Physical Layer Security of Maximal Ratio ... - Semantic Scholar

IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 9, NO. 2, FEBRUARY 2014

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Physical Layer Security of Maximal Ratio Combining in Two-Wave With Diffuse Power Fading Channels Lifeng Wang, Student Member, IEEE, Nan Yang, Member, IEEE, Maged Elkashlan, Member, IEEE, Phee Lep Yeoh, Member, IEEE, and Jinhong Yuan, Senior Member, IEEE

Abstract— This paper advocates physical layer security of maximal ratio combining (MRC) in wiretap two-wave with diffuse power fading channels. In such a wiretap channel, we consider that confidential messages transmitted from a single antenna transmitter to an M-antenna receiver are overheard by an N-antenna eavesdropper. The receiver adopts MRC to maximize the probability of secure transmission, whereas the eavesdropper adopts MRC to maximize the probability of successful eavesdropping. We derive the secrecy performance for two practical scenarios: 1) the eavesdropper’s channel state information (CSI) is available at the transmitter and 2) the eavesdropper’s CSI is not available at the transmitter. For the first scenario, we develop a new analytical framework to characterize the average secrecy capacity as the principal security performance metric. Specifically, we derive new closed-form expressions for the exact and asymptotic average secrecy capacity. Based on these, we determine the high signal-to-noise ratio power offset to explicitly quantify the impacts of the main channel and the eavesdropper’s channel on the average secrecy capacity. For the second scenario, the secrecy outage probability is the primary security performance metric. Here, we derive new closed-form expressions for the exact and asymptotic secrecy outage probability. We also derive the probability of nonzero secrecy capacity. The asymptotic secrecy outage probability explicitly indicates that the positive impact of M is reflected in the secrecy diversity order and the negative impact of N is reflected in the secrecy array gain. Motivated by this, we examine the performance gap between N and N +1 antennas based on their respective secrecy array gains. Index Terms— Physical layer security, two-wave with diffuse power fading, average secrecy capacity, secrecy outage probability, maximal ratio combining.

I. I NTRODUCTION ECURITY and privacy have become increasingly significant concerns in wireless communication networks since the open nature of the wireless medium makes the

S

Manuscript received April 2, 2013; revised October 9, 2013 and December 18, 2013; accepted December 19, 2013. Date of publication January 2, 2014; date of current version January 13, 2014. This work was supported by the Australian Research Council’s Discovery Projects under Grant DP120102607. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Y.-W. Peter Hong. L. Wang and M. Elkashlan are with the School of Electronic Engineering and Computer Science, Queen Mary University of London, London E1 4NS, U.K. (e-mail: [email protected]; [email protected]). N. Yang and J. Yuan are with the School of Electrical Engineering and Telecommunications, The University of New South Wales, Sydney 2052, Australia (e-mail: [email protected]; [email protected]). P. L. Yeoh is with the Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville 3010, Australia (e-mail: [email protected]). Digital Object Identifier 10.1109/TIFS.2013.2296991

wireless transmission vulnerable to eavesdropping and inimical attacks [1]. Conventionally, cryptographic protocols are widely designed and adopted above the physical layer assuming that an error-free physical layer link has already been established [2]. In recent years, the development of decentralized and ad-hoc wireless networks pose great challenges to implement higher-layer key distribution and management in practice [3]. Against this background, physical layer security has emerged as an attractive approach to perform secure transmission in a low complexity manner. The core idea behind this paradigm is to exploit the properties of the wireless channel, such as fading or noise, to transmit confidential messages [4]. While these properties have traditionally been interpreted as impairments, physical layer security takes advantage of these properties to promote secrecy for wireless transmission. Physical layer security was first initiated in [5], in which the wiretap channel was characterized as the fundamental framework to protect information at the physical layer. In this seminal contribution, the secrecy capacity was established as the maximum rate at which the transmitter can reliably send a secret message to the legitimate receiver without being intercepted by an eavesdropper. This result was subsequently generalized to the broadcast channel in [6] and basic Gaussian channel in [7]. Triggered by the rapid advances in multipleinput multiple-output (MIMO) techniques, physical layer security has recently been addressed in MIMO wiretap channels where the transmitter, the receiver, and/or the eavesdropper are equipped with multiple antennas; see, e.g., [8]–[12] and the references therein. Apart from the aforementioned information-theoretic studies, growing research interests have been devoted to examine physical layer security from a practical perspective. To design secure transmission schemes in practice, [13] proposed robust beamforming with artificial noise to mitigate the effect of inaccurate channel state information (CSI) in MIMO wiretap channels. Given the quality of service constraints at the legitimate receiver and the eavesdroppers, the optimum power allocation between transmitted signal and artificial noise was determined in [14]. Considering the availability of partial CSI from the eavesdropper at the transmitter, [15] analyzed the secrecy outage probability in multiple-input singleoutput (MISO) wiretap channels. To facilitate low-complexity implementation, transmit antenna selection was proposed in [16], [17] to promote security with low feedback overhead and

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low computational cost. Based on this scheme, [18] examined the impact of antenna correlation at the receiver and the eavesdropper on the secrecy performance. For the scenario of confidential broadcasting, [19] designed a linear precoder and determined its optimal power allocation across multiple transmit antennas. To provide valuable insights into the secrecy performance in practical fading channels, [20] quantified two secrecy performance metrics, namely the average secrecy rate and the secrecy outage probability, for single antenna wiretap channels. Inspired by this work, [21] took into consideration the single-input multi-output (SIMO) wiretap channel and analyzed the secrecy outage probability with maximal ratio combing (MRC) at the receiver and the eavesdropper for Rayleigh fading. Considering the same channel, [22] characterized the secrecy mutual information for generalized Nakagami-m fading. Different from [20]–[22], this work focuses on SIMO wiretap channels under two-wave with diffuse power (TWDP) fading. The TWDP fading was first modeled in [23] to characterize the propagating scenario where the received signal contains two strong, specular multipath waves. This fading model is of high flexibility as it includes Rayleigh, Rician, and hyper-Rayleigh fading as special cases [24]. In particular, it can be used to describe a link worse than Rayleigh fading [25]. As such, some research attention has been paid to examine the performance of wireless networks under TWDP fading. For example, the average bit error rate was analyzed in [26] for quadrature amplitude modulation and in [27] for noncoherent multiple frequency-shift keying. More recently, the outage probability was derived in [28] for single decodeand-forward (DF) relay networks. In [29], the symbol error rate was derived for multiple DF relay networks. While these papers stand on their own merits, the secrecy performance of the wiretap channel with TWDP fading has not yet been addressed. In this paper, we examine the physical layer security enhancement in the SIMO wiretap channel with TWDP fading. In this wiretap channel, a single antenna transmitter sends confidential information to an M-antenna receiver, while an N-antenna eavesdropper overhears the transmission. To leverage the benefits of multiple antennas, we assume that MRC is applied at the receiver and the eavesdropper. We address two practical eavesdropping scenarios. In the first scenario, we consider that the eavesdropper’s channel state information (CSI) is available at the transmitter. In the second scenario, we consider that the eavesdropper’s CSI is not available at the transmitter. For the first scenario, we characterize the average secrecy capacity as the principal security performance metric. Since the CSI of the eavesdropper is available at the transmitter, the transmitter adapts its transmission rate in order to achieve perfect secrecy. For the second scenario, we characterize the secrecy outage probability as the principal security metric. Since the CSI of the eavesdropper is not available at the transmitter, the transmitter selects a constant secrecy rate and perfect secrecy is not always guaranteed. The novelties and key contributions of this paper are detailed as follows:

