This paper presents the derivation of exact and asymptotic closed-form formulas of the secrecy outage probability (SOP) for space shift keying (SSK) systems over Rayleigh fading channels, especially in the presence of imperfect channel state information (CSI) at a legitimate receiver. Comprehensive analyses of the achievable secrecy performance limit for the SSK systems were performed by judiciously exploiting the derived SOP expressions, which enable us to obtain explicit and useful insights into the effect of channel estimation errors on the secrecy performance for the considered systems. The exactness of our theoretical analyses was validated numerically.

※ The user interface design of www.jsts.org has been recently revised and updated. Please contact inter@theieie.org for any inquiries regarding paper submission.

### Journal Search

- (School of Electronics Engineering, Kyungpook National University, Daegu, Korea seongah@knu.ac.kr)
- (Department of Information and Communication Engineering, Myongji University, Gyeonggi-do, Korea jkkang@mju.ac.kr )
- (Division of IT Convergence Engineering, Hansung University, Seoul, Korea hjlee@hansung.ac.kr)

## 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]}:

##### (1)

$f_{{\lambda _{r}}}\left(\lambda _{r}\right)=\frac{\lambda _{r}^{N_{r}-1}e^{-{\lambda _{r}}/{\rho _{r}}}}{\Gamma \left(N_{r}\right)\rho _{r}^{N_{r}}}$,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:

##### (2)

$F_{{\lambda _{r}}}\left(\lambda _{r}\right)=1-e^{\frac{\lambda _{r}}{\rho _{r}}}\left[\sum _{m=0}^{N_{r}-1}\left(\frac{1}{\rho _{r}}\right)^{m}\frac{\lambda _{r}^{m}}{m!}\right]=1-\frac{\Gamma \left(N_{r},\lambda _{r}/\rho _{r}\right)}{\Gamma \left(N_{r}\right)}$,where $\Gamma \left(\cdot ,\cdot \right)$ is the upper incomplete Gamma function ^{[18]}.

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

##### (3)

$C_{S}=\left\{\begin{array}{ll} C_{B}-C_{E}, & if\,\,\lambda _{B}>\lambda _{E}\\ 0 & \mathrm{o}.\mathrm{w}. \end{array}\right.$,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)

$\Pr \left(C_{S}>0\right)=\Pr \left(\lambda _{B}>\lambda _{E}\right)$ \begin{align*} &=\int _{0}^{\infty }\int _{0}^{\lambda _{B}}f_{{\lambda _{B}}}\left(\lambda _{B}\right)f_{{\lambda _{E}}}\left(\lambda _{E}\right)d\lambda _{E}d\lambda _{B} \\ &=\frac{1}{\rho _{B}\left(N_{B}-1\right)!}\left(\frac{1}{\rho _{B}}\right)^{{N_{B}}-1}\frac{1}{\Gamma \left(N_{E}\right)\rho _{E}^{N_{E}}} \end{align*} $\times \frac{\left(N_{E}-1\right)!}{\frac{1}{\rho _{E}}^{{N_{E}}}}\left[\left(N_{B}-1\right)!\left(\frac{1}{\rho _{B}}\right)^{-{N_{B}}}\right] \\ -\frac{1}{\rho _{B}\left(N_{B}-1\right)!}\left(\frac{1}{\rho _{B}}\right)^{{N_{B}}-1}\frac{1}{\Gamma \left(N_{E}\right)\rho _{E}^{N_{E}}} \\ \times \sum _{k=0}^{N_{E}-1}\frac{\left(N_{E}-1\right)!}{k!}\left(\frac{1}{\rho _{E}}\right)^{-\left({N_{E}}-k\right)} \\ \times \left[\left(k+N_{B}-1\right)!\left(\frac{1}{\rho _{E}}+\frac{1}{\rho _{B}}\right)^{-\left(k+N_{B}\right)}\right]$.## 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:

##### (5)

$$ \begin{aligned} P_{o}\left(R_{S}\right)=& \int_{0}^{\lambda_{B}} f_{\lambda_{E}}\left(\lambda_{E}\right) F_{\lambda_{B}}\left(2^{R_{S}}\left(1+\lambda_{E}\right)-1\right) d \lambda_{E} \\ =& \frac{1}{\Gamma\left(N_{E}\right) \rho_{E}^{N_{E}}} \int_{0}^{\infty} \lambda^{N_{E}-1} e^{-\frac{\lambda}{\rho_{E}}} d \lambda \\ &-\frac{1}{\Gamma\left(N_{E}\right) \rho_{E}^{N_{E}}} \int_{0}^{\infty N_{B}-1} \sum_{m=0}^{-1}\left(\frac{1}{\rho_{B}}\right)^{m} \frac{1}{m !} e^{-\frac{2^{R_{S}-1}}{\rho_{B}}} \lambda^{N_{E}-1} \\ & \times\left(2^{R_{S}}-1+2^{R_{S}} \lambda\right)^{m} e^{-\left(\frac{\lambda}{\rho_{E}}+\frac{2^{R_{S}}}{\rho_{B}} \lambda\right)} d \lambda \\ &(a) \\ =& \frac{1}{\Gamma\left(N_{E}\right) \rho_{E}^{N_{E}}}\left(N_{E}-1\right) !\left(\frac{1}{\rho_{E}}\right)^{-N_{E}} \\ &-\left(\frac{2^{R_{S}}-1}{2^{R_{S}} \rho_{E}}\right)^{N_{E}} e^{-\frac{2^{R_{S}}-1}{\rho_{B}}} \sum_{m=0}^{N_{B}-1}\left(\frac{2^{R_{S}}-1}{\rho_{B}}\right)^{m} \frac{1}{m !} \\ & \times U\left(N_{E}, N_{E}+m+1,\left(2^{R_{S}}-1\right)\left(\frac{1}{2^{R_{S}} \rho_{E}}+\frac{1}{\rho_{B}}\right)\right) \\ &(b) \\ =& 1-\left(\frac{2^{R_{S}}-1}{2^{R_{S}} \rho_{E}}\right)^{N_{E}} e^{-\frac{2^{R_{S}}-1}{\rho_{B}}} \sum_{m=0}^{N_{B}-1}\left(\frac{2^{R_{S}}-1}{\rho_{B}}\right)^{m} \frac{1}{m !} \\ &\times U\left(N_{E}, N_{E}+m+1,\left(2^{R_{S}}-1\right)\left(\frac{1}{2^{R_{S}} \rho_{E}}+\frac{1}{\rho_{B}}\right)\right), \end{aligned} $$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:

##### (6)

$$ \begin{aligned} F_{\lambda_{B}}\left(\lambda_{B}\right)^{\rho_{B}} & \rightarrow \infty \frac{\rho_{B}^{-N_{B}} \lambda_{B}^{N_{B}}}{N_{B} \Gamma\left(N_{B}\right)} \\ & \approx \end{aligned} $$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:

##### (7)

$$ P_{O}^{\infty}=\int_{0}^{\infty} \frac{\lambda^{N_{E}-1} e^{-\frac{\lambda_{E}}{\rho_{E}}}}{\Gamma\left(N_{E}\right) \rho_{E}^{N_{E}}}\left[\frac{\rho_{B}^{-N_{B}}\left(2^{R_{S}}\left(1+\lambda_{E}\right)-1\right)^{N_{B}}}{N_{B} \Gamma\left(N_{B}\right)}\right] d \lambda_{E} $$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.

### 5.1 Error Floor Analysis for High SENR

From the asymptotic SOP in (7), we consider a special case of $\rho _{E}=\alpha \rho _{B}$ for $0<\alpha <1$ of
non-zero secrecy capacity, where the SENR of Eve is linearly proportional to and less
than that of Bob. This can be an appropriate assumption in various secure environments,
such as military or internet of things (IoT) applications ^{[25]}. In this case, for $\rho _{B}\rightarrow \infty $, we have $\rho _{E}\rightarrow
\infty $, which leads to:

##### (8)

$$ \begin{aligned} P_{O}^{\mathrm{EF}} \rho_{E} & \rightarrow \infty\left(2^{R_{S}} \alpha\right)^{N_{B}} \frac{\Gamma\left(N_{B}+N_{E}\right)}{N_{B} \Gamma\left(N_{B}\right) \Gamma\left(N_{E}\right)} \\ & \approx \quad \end{aligned} $$This can be readily obtained by the asymptotic expansion of a confluent hypergeometric
function $U\left(\cdot ,\cdot ,\cdot \right)$ ^{[18]}. From (8), the asymptotic SOP (8) becomes the error floor, which is dependent on the target secrecy rate $R_{S}$ of
the main channel, the antenna numbers of both the legitimate receiver and the eavesdropper
(i.e., $N_{B}$ and $N_{E}$), and the SENR ratio parameter $\alpha $, regardless of
their SENR value.

### 5.2 Error Floor Analysis for High SNR

Similarly, the error floor phenomenon was explored in the high SENR regime $\rho _{B}\rightarrow \infty $ in (7) when the channel estimation errors $\sigma _{e_{B}}^{2}$ and $\sigma _{e_{E}}^{2}$ are independent of SNR $P_{T}/\sigma _{n}^{2}$. Therefore, we consider fixed $\sigma _{e_{B}}^{2}$ and $\sigma _{e_{E}}^{2}$ for the channel estimation errors for Bob and Eve, respectively, and then analyze the error floors of the derived asymptotic SOP.

