Secure Transmission With Antenna Selection in MIMO

6054 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 11, NOVEMBER 2014

Secure Transmission With Antenna Selection inMIMO Nakagami-m Fading Channels

Lifeng Wang, Student Member, IEEE, Maged Elkashlan, Member, IEEE, Jing Huang, Member, IEEE,Robert Schober, Fellow, IEEE, and Ranjan K. Mallik, Fellow, IEEE

Abstract—This paper considers transmit antenna selection(TAS) and receive generalized selection combining (GSC) for se-cure communication in the multiple-input–multiple-output wire-tap channel, where confidential messages transmitted from anNA-antenna transmitter to an NB-antenna legitimate receiverare overheard by an NE-antenna eavesdropper. We assumethat the main channel and the eavesdropper’s channel undergoNakagami-m fading with fading parameters mB and mE , re-spectively. In order to assess the secrecy performance, we presenta new unifying framework for the average secrecy rate and the se-crecy outage probability. We first derive expressions for the prob-ability density function and the cumulative distribution functionof the signal-to-noise ratio with TAS/GSC, from which we deriveexact expressions for the average secrecy rate and the secrecyoutage probability. We then derive compact expressions for theasymptotic average secrecy rate and the asymptotic secrecy outageprobability for two distinct scenarios: 1) the legitimate receiver islocated close to the transmitter, and 2) the legitimate receiver andthe eavesdropper are located close to the transmitter. For thesescenarios, we present new closed-form expressions for several keyperformance indicators: 1) the capacity slope and the power offsetof the asymptotic average secrecy rate, and 2) the secrecy diversityorder and the secrecy array gain of the asymptotic secrecy outageprobability. For the first scenario, we confirm that the capacityslope is one and the secrecy diversity order is mBNBNA. For thesecond scenario, we confirm that the capacity slope and the secrecydiversity order collapse to zero.

Index Terms—Diversity combining, average secrecy rate,Nakagami-m fading, physical layer security, secrecy outageprobability.

I. INTRODUCTION

S ECURE transmission in wireless networks is confrontedwith increasing problems due to the rapid evolution

of future wireless network architectures [1]–[3]. Mobile ter-

Manuscript received October 30, 2013; revised January 29, 2014, April 30,2014, and July 25, 2014; accepted September 23, 2014. Date of publicationSeptember 23, 2014; date of current version November 7, 2014. The material inthis paper was presented in part at the IEEE International Conference on Com-munications, Sydney, Australia, June 2014. The associate editor coordinatingthe review of this paper and approving it for publication was C. Yang.

L. Wang and M. Elkashlan are with the School of Electronic Engineeringand Computer Science, Queen Mary University of London, London E1 4NS,UK (e-mail: [email protected]; [email protected]).

J. Huang is with Qualcomm Technologies Inc., Santa Clara, CA 95051 USA(e-mail: [email protected]).

R. Schober is with the Institute for Digital Communication, Friedrich-Alexander University of Erlangen-Nuremberg, Erlangen 91054, Germany(e-mail: [email protected]).

R. K. Mallik is with the Department of Electrical Engineering, IndianInstitute of Technology Delhi, New Delhi 110 016, India (e-mail: [email protected]).

Digital Object Identifier 10.1109/TWC.2014.2359877

minals are more vulnerable to eavesdropping than their fixedcounterparts. Furthermore, the trend towards network den-sification and multi-layer deployments, as well as the de-velopment of decentralized wireless mesh networks posegreat challenges to the implementation of higher-layer keydistribution and management in practice [4], [5]. Physi-cal layer security is an appealing alternative to resist vari-ous malicious abuses and security attacks. The initial workwas pioneered by Wyner from an information-theoretic per-spective [6]. Stimulated by the development of multiple-input multiple-output (MIMO) communications [7], physicallayer security in MIMO wiretap channels has recently beenaddressed [8]–[11].

The basic concept behind physical layer security is to exploitcharacteristics of wireless channels for transmitting confidentialmessages [12]. In [13], secure connectivity of wireless randomnetworks with multi-antenna transmission in Rayleigh fadingchannels was analyzed to show the connectivity improvement.In [14], the diversity-multiplexing tradeoff (DMT) for MIMOwiretap channel was analyzed. The close relationship betweenthe multi-antenna secrecy communications and cognitive radiocommunications was explored in [15]. Transmit antenna se-lection (TAS) can be adopted to improve information securityat low cost and complexity [16], [17]. In [17], the secrecyoutage probability with receiver side antenna correlation wasderived for Rayleigh fading channels. In [18], maximal-ratiocombining (MRC) and selection combining (SC) were usedto secure the communication in Nakagami-m fading channels.Secrecy mutual information in single-input multiple-output(SIMO) wiretap channels with Nakagami-m fading was exam-ined in [19]. In [20], TAS with receive generalized selectioncombining (GSC) was applied to enable secure transmissionover Rayleigh fading channels. In [21], the secrecy outagewas analyzed in MISO wiretap channels when partial in-formation of the eavesdropper’s channel is known at thetransmitter.

In this paper, we explore antenna selection to secure thetransmission in wiretap channels. We consider the Nakagami-mfading environment due to its versatility in providing a goodmatch to various empirically obtained measurement data [22].Moreover, it includes Rayleigh fading as a special case [19].At the transmitter side, TAS selects the optimal transmit an-tenna which maximizes the instantaneous signal-to-noise ratios(SNRs) at the legitimate receiver. Owing to the fact that thefeedback requirements of TAS are considerably lower com-pared to the so-called closed-loop transmit diversity, it hasbeen applied to the uplink of 4G long term evolution (LTE)

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WANG et al.: SECURE TRANSMISSION WITH ANTENNA SELECTION IN MIMO NAKAGAMI-m FADING CHANNELS 6055

and LTE-Advanced [23]. At the receiver side, GSC selects andcombines the subset of receive antennas with the largest SNRs.This approach offers a performance/implementation tradeoffbetween MRC and SC [24], and reduces the power consump-tion and the cost of RF electronics at the receiver [25]. In non-identically distributed noise, GSC can outperform MRC [26].It is also robust to channel estimation errors since the weakSNR antennas are eliminated [27]. Therefore, research effortshave been devoted to GSC. For example, the performance ofGSC was examined for Rayleigh fading [28] and Nakagami-mfading [29]. In [30], [31], GSC in correlated Nakagami-mfading channels was considered. In [32], the high SNR per-formance of GSC was analyzed in various fading environ-ments. In [33], approximations were presented for the highSNR performance of GSC in relay networks with Nakagami-mfading.

While the aforementioned literature [24]–[33] laid a solidfoundation for the role of GSC in Nakagami-m fading, theimpact of GSC on the wiretap channel in Nakagami-m fadinghas not been investigated yet and the secrecy performance ofTAS/GSC in Nakagami-m fading channels is not well under-stood. In this paper, we address two eavesdropping scenarios:1) Passive eavesdropping and 2) active eavesdropping. Forpassive eavesdropping, we characterize the secrecy outageprobability as the fundamental security metric. For activeeavesdropping, we characterize the average secrecy rate as thefundamental security metric. Our detailed contributions are asfollows.

• We derive a new exact closed-form expression for thecumulative distribution function (CDF) of the SNR withthe GSC. Although CDF expressions were presented in[24], [34] with the aid of the trapezoidal rule, they are notin closed-form and cannot be used to derive the CDF ofthe SNR with TAS/GSC. Hence, we derive new closed-form expressions for the CDF and the probability densityfunction (PDF) of the SNR with TAS/GSC.

• We derive new exact closed-form expressions for theaverage secrecy rate and the secrecy outage probabil-ity using the new CDF and PDF of the SNR withTAS/GSC. Notably, we develop a new comprehensiveanalytical framework for the average secrecy rate. Weaccurately examine the impact of the antenna configu-ration and the channel fading conditions on the averagesecrecy rate.

