**Cooperative Jamming with AF Relay in Power Monitoring and Communication Systems for Mining**

**Wei Meng 1, Yidong Gu 1, Jianjun Bao 1, Li Gan 2, Tao Huang <sup>3</sup> and Zhengmin Kong 2,\***


**Abstract:** In underground mines, physical layer security (PLS) technology is a promising method for the effective and secure communication to monitor the mining process. Therefore, in this paper, we investigate the PLS of an amplify-and-forward relay-aided system in power monitoring and communication systems for mining, with the consideration of multiple eavesdroppers. Explicitly, we propose a PLS scheme of cooperative jamming and precoding for a full-duplex system considering imperfect channel state information. To maximize the secrecy rate of the communications, an effective block coordinate descent algorithm is used to design the precoding and jamming matrix at both the source and the relay. Furthermore, the effectiveness and convergence of the proposed scheme with high channel state information uncertainty have been proven.

**Keywords:** physical layer security; multiple eavesdroppers; full-duplex; underground mine; amplifyand-forward relay

### **1. Introduction**

Underground mining promotes the economy's growth, but the dust and poisonous gases formed during mining make it a dangerous and complex operation. Therefore, a reliable communication system is needed to monitor the mining process and communicate with external management offices to ensure the safety and maximum production of the underground mine. Wireless communication technology is applied to realize information exchange in underground mines due to its simple construction.

However, due to the complex structure of underground mines, there exists significant attenuation of radio wave transmission in wireless communications [1]. To solve these problems, relay-aided wireless communications have been studied to improve the reliability and have also been used to enhance the coverage of a broader range of networks. According to the forwarding protocol adopted by the relay, cooperation relay can be divided into amplify-and-forward (AF) and decode-and-forward (DF) relay [2]. AF is the simplest protocol, and it processes the received signals linearly and then forwards them to the destination [3]. Offering a reasonable trade-off between actual implementation costs and benefits, AF is considered the most promising solution [4].

To guarantee the communication rate in wireless communications, full-duplex (FD) relays are studied in Refs. [5,6]. FD technology allows radios to receive and transmit simultaneously on the same frequency band, which can improve spectrum efficiency [7]. Furthermore, in addition to doubling the spectral efficiency of the physical layer, FD can help to solve the throughput losses due to congestion and large point-to-point delays in existing wireless networks.

In addition, due to the openness and sharing of wireless media, any wireless device connected to the communication system can access messages exchanged through the connection, making wireless channels easy to be eavesdropped on and inject with malicious information [8]. Worse still, relay-aided wireless networks may suffer severe security risks

**Citation:** Meng, W.; Gu, Y.; Bao, J.; Gan, L.; Huang, T.; Kong, Z. Cooperative Jamming with AF Relay in Power Monitoring and Communication Systems for Mining. *Electronics* **2023**, *12*, 1057. https:// doi.org/10.3390/electronics12041057

Academic Editor: Cheng-Chi Lee

Received: 21 January 2023 Revised: 10 February 2023 Accepted: 16 February 2023 Published: 20 February 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

from malicious users since they may eavesdrop on the messages from both the source and the relay. Physical layer security (PLS) can effectively protect the privacy among the transmitter and the legitimate receivers [9]. Shannon conducted pioneering research on secret communications and established the concept of perfect secrecy [10]. Unlike Shannon, Wyner proposed a degraded wiretap channel model in Ref. [11]. After the degraded wiretap channel, the fading wiretap channels and multiple-input-multiple-output wiretap channels have been investigated in Refs. [12,13] and Refs. [14,15], respectively.

The work in Ref. [5] investigated a FD communication system, and the transmission block is divided into an energy harvesting phase and an information transmission phase. Different from Ref. [5], in Ref. [6], an FD is designed to capture energy from the source while forwarding information to the legitimate receivers. With the presence of passive colluding wireless eavesdroppers, Ref. [16] studied the effective secrecy throughput to the physical layer security of in-home and broadband PLC systems. In Ref. [17], the authors investigated the optimal trunk position of FD relay systems with DF and the minimal outage probability criterion considered.

Above all, to the best of our knowledge, the existing contributions fail to ensure secure communications in the challenging FD relay-aided wireless communications scenario in the face of multiple eavesdroppers and imperfect channel state information (CSI). Therefore, in this paper, we propose a PLS scheme of cooperative jamming and precoding for FD-DF relay-assisted wireless communications system considering imperfect CSI, which combines cooperative precoding for legitimate users to improve the quality of legitimate channels and cooperative jamming for illegal users to reduce the quality of eavesdropping channels. Considering the imperfect CSI and multiple eavesdroppers, we use an effective BCD algorithm to design the precoding and jamming matrix at both the source and the relay, in which maximizing the secrecy rate of the FD-AF relay-assisted wireless communications system is emphasized.

This paper is organized as follows. Section 2 describes the system model. The secrecy rate optimization problem is proposed and transformed into a solvable form in Section 3, which also gives the algorithm. Section 4 characterizes the numerical results in different scenarios. Finally, the conclusion is presented in Section 5.

Notation: The **<sup>W</sup>***T*, **<sup>W</sup>***H*, vec(**W**), **W**and tr(**W**) denote the transpose, conjugate transpose, vectorization, Frobenius norm, and trace of the matrix, respectively. ⊗ denote the Kronecker product, and **<sup>W</sup>***<sup>K</sup>* represents the **WW***<sup>H</sup>* along with log|**<sup>E</sup>** <sup>+</sup> **CD**<sup>|</sup> <sup>=</sup> log|**<sup>E</sup>** <sup>+</sup> **DC**<sup>|</sup> . **E** is the identity matrix.

### **2. System Model**

Consider a MIMO wireless system, as shown in Figure 1, where a source, a relay, a user, and two eavesdroppers have *NS*, *NR*, *ND*, and *NE* channels, respectively. We assume that there is no direct link between the source and the user for the long-distance path loss. For simplicity, the eavesdroppers represent all the eavesdroppers eavesdropping the same legitimate in the same time phase. More specifically, in the first time phase, eavesdroppers eavesdrop *E*<sup>1</sup> message from the source, and in the second time phase, eavesdropper eavesdrop *E*<sup>2</sup> message from the relay.

In wireless communications system, messages are transmitted through MIMO wireless communications channels. We describe each path between two nodes by CSI **H***ij*,*<sup>k</sup>* as the matrix of channel coefficients, where *i* ∈ {*S*, *R*}, *j* ∈ {*R*, *D*, *E*1, *E*2}, and *k* = 1, 2 denote the transmitter, receiver, and transmission time phases, respectively. It is worth noting that **H***RR*,1 refers to the self-interference matrix because of self-interference and in the process of transmission **H***ij*,*<sup>k</sup>* stays constant because of the short transmission time.

In this paper, the uncertainty of CSI is taken into consideration, i.e., the CSI of the wireless communications system cannot be perfectly known at the source or the relay due to factors such as the limited capacity of the feedback channel. As a result, the deterministic uncertainty model [18] is introduced to characterize the imperfect CSI, as follows:

$$\mathbf{H}\_{ij,k} \in \mathcal{H}\_{ij,k} = \left\{ \mathbf{H}\_{ij,k} \, \middle| \, \mathbf{H}\_{ij,k} = \mathbf{H}\_{ij,k} + \Delta\_{ij,k'} \, \middle| \, \Delta\_{ij,k} \, \middle| \, \le \delta\_{ij,k} \right\},\tag{1}$$

where Δ*ij*,*<sup>k</sup>* denotes the channel uncertainty as the degree of deviation from the mean CSI **H***ij*,*k*.

