On Physical Layer Security of Double Rayleigh
Fading Channels for Vehicular Communications
Item Type Article
Authors Ai, Yun; Cheffena, Michael; Mathur, Aashish; Lei, Hongjiang
Citation Ai Y, Cheffena M, Mathur A, Lei H (2018) On Physical Layer
Security of Double Rayleigh Fading Channels for Vehicular
Communications. IEEE Wireless Communications Letters: 1–1.
Available: http://dx.doi.org/10.1109/LWC.2018.2852765.
Eprint version Post-print
DOI 10.1109/LWC.2018.2852765
Publisher Institute of Electrical and Electronics Engineers (IEEE)
Journal IEEE Wireless Communications Letters
Rights (c) 2018 IEEE. Personal use of this material is permitted.
Permission from IEEE must be obtained for all other users,
including reprinting/ republishing this material for advertising or
promotional purposes, creating new collective works for resale
or redistribution to servers or lists, or reuse of any copyrighted
components of this work in other works.
Download date 10/08/2022 01:23:18
Link to Item http://hdl.handle.net/10754/628362
2162-2337 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LWC.2018.2852765, IEEE Wireless
Communications Letters
IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. XX, NO. XX, MONTH 2018 1
On Physical Layer Security of Double Rayleigh Fading Channels for
Vehicular Communications
Yun Ai, Member, IEEE , Michael Cheffena, Aashish Mathur, Member, IEEE, and Hongjiang Lei, Member, IEEE
Abstract—In this letter, we study the physical layer secrecy
performance of the classic Wyner’s wiretap model over double
Rayleigh fading channels for vehicular communications links.
We derive novel and closed-form expressions for the average
secrecy capacity (ASC) taking into account the effects of fading,
path loss and eavesdropper location uncertainty. The asymptotic
analysis for ASC is also conducted. The derived expressions can
be used for secrecy capacity analysis of a number of scenarios
including vehicular-to-vehicular (V2V) communications. The ob-
tained results reveal the importance of taking the eavesdropper
location uncertainty into consideration while designing V2V
communication systems.
Index Terms—Physical layer security, average secrecy capacity,
double Rayleigh fading, vehicle-to-vehicle (V2V) communication.
I. INTRODUCTION
P
HYSICAL LAYER SECURITY (PLS) has widely been
considered to be a potential paradigm to enhance com-
munication secrecy against eavesdropping in wireless com-
munication networks [1]. There is an increasing number of
research exploring the physical layer secrecy performance of
communication systems over different fading conditions. The
average secrecy capacity (ASC) over κ–µ and α–µ fading
channels have been derived in [2] and [3], respectively. The
secrecy outage probability (SOP) performance over correlated
composite Nakagami-m/Gamma fading channels was investi-
gated in [4]. In [5], the closed-form expressions for SOP were
derived by taking into consideration both the impact of fading
as well as the eavesdropper’s location uncertainty. The effect
of protected zone around the source node on ASC in a random
wireless network was investigated in [6], which was shown to
lead to significant improvement in security.
Vehicular communications enable the information transmis-
sion between vehicles or with other objects. Vehicle-to-vehicle
(V2V) communications is playing a key role in increasing
road safety and efficiency, which is usually developed as
part of the intelligent transportation system (ITS) and lays
the foundation for future autonomous transport systems [7].
Meanwhile, lack of secure communications between vehicles
might result in abuses or attacks, which can jeopardize ITS
efficiency and threaten driving safety [8]. This motivates the
research on vehicular communications from the perspective of
communication security.
Due to the high mobility of vehicles, the well-known
channel models such as Rayleigh, Rician or Nakagami-m,
Manuscript received February 8, 2018; revised April 24, 2018; accepted
June 29, 2018. Date of publication MONTH DAY, 2018; date of current
version MONTH DAY, 2018. The associate editor coordinating the review
of this paper and approving it for publication was S. Ma.
Y. Ai and M. Cheffena are with the Norwegian University of Science and
Technology, Norway (e-mail: yun.ai@ntnu.no; michael.cheffena@ntnu.no).
