Submitted 7 December 2018 Accepted 13 March 2019 Published 22 April 2019 Corresponding author Kiattisak Maichalernnukul, kiattisak.m@rsu.ac.th Academic editor Shlomi Dolev Additional Information and Declarations can be found on page 18 DOI 10.7717/peerj-cs.186 Copyright 2019 Maichalernnukul Distributed under Creative Commons CC-BY 4.0 OPEN ACCESS On the secrecy performance of transmit- receive diversity and spatial multiplexing systems Kiattisak Maichalernnukul College of Digital Innovation and Information Technology, Rangsit University, Pathum Thani, Thailand ABSTRACT Emerging from the information-theoretic characterization of secrecy, physical-layer security exploits the physical properties of the wireless channel for security purpose. In recent years, a great deal of attention has been paid to investigating the physical-layer security issues in multiple-input multiple-output (MIMO) wireless communications. This paper analyzes the secrecy performance of transmit-receive diversity system and spatial multiplexing systems with zero-forcing equalization and minimum mean- square-error equalization. Specifically, exact and asymptotic closed-form expressions are derived for the secrecy outage probability of such MIMO systems in a Rayleigh fading environment, and the corresponding secrecy diversity orders and secrecy array gains are determined. Numerical results are presented to corroborate the analytical results and to examine the impact of various system parameters, including the numbers of antennas at the transmitter, the legitimate receiver, and the eavesdropper. These contributions bring about valuable insights into the physical-layer security in MIMO wireless systems. Subjects Computer Networks and Communications, Security and Privacy Keywords Physical-layer security, Secrecy outage probability, Transmit-receive diversity, Multiple-Input Multiple-Output, Spatial multiplexing INTRODUCTION Wireless communication systems are intrinsically prone to eavesdropping because of the open nature of the wireless medium. In this context, physical-layer security arising from the information-theoretic analysis of secrecy has attracted a lot of interest so far. This approach indeed takes advantage of the physical characteristics of the radio channel to support secure communications. Groundbreaking works on physical-layer security (Wyner, 1975; Csiszár & Körner, 1978; Leung-Yan-Cheong & Hellman, 1978; Bloch et al., 2008) focused on a basic wiretap channel, where the transmitter, the legitimate receiver, and the eavesdropper possess a single antenna, and established the so-called secrecy capacity. One of their common remarks was that to have a positive secrecy capacity, the channel quality of the transmitter–receiver link has to be better than that of the transmitter-eavesdropper link. Stimulated by advances in multiple-antenna technology for wireless communications, the physical-layer security issues in multiple-input multiple-output (MIMO) wiretap How to cite this article Maichalernnukul K. 2019. On the secrecy performance of transmit-receive diversity and spatial multiplexing sys- tems. PeerJ Comput. Sci. 5:e186 http://doi.org/10.7717/peerj-cs.186 https://peerj.com mailto:kiattisak.m@rsu.ac.th mailto:kiattisak.m@rsu.ac.th https://peerj.com/academic-boards/editors/ https://peerj.com/academic-boards/editors/ http://dx.doi.org/10.7717/peerj-cs.186 http://creativecommons.org/licenses/by/4.0/ http://creativecommons.org/licenses/by/4.0/ http://doi.org/10.7717/peerj-cs.186 1In our context, a MIMO wiretap channel implies that there are multiple antennas at the transmitter, the legitimate receiver, and the eavesdropper. This is generally known as co-located MIMO. For a discussion on its alternative, called distributed or cooperative MIMO, readers are referred to (Dong et al., 2010; He, Man & Wang, 2011; Zou, Wang & Shen, 2013; Wang et al., 2016a). 