1. Introduction

Due to the broadcast nature of wireless media, eavesdroppers within communication range can intercept confidential messages in a network. A wiretap channel model was introduced with secrecy capacity between a transmitter and its intended receiver ^[1]. In recent years, cryptographic protocols have been introduced to provide security in the upper layer under the assumption of an error-free link in the physical layer ^[2]. Thanks to the improvement of multiple-input multiple-output (MIMO) techniques for high-data-rate transmission ^[3], physical-layer security is an attractive alternative to resist various security attacks with multiple-antenna transmission for secure connectivity of a wireless random network ^[4].

Space shift keying (SSK) modulation has become an emerging topic in MIMO systems with low-error performance with a moderate number of antennas, avoidance of inter-channel interference (ICI), relaxed inter-antenna synchronization, and low complexity of transmitters and receivers ^[5-10]. In SSK systems, the index of transmit antennas is used to activate a single antenna among many antennas in order to deliver a message. Robustness of the spatial modulation against the channel estimation errors was introduced ^[6], which led to the active investigation of the impact of imperfect channel state information (CSI) on SSK error performance ^[7-13]. With only a few pilot symbols, SSK systems are known to achieve almost the same performance as that obtained with perfect channel knowledge, even though SSK has lower computational overhead and relaxed adaptive power management (APM) hardware requirements in comparison with conventional MIMO techniques, such as V-BLAST ^[14].

In this paper, we investigate the secrecy outage performance (SOP) of SSK systems when operating over Rayleigh fading channels in the presence of imperfect CSI at a legitimate receiver. To this end, we consider a MIMO multiple-antenna eavesdropper (MIMOME) wiretap channel model ^[15], where a passive eavesdropper (Eve) equipped with $N_{E}$ antennas attempts to intercept communications between an authorized transmitter (``Alice'') and receiver (``Bob'') equipped with $N_{A}$ and $N_{B}$ antennas, respectively. Both Bob and Eve estimate and decode the noisy encoded bits transferred over Rayleigh fading by exploiting the maximum likelihood (ML) detector.

The impact of imperfect channel knowledge on the SOP performance was examined by considering two types of pragmatic assumptions about the variances of channel estimation errors. The first is a special case where the signal-to-estimation-noise ratio (SENR) of Eve is linearly proportional to and less than that of Bob. Second, the channel estimation error variances of both Bob and Eve are assumed to be fixed and independent of the signal-to-noise ratio (SNR). Prior works on the SSK performance with channel imperfectness ^[6-11] do not consider the physical layer secrecy scenario in terms of SOP. However, we investigate the physical layer security performance of a MIMO wiretap channel achieved by SSK systems, especially in the presence of channel estimation error. Moreover, pragmatic closed-form formulas of the error floors were also evaluated, which provide comprehensive and explicit insights into the influence of imperfect CSI on the secrecy performance of SSK systems.

The rest of this paper is organized as follows. The next section describes the system model. Sections 3 and 4, show the derivation of the probability of non-zero secrecy capacity and the secrecy outage probability, respectively, and their asymptotic behaviors are provided in section 5. Section 6 includes performance evaluations of the proposed derivations, and we conclude this paper in the last section.

2. System Model

We consider a MIMOME wiretap system ^[15] consisting of a transmitter (Alice) with $N_{A}$ transmit antennas, a receiver (Bob) with $N_{B}$ receive antennas, and a passive eavesdropper (Eve) with $N_{E}$ receive antennas. The performance analysis for $N_{A}>2$ can be easily derived by applying the union bounding technique ^[16], so $N_{A}=2$ is adopted here without a loss of generality. In addition, we assume that there is no CSI feedback between Alice and Eve, which means that Eve attempts to overhear and extract a confidential message from Alice without any interference with the main channel (i.e., the Alice-Bob channel).

