Design of Low Probability Detection Signal with Application to Physical Layer Security

Abstract

In this work, we mainly focus on low probability detection (LPD) and low probability interception (LPI) wireless communication in cyber-physical systems. An LPD signal waveform based on multi-carrier modulation and an under-sampling method for signal detection is introduced. The application of the proposed LPD signal for physical layer security is discussed in a typical wireless-tap channel model, which consists of a transmitter (Alice), an intended receiver (Bob), and an eavesdropper (Eve). Since the under-sampling method at Bob’s end depends very sensitively on accurate sampling clock and channel state information (CSI), which can hardly be obtained by Eve, the security transmission is initialized as Bob transmits a pilot for Alice to perform channel sounding and clock synchronization by invoking the channel reciprocal principle. Then, Alice sends a multi-carrier information-bearing signal constructed according to Bob’s actual sampling clock and the CSI between the two. Consequently, Bob can coherently combine the sub-band signals after sampling, while Eve can only obtain a destructive combination. Finally, we derived the closed-form expressions of detection probability at Bob’s and Eve’s ends when the energy detector is employed. Simulation results show that the bit error rate (BER) at Alice’s end is gradually decreased with the increase in the signal-to-noise ratio (SNR) in both the AWGN and fading channels. Meanwhile, the BER at Eve’s end is always unacceptably high no matter how the SNR changes.

1. Introduction

Cyber-physical systems (CPSs) are networked systems that integrate computation, communication, and control elements. The principal goal of CPSs is to monitor and (if necessary) change the behavior of a physical process to ensure that it functions correctly, reliably, and efficiently. Nowadays, it has been applied in various domains, such as smart grids, health management, vehicular management, and military applications [1]. As CPSs advance rapidly in the degree of informatization and intelligence, their security issues have attracted both scholarly and industrial attention. Security issues of CPSs cover various aspects, including sensing security, computing security, communication security, and control security [2,3]. For the CPSs that are networked in nature, information sharing and interactions should be built on secure and reliable links among various terminals. As a result, communication security is crucial to CPSs [4,5]. Due to the broadcast nature of radio propagation, secure wireless transmission is a challenge. Malicious attacks on communication systems in CPS are classified as passive attacks and active attacks. Passive attacks are those where the attacker listens to network traffic in order to gain access to sensitive information. Yulong Zou studies the intercept behavior of an industrial wireless sensor network, and propose an optimal sensor scheduling scheme aiming at maximizing the secrecy capacity of wireless transmissions from sensors to the sink [6]. In this paper, we develop a practical countermeasure for passive attacks and propose a physical layer security communication scheme for CPS applications.

A well-designed secure wireless link should have LPD and LPI properties with respect to illegal users [7,8]. The concept of perfect secrecy was first introduced in Shannon’s fundamental paper [9]. He also proposed that security of communication could be guaranteed only when the transmitter and receiver have a certain degree of cooperation, and perfect secrecy could be achieved if a one-time pad protocol were employed. Traditional encryption techniques are based on the complexity of mathematical tasks, such as the computation of discrete logarithms in large finite fields. With the rapid development of computer hardware and computing technologies, such as distributed computing and cloud computing, the security of traditional encryption techniques has become questionable [10]. Quantum communication can provide almost perfect security through the use of quantum laws to detect any possible information leak [11]. However, its application to wireless and mobile communications is confined because the line of sight for the transmission of optical quantum is not always available, particularly in urban areas crowded by large buildings. The classical spread spectrum communication systems have good LPD, and LPI characteristics and are widely used. However, the random and noise-like properties of pseudo-noise spreading sequences are usually deterministic and periodical in actual systems. With the rapid development of blind signal detection techniques [12], the spreading sequences may be cracked by illegal users. Then, the traditional spread spectrum techniques are also not as secure as expected.

Physical layer security is to develop a secure transmission that exploits the physical properties of transceivers without relying on source encryption [13]. Wyner introduced the concept of secrecy capacity over wire-tap channels [14]. In Wyner’s model, the wire-tap channel is a degraded version of the main channel; thus, the eavesdropper can only receive a noisy version of the signal received at the intended receiver. Wyner’s work was extended to single input multiple output (SIMO) systems in the presence of one eavesdropper [15]. Hero proposed an information theoretical framework to investigate information security in wireless multiple-input multiple-output links [16]. Another important line of research is the design of a practical system to achieve near-optimal physical layer security performance [17]. Zheng proposed a low-complexity polar-coded cooperative jamming scheme for the general two-way wire-tap channel, without any constraint on channel symmetry or degradation [18,19,20,21,22]. The research mentioned above is unexceptionally confined to the information-theoretic perspective, which only focuses on the LPI performance. Therefore, the main contribution of our work is to design an LPD signal waveform and investigate its application in physical layer security.

