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Physical Layer Security with Threshold-Based Multiuser Scheduling inMulti-antenna Wireless Networks
Yang, M., Guo, D., Huang, Y., Duong, T. Q., & Zhang, B. (2016). Physical Layer Security with Threshold-BasedMultiuser Scheduling in Multi-antenna Wireless Networks. IEEE Transactions on Communications.https://doi.org/10.1109/TCOMM.2016.2606396
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Download date:26. Jun. 2020
IEEE TRANSACTIONS ON COMMUNICATIONS 1
Physical Layer Security with Threshold-BasedMultiuser Scheduling in Multi-antenna Wireless
NetworksMaoqiang Yang, Student Member, IEEE, Daoxing Guo, Member, IEEE, Yuzhen Huang, Member, IEEE, Trung Q.
Duong, Senior Member, IEEE, and Bangning Zhang
Abstract—In this paper, we consider a multiuser downlinkwiretap network consisting of one base station (BS) equippedwith AA antennas, NB single-antenna legitimate users, and NE
single-antenna eavesdroppers over Nakagami-m fading channels.In particular, we introduce a joint secure transmission schemethat adopts transmit antenna selection (TAS) at the BS andexplores threshold-based selection diversity (tSD) scheduling overlegitimate users to achieve a good secrecy performance whilemaintaining low implementation complexity. More specifically, inan effort to quantify the secrecy performance of the consideredsystem, two practical scenarios are investigated, i.e., Scenario I:the eavesdropper’s channel state information (CSI) is unavailableat the BS, and Scenario II: the eavesdropper’s CSI is availableat the BS. For Scenario I, novel exact closed-form expressionsof the secrecy outage probability are derived, which are validfor general networks with an arbitrary number of legitimateusers, antenna configurations, number of eavesdroppers, andthe switched threshold. For Scenario II, we take into accountthe ergodic secrecy rate as the principle performance metric,and derive novel closed-form expressions of the exact ergodicsecrecy rate. Additionally, we also provide simple and asymptoticexpressions for secrecy outage probability and ergodic secrecyrate under two distinct cases, i.e., Case I: the legitimate user islocated close to the BS, and Case II: both the legitimate user andeavesdropper are located close to the BS. Our important findingsreveal that the secrecy diversity order is AAmA and the slopeof secrecy rate is one under Case I, while the secrecy diversityorder and the slope of secrecy rate collapse to zero under CaseII, where the secrecy performance floor occurs. Finally, when theswitched threshold is carefully selected, the considered schedulingscheme outperforms other well known existing schemes in termsof the secrecy performance and complexity tradeoff.
Index Terms—Physical layer security, multiuser switched di-versity, threshold-based scheduling scheme, secrecy outage prob-ability, ergodic secrecy rate.
I. INTRODUCTION
SECURITY and privacy have attracted enormous attentionin the wireless communications since the inherent broad-
cast nature of radio propagation makes the data transmission
This work was supported by the National Science Foundation of China(No. 61401508) and the Jiangsu Provincial Natural Science Foundation ofChina (No. BK20150719). The work of T. Q. Duong was supported in partby the U.K. Royal Academy of Engineering Research Fellowship under GrantRF1415\14\22.
M. Yang, D. Guo, Y. Huang, and B. Zhang are with the College of Commu-nications Engineering, PLA University of Science and Technology, Nanjing,China (e-mail: [email protected]; [email protected]; yzh [email protected];[email protected]).
T. Q. Duong is with the School of Electronics, Electrical Engineering andComputer Science, Queen’s University Belfast, Belfast BT7 1NN, U.K. (e-mail: [email protected]).
particularly susceptible to malicious attacks [1]. To addressintricate secure problems, many researchers have investigatedvarious cryptographic protocols in the upper layer, from whichan error-free link of physical layer was assumed to guaranteethe reliability of data transmission. As being well-known,the key idea behind traditional cryptographic techniques liesin the complex mathematical operations, which, however,have become increasingly insecure due to the fact that thecomputational ability of eavesdropper becomes more and morepowerful. Motivated by this limitation, physical layer security(PLS) has been introduced as an attractive approach to defendagainst malicious attack and wiretapping by exploiting thedistinct characteristics of different wireless channels. Theconcept of PLS was pioneered by Shannon in [2], and furtherextended by Wyner in [3], where the condition of perfectsecrecy was analyzed in view of an information-theoreticprospective. Later on, various advanced techniques, i.e., mul-tiple antennas, multiuser diversity, and cooperative relaying,have been broadly exploited to improve the PLS of wirelesstransmissions.
In recent years, significant research efforts have been de-voted to incorporating multi-antenna techniques to improve theperformance of wireless communication systems. Specifically,transmit antenna selection (TAS) has been widely investigateddue to the low realization complexity of radio frequency (RF)chains while achieving full diversity [4]–[6]. The authors in [7]analyzed the secrecy performance in multiple-input multiple-output (MIMO) wiretap channels with TAS and differentreceiver combining schemes. In [8], TAS with Alamouticoding and power allocation was addressed in MIMO wiretapchannels. Additionally, TAS and receive generalized selectioncombining (TAS/GSC) was proposed for improving the se-curity in the MIMO wiretap channels [9]. Furthermore, thework [10] investigated the secrecy performance for TAS andmaximal ratio combining (TAS/MRC) system with imperfectfeedback. More recently, TAS was also carried out both atthe multi-antenna primary and secondary transmitter (ST) incognitive wiretap systems [11], [12], where the ST acts as afriendly jammer to confound the eavesdropper and is grantedto share the spectrum of the primary network as a reward.Besides these works, the secrecy performance of antenna-selection-aided MIMO system was addressed in [13] in termsof the probability of zero secrecy capacity and generalizedsecrecy diversity.
On the other hand, multiuser diversity technique has aroused
IEEE TRANSACTIONS ON COMMUNICATIONS 2
much attention from both academia and industry. Differentfrom single user communication systems, multiuser communi-cation systems are more susceptible to malicious attacks [14]–[17]. In [14], the secrecy outage performance of multiuserMIMO systems with TAS and arbitrary number of eaves-droppers was addressed. In [15], the authors investigated thesecrecy performance of multiuser downlink networks with theartificial noise by designing an optimal power allocation tomaximize the total ergodic secrecy capacity of the system. In[16], to increase the secure degrees of freedom of the mainchannel, different opportunistic jammer selection schemeswere proposed in multiuser wiretap networks. In [17], anoptimal user selection scheme for multiuser relaying networkswith cooperative jamming was analyzed in terms of the ergodicsecrecy rate. In addition, the impact of different schedulingschemes on the secrecy performance of multiuser wiretappingnetworks has been investigated in [18]–[20]. Particularly, in[18], the authors have explored the PLS of cognitive radionetworks with different scheduling schemes and made acomprehensive comparison for the achievable secrecy rate.Later, in [19], an on-off opportunistic beamforming for themultiuser downlink channels with a passive eavesdropper wasinvestigated. In [20], two opportunistic scheduling algorithmsof multiuser uplink wiretap networks taking into considerationof the fairness among legitimate users were designed. Albeitthe above-mentioned works have improved our understandingon the impact of user scheduling on the secrecy performance,the main limitation behind these works is the high complexityof the scheduling schemes owing to the continuous estimationfor the channel state information (CSI) of the main channel.As a result, a significant proportion of the air-link resourceand battery life of the mobile terminals are utilized for theCSI feedback instead of valuable data transmission.
