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Secure Transmission Solutions in Energy Harvesting
Enabled Cooperative Cognitive Radio Networks
Mi Xu, Tao Jing, Xin Fan, Yingkun Wen, Yan Huo∗
School of Electronics and Information Engineering, Beijing Jiaotong University, Beijing, China
E-mail: {16120153, tjing, fanxin, 16111024, yhuo}@bjtu.edu.cn
Abstract—In this paper, we investigate secure communica-
tions in energy harvesting enabled cooperative cognitive radio
networks (CCRNs). In such CCRNs, a pair of primary users
(PUs) can only communicate with each other through a relay. To
protect data transmission between PUs, we propose a cooperative
jamming strategy with energy harvesting technology. In the first
phase, a PU as the source (PU-S) broadcasts signals, while
another PU as the destination (PU-D) sends artificial noise (AN)
and all secondary uses (SUs) harvest energy from the received
mixed signals. In the second phase, one SU selected as a relay
(SU-R) uses its harvested energy to forward PU’s signals, while
another SU selected as a jammer (SU-J) employs its harvested
energy to send AN. According to this, we formulate a non-convex
problem aiming to improve the secrecy rate of PUs and divide this
problem into three optimization subproblems. Finally we provide
a feasible joint solution by a two-tiered iterative algorithm.
Numerical results demonstrate the secrecy performance of our
proposed jamming strategy.
Index Terms—Energy harvesting; physical layer security; co-
operative cognitive radio networks; amplify-and-forward relay.
I. INTRODUCTION
A cooperative cognitive radio network (CCRN) is an ef-
ficient solution to utilize spectral resources [1]. In CCRNs,
secondary users (SUs) assist primary users (PUs) to guarantee
secure transmissions, in return, the SUs can be allowed to
opportunistically access to PUs’ spectrum for their own data
transmissions [2]. A new challenge about secure transmission
is arose: eavesdroppers scattered in CCRNs may wiretap and
decode legitimate primary signals due to the openness of
the wireless environment. To address this challenge, coop-
erative jamming is an emerging shielding strategy to pro-
tect signal transmission without compromising the existing
cryptographic-technique-based security protection [3], [4].
Yet, an SU in CCRNs may be considered as a low-end
device without enough energy, which is hard to achieve co-
operative communication. Accordingly, simultaneous wireless
information and power transfer (SWIPT) technology [5], [6]
can be utilized in this paradigm. In this case, SUs can be
powered from wireless signals transmitted by PUs so as to
forward signals of these PUs or send jamming signals. For
example, Gao suggests selecting a low-power SU as a relay to
maximize system throughput with the energy constraint and
the signal to interference plus noise ratio (SINR) requirement
in [7].
∗Yan Huo is the corresponding author.
Some studies introduce SWIPT into cooperative jamming
communication to improve system security [8]–[13]. The
authors in [8] exploit an energy harvesting enabled SU as a
jammer to protect transmissions between PUs and SUs. In [9],
Salem and Hamdi analyze the secrecy capacity of an amplify
and forward (AF) multi-antenna network with an energy
harvesting enabled relay. Then, Xing in [10] presents a feasible
secure transmission strategy in a cooperative wiretap channel
with the aid of a group of energy harvesting enabled helpers.
[11] provides the maximal secrecy rate with the assistance of
several energy harvesting enabled relays which are capable
of simultaneous interference. Furthermore, [12] discusses the
secrecy performance in the scenario of a energy harvesting
enabled relay, while [13] also develops a transmit antenna
selection scheme to analyze the secrecy outage performance.
The above-mentioned studies only consider applying energy
harvesting technology to either a relay or a jammer. Addi-
tionally, these studies focus on secure transmission during the
forward phase in relay networks. To the best of our knowledge,
secure transmission has never been studied for both relays
and jammers jointly in energy harvesting enabled CCRNs.
Motivated by this, we apply energy harvesting technology to a
relay and a jammer simultaneously, and develop a cooperative
jamming strategy to protect the whole transmission. In partic-
ular, our objective is to protect legitimate signals of PUs from
an eavesdropper with the aid of two energy harvesting enabled
SUs. There are three challenges in our work. (1) We assume
that there is no direct link between PUs due to deep fading.
Thus, one of SUs acts as a relay to help PUs to forward signals.
How does the relay find a balance between signal transmit and
energy harvesting? (2) In the presence of an eavesdropper,
how to design a complete secure strategy during the whole
transmission process? (3) How to solve the final non-convex
optimization problem?
