On the secrecy performance of transmitreceive diversity and spatial multiplexing systems
 Published
 Accepted
 Received
 Academic Editor
 Shlomi Dolev
 Subject Areas
 Computer Networks and Communications, Security and Privacy
 Keywords
 Physicallayer security, Secrecy outage probability, Transmitreceive diversity, MultipleInput MultipleOutput, Spatial multiplexing
 Copyright
 © 2019 Maichalernnukul
 Licence
 This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, reproduction and adaptation in any medium and for any purpose provided that it is properly attributed. For attribution, the original author(s), title, publication source (PeerJ Computer Science) and either DOI or URL of the article must be cited.
 Cite this article
 2019. On the secrecy performance of transmitreceive diversity and spatial multiplexing systems. PeerJ Computer Science 5:e186 https://doi.org/10.7717/peerjcs.186
Abstract
Emerging from the informationtheoretic characterization of secrecy, physicallayer 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 physicallayer security issues in multipleinput multipleoutput (MIMO) wireless communications. This paper analyzes the secrecy performance of transmitreceive diversity system and spatial multiplexing systems with zeroforcing equalization and minimum meansquareerror equalization. Specifically, exact and asymptotic closedform 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 physicallayer 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, physicallayer security arising from the informationtheoretic 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 physicallayer security (Wyner, 1975; Csiszár & Körner, 1978; LeungYanCheong & 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 socalled 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 transmittereavesdropper link.
Stimulated by advances in multipleantenna technology for wireless communications, the physicallayer security issues in multipleinput multipleoutput (MIMO) wiretap 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 & Latvaaho, 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 closedform 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 signaltonoise 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 physicallayer security enhancement through transmit antenna selection in a flatfading MIMO channel, and characterized the corresponding performance in terms of the secrecy outage probability and the probability of nonzero secrecy capacity. In the meantime, Ferdinand, Da Costa & Latvaaho (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 transmitreceive 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 transmittereavesdropper link is degraded. This idea enables secret communication (Csiszár & Körner, 1978) and has been extended to more practical MIMO scenarios, e.g., frequencydivision 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, the closedform 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 transmitreceive 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 (highSNR) 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 closedform expressions for the secrecy outage probability of a transmitreceive 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 zeroforcing (ZF) and minimum meansquareerror (MMSE).^{3} It is shown that all exact secrecy outage results simplify to the wellknown 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 transmitreceive 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 ${lim}_{x\to 0}\frac{g\left(x\right)}{x}=0$, and denote $\left(\genfrac{}{}{0.0pt}{}{\cdot}{\cdot}\right)$ as the multinomial coefficient, E[⋅] as the expectation operator, $\frac{\text{d}}{\text{d}x}\left(\cdot \right)$ 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} denote the determinant, transpose, conjugate transpose, inverse, and (i, j)th element of a matrix, respectively, and ϒ(⋅, ⋅) and Γ(⋅, ⋅) 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 $\mathcal{CN}\left(\mathbf{0},\mathbf{K}\right)$ as a zeromean circularlysymmetric complex Gaussian distribution with covariance K (Gallager, 2008, Section 7.8.1), and ${\mathcal{L}}_{\text{max}}\left\{\cdot \right\}$ and $\mathcal{P}\left\{\cdot \right\}$ 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 transmitreceive 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) (1)$C}_{\text{s}}=\left\{\right.\begin{array}{cc}{log}_{2}\phantom{\rule{0.3em}{0ex}}\left(1+{\gamma}_{\text{r}}\right){log}_{2}\phantom{\rule{0.3em}{0ex}}\left(1+{\gamma}_{\text{e}}\right),\hfill & \hfill \text{if}{\gamma}_{\text{r}}>{\gamma}_{\text{e}}\\ 0,\hfill & \hfill \text{if}{\gamma}_{\text{r}}\le {\gamma}_{\text{e}}\\ \hfill \end{array$ where γ_{r} and γ_{e} are the instantaneous received SNRs at the receiver and the eavesdropper, respectively.