A. Mathur is with the Indian Institute of Technology Jodhpur, India (e-mail:
aashishmathur@iitj.ac.in).
H. Lei is with the King Abdullah University of Science and Technology,
Saudi Arabia, and also with Chongqing University of Posts and Telecommu-
nications, China (e-mail: leihj@cqupt.edu.cn).
Digital Object Identifier
which were initially proposed for stationary communication
links, do not fit well for a lot of V2V channel measurements
[9]–[11]. Instead, double Rayleigh fading has been proposed
as a more precise characterization of the dynamic non line-of-
sight (NLOS) vehicular links based on both field measurement
and theoretical analysis in [9]–[11] and the references therein,
where the double-bounce scattering components caused by
scatterers around both the transmitter’s and receiver’s local
environments lead to a cascaded Rayleigh fading processes
[11]. Additionally, it has been shown that double Rayleigh dis-
tribution also fits various measurements in different suburban
outdoor-to-indoor mobile-to-mobile (M2M) communication
scenarios as well as for keyhole channels [12].
Motivated by the latest advances in physical layer security
analysis [2]–[6] and the importance of security in vehicular
communications, we study the secrecy performance of vehic-
ular communications in this letter. Additionally, due to the high
mobility of vehicles, and more importantly, the eavesdropper’s
intention to hide its position, we also take into account of
the uncertainty of the eavesdropper’s location in our analysis.
We represent the location uncertainty with a parameter pair,
which enables us to gain more insights on the impact of
location uncertainty on the secrecy performance compared
to previous results. To the best of the authors’ knowledge,
the secrecy performance of double Rayleigh fading channel
has not been studied from the physical layer perspective, nor
can it be approximated by those of other investigated fading
distributions [2]–[6]. More specifically, we investigate the
secrecy performance of the Wyner’s model in the scenario of
V2V communications under eavesdropper location uncertainty.
Notations: K
v
(·) denotes the modified Bessel function of
second kind with order v [13, Eq. 8.407], C = −ψ(1) is Euler-
Mascheroni constant with ψ(·) being Euler psi function [13,
Eq. 8.367], and G
m,n
p,q
x
a
1
,...,a
p
b
1
,...,b
q
is the Meijer G-function
[13, Eq. 9.343].
II. CHANNEL AND SYSTEM MODELS
We consider the classical Wyner’s wiretap model in our
analysis [3]. Under Wyner’s model, the vehicle S sends confi-
dential information to the desired receiver D (another moving
vehicle in V2V scenario) whereas the eavesdropper vehicle
E tries to intercept the information by decoding its received
signal. It assumed that the communication links for V2V com-
munications experience independent double Rayleigh fading
and the distance for the legitimate communication (between
nodes S and D) is known to S. We also consider the realistic
scenario of eavesdropping that the exact distance for the
eavesdropper link (between nodes S and E) is unknown to
S due to the vehicle mobility and node E’s intention to hide
its exact position; but S has the information that the position
of node E is located between ranges R
1
and R
2
with equal
probability. In V2V scenario, R
1
can be interpreted as the
minimum distance between vehicles to ensure safety or the
2162-2337 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LWC.2018.2852765, IEEE Wireless
Communications Letters
IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. XX, NO. XX, MONTH 2018 2
range of protected zone as proposed in [6] while the upper
limit R
2
is the maximum transmission distance due to S’s
coverage limit.
The received signal at node X, X ∈ {D, E}, is expressed as
y
X
= h
X
x + n, (1)
where x represents the transmitted signal with energy E
s
,
h
X
represents the channel between node S and node X,
n denotes the additive white Gaussian noise (AWGN) with
power spectral density N
0
, which, without loss of generality,
is assumed to be the same for both links. Incorporating the
effects of small-scale fading and path loss, the channel gain
for transmission distance d
X
between nodes S and X can be
further expressed as follows [14]:
h
X
=
g
X
p
1 + d
α
X
, (2)
where g
X
represents the channel gain following double
Rayleigh distribution and α denotes the path loss exponent.
