The frequency spectrum available for the wireless communication is extremely crowded. In order to improve the spectral efficiency, the twoway relay networks have aroused great attention. A simple
N^{th}
bestrelay selection criterion for the opportunistic twoway relay networks is proposed, which can be implemented easily by extending the distributed timer technique in practice, since the proposed criterion is mainly based on the channel gains. The outage performance of the proposed relay selection scheme is analyzed under the outdated channel state information (CSI), and a tight closedform lower bound and asymptotic value of the outage probability over Rayleigh fading channels are obtained. Simulation results demonstrate that the tight closedform lower bound of the outage probability very closely matches with simulated ones in the whole SNR region, and the asymptotic results provide good tight approximations to the simulation ones, especially in the high SNR region.
1. Introduction
I
n recent years, the cooperative diversity
[1]
applied in wireless communication systems has aroused great attention, based on the work of the relay channels
[2]
and the multiple access channels
[3]
. The basic principle of the cooperation is that the singleantenna nodes share their antennas and transmit cooperatively in a way that creates a virtual MIMO system. In comparison with the noncooperative systems, the cooperative systems has more robustness, and can supply higher rate of the data transmission. Therefore, the cooperation is desirable in the practical wireless sensor networks
[4
,
5]
, cognitive radio networks
[6

8]
or ad hoc systems
[9]
, which have rigorous limitations of size, cost and consumption.
Among various cooperative techniques, the opportunistic relay selection
[10
,
11]
, where the best relay is activated, is one of the most efficient schemes to utilize spatial diversity, especially for the cooperative networks with a large number of relays; however, in the conventional oneway relay networks, four time slots are required for the exchange of the one information symbol between two source
S
and
D
nodes with the help of relays, since all terminals are operated in the halfduplex fashion, as shown in
Fig. 1
.
System model for oneway relay networks with one relay node, S and D: transceivers, R: relay, h_{S,R} and h_{R,D} : reciprocal channel response
With the aim of improving the spectral efficiency, the twowayrelaying scheme is proposed in
[12]
, where one or multiple relays are employed to implement reliable bidirectional communications between the two transceivers. The twoway relay network with one relay node is demonstrated in
Fig. 2
, where the simultaneous data exchange is completed between two source nodes
S
and
D
in two time slots. To improve the performance of the twoway relaying system, some new twoway relay selection schemes are presented in
[13

17]
, which adopt either analog network coding (ANC) based on the amplifyandforward (AF) or physical layer network coding (PLNC) based on decodeandforward (DF) protocols, and the outage performance or pairwise error probability of twoway relay networks is also investigated in
[18

20]
.
System model for twoway relay networks with one relay node, S and D: transceivers, R: relay, h_{S,R} and h_{R,D} : reciprocal channel response
On the other hand, most previous researches assume that the channel state information (CSI) is perfect; however, when relay selection is implemented in real systems, there may be a possibility that the channel state of the selected relay at the selection decision phase substantially differs from the actual channel of the data transmission, due to channel variations with time, feekback delay, or the channel estimation error. In other words, the available CSI may be outdated, which leads to the system performance loss. There have been emerging attentions on researching cooperative communication system under the outdated CSI. The oneway relay networks with outdated CSI, for instance, are widely researched
[21

25]
. In very recent years, some efforts are also put into the twoway relay networks with outdated CSI, based on the single relay or multiple relay selection
[26