Transmitreceive diversity system
For the transmitreceive diversity system, the received signal vector at the legitimate receiver, y_{r} ∈ ℂ^{Mr×1}, and that at the passive eavesdropper, y_{e} ∈ ℂ^{Me×1}, depend on the transmitted symbol s ∈ ℂ (with E[s^{2}] = P) according to (2)$\mathbf{y}}_{\text{r}}={\mathbf{H}}_{\text{r}}{\mathbf{w}}_{\text{t}}s+{\mathbf{n}}_{\text{r}$ and (3)$\mathbf{y}}_{\text{e}}={\mathbf{H}}_{\text{e}}{\mathbf{w}}_{\text{t}}s+{\mathbf{n}}_{\text{e}$ respectively, where w_{t} ∈ ℂ^{Mt×1} is the transmit weight (beamforming) vector, and n_{r} and n_{e} are independent circularlysymmetric complexvalued Gaussian noises: ${\mathbf{n}}_{\text{r}}\sim \mathcal{CN}\left(\mathbf{0},{\sigma}_{\text{r}}^{2}{\mathbf{I}}_{{M}_{\text{r}}}\right)$ and ${\mathbf{n}}_{\text{e}}\sim \mathcal{CN}\left(\mathbf{0},{\sigma}_{\text{e}}^{2}{\mathbf{I}}_{{M}_{\text{e}}}\right)$. We focus on a Rayleighfading wiretap channel, meaning that the channel matrices H_{r} and H_{e} have independent identicallydistributed $\mathcal{CN}\left(0,1\right)$ 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}: ${\mathbf{z}}_{\text{r}}^{\u2020}{\mathbf{y}}_{\text{r}}={\mathbf{z}}_{\text{r}}^{\u2020}{\mathbf{H}}_{\text{r}}{\mathbf{w}}_{\text{t}}s+{\mathbf{z}}_{\text{r}}^{\u2020}{\mathbf{n}}_{\text{r}}.$ 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) $\mathbf{w}}_{\text{t}}=\frac{{\mathbf{H}}_{\text{r}}^{\u2020}{\mathbf{z}}_{\text{r}}}{\parallel {\mathbf{H}}_{\text{r}}^{\u2020}{\mathbf{z}}_{\text{r}}\parallel$ and $\mathbf{z}}_{\text{r}}=\mathcal{P}\left\{{\mathbf{H}}_{\text{r}}{\mathbf{H}}_{\text{r}}^{\u2020}\right\$ respectively, and the resultant SNR is (4)$\gamma}_{\text{r,TR}}={\stackrel{\u0304}{\gamma}}_{\text{r}}{\mathcal{L}}_{\text{max}}\left\{{\mathbf{H}}_{\text{r}}{\mathbf{H}}_{\text{r}}^{\u2020}\right\$ where ${\stackrel{\u0304}{\gamma}}_{\text{r}}=\frac{P}{{\sigma}_{\text{r}}^{2}}$ is the average SNR at the receiver. The subscript TR refers to the transmitreceive diversity system, and is sometimes used to avoid confusion between this system and the spatial multiplexing system. Let $\lambda ={\mathcal{L}}_{\text{max}}\left\{{\mathbf{H}}_{\text{r}}{\mathbf{H}}_{\text{r}}^{\u2020}\right\}$, L = min(M_{t}, M_{r}), and K = max(M_{t}, M_{r}). The cumulative distribution function (CDF) of λ is given by Dighe, Mallik & Jamuar (2003) (5)$F}_{\lambda}\left(x\right)=\frac{det\left(\mathbf{S}\left(x\right)\right)}{\left[\prod _{p=1}^{L}\left(Kp\right)!\left(Lp\right)!\right]$ where S(x) is the L × L Hankel matrix with ${\left[\mathbf{S}\left(x\right)\right]}_{ij}=\Upsilon \left(\left{M}_{\text{t}}{M}_{\text{r}}\right+i+j1,x\right).$ By careful inspection of the entries of S(x), this CDF can be rewritten as (6)${F}_{\lambda}\left(x\right)=\sum _{m=1}^{L}\sum _{n=\left{M}_{\text{t}}{M}_{\text{r}}\right}^{\left({M}_{\text{t}}+{M}_{\text{r}}2m\right)m}\frac{{a}_{m,n}}{n!}\Upsilon \left(n+1,mx\right)$ where ${a}_{m,n}=\frac{{c}_{m,n}n!}{{m}^{n+1}\left[{\prod}_{p=1}^{L}\left(Kp\right)!\left(Lp\right)!\right]}$ and c_{m,n} is the coefficient computed by using curve fitting on the plot of $\frac{\text{d}}{\text{d}x}det\left(\mathbf{S}\left(x\right)\right)$ (Dighe, Mallik & Jamuar, 2003). Using Eq. (6) and (Papoulis & Pillai, 2002, Example 51), the CDF of γ_{r,TR} in Eq. (4) is given by (7)${F}_{{\gamma}_{\text{r,TR}}}\left(x\right)=\sum _{m=1}^{L}\sum _{n=\left{M}_{\text{t}}{M}_{\text{r}}\right}^{\left({M}_{\text{t}}+{M}_{\text{r}}2m\right)m}\frac{{a}_{m,n}}{n!}\Upsilon \left(n+1,\frac{mx}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}\right).$
