1

Enhancing Secrecy with Multi-AntennaTransmission in Wireless Ad Hoc Networks

Xi Zhang, Student Member, IEEE, Xiangyun Zhou, Member, IEEE,and Matthew R. McKay, Senior Member, IEEE

Abstract—We study physical-layer security in wireless ad hocnetworks and investigate two types of multi-antenna transmissionschemes for providing secrecy enhancements. To establish securetransmission against malicious eavesdroppers, we consider thegeneration of artificial noise with either sectoring or beamform-ing. For both approaches, we provide a statistical characterizationand tradeoff analysis of the outage performance of the legitimatecommunication and the eavesdropping links. We then investi-gate the networkwide secrecy throughput performance of bothschemes in terms of the secrecy transmission capacity, and studythe optimal power allocation between the information signal andthe artificial noise. Our analysis indicates that, under transmitpower optimization, the beamforming scheme outperforms thesectoring scheme, except for the case where the number oftransmit antennas are sufficiently large. Our study also revealssome interesting differences between the optimal power allocationfor the sectoring and beamforming schemes.

Index Terms—Physical-layer security, ad hoc networks, multi-antenna transmission, artificial noise, power allocation, outageprobability, throughput optimization.

I. INTRODUCTION

INFORMATION security is a prime concern in emergingwireless networks. Traditional security mechanisms, typ-

ically involving cryptographic algorithms, implicitly assumethat any potential eavesdroppers have limited computationalabilities. Rapid development of computing hardware, how-ever, has meant that there is an ever-increasing suscepti-bility of such methods to attack. To address this problemand further strengthen existing wireless security technolo-gies, physical-layer security mechanisms [1–3] have recentlyattracted significant attention. Such mechanisms work byexploiting the physical properties of the wireless mediumin order to provide an additional layer of robustness whichis “information-theoretically secure”. In the past decade,physical-layer security techniques have been proposed in manydifferent communication scenarios; e.g., multi-input multi-output, relay/jammer-assisted, cognitive-radio-enabled, and so

Copyright (c) 2013 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubs-permissions@ieee.org.

X. Zhang and M. R. McKay are with the Department of Electronic andComputer Engineering, the Hong Kong University of Science and Technology,Hong Kong (e-mails: xizhangx@ust.hk, eemckay@ust.hk).

X. Zhou is with the Research School of Engineering, the AustralianNational University, Australia (e-mail: xiangyun.zhou@anu.edu.au).

This paper was presented in part at the IEEE International Workshopon Signal Processing Advances for Wireless Communications, Darmstadt,Germany, June 16–19, 2013. The work of X. Zhang and M. R. McKay wassupported by the Hong Kong Research Grants Council (Grant No. 616312).The work of X. Zhou was supported by the Australian Research Council’sDiscovery Projects funding scheme (Project No. DP110102548).

on (see e.g., [4–9] and the references therein). Very recently,physical-layer security techniques have also been introducedinto large-scale decentralized ad hoc networks, in order toprovide enhanced secrecy performance [10–18].

When characterizing the networkwide secrecy throughputof large-scale networks, the interference due to concurrenttransmissions plays an important role. On the one hand, fromthe point of view of the legitimate receivers, interference isa nuisance which leads to unwanted throughput loss; whilst,on the other hand, to the eavesdroppers, interference makesit more difficult to intercept transmissions (thereby providingenhanced security). In decentralized networks, it is reasonableto assume that the legitimate receivers do not have sophisti-cated multi-user decoding capabilities, and thereby treat inter-ference simply as noise. For the eavesdroppers, however, oneshould typically not make such strong assumptions, since thecapabilities of eavesdroppers are often unknown. This has ledresearchers to design for a worst-case scenario (see e.g., [15–17]), whereby the eavesdroppers are assumed to be capable ofperforming multi-user decoding (e.g., successive interferencecancellation), allowing the interference created by concurrenttransmission of information signals to be potentially resolved.

In order to confuse eavesdroppers with multi-user decod-ability in ad hoc networks, methods have been introducedin [16, 17] based on generating artificial noise (see [19, 20])or cooperative jamming (see [21, 22]). In both cases, the aimis to create non-resolvable interference at the eavesdroppers.Specifically, in [16], the legitimate users which are far awayfrom the intended receiver were selected to emit artificialnoise; while in [17], a certain percentage of legitimate userswere randomly chosen to radiate jamming signals. Note thatwhen the legitimate nodes have only a single antenna, asin [16, 17], some legitimate users must suspend their own mes-sage transmission in order to deliver artificial noise or jammingsignals. In [18], the authors considered the case where thereare many multi-antenna helping jammers, generating artificialinterference but zero-forcing to nearby legitimate receivers.The injected jamming signals can also help to deal witheavesdroppers with multi-user decodability. However, in somecases, such a large number of helping jammers may not beobtainable and when this happens, the designed transmissionscheme might be vulnerable.

A. Our Approach and Contribution

In this paper, we consider two artificial-noise-aided multi-antenna transmission strategies, based on antenna sectoring

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or beamforming, for providing secrecy in wireless ad hocnetworks. Knowing the direction of the intended receiver, withantenna sectoring, an information signal is transmitted towardsthe intended receiver, while artificial noise is simultaneouslyradiated in other sectors. With the channel state informa-tion (CSI) of the intended receiver, through beamforming, arti-ficial noise is injected into the null-space of the intended chan-nel, leaving the intended receiver unaffected. Note that thesetwo transmission schemes do not require any instantaneouschannel knowledge of the eavesdroppers. With the proposedtransmission schemes, non-resolvable interference is created atthe potential eavesdroppers by the introduced artificial noise,allowing us to guarantee a target secrecy throughput, whichappears difficult without using artificial noise. Moreover, as amajor advantage of these schemes, no legitimate users muststop their own message transmission, which is in contrastto the single-antenna transmission approaches [16, 17]. Inparticular, since the artificial-noise generated interference iscreated by the legitimate transmitters themselves, the systemcan work in the absence of helping jammers [18].

Previous studies in [19–30] have clearly shown that smartlyinjecting artificial interference can achieve secrecy enhance-ments in point-to-point scenarios. In such cases, the artificialnoise generated from multi-antenna beamforming results ininterference in some subspaces but no interference in others.Contrarily, in ad hoc networks where transmitters generateartificial noise in a decentralized manner, artificial interfer-ence is created collectively in all subspaces over the entirenetwork. In short, the artificial-noise-aided beamforming andsectoring schemes considered in this paper have a cooperativejamming effect in ad hoc networks, which ensures that everyeavesdropper is subjected to jamming. With these cooperativejamming effects, it is unclear how well the proposed artificial-noise-aided multi-antenna transmission schemes can work inlarge-scale networks, as a means of jointly providing highcommunication performance and security. Our main objectiveis to address this key question.

For both the sectoring and beamforming schemes, wecharacterize the performance of the legitimate link and theeavesdropping link, by deriving new closed-form expressionsfor the connection and secrecy outage probabilities respec-tively. We then illustrate the tradeoff between the connectionand secrecy outage performance. Under constraints on theconnection and secrecy outage probabilities, we quantify theachievable secrecy throughput performance of both schemesin terms of the secrecy transmission capacity. Finally, weinvestigate the optimal power allocation between the infor-mation signal and the artificial noise which maximizes thesecrecy transmission capacity, and compare the correspondingmaximum secrecy transmission capacity of both schemes. Ouranalytical and numerical results suggest that the consideredmulti-antenna transmission schemes can achieve significantsecrecy enhancements in wireless ad hoc networks.