• We derive new expressions for the average secrecy ratein the high SNR regime for two cases: 1) The legitimatereceiver is located close to the transmitter, and 2) thelegitimate receiver and the eavesdropper are located closeto the transmitter. Based on the asymptotic average secrecyrate, we characterize the average secrecy rate in termsof the high SNR slope and high SNR power offset. Weshow that although the high SNR slope is unaffected bythe network parameters, the high SNR power offset isdependent on the system parameters including transceiverantenna configuration and the fading parameters in themain and the eavesdropper’s channels. We reach the inter-esting conclusion that a capacity ceiling is created when

both the legitimate receiver and the eavesdropper are closeto the transmitter.

• In contrast to the prior literature such as [16]–[20], wecarefully investigate the impact of the locations of the legi-timate receiver and the eavesdropper relative to the trans-mitter. As such, we derive new expressions for the secrecyoutage probability in the high SNR regime for two im-portant cases: 1) The legitimate receiver is located closeto the transmitter, and 2) the legitimate receiver and theeavesdropper are located close to the transmitter. Basedon the asymptotic secrecy outage probability, we char-acterize the secrecy outage probability in terms of thesecrecy diversity order and the secrecy array gain. Weshow that when the legitimate receiver is close to thetransmitter, the full secrecy diversity order is achievedand is entirely determined by the antenna configurationand the fading parameters in the main channel. Theimpact of the eavesdropper is only reflected in the se-crecy array gain. We reach the interesting conclusion thatthe secrecy diversity order collapses to zero when boththe legitimate receiver and the eavesdropper are close tothe transmitter.

Notation: In this paper, (·)T denotes the transpose operator,IM denotes the M ×M identity matrix, 0M×N denotes theM ×N zero matrix, E[·] denotes the expectation operator,Fϕ(·) denotes the CDF of random variable (RV) ϕ, fϕ(·)denotes the PDF of ϕ, sgn(·) denotes the signum function, o(·)denotes the higher order terms, and [x]+ = max{x, 0}.

II. SYSTEM MODEL

We consider a MIMO wiretap channel model which consistsof a transmitter (Alice) with NA antennas, a legitimate receiver(Bob) with NB antennas, and an eavesdropper (Eve) withNE antennas. The main channel (Alice-Bob) and the eaves-dropper’s channel (Alice-Eve) are assumed to undergo quasi-static Nakagami-m fading with fading parameters mB and mE ,respectively. In the main channel, Alice selects a single transmitantenna among NA antennas that maximizes the GSC outputSNR at Bob, while Bob combines the LB (1 ≤ LB ≤ NB)strongest receive antennas. In the eavesdropper’s channel, Evecombines the LE (1 ≤ LE ≤ NE) strongest receive antennas.The channel power gain from the pth transmit antenna tothe lB th receive antenna at Bob is denoted as |hp,lB |2 withE[|hp,lB |2] = Ω1, p = 1, · · · , NA, lB = 1, · · · , NB . The chan-nel power gain from the pth transmit antenna to the lE th receiveantenna at Eve is denoted as |gp,lE |2 with E[|gp,lE |2] = Ω2,lE = 1, · · · , NE . Based on GSC, we arrange {|hp,(lB)|2, 1 ≤lB ≤ NB} in descending order as |hp,(1)|2 ≥ |hp,(2)|2 ≥ · · · ≥|hp,(NB)|2, and {|gp,(lE)|2, 1 ≤ lE ≤ NE} in descending orderas |gp,(1)|2 ≥ |gp,(2)|2 ≥ · · · ≥ |gp,(NE)|2. The index of theoptimal transmit antenna is determined as

p∗ = argmax1≤p≤NA

{LB∑

lB=1

∣∣hp,(lB)

∣∣2} . (1)

Secure transmission is achieved by encoding the confidentialmessage block W into a codeword x = [x(1), · · · , x(l), · · · ,


x(L)], where L is the length of x. The codeword is subject to anaverage power constraint 1

L

∑Ll=1 E[|x(l)|2] ≤ P . In the main

channel, at time slot l, the received signal vector is given by

yB(l) = hx(l) + nB(l), (2)

where h = [hp∗,1, hp∗,2, · · · , hp∗,NB]T ∈ CNB×1 is the main

channel vector between transmit antenna p∗ at Alice and theNB receive antennas at Bob, and nB(l) ∼ CNNB×1(0NB×1,δ2BINB

) is the additive white Gaussian noise (AWGN) vector atBob. We denote γ̄B = Ω1

Pδ2B

as the average SNR per antenna at

Bob. Combining the subset of receive antennas with the largestSNRs at Bob results in the instantaneous SNR in the mainchannel as

γB =

LB∑lB=1

γB(lB), (3)

where γB(lB) = |hp∗,(lB)|2 P

δ2B

. In the eavesdropper’s channel, at

time slot l, the received signal vector is given by

yE(l) = gx(l) + nE(l), (4)

where g = [gp∗,1, gp∗,2, · · · , gp∗,NE]T ∈ CNE×1 is the eaves-

dropper’s channel vector between transmit antenna p∗ atAlice and the NE receive antennas at Eve, and nE(l) ∼CNNE×1(0NE×1, δ

2EINE

) is the additive white Gaussian noise(AWGN) vector at Eve. We denote γ̄E = Ω2

Pδ2E

as the average

SNR per antenna at Eve. Combining the subset of receive an-tennas with the largest SNRs at Eve results in the instantaneousSNR in the eavesdropper’s channel as

γE =

LE∑lE=1

γ(lE), (5)

where γ(lE) = |gp∗,(lE)|2 Pδ2E

.

III. NEW STATISTICAL PROPERTIES

In this section, we derive new closed-form expressions forthe PDF and the CDF of γB in the main channel, and the PDFand the CDF of γE in the eavesdropper’s channel, which lay thefoundation for extracting several key secrecy performance indi-cators, namely the high SNR slope, the high SNR power offset,the secrecy diversity order, and the secrecy array gain. Thesestatistics are general in nature and as such are useful for deter-mining the performance of other wireless systems with GSC.

A. CDF and PDF of the SNR in the Main Channel

Theorem 1: The expressions for the CDF and the PDF of γBare derived as

FγB(x)=

(LB

(mB−1)!

(NB

LB

))NA

NA!∑̃

�ρxθρe−ηρx, (6)

fγB(x)=

(LB

(mB−1)!

(NB

LB

))NA

NA!∑̃

�ρxθρ−1e−ηρx(θρ−ηρx),

(7)

where∑̃ Δ

=∑SB

∑S1B

· · ·∑SkB

· · ·∑S|S|B

, SB=

{(nτ,1, · · · , nτ,|S|

)∣∣∣∣ |S|∑k=1

nτ,k = NA

}, |S| is the cardinality of set S , and S denotes a set

of (2mB + 1)-tuples satisfying the condition

S =

{(nΦk,0 · · · , nΦ

k,mB−1, nFk,0, · · · , nF

k,mB

) ∣∣mB−1∑i=0

nΦk,i = LB − 1,

mB∑j=0

nFk,j = NB − LB

},

thereby |S| =(mB+LB−2

mB−1

)(mB+NB−LB

mB

), Sk

B ={(nρk,0, · · · , nρk,mBLB+bF

k

) ∣∣∣mBLB+bFk∑

n=0nρk,n = nτ,k

},

k = 1, · · · , |S|, and �ρ, θρ, and ηρ are respectively given by

�ρ =

|S|∏k=1

⎛⎜⎜⎜⎝(aΦk a

Fk

(n1 − 1)!

(LB)n1

)nτ,k

mBLB+bFk∏

n=0�nρk

,nn

mBLB+bFk∏

n=0nρk,n!