Considering these challenges, we select an SU with SWIPT
as a relay to forward legitimate signals. In this case, the
selected SU can harvest energy from a part of the received
signals and achieve signal processing based on the rest part.
Then, we exploit a SWIPT-enabled cooperative SU as a jam-
mer to protect the whole transmission. Specifically, a primary
receiver may broadcast artificial noise (AN) to protect the first
phase of transmission and to transfer power to the relay and
the jammer. The jammer may interfere with eavesdroppers
using the harvesting energy in the second phase. The main
contributions of our work are summarized as follows.
•We propose a cooperative jamming strategy that can
protect PUs’ transmission from eavesdropping.
•We formulate the secrecy rate optimization problem in a
non-convex form for energy harvesting enabled CCRNs.
•We put forward iterative algorithms to analyze the non-
convex optimization problem from three aspects and pro-
vide a feasible solution in the case of joint optimization.
The rest of the paper is organized as follows. Section II
presents the system model and preliminaries. The problem
formulation and the corresponding solutions based on iterative
algorithms are illustrated in Section III. In Section IV, we
conduct a series of simulation and discussion on our work.
Finally, Section V concludes the paper.
Notations:In this paper, we denote upper-case and lower-
case bold letters as matrices and vectors, respectively. (·)H
and √·represent conjugate transpose and square-root. |·|and
||·||stand for the absolute value and Euclidean norm of a
vector. E[·]is the statistical expectation of random variables.
II. SYSTEM MODEL AND PRELIMINARIES
A. Network Model
A CCRN with a relay, depicted as Fig. 1, is composed of a
pair of PUs and several SUs. One PU with a single antenna as a
source (PU-S) intends to send private signals to its destination
(PU-D). Due to deep fading of the direct link between PU-S
and PU-D, an SU with a single antenna is selected as a relay
(SU-R) to forward these signals by the AF mode.
PU-S
SU-R
E
SU-J
...
SJ
h
DJ
h
PU-D
(a) The first phase.
JD
h
JE
h
PU-S PU-D
SU-R
SU-J
...
E
(b) The second phase.
Fig. 1: Network model.
This network may suffer serious secure challenge during
data transmission due to the open wireless environment and
flexible SUs’ access. An eavesdropper (Eve) with a single
antenna may wiretap and decode legitimate signals. To achieve
secure transmission, PU-S may select another SU with N
antennas to send AN so as to interfere with the reception of
Eve. As a reward, the selected SU-R and SU-J are allowed to
use the PUs channel to transmit their own signals1.
Note that the time slot Tof entire transmission duration
is divided into two phases, as shown in Fig. 2. In the first
phase, PU-S transmits a private signal s1to SU-R and PU-
D broadcasts AN v1to prevent eavesdropping. SU-R may
employ a power splitting policy [14] to split the received signal
1In general, SUs with one or more antennas may also be wiretapped. Yet,
we only select multi-antenna SUs as jammers and study secrecy performance
of PUs in this paper. We will consider other scenarios in CCRNs in future.
PU-S transmits message
PU-D broadcasts AN1
SUs harvest energy
SU-R forwards message
PU-J broadcasts AN2
T
T/2 T/2
(a) Time slot structure.
Energy havester
Message receiver
SU-R
(b) Power splitting.
Fig. 2: Time slot allocation.
and AN into two streams. One stream is used to harvest energy
while the rest is to achieve AF. Nevertheless, SU-J can harvest
energy from both the received signal and AN. Next, using
the harvesting energy, SU-R can forward the received signal
to PU-D while SU-J can broadcast AN v2to achieve secure
transmission in the second phase. Here, we assume that all
channels (as shown in Table I) in both phases experience a
quasi-static Rayleigh block fading (i.e., these channels remain
constant within a time slot and obey independent complex
Gaussian distribution).
TABLE I: List of all channel state information
Variable Definition
hSE The channel gain from PU-S to Eve
hDE The channel gain from PU-D to Eve
hSR The channel gain from PU-S to SU-R
hDR The channel gain from PU-D to SU-R
hRD The channel gain from SU-R to PU-D
hRE The channel gain from SU-R to Eve
hSJ The channel vector from PU-S to SU-J
hDJ The channel vector from PU-D to SU-J
hJD The channel vector from SU-J to PU-D
hJE The channel vector from SU-J to Eve
B. Signal Transmission Model
Here, we provide mathematical expressions of signal trans-
mission in two phases. In the first phase, the received signals
at SU-R and Eve can be expressed as
y(1)
R=(1 −β)hSRs1+(1 −β)hDRv1+n(1)
R,(1)
and
y(1)
E=hSEs1+hDE v1+n(1)
E,(2)
where the first two terms of (1) represent the stream to forward
signals at SU-R. This stream is obtained from the received
signal and AN based on the power splitting ratio β∈[0,1].