Motivated by achieving an LPD signal waveform, we previously proposed an under-sampling spectrum-sparse signal based on active aliasing [23]. In this work, we extend our earlier work to a more practical scenario. Application of the LPD signal for physical layer security is investigated, and a typical wire-tap channel model with three users, namely, the transmitter (Alice), the intended receiver (Bob), and the eavesdropper (Eve), is considered. Since the under-sampling method may be effective only when the sub-band signals are accurately aligned after the sampling process, Alice can shift the central frequencies of the transmitted sub-band signals according to the clock offset between Alice and Bob, to make sure that Bob can collect the signal power on all sub-carriers coherently. Furthermore, a precoding technique based on CSI can be employed to maximize Bob’s SNR at the sampling stage. The sampling clock frequency offset and CSI between Alice and Bob are treated as security keys which can be determined at Alice’s end according to the reciprocal principle. Meanwhile, Alice and Eve do not have a negotiation of compensation for the sampling rate and CSI; Eve can only use incoherent demodulation techniques. Finally, the LPD and LPI performance of the proposed scheme is evaluated by the detection probability of the received signal and BER, respectively.

The rest of this paper is organized as follows. Section 2 presents the construction of the LPD signal waveform, the principle of the signal detection method. Section 3 presents the application of the designed LPD signal for physical layer security. A practical secure transmission scheme based on channel reciprocity is proposed. Section 4 analyses the LPD performance of the designed signal in the Wire-tap channel. Section 5 investigates the signal and information security performance in terms of detection probability and BER at both Bob’s and Eve’s ends by simulations. Finally, the conclusions are drawn in Section 6.

2. LPD Signal Design and Detection Method

2.1. LPD Signal Waveform Design

The basic strategy of LPD signal design is to reduce the level of radio frequency energy; the DSSS signal is an example. In this section, an LPD signal waveform based on multi-carrier modulation is designed. The differences between our design and the traditional multi-carrier modulation method lie in the following aspects: signal structure and receiving method. In our design, signals modulated by the sub-carriers are the same, and the sub carriers should be equally spaced. Furthermore, under-sampling method based on active aliasing is employed for signal detection. The designed LPD signal can be expressed as

$$\begin{array}{c}\hfill x\left(t\right)=\sum _{k=L}^{L+N-1}s\left(t\right)·{\alpha }_{k}exp(k·j{\omega }_{c}t)\end{array}$$

(1)

where the scaled factor ${\alpha }_k$ satisfied power constraint as ${\sum }_{k=1}^N|{\alpha }_k{|}^2=1$, N implies the total number of sub-carriers. Thus, the mean power of signal $x\left(t\right)$ is equivalent to that of signal $s\left(t\right)$, and L is the number of null subcarriers from the zero frequency to the first signal carrier. $s\left(t\right)$ is the original modulated signal with bandwidth ${\omega }_B$. The carrier spacing can be given by ${\omega }_c=D·{\omega }_B$, where D is the ratio between the carrier spacing and baseband width of signal $s\left(t\right)$. The parameter D should be no less than 2, or aliasing may occur between adjacent channels. Moreover, artificial noise can be added over the gaps among useful sub-band signals to enhance the covertness of the transmitted signal. In such a case, D should be determined cautiously to avoid aliasing between artificial noise and useful signals.

The comparison diagram of the spectrum structure between the modulated signal $s\left(t\right)$ and the proposed LPD signal $x\left(t\right)$ is shown in Figure 1. The bandwidth of $x\left(t\right)$ is N times of $s\left(t\right)$ while the power is consistent. As a result, the power spectrum density of signal $x\left(t\right)$ will be significantly reduced, which may be even lower than the background noise provided if N is large enough. Furthermore, $x\left(t\right)$ also performs sparsity in the frequency domain when ${\omega }_c\gg {\omega }_B$. These two characteristics are similar to that of direct sequence spread spectrum and frequency hop** signals.

2.2. Principle of Under-Sampling Method for Intended Receiver

As previously mentioned, the proposed LPD signal has a low power spectrum and is sparse in the frequency domain. Therefore, detecting the LPD signal at the intended receiver becomes a problem. In this part, the under-sampling method based on active aliasing is presented. The sampling rate is determined by the subcarrier spacing, and the sampling and combination for the proposed LPD signal can be finished simultaneously. For simplicity, the principle of the sampling process is explained in frequency domain. The complex-valued signal at the receiver can be given by

$$\begin{array}{c}\hfill r\left(t\right)=\sum _{k=L}^{L+N-1}{h}_{k}s\left(t\right)·{\alpha }_{k}exp(k·j{\omega }_{c}t)+w\left(t\right)\end{array}$$

(2)

where $h_k$ is the channel coefficient over the $kth$ sub-channel determined by the channel environment [24]. For the additive white Gaussian noise (AWGN) channel, channel coefficient $h_k$ is considered to be 1 for all k. For the fading channel, the channel coefficients can be given by $h_k=\left|h_k\right|exp\left(j{\phi }_k\right)$, which means the signal transmitted over the $kth$ sub-channel is scaled by the attenuation factor $\left|h_k\right|$ and phase-shifted by ${\phi }_k$. In this work, $\left|h_k\right|$ is subjected to Rayleigh distribution and ${\phi }_k$ is subjected to uniform distribution. $w\left(t\right)$ is the independent complex additive noise with power spectrum density $N_0$. The sampling process can be modeled as a pulse modulation process, and the sampling pulse is a periodic ideal pulse sequence given by $p\left(t\right)={\sum }_{n=-\infty }^{+\infty }\delta (t-n{T}_s)$, where $\delta \left(t\right)$ is the unit impulse function. The sampling frequency can be calculated by $f_s=1/\phantom{1T_s}{T}_s$.

