EURASIPJournal on Wireless Communications and Networking

RESEARCH Open Access

Opportunistic relaying and jamming with robust design in hybrid full/half-duplex relay system

Zhi Lin*, Yueming Cai, Weiwei Yang and Xiaoming Xu

Abstract

Full duplex (FD) protocol has been widely used in wireless communications, which could transmit and receive signals at the meantime. In this paper, considering the worst-case channel uncertainty, opportunistic relaying and jamming strategy in decode-and-forward (DF) hybrid full/half-duplex (HD) relay system is proposed to enhance security. Specifically, the relay works in FD protocol to receive the confidential signals and transmit jamming signals at the first time slot. Then, the relay switches to HD protocol to transmit the decoded signals. Meanwhile, jammers emit cooperative jamming (CJ) signals to interfere the eavesdropper at two transmission slots. Suppose that imperfect eavesdropper channel condition is considered, we propose a worst-case robust design to obtain distributed jamming weights, which is solved through semi-definite program (SDP). Furthermore, we analyze secrecy rate and secrecy outage performance of the proposed scheme. As a benchmark, a traditional relay selection strategy with HD protocol in distributed relay system is listed for comparison. Simulation results demonstrate that our hybrid scheme with robust design outperforms the traditional relay selection scheme, because the traditional scheme does not consider hybrid FD/HD protocols and robust design based on imperfect channel conditions.

1 Introduction

Due to the broadcast nature of wireless channels which makes communication ubiquitously accessible, security becomes one of the most important issues in wireless communications. Traditionally, communication security is guaranteed through high-layer encryption. With the development of interception technology and computing power, encryption needs to be more complex, leading to higher computation burden. Unlike traditional cryptographic approaches, physical layer security (PLS) takes advantage of the physical characteristics of wireless channels to achieve secure transmission. Wyner pointed out that when the legitimate channel has better propagation conditions than eavesdropping channel, secret transmission is theoretically possible without sharing any key [1].

However, if the channel condition of legitimate user is worse than that of the eavesdropping channels, secrecy rate can be very low or even decline to zero [1]. An

* Correspondence: lz945@sina.cn

College of Communication Engineering, PLA University of Science and Technology, Guanghua Road, Nanjing, China

efficient solution to enhance the legitimate transmission is through cooperative relaying or confusing the eavesdroppers via cooperative jamming. Recently, relay cooperation diversity has attracted more and more attentions, as it can significantly improve the communication coverage area and secrecy performance [2, 3]. To simplify radio hardware in cooperative diversity setups, relay selection strategy is adopted for multiple relay nodes communication. A distributed opportunistic relay selection approach was proposed in cooperative relay system [4]. This relay selection strategy achieves the same secrecy rate performance with low complexity, compared to the scheme that all nodes participate in aiding the communications.

Alternatively, the cooperative nodes can also act as jammers to transmit artificial jamming signals collabora-tively to interfere the eavesdroppers. In [5], the cooperative jamming (CJ) was investigated for Gaussian multiple access and two-way channels. The optimal CJ weights for secrecy rate maximization (SRM) problem in the presence of a single eavesdropper were studied in [6]. Furthermore, hybrid relaying and jamming schemes

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were proposed to combine the advantages of both strategies in [4, 7].

Full-duplex (FD) operation, which always transmits and receives signals in the entire bandwidth, has attracted extensive attention. And a range of theoretical and practical researches have been investigated to take advantages of characteristic of FD protocol to enhance system performance [8, 9]. An interesting work was proposed in [8], where artificial noise (AN) was sent by a multi-antenna FD receiver. In [8], authors aimed to design the optimal jamming vector that maximizes the secrecy rate and mitigates loop interference. To enhance secrecy performance of a multiple-input-single-output-single-eavesdropper (MISOSE) relay system, a joint information beamforming and jamming beamforming strategy were proposed to guarantee both transmitting security and receiving security for a FD base station in [9]. Some works also make use of the advantages of both FD and HD protocols to improve system performance. The hybrid scheme that switches between FD and HD protocols can be employed to enhance secrecy performance [10,11].