•

Eavesdropper’s CSI is available at the transmitter: We derive a new closed-form expression for the exact average secrecy capacity. This expression allows us to quantify the secrecy performance for general operating environments with arbitrary values of SNR and arbitrary number of antennas. Using our derivation method, the exact average secrecy capacity for other fading environments can be obtained in closed form. • Eavesdropper’s CSI is available at the transmitter: We derive a new compact expression for the average secrecy capacity in the high SNR regime. In doing so, we introduce a new analytical framework. Based on this framework, it is easy to obtain the asymptotic average secrecy capacity for other fading environments. We demonstrate that the high signal-to-noise ratio (SNR) slope1 is one. Particularly, the high SNR slope is not affected by M or N. We further characterize the impacts of the main channel and the eavesdropper’s channel on the average secrecy capacity via the high SNR power offset. • Eavesdropper’s CSI is not available at the transmitter: We derive new closed-form expressions for the exact secrecy outage probability and the probability of nonzero secrecy capacity. These expressions enable us to examine the secrecy performance for general operating environments with arbitrary values of SNR and arbitrary number of antennas. • Eavesdropper’s CSI is not available at the transmitter: We derive a new compact expression for the secrecy outage probability in the high SNR regime. This result indicates that the secrecy diversity order is entirely dependent on M. It also indicates that the detrimental effect of the eavesdropper’s channel resides in the secrecy array gain. Furthermore, we quantify the performance gap between N and N +1 antennas via their respective secrecy array gains. Notation: (·)T denotes the transpose operator, (·) H denotes the conjugate transpose operator, · denotes the Euclidean norm, I M denotes the M × M identity matrix, 0 M×N denotes the M × N zero matrix, E [·] denotes the expectation operator, x denotes the greatest integer less than or equal to x, and o (·) denotes the higher order terms. II. S YSTEM M ODEL AND C HANNEL S TATISTICAL P ROPERTIES A. System Model Fig. 1 depicts a SIMO wiretap channel where the transmitter (Alice) encodes her messages and transmits the codewords to the legitimate receiver (Bob), while the malicious eavesdropper (Eve) overhears the transmission. We denote the channel between Alice and Bob as the main channel, and the channel between Alice and Eve as the eavesdropper’s channel. We assume that Alice is equipped with a single antenna, Bob is equipped with M antennas, and Eve is equipped with N 1 We note that the high SNR slope is also known as the maximum multiplexing gain or the number of degrees of freedom [30].

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MRC-combined signal at Bob is written as H H h M x (l) + h M n M (l), y M (l) = h M

(2)

where h M = [h 1 , · · · , h M ]T is the M × 1 main channel vector, and n M (l) ∼ CN M×1 0 M×1 , δ 2M I M is the additive white Gaussian noise (AWGN) vector at Bob. Based on (2), the instantaneous SNR of the main channel is given by γ M = h M 2 P/δ 2M . In the eavesdropper’s channel, the MRC-combined signal at Eve is written as y N (l) = g NH g N x (l) + g NH n N (l),

(3)

is the N × 1 eavesdropper’s where g N = [g1 , · · · , g N  channel vector, and n N (l) ∼ CN N×1 0 N×1 , δ 2N I N is the AWGN vector at Eve. Based on (3), the instantaneous SNR of the eavesdropper’s channel is given by γ N = g N 2 P/δ 2N . ]T

B. Channel Statistical Properties Fig. 1. Illustration of a SIMO wiretap channel, where an N -antenna eavesdropper (Eve) overhears the transmission from a single antenna transmitter (Alice) to an M-antenna legitimate receiver (Bob).

antennas. In this wiretap channel, the instantaneous secrecy capacity C S is defined as

In the wiretap channel, we assume that the main channel and eavesdropper’s channel are subject to TWDP fading. The probability density function (PDF) and the cumulative distribution function (CDF) of γ M are presented in the following lemma. Lemma 1: The PDF and CDF of γ M are given by



CS = CM − CN ,

(1)

where C M = log2 (1 + γ M ) is the instantaneous capacity of the main channel and C N = log2 (1 + γ N ) is the instantaneous capacity of the eavesdropper’s channel. Here, we denote γ M as the instantaneous received SNR of the main channel and γ N as the instantaneous received SNR of the eavesdropper’s channel. It is evident from (1) that C S increases with C M and diminishes with C N . Motivated by this, Bob applies MRC to combine the received signals and maximize the received SNR. This allows Bob to exploit the M-antenna diversity and maximize the probability of secure transmission. On the other hand, Eve applies MRC to exploit the N-antenna diversity and maximize the probability of successful eavesdropping. For this wiretap channel, we take into account two distinct scenarios. In the first scenario, the CSI of the main channel and the eavesdropper’s channel are available at Alice. Based on the CSI of these two channels, Alice calculates C M and C N and then determine C S according to (1). Subsequently, Alice transmits messages to Bob at a secrecy rate no higher than C S . In this scenario, perfect secrecy is always guaranteed. In the second scenario, the CSI of the main channel is available at Alice and the CSI of the eavesdropper’s channel is not known. As such, Alice selects a constant secrecy rate R S to transmit messages to Bob. In this scenario, perfect secrecy is achieved when R S < C S , and is compromised otherwise. To perform secure transmission, Alice encodes the message block w into the codeword x = [x (1) , · · · , x (l) , · · · , x (L)], where L is the length of x.This codeword   is subject to the L E |x (l)|2 ≤ P. We assume average power constraint L1 l=1 that both the main channel and the eavesdropper’s channel are subject to quasi-static fading where the channel coefficients are constant for each transmission block but vary independently between different blocks. At the lth time slot, the

fγM

γ +ϑγ L˜ M ∞  u M,l − γ M 2 M,l  ϑγ M,l k 1  2σ e (γ ) = M 2 γM 2σ 2 2σ 2 l=1 k=0  M+k−1 1 γ × , (4) k! (M + k − 1)! 2σ 2 γ M

and L˜ M ∞ ϑγ  1 1 − 2σ 2γγ  − M,l 2 M 2σ e u e M,l 2M k! l=1 k=0  k M+k−1  i  1 ϑγ M,l γ , × 2σ 2 i ! 2σ 2 γ M

Fγ M (γ ) = 1 −

(5)

i=0

E γ respectively, where γ M = [MM ] is the average per-antenna SNR of the main channel, L˜ M = (2L) M , L is the order of the PDF, u M,l is the lth entry of uM with u M =  a˜ 1 ⊗ · · · a˜ m ⊗ · · · ⊗ a˜ M and a˜ m = a˜ 1 · · · a˜ l · · · a˜ 2L , L a˜ l = a(l+1)/2 , where the first five values of {ai }i=1 are given in Table II of [23], ϑγ M,l = ln(ωγ M,l ), ωγ M,l is the ˜ ˜ ˜ lth entry of ωγ M with  ωγ M = b1 ⊗ · · ·bm ⊗ · · · ⊗ b M · · · exp κ , κ and b˜ m = exp κm,1 · · · exp κm,l m,2L m,l =  (l−1)/2π l 2 2σ , K m is the ratio of the K m 1 + (−1) m cos 2L−1 total specular power to diffuse waves, and m is the relative strength of the two specular components for the mth TWDP branch channel at Bob. Proof: The proof is given in Appendix A. Similar to (4) and (5), the PDF and CDF of γ N are given by

fγN

1 (γ ) = N 2 γN ×



˜

LN  u N,l l=1

2σ 2

e

−



γ γN

+ϑγ N,l

2σ 2

1 γ k! (N + k − 1)! 2σ 2 γ N

∞   ϑγ N,l k k=0

2σ 2

N+k−1 (6)

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and ˜

LN ∞ ϑγ  1 − 2σ 2γγ  1 − N,l 2 N 2σ e u e N,l 2N k! l=1 k=0   i  ϑγ N,l k N+k−1 1 γ . × 2σ 2 i ! 2σ 2 γ N