We observe that by adding extra transmit antennas, theoptimal ratio of power allocated to the information signal thatmaximizes the secrecy transmission capacity of the beamform-ing scheme converges to a certain fraction which is strictly lessthan one, while that for the sectoring scheme keeps increasing

towards one. We show that as the number of transmit antennasgrows large, for both the sectoring and beamforming schemes,the optimized secrecy transmission capacity increases log-arithmically. Our analysis also indicates that, in terms ofthe optimized secrecy transmission capacity, the beamformingscheme outperforms the sectoring scheme for a wide rangeof antenna numbers, at the expense of requiring additionalchannel knowledge. Nevertheless, as the number of transmitantennas becomes sufficiently large, quite interestingly, thesectoring scheme can achieve a better throughput performancethan the beamforming scheme. The performance crossoverhappens at a larger antenna number if the secrecy outageconstraint becomes more stringent.

The rest of the paper is organised as follows. In Section II,we introduce the system model and illustrate the proposedtransmission schemes. In Section III, we characterize theoutage performance of the sectoring scheme. In Section IV,we study the outage performance of the beamforming scheme.In Section V, we analyze and compare the secrecy through-put performance of the sectoring and beamforming schemes.Finally, in Section VI, we conclude this paper.

II. SYSTEM MODEL

The legitimate transmitters and malicious eavesdroppers aremodeled by two independent homogeneous Poisson point pro-cesses (PPPs) on a two dimensional plane R2 with densities λland λe, respectively. The location sets of the legitimate trans-mitters and eavesdroppers are denoted by ΦL and ΦE , respec-tively. Following the widely-used bipolar network model [31,32], we assume that every transmitter has an intended receiverat distance r in a random direction1. Each transmitter isequipped with N transmit antennas, while each receiver (bothlegitimate and malicious) has a single receive antenna. Inaddition to an exponential path loss with parameter α > 2, thewireless channels are assumed to be experiencing independentRayleigh fading. Since we are studying large-scale networkswith uncoordinated concurrent transmissions, the aggregateinterference will be dominant, and the local thermal noise isusually negligible. As done in [13, 14, 17], we omit the thermalnoise and use the signal-to-interference ratio (SIR) as the mainperformance metric.

The total transmit power at each transmitter is denoted by P .Define φ as the ratio of the power of the information signalto the total transmit power. Thus, the power allocated to theinformation signal is PI = Pφ, while the power allocatedto the artificial noise is PA = P (1− φ). In the following,we consider the use of artificial noise in the form of eithersectoring or beamforming, for providing secrecy against themalicious eavesdroppers.

1This bipolar network model is suitable for modeling the case where thedistance from the transmitter to the intended receiver is relatively small,compared with the distances between the transmitters, and it is widely usedin the literature (see e.g., [13, 17, 18, 31, 32]). Generalization can be made bysetting the distance r as a random variable following a certain distribution,and then averaging the network performance over the distribution of r, asmentioned in [31].

3

A. Sectoring with Artificial Noise

With N directional antennas, each transmitter can sendindependent signals in N disjoint sectors, each of thesecovering 2π

N radians with an antenna gain GN . The antennagain usually increases as the spread angle decreases (i.e.,increasing N ). As done in [12], we assume that the side-lobes are suppressed sufficiently and thus can be omittedin later analysis2. We then consider the following schemeto combine sectoring with artificial noise generation: Eachtransmitter sends an information signal in the sector containingits intended receiver, while simultaneously emitting artificialnoise in all other sectors, creating non-resolvable interferenceto the malicious eavesdroppers. Note that the transmitterneeds to know the direction of the intended receiver and thisinformation can be accurately obtained through the discoverymechanisms, such as the “informed discovery” mechanismin [33], where the feedback from the intended receiver isexploited for a high accuracy. We further assume that thetime needed for acquiring and feeding back this directionalinformation is negligible. Note that this sectoring scheme doesnot require the CSI at the transmitter. With the antenna gain,in the intended sector, the information signal is transmittedwith power GNPI , while in each of the other N − 1 sectors,artificial noise is radiated with power GNPA

N−1 .

B. Beamforming with Artificial Noise

With N omnidirectional antennas, the transmitters can per-form artificial-noise-aided beamforming [19, 20]. To do this,the instantaneous CSI of the intended receiver is needed at theassociated transmitter. This information can be obtained viapilot training and we assume that the time needed for the train-ing phase is negligible. With the CSI, the transmitter performsmaximal ratio transmission towards the intended receiver withpower PI . Meanwhile, to confuse the malicious eavesdroppers,artificial noise with total power PA is uniformly injected intothe null space of the intended channel. To be specific, denotingthe intended channel by h, an orthonormal basis is generatedat the transmitter as

[h‖h‖ ,W

], where W is a N × (N − 1)

matrix, the columns of which are mutually orthogonal whilealso being orthogonal to h

‖h‖ . The message vector to betransmitted will then have the following form:

x =h

‖h‖u+ Wv (1)

where u is the information signal, assumed to be complexGaussian distributed with variance PI ; v is the artificial noisevector, the entries of which are complex Gaussian distributedwith zero mean and variance σ2

v = PAN−1 .

2We point out that the analytical results can be generalized to incorporatethe sidelobe leakage signals, by accounting for the eavesdroppers outsidethe main lobe of the intended sector, and carefully evaluating the aggregateartificial noise received by the eavesdroppers. However, the resulting analyticalexpressions become much more complicated, whilst few new insights can beextracted. In fact, it can be shown that, as long as the leakage signals in thesidelobes are not particularly strong, the resulting performance degradationsare insignificant. For simplicity, we focus on the case of perfectly sectorizedantennas with negligible sidelobes.

C. Wiretap Coding and Outage Definition

Here we explain the coding scheme and the outage def-initions. Before transmission, the data is encoded using thewiretap code [1]. The codeword rate and the secret messagerate are denoted by Rb and Rs, respectively. The codewordrate Rb is the actual transmission rate of the codewords, whilethe secrecy rate Rs is the rate of the embedded message.The rate redundancy Re := Rb − Rs is intentionally added,in order to provide secrecy against malicious eavesdropping.More discussions on code construction can be found in [34]. Ifthe channel from the transmitter to its intended receiver cannotsupport the codeword rate Rb, the receiver may not be able todecode the transmitted codeword correctly. We consider thisas a connection outage event.

There are possibly several eavesdroppers trying to interceptthe same transmitter, while the exact number of them isunknown. To minimize the assumption on the eavesdroppers’behavior and design for a worst case, we consider the scenariowhere all eavesdroppers are trying to decode the message fromthe transmitter under consideration. Therefore, if the channelfrom the transmitter to any of the eavesdroppers can supporta data rate larger than the embedded rate redundancy Re, thistransmission fails to achieve perfect secrecy and a secrecyoutage is deemed to occur [35]. As mentioned in [14], thiskind of secrecy outage formulation provides a “strong” secrecyperformance.

Note that we assume the legitimate receivers do not ap-ply multi-user decoding techniques. Thus, the interference atthe legitimate receivers consists of the information signalsand the artificial noise. On the other hand, we assume thatthe eavesdroppers are capable of multi-user decoding, i.e.,resolving concurrent transmissions. In order to design thenetwork parameters to achieve the maximum level of secrecy,as done in [15–17], we consider a worst-case assumption tooverestimate the eavesdroppers’ multi-user decodability: Forthe signal reception at any eavesdropper, only the artificialnoise constitutes the interference, whereas the received infor-mation signals are resolvable and hence are not part of theinterference.

