simultaneously nd the information to all relays,and the signal received at each relay is a superimpod signal.In the cond pha,an optimal single relay node is lected to forward the received signals to two source nodes, and all other relay nodes keep idle.In this paper,we assume that the fading coefficients are constant over one frame and independently vary from one frame to another.Note that the propod strategy requires calculating the instantaneous SNR of both links,which can be realized by training or preambles.Relay lection can be then carried out,and data transmission will u the lected relay until the next channel estimation period comes.For simplicity,we assume perfect channel estimation,and the source and the relay nodes have all the link information.
Let s i,i=1,2denote the symbol that will be transmitted by the source S i.We assume that s i is chon from a constellation of unity power A.The signal received in the k th relay at time t can be expresd as
y r
k =
√
p s h1,r
k
s1+
√
p s h2,r
k
s2+n r
k
(1)
where p s reprents the transmit power at S1,S2,h i,r
k
(i=1,2,k=
1,...,N)stands for the fading coefficient between S i and R k with
zero mean and unit variance,and n r
k
is a zero-mean complex
Gaussian random variable(RV)with two-sided power spectral density
of N0/2per dimension.
Upon receiving the signals,the relay R k then process the received
signal and forwards to two source nodes.Let x r,k be the signal that was
generated by the relay R k,and it is given by
x r
k
=βk y r
k
(2)
whereβk=(p s|h1,r k|2+p s|h2,r k|2+N0)−(1/2)is an amplifica-
tion factor so that the signal transmitted by the relay satisfies the
following power constraint:
E
x
r k
2
≤1.(3)
Then,the relay R k forwards x r
k
to two source nodes.The signal
received by S i,where i=1,2,which is denoted by y
i,k
,
can be
written as
y i,r
k
=
√
p r h i,r
k
x r
k
+n i,r
k
(4)
where p r reprents the transmit power at the relay node.
Combining(1),(2),and(4),after subtracting its own information,
the received signal at each source can be written,respectively,as
y1,r
k
=αk s2+w1,r
k
(5)
y2,r
k
=αk s1+w2,r
k
(6)
whereαk=
√
p s p rβk h1,r
k
h2,r
k
,w1,r
k
=
√
p rβk h1,r
k
n r
k
+n1,r
k
,
and w2,r
k
=
√
p rβk h2,r
k
n r
k
+n2,r
k
.
Finally,the following maximum likelihood(ML)detector can be
applied to recover the received signals:
s1=arg max
s1(t)∈A
y2,r
k
−αk s1
2
s2=arg max
s2(t)∈A
y1,r
k
−αk s2
2.(7)
III.R ELAY S ELECTION FOR T WO-W AY
A MPLIFY-AND-F ORWARD N ETWORKS
In the propod RS-AF scheme,only one best relay is lected
out of N relays to forward the received superimpod signals in the
cond-pha transmission.We assume that,at the beginning of each
transmission,some pilot symbols are transmitted by two source nodes
to assist in the relay lection.One source node(either source S1or
S2)will determine the best relay according to a certain criterion and
broadcast the index of the lected relay to all relays.Then,only the
lected relay,which is known by both source nodes,is active in the
cond pha of transmission,and the rest of the relays will keep idle.
We prent two relay lection methods as follows.
1)O-RS-AF:For O-RS-AF,among all relays,the destination will
lect one relay,which is denoted by R,which has the minimum
destination SER for the ur pair.We have
R=min
k
SER1,r
k
γ1,r
k
|h1,r
k
,h2,r
k
+SER2,r
k
γ2,r
k
|h1,r
k
,
h2,r
k
(8)
where SER i,r
k
(γi,r
k
|h1,r
宁波旅游必去景点k
,h2,r
k
)
,i=1,2reprents the SER at
source nodes S i from the k th relay,which is given h1,r
k
and h2,r
k
.