However, in the above works [8-11], the perfect instantaneous channel state information (CSI) of all links is needed at the nodes which carry out the optimization procedure. Nevertheless, in practical relay communication networks, the perfect CSI is usually unknown and has to be estimated. The mismatch between the real and estimated CSI is caused by some inevitable factors, such as channel estimation errors, feedback delay, and quantization errors. Obviously, the performance of the designs developed in [8-11] could be heavily degraded by the imperfect CSI. Robust algorithm is an effective way to eliminate the influence of estimation errors. Typically, the existing works usually use deterministic uncertainty model (DUM) to characterize imperfect CSI: it assumes that a nominal value of the instantaneous CSI is available but lies in a bounded uncertainty region defined by some norm. The authors in [12] investigated the worst-case robust transmit covariance design problem of secrecy-rate maximization in the presence of multiple eavesdroppers. In [13], cooperative transmission for securing a decode-and-forward (DF) two-hop network was studied, where only the statistical CSI of the eavesdropping channel is available.

To sum up, the researches above only investigate the secrecy performance in FD or HD networks, and little work has been done for hybrid relay networks with opportunistic relay selection. Moreover, cooperative jamming with robust design is not widely considered in multi-relay systems, which can eliminate the influence of estimation errors.

In this paper, we investigate the secrecy performance of opportunistic relaying and jamming in hybrid FD/HD relay network with channel uncertainty. The contributions of the paper are summarized as follows:

• We propose a hybrid FD/HD strategy and opportunistic proactive relay selection approach in our model, where the best relay is selected to switch between FD and HD operation to DF confidential signals. And the other relay nodes send distributed cooperative jamming signals to interfere the receiving signal and interference with noise ratio (SINR) at the eavesdropper.

• In order to eliminate the influence of imperfect CSI to enhance the system security performance, we adopt the worst-case robust design to obtain the CJ beamformer under channel uncertainty. The optimization problem based on robust design is solved by SDP and interior method.

• We derive the secrecy rate and secrecy outage expressions. Furthermore, we compare our scheme with a traditional relay selection scheme. Simulation results clearly show the advantage of our proposed scheme to improve system secrecy performance.

1.1 Notation

Bold uppercase and lowercase letters denote matrices and vectors, respectively. (*)r and (*)H stand for transpose and Hermitian's transpose of a matrix or vector, respectively. IN is the N x N identity matrix. CN(t, a2) denotes the circularly symmetric, complex Gaussian distribution of vectors with mean t and variance a2.

2 System model

The system shown in Fig. 1 consists of a source node S, M cooperative nodes, an intended receiver D, and a legitimate user E, in which cooperative nodes are equipped with both the receive antenna and the transmit antenna and other nodes are equipped with single antenna. In this model, the source S sends private message x, which is destined for intended receiver D and kept secret from receiver E. Before the source transmission, the opportunistic relay R has to be selected, which will

be introduced in Section 3; the other cooperative nodes J;, (1 < i < M - 1) are used to send jamming signals.

In particular, we assume the direct path between S and D is blocked by some obstacle so that there is no effective S-D link. Otherwise, if S-D link exits, system has to separate the useful signals from jamming signals, which will complicate the model. hsr hse, hrd, hre, h,„ hjd, and hje denote the channel coefficients of S-R, S-E, R-D, R-E, R-R, Ji-R, Ji-D, and JrE links, respectively, where h,„ hjd, and hje are stacked in M - 1x1 vector and other channel coefficients are scalars.

Throughout this paper, the following assumptions are adopted: (1) The channels from transmitting nodes to receiving nodes are symmetric, all the channels involved are considered to remain constant during one operation period and are quasi-static. (2) All the noise distributions are zero-mean circular complex Gaussian with unit variance o2. (3) All the channel coefficients experience Rayleigh fading, and the corresponding channel gains are obtained as yi = Pi|hi|2/o2,(i e sr, se, rd, re) which are independently exponentially distributed with mean of Ai,

Yk = Pk|hHf |2; (k e jr,jd,je) and yk~ Erlang(M - 1, Ak). (4) The self-interference (SI) Yrr is well reduced to a tolerable level by efficient SI suppression. (5) The intended user D and legitimate user E employ maximum radio combination (MRC) technology to receive signals.