III. OUTAGE PERFORMANCE OF SECTORING SCHEME

In this section, we study the outage performance of thesectoring scheme. We start by deriving the outage probabilityof the intended links. Then, we characterize the possibility thatthe transmitted message is not secure against the maliciouseavesdroppers.

A. Connection Outage Probability

Here, we derive the connection outage probability pco. Athreshold SIR value for connection outage is defined as βb.We focus on a typical transmitter-receiver pair and placethe receiver at the origin of the coordinate system. FromSlivnyak’s theorem [36], the distribution of all other nodes’locations will not be affected; thus, the obtained statistics canreflect the system performance accurately.

For the typical receiver at the origin, the interfering nodescan be classified into two classes: 1) interferers transmitting

4

information signals towards the typical receiver; 2) interfererssending artificial noise towards the typical receiver. With sucha classification, by [36], the transmitters in ΦL can be dividedinto two independent homogeneous PPPs, which are denotedas ΦI and ΦA with densities 1

N λl and N−1N λl, respectively.

Then, the aggregate interference resulting from the trans-mitters in ΦI and ΦA is given by

II = GNPI∑x∈ΦI

SxoD−αxo

IA =GNPAN − 1

∑x∈ΦA

SxoD−αxo (2)

where Dxo represents the distance from the transmitter at xto the typical receiver at the origin, and Sxo represents thecorresponding channel gain. With independent Rayleigh fad-ing, the channel gains Sxo are independent and exponentiallydistributed with unit mean, i.e., Sxo ∼ Exp (1). The channelgain from the typical transmitter to the typical receiver isdenoted by So, with So ∼ Exp (1). The SIR at the typicalreceiver is given by

SIRo = GNPISor−α (II + IA)

−1 (3)

and the connection outage probability is given by

pco = Pr (SIRo ≤ βb) . (4)

This can be derived in closed-form and is presented in thefollowing theorem:

Theorem 1. The connection outage probability of the sector-ing scheme in (4) is given by

pco=1−exp

(−β

2α

b λlCα,2r2

N

(1+(N−1)

1−2α(φ−1−1

) 2α

))(5)

where

Cα,N := πΓ(N − 1 + 2

α

)Γ(1− 2

α

)Γ (N − 1)

(6)

and Γ (·) is the gamma function.

Proof: See Appendix A.From (5), we observe that as the distance between each

transmitter-receiver pair r or the density of the transmitters λlincreases, the connection outage probability increases towardsone (as expected), with the rate of increase being exponentialin r2 or λl. In the latter case, as λl increases, more transmitters“on average” will be closer to any given legitimate receiver,leading to an increased connection outage probability.

From (5), for a given power allocation ratio, it can be shownthat with a relatively small path-loss exponent, i.e., α = 2 ∼ 4,the connection outage probability decreases when extra direc-tional antennas are added. The improvement comes from twoaspects: 1) the intended sectors shrink and thus the legitimatereceivers are interfered by less information signals; 2) thepower allocated to the artificial noise is distributed in moresectors and thus the legitimate receivers are interfered byrelatively less artificial noise.

B. Secrecy Outage Probability

Here, we characterize the secrecy outage probability pso. Athreshold SIR value for secrecy outage is defined as βe. Asbefore, we focus on a typical transmitter-receiver pair but shiftthe coordinate system to put the transmitter at the origin. Themessage from the typical transmitter is not secure against theeavesdropper at z if SIRz > βe, where SIRz denotes the SIRreceived by the eavesdropper at z. With the sectoring scheme,only the eavesdroppers inside the intended sector of the typicaltransmitter may cause secrecy outage. Those eavesdroppersform a fan-shaped PPP and by [36, Theorem A.1], they canbe mapped as a homogeneous PPP on the whole plane, denotedby ΦZ with density 1

N λe.As done in [15–17], we design for a worst-case scenario

by overestimating the eavesdroppers’ multi-user decodability:The aggregate interference at the eavesdropper side consistsof the artificial noise only. By [36], the transmitters whichare radiating artificial noise towards the eavesdropper at zform a homogeneous PPP ΦA with density N−1

N λl. Hence,the interference seen by the eavesdropper at z is

IA =GNPAN − 1

∑x∈ΦA

SxzD−αxz (7)

where Dxz represents the distance from the transmitter at xto the eavesdropper at z, whilst Sxz ∼ Exp (1) represents thecorresponding channel gain.

The SIR received by the eavesdropper at z is given by

SIRz = GNPISozD−αoz I

−1A (8)

where Doz represents the distance from the typical transmitterto the eavesdropper at z, whilst Soz ∼ Exp (1) represents thecorresponding channel gain.

By taking the complement of the probability that the trans-mitted message is secure against all of the eavesdroppers, thesecrecy outage probability can be expressed as

pso = 1− EΦA

{EΦZ

{ ∏z∈ΦZ

Pr (SIRz < βe|ΦA)

}}. (9)

The following theorem presents closed-form upper and lowerbounds for this quantity:

Theorem 2. The secrecy outage probability of the sectoringscheme in (9) satisfies

pLBso ≤ pso ≤ pUB

so (10)

where

pUBso = 1− exp

(−

πCα,2

λeλl

β2αe (N − 1)

1− 2α (φ−1 − 1)

2α

)(11)

and

pLBso =

πλe

πλe + λlCα,2β2αe (N − 1)

1− 2α (φ−1 − 1)

2α

(12)

with Cα,2 defined in (6).

Proof: See Appendix B.

5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.02

0.04

0.06

0.08

0.1

0.12

Power Allocation Ratio, φ

Secr

ecy

Out

age

Prob

abili

ty,

p so

SimulationUpper BoundLower Bound

N = 3, 4, 5

Fig. 1. The secrecy outage probability bounds of the sectoring scheme in (11)and (12) and the simulation results versus the power allocation ratio. Resultsare shown for the case where α = 4, λl = 0.01, λe = 0.001 and βe = 1.

In the low outage region (i.e., as pso → 0), the upper boundin (11) and the lower bound in (12) share the same leadingorder term:

pUBso ≈

πCα,2

λeλl

β2αe (N − 1)

1− 2α (φ−1 − 1)

2α

≈ pLBso (13)

implying that they both approach the exact secrecy outageprobability as it becomes small. As shown in Fig. 1, the upperbound in (11) gives a very accurate approximation for theentire range of the secrecy outage probabilities shown, whilethe lower bound in (12) gets asymptotically accurate as thesecrecy outage probability becomes small.

From (11), we see that the secrecy outage probabilityincreases with increasing the eavesdroppers’ density λe. In-tuitively, as λe increases, more eavesdroppers “on average”will be closer to any given transmitter, which leads to anincreased secrecy outage probability. More interestingly, thesecrecy outage probability in (11) depends on the densitiesof the transmitter-receiver pairs and the eavesdroppers solelythrough their ratio λe

λl.

From (11), we observe that for a given power allocationratio, the secrecy outage probability can be reduced by addingextra directional transmit antennas. This result follows theintuition that by adding transmit antennas: 1) the intendedsector shrinks and thus less eavesdroppers may cause secrecyoutage; 2) the artificial noise from other transmitters covers alarger region and thus more eavesdroppers are degraded.

IV. OUTAGE PERFORMANCE OF BEAMFORMING SCHEME

In this section, we characterize the outage performanceof the beamforming scheme. As before, we first derive theoutage probability of the intended links. Then, we inspect thepossibility that the transmitted message is not secure againstmalicious eavesdropping.

