Secrecy Cooperative Networks with Outdated Relay Selection over
Correlated Fading Channels
Fan, L., Lei, X., Yang, N., Duong, T. Q., & Karagiannidis, G. K. (2017). Secrecy Cooperative Networks with
Outdated Relay Selection over Correlated Fading Channels.
IEEE Transactions on Vehicular Technology
.
https://doi.org/10.1109/TVT.2017.2669240
Published in:
IEEE Transactions on Vehicular Technology
Document Version:
Peer reviewed version
Queen's University Belfast - Research Portal:
Link to publication record in Queen's University Belfast Research Portal
Publisher rights
Copyright 2017 IEEE.
This work is made available online in accordance with the publisher’s policies. Please refer to any applicable terms of use of the publisher.
General rights
Copyright for the publications made accessible via the Queen's University Belfast Research Portal is retained by the author(s) and / or other
copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated
with these rights.
Take down policy
The Research Portal is Queen's institutional repository that provides access to Queen's research output. Every effort has been made to
ensure that content in the Research Portal does not infringe any person's rights, or applicable UK laws. If you discover content in the
Research Portal that you believe breaches copyright or violates any law, please contact openaccess@qub.ac.uk.
Download date:09. Aug. 2022
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. XX, NO. XX, XXX 2016 1
Impact of Correlated Fading Channels on Multiple
Secure Relaying with Outdated Relay Selection
Lisheng Fan, Xianfu Lei, Nan Yang, Member, IEEE, Trung Q. Duong, Senior Member, IEEE, and George K.
Karagiannidis, Fellow, IEEE
Abstract—In this paper, we study the impact of correlated
fading channels on multiple secure decode-and-forward (DF)
relaying with outdated relay selection, where the information
transmission assisted by the N DF relays from the source to
the destination can be overheard by the eavesdropper in the
network. The eavesdropping channels are correlated with the
main channels, which affects the network security. To enhance
the network security, one best relay is chosen to assist the secure
transmission, which is however maybe outdated in time-varying
channel environments. The impact of both channel correlation
and outdated relay selection on the secrecy performance is
studied by deriving the analytical expression of the secrecy outage
probability (SOP). The asymptotic SOP is also provided with high
main-to-eavesdropper ratio (MER). From the asymptotic SOP, we
find that only the outdated degree of relay selection affects the
network secrecy diversity order, but the channel correlation does
not. Moreover, it is interesting to find that the channel correlation
is beneficial to the transmission security in the high MER regime.
Index Terms—Secure communications, correlated fading chan-
nels, outdated relay selection, secrecy diversity order.
I. INTRODUCTION
Due to the broadcast nature, the wireless transmission
may be overheard by eavesdroppers in the network, and the
severe issue of information leakage arises [1]. To prevent the
wiretap, the encryption algorithm and physical-layer security
(PLS) have been studied in the literature. The pioneering
work of PLS is investigated by Wyner [2], where the wiretap
channel model was firstly proposed. Then researchers extend-
ed to fading channels and studies the important metrics of
secrecy performance, such as secrecy capacity and secrecy
outage probability (SOP) [3]–[5]. As relaying is a promising
technique for the next-generation communications, it is of
vital importance to investigate the PLS of relay networks.
For amplify-and-forward (AF) and decode-and-forward (DF)
relaying, the secrecy performance has been studied by deriv-
ing the analytical expression of SOP [6]–[8]. Moreover, the
asymptotic SOP with high main-to-eavesdropper ratio (MER)
was provided to obtain the insights on the system.
L. Fan is with the School of Computer Science and Educational Software,
Guangzhou University, Guangzhou, China (e-mail: lsfan gzu@126.com).
X. Lei is with the Provincial Key Lab of Information Coding and
Transmission, Southwest Jiaotong University, Chengdu, China (e-mail:
xflei@home.swjtu.edu.cn).
N. Yang is with Australian National University, Canberra ACT 0200,
Australia (e-mail: yangnan1616@gmail.com).
T. Q. Duong is with Queen’s University Belfast, Belfast BT7 1NN, United
Kingdom (e-mail: trung.q.duong@qub.ac.uk).
G. K. Karagiannidis is with Aristotle University of Thessaloniki, Thessa-
loniki 54 124, Greece (e-mail: geokarag@auth.gr).
Manuscript received XXX, XX, 2016; revised XXX, XX, 2016.
