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Journal ArticleDOI

Performance analysis of physical layer security over α–μ fading channel

11 Jan 2016-Electronics Letters (Institution of Engineering and Technology)-Vol. 52, Iss: 1, pp 45-47
TL;DR: The physical layer security over α-μ fading channel is presented and closed-form expressions for the probability of positive secrecy capacity and upper bound of the secrecy outage probability are derived.
Abstract: Recently, many works have focused on analyzing the metrics of physical layer security over different wireless channels, such as additive white Gaussian noise (AWGN), Rayleigh, Rician and Nakagami-m fading distributions. In order to extend the analysis to the general case, α-μ fading channel is considered, which can span the aforementioned cases. For this purpose, the physical layer security over α-μ fading channel is presented in this letter. The closed-form expressions for the probability of positive secrecy capacity and upper bound of the secrecy outage probability are derived. Their accuracies are assessed through comparison of theoretical analysis and simulations results.

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Summary

  • Recently, many works have focused on analyzing the metrics of physical layer security over different wireless channels, such as additive white Gaussian noise (AWGN), Rayleigh, Rician and Nakagami-m fading distributions.
  • In order to extend the analysis to the general case, αμ fading channel is considered, which can span the aforementioned cases.
  • For this purpose, the physical layer security over α-μ fading channel is presented in this letter.
  • The closed-form expressions for the probability of positive secrecy capacity and upper bound of the secrecy outage probability are derived.
  • Their accuracies are assessed through comparison of theoretical analysis and simulations results.

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Performance analysis of physical layer
security over α-µ fading channel
L. Kong, H. Tran and G. Kaddoum
Recently, many works have focused on analyzing the metrics of physical
layer security over different wireless channels, such as additive white
Gaussian noise (AWGN), Rayleigh, Rician and Nakagami-m fading
distributions. In order to extend the analysis to the general case, α-
µ fading channel is considered, which can span the aforementioned
cases. For this purpose, the physical layer security over α-µ fading
channel is presented in this letter. The closed-form expressions for the
probability of positive secrecy capacity and upper bound of the secrecy
outage probability are derived. Their accuracies are assessed through
comparison of theoretical analysis and simulations results.
Introduction: Physical layer security is a promising solution that
addresses the security issue while directly operating at the physical layer
from the information-theoretic viewpoint. Numerous contributions exist
that analyze the secrecy performance over AWGN, Rayleigh, Rician,
Nakagam-m and Weibull fading channels. Performance analysis in terms
of secrecy capacity and outage probability has been investigated [1, 2, 3,
4]. However, to the best knowledge of the authors, there is no previous
work focusing on the general case of fading channels. With regard to
different values of α and µ, the α-µ fading channel can be reduced to
the specific fading channel, such as Rayleigh, Nakagami-m and Weibull
fading distributions by adjusting certain parameters. In this letter, the
secrecy performance over α-µ fading channel is evaluated by the closed-
form expressions for the probability of positive secrecy capacity and
upper bound of secrecy outage probability. Consequently, our theoretical
analysis is confirmed by simulation results.
System model and secrecy performance analysis: A three-node classic
model such as the one shown in Fig. 1 is used here to illustrate a wireless
network with potential eavesdropping. In the wiretap channel model, a
legitimate transmitter (Alice) equipped with a directional antenna wishes
to send secret messages to an intended receiver (Bob) in the presence
of an eavesdropper (Eve), the link between Alice and Bob with fading
coefficient h
m
is called the main channel, while the one between Alice
and Eve with fading coefficient h
w
is named as the wiretap channel.
Both channels undergo the α-µ distribution.
Alice Bob
h
m
h
w
Eve
Fig. 1 Illustration of system model with two legitimate transceivers (Alice and
Bob) and one eavesdropper (Eve).
Recalling that the probability density function (PDF) of the α-µ fading
channel coefficients h
i
, (i {m, w}) is given by [5]
f
h
i
(h) =
α
i
µ
µ
i
i
h
α
i
µ
i
1
ˆ
h
α
i
µ
i
i
Γ (µ
i
)
exp
µ
i
h
α
i
ˆ
h
α
i
i
!
, (1)
where
ˆ
h
i
=
α
q
E
h
α
i
i
is the αroot mean value, α
i
> 0 is an arbitrary
fading parameter, µ
i
> 0 is the inverse of the normalized variance of
h
α
i
i
. The parameter µ
i
is calculated by µ
i
= E
2
h
α
i
i