In Figure 1, during the first time phase, the source sends confidential signals to the relay while *E*<sup>1</sup> eavesdrops on the signals from the source. To interrupt *E*1, the relay emits jamming signals to *E*1. More specifically, the message transmitted by the source is secret data symbol **<sup>S</sup>** ∈ CN (**0**, 1) precoded by the precoding vector **<sup>L</sup>** <sup>∈</sup> <sup>C</sup>*NS*×1. Then, we can formulate the progress at the source as follows:

$$\mathbf{X}\_{\rm s} = \mathbf{L} \mathbf{S}\_{\prime} \tag{2}$$

**Figure 1.** Wireless communication system model with a single relay.

Next, we formulate the messages emitted by the relay. Note that the relay in this time phase only emits jamming to disrupt *E*<sup>1</sup> so the messages can be formulated as follows :

$$\mathbf{X}\_{\mathbb{R}} = \mathbf{J}\_1 \mathbf{Z}\_{1'} \tag{3}$$

where we utilize the jamming precoding vector **J1** <sup>∈</sup> <sup>C</sup>*NR*×<sup>1</sup> and jamming symbol **<sup>Z</sup>**<sup>1</sup> <sup>∈</sup> CN (**0**, 1).

Considering the self-interference of the relay, we can formulate the messages received by the relay:

$$\mathbf{Y}\_{R1} = \mathbf{H}\_{SR,1}\mathbf{L}\mathbf{S} + \mathbf{H}\_{I}\mathbf{I}\_{1}\mathbf{Z}\_{1} + \mathbf{n}\_{R1} \tag{4}$$

where **n***R*<sup>1</sup> is Additive White Gaussian Noise (AWGN) at the relay and **H***<sup>I</sup>* is the selfinterference matrix. Meanwhile, *E*<sup>1</sup> eavesdrops on both of the messages from the source and the relay, so the messages eavesdropped by *E*<sup>1</sup> can be expressed as

$$\mathbf{Y}\_{E1} = \mathbf{H}\_{SE,1}\mathbf{L}\mathbf{S} + \mathbf{H}\_{RE,1}\mathbf{J}\_1\mathbf{Z}\_1 + \mathbf{n}\_{E1} \tag{5}$$

where **n***E*<sup>1</sup> is AWGN at *E*1.

In the second time phase, the source emits the jamming signals **X***S*<sup>2</sup> to *E*<sup>2</sup> where **J2**,**Z**<sup>2</sup> ∈ CN (**0**, 1) represent the jamming precoding vector and jamming symbol, respectively.

$$\mathbf{X}\_{\rm S2} = \mathbf{J}\_2 \mathbf{Z}\_{\rm 2} \tag{6}$$

Then, the relay amplifies the messages it received in the first time phase and forwards them to the user,

$$\mathbf{X}\_{R2} = \mathbf{G}\mathbf{Y}\_{R1} = \mathbf{G} \left( \mathbf{H}\_{SR,1} \mathbf{L} \mathbf{S} + \mathbf{H}\_{I} \mathbf{I}\_{1} \mathbf{Z}\_{1} + \mathbf{n}\_{R1} \right), \tag{7}$$

$$\mathbf{Y}\_{D} = \mathbf{H}\_{RD,2}\mathbf{X}\_{R2} + \mathbf{n}\_{D} = \mathbf{H}\_{RD,2}\mathbf{G}\left(\mathbf{H}\_{SR,1}\mathbf{L}\mathbf{S} + \mathbf{H}\_{I}\mathbf{I}\_{1}\mathbf{Z}\_{1} + \mathbf{n}\_{R1}\right) + \mathbf{n}\_{D},\tag{8}$$

where **<sup>G</sup>** <sup>∈</sup> <sup>C</sup>*NR*×*NR* is the amplifying matrix at the relay and **<sup>n</sup>***<sup>D</sup>* is AWGN at the users, and **X***R*2, **Y***<sup>D</sup>* represent the messages transmitted by the relay and received by the users, respectively.

*E*<sup>2</sup> receive both the signals from the relay and the jamming signals from the source, i.e.,

$$\mathbf{Y}\_{E2} = \mathbf{H}\_{SE,2}\mathbf{J}\_2\mathbf{Z}\_2 + \mathbf{H}\_{RE,2}\mathbf{G} \left(\mathbf{H}\_{SR,1}\mathbf{LS} + \mathbf{H}\_I\mathbf{J}\_1\mathbf{Z}\_1 + \mathbf{n}\_{R1}\right) + \mathbf{n}\_{E2} \tag{9}$$

where **n***E*<sup>2</sup> is AWGN at *E*2.

Above all, to formulate the problem in a mathematical form, we calculate the signalnoise ratio (SNR) at the users, *E*<sup>1</sup> and *E*2, respectively.

$$
\Gamma\_D = (\mathbf{H}\_{RD,2}\mathbf{G}\mathbf{H}\_{SR,1}\mathbf{L})^K \mathbf{Q}\_D^{-1} \tag{10}
$$

$$\text{where } \mathbf{Q}\_D = (\mathbf{H}\_{R\mathbf{D},2}\mathbf{G}\mathbf{H}\_{R\mathbf{E},1}\mathbf{J}\_1)^K + \sigma\_\mathbf{R}^2 (\mathbf{H}\_{R\mathbf{D},2}\mathbf{G})^K + \sigma\_\mathbf{D}^2 \mathbf{E}.$$

$$\Gamma\_{E1} = (\mathbf{H}\_{SE,1}\mathbf{L})^K \mathbf{Q}\_{E1}^{-1} \tag{11}$$

where **Q***E*<sup>1</sup> = (**H***RE*,1**J1**)*<sup>K</sup>* + *σ*<sup>2</sup> *<sup>E</sup>***E**.

$$
\Gamma\_{\rm E2} = \left(\mathbf{H}\_{\rm RE,2}\mathbf{G}\mathbf{H}\_{\rm R1}\mathbf{L}\right)^{K}\mathbf{Q}\_{\rm E2}^{-1} \tag{12}
$$

where **Q***E*<sup>2</sup> = (**H***SE*,2**J2**)*<sup>K</sup>* + (**H***RE*,2**GH***I***J1**)*<sup>K</sup>* + *σ*<sup>2</sup> *<sup>R</sup>*(**H***RE*,2**G**)*<sup>K</sup>* + *<sup>σ</sup>*<sup>2</sup> *<sup>E</sup>***E** and *σ<sup>i</sup>* is the noise amplitude of the corresponding AWGN **n***i*.

Then, we can arrive at the achievable secrecy rate of the legitimate users [11]:

$$R\_D = \log|\mathbf{E} + \Gamma\_D|\_\prime \tag{13}$$

In the non-colluding strategy, each eavesdropper processes messages individually. Therefore, the achievable secrecy rate of the non-colluding [11] eavesdroppers is

$$R\_E = \max\{\log|\mathbf{E} + \Gamma\_{E1}|, \log|\mathbf{E} + \Gamma\_{E2}|\}\tag{14}$$

Finally, we can gain the achievable secrecy rate of the wireless communications system,

$$R\_S = R\_D - R\_E \tag{15}$$

### **3. Optimization Problem Transformation**

In this part, the goal is to maximize the secrecy rate of the communication system. Then according to the system model, we can formulate the optimization problem of the secrecy rate with the transmit power constraint as follows.

$$\max\_{\mathbf{L}, \mathbf{J}\_1, \mathbf{J}\_2, \mathbf{G}} \min\_{\mathbf{H}\_{ij,k} \in \mathcal{H}\_{ij,k}} R\_S \tag{16a}$$

$$\text{s.t. } \|\mathbf{L}\|^2 \lessapprox P\_{\mathbf{S}\prime} \cdot \|\mathbf{J}\_1\|^2 \lessapprox P\_{\mathbf{S}\prime} \|\mathbf{J}\_2\|^2 \lessapprox P\_{\mathbf{R}\prime} \tag{16b}$$

$$\operatorname{tr}((\mathbf{G}\mathbf{H}\_{SR,1}\mathbf{L})^{K} + (\mathbf{G}\mathbf{H}\_{I}\mathbf{J}\_{1})^{K} + \sigma\_{\mathcal{R}}^{2}\mathbf{G}^{K}) \lesssim P\_{\mathcal{R}} \quad \forall \mathbf{H}\_{ij,k} \in \mathcal{H}\_{ij,k} \tag{16c}$$

However, due to the non-convexity of the optimization problem, it is difficult to solve. To deal with the high non-convexity of the function − log|·|, the objective function in (16a) is transformed into an equivalent counterpart through the WMMSE algorithm, which can be solved by the BCD method. The following introduces the WMMSE algorithm.