A. Connection Outage ProbabilityTo facilitate our subsequent analysis, we first derive the

distribution of the interference power resulting from the beam-forming signal in (1).

Lemma 1. For a given realization of the intended chan-nel h, if the transmitted signal x in (1) is received by anon-intended receiver through an unknown channel hz , andif PI 6= PA

N−1 (i.e., φ 6= 1N ), then the resulting interference

power Px is distributed according to the following probabilitydensity function (p.d.f.):

fPx (z) =1

PI

(1− PA

(N − 1)PI

)1−N

e− zPI (14)

×

1− e−(N−1PA− 1PI

)zN−2∑k=0

(N−1PA− 1

PI

)kzk

k!

, z>0 .

Meanwhile, if φ = 1N , Px is gamma distributed with

shape parameter N and scale parameter PI , i.e., Px ∼Gamma (N,PI). Note that the distribution of Px is indepen-dent of the intended channel (i.e., h).

Proof: See Appendix C.With Lemma 1, we now derive the connection outage

probability pco. A threshold SIR value for connection outageis defined as βb. As before, we focus on a typical transmitter-receiver pair, and put the typical receiver at the origin toobserve the network performance. We denote the interferencepower from the transmitter at x to the typical receiver by Pxo,where the path loss is excluded. Note that Pxo admits the p.d.f.in (14). The aggregate interference seen by the typical receiveris given by

IIA =∑x∈ΦL

PxoD−αxo (15)

where Dxo denotes the distance from the transmitter at x tothe typical receiver at the origin.

The channel vector from the typical transmitter to the typicalreceiver is denoted as ho. With Rayleigh fading, the elementsof ho are independent complex Gaussian distributed with zeromean and unit variance. With the beamforming strategy in (1),the SIR at the typical receiver is given by

SIRo = PI‖ho‖2r−αI−1IA (16)

and the connection outage probability is then given by

pco = Pr (SIRo ≤ βb) . (17)

We have the following key theorem:

Theorem 3. The connection outage probability of the beam-forming scheme in (17) admits

pco=1−e−β2αbψ(φ)−e−β

2αbψ(φ)

N−1∑p=1

1

p!

p∑k=1

(2

αβ

2αb ψ(φ)

)kζ(p,k) (18)

where ψ (φ) is defined in (19) and

ζ(p,k)=∑

θ∈comb(p−1p−k)

∏li∈θ

i=1,...,p−k

(li −

2

α(li − i+ 1)

). (20)

Here comb(p−1p−k)

is the set of all distinct subsets of thenatural numbers {1, 2, . . . , p− 1} with cardinality p− k. Foreach subset, the elements are arranged in an increasing orderand li ∈ θ is the i-th element of θ. For p ≥ 1, ζ (p, p) = 1.

Proof: See Appendix D.

6

ψ (φ) =

πλlr2Γ(1− 2

α

) (N−φ−1

N−1

)1−N(

Γ(1 + 2

α

)−(φ−1−1N−1

)1+ 2α ∑N−2

k=0

(N−φ−1

N−1

)k Γ(k+1+ 2α )

Γ(k+1)

)if φ 6= 1

N

πλlr2 Γ(1− 2

α )Γ(N+ 2α )

Γ(N) if φ = 1N

(19)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Power Allocation Ratio, φ

Con

nect

ion

Out

age

Prob

abili

ty,

p co

SimulationExactApproximate

N = 3, 4, 5

Fig. 2. The connection outage probability of the beamforming scheme in (18),the low outage approximation in (21) and the simulation results versus thepower allocation ratio. Results are shown for the case where λl = 0.01,r = 1, α = 4 and βb = 3.

Due to the complexity of the derived expression in (18),observing the effects of varying the key system parameterssuch as the power allocation ratio φ and the number of transmitantennas N seems difficult. Nevertheless, with (18), we mayreadily evaluate the system performance and thereby avoidlarge-scale network simulations. In the low outage region, amuch simpler approximation can be found as follows:

Corollary 1. In the low outage region (i.e., as pco → 0), theconnection outage probability in (18) can be approximated by

pco = β2α

b ψ (φ)Kα,N (21)

where ψ (φ) is defined in (19) and

Kα,N = 1− 2

α

N−1∑p=1

1

p!

p−1∏l=1

(l − 2

α

). (22)

Proof: See Appendix E.As demonstrated in Fig. 2, for outage probabilities of

practical interests, the difference between the exact value andthe low outage approximation in (21) can hardly be seen.

B. Secrecy Outage Probability

Here we derive the secrecy outage probability pso. A thresh-old SIR value for secrecy outage is defined as βe. As before,we focus on a typical transmitter-receiver pair but shift thecoordinate system to put the transmitter at the origin.

As done in [15–17], we design for a worst-case scenarioby overestimating the eavesdroppers’ multi-user decodability:The aggregate interference at the eavesdropper side consistsof the artificial noise only. We denote the channel from thetransmitter at x to the eavesdropper at z by hxz . For the

eavesdropper at z, by (1), the artificial noise received fromthe transmitter at x is given by hHxzWxvx, where Wx and vxconstitute the beamforming matrix and the artificial noisevector for the transmitter at x. Define gxz := hHxzWx and notethat gxz is a N−1 dimensional row vector. The correspondinginterference power is given by ‖gxz‖2σ2

v , where σ2v = PA

N−1 isthe variance of each element of vx. Then, for the eavesdropperat z, the aggregate interference created by the artificial noisefrom the transmitters in ΦL is given by

IA =PA

N − 1

∑x∈ΦL

‖gxz‖2D−αxz (23)

where Dxz denotes the distance from the transmitter at x tothe eavesdropper at z. Since the columns of Wx are unit-normand mutually orthogonal, the elements of gxz are independentcomplex Gaussian distributed with zero mean and unit vari-ance; thus, we know that ‖gxz‖2 ∼ Gamma (N − 1, 1). Notethat the eavesdroppers will also be jammed by the artificialnoise from the typical transmitter. Similar to the discussionsabove, for the eavesdropper at z, the interference powercreated by the typical transmitter is given by PA

N−1‖goz‖2D−αoz ,

where ‖goz‖2 ∼ Gamma (N − 1, 1), and Doz is the distancefrom the typical transmitter to the eavesdropper at z.

Denote ho as the channel from the typical transmitterto the typical receiver, and hoz as the channel from thetypical transmitter to the eavesdropper at z. Then, denote thechannel gain resulting from Rayleigh fading from the typicaltransmitter to the eavesdropper at z by Soz . Due to the factthat the typical transmitter is beamforming towards the typicalreceiver, the channel gain Soz is given by

Soz =

∣∣∣∣hHoz ho‖ho‖

∣∣∣∣2 ∼ Exp (1) . (24)

The SIR for the eavesdropper at z is then given by

SIRz =PISozD

−αoz

PAN−1‖goz‖2D

−αoz + IA

. (25)

By taking the complement of the probability that the trans-mitted message is secure against all of the eavesdroppers, thesecrecy outage probability can be expressed as

pso =1−EΦL

{EΦE

{ ∏z∈ΦE

Pr (SIRz<βe|ΦL)

}}. (26)

We can find closed-form upper and lower bounds as follows:

Theorem 4. The secrecy outage probability of the beamform-ing scheme in (26) satisfies

pLBso ≤ pso ≤ pUB

so (27)

7

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Power Allocation Ratio, φ

Secr

ecy

Out

age

Prob

abili

ty,

p so

SimulationUpper BoundLower Bound

N = 2, 8

Fig. 3. The secrecy outage probability bounds of the beamforming schemein (28) and (29) and the simulation results versus the power allocation ratio.Results are shown for the case where α = 4, λl = 0.01, λe = 0.001and βe = 3.

where

pUBso = 1− exp

−λeλl

π(βe

φ−1−1N−1 + 1

)1−N

Cα,N

(βe

φ−1−1N−1

) 2α

(28)

and

pLBso =

(βeφ−1−1

N−1+1

)1−Nπλe

πλe+λlCα,N

(βe

φ−1−1N−1

) 2α

(29)

with Cα,N defined in (6).