⎞⎟⎟⎟⎠ ,

θρ =

|S|∑k=1

mBLB+bFk∑

n=0

μnnρk,n, ηρ =

|S|∑k=1

mBLB+bFk∑

n=0

νnnρk,n,

where n1 = bΦk + bFk +mB , aΦk , aFk , �n, bFk , bΦk , μn, and νn aredefined in Appendix A.

Proof: The proof is given in Appendix A. �Theorem 2: In the high SNR regime with γB → ∞, the

asymptotic CDF of γB is given by

FγB(x) =

(LB

(NB

LB

))NA(

mB

γ̄B

)mBNBNA

xmBNBNA

((mB − 1)!(mB !)NB−LB (mBNB)!)NA

×

⎛⎝∑SΦB

aΦk

(bΦk +mB(NB − LB) +mB − 1

)!

(LB)bΦk+mB(NB−LB)+mB

⎞⎠NA

, (8)

where SΦB =

{(nΦk,0, · · · , nΦ

k,mB−1

) ∣∣∣mB−1∑i=0

nΦk,i = LB − 1

}.

Proof: The proof is given in Appendix B. �

B. CDF and PDF of the SNR in the Eavesdropper’s Channel

Alice selects the strongest transmit antenna according to thechannel power gains of the main channel, which corresponds toselecting a random transmit antenna for Eve. Hence, similar to


(59) given in Appendix A, the expressions for the CDF and thePDF of γE are respectively derived as

FγE(x) =

LE

(mE − 1)!

(NE

LE

)∑SE

mELE+bFk∑

n=0

aΦk aFk(

bΦk + bFk +mE − 1)!

(LE)bΦk+bF

k+mE

�nxμne−νnx, (9)

fγE(x) =

LE

(mE − 1)!

(NE

LE

)∑SE

mELE+bFk∑

n=1

aΦk aFk(

bΦk + bFk +mE − 1)!

(LE)bΦk+bF

k+mE

�nxμn−1e−νnx(μn−νnx),

(10)

where SE denotes a set of (2mE + 1)-tuples satisfying thecondition

SE =

{(nΦk,0 · · · , nΦ

k,mE−1, nFk,0, · · · , nF

k,mE

) ∣∣mE−1∑i=0

nΦk,i = LE − 1,

mE∑j=0

nFk,j = NE − LE

}.

All the parameters in (9) and (10) are identical to those inTheorem 1 and are calculated accordingly.

IV. AVERAGE SECRECY RATE

In this section, we focus on the active eavesdroppingscenario,1 where the CSI of the eavesdropper’s channel is alsoknown at Alice. Following the wiretap channel in [11], [35],Alice encodes a message block W k into a codeword Xn, andEve receives Y n

w from the output of its channel. The equivoca-tion rate of Eve is Re = H(W k|Y n

w )/n, which is the amountof ignorance that the eavesdropper has about a message W k

[11]. In the active eavesdropping scenario, Alice can adapt theachievable secrecy rate R such that R ≤ Re [11], [35]. Here,We focus on the maximum achievable secrecy rate Cs = Re

[11], [35], which is characterized as [8], [11], [14], [35]

Cs = [CB − CE ]+, (11)

where CB = log2(1 + γB) is the capacity of the main channeland CE = log2(1 + γE) is the capacity of the eavesdropper’schannel. Since the CSI of eavesdropper’s channel is availableto Alice, Alice can transmit confidential messages at a rate Cs,to guarantee perfect secrecy.

In active eavesdropping scenario, the average secrecy rateis essentially a fundamental secrecy performance metric. Wederive new exact and asymptotic expressions for the averagesecrecy rate. Based on the asymptotic expressions, we charac-terize the average secrecy rate in terms of the high SNR slopeand the high SNR power offset, to explicitly capture the impactof arbitrary antennas and channel parameters on the averagesecrecy rate at high SNR [36].

1In this scenario, the eavesdropper is active [35]. Such a scenario is particu-larly applicable in the multicast and unicast networks where the users play dualroles as legitimate receivers for some signals and eavesdroppers for others [12].

A. Exact Average Secrecy Rate

The average secrecy rate is the average of the secrecy rateCs over γB and γE . As such, the exact average secrecy rate isgiven by

C̄s =

∫ ∞

0

∫ ∞

0

CsfγB(x1)fγE

(x2)dx1dx2

=

∫ ∞

0

[∫ ∞

0

CsfγE(x2)dx2

]︸︷︷︸

ω1

fγB(x1)dx1. (12)

We first calculate ω1 in (12) as

ω1 =

∫ x1

0

(log2(1 + x1)− log2(1 + x2)) fγE(x2)dx2. (13)

Using integration by parts, and applying some algebra, wederive (13) as

ω1 = log2(1 + x1)FγE(x1)

−(log2(1+x1)FγE

(x1)−1

ln 2

∫ x1

0

1

1+x2FγE

(x2)dx2

)=

1

ln 2

∫ x1

0

FγE(x2)

1 + x2dx2. (14)

Substituting (14) into (12), and changing the order of integra-tion, we obtain

C̄s =1

ln 2

∫ ∞

0

FγE(x2)

1 + x2

[∫ ∞

x2

fγB(x1)dx1

]dx2

=1

ln 2

∫ ∞

0

FγE(x2)

1 + x2(1− FγB

(x2)) dx2. (15)

Using the new statistical properties in Section III, we calculate(15) as

C̄s =LE

ln 2(mE − 1)!

(NE

LE

)∑SE

mELE+bFk∑

n=0

aΦk aFk

×(bΦk + bFk +mE − 1

)!

(LE)bΦk+bF

k+mE

�n

×[∫ ∞

0

xμn

2

1 + x2e−νnx2dx2

−(

LB

(mB − 1)!

(NB

LB

))NA

NA!∑̃

�ρ

×∫ ∞

0

xμn+θρ2

1 + x2e−(νn+ηρ)x2dx2

]

=LE

ln 2(mE − 1)!

(NE

LE

)∑SE

mELE+bFk∑

n=0

aΦk aFk


)!

(LE)bΦk+bF

k+mE

�n


×[μn!Ψ(μn + 1, μn + 1; νn)

−(

LB

(mB − 1)!

(NB

LB

))NA

NA!∑̃

�ρ(μn + θρ)!

×Ψ(μn + θρ + 1, μn + θρ + 1; νn + ηρ)

], (16)

where Ψ(·, ·; ·) is the confluent hypergeometric function [37,eq. (9.211.4)]. Our new expression for the exact average secrecyrate in (16) applies to arbitrary numbers of antennas, arbitraryfading parameters, and arbitrary average SNRs.

B. Asymptotic Average Secrecy Rate

In order to explicitly examine the performance in the highSNR regime, we proceed to derive the asymptotic averagesecrecy rate. We take into account two realistic scenarios:1) Bob is located close to Alice, which can be mathematicallydescribed as γ̄B → ∞ for arbitrary γ̄E , and 2) Bob and Eve arelocated close to Alice, which can be mathematically describedas γ̄B → ∞ and γ̄E → ∞.

To facilitate the analysis, we rewrite the CDF of γE as

FγE(x) = 1 + χγE

(x), (17)

where

χγE(x) =

LE

(mE − 1)!

(NE

LE

)∑SE

mELE+bFk∑

n=1

aΦk aFk


)!

(LE)bΦk+bF

k+mE

�nxμne−νnx.

1) γ̄B → ∞: In this case, we introduce a new general formto derive the average secrecy rate in the following theorem.

Theorem 3: The asymptotic average secrecy rate is given by

C̄∞s = Δ1 +Δ2, (18)

where

Δ1 =1

ln 2

∫ ∞

0

ln(x1)fγB(x1)dx1 (19)

and

Δ2 =1

ln 2

∫ ∞

0

χγE(x2)

1 + x2dx2. (20)

Proof: The proof is given in Appendix C. �Based on Theorem 3, we calculate the asymptotic average

secrecy rate using the new statistical properties in Section III.Specifically, by substituting (7) into (19), and employing [37,eq. (4.352.1)], Δ1 is derived as

Δ1 = log2(γ̄B)− log2(mB) +1

ln 2

×(

LB

(mB − 1)!

