Outage Constrained Secrecy Throughput Maximization ... - IEEE Xplore

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 5, MAY 2015

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Outage Constrained Secrecy Throughput Maximization for DF Relay Networks Tong-Xing Zheng, Student Member, IEEE, Hui-Ming Wang, Member, IEEE, Feng Liu, and Moon Ho Lee

Abstract—In this paper, we provide a comprehensive study of secrecy transmission in decode-and-forward (DF) relay networks subjected to slow fading. With only channel distribution information (CDI) of the wiretap channels, we aim at maximizing secrecy throughput of the two-hop transmission under a secrecy outage constraint through optimizing transmission region, rate parameters of the wiretap codes and power allocation between the source and relay. We propose fixed transmission parameter scheme (FTPS) and variable transmission parameter scheme (VTPS), which are based on the CDI and instantaneous channel state information of the main channels, respectively. In both schemes, source and relay use the same codeword, and the eavesdropper can use maximum ratio combining (MRC) reception. To improve the secrecy throughput, we further propose VTPS-D1 and VTPS-D2 schemes, where the source and relay either use independent codewords with identical code rates, or different codebooks with different code rates so that the eavesdropper can only decode the two-hop signals individually rather than using MRC. We provide explicit results on the design for all proposed schemes. Numerical results and comparisons on the secrecy throughput of these schemes are presented to reveal their respective superiorities and give some insights into the choice of design scheme.

ception of secrecy capacity. During the past decades, Wyner’s DWTC model has been generalized to various wiretap channel models, and a mass of secrecy transmission schemes have been put forward for wireless transmissions (see [2] and its references). Basically speaking, a positive secrecy capacity/rate can only be achieved when the main (or legitimate) channel (transmitter to the intended receiver) is superior to the wiretap channel (transmitter to the eavesdropper), which can not be always guaranteed due to the fading feature of the wireless channels. However, in [3], it has been first shown that with node cooperation, even when the main channel’s quality is poorer than the wiretap channel’s quality, a positive secrecy capacity may still be achieved. Therefore, cooperative communication becomes a promising technique to enhance the PLS by improving the quality of the main channel and/or degrading that of the wiretap channel.

Index Terms—Physical layer security, relay network, secrecy throughput, secrecy outage, fading, power allocation, optimization.

A serial of secrecy transmission strategies have been proposed based on the idea of node cooperation [4]–[25]. However, there is a rigorous assumption in many works [4]–[12], i.e., the instantaneous channel state information (ICSI) of the eavesdropper is perfectly known at the transmitter. In fact, it is very difficult, if not impossible, to acquire this ICSI in real wiretap scenarios, since the eavesdropper usually passively receives signals. Without the eavesdropper’s ICSI, in [13]–[15], we proposed the target signal-to-noise-ratio (SNR) based cooperative beamforming and jamming schemes for one-way and two-way relay networks. When only the channel distribution information (CDI) of the eavesdropper is available, the ergodic secrecy rate has been maximized in [16], where a cooperative node operates as either a pure jammer or a relay. When the channels suffer slow fading, the secrecy outage performance of the cooperative transmission has been evaluated in [19]–[21], where the secrecy outage refers to the event that confidential messages fail to achieve perfect secrecy [17], [18]. Specifically, the authors in [19] studied the opportunistic use of relays and proposed cooperative jamming and relay chatting schemes, respectively. This idea was generalized to two-way networks in [20]. The authors in [21] maximized the instantaneous secrecy rate under an outage constraint with cooperative jamming over multi-input single-output (MISO) channels. Based on the secrecy outage, secrecy throughput is an appropriate measurement on PLS for slow fading channels. Secrecy throughput is defined as the average secrecy rate

I. I NTRODUCTION

P

HYSICAL Layer Security (PLS), or, information-theoretic security has become a fundamental approach for achieving secure wireless communications with confidential messages since Wyner’s pioneering work [1], where he introduced the degraded wiretap channel (DWTC) model and defined the con-

Manuscript received August 11, 2014; revised December 3, 2014 and January 29, 2015; accepted February 23, 2015. Date of publication March 5, 2015; date of current version May 14, 2015. The work of T.-X. Zheng, H.-M. Wang and F. Liu was partially supported by the NSFC under Grants No. 61102081, and No. 61221063, the Foundation for the Author of National Excellent Doctoral Dissertation of China under Grant 201340, the New Century Excellent Talents Support Fund of China under Grant NCET-13-0458, the Fok Ying Tong Education Foundation under Grant 141063, and the Fundamental Research Funds for the Central University under Grant No. 2013jdgz11. The work of M. H. Lee was partially supported by MEST 2012-002521, NRF, Korea. The associate editor coordinating the review of this paper and approving it for publication was J. Yuan. (Corresponding author: Hui-Ming Wang.) T.-X. Zheng, H.-M. Wang, and F. Liu are with the School of Electronics and Information Engineering, Ministry of Education Key Laboratory for Intelligent Networks and Network Security, Xi’an Jiaotong University, Xi’an 710049, China (e-mail: [email protected]; [email protected]; piziliufeng@ 126.com). M. H. Lee is with the Division of Electronics Engineering, Chonbuk National University, Jeonju 561-756, Korea (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCOMM.2015.2409171

A. Previous Works

0090-6778 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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subject to (s.t.) a secrecy outage probability (SOP) constraint. In [22]–[28], the secrecy throughput of various transmissions has been investigated. More specifically, the authors in [22] derived the achievable secrecy throughput of hybrid automatic retransmission request (HARQ) protocols with a given pair of reliability/secrecy outage probabilities. In [23] and [24], the authors investigated the secrecy throughput in wireless ad hoc networks for both single- and multi-antenna cases. The authors in [25] optimized the secrecy outage capacity under both secrecy outage and channel outage constraints in an orthogonal frequency division multiple access (OFDMA) relay network. In [26], the authors proposed channel-adaptive transmission policies for improving the secrecy throughput in the presence of non-colluding eavesdroppers. In [27], the authors considered the rate parameters optimization in artificial-noise-aided secure multi-antenna transmission, where they proposed both nonadaptive and adaptive encoder schemes. In our recent work [28], we proposed maximizing the secrecy throughput of the primary user in a MISO cognitive radio network s.t. a secrecy outage constraint at the primary user and a throughput constraint at the secondary user. However, none of these works have considered the system design, especially including the rate parameters, for the secrecy throughput maximization in cooperative relaying networks. B. Our Work and Contributions In this paper, we investigate the secure transmission from a source to a destination via the retransmission of a trusted decode-and-forward (DF) relay in the presence of an eavesdropper over slow fading channels. We assume that the ICSI of the eavesdropper is not available, while both the ICSI of the main channels and the CDI of the wiretap channels are known. With the ICSI of the main channels, we adopt the on-off transmission strategy proposed in [29] to avoid secrecy outage due to the bad main channels’ quality, where only when the instantaneous main channel gains exceed the preestablished thresholds, the transmissions occur (Otherwise, the transmissions are suspended). Our objective is to maximize the secrecy throughput s.t. an SOP constraint by optimizing the transmission thresholds, the wiretap code rates and the power allocation between the source and the relay. We provide a comprehensive study on the design of secure DF relaying transmission in this paper. Our main contributions include the explicit parameter design solutions and comprehensive performance analysis for the proposed schemes in the following varies scenarios. 1) We first propose the fixed transmission parameter scheme (FTPS). In FTPS, all parameters are designed based on the CDI of both main channels and wiretap channels, and remain fixed during the transmission. The entire optimization procedure is accomplished off-line, which is of quite low system complexity. We consider the case that the source and relay transmit the same codeword, and the eavesdropper receive the two-hop signals with maximum ratio combining (MRC) method. We formulate the optimization problem, and give the optimal design with some numerical solutions. We also derive a closed-