Proof: See Appendix F.In the low outage region (i.e., as pso → 0), the upper bound

in (28) and the lower bound in (29) share the same leadingorder term:

pUBso ≈ pLead

so :=λeλl

π(βe

φ−1−1N−1 + 1

)1−N

Cα,N

(βe

φ−1−1N−1

) 2α

≈ pLBso (30)

implying that these two bounds both capture the leading orderbehavior of the secrecy outage probability as it becomessmall. The leading order term pLead

so can serve as an accurateapproximation for the secrecy outage probability in the lowoutage region. As shown in Fig. 3, the upper bound in (28)gives a very accurate approximation for the entire range ofthe secrecy outage probabilities shown, while the lower boundin (29) gets asymptotically accurate as the secrecy outageprobability becomes small.

We point out that with the p.d.f. provided in Lemma 1, fol-lowing a similar procedure as that used in deriving Theorem 4,it is not difficult to study the case where the eavesdroppers donot have multi-user decodability and simply treat interferenceas noise. This argument also applies to the sectoring scheme.However, as we mentioned in the introduction, in order toachieve the maximum level of secrecy, assuming eavesdrop-pers with multi-user decodability is a more robust approach.Therefore, in this paper, we retain our focus on this case.

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Power Allocation Ratio, φ

Out

age

Prob

abili

ty

pco

for Sectoring

pso

for Sectoring

pco

for Beamforming

pso

for Beamforming

Fig. 4. The outage probabilities of the sectoring and beamforming schemesin (5), (11), (18) and (28) versus the power allocation ratio. Results are shownfor the case where λl = 0.01, λe = 0.001, r = 1, α = 4, N = 4, βb = 10and βe = 1.

Connection and Secrecy Outage Tradeoff: As shown inFig. 4, for both the sectoring and beamforming schemes, whengiving more power to the information signal (i.e., increasingthe power allocation ratio φ), the connection outage proba-bility pco decreases whilst the secrecy outage probability pso

increases. Hence, there is a tradeoff between the connectionand secrecy outage performance w.r.t. the transmit powerallocation.

V. SECRECY THROUGHPUT PERFORMANCE

In this section, based on the outage probability expressionsobtained in the last two sections, we investigate the network-wide secrecy throughput performance for the sectoring andbeamforming schemes, in terms of the secrecy transmissioncapacity [13].

The secrecy transmission capacity is defined as the achiev-able rate of successful transmission of confidential messagesper unit area with constraints on both the connection andsecrecy outage probabilities. With outage constraints pco = σand pso = ε, the corresponding secrecy transmission capacityis given by

C = (1− σ)λlRs

= (1− σ)λl [Rb −Re]+ (31)

where [x]+

= max {0, x}. Note that the message rate Rs is afunction of σ and ε, since Rb = log2 (1 + βb) is determined bythe connection outage constraint σ, while Re = log2 (1 + βe)is determined by the secrecy outage constraint ε. When-ever Rb−Re is negative, the required secrecy and connectionoutage performance cannot be guaranteed simultaneously, andthe message transmission should be suspended. More discus-sions on the relationship between the transmission rates andthe outage events can be found in Section II-C.

A. Sectoring Scheme

Here we characterize the secrecy transmission capacity ofthe sectoring scheme in the following proposition:

8

Proposition 1. For the sectoring scheme, a tight lower boundto the secrecy transmission capacity in (31) can be given by

CLBSector =(1−σ)λl (32)

×

log2

1 +

( NλlCα,2 r

2 ln( 11−σ )

1+(N−1)1−2α (φ−1−1)

2α

)α2

1 +

(π

Cα,2

λeλl

ln( 11−ε )(N−1)1−

2α (φ−1−1)

2α

)α2

+

with Cα,2 defined in (6).

Proof: From (5) and (11), we solve pco = σ and pUBso = ε

w.r.t. Rb = log2 (1 + βb) and RUBe = log2 (1 + βe), respec-

tively. Then, plugging the obtained Rb and RUBe into (31), we

have the lower bound in (32).Note that (32) is derived based on the secrecy outage

probability upper bound in (11). Since (11) tracks the exactsecrecy outage probability very closely, the lower boundin (32) provides a tight approximation to the actual secrecytransmission capacity. From (32), we observe that the secrecytransmission capacity increases logarithmically as the numberof transmit antennas grows large. The underlying reason isthat under the outage constraints, as the number of transmitantennas grows large, the supported codeword rate Rb in-creases logarithmically, while the required rate redundancy Rediminishes.

By properly adjusting the power allocation ratio φ, we canmaximize the secrecy transmission capacity. Though a generalexpression for the optimal φ seems not available, we still havethe following corollary:

Corollary 2. The optimal power allocation ratio φ∗ of thesectoring scheme, which maximizes the tight secrecy trans-mission capacity lower bound CLB

Sector in (32), is unique andcan be found by numerically solving for the root of the first-order derivative of CLB

Sector w.r.t. φ. This is true even if theobjective function is generally non-concave.

Proof: See Appendix G.For the case where α = 4, setting the derivative of CLB

Sector

in (32) w.r.t. φ to zero gives a cubic equation and solving itprovides a closed-form expression for φ∗. Define:

%=N

λlCα,2 r2ln

(1

1− σ

), ς=

1

ln(

11−ε

) π

Cα,2

λeλl

(33)

κ=%2+ς2+

(%2−ς2+

√((%−ς)2

+1)(

(%+ς)2+1))(

%2−ς2).

Then, the optimal power allocation ratio for α = 4 is

φα=4Sector=

1+ς

23

(2

23%

43ς

13κ

23 +2

43%2ς+2%

23ς

53κ

13

)2

4(N−1)%43κ

23 (%2−ς2)2

−1

. (34)

We observe that as N → ∞, φα=4Sector = 1−O

(N−1

).

In other words, the optimal power allocation ratio increasestowards one as the number of transmit antennas grows large.This observation can be explained by noting that adding extratransmit antennas allows the transmitter to concentrate more

5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Transmit Antenna Number, N

Opt

imal

Pow

er A

lloca

tion

Rat

io,

φ

SectoringBeamforming

α = 3, 4, 5

Fig. 5. The optimal power allocation ratio of the sectoring and beamformingschemes versus the number of transmit antennas. Results are shown for thecase where λl = 0.01, λe = 0.001, r = 1, σ = 0.1 and ε = 0.01.

on the transmission towards the intended receiver, while lessand less eavesdroppers may cause secrecy outage. Hence,more transmit power can be given to the information signal toachieve a better throughput performance, while still satisfyingthe connection and secrecy outage constraints. As can be seenfrom Fig. 5, the observations made from the case where α = 4holds more generally for 3 ≤ α ≤ 5. The numerical resultsin Fig. 6 indicate that with optimized power allocation, thesecrecy transmission capacity increases logarithmically as thenumber of transmit antennas grows large, which agrees withour earlier observation made from (32).