(NB

LB

))NA

NA!∑̃

�̃ρζ1, (21)

where �̃ρ = �ρ

(mB

γ̄B

)−θρand

ζ1 =

⎧⎨⎩0, θρ = 0, η̃ρ = 0,ln(η̃ρ) + C, θρ = 0, η̃ρ > 0,

− (θρ−1)!

(η̃ρ)θρ

, θρ > 0, η̃ρ > 0,(22)

In (22), C is the Euler’s constant [37, eq. (8.367.1)] and η̃ρ =(mB

γ̄B

)−1

ηρ. It is worth noting that �̃ρ and η̃ρ are independent

of γ̄B . We should also note that Δ1 in (21) explicitly quantifiesthe impact of the main channel on the average secrecy rate.

Substituting χγEgiven in (17) into (20), we obtain Δ2

Δ2 =LE

ln 2(mE − 1)!

(NE

LE

)∑SE

mELE+bFk∑

n=1

aΦk aFk


)!

(LE)bΦk+bF

k+mE

�nμn!Ψ(μn + 1, μn + 1, νn), (23)

which explicitly quantifies the impact of the eavesdropper’schannel on the average secrecy rate.

Based on (18), (21), and (23), we derive the asymptoticaverage secrecy rate as

C̄∞s = log2(γ̄B)−log2(mB)+

1

ln 2

(LB

(mB − 1)!

(NB

LB

))NA

×NA!∑̃

�̃ρζ1+LE

ln 2(mE−1)!

(NE

LE

)∑SE

mELE+bFk∑

n=1

aΦkaFk

×(bΦk+ bFk +mE−1

)!

(LE)bΦk+bF

k+mE

�nμn!Ψ(μn+1, μn+1, νr). (24)

Based on (24), we derive two key performance indicators thatdetermine the average secrecy rate at high SNR, namely thehigh SNR slope and the high SNR power offset [36], [38]. Theasymptotic average secrecy rate in (24) can be conveniently re-expressed as [36]

C̄∞s = S∞ (log2(γ̄B)− L∞) , (25)

where S∞ is the high SNR slope in bits/s/Hz/(3 dB) and L∞is the high SNR power offset in 3 dB units. We note that thehigh SNR slope is also known as the maximum multiplexinggain or the number of degrees of freedom [7]. The high SNRpower offset is a more intricate function which depends on thenumber of transmit and receive antennas, as well as the channelcharacteristics [36], [38].

The high SNR slope S∞ is given by

S∞ = limγ̄B→∞

C̄∞S

log2(γ̄B). (26)

Substituting (24) into (26), we obtain

S∞ = 1. (27)

From (27), we see that the eavesdropper’s channel and thenumber of Bob’s receive antennas have no impact on the highSNR slope S∞.


The high SNR power offset L∞ is given by

L∞ = limγ̄B→∞

(log2(γ̄B)−

C̄∞S

S∞

). (28)

Substituting (24) and (27) into (28), we derive L∞ as2

L∞=LB∞(mB , NB , LB , NA)+LE

∞(mE , NE , LE , γ̄E), (29)

where

LB∞(mB , NB , LB , NA) = log2(mB)−

× 1

ln 2

(LB

(mB − 1)!

(NB

LB

))NA

NA!∑̃

�̃ρζ1 (30)

and

LE∞(mE , NE , LE , γ̄E) = −Δ2. (31)

In (30), LB∞ quantifies the contribution of the main channel

to the high SNR power offset. In (31), LE∞ quantifies the

contribution of the eavesdropper’s channel to the high SNRpower offset. We next examine special cases of LB

∞ and LE∞

in which these expressions reduce to more simple forms.Corollary 1: For the special case of Rayleigh fading, where

mB = mE = 1, LB∞ in (30) reduces to

LB∞(1, NB , LB , NA)=− 1

ln 2

(LB

(NB

LB

))NA

NA!∑̃

�̃ρζ1 (32)

and LE∞ in (31) reduces to

LE∞(1, NE , LE , γ̄E) = − 1

ln 2

(NE

LE

)∑SFE

LE∑n=1

aFk �nμn!

×Ψ(μn + 1, μn + 1, νn), (33)

where SFE =

{(nFk,0, n

Fk,1

) ∣∣∣ 1∑j=0

nFk,j = NE − LE

}.

Corollary 2: For the special case of Rayleigh fading withTAS/MRC, where mB = mE = 1, LB = NB , and LE = NE ,LB∞ in (30) reduces to

LB∞(1, NB , NB , NA)=− 1

ln 2NA!

∑S1B

NB∏n=1

(−1

(n−1)!

)nρ1,n

NB∏n=0

nρ1,n!

β, (34)

where S1B =

{(nρ1,0, · · · , nρ1,NB

)|NB∑n=0

nρ1,n = NA

}and

β =

{ln(NA) + C, θρ = 0,

− (θρ−1)!

(NA)θρ, θρ > 0. (35)

From (31), LE∞ reduces to

LE∞(1, NE , NE , γ̄E)=

1

ln 2

NE∑n=1

(1

γ̄E

)n−1

Ψ

(n, n,

1

γ̄E

). (36)

2Here, we explicitly reveal the dependence of the high SNR power offset onmB , NA, NB , LB , mE , NE , LE , γ̄E .

It is clear from (36) that LE∞ is an increasing function of NE . As

such, when the number of antennas at Eve increases, the highSNR power offset also increases, which in turn decreases theaverage secrecy rate.

Corollary 3: For the special case of Rayleigh fading withTAS/SC, where mB = mE = 1, LB = 1, and LE = 1, LB

∞ in(30) reduces to

LB∞(1, NB , 1, NA) = − 1

ln 2(NB)

NANA!∑̃

�̃ρ

× sgn(η̃ρ) (ln(η̃ρ) + C) . (37)

By applying [37, eq. (3.352.4)], LE∞ in (31) reduces to

LE∞(1, NE , 1, γ̄E)=

NE

ln 2

∑SFE

(NE−1)!1∏

j=0

nFk,j !

(−1)nFk,1

×

⎛⎝ sgn(nFk,1

)nFk,1+1

+ 1− sgn(nFk,1

)⎞⎠

×

⎛⎝−e(nF

k,1+1)

γ̄E Ei

⎛⎝−

(nFk,1 + 1

)γ̄E

⎞⎠⎞⎠ ,

(38)

where SFE =

{(nF

k,0, nFk,1)|

1∑j=0

nFk,j = NE − 1

}and Ei(·) is

the exponential integral function defined in [37, eq. (8.211.1)].2) γ̄B → ∞ and γ̄E → ∞: In this case, the average secrecy

rate can be easily obtained based on Theorem 3. We only needto further provide the asymptotic Δ2 with γ̄E → ∞. ObservingΔ1 in (21), the asymptotic Δ2 is derived according to

Δ2 = − (log2(γ̄E)− log2(mE))− Ξ, (39)

where

Ξ =1

ln 2

LE

(mE − 1)!

(NE

LE

)∑SE

mELE+bFk∑

n=1

aΦk aFk


)!

(LE)bΦk+bF

k+mE

�̃n

×((1− sgn(μn))×(C+ln(ν̃n))−sgn(μn)

(μn−1)!

(ν̃n)μn

).

(40)

In (40), �̃n = �n

(mE

γ̄E

)−μn

and ν̃n = νn

(mE

γ̄E

)−1

. We should

note that in (40), Ξ is independent of γ̄E .Substituting (21) and (39) into (18), we derive the asymptotic

average secrecy rate as

C̄∞s = log2

(γ̄Bγ̄E

)− log2

(mB

mE

)

+1

ln 2

(LB

(mB − 1)!

