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IET Communications
Research Article
Secrecy analysis of amplify-and-forward
relaying networks with zero forcing
ISSN 1751-8628
Received on 9th March 2017
Revised 1st June 2017
Accepted on 29th June 2017
E-First on 27th September 2017
doi: 10.1049/iet-com.2017.0215
www.ietdl.org
Abdelhamid Salem1 , Khairi Ashour Hamdi1
1School
of Electrical and Electronic Engineering, the University of Manchester, Manchester, M13 9PL, UK
E-mail: abdelhamid.salem@manchester.ac.uk
Abstract: In this study, the ergodic secrecy capacity and the corresponding outage probability of two-hop amplify-and-forward
relaying system in the presence of a passive eavesdropper are analysed. In order to improve the security, in this study, zeroforcing (ZF) scheme is implemented at different locations in the system. The effect of the ZF-based scheme on the system
security is considered for three different scenarios, based on where the ZF scheme is applied, namely, (i) ZF receivers at the
relay and destination nodes, (ii) ZF precoders at the source and relay nodes, and (iii) ZF precoders/receivers at the relay nodes.
For each scenario, explicit analytical expressions for the ergodic secrecy capacity and secrecy outage probability are derived.
Monte Carlo simulations are also provided to validate the analysis. Results show that increasing the number of source, relay
and/or destination nodes can be favourable or unfavourable to the system security and the significance of this enhancement/
degradation depends on the particular scenario deployed. In addition, the system security improves with increasing the source
and/or relay power.
1 Introduction
The broadcast nature of wireless channels makes wireless networks
more vulnerable to the eavesdropping attack. Therefore, attention
to the issue of the security in wireless communications has
increased rapidly. Physical layer security, which is based on
information theory, has attracted considerable attention in this
context. The concept of physical layer security it was first
developed by Wyner [1], who introduced the wiretap channel for
single point-to-point communication before it was extended to
broadcast channels by Csiszar and Korner in [2]. From these
works, it is reported that achieving secure communications is
possible if the destination channel quality is better than the
eavesdropper channel.
Due to the fact that multiple-input multiple-output (MIMO)
networks have become a main element in the physical layer of
wireless communication, security of such systems has particularly
attracted a considerable amount of attention. For instance, very
recently, the secrecy capacity of a 2 × 2 MIMO system with an
eavesdropper having either one or two antennas is studied in [3, 4],
respectively. Later on, in [5, 6] an optimisation problem is
formulated to solve the secrecy capacity of a general MIMO
scenario with an eavesdropper equipped with multiple antennas.
The authors of [7] showed that antenna selection and combining
techniques enhance the secrecy over such channels.
Moreover, it is reported that the implementation of cooperative
relays can also enhance system secrecy. For instance, it was
presented in [8] that better security can be achieved by simply
forwarding artificial noise by the relay to confuse the eavesdropper.
Different relaying schemes are studied in [9, 10] to maximise the
secrecy capacity, while minimising the total transmit power. Joint
cooperative beamforming and jamming technique are discussed in
[11] in the context of amplify-and-forward (AF) relaying systems
with passive eavesdroppers. All these studies, the authors of [8–11]
considered only single-antenna source and destination nodes. In
contrast, the authors in [12–14] studied the security of MIMO relay
wiretap channels in the presence of an eavesdropper when all the
nodes are equipped with multiple antennas. However, in MIMO
relaying systems, in order to maximise the achievable secrecy
capacity a complex optimization problem, which can be
inconveniently hard to handle, usually needs to be solved [15].
Alternatively and to reduce the complexity, other linear precoding/
IET Commun., 2017, Vol. 11 Iss. 14, pp. 2181-2189
© The Institution of Engineering and Technology 2017
decoding techniques can be effectively implemented as in [15]. In
addition, the authors in [16], studied a multiple-relay MIMO
network using a decode and forward (DF) relaying strategy, and a
joint scheme, which joints the relay selection and the antenna
selection at the base station and the selected relay has been
proposed. In [17] a novel scheme combining relay selection with
artificial noise for AF relaying system in the presence of an
eavesdropper was designed. The secrecy of a distributed relaying
system has been investigated in [18], where all nodes are equipped
with multiple-antennas and the relay nodes cooperate to imitate a
relay node with multiple antennas. In [19] the secrecy rates of
untrusted one-way and two-way half-duplex AF relaying protocols
were considered. The distributed beam-forming for MIMO fullduplex relay networks was considered in [20]. Furthermore, in [21]
the authors studied the cooperative secure transmission for a DF
system, where an opportunistic relay node is adopted to forward
the confidential message and the other relay nodes transmit
artificial noise to confuse the eavesdroppers. In [22] a joint source–
relay precoding scheme was proposed to secure an AF MIMO
wireless relay network in the existence of a multi-antenna
eavesdropper. The authors of [23] investigated the security issue of
AF multiuser peer-to-peer relay networks, where a secure user
transmits the confidential message in the presence of a multiantenna eavesdropper, while the other unclassified users transmit
unclassified messages. The source transmit power and relay beamformer have been designed to maximise the achievable secrecy rate
under the minimum received signal-to-interference-plus-noise ratio
(SINR) requirement at each destination. For more details, we refer
the reader to [24] where an overview of the recent research on
enhancing wireless transmission secrecy via cooperation was
discussed.