With fixed channel estimation error variance s $\sigma _{e_{B}}^{2}$ and $\sigma _{e_{E}}^{2}$, as the SNR $P_{T}/\sigma _{n}^{2}$ increases to $\infty $, the asymptotic SOP in (7) can be written as:

##### (9)

$$ \begin{aligned} P_{O}^{E F} \frac{P_{T}{ }^{2}}{\sigma_{n}} & \rightarrow \infty \frac{\left(2^{R_{S}}-1\right)^{N_{B}+N_{E}}}{2^{R_{S} N_{E}} \Gamma\left(N_{B}\right) N_{B}}\left(\frac{\sigma_{e_{E}}^{2}}{1-\sigma_{e_{E}}^{2}}\right)^{N_{E}}\left(\frac{\sigma_{e_{B}}^{2}}{1-\sigma_{e_{B}}^{2}}\right)^{N_{B}} \\ & \approx \end{aligned} $$ $$ \times U\left(N_{E}, N_{E}+N_{B}+1, \frac{2^{R_{S}}-1}{2^{R_{S}}}+\frac{\sigma_{e_{E}}^{2}}{1-\sigma_{e_{E}}^{2}}\right) $$This is defined as the error floor of SOP. The error floor of SOP in (9) is determined by the number of antennas (i.e., $N_{B}$ and $N_{E}$) and the channel estimation error variance (i.e., $\sigma _{e_{B}}^{2}$ and $\sigma _{e_{E}}^{2}$) at a legitimate receiver and eavesdropper, as well as the target secrecy rate of the main channel (i.e., $R_{S}$).

## 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 $\boldsymbol{N}_{\boldsymbol{B}}$.

## 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.

### ACKNOWLEDGMENTS

This research was supported by the Kyungpook National University Research Fund of 2019.

### REFERENCES

## Author

Seongah Jeong received B.Sc., M.Sc., and Ph. D. degrees from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2010, 2012, and 2015, respectively. She is currently an assistant professor in the School of Electronics Engineering of Kyungpook National University, Daegu, Korea. From 2017 to 2019, she was a Senior Engineer at the Service Standards Lab of the Next-generation Communications Research Center of Samsung Research, Seoul, Korea, where she worked on global UWB/IoT/WiFi standardization for IEEE, the Open Connectivity Foundation (OCF), FiRa Consortium, Car Connectivity Consortium (CCC), etc. She also developed PHY/MAC core technologies for UWB digital key systems, Access, LBS, asset tracking, and D2D in collaboration with global partner companies on practical wireless communication issues. From 2013 to 2014, she was a visiting scholar at the New Jersey Institute of Technology, Newark, NJ, USA. From 2015 to 2016, she was a post-doctoral research fellow with the Information and Electronics Research Institute of KAIST. She was also a post-doctoral research fellow with the John A. Paulson School of Engineering and Applied Sciences at Harvard University, Cambridge, MA, USA, from 2016 to 2017. She has been recognized with awards, including the IEEE Vehicular Technology Society 2021 Jack Neubauer Memorial Award, the Silver Prize (2015) and the Bronze Prize (2014) in the Samsung Human Tech Thesis Award, the KAIST Research Excellence Award (2015), etc. Her research interests include signal processing and optimization for wireless localization and wireless communications, mobile cloud computing, and biology.

Jinkyu Kang is currently a faculty member of the Department of Information and Communication Engineering at Myongji University, Korea. He received a B.Sc. degree in electrical communications engineering in 2009, an M.Sc. degree in electrical engineering in 2011, and a Ph.D. degree in electrical engineering in 2015 from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea. From 2012 to 2013, he was a visiting scholar at the New Jersey Institute of Technology (NJIT), Newark, New Jersey, USA, and he was a post-doctoral research fellow with the Information and Electronics Research Institute at KAIST from 2015 to 2016 and the School of Engineering and Applied Sciences (SEAS), Harvard University, Cambridge, USA, from 2016 to 2017. From 2017 to 2020, he was a staff engineer at Samsung Research (Seoul, Korea), where he was involved in the development of 3GPP 5G NR standardization. His research interests include wireless communication, signal processing, and distributed computing.

Hoojin Lee received his B.S. degree from the School of Electrical Engineering, Seoul National University, Seoul, Korea, in 1997, and his M.S. and Ph.D. degrees in electrical and computer engineering from the University of Texas at Austin, Austin, Texas, USA, in 2002 and 2007, respectively. From 2008 to 2009, he worked as a systems and architecture engineer in the Algorithm and Standards Team, Cellular Products Group of Freescale Semiconductor, Inc., Austin, Texas, USA. Since 2009, he has been working in the Division of IT Convergence Engineering at Hansung University, Seoul, Korea, where he is currently a professor. His current research interests are in the areas of communication theory and physical-layer security of wireless communications.