V
h
α
i
i
, where
E (·) and V (·) are the expectation and variance operators, respectively.
Γ (x) =
R
0
t
x1
e
t
dt is the Euler’s Gamma function. In particular,
when changing the values of α and µ to the following cases: (i) α =
2, µ = 1; (ii) α = 2, µ = m; and (iii) µ = 1, the α-µ fading model can
be simplified such that it follows Rayleigh, Nakagami-m and Weibull
distributions, respectively.
Let g
i
= |h
i
|
2
denote the instantaneous channel power gain with unit
mean. The PDF of g
i
is expressed as [6]
f
g
i
(x) =
α
i
x
α
i
µ
i
2
1
2Ω
α
i
µ
i
2
i
Γ (µ
i
)
exp
"
x
i
α
i
2
#
, (2)
where
i
=
Γ(µ
i
)
Γ(µ
i
+
2
α
i
)
. Therefore, the received signal-to-noise ratio
(SNR) at Bob and Eve receiver sides can be expressed as
γ
i
=
P
i
g
i
N
i
(3)
where P
i
and N
i
are the transmission power and noise power,
respectively. Without loss of generality, we assume N
m
is equal to N
w
in this paper. In addition, since we consider that Alice is equipped with
a directional antenna, then the transmitted powers P
m
and P
w
may be
different because Bob and Eve are present in different locations in the
network.
According to [1, 2, 3, 4], the secrecy capacity for the given network is
given as follows
C
s
= C
m
C
w
=
(
log
2
1+γ
m
1+γ
w
, if γ
m
> γ
w
0, if γ
m
6 γ
w
(4)
where C
m
and C
w
are the capacities of the main channel and the wiretap
channel, respectively.
Therefore, the probability of positive secrecy capacity can be derived
as follows
P r (C
s
> 0) = P r
log
2
1 + γ
m
1 + γ
w
> 0
= P r (γ
m
> γ
w
)
= 1 P r
γ
m
γ
w
< 1
= 1 P r
g
m
g
w
<
P
w
P
m
.
(5)
According to equation (16) in [7], equation (5) is derived as
P r (C
s
> 0) = 1 F
γ
(1) (6)
where F
γ
(x) is the cumulative distribution function (CDF) of x, which is
given as
F
γ
(x) = P r
g
m
g
w
<
P
w
P
m
· x
=
P
w
w
P
m
m
αµ
m
2
x
αµ
m
2
µ
m
β(µ
m
w
)
×
2
F
1
µ
m
+ µ
w
, µ
m
; 1 + µ
m
;
P
w
w
P
m
m
α
2
x
α
2
,
(7)
herein
2
F
1
(., ; .; .) denotes the Gaussian hypergeometric function and
β(., .) is the Beta function.
The outage probability of the secrecy capacity is defined as the
probability that the secrecy capacity C
s
falls below the target secrecy
rate R
s
, i.e.
P
out
(C
s
6 R
s
) = P r
log
2
1 + γ
m
1 + γ
w
6 R
s
= P r
γ
m
6 2
R
s
(1 + γ
w
) 1
= P r (γ
m
6 γ
th
+ γ
th
γ
w
1) , (8)
where γ
th
= 2
R
s
. Due to the complex form of the PDF of α-µ fading
distribution, it is difficult to obtain a closed-form expression for (8).
However, when the target data rate R
s
approaches zero, we can obtain
the upper bound of the outage probability by substituting equation (7)
Accepted in Electronics Letters, 2016
This paper is a postprint of a paper submitted to and accepted for publication in [journal] and is subject to Institution of Engineering
and Technology Copyright. The copy of record is available at IET Digital Library. DOI : 10.1049/el.2015.2160