In the most existing works of the literature, the eavesdrop-
ping channels are assumed to be independent of the main
channels. However, this ideal assumption may not hold in
practice due to many factors such as antenna deployments,
proximity of the legitimate receiver and eavesdropper, and
scattering environments. The impact of channel correlation
between the main and eavesdropping links on the secrecy
performance is studied in [9], [10], where it has been found
that the channel correlation is harmful to the transmission
security in the low MER region. To enhance the transmission
security, the opportunistic selection is an effective technique
to exploit the channel fluctuation between antennas, users
and relays [7], [11]. The selection can be implemented in a
centralized or distributed manner through dedicated feedback
links, which may take some time to complete. In time-varying
channel environments, the channels may vary from the instant
of selection to that of actual data transmission, which causes
the selection based on the outdated channel state information
(CSI) [12] and severely limits the system secrecy performance
[13], [14].
In this paper, we investigate the secure multi-relay networks
with channel correlation between the main and eavesdropping
links, where the information transmission assisted by N DF
relays from the source to the destination may be overheard
by the eavesdropper in the network. To strengthen the secure
transmission, one best relay is selected among N relays, which
is however based on the outdated CSI in time-varying channel
environments. We study the impact of both channel correlation
and outdated relay selection by deriving the analytical and
asymptotic expressions for the SOP. From the asymptotic
result, we find that the secrecy diversity order is equal to
N only in perfect channel state information; otherwise it
degenerates to unity. Moreover, the channel correlation does
not affect the secrecy diversity order, but it can help strength
the secure transmission in high MER region.
Notations: The notation CN(0, σ
2
) denotes a circularly
symmetric complex Gaussian random variable (RV) with
zero mean and variance σ
2
. We use f
X
(·) to represent the
probability density function (PDF) of the RV X. In addition,
I
0
(x) is the modified Bessel function of the first kind of
order zero [15], Pr[·] returns the probability, and E[·] denotes
the statistical average. We use h
A,B
to denote the channel
parameter of the A–B link.
II. SYSTEM MODEL
Fig. 1 shows the system model of a two-phase secure mul-
tiple DF relays network, where the information transmission
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. XX, NO. XX, XXX 2016 2
E
R
N
S
D
Eavesdropping link
Main link
R
1
Fig. 1. A two-phase secure multiple DF relaying network.
from the source S to the destination D can be overheard by
the eavesdropper E. There are no direct links from S to D and
E, and the information transmission are only through the N
relays {R
n
|1 ≤ n ≤ N}. One best relay is chosen among
N available relays to assist the secure transmission. However,
the selection may be outdated in time-varying channel environ-
ments, and the best relay is not always chosen. In addition, due
to the reasons such as antenna deployment and radio scattering,
the channels at the receivers D and E are correlated with
each other, i.e., h
R
n
,D
is correlated with h
R
n
,E
. Due to the
size limitation, the nodes in the network are equipped with a
single antenna, and all links experience time-varying Rayleigh
fading. In the following, we present the two-phase secure data
transmission process and the relay selection criterion for the
considered system.
Suppose that the relay R
n
is used for the two-phase secure
data transmission. In the first phase, the source S sends signal
x
S
to R
n
with transmit power P , and R
n
receives,
y
R
n
=
√
P h
S,R
n
x
S
+ n
R
n
, (1)
where h
S,R
n
∼ CN(0, α) denotes the channel parameter of
the S–R
n
link, and n
R
n
∼ CN(0, σ
2
) is the additive white
Gaussian noise (AWGN) at the relay R
n
. If the relay R
n
can correctly decode the message from the source, i.e., it can
support a target data rate R
t
,
1
2
log
2
1 +
P |h
S,R
n
|
2
σ
2
≥ R
t
, (2)
the relay forwards the message to D in the second phase with
the transmit power P . Accordingly, D and E receive,
y
D
=
√
P h
R
n
,D
x
S
+ n
D
, (3)
y
E
=
√
P h
R
n
,E
x
S
+ n
E
, (4)
where h
R
n
,D
∼ CN(0, β) and h
R
n
,E
∼ CN(0, ε) are the
channel parameters of the
R
n
–D and
R
n
–E links, respectively.
The noise terms n
D
∼ CN(0, σ
2
) and n
E
∼ CN(0, σ
2
) are
the AWGN at D and E, respectively.
We use u
n
= |h
S,R
n
|
2
, v
n
= |h
R
n
,D
|
2
and w
n
= |h
R
n
,E
|
2
to represent the channel gains of the S–R
n
, R
n
–D and R
n
–E
links, respectively. The secrecy outage occurs when the data
rate difference between the main and eavesdropping links falls
below a target secrecy data rate R
s
,
1
2
log
2
1 +
P
σ
2
v
n
−
1
2
log
2
1 +
P
σ
2
w
n
< R
s
, (5)
which is equivalent to
1 +
˜
P v
n
1 +
˜
P w
n
< γ
s
, (6)
where
˜
P = P/σ
2
denotes the transmit SNR, and γ
s
= 2
2R
s
is the secrecy SNR threshold.
