Exact Outage Probability of Cognitive Two-Way Relaying Scheme with Opportunistic Relay Selection under Interference Constraint

Abstract

In this study, the authors analyse the outage performance of a cognitive two-way relaying system under an interference constraint. In the proposed protocol, the best relay which is chosen by an opportunistic relay selection strategy, combines the received packets using XOR operation and then forwards the combined packet to two secondary sources. The authors derive an exact closed-form expression of outage probability over flat and block Rayleigh fading channels. Various Monte-Carlo simulations are presented to verify the theoretical analyses and to compare the performance of the proposed protocol with that of the two-way relaying protocol without network coding.

Full text

Published in IET Communications
Received on 2nd May 2012
Revised on 21st July 2012
doi: 10.1049/iet-com.2012.0235
ISSN 1751-8628
Exact outage probability of cognitive two-way relaying
scheme with opportunistic relay selection under
interference constraint
T.T. Duy H.Y. Kong
Department of Electrical Engineering, University of Ulsan, Ulsan, South Korea
E-mail: hkong@mail.ulsan.ac.kr
Abstract: In this study, the authors analyse the outage performance of a cognitive two-way relaying system under an interference
constraint. In the proposed protocol, the best relay which is chosen by an opportunistic relay selection strategy, combines the
received packets using XOR operation and then forwards the combined packet to two secondary sources. The authors derive
an exact closed-form expression of outage probability over flat and block Rayleigh fading channels. Various Monte-Carlo
simulations are presented to verify the theoretical analyses and to compare the performance of the proposed protocol with that
of the two-way relaying protocol without network coding.
1 Introduction
Cognitive radio is a promising technique for improving
spectrum utilisation efficiency [1 –3]. Conventionally,
cognitive users (secondary users) sense the presence and
absence of licensed users (primary users). If no primary
activity is sensed, then a cognitive user can access a
licensed band. Recently, underlay spectrum-sharing
protocols have been proposed and analysed [4, 5]. In such
schemes, the secondary transmitters must adapt their
transmit power so that the interference created at the
primary user is below the maximum allowable interference.
Owing to the constraint on the transmit power, the
performance of cognitive underlay protocols is severely
degraded in fading environments [6 –10].
One efficient method to improve the performance of the
secondary network is to use cooperative communication
protocols with multiple relays [11– 13]. As we know,
cooperative communication is able to improve the channel
capacity and achieve higher diversity gains in fading
environments [14, 15]. By exploiting the broadcast nature
of a wireless channel, cooperative diversity allows single-
antenna radios to form a virtual antenna array. In
cooperative multi-relay protocols, relay selection is an
important issue for improving system performance. The full
relay selection methods [16, 17] have the advantage of
significantly improving the performance and obtaining the
full diversity order, compared with the partial relay
selection method [18]. In the relay selection method
presented in [17], the best relay is selected to support the
data transmission between the source and the destination,
relying on the channel state information across two hops of
each relay. This method can be called opportunistic relay
selection.
Recently, two-way relaying transmissions have attracted
considerable interest [19 –21]. In such schemes, two
sources exchange their data with help of one or multiple
relays. In digital network coding [19], two sources
separately transmit their packets to relays. Then, the best
relay is chosen to decode the received packets, combine
decoded packets using XOR operation and then broadcast
the combined packet back to both sources. In analog
network coding [20, 21], two sources first broadcast their
signals to relays at the same time. The best relay receives a
superimposed signal, then amplifies the received signal, and
forwards it back to both sources [20, 21]. Hence, the analog
network coding only uses two time slots to relay the signals
whereas the number of time slots is used in the digital
network coding is three.
In this paper, we propose a cognitive two-way relaying
scheme with relay selection under interference constraint. In
the proposed protocol, we use the digital network coding to
relay packets to two secondary sources. The reason is that if
we use the analog network coding in cognitive underlay
network, the interferences caused by the transmission of
two secondary sources at the first time slot are summed up
at the primary receiver. Hence, implementation of such a
protocol requiring a transmit power allocation strategy for
two secondary sources so that the total interference at the
primary user is below the maximum allowable interference
is a difficult work. In the proposed protocol, we also
propose the relay selection methods that choose a best relay
for forwarding the received packets to intended sources by
using network coding. To evaluate the performance of the
2750 IET Commun., 2012, Vol. 6, Iss. 16, pp. 2750–2759
&The Institution of Engineering and Technology 2012 doi: 10.1049/iet-com.2012.0235
www.ietdl.org