The SSK system is applied at Alice, where blocks of 1-bit data are encoded into the index of a single antenna, and then the corresponding antenna to the encoded index is switched on for data communication, while the remaining antennas are silent. The probabilities of transmitting the signals emitted at Alice's two transmit antennas are assumed to be equal. The noisy encoded bits conveyed over Rayleigh fading at Bob and Eve are estimated and decoded using the ML approach, which is referred to as antenna-index coded modulation. Both the main channel between Alice and Bob and Eve's channel between Alice and Eve are assumed to experience slow block fading with fading coefficients that are invariant during one fading block. Also, the block length is assumed to be long enough to allow for capacity-achieving codes within each block. The main channel and Eve's channel are assumed to have the same block length.

The $N_{B}\times 1$ received signal ${y}_{B}$ and the $N_{E}\times 1$ received signal ${y}_{E}$ with SSK modulation are defined as ${y}_{r}=\sqrt{P_{T}}{h}_{r,i}+{n}_{r}$, where $r\in \left\{B,E\right\}$, and $B$ and $E$ are parameters indicating Bob and Eve, respectively. $P_{T}$ is the transmit energy of Alice, and ${h}_{B,i}$ and ${h}_{E,i}$ are $N_{B}\times 1$ and $N_{E}\times 1$ channel vectors from the $i$th transmit antenna to the receive antennas of Bob and Eve for $i\in \left\{1,2\right\}$, which are modeled by Rayleigh fading with zero mean and unit variance $\sigma _{h}^{2}=1$. ${n}_{B}$ and ${n}_{E}$ are $N_{B}\times 1$ and $N_{E}\times 1$ noise vectors, respectively, which are assumed to have i.i.d. $CN\left(0,\sigma _{n}^{2}\right)$ entries. With imperfect CSI at Bob and Eve, without loss of generality, the estimated channels $\hat{{h}}_{B,i}$ and $\hat{{h}}_{E,i}~ $can be expressed as $\hat{{h}}_{r,i}={h}_{r,i}+{e}_{r,i}$ for $r\in \left\{B,E\right\}.$ Furthermore, $\hat{{h}}_{B,i}$ and $\hat{{h}}_{E,i}$ have complex Gaussian variables with zero mean and variance $\sigma _{\hat{h}_{B}}^{2}$ and $\sigma _{\hat{h}_{E}}^{2}$, respectively. ${e}_{B,i}$ and ${e}_{E,i}$ have estimation error with i.i.d. $CN\left(0,\sigma _{e_{B}}^{2}\right)$ and $CN\left(0,\sigma _{e_{E}}^{2}\right)$ entries, respectively.

3. Derivation of Probability of Non-zero Secrecy Capacity

The probability of non-zero secrecy capacity of an SSK system was derived in the presence of channel estimation errors. To this end, ML detection is adopted at Bob and Eve to detect the active transmit antenna index ^[17]. With ML detection, the active transmit antenna index at Bob and Eve can be achieved as $\hat{\mathrm{i}}_{r}=\underset{i=1,\ldots ,~ N_{A}}{\text{argmax}}\left\{D_{r,i}\right\}$, where the decision metric $D_{r,i}$ is defined as $D_{r,i}=\mathbf{\Re }\left\{\boldsymbol{y}_{r}^{\dagger }\sqrt{P_{T}\hat{\boldsymbol{h}}_{r,i}}\right\}-\frac{1}{2}P_{T}\left\| \hat{\boldsymbol{h}}_{r,i}\right\| _{F}^{2}$ for $r\in \left\{B,E\right\}$ ^[11], where $\dagger $ indicates the conjugate transpose. When the estimated channels $\hat{\boldsymbol{H}}_{r}=\left[\hat{\boldsymbol{h}}_{r,1}^{T}~ ~ \hat{\boldsymbol{h}}_{r,2}^{T}\right]^{T}$ at Bob and Eve for $r\in \left\{B,E\right\}$ are obtained, according to the decision rule, the conditional error probabilities of the incorrect index of the active transmit antenna at Bob and Eve are $P_{b}\left(\hat{\boldsymbol{H}}_{r}\right)=Q\left(\sqrt{\lambda }_{r}\right)$, where $r\in \left\{B,E\right\}$. The Q function $Q\left(\cdot \right)$ is defined as $Q\left(x\right)=\int _{0}^{\pi /2}\frac{\exp \left(-\frac{x^{2}}{2\sin ^{2}\theta }\right)}{\pi }d\theta $, and $\lambda _{r}=P_{T}\left\| \hat{\boldsymbol{h}}_{r,1}-~ \hat{\boldsymbol{h}}_{r,2}\right\| ^{2}/\left(2\left(P_{T}\sigma _{e_{r}}^{2}+\sigma _{n}^{2}\right)\right).$ The probability density function (pdf) of $\lambda _{r}$ for $r\in \left\{B,E\right\}$ in $P_{b}\left(\hat{\boldsymbol{H}}_{r}\right)$ can be written as ^[11,16]:

where $\Gamma \left(\cdot \right)$ is the Gamma function, and $\rho _{r}$ represents the SENR at Bob and Eve and is defined as $\rho _{r}=P_{T}\sigma _{\hat{h}_{r}}^{2}\left(P_{T}\sigma _{e_{r}}^{2}+\sigma _{n}^{2}\right)$. Based on (1), the cumulative distribution function (CDF) $F_{{\lambda _{r}}}\left(\lambda _{r}\right)=\int _{0}^{\lambda _{r}}f_{{\lambda _{r}}}\left(\lambda _{r}\right)d\lambda _{r}$ for the channel between Alice and Bob ($r=B$) or between Alice and Eve ($r=E$) can be attained as:

where $\Gamma \left(\cdot ,\cdot \right)$ is the upper incomplete Gamma function ^[18].

The achievable secrecy rate $C_{S}$ is expressed as ^[19]:

where $C_{B}$ and $C_{E}$ denote the capacities of the main channel between Alice and Bob and Eve's channel between Alice and Bob, respectively ^[20-22], which have been derived for an SSK system ^[23,24]. From (3), the exact closed-form expression for the probability of non-zero secrecy capacity can be derived as:

4. Derivation of Secrecy Outage Probability

The SOP was derived as the probability that the instantaneous secrecy capacity is less than a target secrecy rate $R_{S}$ (i.e., $P_{O}\left(R_{S}\right)=Pr(C_{S}<R_{S})).$ We have $\Pr \left\{C_{S}<R_{S}|\lambda _{B}<\lambda _{E}\right\}=1$, so the SOP $P_{O}\left(R_{S}\right)$ can be obtained as:

where $U\left(\cdot ,\cdot ,\cdot \right)$ is a confluent hypergeometric function of the second kind, (a) can be achieved by solving the integration ^[18], and (b) can be obtained from $\Gamma \left(N_{E}\right)=\left(N_{E}-1\right)!$. From the newly derived SOP in (5) and given the target secrecy rate $R_{S}$ of the system, we can expect the guaranteed secrecy level of the main channel in terms of SOP. The probability of non-zero secrecy capacity $\Pr (C_{S}>0)$ in (4) can be obtained by calculating $\Pr \left(C_{S}>0\right)=1-\Pr \left(C_{S}<0\right)=1-P_{O}\left(0\right).$ Thus, the probability of non-zero secrecy capacity can be regarded as a special case of the SOP.