**Lemma 1.** *Define the mean-square error (MSE) matrix*

$$
\widehat{\mathbf{N}} \stackrel{\triangle}{=} \left( \mathbf{TH} \mathbf{-E} \right)^{K} + \mathbf{TRT}^{H} \tag{17}
$$

*where* **R 0***. Then we have*

$$-\log|\mathbf{N}| = \max\_{\mathbf{K}\succ 0} \log|\mathbf{K}| - \text{tr}(\mathbf{K}\mathbf{N}) + \text{tr}(\mathbf{E})\tag{18}$$

$$\log\left|\mathbf{I} + \mathbf{R}^{-1}\mathbf{H}^{K}\right| = \max\_{\mathbf{K}\succ 0, \mathbf{T}} \log|\mathbf{K}| - \text{tr}(\mathbf{K}\widehat{\mathbf{N}}) + \text{tr}(\mathbf{E})\tag{19}$$

To reformulate the parts of − log|·| in the objective function, we apply Lemma 1 on (13) and introduce the MSE matrix **N***<sup>i</sup>* and auxiliary matrices **K***i*, **T***<sup>i</sup>* ,which have been defined in (17) and (19). So, the achievable secrecy rate of the legitimate can be reorganized as

$$\begin{split} R\_D &= \log|\mathbf{E} + \Gamma\_D| = \log\left|\mathbf{E} + (\mathbf{H}\_{RD,2}\mathbf{G}\mathbf{H}\_{SR,1}\mathbf{L})^K \mathbf{Q}\_D^{-1}\right| \\ &= \max\_{\mathbf{K}\_D \succ 0, \mathbf{D}\_D} \log|\mathbf{K}\_D| - \text{tr}(\mathbf{K}\_D \mathbf{N}\_D) + \text{tr}(\mathbf{E}) \end{split} \tag{20}$$

where

$$\mathbf{N}\_D = \left(\mathbf{T}\_D \mathbf{H}\_{RD,2} \mathbf{G} \mathbf{H}\_{SR,1} \mathbf{L} - \mathbf{E}\right)^K + \mathbf{T}\_D \mathbf{Q}\_D \mathbf{T}\_D^H \tag{21}$$

Applying Lemma 1 on (14), the achievable rates of *E*<sup>1</sup> and *E*<sup>2</sup> can be transformed as (22) and (23).

$$\begin{aligned} & -\log|\mathbf{E} + \Gamma\_1| = \log|\mathbf{Q}\_{E1}| - \log\left| \left( \mathbf{H}\_{SE,1} \mathbf{I} \right)^K + \mathbf{Q}\_{E1} \right| \\ &= \underbrace{\log\left| \mathbf{E} + \sigma\_E^{-2} \left( \mathbf{H}\_{RE,1} \mathbf{I}\_1 \right)^K \right|}\_{\mathbf{C}\_{E11}} \underbrace{-\log\left| \mathbf{E} + \sigma\_E^{-2} \left( \left( \mathbf{H}\_{SE,1} \mathbf{I} \right)^K + \left( \mathbf{H}\_{RE,1} \mathbf{J}\_1 \right)^K \right) \right|}\_{\mathbf{C}\_{E12}} \end{aligned} \tag{22}$$

$$\begin{split} & -\log|\mathbf{E} + \Gamma\_{2}| = \log|\mathbf{Q}\_{E2}| - \log\left| \left( \mathbf{H}\_{RE,2}\mathbf{G}\mathbf{H}\_{I}\mathbf{L} \right)^{K} + \mathbf{Q}\_{E2} \right| \\ &= \underbrace{\log\left|\mathbf{E} + \sigma\_{E}^{-2} \left( \left( \mathbf{H}\_{SE,2}\mathbf{J}\_{2} \right)^{K} + \left( \mathbf{H}\_{RE,2}\mathbf{G}\mathbf{H}\_{I}\mathbf{J}\_{1} \right)^{K} + \sigma\_{R}^{2} \left( \mathbf{H}\_{RE,2}\mathbf{G} \right)^{K} \right) \right|}\_{\mathbf{C}\_{E21}} + \\ & - \underbrace{\log\left|\mathbf{E} + \sigma\_{E}^{-2} \left( \left( \mathbf{H}\_{RE,2}\mathbf{G}\mathbf{H}\_{I}\mathbf{L} \right)^{K} + \left( \mathbf{H}\_{SE,2}\mathbf{J}\_{2} \right)^{K} + \left( \mathbf{H}\_{RE,2}\mathbf{G}\mathbf{H}\_{I}\mathbf{J}\_{1} \right)^{K} + \sigma\_{R}^{2} \left( \mathbf{H}\_{RE,2}\mathbf{G} \right)^{K} \right) \right|}\_{\mathbf{C}\_{E22}} \end{split} \tag{23}$$

Then, the auxiliary variables *CE*11, *CE*12, *CE*<sup>21</sup> and *CE*<sup>22</sup> can be rewritten according to Lemma 1 as

$$\mathbf{C}\_{E11} = \max\_{\mathbf{K}\_{E11} > 0, \mathbf{T}\_{E1}} \log |\mathbf{K}\_{E11}| - \text{tr}(\mathbf{K}\_{E11} \mathbf{N}\_{E11}) + \text{tr}(\mathbf{E}) \tag{24}$$

$$\mathbf{C}\_{E12} = \max\_{\mathbf{K}\_{E12} \succeq 0} \log |\mathbf{K}\_{E12}| - \text{tr}(\mathbf{K}\_{E12} \mathbf{N}\_{E12}) + \text{tr}(\mathbf{E}) \tag{25}$$

$$\mathbf{C}\_{E21} = \max\_{\mathbf{K}\_{E21} \succeq 0, \mathbf{T}\_{E2}} \log |\mathbf{K}\_{E21}| - \text{tr}(\mathbf{K}\_{E21} \mathbf{N}\_{E21}) + \text{tr}(\mathbf{E}) \tag{26}$$

$$\mathbf{C\_{E22}} = \max\_{\mathbf{K\_{E2}} \ge 0} \log |\mathbf{K\_{E22}}| - \text{tr}(\mathbf{K\_{E22}} \mathbf{N\_{E2}}) + \text{tr}(\mathbf{E}) \tag{27}$$

where

$$\begin{aligned} \mathbf{N}\_{E11} &= \left(\mathbf{D} \mathbf{T}\_{E1} \mathbf{H}\_{RE,1} \mathbf{J}\_1 - \mathbf{E}\right)^K + \sigma\_E^2 \mathbf{T}\_{E1}^K \\ \mathbf{N}\_{E12} &= \sigma\_E^{-2} \left(\left(\mathbf{H}\_{SE,1} \mathbf{L}\right)^K + \left(\mathbf{H}\_{RE,1} \mathbf{J}\_1\right)^K\right) + \mathbf{E} \\ \mathbf{N}\_{E21} &= \left(\mathbf{T}\_{E21} \mathbf{H}\_{SE,2} \mathbf{J}\_2 \mathbf{X} + \mathbf{T}\_{E22} \mathbf{H}\_{RE,2} \mathbf{G} \mathbf{H}\_I \mathbf{VX} + \\ \sigma\_R \mathbf{T}\_{E23} \mathbf{H}\_{RE,2} \mathbf{G} - \mathbf{E} \right)^K + \sigma\_E^2 \left(\mathbf{T}\_{E21}^K + \mathbf{T}\_{E22}^K + \mathbf{T}\_{E23}^K\right) \end{aligned}$$

$$\mathbf{N}\_{E22} = \sigma\_E^{-2} ( (\mathbf{H}\_{RE,2} \mathbf{G} \mathbf{H}\_I \mathbf{L})^K + (\mathbf{H}\_{SE,2} \mathbf{J}\_2)^K + (\mathbf{H}\_{RE,2} \mathbf{G} \mathbf{H}\_I \mathbf{J}\_1)^K + \sigma\_R^2 (\mathbf{H}\_{RE,2} \mathbf{G})^K) + \mathbf{E}$$

and note the decomposition **T***E*<sup>2</sup> = ! **T***E*<sup>21</sup> **T***E*<sup>22</sup> **T***E*<sup>23</sup> " and **X** = ! 1 **0** " <sup>∈</sup> <sup>C</sup>1×*Nr*.