In an effort to reduce the CSI feedback load of tradi-tional scheduling schemes, the concept of multiuser switched-diversity opportunistic scheduling was firstly proposed in [21],which is to find any acceptable user instead of the bestone by exploiting a threshold-based and ordered schedulingmechanism. The main idea behind switched diversity is totrade off part of desired multiuser diversity gains for con-siderable savings with regard to the CSI feedback require-ments. In recent years, extensive efforts have been devotedto incorporating switched diversity in wireless communicationsystems [22]–[27]. However, few works have addressed thesecure transmission issue of the switched diversity basedopportunistic scheduling scheme. A recent work [28] firstlyaddressed a threshold-based branch selection scheme, namelyswitch-and-examine combining (SEC), to achieve a bettertradeoff between the secrecy outage performance and imple-mentation cost in classical wiretap networks. In [29], [30], amore preferred alternative termed SEC with post-examiningselection (SECps) was investigated in the multiuser downlinkwiretap networks. To be specific, the SEC or SECps schemeexamines the signal-to-noise ratio (SNR) of each branch in oneby one manner with a predefined switched threshold. Once anybranch exceeds the switched threshold, it will be selected fordata transmission. The difference between SEC and SECpsscheme exists in their last step where SEC keeps the last
examined branch, while SECps selects the best one instead.In particular, an improved threshold-based switched diversity(termed tSD) strategy was proposed in [31] and extended todifferent communication scenarios. Different from SEC andSECps scheme, the tSD scheme first examines whether theprevious selected branch exceeds the predetermined switchedthreshold. If this is the case, this branch is kept, otherwise thescheduler switches to the branch that yields the best channelquality.
While the aforementioned works have laid a solid founda-tion for the investigation of threshold-based switched diversityscheduling in multiuser downlink networks, the PLS issues inmultiuser multi-antenna networks with multiple eavesdroppersare still not well understood and the channel fading severity onthe secrecy performance of these networks remains unknown.
With this in mind, in this paper, we introduce a hybridscheme combining TAS with threshold-based selection diver-sity opportunistic scheduling, namely TAS/tSD, in a multiusermulti-antenna wiretap network over Nakagami-m channels.More specifically, we present a comprehensive secrecy perfor-mance analysis of the considered network under two practicaleavesdropping scenarios, i.e., Scenario I: The CSI of eaves-dropping channel is not available at the base station (BS),and Scenario II: The CSI of the eavesdropping channel isavailable at the BS1. The main contributions of this paper aresummarized as follows:
• For Scenario I, we first derive novel exact closed-formexpressions for the secrecy outage probability and non-zero secrecy rate with arbitrary transmit antenna con-figurations and channel fading severity, the number oflegitimate users, and the number of eavesdroppers aswell as the switched threshold, from which the impactof key system parameters on the secrecy performance ofmultiuser wiretap networks with TAS/tSD can be readilyevaluated.
• For Scenario I, in order to achieve more insights, wederive new closed-form approximate expressions of thesecrecy outage probability in high SNR regimes for twodifferent cases, i.e., Case I: the legitimate user is locatedclose to the BS and Case II: both the legitimate userand eavesdropper are located close to the BS. Accordingto the derived expressions, we find that for Case I,the secrecy diversity order is decided by the numberof transmit antennas and the channel fading severity ofmain channel, while for Case II, the secrecy diversityorder collapses to zero and the secrecy performance flooroccurs.
• For Scenario II, we derive new exact closed-form expres-sions for the ergodic secrecy rate, from which we canaccurately examine the impact of the system parameterson the ergodic secrecy rate of the considered networkswith TAS/tSD scheme.
• For Scenario II, we derive new closed-form approximateexpressions of the ergodic secrecy rate in the high SNR
1Similarly as in [1], [9], [33], [34], [40], we consider the case that theCSIs of the eavesdropper’s channels are obtained at the BS, which potentiallydemonstrates that their transmissions can be monitored.
IEEE TRANSACTIONS ON COMMUNICATIONS 3
Fig. 1: System model
regime. Based on the derived results, we characterizethe asymptotic ergodic secrecy rate in terms of the highSNR slope and high SNR power offset. Our findingsdemonstrate that the high SNR slope remains one forCase I, while collapses to zero for Case II. However, thehigh SNR power offset depends on the number of transmitantennas and the fading severity of main channel as wellas the involved parameters of the eavesdropping channel.
• Our results also demonstrate that the considered TAS/tSDscheme achieves a better secrecy performance than theTAS/SECps scheme at the expense of negligible increas-ing complexity. In addition, compared to the best schedul-ing scheme, the TAS/tSD scheduling scheme achieves asimilar secrecy performance, while the average numberof user examinations of TAS/tSD scheme is significantlyreduced.
Notation: We utilize bold lower/upper case symbols torepresent vectors/matrics. We denote |·| the absolute value and(··)
is the binomial coefficient. The notation O (·) representshigher order function, Pr [·] is the probability, the cumulativedistribution function (CDF) and the probability distributionfunction (PDF) of random variable (RV) X are denotedas FX (x) and fX (x), respectively. Ei (·) denotes the one-argument exponential integral function. E [·] stands for theexpectation operator and Γ (·) denotes the Gamma function.
II. SYSTEM AND CHANNEL MODELS
Let us consider a multiuser downlink wiretap network asin Fig. 1, which consists of one BS equipped with AA
antennas, NB single-antenna legitimate users (Bobs), and NE
single-antenna eavesdroppers (Eves). The system consideredin this paper is applicable to practical multiuser scenarios, forinstance, the Internet of Things (IoT) and Wireless SensorNetworks (WSN). The main link and eavesdropper’s link areassumed to be quasi-static independent and non-identicallydistributed (i.n.i.d.) block Nakagami-m fading, and remainunchanged during the coherence time. In addition, each blocktransmission time, being equal to the channel coherence time,
is composed of two parts, i.e., guard interval and data trans-mission interval. During the guard interval phase, BS firstselects a desired legitimate user and transmit antenna pairaccording to the considered TAS/tSD scheme. After that, thedata transmission is completed between the selected legitimateuser and transmit antenna in the following interval.
To achieve secrecy, the BS encodes the message blockw into the codeword x = [x (1) , ..., x (i) , ...x (n)] with1n
∑ni=1 E
[|x (i)|2
]≤ P according to capacity achieving
codebook for the wiretap channel. As such, the receivedinstantaneous SNRs at the k-th legitimate user and at the n-thEve associated with the α-th transmit antenna are expressed,respectively, as
γbα,k =P |hα,k|2
σ2b
(1)
and
γeα,n =P |gα,n|2
σ2e
(2)
where P denotes the transmit power of the BS, hα,k denotesthe channel coefficient between the α-th antenna and the k-thlegitimate user, gα,n represents the channel coefficient betweenα-th antenna and the n-th eavesdropper, σ2
b and σ2e denote
the additive white Gaussian noise (AWGN) variance at eachlegitimate user and eavesdropper, respectively.
In the following, we describe the basic principle of TAS/tSDscheme [31] in detail, from which an acceptable legitimate useris selected by the BS associated with a specific antenna.
• We begin with the first transmit antenna (α = 1) of thetSD scheme. The BS estimates whether the instantaneousSNR γbα,k of the previously selected legitimate user forthis transmit antenna exceeds the predetermined thresholdγT, i.e., if γtSDB,α = γbα,k > γT, the processing procedureof tSD terminates for this transmit antenna.
• Otherwise, once the instantaneous SNR of the previ-ously selected legitimate user is below the threshold, i.e.,γbα,k < γT, the legitimate user with the highest SNRis selected for this transmit antenna, wherein γtSDB,α =
max(γbα,1, γ
bα,2, ..., γ
bα,NB
). The same user scheduling
operation repeats for the remaining transmit antennassequentially.
• As per the TAS scheme, the transmit antenna and legit-imate user pair that results in the largest instantaneousSNR is selected for the following transmission interval.The index of selected transmit antenna α∗ is given by
α∗ = argmax1≤α≤AA
(γtSDB,α
)(3)
Without loss of generality, we assume that the selectionprocedure of legitimate user and transmit antenna pair isdone at a central scheduler (i.e., at the BS). Meanwhile,the requirement of feedback information includes not onlylegitimate user indexes, but also the received SNRs. Differentfrom the conventional application without secrecy constraint,the resulting largest SNR is exploited by the BS for the
IEEE TRANSACTIONS ON COMMUNICATIONS 4
construction of wiretap code2.We also remark that selecting the best antenna at the BS is
optimal for the legitimate channels, however, it corresponds toa random transmit antenna in the perspective of eavesdroppers.As such, the Eves could hardly achieve any additional transmitdiversity from the selected antenna.