To enhance the transmission security, we need to select
one best relay to strength the secure data transmission. Let Ω
denote the candidate set of relays that can successfully decode
the message from the source. Then based on the main channels
only
1
, the relay selection is performed to choose one best relay
R
n
∗
among Ω,
n
∗
= arg max
n∈Ω
v
n
, (7)
which maximizes the received SNR at the destination D.
III. CHANNEL CORRELATION AND OUTDATED RELAY
SELECTION
In this work, we consider the correlated channels between
the receivers D and E, and the correlation between v
n
and w
n
is characterized by [10]
f
w
n
|v
n
(w|v) =
I
0
2
1−ρ
c
ρ
c
vw
βϵ
(1 − ρ
c
)βε
e
−
ρ
c
v
β
+
w
ε
1−ρ
c
, (8)
where ρ
c
∈ [0, 1] is the power correlation coefficient. Specif-
ically, ρ
c
= 0 represents that v
n
is independent of w
n
, while
ρ
c
= 1 denotes the completely linear correlation.
Besides the channel correlation, the outdated relay selection
has a significant impact on the network security. The selection
of (7) can be implemented in a distributed or centralized
manner in practice, through some dedicated feedback channels
[7]. However, due to the limited feedback resources, the
channels may vary from the instant of relay selection to that of
actual data transmission in time-varying channel environments.
Let ˜v
n
∗
and v
n
∗
denote the channels of R
n
∗
–D at the instants
of relay selection and actual data transmission, respectively.
The outdated selection can be characterized by the conditional
PDF f
v
n
∗
|˜v
n
∗
(v| ˜v) [12],
f
v
n
∗
|˜v
n
∗
(v| ˜v) =
1
(1 − ρ
o
)β
e
−
ρ
o
˜v+v
(1−ρ
o
)β
I
0
2
√
ρ
o
v˜v
(1 − ρ
o
)β
, (9)
where ρ
o
∈ [0, 1] denotes the outdated degree of relay
selection. In particular, ρ
o
= 1 denotes that the selection
is based on the perfect CSI, while ρ
o
= 0 represents the
completely outdated relay selection.
In the following, we will study the impact of channel
correlation and outdated relay selection on the network se-
crecy performance by providing the analytical and asymptotic
expressions of secrecy outage probability.
1
The instantaneous channel parameters of eavesdropping links are not
involved in the relay selection criterion, since they are generally hard to obtain
in practice.
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. XX, NO. XX, XXX 2016 3
P
out
=
N
K=1
N
K
e
−
Kγ
t
˜
P α
1 − e
−
γ
t
˜
P α
N−K
1 −
K
k=1
T
m=0
m
i=0
i
j=0
C
km
d
m,ij
(m + j)!(b
1
γ
s
+ b
2
)
−(m+j+1)
. (25)
IV. SECRECY OUTAGE PROBABILITY
A. Analytical Expression
Note that the set Ω may have K(K = 1, 2, ··· , N )
candidates that can correctly decode the message from the
source, we can express the SOP of the considered system as
P
out
=
N
K=1
N
K
Pr
˜
P u
1
≥ γ
t
, ··· ,
˜
P u
K
≥ γ
t
,
˜
P u
K+1
< γ
t
, ··· ,
˜
P u
N
< γ
t
,
1 +
˜
P v
n
∗
1 +
˜
P w
n
∗
< γ
s
, (10)
where γ
t
= 2
2R
t
−1 denotes the SNR threshold of successful
decoding at the relay. Since the random variable u
n
is inde-
pendent of v
n
∗
and w
n
∗
, we can rewrite P
out
as
P
out
=
N
K=1
N
K
Pr
˜
P u
1
≥ γ
t
, ··· ,
˜
P u
K
≥ γ
t
,
˜
P u
K+1
< γ
t
, ··· ,
˜
P u
N
< γ
t
Pr
1 +
˜
P v
n
∗
1 +
˜
P w
n
∗
< γ
s
(11)
=
N
K=1
N
K
e
−
Kγ
t
˜
P α
1 − e
−
γ
t
˜
P α
N−K
Pr
1 +
˜
P v
n
∗
1 +
˜
P w
n
∗
< γ
s
J
K
,
(12)
where the PDF of f
u
n
(u) =
1
α
e
−
u
α
is applied and the
probability J
K
denotes the conditional SOP with the given
K active relays. From the selection criterion in (7) and the
conditional PDF of f
v
n
∗
|˜v
n
∗
(v| ˜v) in (9), the PDF of v
n
∗
is
given by [12]
f
v
n
∗
(v) =
K
k=1
K
k
k(−1)
k−1
β[k(1 − ρ
o
) + ρ
o
]
e
−
kv
[k(1−ρ
o
)+ρ
o
]β
. (13)
From the above f
v
n
∗
(v) and the conditional f
w
n
|v
n
(w|v) in
(8), we obtain the joint PDF of v
n
∗
and w
n
∗
as,
f
v
n
∗
,w
n
∗
(v, w) = f
w
n
∗
|v
n
∗
(w|v)f
v
n
∗
(v)
=
K
k=1
K
k
k(−1)
k−1
βε(1 −ρ
c
)[k(1 − ρ
o
) + ρ
o
]
I
0
2
1 − ρ
c
ρ
c
vw
βε
× e
−
kv
[k(1−ρ
o
)+ρ
o
]β
−(
ρ
c
v
β
+
w
ε
)
1
1−ρ
c
. (14)
Note that the Bessel function I
0
(x) can be expanded by the
series as [15]
I
0
(x) =
∞
m=0
x
2m
4
m
(m!)
