Secure Cooperative Network With Multi-Antenna Full-Duplex Receiver

Abstract

In this paper, physical-layer security of a cooperative wireless system with multi-antenna full-duplex (FD) destination is examined. More specifically, we consider a cooperative transmission scenario in which a source communicates with a destination through a relay in the presence of an eavesdropper. In order to degrade the wiretap channel, FD destination simultaneously transmits an interference signal towards the eavesdropper, while receiving information from the relay. A joint transmit and receive beamformer design is proposed to maximize the signal-to-interference-plus-noise ratio (SINR) at the destination, while guaranteeing that the SINR at the eavesdropper is below a certain threshold. We reformulate optimization problem as a semidefinite relaxation problem in terms of a transmit beamforming matrix. To balance the performance and system complexity, we investigate suboptimum beamforming designs based on zero-forcing criterion. We derive exact closed-form expressions for the average secrecy rates of the proposed suboptimum designs. Our results reveal that optimum and suboptimum beamforming schemes along with friendly jamming significantly improve the secrecy performance. Moreover, simulation results demonstrate that optimum scheme outperforms suboptimum schemes, especially in high transmit power regimes. Index Terms-Average secrecy rate, full-duplex (FD), beam-forming design, cooperative relaying system, zero-forcing (ZF).

Full text

1
Secure Cooperative Network with Multi-Antenna
Full-Duplex Receiver
Fatemeh Jafarian, Student Member, IEEE, Zahra Mobini, Member, IEEE
Mohammadali Mohammadi, Member, IEEE
Abstract—In this paper, physical-layer security of a cooperative
wireless system with multi-antenna full-duplex (FD) destination is
examined. More specifically, we consider a cooperative transmis-
sion scenario in which a source communicates with a destination
through a relay in the presence of an eavesdropper. In order
to degrade the wiretap channel, FD destination simultaneously
transmits an interference signal towards the eavesdropper, while
receiving information from the relay. A joint transmit and
receive beamformer design is proposed to maximize the signal-
to-interference-plus-noise ratio (SINR) at the destination, while
guaranteeing that the SINR at the eavesdropper is below a certain
threshold. We reformulate optimization problem as a semidefinite
relaxation problem in terms of a transmit beamforming matrix.
To balance the performance and system complexity, we inves-
tigate suboptimum beamforming designs based on zero-forcing
criterion. We derive exact closed-form expressions for the average
secrecy rates of the proposed suboptimum designs. Our results
reveal that optimum and suboptimum beamforming schemes
along with friendly jamming significantly improve the secrecy
performance. Moreover, simulation results demonstrate that
optimum scheme outperforms suboptimum schemes, especially
in high transmit power regimes.
Index Terms—Average secrecy rate, full-duplex (FD), beam-
forming design, cooperative relaying system, zero-forcing (ZF).
I. INT ROD UC TI ON
Wireless communication technology has entered almost
every aspect of social life. Smart grids, sensor and cellular
networks, and smart homes are just a few examples of the
wireless systems. Despite their well-known applications and
benefits, the broadcast nature of wireless networks makes
security an ongoing important issue due to the fact that the
information signals can be overheard by both specific users
and eavesdropper nodes [1], [2]. From an information theory
perspective, a secure wireless connection can be realized utiliz-
ing channel features such as fading, noise, and interference [3].
To this end, physical-layer security has been recognized as
a promising approach to enhance the secrecy rate which is
defined as the difference between the instantaneous secrecy
rate of the legitimate channel and that of the eavesdropping
channel [3].
To date, several physical-layer techniques have been intro-
duced such as multiple-input multiple-output (MIMO) trans-
mission, cooperative relaying, and cooperative jamming [1].
It is shown that MIMO transmission can notably improve the
secrecy rate [4], [5] and secrecy outage probability [6]. On one
F. Jafarian, Z. Mobini, and M. Mohammadi are with the Faculty of
Engineering, Shahrekord University, Shahrekord 115, Iran (email: Jafar-
ian@stu.sku.ac.ir and {z.mobini, m.a.mohammadi}@eng.sku.ac.ir).
hand, MIMO technique increases the power of the legitimate
signal due to diversity and on the other hand, it reduces
the amount of overheard signals at the eavesdroppers [7],
[8]. In particular, MIMO transmissions enable beamforming
designs to improve the legitimate channel or degrade the
wiretap channel [9], [10]. There has been a growing body
of research that studies the secure MIMO systems [11]–[14].
In [11] beamforming designs for amplify-and-forward relaying
protocol in two-hop secure communications are provided.
The optimal power allocation strategy that maximizes the
secrecy rate achievable by beamforming in MIMO multiple-
eavesdropper Rayleigh-fading wiretap channels is proposed
in [12]. Many works have also investigated the security of
wireless MIMO cooperative networks in specific application
scenarios such as simultaneous wireless information and power
transfer [13], [14]. Secrecy performance of cooperative relay-
ing with optimal relay selection for different cases of channel
state information (CSI) availability is also investigated in [15].
Another effective way to increase the secrecy rate in
wireless communication systems is to weaken the ability of
eavesdropper nodes to decode the confidential information by
applying controlled interference or an artificial noise [16],
[17], which is known as cooperative jamming. In [18], the
performance of secure wireless communication with different
cooperative schemes including decode-and-forward, amplify-
and-forward, and cooperative jamming has been investigated.
The optimal beamforming scheme with the transmission of
an artificial noise to minimize the secrecy outage probability
has been proposed in [19], [20]. The work in [21] has
studied the cooperative jamming schemes for two-hop relay
networks where inactive nodes in the relay network are used
as cooperative jamming sources to compromise the eaves-
dropper. Physical layer security of a two-way untrusted relay
system, where the two sources exchange information through
an untrusted intermediate relay, using friendly jamming has
been studied in [22]. The secure systems with cooperative
interference signal transmission, however, mainly rely on
external nodes, and hence they may suffer from helpful nodes
mobility, coordination and trust. To overcome this problem, in
some new studies the receiver or destination are considered to
act as a jammer [23], [24].
On a parallel development, recent advances in electronics,
antenna technology, and signal processing provide a full-
duplex (FD) operational mode for nodes, so that they can
transmit and receive information simultaneously [25]. The
potential benefits offered by the application of FD technique
can be exploited in recent wireless networks to satisfy high