After substituting (24)–(27) into (16a), the secrecy rate of the system is equivalently rewritten as

$$\max\_{\mathbf{L}, \mathbf{J}\_1, \mathbf{J}\_2, \mathbf{G}, \mathbf{K}\_i \succ 0, \mathbf{T}\_i} \min\_{\mathbf{H}\_{ij,k} \in \mathcal{H}\_{ij,k}} f(\mathbf{L}, \mathbf{J}\_1, \mathbf{J}\_2, \mathbf{G}, \mathbf{S}\_{\mathbf{i}}, \mathbf{D}\_{\mathbf{i}}) \tag{28}$$

$$\text{s.t.}\,(16c)\tag{29}$$

$$\begin{aligned} f \triangleq \log|\mathbf{K}\_D| - \text{tr}(\mathbf{K}\_D \mathbf{N}\_D) &+ \min\{\log|\mathbf{K}\_{E11}| - \text{tr}(\mathbf{K}\_{E11} \mathbf{N}\_{E11}) + \log|\mathbf{K}\_{E12}| - \text{tr}(\mathbf{K}\_{E12} \mathbf{N}\_{E12}), \\ \log|\mathbf{K}\_{E21}| - \text{tr}(\mathbf{K}\_{E21} \mathbf{N}\_{E21}) + \log|\mathbf{K}\_{E22}| - \text{tr}(\mathbf{K}\_{E22} \mathbf{N}\_{E22}) \end{aligned} \tag{30}$$

where the function *f*(**L**,**J1**,**J2**, **G**, **Ki**, **Ti**) is defined in (30).

To solve the proposed problem and constraint (16c), the slack variables *β<sup>i</sup>* (*i* ∈ {*T*, *E*11, *E*12, *E*21, *E*22, *P*}) are introduced to transform (28) into an optimization problem.

$$\text{tr}(\mathbf{K}\_i \mathbf{N}\_i) \le \beta\_{i\prime} \lor \mathbf{H}\_{ij,k} \in \mathcal{H}\_{ij,k} \tag{31}$$

We can further rewrite the problem (28) as

$$\max\_{\mathbf{L}, \mathbf{J}\_1, \mathbf{J}\_2, \mathbf{G}, \mathbf{K}\_i \succ 0, \mathbf{T}\_i} \text{g}(\mathbf{L}, \mathbf{J}\_1, \mathbf{J}\_2, \mathbf{G}, \mathbf{S}\_{\mathbf{i}}, \mathbf{D}\_{\mathbf{i}}) \tag{32}$$

$$\text{s.t. (16c), (31)}\tag{33}$$

$$\begin{array}{l} \log \stackrel{\Delta}{=} \log |\mathbf{K}\_{D}| - \beta\_{D} + \min \{ \log |\mathbf{K}\_{E11}| - \beta\_{E11} + \log |\mathbf{K}\_{E12}| - \beta\_{E12} \} \\ \log |\mathbf{K}\_{E21}| - \beta\_{E21} + \log |\mathbf{K}\_{E22}| - \beta\_{E22} \end{array} \tag{34}$$

where *g*(**L**,**J1**,**J2**, **G**, **Ki**, **Ti**) is defined in (34), respectively. However, the semi-infinite inequalities (31) are non-convex and need further transformation. In the next step, (31) is transformed into a convex form. In fact, all the inequalities tr(**K***i***N***i*) ≤ *β<sup>i</sup>* can be transformed into a convex form in a similar way. Such as, when *i* = *D*, the semi-definite constraint tr(**K***D***N***D*) can be rewritten as

$$\operatorname{tr}(\mathbf{K}\_{D}\mathbf{N}\_{D}) = \left\| \underbrace{\left\| \begin{pmatrix} \operatorname{vec}(\mathbf{F}\_{D}(\mathbf{T}\_{D}\mathbf{H}\_{RD,2}\mathbf{G}\mathbf{H}\_{SR,1}\mathbf{L} - \mathbf{E})\\ \operatorname{vec}(\mathbf{F}\_{D}\mathbf{T}\_{D}\mathbf{H}\_{RD,2}\mathbf{G}\mathbf{H}\_{RR,1}\mathbf{I})\\ \operatorname{vec}(\sigma\_{R}\mathbf{F}\_{D}\mathbf{T}\_{D}\mathbf{H}\_{RD,2}\mathbf{G})\\ \operatorname{vec}(\sigma\_{D}\mathbf{F}\_{D}\mathbf{T}\_{D}) \end{pmatrix} \right\|\,\tag{35}$$

by applying **<sup>T</sup>***<sup>D</sup>* = **<sup>F</sup>***<sup>H</sup> <sup>D</sup>***F***<sup>D</sup>* and the equality tr(**W***K*) = vec(**W**)<sup>2</sup> . Then we need to extract the uncertain CSI from (35).

$$\phi\_D = \bar{\phi}\_D + \underbrace{\sum\_j \Omega\_{Dj} \text{vec}(\Delta\_j)}\_{\Delta\_D} + \underbrace{\sum\_k a\_k \text{vec}(\Delta\_{k1}) \text{vec}^H(\Delta\_{k2})}\_{\bar{\Delta}\_D} \tag{36}$$

where the identity vec(**ABC**) = \$ **<sup>C</sup>***<sup>T</sup>* <sup>⊗</sup> **<sup>A</sup>** % vec(**B**) is applied and *j* ∈ {*RR*, 1; *RD*, 2; *SR*, 1}. Note that *k*1, *k*2 denote the coupling parts of CSI in *φ<sup>D</sup>* in the **Δ**#*<sup>D</sup>* part. In fact, the uncertainty of the CSI is small enough to make its quadratic forms negligible. As a result, the *φ<sup>D</sup>* can be represented as its asymptotic form as

$$\phi\_D = \bar{\phi}\_D + \underbrace{\sum\_j \Omega\_{Dj} \text{vec}(\Lambda\_j)}\_{\Lambda\_D} \tag{37}$$

where

$$\Phi\_{D} = \begin{bmatrix} \text{vec}(\mathbf{F}\_{D}(\mathbf{T}\_{D}\tilde{\mathbf{H}}\_{RD,2}\mathbf{G}\tilde{\mathbf{H}}\_{SR,1}\mathbf{L} - \mathbf{E})) \\ \text{vec}(\mathbf{F}\_{D}\mathbf{T}\_{D}\tilde{\mathbf{H}}\_{RD,2}\mathbf{G}\tilde{\mathbf{H}}\_{I}\mathbf{1}) \\ \text{vec}(\sigma\_{R}\mathbf{F}\_{D}\mathbf{T}\_{D}\tilde{\mathbf{H}}\_{RD,2}\mathbf{G}) \\ \text{vec}(\sigma\_{D}\mathbf{F}\_{D}\mathbf{T}\_{D}) \end{bmatrix} \tag{38}$$

$$
\boldsymbol{\Omega}\_{DSR,1} = \begin{bmatrix}
\mathbf{L}^T \otimes \mathbf{F}\_D \mathbf{T}\_D \tilde{\mathbf{H}}\_{RD,2} \mathbf{G} \\
\mathbf{0} \\
\mathbf{0} \\
\mathbf{0}
\end{bmatrix} \tag{39}
$$

$$\mathbf{\Omega}\_{DSD,2} = \begin{bmatrix} \left(\mathbf{G}\mathbf{H}\_{SR,1}\mathbf{L}\right)^{T} \otimes \mathbf{F}\_{D}\mathbf{T}\_{D} \\ \left(\mathbf{G}\mathbf{H}\_{RR,1}\mathbf{J}\_{1}\right)^{T} \otimes \mathbf{F}\_{D}\mathbf{T}\_{D} \\ \sigma\_{R}\mathbf{G}^{T} \otimes \mathbf{F}\_{D}\mathbf{T}\_{D} \\ \mathbf{0} \end{bmatrix} \tag{40}$$

$$
\boldsymbol{\Omega}\_{DRR,1} = \begin{bmatrix}
\mathbf{0} \\
\mathbf{J}\_1 \mathbf{^T} \otimes \mathbf{F}\_D \mathbf{D}\_D \mathbf{H}\_{RD,2} \mathbf{G} \\
\mathbf{0} \\
\mathbf{0}
\end{bmatrix} \tag{41}
$$

Then, we exploit the Schur complement lemma to recast the constraint (31) as a matrix inequality by substituting (35) and (37).

$$
\begin{bmatrix}
\delta\_D & \bar{\Phi}\_D^H \\
\bar{\Phi}\_D & \mathbf{E}
\end{bmatrix} \succ - \begin{bmatrix}
0 & \Delta\_D^H \\
\Delta\_D & \mathbf{0}
\end{bmatrix} \tag{42}
$$

To eliminate the **Δ***D*, the sign-definiteness lemma is applied.

