Secured Cooperative Cognitive Radio Networks with Relay Selection

Abstract

In this paper, we propose physical layer security for cooperative cognitive radio networks (CCRNs) with relay selection in the presence of multiple primary users and multiple eavesdroppers. To be specific, we propose three relay selection schemes, namely, opportunistic relay selection (ORS), suboptimal relay selection (SoRS), and partial relay selection (PRS) for secured CCRNs, which are based on the availability of channel state information (CSI) at the receivers. For each approach, we derive exact and asymptotic expressions for the secrecy outage probability. Results show that under the assumption of perfect CSI, ORS outperforms both SoRS and PRS.

Full text

1
Secured Cooperative Cognitive Radio Networks
with Relay Selection
Trung Q. Duong
∗
, Tran Trung Duy
†
, Maged Elkashlan
‡
, Nghi H. Tran
§
, and Octavia A. Dobre
¶
∗
Queen’s University Belfast, UK (e-mail: trung.q.duong@qub.ac.uk)
†
Posts and Telecommunications Institute of Technology, Vietnam (e-mail: trantrungduy@ptithcm.edu.vn)
‡
Queen Mary University of London, London, UK (e-mail:maged.elkashlan@eecs.qmul.ac.uk)
§
University of Akron, U.S.A., (e-mail: nghi.tran@uakron.edu)
¶
Memorial University, Canada, (e-mail: odobre@mun.ca)
Abstract—In this paper, we propose physical layer security
for cooperative cognitive radio networks (CCRNs) with relay
selection in the presence of multiple primary users and multiple
eavesdroppers. To be specific, we propose three relay selec-
tion schemes, namely, opportunistic relay selection (ORS), sub-
optimal relay selection (SoRS), and partial relay selection (PRS)
for secured CCRNs, which are based on the availability of chan-
nel state information (CSI) at the receivers. For each approach,
we derive exact and asymptotic expressions for the secrecy outage
probability. Results show that under the assumption of perfect
CSI, ORS outperforms both SoRS and PRS.
I. INTRODUCTION
Best relay selection (BRS) is widely considered as a promis-
ing technology to enhance the wireless coverage with low
complexity in implementation [1]–[3]. It has attracted great
attention in the research literature [4]–[7]. Particularly, the
performance of BRS has been investigated for single user
with decode-and-forward (DF) [5], amplify-and-forward (AF)
relaying [4], [6], or multiple users [7]. It has been shown that
under an aggregate power constraint, outage-optimal perfor-
mance can be achieved for both DF and AF with BRS, in
other words, the outage probability for BRS is globally optimal
among all possible transmission strategies [3], [4].
On the other hand, in recent years, radio frequency spectrum
has become scarce and expensive wireless resources due to
the ever increasing demand of multimedia services. Having
the capability to mitigate the scarcity of frequency spectrum,
cognitive radio networks, however, may not fulfill this ex-
pectation due to the strict interference power constraint at
primary user. With its potential, BRS has been introduced to
cognitive radio networks. As a matter of fact, it has been
extensively investigated for cognitive radio networks under
spectrum-sharing [8]–[13].
Additionally, physical layer security opens up new dimen-
sions for secured wireless networks [14], for example, a relay
can be used to enhance the security of the legitimate com-
munications [15]. Unlike most previous works, in this paper,
we consider physical layer security of relay networks from
the unlicensed network’s perspective as an efficient design
approach for combating the spectrum scarcity. In doing so, we
take into account the realistic scenario of multiple secondary
relays, multiple PUs, and multiple secondary eavesdroppers.
In addition, we consider both peak interference power con-
straint inflicted by PU and the maximal transmit power at SU
to guarantee the quality of service at the primary network.
Specifically, we introduce several relay selection schemes
based on the availability of channel state information (CSI) at
the receivers. The contributions of this paper are as follows.
• Depending on the availability of CSI’s knowledge, we
propose and compare three relay selection approaches,
namely, opportunistic relay selection (ORS), sub-optimal
relay selection (SoRS), and partial relay selection (PRS).
• We derive new exact and asymptotic expressions in
closed-form for the secrecy outage probability for each
relay selection approach in independently but not identi-
cally distributed (i.n.i.d.) Rayleigh fading channels.
II. SYSTEM AND CHANNEL MODELS
We consider a physical-layer security network where a
secondary source (S) transmits data to a secondary destina-
tion (D) with the help of M secondary relays R
m
, m =
1, 2, . . . , M. We assume that there is no direct link between the
source and destination due to severe path loss and shadowing.
In underlay cognitive network, the secondary source and relays
must adapt their transmit power to satisfy a peak interference
power constraint I
p
at all primary users PU
n
, n = 1, 2, . . . , N.
In addition, the transmit power at SUs must be below a
maximum allowable power P
t
so as not to cause any harmful
interference to the PUs. In the secondary network, there are
eavesdroppers, E
k
, k = 1, 2, . . . , K, attempting to overhear
and decode the source data.
We assume that the channels between any two terminals
are subject to block and flat Rayleigh fading. Each node
has a single half-duplex radio and a single antenna. Due to
half-duplex constraint, a time-division channel allocation is
employed to realize orthogonal channels. To avoid eavesdrop-
pers while combining the data received from the source and
relays, the randomize-and-forward scheme can be used [15].
Let us denote h
SR
m
, h
SP
n
, h
SE
k
, h
R
m
D
, h
R
m
P
n
, and h
R
m
E
k
as the channel power gains of the links S → R
m
, S → P
n
,
S → E
k
, R
m
→ D, R
m
→ P
n
, and R
m
→ E
k
, respectively.
Here, we consider Rayleigh fading channels, which results in
h
XY
, where X ∈ {S, R
m
} and Y ∈ {R
m
, D, P
n
, E
k
}, being
exponential random variables (RVs) with parameters λ
XY
. In
addition, we assume that the primary nodes and eavesdroppers
are located in clusters, which then yields λ
XP
n
= λ
XP
for
all n, and λ
XE
k
= λ
XE
for all k. The operation of the