n(1)
R,n
(1)
E∼CN 0,σ
2is the additive white Gaussian noise
(AWGN) at SU-R and Eve. According to (2), the SINR at Eve
in the first phase is defined as below,
γ(1)
E=PS|hSE|2
PD|hDE|2+σ2,(3)
where PSE[s2
1]and PDE[v2
1]are the transmit power of
PU-S and PU-D, respectively.
In the second phase, we assume that SU-R forwards s2=
ρy(1)
Rand SU-J sends AN v2. Here, ρis an amplification factor
that can be defined as below,
ρ=PR
(1 −β)PS|hSR|2+(1−β)PD|hDR|2+σ2.(4)
Therefore, the received signal at PU-D is as follows,
y(2)
D=hRDρy(1)
R+hJDwv2+n(2)
D,(5)
where PRE[s2
s]and PJE[v2
2]represent the transmit
power of SU-R and SU-J, and we will explain in detail in
next subsection. wdenotes a beamforming vector of SU-J,
i.e., w2=1, and n(2)
D∼CN 0,σ
2is AWGN at PU-D.
Since v1in y(1)
Ris AN transmitted by PU-D itself in the
first phase, we assume that PU-D can easily eliminate these
interference by self-interference cancellation technology. As a
result, (5) can be rewritten as follows,
y(2)
D=hRDρ(1 −β)hSR s1+n(1)
R+hJDwv2+n(2)
D.(6)
In addition, the received signal at Eve is given by
y(2)
E=hREρy(1)
R+hJEwv2+n(2)
E,(7)
where n(2)
E∼CN 0,σ
2is AWGN at Eve.
According to the received signals of the second phase in
(6) and (7), the corresponding SINR at PU-D and Eve can be
calculated by
γ(2)
D=ρ2(1 −β)PS|hRD|2|hSR|2
ρ2|hRD|2σ2+PJ|hJDw|2+σ2,(8)
and
γ(2)
E=
ρ2(1 −β)PS|hRE |2|hSR|2
ρ2(1 −β)PD|hRE |2|hDR|2+ρ2|hRE |2σ2+PJ|hJEw|2+σ2,
(9)
where the sum of PSand PDsatisfies a system power
constraint, i.e., PS+PD≤P. Here, Pdenotes the achievable
maximum power of the whole system.
C. Energy Harvesting
According to the above statement, the energy harvested by
SU-R and SU-J in the first phase are given by
ER=ηβT
2(PS|hSR|2+PD|hDR|2),(10)
EJ=ηT
2(PS||hSJ||2+PD||hDJ ||2),(11)
where η∈[0,1] denotes energy harvesting efficiency. Since
this harvested energy is limited, all of it is used for transmis-
sion in the second phase, i.e., PR=ER
(T/2) and PJ=EJ
(T/2) .
III. PROBLEM FORMULATIONS AND SOLUTIONS
In this section, we first present an optimization problem
and formulate it to achieve secure transmission. For this non-
convex optimization problem, we then divide it into three
tractable subproblems. Finally, we propose a joint optimization
algorithm to find a feasible joint solution.
A. Problem formulation
Under the constraints of energy and power splitting ratio,
our optimization problem aims to maximize the secrecy rate.
The secrecy rate is defined as the difference between the
channel rate of main channel and eavesdropping channel [15],
which can be expressed as
RS=[RD−RE]+,(12)
where [x]+=max{0,x}.RD=1
2log2(1 + γ(2)
D)and RE=
1
2log2(1+γ(1)
E+γ(2)
E)are the channel rate of the main channel
and the eavesdropping channel, respectively. Accordingly, the
optimization problem can be formulated as
(P1) max
w,β,PS,PD
RS
s.t.P
S+PD≤P
0≤β≤1
||w|| =1.
(13)
Since the objective function is non-convex, problem (P1)
is non-convex over PS,PD,βand/or w. We divide problem
(P1) into three tractable sub-problems: (P1.1),(P1.2) and
(P1.3) and to analyze their partial solutions.
Firstly, by fixing PS,PDand β, we can optimize the
beamforming vector wby solving
(P1.1) max
wRS
s.t.||w|| =1.(14)
Secondly, by fixing PS,PDand w, we can get the optimal
power splitting ratio βby solving
(P1.2) max
βRS
s.t.0≤β≤1.