2
(15)
≈
T
m=0
x
2m
4
m
(m!)
2
, (16)
where the truncation error in (16) decays exponentially with T
[10]. Hence with an efficient number of series, we can obtain
an accurate approximation for I
0
(x). In the following, we
ignore the approximation error, and re-express f
v
n
∗
,w
n
∗
(v, w)
as
f
v
n
∗
,w
n
∗
(v, w) =
K
k=1
T
m=0
C
km
v
m
w
m
e
−b
1
v
e
−b
2
w
, (17)
with
C
km
=
K
k
k(−1)
k−1
k(1 − ρ
o
) + ρ
o
ρ
m
c
(1 − ρ
c
)
2m+1
(βε)
m+1
(m!)
2
,
(18)
b
1
=
k
k(1 − ρ
o
) + ρ
o
+
ρ
c
1 − ρ
c
1
β
, (19)
b
2
=
1
(1 − ρ
c
)ε
. (20)
From (17), we can compute J
K
as
J
K
= Pr
v
n
∗
<
γ
s
− 1
˜
P
+ γ
s
w
n
∗
(21)
=
∞
0
γ
s
−1
˜
P
+γ
s
w
0
f
v
n
∗
,w
n
∗
(v, w)dvdw (22)
= 1 −
K
k=1
T
m=0
m
i=0
i
j=0
C
km
d
m,ij
(m + j)!
× (b
1
γ
s
+ b
2
)
−(m+j+1)
, (23)
with
d
m,ij
=
m!
i!
i
j
e
−b
1
(γ
s
−1)/
˜
P
b
m−i+1
1
γ
j
s
γ
s
− 1
˜
P
i−j
. (24)
By applying the result of J
K
into (12), we can obtain the
analytical SOP for the multiple secure relaying with channel
correlation and outdated relay selection, as shown in (25) at
the top of this page. Note that the obtained analytical SOP
consists of elementary functions only, and hence is easily to
be evaluated.
B. Asymptotic Expression
To obtain the insights on the system, we now extend to
derive the asymptotic SOP with high MER. With a large
transmit power P , we can approximate P
out
as
P
out
≃ J
N
(26)
≃ Pr
v
n
∗
w
n
∗
< γ
s
. (27)
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. XX, NO. XX, XXX 2016 4
Let z =
v
n
∗
w
n
∗
, and then the PDF of z is derived as
f
z
(z) =
∞
0
wf
v
n
∗
,w
n
∗
(wz, w)dw (28)
=
N
k=1
N
k
k(−1)
k−1
βε(1 −ρ
c
)[k(1 − ρ
o
) + ρ
o
]
×
kz
[k(1−ρ
o
)+ρ
o
]β
+
ρ
c
z
β
+
1
ε
1
1−ρ
c
kz
[k(1−ρ
o
)+ρ
o
]β
+
ρ
c
z
β
+
1
ε
1
1−ρ
c
2
−
4ρ
c
z
(1−ρ
c
)
2
βε
3/2
,
(29)
where [15, eq. (6.623.2)] is applied in the last equality. For
high MER with β ≫ ε, we can approximate f
z
(z) as
f
z
(z) ≃
N
k=1
N
k
(−1)
k−1
(1 − ρ
c
)λ[k(1 − ρ
o
) + ρ
o
]
{λ + k(1 − ρ
c
)z + ρ
c
z[k(1 − ρ
o
) + ρ
o
]}
2
,
(30)
where λ = β/ε is the MER. From the above f
z
(z), we then
approximate P
out
as
P
out
≃
γ
s
0
f
z
(z)dz (31)
≃
N!