Fγ N (γ ) = 1 −

(7)

i=0

E[γ N ] is the average perrespectively, where γ N = N antenna SNR of the eavesdropper’s channel, L˜ N = (2L) N , u N,l is the lth entry of  u N with u N  = a˜ 1 ⊗ · · · a˜ n ⊗ · · · ⊗ a˜ N and a˜ n = a˜ 1 · · · a˜ l · · · a˜ 2L , a˜ l = a(l+1)/2 , ˜ ωγ N,l is the lth entry b˜ n  of ωγ N with ω  γ N  = b1 ⊗ · · ·  ⊗ · · · ⊗ b˜ N and b˜ n = exp κn,1 · · · exp κn,l · · · exp κn,2L ,

2σ 2 , K n is the ratio κn,l = K n 1 + (−1)l n cos (l−1)/2π 2L−1 of the total specular power to diffuse waves, and n is the relative strength of the two specular components for the nth TWDP branch channel at Eve. III. AVERAGE S ECRECY C APACITY

In this section, we concentrate on the scenario where the CSI of the eavesdropper is known at Alice. This scenario applies to the networks where Eve is an active user and Alice has access to the CSI of Eve. When Eve is not an intended receiver, Alice adapts the transmission rate based on C M and C N in order to guarantee perfect secrecy. Such a scenario is particularly applicable in the multicast and unicast networks where the users play dual roles as legitimate receivers for some signals and eavesdroppers for others [4], [20]. In such a scenario, average secrecy capacity is a pivotal and primary performance metric. Therefore, we first derive a new closedform expression for the exact average secrecy capacity. We then introduce a new analytical framework to derive a compact expression for the asymptotic average secrecy capacity. Based on the asymptotic result, we characterize the high SNR slope and the high SNR power offset which explicitly capture the impact of the channel parameters on the average secrecy capacity at high SNRs. These new closed-form results encompass Rayleigh fading and Rician fading as special cases. To the best knowledge of the authors, the analytical framework and the results presented in this section are new. A. Exact Average Secrecy Capacity The average secrecy capacity is defined as the instantaneous secrecy capacity C S averaged over γ M and γ N . We formulate the average secrecy capacity as ∞ ∞ CS = C S fγ M (γ1 ) f γ N (γ2 )dγ1 dγ2 0 0  ∞ ∞ = C S f γ N (γ2 ) dγ2 fγ M (γ1 ) dγ1 . (8) 0    0 h¯ 1

According to (1), we first express h¯ 1 in (8) as  γ  h¯ 1 = 0 1 log2 (1 + γ1 ) − log2 (1 + γ2 ) f γ N (γ2 ) dγ2 . (9)

Utilizing integration by parts and applying some algebraic manipulations, we derive (9) as γ1 log2 (1 + γ1 ) f γ N (γ2 ) dγ2 h¯ 1 = 0 γ1 − log2 (1 + γ2 ) f γ N (γ2 ) dγ2 0 γ1 = log2 (1 + γ1 ) Fγ N (γ1 ) − log2 (1 + γ2 ) f γ N (γ2 ) dγ2 0

= log2 (1 + γ1 ) Fγ N (γ1 )

 γ1 Fγ N (γ2 ) 1 − log2 (1 + γ1 ) Fγ N (γ1 ) − dγ2 ln 2 0 1 + γ2 γ1 Fγ N (γ2 ) 1 = dγ2 . (10) ln 2 0 1 + γ2 Substituting (10) into (8), we rewrite the average secrecy capacity as  ∞ γ1 Fγ N (γ2 ) 1 CS = dγ2 f γ M (γ1 ) dγ1 . (11) ln 2 0 1 + γ2 0 Changing the order of integration in (11), we obtain

∞  ∞ Fγ N (γ2 ) 1 CS = f γ M (γ1 ) dγ1 dγ2 ln 2 0 1 + γ2 γ2 ∞  Fγ N (γ2 )  1 = (12) 1 − Fγ M (γ2 ) dγ2 . ln 2 0 1 + γ2 It is shown in (12) that C S depends on the statistics of the main channel and the eavesdropper’s channel. Substituting (5) and (7) into (12) and applying [31, eq. (3.351.2)] to solve the resultant integrals, we derive the exact average secrecy capacity given in (13), shown at the bottom of the next is the exponential integral function given page, where E i (α) α  ∞ −x by E i (α) = e 2σ 2 α e x d x. We highlight that our exact 2σ 2 expression in (13) is given in closed-form as it involves finite summations of exponentials, power values, and standard exponential integral functions. B. Asymptotic Average Secrecy Capacity We proceed to derive the asymptotic average secrecy capacity to examine the maximum average achievable secrecy rate in the high SNR regime. To do so, a new analytical framework is introduced from (14) to (30). In this analysis, we consider that the average SNR of the main channel is sufficiently high, i.e., γ M → ∞2 . We maintain the consideration of arbitrary values of the average SNR of the eavesdropper’s channel. In order to gain more insights, we evaluate the high SNR slope and the high SNR power offset, as two key parameters determining the average secrecy capacity in the high SNR regime. These key parameters explicitly capture the impact of antenna configurations and channel fading coefficients on the achievable secrecy rate. Therefore, they have previously been examined for non-secrecy scenarios in [32]–[34]. We commence the asymptotic analysis by presenting the first order expansion of Fγ M (γ ) in the high SNR regime. 2 We note that as γ N → ∞, the probability of successful eavesdropping approaches one. As such, we do not consider γ N → ∞.

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Applying the Taylor   truncated to the kth order  series expansion given by e x = kj =0 x j / j ! + o x k [35] in (5), we derive the first order expansion of Fγ M (γ ) as  L˜ M ∞ ϑγ  1 ϑγ M,l k 1  − M,l 2 2σ u e M,l 2M k! 2σ 2 l=1 k=0  M+k  γ − 2γ 1 γ × e 2σ γ M e 2σ 2 γ M − (M + k)! 2σ 2 γ M  M+k  γ −o 2σ 2 γ M

expression for κ1 as ∞ 1 ∞ ln (γ1 ) f γ M (γ1 )dγ1 κ1 = ln 2 0

 M L˜ M ϑγ

 γ 1  − M,l −M 2 2σ u e + o γ , M,l M 2 M M! 2σ 2 γ M

=

l=1

(14) where o(·) denotes the higher order terms. To facilitate our asymptotic analysis, we rewrite (7) as Fγ N (γ ) = 1 − χγ N (γ ), where ˜

LN ∞ ϑγ  1 1 − 2σ 2γγ  − N,l 2 N 2σ e u e N,l N 2 k!

χγ N (γ ) =

l=1

k=0



 i  ϑγ N,l k N+k−1 1 γ × . 2σ 2 i ! 2σ 2 γ N i=0

It follows that (11) is re-expressed as ∞  γ1 1 − χγ N (γ2 ) 1 CS = dγ2 f γ M (γ1 )dγ1 ln 2 0 1+γ2 0 = κ1 − κ2 ,

(15)

×

∞  k=0

1 κ1 = ln 2 and κ2 =

1 ln 2





∞ 0

ln (1 + γ1 ) fγ M (γ1 )dγ1

1 2 M ln 2

l=1

ϑγ M,l 2σ 2

k

,

(18)

where ψ (·) is the digamma function [36]. To derive the asymptotic expression for κ2 , we change the order of integration in (17) and rewrite κ2 as ∞ χγ N (γ2 ) ∞ 1 f γ M (γ1 )dγ1 dγ2 κ2 = ln 2 0 1+γ2 γ2 ∞  χγ N (γ2 )  1 1 − Fγ M (γ2 ) dγ2 . = (19) ln 2 0 1+γ2 From (14), we find that Fγ M (γ ) ≈ 0 approaches zero when γ M → ∞ and γ M  γ . The condition of γ M  γ is reasonable at high SNRs since γ is the predetermined SNR threshold above which the quality of the received signal exceeds the quality of service requirement. As such, it becomes relatively small when γ M approaches infinity. Applying some algebraic manipulations, we derive the asymptotic expression for κ2 as ∞ χγ N (γ2 ) 1 dγ2 κ2∞ = ln 2 0 1+γ2

