secrecy rate analysis of opportunistic user scheduling in...

Received August 5, 2019, accepted August 28, 2019, date of publication September 2, 2019, date of current version September 18, 2019.

Digital Object Identifier 10.1109/ACCESS.2019.2939048

Secrecy Rate Analysis of Opportunistic UserScheduling in Uplink Networks WithPotential EavesdroppersINKYU BANG1, (Member, IEEE), AND BANG CHUL JUNG 2, (Senior Member, IEEE)1Department of Information and Communication Engineering, Hanbat National University, Daejeon 34158, South Korea2Department of Electronics Engineering, Chungnam National University, Daejeon 34134, South Korea

Corresponding author: Bang Chul Jung ([email protected])

This work was supported in part by the research fund of Hanbat National University in 2019 and in part by the Basic Science ResearchProgram of NRF through the Ministry of Science and ICT under Grant NRF2019R1A2B5B01070697.

ABSTRACT In this paper, we investigate two user scheduling algorithms (optimal user and threshold-baseduser scheduling algorithms) when we consider potential eavesdroppers in an uplink wiretap network. Theoptimal user scheduling (OUS) algorithm selects the user who has the maximum secrecy rate, based onchannel feedback from all users. On the other hand, the threshold-based user scheduling (TUS) algorithmfirst considers the information leakage from the users to potential eavesdroppers and then selects the useramong candidates who satisfy a threshold criterion on the information leakage. The OUS algorithm showsan optimal performance in terms of secrecy rate, but the TUS algorithm can achieve secrecy rate comparablewith theOUS algorithmwith reduced feedback overhead. Formain contributions, wemathematically analyzethe asymptotic behavior of the achievable secrecy rate of two scheduling algorithms when the signal-to-noiseratio (SNR) approaches to infinity. Further, we derive the approximated secrecy rate of the TUS algorithmand propose criteria to determine threshold values which maximize the achievable secrecy rate of the TUSalgorithm.We verify our analytical results through simulations.We perform an extra simulation to investigatethe effect of channel estimation error in the wiretap links on the average secrecy rate. Due to differentscheduling principles in OUS and TUS schemes, the TUS scheme yields robustness against the channelestimation error in the wiretap links, compared with the OUS schemes.

INDEX TERMS Physical-layer security, potential eavesdropper, achievable secrecy rate, multiuser diversity,opportunistic scheduling.

I. INTRODUCTIONThe broadcasting nature of radio signals over the wirelesschannel arises anxiety about data confidentiality from eaves-dropping. To address this problem, cryptographic methodshave been commonly used in upper layers of protocol stacks(e.g., transport layer) in wireless communication systems.In recent years, achieving security at the physical-layer (so-called, physical-layer security) has been considered as oneof the alternatives to improve the conventional crypto-basedsecurity. Physical-layer security is based on a notion ofinformation-theoretic secrecy which exploits the random-ness of wireless channels rather than using computationalhardness [1].

The associate editor coordinating the review of this manuscript andapproving it for publication was Kashif Saleem.

Since Shannon [2] established the fundamental princi-ples of information-theoretic security at physical-layer, manyresearchers have studied physical-layer security to guaran-tee the confidentiality of information over wireless channelsin the presence of eavesdropping attacks. Wyner [3] andCheong and Hellman [4] established the notions of wire-tap channel and secrecy capacity. Subsequent to early stud-ies of the basic principles of the secrecy capacity [3]–[5],the secrecy capacity was investigated in wireless fadingchannel of single-input and single-output (SISO) environ-ments [6]–[8]. Barros and Rodrigues [6], and Bloch et al. [7]characterized the secrecy rate and the secrecy outage prob-ability for the transmission of confidential data over aquasi-static fading channel. Gopala et al. [8] investigatedthe secrecy capacity along with the optimal power andrate allocation strategies. In addition to the analysis of the

127078 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see http://creativecommons.org/licenses/by/4.0/ VOLUME 7, 2019

https://orcid.org/0000-0002-4485-9592

I. Bang, B. C. Jung: Secrecy Rate Analysis of Opportunistic User Scheduling in Uplink Networks

secrecy capacity in wireless fading channel of SISO envi-ronments, the analysis in multiple antennas environmentswas investigated [9]–[11]. Further, physical-layer securityhas been studied in various network settings such as energyharvesting relaying networks, cognitive relaying networks,and multi-hop relaying networks [12]–[14]. Nguyen et al.analyzed and derived the secrecy outage probability whenchannel-aware relay selection schemes are considered inthe underlay cognitive relaying networks with energyharvesting constraints [12]. Liu et al. proposed several relayselection schemes to guarantee secure communication in cog-nitive decode-and-forward relay networks against eavesdrop-ping (e.g., one of the relays performs as a jammer.) [13].Hung et al. considered low-energy adaptive clustering hier-archy (LEACH) networks where a cluster-based multi-hoptransmission employing artificial noises and investigated thesecurity-reliability tradeoff [14].

Particularly, in this paper, we focus on multiuser networksettings. Compared to the achievable secrecy rate analysisin single-user networks [6]–[11], the secrecy rate analysis inmultiuser networks has been less highlighted. In themultiuserdownlink wiretap networks (or wiretap broadcast channel),Pei et al. [15] and Ge et al. [16] derived the secrecy rate inclosed-forms when opportunistic scheduling algorithms wereemployed. Yang et al. investigated a joint secure transmissionscheme employing a combination of the transmit antennaselection and threshold-based user (i.e., a receiver) selectionschemes in multiuser downlink wiretap networks [17]. Espe-cially, they set a threshold value to guarantee themain channelquality between the transmitter and the receiver, and furtheranalyzed the ergodic secrecy rate under the assumption thateavesdroppers’ channel information is available during thescheduling.