III. SECRECY CAPACITY PERFORMANCE ANALYSIS
A. Distributions of Instantaneous SNRs
From (2), the instantaneous signal-to-noise ratio (SNR)
received at node X, X ∈ {D, E}, can be expressed as
γ
X
=
|h
X
|
2
E
s
N
0
=
|g
X
|
2
E
s
(1 + d
α
X
)N
0
. (3)
With g
X
being double Rayleigh distributed, the probability
density function (PDF) f
|g
X
|
2
(x) and cumulative distribution
function (CDF) F
|g
X
|
2
(x) of the random variable (RV) |g
X
|
2
are given by [15]
f
|g
X
|
2
(x)=2K
0
(2
√
x), F
|g
X
|
2
(x)=1 − 2
√
xK
1
(2
√
x). (4)
Since the distance d
D
is a constant known to node S, the
CDF F
γ
D
(γ) of the instantaneous SNR γ
D
can be readily
derived by applying the change of RVs and written as [15]
F
γ
D
(γ)=1−2
s
(1 + d
α
D
)N
0
γ
E
s
K
1
2
s
(1 + d
α
D
)N
0
γ
E
s
. (5)
For instantaneous SNR γ
E
, besides the double Rayleigh
fading, the random location of the node E should also be taken
into account while deriving the distribution functions.
Proposition 1: The CDF F
γ
E
(γ) of the instantaneous SNR
γ
E
received by node E given that its distance to node S is
any distance between R
1
and R
2
with equal probability, can
be closely approximated numerically by the expression in (6)
at the top of next page, where w
ι
and t
ι
are the weights and
abscissas detailed in [16, Eq. 25.4.30].
Proof : Please refer to Appendix A.
B. Average Secrecy Capacity
The instantaneous secrecy capacity of the considered system
is defined as C
s
= max{ln(1 + γ
D
) −ln(1 + γ
E
), 0} [2]. The
average secrecy capacity C
s
can be obtained from [17]
C
s
=
Z
∞
0
F
γ
E
(γ)
1 + γ
[1 − F
γ
D
(γ)] dγ. (7)
Substituting (5) and (6) into (7) and utilizing the fol-
lowing transformations:
1
1+x
= G
1,1
1,1
( x|
0
0
), K
v
(x) =
1
2
G
2,0
0,2
x
2
4
v
2
,−
v
2
[13, Eq. 9.3], and the integral of product
of Meijer G-functions [18, Eq. (07.34.21.0013.01)], we can
obtain the exact expression for ASC in (8) on next page.
Remark 1: Observing (8), the ASC under the eavesdropper
location uncertainty is bounded by the first term, which is only
related to the legitimate node D. Indeed, the ASC is always
less than the capacity of the legitimate channel capacity. The
second term of (8) demonstrates the impact of the location
uncertainty of node E on the ASC, where the parameter
R
2
−R
1
2
represents the mean uncertainty range and
R
2
+R
1
2
denotes the
mean distance to the source node S. From (8), the effects of
eavesdropper’s location uncertainty on the ASC can be readily
evaluated.
The above derived expressions provide the average secrecy
capacity and distribution of SNR γ
E
in general cases. Next, we
consider the following special cases to provide more insights.
Lemma 1: When the node S gains information on the exact
position of node E, i.e., mathematically R
1
= R
2
→ d
E
in
(8), the ASC C
s
can be expressed as
C
s
=
a
2
· G
3,1
1,3
a
2
4
−
1
2
1
2
,−
1
2
,−
1
2
−
ab
4
· G
1,1:2,0:2,0
1,1:0,2:0,2
−1
−1
−
1
2
, −
1
2
−
1
2
, −
1
2
a
2
4
,
b
2
4
, (9)
where a = 2
q
(1+d
α
D
)N
0
E
s
and b = 2
q
(1+d
α
E
)N
0
E
s
.
Proof : When d
E
is known to node S, the CDF F
γ
E
(γ)
of the SNR γ
E
will have the same form as that in (5).
Substituting the CDFs of γ
D
and γ
E
into (7) and utilizing
the equalities [18, Eq. (07.34.21.0013.01)] and the solution to
the integral of product of three Meijer-G functions in terms of
extended generalized bivariate Meijer-G function (EGBMGF)
[18, Eq. (07.34.21.0081.01)], we can obtain the result in (9).