security requirements. The authors in [26] have investigated
physical-layer security of a two-tier decentralized wireless
network using FD jamming techniques. The work in [27] has
considered FD communications in a multi-user multiple-input
single-output wireless non-cooperative network with receiver-
based friendly jamming. In [28] jammer selection and FD
users were used in heterogeneous cellular networks and signifi-
cant improvements of the system security was shown. Antenna
selection and power allocation for a wiretap channel with
multiple-antenna FD destination have been studied in [29].
In [30], secrecy rate expressions for a wireless network with
joint wireless-powered FD relay transmission and friendly
jamming have been developed.
Motivated by all above, in this paper, the advantage of using
MIMO FD destination in a cooperative system including a
source and a relay in the presence of an active eavesdropper
is examined. The destination node receives information from
the decode-and-forward relay while simultaneously sends a
jamming signal to the eavesdropper. In an FD wireless sys-
tem, the interference signal also affects the receiver circuits
of an FD node, which is called self-interference (SI) [31].
Accordingly, we suggest transmit and receive beamforming
designs to completely remove SI in the destination node and
hence, our system is not interference limited. We note that
an active eavesdropper plays dual roles in the network, i.e.,
it works as an eavesdropper for some signals and legitimate
user for others [18]. Active eavesdropper exchanges signaling
messages with the nodes in the network and its CSI is
available.
Two potential application scenarios for our proposed secure
wireless relaying system are 1) a secure remote monitoring
system using cooperative relaying where a monitor sends
confidential messages to the FD multiple antenna monitoring
center, and 2) a secure wireless sensor network where sensing
data from the remote sensors are collected by a destination
with the help of a relay node while an eavesdropper tries
to intercept the transmitted sensing information. Using FD
operation and multiple antenna transmission, the destination
can significantly degrade the relay-eavesdropper link by trans-
mitting artificial noise when it receives the information from
the relay.
The main contributions of this paper can be outlined as
follows:
•Different suboptimal transmit and receive beamformer
vectors are designed based on zero-forcing (ZF) crite-
rion, namely transmit zero forcing (TZF), and receive
zero forcing (RZF) to maximize the received signal-to-
interference-plus-noise ratio (SINR) at the destination
such that the SI is completely cancelled out. An optimum
receiver and transmit destination beamformer scheme
using semidefinite relaxation (SDR) approach is also
proposed, where the aim is maximizing the SINR at
the destination, while guaranteeing that the SINR at the
eavesdropper is below a certain threshold. We observe
that optimum scheme outperforms suboptimal schemes
especially for high transmit power. Further, the secrecy
performance gap between the optimum and suboptimal
schemes decreases with increasing the antenna numbers.
ŚŚ
ŚŚ
Information signal
Jamming signal
Source Relay
Destination
Eavesdropper
,
EE
^Z
Fig. 1. Considered secure relay system with multi-antenna FD destination.
•Average secrecy rates of the system for different subopti-
mal beamforming schemes are derived. For comparison,
the results of maximum ratio combining (MRC) and max-
imal ratio transmission (MRT) beamformer at the desti-
nation input and output are also presented. We observe
that for TZF and RZF schemes, the average secrecy rate
increases significantly with the increase in the number of
transmit and receive antennas. Simulation results suggest
that when the distance between eavesdropper and relay is
low to moderate, MRC/MRT scheme outperforms other
suboptimal schemes. However, as this distance increases,
TZF outperforms MRC/MRT.
Notation: We use bold upper case letters to denote matrices,
bold lower case letters to denote vectors. The superscripts (·)T,
(·)∗,(·)†, and (·)−1stand for transpose, conjugate, conjugate
transpose, and matrix inverse, respectively; the Euclidean norm
of the vector is denoted by ∥ · ∥; the trace and the rank
of a matrix are denoted by tr(·)and rank(·), respectively;
Pr(·)denotes the probability; E{·},fX(·)and FX(·)denote
the expectation, the probability density function (pdf), and
cumulative distribution function (cdf) of the random variable
(RV) X, respectively; and CN(µ, σ2)denotes a circularly
symmetric complex Gaussian RV Xwith mean µand variance
σ2;Γ(a)is the Gamma function; γ(a, x)and Γ(a, x)are
upper incomplete and lower incomplete Gamma function [32,
Eq. (8.350)], and Ei(x)is the exponential integral [32, Eq.
(8.211.1)].
II. SYSTEM MO DE L
Consider a secure relaying system with one source (S),
one decode-and-forward relay (R), one FD destination (D),
and one active eavesdropper (E) as shown in Fig. 1. We
assume a single-antenna relaying system with multiple antenna
destination which is equipped with NRantennas for reception
and NTantennas for transmission. This model with the single
antenna source, relay, and eavesdropper nodes1facilities sys-
tem analysis and the derived expressions are useful to obtain
design insights.
1The performance analysis of various transmission schemes under a multi-
antenna source, relay, and eavesdropper nodes implementation is an interesting
direction for future research.