Due to the fact that each eavesdropper has received thesignal from BS, we consider the most powerful colludingeavesdropper attacking scenario, where the eavesdroppers canshare their available observations to decode the confidentialmessages. From the eavesdropper’s design perspective, thecolluding eavesdropping scenario yields the best possibleperformance of illegitimate link while it represents the worstcase scenario in the viewpoint of secure transmission [35]–[37]. Additionally, all Eves are assumed to be perfectly col-luded where the inter-eavesdropper channels are considerederror-free. In doing so, the maximal ratio combining (MRC)scheme is adopted among the eavesdroppers to achieve thebest wiretapping performance.
III. SECRECY PERFORMANCE ANALYSIS
In this section, we present a comprehensive analysis on thesecrecy performance of the considered system with TAS/tSDscheme. Before delving into the detailed analysis, we firstpresent the statistical characteristics of legitimate channel andeavesdropping channel.
A. Preliminaries
Based on the aforementioned TAS/tSD scheme, it is obviousthat the events of selecting an acceptable user are mutuallyexclusive for a specific transmit antenna α, therefore, the CDFof the end-to-end instantaneous SNR γtSDB,α is given by [31]
FγtSDB,α
(γ) = Pr[γtSDB,α ≤ γ
]=
NB∑k=1
Pr[γtSDB,α = γbα,k&γ
bα,k ≤ γ
](4)
Besides, each legitimate link is assumed to follow indepen-dent and identically distributed (i.i.d.) Nakagami-m distribu-tion3. Following the same steps as developed in [31], thus theCDF of γtSDB,α can be rewritten as
FγtSDB,α
(γ) =
Fγb
α,k(γ)− Fγb
α,k(γT)
+Fγbα,k
(γT)[Fγb
α,k(γ)
]NB−1
, γ ≥ γT[Fγb
α,k(γ)
]NB
, γ < γT
(5)
where
Fγbα,k
(γ) = 1− exp
(−mB
γ̄Bγ
)mB−1∑k=0
1
k!
(mB
γ̄Bγ
)k(6)
2Similar as in [31], we assume in the following analysis that there is nofeedback error or delay in the transmission, while each legitimate user hasperfect CSI of its own.
3Similar to the work in [23], [32], we assume the i.i.d. fading channelswhere legitimate users are located approximately the same distance fromthe BS, or they are distributed in the whole cluster and slow power controlprotocol is employed.
and
fγbα,k
(γ) =
(mA
γ̄B
)mB γmB−1
Γ (mB)exp
(−mB
γ̄Bγ
)(7)
denote the CDF and PDF of γbα,k, respectively. Moreover,
γ̄B = E[γbα,k
]represents the average SNR of each legitimate
link and mB is the fading severity of the legitimate channel.Let us define γB as the end-to-end instantaneous SNR of
TAS/tSD scheme. According to the principle of TAS, γB canbe presented as γB = max
(γtSDB,α
). Therefore, the CDF of γB
is given by
FγB (γ) =[FγtSD
B,α(γ)
]AA
(8)
Moreover, resorting to binomial theorem [41, Eq.(1.111)], theCDF of γB can be rewritten as (9).
On the other hand, since MRC scheme is adopted at thecolluding eavesdroppers, the end-to-end instantaneous SNR ofthe colluding eavesdroppers is γE =
∑NE
n=1 γeα∗,n, where α∗
represents the selected antenna at the BS. Thus, the CDF andPDF of γE are, respectively, expressed as
FγE (γ) = 1− exp
(−mE
γ̄Eγ
)NEmE−1∑k=0
1
k!
(mE
γ̄Eγ
)k(10)
and
fγE (γ) =
(mE
γ̄E
)NEmE γNEmE−1
Γ (NEmE)exp
(−mE
γ̄Eγ
)(11)
where γ̄E = E[γeα∗,n
]denotes the average SNR of eaves-
dropper’s channel and mE denotes the fading severity of thewiretap channel.
The instantaneous secrecy rate Cs of the considered systemis given by
Cs = [CB − CE]+= [log (1 + γB)− log (1 + γE)]
+ (12)
where
[u]+= max (u, 0) =
{u, u > 00, u ≤ 0
(13)
while CB = log (1 + γB) and CE = log (1 + γE) repre-sent the instantaneous rate of the legitimate channel andwiretap channel, respectively. According to [34], [38], whenthe eavesdropper’s CSI is not available at the BS, i.e., thepassive eavesdropping scenario, BS has no choice but toassume the instantaneous rate of the eavesdropping channelas C̃E = CB − Rs to achieve secure transmission, whereRs denotes a constant secrecy rate chosen by the BS. Then,the BS constructs the wiretap codes by exploiting CB andC̃E. If Rs ≤ Cs (i.e., C̃E ≥ CE), the codewords insure aperfect secrecy. Otherwise, if Rs > Cs (i.e., C̃E < CE),the confidential data can be overheard by the eavesdroppersand the secrecy is compromised. Hence, the secrecy outageprobability is adopted as a useful and well-acceptable secrecyperformance metric under this scenario.
IEEE TRANSACTIONS ON COMMUNICATIONS 5
FγB (γ) =
AA∑q=0
(AA
q
)[Fγb
α,k(γT)
]q q∑q1=0
(qq1
)(−1)
q1(NB−1)(q−q1)+AA−q∑
q2=0
((NB−1)(q−q1)+AA−q
q2
)(−1)
q2
× exp(−mBq2γ
γ̄B
)ΘB,q2
(mB
γ̄B
)ϕB
γϕB , γ ≥ γTAANB∑q=0
(AANB
q
)(−1)
qexp
(−mBqγ
γ̄B
)ΘB,q
(mB
γ̄B
)ϕB
γϕB , γ < γT
(9)
where
ΘB,q =
q∑n1=0
n1∑n2=0
...
nmB−2∑nmB−1=0
q!
nmB−1!
mB−1∏t=1
[(t!)
nt+1−nt
(nt−1 − nt)!
], n0 = q, nmB = 0, ϕB =
mB−1∑p=1
np
B. Secrecy Outage Probability
According to [1], [34], the secrecy outage probability isdefined as Pout (Rs) = Pr (Cs < Rs). Mathematically, thesecrecy outage probability can be formulated as
Pout (Rs)=Pr (Cs < Rs|γB > γE) Pr (γB > γE)
+Pr (γB < γE) (14)
Following the detailed algebraic manipulations in [1], we have
Pout (Rs) =
∫ ∞
0
∫ 2Rs (1+y)−1
0
fγB (x) fγE (y) dxdy
=
∫ ∞
0
FγB(2Rs (1 + y)− 1
)fγE (y) dy (15)
where fγB(x) is the PDF of γB. Due to the fact that the
switched threshold γT is incorporated in the CDF of γBas shown in (9), there exists relationship between the term2Rs (1 + y)− 1 and γT in (15), i.e., 2Rs (1 + y)− 1 ≥ γT or2Rs (1 + y) − 1 < γT. To facilitate the analysis in (15), herewe introduce a bound point as H(γT) = 2−Rs (1 + γT) − 1.As such, the secrecy outage probability can be converted to apiecewise one with respect to the bound point as
Pout (Rs) =
∫ H(γT)
0FγB
(Φy
)fγE (y) dy +
∫∞H(γT)
FγB
(Φy
)×fγE (y) dy, H(γT) ≥ 0∫∞
0FγB
(Φy
)fγE (y) dy, H(γT) < 0
(16)
where Φy = 2Rs (1 + y) − 1. In what follows, by sub-stituting (7) and (9) into (16) and utilizing the results in[41, Eqs. (1.111), (3.351.2) and (3.351.3)], the exact closed-form expression of secrecy probability can be obtained in thefollowing theorem.
Theorem 1. The secrecy outage probability of multiuser multi-antenna wiretap networks with TAS/tSD scheme is given as(17), where γ (·, ·) and Γ (·, ·) denote the lower and upper in-complete Gamma function [41, Eqs. (8.350.1) and (8.350.2)],respectively.