ϑ k γ N,l   L˜ N ∞ N+k−1 ϑγ N,l   ζ γ N,i 1  2σ 2 − 2 , = N u N,l e 2σ 2 ln 2 k! i! k=0

i=0

0

(16)

χγ N (γ2 ) f γ M (γ1 )dγ2 dγ1 . 1+γ2

where   ζ γ N,i (17)

We next derive the asymptotic expressions for κ1 and κ2 . In the high SNR regime with γ M → ∞, we have ln (1 + γ1 ) ≈ ln (γ1 ). As such, we apply [31, eq. (4.352.1)] and perform some algebraic manipulations to derive the asymptotic

CS =



(20)

∞ γ1

0

M ϑγ  1 − M,l 2σ 2 u e M,l 2 M ln 2

ψ (M + k) k!

l=1

where

L˜

 = log2 2σ 2 γ M +

Fγ M (γ ) = 1 −

L˜ M  ∞  u M,l l=1 k=0



k!

e

ϑγ − M,l 2σ 2



×

1 γN

 −

q−1  (q − 1)! t =0

t!

1 2 2σ γ N 

i

q=1

t

1 2σ 2 γ

i−q i    i 1 − 2 + 2σ γ N q .

(21)

N

⎡ k M+k−1   i  i−q i    1 1 1 i 1 ⎣ Ei + − 2 − 2 i! γM 2σ γ M 2σ γ M q q=1

i=0

 k N+k1 −1 L˜ N  ∞ ϑ  u N,l1 − γ N,l21 ϑγ N,l1 1  1 2σ e  j  i 2 2σ 2 N γ M l1 =1 k1 =0 k1 ! j! γ N t =0 j =0 ⎛ ⎞⎤    i+ j    φ−η  φ i+ j η−1 i+ j −η  1 1 1 i + j 1 1 1 1 − 1)! (η ⎠⎦ . (13) − 2 − 2 × ⎝ Ei + + + γM γN 2σ 2σ φ! γM γN 2σ 2 η ×

q−1 

(q − 1)! t!

1 2 2σ γ M

t

ϑγ M,l 2σ 2

= Ei



−

1

η=1

φ=0

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Finally, by substituting κ1∞ and κ2∞ into (15), the asymptotic ∞ average secrecy capacity C S is derived as ∞ ∞ χγ N (γ2 ) 1 1 ∞ CS = ln (γ1 ) f γ M (γ1 )dγ1 − dγ2 . ln 2 0 ln 2 0 1+γ2 (22) We see that (22) provides a general form to obtain the asymptotic average secrecy capacity. By employing (22), the asymptotic average secrecy capacity for other fading environments can be easily derived based on their PDFs and CDFs. According to (18) and (20), the asymptotic average secrecy capacity for TWDP fading is ∞ CS



= log2 2σ γ M 2



L˜

M ϑγ 1  − M,l u M,l e 2σ 2 + M 2 ln 2

l=1



∞ 

k

˜

LN ψ (M + k) ϑγ M,l 1  × − u N,l k! 2σ 2 2 N ln 2 k=0 l=1

ϑ k γ N,l ∞ N+k−1 ϑγ N,l   1   2σ 2 − 2 ζ γ N,i . × e 2σ (23) k! i! k=0

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 first express the high SNR slope as ∞

CS  . γ M →∞ log2 γ M

S∞ = lim

(25)

Substituting (22) into (25), we obtain S∞ = 1.

(26)

From (26), we conclude that the number of antennas at Bob and Eve have no impact on the high SNR slope. We next express the high SNR power offset L∞ as   ∞   CS log2 γ M − . (27) L∞ = lim γ M →∞ S∞ It is clear from (27) that the effects of the main channel and the eavesdropper’s channel on the asymptotic average secrecy capacity reside in L∞ . Substituting (22) and (26) into (27), we derive L∞ as M N + L∞ , L∞ = L∞

(28)

where

 = − log2 2σ 2 −

L˜

M ϑγ 1  − M,l 2σ 2 u e M,l 2 M ln 2 l=1  ∞  ψ (M + k) ϑγ M,l k × k! 2σ 2

k=0

Average secrecy capacity versus γ M for γ N = 10 dB and N = 2.

and N = κ ∞. L∞ 2

i=0

To gather deep insights, we evaluate the high SNR slope and the high SNR power offset, as two key parameters determining the average secrecy capacity in the high SNR regime. Conveniently, we rewrite the asymptotic average secrecy capacity in (23) in a general form as     ∞ C S = S∞ log2 γ M − L∞ , (24)

M L∞

Fig. 2.

(29)

(30)

Based on (28), (29), and (30), we find that the high SNR power offset is independent of γ M . We conclude that the contribution of the main channel to L∞ is characterized by M and the contribution of the eavesdropper’s channel to L L∞ ∞ N . We highlight that L M assesses the is characterized by L∞ ∞ benefits of M on the average secrecy capacity. Specifically, M decreases as M increases, and as such the average secrecy L∞ N quantifies the loss capacity increases. On the other hand, L∞ of average secrecy capacity due to eavesdropping. Specifically, N increases with N, and as such the average secrecy capacity L∞ decreases. We highlight that our analytical framework serves as a useful and general tool to evaluate the asymptotic average secrecy capacity. Specifically, the asymptotic average secrecy capacity for other fading environments can be derived based on the corresponding PDFs and CDFs. C. Numerical Examples Fig. 2 depicts the average secrecy capacity versus γ M for different M in TWDP fading channels. We set K = 3 dB and  = 1. The exact and asymptotic average secrecy capacity results are obtained from (13) and (23), respectively. Evidently, the exact curves match precisely with Monte Carlo simulations and the asymptotic curves well approximate the exact ones in the high SNR regime. We first see that the curves for different M have the same secrecy capacity slope, which is indicated by (26). We also see that the average secrecy capacity increases with increasing M. This can be explained by the fact that increasing M brings about additional power gains via MRC. M in (29) decreases with increasing M and It follows that L∞ accordingly the high SNR power offset L∞ decreases. Fig. 3 depicts the average secrecy capacity versus γ M for different N in TWDP fading channels. We set K = 3 dB and  = 1. We see that the average secrecy capacity

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the high SNR slope, S∞ , is constant unity for Rayleigh and Rician fading. As such, we provide simplified expressions for M and L N in the following two remarks. L∞ ∞ M in (29) reduces to Remark 1: For Rayleigh fading, L∞ M = −ψ (M) log2 e L∞

and

N L∞

(31)

in (30) reduces to N L∞ =

N−1  1  1  ζ γ N,i . ln 2 i!

(32)

i=0

Fig. 3.

Average secrecy capacity versus γ M for γ N = 10 dB and M = 4.