proposed protocol is split into two time slots. In the first time
slot, the source (S) broadcasts its data to the best relay R
b
,
b = 1, 2, . . . , M. Then, this relay decodes and forwards the
decoded data to the destination (D) in the second time slot.
III. PERFORMANCE ANALYSIS
Under interference constraint at K PUs, the transmit power
of node X is given as
P
X
= min

P
t
,
I
p
γ
XP

= P
t
min

1,
µ
γ
XP

, (1)
where γ
XP
= max
n=1,2,...,N
(h
XP
n
) and µ = I
p
/P
t
is a positive
constant. The signal-to-noise ratio (SNR) of the X → Y links,
with X ∈ {S, R
m
} and Y ∈ {R
m
, D, E
k
}, can be given as
Ψ
XY
=
P
t
min

1,
µ
γ
XP

h
XY
N
0
= Q
t
min

1,
µ
γ
XP

h
XY
, (2)
where N
0
is power of the additive noise at node Y, and Q
t
=
P
t
/N
0
is the average SNR. The secrecy rate of the X → Y
links, where X ∈ {S, R
m
} and Y ∈ {R
m
, D}, is given as
I
XY
=


1
2
log
2


1 + Q
t
min

1,
µ
γ
XP

h
XY
1 + Q
t
min

1,
µ
γ
XP

γ
XE




+
, (3)
where γ
XE
= max
k=1,2,...,K
(h
XE
k
) and (x)
+
= max (x, 0).
A. Opportunistic Relay Selection (ORS) Method
Assuming that the CSI of the X → E links are perfectly
known at the nodes X, X ∈ {S, R
m
}, we propose an oppor-
tunistic relay selection method (ORS) based on the max-min
strategy such that
R
b
= arg max
m=1,2,...,M
min (I
SR
m
, I
R
m
D
) . (4)
From (3) and (4), the secrecy outage probability of the OPR
protocol is formulated as
P
out
ORS
= Pr

max
m=1,2,...,M
min (W
1m
, W
2m
) < ρ

, (5)
where ρ = 2
R
th
, R
th
is a predetermined threshold, and
W
1m
=
1 + Q
t
min(1, µ/γ
SP
)h
SR
m
1 + Q
t
min(1, µ/γ
SP
)γ
SE
,
W
2m
=
1 + Q
t
min(1, µ/γ
R
m
P
)h
R
m
D
1 + Q
t
min(1, µ/γ
R
m
P
)γ
R
m
E
.
Lemma 1: The cumulative distribution function (CDF)
F
W
2m
(z) is given in an exact closed-form as
F
W
2m
(z) = 1 −
K

k=1
(−1)
k+1
C
k
K
kλ
R
m
E
kλ
R
m
E
+ λ
R
m
D
z
×

exp (−λ
R
m
D
(z − 1)/Q) (1 − exp (−λ
R
m
P
µ))
N
+
N

n=1
(−1)
n+1
C
n
N
nλ
R
m
P
Qµ
nλ
R
m
P
Qµ + λ
R
m
D
(z − 1)
× exp(−nλ
R
m
P
µ − λ
R
m
D
(z − 1)/Q)