proposed protocol, we derive the exact closed-form
expression of outage probability. Then, various Monte-
Carlo simulations are presented to verify the mathematical
expressions. The results show that the simulation and
theoretical results are in good agreement. In addition, the
proposed protocol obtains better performance than a two-
way relaying protocol in which the network coding is not
used.
The rest of this paper is organised as follows. The system
model and the proposed schemes are described in Section 2.
In Section 3, the performance evaluation of the protocols is
analysed. The simulation results are presented in Section 4.
Finally, conclusions are drawn in Section 5.
2 System model
Fig. 1 presents the system model for the cognitive two-way
relaying scheme under an interference constraint. In this
figure, there are two secondary sources S
1
and S
2
attempting
to transmit their packet to each other. We assume that there
is no direct link between the sources and hence the data
transmission between these sources is realised with the help
of Mavailable relays, for example, R
1
,R
2
,...,R
M
.Itis
assumed that each node is equipped with a single antenna
and operates in half-duplex mode.
For ease of analysis, we assume the distances between each
source and all relays are same. As presented in Fig. 1,we
denote d
1
,d
2
,d
3
,l
1
and l
2
as the distances between links
S
1
–R
i
,S
2
–R
i
,PR–R
i
,S
1
–PR and S
2
–PR, respectively,
where i[{1, 2, ...,M}. We also denote h
1,i
,h
2,i
,h
3,i
,g
1
and g
2
as Rayleigh fading channel coefficients of links S
1
–
R
i
,S
2
–R
i
,PR–R
i
,S
1
–PR and S
2
–PR, respectively. We
assume that all the channels are modelled by flat and block
fading channels in which the channel coefficients remain
constant during a transmission of a packet but change
independently from one packet to another [21]. We set
c
a,i
¼|h
a,i
|
2
and
w
b
¼|g
b
|
2
,a[{1, 2, 3}and b[{1, 2};
similar to [6, 7],
c
a,i
and
w
b
are exponential random
variables with parameters
l
a=d
b
aand Vb=l
b
b,
respectively, where
b
is path-loss exponent.
The operation of the proposed protocol, called PR1, is split
into time slots as follows. In the first time slot, the source S
1
broadcasts its packet L
1
to all the relays. Next, the packet L
2
of
the source S
2
is sent to all relays in the second time slot. In the
third time slot, a relay is chosen to transmit the combined
packet to the sources S
1
and S
2
by using XOR operation.
The relay selection method will be presented in detail in the
next section.
For a baseline, we compare the performance of the
proposed protocol with the two-way relaying protocol
without network coding (named PR0). In the PR0 protocol,
the operation of data transmission is divided into four time
slots. In the first time slot, the source S
1
broadcasts the
packet L
1
to all the relays. In the second time slot, one of
relays which decodes successfully is chosen to forward this
packet to the source S
2
. Similarly, in the third and fourth
time slots, the source S
2
transmits the packet L
2
to all the
relays and one of the successful relays is selected to
transmit the packet L
2
to the source S
1
.
3 Performance analysis
In the underlay approaches [4 –13], the secondary sources and
secondary relays must adapt their transmit power so that the
interference caused at the PR is below a threshold I
PR
. The
threshold I
PR
which is given by the PR is maximum
tolerable interference level at which the PR can still
maintain the reliable decoding. In [22], the authors
proposed a method to determine the interference level I
PR
in cognitive radio networks.
Similar to [5, Eq. (1)], the received signal-to-noise ratios
(SNR) at relay R
i
of the transmissions from the sources S
1
and S
2
are, respectively, given as
g
1,i=IPR|h1,i|2
N0|g1|2=Q
c
1,i
w
1
and
g
2,i=IPR|h2,i|2
N0|g2|2=Q
c
2,i
w
2
(1)
where N
0
is the variance of additive white Gaussian noise at
the secondary receivers and Q¼I
PR
/N
0
.
Therefore the achievable rates of the links S
1
–R
i
and S
2
–R
i
are formulated as
B1,i=1
3log2(1 +
g
1,i) and B2,i=1
3log2(1 +
g
2,i) (2)
where the ratio 1/3 represents the fact that data transmission is
split into three time slots.
We assume that relay R
i
decodes the packet L
1
(L
2
)
successfully if the data rate B
1,i
(B
2,i
) is larger than the target
rate Bof the system. Let us denote D
1
as a set of relays
decoding the packet L
1
successfully. Since D
1
is a random
decoding relay set, the size of D
1
is a random variable, that
is, |D
1
|¼k
1
,k
1
[{0, 1, ...,M}. Without loss of
generality, we assume that D1={R1,R2,...,Rk1}.
Now, it is noted from (1) that the random variables
g
1,i
are
not independent because they contain a common random
variable
w
1
. Hence, unlike [5, Eq. (6)], we find another
method to calculate the probability of the set D
1
, which can
be expressed as follows
Pr[D1]=Pr B1,1 .B,B1,2 .B,...,B1,k1
.B,
B1,k1+1,B,B1,k1+2,B,...,B1,M,B

(3)
Fig. 1 System model for cognitive two-way relaying scheme under
interference constraint
IET Commun., 2012, Vol. 6, Iss. 16, pp. 2750–2759 2751
doi: 10.1049/iet-com.2012.0235 &The Institution of Engineering and Technology 2012
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Substituting (2) into (3), we obtain (4) as
Pr[D1]=Pr min
t=1,2,...,k1
Q
r
1
c
1,t

.
w
1≥max
v=k1+1,...,M
Q
r
1
c
1,v

(4)
where
r
1
¼2
3B
21.
Setting Z1=mint=1,2,...,k1(Q
c
1,t/
r
1) and T1=
maxv=k1+1,...,M(Q
c
1,v/
r
1), the cumulative density functions
(CDF) of Z
1
and T
1
are, respectively, expressed as
FZ1(z)=Pr[Z1≤z]=1−exp −k1
l
1
r
1z
Q

(5)
FT1(t)=Pr[T1≤t]=1−exp −
l
1
r
1t
Q

M−k1
=
M−k1
v=0
(−1)vCv
M−k1exp −v
l
1
r
1t
Q
 (6)
where Cx
y=y!/x!(y−x)!with x≤y.
Combining (4)–(6), we can calculate Pr[D
1
]as
Pr[D1]=Pr[Z1.
w
1≥T1]
=+1
0
V1exp (−V1x)(1 −FZ1(x))FT1(x)dx
=
M−k1
v=0
(−1)vCv
M−k1
V1Q
V1Q+(v+k1)
l
1
r
1
(7)
We note that when |D
1
|¼k
1
, there are Ck1
Mpossible cases for
the decoding set D
1
. Hence, from (7), it can be obtained the
probability Pr[|D
1
|¼k
1
]as
Pr[|D1|=k1]=Ck1
M
M−k1
v=0
(−1)vCv
M−k1
V1Q
V1Q+(v+k1)
l
1
r
1
(8)
Similarly, if we denote D
2
as a set of relays decoding the
packet L
1
successfully, the probability Pr[|D
2
|¼k
2
],
k
2
[{0, 1, ...,M}, is calculated as
Pr[|D2|=k2]=Ck2
M
M−k2
v=0
(−1)vCv
M−k2
V2Q
V2Q+(v+k2)
l
2
r
1
(9)
If we denote D¼D
1
>D
2
, the set Dis the set of relays
successfully decoding both packets L
1
and L
2
. Now, we
derive the probability Pr[|D|¼k], k[{0, 1, ...,M},
k≤k
1
,k≤k
2
. Without loss of generality, we assume that
D¼{R
1
,R
2
,...,R
k
},D
1
¼{R
1
,R
2
,...,R
k
,R
k+1
,...,
R
k+t
}D
2
¼{R
1
,R
2
,...,R
k
,R
k+t+1
,...,R
k+t+v
},
0≤t≤M2k,0≤v≤M2k2t. Similar to (4)–(7), the
probability of the decoding set Dcan be calculated as
Pr[D]=Pr min
z=1,...,k+t
Q
r
1
c
1,z