28]
. In
[27]
, the outage performance of the twoway AF relaying system with outdated CSI is studied. In
[28]
, a multiple relay selection scheme for AF based on twoway relay networks with outdated CSI is proposed.
For the schemes of the single relay selection, the selected best relay, however, might be unavailable due to various reasons in the real system, such as load balancing and scheduling issues; hence, the second best relay, the third best relay, or more generally, the
N^{th}
best relay has to be adopted, in order to guarantee the networks operate well. Hence, the scheme of the
N^{th}
bestrelay selection is proposed to adopt the
N^{th}
best relay, when the best relay may be unavailable, in the opportunistic twoway relay networks. In the paper, it is assumed that the CSI in the data transmission phase differs (i.e., is outdated) from that in the relay selection phase, while the CSI is invariant in each phase separately, from a practical point of view. The impact of the outdated CSI on the network is discussed over Rayleigh fading channels by obtaining a tight closedform lower bound of the outage probability. Simulation results are also given to support the analytical results, which show that the analytical results are very close to the simulated ones in the whole SNR regime, and the asymptotic results well match the simulated ones, especially in the high SNR regime.
The rest of the paper is organized as follows. Section 2 characterizes the system model, and the proposed relay selection scheme. The outage performance under outdated CSI is analyzed in Section 3. In Section 4, simulation results are demonstrated. The conclusion of the paper is given in Section 5.
2. Relay Selection Scheme
 2.1 System Model
Consider an ANCbased opportunistic twoway relay network consisting of two transceivers and
M
relays, denoted by
S
,
D
and
R_{m}
,
m
=0,1,...,
M
1, respectively, as illustrated in
Fig. 3
. All nodes are singleantenna units and operate in a halfduplex mode. No direct communication is assumed between the two transceivers
S
and
D
, due to the poor quality of the channel. In order to analyze conveniently, it is assumed that the distances
d
_{S,Rm}
between node
S
and any relay
R_{m}
are the same, and then we can obtain that the distributions of the channel response
h
_{S,Rm}
are identical, denoted as
with the path loss
α
. Similarly,
is acquired, where
d
_{Rm,D}
is the distances between node
D
and any relay
R_{m}
. Without loss of generality, the normalized distances
d
_{S,Rm}
and
d
_{Rm,D}
are employed, satisfying
d
_{S,Rm}
+
d
_{Rm,D}
=1.
System model of opportunistic twoway relay network with M relay nodes, S and D: transceivers, R_{m} (m=0,1,...,M1): relays, h_{S,Rm} and h_{Rm,D} : reciprocal channel response, normalized distance d_{S,Rm}+d_{Rm,D}=1
The relay selection phase in the proposed opportunistic twowayrelaying system consists of two steps. In the first time slot, the
S
and
D
transit their signal to the relays, simultaneously, as shown in
Fig. 3
. Let
x_{S}
and
x_{D}
be the information symbols transmitted by the
S
and
D
, respectively. With perfect synchronization, the received signal
y_{m}
of the
m^{th}
relay can be represented as
where
P_{S}
and
P_{D}
are the transmit powers of the
S
and
D
, respectively, and
n_{m}
~
CN
(0,
N
_{0}
) is the complex Gaussian noise with the zero mean and variance
N
_{0}
.
At the second time slot, the
m^{th}
relay amplifies and retransmits the received signal to the
S
and
D
. Then the received signals at the
S
and
D
are given, respectively, by
where the amplifying coefficient
P_{R}
is the transmission power of
R_{m}
, and
n_{mS}
and
n_{mD}
are the complex Gaussian noise with zero mean and variance
N
_{0}
, that is,
n_{mS}
~
CN
(0,
N
_{0}
) and
n_{mD}
~
CN
(0,
N
_{0}
).
Assuming that the selfinterference can be removed completely, the signals
y
^{'}
_{S}
and
y
^{'}
_{D}
after the selfinterference cancellation are written, respectively, as
The residual signals
y_{S}
and
y_{D}
can be uesd to decode the information symbols
x_{S}
and
x_{D}
at
D
and
S
, respectively. According to (5) and (6), the endtoend SNRs of the links
D
→
R_{m}
→
S
and
S
→
R_{m}
→
D
are given, respectively, as
 2.2 Proposed Relay Selection Scheme
The optimal relay selection criterion, which maximizes the minimum of the two endtoend SNRs, is written as
[29]
:
It can be observed from (9) that the optimal relay selection criterion is based on instantaneous endtoend SNRs
γ_{DmS}
and
γ_{SmD}
, and requires continuous CSI feedback from all the links of the network, in other words, the criterion can only be implemented in a centralized manner, which is difficult to be realized in practice. Since the endtoend SNRs
γ_{DmS}
and
γ_{SmD}
are difficult to be analyzed straightforwardly, the more tractable results are obtained, based on (7) and (8), which can be expressed as
At the high SNR region,
for (10) and (11) are satisfied, respectively, so we have
Taking the reciprocal of (12) and (13) on both sides, the tight upper bound of
γ_{DmS}
and
γ_{SmD}
are given, respectively, by
Substituting (14) and (15) into (9), and considering that the
N^{th}
best relay has to be selected, then the proposed
N^{th}
best relay selection scheme can be represented as:
where
Ψ
=min{
P_{D}
/(
P_{R}
+
P_{S}
),1} ,
Φ
=min{
P_{S}
/(
P_{R}
+
P_{D}
),1} , and the argument expression arg
denotes a value of the argument
m
, from the set {0,1,...,
M
1}, who make the function
F
(
m
) to attain its
N^{th}
largest value.
Observing from (16), the proposed
N^{th}
bestrelay selection is mainly based on the channel gains ｜
h
_{Rm,D}
｜ and ｜
h
_{S,Rm}
｜ , which can be implemented easily in practice, by extending the distributed timer technique in
[13]
. Specifically, each source node broadcasts one pilot to the relay nodes, so that relays can estimate the channel quality. Then each relay starts a timer, whose duration is inversely proportional to its estimated channel quality. The relay node, whose timer is expired, notifies the other relays via a flag packet. There is a onebit message in the flag packet, to demonstrate the relay is busy or idle. Once
N
flag signals are sent, the last relay that sends the flag signal is the
N^{th}
best relay, denoted as
R
_{m*}
, and the other relays keep silent. Note that a practical method to choose
N
is to adopt the number of the relay, who first sends the idle message.
3. Outage Probability Analysis under Outdated CSI
 3.1 Outdated CSI Model
In the paper, we assume that the CSI in the relay selection phase differs from that in the data transmission phase, while the CSI remains the same in each phase separately.
At the data transmission phase, let
denote the channel coefficients between
S
and
R_{m}
, and
between
R_{m}
and
D
. Since slowly timevarying Rayleigh fading channels are adopted, the channel coefficients
and
can be modeled as two independent firstorder GaussMarkov processes
[21
,
30]
:
where
ρ
_{S,Rm}
and
ρ
_{Rm,D}
is the correlation coefficient to describe the impact of the outdated CSI on the
S