(NB

LB

))NA

NA!∑̃

�̃ρζ1 − Ξ. (41)


Fig. 1. Average secrecy rate for NA = 2, NB = 4, NE = 3, LE = 2,mB = mE = 2, and γ̄E = 10 dB.

From (41), we see that for a fixed ratio of γ̄B and γ̄E , theaverage secrecy rate is a constant value at high SNR. Accordingto (26), the high SNR slope S∞ is zero. This new result showsthat when the eavesdropper is located close to the transmitter,increasing the transmit power does not have an impact on theaverage secrecy rate.

C. Numerical Examples

Fig. 1 plots the average secrecy rate versus γ̄B for differentLB . The exact curves are obtained from (16). Considering thescenario where Bob is located close to Alice, the asymptoticaverage secrecy rate curves are obtained from (24). We see thatthe exact curves are well-validated by Monte Carlo simulations3

marked with ‘•’. We also see that the asymptotic curves wellapproximate the exact curves at high SNR. As suggested by(27), the high SNR slope is independent of LB , which is alsoindicated by the parallel slopes of the asymptotes. The averagesecrecy rate increases with the number of selected antennasLB . This is because LB

∞ in (30) decreases with increasing LB

and accordingly the high SNR power offset L∞ decreases.Notably, the performance difference diminishes for large LB .This confirms that TAS/GSC provides a performance tradeoffbetween TAS/MRC and TAS/SC in MIMO wiretap channels.

Fig. 2 plots the average secrecy rate versus γ̄B for differentLE . The parallel slopes of the asymptotes confirm that the highSNR slope is independent of LE . The average secrecy ratedecreases with the number of selected antennas LE . This is dueto the fact that LE

∞ in (31) increases with LE and accordinglythe high SNR power offset L∞ increases.

Fig. 3 plots the high SNR power offset for several cases.Here, (a) and (b) represent the impact of increasing Eve’s an-tennas for different LE , and (c) and (d) represent the impact ofincreasing Bob’s antennas for different LB . The power offset isobtained using (29). We see that the power offset increases withincreasing NE and LE , which in turn decreases the averagesecrecy rate, as shown in (25). We also see that the power offsetdecreases with increasing NB and LB , which in turn increasesthe average secrecy rate.

3In this paper, Monte Carlo simulated results are numerically computedbased on system parameters given in each figure, which validate the accuracyof our analytical results.

Fig. 2. Average secrecy rate for NA = 2, NB = 4, NE = 4, LB = 2,mB = mE = 2, and γ̄E = 10 dB.

Fig. 3. High SNR power offset with mB = mE = 2 and γ̄E = 10 dB forfour cases: (a) NA = 2, NB = 4, NE = N , LB = 2, LE = 4, (b) NA =2, NB = 4, NE = N , LB = 2, LE = 2, (c) NA = 4, NB = N , NE = 3,LB = 2, LE = 2, and (d) NA = 4, NB = N , NE = 3, LB = 4, LE = 2.

Fig. 4. Average secrecy rate for NA = 2, NB = 4, NE = 3, mB = mE =2, and LE = 2.

Fig. 4 plots the average secrecy rate versus γ̄B for differentLB . The exact curves are obtained from (16). Considering thescenario where Bob and Eve are located close to Alice, weset γ̄B

γ̄E

∣∣∣dB

= 10 dB, and the asymptotic curves are obtained

from (41). Observe that the average secrecy rate increases withincreasing LB . As predicted by (41), the average secrecy rateconverges to a finite limit at high SNR, which proves that thehigh SNR slope collapses to zero.


Fig. 5. Average secrecy rate for NA = 2, NB = 4, NE = 4, mB = mE =2, and LB = 2.

Fig. 5 plots the average secrecy rate versus γ̄B for different

LE . Here, we set γ̄B

γ̄E

∣∣∣dB

= 10 dB. We see that the average

secrecy rate decreases with increasing LE . As reflected in (41),the average secrecy rate approaches a constant at high SNR.

V. SECRECY OUTAGE PROBABILITY

In this section, we concentrate on passive eavesdroppingscenario, where the CSI of the eavesdropper’s channel is notknown at Alice. In such a scenario, Alice has no choice butto encode the confidential data into codewords of a constantrate Rs [35], if Rs ≤ Cs (Cs has been defined in (11)), perfectsecrecy can be achieved. Otherwise, if Rs > Cs, information-theoretic security is compromised. In other words, unlike theactive eavesdropping scenario, perfect secrecy cannot be guar-anteed in the passive eavesdropping scenario, since Alice hasno information about the eavesdropper’s channel. Motivatedby this, we adopt the secrecy outage probability as a usefulperformance measure. We derive new closed-form expressionsfor the exact and the asymptotic secrecy outage probability.Based on the asymptotic expressions, we present two keyperformance indicators, namely the secrecy diversity order andthe secrecy array gain.

A. Exact Secrecy Outage Probability

A secrecy outage is declared when the secrecy rate Cs is lessthan the expected secrecy rate Rs. As such, the secrecy outageprobability is derived as

Pout(Rs) = Pr(Cs < Rs)

=

∫ ∞

0

fγE(x2)FγB

(2Rs(1 + x2)− 1

)dx2. (42)

Substituting (6) and (10) into (42), and applying the binomialexpansion [37, eq. (1.111)] and [37, eq. (3.351.3)], we obtain

Pout(Rs) =LE

(mE − 1)!

(NE

LE

)∑SE

mELE+bFk∑

n=1

aΦk aFk


)!

(LE)bΦk+bF

k+mE

�n

×(

LB

(mB − 1)!

(NB

LB

))NA

NA!

×∑̃

�ρ

θρ∑q=0

(θρq

)2Rsq(2Rs−1)θρ−qe−ηρ(2

Rs−1)

×(

μnΓ(q + μn)

(ηρ2Rs + νn)q+μn− νn(q+μn)!

(ηρ2Rs+νn)q+μn+1

).

(43)

Our new expression for the exact secrecy outage probabilityin (43) applies to arbitrary numbers of antennas at Bob andEve, arbitrary fading parameters, and arbitrary average SNRsin the main and eavesdropper’s channels. As shown in [17], theprobability of positive secrecy can be evaluated as 1− Pout(0).

B. Asymptotic Secrecy Outage Probability

In this subsection, we turn our attention to the asymptoticsecrecy outage probability. We consider the following twoscenarios.

1) γ̄B → ∞: In this case, Bob is located close to Alice.We substitute (8) and (10) into (42), and derive the asymptoticsecrecy outage probability as

P∞out(Rs) = (Gaγ̄B)

−Gd + o(γ̄−Gd

B

), (44)

where the secrecy diversity order is

Gd = mBNBNA (45)

and the secrecy array gain is

Ga=

⎡⎢⎣ LE

(mE − 1)!

(LB

(NB

LB

))NA

(mB)mBNBNA

(Γ(mB)(mB !)NB−LB (mBNB)!)NA

×(NE

LE

)⎛⎝∑SΦB

aΦk

(bΦk +mB(NB−LB)+mB−1

)!


⎞⎠NA

×∑SE

mELE+bFk∑

n=1

aΦk aFk

(bΦk + bFk +mE − 1

)!

(LE)bΦk+bF

k+mE

�n

×mBNBNA∑

q=0

(mBNBNA

q

)(Γ(q+μn)μn−(q + μn)!)

× 2Rsq(2Rs − 1)mBNBNA−q

(νn)q+μn

]− 1mBNBNA

. (46)

Based on (45) and (46), we find that the secrecy diversityorder is entirely determined by the antenna configuration andthe fading parameters in the main channel. The impact ofthe eavesdropper’s channel is only reflected in the secrecyarray gain.