Similarly, the eavesdropper can estimate the symbol s as $\mathbf{z}}_{\text{e}}^{\u2020}{\mathbf{y}}_{\text{e}}={\mathbf{z}}_{\text{e}}^{\u2020}{\mathbf{H}}_{\text{e}}{\mathbf{w}}_{\text{t}}s+{\mathbf{z}}_{\text{e}}^{\u2020}{\mathbf{n}}_{\text{e}$ where the receive weight vector $\mathbf{z}}_{\text{e}}=\frac{{\mathbf{H}}_{\text{e}}{\mathbf{w}}_{\text{t}}}{\parallel {\mathbf{H}}_{\text{e}}{\mathbf{w}}_{\text{t}}\parallel$ is chosen to maximize the SNR of the estimate, yielding (8)$\gamma}_{\text{e,TR}}={\stackrel{\u0304}{\gamma}}_{\text{e}}\parallel {\mathbf{H}}_{\text{e}}{\mathbf{w}}_{\text{t}}{\parallel}^{2$ where ${\stackrel{\u0304}{\gamma}}_{\text{e}}=\frac{P}{{\sigma}_{\text{e}}^{2}}$ is the average SNR at the eavesdropper. The probability density function (PDF) of γ_{e,TR} in Eq. (8) is given by Maichalernnukul (2018) (9)${f}_{{\gamma}_{\text{e,TR}}}\left(x\right)=\frac{{x}^{{M}_{\text{e}}1}{e}^{\frac{x}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}}}{\left({M}_{\text{e}}1\right)!{{\stackrel{\u0304}{\gamma}}_{\text{e}}}^{{M}_{\text{e}}}}.$
Spatial multiplexing system
Unlike the transmitreceive 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 s_{i} ∈ ℂ (with E[s_{i}^{2}] = P) . Let s = [s_{1}, s_{2}, …, s_{Mt}]^{T}. The received signal vectors at the legitimate receiver and the passive eavesdropper are given, respectively, by $\mathbf{y}}_{\text{r}}={\mathbf{H}}_{\text{r}}\mathbf{s}+{\mathbf{n}}_{\text{r}$ where H_{r} and n_{r} are defined in Eq. (2), and $\mathbf{y}}_{\text{e}}={\mathbf{H}}_{\text{e}}\mathbf{s}+{\mathbf{n}}_{\text{e}$ 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) $\mathbf{W}}_{\text{r,ZF}}={\left({\mathbf{H}}_{\text{r}}^{\u2020}{\mathbf{H}}_{\text{r}}\right)}^{1}{\mathbf{H}}_{\text{r}}^{\u2020$ and ${\mathbf{W}}_{\text{r,MMSE}}={\left({\mathbf{H}}_{\text{r}}^{\u2020}{\mathbf{H}}_{\text{r}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}{\mathbf{I}}_{{M}_{\text{t}}}\right)}^{1}{\mathbf{H}}_{\text{r}}^{\u2020}.$ It is noteworthy that as the average SNR at the receiver grows very large, i.e., ${\stackrel{\u0304}{\gamma}}_{\text{r}}\to \infty $, W_{r,MMSE} approaches W_{r,ZF}. Left multiplying y_{r} by W_{r,ZF} and W_{r,MMSE}, we obtain the ith symbol estimate (i = 1, 2, …, M_{t}), the SNRs of which are, respectively, (Jiang, Varanasi & Li, 2011) (10)$\gamma}_{\text{r,ZF},i}=\frac{{\stackrel{\u0304}{\gamma}}_{\text{r}}}{{\left[{\left({\mathbf{H}}_{\text{r}}^{\u2020}{\mathbf{H}}_{\text{r}}\right)}^{1}\right]}_{ii}$ and (11)${\gamma}_{\text{r,MMSE},i}=\frac{{\stackrel{\u0304}{\gamma}}_{\text{r}}}{{\left[{\left({\mathbf{H}}_{\text{r}}^{\u2020}{\mathbf{H}}_{\text{r}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}{\mathbf{I}}_{{M}_{\text{t}}}\right)}^{1}\right]}_{ii}}1.$ The CDFs of γ_{r,ZF,i} and γ_{r,MMSE,i} are given, respectively, by Chen & Wang (2007) (12)$F}_{{\gamma}_{\text{r,ZF}}}\left(x\right)=1{e}^{\frac{x}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}}\sum _{m=0}^{{M}_{\text{r}}{M}_{\text{t}}}\frac{{x}^{m}}{m!{\stackrel{\u0304}{\gamma}}_{\text{r}}^{m}$ and Smith (2007) (13)$F}_{{\gamma}_{\text{r,MMSE}}}\left(x\right)=1\frac{{e}^{\frac{x}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}}}{{\left(x+1\right)}^{{M}_{\text{t}}1}}\sum _{m=0}^{{M}_{\text{r}}1}{d}_{m}{x}^{m$ where ${d}_{m}={\sum}_{n=max\left(0,m{M}_{\text{t}}+1\right)}^{m}\left(\genfrac{}{}{0.0pt}{}{{M}_{\text{t}}1}{mn}\right)\frac{1}{n!{\stackrel{\u0304}{\gamma}}_{\text{r}}^{n}}$. 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 (14)$f}_{{\gamma}_{\text{e,ZF}}}\left(x\right)=\frac{{x}^{{M}_{\text{e}}{M}_{\text{t}}}{e}^{\frac{x}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}}}{\left({M}_{\text{e}}{M}_{\text{t}}\right)!