It is assumed that the Dand Eare located far away from
the S. Due to the long distances between the Sand Dand E,
there is a higher probability that these links suffer from strong
blockage and shadowing, compared to the links between the S
and R, and hence there is no direct link from the Sto the D
or to E[33]. Therefore, Scommunicates with Dwith the aid
of R, while Etries to overhear the transmitted message from
R. On the other hand, the FD destination transmits jamming
signal to create interference at E, while receiving the signals
from the R[29].
All links are modeled as block Rayleigh fading channels,
i.e., the channel remains static for one coherence interval and
changes independently in different intervals. Moreover, we
assume that the links experience both large-scale path loss and
small-scale fading. For the large-scale path loss, we assume
the path loss model d−µ
i,j where µ > 2is the path loss exponent
and di,j is the distance between the transmitter iand receiver
j, with i∈ {S, R, D}and j∈ {R, D, E }. The small-scale
fading for S-Rand R-Elink are denoted as hSR and hRE ,
respectively. Moreover, the NR×1small-scale fading vector
of the R-Dlink and the 1×NTsmall-scale fading vector
of the D-Elink are denoted as hRD and hDE , respectively.
Furthermore, similar to [25], [34], [35], we model the elements
of the NR×NTSI channel at the FD destination, HSI, as
independent identically distributed (i.i.d) CN (0, σ 2
SI)RVs. We
also assume that perfect CSI is available at the destination2.
A. Transmission Protocol
We adopt a two-hop cooperative transmission protocol be-
tween Sand Das follows:
During the first hop, Stransmits its signal xswith
E|xs[n]|2= 1 to R, and Rreceives
yR[n] = ps
dµ
SR
hSR [n]xs[n] + nR[n],(1)
where psis the source transmit power, and nRis the additive
white Gaussian noise (AWGN) at Rwith EnRn†
R=σ2
R.
The relay first decodes received signal and then, at the second
hop, Rtransmits xR[n] = √prˆxs[n], where pris the relay
transmit power and ˆxs[n]is the estimate of the source signal at
the R. At the same time, the FD destination transmits jamming
signal xJ[n], with E|xJ[n]|2= 1 while receiving signal
from R. Therefore, the received signals at Dand Ecan be
expressed as
yD[n]=pr
dµ
RD
w†
rhRD ˆxs[n]+√pdw†
rHSIwtxJ[n]+w†
rnD[n],
(2)
and
yE[n] = pr
dµ
RE
hRE ˆxs[n] + pd
dµ
DE
hDE wtxJ[n] + nE[n],
(3)
2It would be interesting to extend the beamforming designs and analytical
results to the practical imperfect CSI case and also under the passive
eavesdropping where the CSI of the eavesdropper’s channel is not available
at the destination node.
respectively, where wrand wtare NR×1receive beamformer
vector and NT×1transmit beamformer vector at D, respec-
tively, pdis the transmit power at D,nD[n]is the NR×1
AWGN vector at Dwith EnDn†
D=σ2
DINRand nE[n]is
the AWGN at Ewith EnEn†
E=σ2
E. Therefore, the SINR
at Dcan be written as
γD= min 
ps|hSR |2
dµ
SR σ2
R
,
pr
dµ
RD |w†
rhRD|2
pd|w†
rHSIwt|2+σ2
D
.(4)
Moreover, the received SINR at Ecan be expressed as
γE=
pr
dµ
RE |hRE |2
pd
dµ
DE |hDE wt|2+σ2
E
.(5)
III. RECEIVE AND TRANSMIT BEAMFORMER DESIGN
From (4) and (5), it is evident that the received SINRs
at the destination and eavesdropper are dependent on the
beamforming design at the destination. Therefore, in this
section, we study the optimum and suboptimum destination
beamforming designs.
A. Optimum Beamformer Design
With optimum beamformer design, we aim to maximize the
received SINR at the destination, while guaranteeing that the
SINR at the eavesdropper is smaller than a certain threshold.
In this case, the optimization problem can be mathematically
formulated as
max
wr,wt
γD(wr,wt,)
s.t. γE(wt)≤γth
∥wr∥=∥wt∥= 1.(6)
By invoking (4) and (5), the optimization problem (6) can be
written as
max
wr,wt
min 
ps|hSR |2
dµ
SR σ2
R
,
pr
dµ
RD |wr†hRD|2
pd|w†
rHSIwt|2+σ2
D