**Lemma 2.** *Defined matrix* **<sup>U</sup>** *and* {**P***i*, **<sup>Q</sup>***i*}, *<sup>i</sup>* <sup>∈</sup> {1, 2, . . . , *<sup>N</sup>*} *with* **<sup>U</sup>** <sup>=</sup> **<sup>U</sup>***H, the semi-infinite Linear Matrix Inequality (LMI) of the form*

$$\mathbf{U} \succ \sum\_{i}^{N} \left( \mathbf{P}\_{i}^{H} \mathbf{Y}\_{i} \mathbf{Q}\_{i} + \mathbf{Q}\_{i}^{H} \mathbf{Y}\_{i}^{H} \mathbf{P}\_{i} \right) \quad \|\mathbf{Y}\_{i}\| \leq \delta\_{i} \tag{43}$$

Holds if and only if there exist nonnegative real numbers *λ*1, *λ*2,..., *λ<sup>N</sup>* such that

$$
\begin{bmatrix}
\mathbf{U} - \sum\_{i=1}^{N} \lambda\_i \mathbf{Q}\_i^H \mathbf{Q}\_i & -\delta\_1 \mathbf{P}\_1^H & \cdots & -\delta\_N \mathbf{P}\_N^H \\
\vdots & \vdots & \ddots & \vdots \\
\end{bmatrix} \succ \mathbf{0} \tag{44}$$

Appropriately choose the parameters below

$$\mathbf{U}\_D = \begin{bmatrix} \beta\_D & \bar{\Phi}\_D^H \\ \bar{\Phi}\_D & \mathbf{I} \end{bmatrix} \tag{45}$$

$$\mathbf{Q}\_{D1} = \mathbf{Q}\_{D2} = \mathbf{Q}\_{D3} = [-1\mathbf{0}] \tag{46}$$

$$\mathbf{P}\_{D1} = \begin{bmatrix} \mathbf{0} & \mathbf{0}\_{DSR,1}^H \end{bmatrix} \tag{47}$$

$$\mathbf{P}\_{D2} = \begin{bmatrix} \mathbf{0} & \mathbf{0}\_{DSD,2}^H \end{bmatrix} \tag{48}$$

$$\mathbf{P}\_{D3} = \begin{bmatrix} \mathbf{0} & \mathbf{0}\_{DRR,1} \end{bmatrix} \tag{49}$$

Apply Lemma 2 to transform (42) as

$$
\begin{bmatrix}
\begin{bmatrix}
\beta\_D - \lambda\_{D1} - \lambda\_{D2} - \lambda\_{D3} & \bar{\phi}\_D^H \\
\bar{\phi}\_D & \mathbf{E}
\end{bmatrix} & \Theta\_D^H \\
\mathbf{\Theta}\_D & \text{diag}(\lambda\_{D1} \mathbf{E}, \lambda\_{D2} \mathbf{I}, \lambda\_{D3} \mathbf{I})
\end{bmatrix} \succ 0 \tag{50}
$$

where **<sup>Θ</sup>***<sup>D</sup>* <sup>=</sup> <sup>−</sup>[*δDSR*,1**P***<sup>T</sup> <sup>D</sup>*1, *<sup>δ</sup>DSD*,2**P***<sup>T</sup> <sup>D</sup>*2, *<sup>δ</sup>DRR*,1**P***<sup>T</sup> <sup>D</sup>*3'] *T* . Similarly, the other constraint tr(**K***i***N***i*) ≤ *β<sup>i</sup>* is written as follows.

$$\begin{bmatrix} \begin{bmatrix} \beta\_i - \sum\_{k=l}^{j} \lambda\_k & \bar{\phi}\_i^H\\ \bar{\phi}\_i & \mathbf{E} \end{bmatrix} & \mathbf{O}\_i^H\\ \mathbf{O}\_i & \mathbf{E}\_i \end{bmatrix} \succ \mathbf{0} \tag{51}$$

By assembling all the components, the problem can now be written as

$$\max\_{\mathbf{L}, \mathbf{J}\_1, \mathbf{J}\_2, \mathbf{G}, \mathbf{F}\_i \succeq 0, \mathbf{T}\_i, \lambda\_i \succeq 0, \beta\_i} h(\mathbf{L}, \mathbf{J}\_1, \mathbf{J}\_2, \mathbf{G}, \mathbf{F}\_{\mathbf{i}}, \mathbf{T}\_{\mathbf{i}}, \lambda\_{\mathbf{i}}, \beta\_{\mathbf{i}}) \tag{52}$$

$$\text{s.t.}\ (16c),\ (50),\ (51)\tag{53}$$

$$\begin{array}{ll} \text{In} \stackrel{\Delta}{=} 2\log|\mathbf{F}\_{D}| - \beta\_{D} + \min\{2\log|\mathbf{F}\_{E11}| - \beta\_{E11} + 2\log|\mathbf{F}\_{E12}| - \beta\_{E12} \\ \text{2\log|\mathbf{F}\_{E21}| - \beta\_{E21} + 2\log|\mathbf{F}\_{E22}| - \beta\_{E22} \end{array} \tag{54}$$

where the function *h*(**L**,**J1**,**J2**, **G**, **Fi**, **Ti**, *λi*, *βi*) is defined in (54). The problem (52) remains non-convex. However, it becomes a convex optimization problem when fixing some of the optimization variables. In other words, after proper manipulations, its sub-problems can become convex, which are readily solvable. Therefore, a BCD algorithm is employed to solve the nonconvex problem (52), which is summarized in Algorithm 1.

### **Algorithm 1** AN-BF scheme to solve the optimization problem

**input** *l*= 0, precoding vector**L**=**L**(0); jamming precoding vector**J1**=**J1** (0) ,**J2**=**J2** (0) ; **F***<sup>i</sup>* = **<sup>F</sup>**(0) *<sup>i</sup>* , **<sup>G</sup>** <sup>=</sup> **<sup>G</sup>**(0); **repeat** 1: Begin BCD to deal with the (52) with **L**=**L**(l−1),**J1**=**J1** (l−1) ,**J2**=**J2** (l−1) ; **F***<sup>i</sup>* = **<sup>F</sup>**(l−1) *<sup>i</sup>* , **<sup>G</sup>** <sup>=</sup> **<sup>G</sup>**(l−1), and gain the **<sup>D</sup>**(l) *i* ; 2: Solve (52) with **L**=**L**(l−1),**J1**=**J1** (l−1) ,**J2**=**J2** (l−1) ; **<sup>D</sup>***<sup>i</sup>* <sup>=</sup> **<sup>D</sup>**(l) *<sup>i</sup>* , **<sup>G</sup>** <sup>=</sup> **<sup>G</sup>**(l−1), and gain the **<sup>F</sup>**(l) *<sup>i</sup>* ; 3: Solve (52) to attain **J1** (l) ,**J2** (l) and**L**(l) with **<sup>D</sup>***<sup>i</sup>* <sup>=</sup> **<sup>D</sup>**(l) *<sup>i</sup>* , **<sup>G</sup>** <sup>=</sup> **<sup>G</sup>**(l−1), **<sup>F</sup>***<sup>i</sup>* <sup>=</sup> **<sup>F</sup>**(l) *i* ; 4: Solve (52) to gain **G**(l) with **L**=**L**(l),**J1**=**J1** (l) ,**J2**=**J2** (l) , **<sup>F</sup>***<sup>i</sup>* <sup>=</sup> **<sup>F</sup>**(l) *<sup>i</sup>* , **<sup>D</sup>** <sup>=</sup> **<sup>D</sup>**(l); **until** *y*(*l*) <sup>−</sup> *<sup>y</sup>*(*l*−1) <sup>≤</sup> *<sup>ε</sup>*.