The SER that was conditioned on the instantaneous received SNR can be written as [24]
SER i,r k γi,r k |h 1,h 2
=Q
√
cγi,r k
(9)
where Q (·)is the Gaussian-Q function,Q (x )=(1/√2π) ∞x exp(−t 2
/2)dt ,c is a constant determined by the modulation ,c =2for binary pha-shift keying (BPSK)constellation,and γi,r k stands for the destination SNR,which is calculated as
γi,r k
=
|αk |2Var w i,r k
.(10)
2)S-RS-AF:The aforementioned O-RS-AF scheme is very dif-ficult to analyze.In this ction,we propo the suboptimal relay lection amplify-and-forward (S-RS-AF)scheme.It is well known that the sum SERs of two source ,SER 1,r k +SER 2,r k ,is typically dominated by the SER of the worst ur.As a result,for low complexity,the relay node,which minimizes the maximum SER of two urs,can be lected to achieve the near-optimal SER performance.We refer to such a lectio
n criterion as the min–max lection criterion.Let R denote the lected relay.Then,the min–max lection can be formulated as follows:R =min k
max SER 1,r k γ1,r k |h 1,r k ,h 2,r k
SER 2,r k
γ2,r k |h 1,r k ,h 2,r k
(11)
which can further be formulated using the effective SNRs as
γR =max k
min γ1,r k ,γ2,r k
(12)
where k =1,...,N .
IV .P ERFORMANCE A NALYSIS
A.Asymptotic SER of the RS-AF Scheme
In this ction,we derive the analytical average SER of the propod
RS-AF schemes bad on the min–max criterion.As aforementioned,the O-RS-AF scheme is very difficult to analyze.As will later be shown,the min–max lection scheme propod in Section III-B has almost the same performance as the O-RS-AF scheme.Therefore,in this ction,we will instead analyze the performance of the S-RS-AF scheme.
Now,let us first calculate the probability density function (pdf)of γR in (12).Becau γ1,r k and γ2,r k are identically distributed,they have the same pdf and cumulative distribution function (cdf),denoted by f γk (x )and F γk (x ),respectively.Without loss of generality,we u γ1,r k for derivations,which can be written as
γ1,r k =
|αk |2
Var {w 1,r k }
声律启蒙十一真
=ψr ψs h 1,r k 2 h 2,r k 2
ψr h 1,r k
2+ψs h 2,r k
2
+1
≈ψr ψs h 1,r k 2 h 2,r k 2
ψr h 1,r k
2+ψs h 2,r k
2
(13)
where Var {w 1,r k }=p r β2N 0|h 1,r k |2+N 0,ψs Δ
=(p s /N 0(1+λ)),
and ψr Δ
=(p r /N 0).For convenience,we assume that p s =λp r ,λ>0.
Define γmin
k Δ
=min {γ1,r k ,γ2,r k }.Let f γmin k (x )and F γmin k
(x )rep-rent its pdf and cdf,respectively.Then,the pdf of γR can be calculated by using order statistics as [25]f γR (x )=Nf γmin k
(x )F N −1
γmin k
(x )
=2Nf γk (x )
1−F γk (x )
1−
1−F γk (x )
2 N −1
(14)
where f γmin k (x )=2f γk (x )(1−F γk (x )),F γmin k
(x )=1−(1−F γk (x ))2,
and by upper bounding (13)with a harmonic mean,f γk (x )can be obtained by [26]
f γk (x )=
2x exp (−x (ψ−1r
+ψ−1
s ))ψr ψs
× ψr +ψs √ψr ψs
×K 1
2x √
ψr ψs
+2K 0
2x √
ψr ψs
U (x )
(15)
where K 0(·)and K 1(·)denote the zeroth-and first-order modified Besl functions of the cond kind,respectively,and U (·)is the unit step function.At high SNR,when z approaches zero,the K 1(z )function converges to 1/z [27],and the value of the K 0(·)function is comparatively small,which could
be ignored for asymptotic analysis.Hence,at high SNR,f γk (x )in (15)can be reduced as
lim x →0f γk (x )=ψ2exp −ψ
2
x
(16)
where ψΔ
=2(ψ−1r +ψ−1
s ).Its corresponding cdf can be written as
F γk (x )=1−exp −ψ
2
x
.
(17)
The pdf of γR can then approximately be calculated as
f γR (x )=Nψexp(−ψx )[1−exp(−ψx )]N −1.
(18)
Using the fact that lim χ→01−exp(−χ)=χ,the cdf of γR can approximately be written as
lim ψ→0
F γR (x )=
lim ψ→0
(1−exp(−ψx ))
N
=(ψx )N .