. (6)
See Appendix A.
Theorem 1: The outage probability of ORS is given in (7),
which is shown at the top of the next page. See Appendix B.
Lemma 2: An asymptotic expression for P
out
ORS
at high Q
t
value is given by
P
out
OPR
Q
t
→+∞
≈
1 +
M

m=1
(−1)
m
M

u
1
,u
2
,...,u
m
=1
u
1
<u
2
<...<u
m
m

t=1

K

k=1
(−1)
k
kC
k
K
λ
R
m
E
kλ
R
m
E
+ λ
R
m
D
ρ

K

k=1
(−1)
k+1
kC
k
K
λ
SE
kλ
SE
+
m

t=1
λ
SR
u
t
ρ
. (8)
The proof is omitted here due to space limit.
B. Sub-optimal Relay Selection (SoRS) Protocol
When the CSI of the S → E, R
m
→ E links is not available
at the source and relay R
m
but only the average channel
information, the sub-optimal SoRS method is proposed as
R
b
= arg max
m=1,2,...,M

min

h
SR
m
E {γ
SE
}
,
h
R
m
D
E {γ
R
m
E
}

, (9)
where E {γ
SE
} = 1/λ
SE
and E {γ
R
m
E
} = 1/λ
R
m
E
is the
average value of γ
SE
and γ
R
m
E
, respectively. Hence, the
secrecy outage probability of the SoRS protocol is given as
P
out
SoRS
= Pr (min (W
3b
, W
4b
) < ρ) , (10)
where
W
3b
=
1 + Q
t
min (1, µ/γ
SP
) h
SR
b
1 + Q
t
min (1, µ/γ
SP
) γ
SE
W
4b
=
1 + Q
t
min (1, µ/γ
R
b
P
) h
R
b
D
1 + Q
t
min (1, µ/γ
R
b
P
) γ
R
b
E
.
In addition, when Q
t
≫ 1, we obtain W
3b
≈ h
SR
b
/γ
SE
and
W
4b
≈ h
R
b
D
/γ
R
b
E
. Hence, an asymptotic expression for (10)
at high Q
t
is given as
1
P
out
SoRS
≈ P
asym
SoRS
= Pr

min

h
SR
b
γ
SE
,
h
R
b
D
γ
R
b
E

< ρ

. (11)
Theorem 2: The closed-form expression for P
asym
SoRS
in (11)
is given in (12). See Appendix C.
C. Partial Relay Selection (PRS) Protocol
In this protocol, only the quality of the source-relay channel
is used to choose the best relay as follows:
R
b
= arg max
m=1,2,...,M
(h
SR
m
) (13)
Hence, the secrecy outage probability can be formulated as
P
out
PRS
= Pr (min (W
5b
, W
6b
) < ρ) , (14)
= 1 − Pr (W
5b
≥ ρ)
  
J
1
Pr (W
6b
≥ ρ)
  
J
2
,
1
The exact expression of outage probability for SoRS can be obtained in
a similar manner as in ORS. However, it is mathematically intractable and
omitted here due to space limit.

P
out
ORS
= 1 +
M

m=1
(−1)
m
M

u
1
,...,u
m
=1
u
1
<...<u
m
m

t=1

1 − F
W
2u
t
(ρ)

K

k=1
(−1)
k+1
kC
k
K
λ
SE
kλ
SE
+
m

t=1
λ
SR
u
t
ρ

exp

−
m

t=1
λ
SR
u
t
(ρ − 1)
Q
t

× (1 − exp (−λ
SP
µ))
N
+
N

n=1
(−1)
n+1
nC
n
N
λ
SP
Q
t
µ
nλ
SP
Q
t
µ +
m

t=1
λ
SR
u
t
(ρ − 1)
exp

−nλ
SP
µ −
m

t=1
λ
SR
u
t
(ρ − 1)
Q
t


. (7)
P
asym
SoRS
=
M

l=1
K

k
1
=0
K

k
2
=0
(−1)
k
1
+k
2
C
k
1
K
C
k
2
K

Ω
1l
Ω
2l
ρ
2
(Ω
2l
ρ + k
2
) (Ω
1l
ρ + Ω
2l
ρ + k
1
+ k
2
)
+
Ω
1l
Ω
2l
ρ
2
(Ω
1l
ρ + k
1
) (Ω
1l
ρ + Ω
2l
ρ + k
1
+ k
2
)
+
M−1