### **4. Results**

In this section, numerical simulations are provided to evaluate the performance of the proposed scheme in terms of the average secrecy rate. In this part, we consider a wireless communications system with *NS* = *NR* = *ND* = *NE* = *N* = 2. Besides, for simplicity, the CSI uncertainty bound *δij*,*<sup>k</sup>* is represented as the corresponding determinant of mean CSI multiplied by one certain coefficient, or *δij*,*<sup>k</sup>* = *μ* **H***ij*,*<sup>k</sup>* .

Figure 2 portrays the average secrecy rate versus numbers of iterations with *PS* = *PR* = *P* = 10 dB. By the proposed scheme, the average secrecy rate always converges within about 40 iterations. It indicates that the CSI uncertainty has a destructive effect on the secrecy rate and the BCD algorithm converges faster with larger uncertainty. Additionally, the proposed scheme achieves a better average secrecy rate with more ports of legitimate users and fewer ports of eavesdroppers, which is especially obvious in small uncertainty scenarios. It can be explained that the number of ports suggests the ability to receive or intercept the information.

Figure 3 shows the impact of a different transmit power of the proposed scheme. It can be observed that the average secrecy rate increases with the increase of transmitting power. In addition, it is observed that the security rate does not improve significantly when the transmitted power is more than 10 dB under the condition of more ports of eavesdroppers and greater CSI uncertainty. It can be explained that the increase in transmitting power increases the capacity of not only legitimate users but also eavesdroppers, resulting in a slight change in the security rate.

We compare the proposed schemes with a similar one without jamming by presenting the numerical results in Figure 4. Our proposed scheme achieves better performance in terms of the average secrecy rate, especially with lower uncertainty and higher transmit power. Therefore, to some extent, jamming can disturb the interception of eavesdroppers even with higher uncertainty.

**Figure 2.** Average secrecy rate versus the number of iterations, a comparison of different ports number and CSI uncertainty.

**Figure 3.** Average secrecy rate versus power constraint, a comparison of different antenna numbers and CSI uncertainty.

**Figure 4.** Average secrecy rate versus power constraint comparison of different schemes.

### **5. Discussion**

In this paper, the precoding jamming scheme has been proposed to enhance the security of AF relay-aided power monitoring and communication systems, where the CSI uncertainty and colluding eavesdroppers are considered. Such a system can be used in an underground mining process to guarantee the communication with management offices to ensure the safety. The scheme combined cooperative precoding for users and cooperative jamming for eavesdroppers. Numerical results have shown that the proposed scheme outperforms the scheme without jamming. Furthermore, the effectiveness of the proposed scheme with high CSI uncertainty has been proven.

**Author Contributions:** Methodology, W.M.; writing—original draft, Y.G. and L.G.; formal analysis, J.B.; validation, T.H. and Z.K.; writing—review and editing, T.H. and Z.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the National Natural Science Foundation of China under Grant 62173256.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**


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**Lintao Li 1,\*, Jiayi Lv 2, Xin Ma 1, Yue Han <sup>3</sup> and Jiaqi Feng <sup>1</sup>**


**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.

**Keywords:** communication system security; physical layer; wireless communication; precoding; wire-tap channel

### **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

**Citation:** Li, L.; Lv, J.; Ma, X.; Han, Y.; Feng, J. Design of Low Probability Detection Signal with Application to Physical Layer Security. *Electronics* **2023**, *12*, 1075. https://doi.org/ 10.3390/electronics12051075

Academic Editor: Christos J. Bouras

Received: 31 January 2023 Revised: 14 February 2023 Accepted: 17 February 2023 Published: 21 February 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

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 [18–22]. 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 undersampling 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 multicarrier 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

$$\mathbf{x}(t) = \sum\_{k=L}^{L+N-1} s(t) \cdot \boldsymbol{\alpha}\_k \exp(k \cdot j\omega\_c t) \tag{1}$$

where the scaled factor *α<sup>k</sup>* satisfied power constraint as ∑*<sup>N</sup> <sup>k</sup>*=<sup>1</sup> |*αk*| <sup>2</sup> = 1, *N* implies the total number of sub-carriers. Thus, the mean power of signal *x*(*t*) is equivalent to that of signal *s*(*t*), and *L* is the number of null subcarriers from the zero frequency to the first signal carrier. *s*(*t*) is the original modulated signal with bandwidth *ωB*. The carrier spacing can be given by *ω<sup>c</sup>* = *D* · *ωB*, where *D* is the ratio between the carrier spacing and baseband width of signal *s*(*t*). 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*(*t*) and the proposed LPD signal *x*(*t*) is shown in Figure 1. The bandwidth of *x*(*t*) is *N* times of *s*(*t*) while the power is consistent. As a result, the power spectrum density of signal *x*(*t*) will be significantly reduced, which may be even lower than the background noise provided if *N* is large enough. Furthermore, *x*(*t*) also performs sparsity in the frequency domain when *ω<sup>c</sup> ωB*. These two characteristics are similar to that of direct sequence spread spectrum and frequency hopping 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

$$r(t) = \sum\_{k=L}^{L+N-1} h\_k s(t) \cdot a\_k \exp(k \cdot j\omega\_c t) + w(t) \tag{2}$$

where *hk* 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 *hk* is considered to be 1 for all *k*. For the fading channel, the channel coefficients can be given by *hk* = |*hk*| exp(*jϕk*), which means the signal transmitted over the *kth* sub-channel is

scaled by the attenuation factor |*hk*| and phase-shifted by *ϕk*. In this work, |*hk*| is subjected to Rayleigh distribution and *ϕ<sup>k</sup>* is subjected to uniform distribution. *w*(*t*) 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*(*t*) = ∑+<sup>∞</sup> *<sup>n</sup>*=−<sup>∞</sup> *<sup>δ</sup>*(*<sup>t</sup>* <sup>−</sup> *nTs*), where *<sup>δ</sup>*(*t*) is the unit impulse function. The sampling frequency can be calculated by *fs* = 1/*Ts*.

**Figure 1.** Comparison diagram of the spectrum between *s*(*t*) and *x*(*t*). (**a**) Spectrum of modulated signal *s*(*t*). (**b**) Spectrum of the proposed LPD signal *x*(*t*).

The frequency representation of the proposed sampling process is illustrated in Figure 2. As shown in Figure 2a, signal *X*(*jω*) consists of *N* sub-band signals *Sk*(*jω*) with sub-band spacing *ωc*, and the total bandwidth of *X*(*jω*) is *Nωc*. 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*(*jω*) and *P*(*jω*). For each sub-band signal *Sk*(*jω*), a replica of *Sk*(*jω*) remains at each integer multiple of *ωs*. If the sampling rate is chosen as *fs* = *fc*, replicas of sub-band signals *Sk*(*jω*) may be aligned and added coherently, as shown in Figure 2c. The mean power of the sampled signal increases by *N* times. Consequently, *S*(*jω*) 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 *fs* = *fc*, 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)*ωc*,(*L* + *N* − 1/2)*ωc*], and the bandwidth of the pass band is *Nωc*. Then, frequency contents out of the pass band will be filtered out by the analog BPF. The SNR of signal *y*(*t*) can be given by *SNRy* = *Ps*/*NN*0*ωc*, where *Ps* 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 *pm*(*t*) = ∑+<sup>∞</sup> *<sup>n</sup>*=−<sup>∞</sup> *<sup>δ</sup>*(*<sup>t</sup>* <sup>−</sup> *nTs* <sup>−</sup> *<sup>m</sup>*Δ*Ts*) where *<sup>m</sup>* <sup>=</sup> 0, 1, 2, ··· , *<sup>M</sup>* <sup>−</sup> 1 and <sup>Δ</sup> <sup>=</sup> 1/*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 *ωB*. It's important to note that artificial noise (if it exists) will be filtered out by the LPF.