{\stackrel{\u0304}{\gamma}}_{\text{e}}^{{M}_{\text{e}}{M}_{\text{t}}+1}$ and (15)${f}_{{\gamma}_{\text{e,MMSE}}}\left(x\right)=\frac{{e}^{\frac{x}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}}}{{\left(x+1\right)}^{{M}_{\text{t}}}}\sum _{m=0}^{{M}_{\text{e}}1}{g}_{m}\left[\frac{{x}^{m+1}}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}+\left({M}_{\text{t}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}m1\right){x}^{m}m{x}^{m1}\right]$ 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 (16)${P}_{\text{out}}\left(R\right)=\text{Pr}\left\{{C}_{\text{s}}<R\right\}=\text{Pr}\left\{{\gamma}_{\text{r}}<{2}^{R}{\gamma}_{\text{e}}+{2}^{R}1\right\}={\int}_{0}^{\infty}{f}_{{\gamma}_{\text{e}}}\left(v\right){F}_{{\gamma}_{\text{r}}}\phantom{\rule{0.3em}{0ex}}\left({2}^{R}v+{2}^{R}1\right)\text{d}v.$
Transmitreceive diversity system
From Eqs. (7), (9) and (16), we can derive the exact secrecy outage probability for the transmitreceive diversity system as follows: (17)$P}_{\text{out,TR}}\left(R\right)=\frac{1}{\left({M}_{\text{e}}1\right)!{{\stackrel{\u0304}{\gamma}}_{\text{e}}}^{{M}_{\text{e}}}}\sum _{m=1}^{L}\sum _{n=\left{M}_{\text{t}}{M}_{\text{r}}\right}^{\left({M}_{\text{t}}+{M}_{\text{r}}2m\right)m}\frac{{a}_{m,n}}{n!}{\int}_{0}^{\infty}{v}^{{M}_{\text{e}}1}{e}^{\frac{v}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}}\times \Upsilon \left(n+1,\frac{\left({2}^{R}v+{2}^{R}1\right)m}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}\right)\text{d}v=\frac{1}{\left({M}_{\text{e}}1\right)!{{\stackrel{\u0304}{\gamma}}_{\text{e}}}^{{M}_{\text{e}}}}\sum _{m=1}^{L}\sum _{n=\left{M}_{\text{t}}{M}_{\text{r}}\right}^{\left({M}_{\text{t}}+{M}_{\text{r}}2m\right)m}{a}_{m,n}\left[\right.{\int}_{0}^{\infty}{v}^{{M}_{\text{e}}1}{e}^{\frac{v}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}}\text{d}v\phantom{\rule{10.00002pt}{0ex}}{e}^{\frac{\left({2}^{R}1\right)m}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}}\sum _{k=0}^{n}{\left(\frac{m}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}\right)}^{k}\sum _{l=0}^{k}\frac{{2}^{lR}{\left({2}^{R}1\right)}^{kl}}{l!\left(kl\right)!}{\int}_{0}^{\infty}{v}^{l+{M}_{\text{e}}1}{e}^{\left(\frac{{2}^{R}m}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\right)v}\text{d}v\left]\right.=1\frac{1}{\left({M}_{\text{e}}1\right)!{{\stackrel{\u0304}{\gamma}}_{\text{e}}}^{{M}_{\text{e}}}}\sum _{m=1}^{L}\sum _{n=\left{M}_{\text{t}}{M}_{\text{r}}\right}^{\left({M}_{\text{t}}+{M}_{\text{r}}2m\right)m}{a}_{m,n}{e}^{\frac{\left({2}^{R}1\right)m}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}}\sum _{k=0}^{n}{\left(\frac{m}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}\right)}^{k}\times \sum _{l=0}^{k}\frac{\left(l+{M}_{\text{e}}1\right)!{2}^{lR}{\left({2}^{R}1\right)}^{kl}}{l!\left(kl\right)!{\left(\frac{{2}^{R}m}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\right)}^{l+{M}_{\text{e}}}$ where the second equality is obtained by using (Gradshteyn & Ryzhik, 2000, Equations (1.111) and (8.352.1)), and the last equality is obtained by using (Gradshteyn & Ryzhik, 2000, Equation (3.351.3)) and (Maaref & Aïssa, 2005, Equation (11)). For the special case of M_{t} = M_{r} = M_{e} = 1, the secrecy outage probability expression in Eq. (17) reduces to (18)$P}_{\text{out,TR}}\left(R\right)=1\frac{{\stackrel{\u0304}{\gamma}}_{\text{r}}{e}^{\frac{{2}^{R}1}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}}}{{\stackrel{\u0304}{\gamma}}_{\text{r}}+{2}^{R}{\stackrel{\u0304}{\gamma}}_{\text{e}}$ which agrees exactly with a result given in (Bloch et al., 2008, Equation (9)).