In the multiuser uplink wiretap networks (or wiretap mul-tiple access channel), the asymptotic behavior of the secrecyrate has been studied when the number of legitimate userstends to infinity [18]–[21]. In [18] and [19], Jin et al. inves-tigated secrecy rate scaling in terms of the number of usersin a cell to achieve the optimal multi-user diversity when asingle-cell and multi-cell uplink wiretap networks are con-sidered, respectively. In [20] and [21], Bang et al. analyzedthe effect of multiple antennas and artificial noise, respec-tively, on the secrecy rate to achieve the optimal multiuserdiversity in a single-cell uplink wiretap network. Further,Ge et al. proposed cumulative distribution function (CDF)-based scheduling and derived the closed-form expressions ofthe secrecy rate in [22].

In short, in multiuser uplink wiretap networks, the analysisof the secrecy rate was already investigated in terms ofsecrecy rate scaling or the closed-form expression. However,we further notice two things in previous work related tosecrecy rate analysis in multiuser uplink wiretap networks;(1) The exact closed-form expression was derived in [22]but the proposed scheduling scheme might be vulnera-ble to channel estimation errors on wiretap links (i.e., thewireless link between transmitter and eavesdropper);

(2) A threshold-based user scheduling scheme which wascommonly considered in [18]–[21] is robust to channel esti-mation errors on wiretap links. However, when the number ofusers is finite, the achievable secrecy rate of this schedulingscheme has not been investigated yet. Accordingly, still,it is important to derive the achievable secrecy rate evenin a single cell uplink wiretap network when we considervarious scheduling schemes such as the threshold-based userscheduling scheme in [18]–[21]. This will be helpful to fillthe gap between secrecy rate analysis of different schedulingschemes and to fully understand the characteristics of thesecrecy rate in uplink multiuser networks.

In this paper, we investigate two user scheduling algo-rithms in a single cell uplink wiretap network: optimaluser scheduling (OUS) and threshold-based user scheduling(TUS) algorithms. The uplink wiretap network consists ofa single desired receiver and a finite number of users (i.e.,transmitters) including potential eavesdroppers and we ana-lyze various aspects of two user scheduling algorithms. Themain contributions of this paper are summarized as follows:• We provide the approximated ergodic secrecy rate ofthe threshold-based user scheduling algorithm, proposecriteria of threshold values to maximize the secrecy rate,and validate the analytical results through simulations(see Sections IV and V).

• We mathematically analyze asymptotic behavior of theachievable secrecy rate of two scheduling algorithms asthe signal-to-noise ratio (SNR) approaches to infinity(see Sections III and IV).

• We investigate the impact of wiretap links’ channelestimation errors and the effect of multiple anten-nas at the receiver, on the secrecy rate of two userscheduling algorithms, respectively, through simulations(see Section VI).

The rest of this paper is organized as follows. In Section II,the overall system model is presented. In Section III, OUSalgorithm is introduced and its analytical results are provided.Similarly, TUS algorithm is introduced, its analytical resultsare provided, and, additionally, criteria of threshold valuesto maximize the secrecy rate is proposed in Section IV. Theperformance of OUS and TUS in terms of the secrecy rate isevaluated in Section V. In Section VI, we additionally discussfeatured issues in applying our proposed scheduling schemes.Finally, conclusive remarks and future work are provided inSection VII.

II. SYSTEM MODELAs described in Fig. 1, we consider a multiuser SISO uplinkwiretap network which consists of a single desired receiverand N legitimate users (transmitters) including K potentialeavesdroppers (i.e., K < N ) [23], [24]. From the perspectiveof the system, a total of N users are considered during eachscheduling time slot, regardless of the number of potentialeavesdroppers. However, the system properly sets the num-ber of potential eavesdroppers before a specific schedul-ing algorithm is running. For example, all unscheduled

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FIGURE 1. An example of multiuser SISO uplink network: a single receiverand N users (index n) including K potential eavesdroppers (index k).

users are considered as potential eavesdroppers if we setK = N − 1 [24]. On the other hands, only one user canbe considered as a potential eavesdropper (e.g., a user nearthe scheduled user) if we set K = 1 [25]. We assume thatthe potential eavesdroppers operate without any cooperationamong them (i.e., the non-colluding eavesdroppers model).Throughout the paper, we use terms ‘desired’ and ‘wire-tap’ links to indicate transmission links from a scheduleduser to the desired receiver and a potential eavesdropper,respectively.

Let hn ∈ C denotes a channel fading coefficient fromuser n to the desired receiver for n ∈ N , {1, · · · ,N } andis assumed to be a complex Gaussian random variables withzero mean and variance σ 2hn , i.e., hn ∼ CN (0, σ

2hn ). Similarly,

let gnk ∈ C denotes a channel fading coefficient from user nto potential eavesdropper k and is assumed to be a complexGaussian random variables with zero mean and variance σ 2gnk ,i.e., gnk ∼ CN (0, σ 2gnk ). For analytical tractability, we assumethat hn and gnk are independent and identically distributed(i.i.d.), i.e., σ 2hn = σ

2h and σ

2gnk = σ

2g ∀n, k .

1

We consider a time slot based system where a single useris scheduled in one time slot (or one scheduling slot) tosecurely send data to the desired receiver against potentialeavesdroppers. As shown in Fig. 2, one time slot is split intomultiple mini-slots and each user transmits a pilot duringone mini-slots for channel estimation purpose (total N mini-slots). Accordingly, the local channel state information (CSI)including both desired and wiretap links is available at eachuser (i.e., hn and gnk ∀k for only user n). Note that a specificchannel feedback method after local CSI estimation at eachuser mainly relies on a user scheduling algorithm.We assumethat the signaling overhead for the channel estimation andfeedback is negligible.

1For example, we can set σ 2hn = σ2h to consider equal-distance users

(or equivalently using proper power control at each user) and set σ 2g =

max∀k{σ 2gnk

}to consider the worst case of secrecy performance.

FIGURE 2. An example of a time slot which consists of multiple mini-slotsfor channel estimations, channel feedbacks, and data transmission.