It is worth mentioning that other secrecy performancemetrics can be conveniently calculated from (17). For instance,the probability of positive secrecy can be easily evaluated bysetting Rs = 0 into (17), i.e., Pr (Cs > 0) = 1− Pout (0).
In general, due to the complexity of the involved expres-sions, it is difficult to explore the effect of system parameterson the secrecy performance from (17). In order to achievemore insights from (17), we provide an asymptotic secrecyoutage analysis in the high SNR regime via the following keycorollaries.
Corollary 1. When γ̄B → ∞ and γ̄E is fixed, the asymptoticsecrecy outage probability of multiuser multi-antenna wiretapnetworks with TAS/tSD scheme is given by
Pout (Rs)=(Ξ · γ̄B)−Ψ+O
(γ̄−ΨB
), (18)
where the secrecy diversity order is Ψ = AAmB, and thesecrecy array gain Ξ is given by (19).
Proof: A detailed proof is provided in Appendix A.From Corollary 1, we highlight that the secrecy diversity
gain is determined by the number of transmit antennas at theBS and the fading severity of the legitimate channel, which isindependent of the number of legitimate users or eavesdrop-pers, (i.e., NB or NE), and the average SNR of eavesdroppingchannel γ̄E. However, the effect of these involved parameterson the secrecy outage performance can be described by thesecrecy array gain.
In what follows, we consider the case of γ̄B → ∞ andγ̄E → ∞ with a fixed main-to-eavesdropper ratio (MER), i.e.,γ̄Bγ̄E
= λbe.
Corollary 2. In the case of γ̄B → ∞ and γ̄E → ∞ withγ̄Bγ̄E
= λbe, both legitimate user and eavesdropper are locatedclose to the BS, the asymptotic secrecy outage probability withTAS/tSD scheme is derived as
P∞out (Rs) ≈
AA∑q=0
(AA
q
)(−1)
qΘB,q
(mB2
Rs
mE
)ϕB
(λbe)ϕB
×(mB2
Rsq
mEλbe + 1
)−(NEmE+ϕB)Γ (NEmE + ϕB)
Γ (NEmE)
(20)
Proof: Based on (17), the asymptotic secrecy outageprobability can be easily derived after some algebraic manip-ulations.
According to (20), we confirm that the secrecy outageprobability approaches a constant at high SNR value, whichimplies that the secrecy diversity gain is not achievable for
IEEE TRANSACTIONS ON COMMUNICATIONS 6
Pout (Rs) =
(mE
γ̄E
)NEmE1
Γ(NEmE)
{AANB∑q=0
(AANB
q
)(−1)
qexp
(−mBq
γ̄B
(2Rs − 1
))ΘB,q
(mB(2Rs−1)
γ̄B
)ϕB ϕB∑s=0
(ϕB
s
)(2Rs
2Rs−1
)s×(mE
γ̄E+ mB2Rsq
γ̄B
)−(NEmE+s)
γ(NEmE+s,
(mE
γ̄E+mB2Rsq
γ̄B
)H(γT)
)+
AA∑q=0
(AA
q
)[Fγb
α,k(γT)
]q q∑q1=0
(qq1
)×(−1)
q1[(NB−1)(q−q1)+AA−q]∑
q2=0
((NB−1)(q−q1)+AA−q
q2
)(−1)
q2 exp(−mBq2
γ̄B
(2Rs − 1
))ΘB,q2
(mB(2Rs−1)
γ̄B
)ϕB
×ϕB∑s=0
(ϕB
s
)(2Rs
2Rs−1
)sΓ(NEmE+s,
(mB2Rsq2
γ̄B+mE
γ̄E
)H(γT)
)(mB2Rsq2
γ̄B+ mE
γ̄E
)−(NEmE+s)},H(γT) ≥ 0
(mE
γ̄E
)NEmE1
Γ(NEmE)
{AA∑q=0
(AA
q
)[Fγb
α,k(γT)
]q q∑q1=0
(qq1
)(−1)
q1[(NB−1)(q−q1)+AA−q]∑
q2=0
((NB−1)(q−q1)+AA−q
q2
)× (−1)
q2 exp(−mBq2
γ̄B
(2Rs−1
))ΘB,q
(mB(2Rs−1)
γ̄B
)ϕB ϕB∑s=0
(ϕB
s
)(2Rs
2Rs−1
)s(mB2Rsq2
γ̄B+mE
γ̄E
)−(NEmE+s)
× Γ (NEmE+s)
}, H(γT) < 0
(17)
Ξ =
[1
Γ(NEmE)
(mB
mB
mB !
)AA AA∑q=0
(AA
q
)(−1)
q(γT)
mBq(2Rs − 1
)mB(AA−q)mB(AA−q)∑q1=0
(mB(AA−q)
q1
)×(
2Rs
2Rs−1
)q1(mE
γ̄E
)−q1Γ(NEmE+q1,
mE
γ̄EH(γT)
)]−1
,H(γT) ≥ 0[1
Γ(NEmE)
(mB
mB
mB !
)AA AA∑q=0
(AA
q
)(−1)
q(γT)
mBq(2Rs − 1
)mB(AA−q)mB(AA−q)∑q1=0
(mB(AA−q)
q1
)×(
2Rs
2Rs−1
)q1(mE
γ̄E
)−q1Γ (NEmE+q1)
]−1
, H(γT) < 0
(19)
this particular case. Besides, it is noteworthy that increasingthe transmit power at the BS does not improve the secrecyperformance.
C. Ergodic Secrecy Rate
In this subsection, we concentrate on the scenario where theCSI of the eavesdropping channel is available at BS. As such,the BS adjusts the transmission rate adaptively with the wiretapcoding scheme. More specifically, any average transmissionrate below the ergodic secrecy rate of the channel is achievablein principle [1], [9]. Different from the Scenario I, here we takethe ergodic secrecy rate as a principal metric to evaluate thesecrecy performance of the considered system with TAS/tSDscheme.
According to [9], the ergodic secrecy rate is given by
Cs =
∫ ∞
0
∫ ∞
y
[log2 (1 + x)− log2 (1 + y)]
×fγB (x) fγE (y) dxdy
(21)
To solve the above double integral, we follow the similarprocedures as introduced in [9]. First, the inner integral canbe evaluated with the adoption of integration by parts, andthen applying some mathematical manipulations, the ergodic
secrecy rate can be further written as in [9, Eq. (15)]
Cs =1
ln 2
∫ ∞
0
FγE (y)
1 + y[1− FγB (y)] dy (22)
Moreover, by taking the switched threshold γT into account,the ergodic secrecy rate can be re-expressed as
Cs =1
ln 2
[∫ γT
0
FγE (y)
1 + y[1− FγB (y)] dy
+
∫ ∞
γT
FγE (y)
1 + y[1− FγB (y)] dy
] (23)
Now, by substituting (9) and (10) into (23), resorting to thenew derived formulae of integration (44), (46) and (48) inAppendix B and performing some algebraic manipulations, theergodic secrecy rate can be derived in the following theorem.Theorem 2. The exact ergodic secrecy rate of multiuser multi-antenna wiretap networks with TAS/tSD scheme is given as
Cs =1
ln 2
(C1 + C2
)(24)
where C1 and C2 are expressed in (25) and (26), respectively.
In what follows, to evaluate the impact of key systemparameters on the ergodic secrecy rate, we also look into theergodic secrecy rate in the high SNR regime. We first considerthe asymptotic ergodic secrecy rate in the case of γ̄B → ∞and a fixed γ̄E.
IEEE TRANSACTIONS ON COMMUNICATIONS 7
C1 =
{−AANB∑q=1
(AANB
q
)(−1)
qΘB,q
(mB
γ̄B
)ϕB
exp(mBqγ̄B
) ϕB∑k=0
(ϕB
k
)(−1)
ϕB−k(mBqγ̄B
)−k [Γ(k, mBq
γ̄B
)−Γ
(k, mBq
γ̄B(γT + 1)
)]NEmE−1∑k=0
1k!