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: (19)$P}_{\text{out,ZF}}\left(R\right)={\int}_{0}^{\infty}{f}_{{\gamma}_{\text{e,ZF}}}\left(v\right)\text{d}v\frac{{e}^{\frac{{2}^{R}1}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}}}{\left({M}_{\text{e}}{M}_{\text{t}}\right)!{{\stackrel{\u0304}{\gamma}}_{\text{e}}}^{{M}_{\text{e}}{M}_{\text{t}}+1}}\sum _{m=0}^{{M}_{\text{r}}{M}_{\text{t}}}\frac{1}{m!{\stackrel{\u0304}{\gamma}}_{\text{r}}^{m}}\times {\int}_{0}^{\infty}{\left({2}^{R}v+{2}^{R}1\right)}^{m}{v}^{{M}_{\text{e}}{M}_{\text{t}}}{e}^{\left(\frac{{2}^{R}}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\right)v}\text{d}v=1\frac{{e}^{\frac{{2}^{R}1}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}}}{\left({M}_{\text{e}}{M}_{\text{t}}\right)!{{\stackrel{\u0304}{\gamma}}_{\text{e}}}^{{M}_{\text{e}}{M}_{\text{t}}+1}}\sum _{m=0}^{{M}_{\text{r}}{M}_{\text{t}}}\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{r}}^{m}}\sum _{n=0}^{m}\frac{{2}^{nR}{\left({2}^{R}1\right)}^{mn}}{n!\left(mn\right)!}\times {\int}_{0}^{\infty}{v}^{n+{M}_{\text{e}}{M}_{\text{t}}}{e}^{\left(\frac{{2}^{R}}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\right)v}\text{d}v=1\frac{{e}^{\frac{{2}^{R}1}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}}}{\left({M}_{\text{e}}{M}_{\text{t}}\right)!{\left(\frac{{2}^{R}{\stackrel{\u0304}{\gamma}}_{\text{e}}}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}+1\right)}^{{M}_{\text{e}}{M}_{\text{t}}+1}}\times \sum _{m=0}^{{M}_{\text{r}}{M}_{\text{t}}}\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{r}}^{m}}\sum _{n=0}^{m}\frac{{2}^{nR}{\left({2}^{R}1\right)}^{mn}\left(n+{M}_{\text{e}}{M}_{\text{t}}\right)!}{n!\left(mn\right)!{\left(\frac{{2}^{R}}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\right)}^{n}$ where the second equality is obtained by using (Gradshteyn & Ryzhik, 2000, Equation (1.111)) and (Papoulis & Pillai, 2002, Equation (418)), and the last equality is obtained by using (Gradshteyn & Ryzhik, 2000, Equation (3.351.3)). For the special case of M_{t} = M_{r} = M_{e} = 1, Eq. (19) simplifies to Eq. (18).
Meanwhile, the secrecy outage probability for the spatial multiplexing system with MMSE equalization can be derived from Eqs. (13), (15) and (16) as follows: (20)${P}_{\text{out,MMSE}}\left(R\right)={\int}_{0}^{\infty}{f}_{{\gamma}_{\text{e,MMSE}}}\left(v\right)\text{d}v\frac{{e}^{\frac{{2}^{R}1}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}}}{{2}^{\left({M}_{\text{t}}1\right)R}}\sum _{m=0}^{{M}_{\text{e}}1}{g}_{m}\sum _{n=0}^{{M}_{\text{r}}1}{d}_{n}\phantom{\rule{10.00002pt}{0ex}}\times \left[\right.{\int}_{0}^{\infty}\frac{{\left({2}^{R}v+{2}^{R}1\right)}^{n}{e}^{\left(\frac{{2}^{R}}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\right)v}}{{\left(v+1\right)}^{2{M}_{\text{t}}1}}\times \left[\right.\frac{{v}^{m+1}}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}+\left({M}_{\text{t}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}m1\right){v}^{m}m{v}^{m1}\left]\right.\text{d}v\left]\right.=1\frac{{e}^{\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}}}{{2}^{\left({M}_{\text{t}}1\right)R}}\sum _{m=0}^{{M}_{\text{e}}1}{g}_{m}\sum _{n=0}^{{M}_{\text{r}}1}{d}_{n}\sum _{k=0}^{n}\left(\genfrac{}{}{0.0pt}{}{n}{k}\right){\left(1\right)}^{k}{2}^{\left(nk\right)R}\phantom{\rule{10.00002pt}{0ex}}\times \left[\right.\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\sum _{{l}_{1}=0}^{m+1}\left(\genfrac{}{}{0.0pt}{}{m+1}{{l}_{1}}\right){\left(1\right)}^{{l}_{1}}{\int}_{1}^{\infty}{v}^{m+nk{l}_{1}2{M}_{\text{t}}+2}{e}^{\left(\frac{{2}^{R}}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\right)v}\text{d}v\phantom{\rule{10.00002pt}{0ex}}+\left({M}_{\text{t}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}m1\right)\sum _{{l}_{2}=0}^{m}\left(\genfrac{}{}{0.0pt}{}{m}{{l}_{2}}\right){\left(1\right)}^{{l}_{2}}{\int}_{1}^{\infty}{v}^{m+nk{l}_{2}2{M}_{\text{t}}+1}{e}^{\left(\frac{{2}^{R}}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\right)v}\text{d}v\phantom{\rule{10.00002pt}{0ex}}+m\sum _{{l}_{3}=0}^{m1}\left(\genfrac{}{}{0.0pt}{}{m1}{{l}_{3}}\right){\left(1\right)}^{{l}_{3}}{\int}_{1}^{\infty}{v}^{m+nk{l}_{3}2{M}_{\text{t}}}{e}^{\left(\frac{{2}^{R}}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\right)v}\text{d}v\left]\right.