When user n is scheduled, the received signals atthe desired receiver and at potential eavesdropper k areexpressed, respectively, as

y = hnxn + z,

yk = gnkxn + zk ,

where xn denotes the desired data symbol for user n with anaverage power constraint E [xn] ≤ P, and z and zk denote thecircularly symmetric complex additive white Gaussian noises(AWGNs) with zero mean and variance σ 2. We define thetransmit SNR as ρ , P

σ 2.

The achievable secrecy rate of user n is obtained as

Cn =[log

(1+ |hn|2ρ

)− log

(1+max

k∈K|gnk |2ρ

)]+, (1)

where the first and second terms in right-hand side representthe achievable rate of the desired and wiretap links, respec-tively, and [x]+ = max {x, 0}.The achievable rate of the wiretap link (i.e., information

leakage from the scheduled user to potential eavesdroppers)in (1) is formulated into a maximum of each eavesdropper’sachievable rate due to a noncooperation assumption. Further,the achievable secrecy rate of the scheduled user in (1) highlydepends on user scheduling schemes. Throughout the paper,we focus on the centralized scheduling in which the receiverexplicitly determines the scheduled user even though the userscheduling algorithms can be implemented in a distributedmanner by exploiting backoff timer as in [26].

III. OPTIMAL USER SCHEDULING SCHEMEIn this section, we introduce an optimal user scheduling(OUS) scheme in terms of the achievable secrecy rate.In order to maximize the achievable secrecy rate, a serveduser has to be selected by taking (1) into consideration.Accordingly, the selected user index of the OUS scheme isgiven by

n? = argmaxn∈N

log 1+ |hn|2ρ1+max

k∈K|gnk |2ρ

. (2)The OUS requires N + KN mini-slots for channel estima-

tion (N mini-slots) and feedback (KN mini-slots) since eachuser has to report CSI of K wiretap links to the receiver.

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For the scheduled user n?, the ergodic secrecy rate of theOUS scheme is obtained as

C̄n? = E

log 1+ |hn? |2ρ1+max

k∈K|gn?k |2ρ

∣∣∣∣∣∣Cn? ≥ 0Pr {Cn? ≥ 0}

≈ E

log 1+ |hn? |2ρ1+max

k∈K|gn?k |2ρ

=

∫∞

0log (z) dFZn? (z) , (3)

where the approximation holds from the fact thatPr {Cn? ≥ 0} ≈ 1 for sufficiently large N . FZn? (z) is thecumulative density function (CDF) of a random variable Zn?which is defined as

Zn? ,1+ |hn? |2ρ

1+maxk∈K|gn?k |2ρ

= maxn∈N

|hn|2 + 1/ρmaxk∈K|gnk |2 + 1/ρ

.In order to calculate the ergodic secrecy rate of the OUS

scheme, we first need to derive the CDF of Zn? . For notationalsimplicity, we use the following notations given byX = |hn|2,Y = max

k∈K|gnk |2, and, Z =

|hn|2+1/ρmaxk∈K|gnk |2+1/ρ

=X+1/ρY+1/ρ .

Both |hn|2 and |gnk |2 follow an exponential distribution.Further, Y is the maximum of K i.i.d. exponential randomvariables. Thus, the CDF of Z is obtained using the relation-ship among X , Y , and Z , and it is given by [27]

FZ (z) =∫∞

1/ρ

∫ zy1/ρ

fX

(x −

1ρ

)fY

(y−

1ρ

)dxdy

= 1+K∑i=1

(Ki

)(−1)i

σ 2h i

σ 2g z+ σ2h ie−

z−1σ2h ρ ,

where fX (x) and fY (y) are probability density functions(PDFs) of X and Y , respectively.

Since Zn? is a maximum of N i.i.d. random variable Z , theCDF of Zn? is obtained as follows:

FZ ? (z) =

(1+

K∑i=1

(Ki

)(−1)i

σ 2h i

σ 2g z+ σ2h ie−

z−1σ2h ρ

)N. (4)

Using (3) and (4), the ergodic secrecy rate of the OUSscheme can be calculated as follows:

C̄n? = N∫∞

0log (z)

(1+

K∑i=1

(Ki

)(−1)i σ 2h i

σ 2g z+ σ2h ie−

z−1σ2h ρ

)N−1

×

K∑i=1

(Ki

)(−1)i+1 σ 2h σ

2g ρi+σ

2g iz+σ

2h i

2

ρ(σ 2g z+σ

2h i)2 e−z−1σ2h ρ

dz.(5)

Note that (5) generally cannot be expressed as a closedform but it can be evaluated through numerical calcula-tions [28]. Additionally, for the special case of K = 1 andρ →∞, we derive the closed-form expression of (5).

Proposition 1 (Secrecy performance of the OUS schemefor K = 1 and high SNR): In the case of K = 1 and ρ →∞,the closed form of (5) is obtained as follows:

C̄∞n? =N∑i=1

(Ni

)(−1)i

(γ + ψ (i)+ log

(σ 2g

σ 2h

)), (6)

where γ ≈ 0.577216 is Euler’s constant, and ψ(x) is thedigamma function defined as ψ(x) , ddx ln0(x) where 0(x)is the gamma function defined as 0(x) ,

∫∞

0 tx−1e−tdt.

Proof: See Appendix A.

IV. THRESHOLD-BASED USER SCHEDULING SCHEMEIn this section, we introduce a threshold-based user schedul-ing (TUS) scheme discussed in [18]–[21]. The basic ideaof the TUS scheme is to select a user in order to preventinformation leakage from a user against eavesdroppers byusing an appropriate threshold value (i.e., applying at thewiretap links). It is worth noting that the threshold-basedselection scheme in [17] mainly considers the desired link(i.e., CSI between the transmitter and the receiver) insteadof the wiretap link. This difference results in a new analysisin this paper, compared with the results in [17]. The overallscheduling process of the TUS scheme in a certain time slotis described as follows:• Step 1: Each user estimates its expected informa-tion leakage based on CSI from K eavesdroppers,i.e., max

k∈K|gnk |2.

• Step 2: Only users who satisfy the following thresholdcriterion (C1) transmit a feedback message, i.e., |hn|2,to the receiver.