v=1
(−1)
v
M

j
1
,...,j
v
=1,̸=l
j
1
<j
2
<...<j
v
Ω
1l
Ω
2l
ρ
2
(Ω
2l
ρ + k
2
)
−1

Ω
1l
ρ
+ Ω
2l
ρ
+
k
1
+
k
2
+
m

t=1
Ω
j
t
ρ

−1
+
M−1

v=1
(−1)
v
M

j
1
,...,j
v
=1,̸=l
j
1
<j
2
<...<j
v
Ω
1,l
Ω
2,l
ρ
2
(Ω
1l
ρ + k
1
)
−1

Ω
1l
ρ + Ω
2l
ρ + k
1
+ k
2
+
m

t=1
Ω
j
t
ρ

−1

. (12)
where
W
5b
=
1 + Q
t
min (1, µ/γ
SP
) h
SR
b
1 + Q
t
min (1, µ/γ
SP
) γ
SE
W
6b
=
1 + Q
t
min (1, µ/γ
R
b
P
) h
R
b
D
1 + Q
t
min (1, µ/γ
R
b
P
) γ
R
b
E
.
Theorem 3: The exact closed-form of secrecy outage proba-
bility for PRS is readily obtained from (15) and (16) by taking
into account (14). See Appendix E.
Lemma 3: The asymptotic expression for secrecy outage
probability of the PRS scheme is given in (17). The proof is
neglected here due to space limit.
IV. SIMULATION RESULTS
In this section, we present various Monte Carlo simulations
to verify the theoretically derived results. In a two-dimensional
topology, we assume that the co-ordinates of the source (S),
the destination (D), the relay (R
m
), the primary users (P
n
)
and the eavesdropper (E
k
), are (0; 0), (1; 0), (x
m
; 0), (x
P
; y
P
)
and (x
E
; y
E
), respectively. To take the path-loss into account,
we assume λ
XY
= E

|h
XY
|
2

= d
β
X,Y
, where X ∈ {S, R
m
},
Y ∈ {R
m
, D, P
n
, E
k
}, and β is path-loss exponent. In the
simulations, we assume that the path-loss exponent is 3.
In Fig. 1, we present the secrecy outage probability as a
function of Q
t
in dB. In this scenario, we assume that there are
3 secondary relays, 1 primary user and 1 eavesdropper. The
nodes PU and E are placed at {−0.5; −1} and {0.9; 0.3},
respectively, while the position of the relays are {0.4; 0},
{0.5; 0} and {0.6; 0}. It is also assumed that R
th
= 0.4 and
µ = 1. As we can see, the performance of ORS is better than
that of ORS and SoRS, while SoRS outperforms ORS.
In Fig. 2, we investigate the effect of the number of the
primary users and eavesdroppers on the secrecy outage per-
formance of the ORS methods. In this figure, we consider the
i.i.d networks with 4 relays placed at the positions {0.4; 0}.
Fig. 1. Secrecy outage probability as a function of SNR (Q
t
) in dB.
The remaining parameters are set to {x
P
, y
P
} = {−0.5; −1},
{x
E
, y
E
} = {0.5; 0.5}, R
th
= 0.4, and µ = 1. As can
be observed, the outage performance of ORS significantly
improves when the number of eavesdroppers (K) decreases.
In addition, at high Q
t
, the outage probability is independent
of Q
t
as well as the position and the number of primary users.
V. CONCLUSIONS
We have proposed three relay selection schemes for securing
the fidelity of cooperative cognitive radio networks. We have
derived exact and asymptotic expressions for secrecy outage
probability, that readily enable us to compare the three ap-
proaches. Our results show that ORS outperforms SoRS and
PRS, provided that perfect CSI is known at the receivers.

J
1
=
K

k=1
(−1)
k+1
C
k
K
kλ
SE
kλ
SE
+ λ
SR
m
ρ

exp

−
λ
SR
m
(ρ − 1)
Q
t

(1 − exp (−λ
SP
ρ))
N
+
N

n=1
(−1)
n+1
C
n
N
×
nλ
SP
Q
t
µ exp (−nλ
SP
µ − λ
SR
m
(ρ − 1) /Q
t
)
nλ
SP
Q
t
µ + λ
SR
m
(ρ − 1)

+
M

m=1,m̸=b
(−1)
m
M

j
1
,j
2
,...,j
m
=1,̸=b
j
1
<j
2
<...<j
m
λ
SR
m
λ
SR
m
+
m

t=1
λ
SR
j
t
×
K

k=1
(−1)
k+1
C
k
K
kλ
SE
kλ
SE
+

λ
SR
m
+
i

t=1
λ
SR
j
t

ρ

exp

−

λ
SR
m
+
i

t=1
λ
SR
j
t
(ρ − 1)