s.t.
pr
dµ
RE |hRE |2
pd
dµ
DE |hDE wt|2+σ2
E≤γth
∥wr∥=∥wt∥= 1.(7)
We observe that only the second term inside the min operator
in (7) depends on wr. Accordingly, for a given wt, the
optimum receive beamformer, wr, is obtained by maximizing
|wr†hRD|2
|w†
rHSIwt|2pd+σ2
D
. This means that wris the solution of the
generalized Rayleigh quotient
max
∥wr∥=1
w†
rhRDh†
RDwr
w†
rAwr
,(8)
where A=pdHSIwtw†
tH†
SI +σ2
DI. Therefore, the optimum
wrcan be derived as
wr=A−1hRD
∥A−1hRD∥.(9)

By substituting wrinto γDand using Sherman-Morrison
formula, γDcan be written as
γD=pr
dµ
RD
h†
RDpdHSI wtw†
tH†
SI +B−1
hRD
=pr
dµ
RD h†
RDB−1hRD −pd|h†
RDB−1HSI wt|2
(1 + pdw†
tH†
SIB−1HSI wt)
=pr
dµ
RD h†
RDB−1hRD −pd|h†
RDB−1HSIwt|2
w†
t(I+pdH†
SIB−1HSI )wt,
(10)
where B=σ2
DI. To this end, the optimization problem (7)
can be re-written as
max
∥wt∥=1 
pr
σ2
Ddµ
RD ∥hRD∥2−
pr
dµ
RD
pd|h†
RDB−1HSI wt|2
w†
t(I+pdH†
SIB−1HSI )wt

s.t. hDE wtw†
th†
DE ≥q, (11)
where q=pr|hRE |2dµ
DE
dµ
RE γthpd−σ2
Edµ
DE
pd. If we let
pd|h†
RDB−1HSI wt|2
w†
t(I+pdH†
SIB−1HSI )wt
=pdw†
tH†
SIB−1hRD h†
RDB−1HSI wt
and define wt,wt=F1
2wt, where F=(I+pdH†
SIB−1HSI ),
then (11) can be expressed as
min
∥wt∥=1 F−1
2w†
tH†
SIB−1hRD h†
RDB−1HSI F−1
2wt
s.t. hDE F−1
2wtF−1
2w†
th†
DE ≥q. (12)
The optimization problem (12) does not have a closed-form
solution and also it has not convex form. By defining variable
Wt,wtw†
tand relaxing the rank-one constraint of Wt, the
problem (12) can be expressed as the following SDR problem
min
∥wt∥=1 trWtF−1
2H†
SIB−1hRD h†
RDB−1HSI F−1
2
s.t. trhDE F−1
2WtF−1
2h†
DE ≥q
trWt= 1,
Wt≥0.(13)
The problem (13) has standard form and also has only
two inequalities which can be efficiently solved using CVX
software [36]. There is a rank-one optimum solution of Wt
in accordance with Shapiro-Barvinok-Pataki rank reduction
proposition [37], and hence the optimum wtof the original
problem is the eigenvector depending on the non-zero eigen-
value of Wt.
B. Suboptimum Beamforming Design
The optimal design maximizes the received SINR at desti-
nation while guaranteeing that the SINR at the eavesdropper
is smaller than a certain threshold. However, the optimal
design has a high computational and implementation com-
plexity. To further reduce system complexity, here, we present
different suboptimal beamforming designs based on ZF self-
interference suppression, namely TZF and RZF. Moreover, we
consider the MRC/MRT scheme as a benchmark. MRT/MRC
beamformer is suitable for low complexity FD systems as
it does not need to estimate the SI channel [35]. Moreover,
MRT/MRC beamformer is preferred for half-duplex operation,
and hence it is interesting to characterize its performance in
the FD case.
1) TZF Scheme: We assume destination uses several trans-
mitting antennas to completely remove the SI. Moreover,
MRC is used at the destination input, so that the receive
beamformer vector is equal to wr=hRD
∥hRD∥. The optimum
transmit beamforming vector wtis obtained by solving the
following optimization problem
max |hDE wt|2
s.t. h†
RDHSIwt= 0.(14)
The transmit beamforming vector for the TZF scheme is ob-
tained as wZF
t=Bh†
DE
∥Bh†
DE ∥, where B=I−H†
SIhRD h†
RDHSI
∥h†
RDHSI ∥2[35].
By replacing wrand wZF
tin (4) and (5), the SINR at
destination and eavesdropper are obtained as
γTZF
D= min ps|hSR |2
dµ
SR σ2
R
,pr∥hRD∥2
dµ
RDσ2
D,(15)
and
γTZF
E=
pr
dµ
RE |hRE |2
pd
dµ
DE |hDE wZF
t|2+σ2
E
,(16)
respectively.
2) RZF Scheme: As an alternative solution, the transmit
beamforming vector can be set using the MRT principle, i.e.,
wt=h†
DE
∥hDE ∥. Moreover, the optimum receive beamforming
vector wris designed with the ZF criterion w†
rHSIh†
DE = 0.
To ensure the feasibility of the RZF, destination should be
equipped with NR>1antennas. Substituting wtinto (4) the
optimal receive beamforming vector wris the solution of the
following problem:
max |w†
rhRD|2
s.t. w†
rHSIh†
DE = 0.(17)
The receive beamforming vector can be obtained as wZF
r=
DhRD
∥DhRD∥, where D≃I−HSI h†
DE hDE H†
SI
∥hDE HSI∥2. By replacing wt
and wZF
rin (4) and (5), the SINR at the destination and
eavesdropper can be expressed as
γRZF
D= min 
ps|hSR |2
dµ
SR σ2
R
,pr|wZF
r
†hRD|2
dµ
RDσ2
D
,(18)
and
γRZF
E=
pr
dµ
RE |hRE |2
pd
dµ
DE ∥hDE ∥2+σ2
E
,(19)
respectively.
3) MRC/MRT Scheme: In the MRC/MRT scheme, wrand
wtare obtained as
wMRC
r=hRD
∥hRD∥,wMRT
t=h†
DE
∥hDE ∥.(20)
It is worth mentioning that in the absence of SI, TZF and
RZF schemes have the same performance with the MRC/MRT
scheme. But the MRC/MRT scheme does not have the optimal