(C1) maxk∈K|gnk |2 ≤ ηL,

where ηL is the predetermined positive threshold value,which can be determined through simulation or our pro-posed methods.

• Step 3: After collecting feedback from only selectedusers, the receiver schedules the user (n∗) who has thelargest |hn|2. Then, the scheduled user transmit its data.

The TUS requires N + 1 to 2N mini-slots for channelestimation (N mini-slots) and feedback (1 ∼ N mini-slots) since selected users only send an indication insteadof K wiretap links’ CSI.

A. ERGODIC SECRECY RATE OF THE TUS SCHEMENowwe focus on deriving the ergodic secrecy rate of the TUSscheme summarized in the following theorem.Theorem 1: For the scheduled user n∗ and a given thresh-

old value ηL, the ergodic secrecy rate of the TUS scheme isgiven by (7), as shown at the top of next page, where pη is aprobability that a certain user satisfies the threshold criterion

(C1), given by pη ,(1− exp

(−ηLσ 2g

))Kand E1 (x) ,∫

∞

xexp(−t)

t dt.

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C̄n∗ (ηL) =N∑n=1

(Nn

)pnη(1− pη

)N−n n∑i=1

(ni

)(−1)i+1 e

iσ2h ρ E1

(i

σ 2h ρ

)

−1pη

K∑i=1

(Ki

)(−1)i+1

(e

iσ2g ρ

(E1

(i

σ 2g ρ

)− E1

(i

σ 2g ρ+iηLσ 2g

))− e−iηLσ2g log (1+ ηLρ)

), (7)

Proof: Similar to the OUS scheme, for the scheduleduser n∗ and a given threshold value ηL, the ergodic secrecyrate of the TUS scheme is derived as

C̄n∗ (ηL) ≈ E[log

(1+|hn∗ |2ρ

)−log

(1+max

k∈K|gn∗k |2ρ

)]= C̄d (ηL)− C̄w (ηL) , (8)

where the approximation holds with the same assumptionin (3). Here, we define C̄d (ηL) , E

[log

(1+ |hn∗ |2ρ

)]and

C̄w (ηL) , E[log

(1+max

k∈K|gn∗k |2ρ

)], respectively, for

notation simplicity during the proof.Next, we need to obtain C̄d (ηL) and C̄w (ηL) in (8) for a

given threshold value ηL. In the TUS scheme, the number ofusers who transmit their feedback message varies dependingon a value of ηL. Let Nc denote the number of users whosatisfy the threshold criterion (C1) in a certain time slot.On condition of a given Nc, |hn∗ |2 is the maximum of Nci.i.d. exponential random variables since each |hn|2 follows anexponential distribution independent of |gnk |2. Thus, we canobtain C̄d (ηL;Nc) given by

C̄d (ηL;Nc) = E[log

(1+ |hn∗ |2ρ

)|Nc]

=

∫∞

0log (1+ ρx) dF|hn∗ |2 (x)

=

Nc∑i=1

(Nci

)(−1)i+1 e

iσ2h ρ E1

(i

σ 2h ρ

), (9)

where the last equality holds from [29, (4.337.2)] and E1 (x)is the exponential integral function [29].

Accordingly, C̄d (ηL) in (8) is obtained through theexpected value of all possible conditional expectation in (9)when Nc = n for n ∈ {1, · · · ,N } and it it given by

C̄d (ηL) = E[log

(1+ |hn∗ |2ρ

)]=

N∑n=1

(Nn

)pnη(1−pη

)N−n E [log (1+|hn∗ |2ρ) |Nc]=

N∑n=1

(Nn

)pnη(1− pη

)N−n C̄d (ηL; n) , (10)where pη is a probability that a certain user satis-fies the threshold criterion (C1), given by pη ,(1− exp

(−ηLσ 2g

))K.

To obtain C̄w (ηL) in (8), we first need to derive the CDFof Y ∗ = max

k∈K|gn∗k |2. Since the scheduled user (n∗) always

satisfies the threshold criterion (C1), the CDF of Y ∗ is themaximum of K i.i.d. truncated exponential random vari-ables [27]. Thus, C̄w (ηL) is given by (11), as shown at thebottom of the next page. Finally, substituting (10) and (11)into (8) yields the ergodic secrecy rate of the TUS scheme(i.e., C̄d (ηL)− C̄w (ηL)) and it is given by (7).

Note that we derive the ergodic secrecy rate of the TUSscheme for general parameters such as N , K , ρ, σ 2h , σ

2g ,

and ηL.Corollary 1: [Secrecy performance of TUS in high SNR]

For ρ → ∞, the ergodic secrecy rate of the TUS schemeC̄n∗ (ηL) is reduced to (12), as shown at the bottom of thenext page, where γ ≈ 0.577216 is Euler’s constant.

Proof: See Appendix B.

B. DETERMINATION OF THRESHOLD VALUEFor given system parameters (N , K , ρ, σ 2h , and σ

2g ),

the ergodic secrecy rate of the TUS scheme is a functionof ηL. Therefore, we propose two methods to determinethe threshold value: an optimal-determination method and apredictable-determination method.

1) OPTIMAL-DETERMINATION METHODThe optimal-determination method is to set the thresholdvalue by using a linear search algorithm on (7) for all ηL.Thus, the threshold value set by the optimal-determinationmethod is given by

ηoptL = argmax

∀ηL

C̄n∗ (ηL) . (13)

Definitely, the optimal-determination method guarantees theoptimal secrecy performance of the TUS scheme. However,it inherently incurs a computational cost since it requires anexhaustive search for all ηL.