.
w
1≥max
w=k+t+1,...,M
Q
r
1
c
1,w

×Pr min
z=1,...,k,k+t+1,...,k+t+v
Q
r
1
c
2,z

.
w
2

≥max
w=k+1,...,k+t,k+t+v+1,...,M
Q
r
1
c
2,w

=
M−k−t
z=0
(−1)zCz
M−k−t
V1Q
V1Q+(z+k+t)
l
1
r
1

×
M−k−v
w=0
(−1)wCw
M−k−v
V2Q
V2Q+(w+k+v)
l
2
r
1

(10)
In addition, we can observe that there are Ck
Mpossible choices
for event |D|¼k. After this choice, there are Ct
M−kpossible
choices of relays which only successfully decode the packet
L
1
. Finally, there remains Cv
M−k−tpossible choices of relays
which only successfully decode the packet L
2
. Hence, the
probability Pr[|D|¼k] can be calculated as
Pr[|D|=k]=Ck
M
M−k
t=0
M−k−t
v=0
Ct
M−kCv
M−k−tPr[D]
=Ck
M
M−k
t=0
M−k−t
v=0
Ct
M−kCv
M−k−t
×
M−k−t
z=0
(−1)zCz
M−k−tV1Q
V1Q+(z+k+t)
l
1
r
1

×
M−k−t
w=0
(−1)wCw
M−k−tV2Q
V2Q+(w+k+v)
l
2
r
1

(11)
Next, similar to (1), the instantaneous received SNR at the
sources S
1
and S
2
because of the transmission of the relay
R
i
are, respectively, given as
x
1,i=IPR|h1,i|2
N0|h3,i|2=Q
c
1,i
c
3,i
and
x
2,i=IPR|h2,i|2
N0|h3,i|2=Q
c
2,i
c
3,i
(12)
As mentioned above, all the channels which are modelled by
flat and block fading channels, change after each time slot.
Therefore it is noted that the values of h
1,i
and h
2,i
in (1)
are different from those in (12). However, for ease of
presentation, we will not use an indicator of time in this paper.
Next, we consider two cases as follows: In the first case, no
relay can decode successfully both packets L
1
and L
2
, that is,
D¼{Ø},D
1
¼{R
k+1
,...,R
k+t
}D
2
¼{R
k+t+1
,...,R
k+t+v
}.
In this case, only one relay is selected to transmit the packet
L
1
(or L
2
) to the source S
1
(or S
2
). Hence, we propose a
relay selection strategy as follows
Rj:max
z=k+1,k+2,...,k+t;
w=k+t+1,...,k+t+v
(
x
1,w,
x
2,z) (13)
2752 IET Commun., 2012, Vol. 6, Iss. 16, pp. 2750–2759
&The Institution of Engineering and Technology 2012 doi: 10.1049/iet-com.2012.0235
www.ietdl.org

Equation (13) implies that the relay having the best channel
to its intended receiver (S
1
or S
2
) is chosen to forward the
successfully decoded packet in the third time slot. It is also
noted that the relay selection processes in this paper can be
executed in a distributed manner [16].
Now, setting Z2=maxz=k+1,k+2,...,k+t(
x
2,z) and T2=
maxw=k+t+1,...,k+t+v(
x
1,w), combining (12) and [5, Eq. (3)],
the CDFs FZ2(z) and FT2(t) can be derived as
FZ2(x)=Pr max
z=k+1,k+2,...,k+t(
x
2,z),x

=
l
2x
l
2x+
l
3Q

t
(14)
FT2(x)=Pr max
w=k+t+1,...,k+t+v(
x
1,w),x

=
l
1x
l
1x+
l
3Q

v
(15)
In addition, from (13), we can observe that the probability that
the source S
2
cannot receive the packet L
1
successfully can be
formulated as follows
pout
1=Pr[Z2,T2]+Pr Z2≥T2,1
3log2(1 +Z2),B

=Pr[Z2,T2]+Pr[T2≤Z2,
r
1]
(16)
In (16), the term Pr[Z
2
,T
2
] can be calculated as
Pr[Z2,T2]=+1
0
fT2(x)FZ2(x)dx(17)
where fT2(t) is the probability density function (PDF) of T
2
and is given as
fT2(x)=∂FT2(x)
∂x=v
l
v
1
l
3Qxv−1
(
l
1x+
l
3Q)v+1(18)
Substituting (14) and (18) into (17), we have
Pr[Z2,T2]=+1
0
v
l
v
1
l
t
2
l
3Qxv+t−1
(
l
1x+
l
3Q)v+1(
l
2x+
l
3Q)tdx(19)
From (19), we consider more two cases as follows:
Case 1: If
l
1
¼
l
2
¼
l
, (19) can be calculated as
Pr[Z2,T2]=+1
0
v
l
v+t
l
3Qxv+t−1
(
l
x+
l
3Q)v+t+1dx
=v
l
3Q+1
l
3Q
(x−
l
3Q)v+t−1dx
xv+t+1
=v
l
3Q
v+t−1
a=0
(−1)aCa
v+t−1(
l
3Q)a+1
l
3Q
dx
xa+2
=v
v+t−1
a=0
(−1)aCa
v+t−1
a+1(20)
Case 2: If
l
1
=
l
2
, using Appendix 1, we obtain (see (21))
Next, we consider the probability Pr[T
2
≤Z
2
,
r
1
] in (16).
Similarly, we have
Pr[T2≤Z2,
r
1]=
r
1
0
fT2(x)dx
r
1
x
fZ2(y)dy
=FT2(
r
1)FZ2(
r
1)−
r
1
0
fT2(x)FZ2(x)dx
(22)
Substituting (14), (15) and (18) into (22), we can obtain
Pr[T2≤Z2,
r
1]=
l
2
r
1
l
2
r
1+
l
3Q