B. Beamforming Scheme

Now we study the secrecy transmission capacity of thebeamforming scheme. Generally speaking, the secrecy trans-mission capacity can be computed by solving pco = σw.r.t. Rb = log2 (1 + βb) from (18) and solving pso = εw.r.t. Re = log2 (1 + βe) from (26), and then plugging theobtained results into (31). However, the equation pco = σseems not analytically solvable due to the existence of multiplesummations. Note that the typical value of the connectionoutage constraint σ is expected to be small (e.g., below 0.1).This allows us to use the low outage approximation providedin (21). Letting pco = σ, recalling that Rb = log2 (1 + βb),the supported codeword rate Rb can be approximated by

Rb = log2

[1 +

(σ

ψ (φ)Kα,N

)α2

](35)

where ψ (φ) and Kα,N are defined in (19) and (22), respec-tively.

Then, we need to solve pso = ε w.r.t. Re = log2 (1 + βe).Note that the secrecy outage constraint ε is also expected to besmall (e.g., below 0.1), this allows us to use the leading-orderlow outage approximation of the secrecy outage probabilityin (30). In the special case of α = 4 and N = 2, bysolving pLead

so = ε, which turns to be a cubic equation, anapproximation to the required rate redundancy Re can be

9

obtained as follows:

Re=log2

1+φ

1−φ

(

108πλeελlCα,2

+12

√(9πλe

ελlCα,2

)2

+12

) 23

−12

63

√108πλeελlCα,2

+12

√(9πλe

ελlCα,2

)2

+12

2 (36)

where Cα,2 is defined in (6). Plugging (35) and (36) into (31),we have a closed-form approximation for the secrecy trans-mission capacity of the beamforming scheme when α = 4and N = 2:

CApproxBeam = (1− σ)λl

[Rb − Re

]+. (37)

Unfortunately, it seems that we cannot extend the analyticalresults to the case where N ≥ 3, due to the fact that thereis not a general algebraic solution for the quintic (i.e., fifthorder) equation and above (see Abel’s impossibility theoremin [37], an alternative proof to it written in English canbe found in [38]). Hence, for these cases, we study thesecrecy transmission capacity numerically. From (30), bysolving pLead

so = ε w.r.t. Re = log2 (1 + βe), an approximationto the required rate redundancy can be obtained and we denoteit as Re. Then, plugging Rb in (35) and Re into (31), wehave an approximation for the secrecy transmission capacityof the beamforming scheme, which can be written as (37).Note that this approximation is very accurate for small out-age constraints (i.e., σ and ε), since the outage probabilityapproximations in (21) and (30) are both very accurate in thelow outage region. We then numerically optimize the powerallocation to maximize the secrecy transmission capacity.

As illustrated in Fig. 5, as the number of transmit anten-nas N grows large, the optimal power allocation ratio thatmaximizes the secrecy transmission capacity of the beamform-ing scheme converges to a certain value, which is strictly lessthan one. Here we explain why the optimal power allocationratio does not increase to one. Note that the density ofthe eavesdroppers which may cause secrecy outage does notchange with increasing N . More importantly, as N increases,the power of the emitted artificial noise decreases, and theincrements of the artificial interference brought by addingextra transmit antennas will be neutralized. In such cases,giving too much power to the information signal will breakthe secrecy outage constraint and thus the optimal powerallocation ratio will not increase to one. As demonstratedin Fig. 6, with optimized transmit power allocation, as Ngrows large, the maximum secrecy transmission capacity ofthe beamforming scheme increases logarithmically, which issimilar to the sectoring scheme.

C. Sectoring versus Beamforming

Here we compare the sectoring and beamforming schemes,in terms of the optimal power allocation ratio that maximizesthe secrecy transmission capacity and the corresponding max-imum secrecy transmission capacity.

Regarding the optimal power allocation ratio:• As shown in Fig. 5, as the number of transmit antennas N

increases, the optimal power allocation ratio for the

5 10 15 20 25 30

0.01

0.02

0.03

0.04

0.05

0.06

Transmit Antenna Number, N

Max

imum

Sec

recy

Tra

nsm

issi

on C

apac

ity,

C

SectoringBeamforming

α = 4

α = 3

α = 5

Fig. 6. The maximum secrecy transmission capacity of the sectoring andbeamforming schemes versus the number of transmit antennas. Results areshown for the case where λl = 0.01, λe = 0.001, r = 1, σ = 0.1 andε = 0.01.

beamforming scheme converges to a certain value whichis less than one, while that for the sectoring schemekeeps increasing towards one. This difference can beexplained by noting that for the beamforming scheme,with increasing N , the equivalent density of eavesdrop-pers which may cause secrecy outage remains the same,while that for the sectoring scheme decreases inverselylinearly. Hence, the reduced secrecy outage probabilityof the sectoring scheme allows the transmitters to givemore transmit power to the information signal, while stillsatisfying the secrecy outage constraint, which is not truefor the beamforming scheme.

• As shown in Fig. 5, when the path-loss becomes moresevere (i.e., as α increases), the optimal power allocationratio for the beamforming scheme decreases. More inter-estingly, with increasing α and N , there is twist in theoptimal power allocation ratio of the sectoring scheme.The underlying reason is that with increasing N , theoptimal power allocation ratio of the sectoring schemeincreases faster in a high path-loss environment.

Regarding the maximum secrecy transmission capacity:• As shown in Fig. 6, the beamforming scheme outper-

forms the sectoring scheme for a wide range of antennanumbers, in terms of achieving a larger secrecy trans-mission capacity. This makes sense because that havingmore channel knowledge at the transmitters, such thatbeamforming can be performed, is generally better thanhaving directional transmit antennas, which are neededfor the sectoring scheme.

• However, when N becomes sufficiently large, the sec-toring scheme can achieve a larger secrecy transmissioncapacity, compared with the beamforming scheme. Thereason behind this is that with increasing N , the trans-mitter of the sectoring scheme becomes quite capableof concentrating the transmission towards the intendedreceiver while less and less eavesdroppers may causesecrecy outage. As can be seen from Fig. 5, the cor-responding optimal power allocation ratio also increases

10

with increasing N , giving more transmit power to theinformation signals to achieve a better throughput perfor-mance. In addition, the performance crossover betweenthe sectoring and beamforming schemes happens at alarger N if the secrecy outage constraint becomes morestringent (i.e., reducing ε).

VI. CONCLUSION

In this paper, we studied physical-layer security in wirelessad hoc networks and investigated two types of multi-antennatransmission schemes. In particular, in order to jam the eaves-droppers over the entire network, we combined artificial noisewith either sectoring or beamforming. For these two schemes,we provided closed-form expressions for the connection andsecrecy outage probabilities and showed the tradeoff betweenthem. We then quantified the secrecy throughput performancefor both schemes in terms of the secrecy transmission capacity.Our results indicated that the proposed transmission schemescan provide significant secrecy enhancements over single-antenna methods. Our analysis also shed light on the behaviorof the optimal power allocation between the information signaland the artificial noise for achieving the maximum secrecythroughput.