(15)
Thirdly, by fixing βand w, (P1) can become a power
allocation optimization problem as below
(P1.3) max
PS,PD
RS
s.t.P
S+PD≤P.
(16)
B. Problem analysis and proposed solutions
1) The solution of (P1.1): To maximize secrecy rate in
(P1.1), we can apply zero-forcing (ZF) beam-forming tech-
nique [16] to obtain the optimal w. Thus, the AN sent by SU-J
may not interfere with PU-D. The ZF beam-forming vector w
can be obtained by solving optimization problem (P1.1).
(P1.1)max
w|hJEw|2
s.t.hJDw=0
||w|| =1.
(17)
The solution to (P1.1)is given by lemma 1 as follows.
Lemma 1. The optimal wfor problem (P1.1)can be ex-
pressed as follows
w=(I−hH
JDhJD
||hJD||2)hH
JE
||(I−hH
JDhJD
||hJD||2)hH
JE||
.(18)
Proof: Please see Appendix A.
2) The solution of (P1.2): Since hJDw=0, we can rewrite
the SINR of PU-D as
γ(2)
D=ηβ(1 −β)PS|hRD |2|hSR|2
ηβ|hRD |2σ2+(1−β)σ2+σ4
PS|hSR|2+PD|hDR|2
.(19)
By fixing PS=¯
PS,P
D=¯
PD,(P1.2) can be transformed
into (P1.2)as follows
(P1.2)max
β{f1(β)−f2(β)}
s.t.0≤β≤1,
(20)
where
f1(β)=1
2[log2(−Aβ2+aβ +C)+log
2(−Fβ2+bβ +H)],
f2(β)=1
2[log2(Bβ +C)+log
2(−cβ2+dβ +e)],
A=η¯
PS|hRD|2|hSR|2,
B=η|hRD|2σ2−σ2,
C=σ2+σ4
¯
PS|hSR|2+¯
PD|hDR|2,
D=1+ ¯
PS|hSE|2
¯
PD|hDE|2+σ2
,
E=η¯
PS|hRE|2|hSR|2,
F=η¯
PS|hRE|2|hDR|2,
G=η|hRD|2σ2−PJ|hJE w|2−σ2,
H=PJ|hJEw|2+σ2+PJ|hJEw|2σ2+σ4
¯
PS|hSR|2+¯
PD|hDR|2,
a=A+B, b =F+G, c =DF +E,
d=DF +DG +E, e =DH.
Here, (P1.2)is still a non-convex problem due to the
logarithmic objective function. Consequently, we apply the
difference of two convex functions (D.C.) approximation pro-
gramming [17] to convert it into a convex one.
Specifically, assuming that ˜
βis a feasible solution of (20),
we replace f2(β)approximately with its first-order Taylor
series expansion at the feasible solution f2(˜
β), i.e., (20) can
be rewritten as
(P1.2)max
β{f1(β)−f2(˜
β)−(β−˜
β)f2(˜
β)}
s.t.0≤β≤1,
(21)
where f2(˜
β)is a gradient of f2(β)at ˜
β, which is given by
f2(˜
β)=1+ ˜
β
2ln2 B
˜
βB +C+−2c˜
β+d
−c˜
β2+d˜
β+e.(22)
As a result, (P1.2)is a convex programming problem. It
can be solved by some optimization packages (such as CVX
[18]). We can obtain the optimal βby iterating over problem
(P1.2), which is summarized by Algorithm 1.
Algorithm 1 An iterative algorithm for optimal β
Initialization:
Set ˜
βas the value which is feasible to problem (P1.2);
Iteration:
Repeat: set i=i+1
1: Solve (P1.2)with ˜
βand obtain the optimal values β∗;
2: Update ˜
β=β∗;
Until: Satisfy the given convergence threshold or reach
the maximum iteration number.
3) The solution of (P1.3): Assuming the optimal value
obtained in III-B2 is β∗, we can fix β=β∗to solve
(P1.3). Here, we employ simulated annealing (SA) algorithm
to obtain optimal PSand PD. The specific steps are illustrated
in Algorithm 2. As proved in [19], the convergence of SA
algorithm can be guaranteed.