Lemma 2: When the node S has information on the position
of node E and the transmit SNR
E
s
N
0
→ ∞, the asymptotic
ASC C
s
can be calculated as
C
s
= ln(µ)[1−
√
ρ
1
G
2,2
2,2
( ρ
1
|a
1
)] + G
2,2:2,0:2,0
3,3:0,2:0,2
a
2
|b|c|
1, ρ
1
· π
√
ρ
1
+π
√
ρ
1
G
2,2:2,0:2,0
3,3:0,2:0,2
a
2
−1|b|c|
1, ρ
1
−ln(ν)
√
ρ
2
· G
2,2
2,2
( ρ
2
|a
1
) + π
√
ρ
2
G
2,2:2,0:2,0
3,3:0,2:0,2
a
2
|b|c|
1, ρ
2
+ π
√
ρ
2
G
2,2:2,0:2,0
3,3:0,2:0,2
a
2
− 1|b|c|
1, ρ
2
− 2C, (10)
where a
1
=
−0.5, −0.5
0.5, −0.5
, a
2
=
−0.5, −0.5, −1
−0.5, −0.5, −1
, b =
−
0, 0
,
c =
−
0.5, −0.5
, µ =
1
(1+d
α
D
)
, ν =
1
(1+d
α
E
)
, ρ
1
=
µ
ν
, ρ
2
=
ν
µ
.
Proof : Please refer to Appendix B.
Remark 2: Rewriting (15) leads to the following expression:
C
s
= ln(ρ
1
) ·
Z
∞
0
Z
∞
y
ρ
1
2K
0
(2
√
y) · 2K
0
(2
√
x)dxdy
+
Z
∞
0
Z
∞
y
ρ
1
2K
0
(2
√
y) · 2K
0
(2
√
x) ln
x
y
dxdy. (11)
Noticing that both integrals in (11) are consistent, we can
conclude that the asymptotic ASC follows the scaling law of
2162-2337 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LWC.2018.2852765, IEEE Wireless
Communications Letters
IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. XX, NO. XX, MONTH 2018 3
F
γ
E
(γ)
∼
=
1 −
√
γ
R
2
+ R
1
N
0
E
s
1
2
L
X
ι=0
w
ι
·
R
2
− R
1
2
t
ι
+
R
2
+ R
1
2
·
h
1 +
R
2
− R
1
2
t
ι
+
R
2
+ R
1
2
α
i
1
2
· G
2,0
0,2
N
0
γ
E
s
h
1 +
R
2
− R
1
2
t
ι
+
R
2
+ R
1
2
α
i
1
2
,−
1
2
. (6)
C
s
∼
=
s
(1 + d
α
D
)N
0
E
s
G
3,1
1,3
(1 + d
α
D
)N
0
E
s
−
1
2
1
2
,−
1
2
,−
1
2
−
N
0
E
s
R
2
+ R
1
L
X
ι=0
w
ι
R
2
− R
1
2
t
ι
+
R
2
+ R
1
2
h
1 +
R
2
− R
1
2
t
ι
+
R
2
+ R
1
2
α
i
1
2
· (1 + d
α
D
)
1
2
· G
1,1:2,0:2,0
1,1:0,2:0,2
−1
−1
−
1
2
, −
1
2
−
1
2
, −
1
2
N
0
E
s
h
1 +
R
2
− R
1
2
t
ι
+
R
2
+ R
1
2
α
i
,
N
0
E
s
(1 + d
α
D
)
. (8)
0 10 20 30 40 50 60 70 80
0
0.5
1
1.5
2
2.5
3
3.5
= 2.2, Analytical
= 3.2, Analytical
= 2.2, Asymptotic
= 3.2, Asymptotic
Simulation
Fig. 1: The ASC vs. E
s
/N
0
for fixed d
E
and varying α values.