/Q
t

(1 − exp (−λ
SP
ρ))
N
+
N

n=1
(−1)
n+1
C
n
N
nλ
SP
Q
t
µ exp

−nλ
SP
µ −

λ
SR
m
+
i

t=1
λ
SR
j
t

(ρ − 1) /Q
t

×

nλ
SP
Q
t
µ +

λ
SR
m
+
i

t=1
λ
SR
j
t

(ρ − 1)

−1

. (15)
J
2
=
K

k=1
(−1)
k+1
C
k
K
kλ
RE
kλ
RE
+ λ
R
m
D
ρ

exp [−λ
R
m
D
(ρ − 1) /Q
t
] [1 − exp (−λ
RP
µ)]
N
+
N

n=1
(−1)
n+1
C
n
N
×
nλ
RP
Q
t
µ
nλ
RP
Q
t
µ + λ
R
m
D
(ρ − 1)
exp [−nλ
RP
µ − λ
R
m
D
(ρ − 1) /Q
t
]

. (16)
P
out
PRS
Q
t
→+∞
≈
1 −
M

m=1

K

k=1
(−1)
k+1
kC
k
K
λ
SE
kλ
SE
+ λ
SR
m
ρ
+
M

i=1,̸=m
(−1)
i
M

j
1
,...,j
i
=1,̸=m
j
1
<...<j
i
λ
SR
m
λ
SR
m
+
i

t=1
λ
SR
j
t
×
K

k=1
(−1)
k+1
kC
k
K
λ
SE

kλ
SE
+ (λ
SR
m
+
i

t=1
λ
SR
j
t
)ρ

−1


K

k=1
(−1)
k+1
C
k
K
kλ
RE
kλ
RE
+ λ
R
m
D
ρ

. (17)
Fig. 2. Secrecy outage probability of the ORS scheme.
APPENDIX A: PROOF OF LEMMA 1
Let x = γ
R
m
E
and y = γ
R
m
P
. The CDF F
W
2m
(z) (with
z > 1) conditioned on x and y is obtained as
F
W
2m
(z|x, y) = Pr

1 + Q
t
min (1, µ/y) h
R
m
D
1 + Q
t
min(1, µ/y)x
< z

,
= 1 − exp

−
λ
R
m
D
(z − 1)
Q
t
min(1, µ/y)
− λ
R
m
D
zx

. (A.1)
Then, the CDF F
W
2m
(z) can be formulated as
F
W
2,m
(z) =

+∞
0
f
γ
R
m
P
(y)

+∞
0
f
γ
R
m
E
(x)F
W
2m
(z|x, y)dx
  
J
1
(y)
dy. (A.2)
In addition, the probability density function (PDF) of γ
XY
,
X ∈ {S, R
m
} and Y ∈ {P, E}, can be given as
f
γ
XY
(z) =
T

t=1
(−1)
t+1
C
t
T
tλ
XY
exp (−tλ
XY
z) , (A.3)
where C
t
T
=
T !
t!(T −t)!
, and T =

N if Y ≡ P
K if Y ≡ E
. From
(A.3), and (A.1), the integral J
1
is given by
J
1
(y) = 1 −
K

k=1
(−1)
k
C
k
K
kλ
R
m
E
exp(−
λ
R
m
D
(z−1)
Q
t
min(1,µ/y)
)
kλ
R
m
E
+ λ
R
m
D
z
.
(A.4)
Then substituting (A.3) and (A.4) into (A.2), and after some
manipulation, we obtain (6), which completes the proof.

APPENDIX B: PROOF OF THEOREM 1
Let x = γ
SP
and y = γ
SE
, the CDF F
W
1m
(z) conditioned
on x and y can be given as
F
W
1m
(z|x, y) = 1 − exp

−
λ
SR
m
(z − 1)
Q
t
min(1, µ/x)
− λ
SR
m
zy

.
(B.1)
Next, let us denote T
m
= min (W
1m
, W
2m
), the CDF F
T
m
(z)
conditioned on x and y is given by
F
T
m
(z|x, y) = 1 − (1 − F
W
2m
(z))
× exp

−λ
SR
m
(
(z − 1)
Q
t
min(1, µ/x)
+ zy)

. (B.2)
Combining (5) and (B.2), we obtain the probability condi-
tioned P
out
ORS
(x, y) on x and y as
P
out
ORS
(x, y) = 1 +
M

m=1
(−1)
m
M

u
1
,u
2
,...,u
m
=1
u
1
<u
2
<...<u
m
m

t=1
(1 − F
W
2u
t
(ρ))
× exp

−
m

t=1
λ
SR
u
t
(
ρ − 1
Q
t
min(1, µ/x)
+ ρy)