Unlike other studies, in this paper we analyse the security of
MIMO AF relaying systems with zero-forcing (ZF) processing for
various scenarios based on the ZF design strategy [We select ZF
scheme and not others such as minimum mean square error
(MMSE), because ZF is simple and ease of implementation.]. In
light of this, the ergodic secrecy capacity and secrecy outage
probability of the cooperative MIMO system are investigated for
the following scenarios, (i) when ZF receivers are implemented at
the relay and destination nodes, (ii) when the ZF precoders are
implemented at the source and relay nodes, and (iii) when the ZF
precoders/receivers are implemented at the relay nodes.
2181
can be straight forward applied to the cases where the nodes are
antennas.
Practical example of this model is the communication between
the base station and users through relay station, in up-link (the first
scenario, ZF receivers at the relay and destination nodes) or downlink (the second scenario, ZF precoders at the source and relay
nodes) or multi-pair communication through relay station (the third
scenario, ZF precoders/receivers at relay nodes).
Therefore, the received signal vector at the relay nodes,
yr = [y1, …, yNr]T, can be expressed as
yr = as G1Ws x + nr
Fig. 1 Block diagram of a two-hop AF relay system with ZF processing in
the presence of one eavesdropper
Throughout this paper, Monte Carlo simulations are presented to
confirm the correctness of the analysis. The results reveal that
increasing the number of source, relay and/or destination nodes can
be generally favourable or unfavourable to the system security
depending on the practical scenario. It will also be demonstrated
that increasing the source/relay power can considerably enhance
the secrecy capacity and secrecy outage probability of the systems
under consideration.
The notations used in this paper are: Bold uppercase and bold
lowercase letters denote matrices and vectors, respectively.
Transpose operation, conjugate operation and conjugate transpose
are denoted by . T, . ∗ and . H, respectively. The notation ∥ . ∥
denotes Euclidean norm and . represents the absolute value of a
scalar. log . represents logarithm of base-2; metric, complex
Gaussian distribution with mean μ and variance σ 2 is denoted by
CN μ, σ 2 ; I identity matrix and det A denotes the determinant of
matrix A. diag{a} is a diagonal matrix whose diagonal elements
are the elements of the vector a; E . denotes expectation; Tr . is
the trace of a matrix; A k is the kthcolumn in matrix A and A k, k is
the element k, k in matrix A.
2 System model
We consider a MIMO AF relay system consisting of Ns source
nodes transmitting independent messages to Nd destination nodes
via Nr relay nodes in the presence of a single antenna passive
eavesdropper node to eavesdrop a specific message in the system,
as shown in Fig. 1. As indicated in the figure, the channels between
the nodes are denoted as G1 ∼ CNNr, Ns 0Nr × Ns, INr × Ns ,
G2 ∼ CNNd, Nr 0Nd × Nr, INd × Nr , and h ∼ CN1, Nr 01 × Nr, INr .
In order to focus our study on the cooperative phase, i.e. relayto-destination link, we assume that all communications are
performed through the relaying nodes and that there are no direct
links between the source and destination/eavesdropper due to the
deep shadowing. This assumption is well studied in the literature
for the relay systems [25–27]. The assumption refers to the systems
where the source communicates with the relay by a local
connection [28], and it also refers to the relay systems in which the
broadcast phase is secure. This situation occurs in systems where
the source and the relay are located in one cluster, while the
destination and the eavesdropper are located in another and the
communication can only rely on the relay. Therefore, transmission
between the source nodes and destination nodes is achieved as
follows. In the first phase, the source nodes transmit their
independent messages to the relay, and in the second phase the
relay forward the received messages to the legitimate destination
nodes.
In this paper, we also assume that there is full cooperation
between the nodes where the ZF is performed, i.e. the cooperating
nodes exchange all the information with each other. Our analysis
2182
(1)
where Ws is the Ns × Ns source weight matrix, x is the Ns × 1
transmitted signal vector with variance INs, nr is an Nr × 1 additive
white Gaussian noise (AWGN) vector at the relay node with
variance INrσr2 and as is the source normalisation constant which is
designed to constrain the total transmit power at the source Ps
[29, 30], and is given by
as =
Ps
Tr E WsWsH
.