form expression of the maximum secrecy throughput at high SNR regime, which is very close to the exact value. 2) We then propose the variable transmission parameter scheme (VTPS). In VTPS, all parameters are designed based on the ICSI of main channels along with the CDI of wiretap channels, and are adjusted real-time during the transmission. We still consider the identical-codeword case at source and relay where the eavesdropper can adopt MRC reception. We give analytical solutions of the optimal parameters as well as an accurate integral representation of the maximum secrecy throughput. We also derive a closed-form expression of the maximum secrecy throughput at high SNR regime for the dominant wiretap link case. 3) To further improve the secrecy throughput, we propose two adaptive schemes with more flexibility compared with VTPS, namely, VTPS-D1 and VTPS-D2, respectively. In VTPS-D1, both source and relay transmit independent codewords with the same codeword rates; while in VTPS-D2, they use different codebooks with different codeword rates. In both cases, the eavesdropper can only decode the two-hop signals individually instead of using the MRC. We derive solutions to the secrecy throughput maximization problem for both schemes. By comparing the secrecy throughput of the proposed schemes, we provide some insights into the choice of the design scheme. We have to emphasize that our design results based on the secrecy outage constrained framework are new for DF relay networks. In both FTPS and VTPS, the eavesdropper can combine the two-hop signals, which is distinguished from [25], where only the stronger one is considered. Due to the double information leakage to the eavesdropper, our case presents a more severe scenario on the secrecy. The analysis on the secrecy outage becomes more complicated, which makes the parameter optimization problem much more difficult. To the best of our knowledge, the above problems have not been addressed by any other existing works. Moreover, we are the first to design secure transmission for considering different codewords, different codebooks and different rate parameters at source and relay in DF networks. C. Organization and Notations The remainder of this paper is organized as follows. In Section II, we describe the system model and the optimization problems. In Sections III, IV, and V, we study the secrecy throughput optimization with FTPS, VTPS, VTPS-D1 and VTPS-D2, respectively. In Section VI, we conclude our work. We use the following notations in this paper: i → j denotes the link from node i to node j; CN (μ, σ 2 ) and Exp(λ) denote the circularly symmetric complex Gaussian distribution with mean μ and variance σ 2 , and the exponential distribution with parameter λ, respectively; R denotes the two-dimensional real number domain; ∅ denotes the null set; P{·} denotes the probability and EA (·) denotes the expectation with respect to (w.r.t.) A; log(·) and ln(·) denote the base-2 and natural logarithms, respectively. ⇔ represents an equivalence relation.

ZHENG et al.: OUTAGE CONSTRAINED SECRECY THROUGHPUT MAXIMIZATION FOR DF RELAY NETWORKS

II. S YSTEM M ODEL AND P ROBLEM D ESCRIPTION We consider the confidential message delivery from the source (S) to the destination (D) in the presence of an eavesdropper (E). Due to the deep fading of the direct link S → D, a trusted DF relay (R) helps S to forward the messages to D. Each node in the system works in a half-duplex mode and is equipped with a single antenna. The wireless link is subjected to quasi-static Rayleigh fading together with a large-scale path loss governed by the exponent α. The channel (power) gain coefficient w.r.t. i → j can be expressed as gij = |hij |2 d−α ij , i ∈ {S, R}, j ∈ {D, R, E},

(1)

where hij ∼ CN (0, 1) and dij are the Rayleigh fading coefficient and the path distance w.r.t. i → j, respectively. We know gij ∼ Exp(G−1 ij ), with its probability density function (PDF) Δ

denoted as fij (x), where Gij =

d−α ij .

In the following, we let

Δ

g = (gSR , gRD ) ∈ R represent the instantaneous gains of the main channels (S → R and R → D). We assume that the ICSI of the main channels as well as the CDI of the wiretap channels are perfectly known, while the ICSI of the wiretap channels is not available. We consider an aggregate power constraint P0 on S and R. We denote the power allocated to S and R as P1 = γP0 and P2 = (1 − γ)P0 , respectively, where γ ∈ (0, 1) represents the power allocation factor. We assume that the thermal noise n0 at each receiver node is additive white Gaussian noise (AWGN) with n0 ∼ CN (0, 1), such that P0 can be also treated as the normalized transmit SNR. Due to the ignorance of the direct link S → D, D only receives signals from R,1 and the channel capacities of R and D can be expressed as CR = log(1 + P1 gSR ) and CD = log(1 + P2 gRD ), respectively.

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identical.2 In that case, E can only decode the two-hop signals individually. In other words, if only the two hops are both secured, the complete transmission is secured. The overall SOP can be characterized as a combination of the individual SOP of each hop, and is expressed as Δ

Ps = 1 − (1 − Ps,1 )(1 − Ps,2 ),

(3)

Δ

where Ps,i = P{CE,i > Rt,i − Rs } for i = 1, 2, with CE,1 = log(1 + P1 gSE ) and CE,2 = log(1 + P2 gRE ). B. On-Off Transmission To guarantee that both R and D are able to decode the messages correctly, the code rates should satisfy 0 < Rs ≤ min{Rt,1 , Rt,2 }, Rt,1 ≤ CR , Rt,2 ≤ CD .

(4)

To avoid an undesired capacity outage (i.e., Rt,1 > CR , or Rt,2 > CD ) or an unacceptable high secrecy outage (i.e., Ps > , where ∈ [0, 1] is a prescribed outage threshold), we adopt the on-off transmission strategy [29], where the transmitter decides whether to transmit signals based on the ICSI of the main channels. Specifically, S and R transmit signals bearing the confidential messages only when their respective channel gains gSR and gRD exceed the pre-established transmission thresholds μ1 and μ2 ; Otherwise, S and R remain silent. Under such an on-off strategy, we define the set of those main channel gains g’s satisfying the transmission condition as the transmission region, which can be characterized as Δ

T = {g : gSR ≥ μ1 , gRD ≥ μ2 }.

(5)

The corresponding transmission probability is given by Pt = P{g ∈ T }.

(6)

A. Secrecy Outage To guarantee the secrecy transmission, we employ the wellknown Wyner’s wiretap encoding scheme [17], [18]. The rates of the transmitted codewords and the confidential messages are denoted as Rt and Rs , and the difference between them, Δ denoted as the rate redundancy Re = Rt − Rs , reflects the cost of the secrecy transmission against eavesdropping. Once the capacity of the wiretap channels CE exceeds the rate redundancy Re , perfect secrecy fails and a secrecy outage occurs. The SOP Δ can be defined as [18]Ps = P{CE > Re }. In the relaying transmission system, when S and R use the same codeword, E can combine the signals from S and R with MRC reception, and its channel capacity is given by CE = log(1 + P1 gSE + P2 gRE ). The SOP can be expressed as Δ

Ps = P{CE > Rt − Rs }.

(2)

When S and R use independent codewords or even different codebooks, their codeword rates Rt,1 and Rt,2 need not to be 1 In fact, we present pessimistic results on secrecy compared with the scenario

where the direct link from S to D exists.

C. Optimization Objective We evaluate the secrecy throughput, which is defined as the average of the secrecy rate, i.e., 1 (7) Φ = Eg Rs (g) . 2 Note that, the wiretap code rates could be designed according to the instantaneous main channel gains g. Only when g ∈ T , S and R transmit messages with Rs (g); Otherwise, the transmissions are suspended, and we set Rs (g) = 0. We aim at maximizing the secrecy throughput s.t. an SOP constraint (Ps ≤ ) through optimizing the transmission region T , the power allocation γ, and the wiretap code rates, including the codeword rates Rt,1 , Rt,2 and the secrecy rate Rs . To achieve our objective, we propose four schemes with different complexity and flexibility, namely, FTPS, VTPS, VTPS-D1, and VTPS-D2, respectively. A comprehensive study of these schemes is performed in the following sections. 2 To guarantee consistent transmission of the secrecy message, R should be s identical in both hops, while transmission rates Rt,1 and Rt,2 in two hops can be different. This is a unique feature of secure DF relay communication.

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III. F IXED T RANSMISSION PARAMETER S CHEME In this section, we investigate the secrecy throughput maximization with FTPS. We consider the case where S and R transmit the same codewords with the consistent codeword rate Rt and the secrecy rate Rs . In this case, E can combine the signals from S and R with MRC reception. A. Optimization Problem Formulation Since Rs is independent of the main channel gains g, the secrecy throughput defined in (7) can be rewritten as ΦF =

1 Rs Pt , 2

(8)

where Pt can be derived from (5) and (6), and is given by μ1 μ2 − . (9) Pt = exp − GSR GRD The SOP Ps can be defined in (2), and is given by 2Rt −Rs − 1 Ps = P η E > = F (x)| 2Rt −Rs −1 x= P P0 x 0 x ⎧ α exp − −α exp − 2 α1 α2 ⎨ 1 , α1 = α2 2 α1 −α = (10) ⎩ 1 + x exp − x , α =α α1

1

α1

Δ

2

Δ

Proof: With the monotonically decreasing feature of the CCDF F (x) w.r.t. x, the SOP constraint (12d) yields Rt −Rs −1 2Rt −Rs − 1 2 ≥ F −1 (), (14) ≤⇒ F P0 P0 Δ

where F −1 () is the inverse function of F (x). Define κ(γ) = F −1 (), then (13) can be directly given from (14). Remark 1: Due to the transcendental equation F (x) = , κ(γ) is an implicit function of γ, which can be calculated with Newton method. Remin in (13) represents the minimum rate redundancy against eavesdropping under an SOP constraint. In the objective function (12a), Pt given in (9) is an explicit function of μ1 and μ2 . Meanwhile, μ1 and μ2 are coupled with Rt and γ by (12b), and further coupled with Rs by (13). Therefore, Pt is a function of all parameters μ1 , μ2 , γ, Rt , and Rs . To maximize ΦF in (12a), we always have max Rs Pt ⇔ max Rs max max Pt . (15) μ1 ,μ2 ,γ,Rt ,Rs

0 < Rs ≤ Rt ≤ min {log(1 + P1 μ1 ), log(1 + P2 μ2 )} . (11)

The above equivalence relation implies that the entire optimization procedure can be decomposed into three steps: Firstly, for given Rs and γ, we optimize μ1 , μ2 and Rt to maximize Pt ; Next, for a fixed Rs , we derive the optimal γ to maximize Pt further; Finally, we design Rs to maximize ΦF . In the following, we discuss them step by step. B. Optimal Solutions We solve the problem (12a) with three steps. 1) Step 1: For given γ and Rs , we determine μ1 , μ2 and Rt to maximize Pt . This sub-problem can be formulated as max Pt (Rs , γ),

1 Rs Pt , μ1 ,μ2 ,γ,Rt ,Rs 2 s.t. Rt ≤ min {log(1 + P1 μ1 ), log(1 + P2 μ2 )} , 0 ≤ Rs ≤ Rt , Ps ≤ , 0 < γ < 1, 0 ≤ μ1 , 0 ≤ μ2 .