2For this kind of channel, the channel gains are allowed to change from channel use to channel use (Poor & Schaefer, 2017). channels1 have been recently explored in the literature (Goel & Negi, 2008; Khisti & Wornell, 2010; Oggier & Hassibi, 2011; Mukherjee & Swindlehurst, 2011; Yang et al., 2013; Ferdinand, Da Costa & Latva-aho, 2013; Lin, Tsai & Lin, 2014; Wang, Wang & Ng, 2015; Schaefer & Loyka, 2015; Wang et al., 2016b; Maichalernnukul, 2018). A brief overview of these works is provided in the following subsection. Related works In Khisti & Wornell (2010), a closed-form expression for the secrecy capacity of the Gaussian MIMO wiretap channel was derived from solving a minimax problem. Meanwhile, the problem of computing the perfect secrecy capacity of such a channel was analytically investigated in Oggier & Hassibi (2011). By relaxing the assumption of perfect channel state information (CSI) used in Khisti & Wornell (2010), Oggier & Hassibi (2011), Schaefer & Loyka (2015) studied the secrecy capacity of the compound Gaussian MIMO wiretap channel. In Mukherjee & Swindlehurst (2011), a few beamforming schemes were proposed to improve the secrecy capacity of the Gaussian MIMO wiretap channel in the presence of CSI errors. With the objective of achieving perfect secrecy at the physical layer, MIMO precoding and postcoding designs using the signal-to-noise ratio (SNR) criterion were presented in Lin, Tsai & Lin (2014). In all aforementioned works, the channel was assumed to be fixed over the whole transmission time. More precisely, the channel gains for the Gaussian MIMO wiretap channel are constant. This is rarely practical for the wireless medium as multipath propagation normally makes transmission conditions vary with time (Poor & Schaefer, 2017). Such variation is called fading. In (Yang et al., 2013; Ferdinand, Da Costa & Latva- aho, 2013; Maichalernnukul, 2018), the secrecy capacity of the fading MIMO wiretap channel2 was characterized. Specifically, Yang et al. (2013) focused on the physical-layer security enhancement through transmit antenna selection in a flat-fading MIMO channel, and characterized the corresponding performance in terms of the secrecy outage probability and the probability of non-zero secrecy capacity. In the meantime, Ferdinand, Da Costa & Latva-aho (2013) analyzed the secrecy outage probability of orthogonal space–time block code (OSTBC) MIMO systems when the transmitter–receiver and transmitter- eavesdropper links experience different kinds of fading. In contrast to space–time coding (which is based on transmit diversity), transmit beamforming and receive combining (which is based on transmit-receive diversity) achieve additional array gain (Tse & Viswanath, 2005). Besides, Goel & Negi (2008) showed that multiple transmit antennas can be deployed to generate artificial noise, such that only the transmitter-eavesdropper link is degraded. This idea enables secret communication (Csiszár & Körner, 1978) and has been extended to more practical MIMO scenarios, e.g., frequency-division duplex systems (Wang, Wang & Ng, 2015) and heterogeneous cellular networks (Wang et al., 2016b). More recently, in Maichalernnukul (2018), the average secrecy capacity of transmit- receive diversity systems in the fading MIMO wiretap channel and its upper bound were derived in closed form. Nevertheless, the corresponding secrecy outage probability has not been investigated yet. There are two reasons why we should study this performance. First, Maichalernnukul (2019), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.186 2/21 https://peerj.com http://dx.doi.org/10.7717/peerj-cs.186 3The rationale for using these ‘‘classical’’ detection techniques for the spatial multiplexing MIMO systems is twofold. First, the ZF and MMSE detectors are the basic building