In (31), ψ (M) can be expressed as ψ (M) = −C +  M−1 1 k=1 k [31, eq. (8.365.4)], where C is the Euler’s constant [31, eq. (8.367.1)]. We confirm that ψ (M) is an increasing function of M. As such, an increase in M decreases M and thus improves the average secrecy capacity. We L∞ N and thus also confirm that an increase in N increases L∞ degrades the average secrecy capacity. Furthermore, we note N = 0 and that when the eavesdropper is in absence, we have L∞ M M L∞ = L∞ . In this case, L∞ in (31) reduces to the high SNR power offset of the SIMO Rayleigh fading channel, equivalent to [32, eq. (15)] with a single transmit antenna. Remark 2: For Rician fading, K is the Rician K -factor M in (29) reduces to and 2σ 2 = K 1+1 . In this case, L∞  ∞ e−M K  (M K )k 1 M − ψ (M + k) (33) = −log2 L∞ 1+K ln 2 k! k=0

N in (30) reduces to and L∞ N = L∞

∞ N+k−1  1 −N K  (N K )k  1  e ζ γ N,i . ln 2 k! i! k=0

(34)

i=0

N L∞

Fig. 4. Solid line shows the high SNR power offset versus M for γ N = 10 dB and N = 2. Dash line shows the high SNR power offset versus N for γ N = 10 dB and M = 2. N decreases with increasing N. This is due to the fact that L∞ in (30) increases with increasing N and accordingly the high SNR power offset L∞ increases. We see in Figs. 2 and 3 that the asymptotic curves match the exact curves in the regime of medium and high γ M . This is not surprising since the asymptotic result is derived under the consideration of γ M → ∞. Fig. 4 depicts the high SNR power offset for different M and N in TWDP fading channels. We set K = 3 dB and  = 1. We first see that for fixed N = 2, increasing M decreases L∞ , which increases the average secrecy capacity. We also see that for fixed M = 2, increasing N increases L∞ , which decreases the average secrecy capacity.

D. Special Cases We next present results for the special cases of Rayleigh fading and Rician fading. Observing (26), we confirm that

Taking the derivative of with respect to K , we confirm d LN N is an increasing function that d K∞ ≥ 0. This indicates that L∞ of K . As such, when the Rician K -factor of the eavesdropper’s channel increases, the high SNR power offset increases and the average secrecy capacity decreases. IV. S ECRECY O UTAGE P ROBABILITY In this section, we focus on the scenario where the CSI of the eavesdropper is not known at Alice. In this scenario, Alice transmits confidential information at a constant secrecy rate. Perfect secrecy is only guaranteed when the secrecy rate is lower than the instantaneous secrecy capacity, otherwise, perfect secrecy is compromised. In such a scenario, secrecy outage probability is a useful security metric to characterize the probability that perfect secrecy is compromised [20]. Motivated by this, we derive new exact and asymptotic closedform expressions for the secrecy outage probability. Based on the asymptotic expression, we examine two key performance parameters governing secrecy outage probability in the high SNR regime, namely secrecy diversity order and secrecy array gain. We further derive the probability of non-zero secrecy capacity. This essentially represents the probability of existence of positive secrecy. These new closed-form results encompass Rayleigh fading and Rician fading as special cases.

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A. Exact Secrecy Outage Probability We first concentrate on the secrecy outage probability. Given the expected secrecy rate R S , a secrecy outage is declared when the instantaneous secrecy capacity C S drops below R S . As such, the secrecy outage probability is given by Pout (R S ) = Pr (C S < R S ) ∞ 2 R S (1+γ2 )−1 = f γ M (γ1 ) f γ N (γ2 ) dγ1 dγ2 0 ∞ 0

 = fγ N (γ2 ) Fγ M 2 R S (1+γ2 ) − 1 dγ2 . (35) 0

Substituting the PDF of γ N given in (6) and the CDF of γ M given in (5) into (35), we re-express the secrecy outage probability as Pout (R S ) = 1 − M+k−1 

×

×

1 i!

i=0 ∞ −

e

L˜ M  ∞   ϑγ M,l k u M,l e− 2σ 2 k!

1 2 M+N γ N

l=1 k=0

L˜ N 

1 2σ 2 l1 =1  γ2

2σ 2 γ N

0





∞  k1 =0

ϑγ N,l1

k1

−

2 R S (1+γ2 )−1 − 2σ 2 γ M

γ2 e 2σ 2 γ N i  RS 2 (1 + γ2 ) − 1 × dγ2 . (36) 2σ 2 γ M  

j =0  ∞ − 2 R S + 1 γ2 γ γ

N

× =e

M

e

0 R S −1 2σ 2 γ M

−2

We now derive the asymptotic secrecy outage probability as γ M → ∞. This expression allows us to examine the secrecy performance in the high SNR regime via two parameters, namely the secrecy diversity order and the secrecy array gain. Substituting (14) into (35) and performing algebraic manipulations, the asymptotic secrecy outage probability is derived as

 −G d  ∞ d + o γ −G , (39) Pout (R S ) = G a γ M M Gd = M

(γ2 )

N

N+k1 + j −1

dγ2

i− j i   R S  i 2 −1 2σ 2σ 2 j

∞  

Pout (R S ) = 1 −

e

R S −1 2σ 2 γ M

−2

2 M+N

j − 1)!

.

ϑγ N,l1

k

C. Probability of Non-Zero Secrecy Capacity According to (1), the non-zero secrecy capacity is achieved when γ M > γ N . As such, the probability of non-zero secrecy capacity is given by Pr (C S > 0) = Pr (γ M > γ N ) ∞ γ1 f γ M (γ1 ) f γ N (γ2 )dγ1 dγ2 = 0 ∞ 0 = f γ M (γ1 ) Fγ N (γ1 ) dγ1 . (42) 0

˜

2σ 2

ϑγ M,l 2σ 2

k!

M+k−1  i=0

ϑγ

N,l 1 L˜ N  ∞  ϑγ N,l1 k1 u N,l1 e− 2σ 2 1 i! 2σ 2 k1 ! (N + k1 − 1)!

l1 =1 k1 =0

i− j R S j N+k1 + j −i j i   R S  2 γM γ N (N + k1 + j − 1)! i 2 −1 . ×   N+k1 + j 2 2σ j 2 RS γ + γ j =0

l1 =1

It is evident from (40) that the secrecy diversity order is solely dependent on M and is independent of N. Hence, the secrecy diversity order increases with the number of antennas at Bob. It is also evident from (41) that the eavesdropper’s channel exerts a negative effect on the secrecy array gain. As such, increasing the number of antennas at Eve decreases the secrecy array gain and thus degrades the secrecy outage probability.

(37)

LM  ∞   ϑγ M,l k u M,l e− l=1 k=0

l=1

M 

i  M 1 RS 2 × γ N 2σ 2 k! (N +k − 1)! i k=0 i=0 − M1  RS M−i 2 −1 × . (41) (N +k − 1+i)! 2σ 2

2

j =0 N+k1 + j −i j +1 R j S 2 γM γ N (N + k1 + ×   N+k1 + j R 2 Sγ N + γ M

(40)

and the secrecy array gain is given by ⎡ L˜ N L˜ M ϑγ N,l ϑγ   1 1 − M,l − G a = ⎣ M+N u M,l e 2σ 2 u N,l1 e 2σ 2 2 M!

ϒ

By applying the binomial expansion given by [31, eq. (1.111)] and useful formula [31, eq. (3.351.3)] given by  ∞ na −μx x e d x = n!μ−n−1 , we derive ϒ as 0 RS  − 2 2 −1 2 R S i RS 2σ γ M ∞ γ2

e − 2γ + γ 1 γM γ2  N+k1 −1 M N 2σ 2 e ϒ =  (N+k −1) 1 2σ 2 0 γN  i γ2 2 RS − 1 × + dγ2 2σ 2 2σ 2 2 R S RS − 2 −1 R S i i− j i   R S e 2σ 2 γ M 2γ  i 2 −1 M 2 =  (N+k −1) 2σ 1 2σ 2 2 R S j γ

B. Asymptotic Secrecy Outage Probability

where the secrecy diversity order is given by

ϑγ N,l 1 2σ 2

u N,l1 e k1 ! (N + k1 − 1)!

2σ 2 N+k1 −1

ϑγ M,l 2σ 2

Substituting (37) into (36), we obtain the exact secrecy outage probability given in (38), shown at the bottom of the page. This exact expression is derived in closed-form. It consists of finite summations of exponential functions and power functions.