. (B.3)
In addition, we have
P
out
ORS
(x, y)=

+∞
0
f
γ
SP
(x)

+∞
0
f
γ
SE
(y)P
out
OPR
(x, y)dydx. (B.4)
By substituting (A.3) and (B.3) into (B.4), after some manip-
ulations, we obtain (7), which concludes the proof.
APPENDIX C: PROOF OF THEOREM 2
From (11), we can rewrite P
asym
SoRS
as follows:
P
asym
SoRS
= 1 − Pr

min(
E {γ
SE
}U
1
γ
SE
,
E {γ
R
b
E
}U
2
γ
R
b
E
) ≥ ρ

(C.1)
where U
1
= h
SR
b
/E {γ
SE
} and U
2
= h
R
b
D
/E {γ
R
b
E
}.
From (C.1), we obtain the outage probability conditioned on
U
1
and U
2
as
P
asym
SoRS
(u
1
, u
2
) =
= 1 − Pr

γ
SE
≤
E {γ
SE
} u
1
ρ
, γ
R
b
E
≤
E {γ
R
b
E
} u
2
ρ

,
= 1 −
K

k
1
=0
K

k
2
=0
(−1)
k
1
+k
2
C
k
1
K
C
k
2
K
× exp

−
k
1
u
1
ρ

exp

−
k
2
u
2
ρ

. (C.2)
In addition, the outage probability P
asym
SoRS
can be given as
P
asym
SoRS
=
 
f
U
1
,U
2
(u
1
, u
2
)P
asym
SoRS
(u
1
, u
2
)du
1
du
2
, (C.3)
where f
U
1
,U
2
(u
1
, u
2
) is common PDF of U
1
and U
2
. By using
Appendix D, we obtain f
U
1
,U
2
(u
1
, u
2
) as presented in (C.4)
at the top of next page, with Ω
1m
= λ
SR
m
/λ
SE
, Ω
2m
=
λ
R
m
D
/λ
R
m
E
and Ω
m
= Ω
1m
+ Ω
2m
.
Substituting (C.4) into (C.2), we can obtain P
asym
SoRS
as in (12),
which finalizes the proof.
APPENDIX D: DERIVATION OF f
U
1
,U
2
(u
1
, u
2
)
Let us denote Φ
1m
= h
SR
m
/E {γ
SE
}, Φ
2m
=
h
R
m
D
/E {γ
R
m
E
}, and Φ
m
= min (Φ
1m
, Φ
2m
), U
b
=
max
m=1,2,...,M
(Φ
m
), and Z
b
= max
m=1,2,...,M, ̸=b
(Φ
m
), we can
formulate the CDF F
U
b
(z) as
F
U
b
(z) =

z
0
Ω
b
exp (−Ω
b
t) F
Z
b
(z) dt, (D.1)
where the CDF F
Z
b
(z) is given as
F
Z
b
(z)=1+
M−1

m=1
(−1)
m
M

j
1
,...,j
m
=1,̸=b
j
1
<...<j
m
exp(−
m

t=1
Ω
j
t
z). (D.2)
Substituting (D.2) into (D.1), we obtain (D.3) as
F
U
b
(z) = 1 − exp (−Ω
b
z) +
M−1

m=1
(−1)
m
M

j
1
,...,j
m
=1,̸=b
j
1
<...<j
m




Ω
b
Ω
b
+
m

t=1
Ω
j
t

1 − exp(−(Ω
b
+
m

t=1
Ω
j
t
)z)





. (D.3)
From (D.3), the corresponding PDF of U
b
is given by
f
U
b
(z) = Ω
b
exp (−Ω
b
z) +
M−1

m=1
(−1)
m
M

j
1
,...,j
m
=1,̸=b
j
1
<...<j
m
Ω
b
exp(−(Ω
b
+
m

t=1
Ω
j
t
)z). (D.4)
The joint CDF F
U
1
,U
2
(u
1
, u
2
) can be given as
F
U
1
,U
2
(u
1
, u
2
) =
M

l=1

+∞
0
G (t)
∂t
f
U
l
(t)
f
Φ
l
(t)
dt, (D.5)
where G(t) can be obtained as
G (t) = Pr (Φ
1l
< u
1
, Φ
2l
< u
2
, min (Φ
1l
, Φ
2l
) < t) =