(12a) (12b) (12c) (12d) (12e)

Since Ps is irrespective to the instantaneous channel gains g, (12d) also reflects an average SOP constraint. Before proceeding, we first transform the SOP constraint (12d) into a more explicit form in the following lemma, which will be repeatedly exploited in the following sections. Lemma 1: For any given γ and Rt , the secrecy rate Rs that satisfies the SOP constraint (12d) is restricted by Rs ≤ Rt − Remin , Δ

(13)

where Remin = log(1 + P0 κ(γ)), and κ(γ) is the unique root of the equation F (x) = , with F (x) defined in (10).

(16a)

μ1 ,μ2 ,Rt

s.t. Rt ≤ min {log(1 + P1 μ1 ), log(1 + P2 μ2 )} , (16b)

According to the above discussions, the problem for the secrecy throughput maximization can be formulated as below ΦF =

μ1 ,μ2 ,Rt

Δ

where ηE = γgSE + (1 − γ)gRE , α1 = γGSE , α2 = (1 − γ)GRE , and F (x) is the complementary cumulative distribution function (CCDF) of ηE . Note that, although Ps in (10) has two distinguished expressions, we can easily prove that they are actually uniform when α1 → α2 . For convenience, we only discuss the first expression, i.e., γGSE = (1 − γ)GRE . To avoid a capacity outage of the main channels, for any given μ1 and μ2 , the constraint (4) can be revised as

max

γ

Rs

Rt ≥ Rs + log (1 + P0 κ(γ)) ,

(16c)

0 ≤ μ1 , 0 ≤ μ2 ,

(16d)

and the solution is presented in the following proposition. Proposition 1: For any given γ and Rs , the maximum transmission probability Pt (Rs , γ) in (16a) is given by (1+P0 κ(γ)) T −1 (1+P0 κ(γ)) T −1 Pt (Rs , γ) = exp − − , γP0 GSR (1−γ)P0 GRD (17) Δ

where T = 2Rs , and the optimal parameters are designed as Rt∗ = Rs +log (1+P0 κ(γ)) , μ∗1 =

(1+P0 κ(γ)) T −1 , γP0

μ∗2 =

(18) (1+P0 κ(γ)) T −1 . (1−γ)P0

(19)

Proof: From (9), it is clear that to achieve a higher Pt (Rs , γ), μ1 and μ2 should be set as smaller as possible. ConRt Rt strained by (16b), we have μmin = 2 P1−1 and μmin = 2 P2−1 . 1 2 Obviously, Rt should also be set at its minimum, and from (16b), we have Rtmin = Rs + log(1 + P0 κ(γ)). Plugging Rtmin into μmin and μmin 1 2 , we complete the proof.


Fig. 1. Transmission probability vs. power allocation factor for different system total powers, with Rs = 0.1 bits/s/Hz, = 0.3, α = 4, dSR = dRD = 1, dSE = 1.5, and dRE = 1.2 (normalized distance).

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Fig. 2. Optimal power allocation vs. system power for different secrecy rates and SOP constraints, with α = 4, dSR = dRD = 1, dSE = 1.5, and dRE = 1.2.

2) Step 2: For a given Rs , we optimize γ to maximize Pt (Rs , γ) in (17). This sub-problem is formulated as max γ

Pt (Rs , γ),

s.t. 0 < γ < 1, Rs > 0,

(20)

with the optimal γ ∗ given in the following proposition. Proposition 2: For a given Rs , Pt (Rs , γ) in (17) is a quasiconcave function3 of γ, and the optimal γ ∗ that maximizes Pt (Rs , γ) is the unique root of the following equation

−1)γ γ P0 (1 + λ(γ)) + (T T κ(γ) G

= RD , (21) (T −1)(1−γ) GSR (1 − γ) − P λ(γ) T κ(γ)

Δ

0

Δ

κ(γ) κ(γ) 1 1 where λ(γ) = β(γ) + e−β(γ) with β(γ) = γG − (1−γ)G −1 SE RE (here, β(γ) = 0 for the case γGSE = (1 − γ)GRE ). Proof: Please see Appendix A. Proposition 2 is well verified by numerical examples in Fig. 1. We see that, for a given P0 , Pt (Rs , γ) first increases and then decreases with the rise of γ, and there exists a unique γ ∗ that maximizes Pt (Rs , γ). Besides, Pt (Rs , γ) rises with an increasing P0 , and levels off for high enough P0 ’s. The reason

can be seen in (17)as P0 → ∞, Pt (Rs , γ) converges to

κ(γ)T exp − κ(γ)T γGSR − (1−γ)GRD , which is solely determined by γ, and is irrespective to P0 . From (21), we can numerically calculate the root γ ∗ with efficient iteration algorithms, e.g., interior point method or Newton iteration method. Fig. 2 shows how the optimal γ ∗ is influenced by P0 , Rs and . We see that, γ ∗ rises with an increasing P0 . The reason is that, a higher P0 enlarges CE , which is mainly determined by the second hop (since dRE < dSE ), such that we should allocate less power to R to suppress the growth of CE . For a given P0 and , a higher Rs leads to a lower γ ∗ . It is because that the maximum Rs is primarily 3 For a given R , with the rise of γ from 0 to 1, P (R , γ) first monotonically s t s increases and then monotonically decreases.

Fig. 3. Transmission probability vs. secrecy rate for different SOP constraints, with P0 = 20 dBm, α = 4, dSR = dRD = 1, dSE = 1.6, and dRE = 1.4.

bottlenecked by the second hop, hence, more power should be given to R to support a higher Rs . We also find that γ ∗ tends to a constant as P0 → ∞, which is irrespective to Rs . Besides, this constant increases as decreases, since to satisfy a harsher SOP constraint, more power should be allocated to S to weaken the impact of E. Substituting γ ∗ into (17), we obtain the maximum transmission probability Pt∗ (Rs ). Fig. 3 illustrates Pt∗ (Rs ) versus Rs for different ’s. For a given , Pt∗ (Rs ) decreases rapidly with an increasing Rs (approximately exponentially decreases w.r.t. 2Rs , see (17)). For a given Rs , Pt∗ (Rs ) reduces as decreases. We can infer that Pt∗ (Rs ) reduces to 0 as → 0, which implies that without the ICSI of the wiretap channels, to achieve secrecy transmission with a zero SOP is impossible. 3) Step 3: Having obtained the maximum Pt∗ (Rs ) for given Rs ’s, the optimal Rs∗ that maximizes ΦF in (12a) is given by Rs∗ = arg max Rs Pt∗ (Rs ), Rs >0

(22)

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Fig. 4. Secrecy throughput vs. secrecy rates for different SOP constraints, with P0 = 30 dBm, α = 4, dSR = dRD = 1, dSE = 1.8, and dRE = 1.6.

and the maximum secrecy throughput Φ∗F is Φ∗F =

1 ∗ ∗ ∗ R P (R ) . 2 s t s

Δ

(23)

Since Pt∗ (Rs ) monotonically decreases with a rising Rs , intuitively, Rs Pt∗ (Rs ) should be a first increasing then decreasing function of Rs , and there is a unique maximum of Rs Pt∗ (Rs ). To show this, we demonstrate numerical examples of the secrecy throughput ΦF versus Rs for given ’s in Fig. 4. We see that with the increase of Rs , ΦF first monotonically rises and then monotonically decreases, and there exists a unique Rs that maximizes ΦF . However, since it is quite complicated to obtain a closed-form expression on Pt∗ (Rs ), we can hardly prove the quasi-concavity of ΦF w.r.t. Rs , and we can not perform an analytical optimization of ΦF either. Therefore, we can only numerically search for the optimal Rs∗ with (22). However, we can derive the closed-form expressions on both the optimal secrecy rate Rs∗ and the maximum secrecy throughput Φ∗F at high SNR regime. C. High-SNR Approximations At high SNR regime, the solution to the problem (12a) is provided in the following proposition. Proposition 3: At high SNR regime (P0 → ∞), the maximum secrecy throughput in (12a) is given by 1 (24) ΦF = Rs exp f1 (γ ) − f2 (γ )2Rs , 2 where γ is the unique root of the following equation γ (1 + λ(γ)) GRD + = 0, (1 − γ)λ(γ) GSR with λ(γ) defined in (21), Rs is given by ln f3 (γ ) Rs = log , W (ln f3 (γ ))

Fig. 5. Secrecy throughput vs. system total power for different SOP constraints, with α = 4, dSR = dRD = 1, dSE = 1.8, and dRE = 1.5.