At time slot t, source S sends private message x(t), which is destined for opportunistic relay R and kept secret from eavesdropper E, R receive x(t) from S. Meanwhile, opportunistic relay R and cooperative nodes Ji transmit the jamming signal zr(t) and j(t), respectively. Since we assume that there is no direct link of S-D, legitimate destination cannot receive any useful information except jamming signal at time slot t; thus, we can neglect the receiving signal at D. The receiving signals at the relay R and eavesdropper E are given by

yr(t) = PPSffi(t)x(t) + vPRhrr(t)zr(t) n s

+ (t)frj(t) + nr (t) (1)

ye(t) = pP^;hse(t)x(t) + PPRMtK (t)

^^PJh^ (t)f rj(t) + «e (t), (2)

where x(t) and j(t) denote the transmitted useful signals and jamming signals with £{|x(t)|2,|/(t)|2} = 1, respectively. The term nr and ne represent naturally occurring noise at best relay R and eavesdropper E. At time slot t, the distributed jamming beamforming weight is stacked in vector fr =[/Ti,fr2, •",fr(M- i)]T, and since the beam-forming weight only determine the transmit direction, we normalize one-dimensional vector fr as |f^fr | = 1. We assume that there exist per-node power constraints of all the nodes. PS denotes the consumed power of the source, PR denotes the consumed power of the

opportunistic relay, while Pji(1 < i < M - 1) denotes the consumed power of cooperative node Ji, respectively. And to relax the complexity, we assume PJi = Pj, (1 < i < M - 1). Thus, we can regard the cooperative jamming nodes as a multi-antenna jammer to emit weighted jamming signals to interfere eavesdropper.

At time slot t +1, the chosen relay R switches to HD model and only transmits the previously decoded x(t) to D. At the same time, S transmits the jamming signal zs(t + 1), and cooperative nodes Ji still emit jamming signals j(t + 1). The receiving signals at D and E equal

yd (t + 1) = vPhd (t + 1)x(t)

+ hjd (t + 1)f dj(t + 1) + «d (t + 1)

ye(t + 1) = PPSffit + 1)Zs(t + 1) + PPffihre(t + 1)x(t)

+ VPJhjH (t + 1)fdj(t + 1) + «e(t + 1).

While at time slot t + 1, the distributed jamming beamforming weight is stacked in vector fd = [/d1,/d2, •",

/d(M -1)] .

Regarding the available CSI in a wireless communication system, the receiver usually estimates the channel using a training sequence (pilot symbols). At the transmitter, the CSI can be obtained through a feedback channel or from previous received signals, exploiting the channel reciprocity in time division duplexing (TDD) (see [14] for an overview of different channel estimation strategies). In this paper, we assume that the CSI hje is partially known at jammers, and this situation is reasonable while E is a legitimate user but also a potential active eavesdropper in wireless network [15-17]. Since the wireless system tries to enhance secrecy performance through signal processing at distributed jammers, thus, the system requires the receiving nodes feedback corresponding CSI to jammers. The potential eavesdropper additionally exchanges messages with the relay, appearing as a legitimate user. The goal of E is not only to intercept private message but also to feedback false training signals to interfere the estimation on S-E and Ji-E links.

In this correspondence, we consider the additive model for the eavesdropping channel state information (ECSI) available at jamming nodes as follows, which will be used in Section 3.

hje = hje + eje (5)

where hje denotes estimation of the channel hje and eje-denotes channel uncertainty. Since we assume the ECSI at jamming nodes are imperfect, we consider DUM, where the error is deterministically bounded,

i.e., eje e №{e: HeH < e}, where e is assumed to be the upper bound on the channel uncertainty.