(2)
Consequently, the received signal vector at the destination
nodes, yd = [y1, …, yNd]T, can be written as
yd = as ar Wd G2WrG1Ws x + ar Wd G2Wrnr + Wd nd
(3)
where Wd is the Nd × Nd destination weight matrix, Wr is the
Nr × Nr relay weight matrix, nd is an Nd × 1 AWGN vector at the
destination node with variance INdσd2 and ar is the relay
normalisation constant which is designed to constrain the relay
transmit power Pr , and is expressed as [29, 30]
ar =
(Pr /σr2) Tr E WsWsH
(Ps /σr2) Tr E Q + Tr E WsWsH Tr E WrWrH
(4)
where Q = Wr G1 Ws WsH G1H WrH. On the other hand, the received
signal at the eavesdropper is
ye = asarh WrG1Ws x + arhWrnr + ne
(5)
where ne is the AWGN at the eavesdropper with variance σe2.
Assuming that full-channel state information (CSI) is unknown at
the transmitter, the ergodic secrecy capacity can be obtained as [31,
page 4692] [32, Eq. (5)]
C̄s = E Cd − E Ce
+
(6)
where Cd and Ce are the destination and eavesdropper capacities
given by Cd = 1/2 log 1 + γd and Ce = 1/2 log 1 + γe ,
respectively, with γd and γe denote the SINRs at the destination and
eavesdropper, respectively. The factor 1/2 is resulted from the fact
that two time slots are required for transmitter-to-receiver data
transmission [9]. It should be highlighted here that when the
transmitter has only the CSI of the main channel, (6) will represent
the lower bound of secrecy capacity [33, 34].
In case the transmitter either knows both channels or only the
legitimate channel the power allocation scheme can be applied to
further improve the secrecy performance. In this case, non-zero
ergodic secrecy capacity can be attained by allocating the power
over the channel conditions for which the destination channel is
better than the eavesdropper channel [31].
The other performance measure that we will be looking into in
this paper is the secrecy outage probability Pout which is defined
as the probability that the secrecy capacity is less than the target
secrecy rate R0 and is expressed mathematically as [35, 36]
IET Commun., 2017, Vol. 11 Iss. 14, pp. 2181-2189
© The Institution of Engineering and Technology 2017
Pout Rs = Pr Cs < R0 .
(7)
Conventionally, to achieve optimal secrecy capacity of the system
under consideration, a complex optimisation problem should be
solved. However, in order to reduce complexity, we implement
linear ZF scheme instead at various location as in the following
sections.
3 Scenario 1: ZF at the relay and destination
nodes
In this scenario, we analyse the secrecy capacity and secrecy
outage probability when ZF receivers are implemented at the relay
and destination nodes. It is assumed that the relay and destination
know G1 and G2, respectively. It is also assume that Nr > Ns and
Nd > Nr. According to [30, 37], the weights at the nodes are given
by
Ws = INs
Wr = P G1H G1
−1
G1H
G2H
γdk =
C̄s1 = E Cd − E Ce
E Ce =
ar2 G1H G1
−1
k, k
Wd WdH σd2 k, k
(9)
2
+ σd2 G2H G2
.
−1
∑
i = 1, i ≠ k
N
as ar ∑i =s 1, i ≠ k
2
2
(10)
Ns
(17)
(18)
1 + γdk
<υ
1 + γek
(19)
(20)
Pout = 1 − Pr γdk > υ + υ γek − 1 ∣ γek = Φ
∫
−∫
(1 − υ)/ υ
0
∞
(1 − υ)/ υ
2
2
2
.
2
2
hWr g1i + ar ∥ hWr ∥ σr + σe
Ps
Ns
E Cd =
(13)
1
2 ln 2
×
∫
0
∞
IET Commun., 2017, Vol. 11 Iss. 14, pp. 2181-2189
© The Institution of Engineering and Technology 2017
f γek Φ dΦ
(22)
F̄ γdk υ + υ Φ − 1 f γek Φ dΦ
2 zc/a Nd − Nr + 1 e−(z / a) c + b
Γ Nd − Nr + 1
×
×
Nr − Ns
∑
zb/a
p!
p=0
((q − p + 1)/2)
b
c
p Nd − Nr + p
∑
Nd − Nr + p
q
q=0
Jp − q − 1
(23)
2z bc
a
where F̄ γdk is the complementary cumulative distribution function
(CCDF) of γdk, which is given by (23), shown at the top of the next
page, where z = υ + υ Φ − 1, a = PsPr /σs2σr2, b = Ns(Pr /σr2), and
c = (Ps /σs2)Nr + (Ns2 /(Nr − Ns)) and f γek is the probability density
function (PDF) of γek and given by
2 γr z
1
1 − e− z γrs
z
(1 + N d − N r)/2
2z
F̄ γdk z =
(12)
The normalisation constants at the source and relay nodes of
this scenario are determined simply by substituting the weights (8)
into (2) and (4) to obtain the following:
(21)
and consequently
hWr g1i xi + ar hWr nr + ne (11)
as2 ar2 h Wr g1k 2
as =
0
Pout = Pr γdk < υ + υ γek − 1 .