(25)

(26)

f1 (γ ), f2 (γ ) and f3 (γ ) are positive functions of γ , with Δ Δ f1(γ) = γ P01GSR + (1−γ)P1 0 GRD , f2 (γ ) = (1+P0 κ(γ))f1 (γ),

1

f3 (γ ) = 2 f2 (γ ) ln 2 , and W (x) is the Lambert-W function [30, Sec. 4-13]. Proof: Please see Appendix B. From (25), we find that the optimal γ is irrespective to P0 or Rs at high SNR regime, and is mainly influenced by (as λ(γ), β(γ) and κ(γ) are actually functions of ) as well as the relative distances between nodes. Recalling Fig. 2, γ is just the converged value of the optimal γ ∗ as P0 → ∞. Remark 2: We can similarly prove that, the optimal γ at low √ √ , which relies only SNR regime (P0 → 0) is γ ◦ = √G G+RD GRD SR on the relative distances between the legitimate nodes. That is the reason why γ ∗ tends to 0.5 as P0 → 0 (since dSR = dRD ) in Fig. 2. Similar to Proposition 3, we can give the optimal secrecy rate Rs◦ and the maximum secrecy throughput Φ◦F at low SNR regime, which have the identical forms of Rs and ΦF , simply with γ replaced by γ ◦ . With the high-SNR power allocation, the optimal secrecy rate and the maximum secrecy throughput can be directly derived from the closed-form expressions instead of numerical search and calculation. In Fig. 5, we depict the exact Φ∗F , the highSNR ΦF as well as the low-SNR Φ◦F . We see that, Φ∗F increases with the rise of P0 , and levels off at high enough P0 ’s. ΦF is very approximate to Φ∗F at high P0 region, which implies that, the high-SNR design could be a robust alternative to the optimal scheme, since the power allocation γ is not sensitive to P0 . Interestingly, ΦF is also quite close to Φ∗F at low SNR region. We can explain it like this: Recalling Fig. 1, when P0 is extremely small, the maximum Pt (or ΦF ∝ Pt ) maintains low and smooth within a relatively wide region of γ near the optimal γ ∗ , such that even though γ differs from γ ∗ , the difference between ΦF and Φ∗F is almost negligible. Similarly, Φ◦F approaches Φ∗F at low SNR region. IV. VARIABLE T RANSMISSION PARAMETER S CHEME In this section, we investigate the secrecy throughput maximization with VTPS. Similar to FTPS, we still consider the case where S and R transmit the same codewords, and E can


adopt the MRC reception. Since the transmission parameters are designed based on the ICSI of the main channels, they are functions of g. From (7), we have the following relationship, max ΦV ⇔ max Eg [Rs (g)] ⇔ Eg [max (Rs (g))] ,

(27)

which implies that, if only we maximize Rs (g) under each g, the overall secrecy throughput ΦV is naturally maximized. Therefore, in the following, we concentrate on the secrecy rate maximization under given channel gains g. A. Optimal Solutions Considering the constraints for capacity outage, secrecy outage, and aggregate power, the problem for instantaneous secrecy rate maximization can be formulated as follows max

T ,γ,Rt

Rs (g),

s.t. g ∈ T , T = ∅,

(28a) (28b)

Rt (g) ≤ min{CR , CD },

(28c)

0 ≤ Rs (g) ≤ Rt (g),

(28d)

Ps (g) ≤ ,

(28e)

0 < γ(g) < 1,

(28f)

where Ps (g) is given in (10). Since Ps (g) depends on the instantaneous channel gains g, (28e) is actually a conditioned SOP constraint. According to Lemma 1, (28e) is equivalent to Rs (g) ≤ Rt (g) − log (1 + P0 κ(γ)) .

(29)

The solution to (28a) is given in the following proposition. Proposition 4: For given main channel gains g, the SOP constrained maximum secrecy rate in (28a) is given by 1+γ ∗ (g)P0 gSR log ∗ 1+P0 κ(γ ∗ ) , g ∈ T , Rs (g) = (30) 0, g ∈ T , with the optimal parameters designed as follows gSR gRD κ1 , gRD > κ2 } , (35) gRD , g ∈ T o, γ o (g) = gSR +gRD (36) 0, g ∈ T o , transmiswhere κ1 = GSE ln 1, κ2 = GRE ln 1 , the partial gRD κ2 i and T ii = sion regions T = g : gSR > κ1 , gSR > κ1 SR g : gRD > κ2 , ggRD > κκ12 . Proof: Please see Appendix D. Compared with the transmission region in (31), the new transmission region in (35) provides a more compact and intuitive result, which makes the transmission decision much more straightforward. For example, when gSR < κ1 or gRD < κ2 , the transmissions must be suspended. Remark 3: Compared with MRC reception, DWL case represents an optimistic design on secrecy, where the rate leakage to E is determined by the stronger wiretap link. The design for the DWL case is more applicable to the scenario where E locates much closer to S (or R) than to the other. The transmission probability can be calculated from (35) GSE 1 GRE 1 ln − ln Pto = P{g ∈ T o } = exp − . (37) GSR GRD 4 Without the ICSI of the wiretap channels, the secrecy outage performance is mainly influenced by the average SNR of the wiretap links. Note that, the large-scale path loss plays a primary role causing signal attenuation, hence the relative distances between nodes are very critical to the secrecy.

1748


Fig. 6. Secrecy throughput vs. system total power for different (dSE , dRE ) pairs, with = 0.3, α = 6, dSR = 1, and dRD = 0.8.

Fig. 7. Transmission regions for FTPS and VTPS. The transmission regions for FTPS, VTPS and DWL case are given from (19), (31), and (35).

The secrecy throughput can be calculated from (7) and (34) 1 o o ΦV = R (g)fgSR (x)fgRD (y)dxdy. (38) 2 s To

Although the analytical solution of (38) is complicated to obtain, we can derive a closed-form expression at high SNR regime. Considering P0 → ∞, from (34), we have Rs (g) = when g ∈ T i and Rs (g) = log gκRD when g ∈ T ii . log gκSR 1 2 Therefore, the maximum secrecy throughput can be given by 1 ΦV = R (g)fgSR (x)fgRD (y)dxdy 2 s To

=

GSE 1 GRE 1 1 Ei ln + ln , 2 ln 2 GSR GRD

(39)

Δ ∞ −t where Ei(x) = x e t dt. We see that, at high SNR regime, the maximum secrecy throughput converges to a constant which depends only on the SOP constraint and the relative distances between nodes. In other words, increasing power may not be quite effective to enhance the secrecy throughput. Fig. 6 depicts the exact maximum secrecy throughput Φ∗V of VTPS and its approximation ΦoV of the DWL case. We see that, in both figures, ΦoV approaches Φ∗V for a wide region of P0 , which has verified the accuracy of the results for the DWL case. Besides, ΦoV converges to ΦV when P0 becomes large enough, which indicates that ΦV is exactly a lower-complexity alternative to ΦoV at high SNR regime. We also find that, the secrecy throughput performance is more degraded when dSE < dRE compared with dSE > dRE . The reason is that, as dSR > dRD , more power is likely to be allocated to S according to (32) and (36), and when E is closer to S than to R, its capacity becomes higher, which consequently results in a more serious reduction of the secrecy throughput. Fig. 7 illustrates the relation between the transmission regions of FTPS and VTPS. The DWL region with reference point A(κ1 , κ2 ) indicates that, if gSR < κ1 or gRD < κ2 , the

Fig. 8. Relative secrecy throughput gain of VTPS over FTPS vs. system power for different SOP constraints, with α = 4, dSR = dRD = 1, dSE = 1.6, and dRE = 1.4.

transmissions are suspended. Each solid line represents the bound of the transmission region with the same maximum Rs achieved in VTPS. The value of Rs increases along the dash arrow, which indicates that, for better main channels’ quality, a higher Rs can be obtained. Each dash line represents the bound of the maximum transmission region achieved under a certain Rs requirement in FTPS. For both FTPS and VTPS, the direction of the dash arrow indicates the transmission region, while the opposite direction indicates the suspended region. From Fig. 7, we find that, for a fixed Rs , the transmission region for FTPS is a portion of that for VTPS. In fact, the FTPS can be regarded as a special case of VTPS with fixed parameters. Each point on the solid line w.r.t. a given Rs implies a specific power allocation for FTPS. Only the point along the dash arrow is the optimal that provides a maximum transmission region, or, maximum transmission probability. In Fig. 8, we show the relative secrecy throughput gain ΔΦ of VTPS over FTPS. As expected, VTPS significantly


outperforms FTPS. ΔΦ becomes larger with a decreasing P0 or a decreasing , which indicates that the superiority of VTPS over FTPS becomes more remarkable under a lower system power or a harsher SOP constraint; while on the contrary, the FTPS may be considered due to its much lower complexity.