(mE
γ̄E
)k AANB∑q=1
(AANB
q
)(−1)
qΘB,q
(mB
γ̄B
)ϕB
exp(mBqγ̄B
+ mE
γ̄E
)×
ϕB+k∑k1=0
(ϕB+kk1
)(−1)
ϕB+k−k1(mBqγ̄B
+ mE
γ̄E
)−k1 [Γ(k1,
mBqγ̄B
+ mE
γ̄E
)− Γ
(k1,
(mBqγ̄B
+ mE
γ̄E
)(γT + 1)
)]} (25)
C2 =
{−
AA∑q0=1
(AA
q0
)(−1)
q0ΘB,q0
(mB
γ̄B
)ϕB
exp(mBq0γ̄B
) ϕB∑k1=0
(ϕB
k1
)(−1)
ϕB−k1(mBq0γ̄B
)−k1Γ(k1,
mBq0γ̄B
(γT + 1))
+NEmE−1∑k=0
1k!
(mE
γ̄E
)k AA∑q0=1
(AA
q0
)(−1)
q0ΘB,q0
(mB
γ̄B
)ϕB
exp(mBq0γ̄B
+ mE
γ̄E
) k+ϕB∑k1=0
(AA
k1
)(−1)
k+ϕB−k1
×(mBq0γ̄B
+ mE
γ̄E
)−k1Γ(k1,
(mBq0γ̄B
+ mE
γ̄E
)(γT + 1)
)−
AA∑q=1
(AA
q
)[Fγb
α,k(γT)
]q q∑q1=0
(qq1
)(−1)
q1[(NB−1)(q−q1)+AA−q]∑
q2=0
×((NB−1)(q−q1)+AA−q
q2
)(−1)
q2ΘB,q2
(mB
γ̄B
)ϕB
exp(mBq2γ̄B
) ϕB∑k1=0
(ϕB
k1
)(−1)
ϕB−k1(mBq2γ̄B
)−k1Γ(k1,
mBq2γ̄B
(γT + 1))
+NEmE−1∑k=0
1k!
(mE
γ̄E
)k AA∑q=1
(AA
q
)[Fγb
α,k(γT)
]q q∑q1=0
(qq1
)(−1)
q1[(NB−1)(q−q1)+AA−q]∑
q2=0
((NB−1)(q−q1)+AA−q
q2
)(−1)
q2ΘB,q2
×(mB
γ̄B
)ϕB
exp(mBq0γ̄B
+ mE
γ̄E
) k+ϕB∑k1=0
(k+ϕB
k1
)(−1)
k+ϕB−k1(mBq0γ̄B
+ mE
γ̄E
)−k1Γ(k1,
(mBq0γ̄B
+ mE
γ̄E
)(γT + 1)
)}
(26)
Corollary 3. When γ̄B → ∞ and γ̄E is fixed, the asymp-totic ergodic secrecy rate of multiuser multi-antenna wiretapnetworks with TAS/tSD scheme is given by
C∞s,1 = ∆∞
1 +∆∞2 (27)
where ∆∞1 and ∆∞
2 is provided by (28) and (29), respectively.
Proof: A detailed proof is provided in Appendix C.In order to gain more insights, we also provide two metrics
i.e., the high SNR slope and the high SNR power offset, todescribe the asymptotic behavior of the ergodic secrecy ratein the high SNR regime. We adopt a general form to rewritethe asymptotic ergodic secrecy rate as [1]
C∞s,1 = S∞ (log2γ̄B − ζ∞) (30)
where S∞ denotes the high SNR slope in bits/s/Hz (3dB) andζ∞ is the high SNR power offset in 3dB units. According to[1], the high SNR slope is given by
S∞ = limγ̄B→∞
C∞s,1
log2γ̄B(31)
Substituting (27) into (31) and performing some algebraicmanipulations, we have
S∞ = 1 (32)
From (32), we find that the key parameters, such as the numberof legitimate users NB and of eavesdroppers NE as well as theswitched threshold γT, have no impact on the high SNR slope.Next, we turn our attention to the high SNR power offset ζ∞,which can be easily derived as follows:
ζ∞ = limγ̄B→∞
(log2γ̄B − C
∞s,1
S∞
)(33)
It is noted that (33) reflects the impact of main channel andeavesdropping channel on the ergodic secrecy rate. Therefore,inserting (30) and (31) into (33), ζ∞ can be further expressedas
ζ∞ = ζ∞ (AA,mB) + ζ∞ (NE,mE, γ̄E) , (34)
where
ζ∞ (AA,mB) =−AA
Γ(mB)
AA−1∑q=0
(AA−1q
)(−1)
qΘB,qΓ (mB + ϕB)
×(1 + q)−(mB+ϕB)
[ψ(mB+ϕB)
ln 2 − log2 (mB (1 + q))]
(35)
and
ζ∞ (NE,mE, γ̄E) = ∆∞2 (36)
Based on the analysis above, we can draw the conclusionthat the high SNR offset is independent of γ̄B. Besides, thepositive impact of main channel on ergodic secrecy rate ischaracterized by ζ∞ (AA,mB), which is related to the key sys-tem parameters of the legitimate channel, i.e., AA and mB. Onthe other hand, the negative impact of eavesdropper’s channelis characterized by ζ∞ (NE,mE, γ̄E), which is associated withthe parameters of eavesdropping channel, i.e., NE,mE and γ̄E.In addition, ζ∞ (NE,mE, γ̄E) also explicitly quantifies the lossof ergodic secrecy rate due to the behavior of the wiretappingat Eves.
In the sequel, we take into account the case of γ̄B → ∞and γ̄E → ∞ with a fixed MER γ̄B
γ̄E= λbe and provide the
following corollary.
Corollary 4. The asymptotic ergodic secrecy rate at γ̄B →∞ and γ̄E → ∞ with γ̄B
γ̄E= λbe of multiuser multi-antenna
wiretap networks with TAS/tSD scheme is given by (37).
IEEE TRANSACTIONS ON COMMUNICATIONS 8
∆∞1 = AA
Γ(mB)
AA−1∑q=0
(AA−1q
)(−1)
qΘB,qΓ (mB + ϕB) (1 + q)
−(mB+ϕB)[ψ(mB+ϕB)
ln 2 − log2 (mB (1 + q))]+ log2 (γ̄B) (28)
∆∞2 = 1
ln 2
NEmE−1∑k=0
1k!
(mE
γ̄E
)k [(−1)
k−1exp
(mE
γ̄E
)Ei
(−mE
γ̄E
)+
k∑k1=1
(k1 − 1)!(−1)k−k1
(mE
γ̄E
)−k1]
(29)
Proof: A detailed proof is provided in Appendix D.As can be observed from (37) that a rate ceiling exists in
this particular case. Once again, it demonstrates that increasingthe transmit power at the BS does not have a positive impacton the ergodic secrecy rate when the eavesdropper is locatedclose to the transmitter.
D. Average Number of User Examinations
Compared with the best scheduling scheme, i.e., SC scheme,where all the NB legitimate users are examined to selectthe best one, tSD scheme only examines the legitimate useradopted in the previous time slot. Once it is acceptable, thereis no need to check out the remaining (NB − 1) legitimateusers. Hence, the average number of user examinations canbe expressed as [31]:
NavgTAS/tSD =
AA∑α=1
[1 + (NB − 1)Fγb
α,k(γT)
](38)
For the SECps scheme, it examines the SNRs of additionalmain links only if necessary. Once the SNR of the k-thlegitimate user is acceptable, there is no need to check outthe remaining SNRs of (NB − k) legitimate users. Hence, theaverage number of user examinations can be characterized as[22]
NavgTAS/SECps =
AA∑α=1
NB−1∑k=0
[Fγb
α,k(γT)
]k(39)
On the other hand, since 0 ≤ Fγbα,k
(γT) ≤ 1, we have[Fγb
α,k(γT)
]k< Fγb
α,k(γT) , k ∈ {1, 2, ..., NB − 1} (40)
By jointly taking (38), (39) and (40) into account, we have
NavgTAS/SECps ≤ Navg
TAS/tSD ≤ AANB (41)
As a result, the implementation cost of TAS/tSD scheme ishigher than that of TAS/SECps scheme, while it is lower thanTAS/SC scheme.