(1 − ρ
c
)γ
s
λ
N
, If ρ
o
= 1
(1 − ρ
c
)γ
s
λ
N
k=1
N
k
k(−1)
k−1
k(1 − ρ
o
) + ρ
o
, If ρ
o
< 1
.
(32)
where we apply the approximation of (1 + x)
−1
=
N
k=0
(−1)
k
x
k
[15] from (31) to (32). From this asymptotic
result, we have the following remarks on the network security:
Remark 1: For a given number of relays, the network
secrecy diversity order depends on the outdated degree of relay
selection, but not on the channel correlation.
Remark 2: The network secrecy diversity order is equal to
N when ρ
o
= 1, indicating that the network security can be
rapidly enhanced by increasing the number of relays in the
perfect CSI environments.
Remark 3: As long as the relay selection is outdated, the
network secrecy diversity order degenerates to unity. This is
because there exists a possible wrong relay selection with an
outdated CSI. This incorrect selection limits the entire network
secrecy performance.
Remark 4: The channel correlation between the main and
eavesdropping channels is beneficial to the transmission secu-
rity in high MER region
2
. With higher channel correlation, the
destination has more information about the channel fluctuation
of eavesdropping links. In particular, for the completely linear
correlation with ρ
c
= 1, the fluctuation of eavesdropping
channels is completely accordance with that of main channels
and hence the destination has the perfect information about the
channel fluctuation of eavesdropping links, making v
n
∗
/w
n
∗
equal to β/ε. This helps the network suppress the wiretap
perfectly, leading to a zero SOP.
2
Note that in the low MER region, the channel correlation is however
harmful to the secure transmission [9], [10].
0 5 10 15 20 25 30 35 40
10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
P (dB)
Secrecy outage probability
Analysis
Asymptotic
Simulation
N=3
λ=20 dB
D=0.5
σ=1
ρ
c
=0.3
ρ
c
=0.5
ρ
c
=0.7
ρ
c
=0.3
ρ
c
=0.5
ρ
c
=0.7
ρ
o
=0.5
ρ
o
=1.0
Fig. 2. Secrecy outage probability versus the transmit power P .
V. NUMERICAL AND SIMULATION RESULTS
In this section, we provide some numerical and simulation
results to verify the proposed studies for the multiple secure
relaying with channel correlation and outdated relay selection.
All links in the network experience Rayleigh fading, and we
adopt the path-loss model with exponent of four to measure the
average channel gains of main links. Without loss of generality,
we normalize the distance between the source and destination
to unity, where the relays are in between of them. Let D denote
the distance between the source and relays, and accordingly,
α = D
−4
and β = (1 − D)
−4
are set. The target data rate
R
t
of the first hop is 1 bps/Hz, and hence the associated γ
t
is
3. The secrecy data rate R
s
is 0.2 bps/Hz, so that the secrecy
SNR threshold γ
s
is 1.32.
Fig. 2 demonstrates the numerical and simulated secrecy
outage probabilities versus the transmit power P, where N =
3, λ = 20dB and D = 0.5. Several cases of channel correlation
are considered with ρ
c
= 0.3, 0.5 and 0.7. Both perfect and
outdated CSI environments are studied with ρ
o
= 1.0 and
0.5, respectively. As observed from the figure, for different
values of ρ
c
and ρ
o
, the analytical result matches well with
the simulation one, and the asymptotic result converges to
the exact one when P is large. This validates the derived
analytical and asymptotic expressions of the SOP. Moreover,
in both perfect and outdated CSI environments, the secrecy
performance becomes better with larger ρ
c
, as higher channel
correlation helps the destination have more information about
the channel fluctuation of eavesdropping links. The secrecy
performance also improves with larger ρ
o
, as better CSI
helps select the best relay to assist the secure transmission.
Furthermore, the secrecy performance improves with larger P .
But the improvement is saturated in high P region, as the fixed
MER becomes the bottleneck of the secrecy performance.
Fig. 3 illustrates the secrecy outage probability versus MER
with different values of ρ
c
and ρ
o
, where P = 40dB and
N varies from 1 to 3. We can see from this figure that for
different values of N, ρ
c
and ρ
o
, the analytical result fits well
with the simulation one, and the asymptotic result converges
to the simulation one in the high MER region. Moreover,