N

M

(38)

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255

Substituting (4) and (7) into (42) yields Pr (C S > 0) = 1 −

×

L˜ M N+k−1  

1 2 M+N γ M

˜

LN 

u N,l e

l=1

γ N,l 2σ 2

∞ 



k=0 k1

∞ 

ϑγ M,l 1 − 2σ 2

ϑ

ϑγ − N,l 2σ 2

ϑγ M,l 2σ 2

k

k!

1

u M,l1 e k1 !2σ 2 (M + k1 − 1)!i ! i=0 l1 =1 k1 =0  M+k1 −1  i γ 1 ∞ γM γM − M2 γ + γ 1 2σ M N e dγ M × 2σ 2 γ M 2σ 2 γ N 0 



.





(43) By employing [31, eq. (3.351.3)],  in (43) is evaluated as  −i

 ∞ 1 1 2σ 2 γ N M+k1 −1+i −γ M γ M + γ N  =  (M+k −1) e dγ M (γ M ) 1 0 γM   M+k1  (i+1) (M + k1 − 1 + i)! = 2σ 2 γ N γM (44)   M+k1 +i . γM +γN

Fig. 5. Secrecy outage probability versus γ M with γ N = 10 dB, N = 2, and  = 1.

Substituting (44) into (43), we derive the probability of nonzero secrecy capacity as Pr (C S > 0) = 1 −

LN  ∞   ϑγ N,l k u N,l e− ˜

1 2 M+N

l=1 k=0

2σ 2

ϑγ N,l 2σ 2

k!

L˜ M  ∞ N+k−1    M + k1 − 1 + i  ϑγ M,l k1 1 × i 2σ 2 l1 =1 k1 =0

×

i=0

ϑγ M,l 1 − 2σ 2

1 u M,l1 e γ iM γ M+k N  M+k1 +i .  k1 ! γ M +γ N

(45)

For the special case of Rayleigh fading where no specular waves exist in the main channel and eavesdropper’s channel, (45) reduces to [21, eq. (3)]. This highlights the validity and generality of our result. D. Numerical Examples Fig. 5 depicts the secrecy outage probability versus γ M for different M in TWDP fading channels. We set R S = 1 bit/s/Hz and K = 3 dB. The exact and asymptotic secrecy outage probability results are obtained from (38) and (39), respectively. Precise agreement can be seen between the exact curves and the Monte Carlo simulations. We also see that the asymptotic curves accurately predict the secrecy diversity order and the secrecy array gain. We observe that the secrecy outage probability decreases dramatically with increasing M. This can be explained by the fact that M increases the secrecy diversity order. Fig. 6 depicts the secrecy outage probability versus γ M for different N in TWDP fading channels. We set R S = 1 bit/s/Hz and K = 3 dB. We see that the secrecy outage probability curves are parallel in the high SNR regime. This is due to the fact that the secrecy diversity order is independent of N as indicated by (40). We also see that the secrecy outage probability increases with N. This is explained by the fact

Fig. 6. Secrecy outage probability versus γ M with γ N = 10 dB, M = 4, and  = 1.

that the secrecy array gain decreases with increasing N as indicated by (41). E. Special Cases We next examine results for the special cases of Rayleigh fading and Rician fading. Based on (40), we confirm that the secrecy diversity order is maintained at M for Rayleigh and Rician fading. We then offer the following two remarks to present simplified expressions for the exact secrecy outage probability and the secrecy array gain. Remark 1: For Rayleigh fading, the exact secrecy outage probability in (38) reduces to − M−1 i  i− j e γ M   i RS 2 −1 Pout (R S ) = 1 − j (N − 1)! 2 R S −1

i=0 j =0 N+ j −i j R j 2 S γM γ N (N + j − ×  N+ j  i ! 2 RS γ N + γ M

1)!

.

(46)

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The secrecy array gain in (41) reduces to Ga =

 1

 M  M  2 R S i 2 R S −1 M−i γ iN (N+i) − M i=0

M!(N)

i

.

(47)

Combining the first two terms of [21, eq. (6)], we find that the exact secrecy outage probability in [21] can be simplified as (46). From (47), we see that the secrecy array gain decreases with increasing N and γ N . Remark 2: For Rician fading, the exact secrecy outage probability in (38) reduces to R S −1 −N K 2σ 2 γ M

−2

Pout (R S ) = 1−e

M+k−1 ∞  (M K )k e−M K  k=0

×

∞  k1 =0

(N K ) i !k1! (N +k1 −1)! k1

N+k +j−i

×

k!

i=0

i−j i   R S  i 2 −1 j =0

j

2σ 2

j

2 RS j γ M 1 γ N (N +k1 + j −1)! .   N+k1+j R 2 S γ N +γ M

(48)

where  is given by  M−i R i M M  R S −1 2 S  (N + i ) γ iN i=0 i i 2 . (52)  =     M M R S − 1 M−i 2 R S i  (N + i ) γ i N i=0 N i 2

The secrecy array gain in (41) reduces to  ∞ M   M RS i e−(M+N) K  (N K )k 2 Ga = M! k! (N + k − 1)! i k=0

Fig. 7. SNR gap versus N for γ N = 10 dB and M = 4 in three different fading scenarios.

i=0

− 1

  M−i M RS i × 2 − 1 (K + 1)  (N + k + i ) γ N . (49) Taking the derivative of G a with respect to K , we confirm that d Ga d K > 0, which indicates that G a is an increasing function of the Rician K -factor. This confirms that the secrecy array gain increases with K .

From (51), increasing N to N + 1 antennas results in an SNR loss of 10 M log10 (1 +  ) dB. Remark 4: For Rician fading, based on (49) and (50), the SNR gap between N and N + 1 antennas is characterized as  1 10K 10 G a (N + 1) log10 e − log10 1 + , (53) = G a (N) dB M M 2 where 1 =

k=0 j =0

F. Performance Gap In this subsection, we evaluate the performance loss when the number of antennas at Eve increases from N to N + 1. As indicated by (40) and (41), increasing the number of antennas at Eve only impacts the secrecy array gain. As such, we derive the SNR gap between N and N + 1 antennas as a simple ratio of their respective secrecy array gains. Motivated by this, we define the SNR gap between N and N + 1 antennas as  G a (N + 1) G a (N + 1) = 10log10 . (50) G a (N) dB G a (N) For TWDP fading channels, the SNR gap between N and N + 1 antennas is calculated using (41) together with (50). For the special cases of Rayleigh fading and Rician fading, we proceed to provide some useful insights in the following remarks. Remark 3: For Rayleigh fading, based on (47) and (50), the SNR gap between N and N + 1 antennas is characterized as G a (N + 1) G a (N)

∞  k−1   k

=− dB

10 log10 (1 +  ) , M

(51)

M   M−i N j K k  M RS 2 −1 j k! (N + k)! i i=0

× (K + 1) M−i 2 R S i (N + k + i )!γ iN ∞ M   M−i  (N K )k  M R S i 2 −1 + k! (N + k)! i k=0

i=0

× (K + 1) M−i 2 R S i (N + k − 1 + i )!γ iN

(54)

and 2 =

∞  k=0

M 

 M−i  M (N K )k 2 RS − 1 k! (N + k − 1)! i

×(K + 1)

i=0

M−i R S i

2

(N + k − 1 + i )!γ iN .

(55)

From (53), increasing N to N + 1 antennas results in an SNR

1 10K 1 + loss of 10 log 10 M 2 − M log10 e dB. For the special case of non-line-of-sight with K = 0, (53) reduces to (51). Fig. 7 depicts the SNR gap between N and N + 1 antennas for three different fading scenarios: 1) TWDP fading with K = 3 dB and  = 1, 2) Rician fading with K = 3 dB and  = 0, and 3) Rayleigh fading. We set R S = 1 bit/s/Hz. We see that the SNR gap diminishes with increasing N. We also see that the SNR gap for all three fading scenarios approach the same value for large N.

