On the secrecy performance of transmit-receive diversity and spatial multiplexing systems

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.


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 1 In 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). 2 For this kind of channel, the channel gains are allowed to change from channel use to channel use (Poor & Schaefer, 2017). channels 1 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 & Latvaaho, 2013;Maichalernnukul, 2018), the secrecy capacity of the fading MIMO wiretap channel 2 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 transmittereavesdropper 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 transmitreceive 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, 3 The 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 M t , M r , and M e , 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 M t M r , while the spatial multiplexing systems with ZF equalization and MMSE equalization yield the same secrecy diversity order of M r −M t +1. All of these secrecy diversity orders turn out to be independent of M e .

Notation and organization
Throughout this paper, we write a function g (x) of variable x as o(x) if lim x→0 g (x) x = 0, and denote · · as the multinomial coefficient, E[·] as the expectation operator, d dx (·) as the first derivative operator with respect to variable x, · as the Euclidean norm of a vector, and I N as the identity matrix of size N × N . Moreover, det(·), (·) T , (·) † , (·) −1 , and [·] ij 4 This assumption holds, for example, if the receiver and eavesdropper are able to perfectly estimate H r and H e , respectively, and the receiver sends H r 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 (·,·) 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 L max {·} 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 M t , M r , and M e antennas, respectively. The instantaneous secrecy capacity of these systems is given by (Bloch et al., 2008, Lemma 1) where γ r and γ e 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, y r ∈ C M r ×1 , and that at the passive eavesdropper, y e ∈ C M e ×1 , depend on the transmitted symbol s ∈ C (with E[|s| 2 ] = P) according to and respectively, where w t ∈ C M t ×1 is the transmit weight (beamforming) vector, and n r and n e are independent circularly-symmetric complex-valued Gaussian noises: n r ∼ CN (0,σ 2 r I M r ) and n e ∼ CN (0,σ 2 e I M e ). We focus on a Rayleigh-fading wiretap channel, meaning that the channel matrices H r and H e have independent identically-distributed CN (0,1) entries. In addition, we assume that the three terminals know H r , but H e is available only at the eavesdropper. 4 The receiver estimates the symbol s by applying the receive weight (combining) vector z r to the received signal vector y r : The optimal choices of w t and z r in the sense of maximizing the SNR of this estimate (i.e., the instantaneous received SNR) are given by Dighe, Mallik & Jamuar (2003) respectively, and the resultant SNR is 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.
By careful inspection of the entries of S(x), this CDF can be rewritten as and c m,n is the coefficient computed by using curve fitting on the plot of d dx det(S(x)) (Dighe, Mallik & Jamuar, 2003). Using Eq. (6) and (Papoulis & Pillai, 2002, Example 5-1), the CDF of γ r,TR in Eq. (4) is given by Similarly, the eavesdropper can estimate the symbol s as (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,...,M t ) at the transmitter is used to transmit the symbol where H e and n e are defined in Eq. (3). We assume that the receiver and the eavesdropper know H r and H e , respectively, and the numbers of antennas at these two terminals (M r and M e ) are no less than the number of antennas at the transmitter (M t ). The assumption on M t , M r , and M e 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 y r . These matrices are given by Tse & Viswanath (2005) It is noteworthy that as the average SNR at the receiver grows very large, i.e.,γ r → ∞, W r,MMSE approaches W r,ZF . Left multiplying y r by W r,ZF and W r,MMSE , we obtain the ith symbol estimate (i = 1,2,...,M t ), the SNRs of which are, respectively, (Jiang, Varanasi & Li, 2011) and The CDFs of γ r,ZF,i and γ r,MMSE,i are given, respectively, by Chen & Wang (2007) and Smith (2007) The symbol index i is omitted from Eqs. (12) and (13) because all the elements of H r 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., γ e,ZF,i and γ e,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 γ e,ZF,i and γ e,MMSE,i , respectively, as and where g m is similar to d m , 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

Spatial multiplexing system
From Eqs. (12), (14) and (16), we can derive the exact secrecy outage probability for the spatial multiplexing system with ZF equalization as follows: where the second equality is obtained by using ( (15) and (16) as follows: where the second equality is obtained by changing the limits of integration and using

ASYMPTOTIC SECRECY OUTAGE PROBABILITY
In this section, we focus on deriving the asymptotic secrecy outage probability of the aforementioned systems asγ r → ∞. This expression enables one to analyze the secrecy performance in the high-SNR regime through two performance indicators: secrecy diversity order and secrecy array gain (Yang et al., 2013). The secrecy diversity order indicates the slope of the secrecy outage probability versusγ r curve at high SNR in a log-log scale, whereas the secrecy array gain indicates the shift of the curve with respect to the benchmark secrecy outage curve.