APPENDIX

A. Proof of Theorem 1

Here we derive the connection outage probability in (5).By (2), the total interference seen by the typical receiver atthe origin is II + IA. Since II and IA are two independentshot noise processes, by [31, eq. (8)], the Laplace transformof the p.d.f. of II + IA is given by

LII+IA(s)=exp

(−λlCα,2G

2α

N

N

(P

2α

I +(N−1)1−2αP

2α

A

)s

2α

)(38)

where Cα,2 is defined in (6).By (3) and (4), we have

pco = 1− Pr

(So ≥

βbrα

GNPI(II + IA)

)= 1−

∫ ∞0

exp

(− βbr

α

GNPIz

)fII+IA (z) dz

= 1− LII+IA

(βbr

α

GNPI

)(39)

where fII+IA (·) is the p.d.f. of II+IA and the last line comesfrom the definition of the Laplace transform. Then, pluggingin the Laplace transform in (38) leads to the results in (5).

B. Proof of Theorem 2

We first derive the secrecy outage probability upper boundin (11). By [31, eq. (8)], the Laplace transform of the p.d.f.of the aggregate artificial noise IA in (7) is given by

LIA (s) = exp

(−λlCα,2G

2α

N

N(N − 1)

1− 2α P

2α

A s2α

)(40)

where Cα,2 is defined in (6).

By applying the probability generating functional of aPPP (see Definition A.5 in [36]) to the inner expectation of (9),we have

pso =1−EΦA

{exp

(−λeN

∫R2

Pr(SIRz>βe|ΦA)dz

)}. (41)

Invoking the bounding technique used in [13, 39] (i.e., apply-ing Jensen’s inequality), we get the following upper bound:

pso≤pUBso :=1−exp

(−λeN

∫R2

Pr (SIRz > βe) dz

)=1−exp

(−λeN

∫R2

LIA(βeD

αoz

GNPI

)dz

). (42)

Plugging in (40) and changing to a polar coordinate system toevaluate the integral yields the results in (11).

By considering the nearest eavesdropper only, we nowderive the secrecy outage probability lower bound in (12).Denote the location of the nearest eavesdropper by n. By [40,eq. (2)], the distance between the typical transmitter and thenearest eavesdropper in ΦZ , denoted by Don, is distributedaccording to the following p.d.f.:

fDon (z) =2πλeN

z e−πλeN z2 , z > 0 . (43)

The received SIR at this eavesdropper is given by

SIRn = GNPISonD−αon I

−1A (44)

where Son ∼ Exp (1), IA is the aggregate artificial noise,defined in (7).

By (44), the secrecy outage probability in (9) can be lowerbounded by

pso ≥ pLBso := Pr (SIRn > βe)

= EDon,IA[exp

(−βeD

αon

GNPIIA

)]= EDon

[LIA

(βeD

αon

GNPI

)]. (45)

Using (40) and (43) to evaluate the last expectation yields theresults in (12).

C. Proof of Lemma 1

Here we derive the interference distribution in (14). Fora given realization of the intended channel h and with thebeamforming strategy in (1), the covariance matrix of thetransmitted signal vector x is

Cx = PIhhH

‖h‖2+

PAN − 1

WWH . (46)

Noting that WWH = I− hhH

‖h‖2 , we can rewrite the covariancematrix as

Cx =PA

N − 1I +

(PI −

PAN − 1

)hhH

‖h‖2. (47)

If this signal x is received by a non-intended receiver throughan unknown channel hz , the corresponding interference poweris given by

Px =PA

N − 1‖hz‖2 +

(PI −

PAN − 1

) ∣∣∣∣hHz h

‖h‖

∣∣∣∣2 . (48)

11

For a given h 6= 0 and with a random hz , the interferencepower Px can be interpreted as the weighted sum of thesquared magnitude of hz and the squared magnitude of theprojection of hz on the direction of h. Hence, the amplitudeof h is irrelevant and only its direction matters; more impor-tantly, with the ergodicity of hz over the symmetric complexspace, the statistical distribution of Px becomes independent ofthe direction of h. For this reason, we set h = [1, 0, . . . , 0]

T tocharacterize the distribution of Px. Denote the elements of hzas hzi with i = 1, . . . , N . The interference power Px can beexpressed as

Px = PI |hz1|2 +PA

N − 1

N∑i=2

|hzi|2 . (49)

We first consider the case where PI 6= PAN−1 (i.e., φ 6= 1

N ).

Note that PI |hz1|2 ∼ Exp(

1PI

)and PA

N−1

∑Ni=2 |hzi|

2 ∼

Gamma(N − 1, PA

N−1

)are mutually independent. The p.d.f.

of Px can be computed as

fPx (z) =1

PI

(1− PA

(N − 1)PI

)1−N

e− zPI

× γ(N − 1,

(N − 1

PA− 1

PI

)z

), z > 0 (50)

where γ (·, ·) is the regularized lower incomplete gammafunction. Plugging in the series representation of γ (·, ·) yieldsthe results in (14). Note that the expression above does nothold for PI = PA

N−1 (i.e., φ = 1N ). From (49), it is clear that

when φ = 1N , Px ∼ Gamma (N,PI).

D. Proof of Theorem 3

Here we derive the connection outage probability in (18).By [31, eq. (8)], the Laplace transform of the p.d.f. of theaggregate interference IIA in (15) is given by

LIIA (s) = exp

(−λlπE

[P

2αx

]Γ

(1− 2

α

)s

2α

). (51)

If PI 6= PAN−1 (i.e., φ 6= 1

N ), by (14), E[P

2αx

]is given in (52);

and if PI = PAN−1 (i.e., φ = 1

N ), we have

E[P

2αx

]= P

2α

I

Γ(N + 2

α

)Γ (N)

. (53)

In (16), ‖ho‖2 ∼ Gamma (N, 1) and its complementarycumulative distribution function takes the form in [41, eq. (9)],with N = {1}, K = {1, . . . , N − 1} and ank = 1

k! . Hence,with the Laplace transformation obtained in (51), we caninvoke [41, Theorem 1] to characterize the connection outageprobability as follows:

pco = 1−N−1∑p=0

[(−s)p

p!

dp

dspLIIA (s)

]s=

βbrα

PI

. (54)

By convention, the zero-th order derivative denotes the func-tion itself. Plugging (51) into (54), we have the results in (18).

E. Proof of Corollary 1Here we derive a low outage approximation for the con-

nection outage probability in (18). From (18) and (19), it isclear that when ψ (φ) → 0, the connection outage probabil-ity pco → 0. Note that the quantity ψ (φ) couples many systemparameters like r and λl, and more importantly, φ and N . Bymaking ψ (φ) small, we are effectively studying all cases ofsystem parameters that may lead to a small pco, including butnot limited to the asymptotic regions where r → 0 or λl → 0.We expand (18) around ψ (φ) = 0 as follows:

pco=β2αb ψ(φ)−e

−β2αbψ(φ)β

2αb ψ(φ)

2

α

N−1∑p=1

1

p!ζ(p, 1)+O

(ψ(φ)2

)=β

2αb ψ (φ)−β

2αb ψ (φ)

2

α

N−1∑p=1

1

p!

p−1∏l=1

(l− 2

α

)+O

(ψ (φ)2

)(55)

where ζ (·, ·) is defined in (20). Ignoring the high order termsgives the results in (21).