In order to characterize the impact of GSC on the secrecyoutage probability, we quantify the secrecy outage tradeoffbetween LB + l and LB , l = 1, · · · , NB − LB . From (45), we


confirm that LB + l and LB have the same secrecy diversityorder. As such, one can conclude that the SNR gap betweenLB + l and LB is strictly determined by their respective se-crecy array gains and is expressed as

Ga(LB + l)

Ga(LB)=

[(mB !)

l(LB + l)(

NB

LB+l

)LB

(NB

LB

)

×

∑SΦl

B

aΦl

k

(bΦ

l

k+mB(NB−LB−l)+mB−1

)!

(LB+l)bΦ

l

k+mB(NB−LB−l)+mB∑

SΦB

aΦk(bΦk +mB(NB−LB)+mB−1)!

(LB)bΦk

+mB(NB−LB)+mB

⎤⎥⎥⎥⎥⎦− 1

mBNB

(47)

where SΦl

B satisfies the condition

SΦl

B =

{(nΦl

k,0, · · · , nΦl

k,mB−1

)∣∣∣mB−1∑i=0

nΦl

k,i = LB + l − 1

}.

Corollary 4: For the special case of Rayleigh fading, thesecrecy diversity order in (45) reduces to NBNA and thesecrecy array gain in (46) reduces to

Ga =

⎡⎣(NE

LE

)(2Rs − 1)NBNA

(LB !)NA(LB)NA(NB−LB)

∑SFE

×LE∑n=1

aFk �n

NBNA∑q=0

(NBNA

q

)(2Rs

2Rs − 1

)q

× (Γ(q + μn)μn − (q + μn)!)

(νn)q+μn

]− 1NBNA

. (48)

Based on (48), we confirm that Ga(LB+1)Ga(LB) > 1. This proves that

the secrecy array gain is an increasing function of LB . It followsthat the SNR gap between LB + l and LB in (47) reduces to

Ga(LB+l)

Ga(LB)=

⎡⎢⎣ (LB)lLB !

(LB+l)!(1+ l

LB

)NB−LB−l

⎤⎥⎦− 1

NB

. (49)

Based on (49), we confirm that(

Ga(LB+1+l)Ga(LB+1)

)/(

Ga(LB+l)Ga(LB)

)< 1. This proves that the SNR gap is a decreasing

function of LB .

Fig. 6. Secrecy outage probability for NB = 3, NE = 3, LE = 2, mB =1, mE = 2, and γ̄E = 10 dB.

2) γ̄B → ∞, γ̄E → ∞: In this case, both Bob and Eve arelocated close to Alice. Based on (43), the asymptotic secrecyoutage probability is derived as

P∞out(Rs) = lim

γ̄B→∞,γ̄E→∞Pout(Rs)

=LE

(mE − 1)!

(NE

LE

)∑SE

mELE+bFk∑

n=1

aΦk aFk

× �̃n

(bΦk + bFk +mE − 1

)!

(LE)bΦk+bF

k+mE

×(

LB

(mB − 1)!

(NB

LB

))NA

NA!

×∑̃

�̃ρ

(mB γ̄EmE γ̄B

)θρ 2Rsθρ(η̃ρ2Rs

mB γ̄E

mE γ̄B+ν̃n

)θρ+μn

×(μnΓ(θρ + μn)−

ν̃n(θρ + μn)!

η̃ρ2RsmB γ̄E

mE γ̄B+ ν̃n

). (50)

For a fixed ratio of γ̄B and γ̄E , (50) confirms that the secrecyoutage probability approaches a constant at high SNR, whichimplies that the secrecy diversity order is zero. Once again, thisresult shows that increasing the transmit power does not havean impact on the secrecy outage probability.

C. Numerical Examples

Fig. 6 plots the secrecy outage probability versus γ̄B fordifferent LB and NA. The expected secrecy rate is Rs =1 bit/s/Hz. The exact curves are obtained from (43). Consid-ering Bob is located close to Alice, the asymptotic curves areobtained from (44). The exact curves are in precise agree-ment with the Monte Carlo simulations. We also see thatthe asymptotic curves accurately predict the secrecy diversityorder and the secrecy array gain. According to (45), we seethat the secrecy diversity order increases with NA, which in


Fig. 7. SNR gap for mB = 2 and NB = 10.

Fig. 8. Secrecy outage probability for NB = 3, NE = 3, mB = 1, mE =2, and LE = 2.

turn decreases the secrecy outage probability. We also see thatthe secrecy outage probability decreases with LB , due to anincrease in the secrecy array gain.

Fig. 7 plots the SNR gap versus LB for different l. TheSNR gap between LB + l and LB is obtained from (47). Theexpected secrecy rate is Rs = 1 bit/s/Hz. On the one hand,the SNR gap increases with increasing l. This confirms that thesecrecy array gain is an increasing function of LB . On the otherhand, the SNR gap decreases with increasing LB . This showsthat the SNR gap is a decreasing function of LB .

Fig. 8 plots the secrecy outage probability versus γ̄B fordifferent LB and NA. The expected secrecy rate is Rs =1 bit/s/Hz. The exact curves are obtained from (43). Consid-ering both Bob and Eve are located close to Alice, we setγ̄B

γ̄E

∣∣∣dB

= 10 dB and the asymptotic curves are obtained from

(50). Observe that the secrecy outage probability decreaseswith LB and NA. As suggested by (50), the secrecy outageprobability approaches a constant at high SNR, which confirmsthat the secrecy diversity order is zero.

VI. CONCLUSION

We analyzed TAS/GSC for physical layer security in MIMOwiretap channels. In doing so, we derived new analyticalexpressions of the statistical properties on the SNR withTAS/GSC in Nakagami-m fading. With the aid of these results,we first presented new closed-form expressions for the exactand the asymptotic average secrecy rate. Using these expres-

sions, we precisely characterized the high SNR slope and thehigh SNR power offset. We then presented new closed-formexpressions for the exact and the asymptotic secrecy outageprobability, which concisely characterized the secrecy diversityorder and the secrecy array gain. Several key observations weredrawn based on the locations of the legitimate receiver andthe eavesdropper relative to the transmitter. We showed that acapacity ceiling and an outage floor were created when boththe legitimate receiver and the eavesdropper were close to thetransmitter. Massive MIMO [39] could be used to cope withthis issue. Moreover, our results provide a useful analyticalguideline for the more general scenarios of: 1) Cooperativejamming [40] to confound the eavesdroppers, 2) imperfectchannel knowledge, and 3) multiple destinations and multipleeavesdroppers.

APPENDIX APROOF OF THEOREM 1

We first present the PDF and the CDF of the SNR of a singlebranch in the main channel with Nakagami-m fading as [41]

f(x) =xmB−1

(mB − 1)!

(mB

γ̄B

)mB

e−mB

γ̄Bx (51)

and

F (x) = 1− e−x

mBγ̄B

mB−1∑j=0

(xmB

γ̄B

)j

j!, (52)

respectively. The marginal moment generating function (MGF)of (51) is given by [24]

Φ(s, x) =

(mB

γ̄B

)mB mB−1∑i=0

xie−(s+

mBγ̄B

)x

i!(s+ mB

γ̄B

)mB−i. (53)

As shown in [24], [34], the MGF expression for the SNR γ afterGSC is expressed as

Φγ(s) = LB

(NB

LB

)∫ ∞

0

e−sxf(x)

× (Φ(s, x))LB−1 (F (x))NB−LB dx. (54)

Here, the MGF is defined as Φγ(s) = E[e−γs]. In order toevaluate the integral in (54), we will rewrite (Φ(s, x))LB−1 and(F (x))NB−LB .

Based on (53), using the multinomial theorem [42], werewrite (Φ(s, x))LB−1 as

(Φ(s, x))LB−1 =

(mB

γ̄B

)mB(LB−1)∑SΦB

aΦk

×(s+

mB

γ̄B

)bΦk−mB(LB−1)

xbΦk e−cΦ

kx, (55)


where SΦB =

{(nΦk,0, · · · , nΦ

k,mB−1

) ∣∣∣mB−1∑i=0

nΦk,i = LB − 1

},

aΦk = (LB−1)!mB−1∏i=0

nΦk,i

!

mB−1∏i=0

(1i!