blocks of advanced MIMO communication architectures (e.g., layered space–time architectures (Foschini, 1996; Seethaler, Artés & Hlawatsch, 2004) and joint transmit-receive equalizers (Palomar & Lagunas, 2003; Jiang, Li & Hager, 2005)), and have been extensively addressed in the MIMO literature (Jankiraman, 2004; Biglieri et al., 2007; Heath Jr & Lozano, 2018). Second, they have low computational complexity compared to the (optimum) maximum likelihood (ML) detector, and their performance can be very close to the ML performance for a well-conditioned MIMO channel, i.e., its condition number is near to unity (see Seethaler, Artés & Hlawatsch (2005) for more details). the closed-form results of Maichalernnukul (2018) are complicated, and from these results, it is not clear how the system parameters (e.g., the numbers of antennas at the transmitter, the legitimate receiver, and the eavesdropper) affect the secrecy performance. In fact, quantifying the secrecy outage probability at high SNR in terms of two parameters, namely secrecy diversity order and secrecy array gain, can provide insights into this effect (Yang et al., 2013). Second, it was shown in Bashar, Ding & Li (2011) that although transmit beamforming in the transmit-receive diversity systems maximizes the achievable capacity of the main channel (i.e., that for the transmitter–receiver link), they still have secrecy outages at an arbitrary target secrecy rate. The first objective of our work is to present the exact and asymptotic (high-SNR) analysis of the secrecy outage probability of these systems. It is well known that the multiple antennas of MIMO systems can be exploited to obtain spatial multiplexing, i.e., transmission of independent data streams in parallel (Tse & Viswanath, 2005). This leads to an increase in the data rate. While several key performance metrics of spatial multiplexing MIMO systems, e.g., error probability, outage and ergodic capacity, have been extensively studied in the literature (Chen & Wang, 2007; Smith, 2007; Ordóñez et al., 2007; Kumar, Caire & Moustakas, 2009; Jiang, Varanasi & Li, 2011), little is known about the secrecy performance of these systems in the fading MIMO wiretap channel. The second objective of our work is to fill this knowledge gap by providing a relevant secrecy outage probability characterization. Contributions The main contributions of this work are summarized as follows: • We derive exact and asymptotic closed-form expressions for the secrecy outage probability of a transmit-receive diversity system in the fading MIMO wiretap channel. We also do the same for the secrecy outage probability of spatial multiplexing systems with linear equalization, especially zero-forcing (ZF) and minimum mean-square-error (MMSE).3 It is shown that all exact secrecy outage results simplify to the well-known result (Bloch et al., 2008, Equation (9)) for the case where the transmitter, the legitimate receiver, and the eavesdropper have a single antenna. • We determine the secrecy diversity order and secrecy array gain that the above systems achieve, and discuss the impact of the numbers of antennas at the transmitter, the legitimate receiver, and the eavesdropper, denoted as Mt, Mr, and Me, respectively, on the system secrecy and complexity. Through numerical results, it is verified that the transmit-receive diversity system attains a secrecy diversity order of MtMr, while the spatial multiplexing systems with ZF equalization and MMSE equalization yield the same secrecy diversity order of Mr−Mt+1. All of these secrecy diversity orders turn out to be independent of Me. Notation and organization Throughout this paper, we write a function g(x) of variable x as o(x) if limx→0 g(x) x =0, and denote ( · · ) as the multinomial coefficient, E[·] as the expectation operator, ddx (·) as the first derivative operator