Algorithm 2 The SA algorithm of (P1.3)
Initialization:
Set initial value P(0)
S,P(0)
D,m=0,t =t0;
Set the cooling ratio μ,0<μ<1;
Iteration:
1: for i=1:imax do
2: Calculate R(i−1)
S(P(i−1)
S,P(i−1)
D);
3: Generate (P(i)
S,P(i)
D)based on (P(i−1)
S,P(i−1)
D)and
then calculate R(i)
S;
4: Compute ΔRS=R(i)
S−R(i−1)
S;
5: if ΔRS<0then
6: (P(i−1)
S,P(i−1)
D)=(P(i)
S,P(i)
D)with exp(ΔRS
t);
7: m=m+1;
8: else
9: (P(i−1)
S,P(i−1)
D)=(P(i)
S,P(i)
D);
10: m=0;
11: end if
12: if m>m
max then
13: Update (P(i−1)
S,P(i−1)
D)=(P(i)
S,P(i)
D);
14: stop
15: end if
16: Set t=μt;
17: if t≤tmin then
18: break
19: end if
20: end for
21: return (P(i−1)
S,P(i−1)
D).
4) The algorithm of (P1): The above three sub-problems
have discussed problem (P1) from different aspects. Next, we
propose an outer iterative algorithm to provide a feasible joint
solution of (P1). The specific steps of the iterative algorithm
are shown in Algorithm 3, where εis convergence precision
and imax represents the maximum iteration number. Since the
iteration process can ensure that the secrecy rate is monotoni-
cally increasing (or at least nondecreasing), Algorithm 3 will
converge to a feasible solution.
S
D
E
R
J
m
m
0246
4
8
6
2
8
Fig. 3: Nodes distribution.
15 20 25 30 35
P(dBm)
0
1
2
3
4
5
Secrecy rate (bps/Hz)
proposed algorithm(with SU-J)
proposed algorithm(without SU-J)
random algorithm(with SU-J)
Fig. 4: Secrecy rate v.s. transmit power.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.5
1
1.5
2
2.5
3
3.5
4
Secrecy rate (bps/Hz)
proposed algorithm(with SU-J)
proposed algorithm(without SU-J)
random algorithm(with SU-J)
Fig. 5: Secrecy rate v.s. η.
12345678910
The number of iterationV
0
1
2
3
4
5
6
7
8
9
10
Computational Time (s)
Without SU-J
With SU-J
Fig. 6: Computational time analysis.
Algorithm 3 An outer iterative algorithm of (P1)
Initialization:
Set P(0)
S=P
2;P(0)
D=P
2;i=0;
Solve (P1.1) to obtain optimal w∗and R(0)
S;
Iteration:
Repeat: set i=i+1
1: Fix w=w∗,¯
PD=P(i−1)
D, update β(i)by Algorithm 1;
2: Fix w=w∗,¯
β=β(i), update P(i)
Dby Algorithm 2;
3: Update R(i)
S;
Until: 0≤R(i)
S−R(i−1)
S≤εor i≥imax.
IV. NUMERICAL RESULTS AND ANALYSIS
In this section, numerical results are presented to evaluate
the secrecy performance of our algorithms in a typical CCRN.
This network consists of a pair of PUs, two energy harvest-
ing enabled SUs, and an eavesdropper. All those nodes are
distributed as shown in Fig. 3, where signals of PU-S can be
received within a circle with the radius 6m. For simplicity,
unless otherwise noted, the simulation parameters are set as
follows. The maximum transmit power and noise variance of
the system are P=25dBm and σ2=−50 dBm, respectively.
The number of antennas for the jammer is defined as N=6.
The energy harvesting efficiency ηis 0.8. In addition, all
channels follow the identically distributed complex Gaussian
distribution with zero mean and variance of d(−α), where dis
the Euclidean distance between two nodes, and αrepresents
the path loss factor that is set to be 3.
A. The effect of different parameters on secrecy rate.
We study the impact of different parameters on the secrecy
rate based on two benchmark schemes. First, we still apply
our proposed algorithm, but remove the jammer (as the model
of [9]), i.e., SU-J dose not send jamming signals in the second
phase. Second, we remain SU-J, but select βand PS(PD=
P−PS)randomly, i.e., not to optimize these parameters. We
call two benchmark schemes as proposed algorithm (without
SU-J) and random algorithm (with SU-J), respectively.
Clearly, by increasing the maximum transmit power P, the
secrecy rate of the three algorithms increases gradually as
shown in Fig. 4. It also can be notably seen that the proposed
algorithm (with SU-J) outperforms the other two algorithms.
Then, we investigate the impact of energy harvesting coeffi-
cient ηon the secrecy rate of the three algorithms in Fig. 5. The
secrecy rate of three algorithms grows with the growth of η.