where a = as ar, g1k and g1i are the kth and the ith columns in the
matrix G1, respectively. Hence, the SINR of the kth transmitted
signal at the eavesdropper can be given as
γek =
−1 + Ns + ψ
where υ = 22R0. In addition,
Pout = 1 −
Ns
0
which can also be expressed in terms of the destination and
eavesdropper SINRs as
k, k
On the other hand, the received signal at the eavesdropper of
the kth transmitted signal is expressed as
yek = a h Wr g1k xk + a
1
−ψ
2ln 2
By conditioning on γek, (20) can be written as
as2 ar2
σ
k, k r
(15)
Pout = Pr Cd − Ce < R0
where A = Wd G2 Wr G1 Ws and B = Wd G2 Wr. Now substituting
(8) into (9) yields
γdk =
+
E Cd and E Ce are given by (16) and (17), shown at the top of
the next page, respectively, where γrs = ar2 as2 /σd2 , γr = ar2 σr2 /σd2 and
ψ 0 . is the Poly-gamma function. (see (16))
Pout = Pr
as ar A A
ar2 B BH σr2 +
and
Ns /(Nr − Ns) = Tr E
[38, Lemma 1]. The ergodic secrecy
capacity of this scenario can be obtained as
H
2
Nr = Tr E Q ,
WrWrH
(8)
where P is an INr × Ns matrix to ensure that the Nr signals are
transmitted at the relay nodes. From (3), the SINR of the kth
transmitted signal at the destination can be written as follows:
2
Ns = Tr E WsWsH ,
where
(14)
Proof: The proof is provided in Appendix 1. □
The secrecy outage probability of this scenario can be obtained
as follows. Based on (7), Pout can also be written as
−1
Wd = G2H G2
(Pr /σr2) Ns Nr − Ns
(Ps /σr2) Nr Nr − Ns + Ns2
ar =
(1 + N r − N s)/2
J1 + Nr − Ns 2 γr z
Γ Nr − Ns + 1
J1 + N d − N r 2 z
dz .
Γ Nd − Nr + 1
(16)
2183
3.1 Numerical results
In this subsection, numerical results of the secrecy capacity and
secrecy outage probability for system 1 are presented. In all our
evaluations from this point onward, Monte Carlo simulations are
included and the channel coefficients are randomly generated in
each simulation run. It should also be mentioned that, the integrals
are efficiently evaluated by using numerical integration. To start
with, Fig. 2 depicts the ergodic secrecy capacity as a function of Ns
for Pr = 2, 4 , 6, and 8 dBw when Ps = 10 dBw, Nd = 50, Nr = 42
and noise power at all nodes is set σr2 = σd2 = σe2 = 10 dBm. One can
see that the analytical results obtained from (15) and the simulated
ones are in good agreement. It is also apparent that the secrecy
capacity degrades with increasing Ns and this is because increasing
Ns leads to decrease the normalisation constant at the source as. In
addition, the secrecy capacity enhances as Pr is increased, and this
enhancement becomes less significant when Ns is larger than 35.
To illustrate the impact of Nr and Ps on the secrecy capacity, we
plot in Fig. 3 the ergodic secrecy capacity versus Nr for several
values of Ps when Pr = 2 dBw, Ns = 10, Nd = 50 and the noise
power at all nodes is again set as σr2 = σd2 = σe2 = 10 dBm. The first
observation one can see here is that as Ps increases the ergodic
secrecy capacity enhances and this enhancement becomes less
significant when Nr is relatively high. The other interesting
observation from this figure is the fact that for each value of Ps,
there exist an optimal Nr value that maximises the ergodic secrecy
capacity. This can be explained by the fact that, increasing Nr will
lead to increase ar to an optimal value after that increasing Nr will
result in smaller values of ar. Therefore, adding more relay nodes
might be detrimental to the system performance.
As for the secrecy outage probability, it is plotted in Fig. 4
versus the threshold value of the secrecy rate for different values of
Nd when Ps = − 13 dBw, Pr = − 13 dBw, Nr = 11, Ns = 10, and
σr2 = σd2 = σe2 = 0 dBm. It is clearly visible that the analytical results
agree well with the simulated ones. It is also observed that this
probability deteriorates as Nd is increased which is due to the fact
that increasing Nd will result in increasing the diversity in the
second phase. Similar trend is observed in [30], in terms of outage
probability.