1749

where T (γ(g)) is the unique root of the equation Y (T ) = , with Y (T ) expressed as 1+(1−γ)P gRD −T 1+γP gSR −T − T (1−γ)P0 G − T γP0 G 0 0 SE RE Y (T ) = 1− 1−e 1−e , (44) and the optimal parameters are designed as below

V. VTPS W ITH D IFFERENT C ODEWORDS OR C ODEBOOKS To further improve the secrecy throughput, S and R can use different codewords or even different codebooks. In these cases, E can no longer adopt the MRC method to receive signals from S and R, but only individually decodes the messages in each hop, which degrades its wiretap capability greatly. Therefore, the overall SOP turns out to be a combination of the individual SOP of each hop, just as defined in (3) with Rt,1 (g)−Rs (g) −1 2 , (40) Ps,1 (g) = exp − γ(g)P0 GSE

2Rt,2 (g)−Rs (g) − 1 Ps,2 (g) = exp − (1 − γ(g)) P0 GRE

.

(41)

Then the overall SOP Ps (g) can be given by R (g) 2 t,1 −T (g) − T (g)γ(g)P G 0 SE Ps (g) = 1 − 1 − e × 1−e

2

Rt,2 (g)

−T (g) 0 GRE

− T (g)(1−γ(g))P

,

(42)

Δ

with T (g) = 2Rs (g) . Note that, for the case with identical codeword rates, Rt (g) = Rt,1 (g) = Rt,2 (g). In the following two subsections, we consider 1) S and R use independent codewords (or different codebooks) with identical Rt ; 2) S and R use different codebooks with different Rt ’s. For convenience, we name them VTPS-D1 and VTPSD2, respectively.5 Intuitively, both VTPS-D1 and VTPS-D2 will outperform VTPS due to the degradation of eavesdropper’s wiretap capability. Besides, VTPS-D2 should be superior to VTPS-D1, since Rt is bottlenecked by the weaker-quality link of the main channels in VTPS-D1, while having different Rt ’s in VTPS-D2 can potentially support a higher Rs . A. Optimization of VTPS-D1 Similar to VTPS, the optimization problem of VTPS-D1 can be formulated as (28a), simply with Ps (g) given in (42), where Rt,1 (g) = Rt,2 (g) = Rt (g). In the following proposition, we give the solution to the new problem for VTPS-D1. Proposition 6: For given main channel gains g, the SOP constrained maximum secrecy rate in VTPS-D1 is log T (γ ∗ (g)) , g ∈ T , Rs∗ (g) = (43) 0, g ∈ T , 5 The

extension of FTPS can be performed in a similar way, which is omitted in the paper.

T = ∗

g

g:e

γ (g) =

g

− GSR

SE

+e

gRD gSR +gRD ,

0,

− GRD

RE

−e

g

− GSR

SE

g

e

− GRD

RE

< , (45)

g∈T, g ∈ T ,

(46)

Rt∗ (g) = log (1 + γ ∗ (g)P0 gSR ) .

(47)

Proof: Please see Appendix E. We provide a convenient decision criterion for transmission in (45). When g ∈ T , we obtain the maximum secrecy rate Rs∗ (g) through solving the equation Y (T ) = with γ = gRD ∗ gSR +gRD . The maximum secrecy throughput ΦV 1 , can be calculated according to (27). Note that, we obtain a closed-form power allocation γ ∗ (g) independent of P0 , which is just the same as that of VTPS. The optimal γ ∗ (g) concurrently maximizes the secrecy rate and the capacity of the two-hop main channels, which implies that to enlarge Rs∗ , we should maximize Rt first. Remark 4: Transmission region T in (45) actually implies that, for any g ∈ T , a positive Rs that satisfies the SOP constraint can be always obtained with the optimal γ ∗ . However, for some other γ’s, it may not be true. In fact, for the existence of such a positive Rs , there is a feasible region of γ, which is characterized by (γmin , γmax ). According to Appendix E, γmin and γmax can be calculated by solving the equation Ps (g)|Rs (g)=0 = , with Rt (g) substituted by log(1+γP0 gSR ) and log(1 + (1 − γ)P0 gRD ), respectively. B. Optimization of VTPS-D2 In VTPS-D2, S and R use different codebooks with codeword rates Rt,1 (g), and Rt,2 (g). Similar to VTPS-D1, the optimization problem can be formulated as follows max

T ,γ,Rt,1 ,Rt,2

Rs (g),

(48a)

s.t. (28b), (28e), (28f),

(48b)

Rt,1 (g) ≤ CR , Rt,2 (g) ≤ CD ,

(48c)

0 ≤ Rs (g) ≤ min {Rt,1 (g), Rt,2 (g)} ,

(48d)

with the solution presented in the following proposition. Proposition 7: For given main channel gains g, the SOP constrained maximum secrecy rate in (48a) is given by Rs∗ (g)

=

log T (γ ∗ (g)) , 0,

g∈T, g ∈ T ,

(49)

1750


where T (γ(g)) has already been defined in Proposition 6. The optimal parameters are designed as follows g g g g − SR − RD − SR − RD T = g : e GSE + e GRE − e GSE e GRE < , (50) γ (g), g ∈ T , γ ∗ (g) = (51) 0, g ∈ T , ∗ (g) = log (1 + γ ∗ (g)P0 gSR ) , Rt,1

(52)

∗ Rt,2 (g) = log (1 + (1 − γ ∗ (g)) P0 gRD ) ,

(53)

with γ (g) in (51) the unique root of the following equation ψ1 (γ) − ψ2 (γ) = 0,

(54)

where ψ1 (γ) =

ψ2 (γ) =

e

e

−

1+(1−γ)P0 gRD −T T (1−γ)P0 GRE

(1 − γ)2 GRE 1+γP gSR −T − T γP0 G 0

SE

γ 2 GSE

1+γP0 gSR −T − 1 − e T γP0 GSE , (55)

1−e

−

Fig. 9. Secrecy rate vs. power allocation for different power, with α = 4, = 0.3, gSR = gRD = 2, dSR = dRD = 1, dSE = 1.5, and dRE = 1.2.

−T

1+(1−γ)P0 gRD T (1−γ)P0 GRE

. (56)

Proof: Please see Appendix F. Note that, the transmission region T in VTPS-D2 is the same as that in VTPS-D1. For any g ∈ T , we can efficiently obtain the optimal γ ∗ (g) through Newton method according to (54), and determine the code rates shown in (49), (52), and (53), respectively. The maximum secrecy throughput, denoted as Φ∗V 2 , can be numerically calculated according to (27). Different from (46), the optimal γ ∗ (g) in (51) is an implicit function of P0 , which makes the relation between the maximum Rs∗ (g) and the power P0 not such straightforward. However, in the following corollary, we reveal how the maximum Rs∗ (g) varies w.r.t. P0 . Corollary 1: For any given g ∈ T , the maximum Rs∗ (g) in (49) is a monotonically increasing function of P0 , and converges to a constant Rs as P0 → ∞, where Rs is the unique root of the equation below e

−

gSR 2Rs GSE

+e

−

gRD 2Rs GRE

−e

−

gSR 2Rs GSE

e

−

gRD 2Rs GRE

= .