=1\frac{{e}^{\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}}}{{2}^{\left({M}_{\text{t}}1\right)R}}\sum _{m=0}^{{M}_{\text{e}}1}{g}_{m}\sum _{n=0}^{{M}_{\text{r}}1}{d}_{n}\sum _{k=0}^{n}\left(\genfrac{}{}{0.0pt}{}{n}{k}\right){\left(1\right)}^{k}{2}^{\left(nk\right)R}\phantom{\rule{10.00002pt}{0ex}}\times \left[\right.\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\sum _{{l}_{1}=0}^{m+1}\left(\genfrac{}{}{0.0pt}{}{m+1}{{l}_{1}}\right)\frac{{\left(1\right)}^{{l}_{1}}\Gamma \left(m+nk{l}_{1}2{M}_{\text{t}}+3,\frac{{2}^{R}}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\right)}{{\left(\frac{{2}^{R}}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\right)}^{m+nk{l}_{1}2{M}_{\text{t}}+3}}\phantom{\rule{10.00002pt}{0ex}}+\left({M}_{\text{t}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}m1\right)\sum _{{l}_{2}=0}^{m}\left(\genfrac{}{}{0.0pt}{}{m}{{l}_{2}}\right)\frac{{\left(1\right)}^{{l}_{2}}\Gamma \phantom{\rule{0.3em}{0ex}}\left(m+nk{l}_{2}2{M}_{\text{t}}+2,\frac{{2}^{R}}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\right)}{{\left(\frac{{2}^{R}}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\right)}^{m+nk{l}_{2}2{M}_{\text{t}}+2}}\phantom{\rule{10.00002pt}{0ex}}+m\sum _{{l}_{3}=0}^{m1}\left(\genfrac{}{}{0.0pt}{}{m1}{{l}_{3}}\right)\frac{{\left(1\right)}^{{l}_{3}}\Gamma \left(m+nk{l}_{3}2{M}_{\text{t}}+1,\frac{{2}^{R}}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\right)}{{\left(\frac{{2}^{R}}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\right)}^{m+nk{l}_{3}2{M}_{\text{t}}+1}}\left]\right.$ where the second equality is obtained by changing the limits of integration and using (Gradshteyn & Ryzhik, 2000, Equation (1.111)) and (Papoulis & Pillai, 2002, Equation (418)), and the last equality is obtained by using (Gradshteyn & Ryzhik, 2000, Equation (3.381.3)). For the special case of M_{t} = M_{r} = M_{e} = 1, Eq. (20) reduces to Eq. (18).
Asymptotic Secrecy Outage Probability
In this section, we focus on deriving the asymptotic secrecy outage probability of the aforementioned systems as ${\stackrel{\u0304}{\gamma}}_{\text{r}}\to \infty $. This expression enables one to analyze the secrecy performance in the highSNR 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 versus ${\stackrel{\u0304}{\gamma}}_{\text{r}}$ 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.
Transmitreceive diversity system
First, we look for a firstorder expansion of Eq. (5), which will be immediate from a firstorder 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 firstorder Taylor expansion of det(S(x)) around x = 0 is (21)$det\left(\mathbf{S}\left(x\right)\right)=\left[\right.\prod _{p=1}^{L}\frac{\left(Kp\right)!{\left[\left(Lp\right)!\right]}^{2}}{\left({M}_{\text{t}}+{M}_{\text{r}}p\right)!}\left]\right.{x}^{{M}_{\text{t}}{M}_{\text{r}}}+o\left({x}^{{M}_{\text{t}}{M}_{\text{r}}}\right).$ Substituting Eq. (21) into Eq. (5) yields (22)${F}_{\lambda}\left(x\right)=\left[\right.\prod _{p=1}^{L}\frac{\left(Lp\right)!}{\left({M}_{\text{t}}+{M}_{\text{r}}p\right)!}\left]\right.{x}^{{M}_{\text{t}}{M}_{\text{r}}}+o\left({x}^{{M}_{\text{t}}{M}_{\text{r}}}\right).$ Using Eq. (22) and (Papoulis & Pillai, 2002, Example 51), the firstorder expansion of the CDF of γ_{r,TR} is given by (23)${F}_{{\gamma}_{\text{r,TR}}}\left(x\right)=\left[\right.\prod _{p=1}^{L}\frac{\left(Lp\right)!}{\left({M}_{\text{t}}+{M}_{\text{r}}p\right)!}\left]\right.{\left(\frac{x}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}\right)}^{{M}_{\text{t}}{M}_{\text{r}}}+o\left({\left(\frac{x}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}\right)}^{{M}_{\text{t}}{M}_{\text{r}}}\right).