Fig. 2 Ergodic secrecy capacity versus Ns for various values of Pr
Fig. 3 Ergodic secrecy capacity versus Nr for various values of Ps
4 Scenario 2: ZF at the source and relay nodes
In this section, the ergodic secrecy capacity and the corresponding
secrecy outage probability are analysed when the ZF precoders are
implemented at the source and relay nodes. Similarly, as in the
previous section, it is assumed here that the source nodes know G1
and the relay nodes know G2, and that Ns > Nr, Nr > Nd. The
weights at the legitimate nodes in this scenario are [30, 37]
Ws = G1H G1 G1H
−1
Wr = G2H G2 G2H
−1
P1
P2
Wd = I N d
where P1 is an INr × Ns matrix to ensure that Nr out of Ns messages
are sent at the source, and P2 is an INd × Nr matrix to also ensure that
Nd out of Nr messages are sent at the relay.
By substituting (25) into (9), we can obtain
Fig. 4 Secrecy outage probability of system 1
f γek Φ = − 1 + Φ
− Ns
1 − Ns .
(24)
Proof: The proof is provided in Appendix 2. □
Finally, the secrecy outage probability is obtained by
substituting (23) and (24) into (22).
2184
(25)
γdk =
as2 ar2
.
ar σr2 + σd2
2
(26)
On the other hand, the received signal at the eavesdropper node
of the kth signal is
IET Commun., 2017, Vol. 11 Iss. 14, pp. 2181-2189
© The Institution of Engineering and Technology 2017
where
Nr /(Ns − Nr) = Tr E WsWsH ,
WrWrH
Nd /(Nr − Nd) = Tr E Q ,
and Nd /(Nr − Nd) = Tr E
[38].
The ergodic secrecy capacity of this scenario can be obtained as
C̄s2 = E Cd − E Ce
+
(31)
E Cd and E Ce are given by (32) and (33), respectively, where
N1 = Nr − Nd + 1, N2 = Nr − Nd + 2, and N = Nr − Nd
E Cd =
1
log 1 + γdk .
2
(32)
(see (33))
Proof: The proof is provided in Appendix 3. □
As for the secrecy outage probability of this scenario, the PDF
of the SINR at the eavesdropper, f γek Φ , in (22) cannot be easily
expressed in a closed-form. However, numerical solution of this
problem does not introduce any computational difficulties and
hence, only simulation results are presented below:
Fig. 5 Ergodic secrecy capacity versus Nd for various values of Pr
4.1 Numerical results
Fig. 6 Ergodic secrecy capacity versus Nr for various values of Ps
yek = a h Wr k xk + a
Nd
∑
i = 1, i ≠ k
h Wr i xi + ar hWr nr + ne
(27)
where a = asar. Therefore, the SINR at the eavesdropper of the kth
message is given by
γek =
N
as ar ∑i =d 1, i ≠ k
2
2
as2 ar2 h Wr
h Wr
i
2
k
2
+ ar ∥ hWr ∥2 σr2 + σe2
2
(28)
Again, by substituting (25) into (2) and (4), we can find the
normalisation constants as
as =
ar =
Ps Ns − Nr
Nr
(Pr /σr2) Nr Nr − Nd
.
(PS /σr2)Nd Ns − Nr + Nr Nd
E Ce =
1
ln 2
−
∫
∞
Nd − 1
0
(1/2) + N1 − (N2 /2)
Nd
IET Commun., 2017, Vol. 11 Iss. 14, pp. 2181-2189
© The Institution of Engineering and Technology 2017
We discuss here the secrecy capacity and secrecy outage
probability of system 2. Fig. 5 presents the ergodic secrecy
capacity versus Nd for Pr = 2, 4, 6, and 8 dBw when Ns = 50,
Nr = 42, Ps = 10 dBw and σr2 = σd2 = σe2 = 10 dBm. From this
figure, we can observe that, the secrecy capacity decreases as the
number of destination nodes increases. However, this decreasing
becomes more significant when Nd is larger than 25. It is also noted
that when Nd approaches Nr, i.e. 42, the ergodic secrecy capacity
seriously deteriorates. The last remark on these results is that
increasing Pr will always result in enhancing the secrecy capacity.
Similar to the previous system, and in order to show the impact of
Nr on this capacity of this system, we plot in Fig. 6 the ergodic
secrecy capacity as a function of Nr for various values of Ps when
Nd = 10, Ns = 45, Pr = 2 dBw and σr2 = σd2 = σe2 = 10 dBm. In
general, it is evident that as Ps increases the secrecy is enhanced
and this is because increasing Ps leads to increase the interference
power at the eavesdropper. It is also worthwhile pointing out that
for each value of Ps there exists an optimal Nr value that will
maximise the ergodic secrecy capacity and therefore the latter
should be carefully chosen. This could be explained by the fact that
increasing Nr will increase the normalisation constant at the relay
ar while decreasing the normalisation constant at the source as.
Therefore, as in the previous system, adding more relay nodes
might not be always beneficial to the system secrecy.
Furthermore, the secrecy outage probability results of system 2
are presented in Fig. 7 when Nr = 10, Nd = 5, Ns = 15, 20, 25,
Pr = 5 dBw, Ps = 6 dBw, and σr2 = σd2 = σe2 = 30 dBm. It is clear
that this probability decreases as the number of the source nodes
increases, since increasing Ns leads to increase the normalisation
constant at the source as.