(57)

Proof: For any fixed γ irrespective to P0 , recalling ∂Y ∂Y Y (T ) = in (44), we prove that dT dP0 = − ∂P0 / ∂T > 0, i.e., Rs (γ) is an increasing function of P0 . (1) (2) Consider P0 > P0 , and denote the optimal γ that maxi(1) mizes Rs as γ and γ (2) , then we have

(a) (b) (1) (1) (2) Rs P0 , γ (1) ≥ Rs P0 , γ (2) > Rs P0 , γ (2) , (58) (1)

where (a) holds because γ (1) is optimal for P0 , and (b) is due to the monotonicity of Rs (γ) w.r.t. P0 . Then, we conclude that with a larger P0 , the maximum Rs∗ becomes higher. As P0 → ∞, (44) can be simplified as (57). Solving (57), we can obtain the converged Rs . Corollary 1 indicates that, although Rs∗ can be improved through increasing P0 , it levels off at high enough P0 region. The underlying reason is that, increasing power improves not only the main channels, but also the wiretap channels.

RD Setting γ(g) = gSRg+g in (51), VTPS-D2 degenerates into RD VTPS-D1. In other words, the optimal solution to VTPS-D1 is just sub-optimal to VTPS-D2. Note that, the converged Rs in (57) is independent of γ. Interesting, although the optimal power allocation of VTPS-D1 and VTPS-D2 may be different, their high-SNR maximum secrecy rates stay the same. Fig. 9 depicts the maximum secrecy rates Rs∗ ’s of VTPS-D1 and VTPS-D2 versus γ for different P0 ’s. We see that, for a given P0 , Rs∗ first rises and then decreases with the increase of γ.6 The underlying reason is that, with more power allocated to S, the capacity of E reduces as E is closer to R than to S; After γ exceeds the optimal γ ∗ , allocating more power to S will reduce the main channel capacity of the second hop, by which Rs∗ is bottlenecked. As shown in Corollary 1, Rs∗ becomes higher with an increasing P0 , and converges to a constant as P0 → ∞. We also observe that, with the increase of P0 , the optimal γ ∗ of VTPS-D2 decreases. It is because that, Rs∗ is limited by the second hop, such that to achieve a higher Rs∗ , more power should be given to R. While for VTPS-D1, the optimal γ ∗ keeps fixed, just as shown in (46). As expected, the optimal point of VTPS-D1 stops on the curve of VTPS-D2, since VTPS-D1 is a sub-optimal choice of VTPS-D2. In Fig. 10, we compare the secrecy throughput performance of the proposed adaptive schemes by evaluating the relative secrecy throughput gain. The left-hand figure shows the relative gain ΔΦ1 of VTPS-D1 over VTPS. As expected, due to the degradation of eavesdropper’s wiretap capability, VTPS-D1 achieves a considerable improvement of secrecy throughput over VTPS. Besides, the improvement is more remarkable for a higher power or a lower tolerable SOP. The right-hand figure depicts the relative throughput gain ΔΦ2 of VTPS-D2 over VTPS-D1. We find that, VTPS-D2 is always superior to VTPSD1 due to a more flexible design with different codeword rates at S and R, and the advantage is more highlighted under a lower

6 As pointed out in Remark 4, there is a feasible region of γ ∈ (γ min , γmax ) for g ∈ T in VTPS-D1.


1751

The first-order derivative of J(γ) w.r.t. γ is given by

dκ(γ) P T γ − κ(γ) − (T − 1) 0 dγ dJ(γ) = 2 dγ P0 γ GSR

P0 T (1 − γ) dκ(γ) + κ(γ) + (T − 1) dγ + . P0 (1 − γ)2 GRD

(60)

First, we calculate dκ(γ) dγ through the derivative rule for implicit functions from the transcendental equation F (κ(γ)) = as ∂F (κ(γ)) ∂F (κ(γ)) dκ(γ) =− , (61) dγ ∂γ ∂κ where Fig. 10. Relative secrecy throughput gain vs. system power for different SOP constraints, with α = 4, dSR = dRD = 1, dSE = 1.6, and dRE = 1.2.

power and a harsher SOP constraint. Whereas, at high-SNR regime, VTPS-D1 and VTPS-D2 provide an extremely close secrecy throughput performance.

∂F (κ(γ)) ∂κ(γ)

and

−

A PPENDIX A P ROOF OF P ROPOSITION 2 In this appendix, we derive the optimal γ ∗ that maximizes Pt (Rs , γ) in (20). Introduce an auxiliary function J(γ) =

(1 + P0 κ(γ)) T − 1 (1 + P0 κ(γ)) T − 1 + , (59) γP0 GSR (1 − γ)P0 GRD

and we have Pt (Rs , γ) = exp(−J(γ)). Clearly, to maximize Pt (Rs , γ) is equivalent to minimize J(γ).

can be respectively given by

κ(γ)

κ(γ)

−

e γGSE − e (1−γ)GRE ∂F (κ(γ)) =− < 0, ∂κ(γ) γGSE − (1 − γ)GRE κ(γ) κ(γ) κ(γ) ∂F (κ(γ)) = ∂γ

− Hence,

dκ(γ) dγ

e

−

γGSE

γ

+

e

−

(1−γ)GRE

1−γ

(γGSE − (1 − γ)GRE )2

.

(63)

can be given from (61)

dκ(γ) = dγ

κ(γ) −β(γ) + κ(γ) γ e 1−γ e−β(γ) − 1

+

κ(γ) , γ(1 − γ)β(γ)

(64)

Δ κ(γ) κ(γ) γGSE − (1−γ)GRE (here, we assume β(γ) = Δ 1 1 λ(γ) = β(γ) + e−β(γ) , and dκ(γ) dγ can be rewritten −1

where β(γ) = Define

(62)

γGSE − (1 − γ)GRE κ(γ) κ(γ) − (1−γ)G − γG SE RE −e GSE GRE e

VI. C ONCLUSION In this paper, we provide a comprehensive study on the design of secrecy transmission in DF relay networks for maximizing the secrecy throughput under an SOP constraint. We investigate four schemes, namely, FTPS and VTPS, VTPSD1 and VTPS-D2, respectively. For all these schemes, we provide explicit design solutions on the optimal parameters and derive some analytical high-SNR approximate results. We compare the secrecy throughput performance among the proposed schemes, and conclude that: 1) Due to the adaptive design, VTPS significantly outperforms FTPS though at the cost of complexity. The superiority is more dominant under a lower power or a smaller SOP threshold. 2) Because of the degradation of eavesdropper’s decode capability, both VTPSD1 and VTPS-D2 considerably improve the secrecy throughput performance compared with VTPS. Besides, their advantages over VTPS are more remarkable under a higher power or a harsher SOP constraint. 3) With a more flexible design with different codeword rates at S and R, VTPS-D2 is always superior to VTPS-D1, and the secrecy throughput gain of VTPS-D2 over VTPS-D1 is more highlighted under a higher level SOP constraint or a lower power.

∂F (κ(γ)) ∂γ

dκ(γ) κ(γ) = dγ γ

1+

λ(γ) 1−γ

0). as

.

(65)

Plugging (65) into (60) yields dJ(γ) GRD = J2 (γ) J1 (γ) − , dγ GSR Δ

Δ

(66) Δ

κ(γ)T ϕ2 (γ) 1 (γ) where J1 (γ) = ϕ 2 (γ) = P0 γ 2 (1−γ)2 GRD , with ϕ1 (γ) = ϕ2 (γ) and J

Δ −1)γ (T −1)(1−γ) and ϕ γ P0 (1+λ(γ))+ (T (γ) = (1−γ) − 2 T κ(γ) T κ(γ) P0 λ(γ) . We can easily prove that −1 < λ(γ) < 0 always holds, hence, we have ϕ1 (γ), ϕ2 (γ) > 0. We can readily see that J2 (γ) > 0 always holds, and from the expressions of ϕ1 (γ) and ϕ2 (γ), we have J1 (γ) ∈ (0, +∞) for γ ∈ (0, 1). Evidently, if J1 (γ) monotonically increases w.r.t. γ, there must exist a unique γ ∗ that makes dJ(γ) dγ first negative and then positive after γ exceeds γ ∗ . That is, we can prove that J(γ) is a quasi-convex function of γ, i.e., J(γ) first monotonically decreases and then monotonically increases with the rise of γ.