IV. SIMULATIONS AND DISCUSSION
In this section, we present some numerical results to validatethe aforementioned secrecy analysis and analyze the jointimpact of key parameters on the secrecy performance of theconsidered system.
Fig. 2 presents the secrecy outage probability and asymp-totic secrecy outage probability versus different average le-gitimate user’s SNR γ̄B in Case I. It is observed from thefigure that the theoretical results of secrecy outage probability
0 5 10 15 20 25 30 35 40 45
10−5
10−4
10−3
10−2
10−1
100
γ̄B(dB)
Pout(R
s)
Rs = 1
Rs = 4
Anal.AA= 1,mB=1 in Eq. (19)
Anal.AA= 2,mB=1 in Eq. (19)
Anal.AA= 2,mB=2 in Eq. (19)
Asymptotic Result in Eq. (20)
Simulation Result
Fig. 2: Secrecy outage probability in Case I versus differentγ̄B for γ̄E = 0dB, γT = 10dB, mE = 1 and NB = NE = 2.
in (17) match precisely with the Monte Carlo simulations andthe high SNR curves given by (18) agree very well with theexact ones in the high SNR regime and accurately predict thesecrecy diversity order and secrecy array gain. Furthermore,increasing the number of antennas at the BS and the channelfading severity parameter of legitimate channel have positiveimpact on the secrecy performance. This is due to the fact thatmore number of transmit antennas results in larger transmitdiversity gains and the higher mB means the better channelquality of legitimate channel. Additionally, as can be expected,the secrecy outage probability of TAS/tSD scheme degradeswith the increment of the predefined secrecy rate Rs, andincreasing Rs does not influence the secrecy diversity orderas indicated by the parallel slopes of the asymptotes.
Fig. 3 illustrates the secrecy outage probability of thesystem with different NB. As illustrated in the figure, weobserve that the secrecy performance can be improved asincreasing NB for the particular case when γB ≤ γT. However,when γB ≥ γT, increasing the number of legitimate usersNB has marginal impact on the secrecy outage performancedue to no additional secrecy diversity order. Moreover, thebetter secrecy performance can be achieved with the largerpredefined switched threshold γT, while the TAS/tSD schemeand TAS/SECps scheme turn into the TAS/SC scheme as γT
IEEE TRANSACTIONS ON COMMUNICATIONS 9
C∞s,2 = log2 (λbe) +AA
AA−1∑q=0
(AA−1q
)(−1)
qΘB,q
Γ(mB+ϕB)Γ(mB) (1 + q)
−(mB+ϕB)[ψ(mB+ϕB)
ln 2 − log2 (mB (1 + q))]
−ψ(NEmE)ln 2 + log2 (mE)
(37)
2 4 6 8 10 12 14 16
10−5
10−4
10−3
10−2
10−1
100
NB
Pout(R
s)
γ̄B = 10dB
γ̄B = 15dB
γT = 10dBγT = 12dB
TAS/tSD schemeTAS/SECps schemeTAS/SC scheme
Fig. 3: Secrecy outage probability versus NB with differentγ̄B and γT for AA = 2, mB = mE = 1, NE = 2, Rs = 1 andγ̄E = 0dB.
0 10 20 30 40 50 60
10−5
10−4
10−3
10−2
10−1
γ̄B(dB)
Pout(R
s)
γT = 10dB γT = 20dB γT = 30dB
Anal. Result in Eq.(17) (λbe = 10dB)
Anal. Result in Eq.(17) (λbe = 12dB)
Asymptotic Result in Eq.(20)
Simulation Result
Fig. 4: Secrecy outage probability in Case II versus γ̄B forAA = 3, NB = 4, NE = 2,mB = 2,mE = 1, Rs = 1 withdifferent γT and λbe.
approaches to infinite.Fig. 4 shows the secrecy outage probability versus γ̄B with
0 5 10 15 20 25 30 35 400
2
4
6
8
10
12
γ̄B(dB)
CS(bit/s/Hz)
Anal. AA = 1, NB = 2, mB = 1, NE = 2 in Eq.(24)
Anal. AA = 2, NB = 2, mB = 1, NE = 2 in Eq.(24)
Anal. AA = 2, NB = 8, mB = 1, NE = 2 in Eq.(24)
Anal. AA = 2, NB = 8, mB = 2, NE = 2 in Eq.(24)
Anal. AA = 1, NB = 2, mB = 1, NE = 6 in Eq.(24)
Asymtotic Result in Eq.(27)
Simulation Result
6 8 10 120.5
1
1.5
2
2.5
3
Fig. 5: Ergodic secrecy rate versus γ̄B with different AA, NB
and mB settings as well as γT = 10dB, NE = 2 and mE = 2.
different γT and λbe in Case II. As shown obviously inthe figure, the larger switched threshold γT results in bettersecrecy performance since more legitimate users have beenexamined. It can also be observed that at a fixed MER λbe, thesecrecy outage probability improves with γ̄B in the low SNRarea, however the improvement tends to be saturated in themedium and high SNR area. This is attributed to the fact thatthe MER is the bottleneck of the secrecy outage probability. Inaddition, we can see that the secrecy outage probability floordecreases with the increment of MER.
Fig. 5 depicts the ergodic secrecy rate of the system withTAS/tSD scheme against different NB in Case I. It can beobserved that the curves of ergodic secrecy rate generated by(24) are in exact agreement with the Monte Carlo simulationsand the curves of asymptotic ergodic secrecy rate in (27) wellapproximate the analytical ones in the high SNR regimes. Wealso observe that, both the number of legitimate users NB
and antenna configurations contribute to the improvement ofergodic secrecy rate in the low γ̄B regime, where the TAS/tSDscheme is equal to NB-legitimate users TAS/SC scheme andmultiuser diversity gain is achieved. However, in the high γ̄Bregime, BS intends to schedule the first user, and thus theergodic secrecy rate is independent of the number of legitimateusers NB. From another point of view, the high power offsetζ∞ in (34) is not relevant to the number of legitimate usersNB under this case. It needs to be pointed out that, in contrastwith the secrecy outage probability, increasing the fadingseverity parameter of the legitimate channel slightly degrades
IEEE TRANSACTIONS ON COMMUNICATIONS 10
0 5 10 15 20 25 30 35 400
2
4
6
8
10
12
14
γ̄B(dB)
Cs(bit/s/Hz)
γT = 10dB
γT = 20dB
γT = 30dB
TAS/tSD scheme
TAS/SECps scheme
TAS/SC scheme
Asymtotic Result
19 20 21
5.5
6
6.5
7
Fig. 6: Ergodic secrecy rate versus γ̄B with different γT forAA = 2, NB = 8, NE = 2, mB = mE = 1 and γ̄E = 0dB.
10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1
γ̄B(dB)
S∞
47.8 48 48.2
0.618
0.62
0.622
0.624
43 44 450.72
0.73
0.74
0.75
AA=1, mB=1, NE = 2, mE = 1, γ̄E = 0dB
AA=2, mB=1, NE = 2, mE = 1, γ̄E = 0dB
AA=2, mB=2, NE = 2, mE = 1, γ̄E = 0dB
AA=2, mB=2, NE = 8, mE = 1, γ̄E = 0dB
AA=2, mB=2, NE = 8, mE = 2, γ̄E = 0dB
AA=2, mB=2, NE = 8, mE = 2, γ̄E = 10dB
Fig. 7: High SNR slope S∞ versus γ̄B with different systemparameters.
the ergodic secrecy rate. In addition, as can be expected,increasing the number of eavesdroppers can considerablyimprove the wiretapping capability of colluding eavesdropper,which leads to the increment of the high power offset ζ∞, andthus degrades secrecy performance of the considered systems.