t
l
1
r
1
l
1
r
1+
l
3Q

v
−
r
1
0
v
l
v
1
l
t
2
l
3Qxv+t−1
(
l
1x+
l
3Q)v+1(
l
2x+
l
3Q)tdx
(23)
Similar to calculate Pr[Z
2
,T
2
], we consider two cases as
follows (see (24))
Case 1: If
l
1
¼
l
2
¼
l
, (23) can be calculated as
Pr[Z2,T2]=v
l
v
1
l
2
l
3Q
×
v−1
a=0
v−1−a
b=0
(−1)a+b
a+b+1Ca
v+t−1Cb
v−1−a
(
l
2−
l
1)b
l
v−a
1
l
a+b+1
2
l
3Q
+
v+t−1
a=v
(−1)aCa
v+t−1(
l
3Q)a
v+1
b=2
mb
(b−1)
l
1(
l
2
l
3Q)b−1+
1+a−v
b=2
nb
(b−1)(
l
3Q)b−1+n1ln
l
2
l
1

⎛
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎠
(21)
Pr[T2≤Z2,
r
1]=
lr
1
lr
1+
l
3Q

v+t
−
r
1
0
v
l
v+t
l
3Qxv+t−1
(
l
x+
l
3Q)v+t+1dx=
lr
1
lr
1+
l
3Q

v+t
−v
v+t−1
a=0
(−1)aCa
v+t−1
a+11−
l
3Q
lr
1+
l
3Q

a+1
 (24)
IET Commun., 2012, Vol. 6, Iss. 16, pp. 2750–2759 2753
doi: 10.1049/iet-com.2012.0235 &The Institution of Engineering and Technology 2012
www.ietdl.org

Case 2: If
l
1
=
l
2
, similarly to (45), we can calculate (23) as
follows
Pr[T2≤Z2,
r
1]=
l
2
r
1
l
2
r
1+
l
3Q

t
l
1
r
1
l
1
r
1+
l
3Q

v
−v
l
v
1
l
2
l
3Q
v−1
a=0
(−1)aCa
v+t−1(
l
3Q)aI3

+
v+t−1
a=v
(−1)aCa
v+t−1(
l
3Q)aI4
(25)
where integrals I
3
and I
4
can be calculated, similarly to
(46)–(49), as
I3=
l
2
r
1+
l
3Q
l
3Q
xv−1−adx
(
l
1x+
l
3Q(
l
2−
l
1))v+1
=
v−1−a
b=0
(−1)bCb
v−1−a
a+b+1
(
l
2−
l
1)b
l
v−a
1
l
a+b+1
2(
l
3Q)a+1
1−
l
3Q
l
1
r
1+
l
3Q

a+b+1

(26)
I4=
l
2
r
1+
l
3Q
l
3Q
dx
(
l
1x+
l
3Q(
l
2−
l
1))v+1x1+a−v
=n1ln
l
2
r
1+
l
3Q
l
1
r
1+
l
3Q

+
v+1
b=2
mb
(b−1)
l
1(
l
2
l
3Q)b−11−
l
3Q
l
1
r
1+
l
3Q

b−1

+
1+a−v
b=2
nb
(b−1)(
l
3Q)b−11−
l
3Q
l
2
r
1+
l
3Q

b−1

(27)
Since we finish calculating the probabilities Pr[Z
2
,T
2
] and
Pr[T
2
≤Z
2
,
r
1
], from (11) and (16), the outage
probability at the source S
2
, in the case that there is no
relay which can decode both packets successfully, is given as
PPR1,1
S2=
M
t=0
M−t
v=0
Ct
MCv
M−t
M−t
z=0
(−1)zCz
M−tV1Q
V1Q+(z+t)
l
1
r
1

×
M−t
w=0
(−1)wCw
M−tV2Q
V2Q+(w+v)
l
2
r
1

pout
1
(28)
Next, we consider the outage probability at the source S
2
when |D|¼k.0. In this case, the best relay selection
strategy is chosen by the opportunistic strategy as [17]
Rj: min (
x
1,j,
x
2,j)=max
i=1,2,...,kmin (
x
1,i,
x
2,i) (29)
Let us denote W
i
¼min(
x
1,i
,
x
2,i
), then using (12), the CDF
FWi(x) is given as (see (30))
In (30), the probability Pr[
c
2,i
≤min(
c
1,i
,(x/Q)
c
3,i
)] can
be given as
Pr
c
2,i≤min
c
1,i,x
Q
c
3,i

=+1
0
f
c
2,i(t)Pr min
c
1,i,x
Q
c
3,i

≥t

dt
=+1
0
l
2exp (−
l
2t) exp −
l
1+Q
x
l
3

t

dt
=
l
2x
(
l
1+
l
2)x+
l
3Q
(31)
Also, we can easily calculate the probability Pr[
c
1,i
,
min(
c
2,i
,(x/Q)
c
3,i
)] in (30) as
Pr
c
1,i,min
c
2,i,x
Q
c
3,i