The frequency representation of the proposed sampling process is illustrated in Figure 2. As shown in Figure 2a, signal $X\left(j\omega \right)$ consists of N sub-band signals $S_k\left(j\omega \right)$ with sub-band spacing ${\omega }_c$, and the total bandwidth of $X\left(j\omega \right)$ is $N{\omega }_c$. The frequency domain representation of the sampling function is illustrated in Figure 2b. The spectrum of the sampled signal can be represented as a convolution of $X\left(j\omega \right)$ and $P\left(j\omega \right)$. For each sub-band signal $S_k\left(j\omega \right)$, a replica of $S_k\left(j\omega \right)$ remains at each integer multiple of ${\omega }_s$. If the sampling rate is chosen as $f_s=f_c$, replicas of sub-band signals $S_k\left(j\omega \right)$ may be aligned and added coherently, as shown in Figure 2c. The mean power of the sampled signal increases by N times. Consequently, $S\left(j\omega \right)$ can be recovered from the sampled signal with an ideal low-pass filter. Otherwise, these sub-band signals would not be aligned as shown in Figure 2d if $f_s\ne f_c$, aliasing between adjacent sub-band signals can hardly be eliminated.

2.3. Practical Receiver Design

As illustrated in the last section, the feasibility of the under-sampling method has been proved. However, the proposed LPD signal waveform does not exhibit a constant envelope. The sampling phase plays an important role in the sampling process. Here, a practical receiver-based on a multiphase clock [25,26] is presented. The block diagram of the receiver is shown in Figure 3. At the front end of the receiver, the pass band of the analog band pass filter (BPF) is $[(L-1/2){\omega }_c,(L+N-1/2){\omega }_c]$, and the bandwidth of the pass band is $N{\omega }_c$. Then, frequency contents out of the pass band will be filtered out by the analog BPF. The SNR of signal $y\left(t\right)$ can be given by $SN{R}_y=P_s/N{N}_0{\omega }_c$, where $P_s$ denotes the average transmit power. Thereafter, the multiphase clock, which can produce several sampling clocks with the same frequency but different phases, are employed. They can be modeled as $p_m\left(t\right)={\sum }_{n=-\infty }^{+\infty }\delta (t-n{T}_s-m\Delta T_s)$ where $m=0,1,2,\cdots ,M-1$ and $\Delta =1/\phantom{1M}M$. Thus, a total of M sampled signals can be obtained. Comparing the mean power of these sampled signals, we can select the sampled signal with the maximum mean power as input for the LPF. The LPF is considered to be an ideal LPF with cut-off frequency ${\omega }_B$. It’s important to note that artificial noise (if it exists) will be filtered out by the LPF.

For the noise component, $w\left(t\right)$ can be written by summation of N sub-band noise elements as $w\left(t\right)={\sum }_{k=0}^{N-1}{w}_k\left(t\right)exp[(k+L)·j{\omega }_{c}t]$, where $w_k\left(t\right)$ is an independent zero-mean band-limited AWGN with bandwidth ${\omega }_c$ and power spectrum density $N_0$. After sampling, these sub-band noises are added incoherently, and the power spectrum density becomes $N{N}_0$. Following that, the SNR of signal $\tilde{s}\left[n\right]$ can be given by

$$\begin{array}{c}\hfill SN{R}_d=\frac{E\left[{\tilde{s}}^2\left[n\right]\right]}{N{N}_0{\omega }_B}=\frac{{\left|{\displaystyle \sum _{k=L}^{L+N-1}}\sqrt{1/N}·exp(j2\Delta k\pi )\right|}^2{P}_s}{N{N}_0{\omega }_B}\end{array}$$

(3)

and then the receiving gain can be achieved as

$$\begin{array}{c}\hfill \eta =\frac{SN{R}_d}{SN{R}_y}=D·{\left|\sum _{k=L}^{L+N-1}\sqrt{1/N}exp(j2\Delta k\pi )\right|}^2\end{array}$$

(4)

As a result, the maximum receiving gain becomes $ND$ if the sampling phase is synchronized perfectly when $\Delta =0$. Followed by the LPF, signal $\tilde{s}\left[n\right]$ can be demodulated in traditional ways.

2.4. Complexity Analysis of Receiver

In this section, the complexity of the proposed receiver is investigated. For a spectrum sparse signal with bandwidth $ND{f}_B$, the wideband bandpass filter is used for signal extraction. Different from the traditional receiver, the multiphase clock should be employed to obtain L-sampled copies. The sampled signal, which has the highest power, is chosen for processing in the following steps. Therefore, a total of M analog to digital converters(ADCs) is needed. Assuming the power of the sampled signal is calculated over Q samples, the selection combining step consumes $QM$ times multiplier, $(Q-1)M$ times add operation, and $lo{g}_2\left(L\right)$ times comparison operation.