3 Opportunistic relaying and jamming

3.1 Opportunistic relaying

Traditionally, there are two strategies of opportunistic relay selection: reactive relay selection and proactive relay selection (Fig. 2). Specifically, the reactive strategy first chooses the nodes to form a collection, which can successfully decode signals between the links of S-R. Then, the best relay is selected from the collection that maximizes the instantaneous channel gains of R-D link. Conversely, the proactive selection would choose the best relay prior to the source transmission, which depends on instantaneous channel gains of both S-R and R-D links. In this paper, we adopt proactive selection strategy in the hybrid FD/HD relay system.

In opportunistic proactive selection, the best relay R* is selected from a collection of M possible cooperative nodes prior to the source transmission. It requires that each relay should know its own instantaneous channel gains of S-R and R-D links. The best relay is chosen to maximize the minimum of the weighted channel gains between S-R and R-D links for all the M relays.

R* = argmax Wk, k = 1,...,M

Wk = min{ ZYsk, (1-Z)VkA, (6j

Reactive relay selection

1st time slot m 2nd time slot

\ ¡ i

! ® ® ! ® j

\Ys) / \ CîC> /

M relays M relays

Proactive relay selection

■ 1st time slot 2nd time slot

/ Best relay

(sy [ (r) j

\ Cri /

M relays

Fig. 2 Relay selection strategy: opportunistic relay selection can be performed proactively before the source transmission or reactively after the source transmission [21]. The grade band indicates when relay selection occurs

where Z denotes power allocation at the source. In this case, the communication through the best relay would fail to outage while either S - R or R - D links occur outage.

Since the minimum of two independent exponential distributed variables also follows exponential distribution with the sum of the two parameters, which is shown as follows:

MZu+IOT) • <7>

From (7), we obtain the outage probability of opportunistic proactive relay system as follows:

poutage — Pr / Wk < Noi22Rd)\

P°pp-pro = Prf Wk < ptot J

f No(22R-1)1 = Pr^ max Wk < -'- \

{ ptot )

M f NO(22R-1)"| M r i NO(22R-1) ( 1 1

= 0 i1- exn "-po- {TTk +

In proactive selection strategy, selecting a single relay before the source transmission could potentially results in degraded performance. On the other hand, selecting a single relay for information forwarding before source transmission simplifies the receiver design and the overall network operation, since proactive selection is equivalent to routing. Although active selection is more precise because it depends on instantaneous CSI during the communication, it would cost additional spectrum during transmission to select best relay. Conversely, proactive selection can facilitate the receiver operation and system design while all the cooperative nodes stay idle except the opportunistic relay during the transmission. Thus, our opportunistic proactive relay strategy can be viewed as energy efficient.

3.2 Cooperative jamming with robust design

Unlike traditional opportunistic relay selection, except the chosen relay, the remaining nodes would stay idle. In our proposed scheme, the other cooperative nodes operate as jammers to suppress the eavesdropping process.

If the ECSI is known (for instance, the eavesdropper is an active user in the wireless network, it has to feed back its CSI), the source transmits designed CJ beamformer to enhance the security of system, which suppresses or eliminates the information leakage to eavesdropper. During the two time slots, cooperative jamming beam-former depends on J-R and J-D links, respectively; thus,

jamming signals would have no influence on the chosen relay and legitimate D.

First, under the assumption that ECSI is available, we apply zero-forcing (ZF) algorithm on the jamming beam-forming design, which requires that hjHfr = 0. At time slot t, we assume that beamformer fr is a unit-normalized one-dimensional vector for the cooperative jamming. For the case of perfect CSI, the constraint of ZF algorithm is given by

min (hje + e/e)HQ* (hje + eje)

Using the Lagrange theorem, the problem (13) can be written as

L( e/e, A) = eje (Q* + AIM e/e + 2Re(eH Q*hje) i ~ ^ * ~ 1„2

+ hj Q*hje-Ae2

max jhH f r I

f I ''

s.t. hjHfr = 0, jff fr I = 1.

where A > 0. When eje = —hH Q* (Q* + AIM) 1, the minimum of L(eje, A) is achieved. The problem (13) is written as

The solution of (9) is referred as the null-steering beamformer, which is written as

(Im-H)hj |(Im—H)hjÉ

where H

hjr ( hjr hjr

jr is the orthogonal matrix

onto the subspace spanned by hjr. On the other hand, the optimal CJ beamformer fd at time slot t + 1 can be obtained with the same way.