(19)
The average SER can be then derived by averaging over the Rayleigh fading channels as
SER RS =E [SER (γR |h 1,h 2)]=E [Q (√
cγR )].
(20)
By introducing a new RV with standard normal distribution X ∼N (0,1),the average SER can be rewritten as [7]
SER RS =P {X >
√
cγR }
=P
γR
<
X 2
c
=E
F γR
X 2c =
∞
F γR
X 2
F X (x )dx.
(21)
Recalling(19)and X∼N(0,1),(21)can further be written as
SER RS=
1
√
2π
ψ
c
N∞
x2N exp
−x
2
2
dx.(22)
Becau ∞
t2n exp(−kt2)dt=((2n−1)!!/2(2k)n)
π/k[28],
we canfinally obtain
告的组词
SER RS=(2N−1)!!
2
ψ
c
N
(23)
where(2n−1)!!Δ=
n
k=1
2k−1=((2n−1)!/n!2n).
It clearly indicates in(23)that a diversity order of N can be achieved for the propod RS-AF scheme in a bidirectional relay network with two sources and N relays.Note that,for other types of channels, e.g.,Nakagami-m and Rician fading channels,we may merely u the similar approach to derive the cdf ofγR in(19),and by(20),the analytical SER can be obtained.
B.SER Comparison With the AP-AF Scheme
In this ction,wefirst derive the SER of AP-AF.Note that thefirst pha of RS-AF and of AP-AF are the same.However,in the AP-AF scheme,all the relay nodes are ud to forward the received signals over mutually orthogonal channels,as shown by the dashed lines in Fig.1.As a result,the effective SNR at the source node becomes
γAP,i=
N
k=1
γi,r
k
(24)
whereγAP,i reprents the effective SNR at the i th source node.For fairness,the total transmit energy and equal power division among relay nodes are assumed for both systems.
By using a general result bad on[4],[6],[7],and[29],the SER in(23)can be approximated in the high-SNR regime by considering a first-order expansion of the cdf ofγAP,i.In particular,if thefirst-order expansion of the cdf ofγAP,i can be written in the form
Fγ
AP,i (x)=μγN AP,i+o
γN+ε
AP,i
,ε>0(25)
whereμreprents a constant value,at high SNR,the asymptotic average SER of AP-AF can be written as[6],[7]
SER AP,i=(2N−1)!!
2N!c N
∂N Fγ
i,r k
∂γN
2022流星雨i,r k
.(26)
Bad on[6],we can get
∂N Fγ
i,r k ∂γN
i,r k =
N
k=1
fγ
i,r k
(0)(27)
where fγ
i,r k (0)=Nψ−1
r
+ψ−1
s
.
Substituting(27)into(26)yields
SER AP,i=(2N−1)!!
2N!c N
Nψ−1
r
+ψ−1
s
N
.(28)
Comparing(23)with(28),we canfinally obtain
SER RS
AP,i =N!
连云港特色美食1+2λ春饼蒸多长时间
N
.(29)
We can easily prove that the ratio in(29)is always smaller than1 for all N>1.It clearly indicates that RS-AF obtains better SER than AP-AF,and this gain gets larger when the number of relay nodes increas.Note that the major difference between RS-AF and AP-AF is that RS-AF utilizes all the tra
nsmit power in the best relay,whereas AP-AF equally splits the transmit power into every relay node.In RS-AF,there exists a relay node determination process,but in AP-AF, there is none.
V.T RANSMIT P OWER A LLOCATION
In this ction,we prent how we can allocate power to both sources and the relay subject to the total transmission power con-straint.It is shown in(23)that the asymptotic SER of the propod RS-AF scheme nonlinearly depends on p s and p r.Hence,when the total transmit power isfi,2p s+p r=p,the PA problem over Rayleigh channels can be formulated to minimize the asymptotic SER in(23)as
min SER RS
<2p s+p r=p
0<p s<p
0<p r<p.(30) The PA problem is tofind p s such that the SER in(23)is minimized subject to the power constraint by solving the following optimization problem:
两腮长痘L(p s)=SER RS+ξ(2p s+p r−p)(31)
whereξis a positive Lagrange multiplier.The necessary condition for the optimality is found by tting the derivatives of the Lagrangian in (31)with respect to p s and p r equal to zero,respectively.We can get
∂L(p s)
∂p s
=
∂SER RS
∂p s
+2ξ=0
∂L(p r)
∂p r
=
∂SER RS
∂p r
+ξ=0.(32)
Integrating the power constraint2p s+p r=p and SER RS given in (23)into(32),we can obtain
p s=
p
4
p r=
p
2
(33) which indicates that the power allocated in the relay should be equal to the total transmit power
at both sources to compensate for the energy ud to broadcast the combined information in one time slot,regardless of the number of relays.