(29)
5 Scenario 3: ZF at the relay node
(30)
In this scenario, the ergodic secrecy capacity and secrecy outage
probability are analysed when ZF precoders/receivers are deployed
at the relay node only. It is assumed here that the relay node knows
both G1 and G2, and that Nr > Ns and Ns = Nd = N. The weights at
the legitimate nodes of this scenario are [30, 37]
(1/2) + N1 − (N2 /2)
Z (1/2) −1 + N2 J N2 − 1, 2
z N!
Z (1/2) −1 + N2 J N2 − 1, 2 Nd Z
dz .
z N!
Nd − 1 Z
(33)
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Dik, j = {
Nr − N + i − 2 !
Nr − N + i + j − 3 !,
j=k
Nr − N + i − 1 !
Nr − N + i + j − 2 !,
j≠k
(40)
At high SNR, last equation (39) is reduced to
ar =
(Pr /σr2) Nr − Nd
.
(Ps /σr2)
(41)
The ergodic secrecy capacity of this scenario can be obtained as
C̄s3 = E Cd − E Ce
E Cd =
1
2ln 2
∫
0
∞
+
1
1 − e− zt e− zb
z
2 z(1 + Nr − Ns)/2 J 1 + Nr − Ns, 2 z
dz
Γ Nr − Ns + 1
Fig. 7 Secrecy outage probability of system 2
Ws = INs
Wr = G2H G2 G2H
−1
G1 G1H
−1
G1H
(34)
Wd = INd .
Substituting these weights into (9) produces
γdk =
as2 ar2
ar2 G1H G1
−1
2
σ
k, k r
+ σd2
.
(35)
On the other hand, the received signal at the eavesdropper node
of the kth message is given by
yek = a hWr G1 k xk
+a
Nd
∑
i = 1, i ≠ k
hWr G1 i xi + arhWr nr + ne
(36)
where a = as ar. Consequently, the SINR at the eavesdropper node
of the kth transmitted signal can be written as
γek =
N
as2 ar2 hWr G1
as2 ar2 ∑i =d 1, i ≠ k hWr G1
i
2
k
2
+ ar2∥ hWr ∥2 σr2 + σe2
. (37)
Now, by substituting the weights (34) into (2), the normalisation
constants at the source node becomes
as =
Ps
Ns
(38)
where Ns = Tr E WsWsH . Similarly, substituting these weights
into (4) yields [30, 39]
ar =
(Pr /σr2) Nr − Nd Ξ
N
(Ps /σr2) Ξ + Nr − Nd ∑ k = 1det Dk
(39)
N
while Ξ = ∏l = 1 Nr − l ! N − l ! Nr − l ! and Dk is N × N matrix
the elements of which are given by
E Ce =
1
2 ln 2
−
2186
∫
∞
2 Nd − 1
0
N2
1
+ N1 −
2
2
2 Nd
(1/2) + N1 − (N2 /2)
(42)
(43)
(see (44))
E Cd and E Ce are given by (43) and (44), respectively, where
t = as2 /σr2 and b = σd2 /σr2 ar2.
Proof: The proof is provided in Appendix 4. □
Similarly as in the previous scenario, the secrecy outage
probability cannot be expressed in a closed-form. Hence, only
numerical results will be presented in the following section using
software tools.
5.1 Numerical results
Fig. 8 shows the ergodic secrecy capacity as function of Ns and Nd
for Pr = 2, 4, 6, and 8 dBw when Nr = 50, Ps = 10 dBw, and
σr2 = σd2 = σe2 = 10 dBm. As we can see from this result, the ergodic
secrecy capacity deteriorates as Ns and Nd are increased for all
values of Pr. The other observation is that, for given values of Ns
and Nd, increasing Pr results in enhancing the secrecy capacity and
this enhancement becomes less significance when Pr is larger than
6 dBw.
Moreover, the corresponding secrecy outage probability for this
system is shown in Fig. 9 for different Nr values with Ns = Nd = 5,
Ps = 4 dBw, Pr = 3 dBw, and σr2 = σd2 = σe2 = 30 dBm. It is seen that
this probability degrades as Nr is increased and this is because
increasing Nr will generally increase the system diversity in the
first phase, as it is found in [30], in terms of outage probability.
6 Conclusion
In this paper, the ergodic secrecy capacity and the secrecy outage
probability of MIMO AF relay systems in the existence of a
passive eavesdropper are analysed for different scenarios: (i) when
ZF receivers are implemented at the relay and destination nodes,
(ii) when the ZF precoders are implemented at the source and relay
nodes, and (iii) when the ZF precoders/receivers are implemented
at the relay nodes. In each scenario, the ergodic secrecy capacity
and secrecy outage probability are investigated. Results showed
that the number of source, relay and/or destination nodes can
control on both the ergodic secrecy capacity and secrecy outage
probability, based on the ZF design strategy. It was also shown that
the ergodic secrecy capacity and secrecy outage probability can be
further improved by increasing the source and/or the relay transmit
power.