However, it is impractical for the relay to obtain perfect ECSI in some cases (channel estimation errors, feedback delay, and quantization errors), and we cannot optimize system performance without knowledge of the eavesdropper's CSI. Thus, we follow the robust approach of [12] to find the optimal CJ beamformer under imperfect CSI. And the optimization problem becomes

max min (h;e + ej

s.t. hjr Qz h;V = 0,

where Q = frf^ denotes jamming matrix; problem (11) can be transformed to

Q s. t.

max tr [(Qz—?)hjrh^] —Ke2

Qz kIm + Qz. Qz> 0, k>0 ? > Qz (kIm + Qz )HQz

hjr Qz hjr = 0.

Proof See Appendix.

Problem (12) is a SDP with a set of linear matrix inequality (LMI) constraints. Note that eje is not explicit in (12), the optimal robust matrix Q* is derived from the hidden eje, which can be obtained through the following problem:

max lj Q:hje—Ae2 — h" Q* ( Q* + AIm) Q*H hje

s.t. Q* + AIm> 0, Q*Hhje eR(Q* + AIM,

where A is solution of the following problem:

Q*H hj

hHQ* hjHQ*hje—Ae2—n

Problem (16) can be efficiently solved by well-studied interior-point algorithm-based package SeDuMi [18]. After Qz* and eje are obtained, then, we can get the optimal fd. Meanwhile, the optimal fr can be obtained with the same approach.

4 Performance analysis

4.1 Secrecy rate

The hybrid FD/HD technology is adopted at the relay to strengthen the interference on eavesdropper to enhance system secrecy performance. However, the whole transmission is divided into two time slots; the source uses two time slots to transmit a same data packet. Thus, the capacity is computed by a one-half coefficient, which is given by

Csr = 2 log2 I 1 +

S |«sr I

Pr i hrr i2 + Pj

hjr f r

1, Ps Ihsr I2

= 2log^ 1 + P IU |2, 2

2 \ Pr hrr +ff2

RI hrr I Y sr

= ï'-H1 + Yrr + 1

Since jammers emit cooperative jamming signals in the null space of J-R and J-E links, respectively, jamming signals have no effect on the relay and destination D. The channel capacity of R-D link is given by

A> 0,n

1 I PR\hrd Crd = X log2 1 + ---¡2

2 I r, L». 1

pj ihjd fd I + a2

ii Lp_rM_ = 2log^1

= 2log2(1 + Yrd) •

Phfd = P(Chfd < rth) (1 + X 2R = p [y^y < 22Rth

= Zz(22RtY)

« (1+y)22Rth-1

fx (x)fy (y)dxdy•

The cumulative distribution function (CDF) of X can be obtained as

Thus, the capacity of main channel is

CD = min(CSr, Crd)

g2 . r

= 2 log2 + min(~ + 1

On the other hand, from (1)-(4), the capacity of eavesdropping channel is obtained as /

CE = 2 log2

S \hse\

\ PR\hre\2 + Pj

Pr \ hre

h » f

PS \ hSe \2 + Pjlh » f dl +

£fdl + a2 J

= 1log2 1 +

Yre + Yjer + 1 Yse + Yjed + 1 >

Fx (x) = 1-

Asr ~b ArrX

Thus, the probability density function (PDF) of X is given as

-A^r^l Xsr + Ard

fx (x) = e XsrX'ä

(Asr + Arrx)Ard (Asr —+ ArrX)

Since Yjer = PJ \hje fr I M Yjed = PJ \hje fd I /ff2 > which follow Erlang distribution of M - 1 orders with parameter A;e. The PDF and CDF of jjer and jjed can be expressed as

f (t)= Aj

(M-2)!

e-V t>0

F(t) = E

jf ,et

= i-S^MLe-^ t>0.