=
l
1x
(
l
1+
l
2)x+
l
3Q(32)
Substituting (31) and (32) into (30), we obtain
FWi(x)=(
l
1+
l
2)x
(
l
1+
l
2)x+
l
3Q(33)
Hence, if W
j
¼max(W
i
), the CDF FWj(x) is expressed as
FWj(x)=(
l
1+
l
2)x
(
l
1+
l
2)x+
l
3Q

k
(34)
Differentiating (33) and (34) with respect to x, the
corresponding PDFs fWi(x) and fWj(x) are, respectively,
obtained by (35) and (36) as follows
fWi(x)=∂FWi(x)
∂x=(
l
1+
l
2)
l
3Q
((
l
1+
l
2)x+
l
3Q)2(35)
fWj(x)=
∂FWj(x)
∂x=k(
l
1+
l
2)k
l
3Qxk−1
((
l
1+
l
2)x+
l
3Q)k+1(36)
Since the achievable rate of the link between the chosen relay
R
j
and the source S
2
is (1/3)log
2
(1 +
x
2,j
), the outage
probability at the source S
2
can be formulated as
Pr 1
3log2(1 +
x
2,j),B

=Pr[
x
2,j,
r
1] (37)
Similar to [17, Eq. (16)], the probability Pr[
x
2,j
,
r
1
] can be
FWi(x)=Pr[min (
x
1,i,
x
2,i),x]=Pr
c
1,i≥
c
2,i,
c
2,i
c
3,i
,x
Q

+Pr
c
1,i,
c
2,i,
c
1,i
c
3,i
,x
Q

=Pr
c
2,i≤min
c
1,i,x
Q
c
3,i

+Pr
c
1,i,min
c
2,i,x
Q
c
3,i

(30)
2754 IET Commun., 2012, Vol. 6, Iss. 16, pp. 2750–2759
&The Institution of Engineering and Technology 2012 doi: 10.1049/iet-com.2012.0235
www.ietdl.org

formulated as
Pr[
x
2,j,
r
1]=+1
0
∂Pr[
x
2,i,
r
1, min(
x
1,i,
x
2,i),z]
∂z
fWj(z)
fWi(
r
1)dz
(38)
Using Appendix 2, we have (see (39))
Now, using (11), we obtain the outage probability at the
source S
2
when |D|¼k.0as
PPR1,2
S2=Ck
M
M−k
t=0
M−k−t
v=0
Ct
M−kCv
M−k−t
×
M−k−t
z=0
(−1)zCz
M−k−tV1Q
V1Q+(z+k+t)
l
1
r
1

×
M−k−t
w=0
(−1)wCw
M−k−tV2Q
V2Q+(w+k+v)
l
2
r
1

Pr[
x
2,j,
r
1]
(40)
From (28) and (40), the average outage probability at the
source S
2
is calculated as PPR1
S2=PPR1,1
S2+PPR1,2
S2. It is noted
that the outage probability at the source S
1
is easily
obtained by replacing
l
1
with
l
2
and vice versa in the
expression of PPR1
S2.
For the PR0 protocol, it is also assumed that at the end of
the first time slot, the set of successful relay is
D1={R1,R2,...,Rk1
}. The strategy of the best relay
selection is expressed as follows:
Rb:
g
2,b=max
i=1,2,...,k1
Q
c
2,i
c
3,i
 (41)
Since the data transmission in the PR0 protocol is split into
four time slots, the achievable rates of the links S
1
–R
i
and
R
b
–S
2
are (1/4)log
2
(1 +Q(
c
1,i
/
w
1
)) and (1/4)log
2
(1 +
g
2,b
).
Therefore the outage probability at the source S
2
in each
case of the set D
1
is formulated as follows:
PPR0
S2=
D1
Pr[D1]Pr 1
4log2(1 +
g
2,b),B|D1

(42)
Combining (8) and Pr[(1/4) log2(1 +
g
2,b),B|D1]=
(((
l
2
r
0)/(
l
2
r
0+V2Q)))k1with
r
0
¼2
4B
21, we obtain the
average outage probability at the source S
2
as
PPR0
S2=
M
k1=0
Ck1
M
M−k1
v=0
(−1)vCv
M−k1
V1Q
V1Q+(v+k1)
l
1
r
0
l
2
r
0
l
2
r
0+
l
3Q

k1
(43)
From (43), if we replace
l
1
with
l
2
and vice versa in the
expression of PPR0
S2, we can obtain the outage probability at
the source S
1
.
4 Simulation results
In this section, we describe the Monte-Carlo simulations
performed to evaluate and compare the performances of the
presented protocols above. Theoretical results were obtained
from the equations derived above. In all simulations, we
assume that a path-loss exponential
b
equal to 3.
In a two-dimensional plane, assume that the co-ordinates of
S
1
,S
2
,R
i
and PR are (0,0), (1,0), (x
1
, 0) and (x
2
,y
2
),
respectively, where 0 ,x
1
,x
2
,1. Therefore d
1
¼x
1
,
d2 ¼12d1, d3=
(x1−x2)2+y2
2
,l1=
x2
2+y2
2
and
l2=
(1 −x2)2+y2
2
.
In Fig. 2, we present the probability Pr[|D|¼k]asa
function of the value Qin dB. In this figure, the parameters
Fig. 2 Probability Pr[|D|¼k] as a function of Q in dB when
x
1
¼0.5, B ¼1, M ¼3x
2
¼0.5 and y
2
¼0.25
Pr[
x
2,j,
r
1]=k(
l
1+
l
2)k−1
l
3Q
×
(
l
1+
l
2)
k−1
m=0
(−1)mCm
k−1
(m+1)
1
(
l
1+
l
2)k−1
l
3Q1−
l
3Q
(
l
1+
l
2)
r
1+
l
3Q

m+1

−
l
1
2
l=1
a
l
l
k−1
1
(−1)k−lCk−l
k−1(
l
2
r
1+
l
3Q)k−lln (
l
1+
l
2)
r
1+
l
3Q
l
2
r
1+
l
3Q