Transmit-receive diversity system
First, we look for a first-order expansion of Eq. (5), which will be immediate from a first-order expansion of det(S(x)). Following the approach outlined in (McKay, 2006, Appendix B.7) and using (Kalman, 1984, Equations (1) and (2)), it is straightforward to show that the first-order Taylor expansion of det(S(x)) around x = 0 is Substituting Eq. (21) into Eq. (5) yields Using Eq. (22) and (Papoulis & Pillai, 2002, Example 5-1), the first-order expansion of the CDF of γ r,TR is given by Using Eqs. (9), (16) and (23), and following the same procedure as used in Eq. (17), an asymptotic expression for P out,TR (R) withγ r → ∞ is obtained as where the secrecy diversity gain is and the secrecy array gain is It is clear from Eq. (25) that the secrecy diversity order is dependent on M t and M r , and independent of M e . It can also be seen from Eq. (26) that the eavesdropper channel has an adverse impact on the secrecy array gain. Accordingly, increasing the number of antennas at the eavesdropper lessens the secrecy array gain, thereby rising the secrecy outage probability.

Spatial multiplexing system
Applying (Gradshteyn & Ryzhik, 2000, Equation (1.211.1)) to the exponential function in Eq. (12) and performing some algebraic manipulations, the first-order expansion of the CDF of γ r,ZF,i can be derived as Using Eqs. (14), (16) and (27), and following the same procedure as used in Eq. (19), an asymptotic expression for P out,ZF (R) withγ r → ∞ is obtained as where and Adopting the same steps as for deriving the first-order expansion of F γ r,ZF (x), we obtain Using Eqs. (15), (16) and (31), and following the same procedure as used in Eq. (20), an asymptotic expression for P out,MMSE (R) withγ r → ∞ is obtained as and It is obvious from Eqs. (29) and (33) that the secrecy diversity orders of the spatial multiplexing systems with ZF equalization and MMSE equalization are dependent on M t and M r , and independent of M e . It can also be observed from Eqs. (30) and (34) that increasing M e decreases the corresponding secrecy array gains.