0 1 5 10 15 20
0
1
2
4
6
8
10
12
0 2 4 6 8 10 12
0
1
2
4
6
8
10
12
Fig. 2: The asymptotic ASC vs. d
E
/d
D
and ln((1 + d
α
E
)/(1 + d
α
D
)).
2 2.5 3 3.5 4 4.5 5 5.5 6
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Fig. 3: The ASC vs. path loss exponent α with fixed E
s
/N
0
values.
20
0.7
6
15
0.8
4
0.9
10
1
2
5
0
Fig. 4: ASC vs. uncertainty range
R
2
−R
1
2
and mean uncertainty distance
R
2
+R
1
2
.
Θ(ln(ρ
1
)) (namely Θ
ln
1+d
α
D
1+d
α
E
) as ρ
1
increases and thus
depends only on the relative position of the nodes D and E.
Remark 3: When the distances d
D
and d
E
are equal, namely
ρ
1
= ρ
2
= 1 in (10), the asymptotic ASC will always be a
constant equaling 1 nat/s/Hz, which is different from other
fading distributions [3, Figs. 1, 2]. This also reaffirms our
statement on scaling law of Θ
ln
1+d
α
D
1+d
α
E
since ln(1) = 0.
IV. NUMERICAL RESULTS AND DISCUSSION
For the purpose of numerical evaluation, we adopt the
following simulation parameters: unless stated otherwise, node
D is d
D
= 3 m away from node S, node E is located
somewhere between R
1
= 4 m and R
2
= 12 m away from S.
Figure 1 shows the ASC as a function of the transmit SNR
E
s
/N
0
for different values of path loss exponent assuming that
the location of E is known to S. It can be observed that the
ASC performance can be improved by increasing the transmit
SNR within the low transmit SNR region, whereas the transmit
SNR has a limited impact on the ASC in the high transmit
SNR region, which is clearly independent of the transmit SNR
and is theoretically verified in Lemma 2. This reveals that the
secrecy capacity limit does not depend on the transmit SNR
but the relative locations of the nodes, which is confirmed in
Fig. 2. Figure 2 illustrates the ASC versus the ratio
d
E
d
D
in the
first subplot as well as the scaling law of the asymptotic ASC
with respect to ln
1+d
α
D
1+d
α
E
in the second subplot.
Figure 3 depicts the ASC versus the path loss exponent
with fixed values of input SNR. It is clear that a monotonic
increasing or decreasing relationship does not necessarily hold
between the ASC and the path loss exponent for fixed E
s
/N
0
values. This is due to the fact that secrecy capacity depends
on the capacity difference of the legitimate and eavesdropper
channels even though greater values of path loss exponent
indicate less capacity for both channels.
Figure 4 illustrates the impact of the eavesdropper location
uncertainty on the ASC performance. It is concluded that
when the uncertainty on E’s location decreases (i.e., smaller
values of
R
2
−R
1
2
), the uncertainty of node E’s received SNR
decreases; therefore, the ASC increases, which is also in
accordance with the results in [6]. Also, as the node E is
statistically located further (i.e., greater values of
R
2
+R
1
2
),
the ASC increases. Additionally, it is observed that under the
condition of eavesdropper location uncertainty, the uncertainty
range
R
2
−R
1
2
poses a greater impact on the ASC when the
mean uncertainty distance
R
2
+R
1
2
is small. The results imply
the importance of taking into account the uncertainty of
2162-2337 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LWC.2018.2852765, IEEE Wireless
Communications Letters
IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. XX, NO. XX, MONTH 2018 4
eavesdropper’s location while designing V2V communication
systems with PLS.
V. CONCLUSION
In this letter, the physical layer secrecy capacity over double
Rayleigh fading channels were derived and analyzed. The
obtained results can be used for the secrecy capacity analysis
for V2V and M2M communications as well as for keyhole
channels.