There are also two other possible architectures of receivers for the designed LPD signal. The first receiver architecture uses parallel demodulators for each subcarrier and post-detection combining to recover the signal $s\left(t\right)$. Each demodulator needs a narrow band filter and ADC. The complexity and power consumption of the receiver will grow in direct proportion to the subcarrier number N. The second receiver architecture uses direct baseband sampling or radio frequency bandpass sampling method. The sampling rate should be at least twice the bandwidth of the LPD signal as $2ND{f}_B$ that performance requirements for ADCs will be ultra high. As mentioned above, the proposed under-sampling detection method has lower implementation complexity and hardware requirements.

3. Design of Physical Layer Security Communication System Using Proposed LPD Signal

The analyses in Section 2 show that the designed LPD signal can be exactly detected if and only if the sampling rate is synchronized perfectly. Otherwise, a different sampling rate may lead to a destructive combination after sampling. Then, the intrinsic sampling clock offset between the transmitter and the receiver can be used for secure transmission. In this section, the application of the designed LPD signal for physical layer security is discussed.

3.1. Wireless-Tap Channel Model

In this work, the typical wire-tap channel models consisting of Alice, Bob, and Eve are considered. The secure transmission model is shown in Figure 4, and details of the transmission protocol are presented as follows:

• Bob sends a transmission request signal to Alice, followed by pilot signals in the same frequency that Alice is going to use for secure data transmission.
• Alice estimates the sampling clock offset and CSI via the pilot signals.
• According to the sampling clock offset between Alice and Bob, Alice shifts the central frequencies of sub-band signals to ensure that they are aligned at Bob’s side after sampling.
• As the CSI of the channel from Bob to Alice has become known, the CSI from Alice to Bob can also be informed according to the channel reciprocal principle. Then, a precoding scheme is employed to enhance both capacity and security.
• Alice securely transmits the modified LPD signal to Bob.

Details of the sampling clock compensation and precoding scheme for security enhancement are presented in this section. These two methods exploit the physical properties of the sampling clock and channel characteristics between Alice and Bob, respectively. For simplification, the sampling phase offset is considered to be $\Delta =0$ in what follows unless stated otherwise.

3.2. Sampling Clock Offset Compensation

The sampling clock offset for the same frequency ${\omega }_c$ between Alice and Bob is defined as ${\kappa }_{\omega }={\omega }_{B,c}-{\omega }_{A,c}$, where ${\omega }_{A,c}$ and ${\omega }_{B,c}$ indicate the actual clock frequency of Alice and Bob, respectively. The sampling clock offset can be estimated nearly perfectly only if the SNR is sufficiently high or the number of pilot symbols is sufficiently large. According to the secure transmission protocol, ${\kappa }_{\omega }$ can be estimated in step 2. Then, the LPD signal is modified as

$$\begin{array}{c}\hfill x_s\left(t\right)=\sum _{k=L}^{L+N-1}s\left(t\right)·{\alpha }_{k}exp\left[k·j({\omega }_{A,c}+{\kappa }_{\omega })t\right]\end{array}$$

(5)

The central frequencies of sub-band signals are shifted according to ${\kappa }_{\omega }$, which is considered to be a shared key between Alice and Bob. As the sampling clock offset has been compensated at the transmitter, the sampling clock synchronization between Alice and Bob would be realized. When Bob sampled the received signal with sampling clock ${\omega }_{B,c}$, sub-band signals in the transmitted signal would be aligned naturally, as shown in Figure 3. Then, modulated signal $s\left(t\right)$ may be recovered. Taking the weighted factor ${\alpha }_k$ and the channel coefficients $h_k$ into account, the SNR of sampled signal $\tilde{s}\left[n\right]$ at Bob is given by

$$\begin{array}{c}\hfill SN{R}_{B,d}=\frac{{\left|{\displaystyle \sum _{k=L}^{L+N-1}}{\alpha }_k{h}_{k}exp(j2\Delta k\pi )\right|}^2·P_s}{N{N}_0{\omega }_B}\end{array}$$

(6)

For Eve, the sampling frequency can hardly be the same as that of Bob. Although the receiving method is known to Eve, sub-band signals cannot always be aligned at the baseband after sampling. The spectrum of the sampled signal at Eve would be the same as that in Figure 3. The sampled signal is a summation of sub-band signals with different carrier frequency offsets, which can hardly be eliminated. Such a sampled signal cannot be used for demodulation, and interception by Eve cannot be realized.