into equation (8), to get the following relationship
P
out
(C
s
6 R
s
) = P r (γ
m
6 γ
th
+ γ
th
γ
w
1)
6 P r (γ
m
6 γ
th
γ
w
)
6 P r
γ
m
γ
w
6 γ
th
6 P r
g
m
g
w
6
P
w
P
m
· γ
th
6 F
γ
(γ
th
) . (9)
Numerical Analysis: Fig. 2 shows the simulation and analysis results of
the probability of positive secrecy capacity versus the transmission power
P
m
over α-µ fading channel for selected power values of eavesdropper
P
w
provided that α = 2 and µ
m
= µ
w
= 1 (Rayleigh fading). One can
observe that the analytical and simulation results are in perfect match for
any given set of parameters. In addition, for the case of fixed values of
P
w
, the larger P
m
the higher the probability of positive secrecy capacity.
In Fig. 3, the probability of positive secrecy capacity in terms of different
values of α and µ for fixed P
w
= 10 dB is illustrated. Here, a similar
conclusion is obtained to that of Fig. 2.
Similarly, Fig. 4 and Fig. 5 show the simulation and analysis results
of the upper bound of the outage probability of physical layer security
over α-µ fading channel with regard to two cases: (i) fixed α = 2, µ
m
=
µ
w
= 1 while varying P
w
; (ii) fixed P
w
while changing the values of α
and µ. Here, we fix the target data rate as R
s
= 0.01 bps. We can easily
draw the same conclusion about the accuracy of our derived expression
for the upper bound of outage probability, i.e. analytical derivations are
verified by the simulation results.
−10 −5 0 5 10 15 20 25
10
−4
10
−3
10
−2
10
−1
10
0
P
m
(dB)
P
r
(C
s
>0)
Simulation
Analysis
P
w
= 20, 10, 0, −10 (dB)
Fig. 2 The probability of positive secrecy capacity versus P
m
for selected
values of P
w
values with fixed values of α = 2 and µ
m
= µ
w
= 1.
−10 −5 0 5 10 15 20 25
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
P
m
(dB)
P
r
(C
s
>0)
α = 2, µ
m
= µ
w
= 1
α = 2, µ
m
= µ
w
= 2
α = 3, µ
m
= µ
w
= 2
α = 5, µ
m
= µ
w
= 3
α = 2, µ
m
= 1, µ
w
= 3
α = 2, µ
m
= 3, µ
w
= 2
Fig. 3 The probability of positive secrecy capacity versus P
m
for different
values of α and µ
i
and a fixed value of P
w
= 10 dB. The solid and circle (o)
lines correspond to the simulation and analysis results, respectively.
Conclusion: In this letter, we derive closed-form expressions for the
probability of positive secrecy capacity and upper bound of outage
probability for physical layer security over α-µ fading channels.
For verification and correctness measures, the derived closed-form
expressions are validated by simulation results.
−10 −5 0 5 10 15 20 25
10
−4
10
−3
10
−2
10
−1
10
0
P
m
(dB)
P
out
(C
s
<R
s
)
Simulation
Analysis
P
w
= 20, 10, 0, −10 (dB)
Fig. 4 The upper bound of secrecy outage probability versus P
m
for selected
values of P
w
with fixed values of α = 2 and µ
m
= µ
w
= 1.
−10 −5 0 5 10 15 20 25
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
P
m
(dB)
P
out
(C
s
<R
s
)
α = 2, µ
m
= µ
w
= 1
α = 2, µ
m
= µ
w
= 2
α = 3, µ
m
= µ
w
= 2
α = 5, µ
m
= µ
w
= 3
α = 2, µ
m
= 1, µ
w
= 3
α = 2, µ
m
= 3, µ
w
= 2
Fig. 5 The upper bound of secrecy outage probability versus P
m
for different
values of α and µ
i
and a fixed value of P
w
= 10 dB. The solid and circle (o)
lines correspond to simulation and analysis results, respectively.
Acknowledgment: This work has been supported by the ETS’s
research chair of physical layer security in wireless networks.
L. Kong, H. Tran and G. Kaddoum (Department of Electrical
Engineering, University of Quebec, ETS, Montreal, Quebec, Canada )
E-mail: long.kong.1@ens.etsmtl.ca
References
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Information-Theoretic Security’, IEEE Trans. Information Theory, 2008,
54, (6), pp. 2515-2534
2 Liu, X.: ‘Probability of strictly positive secrecy capacity of the Rician-
Rician fading channel’, IEEE Wireless Commun. Lett., 2013, 2, (1), pp.
50-53
3 Sarkar, M.Z.I.; Ratnarajah, T.; Sellathurai, M.: ‘Secrecy capacity of
Nakagami-m fading wireless channels in the presence of multiple
eavesdroppers’, 2009 Conference Record of the Forty-Third Asilomar
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Weibull fading channel’, IEEE Global Communications Conference
(GLOBECOM), 2013, Atlanta, GA, USA, pp.659-664
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stacy distribution’, IEEE Trans. Vehicular Technology, 2007, 56, (1), pp.
27-34
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Signals over α-µ Fading Channels’, IEEE Commun. Lett., 2008, 12, (9),
pp.675-677
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8th International Symposium on Wireless Communication Systems
(ISWCS), 2011, Aachen, Germany, pp.477-481
2
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Q1. What are the contributions mentioned in the paper "Performance analysis of physical layer security over α-μ fading channel" ?

For this purpose, the physical layer security over α-μ fading channel is presented in this letter.