$ Using Eqs. (9), (16) and (23), and following the same procedure as used in Eq. (17), an asymptotic expression for P_{out,TR}(R) with ${\stackrel{\u0304}{\gamma}}_{\text{r}}\to \infty $ is obtained as (24)${P}_{\text{out,TR}}^{\infty}\left(R\right)={\left({A}_{\text{TR}}{\stackrel{\u0304}{\gamma}}_{\text{r}}\right)}^{{D}_{\text{TR}}}+o\left({\stackrel{\u0304}{\gamma}}_{\text{r}}^{{D}_{\text{TR}}}\right)$ where the secrecy diversity gain is (25)$D}_{\text{TR}}={M}_{\text{t}}{M}_{\text{r}$ and the secrecy array gain is (26)${A}_{\text{TR}}=\left[\right.\frac{1}{\left({M}_{\text{e}}1\right)!}\left[\right.\prod _{p=1}^{L}\frac{\left(Lp\right)!}{\left({M}_{\text{t}}+{M}_{\text{r}}p\right)!}\left]\right.\sum _{n=0}^{{M}_{\text{t}}{M}_{\text{r}}}\left(\genfrac{}{}{0.0pt}{}{{M}_{\text{t}}{M}_{\text{r}}}{n}\right)\times \left(n+{M}_{\text{e}}1\right)!{2}^{nR}{\left({2}^{R}1\right)}^{{M}_{\text{t}}{M}_{\text{r}}n}{\stackrel{\u0304}{\gamma}}_{\text{e}}^{n}{\left]\right.}^{\frac{1}{{M}_{\text{t}}{M}_{\text{r}}}}.$ 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 firstorder expansion of the CDF of γ_{r,ZF,i} can be derived as (27)${F}_{{\gamma}_{\text{r,ZF}}}\left(x\right)=\frac{{x}^{{M}_{\text{r}}{M}_{\text{t}}+1}}{\left({M}_{\text{r}}{M}_{\text{t}}+1\right)!{\stackrel{\u0304}{\gamma}}_{\text{r}}^{{M}_{\text{r}}{M}_{\text{t}}+1}}+o\left({\left(\frac{x}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}\right)}^{{M}_{\text{r}}{M}_{\text{t}}+1}\right).$ Using Eqs. (14), (16) and (27), and following the same procedure as used in Eq. (19), an asymptotic expression for P_{out,ZF}(R) with ${\stackrel{\u0304}{\gamma}}_{\text{r}}\to \infty $ is obtained as (28)${P}_{\text{out,ZF}}^{\infty}\left(R\right)={\left({A}_{\text{ZF}}{\stackrel{\u0304}{\gamma}}_{\text{r}}\right)}^{{D}_{\text{ZF}}}+o\left({\stackrel{\u0304}{\gamma}}_{\text{r}}^{{D}_{\text{ZF}}}\right)$ where (29)${D}_{\text{ZF}}={M}_{\text{r}}{M}_{\text{t}}+1$ and (30)${A}_{\text{ZF}}={\left[\frac{\sum _{n=0}^{{M}_{\text{r}}{M}_{\text{t}}+1}\left(\genfrac{}{}{0.0pt}{}{{M}_{\text{r}}{M}_{\text{t}}+1}{n}\right){2}^{nR}{\left({2}^{R}1\right)}^{{M}_{\text{r}}{M}_{\text{t}}+1n}\left(n+{M}_{\text{e}}{M}_{\text{t}}\right)!{\stackrel{\u0304}{\gamma}}_{\text{e}}^{n}}{\left({M}_{\text{r}}{M}_{\text{t}}+1\right)!\left({M}_{\text{e}}{M}_{\text{t}}\right)!}\right]}^{\frac{1}{{M}_{\text{r}}{M}_{\text{t}}+1}}.$
Adopting the same steps as for deriving the firstorder expansion of F_{γr,ZF}(x), we obtain (31)${F}_{{\gamma}_{\text{r,MMSE}}}\left(x\right)=\frac{{x}^{{M}_{\text{r}}}}{\left({M}_{\text{r}}{M}_{\text{t}}+1\right)!{\stackrel{\u0304}{\gamma}}_{\text{r}}^{{M}_{\text{r}}{M}_{\text{t}}+1}{\left(x+1\right)}^{{M}_{\text{t}}1}}+o\phantom{\rule{0.3em}{0ex}}\left({\left(\frac{x}{{\stackrel{\u0304}{\gamma}}_{\text{r}}}\right)}^{{M}_{\text{r}}{M}_{\text{t}}+1}\right).$ Using Eqs. (15), (16) and (31), and following the same procedure as used in Eq. (20), an asymptotic expression for P_{out,MMSE}(R) with ${\stackrel{\u0304}{\gamma}}_{\text{r}}\to \infty $ is obtained as (32)${P}_{\text{out,MMSE}}^{\infty}\left(R\right)={\left({A}_{\text{MMSE}}{\stackrel{\u0304}{\gamma}}_{\text{r}}\right)}^{{D}_{\text{MMSE}}}+o\left({\stackrel{\u0304}{\gamma}}_{\text{r}}^{{D}_{\text{MMSE}}}\right)$ where (33)${D}_{\text{MMSE}}={M}_{\text{r}}{M}_{\text{t}}+1$ and (34)${A}_{\text{MMSE}}=\left[\right.\frac{{e}^{\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}}{2}^{\left({M}_{\text{r}}{M}_{\text{t}}+1\right)R}}{\left({M}_{\text{r}}{M}_{\text{t}}+1\right)!}\sum _{m=0}^{{M}_{\text{e}}1}{g}_{m}\sum _{n=0}^{{M}_{\text{r}}}\left(\genfrac{}{}{0.0pt}{}{{M}_{\text{r}}}{n}\right)\frac{{\left(1\right)}^{n}{\stackrel{\u0304}{\gamma}}_{\text{e}}^{mn+{M}_{\text{r}}2{M}_{\text{t}}+1}}{{2}^{nR}}\phantom{\rule{10.00002pt}{0ex}}\times \left[\right.