2) PREDICTABLE-DETERMINATION METHODThe predictable-determination method is to set the thresholdvalue by exploiting the basic principles of the TUS scheme.In the TUS protocol, if there is no user satisfying the thresholdcriterion (C1), the corresponding time slot would be wasted.Let p0 denote the probability that there exists at least oneuser satisfying the threshold criterion (C1) and it is expressed

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as follows:

p0 = 1−

1− (1− exp(−ηLσ 2g

))KN . (14)In order to regulate the number of wasted time slots under

a certain level, p0 should be greater than some constant,i.e., p0 ≥ 1−�0 where �0 represents the wasted time slot ratio.Using p0 = 1−�0, the threshold value set by the predictable-determination method is obtained as follows:

ηpreL = −σ

2g log

(1−

(1− �

1N0

) 1K). (15)

Note that �0 is the control parameter which we can deter-mine. We empirically choose �0 between 10−5 and 10−3.The predictable-determination method does not provide anoptimal threshold value for C̄n∗ (ηL). However, it is directlycalculated from (15) and provides quite good secrecy perfor-mance.

V. PERFORMANCE EVALUATIONIn this section, we evaluate the performance of OUS and TUSschemes in terms of the average secrecy rate through our anal-ysis and simulations, compared with a conventionalMaxSNRuser scheduling scheme. The MaxSNR scheme selects auser having the largest value of SNR on the desired link(i.e., |hn|2). In addition, for ρ →∞, the ergodic secrecy rateof the MaxSNR scheme can be calculated as

C̄∞MaxSNR =N∑i=1

(Ni

)(−1)i

(γ + log

(i

σ 2h

))

−

K∑i=1

(Ki

)(−1)i

(γ + log

(iσ 2g

)), (16)

where γ ≈ 0.577216 is Euler’s constant and the detailedderivation is provided in Appendix C.

Note that the MaxSNR scheme does not utilize CSI ofwiretap links, (i.e., gnk ) and thus it shows a baseline secrecyperformance when gnk is unavailable. We use (16) for com-parison in high SNR regime.

FIGURE 3. Average achievable secrecy rate for varying threshold value ηLwhen �0 = 10−3, ρ = 10 dB, σ2h = 1, σ

2g = 1, and various sets of N and K .

A. NUMERICAL RESULTSFig. 3 shows the average achievable secrecy rate for vary-ing threshold values ηL. For comparison, we consider fourdifferent (N ,K ) pairs. For each line, a star-shaped markerindicates ηoptL , and its corresponding average secrecy rate.Similarly, a circle marker in each line indicates ηpreL andits corresponding average secrecy rate. As N increases forfixed K , values of both ηoptL and η

preL decrease since the

criterion (C1) can be satisfied with high probability evenfor a small threshold value when the number of users in thesystem increases. On the contrary, threshold values of bothmethods increase when K increases for fixed N . The gapbetween the secrecy rate of ηpreL and that of η

optL is negli-

gible. Thus, using ηpreL instead of ηoptL is one of reasonable

alternatives.Fig. 4 shows the average achievable secrecy rate for vary-

ing SNRs. System parameters are set to N = 30, K = 1,σ 2h = 1, and σ

2g = 1. For determining η

preL , �0 = 10

−3

is used. As the SNR (ρ) increases, the secrecy rates of allschemes increase but finally converge to certain values sincean increase in the transmit power increases the achievable rateof desired channel and that of wiretap channel at the sametime. The analytical results for the average secrecy rate of the

C̄w (ηL) =1pη

K∑i=1

(Ki

)(−1)i+1

(e

iσ2g ρ

(E1

(i

σ 2g ρ

)− E1

(i

σ 2g ρ+iηLσ 2g

))− e−iηLσ2g log (1+ ηLρ)

), (11)

C̄∞n∗ (ηL) =N∑n=1

(Nn

)pnη(1− pη

)N−n n∑i=1

(ni

)(−1)i

(γ + log

i

σ 2h

)

−

K∑i=1

(Ki

)(−1)i

pη

(γ + log

(iσ 2g

)+ E1

(iηLσ 2g

)+ e−iηLσ2g log ηL

), (12)

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FIGURE 4. Average achievable secrecy rate for varying SNR when�0 = 10−3, N = 30, K = 1, σ2h = 1, and σ

2g = 1.

TUS scheme, including analytical convergence points in (12)when ρ →∞, matches well with simulation results.Fig. 5 shows the average achievable secrecy rate for vary-

ing the number of users. The system parameters are set asK = 3, ρ = 10 dB, σ 2h = 1, and σ

2g = 1. We use

�0 = 10−4 to determine ηpreL . For all three schemes, the aver-

age secrecy rates increase as N increases since an increasein the number of users in the system provides additionalmultiuser diversity which contributes to enhancing secrecyperformance. OUS and TUS schemes outperform the conven-tional scheduling scheme because those schemes effectivelytake the wiretap channel state of the desired users, i.e., |gnk |2,into consideration. The OUS scheme yields the best perfor-mance among three schemes since it always guarantees theoptimal user selection in terms of the secrecy rate. However,the OUS scheme requires feedback messages from all users,whereas the TUS scheme only requires feedback messagesfrom users satisfying the threshold criterion (C1). In addition,the average secrecy rate of the TUS scheme with ηpreL and thatwith ηoptL are almost similar for all ranges of N .Fig. 6 shows the average achievable secrecy rate for vary-

ing the number of potential eavesdroppers. System parame-ters are set as N = 100, ρ = 10 dB, σ 2h = 1, and σ

2g = 1.

We use �0 = 10−4 to determine ηpreL . For a small number

of potential eavesdroppers (K ≤ 5), the average secrecy rateof the TUS scheme with ηpreL and that with η

optL are almost

similar. However, the secrecy performance gap between theTUS scheme with ηpreL and that with η

optL increases when

K increases due to the fact that �0 = 10−4 is not appropriateto reflect the effect of potential eavesdroppers in the system.For all three schemes, the average secrecy rates decrease asK increases since the information leakage of the desired usersincreases.

Fig. 7 shows the average achievable secrecy rate whenwe consider a relative channel-quality ratio between a mean

FIGURE 5. Average achievable secrecy rate for varying the number ofusers when �0 = 10−4, K = 3, ρ = 10 dB, σ2h = 1, and σ

2g = 1.