**Figure 2.** Frequency domain representation of proposed sampling method. (**a**) Spectrum of *x*(*t*). (**b**) Spectrum of sampling function. (**c**) Spectrum of sampled signal with *fs* = *fc*. (**d**) Spectrum of sampled signal with *fs* = *fc*.

For the noise component, *w*(*t*) can be written by summation of *N* sub-band noise elements as *w*(*t*) = ∑*N*−<sup>1</sup> *<sup>k</sup>*=<sup>0</sup> *wk*(*t*) exp[(*k* + *L*) · *jωct*], where *wk*(*t*) is an independent zeromean band-limited AWGN with bandwidth *ω<sup>c</sup>* and power spectrum density *N*0. After sampling, these sub-band noises are added incoherently, and the power spectrum density becomes *NN*0. Following that, the SNR of signal #*s*[*n*] can be given by

$$SNR\_d = \frac{E\left[\overline{\mathbf{s}}^2[n]\right]}{NN\_0\omega\_B} = \frac{\left|\sum\_{k=L}^{L+N-1} \sqrt{1/N} \cdot \exp(j2\Delta k\pi)\right|^2 P\_s}{NN\_0\omega\_B} \tag{3}$$

and then the receiving gain can be achieved as

$$\eta = \frac{SNR\_d}{SNR\_y} = D \cdot \left| \sum\_{k=L}^{L+N-1} \sqrt{1/N} \exp(j2\Delta k\pi) \right|^2 \tag{4}$$

As a result, the maximum receiving gain becomes *ND* if the sampling phase is synchronized perfectly when <sup>Δ</sup> <sup>=</sup> 0. Followed by the LPF, signal #*s*[*n*] can be demodulated in traditional ways.

**Figure 3.** Block diagram of receiver.

### *2.4. Complexity Analysis of Receiver*

In this section, the complexity of the proposed receiver is investigated. For a spectrum sparse signal with bandwidth *NDfB*, 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 *log*2(*L*) 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*(*t*). 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 2*NDfB* 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:


**Figure 4.** Secure communication system model.

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 Δ = 0 in what follows unless stated otherwise.

### *3.2. Sampling Clock Offset Compensation*

The sampling clock offset for the same frequency *ω<sup>c</sup>* between Alice and Bob is defined as *κω* = *ωB*,*<sup>c</sup>* − *ωA*,*c*, where *ωA*,*<sup>c</sup>* and *ωB*,*<sup>c</sup>* 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, *κω* can be estimated in step 2. Then, the LPD signal is modified as

$$\alpha\_{\boldsymbol{s}}(t) = \sum\_{k=L}^{L+N-1} s(t) \cdot \alpha\_{k} \exp[k \cdot j(\omega\_{A,\boldsymbol{c}} + \kappa\_{\boldsymbol{\omega}})t] \tag{5}$$

The central frequencies of sub-band signals are shifted according to *κω*, 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 *ωB*,*c*, sub-band signals in the transmitted signal would be aligned naturally, as shown in Figure 3. Then, modulated signal *s*(*t*) may be recovered. Taking the weighted factor *α<sup>k</sup>* and the channel coefficients *hk* into account, the SNR of sampled signal #*s*[*n*] at Bob is given by

$$SNR\_{B,d} = \frac{\left| \sum\_{k=L}^{L+N-1} a\_k h\_k \exp(j2\Delta k \tau) \right|^2 \cdot P\_s}{NN\_0 \omega\_B} \tag{6}$$

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

$$SNR\_{B,d} = \frac{\left| \begin{array}{c} \sum\limits\_{k=L}^{L+N-1} \alpha\_k h\_k \\ k=L \end{array} \right|^2 \cdot P\_s}{NN\_0 \omega\_B} \tag{7}$$

when Δ = 0. Weighted factor *α<sup>k</sup>* and channel fading coefficient *hk* can be written as 1 × *N* vectors by *α* = [*αL*, *αL*<sup>+</sup>1, ··· , *αL*+*N*−1] and **H** = [*hL*, *hL*+1, ··· , *hL*+*N*−1]. The Cauchy– Schwarz inequality [27] states that for all vectors *α* and **H** of an inner product space, the following equation holds true:

$$|<\mathfrak{a}, \mathbf{H}>|^2 \le ||\mathfrak{a}||^2 \cdot ||\mathbf{H}||^2\tag{8}$$

where < *α*, **H** > denotes the inner product of vectors *α* and **H**, and the notion || · || denotes the Euclidean norm. Moreover, the equality holds only when *α* and **H** are linearly dependent. It can be written by *α* = *λ***H**∗, where *λ* is a nonzero constant. The superscript ∗ denotes a conjugate operation. In addition, the weighted factor is constrained by ∑*L*+*N*−<sup>1</sup> *<sup>k</sup>*=*<sup>L</sup>* |*αk*| <sup>2</sup> = 1. In order to make the equality in Equation (8) hold true, the weighted factor *α<sup>k</sup>* should be given by *α<sup>k</sup>* = *h*<sup>∗</sup> *<sup>k</sup>*/||**H**||. Then, the SNR of the sampled signal at Bob can be given by *SNRB*,*<sup>d</sup>* <sup>=</sup> ||**H**||<sup>2</sup> · *Ps*/(*NN*0*ωB*). 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 *η* versus sampling phase offset Δ is shown in Figure 5. We assumed that channel coefficients for sub-band signals are independent, identically distributed, and subject to Rayleigh distribution. Let ˆ *hk* denote the estimates of *hk* that can be written by ˆ *hk* = *hk* + *h<sup>e</sup> <sup>k</sup>*. Two different scenarios are explored: (1) perfect CSI, the channel coefficients are perfectly known as *h<sup>e</sup> <sup>k</sup>* = 0; (2) imperfect CSI, *<sup>h</sup><sup>e</sup> <sup>k</sup>* is supposed to be a Gaussian random variable with zero mean, and the estimation error is defined by *<sup>ρ</sup>* <sup>=</sup> *<sup>E</sup>*{|*h<sup>e</sup> k*| 2 \* |*hk*| 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 undersampling method.

**Figure 5.** Receiving gain *η* versus sampling phase offset Δ over fading channel.

### **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 H<sup>0</sup> and H<sup>1</sup> defined as

$$\begin{cases} \ \mathcal{H}\_0: r = w \\ \ \mathcal{H}\_1: r = x + w \end{cases} \tag{9}$$

where H<sup>0</sup> represents the null hypothesis, and H<sup>1</sup> 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*int. Users are to detect whether H<sup>0</sup> or H<sup>1</sup> is true based on the test statistic *V*.

The performance of the ED method is always evaluated by two probabilities, *Pd* and *Pf a*. *Pd* implies the probability of detection that H<sup>1</sup> is accepted when H<sup>1</sup> is true, while *Pf a* is the false alarm probability that H<sup>1</sup> is assumed when H<sup>0</sup> is true. The probability density function of normalized decision statistic *Y* = 2*V*/*N*<sup>0</sup> has a central chi-square distribution with *v* = 2*T*int*W* degrees of freedom when H<sup>0</sup> is true. It can be written by

$$P\_{\mathcal{H}\_0}(Y) = \frac{1}{2^{v/2}\Gamma(v/2)} y^{(v-2)/2} e^{-Y/2} \tag{10}$$

where Γ(*u*) is Gamma function defined by Γ(*u*) = , <sup>∞</sup> <sup>0</sup> *t <sup>u</sup>*−<sup>1</sup> exp(−*t*)*dt*.