(exp(−Ω
1l
u
1
)−exp(−Ω
1l
t))
× (exp(−Ω
2l
t)−exp(−Ω
2l
u
2
)) ; if t<min (u
1
, u
2
)
(1−exp(−Ω
1l
u
1
))
× (1−exp(−Ω
2l
u
2
)) ; if min (u
1
, u
2
)≤t
. (D.6)
From (D.5) and (D.6), we can obtain
F
U
1
,U
2
(u
1
, u
2
)=







M

l=1

u
1
0
∂G(t)
∂t
f
U
b
(t)
f
Φ
l
(t)
dt; ifu
1
< u
2
M

l=1

u
2
0
∂G(t)
∂t
f
U
b
(t)
f
Φ
l
(t)
dt; ifu
1
≥ u
2
. (D.7)
For the case u
1
< u
2
and u
1
≥ u
2
, by substituting the PDF
f
Φ
m
(z), (D.4) into (D.7), we can obtain (D.8) as shown at
the top of next page, respectively. For the case, u
1
≥ u
2
,
F
U
1
,U
2
(u
1
, u
2
) can be similarly obtain by interchangeably
replacing u
1
by u
2
and Ω
1
by Ω
2
in (D.8). Since the
joint PDF f
U
1
,U
2
(u
1
, u
2
) is given as f
Φ
1,b
,Φ
2,b
(u
1
, u
2
) =
∂
2
F
Φ
1,b
,Φ
2,b
(u
1
,u
2
)
∂u
1
∂u
2
, from (D.8), we can obtain (C.4).

f
U
1
,U
2
(u
1
, u
2
)=





















M

l=1



Ω
1l
Ω
2l
exp (−Ω
1l
u
1
) exp (−Ω
2l
u
2
) +
M−1

v =1
(−1)
m
M

j
1
,j
2
...,j
v
=1,̸=v
j
1
<j
2
<...<j
v
Ω
1l
Ω
2l
exp(−Ω
2l
u
2
) exp(−(Ω
1l
+
m

t=1
Ω
j
t
)u
1
)



; if u
1
< u
2
M

l=1



Ω
1l
Ω
2l
exp (−Ω
1l
u
1
) exp (−Ω
2l
u
2
) +
M−1

v =1
(−1)
m
M

j
1
,j
2
...,j
v
=1,̸=v
j
1
<j
2
<...<j
v
Ω
1l
Ω
2l
exp (−Ω
1l
u
1
) exp(−(Ω
2l
+
m

t=1
Ω
j
t
)u
2
)



; if u
1
≥ u
2
. (C.4)
F
U
1
,U
2
(u
1
, u
2
) =
M

l=1
1 − exp(−Ω
l
u
1
) − exp(−Ω
2l
u
2
) (1 − exp(−Ω
1l
u
1
)) − exp(−Ω
1l
u
1
) (1 − exp(−Ω
2l
u
1
))
+
M−1

v =1
(−1)
v
M

j
1
,...,i
v
=1,̸=l
j
1
<j
2
<...<j
l
Ω
l
Ω
l
+
m

t=1
Ω
j
t

1 − exp(−(Ω
l
+
m

t=1
Ω
j
t
)u
1
)

−
Ω
1l
exp (−Ω
2l
u
2
)
Ω
1l
+
m

t=1
Ω
j
t

1 − exp(−(Ω
1l
+
m

t=1
Ω
j
t
)u
1
)

−
Ω
2l
exp (−Ω
1l
u
1
)
Ω
2l
+
m

t=1
Ω
j
t

1 − exp(−(Ω
2l
+
m

t=1
Ω
j
t
)u
1
)

. (D.8)
APPENDIX E: PROOF OF THEOREM 3
Similarly we set x = γ
SE
and y = γ
SP
and The term J
1
in
(14) conditioned on x and y can be obtained as
J
1
(x, y) = Pr

h
SR
b
≥
ρ − 1
Q min (1, µ/y)
+ ρx, h
SR
b
≥ Z

,
=

+∞
ρ−1
Q min(1,µ/y)
+ρx
λ
SR
b
exp (−λ
SR
b
z)F
Z
(z) dz, (E.1)
where F
Z
(z) is the CDF of Z, which is given as
F
Z
(z) = 1+
M

m=1,̸=b
(−1)
m
M

j
1
,...,j
m
=1,̸=b
j
1
<...<j
m
exp(−
m

t=1
λ
SR
j
t
z).
(E.2)
Substituting (E.2) into (E.1) to obtain (E.3) as follows:
J
1
(x, y) = exp