Fig. 6 shows that the ergodic secrecy rate improves with theincrease of the switched threshold γT. This is due to the factthat the scheduler has more opportunities to adopt TAS/SCscheme as the switched threshold increases, thus it yields abetter performance at the expense of higher complexity. Thatis to say, when γT comes close to infinity, the TAS/tSD scheme
0 10 20 30 40 50 601
1.5
2
2.5
3
3.5
4
4.5
γ̄B(dB)
CS(bit/s/Hz)
γT = [10,20,30]dB
γT = [10,20,30]dB
Anal. Result in Eq.(24) (λbe = 10dB)
Anal. Result in Eq.(24) (λbe = 12dB)
Asym. Result in Eq.(37)
Simulation Result
Fig. 8: Ergodic secrecy rate in Case II versus γ̄B for AA =2, NB = 2, NE = 2,mB = 2,mE = 1 with different γT andλbe.
completely turns into the TAS/SC scheme. In addition, it is ob-served that the TAS/tSD scheme outperforms the TAS/SECpsscheme regardless of switched threshold γT. Fig. 7 presentsthe impact of key system parameters AA,mB, NE,mE and γ̄Eon the high SNR slope S∞. As illustrated in the figure, the keysystem parameters of the legitimate channel AA and mB havea positive impact on the high SNR slope. However, when theparameters related with eavesdropping channel, i.e., NE, mE
or γ̄E increases, the speed of convergence to 1 slows down.This is because that the second term on the right hand side of(34) has a great impact on the high SNR slope S∞.
Fig. 8 shows the ergodic secrecy rate versus γ̄B withdifferent γT and λbe in Case II. It is shown in Fig. 8 thatthe exact curves of (24) are still in good agreement withthe Monte Carlo simulations, and the asymptotic results in(20) predict the exact ones precisely in the high SNR area.Similar to the secrecy outage probability in Case II, we findthat the increment of switched threshold γT contributes to theimprovement of the ergodic secrecy rate. It can be observedthat at a fixed MER λbe, the ergodic secrecy rate increaseswith γ̄B in the low SNR area, but tends to a rate ceiling inthe medium and high SNR area. This is due to the fact thatthe MER is the bottleneck of the ergodic secrecy rate underthis case. Furthermore, we also find that increasing the MERleads to the reduction of rate ceiling.
Fig. 9 presents the saving percentage of the number oflegitimate user estimations, i.e.,
(1−Navg
TAS/tSD/AA/NB
).
As illustrated in this figure, the saving percentage of TAS/tSDscheme reduces to zero as the switched threshold increases. Byjointly considering Fig. 3, Fig. 6 and Fig. 9, we find that theTAS/tSD scheme achieves better secrecy performance than theTAS/SECps scheme at the expense of negligible complexitycost, and achieves a similar performance as TAS/SC scheme
IEEE TRANSACTIONS ON COMMUNICATIONS 11
−10 −5 0 5 10 15 20 25 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
γT(dB)
Num
ber
of e
xam
ined
use
r re
duct
ion
perc
enta
ge
γ̄B=10dB
γ̄B=15dB
NB=2
NB=6
TAS/SC schemeTAS/tSD schemeTAS/SECps scheme
Fig. 9: The reduced percentage of number of user examinationsversus switched thresholds γT with different NB and γ̄B,where AA = 2 and mB = 2.
while maintaining a lower number of user estimations loadcompared with TAS/SC scheme.
V. CONCLUSIONS
In this paper, we have presented a comprehensive secrecyperformance analysis of multiuser multi-antenna wiretap net-works with the TAS/tSD scheme. Particularly, the secrecyperformance of two practical scenarios was addressed withrespect to the availability of the CSI of the eavesdroppingchannel at the BS. When the BS has the full knowledge of theeavesdropper’s CSI, we have derived closed-form exact ex-pressions of the secrecy outage probability and the probabilityof non-zero secrecy capacity, which provides a fast and effi-cient way to evaluate the secrecy performance of the system.Moreover, simple and informative high SNR approximationsfor the secrecy outage probability were derived under twodistinct cases, which enable us to exploit more insights aboutthe impact of the key parameters on the secrecy performance.For the case of eavesdropper’s CSI is not available at theBS, we have investigated in detail the ergodic secrecy rateachieved by the system with TAS/tSD scheme. In doing so,novel closed-form expressions for the exact and asymptoticergodic secrecy rate were derived. Our results demonstratethat when the switched threshold being carefully selected, theTAS/tSD scheme outperforms the TAS/SECps scheme in termsof the secrecy performance at the expense of little complexitycost, while maintaining a lower number of user estimationsload compared with TAS/SC scheme.
APPENDIX AA DETAILED DERIVATION FOR COROLLARY 1
In the high SNR regime, i.e., γ̄B → ∞ with a fixed γ̄E, theCDF of γbα,k can be approximated as
Fγbα,k
(x) ≈ mBmB
(mB)!
(x
γ̄B
)mB
(42)
By inserting (42) into (8) and using the law of binomialtheorem, we have
FγB(x) =
AA∑q=0
(AA
q
)[mB
mB
(mB)!
(γTγ̄B
)mB]q q∑
q1=0
(qq1
)(−1)
q1
×[mB
mB
(mB)!
(xγ̄B
)mB](NB−1)(q−q1)+AA−q
, x ≥ γT[mB
mB
(mB)!
(xγ̄B
)mB]AANB
, x < γT
(43)
As such, by plugging (43) into (16) and neglecting thehigher order terms, with the help of [41, Eqs. (3.381.3) and(3.351.3)], the desired asymptotic outage probability Pout (Rs)can be easily derived as (18) after some simple mathematicalmanipulations.
APPENDIX BA DERIVATION FOR SOME INTEGRAL EQUATIONS
In this Appendix, we present some useful results on thesolution of some integrals, which will be frequently adoptedto evaluate some performance metrics.
Consider the following first integral∫ v
0
exp (−ux) xk
1 + xdx (44)
= exp (u)
k∑k1=0
(k
k1
)(−1)
k−k1
× u−k1 [Γ (k1, u)− Γ (k1, u (v + 1))]
Proof: Let y = 1 + x, then we have∫ v
0
exp (−ux) xk
1 + xdx =
∫ v+1
1
exp (−u (y − 1)) (y − 1)k
ydy
(45a)
By adopting the binomial theorem, the above expression canbe further expanded as∫ v
0
exp (−ux) xk
1 + xdx (45b)
= exp (u)k∑
k1=0
(k
k1
)(−1)
k−k1∫ v+1
1
exp (−uy) yk1−1dy
= exp (u)
k∑k1=0
(k
k1
)(−1)
k−k1
×(∫ ∞
1
exp (−ux)xk1−1dx−∫ ∞
v+1
exp (−ux)xk1−1dx)
(45c)
IEEE TRANSACTIONS ON COMMUNICATIONS 12
With the aid of variable substitution operation y = ux, andresorting to the upper incomplete gamma function in [41, Eq.(8.350.2)] as Γ (α, µ) =
∫∞µe−ttα−1dt, we obtain∫ v+1
1
exp (−ux)xk1−1dx=u−k1 [Γ (k1, u)− Γ (k1, u (v+1))]
(45d)
Finally, by substituting (45d) into (45b), therefore we completethe proof of (44).
Consider the following second integral∫ ∞
v
exp (−ux)1 + x
dx = exp (u) Γ (0, u (1 + v)) (46)
Proof: Similarly, with the operation of variable substitu-tion y = 1 + x, we have∫ ∞
v
exp (−ux)1 + x
dx =
∫ ∞
1+v
exp (−u (y − 1))
ydy (47a)
and then let z = uy, we have∫ ∞
v
exp (−ux)1 + x
dx = exp (u)
∫ ∞
u(1+v)
exp (−z)z
dz (47b)
Resorting to the upper incomplete gamma function in [41, Eq.(8.350.2)], the desired result can be derived.
Consider the following third integral∫ ∞
v
exp (−ux) xk
1 + xdx
= exp (u)k∑
k1=0
(k
k1
)(−1)k−k1
[1
uk1Γ (k1, u (v + 1))
](48)
Proof: Similar to the proofs above, arming with variablesubstitution y = 1 + x, z = uy and resorting to the upperincomplete gamma function in [41, Eq. (8.350.2)], we obtainthe important integral equations.