R_{m}
and
R_{m}

D
links, respectively,
ξ
_{S,Rm}
and
ξ
_{Rm,D}
are independent white complex Gaussian processes written as
respectively. It is assumed that
ρ
_{S,Rm}
=
ρ
_{Rm,D}
=
ρ
in the paper. From the Jakes’ model,
ρ
=
J
^{2}
_{0}
(2
πf_{d}τ
)∈[0,1] , where
J
_{0}
(·) is the zeroorder Bessel function of the first kind,
f_{d}
is the Doppler frequency,
τ
denotes the delay between the instants of the relay selection and the data transmission.
Let
denote the channel gains of the m
S
–
R_{m}
and
R_{m}
–
D
links at the relay selection phase, respectively, while
and
denote the channel gains of the two links at the data transmission phase from the selected relay, respectively.
 3.2 Outage Probability Analysis
In this subsection, the outage probability of the proposed scheme with outdated CSI is discussed.
The outage event occurs when either
γ_{Sm*D}
or
γ_{Dm*S}
falls below the threshold, given by:
where the threshold
γ_{th}
= 2
^{2υ}
1 , and
ν
is the target rate.
In the presence of the outdated CSI, the probability density function (PDF) of the channel gain
u
_{1}
for the
S

R_{m*}
link, at the relay selection phase, is written as
[31]
:
Then the PDF of
u
conditioned on
u
_{1}
, for the
S

R_{m*}
link at the data transmission phase, takes the following expression
[31]
:
where
I
_{0}
(·) stands for the zeroorder modified Bessel function of the first kind. Based on (19) and (20), the PDF of
u
is written as (see the Appendix for the proof)
where
Similarly, the PDF of the channel gain
v
for the
R_{m*}