with respect to variable x, ‖·‖ as the Euclidean norm of a vector, and IN as the identity matrix of size N ×N . Moreover, det(·), (·)T, (·)†, (·)−1, and [·]ij Maichalernnukul (2019), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.186 3/21 https://peerj.com http://dx.doi.org/10.7717/peerj-cs.186 4This assumption holds, for example, if the receiver and eavesdropper are able to perfectly estimate Hr and He, respectively, and the receiver sends Hr to the transmitter through a noiseless broadcast channel, which can be heard by the eavesdropper (Goel & Negi, 2008). denote the determinant, transpose, conjugate transpose, inverse, and (i,j)-th element of a matrix, respectively, and ϒ(·,·) and 0(·,·) are the lower and upper incomplete gamma functions defined in (Gradshteyn & Ryzhik, 2000, Equation (8.350.1)) and (Gradshteyn & Ryzhik, 2000, Equation (8.350.2)), respectively. We also denote CN(0,K) as a zero-mean circularly-symmetric complex Gaussian distribution with covariance K (Gallager, 2008, Section 7.8.1), and Lmax{·} and P{·} as the largest eigenvalue of a square matrix and the associated eigenvector, respectively. The layout of the paper is as follows. ‘System Model’ describes the system model of interest. ‘Exact Secrecy Outage Probability’ and ‘Asymptotic Secrecy Outage Probability’ present exact and asymptotic analysis of the corresponding secrecy outage probability, respectively. ‘Numerical Results’ provides the numerical results of theoretical analysis and simulations, followed by the conclusion given in ‘Conclusion’. SYSTEM MODEL In this section, we consider transmit-receive diversity and spatial multiplexing systems where the transmitter, the legitimate receiver, and the passive eavesdropper are equipped with Mt, Mr, and Me antennas, respectively. The instantaneous secrecy capacity of these systems is given by (Bloch et al., 2008, Lemma 1) Cs= {log2(1+γr)−log2(1+γe), if γr >γe 0, if γr≤γe (1) where γr and γe are the instantaneous received SNRs at the receiver and the eavesdropper, respectively. Transmit-receive diversity system For the transmit-receive diversity system, the received signal vector at the legitimate receiver, yr ∈CMr×1, and that at the passive eavesdropper, ye ∈CMe×1, depend on the transmitted symbol s∈C (with E[|s|2]=P) according to yr=Hrwts+nr (2) and ye=Hewts+ne (3) respectively, where wt∈CMt×1 is the transmit weight (beamforming) vector, and nr and ne are independent circularly-symmetric complex-valued Gaussian noises: nr∼CN(0,σ2r IMr) and ne∼CN(0,σ2e IMe). We focus on a Rayleigh-fading wiretap channel, meaning that the channel matrices Hr and He have independent identically-distributed CN(0,1) entries. In addition, we assume that the three terminals know Hr, but He is available only at the eavesdropper.4 The receiver estimates the symbol s by applying the receive weight (combining) vector zr to the received signal vector yr: z†ryr=z † rHrwts+z † rnr. Maichalernnukul (2019), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.186 4/21 https://peerj.com http://dx.doi.org/10.7717/peerj-cs.186 The optimal choices of wt and zr in the sense of maximizing the SNR of this estimate (i.e., the instantaneous received SNR) are given by Dighe, Mallik & Jamuar (2003) wt= H†rzr ‖H†rzr‖ and zr=P{HrH†r} respectively, and the resultant SNR is γr,TR= γ̄rLmax{HrH†r} (4) where γ̄r = P σ2r is the average SNR at the receiver. The subscript TR refers to the transmit-receive diversity system, and is sometimes used to avoid confusion between this system and the spatial multiplexing system. Let λ=Lmax{HrH † r}, L=min(Mt,Mr), and K =max(Mt,Mr). The cumulative distribution function (CDF) of λ is given by Dighe, Mallik & Jamuar (2003) Fλ(x)= det(S(x))[∏L p=1(K −p)!(L−p)! ] (5) where S(x) is the L×L Hankel matrix with [S(x)]ij =ϒ(|Mt−Mr|+i+j−1,x). By careful inspection of the entries of S(x), this CDF can be rewritten as Fλ(x)= L∑ m=1 (Mt+Mr−2m)m∑ n=|Mt−Mr| am,n n! ϒ(n+1,mx) (6) where am,n = cm,nn! mn+1 [∏L p=1(K−p)!