Similarly, it can also be observed that our proposed algorithm
(with SU-J) is better than others. In addition, we present the
computational time2of Algorithm 13in Fig. 6. It shows that
the security performance is improved at the expense of a little
computational time.
Numerical results demonstrate that our proposed algorithm
(with SU-J) can not only achieve a certain non-zero secrecy
rate, but also have a better performance.
B. Convergence of the proposed algorithms
In this subsection, the convergence of three algorithms is
verified by numerical results in Fig. 7. Specifically, Fig. 7(a)
and Fig. 7(b) depict the convergence of Algorithm 1 and
Algorithm 2, respectively. The secrecy rate is monotonically
increasing at each iteration, and there is an upper bound
because of the power constraint. Besides, the secrecy rate
increases when increasing the maximum transmit power. Note
that we set PS=PD=P
2in Fig. 7(a).
Fig. 7(d) illustrates the convergence of Algorithm 3. In this
figure, it can be seen that Algorithm 3 converges after one
iteration. This phenomenon can be explained from the fact that
βin Algorithm 3 always converges to a same value (as shown
in Fig. 7(c)), regardless of how Pchanges.
V. C ONCLUSION AND FUTURE WORK
In this paper, we study secure transmission in CCRNs where
energy harvesting enabled SUs (may be selected as a relay or
jammer) can assist PUs to improve the secrecy rate. For a
two-phase transmission, we propose a cooperative jamming
strategy that PU-D and SU-J send AN to jam Eve in the
first phase and the second phase, respectively. Further, upon
this jamming strategy, we propose a optimization problem
aiming to improve the secrecy rate of PUs. Considering the
non-convexity of the optimization problem, we first divide it
into three tractable sub-problems and then get a feasible joint
2The results are simulated on a computer with Intel(R) Core(TM) i5-6600
CPU@3.30GHz processor and 16GB RAM.
3Since the computational time of Algorithm 2 is very short, we only
analyze the computational time of Algorithm 1. In addition, we only give
the computational time of 10 iterations, due to the fact that Algorithm 1
converges after 10 iterations as shown in Fig. 7.
0102030
The number of iterations
0.5
1
1.5
2
2.5
3
3.5
Secrecy rate (bps/Hz)
P=20dBm
P=25dBm
P=30dBm
(a) Convergence of algorithm 1.
0 40 80 120 160 200
The number of iterations
1.5
2
2.5
3
3.5
4
4.5
Secrecy rate (bps/Hz)
P=20dBm
P=25dBm
P=30dBm
(b) Convergence of algorithm 2.
0 5 10 15 20 25 30
The number of iterations
0.3
0.4
0.5
0.6
0.7
0.8
P=20dBm
P=25dBm
P=30dBm
(c) Convergence of βin algorithm 1.
123456
The number of iterations
0
1
2
3
4
5
Secrecy rate (bps/Hz)
P=20dBm
P=25dBm
P=30dBm
(d) Convergence of algorithm 3.
Fig. 7: Convergence of proposed algorithms.
solution by a two-tiered iterative algorithm. Finally, numerical
results demonstrate the effectiveness of our proposed jamming
strategy. In future work, we will consider secure transmission
in more practical CCRNs with imperfect CSI.
APPENDIX A. PROOF O F LEMMA 1
To guarantee hJDw=0,wis defined as w=Hu, where
||u|| =1.H∈CN×Nis the null space of hJD which can be
expressed as
H=I−hH
JDhJD
||hJD||2.(23)
Here, HH=Hand HH =H. Thus, (P1.1) can be converted
into following form
max
u|hJEHu|2.(24)
To achieve the same direction between uand (hJEH)H,u
should obey the following form
u=(hJEH)H
||(hJEH)H|| =HHhH
JE
||HHhH
JE|| =HhH
JE
||HhH
JE||.(25)
Thus, according to (23) and (25), we can get the optimal w
which can be further expressed as
w=Hu == HhH
JE
||HhH
JE|| =(I−hH
JDhJD
||hJD||2)hH
JE
||(I−hH
JDhJD
||hJD||2)hH
JE||
.(26)
The proof of Lemma 1 is completed.
ACKNOWLEDGMENTS
We are very grateful to all reviewers who have helped
improve the quality of this paper. This work was supported
by the National Natural Science Foundation of China (Grant
No. 61471028 and 61572070), and the Fundamental Research
Funds for the Central Universities (Grant No. 2017JBM004).
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