FIGURE 6. Average achievable secrecy rate for varying the number ofeavesdroppers when �0 = 10−4, N = 100, ρ = 10 dB, σ2h = 1, and σ

2g = 1.

channel-quality of the desired link and that of the wiretap

link, which is defined as λ ,σ 2g

σ 2h. System parameters are

set as N = 30, σ 2h = 1, and σ2g = 1. We utilize (6), (12),

and (16) to obtain results. Note that λ equivalently impliesthe relative distance ratio since values of σ 2h and σ

2g are

inversely proportional to distances of main and wiretap links,respectively. As λ increases, the secrecy rates of all schemesdecrease and finally converge to zero since a large value ofλ implies that scheduled user is located closer to potentialeavesdroppers rather than the desired receiver, i.e., σ 2g � σ

2h .

On contrary to this, the secrecy rates of all schemes increaseas λ decreases.

VI. DISCUSSIONIn this section, we discuss some issues in applying our pro-posed scheduling schemes (OUS and TUS) such as the impact

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FIGURE 7. Average achievable secrecy rate for varying the relativechannel-quality ratio λ when N = 30, K = 1, and ρ →∞.

of channel estimation errors, multiple antennas, and lots ofeavesdroppers. Here, we also provide a summary of CSIrelated properties of OUS and TUS.

A. IMPACT OF CHANNEL ESTIMATION ERRORS ONAVERAGE SECRECY RATEWe investigate the effect of imperfect CSI between the userand the potential eavesdroppers on the secrecy rate. We con-sider estimated CSI of the wiretap link between user n andpotential eavesdropper k as follows:

ĝnk = gnk + ge, (17)

where gnk ∼ CN (0, σ 2g ) and ge ∼ CN (0, σ 2e ) denote theoriginal CSI and the channel estimation error of the wiretaplink, respectively. The Gaussian error is commonly used inmodeling the channel estimation error [20].

Fig. 8 shows average secrecy rate when σ 2e of ge in (17)varies from 0 (i.e., no channel estimation error) to 1. Also,we have set other system parameters as N = 30, K = 1,σ 2h = 1, and σ

2g = 1. Interestingly, the TUS scheme

shows a better performance in terms of secrecy rate than theOUS scheme when channel estimation error increases. Also,the secrecy performance degradation in the OUS scheme iseven worse than our expectation. For example, the MaxSNRscheme outperforms the OUS scheme when channel estima-tion error is quite large (e.g., σ 2e = 0.6).

2 Please note thatsevere secrecy performance degradation in the OUS schemecomes from the high dependency of CSI of the wiretap linksin its scheduling policy. Suppose ĝnk is completely differentfrom the original gnk , the OUS scheme does not guarantee theoptimal secrecy performance anymore since it may select thescheduled user which might have the poor channel condition.

2TheMaxSNR scheme shows the same secrecy performance regardless ofchannel estimation error since it only requires CSI of the desired link.

FIGURE 8. Average secrecy rate when σ2e of ge in (17) varies from 0(i.e., no channel estimation error) to 1, we set N = 30, K = 1,σ2h = 1, and σ

2g = 1.

However, in the TUS scheme, at least the secrecy perfor-mance of theMaxSNR scheme can be guaranteed even in thissituation. If we set a threshold value of the TUS as large aspossible (i.e., ηL → ∞), then the TUS operates the sameas the MaxSNR scheme since all users report indications ofthe threshold condition and the receiver selects the scheduleduser using only the CSI of the desired link, which is exactlythe same as the MaxSNR scheme.

B. APPLYING MULTIPLE ANTENNA ON OUS AND TUSWe investigate the effect of multiple antennas at thereceiver which employs a maximum ratio combining (MRC)technique.3 We still consider that transmitters includingpotential eavesdroppers equip with a single antenna.

If we consider MRC at the receiver, only a distribu-tion of the desired channel (i.e., |hn|2 in (1)) changesfrom the exponential distribution to Chi-squared distribution.Thus, the analysis in this paper can be extended. However,the extended analysis would not be straightforward and not beeasily tractable. We leave this issue for future work. Instead,we provide simulation results.

Fig. 9 shows average secrecy rate when the number ofantennas (M ) at the receiver varies from 1 (a single antenna)to 16. Additionally, we have set other system parameters asN = 30, ρ = 30 dB, K = 1, σ 2h = 1, and σ

2g = 1. As we

expect, the secrecy rate of all scheduling schemes is improvedwhen the number of antennas at the receiver increases. Inter-estingly, as the number of antennas at the receiver increases,the performance gap between different scheduling schemes(e.g., OUS and TUS) also increases. However, it seems thatthe ratio of secrecy rate of TUS and that of OUS keeps

3Including theMRC technique, several multiple antennas techniques (e.g.,zero-forcing technique) can be applied. However, we limit our focus to theMRC technique.

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TABLE 1. The summary of CSI related information on OUS and TUS.

FIGURE 9. Average achievable secrecy rate for varying the number ofantennas (M) at the receiver when N = 30, ρ = 30 dB, K = 1, σ2h = 1, andσ2g = 1.

constant. To confirm this, we newly define the ratio of secrecyrate of TUS (or MaxSNR) and that of OUS as optimality.Fig. 10 shows the optimality which we defined as the ratio

of secrecy rate of TUS (or MaxSNR) and that of OUS whenwe have setN = 30, ρ = 30 dB,K = 1, σ 2h = 1, σ

2g = 1, and

varying M (1 ∼ 16), which corresponds to Fig. 9. For all M ,optimality of TUS shows approximately 89% but optimalityof MaxSNR increases when the number of antennas at thereceiver increases.

C. A CASE OF TOO MANY EAVESDROPPERSFrom our analysis such as (6), (7), and (12), we can obtain thesecrecy rate of an extreme case where there exist too manypotential eavesdroppers (e.g., K = N − 1). In the case of toomany eavesdroppers existing, multiuser diversity from userschedulingmight not be enough to guarantee a certain level ofthe secrecy rate. In this situation, wemight combine OUS andTUS with other techniques such as multiple antennas [30],artificial noise [31], and full duplexing [32] to additionallyimprove secrecy performance. As discussed in Section VI-Band shown in Fig. 9, using the simple MRC-based multipleantennas technique can effectively increase the secrecy rate.