APPENDIX A: PROOF OF PROPOSITION 1
We first derive the CDF of RV |h
E
|
2
with location uncer-
tainty of E for S. The CDF F
|h
E
|
2
(x) of the RV |h
E
|
2
is related
to the CDF F
|g
E
|
2
(·) of RV |g
E
|
2
shown in (4) as follows [5]:
F
|h
E
|
2
(x) =
2
R
2
2
− R
2
1
Z
R
2
R
1
zF
|g
E
|
2
(x(1+z
α
))dz =1−
4
R
2
2
− R
2
1
·
Z
R
2
R
1
z
p
x(1 + z
α
)K
1
(2
p
x(1 + z
α
)) dz. (12)
Expressing the modified Bessel function using Meijer G-
function [13, Eq. 9.34] and applying the Gauss-Legendre
quadrature technique [16, Eq. 25.4.30], the CDF F
|g
E
|
2
(·) in
(12) can be approximated by
F
|h
E
|
2
(x) = 1−
2
√
x
R
2
2
−R
2
1
Z
R
2
R
1
z
√
1 + z
α
G
2,0
0,2
(x(1 + z
α
)|
1
2
,−
1
2
)
| {z }
g(z)
dz
∼
=
1 −
√
x
R
2
+ R
1
L
X
ι=0
w
ι
· g
R
2
− R
1
2
t
ι
+
R
2
+ R
1
2
, (13)
where w
ι
and t
ι
, (ι = 1, . . . , L), are the weights and
zeros of the L-order Legendre polynomial [16, Eq. 25.4.30].
An arbitrarily accurate approximation can be achieved by
choosing the appropriate value of L. It should be noted that
the modified Bessel function in (12) can also be expressed
as K
1
(z) = 2
√
πe
−z
zU(1.5, 3, 2z) with U(·, ·, ·) being the
Kummer hypergeometric function [13, Eq. 9.2]. The existence
of the exponential term e
−2
√
x(1+z
α
)
in K
1
(2
p
x(1 + z
α
))
implies that (13) can converge rapidly as L increases and only
limited terms are required to get satisfactory accuracy.
Finally, utilizing the first equality of (3) in (13), we obtain
the CDF F
γ
E
(γ) of the SNR γ
E
as in Proposition 1.
APPENDIX B: PROOF OF LEMMA 2
The ASC can be alternatively expressed as [3]
C
s
=
Z
∞
0
f
γ
E
(γ
E
)
Z
∞
γ
E
ln
1 + γ
D
1 + γ
E
f
γ
D
(γ
D
) dγ
D
dγ
E
. (14)
For notational simplicity, we make the following changes of
RVs: γ
t
= E
s
/N
0
, µ = 1/(1 + d
α
D
), ν = 1/(1 + d
α
E
), ρ
1
=
µ
ν
,
ρ
2
=
ν
µ
, x = |h
D
|
2
, and y = |h
E
|
2
; and the SNRs can be
written as: γ
D
= γ
t
µx and γ
E
= γ
t
νy. Then, the ASC can
be expressed after some mathematical manipulations as
C
s
= 4
Z
∞
0
K
0
(2
√
y)
Z
∞
ρ
2
y
ln
1 +
µx − νy
1
γ
t
+ νy
K
0
(2
√
x) dxdy
γ
t
→∞
≈ 4
Z
∞
0
Z
∞
ρ
2
y
K
0
(2
√
y) ln(µx)K
0
(2
√
x) dxdy −4
Z
∞
0
Z
∞
ρ
2
y
K
0
(2
√
y) ln(νy)K
0
(2
√
x) dxdy. (15)
Denoting the first integral in (15) by I
1
and second one by
I
2
; after changing order of integration for I
1
and applying the
equality [13, Eq. 6.561.4], we can have
I
1
=4
Z
∞
0
ln(µx)K
0
(2
√
x)
h
1
2
−
√
ρ
1
xK
1
(2
√
ρ
1
x)
i
dx, (16)
I
2
=4
Z
∞
0
ln(νy)K
0
(2
√
y)
√
ρ
2
yK
1
(2
√
ρ
2
y) dy. (17)
Finally, representing the functions in the (16) and (17)
through Meijer-G functions and applying integrals of prod-
uct of Meijer G-functions in [18, Eqs. (07.34.21.0081.01),
(07.34.21.0013.01)], we obtain the asymptotic ASC in (10).