-2 -2 where Yjer = Pjh%fr /a2, y)ed = Pjh%id /a2.

Thus, the secrecy rate of this hybrid FD/HD system can be written as

To simplify the deduce, we assume yre + jjer » 1, yse +

Yse j y_re

yre +Yjer yse +yjed

Yjed » 1. Then, y is written as y = —Yfe— + Y rZ . Thus,

' Ï 7er I QP ' / if/1

the PDF of Y is obtained as

Chfd=2

log2 + min

- logJ1 +

Yrr + 1 '

Yse ï Y)

Yre + Yjer + 1 Yse + Yjed +

where [G]+ = max(G, 0).

r / \ AseAre

fy (y) =

\Aje Are

\Aje Ase

,, > / |Aje Are| |Aje A,

' ^ (k + ^ () , , )k+2 + (, , , s

k=M-1 \ \Aje + AseZ) [Aje + ArezJ

k+2 I ' (27)

Substituting (24) and (27) into (22), the secrecy outage probability is given by

4.2 Secrecy outage probability

Secrecy outage refers to the event that the secrecy rate falls below a prescribed transmission rate, equals P{CHFD < Rth}. In this part, we will analyze tine

outage performance. At first, letting X = min , Yr^j,

Y = Yrejr4 Yserfcrrand Z = "i+X,the secrecy outage

probability of the transmission is written as

Phfd = fy (y)

(Asr +Xri)(2Rth (y+1)-l) e xsrArd

Arr (2Rth (y + 1)-^ + As,

4.3 Comparison with existing work

As shown above, an opportunistic relaying strategy with CJ is proposed in distributed relay networks. In

comparison, cooperative transmission for securing a DF dual-hop network where massive cooperative nodes coexist with a potential single eavesdropper is investigated. In [19], it adopts opportunistic relay strategy and works in HD model. Besides, it assumes instantaneous perfect CSI of the eavesdropper's channel is available.

In [19], a conventional relay selection strategy is proposed during the second phase of the protocol. The existing solutions are summarized as follows:

(1) Conventional selection (CS): This solutions does not take the eavesdropper channels into account, and the relay node is selected based on the instantaneous CSI of the S-D links. Although it is an effective solution for non-eavesdropper environments, it cannot support systems with secrecy constraints. The conventional selection is written as

arg max

Ri i=1,...,M

(2) Optimal selection (OS): The optimal selection scheme takes the relay-eavesdropper links into account and decides the relay node based on the knowledge of both R-D and R-E links. The optimal selection maximizes the secrecy capacity and is given as

R* arg max

Ri ,i=1,...,M

Yr,d Yr,e i

(3)Suboptimal selection (SS): The suboptimal selection consists of a practical implementation of the optimal selection as it avoids the instantaneous estimate of the R-E links by deciding the appropriate relay based on the knowledge of expectation of R-E CSI. It is a solution which efficiently fills the gap between optimal and conventional selection with a low implementation/complexity overhead. The suboptimal selection is expressed as

The authors in [19] deal with relay selection in cooperative networks with secrecy constraints. The proposed scheme in [19] enables an opportunistic selection of relay nodes to increase secrecy performance. The selected relay assists the source to deliver confidential information to destination via DF strategy. The proposed selection technique protects the primary destination against interference and eavesdropping. However, the approach is analyzed based on instantaneous perfect knowledge of the eavesdropper channels, which is assumed imperfect.

Unlike the traditional relay selection strategy in [19], after the best relay is selected, it would not select the optimal jammer to transmit artificial jamming signals. Our proposed scheme selects the best relay to switch between FD and HD operation to deteriorate the receiving SINR at the eavesdropper. At the first time slot, the best relay works in FD protocol to receive useful signals from source and emit artificial jamming signals to interfere eavesdropper. At the second time slot, source transmits artificial jamming signals and relay switch to HD protocol to DF the confidential signals. It can simultaneously avoid the interference at destination from the source and degrade the receiving at eavesdropper.