APPENDIX CA DETAILED DERIVATION FOR COROLLARY 3
Before probing into the analysis of the asymptotic ergodicsecrecy rate, we first rewrite the CDF of γE as FγE (y) =1− φγE (y), where
φγE (y) = exp(−mEy
γ̄E
)NEmE−1∑k=0
1k!
(mEyγ̄E
)k(49)
To this end, with the help of (49), the ergodic secrecy rate canbe re-expressed as
Cs,1 =1
ln 2
∫ ∞
0
∫ x
0
1− φγE (y)
1 + ydyfγB (x) dx = ∆1 +∆2
(50)
where
∆1 =1
ln 2
∫ ∞
0
ln (1 + x)fγB (x) dx (51)
and
∆2 = − 1
ln 2
∫ ∞
0
∫ x
0
φγE (y)
1 + yfγB (x) dydx (52)
Now, in the following, we discuss the characteristics of ∆1
and ∆2 in the high SNR regime, respectively. According tothe basic idea of tSD scheme, when γ̄B → ∞, the probabilitythat each SNR of the main link γbk exceeds the predefinedthreshold γT approaches one. That is to say, the instantaneousSNR of the main channel reduces to γB = γbα,k, the PDF ofwhich can be characterized by
fγB (x) ≈ AA
Γ (mB)
AA−1∑q=0
(AA − 1
q
)(−1)
qΘB,q
(mB
γ̄B
)mB+ϕB
×xmB+ϕB−1 exp
(−mB (q + 1)
γ̄Bx
)(53)
As x → ∞, we have ln (1 + x) ≈ x, thus, ∆1 asymptoti-cally turns into
∆∞1 =
∫ ∞
0
[log2 (γ̄B)+log2
(x
γ̄B
)]fγB (x) dx
= log2 (γ̄B) +
∫ ∞
0
log2
(x
γ̄B
)fγB (x) dx (54)
Now, by utilizing the PDF of γB and resorting to [41,Eq.(4.352.1)], we have ∆∞
1 as (28), where ψ (·) denotes thedigamma function [42].
In what follows, by exchanging the order of integration in∆2, therefore, ∆2 can be rewritten as
∆2 = − 1
ln 2
∫ ∞
0
∫ x
0
φγE (y)
1 + yfγB (x) dydx
= − 1
ln 2
∫ ∞
0
φγE (y)
1 + y(1− FγB (x)) dy (55)
Depending on the CDF expression in (8), as γ̄B → ∞, wehave FγB (x) → 0. Hence, ∆2 can be further expressed as
∆∞2 = − 1
ln 2
∫ ∞
0
φγE (y)
1 + ydy (56)
As such, by utilizing (49) and resorting to [41, Eq. (3.353.5)],the asymptotic expression of ∆2 can be derived as (29) aftersome mathematical manipulations. Finally, summing up ∆∞
1
and ∆∞2 yields the asymptotic ergodic secrecy rate C∞
s,1.
APPENDIX DA DETAILED DERIVATION FOR COROLLARY 4
When γ̄B → ∞ and γ̄E → ∞ with γ̄Bγ̄E
= λbe, theasymptotic ergodic secrecy rate can be conveniently achievedaccording to the proof of Corollary 3 in Appendix E. To bespecific, what we need to do is just considering the asymptoticbehavior for ∆∞
2 as γ̄B → ∞. Following the similar proceduredeveloped in [40] and observing ∆∞
1 in (54), the asymptoticexpression for ∆∞
2 is given by
∆∞2,2 = −log2 (γ̄E)−
∫ ∞
0
log2
(y
γ̄E
)fγE (y) dy (57)
By plugging the PDF of γB in (53) into (57) and capitalizingon the equation [41, Eq.(4.352.1)], we have
∆∞2,2 = −log2 (γ̄E)−
ψ (NEmE)
ln 2+ log2 (mE) (58)
IEEE TRANSACTIONS ON COMMUNICATIONS 13
Therefore, by summing up the new asymptotic result (58) for∆∞
2 and (28) into (50), we eventually obtain (37) and completethe proof.
ACKNOWLEDGMENTS
The authors wish to thank the reviewers for valuable andconstructive suggestions that improved and clarified the paper.
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IEEE TRANSACTIONS ON COMMUNICATIONS 14
Maoqiang Yang received his B.S. degree fromSouth China University of Technology, Guangzhou,China, in 2010 and the M.S. degree from PLA Uni-versity of Science and Technology, Nanjing, China,in 2013. He is currently working toward his Ph.D.degree in PLA University of Science and Tech-nology. His research interests focus on nonlinearsignal processing, MIMO communications systems,multiuser communication systems, cooperative com-munications and physical layer security.
Daoxing Guo received the B.S. degree, M.S. degreeand ph.D. degree from Institute of CommunicationsEngineering (ICE), Nanjing, China, in 1995, 1999and 2002, respectively. He is currently a Full Profes-sor and also a Ph.D. Supervisor with PLA Universityof Science and Technology. He has authored andcoauthored more than 40 conference and journalpapers and has been granted over 20 patents inhis research areas. He has served as a reviewer forseveral journals in communication field. His currentresearch interests include satellite communications
systems and Transmission technologies, communication anti-jamming tech-nologies, communication anti-interception technologies including physicallayer security and so on.
Yuzhen Huang received his B.S. degree in Com-munications Engineering, and Ph.D. degree in Com-munications and Information Systems from Collegeof Communications Engineering, PLA Universityof Science and Technology, in 2008 and 2013 re-spectively. He has been with College of Commu-nications Engineering, PLA University of Scienceand Technology since 2013, and currently as anAssistant Professor. His research interests focus onchannel coding, MIMO communications systems,cooperative communications, physical layer security,
and cognitive radio systems. He has published nearly 20 research papers ininternational journals and conferences such as IEEE TCOM, IEEE TVT, IEEECL, WCNC, etc. He and his coauthors have been awarded a Best Paper Awardat the WCSP 2013. He received an IEEE COMMUNICATIONS LETTERSexemplary reviewer certificate for 2014.
Trung Q. Duong (S’05, M’12, SM’13) received hisPh.D. degree in Telecommunications Systems fromBlekinge Institute of Technology (BTH), Sweden in2012. Since 2013, he has joined Queen’s UniversityBelfast, UK as a Lecturer (Assistant Professor). Hiscurrent research interests include physical layer se-curity, energy-harvesting communications, cognitiverelay networks. He is the author or co-author of morethan 200 technical papers published in scientificjournals (105 articles) and presented at internationalconferences.
Dr. Duong currently serves as an Editor for the IEEE TRANSACTIONSON COMMUNICATIONS, IEEE COMMUNICATIONS LETTERS, IET COM-MUNICATIONS, WILEY TRANSACTIONS ON EMERGING TELECOMMUNICA-TIONS TECHNOLOGIES, and ELECTRONICS LETTERS. He has also servedas the Guest Editor of the special issue on some major journals includingIEEE JOURNAL IN SELECTED AREAS ON COMMUNICATIONS, IET COM-MUNICATIONS, IEEE WIRELESS COMMUNICATIONS MAGAZINE, IEEECOMMUNICATIONS MAGAZINE, EURASIP JOURNAL ON WIRELESS COM-MUNICATIONS AND NETWORKING, EURASIP JOURNAL ON ADVANCESSIGNAL PROCESSING. He was awarded the Best Paper Award at the IEEEVehicular Technology Conference (VTC-Spring) in 2013, IEEE InternationalConference on Communications (ICC) 2014. He is the recipient of prestigiousRoyal Academy of Engineering Research Fellowship (2015-2020)
Bangning Zhang received the B.S. degree and M.S.degree in Institute of Communications Engineering(ICE), Nanjing, China, in 1984 and 1987, respec-tively. He is currently a Full Professor and the Headof College of Communications Engineering. He hasauthored and coauthored more than 80 conferenceand journal papers and and has been granted over20 patents in his research areas. He has served as areviewer for several journals in communication field.His current research interests include communicationanti-jamming technologies, microwave technologies,
satellite communications systems, cooperative communications and physicallayer security.