¯
RTZF
0=1
ln 2
NR−1

m=0
1
(2a1)mm!(−1)m−1e1
a0+1
2a1Ei −1
a0−1
2a1+
m

k=1
(k−1)!(−1)m−k1
a0
+1
2a1−k
+(−1)meϱEi (−ϱ)
−
m

k=1
(k−1)!(−1)m−k(ϱ)
−k−
NT−2

k2=0
B
a2
eϱEi (−ϱ) +
NT−2

k2=0
k2+1

i=1
Ai
a2
(ϱ)i−1a2
a3i
Γi−1,a2
a3
ϱ,(28)
performance in the presence of SI, and it is suitable for half-
duplex or FD systems with low SI.
By substituting wMRC
rand wMRT
tinto (4) and (5), the SINR
at the destination and eavesdropper nodes are, respectively,
given by
γMRC
D= min 
ps|hSR |2
dµ
SR σ2
R
,
pr
dµ
RD ∥hRD∥2
pd|w†
rHSIwt|2+σ2
D
,(21)
and
γMRC
E=
pr
dµ
RE |hRE |2
pd
dµ
DE ∥hDE ∥2+σ2
E
.(22)
IV. AVE RAG E SEC RE CY RATE ANALYS IS
In this section, we study the secrecy rate performance
which is a fundamental metric to evaluate the security of
wireless communications under active eavesdropping. For a
given channel realization, the instantaneous secrecy rate of
different beamforming designs is given by [2]
Ri
0=⌊Ci
D−Ci
E⌋+,(23)
where ⌊x⌋+= max(x, 0),i∈ {TZF,RZF,MRC},Ci
D=
log2(1 + γi
D), and Ci
E= log2(1 + γi
E). The instantaneous
secrecy rate is different for various channel realizations and
hence to study the security in a long-term basis, average
secrecy rate can be considered as another performance cri-
terion [2] as
¯
Ri
0=E{⌊Ci
D−Ci
E⌋+}.(24)
The average secrecy rate in (24) can be written as [30]
¯
Ri
o=1
ln 2 ∞
0
Fγi
E(x)
1 + x1−Fγi
D(x)dx. (25)
Therefore, the remaining key task is to characterize the exact
cdf of the received SINR at the destination and eavesdropper.
In the sequel, we derive the cdfs Fγi
Dand Fγi
Efor the
considered TZF, RZF, and MRC/MRT schemes. Derivation
of the corresponding cdfs of the optimum scheme is difficult.
Therefore, we have resorted to simulations for evaluating the
average secrecy rate of the optimum scheme in Section V.
The following proposition presents the exact expressions for
the cdfs of the γDand γEfor the TZF scheme.
Proposition 1. The cdfs of γDand γEfor the TZF scheme
can be derived as
FγTZF
D(x) = 1 −e−x
a0
Γ(NR,x
2a1)
Γ(NR),(26)
and
FγTZF
E(x) = 1 −e−x
a21−a3
a2
x
NT−2

k2=0 1 + a3
a2
x−k2−1,
(27)
respectively, where a0,ps
dµ
SR σ2
R
,a1,pr
dµ
RDσ2
D
,a2,pr
dµ
RE σ2
E
,
and a3,pd
dµ
DE σ2
E
.
Proof: See Appendix A.
From (26), it is observed that FγTZF
D(x)decreases with the
increase in the number of received antennas. Also, we see that
FγTZF
E(x)increases with the destination power. It is intuitive
because increasing the destination power improves the jam-
ming signal and reduces the received SINR at eavesdropper.
For the average secrecy rate of the TZF scheme, we have
the following key result:
Proposition 2. The average secrecy rate achieved by the
TZF scheme for NR> NT−1is given by (28) at the top
of the page where Ai=a2
a3(k2+1−i)!
∂(k2+1−i)
∂x(k2+1−i)
xm+1
(1+x)|x=−a2
a3
,
B= (−1)m+1(1 −a3
a2)−k2−1, and ϱ=1
a0+1
2a1+1
a2.
Proof: See Appendix B.
Proposition 3. The cdfs of γDand γEfor the RZF scheme
can be expressed as
FγRZF
D(x) = 1 −e−x
a0
Γ(NR−1,x
2a1)
Γ(NR−1) ,(29)
and
FγRZF
E(x) = 1 −e−x
a21−a3
a2
x
NT−1