3.3. Precoding Scheme for Fading Channel

According to the proposed secure transmission protocol, the channel fading coefficients from Bob to Alice can be estimated by Alice. Then, the channel coefficient from Alice to Bob can also be known based on the channel reciprocal principle. Therefore, a precoding scheme that exploits the channel characteristics could be employed to improve the receiving gain at Bob. Meanwhile, the precoding scheme can also optimize the power allocation of sub-band signals. As stated, the SNR of the sampled signal at Bob is given by

$$\begin{array}{c}\hfill SN{R}_{B,d}=\frac{{\left|{\displaystyle \sum _{k=L}^{L+N-1}}{\alpha }_k{h}_k\right|}^2·P_s}{N{N}_0{\omega }_B}\end{array}$$

(7)

when $\Delta =0$. Weighted factor ${\alpha }_k$ and channel fading coefficient $h_k$ can be written as $1\times N$ vectors by $\mathbf{\alpha }=[{\alpha }_L,{\alpha }_{L+1},\cdots ,{\alpha }_{L+N-1}]$ and $\mathbf{H}=[h_L,h_{L+1},\cdots ,h_{L+N-1}]$. The Cauchy–Schwarz inequality [27] states that for all vectors $\mathbf{\alpha }$ and $\mathbf{H}$ of an inner product space, the following equation holds true:

$$\begin{array}{c}\hfill |<\mathbf{\alpha },\mathbf{H}>{|}^2\le {||\mathbf{\alpha }||}^2·||\mathbf{H}|{|}^2\end{array}$$

(8)

where $<\mathbf{\alpha },\mathbf{H}>$ denotes the inner product of vectors $\mathbf{\alpha }$ and $\mathbf{H}$, and the notion $\left|\right|·\left|\right|$ denotes the Euclidean norm. Moreover, the equality holds only when $\mathbf{\alpha }$ and $\mathbf{H}$ are linearly dependent. It can be written by $\mathbf{\alpha }=\lambda {\mathbf{H}}^{*}$, where $\lambda$ is a nonzero constant. The superscript ∗ denotes a conjugate operation. In addition, the weighted factor is constrained by ${\sum }_{k=L}^{L+N-1}|{\alpha }_k{|}^2=1$. In order to make the equality in Equation (8) hold true, the weighted factor ${\alpha }_k$ should be given by ${\alpha }_k=h_k^{*}/\left|\right|\mathbf{H}\left|\right|$. Then, the SNR of the sampled signal at Bob can be given by $SN{R}_{B,d}={\left|\right|\mathbf{H}\left|\right|}^2·P_s/\left(N{N}_0{\omega }_B\right)$. We can conclude that the optimal power allocation strategy for the frequency selective fading channel is to make the SNR over each sub-band identical.

Receiving gain $\eta$ versus sampling phase offset $\Delta$ is shown in Figure 5. We assumed that channel coefficients for sub-band signals are independent, identically distributed, and subject to Rayleigh distribution. Let ${\widehat{h}}_k$ denote the estimates of $h_k$ that can be written by ${\widehat{h}}_k=h_k+h_k^e$. Two different scenarios are explored: (1) perfect CSI, the channel coefficients are perfectly known as $h_k^e=0$; (2) imperfect CSI, $h_k^e$ is supposed to be a Gaussian random variable with zero mean, and the estimation error is defined by $\rho =E\{|h_k^e{|}^2/\phantom{|h_k^e{|}^2|h_k{|}^2}|h_k{|}^2\}$.

The simulation results show that the sampling phase offset plays an important role in the proposed scheme. It reveals that the accuracy requirement for sampling phase offset is higher with the increase in sub-carrier number N. In addition, an estimation error of CSI may result in inaccurate precoding on Alice’s side. It may lead to a performance loss of receiving gain on Bob’s side, but will not influence the effectiveness of the under-sampling method.

4. Performance Evaluation

In this section, we will investigate the signal security performance in terms of probability of detection at Bob’s and Eve’s ends. It is assumed that Bob uses the proposed under-sampling method, while Eve can only use the energy detection method because of the sampling clock offset and channel differences.

4.1. Energy Detection Method

The signal detection problem can be modeled as a binary hypothetical testing problem with hypotheses ${\mathcal{H}}_0$ and ${\mathcal{H}}_1$ defined as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathcal{H}}_0:r=w\hfill \\ {\mathcal{H}}_1:r=x+w\hfill \end{array}\right.\end{array}$$

(9)

where ${\mathcal{H}}_0$ represents the null hypothesis, and ${\mathcal{H}}_1$ represents the alternative hypothesis that a useful signal exists. The energy of the received signal is calculated in a bandwidth of W Hz over a period of $T_{\mathrm{int}}$. Users are to detect whether ${\mathcal{H}}_0$ or ${\mathcal{H}}_1$ is true based on the test statistic V.

The performance of the ED method is always evaluated by two probabilities, $P_d$ and $P_{fa}$. $P_d$ implies the probability of detection that ${\mathcal{H}}_1$ is accepted when ${\mathcal{H}}_1$ is true, while $P_{fa}$ is the false alarm probability that ${\mathcal{H}}_1$ is assumed when ${\mathcal{H}}_0$ is true. The probability density function of normalized decision statistic $Y=2V/\phantom{2V{N}_0}{N}_0$ has a central chi-square distribution with $v=2T_{\mathrm{int}}W$ degrees of freedom when ${\mathcal{H}}_0$ is true. It can be written by

$$\begin{array}{c}\hfill P_{{\mathcal{H}}_0}\left(Y\right)=\frac{1}{2^{v/\phantom{v2}2}\Gamma \left(v/\phantom{v2}2\right)}{y}^{(v-2)/\phantom{(v-2)2}2}{e}^{-Y/\phantom{-Y2}2}\end{array}$$

(10)

where $\Gamma \left(u\right)$ is Gamma function defined by $\Gamma \left(u\right)={\int }_0^{\infty }{t}^{u-1}exp(-t)dt$.