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V. C ONCLUSION We analyzed the physical layer security of MRC systems in TWDP fading channels. Two practical scenarios were taken into account, depending on whether or not the CSI of the eavesdropper is known at the transmitter. For the first scenario where Eve’s CSI is not known, we derived new expressions for the exact and asymptotic average secrecy capacity. Based on these, we demonstrated that the high SNR slope is one. We further characterized the joint impacts of the main channel and the eavesdropper’s channel on the average secrecy capacity via the high SNR power offset. For the second scenario where Eve’s CSI is known, we derived new expressions for the exact and asymptotic secrecy outage probability. Based on these, we showed that the secrecy diversity order is solely dependent on the number of receive antennas at the legitimate receiver and independent of the number of antennas at the eavesdropper. We further examined the performance loss by presenting the SNR gap between N and N + 1 antennas. Based on the SNR gap, we accurately determined the loss of secrecy array gain with increasing number of antennas at the eavesdropper. A PPENDIX A P ROOF OF L EMMA 1 To derive the statistics of γ M , we first write γ M as γ M = γ M H (ϒ M ), where H (ϒ M ) is the sum of TWDP fading power variables. Using [28, eq. (26)], the PDF of H (ϒ M ) is given by L˜ M ∞  u M,l − x+ϑγ2M,l  ϑγ M,l k 1  e 2σ f H (ϒ M ) (x) = M 2 2σ 2 2σ 2 l=1 k=0

x  M+k−1 1 × , (56) k! (M + k − 1)! 2σ 2 Applying

the  probability theory given by f γ M (γ ) = γ f H (ϒ M ) γ × γ1 , we obtain the PDF of γ M in (4). Based M M on (4), the CDF of γ M is derived as γ f γ M (x)d x Fγ M (γ ) = 0

L˜ M ∞  ϑγ  ϑγ M,l k 1  − M,l 2 u M,l e 2σ = M 2 2σ 2 l=1 k=0 γ 1 2σ 2 γ M × e−x x M+k−1 d x. k! (M + k − 1)! 0

(57)

Employing [31, eq. (3.351.1)] and Taylor series expansion ϑγ M,l

k ∞  1 ϑγ M,l , we obtain the desired result given by e 2σ 2 = k! 2σ 2 in (5).

k=0

R EFERENCES [1] Y.-S. Shiu, S. Y. Chang, H.-C. Wu, S. C.-H. Huang, and H.-H. Chen, “Physical layer security in wireless networks: A tutorial,” IEEE Commun. Mag., vol. 18, no. 2, pp. 66–74, Apr. 2011. [2] H. V. Poor, “Information and inference in the wireless physical layer,” IEEE Commun. Mag., vol. 19, no. 1, pp. 40–47, Feb. 2012. [3] J. Huang and A. L. Swindlehurst, “Cooperative jamming for secure communications in MIMO relay networks,” IEEE Trans. Signal Process., vol. 59, no. 10, pp. 4871–4884, Oct. 2011.

[4] L. Dong, Z. Han, A. P. Petropulu, and H. V. Poor, “Improving wireless physical layer security via cooperating relays,” IEEE Trans. Signal Process., vol. 58, no. 3, pp. 1875–1888, Mar. 2010. [5] A. D. Wyner, “The wire-tap channel,” Bell Syst. Tech. J., vol. 54, no. 8, pp. 1355–1387, Oct. 1975. [6] I. Csiszár and J. Körner, “Broadcast channels with confidential messages,” IEEE Trans. Inf. Theory, vol. 24, no. 3, pp. 339–348, May 1978. [7] S. K. Leung-Yan-Cheong and M. E. Hellman, “The Gaussian wiretap channel,” IEEE Trans. Inf. Theory, vol. 24, no. 4, pp. 451–456, Jul. 1978. [8] S. Shafiee and S. Ulukus, “Achievable rates in Gaussian MISO channels with secrecy constraints,” in Proc. IEEE ISIT, Nice, France, Jun. 2007, pp. 2466–2470. [9] A. Khisti and G. W. Wornell, “Secure transmission with multiple antennas—Part I: The MISOME wiretap channel,” IEEE Trans. Inf. Theory, vol. 56, no. 7, pp. 3088–3104, Jul. 2010. [10] A. Khisti and G. W. Wornell, “Secure transmission with multiple antennas—Part : The MIMOME wiretap channel,” IEEE Trans. Inf. Theory, vol. 56, no. 11, pp. 5515–5532, Nov. 2010. [11] F. Oggier and B. Hassibi, “The secrecy capacity of the MIMO wiretap channel,” IEEE Trans. Inf. Theory, vol. 57, no. 8, pp. 4961–4972, Aug. 2011. [12] X. Zhou, R. K. Ganti, and J. G. Andrews, “Secure wireless network connectivity with multi-antenna transmission,” IEEE Trans. Wireless Commun., vol. 10, no. 2, pp. 425–430, Feb. 2011. [13] A. Mukherjee and A. L. Swindlehurst, “Robust beamforming for security in MIMO wiretap channels with imperfect CSI,” IEEE Trans. Signal Process., vol. 59, no. 1, pp. 351–361, Jan. 2011. [14] N. Romero-Zurita, M. Ghogho, and D. McLernon, “Outage probability based power distribution between data and artificial noise for physical layer security,” IEEE Signal Process. Lett., vol. 19, no. 2, pp. 71–74, Feb. 2012. [15] S. Gerbracht, C. Scheunert, and E. A. Jorswieck, “Secrecy outage in MISO systems with partial channel information,” IEEE Trans. Inf. Forensics Security, vol. 7, no. 2, pp. 704–716, Apr. 2012. [16] H. Alves, R. D. Souza, M. Debbah, and M. Bennis, “Performance of transmit antenna selection physical layer security schemes,” IEEE Signal Process. Lett., vol. 19, no. 6, pp. 372–375, Jun. 2012. [17] N. Yang, P. L. Yeoh, M. Elkashlan, R. Schober, and I. B. Collings, “Transmit antenna selection for security enhancement in MIMO wiretap channels,” IEEE Trans. Commun., vol. 61, no. 1, pp. 144–154, Jan. 2013. [18] N. Yang, H. A. Suraweera, I. B. Collings, and C. Yuen, “Physical layer security of TAS/MRC with antenna correlation,” IEEE Trans. Inf. Forensics Security, vol. 8, no. 1, pp. 254–259, Jan. 2013. [19] G. Geraci, M. Egan, J. Yuan, A. Razi, and I. B. Collings, “Secrecy sumrates for multi-user MIMO regularized channel inversion precoding,” IEEE Trans. Commun., vol. 60, no. 11, pp. 3472–3482, Nov. 2012. [20] M. Bloch, J. Barros, M. R. D. Rodrigues, and S. W. McLaughlin, “Wireless information-theoretic security,” IEEE Trans. Inf. Theory, vol. 54, no. 6, pp. 2515–2534, Jun. 2008. [21] F. He, H. Man, and W. Wang, “Maximal ratio diversity combining enhanced security,” IEEE Commun. Lett., vol. 15, no. 5, pp. 509–511, May 2011. [22] M. Z. I. Sarkar and T. Ratnarajah, “On the secrecy mutual information of Nakagami-m fading SIMO channel,” in Proc. IEEE ICC, Cape Town, South Africa, May 2010, pp. 1–5. [23] G. D. Durgin, T. S. Rappaport, and D. A. de Wolf, “New analytical models and probability density functions for fading in wireless communications,” IEEE Trans. Commun., vol. 50, no. 6, pp. 1005–1015, Jun. 2002. [24] J. Frolik, “A case for considering hyper-Rayleigh fading channels,” IEEE Trans. Wireless Commun., vol. 6, no. 4, pp. 1235–1239, Apr. 2007. [25] S. H. Oh, K. H. Li, and W. S. Lee, “Performance of BPSK pre-detection MRC systems over two-wave with diffuse power fading channels,” IEEE Trans. Wireless Commun., vol. 6, no. 8, pp. 2772–2775, Aug. 2007. [26] H. A. Suraweera, W. S. Lee, and S. H. Oh, “Performance analysis of QAM in a two-wave with diffuse power fading environment,” IEEE Commun. Lett., vol. 12, no. 2, pp. 109–111, Feb. 2008. [27] S. Haghani, “Bit error rate of noncoherent MFSK with S + N selection combining in two wave with diffuse power fading,” in Proc. Global Telecommun. Conf., Houston, TX, USA, Dec. 2011, pp. 1–6. [28] Y. Lu, X. Wang, and N. Yang, “Outage probability of cooperative relay networks in two-wave with diffuse power fading channels,” IEEE Trans. Commun., vol. 60, no. 1, pp. 42–47, Jan. 2012. [29] Y. Lu and N. Yang, “Symbol error rate of decode-and-forward relaying in two-wave with diffuse power fading channels,” IEEE Trans. Wireless Commun., vol. 11, no. 10, pp. 3412–3417, Oct. 2012.