Meanwhile, the decision statistic obeys a non-central chi-square distribution with *v* degrees of freedom and non-central parameter *λ* = 2*E*/*N*<sup>0</sup> when H<sup>1</sup> is true. The *E* implies the signal energy in the time period *T*int. The PDF can be written by

$$P\_{\mathcal{H}\_1}(Y) = \frac{1}{2} \left(\frac{Y}{\lambda}\right)^{(v-2)/4} e^{-(Y+\lambda)/2} I\_{(v-2)/2} \left(\sqrt{Y\lambda}\right) \tag{11}$$

where *In*(*u*) is the Bessel function of the first kind of order *n*. Therefore, the performance of ED can be described by

$$P\_{fa} = \int\_{2V\_T/N\_0}^{\infty} P\_{\mathcal{H}\_0}(Y)dY \tag{12}$$

and

$$P\_d = \int\_{2V\_T/N\_0}^{\infty} P\_{\mathcal{H}\_1}(Y)dY \tag{13}$$

where *VT* denotes the decision threshold.

According to the central limit theorem, *<sup>P</sup>*H<sup>0</sup> (*Y*) and *<sup>P</sup>*H<sup>1</sup> (*Y*) will converge to a Gaussian distribution as *v* goes to infinity. The approximated PDF can be written by

$$P\_{\mathcal{H}\_0}(Y) \approx \frac{1}{\sqrt{2\pi}\sigma\_w} e^{-\frac{(Y-\mu\_W)^2}{2\sigma\_w^2}} \tag{14}$$

$$P\_{\mathcal{H}\_1}(Y) \approx \frac{1}{\sqrt{2\pi}\sigma\_{sw}}e^{-\frac{(Y-\mu\_{sw})^2}{2\sigma\_{sw}^2}}\tag{15}$$

where *μ<sup>w</sup>* = 2*T*int*W*, *σ*<sup>2</sup> *<sup>w</sup>* = 4*T*int*W*, *μsw* = 2*T*int*W* + 2*E*/*N*<sup>0</sup> and *σ*<sup>2</sup> *sw* = 4*T*int*W* + 8*E*/*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

$$\begin{split} P\_{fa} &= \frac{1}{\sqrt{2\pi}\sigma\_{\text{w}}} \int\_{\psi}^{\infty} \exp\left(-\frac{\left(Y - \mu\_{\text{w}}\right)^{2}}{2\sigma\_{\text{w}}^{2}}\right) dY\\ &= Q\left(\frac{\psi - \mu\_{\text{w}}}{\sigma\_{\text{w}}}\right) \end{split} \tag{16}$$

$$\begin{split} P\_d &= \frac{1}{\sqrt{2\pi}\sigma\_{sw}} \int\_{\psi}^{\infty} \exp\left(-\frac{(Y-\mu\_{sw})^2}{2\sigma\_{sw}^2}\right) dY\\ &= Q\left(\frac{\psi-\mu\_{sw}}{\sigma\_{sw}}\right) \end{split} \tag{17}$$

where *Q*(*u*) is Q function defined by *Q*(*u*) = <sup>√</sup><sup>1</sup> 2*π* , <sup>+</sup><sup>∞</sup> *u* exp(*x*2) <sup>2</sup> *dx*. Given a predetermined false alarm probability *P*ˆ *f a*, the decision threshold can be calculated by

$$
\psi^\* = \sigma\_w Q^{-1}(\hat{P}\_{fa}) + \mu\_w \tag{18}
$$

where *Q*−1(*u*) is inverse function of Q(u). Substituting Equation (18) for Equation (17), we can get

$$\begin{split} P\_d &= Q \left( \frac{\psi^\* - \mu\_{sw}}{\sigma\_{sw}} \right) \\ &= Q \left( \frac{\sigma\_w Q^{-1} (\hat{P}\_{fa}) + \mu\_w - \mu\_{sw}}{\sigma\_{sw}} \right) \end{split} \tag{19}$$

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

$$(E/N\_0)\_{Bob} = \frac{||\mathbf{H}||\_2^2 E\_s}{NN\_0} \tag{20}$$

where *Es* denotes the average symbol energy of *s*(*t*). As a result, the decision threshold at Bob's end is *ψ*∗ *Bob* <sup>=</sup> <sup>2</sup>*Q*−1(*P*<sup>ˆ</sup> *f a*) + 2, and the detection probability is

$$\begin{split} P\_{d,Bob} &= Q \left( \frac{\Psi\_{Bob}^{\*} - \mu\_{\rm sw}}{\sigma\_{\rm sw}} \right) \\ &= Q \left( \frac{Q^{-1} (\mathcal{P}\_{fa}) - (E/N\_0)\_{Bob}}{\sqrt{1 + 2 \cdot (E/N\_0)\_{Bob}}} \right) \end{split} \tag{21}$$

For the illegal user Eve, the time–bandwidth product is approximated as *T*int*W* = *ND*. Furthermore, the *E*/*N*<sup>0</sup> at Eve's end can be written by

$$(E/N\_0)\_{\text{Evv}} = \frac{\left| \sum\_{k=1}^{N} h\_k \mathcal{g}\_k \right|^2}{\left\| \mathbf{H} \right\|\_2^2} \cdot \frac{E\_s}{N N\_0} \tag{22}$$

where *gk* is the channel coefficient of the wire-tap channel that is independent of *hk*. In AWGN channel, *gk* = 1(*k* = 1, 2, ··· , *N*) and Equation (22) is simplified as (*E*/*N*0)*Eve* = *Es* \* (*NN*0). Furthermore, the decision threshold at Eve's end can be given by *ψ*∗ = 2 <sup>√</sup>*NDQ*−1(*P*<sup>ˆ</sup> *f a*) + 2*ND* and the detection probability can be written by

$$\begin{split} P\_{d, \text{Evv}} &= Q \left( \frac{\Psi^\* - \mu\_{\text{sv}}}{\sigma\_{\text{sw}}} \right) \\ &= Q \left( \frac{\sqrt{ND}Q^{-1}(P\_{fa}) - (E/N\_0)\_{\text{Evv}}}{\sqrt{ND + (E/N\_0)\_{\text{Evv}}}} \right) \end{split} \tag{23}$$

### **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*(*t*) is assumed to be a BPSK-modulated signal with a bandwidth of 10MHz, which means *fB* = 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 *Pf a* = 1*e* − 3.

**Figure 6.** Diagram of the system model in simulations.

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.

**Figure 7.** Signal detection performance of Bob and Eve over AWGN channel.

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 *α<sup>k</sup>* and channel coefficients *gk* 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.

**Figure 8.** Signal detection performance of Bob and Eve over fading 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 *SNRy* at Bob and Eve are illustrated in Figure 9. The sampling phase offset at Bob is set as Δ = 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 *γ* in the figure is defined as *γ* = *Px*/*Pj*, where *Px* denotes the transmit power of useful signals and *Pj* denotes the transmit power of artificial jamming signals. As a result, the proposed secure communication scheme is proven effective in the AWGN channel.

**Figure 9.** BER performance of Bob and Eve over AWGN channel for *D* = 5 and *N* = 4.

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 *ρ* 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 *ξ <sup>f</sup>* is also given in this figure, where *ξ <sup>f</sup>* denotes the sampling frequency offset between Alice and Bob. Simulation results show that the BER at Eve is about 0.5 even *ξ <sup>f</sup>* = 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.

**Figure 10.** BER performance of Bob and Eve over fading channel for *D* = 5 and *N* = 4.

### **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.

**Author Contributions:** Conceptualization, L.L.; formal analysis, L.L., J.L., and Y.H.; supervision, L.L.; writing—original draft, X.M., Y.H., and J.F.; writing—review and editing, L.L. and J.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Fundamental Research Funds for the Central Universities (FRF-TP-19-052A1) and the Foundation of Beijing Engineering and Technology Center for Convergence Networks and Ubiquitous Services.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflicts of interest.

### **References**


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