+

k−1
q=0
(−1)qCq
k−1
k−q−l(
l
2
r
1+
l
3Q)q[((
l
1+
l
2)
r
1+
l
3Q)k−q−l−(
l
2
r
1+
l
3Q)k−q−l]
⎛
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎠
−
l
1
k−1
l=1
b
l
(
l
1+
l
2)k−1
(−1)k−lCk−l
k−1(
l
3Q)qln (
l
1+
l
2)
r
1+
l
3Q
l
3Q

+

k−1
q=0
(−1)qCq
k−1
k−q−l(
l
3Q)q[((
l
1+
l
2)
r
1+
l
3Q)k−q−l−(
l
3Q)k−q−l]
⎛
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎠
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
(39)
IET Commun., 2012, Vol. 6, Iss. 16, pp. 2750–2759 2755
doi: 10.1049/iet-com.2012.0235 &The Institution of Engineering and Technology 2012
www.ietdl.org

are set as x
1
¼x
2
¼0.5, y
2
¼0.25, B¼1 and M¼3. We
can see from this figure that when k¼0, the probability
Pr[|D|¼0] decreases with increasing Q. In a contrary, the
probability Pr[|D|¼3] increases with increasing Q.In
addition, when k¼1 and k¼2, with increasing Q,the
probability Pr[|D|¼k] increases to the maximum value and
then decreases to zero.
Figs. 3 and 4present the outage probability as a function of
Qin dB. In Fig. 3, we also consider the symmetric network,
that is, x
1
¼x
2
¼0.5, in which the outage probabilities at the
sources S
1
and S
2
are same. Here, we fix Band y
2
as 1 and
0.25, respectively, while changing the value of M.Aswe
can see, the protocols obtain better performance when Mis
higher. This is because they obtain higher diversity gain
with the increasing of M.InFig. 4, we investigate the effect
of the target rate on the outage performance. In this
simulation, an asymmetric network is considered, that is,
x
1
¼0.4, x
2
¼0.5, y
2
¼0.25 and M¼3. It can be
observed from this figure that the outage probability at the
sources S
1
and S
2
decreases with the decreasing of the
target rate. In addition, for both protocols, the outage
probability at the source S
1
is lower than that at the source
S
2
. It is because of the fact that relays are nearer the source
S
1
than S
2
.
In Fig. 5, we present the sum outage probability of the PR0
and PR1 protocols as a function of the target rate. In this
figure, the symmetric network with x
1
¼0.5 is simulated. In
addition, we assume that the PR is placed at the position of
(0.2, 0.25) and the number of relays Mis 3. Similar to
Fig. 4, we can see that the sum outage probability increases
with increasing of target rate. In addition, at low value of B,
the performances of the PR0 and PR1 protocols are quite
similar but at high values of B, the proposed protocol
outperforms the PR0 protocol.
In Figs. 6 and 7, we investigate the effect of the distances
on the performances of the PR0 and PR1 protocols. In
Fig. 6, we set the parameters as follows: M¼3, B¼1 and
Q¼0 dB. We assume that the position of the PU is (0.5,
0.25). As we can see, the sum outage probability of both
protocols has a maximum value at x
1
¼0.5. This is because
Fig. 3 Outage probability at S
1
and S
2
as a function of Q in dB
when x
1
¼0.5, B ¼1, x
2
¼0.5 and y
2
¼0.25
Fig. 4 Outage probability at S
1
and S
2
as a function of Q in dB
when x
1
¼0.4, M ¼3, x
2
¼0.5 and y
2
¼0.25
Fig. 5 Sum outage probability as a function of the target rate B
when x
1
¼0.5, M ¼3, x
2
¼0.2 and y
2
¼0.25
Fig. 6 Sum outage probability as a function of the distance d
1
(x
1
)
when M ¼3, B ¼1, Q ¼0 dB, x
2
¼0.5 and y
2
¼0.25
Fig. 7 Sum outage probability as a function of the distance d
1
(x
1
)
when M ¼3, B ¼1, Q ¼0 dB, x
2
¼0.3 and y
2
¼0.25
2756 IET Commun., 2012, Vol. 6, Iss. 16, pp. 2750–2759
&The Institution of Engineering and Technology 2012 doi: 10.1049/iet-com.2012.0235
www.ietdl.org

at this position, the distance between relays and the PU is
nearest and hence, the power of relays is smaller because of
the maximum interference constraint. Therefore it can also
be observed that the sum outage probability decreases when
relays are near the source S
1
or the source S
2
.InFig. 7,we
change the position of PR, that is, x
2
¼0.3. In this figure,
the maximum value of the sum outage probability is
obtained at d
1
¼x
1
¼0.35 and the minimum one is at
d
1
¼x
1
¼1. Therefore we can see that the performance of
the protocols is better when the relays are further from the
PR. In addition, from Figs. 6 and 7, it can be observed that
the proposed protocol outperforms the PR1 protocol.
Finally, it can be observed from Figs. 2–7that the
simulation and theoretical results are in good agreement.
This validates the correction of the theoretical analyses.
5 Conclusions
In this paper, we proposed a network coded protocol for a
cognitive two-way relaying system under an interference
constraint. In the proposed protocol, the best relay is chosen
by using the opportunistic relay selection strategy and it
forwards the combined packet to two secondary sources. To
evaluate and compare the performance of the proposed
protocol with one that does not use network coding, we
derived the exact closed-form expressions of outage
probability. Then, various Monte-Carlo simulations were
presented to verify the mathematical expressions. The
results show that the simulation results match very well
with the theoretical results. In addition, the proposed
protocol PR1, which uses network coding, outperforms the
PR0 protocol without network coding, in terms of the sum
outage probability while using fewer time slots.
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7 Appendix 1
7.1 Derivation of (21)
From (19), we have
Pr[Z2,T2]=v
l
v
1
l
2
l
3Q+1
l
3Q
(x−
l
3Q)v+t−1
xt(
l
1x+
l
3Q(
l
2−
l
1))v+1dx
=v
l
v
1
l
2
l
3Q
v+t−1
a=0
(−1)aCa
v+t−1(
l
3Q)a
×+1
l
3Q
xv−1−adx
(
l
1x+
l
3Q(
l
2−
l
1))v+1
(44)
In addition, we can rewrite (44) in the following form (see (45))
To calculate the probability Pr[Z2 ,T2] in (45), we have
to calculate the integrals I
1
and I
2
. First, considering the
Pr[Z2,T2]=v
l
v
1
l
2
l
3Q×