Besides, the instantaneous knowledge of all links and average knowledge of the eavesdropper links are considered in [19]. However, in practical, the perfect CSI is usually unknown and has to be estimated. The performance of the designs based on imperfect CSI will be heavily degraded. In our proposed scheme, we propose a worst-case robust design to obtain the CJ beamformer under channel uncertainty to improve secrecy performance, while traditional relay selection scheme [19] is based on estimated ECSI. Our robust algorithm can be viewed as an effective way to eliminate the influence of estimation

R = arg max <

£| yr,

In this comparison, we assume opportunistic relay selection strategy in [19] adopts reactive relay selection and OS, which is optimal in the above three strategy. However, due the effect of estimation errors, we assume the instantaneous knowledge of the eavesdropping channel is imperfect.

The difference between our proposed scheme and relay selection scheme in [19] can be described as follows:

5 Numerical analysis

In this section, we present simulation results to illustrate the performance analysis mentioned in Section 4 and compare our proposed scheme with [19]. For convenience, we assume the noise power o2 = 1 and number of cooperative nodes is M = 5. To simplify the deducing, we consider Gaussian's noise in the channel estimation and define spherical uncertainty regions with a radius equal to e = g||h;e||, 0<g<1. Note that for these uncertainty regions, hje = hje + e;e^0, Ve;e e

In Fig. 3, we present the secrecy rate comparison against transmit SNR between our proposed scheme

_e_ Nonrobust-TRS (yae=20dB,yre=10dB) —e— NonrDbust-TRS(yse=yre=15dB) —b— Robust-HCJ (ysa=20dB,yre=10dB) —B— Robust-HCJ (ym=yre=15dB)

Transmit SNR (dB)

Fig. 3 Secrecy rate versus transmit SNR ysr = yrdl the channelerror parameter is set g = 0.3

2.7 2.6

Nonrobust-TRS (Y^OdB.Y^IOdB) Nonrabust-TRS (ym=yra=15dB)

- Robust-HCJ (Y^OdB.Y^IOdB)

- Robust-HCJ (y =y =15dB)

0.2 0.4 0.6 0.8

Channel Error Parameter g

Fig. 5 Secrecy rate versus channelerror parameter g, the transmit SNR is fixed as ysr = yrd = 40 dB

(HCJ) and traditional relay selection scheme (TRS), where we let ysr = yrd and yrr = 5 dB, assuming that the channel error parameter is g = 0.3. It can be intuitively shown that, when S-E and R-E channels have similar gains (yse = yre = 15 dB), secrecy rate is higher than the situation when one of the eavesdropping channels is weak (yse = 20 dB, yre = 15 dB). HCJ scheme far outperforms TRS scheme because of cooperative jamming based on robust design and hybrid FD/HD protocols.

Figure 4 presents the secrecy outage probability of different methods versus secrecy rate threshold, where we let channel error parameter g =0.3, ysr = yrd = 40 dB and yrr = 5 dB. As seen in the figure, TRS scheme yields higher outage probability than HCJ scheme. When the secrecy rate threshold increases, outage probability of HCJ and TRS schemes would

get close since the secrecy rate of two schemes is far less than the threshold rate.

Figure 5 plots the secrecy rate for the legitimate destination as a function of the channel error parameter g, where we let ysr = yrd = 40 dB and yrr = 5 dB. It can be observed from Fig. 5 that the secrecy rate of the TRS scheme is deeply affected by the channel errors, while the influence on HCJ is much less. Furthermore, when the channel error bound e increases, the robust HCJ scheme guarantees better secrecy rate, while the non-robust design violates decline quite seriously especially for large g.