Meanwhile, the decision statistic obeys a non-central chi-square distribution with v degrees of freedom and non-central parameter $\lambda =2E/\phantom{2E{N}_0}{N}_0$ when ${\mathcal{H}}_1$ is true. The E implies the signal energy in the time period $T_{\mathrm{int}}$. The PDF can be written by

$$\begin{array}{c}\hfill P_{{\mathcal{H}}_1}\left(Y\right)=\frac{1}{2}{\left(\frac{Y}{\lambda }\right)}^{(v-2)/\phantom{(v-2)4}4}{e}^{-(Y+\lambda )/\phantom{-(Y+\lambda )2}2}{I}_{(v-2)/\phantom{(v-2)2}2}\left(\sqrt{Y\lambda }\right)\end{array}$$

(11)

where $I_n\left(u\right)$ is the Bessel function of the first kind of order n. Therefore, the performance of ED can be described by

$$\begin{array}{c}\hfill P_{fa}={\int }_{2V_T/\phantom{2V_T{N}_0}{N}_0}^{\infty }{P}_{{\mathcal{H}}_0}\left(Y\right)dY\end{array}$$

(12)

and

$$\begin{array}{c}\hfill P_d={\int }_{2V_T/\phantom{2V_T{N}_0}{N}_0}^{\infty }{P}_{{\mathcal{H}}_1}\left(Y\right)dY\end{array}$$

(13)

where $V_T$ denotes the decision threshold.

According to the central limit theorem, $P_{{\mathcal{H}}_0}\left(Y\right)$ and $P_{{\mathcal{H}}_1}\left(Y\right)$ will converge to a Gaussian distribution as v goes to infinity. The approximated PDF can be written by

$$\begin{array}{c}\hfill P_{{\mathcal{H}}_0}\left(Y\right)\approx \frac{1}{\sqrt{2\pi }{\sigma }_w}{e}^{-\frac{{(Y-{\mu }_w)}^2}{2{\sigma }_w^2}}\end{array}$$

(14)

$$\begin{array}{c}\hfill P_{{\mathcal{H}}_1}\left(Y\right)\approx \frac{1}{\sqrt{2\pi }{\sigma }_{sw}}{e}^{-\frac{{(Y-{\mu }_{sw})}^2}{2{\sigma }_{sw}^2}}\end{array}$$

(15)

where ${\mu }_w=2T_{\mathrm{int}}W$, ${\sigma }_w^2=4T_{\mathrm{int}}W$, ${\mu }_{sw}=2T_{\mathrm{int}}W+2E/\phantom{E{N}_0}{N}_0$ and ${\sigma }_{sw}^2=4T_{\mathrm{int}}W+8E/\phantom{E{N}_0}{N}_0$.

4.2. Detection Performance at Bob’s and Eve’s Ends

In order to verify the signal security of the designed waveform, detection performance at Bob’s and Eve’s ends will be analyzed in this section. It is assumed that both Bob and Eve use the ED method. However, the under-sampling method was employed at Bob’s end owing to the negotiation with Alice, and the sampling clock offset and CSI can be perfectly known. Moreover, the constant false alarm rate algorithm is applied.

From Equations (14) and (15), we can conclude that

$$\begin{array}{cc}\hfill P_{fa}& =\frac{1}{\sqrt{2\pi }{\sigma }_w}{\int }_{\psi }^{\infty }exp\left(-\frac{{(Y-{\mu }_w)}^2}{2{\sigma }_w^2}\right)dY\hfill \\ \hfill & =Q\left(\frac{\psi -{\mu }_w}{{\sigma }_w}\right)\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill P_d& =\frac{1}{\sqrt{2\pi }{\sigma }_{sw}}{\int }_{\psi }^{\infty }exp\left(-\frac{{(Y-{\mu }_{sw})}^2}{2{\sigma }_{sw}^2}\right)dY\hfill \\ \hfill & =Q\left(\frac{\psi -{\mu }_{sw}}{{\sigma }_{sw}}\right)\hfill \end{array}$$

(17)

where $Q\left(u\right)$ is Q function defined by $Q\left(u\right)=\frac{1}{\sqrt{2\pi }}{\int }_u^{+\infty }\frac{exp\left(x^2\right)}{2}dx$. Given a predetermined false alarm probability ${\widehat{P}}_{fa}$, the decision threshold can be calculated by

$$\begin{array}{c}\hfill {\psi }^{*}={\sigma }_w{Q}^{-1}({\widehat{P}}_{fa})+{\mu }_w\end{array}$$

(18)

where $Q^{-1}\left(u\right)$ is inverse function of Q(u). Substituting Equation (18) for Equation (17), we can get

$$\begin{array}{cc}\hfill P_d& =Q\left(\frac{{\psi }^{*}-{\mu }_{sw}}{{\sigma }_{sw}}\right)\hfill \\ \hfill & =Q\left(\frac{{\sigma }_w{Q}^{-1}\left({\widehat{P}}_{fa}\right)+{\mu }_w-{\mu }_{sw}}{{\sigma }_{sw}}\right)\hfill \end{array}$$