F. Proof of Theorem 4We first derive the secrecy outage probability upper bound

in (28). By [31, eq. (8)], the Laplace transform of the p.d.f.of the aggregate artificial noise IA in (23) is given by

LIA (s) = exp

(−λlCα,N

(PA

N − 1

) 2α

s2α

)(56)

where Cα,N was defined in (6).Using the generating functional of a PPP (see Definition A.5

in [36]), by (26), we have

pso = 1− EΦL

{exp

[−λe

∫R2

Pr (SIRz > βe|ΦL) dz

]}≤ 1− exp

[−λe

∫R2

Pr (SIRz > βe) dz

](57)

where the second line is obtained by applying Jensen’s in-equality. By (25), the probability inside the integral can beevaluated as

Pr (SIRz > βe) (58)

= E‖goz‖2,IA

{exp

(−βeD

αoz

PI

(PA

N − 1‖goz‖2D−αoz + IA

))}=

(βeφ−1 − 1

N − 1+ 1

)1−N

LIA(βeD

αoz

PI

)=

(βeφ−1 − 1

N − 1+ 1

)1−N

exp

(−β

2αe D

2ozλlCα,N

(φ−1 − 1

N − 1

) 2α

).

Plugging (58) into (57), after changing to a polar coordinatesystem, evaluating the integral gives the results in (28).

By considering the nearest eavesdropper only, we nowderive the secrecy outage probability lower bound in (29).Denote the location of the nearest eavesdropper by n. By [40,eq. (2)], the distance between the typical transmitter and thenearest eavesdropper in ΦE , denoted by Don, is distributedaccording to the following p.d.f.:

fDon (z) = 2πλez e−πλez2

, z > 0 . (59)

The received SIR at this eavesdropper is given by

SIRn =PISonD

−αon

PAN−1‖gon‖2D

−αon + IA

(60)

12

E[P

2αx

]=

1

PI

(1− PA

(N−1)PI

)1−N(P

1+ 2α

I Γ

(1+

2

α

)−(

PAN−1

)1+ 2αN−2∑k=0

(1− PA

(N−1)PI

)k Γ(k + 1 + 2

α

)Γ (k + 1)

)(52)

where Son ∼ Exp (1), ‖gon‖2 ∼ Gamma (N − 1, 1) and IAis the aggregate artificial noise, defined in (23).

Then, the secrecy outage probability in (26) can be lowerbounded by

pso ≥ pLBso

:= EDon,IA,‖gon‖2[Pr(SIRn > βe|Don, IA, ‖gon‖2

)]=

(βeφ−1 − 1

N − 1+ 1

)1−N

EDon,IA[exp

(−D

αonβePI

IA

)]=

(βeφ−1 − 1

N − 1+ 1

)1−N

EDon[LIA

(DαonβePI

)]. (61)

Plugging in (56) and (59) to evaluate the last expectation leadsto the results in (29).

G. Proof of Corollary 2

Here we show that the optimal power allocation ratio thatmaximizes the secrecy transmission capacity lower boundin (32) is unique. Define the following quantities:

x := (N − 1)1− 2

α(φ−1 − 1

) 2α

δ :=α

2. (62)

With the notation defined in (33), the lower bound in (32) canbe expressed as

CLBSector = (1− σ)λl

log2

1 +(

%1+x

)δ1 +

(ςx

)δ

+

. (63)

We assume that % > ς , such that a positive CLBSector can be

achieved by choosing x from(

ς%−ς ,∞

). More discussions

on the feasibility of a positive CLBSector can be found in [42].

Then, by the monotonicity of the logarithm function, maximiz-ing CLB

Sector is equivalent to maximizing the following function:

f (x) =

(1 +

(%

1 + x

)δ)(1 +

( ςx

)δ)−1

. (64)

The derivative of f (x) w.r.t. x is given by

d

dxf (x) =

ςδ + %δςδ

(1+x)δ+1 − %δ(

x1+x

)δ+1

δ−1xδ+1(

1 +(ςx

)δ)2 (65)

where the denominator is always positive. When increasing xfrom ς

%−ς to infinity, the first two terms in the numeratordecreases from ςδ +%−1ςδ (%− ς)δ+1 to ςδ and the third termincreases from %−1ςδ+1 to %δ monotonically. Since % > ς ,the derivative of f (x) is first positive and then negativeon x ∈

(ς%−ς ,∞

). When increasing φ from zero to one, the

value of x decreases from infinity to zero monotonically. This

implies that the derivative of CLBSector w.r.t. φ is first positive

and then negative with increasing φ, and thus the optimal valueof φ is unique.

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Xi Zhang (S’11) received the B.E. degree in com-munication engineering from the University of Elec-tronic Science and Technology of China in 2010.He is currently working toward the Ph.D. degreein electronic and computer engineering at the HongKong University of Science and Technology. Hisresearch interests are in the fields of wireless com-munication and signal processing techniques, includ-ing physical-layer security, ad hoc networking, andrandom matrix theory.

Xiangyun Zhou (S’08-M’11) is a lecturer at theAustralian National University (ANU), Australia.He received the B.E. (hons.) degree in electron-ics and telecommunications engineering and thePh.D. degree in telecommunications engineeringfrom the ANU in 2007 and 2010, respectively. FromJune 2010 to June 2011, he worked as a postdoctoralfellow at UNIK - University Graduate Center, Uni-versity of Oslo, Norway. His research interests arein the fields of communication theory and wirelessnetworks.

Dr. Zhou serves on the editorial board of the following journals: IEEECommunications Letters, Security and Communication Networks (Wiley), andAd Hoc & Sensor Wireless Networks. He has also served as the TPC memberof major IEEE conferences. Currently, he is the Chair of the ACT Chapterof the IEEE Communications Society and Signal Processing Society. He is arecipient of the Best Paper Award at the 2011 IEEE International Conferenceon Communications.

Matthew R. McKay (S’03-M’07-SM’13) receivedthe combined B.E. degree in electrical engineeringand the B.IT. degree in computer science fromthe Queensland University of Technology, Australia,in 2002, and the Ph.D. degree in electrical engi-neering from the University of Sydney, Australia,in 2007. He then worked as a Research Scientist atthe Commonwealth Science and Industrial ResearchOrganization (CSIRO), Sydney, prior to joining thefaculty at the Hong Kong University of Science andTechnology (HKUST) in 2007, where he is currently

the Hari Harilela Associate Professor of Electronic and Computer Engineering.He is also a member of the Center for Wireless Information Technology atHKUST, as well as an affiliated faculty member with the Division of Biomed-ical Engineering. His research interests include communications and signalprocessing; in particular the analysis and design of MIMO systems, randommatrix theory, information theory, wireless ad-hoc and sensor networks, andphysical-layer security.

Dr. McKay was awarded the University Medal upon graduating fromthe Queensland University of Technology. He and his coauthors have beenawarded a Best Student Paper Award at IEEE ICASSP 2006, Best Stu-dent Paper Award at IEEE VTC 2006-Spring, Best Paper Award at ACMIWCMC 2010, Best Paper Award at IEEE Globecom 2010, Best Paper Awardat IEEE ICC 2011, and was selected as a Finalist for the Best Student PaperAward at the Asilomar Conference on Signals, Systems, and Computers 2011.In addition, he received the 2010 Young Author Best Paper Award by theIEEE Signal Processing Society, the 2011 Stephen O. Rice Prize in the Fieldof Communication Theory by the IEEE Communication Society, and the 2011Young Investigator Research Excellence Award by the School of Engineeringat HKUST. Dr. McKay serves on the editorial boards of the IEEE Transactionson Wireless Communications and the mathematics journal, Random Matrices:Theory and Applications. In 2011, he served as the Chair of the Hong KongChapter of the IEEE Information Theory Society, whilst previously serving asthe Vice-Chair and the Secretary. He has also served on the technical programcommittee for numerous international conferences, as well as the PublicationsChair for IEEE SPAWC 2009, Publicity Chair for IEEE SPAWC 2012, andPoster Chair for IEEE CTW 2013.