(L−p)! ] and cm,n is the coefficient computed by using curve fitting on the plot of ddx det(S(x)) (Dighe, Mallik & Jamuar, 2003). Using Eq. (6) and (Papoulis & Pillai, 2002, Example 5-1), the CDF of γr,TR in Eq. (4) is given by Fγr,TR(x)= L∑ m=1 (Mt+Mr−2m)m∑ n=|Mt−Mr| am,n n! ϒ ( n+1, mx γ̄r ) . (7) Similarly, the eavesdropper can estimate the symbol s as z†eye=z † eHewts+z † ene where the receive weight vector ze= Hewt ‖Hewt‖ is chosen to maximize the SNR of the estimate, yielding γe,TR= γ̄e‖Hewt‖ 2 (8) Maichalernnukul (2019), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.186 5/21 https://peerj.com http://dx.doi.org/10.7717/peerj-cs.186 where γ̄e = P σ2e is the average SNR at the eavesdropper. The probability density function (PDF) of γe,TR in Eq. (8) is given by Maichalernnukul (2018) fγe,TR(x)= xMe−1e− x γ̄e (Me−1)!γ̄ Me e . (9) Spatial multiplexing system Unlike the transmit-receive diversity system, the spatial multiplexing system allows the simultaneous transmission of different symbols, i.e., the ith antenna (i=1,2,...,Mt) at the transmitter is used to transmit the symbol si∈C (with E[|si|2]=P) . Let s=[s1,s2,...,sMt]T. The received signal vectors at the legitimate receiver and the passive eavesdropper are given, respectively, by yr=Hrs+nr where Hr and nr are defined in Eq. (2), and ye=Hes+ne where He and ne are defined in Eq. (3). We assume that the receiver and the eavesdropper know Hr and He, respectively, and the numbers of antennas at these two terminals (Mr and Me) are no less than the number of antennas at the transmitter (Mt). The assumption on Mt, Mr, and Me is necessary for the theoretical analysis hereafter. In order for the receiver to estimate s, the ZF or MMSE receive weight (equalizing) matrix is applied to yr. These matrices are given by Tse & Viswanath (2005) Wr,ZF= ( H†rHr )−1 H†r and Wr,MMSE= ( H†rHr+ 1 γ̄r IMt )−1 H†r. It is noteworthy that as the average SNR at the receiver grows very large, i.e., γ̄r →∞, Wr,MMSE approaches Wr,ZF. Left multiplying yr by Wr,ZF and Wr,MMSE, we obtain the ith symbol estimate (i=1,2,...,Mt), the SNRs of which are, respectively, (Jiang, Varanasi & Li, 2011) γr,ZF,i= γ̄r[( H†rHr )−1] ii (10) and γr,MMSE,i= γ̄r[( H†rHr+ 1 γ̄r IMt )−1] ii −1. (11) The CDFs of γr,ZF,i and γr,MMSE,i are given, respectively, by Chen & Wang (2007) Fγr,ZF(x)=1−e − x γ̄r Mr−Mt∑ m=0 xm m!γ̄mr (12) Maichalernnukul (2019), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.186 6/21 https://peerj.com http://dx.doi.org/10.7717/peerj-cs.186 and Smith (2007) Fγr,MMSE(x)=1− e− x γ̄r (x+1)Mt−1 Mr−1∑ m=0 dmx m (13) where dm= ∑m n=max(0,m−Mt+1) (Mt−1 m−n ) 1 n!γ̄nr . The symbol index i is omitted from Eqs. (12) and (13) because all the elements of Hr are statistically independent and identically distributed. Similarly, the eavesdropper performs ZF or MMSE equalization, and the resulting SNRs of the ith symbol estimate (i.e., γe,ZF,i and γe,MMSE,i) can be expressed, respectively, as Eqs. (10) and (11) with the subscript r being replaced by the subscript e. Replacing the subscript r with the subscript e in Eqs. (12) and (13), and taking the derivative of these equations with respect to x, we obtain the PDFs for γe,ZF,i and γe,MMSE,i, respectively, as fγe,ZF(x)= xMe−Mte− x γ̄e (Me−Mt)!γ̄ Me−Mt+1 e (14) and fγe,MMSE(x)= e− x γ̄e (x+1)Mt Me−1∑ m=0 gm [ xm+1 γ̄e + ( Mt+ 1 γ̄e −m−1 ) xm−mxm−1 ] (15) where gm is similar to dm, except that the subscript r is replaced by the subscript e. EXACT SECRECY OUTAGE PROBABILITY The secrecy outage probability is defined as the probability that the instantaneous secrecy capacity is less than a target secrecy rate R > 0 (Bloch et al., 2008). From Eq. (1), this performance metric can be expressed as Pout(R)=Pr{Cs