v−1
a=0
(−1)aCa
v+t−1(
l
3Q)a+1
l
3Q
xv−1−adx
(
l
1x+
l
3Q(
l
2−
l
1))v+1
 !
I1
+
v+t−1
a=v
(−1)aCa
v+t−1(
l
3Q)a+1
l
3Q
dx
(
l
1x+
l
3Q(
l
2−
l
1))v+1x1+a−v
 !
I2
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(45)
IET Commun., 2012, Vol. 6, Iss. 16, pp. 2750–2759 2757
doi: 10.1049/iet-com.2012.0235 &The Institution of Engineering and Technology 2012
www.ietdl.org

integral I
1
, we have
I1=+1
l
2
l
3Q
(x−
l
3Q(
l
2−
l
1))v−1−adx
l
v−a
1xv+1
=
v−1−a
b=0
(−1)bCb
v−1−a
a+b+1
(
l
2−
l
1)b
l
v−a
1
l
a+b+1
2(
l
3Q)a+1
(46)
Now, considering the integral I
2
; we first rewrite the integrand
of I
2
into the partial expansion as follows
1
(
l
1x+
l
3Q(
l
2−
l
1))v+1x1+a−v
=
v+1
b=1
mb
(
l
1x+
l
3Q(
l
2−
l
1))b+
1+a−v
b=1
nb
xb
(47)
where
mb=(−1)a+1−v
(v+1−b)!"v+1−b
c=1(a−v+c)
l
a+1−v
1
[
l
3Q(
l
2−
l
1)]a+2−b,nb
=(−1)a−b−v+1"a−b−v+1
c=1(v+c)
l
a−b−v+1
1
(a−b−v+1)![
l
3Q(
l
2−
l
1)]a+2−b
(48)
Using (47) and (48), we can calculate the integral I
2
as
I2=+1
l
3Q
dx
(
l
1x+
l
3Q(
l
2−
l
1))v+1x1+a−v
=+1
l
3Q
m1dx
l
1x+
l
3Q(
l
2−
l
1)++1
l
3Q
n1dx
x
+
v+1
b=2+1
l
3Q
mb
(
l
1x+
l
3Q(
l
2−
l
1))b+
1+a−v
b=2+1
l
3Q
nb
xbdx
=
v+1
b=2
mb
(b−1)
l
1(
l
2
l
3Q)b−1+
1+a−v
b=2
nb
(b−1)(
l
3Q)b−1
+n1ln
l
2
l
1
 (49)
Now, substituting (46) and (49) into (45), we can obtain
Pr[Z2 ,T2] in the case that
l
1=
l
2 as presented in (21).
8 Appendix 2
8.1 Derivation of (39)
Substituting (35) and (36) into (38), we have
Pr[
x
2,j,
r
1]=k(
l
1+
l
2)k−1
+1
0
∂Pr[
x
2,i,
r
1, min (
x
1,i,
x
2,i),z]
∂z
zk−1dz
((
l
1+
l
2)z+
l
3Q)k−1
(50)
Next, we can rewrite the term Pr[
x
2
,i ,
r
1, min(
x
1
,i,
x
2
,i) ,z] in (50) as
Pr[
x
2,i,
r
1, min (
x
1,i,
x
2,i),z]
=Pr Q
c
2,i
c
3,i
,
r
1,min Q
c
1,i
c
3,i
,Q
c
2,i
c
3,i

,z

=Pr
c
3,i.max Q
r
1
c
2,i,Q
z
c
1,i

,
c
1,i,
c
2,i

 !
J1
+Pr
c
3,i.max Q
r
1
c
2,i,Q
z
c
2,i

,
c
1,i≥
c
2,i

 !
J2
(51)
Considering the probability J
1
in (51), it can be formulated as
(see (52))
Now, to calculate the probability J
3
, we consider two more
cases as follows:
Case 1: if z ≥
r
1, we can obtain J
3
by
J3=t
r
1
Q
0
l
2exp(−
l
2v)dvv
0
l
1exp(−
l
1w)dw
=
l
1
l
1+
l
2
−exp −
l
2
t
r
1
Q

+
l
2
l
1+
l
2
exp −(
l
1+
l
2)t
r
1
Q

(53)
Substituting (53) into (52), we obtain
J1=
l
1
l
1+
l
2
−
l
3Q
l
2
r
1+
l
3Q
+
l
2
l
1+
l
2
l
3Q
(
l
1+
l
2)
r
1+
l
3Q(54)
Case 2: If z ,
r
1, we calculate the probability J
3
as
J3=zt/Q
0
l
1exp (−
l
1v)dvt
r
1/Q
v
l
2exp (−
l
2w)dw
=
l
2
l
1+
l
2
+exp −(
l
1z+
l
2
r
1)t
Q

−exp −
l
2
t
r
1
Q

−
l
1
l
1+
l
2
exp −(
l
1+
l
2)t
r
1
Q
 (55)
Therefore substituting (55) into (52), we have
J1=
l
1
l
1+
l
2
−
l
3Q
l
2
r
1+
l
3Q+
l
3Q
l
1z+
l
2
r
1+
l
3Q
−
l
1
l
1+
l
2
l
3Q
(
l
1+
l
2)z+
l
3Q
(56)
J1=+1
0
l
3exp (−
l
3t)Pr max Q
r
1
c
2,i,Q
z
c
1,i