−λ
SR
b
(
ρ − 1
Q
t
min (1, µ/y)
+ ρx)

+
M

m=1,
m̸=b
(−1)
m
M

j
1
,...,j
m
=1,̸=b
j
1
<...<j
m
λ
SR
b
λ
SR
b
+
m

t=1
λ
SR
j
t
× exp

−(λ
SR
m
+
m

t=1
λ
SR
j
t
)(
ρ − 1
Q
t
min (1, µ/y)
+ ρx)

. (E.3)
Using the fact that J
1
=

+∞
0
f
γ
SP
(y)


+∞
0
J
1
(x, y)f
γ
SE
(x) dx

dy and (A.3),
we can obtain (15). Now, considering the probability J
2
in
(14). Similarly as in (A.1)-(A.4), we can get (16).
REFERENCES
[1] A. Bletsas, A. Khisti, D. P. Reed, and A. Lippman, “A simple cooperative
diversity method based on network path selection,” IEEE J. Sel. Areas
Commun., vol. 24, no. 3, pp. 659–672, Mar. 2006.
[2] A. Bletsas, H. Shin, M. Z. Win, and A. Lippman, “Cooperative diversity
with opportunistic relaying,” in Proc. IEEE WCNC, Las Vegas, NV, Apr.
2006, pp. 1034–1039.
[3] A. Bletsas, H. Shin, and M. Z. Win, “Cooperative communications with
outage-optimal opportunistic relaying,” IEEE Trans. Wireless Commun.,
vol. 6, no. 9, pp. 3450–3460, Sep. 2007.
[4] ——, “Outage optimality of opportunistic amplify-and-forward relay-
ing,” IEEE Commun. Lett., vol. 11, no. 3, pp. 261–263, 2007.
[5] T. Q. Duong, V. N. Q. Bao, and H.-J. Zepernick, “On the performance of
selection decode-and-forward relay networks over Nakagami-m fading
channels,” IEEE Commun. Lett., vol. 13, no. 3, pp. 172–174, Mar. 2009.
[6] Y. Zhao, R. Adve, and T. J. Lim, “Symbol error rate of selection amplify-
and-forward relay systems,” IEEE Commun. Lett., vol. 10, no. 11, pp.
475–478, Nov. 2006.
[7] E. Beres and R. S. Adve, “Selection cooperation in multisource coop-
erative networks,” IEEE Trans. Wireless Commun., vol. 7, no. 1, pp.
118–127, 2008.
[8] J. Lee, H. Wang, J. G. Andrews, and D. Hong, “Outage probability
of cognitive relay networks with interference constraints,” IEEE Trans.
Wireless Commun., vol. 10, no. 2, pp. 390–395, Feb. 2011.
[9] M. Xia and S. A
¨
ıssa, “Cooperative AF relaying in spectrum-sharing
systems: Outage probability analysis under co-channel interferences and
relay selection,” IEEE Trans. Commun., vol. 60, no. 11, pp. 3252–3262,
Dec. 2012.
[10] H. Tran, T. Q. Duong, and H.-J. Zepernick, “Performance analysis of
cognitive relay networks under power constraint of multiple primary
users,” in Proc. IEEE GLOBECOM, Houston, TX, Dec. 2011, pp. 1–6.
[11] H. Ding et al., “Asymptotic analysis of cooperative diversity systems
with relay selection in a spectrum sharing scenario,” IEEE Trans. Veh.
Technol., vol. 60, no. 2, pp. 457–472, Feb. 2011.
[12] V. N. Q. Bao, T. Q. Duong, D. B. da Costa, G. C. Alexandropoulos,
and A. Nallanathan, “Cognitive amplify-and-forward relaying with best
relay selection in non-identical Rayleigh fading,” IEEE Commun. Lett.,
vol. 60, no. 3, pp. 757–759, Mar. 2013.
[13] K. J. Kim, T. Q. Duong, and X.-N. Tran, “Performance analysis of
cognitive spectrum-sharing single-carrier systems with relay selection,”
IEEE Trans. Signal Process., vol. 60, no. 12, pp. 6435–6449, Dec. 2012.
[14] M. Bloch, J. Barros, M. R. D. Rodrigues, and S. W. McLaughlin, “Wire-
less information-theoretic security,” IEEE Trans. Inf. Theory, vol. 54,
no. 6, pp. 2515–2534, Jun. 2008.
[15] J. Mo, M. Tao, and Y. Liu, “Relay placement for physical layer security:
A secure connection perspective,” IEEE Commun. Lett., vol. 16, no. 6,
pp. 878–881, Jun. 2012.