Z (1/2) −1 + N2 J N2 − 1, 2
z N!
Z (1/2) −1 + N2 J N2 − 1, 2 Nd Z
dz
z N!
Nd − 1 Z
(44)
IET Commun., 2017, Vol. 11 Iss. 14, pp. 2181-2189
© The Institution of Engineering and Technology 2017
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
Fig. 8 Ergodic secrecy capacity versus Ns /Nd for various values of Pr
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
Fig. 9 Secrecy outage probability of system 3
7 Acknowledgments
This paper was an extension of the authors' previous work
published in [41]. The authors extend their work by studying the
impact of other parameters on the secrecy capacity and considering
the secrecy outage probability.
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[47]
[48]
[49]
γek =
2
γek =
This appendix derives the ergodic capacities at the legitimate
receiver and eavesdropper nodes in scenario 1.
To begin with, (10) can be rewritten as
γrs
γr X + Y
γdk =
k, k
X = G1H G1
−1
k, k
, and
. Using lemma 1 in [40], the destination ergodic
1
2 ln 2
∫
0
1
ℳR z − ℳ γrs + R z dz
z
ℳR z =
∫
∞
−∞
− zr
e
f R r dr
(47)
while f R r is the PDF of R. Since X and Y are independent
random variables, we can write [42]
ℳR z = ℳγr X z ℳY z
(49)
∫
0
∞
1
1 − e− z γrs ℳR z dz .
z
ℳX z =
∫
0
e− zx
−1/ x
e
dx
Γ N r − N s + 1 X Nr − Ns + 2
(51)
where Γ . is the Gamma function. Using the identities in [42, 43],
we can get
ℳγr X (z) =
2 γr z
(1 + N r − N s)/2
J1 + Nr − Ns 2 γr z
Γ Nr − Ns + 1
(52)
where J . denotes the modified Bessel function of the second
kind. Similarly, the MGF of Y can be found to be
ℳY z =
2 z(1 + Nd − Nr)/2 J1 + Nd − Nr 2 z
.
Γ Nd − Nr + 1
(53)
Substituting (52) and (53) into (48) and then into (50), we find
the destination ergodic capacity. Now to derive the eavesdropper
ergodic capacity, (12) can be written as
2188
0
(55)
i
2
. Using lemma 1 in [40],
1
ℳY z − ℳ X + Y z dz .
z
(56)
− Ns − 1
− Ns
(57)
.
(58)
Now by substituting (57) and (58) into (56), the eavesdropper
ergodic capacity can be found as in (17).
9 Appendix 2
To drive the CDF of SINR at the destination for scenario 1, we first
rewrite SINR at the destination node as
a
bX + cY
γdk =
a = as2 ar2,
Y = G2H G2
−1
k, k
b = ar2σr2,
c = σd2 ,
(59)
X = G1H G1
−1
k, k
,
and
. The CDF of γdk is given by
a
≤z
bX + cY
(60)
= Pr χ a Υ − cz ≤ z bΥ
(50)
By using the PDF of X presented in [30, 37], we can derive the
MGF of X as follows [42]:
∞
∞
Fγdk z = Pr
Therefore, (46) becomes
(54)
as2 ar2 X
as2 ar2 Y
ℳX + Y z = z + 1
where
ℳ γrs + R z = e− z γrsℳR z .
.
Since Y and X + Y both have Gamma distribution, their MGFs
are given by, respectively,
(48)
and
1
2 ln 2
∫
1
2 ln 2
E Ce =
(46)
where R = γr X + Y and ℳR z denotes the moment generating
function (MGF) of R and calculated as
E Cd =
+ ar ∥ hWr ∥2 σr2 + σe2
ℳY z = z + 1
∞
2
2
N
capacity can be expressed as
E Cd =
k
k
where X = h k 2, Y = ∑i =s 1, i ≠ k h
the ergodic eavesdropper capacity is
(45)
where γrs = ar2 as2 /σd2 , γr = ar2 σr2 /σd2 ,
−1
h
2
In interference limited systems, the noise power can be
neglected in comparison with the interference power [44–46],
which represents the worst case scenario and produces the upper
bound of the eavesdropper capacity; hence, (54) becomes
8 Appendix 1
Y = G2H G2
as2 ar2 h
N
as ar ∑i =s 1, i ≠ k
2
(61)
while χ = 1/ X and Υ = 1/Y. We get
Fγdk z =
∫
0
∞
z by
a y − cz
Fχ
f Υ y dy .
(62)
Simply we can rewrite (62) as
Fγdk z = 1 −
∫
∞
cz / a
F̄ χ
z by
a y − cz
f Υ y dy .