The SER improvement using OPA compared with EPA can be calculated as
SER opt
SER equ
=
8
9
N
(34)
which shows that the improvement exponentially increas with the number of relay nodes.
VI.S IMULATION R ESULTS
In this ction,we provide simulation results for the propod RS-AF scheme.For symmetrical reasons,both source nodes should have the same SER,and thus,it would be sufficient to examine only one source node.We include the AP-AF scheme for comparison.All simulations are performed for a BPSK modulation over the Rayleigh fading channels.For simplicity,we assume that the total energy p=3
Fig.2.Simulated SER performance by the O-RS-AF and S-RS-AF methods, where p s=p r=1
.
Fig.3.Simulated SER performance by the propod RS-AF and the AP-AF schemes,where p s=p r=1.
and S1,S2,and R k(k=1,...,N)have the same noi variance N0. The SNRψs can then be calculated asψs=p s/N0.
A.Simulated Results
In Fig.2,we compare the O-RS-AF and S-RS-AF(min–max) methods,where p s=p r=1.Bad on thefigure,the propod S-RS-AF approach has almost the same SER as the O-RS-AF scheme. In particular,when the number of relay nodes increas,we can obrve almost no difference between the two methods,which indicates that the min–max relay lection achieves near-optimal single-relay-lection performance.We can also e in Fig.2that the performance improves when the number of relay increas.Note that, for convenience,we u RS-AF to replace S-RS-AF in the rest of this paper.
Fig.3compares the simulated SER performance of our propod RS-AF scheme and the AP-AF schemes for N=2,3,4relay
nodes.Fig.4.Analytical and simulated SER performance by the propod RS-AF scheme,where p s=p r=1,and N=1,2,3,4.
For RS-AF,p s=p r=1,whereas for AP-AF,p s=1,and p r= 1/N.It can be obrved that the propod scheme has much better performance than the AP-AF scheme.This result can be verified by the theoretical analysis given in(29).In particular,asλ=p s/p r=1,(29) can be reduced to N!(3/(1+2N))N.Correspondingly,Fig.3shows that,at high ,SNR=20dB,the RS-AF scheme has a better SER of a factor about0.6,0.5,and0.3over AP-AF with N=2,3,4, respectively,which also indicates that the SER gain increas with the number of relay nodes.
B.Analytical Results
In Fig.4,we compare the analytical and simulated SER perfor-mance of the propod RS-AF scheme.Thisfigure shows that,at high SNR,the asymptotic analytical SER given by(23)is converged to the simulated result using O-RS-AF.This ca verifies the derived analytical expressions.
C.PA
In Fig.5,we examine the SER performance of the RS-AF scheme using OPA with p s=p/4and p r=p/2
subject to the total power constraint for N=1,...,4.The EPA results are provided for compar-ison with p s=p r=p/3.In Fig.5,it can be obrved that,with OPA, the propod scheme obtains better performance gain compared with the EPA scheme at high SNR,and this improvement exponentially increas with the number of relay nodes,satisfying(34).
In Fig.6,we plot the SER curves in terms ofλ=p s/p r defined in(13)using different noi variance,and the number of relays is t to two.Fig.6shows that the best performance is obtained when λ=0.5.In other words,0p s=p/4and p r=p/2are the optimum power ttings between the sources and the relay,which further verify the PA approach introduced in Section V.
VII.C ONCLUSION
In this paper,we have propod a joint relay lection and ANC over two-way relay channels.A simple min–max relay lection method is propod,which achieves almost the same performance as the OS-RS-AF scheme.We derived the asymptotic SER expression of the RS-AF scheme,which is verified through simulations.We showed,