D. SUMMARYIn this subsection, we summarize some featured informationof our proposed scheduling schemes, especially related to CSI(e.g., feedback and estimation errors).

FIGURE 10. Optimality for varying the number of antennas (M) at thereceiver when N = 30, ρ = 30 dB, K = 1, σ2h = 1, and σ

2g = 1.

Table 1 shows the summary of CSI related informationon OUS and TUS. For the required number of mini-slotsfor CSI feedback, as we discussed in Sections III and IV,the OUS requires N + KN mini-slots whereas the TUS onlyrequires N + 1 to 2N mini-slots. For CSI feedback timecomplexity, we consider processing time complexity in termsof the required number of mini-slots for CSI feedback whenK = N − 1. It can be expressed in a big-O notation whichdescribes the limiting behavior of a function when the argu-ment tends towards a particular value or infinity [33]. TheOUS shows a quadratic characteristic since O(N + KN ) =O(N 2) when K = N − 1, but TUS only shows a linearcharacteristic since the required number of mini-slots for CSIfeedback does not depend on the number of potential eaves-droppers. Both OUS and TUS are centralized schedulingschemes but they can be implemented in the distributed man-ner, as we discussed in Section II. Interestingly, the TUS isrobust to channel estimation errors on wiretap links whereasthe OUS is not, as we discussed in Section VI-A.

VII. CONCLUSIONIn this paper, we investigated two user scheduling algorithms(OUS and TUS schemes) in uplink wiretap networks whenwe consider the potential eavesdropping scenario. The OUSscheme achieves the optimal secrecy rate but it requires feed-back from all legitimate users. The TUS scheme shows asuboptimal secrecy performance comparable to OUS scheme

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while reducing the feedback overhead using the thresholdvalue. We analyzed the approximated secrecy rate of twoscheduling algorithms, including asymptotic behavior of theachievable secrecy rate when SNR tends to infinity. Further,we performed additional simulations to investigate the impactof channel estimation error in the wiretap links, and the effectof multiple antennas at the receiver employing MRC, on theaverage secrecy rate, respectively. Due to different schedulingprinciples in OUS and TUS schemes, the TUS scheme yieldsrobustness against the channel estimation error in the wiretaplinks, compared with the OUS scheme. For both OUS andTUS schemes, the secrecy performance is improved whenwe consider multiple antennas at the receiver. Instead of thesecrecy rate analysis, analyzing the secrecy outage probabil-ity of various user scheduling algorithms in uplink wiretapnetworks remains for future work. We leave the diversityorder analysis of the secrecy outage probability with bothOUS and TUS schemes as a further study.

APPENDIX APROOF FOR PROPOSITION 1From (4), we have the PDF of Zn? given by

fZ ? (z) =ddzFZ ? (z)

= N

(1+

K∑i=1

(Ki

)(−1)i σ 2h i

σ 2g z+ σ2h ie−

z−1σ2h ρ

)N−1

×

K∑i=1

(Ki

)(−1)i+1

σ 2h σ2g ρi+ σ

2g iz+ σ

2h i

2

ρ(σ 2g z+ σ

2h i)2 e− z−1σ2h ρ

.(18)

Using L’Hopital’s Rule, in the case of K = 1 and ρ →∞,we can get the PDF of Zn? as follows:

fZ ? (z)=N∑i=1

(Ni

)(−1)i+1 i

(σ 2h

σ 2g

)i (1

σ 2g z+ σ2h

)i+1. (19)

Therefore, by using (19), we can get the result inProposition 1 as follows:

C̄∞n? =∫∞

0log (z) fZn? (z) dz (20)

=

∫∞

0log (z)

N∑i=1

(Ni

)(−1)i+1 i

×

(σ 2h

σ 2g

)i (σ 2h

σ 2g z+ σ2h

)i+1dz

=

N∑i=1

(Ni

)(−1)i+1 i

(σ 2h

σ 2g

)i×

∫∞

0

log (z)(z+ σ 2h /σ

2g

)i+1 dz=

N∑i=1

(Ni

)(−1)i

(γ + ψ (i)+ log

(σ 2g

σ 2h

)), (21)

where the last equality holds from [29, (4.253.6)], andγ and ψ(x) are Euler’s constant and the digamma function,respectively, defined in (6).

APPENDIX BPROOF FOR COROLLARY 1For the scheduled user (n∗), Let us denote Y ∗ = max

k∈K|gn∗k |2.

Thus, Y ∗ is the maximum of K i.i.d. truncated exponentialrandom variables. From [27], the CDF of Y ∗ is given by

FY ∗ (y) =

1− exp(−

yσ 2g

)1− exp

(−ηLσ 2g

)K

, (22)

where y ∈ [0, ηL].Using a differentiation of FY ∗ (y) in (22) and the binomial

theorem, the PDF of Y ∗ is given by

fY ∗ (y) =Kσ 2g pη

K−1∑i=0

(K − 1i

)(−1)i e

−(i+1)yσ2g . (23)

When ρ →∞, (8) is reduced to

C̄∞n∗ (ηL) ≈ limρ→∞E

log 1+ |hn∗ |2ρ1+max

k∈K|gn∗k |2ρ

= E

log |hn∗ |2maxk∈K|gn∗k |2

=E

[log

(|hn∗ |2

)]︸︷︷︸

desired link

−E[log

(maxk∈K|gn∗k |2

)]︸︷︷︸

wiretap link

, (24)

where the approximation holds from the fact that

Prob{|hn∗ |2 ≥ max

k∈K|gn∗k |2

}≈ 1 for sufficiently large N .