{\stackrel{\u0304}{\gamma}}_{\text{e}}\sum _{{k}_{1}=0}^{m+1}\left(\genfrac{}{}{0.0pt}{}{m+1}{{k}_{1}}\right){\left(\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\right)}^{{k}_{1}}\Gamma \phantom{\rule{0.3em}{0ex}}\left(mn{k}_{1}+{M}_{\text{r}}2{M}_{\text{t}}+3,\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\right)\phantom{\rule{10.00002pt}{0ex}}+{\stackrel{\u0304}{\gamma}}_{\text{e}}\left({M}_{\text{t}}+\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}m1\right)\sum _{{k}_{2}=0}^{m}\left(\genfrac{}{}{0.0pt}{}{m}{{k}_{2}}\right){\left(\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\right)}^{{k}_{2}}\Gamma \phantom{\rule{0.3em}{0ex}}\left(mn{k}_{2}+{M}_{\text{r}}2{M}_{\text{t}}+2,\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\right)\phantom{\rule{10.00002pt}{0ex}}m\sum _{{k}_{3}=0}^{m1}\left(\genfrac{}{}{0.0pt}{}{m1}{{k}_{3}}\right){\left(\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\right)}^{{k}_{3}}\Gamma \phantom{\rule{0.3em}{0ex}}\left(mn{k}_{3}+{M}_{\text{r}}2{M}_{\text{t}}+1,\frac{1}{{\stackrel{\u0304}{\gamma}}_{\text{e}}}\right)\left]\right.{\left]\right.}^{\frac{1}{{M}_{\text{r}}{M}_{\text{t}}+1}}.$ 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 closedform expressions derived above to generate the theoretical results, and adopt the Monte Carlo method to generate the simulation results. Remember that ${\stackrel{\u0304}{\gamma}}_{\text{r}}$ and ${\stackrel{\u0304}{\gamma}}_{\text{e}}$ are the average SNRs at the legitimate receiver and the passive eavesdropper, respectively. Unless otherwise indicated, the SNR ${\stackrel{\u0304}{\gamma}}_{\text{e}}$ 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 transmitreceive diversity system (computed with Eq. (17)) and its simulation counterpart (labeled with “simu.”) against ${\stackrel{\u0304}{\gamma}}_{\text{r}}$. As seen in the figure, the theoretical and simulation results match perfectly. For a given ${\stackrel{\u0304}{\gamma}}_{\text{r}}$, 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 morebalanced 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 given ${\stackrel{\u0304}{\gamma}}_{\text{r}}$, 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 transmitreceive diversity system derived in Eqs. (24)–(26) at a fixed ${\stackrel{\u0304}{\gamma}}_{\text{e}}$ (i.e., ${\stackrel{\u0304}{\gamma}}_{\text{e}}=10$ dB). The exact and asymptotic secrecy outage curves are labeled with “exact” and “asym.”, respectively. As ${\stackrel{\u0304}{\gamma}}_{\text{r}}$ 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, where we plot the simulated secrecy outage curves in the case of M_{r} − M_{t} = 0, M_{e} − M_{t} = 0 and that of M_{r} − M_{t} = 2, M_{e} − M_{t} = 4.
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 fixed ${\stackrel{\u0304}{\gamma}}_{\text{e}}$ (i.e., ${\stackrel{\u0304}{\gamma}}_{\text{e}}=10$ dB). As ${\stackrel{\u0304}{\gamma}}_{\text{r}}$ 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. (30) and (34).
Finally, it is interesting to compare the computational complexity of all three systems. To this end, we express such complexity in terms of the number of floatingpoint operations (flops), and the relevant calculations are summarized as follows:^{5} (1) the number of flops required to compute z_{r} (via power iteration (Golub & Van Loan, 2013, Section 7.3)), w_{t}, and z_{e} for the transmitreceive 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 transmitreceive diversity system has the highest computational complexity, even with N = 1.
System  Number of Flops 

TransmitReceive Diversity  $2{M}_{\text{t}}{M}_{\text{r}}^{2}+2{M}_{\text{t}}{M}_{\text{r}}+2{M}_{\text{t}}{M}_{\text{e}}+2{M}_{\text{t}}+\left(2N1\right){M}_{\text{r}}^{2}+2N{M}_{\text{r}}+2{M}_{\text{e}}$ 
Spatial Multiplexing with ZF  $2{M}_{\text{t}}^{2}+4{M}_{\text{t}}{M}_{\text{r}}+4{M}_{\text{t}}{M}_{\text{e}}{M}_{\text{r}}{M}_{\text{e}}+2$ 
Spatial Multiplexing with MMSE  $2{M}_{\text{t}}^{2}+4{M}_{\text{t}}{M}_{\text{r}}+4{M}_{\text{t}}{M}_{\text{e}}{M}_{\text{r}}{M}_{\text{e}}+4$ 
Conclusion
We have presented exact and asymptotic analysis of the secrecy outage probability of the transmitreceive diversity system and spatial multiplexing systems with ZF equalization and MMSE equalization in a Rayleighfading MIMO wiretap channel. This asymptotic analysis has shown that the transmitreceive 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}. Numerical results based on both theoretical analysis and simulations have demonstrated how M_{t}, M_{r}, and M_{e} affect the secrecy performance of such MIMO systems.