In (10), The Euler constant C comes from the integral
in (16): (a)
R
∞
0
ln(x)K
0
(2
√
x) = −C. To prove the rela-
tion, we first consider the Mellin transform of K
0
(2
√
t): (b)
R
∞
0
t
s−1
K
0
(t) dx = 2
s−1
Γ(
s
2
)
2
[13, Eq. 12.43.18]. Differ-
entiating the equality (b) and setting s = 2 leads to (c)
R
∞
0
t ln(t)K
0
(t) dt = −C + ln(2). Finally, substituting t =
2
√
x in (c) and applying the equality:
R
∞
0
K
0
(2
√
x) dx = 0.5,
we can obtain the equality after some straightforward algebra.
REFERENCES
[1] A. Hyadi et al., “An overview of physical layer security in wireless
communication systems with CSIT uncertainty,” IEEE Access, vol. 4,
pp. 6121–6132, Sept. 2016.
[2] N. Bhargav et al., “Secrecy capacity analysis over κ–µ fading channels:
Theory and applications,” IEEE Trans. Commun., vol. 64, no. 7, pp.
3011–3024, Jul. 2016.
[3] H. Lei et al., “Secrecy capacity analysis over α-µ fading channels,”
IEEE Commun. Lett., vol. 21, no. 6, pp. 1445–1448, Jun. 2017.
[4] G. C. Alexandropoulos and K. P. Peppas, “Secrecy outage analysis
over correlated composite Nakagami-m/Gamma fading channels,” IEEE
Commun. Lett., vol. 22, no. 1, pp. 77–80, Jan. 2018.
[5] D. S. Karas et al., “Physical layer security with uncertainty on the
location of the eavesdropper,” IEEE Wireless Commun. Lett., vol. 5,
no. 5, pp. 540–543, Oct. 2016.
[6] W. Liu et al., “On ergodic secrecy capacity of random wireless networks
with protected zones,” IEEE Trans. Veh. Technol., vol. 65, no. 8, pp.
6146–6158, Aug. 2016.
[7] L. Li et al., “A survey of traffic control with vehicular communications,”
IEEE Trans. Intell. Transp. Syst., vol. 15, no. 1, pp. 425–432, Feb. 2014.
[8] M. Azees et al., “Comprehensive survey on security services in vehicular
ad-hoc networks,” IET Intell. Transp. Syst., vol. 10, no. 6, Aug. 2016.
[9] A. S. Akki and F. Haber, “A statistical model of mobile-to-mobile land
communication channel,” IEEE Trans. Veh. Technol., vol. 35, no. 1, pp.
2–7, Feb. 1986.
[10] V. Erceg et al., “Comparisons of a computer-based propagation pre-
diction tool with experimental data collected in urban microcellular
environments,” IEEE J. Sel. Areas Commun., vol. 15, no. 4, May 1997.
[11] I. Z. Kovacs, “Radio channel characterisation for private mobile radio
systems-mobile-to-mobile radio link investigations,” Ph.D. dissertation,
Aalborg Universitet, 2002.
[12] D. Chizhik et al., “Multiple-input-multiple-output measurements and
modeling in Manhattan,” IEEE J. Sel. Areas Commun., vol. 21, no. 3,
pp. 321–331, Apr. 2003.
[13] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and
Products. Academic Press, 2007.
[14] A. Tukmanov et al., “Outage performance analysis of imperfect-CSI-
based selection cooperation in random networks,” IEEE Trans. Com-
mun., vol. 62, no. 8, pp. 2747–2757, Aug. 2014.
[15] M. Seyfi et al., “Relay selection in dual-hop vehicular networks,” IEEE
Signal Process. Lett., vol. 18, no. 2, pp. 134–137, Feb. 2011.
[16] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions:
With Formulas, Graphs, and Mathematical Tables. Courier Corporation,
1964, vol. 55.
[17] H. Lei et al., “On secrecy performance of mixed RF-FSO systems,”
IEEE Photon. J., vol. 9, no. 4, pp. 1–14, Aug. 2017.
[18] The Wolfram Functions Site. [Online]. Available: http://functions.
wolfram.com/