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[30] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge, U.K.: Cambridge Univ. Press, 2005. [31] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. San Diego, CA, USA: Academic, 2007. [32] A. Lozano, A. Tulino, and S. Verdú, “High-SNR power offset in multiantenna communication,” IEEE Trans. Inf. Theory, vol. 51, no. 12, pp. 4134–4151, Dec. 2005. [33] S. Jin, M. McKay, C. Zhong, and K.-K. Wong, “Ergodic capacity analysis of amplify-and-forward MIMO dual-hop systems,” IEEE Trans. Inf. Theory, vol. 56, no. 5, pp. 2204–2224, May 2010. [34] R. H. Y. Louie, M. R. McKay, and I. B. Collings, “New performance results for multiuser optimum combining in the presence of Rician fading,” IEEE Trans. Commun., vol. 57, no. 8, pp. 2348–2358, Aug. 2009. [35] M. Spivak, Calculus, 3rd ed. Houston, TX, USA: Publish or Perish, 1994. [36] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, 9th ed. New York, NY, USA: Dover, 1970. Lifeng Wang (S’12) is working toward the Ph.D. degree in electronic engineering at Queen Mary University of London. Before that, he received the M.S. degree in electronic engineering from the University of Electronic Science and Technology of China in 2012 and the B.S. degree in mathematics from the North China University of Water Resources and Electric Power in 2009. His research interests include MIMO, cognitive radio, and physical layer security.

Nan Yang (S’09–M’11) received the B.S. degree in electronics from China Agricultural University, Beijing, China, in 2005, and the M.S. and Ph.D. degrees in electronic engineering from the Beijing Institute of Technology in 2007 and 2011, respectively. From 2008 to 2010, he was a Visiting Ph.D. Student with the School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, Australia. From 2010 to 2012, he was a Post-Doctoral Research Fellow with the Wireless and Networking Technologies Laboratory, Commonwealth Scientific and Industrial Research Organization, Marsfield, Australia. Since 2012, he has been with the School of Electrical Engineering and Telecommunications, University of New South Wales, where he is currently a Post-Doctoral Research Fellow. His general research interests include communications theory and signal processing, with specific interests in collaborative networks, multiple-antenna systems, network security, and distributed data processing. Dr. Yang received the Exemplary Reviewer Certificate of the IEEE C OM MUNICATIONS L ETTERS in 2012 and the Best Paper Award at the IEEE 77th Vehicular Technology Conference (VTC-Spring) in 2013. He is currently serving as the Editor of the Transactions on Emerging Telecommunications Technologies. He has served as Technical Program Committee Member for several IEEE conferences, such as the IEEE International Conference on Communications and the IEEE Global Communications Conference.

Maged Elkashlan (M’06) received the Ph.D. degree in electrical engineering from the University of British Columbia, Canada, in 2006. From 2006 to 2007, he was with the Laboratory for Advanced Networking, University of British Columbia. From 2007 to 2011, he was with the Wireless and Networking Technologies Laboratory, Commonwealth Scientific and Industrial Research Organization, Australia. He held an adjunct appointment at the University of Technology Sydney, Australia. In 2011, he joined the School of Electronic Engineering and Computer Science, Queen Mary University of London, U.K., as an Assistant Professor. He holds visiting faculty appointments at the University of New South Wales,

Australia, and Beijing University of Posts and Telecommunications, China. His research interests fall into the broad areas of communication theory, wireless communications, and statistical signal processing for distributed data processing, large scale MIMO, millimeter wave communications, cognitive radio, and network security. Dr. Elkashlan currently serves as an Editor of the IEEE T RANSAC TIONS ON W IRELESS C OMMUNICATIONS, the IEEE T RANSACTIONS ON V EHICULAR T ECHNOLOGY, and the IEEE C OMMUNICATIONS L ETTERS . He serves as the Lead Guest Editor of the special issue on “Green Media: The Future of Wireless Multimedia Networks” of the IEEE W IRELESS C OMMUNICATIONS M AGAZINE, Lead Guest Editor of the special issue on “Millimeter Wave Communications for 5G” of the IEEE C OMMUNICATIONS M AGAZINE, and Guest Editor of the special issue on “Location Awareness for Radios and Networks” of the IEEE J OURNAL ON S ELECTED A REAS IN C OMMUNICATIONS. He received the Best Paper Award at the IEEE Vehicular Technology Conference (VTC-Spring) in 2013. He received the Exemplary Reviewer Certificate of the IEEE C OMMUNICATIONS L ETTERS in 2012.

Phee Lep Yeoh received the B.E. degree with University Medal from the University of Sydney, Australia, in 2004, and the Ph.D. degree from the University of Sydney, Australia, in 2012. From 2004 to 2008, he was with Telstra Australia as a Radio Network Design and Optimization Engineer. From 2008 to 2012, he was with the Telecommunications Laboratory, University of Sydney and the Wireless and Networking Technologies Laboratory, Commonwealth Scientific and Industrial Research Organization (CSIRO), Australia. In 2012, he joined the Department of Electrical and Electronic Engineering, University of Melbourne, Australia. He is a recipient of the Australian Research Council Discovery Early Career Researcher Award, the University of Sydney Postgraduate Award, the Norman I Price Scholarship, and the CSIRO Postgraduate Scholarship. His research interests include heterogeneous networks, largescale MIMO, cooperative communications, and cognitive networks.

Jinhong Yuan (M’02–SM’11) received the B.E. and Ph.D. degrees in electronics engineering from the Beijing Institute of Technology, Beijing, China, in 1991 and 1997, respectively. From 1997 to 1999, he was a Research Fellow with the School of Electrical Engineering, University of Sydney, Sydney, Australia. In 2000, he joined the School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, Australia, where he is currently a Telecommunications Professor with the School. He has published two books, three book chapters, over 200 papers in telecommunications journals and conference proceedings, and 40 industrial reports. He is a co-inventor of one patent on MIMO systems and two patents on low-density-parity-check codes. He has co-authored three Best Paper Awards and one Best Poster Award, including the Best Paper Award from the IEEE Wireless Communications and Networking Conference, Cancun, Mexico, in 2011, and the Best Paper Award from the IEEE International Symposium on Wireless Communications Systems, Trondheim, Norway, in 2007. He is currently serving as an Associate Editor for the IEEE T RANSACTIONS ON C OMMUNICATIONS. He serves as the IEEE NSW Chair of Joint Communications/Signal Processions/Ocean Engineering Chapter. His current research interests include error control coding and information theory, communication theory, and wireless communications.