(63)
From [37, 47, 48], the CDF of χ and the PDF of Υ are given,
respectively, by
Fχ x =
fΥ y =
γ Nr − Ns + 1, x
Γ Nr − Ns + 1
(64)
yNd − Nre− y
Γ Nd − Nr + 1
(65)
where γ , is incomplete Gamma function. Now, by changing
variables and substituting (64) and (65), we can get the CDF and
then the CCDF of γdkas in (23).
IET Commun., 2017, Vol. 11 Iss. 14, pp. 2181-2189
© The Institution of Engineering and Technology 2017
To find the PDF of the SINR at the eavesdropper, we first
derive the CDF, Fγek, which is defined as
Fγek = Pr γek ≤ Φ .
where N1 = Nr − Nd + 1, N2 = Nr − Nd + 2, and N = Nr − Nd.
Similarly, ℳΥ z is obtained as in
(66)
ℳΥ z
Substituting (55) into (66) we get
Fγek = Pr X ≤ Y Φ .
=
(67)
Fγek = Pr X ≤ Y Φ ∣ Y = y
∫
∞
0
∫
∞
0
Nd − 1 Z
FX yΦ f Y y dy .
By substituting ℳΥ z and ℳ β z into (75), the eavesdropper
ergodic capacity can be found.
(69)
11 Appendix 4
1 − e− yΦ
y Ns − 2 e − y
dy .
Γ Ns − 1
Fγek = 1 − 1 + Φ
1 − Ns
.
(71)
− Ns
1 − Ns
t
γdk =
where
The PDF of γek is found by simply differentiating Fγek, therefore
f γek Φ = − 1 + Φ
To derive the destination ergodic capacity of the third scenario,
(35) can be rewritten as
(70)
Using the identities in [43], (70) can be simplified to
t = as2 /σr2
H
ϕ = G1 G1
−1
k, k
−1
H
G1 G1
k, k
b = σd2 /σr2 ar2.
and
(77)
+b
Now,
by
substitution
and using lemma 1 in [40], the ergodic capacity
at the destination can be expressed as
E Cd =
(72)
1
2ln 2
∫
0
∞
1
1 − e− zt e− zbℳϕ z dz .
z
(78)
In order to find the MGF of ϕ, we can follow the same steps
used to derive ℳX z in Appendix 1. Hence
10 Appendix 3
Since γdk is not random variable, E Cd can be simply expressed as
1
log 1 + γdk .
2
E Cd =
(73)
In order to derive the eavesdropper ergodic capacity in
interference limited systems, which represents the worst-case
scenario and gives the upper bound of the eavesdropper capacity,
we can write (28) as
as2 ar2 X
γek = 2 2
as ar Υ
N
∑i =d 1, i ≠ k
2
1
E Ce =
2 ln 2
∫
0
∞
2
1
ℳΥ z − ℳ β z dz .
z
(75)
1
+ N1 −
N2
2
Z (1/2) −1 + N2 J N2 − 1, 2 Nd Z
N!
IET Commun., 2017, Vol. 11 Iss. 14, pp. 2181-2189
© The Institution of Engineering and Technology 2017
2 z(1 + Nr − Ns)/2 J 1 + Nr − Ns, 2 z
.
Γ Nr − Ns + 1
(79)
Now, to calculate the eavesdropper ergodic capacity in
interference limited systems, which represents the worst scenario
and produces the upper bound of the eavesdropper capacity, (37)
can be simplified as
γek =
where
By using the PDF of the random variable β derived in [49], its
MGF is found to be
2 Nd2
ℳϕ z =
(74)
where X = h Wr k , Υ =
h Wr i , β = X + Υ. Aain,
using lemma 1 in [40], we can express the eavesdropper ergodic
capacity as
ℳβ z =
Z (1/2) −1 + N2 J N2 − 1, 2
N!
(68)
Since X and Y have exponential and Gamma distributions,
respectively, (69) becomes
Fγek =
(1/2) + N1 − (N2 /2)
.
By conditioning on Y, we get
=
2 Nd − 1
as2 ar2 h Wr1
2
as2 ar2 ∑ i = 1, i ≠ k Nd h Wr1
Wr1 = G2H G2 G2H
N
∑i =d 1, i ≠ k
k
−1
.
Let
i
(80)
2
ζ = h Wr 1
k
2
,
2
h Wr1 i and again by using lemma 1 [40], the
ϱ=
eavesdropper ergodic capacity can written as
E Ce =
1
2 ln 2
∫
0
∞
1
ℳϱ z − ℳτ z dz
z
(81)
where τ = ϱ + ζ; ℳϱ z and ℳτ z are exactly identical to ℳΥ z
and ℳ β z derived in Appendix 3, respectively. Substituting these
values in (81), the eavesdropper ergodic capacity for this scenario
can be easily obtained.
(76)
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