k2=0 1 + a3
a2
x−k2−1,
(30)
respectively.
Proof: The proof is similar to Proposition 1 and thus
omitted.
From Proposition 1 and Proposition 3 we observe that
γRZF
Dand γRZF
Ehave the same statistics as γTZF
Dand γTZF
E,
respectively. Therefore, the exact average secrecy rate of the
RZF scheme can be obtained from (28) by replacing (NT−2)
and NRwith (NT−1) and (NR−1), respectively. We omit
the final result for the sake of brevity.
For the MRC/MRT scheme, we have the following key
result:
Proposition 4. The cdfs of γDand γEfor the MRC/MRT

0 5 10 15 20 25 30 35 40
ps = pr = p (dBm)
0
0.5
1
1.5
2
2.5
3
3.5
4
Average Secrecy Rate (bps/HZ)
Simulation, TZF, [NR=3, NT=3]
Simulation, RZF, [NR=3, NT=3]
Simulation ,MRC, [NR=3, NT=3]
Simulation , TZF, [NR=2, NT=3]
Simulation ,RZF, [NR=3, NT=2]
Simulation ,MRC, [NR=3, NT=2]
Simulation, MRC, [NR=2, NT=3]
Simulation ,TZF, [NR=3, NT=2]
Simulation , RZF, [NR=2, NT=3]
Simulation , MRC, [NR=2, NT=2]
Simulation , TZF, [NR=2, NT=2]
Simulation , RZF, [NR=2, NT=2]
Analytical
Fig. 2. Average secrecy rate of different suboptimal beamforming schemes
versus pfor different antenna configurations.
scheme can be expressed as
FγMRC
D(x)=1−e−x
a0
Γ(NR)

ΓNR,x
a1−e1
c2
ΓNR,1
c2+x
a1
1 + a1
c2xNR

,
(31)
and
FγMRC
E(x) = 1 −e−x
a21−a3
a2
x
NT−1

k2=0 1 + a3
a2
x−k2−1,
(32)
respectively, where c2,pd
σ2
SI
σ2
D
.
Proof: See Appendix C.
To this end, the average secrecy rate of the MRC/MRT
scheme can be obtained by substituting (31) and (32) into (25).
Remark 1. It is interesting to investigate the secrecy rate
performance of the considered system when the number of
transmitting and receiving antennas is large. Using the law
of large numbers [38], we can represent that when NT→
∞and NR→ ∞, the received SINR at the destination for
TZF, RZF, and MRC/MRT beamforming schemes is given by
γD≃ps|hSR |2
dµ
SR σ2
R
. Also, the received SINR at the eavesdropper
approaches to zero, i.e., γE≃0. Thus, FγE(x)=1and
FγD(x) = 1 −e−x
a0. Accordingly, the average secrecy rate
can be written as
¯
Ro=1
ln 2 ∞
0
e−x
a0
1 + xdx. (33)
With the help of integral identity [32, Eq. (3.354,4)], ¯
Rois
obtained as
¯
Ro=−1
ln 2 e−1
a0Ei−1
a0.(34)
Remark 2. From Propositions 1-4, it is observed that the
average secrecy rate of MRC/MRT scheme depends on the SI
strength, σ2
SI, while ¯
RTZF
oand ¯
RRZF
oare independent of the
σ2
SI.
5 10 15 20 25 30 35 40
pd (dBm)
0
0.5
1
1.5
2
2.5
Average Secrecy Rate (bps/HZ)
Simulation, TZF, [NR=4, NT=2]
Simulation, RZF, [NR=4, NT=2]
Simulation, MRC, [NR=4, NT=2]
Simulation, TZF, [NR=2, NT=4]
Simulation, MRC, [NR=2, NT=4]
Simulation, RZF, [NR=2, NT=4]
Analytical
Fig. 3. Average secrecy rate of different beamforming schemes versus
destination transmit power.
V. NUMERICAL RE SU LTS AND DISCUSSION
In this section, the performance of the proposed secure
cooperative system is evaluated and the accuracy of the
theoretical results is verified. Unless otherwise stated, the noise
variance σ2
D=σ2
E=σ2
R=−90 dBm and σ2
SI =−60 dBm.
The position of source, destination, relay and eavesdropper are
(0,0),(400m,0),(100m,0), and (120m,50m), respectively.
Moreover, the path-loss coefficient µis set to 3.
In Fig. 2, the average secrecy rate for TZF, RZF, and
MRC/MRT beamforming schemes is plotted versus the trans-
mit power ps=pr=p, where the analytical curves are
based on Propositions 1-4. The solid curves are used to denote
the analytical results, and the markers without line show the
simulation results. As shown, the average secrecy rate for all
schemes increases with increasing power. By comparison, it
can be observed that TZF (NR= 2, NT= 3) and RZF
(NR= 3, NT= 2) have the same performance, which is
in line with the results derived in Section IV. For both TZF
and RZF schemes, average secrecy rate increases by growing
the number of transmit and receive antennas. However, the
effect of increasing the number of receiving antennas is more
prominent.
Fig. 3 depicts the average secrecy rate of TZF, RZF, and
MRC/MRT beamforming schemes versus destination power.
As shown, in the RZF and TZF schemes, the average secrecy
rate improves with increasing the destination power. However,
in the MRC/MRT scheme, the average secrecy rate first
increases and then it remains constant. In other words, the
average secrecy rate for the MRC/MRT scheme shows a
floor at high transmit power. This is intuitive since with high
destination power the SI will be maximal which reduces the
SINR at the destination. As it can be observed from Fig. 2
and Fig. 3, there is a close match between the analytical and
simulation curves which verifies the accuracy of the analytical
results derived in this paper.
In Fig. 4, the average secrecy rate is plotted versus the
eavesdropper location where the y-coordinate of the eaves-
dropper is fixed as 50m and its x-coordinate is within the range
of [150m, 400m]. As expected, as eavesdropper gets closer to