5. Asymptotic Analysis

An asymptotic analysis was performed in the high-SENR regime with large enough $\rho _{B}$ (i.e., $\rho _{B}\rightarrow \infty $) for the SOP in (5) to provide the meaningful insights. To this end, the CDF $F_{{\lambda _{B}}}\left(\lambda _{B}\right)$ in (2) was simplified for $\rho _{B}\rightarrow \infty $ as:

With the expression in (5) and the CDF $F_{{\lambda _{B}}}\left(\lambda _{B}\right)$ in (6), by letting $\rho _{B}\rightarrow \infty $, the asymptotic SOP can be derived as:

where (a) can be obtained by solving the integration ^[18]. In the following, we consider two types of error floor obtained from the asymptotic SOP (7). The first is a special case of $\rho _{E}=\alpha \rho _{B}$ for $0<\alpha <1$ (i.e., the SENR of EVE is linearly proportional to and less than that of Bob). The second assumes fixed channel estimation error variances $\sigma _{e_{B}}^{2}$ and $\sigma _{e_{E}}^{2}$, which correspond to the case where the energy of pilot signals is independently set regardless of energy $P_{T}$ of the data signals.

6. Numerical Results

We evaluated the performance of the exact SOP in (5), its asymptotic SOP in (7), and error floors (8) and (9) for high SENR and high SNR in SSK systems. Unless stated otherwise, it is assumed that the target secrecy rate is $R_{S}=1$ with the number of antennas at Alice $N_{A}=2$. For the error floor evaluation in (8) with high SENR, the SENR ratio parameter is set to $\alpha =0.01$(i.e., $\rho _{E}=0.01\rho _{B}$), while we set $\sigma _{e_{B}}^{2}=0.001$ and $\sigma _{e_{E}}^{2}=0.1$ for the error floor evaluation in (9) for high SNR.

Fig. 1 and 2 show the SOP, corresponding asymptotic SOP, and error floor as function of SENR ($\rho _{B}$) and of SNR $P_{T}/\sigma _{n}^{2}$ with various numbers of antennas $N_{E}$ at Eve and $N_{B}$ at Bob, respectively (i.e., $N_{B}=2,4$ and $N_{E}=2,3,4,8$). It is obvious that all SOP curves obtained from the exact expression (5) asymptotically converge to the lines obtained from the asymptotic SOP expression (7), error floor expression (8) with the increase of SENR in Fig. 1, and expression (9) with the increase of SNR in Fig. 2. Furthermore, Fig. 1 and 2 show that the slopes of SOP curves predominantly depend on the number of antennas at the legitimate receiver rather than that of the eavesdropper. That is, as the number of legitimate receivers increases, the SOP dramatically decreases with the increase of SENR or SNR.

Fig. 3 shows the SOP, corresponding asymptotic SOP, and error floor as function of SNR $P_{T}/\sigma _{n}^{2}$ with the different channel estimation error performance of Eve, which has a similar tendency to that of SOP as a function of SENR, so it is omitted here. We consider $N_{B}=4$ and $N_{E}=2$. Fig. 3 shows that the SOP degrades as the channel estimation performance of the eavesdropper improves, which is limited to the case where the channel estimation performance of the legitimate receiver is worse than the eavesdropper in terms of SENR.

Fig. 1. Secrecy outage probability (SOP) as a function of $\boldsymbol{\rho }_{\boldsymbol{B}}$ with the different $\boldsymbol{N}_{\boldsymbol{E}}$ and $\boldsymbol{N}_{\boldsymbol{B}}$.

Fig. 2. Secrecy outage probability (SOP) as a function of SNR with the different $\boldsymbol{N}_{\boldsymbol{E}}$ and $\boldsymbol{N}_{\boldsymbol{B}}$.

Fig. 3. Secrecy outage probability (SOP) as a function of SNR with the different $~ \boldsymbol{\sigma }_{\boldsymbol{e}_{\boldsymbol{E}}}^{2}$.

5. Concluding Remarks

We have comprehensively studied the achievable performance of SSK systems with the imperfect CSI at a receiver by deriving the closed-form expression of the SOP and evaluating its asymptotic behavior. Furthermore, we have also provided explicit insights into the impact of channel estimation errors on the performance and optimization for the SSK systems through analytical calculation of the exact error floors of the corresponding SSK systems. Numerical results from several simulations were presented, which obviously verified the exactness of our theoretical analysis. Future work may include extensions with various transmission schemes and comparisons with experimental results.