In Fig. 6, we present the secrecy rate comparison for different transmit antennas M and assume ysr = yrd = 40 dB and yrr = 5 dB. Since the increase of cooperative nodes can provide a better relay as the best relay to DF useful signals and it generates stronger interference to

Nonrobust-TRS (yBa=20dB,yre=10dB) E Nonrobust-TRS (ym=yre=15dB) : ~b_ Robust-HCJ (yse=20dB,yre=10dB) --a— Robust-HCJ (y =y =15dB) ~

Secrecy rate threshold (bit/s/Hz)

Fig. 4 Secrecy outage probability versus secrecy rate threshold, the channelerror parameter is set g = 0.3

Nonrobust-TRS (ys9=20dB,l Nonrobust-TRS (Y^Y^IidB) . Robust-HCJ (yae=20dB,yre=10dB) . Robust-HCJ (y =y =15dB)

Fig. 6 Secrecy rate versus number of cooperative nodes M, the transmit SNR is fixed as ysr = yrd = 40 dB and the channelerror parameter is set g = 0.3

confuse eavesdropper, the secrecy rate of the two schemes gets better while the number of nodes increases. Not surprisingly, the secrecy rate performance of hybrid scheme is better than the traditional one under the same condition. Moreover, we observe that the secrecy rate achieved by TRS scheme increased slowly compared with our proposed scheme, while the cooperative nodes increase.

Finally, we compare the computational complexity of our proposed scheme and TRS scheme. For the sake of notational simplicity, we evaluate how the computational complexity of both schemes scale with the number of nodes M. In our proposed scheme, the complexity of the opportunistic relay selection is O(M). The complexity order of obtain fr and fd through solving the SDP problem (11)-(16) is O(M5) [20]. Thus, the computational complexity of our proposed scheme is O(M5). For the TRS scheme, the major computation task is how to obtain best relay R based on imperfect CSI (30), which has complexity order of O(M). Thus, the computational complexity of traditional CJ scheme is O(M). Based on the analysis above, we find that our proposed scheme has a higher complexity than traditional one. However, the proposed scheme outperforms TRS scheme both on secrecy rate and secrecy outage probability.

6 Conclusions

In this paper, we addressed a hybrid opportunistic relaying and jamming strategy in multi-relay system under imperfect channel estimation. A proactive opportunistic relay selection is proposed, in which the criteria are based on the channel conditions. To enhance secrecy performance of cooperative jamming under channel uncertainty, we transform the optimization problem into a SDP with some LMI constraints, which is solved by the interior point approach. Furthermore, we derive the secrecy rate and deduce the outage probability. Since our proposed scheme takes advantages of cooperative jamming with robust design and increases interference on eavesdropper through hybrid FD/HD protocols, simulation results demonstrated that the proposed scheme leads to significant improvement than TRS strategy in distributed relay system.

Appendix

The problem (11) can be transformed to

Qz (he

s.t. e™ Qz eje + 2Re -eH eje + e2 >0.

(j Qz he) + j Qz hje-U>0

According to the S-procedure, there exists an eje satisfying above inequalities only if there exists a k > 0 such that

kIm + Qz

Q z hje

hjH Qz hjH Qz hje -KE -u

Applying the Schur complement, (34) can be expressed as

-Ke2 4 hjQzhje-hjQz(kIm 4 Qz)fQzhje>u (35) Then, the problem (35) becomes Qmax -Ke2 4 hjHQzhjeQz (kIm 4 QzQzhje,

Qz ,K>0

which equals to

Qmaxr -Ke2 + h™ Qzhje -hj™ ^hje

Qz;K >0;S

s.t. Qz (kIm + Qz )+Qz < \

Therefore, we use the Schur complement to transform above constraints into LMI and the problem (11) can be written

max tr [(Qz-^jhjd] -Ke2

\ Qz Qz k Im + Qz

Qz > 0; K>0 \ > Qz(*Im + Qz)HQz

hjd Qz hjd = 0;

The constraints of (32) can be rewritten as

and the proof is completed. Competing interests

The authors declare that they have no competing interests. Acknowledgements

This work is supported by the National Natural Science Foundation of China (No.61371122 and No. 61471393) and the China Postdoctoral Science Foundation under a Special Financial Grant No. 2013T60912.

Received: 1 September 2015 Accepted: 24 April 2016 Published online: 10 May 2016

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