150 200 250 300 350 400
x-Coordinate of the eavesdropper
1
1.5
2
2.5
3
3.5
4
Average Secrecy Rate (bps/HZ)
TZF, [NR=2, NT=3]
MRC, [NR=2, NT=3]
RZF, [NR=2, NT=3]
Fig. 4. Average secrecy rate versus x-coordinates of the eavesdropper.
5 10 15 20 25 30 35 40
ps = pr = p (dBm)
0
0.5
1
1.5
2
2.5
3
Average Secrecy Rate (bps/HZ)
Optimum, [ NR=4, NT=2]
TZF, [NR=4, NT=2]
Optimum, [ NR=2, NT=2]
TZF, [NR=2, NT=2]
Fig. 5. Average secrecy rate comparison between the optimum and TZF
beamforming scheme for different antenna numbers at the destination.
the destination, average secrecy rate of all schemes increase. In
addition, when the distance between eavesdropper and relay is
low, MRC/MRT scheme outperforms other schemes. However,
as this distance increases TZF has the best performance.
Fig. 5 compares the average secrecy rate due to optimum
and TZF beamforming schemes. We observe that optimum
scheme outperforms TZF scheme especially for high transmit
power. Further, the secrecy performance gap between optimum
and TZF schemes decreases with increasing the number of
antennas.
In Fig. 6, we examine the effect of SI strength on the average
secrecy rate for the optimum and MRC/MRT schemes. As
expected SI does not affect the optimum scheme, hence its
average secrecy rate remains constant. When the SI strength
is low, MRC/MRT and optimum schemes have the same per-
formance, while as SI strength increases, the average secrecy
rate of the MRC/MRT scheme significantly decreases.
VI. CONCLUSION
We have considered physical-layer security of a cooperative
wireless system with a multi-antenna FD destination which
simultaneously receives information from relay and transmits
a jamming signal to confound the eavesdropper. We designed
-90 -80 -70 -60 -50 -40 -30
2
SI (dBm)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Average Secrecy Rate(b/s/HZ)
Optimum, [NR=3, NT=4]
MRC, [NR=3, NT=4]
Optimum, [NR=2, NT=4]
MRC, [NR=2, NT=4]
Fig. 6. Average secrecy rate versus σ2
SI for optimum and MRC/MRT
beamforming schemes.
optimum transmit and receive beamformer at the destination
to maximize the SINR at the destination, while guaranteeing
that the SINR at the eavesdropper is below a certain thresh-
old. Moreover, ZF-based beamforming designs have been
proposed to completely cancel the SI, which can balance
the performance and system complexity. We presented exact
closed-form expressions for the average secrecy rate of the
ZF-based beamforming designs. Our results suggest that the
average secrecy rate can be significantly improved via multi-
antenna FD receiver with optimum and ZF-based beamforming
designs. Moreover, when the distance between eavesdropper
and relay is small, MRC/MRT scheme outperforms other ZF-
based beamforming schemes.
APPENDIX A
PROO F OF PROPOSITION 1
By invoking γTZF
Din (15), FγTZF
D(x)can be expressed as
FγTZF
D(x) = Prmin ps|hSR |2
dµ
SR σ2
R
,pr∥hRD∥2
dµ
RDσ2
D< x.(35)
For the notational convenience, let us denote Z1,|hSR|2,
and Y1,∥hRD∥2. Hence, The cdf of γTZF
Din (35) can be
written as
FγTZF
D(x) = Pr (min(a0Z1, a1Y1)< x)
= 1 −1−PrZ1<x
a01−PrY1<x
a1
= 1 −1−FZ1x
a01−FY1x
a1.(36)
We note that Z1is an exponential distributed RV and
Y1is a Chi-square distributed RV with 2NRdegrees-of-
freedom. Therefore, by substituting FZ1(x
a0)=1−e−x
a0and
FY1(x
a1)=Γ(NR,x
2a1)
Γ(NR)into (36), FγTZF
D(x)can be obtained as
in (26).
Furthermore, by invoking (16), FγTZF
E(x)can be expressed