(19)

For the intended user Bob, the time–bandwidth product is approximated as $T_{\mathrm{int}}W=1$ when the under-sampling method is employed. Furthermore, the ratio of instance symbol energy and power spectrum density is

$$\begin{array}{c}\hfill {\left(E/\phantom{E{N}_0}{N}_0\right)}_{Bob}=\frac{{∥\mathbf{H}∥}_2^2{E}_s}{N{N}_0}\end{array}$$

(20)

where $E_s$ denotes the average symbol energy of $s\left(t\right)$. As a result, the decision threshold at Bob’s end is ${\psi }_{Bob}^{*}=2Q^{-1}({\widehat{P}}_{fa})+2$, and the detection probability is

$$\begin{array}{cc}\hfill P_{d,Bob}& =Q\left(\frac{{\psi }_{Bob}^{*}-{\mu }_{sw}}{{\sigma }_{sw}}\right)\hfill \\ \hfill & =Q\left(\frac{Q^{-1}({\widehat{P}}_{fa})-{\left(E/\phantom{E{N}_0}{N}_0\right)}_{Bob}}{\sqrt{1+2·{\left(E/\phantom{E{N}_0}{N}_0\right)}_{Bob}}}\right)\hfill \end{array}$$

(21)

For the illegal user Eve, the time–bandwidth product is approximated as $T_{\mathrm{int}}W=ND$. Furthermore, the $E/\phantom{E{N}_0}{N}_0$ at Eve’s end can be written by

$$\begin{array}{c}\hfill {\left(E/\phantom{E{N}_0}{N}_0\right)}_{Eve}=\frac{{\left|{\displaystyle \sum _{k=1}^N}{h}_k{g}_k\right|}^2}{{∥\mathbf{H}∥}_2^2}·\frac{E_s}{N{N}_0}\end{array}$$

(22)

where $g_k$ is the channel coefficient of the wire-tap channel that is independent of $h_k$. In AWGN channel, $g_k=1(k=1,2,\cdots ,N)$ and Equation (22) is simplified as ${\left(E/\phantom{E{N}_0}{N}_0\right)}_{Eve}=E_s/\phantom{E_s\left(N{N}_0\right)}\left(N{N}_0\right)$. Furthermore, the decision threshold at Eve’s end can be given by ${\psi }^{*}=2\sqrt{ND}{Q}^{-1}({\widehat{P}}_{fa})+2ND$ and the detection probability can be written by

$$\begin{array}{cc}\hfill P_{d,Eve}& =Q\left(\frac{{\psi }^{*}-{\mu }_{sw}}{{\sigma }_{sw}}\right)\hfill \\ \hfill & =Q\left(\frac{\sqrt{ND}{Q}^{-1}\left({\widehat{P}}_{fa}\right)-{\left(E/\phantom{E{N}_0}{N}_0\right)}_{Eve}}{\sqrt{ND+{\left(E/\phantom{E{N}_0}{N}_0\right)}_{Eve}}}\right)\hfill \end{array}$$

(23)

5. Simulation Results

In this section, a number of experiments are designed to evaluate both the reliability and security of the proposed secure transmission system. The receiving gain and BER are chosen as indicators to assess the feasibility and security of the proposed physical layer security communication system. The receiving gain, which was defined in section II implies the phenomenon of SNR improvement caused by the under-sampling method on Bob’s side. For secure wireless communication systems, it is desired that the BER at Bob’s side is decreased rapidly with the increase in received SNR, while the BER at Eve’s side is always unacceptably high. To illustrate the robustness of the proposed physical layer security communication system, simulations are conducted over both AWGN and fading channels. In simulations, the signal $s\left(t\right)$ is assumed to be a BPSK-modulated signal with a bandwidth of 10MHz, which means $f_B=10$ MHz. Furthermore, the parameter L is set as $L=1$. It is noticed that all simulations in this work are implemented using Matlab. The diagram of system model simulations is shown in Figure 6.

5.1. LPD Performance

The objective of LPD property is to guarantee the covertness of the signal waveform, which means Bob can detect the signals transmitted by Alice, while Eve can hardly detect the presence of the transmit signals. In this section, we will investigate the detection performance at Bob’s and Eve’s ends in both AWGN and fading channels. The detection method is as described in the last section, and the predetermined false alarm rate is $P_{fa}=1e-3$.

Simulation results in Figure 7 show that detection probability at Bob’s end is always superior to Eve’s when the channel signal-to-noise ratio is less than 10dB over the AWGN channel. There exists a security region depicted by the SNR, in which Bob’s detection probability is approaching 1, while that of Eve’s is at a low level. For example, when the SNR is in the [−3,4] (dB) interval, the detection probability of Bob is close to 1, while the detection probability of Eve is always lower than 0.1 given $N=10$ and $D=4$. In practical applications, Alice can adjust the transmit power so that the received SNR is always in this region, thereby ensuring the covertness of the signal.