)nΦk,i , bΦk =

mB−1∑i=0

nΦk,ii, and cΦk =

(LB − 1)(s+ mB

γ̄B

).

Based on (52), we proceed to employ the multinomial theo-rem to express (F (x))NB−LB as

(F (x))NB−LB =∑SFB

aFk

(mB

γ̄B

)bFk

xbFk e−cF

kx, (56)

where SFB =

{(nFk,0, · · · , nF

k,mB

) ∣∣∣mB∑j=0

nFk,j = NB − LB

},

aFk = (NB−LB)!mB∏j=0

nFk,j

!

mB−1∏j=0

(−1j!

)nFk,j+1

, bFk =mB−1∑j=0

jnFk,j+1, and

cFk = mB

γ̄B

mB∑j=1

nFk,j .

Substituting (51), (55), and (56) into (54), and applying [37,eq. (3.351.3)], Φγ(s) is derived as

Φγ(s) =LB

(mB − 1)!

(NB

LB

)×(mB

γ̄B

)mBLB∑SΦB

∑SFB

aΦk aFk

(mB

γ̄B

)bFk

×Γ(bΦk +bFk +mB

) (s+mB

γ̄B

)bΦk−mB(LB−1)

(s+cΦk +cFk +mB

γ̄B

)bΦk+bF

k+mB

. (57)

Let Fγ(x) denote the CDF of γ, the Laplace transform of Fγ(x)is given by L[Fγ(x)] = Φγ(s)/s [25]. Therefore, the Laplacetransform of the CDF of γ is

L[Fγ(x)]=LB

(mB−1)!

(NB

LB

)×(mB

γ̄B

)mBLB∑SΦB

∑SFB

aΦk aFk

(mB

γ̄B

)bFk

×Γ(bΦk +bFk +mB

)(LB)

bΦk+bF

k+mB

(s+mB

γ̄B

)bΦk−mB(LB−1)

s(s+

cFk

LB+mB

γ̄B

)bΦk+bF

k+mB

.

(58)

Using a partial fraction expansion [37, eq. (2.102)], we canrewrite (58) in an equivalent form. Then, taking the inverseLaplace transform of L[Fγ(x)] to obtain

Fγ(x) =LB

(mB − 1)!

(NB

LB

)∑S

mBLB+bFk∑

n=0

aΦk aFk

×Γ(bΦk + bFk +mB

)(LB)

bΦk+bF

k+mB

�nxμne−νnx, (59)

where S denotes

S =

{(nΦk,0 · · · , nΦ

k,mB−1, nFk,0, · · · , nk,mF

B

) ∣∣∣mB−1∑i=0

nΦk,i = LB − 1,

mB∑j=0

nFk,j = NB − LB

⎫⎬⎭ ,

and we define �n as

�n=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(mB

γ̄B

)μn

(1

LB

mB∑j=1

nFk,j+1

)−n1

n=0(mB

γ̄B

)μn

(Υ1+Υ2−

1−sgn(cFk )(n−1)!

)1≤n≤mB(LB−1)

− bΦk(mB

γ̄B

)μn

(Υ3+Υ4−

1−sgn(cFk )(n−1)!

)mB(LB−1)−bΦk

<n≤mBLB+bFk(60)

with n1 = bΦk + bFk +mB ,

Υ1 = −sgn

(cFk)

(n− 1)!

⎛⎝ 1

LB

mB∑j=1

nFk,j + 1

⎞⎠−n1

,

Υ2 =sgn

(cFk)

(n− 1)!(−1)1−n2

n1∑l=1

(l − n2 − 1

l − 1

)⎛⎝ 1

LB

mB∑j=1

nFk,j + 1

⎞⎠−(n1−l+1)⎛⎝ 1

LB

mB∑j=1

nFk,j

⎞⎠n2−l

,

Υ3 = −sgn

(cFk)

(n2 − 1)!

⎛⎝ 1

LB

mB∑j=1

nFk,j + 1

⎞⎠−(n1−n2+1)

,

Υ4 =sgn

(cFk)

(n2 − 1)!

mB(LB−1)−bΦk∑

l=1

(−1)l+1

(n1 − n2 + l − 1

l − 1

)⎛⎝ 1

LB

mB∑j=1

nFk,j

⎞⎠−(n1−n2+l)

,

where n2 = n−mB(LB − 1) + bΦk . In (59), we also have

μn =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

0 n = 0n− 1 1 ≤ n

≤ mB(LB − 1)− bΦkn− sgn

(cFk)(

mB(LB − 1)− bΦk)− 1 mB(LB − 1)− bΦk

< n ≤ mBLB + bFk

and

νn =

⎧⎪⎨⎪⎩0 n = 0mB

γ̄B1 ≤ n ≤ mB(LB − 1)− bΦk(

cFk

LB+ mB

γ̄B

)mB(LB − 1)−bΦk <n≤mBLB+bFk

The CDF of γB with TAS and GSC is given by FγB=

(Fγ(x))NA . Based on (59), and employing the multinomial

theorem, we derive the CDF of γB as (6). Taking the derivativeof the CDF in (6), we obtain the PDF of γB as (7).


APPENDIX BPROOF OF THEOREM 2

We start with the asymptotic CDF for the SNR of a singlebranch of the main channel. In the high SNR regime with γ̄B →∞, applying the Taylor series expansion truncated to the kth

order given by ex =k∑

j=0

xj

j! + o(xk) in (52), we obtain

F (x)=1−e−xmB

γ̄B

⎛⎝exmBγ̄B −

(xmB

γ̄B

)mB

mB !−o

((xmB

γ̄B

)mB)⎞⎠

=

(xmB

γ̄B

)mB

mB !+ o(xmB ). (61)

Substituting (51), (55), and (61) into (54) yields

Φγ(s) =LB

(mB − 1)!(mB !)NB−LB

×(NB

LB

)(mB

γ̄B

)mBNB ∑SΦB

aΦk

×(bΦk +mB(NB−LB)+mB−1

)!

(LB)bΦk+mB(NB−LB)+mB (s+mB

γ̄B)mBNB

. (62)

It is shown in Appendix A that L[Fγ(x)] = Φγ(s)/s. Takingthe inverse Laplace transform of L[Fγ(x)], Fγ is derived as

Fγ(x)

=LB

(mB − 1)!(mB !)NB−LB

(NB

LB

)

×(mB

γ̄B

)mBNB ∑SΦB

aΦk

(bΦk +mB(NB−LB)+mB−1

)!

(LB)bΦk+mB(NB −LB)+mB

×

⎛⎜⎝(mB

γ̄B

)−mBNB

−mBNB∑n=1

(mB

γ̄B

)−(mBNB−n+1)

(n−1)! xn−1e−mB

γ̄Bx

⎞⎟⎠.(63)

Still employing the Taylor series expansion truncated to the kth

order given by ex =k∑

j=0

xj

j! + o(xk) in (63), we rewrite (63) as

Fγ(x) =LB

(NB

LB

) (mB

γ̄B

)mBNB

xmBNB

(mB − 1)!(mB !)NB−LB (mBNB)!

×∑SΦB

aΦk

(bΦk +mB(NB − LB) +mB − 1

)!


. (64)

Based on (64), the asymptotic expression for the CDF of γB isFγB

(x) = (Fγ(x))NA and the final expression is shown in (8).