¯
RT ZF
o=1
ln 2 ∞
0
e−x
a0
Γ(NR,x
2a1)
Γ(NR)(1 + x)dx
  
I1
−1
ln 2 ∞
0
e−x(1
a0+1
a2)Γ(NR,x
2a1)
Γ(NR)(1 + x)dx
  
I2
+1
ln 2
a3
a2∞
0xe−x
a2NT−2
k2=0 (1 + a3
a2x)−k2−1
1 + xe−x
a0
Γ(NR,x
2a1)
Γ(NR)dx
  
I3
.(39)
as
FγTZF
E(x) = Pr(γTZF
E< x)
= 1 −∞
x
a2
FY2a2z2−x
a3xfZ2(z2)dz2,(37)
where Y2,|hDE wZF
t|2, and Z2,|hRE |2. Note that
Z2is exponentially distributed RV and Y2has Chi-square
distribution with 2(NT−1) degrees-of-freedom. Therefore,
by substituting fZ2(z2)=1
σ2
R
e−z2
σ2
Rand FY2(a2z2−x
a3x)=1−
e−a2z2−x
a3xσ2
ENT−2
k2=0 (1
k2!)(a2z2−x
a3xσ2
E
)k2into (37), FγTZF
Ecan be ex-
pressed as
FγTZF
E(x)= 1−e−x
σ2
Ra2+
NT−2

k2=0
1
k2!σ2
R
e
1
a3σ2
Ea2
a3σ2
Exk2
×∞
x
a2
e−z2a2
a3xσ2
E
+1
σ2
Rz2−x
a2k2
dz2.(38)
To this end, with the help of integral identity [32, Eq.
(3.382,2)], the desired result in (27) is obtained, which com-
pletes the proof.
APPENDIX B
PROO F OF PROPOSITION 2
By substituting FγTZF
D(x)and FγTZF
E(x)into (25), the aver-
age secrecy rate of the TZF can be expressed as (39) at the
top of the page.
For the integral terms in (39) we can obtain closed-
form solutions as follows. Applying the series expansion of
Γ(a, x)[32, Eq. (8.352.4)] we have
I1=1
ln 2
NR−1

m=0
1
(2a1)mm!∞
0
xme−x(1
a0+1
2a1)
(1 + x)dx
=1
ln 2
NR−1

m=0
1
(2a1)mm!(−1)m−1e1
a0+1
2a1Ei
−1
a0−1
2a1
+
m

k=1
(k−1)!(−1)m−k1
a0
+1
2a1−k,(40)
where we have used the relationship [32, Eq. (3.353.5)].
Similarly, I2can be evaluated as
I2=
NR−1

m=0
1
(2a1)mm! ln 2 (−1)m−1eϱEi (−ϱ)
+
m

k=1
(k−1)!(−1)m−kϱ−k,(41)
where ϱ=1
a0+1
2a1+1
a2. Now, we turn our attention to I3.
By applying the series expansion of Γ(a, x)[32, Eq. (8.352.4)]
and then using partial fractions, we can write
I3=1
ln 2
NR−1

m=0
NT−2

k2=0
1
a2
1
(2a1)mm!
×∞
0
e−ϱxk2+1

i=1
Ai
(1 + a3
a2x)i+B
1 + xdx. (42)
To this end, with the help of the integral identities [32, Eq.
(3.352.4) and Eq. (8.352.4)] we have
I3=1
ln 2
NR−1

m=0
NT−2

k2=0
1
a2(2a1)mm!k2+1

i=1
Aiϱi−1
a2
a3i
×Γi−1,ϱ
a3
a2−BeϱEi (ϱ).(43)
Finally, substituting (40), (41) and (43) into (39), after some
manipulations, average secrecy rate of the TZF scheme is
derived as (28).
APPENDIX C
PROO F OF PROPOSITION 4
With the help of γMRC
Din (21), FγMRC
D(x)can be written
as
FγMRC
D(x)=Pr 
min 
ps|hSR |2
dµ
SR σ2
R
,
pr
dµ
RD ∥hRD∥2
pd|w†
rHSIwt|2+σ2
D
< x
.
(44)
For the notation convenience, let us define U,|w†
rHSIwt|2.
FγMRC
D(x)can be expressed as
FγMRC
D(x) = Prmin a0Z1,a1Y1
c2U+ 1< x
= 1−1−FZ1x
a01−FUa1y1−x
c2x,(45)
where Ufollows exponential distribution [39]. Therefore, by
substituting FU(a1y1−x
c2x)=1−∞
0(1 −e−a1y1−x
c2x)fY1(y1)dy1
and FZ1(x
a0)=1−e−x
a0into (45), FγMRC
D(x)can be written as
FγMRC
D(x) = 1 −e−x
a0∞
x
a1
yNR−1
1e−y1
Γ(NR)dy1
+e−x
a0e1
c2∞
x
a1
yNR−1
1e−y1(1+ a1
c2x)
Γ(NR)dy1.(46)

Now, with the help of integral identity [32, Eq. (3.381.3)],
FγMRC
D(x)can be obtained as (31). Moreover, since γMRC
E=
γRZF
E,FγMRC
E(x)can be readily obtained by using the similar
steps as in Proposition 1.
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