Furthermore, the range of the security region increases with the sub-carrier number N. This means a larger bandwidth may always lead to stronger security in signal covertness. Such a conclusion is completely consistent with how the larger the spread spectrum ratio is in DSSS, the better the security is in the direct sequence spread spectrum communication system.

Simulation results in Figure 8 show that Bob’s detection performance in fading channel is basically the same as that in the AWGN channel, and the precoding scheme is proved to be effective. However, for Eve, the weighted factor ${\alpha }_k$ and channel coefficients $g_k$ are completely independent, and the SNR at Eve’s side is significantly reduced. Therefore, the security region is wider than that in the AWGN channel.

5.2. Comparison of BER Performance between Bob and Eve

The objective of the proposed physical layer security communication scheme is to simultaneously guarantee the LPD and LPI properties of wireless links. On the one hand, Bob can detect and demodulate the signals transmitted by Alice, while Eve can hardly detect the presence of the transmitted signals. On the other hand, although Eve can detect the transmitted signal, he can hardly extract useful information.

Arguably, BER is an effective and useful measure for both reliability and security. We hope the BER at Bob’s side is as low as possible; meanwhile, the BER at Eve’s side is (very close to) 0.5, so he essentially cannot recover any information transmitted by Alice. Simulation results demonstrate that the proposed scheme can guarantee that the BER at Eve will always be unacceptably high regardless of the received SNR, while the BER at Bob will be decreased significantly as the received SNR increases.

For the AWGN channel, the security of the proposed communication system is mainly determined by the sampling clock frequency offset between Bob and Eve. According to the communication protocol proposed in Section III, Bob can increase the transmit power or length of pilot signals in order to improve the estimation accuracy. In this way, the sampling clock offset can be estimated nearly perfectly as the SNR of the pilot signal is sufficiently high or the number of pilot symbols is sufficiently large. Meanwhile, the sampling frequency between Alice and Eve can hardly be synchronized because they have no negotiation for sampling frequency synchronization. The BER performance of Bob and Eve is shown in Figure 9. The parameters are set as $D=5$ and $N=4$; thus, the sampling frequency is 50 MHz. As the accuracy of the sampling clocks is always at PPM(parts per million) level, we can reasonably assume that the sampling clock offset between Alice and Eve is 1 Hz. The BER versus $SN{R}_y$ at Bob and Eve are illustrated in Figure 9. The sampling phase offset at Bob is set as $\Delta =0,1/8,1/16$. A significant improvement in BER performance can be achieved when the sampling phase offset decreases. Simulation results show that the BER at Bob decreases rapidly as the SNR increases. Meanwhile, the BER at Eve stays at a high level, and decreases very slowly with the increase in SNR that he can hardly intercept useful information. When some artificial jamming signals are added to the LPD signal, simulation results in Figure 9 show that Bob can still detect and demodulate the LPD signal. The parameter $\gamma$ in the figure is defined as $\gamma =P_x/P_j$, where $P_x$ denotes the transmit power of useful signals and $P_j$ denotes the transmit power of artificial jamming signals. As a result, the proposed secure communication scheme is proven effective in the AWGN channel.

Next, the BER performance of the proposed secure communication scheme over fading channels is shown in Figure 10. For Bob, both perfect CSI and imperfect scenarios are investigated. For the imperfect CSI scenario, the estimation error $\rho$ is assumed to be 0.2 and 0.4. It is not surprising that the BER performance loss is induced by the increase in estimation error under the same channel condition. The results have clearly demonstrated that Bob can detect and demodulate the LPD signal effectively. The BER performance at Eve with different ${\xi }_f$ is also given in this figure, where ${\xi }_f$ denotes the sampling frequency offset between Alice and Bob. Simulation results show that the BER at Eve is about 0.5 even ${\xi }_f=0$, which means the sampling clock offset between Alice and Eve does not exist. It reveals that Eve can hardly extract useful information only because he has different channel coefficients. It can be seen that Eve will obtain a BER of about 0.5 no matter how the SNR changes. As a result, the proposed secure communication scheme is also proven effective in the fading channel.

6. Conclusions

In this work, a physical layer security communication scheme has been proposed for CPS applications. First, a structured LPD signal waveform is designed, and the detection method for the LPD signal is proposed. Analysis shows that the maximum receiving gain is given by $ND$ and decreased with the increase in sapling phase offset. Then, a wireless wire-tap channel is presented, and a secure transmission protocol is proposed. The channel reciprocal principle is applied to achieve the sampling clock offset and CSI between Alice and Bob. Based on such information, the sampling clock compensation method and precoding scheme, which can maximize Bob’s SNR at the sampling stage, are proposed. To demonstrate the LPD property, detection probability at both Bob’s and Eve’s ends are derived with the energy detector model. Simulation results show that there exists a specific SNR interval where Bob’s detection probability is approaching 1, while Eve’s is well below 0.1. The range is approximately 7 dB and 17 dB in AWGN and fading channel, respectively, when $N=10$ and $D=4$. In addition, simulation results in AWNG and fading channel also show that the BER at Bob’s end is always decreased with the increase in SNR or the number of sampling phases, while Eve’s BER has always been around 0.5 regardless of the SNR. As a result, both the effectiveness and security of the proposed scheme are verified.