APPENDIX CPROOF OF THEOREM 3

Substituting (14) into (12), we rewrite the average secrecyrate as

C̄s =1

ln 2

∫ ∞

0

[∫ x1

0

FγE(x2)

1 + x2dx2

]fγB

(x1)dx1. (65)

Substituting (17) into (65), we transform (65) as

C̄s =1

ln 2

∫ ∞

0

ln(1 + x1)fγB(x1)dx1︸︷︷︸

ω2

+1

ln 2

∫ ∞

0

∫ x1

0

χγE(x2)

1 + x2fγB

(x1)dx2dx1︸︷︷︸ω3

. (66)

In the high SNR regime with γ̄B → ∞, ln(1 + x1) ≈ ln(x1),thereby the asymptotic expression for ω2 can be written as Δ1

in (19). Changing the order of integration in ω3, we rewrite

ω3 =1

ln 2

∫ ∞

0

χγE(x2)

1 + x2

∫ ∞

x2

fγB(x1)dx1dx2

=1

ln 2

∫ ∞

0

χγE(x2)

1 + x2(1− FγB

(x2)) dx2. (67)

According to (8), when γ̄B → ∞, FγB(x2) ≈ 0. Hence, the

asymptotic expression for ω3 can be expressed as Δ2 in (20).Based on (19), (20), and (66), we derive the asymptotic expres-sion for the average secrecy rate as (18).

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Lifeng Wang (S’12) received the M.S. degree inelectronic engineering from the University of Elec-tronic Science and Technology of China, Chengdu,China, in 2012. He is currently working toward thePh.D. degree in electronic engineering with QueenMary University of London, London, U.K.

His research interests include physical layer se-curity, millimeter-wave communications, and 5GHetNets.

Maged Elkashlan (M’06) received the Ph.D. degreein electrical engineering from The University ofBritish Columbia, Vancouver, Canada, in 2006. From2007 to 2011, he was with the Wireless and Network-ing Technologies Laboratory, Commonwealth Scien-tific and Industrial Research Organization, Clayton,Australia. During this time, he held an adjunct ap-pointment at the University of Technology, Sydney,Australia. In 2011, he joined the School of ElectronicEngineering and Computer Science, Queen MaryUniversity of London, London, U.K., as an Assistant

Professor. He also holds visiting faculty appointments at the University ofNew South Wales, Sydney, Australia, and Beijing University of Posts andTelecommunications, Beijing, China. His current research interests includedistributed information processing, security, cognitive radio, and 5G HetNets.

Dr. Elkashlan is an Editor of the IEEE TRANSACTIONS ON WIRELESSCOMMUNICATIONS, the IEEE TRANSACTIONS ON VEHICULAR TECHNOL-OGY, and the IEEE COMMUNICATIONS LETTERS. He also serves as the LeadGuest Editor for the special issue on “Green Media: The Future of WirelessMultimedia Networks” of the IEEE Wireless Communications Magazine andfor the special issue on “Millimeter Wave Communications for 5G” of the IEEECommunications Magazine and the Guest Editor for the special issue on “En-ergy Harvesting Communications” of the IEEE Communications Magazine andfor the special issue on “Location Awareness for Radios and Networks” of theIEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS. His researchhas won several academic awards, including Best Paper Awards at IEEE ICC2014, CHINACOM 2014, and IEEE VTC-Spring 2013. He received the Exem-plary Reviewer Certificate of the IEEE COMMUNICATIONS LETTERS in 2012.

Jing Huang (S’10–M’13) received the B.S. degreein communication engineering from Jilin University,Changchun, China, in 2006, the M.S. degree in com-munications and information systems from BeijingUniversity of Posts and Telecommunications, Beijing,China, in 2009, and the Ph.D. degree in electricaland computer engineering from the University ofCalifornia, Irvine, CA, USA, in 2013.

He was an Intern at Broadcom Corporation,Sunnyvale, CA, during the spring and summer of2012. He is currently with Qualcomm Technologies

Inc., Santa Clara, CA. His research interests include cooperative communica-tions, applied signal processing, and radio resource management in wirelessnetworks.


Robert Schober (S’98–M’01–SM’08–F’10) wasborn in Neuendettelsau, Germany, in 1971. He re-ceived the Diplom (Univ.) and Ph.D. degrees in elec-trical engineering from the University of Erlangen-Nuremberg (FAU), Erlangen, Germany, in 1997 and2000, respectively. From May 2001 to April 2002,he was a Postdoctoral Fellow at the University ofToronto, Toronto, Canada, sponsored by the GermanAcademic Exchange Service (DAAD). Since May2002, he has been with The University of BritishColumbia (UBC), Vancouver, Canada, where he is

currently a Full Professor. Since January 2012, he has been also an Alexandervon Humboldt Professor and the Chair for Digital Communication at FAU,Erlangen, Germany. His research interests fall into the broad areas of commu-nication theory, wireless communications, and statistical signal processing.

Dr. Schober is a Fellow of The Canadian Academy of Engineering andThe Engineering Institute of Canada. He is currently the Editor-in-Chief ofthe IEEE TRANSACTIONS ON COMMUNICATIONS. He was a recipient ofseveral awards for his work, including the 2002 Heinz Maier-Leibnitz Awardof the German Science Foundation (DFG), the 2004 Innovations Award of theVodafone Foundation for Research in Mobile Communications, the 2006 UBCKillam Research Prize, the 2007 Friedrich Wilhelm Bessel Research Awardof the Alexander von Humboldt Foundation, the 2008 Charles McDowellAward for Excellence in Research from UBC, a 2011 Alexander von HumboldtProfessorship, and a 2012 NSERC E.W.R. Steacie Fellowship. He was also arecipient of the best paper awards from the German Information TechnologySociety (ITG); the European Association for Signal, Speech and Image Pro-cessing (EURASIP); IEEE WCNC 2012; IEEE Globecom 2011; IEEE ICUWB2006; the International Zurich Seminar on Broadband Communications; andEuropean Wireless 2000.

Ranjan K. Mallik (S’88–M’93–SM’02–F’12) re-ceived the B.Tech. degree from the Indian Instituteof Technology, Kanpur, India, in 1987 and the M.S.and Ph.D. degrees from the University of SouthernCalifornia, Los Angeles, CA, USA, in 1988 and1992, respectively, all in electrical engineering. FromAugust 1992 to November 1994, he was a Scien-tist at the Defence Electronics Research Laboratory,Hyderabad, India, working on missile and EWprojects. From November 1994 to January 1996,he was a Faculty Member of the Department of

Electronics and Electrical Communication Engineering, Indian Institute ofTechnology, Kharagpur, India. From January 1996 to December 1998, hewas with the faculty of the Department of Electronics and CommunicationEngineering, Indian Institute of Technology, Guwahati, India. Since December1998, he has been with the faculty of the Department of Electrical Engineering,Indian Institute of Technology, Delhi, New Delhi, India, where he is currentlya Professor. His research interests are in diversity combining and channelmodeling for wireless communications, space-time systems, cooperative com-munications, multiple-access systems, power line communications, differenceequations, and linear algebra.

Dr. Mallik is a member of the IEEE Communications, IEEE InformationTheory, and IEEE Vehicular Technology Societies; the American MathematicalSociety; and the International Linear Algebra Society. He is a Fellow ofthe Indian National Academy of Engineering; the Indian National ScienceAcademy; The National Academy of Sciences, India (Allahabad); the IndianAcademy of Sciences, Bangalore; The World Academy of Sciences; The Insti-tution of Engineering and Technology, U.K.; and The Institution of Electronicsand Telecommunication Engineers, India. He is a Life Member of the IndianSociety for Technical Education and an Associate Member of The Institutionof Engineers (India). He is an Area Editor of the IEEE TRANSACTIONS ON

WIRELESS COMMUNICATIONS. He has been a recipient of the Hari OmAshram Prerit Dr. Vikram Sarabhai Research Award in the field of Electronics,Telematics, Informatics, and Automation and the Shanti Swarup BhatnagarPrize in Engineering Sciences. He is a member of Eta Kappa Nu.

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Secure Transmission With Antenna Selection in MIMO