NUMERICAL RESULTS
In this section, we validate the preceding theoretical analysis and investigate the effect of the various system parameters. For these purposes, theoretical and simulation results are obtained by using MATLAB. Specifically, we use the closed-form expressions derived above to generate the theoretical results, and adopt the Monte Carlo method to generate the simulation results. Remember thatγ r andγ e are the average SNRs at the legitimate receiver and the passive eavesdropper, respectively. Unless otherwise indicated, the SNR γ e is set to 10 dB, and the target secrecy rate R is set to 1 bit/s/Hz. Figure 1 shows the theoretical secrecy outage probability of the transmit-receive diversity system (computed with Eq. (17)) and its simulation counterpart (labeled with ''simu.'') againstγ r . As seen in the figure, the theoretical and simulation results match perfectly. For a givenγ r , when M t + M r = 4 and M e = 2, the secrecy outage probability with M t = 2 and M r = 2 is lower than that with M t = 3 and M r = 1. This is consistent with the fact that for a fixed total number of antennas at the transmitter and legitimate receiver (M t + M r ), a more-balanced antenna configuration provides a larger diversity gain (Dighe, Mallik & Jamuar, 2003;Maaref & Aïssa, 2005). Specifically, from Eq. (25), we have D TR = 4 for M t = 2 and M r = 2, and D TR = 3 for M t = 3 and M r = 1. However, when M t M r = 12 and M e = 3, the secrecy outage probability with M t = 4 and M r = 3 is higher than that with M t = 6 and M r = 2. The reason is that for the same product of M t and M r , an increase in M t + M r yields a performance enhancement (Dighe, Mallik & Jamuar, 2003). Figure 2 depicts the theoretical secrecy outage probability of the aforementioned system for different combinations of M t , M r , and M e . We observe that when (M t ,M r ) is kept fixed (i.e., at (2,1), (4,2), or (6,3)), the larger M e is, the smaller the array gain (as discussed in Eq. (26)), which worsens the secrecy outage performance. Furthermore, it can be seen that for a givenγ r , the secrecy outage probability with (M t ,M r ,M e ) = (2,1,1) is higher than that with (M t ,M r ,M e ) = (4,2,2). Meanwhile, the secrecy outage probability with (M t ,M r ,M e ) = (4,2,2) is higher than that with (M t ,M r ,M e ) = (6,3,3). The same performance trend occurs when (M t ,M r ,M e ) increases from (2,1,2) to (6,3,6) or from (2,1,3) to (6,3,9). These results reveal that adding M t and M r proportionally to M e is advantageous. Figure 3 verifies the asymptotic secrecy outage probability of the transmit-receive diversity system derived in Eqs. (24)-(26) at a fixedγ e (i.e.,γ e = 10 dB). The exact and asymptotic secrecy outage curves are labeled with ''exact'' and ''asym.'', respectively. Asγ r grows, the asymptotic curves approach the exact ones for different values of M t , M r , and M e . It can also be observed that the secrecy diversity gain is M t M r , as predicted by Eq. (25), and the secrecy array gain diminishes with increasing M e , as predicted by Eq. (26). Figure 4 compares the theoretical secrecy outage results for the spatial multiplexing systems with ZF equalization (computed with Eq. (19)) and MMSE equalization (computed with Eq. (20)), and their simulation counterparts. The theoretical and simulation results agree well, and both kinds of systems exhibit similar secrecy outage performance. Indeed, the spatial multiplexing system with MMSE equalization achieves lower secrecy outage probability when the number of antennas at the eavesdropper is more than that at the receiver, as illustrated in Fig. 5. In addition, most noteworthy in Eq. (19) is the fact that, when the values of (M r −M t ) and (M e −M t ) are fixed, the secrecy outage probability of the spatial multiplexing system with ZF equalization remains the same regardless of the value of M t that is used. This fact is confirmed by Fig. 6,  Figures 7 and 8 verify the asymptotic secrecy outage probability of the spatial multiplexing system with ZF equalization derived in Eqs. (28)-(30) and that of the spatial multiplexing system with MMSE equalization derived in Eqs. (32)-(34), respectively, at a fixedγ e (i.e.,γ e = 10 dB). Asγ r increases, the asymptotic curves tend towards the exact ones for different values of M t , M r , and M e . It can also be noticed that the secrecy diversity gains of the two systems are M r − M t + 1, as predicted by Eqs. (29) and (33), and the corresponding secrecy array gains lessen with growing M e , as predicted by Eqs. 3)), w t , and z e for the transmit-receive diversity system; (2) the number of flops required to compute W r,ZF and W e,ZF for the spatial multiplexing system with ZF equalization; and (3) the number of flops required to compute W r,MMSE and W e,MMSE for the spatial multiplexing system with MMSE equalization. The results are given in Table 1, where N is the number of iterations used in the power iteration method. 6 Figure 9 shows the system complexity as a function of M t for M t = M r = M e and for M r = M e = 2M t . From this figure, we see that the computational complexity of the spatial multiplexing system with ZF equalization is comparable to that of the spatial multiplexing system with MMSE equalization, while the transmit-receive diversity system has the highest computational complexity, even with N = 1.

CONCLUSION
We have presented exact and asymptotic analysis of the secrecy outage probability of the transmit-receive diversity system and spatial multiplexing systems with ZF equalization and MMSE equalization in a Rayleigh-fading MIMO wiretap channel. This asymptotic analysis has shown that the transmit-receive diversity system achieves a secrecy diversity order of M t M r , whereas the two spatial multiplexing systems offer the same secrecy diversity order of M r − M t + 1. Interestingly, all of these secrecy diversity orders do not rely on M e .