,t,
c
1,i,
c
2,i

 !
J3
dt(52)
2758 IET Commun., 2012, Vol. 6, Iss. 16, pp. 2750–2759
&The Institution of Engineering and Technology 2012 doi: 10.1049/iet-com.2012.0235
www.ietdl.org

Similarly, the probability J
2
can be calculated as follows
J2=
l
2
r
1
(
l
1+
l
2)
r
1+
l
3Qif z≥
r
1
l
2z
(
l
1+
l
2)z+
l
3Qif z,
r
1
⎧
⎪
⎪
⎨
⎪
⎪
⎩
(57)
Substituting (56) and (57) into (51), we obtain the probability
Pr[
c
2
,i ,x, min(
c
1
,i,
c
2
,i) ,z]. Next, differentiating
Pr[
c
2
,i ,x, min(
c
1
,i,
c
2
,i) ,z] with respect to zand then
substituting the obtained result into (50), we obtain (see (58))
In (58), the integral J
4
can be calculated as
J4=1
(
l
1+
l
2)k−1(
l
1+
l
2)
r
1+
l
3Q
l
3Q
(z−
l
3Q)k−1dz
zk+1
=
k−1
m=0
(−1)mCm
k−1(
l
3Q)m
(
l
1+
l
2)k−1(
l
1+
l
2)
r
1+
l
3Q
l
3Q
dz
zm+2
=
k−1
m=0
(−1)mCm
k−1
(m+1)
1
(
l
1+
l
2)k−1
l
3Q
1−
l
3Q
(
l
1+
l
2)
r
1+
l
3Q

m+1

(59)
Next, we calculate the integral J
5
in (58). Also, we first
rewrite the integrand of J
5
into the partial expansion as
follows
1
(
l
1z+
l
2
r
1+
l
3Q)2((
l
1+
l
2)z+
l
3Q)k−1
=
a
1
(
l
1z+
l
2
r
1+
l
3Q)+
a
2
(
l
1z+
l
2
r
1+
l
3Q)2
+
k−1
l=1
b
l
((
l
1+
l
2)z+
l
3Q)l
(60)
where
a
1=(−1)k−1(k−1)(
l
1+
l
2)
l
k−1
1
l
k
2(
l
3Q+(
l
1+
l
2)
r
1)k
a
2=(−1)k−1
l
k−1
1
l
k−1
2(
l
3Q+(
l
1+
l
2)
r
1)k−1
and
b
l=(−1)k−l−1(k−l)(
l
1+
l
2)2
l
k−l−1
1
l
k−l+1
2(
l
3Q+(
l
1+
l
2)
r
1)k−l+1
Hence, the integral J
5
can be rewritten as (see (61))
From (61), after some manipulation, we obtain J
5
as
(see (62))
Substituting (59) and (62) into (58), the outage probability
Pr[
x
2
,j ,
r
1] can be given as in (39).
Pr[
x
2,j,
r
1]=k(
l
1+
l
2)k−1
l
3Q
×(
l
1+
l
2)
r
1
0
zk−1dz
((
l
1+
l
2)z+
l
3Q)k+1
 !
J4
−
l
1
r
1
0
zk−1dz
(
l
1z+
l
2
r
1+
l
3Q)2((
l
1+
l
2)z+
l
3Q)k−1
 !
J5
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
(58)
J5=
2
l=1
a
lx
0
zk−1dz
(
l
1z+
l
2
r
1+
l
3Q)l+
k−1
l=1
b
lx
0
zk−1dz
((
l
1+
l
2)z+
l
3Q)l=
2
l=1
a
l
l
k−1
1(
l
1+
l
2)
r
1+
l
3Q
l
1z+
l
3Q
(z−
l
2
r
1−
l
3Q)k−1dz
zl
+
k−1
l=1
b
l
(
l
1+
l
2)k−1(
l
1+
l
2)
r
1+
l
3Q
l
3Q
(z−
l
3Q)k−1
zldz=
2
l=1
a
l
l
k−1
1
k−1
q=0
(−1)qCq
k−1(
l
2
r
1+
l
3Q)q(
l
1+
l
2)
r
1+
l
3Q
l
2
r
1+
l
3Q
zk−1−q−ldz
+
k−1
l=1
b
l
(
l
1+
l
2)k−1
k−1
q=0
(−1)qCq
k−1(
l
3Q)q(
l
1+
l
2)
r
1+
l
3Q
l
3Q
zk−1−q−ldz
(61)
J5=
2
l=1
a
l
l
k−1
1
(−1)k−lCk−l
k−1(
l
2
r
1+
l
3Q)k−lln (
l
1+
l
2)
r
1+
l
3Q
l
2
r
1+
l
3Q

+

k−1
q=0
(−1)qCq
k−1
k−q−l(
l
2
r
1+
l
3Q)q[((
l
1+
l
2)
r
1+
l
3Q)k−q−l−(
l
2
r
1+
l
3Q)k−q−l]
⎛
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎠
+
k−1
l=1
b
l
(
l
1+
l
2)k−1
(−1)k−lCk−l
k−1(
l
3Q)qln (
l
1+
l
2)
r
1+
l
3Q
l
3Q

+

k−1
q=0
(−1)qCq
k−1
k−q−l(
l
3Q)q[((
l
1+
l
2)
r
1+
l
3Q)k−q−l−(
l
3Q)k−q−l]
⎛
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎠
(62)
IET Commun., 2012, Vol. 6, Iss. 16, pp. 2750–2759 2759
doi: 10.1049/iet-com.2012.0235 &The Institution of Engineering and Technology 2012
www.ietdl.org