For the desired link in (24), E[log

(|hn∗ |2

)]is derived as

E[log

(|hn∗ |2

)]=

N∑n=1

(Nn

)pnη(1− pη

)N−n E [log (|hn∗ |2) |Nc = n]=

N∑n=1

(Nn

)pnη(1− pη

)N−n×

n∑i=1

(ni

)(−1)i+1

i

σ 2h

∫∞

0log (x) e

−ixσ2h dx

=

N∑n=1

(Nn

)pnη(1− pη

)N−n×

n∑i=1

(ni

)(−1)i

(γ + log

(i

σ 2h

)), (25)

where the last equality holds from [29, (4.331.1)] andγ denotes Euler’s constant.

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For the wiretap link in (24), E[log

(maxk∈K|gn∗k |2

)]is

obtained as

E[log

(maxk∈K|gn∗k |2

)]=

∫ ηL0

log(y)fY ∗ (y) dy (26)

=Kσ 2g pη

K−1∑i=0

(K − 1i

)(−1)i

∫ ηL0

log (y) e−(i+1)yσ2g dy (27)

=

K∑i=1

(Ki

)(−1)i

pη

(γ + log

(iσ 2g

)

+E1

(iηLσ 2g

)+ e−iηLσ2g log ηL

), (28)

where the last equality holds from from [29, (4.331.1)] andthe definition of the exponential integral function.

Finally, by substituting (25) and (28) into (24), the resultin Corollary 1 is obtained.

APPENDIX CDERIVATION OF C̄∞MaxSNRFor the MaxSNR scheme, the scheduled user index is deter-mined as

n̂ = argmaxn∈N

{|hn|2

}. (29)

Thus, for ρ →∞, the ergodic secrecy rate of theMaxSNRscheme is given by

C̄∞MaxSNR ≈ limρ→∞E

log 1+ |hn̂|2ρ1+max

k∈K|gn̂k |2ρ

= E

log |hn̂|2maxk∈K|gn̂k |2

= E

[log

(|hn̂|

2)]

︸︷︷︸desired link

−E[log

(maxk∈K|gn̂k |

2)]

︸︷︷︸wiretap link

, (30)

where the approximation holds from the fact that

Prob{|hn̂|2 ≥ max

k∈K|gn̂k |2

}≈ 1 for sufficiently large N .

For the desired link in (30), E[log

(|hn̂|2

)]is obtained as

E[log

(|hn̂|

2)]=

N∑i=1

(Ni

)i

σ 2h(−1)i+1

×

∫∞

0log (x) exp

(−ix

σ 2h

)dx

=

N∑i=1

(Ni

)(−1)i

(γ + log

(i

σ 2h

)), (31)

where the last equality holds from [29, (4.331.1)] andγ denotes Euler’s constant.

Similar to E[log

(|hn̂|2

)], E

[log

(maxk∈K|gn̂k |2

)]is given

by

E[log

(maxk∈K|gn̂k |

2)]=

K∑i=1

(Ki

)(−1)i

(γ + log

(iσ 2g

)).

(32)

Finally, by plugging (31) and (32) into (30), C̄∞MaxSNR in(16) is obtained.

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INKYU BANG (S’11–M’17) received the B.S.degree in electrical and electronic engineeringfrom Yonsei University, Seoul, South Korea,in 2010, and the M.S. and Ph.D. degrees in electri-cal engineering from the Korea Advanced Institutefor Science and Technology (KAIST), Daejeon,South Korea, in 2012 and 2017, respectively.

He was a Research Fellow with the Depart-ment of Computer Science from the National Uni-versity of Singapore (NUS), from 2017 to 2019.

In 2019, he was a Senior Researcher with the Agency for Defense Devel-opment (ADD). He has been an Assistant Professor with the Departmentof Information and Communication Engineering, Hanbat National Univer-sity, since 2019. His current research interests include information-theoreticsecurity (physical-layer security), wireless system security, the Internet ofThings, simultaneous wireless information and power transfer (SWIPT), andmachine learning application in wireless communications.

BANG CHUL JUNG (S’02–M’08–SM’14)received the B.S. degree in electronics engineer-ing from Ajou University, Suwon, South Korea,in 2002, and the M.S. and Ph.D. degrees in elec-trical and computer engineering from the KoreaAdvanced Institute for Science and Technology(KAIST), Daejeon, South Korea, in 2004 and2008, respectively.

He was a Senior Researcher/Research Professorwith the KAIST Institute for Information Tech-

nology Convergence, Daejeon, from 2009 to 2010. From 2010 to 2015, hewas a Faculty Member with Gyeongsang National University, Tongyeong,South Korea. He is currently a Professor with the Department of ElectronicsEngineering, Chungnam National University, Daejeon. His current researchinterests include wireless communications, statistical signal processing,information theory, interference management, radar signal processing, spec-trum sharing, multiple antennas, multiple access techniques, radio resourcemanagement, machine learning, and deep learning.

Dr. Jung was a recipient of the 5th IEEE Communication Society Asia–Pacific Outstanding Young Researcher Award, in 2011, the KICS HaedongYoung Scholar Award, in 2015, and the 29th KOFST Science and Tech-nology Best Paper Award, in 2019. He has been an Associate Editor of theIEICE Transactions on Fundamentals of Electronics, Communications, andComputer Sciences, since 2018.

VOLUME 7, 2019 127089

INTRODUCTIONSYSTEM MODELOPTIMAL USER SCHEDULING SCHEMETHRESHOLD-BASED USER SCHEDULING SCHEMEERGODIC SECRECY RATE OF THE TUS SCHEMEDETERMINATION OF THRESHOLD VALUEOPTIMAL-DETERMINATION METHODPREDICTABLE-DETERMINATION METHOD

PERFORMANCE EVALUATIONNUMERICAL RESULTS

DISCUSSIONIMPACT OF CHANNEL ESTIMATION ERRORS ON AVERAGE SECRECY RATEAPPLYING MULTIPLE ANTENNA ON OUS AND TUSA CASE OF TOO MANY EAVESDROPPERSSUMMARY

CONCLUSIONREFERENCESBiographiesINKYU BANGBANG CHUL JUNG

secrecy rate analysis of opportunistic user scheduling in...

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