1752


The above γ ∗ is the optimal value that minimizes J(γ), and RD can be obtained by setting J1 (γ) = G GSR . Now, the optimization problem (20) is turned into the proof of the monotonicity of J1 (γ) w.r.t γ. Calculating the derivative of ϕ1 (γ) w.r.t. γ, we have ⎞

⎛ dκ(γ) κ(γ)−γ dγ dλ(γ) T −1 dϕ1 (γ) ϕ1 (γ) ⎠. = +γ ⎝P0 + dγ γ dγ T κ2 (γ)

where f1 (γ ) and f2 (γ ) have been defined in Proposition 3. From (25), we can easily see that γ does not depend on Rs , such that f1 (γ ) and f2 (γ ) are independent of Rs . Now, we derive the optimal Rs that maximizes ΦF (Rs ). The first-order derivative of ΦF (Rs ) w.r.t. Rs is given by dΦF (Rs ) 1 f1 (γ )−f2 (γ )2Rs 1−f2 (γ )Rs 2Rs ln 2 . (74) = e dRs 2

(67) 1 (γ) in (65) into (67), dϕdγ is rewritten as dϕ1 (γ) dλ(γ) ϕ1 (γ) (T − 1)λ(γ) = + γ P0 − . (68) dγ γ dγ T (1 − γ)κ(γ)

Substituting

dκ(γ) dγ

Recall the expression of λ(γ), and dλ(γ) dγ can be given by dλ(γ) e−β(γ) dβ(γ) 1 = . (69) 2 − 2 −β(γ) dγ β (γ) dγ e −1

f2 (γ )Rs 2Rs ln 2 = 1. Δ

1

Substituting

dκ(γ) dγ

(70)

T T = 2 f2 (γ ) ln 2 .

1 (γ) Since λ(γ) < 0, from (68), we have proven that dϕdγ > 0, i.e., ϕ1 (γ) monotonically increases w.r.t. γ. Similarly, we prove that ϕ2 (γ) monotonically decreases w.r.t. γ. Given that both ϕ1 (γ) and ϕ2 (γ) are positive, then we can infer that, J1 (γ) = ϕ1 (γ) ϕ2 (γ) is a monotonically increasing function of γ. By now, we have proven the monotonicity of J1 (γ) w.r.t. γ. Therefore, we conclude that J(γ) is a quasi-convex function of γ. In other words, Pt (Rs , γ) is a quasi-concave function of γ, and the optimal γ ∗ that maximizes Pt (Rs , γ) is the unique root RD of the equation J1 (γ) = G GSR .

A PPENDIX C P ROOF OF P ROPOSITION 4 From (28d) and (29), to maximize Rs (g), Rt (g) should first be set at its maximum. From (28c), we have Rtmax (g) = min{CR , CD }.

When P0 → ∞, (21) can be approximately transformed into (25), with which we can derive the optimal γ that maximizes Pt (Rs , γ) in (17). Substituting γ into Pt (Rs , γ) yields (72)

and the corresponding secrecy throughput in (23) is given by (73)

(78)

The maximum Rs (g) can be given from (29)7 + + 1+P1 gSR 1+P2 gRD Rs (g) = min log , log , (79) 1+P0 κ(γ) 1+P0 κ(γ) Δ

where [x]+ = max(x, 0) with x a real number. From (79), we see that, to afford a positive Rs (g) that satisfies the SOP constraint for a given γ(g), the channel gains g should satisfy gSR >

A PPENDIX B P ROOF OF P ROPOSITION 3

1 ΦF (Rs ) = Rs exp f1 (γ ) − f2 (γ )2Rs , 2

1

Further define f3 (γ ) = 2 f2 (γ ) ln 2 , then T can be derived as ln f3 (γ ) ln f3 (γ ) ⇒ R T = = log . (77) s W (ln f3 (γ )) W (ln f3 (γ ))

(71)

The right-hand-side (RHS) inequation in (71) holds for the fact that −1 < λ(γ) < 0. Plugging (71) into (69), we have dλ(γ) dγ > 0.

Pt (Rs ) = exp f1 (γ ) − f2 (γ )2Rs ,

(76)

Substitute (77) into (73), and we obtain the final ΦF in (24).

into (70) yields

dβ(γ) κ(γ)λ(γ) κ (γ)(1 + λ(γ)) = 2 − < 0. dγ γ (1 − γ)GSE γ(1 − γ)2 GRE

(75)

Define T = 2Rs , and (75) can be rewritten as

−β(γ)

e We can simply prove that (e−β(γ) − β 21(γ) < 0 always −1)2 holds, and from the expression of β(γ), we have

γ dκ(γ) (1 − γ) dκ(γ) dβ(γ) dγ − κ(γ) dγ + κ(γ) = − . 2 dγ γ GSE (1 − γ)2 GRE

Rs

Note that ef1 (γ )−f2 (γ )2 > 0, and 1 − f2 (γ )Rs 2Rs ln 2 monotonically decreases w.r.t. Rs , then we can easily conclude F (Rs ) is first positive and then negative when Rs inthat dΦdR s creases from 0 to ∞. It implies that ΦF (Rs ) first increases and then decreases. Hence, there is a unique Rs that maximizes ΦF (Rs ), and the optimal Rs can be obtained by setting dΦF (Rs ) = 0, i.e., it is the root of the following equation dRs

κ(γ) , γ(g)

gRD >

κ(γ) . 1 − γ(g)

(80)

With (80), we can determine the transmission region T from κ(γ) (5), with μ1 and μ2 replaced by κ(γ) γ(g) and 1−γ(g) , respectively. With (78)–(80), we now maximize Rs (g) by optimizing γ(g). From (79), the optimization problem is equivalent to 1 + γ(g)P0 gSR 1 + (1 − γ(g)) P0 gRD Δ , max Q(γ) = min γ 1 + P0 κ(γ) 1 + P0 κ(γ) s.t. (80), 7 We

0 < γ(g) < 1.

use κ(γ) instead of κ(γ(g)) for notation simplicity here.

(81)


We see that, for given main channel gains g, once any γ violates (80), the transmissions are suspended. Next, we only consider the case when (80) is established. RD , we have Q(γ) = If γgSR ≤ (1 − γ)gRD , i.e., γ ≤ gSRg+g RD 1+γP0 gSR 1+P0 κ(γ) .

The first-order derivative of Q(γ) is given by P0 Q1 (γ) dQ(γ) = , dγ (1 + P0 κ(γ))2

(82)

Δ

where Q1 (γ) = gSR (1+P0 κ(γ)) − (1 + γP0 gSR ) dκ(γ) dγ . Since γgSR > κ(γ), we have gSR (1+P0 κ(γ)) > κ(γ) γ +P0 gSR κ(γ). Substituting this inequation into Q1 (γ) yields Q1 (γ) >

Recall

dκ(γ) dγ

1 P0 gSR + γ

dκ(γ) κ(γ) − γ dγ

.

(83)

Q1 (γ) >

P0 gSR +

1 γ

−κ(γ)λ(γ) 1−γ

is given by T i = g : gSR > κ1 , ggRD > SR

.

(84)

A PPENDIX D P ROOF OF P ROPOSITION 5 Recall (10), which gives an explicit relation between κ(γ) and γ, and we give an approximate κ(γ) for the DWL case in the following lemma. Lemma 2: For the DWL case with a given γ, κ(γ) is approximated by κo (γ), which is given by (85)

where κ1 = GSE ln 1 and κ2 = GRE ln 1 . Proof: For a certain γ, the average SNR’s of both wiretap links are α1 P0 and α2 P0 , with α1 and α2 defined in (10). Considering the average SNR of one wiretap link is much stronger than the other, e.g., α1 α2 . By ignoring

the influence of . Since κ(γ) is α2 in (10), we have F (κ(γ)) ≈ exp − κ(γ) α1 the root of the equation F (x) = , i.e., F (κ(γ)) = , then we κ(γ)

0 gSR we have Q(γ) = 1+γP 1+γP0 κ1 , and max Q(γ) ⇔ max γ; if γ > gRD gSR +gRD , we have max Q(γ) ⇔ min γ. In that case, the maxiRD . Similarly, for mum Q(γ) is achieved when γ = γ o = gSRg+g RD the curve of Q(γ) with κo (γ) = (1 − γ)κ2 , the optimal γ that RD . maximizes Q(γ) is still γ o = gSRg+g RD We have proven that, for both curves of Q(γ) with κo (γ) = RD . In γκ1 and κo (γ) = (1 − γ)κ2 , the optimal γ is γ o = gSRg+g RD fact, the real curve of Q(γ) is a combination of the above two 2 , i.e., ggRD > κκ21 , κo (γ) = curves (see (85)). When γ o > κ1κ+κ 2 SR o

> 0, i.e., when γ ≤ Since λ(γ) < 0, then Q1 (γ) > 0 ⇒ dQ(γ) dγ gRD , Q(γ) monotonically increases w.r.t. γ. gSR +gRD RD , Q(γ) Similarly, we can prove that, when γ > gSRg+g RD monotonically decreases w.r.t. γ. Therefore, the maximum RD . Q(γ) is achieved when γ = gSRg+g RD By now, we have proven that, once there exist some γ’s that satisfy (80), the optimal γ that maximizes Rs (g) in (79) is γ ∗ = gRD ∗ gSR +gRD . Substituting γ into (78)–(80), we can finally obtain the optimal results presented in Proposition 4.