D
link, at the data transmission phase, can be obtained as
where
Substituting (21) and (24) into (18), the tight lower bound of the outage probability is given by
According to the Taylor’s series approximation
, the asymptotic expression of the outage probability at the high SNR region is represented as:
Simulation results later will demonstrate the tight lower bound of the outage probabilty in (26) very closely matches with simulated ones in the whole SNR region, and the asymptotic results in (27) provide good tight approximations to the simulation ones, especially in the high SNR region.
4. Simulation Results
To evaluate the performance of the proposed scheme, some simulation results are presented in the section. The number of the relays is
M
=10, the path loss
α
=3, the threshold
γ_{th}
=0 dB, the normalized distance
d
_{S,Rm}
=0.2 and
d
_{Rm,D}
=0.8 . For simplicity, the equal power allocation scheme is only considered, that is,
P_{S}
=
P_{D}
=
P_{R}
. The Equation (17) is used as the outdated CSI model in the simulation, where the correlation coefficients are assumed as
ρ
_{S,Rm}
=
ρ
_{Rm,D}
=
ρ
.
Fig. 4
shows the analytical and simulated results of outage probability versus SNR, under the perfect CSI with the channel correlation coefficient
ρ
=1, in the case of
N
∈{1,2,4} . Note that
N
=1 is corresponding to the existing optimal relay selection scheme
[29]
, which is a special case included in our proposed scheme. It is shown that the outage performance becomes worse as
N
increases, since the best relay is unavailable and the
N^{th}
best relay is chosen under outdated CSI. It also can be observed that the analytical results are very close to the simulated ones in the whole SNR region. Furthermore, it is worth noting that the curves have the same slopes for different values of
N
, since the diversity order of the twowayrelaying system is equal to the number of the relays when the CSI is perfect with
ρ
=1.
Outage probability vs. total SNR, ρ=1 and N∈{1,2,4}
Fig. 5
plots the analytical, asymptotic and simulated results of outage probability versus SNR, under the outdated CSI with
ρ
=0.5, in the case of
N
∈{1,2,4}. As observed from the figure, the analytical results very closely match with simulated ones in the whole SNR region, and the asymptotic results provide good tight approximations to the simulation ones, especially in the high SNR region. It also can be found that the outage performance is worse in
Fig. 5
when
ρ
=0.5, in comparison with that in
Fig. 4
when
ρ
=1. Moreover, the curves have the same slopes for different values of
N
, since the diversity order of the opportunistic twowayrelaying system is equal to one when
ρ
is less than one.
Outage probability vs. total SNR, ρ=0.5 and N∈{1,2,4}
In order to depict the great impact of the outdated CSI on the system performances,
Fig. 6
demonstrates the outage probability as a function of
ρ
, where
N
=2 and different values SNR={20dB，30dB，40dB}. As expected, better results are obtained by increasing the value of
ρ
, since larger
ρ
means better quality of the channel.
Fig. 7
describes the outage probability as a function of
ρ
, for SNR=20dB and different values
N
∈{1,2,3,4}. It also can be seen that the outage performance becomes better when
ρ
increases. For
N
=1 as an example, the outage probability are around 10
^{3}
and 10
^{4}
when
ρ
is equal to 0.8 and 0.9, respectively. In addition, it also shows that the outage performance becomes worse when
N
is increased.
Outage probability vs. ρ, SNR={20dB，30dB，40dB} and N=2
Outage probability vs. ρ, N∈{1,2,3,4} and SNR=20dB
5. Conclusion
The opportunistic twoway relay selection, where the best relay is chosen, is one of the most efficient schemes to utilize spatial diversity. The selected best relay, however, might be unavailable due to various reasons in the real system, such as load balancing and scheduling issues. In this paper, the simple
N^{th}
bestrelay selection scheme for the opportunistic twoway relay networks is proposed, which is mainly based on the channel gains and can be implemented easily in practice. The outage performance of the proposed scheme is discussed with the outdated CSI, and the closedform lower bound of the outage probability and its approximation in the high SNR region are derived. Numerical and simulation results have verified the theoretical analyses. Note that the CSI among the relays is required to be considered to improve the system performance. Besides, balancing tasks that the best relays have to listen to the network to send the flag packet while they are busy with their own normal work, should also be investigated. These problems mentioned above are left for future research.
BIO
Jinglan Ou received the Ph.D. degree in circuits and systems from Chongqing University, Chongqing, China, in 2011. She joined the faculty of Chongqing University in 2006, where she is currently a Lecturer and tutor of master students. Her major research interest includes cooperative communications and OFDMbased wireless communications.
Email: oujinglan2013@gmail.com, oujinglan@cqu.edu.cn.
Haowei Wu received the Ph.D. degree in circuits and systems from Chongqing University, Chongqing, China, in 2012. He joined the faculty of Chongqing University in 2006, where he is currently an Associate Professor and tutor of master students. His major research interest includes OFDM, physicallayer network coding and cooperative communications.
Email: wuhaowei2013@gmail.com, wuhaowei@cqu.edu.cn.
Qi Wang is currently working towards his M.S. degree in College of Communication Engineering, Chongqing University. His major research interest includes cooperative communications and wireless communications.
Email: 1182631495@qq.com
Yutao Zou is currently working towards his M.S. degree in College of Communication Engineering, Chongqing University. His major research interest includes cooperative communications and MIMO systems.
Email:494405235@qq.com
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