κo (γ) = max {γκ1 , (1 − γ)κ2 } ,

Substituting κo (γ) into (79), we obtain the explicit form of Rs (g). To derive the maximum Rs (g) in (79), we should maximize Q(γ) in (81), simply with κ(γ) replaced by κo (γ). Similar to Proposition 4, we consider the case when γgSR > κo (γ) and (1 − γ)gRD > κo (γ) simultaneously hold, otherwise, the transmissions are off. For the entire curve of Q(γ) RD , with κo (γ) = γκ1 , if γgSR ≤ (1 − γ)gRD , i.e., γ ≤ gSRg+g RD

P0 gSR γ o κ1 , we have Rso (g) = log 1+γ 1+γ o P0 κ1 . To achieve a positive o condition Rs (g), gSR > κ1is required, and the transmission

in (65), then (83) can be rewritten as

1753

have e− α1 ≈ ⇒ κ(γ) ≈ α1 ln 1 . Similarly, when α1 α2 , κ(γ) is approximated by we have κ(γ) ≈ α2 ln 1 . Therefore, κo (γ) = max α1 ln 1 , α2 ln 1 .

κ2 κ1 . Similarly, when 1+(1−γ o )P0 gRD ii 1+(1−γ o )P0 κ2 , and T =

gRD < κκ21 , we have Rso (g) = log gSR SR g : gRD > κ2 , ggRD > κκ12 . Combining T i and T ii , the overall transmission region is given in (35).

A PPENDIX E P ROOF OF P ROPOSITION 6 Similar to Appendix C, we get a maximum Rt (g) in (78). Recalling (42), we find Ps (g) monotonically rises with an increasing Rs (g) for a given γ. In that case, the maximum Rs (g) that satisfies the SOP constraint is achieved when Ps (g) = . Denote T (γ) as the unique root of the equation Ps = , then we derive the optimal γ(g) to maximize T (γ). RD , and substitute Rtmax into Ps = , Consider γ < gSRg+g RD then the derivative of T (γ) w.r.t. γ can be given by ∂Ps dT (γ) =− dγ ∂γ We easily have

∂Ps ∂T

> 0, and

∂Ps ∂γ

∂Ps . ∂T

(86)

can be calculated by

1+γP0 gSR −T T −1 ∂Ps − =− 2 e T γP0 GSE ∂γ T γ P0 GSE

0 gSR −T 1 + P0 gSR − T − T1+γP − e (1−γ)P0 GRE 2 T (1 − γ) P0 GRE

1−e

−T

1+γP g

0 SR − T (1−γ)P G 0

RE

1+γP gSR −T − T γP0 G 0 SE 1−e . (87)

max

Since 1 < T = 2Rs ≤ 2Rt

= 1 + γP0 gSR , we have

∂Ps ∂γ

0, i.e., T (γ) monotonically increases w.r.t. RD . Quite similarly, we can prove that T (γ) γ when γ < gSRg+g RD RD . Clearly, monotonically decreases w.r.t. γ when γ ≥ gSRg+g RD gRD ∗ the optimal γ that maximizes T (γ) is γ = gSR +gRD .

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Plugging γ = γ ∗ into Ps , we get Y (T ) in (44). The maximum Rs∗ = log T (γ ∗ ) should be positive, i.e., T (γ ∗ ) > 1. It is equivalent to Y (T (γ ∗ )) = > Y (1), where Y (1) reflects the minimum achievable SOP Psmin (g) for a positive Rs . Hence, the transmission region can be given by T = {g : Y (1) < } , g

− GSR

where Y (1) = e

SE

g

+e

− GRD

RE

−e

g

− GSR

SE

(88) g

e

− GRD

RE

.

A PPENDIX F P ROOF OF P ROPOSITION 7 From (48d), to maximize Rs (g), we should maximize both Rt,1 (g) and Rt,2 (g). From (48c), for a given γ(g), we have max Rt,1 (g) = CR = log (1 + γ(g)P0 gSR ) ,

(89)

max Rt,2 (g) = CD = log (1 + (1 − γ(g)) P0 gRD ) .

(90)

Next, we determine the transmission region T . Substitute max max (g) and Rt,2 (g) into Ps (g) in (42), then Ps (g) can be Rt,1 transformed as Y (T ) in (44). Similar to the proof of (88), we obtain the transmission region T shown in (50). We derive the derivative of T (γ) w.r.t. γ from (44) as dT (γ) ∂Y ∂Y =− / , dγ ∂γ ∂T where

∂Y ∂T

> 0 always holds, and

∂Y ∂γ

(91)

is given by

T −1 ∂Y = (ψ1 (γ) − ψ2 (γ)) , ∂γ T P0

(92)

with ψ1 (γ) and ψ2 (γ) given in (55) and (56), respectively. We can readily observe that ψ1 (γ) and ψ2 (γ) are monotonically increasing and decreasing functions of γ, respectively. Hence, ∂Y ∂γ in (92) is an increasing function of γ. Since ∂Y ∂Y ∂Y ∂γ |γ=0 = −∞, and ∂γ |γ=1 = +∞, we see that, ∂γ is fist negative and then positive with the increase of γ, and the zerocrossing point of ∂Y ∂γ exists when ψ1 (γ) = ψ2 (γ).

From (91), we can finally conclude that dTdγ(γ) is a first positive and then negative function of γ. Therefore, T (γ) is a quasi-concave function of γ, and the optimal γ ∗ that maximizes T (γ) is the unique root of the equation ψ1 (γ) = ψ2 (γ). R EFERENCES [1] A. D. Wyner, “The wire-tap channel,” Bell Syst. Tech. J., vol. 54, no. 8, pp. 1355–1387, Oct. 1975. [2] A. Mukherjee, S. Fakoorian, J. Huang, and A. Swindlehurst, “Principles of physical layer security in multiuser wireless networks: A survey,” IEEE Commun. Surveys Tuts., vol. 16, no. 3, pp. 1550–1573, 3rd Quart. 2014. [3] L. Lai and H. El Gamal, “The relay-eavesdropper channel: Cooperation for secrecy,” IEEE Trans. Inf. Theory, vol. 54, no. 9, pp. 4005–4019, Sep. 2008. [4] I. Krikidis, J. Thompson, and S. Mclaughlin, “Relay selection for secure cooperative networks with jamming,” IEEE Trans. Wireless Commun., vol. 8, no. 10, pp. 5003–5011, Oct. 2009. [5] J. Li, A. P. Petropulu, and S. Weber, “Secrecy rate optimization under cooperation with perfect channel state information,” in Proc. 43rd Asilomar Conf. Signal, Syst. Comput., Nov. 2009, pp. 824–828.

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Tong-Xing Zheng (S’14) received the B.S. degree from School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an, China, in 2010. He is currently pursuing the Ph.D. degree in the School of Electronics and Information Engineering, Xi’an Jiaotong University. His research interests are in the area of cooperative communications systems and physical-layer security of wireless communications.

Feng Liu received the B.S. degree from School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an, China, in 2013. Currently, he is pursuing the M.S. degree in the School of Electronics and Information Engineering, Xi’an Jiaotong University. His research interests are in the area of cooperative communications and physical layer security.

Hui-Ming Wang (S’07–M’10) received the B.S. and Ph.D. degrees in electrical engineering from Xi’an Jiaotong University, Xi’an, China, in 2004 and 2010, respectively. He is currently a Full Professor with the Department of Information and Communications Engineering, Xi’an Jiaotong University, and also with the Ministry of Education Key Laboratory for Intelligent Networks and Network Security, China. From 2007 to 2008, and 2009 to 2010, he was a Visiting Scholar at the Department of Electrical and Computer Engineering, University of Delaware, USA. His research interests include cooperative communication systems, physicallayer security of wireless communications, MIMO, and space-timing coding. Dr. Wang received the National Excellent Doctoral Dissertation Award in China in 2012, a Best Paper Award of International Conference on Wireless Communications and Signal Processing, 2011, and a Best Paper Award of IEEE/CIC International Conference on Communications in China, 2014.

Moon Ho Lee received the Ph.D. degree from Chonnam National University, Korea in 1984, and from the University of Tokyo, Japan, in 1990, both in electrical engineering. He is a Professor and former Chair of the Department of Electronics Engineering in Chonbuk National University, Korea. He was with the University of Minnesota, USA, from 1985 to 1986 as a Postdoctoral Researcher. He was with Namyang MBC Broadcasting as a Chief Engineer from 1970 to 1980, after which he joined Chonbuk National University as a Professor. He has made significant original contributions in the areas of mobile communication code design, channel coding, and multi-dimensional source and channel coding. He has 116 patents. Dr. Lee is a member of the National Academy of Engineering in Korea and a Foreign Fellow of the Bulgaria Academy of Sciences. He is the inventor of Jacket Matrix and its entry on Wikipedia has been cited over 95,559 times as of December 2014.