
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

38594318
58735
10.1038/s41598-024-58735-y
Article
Jamming precoding in AF relay-aided PLC systems with multiple eavessdroppers
Kong Zhengmin 1
Cui Jiaxing 1
Ding Li liding@whu.edu.cn

1
Huang Tao 2
Yan Shihao 3
1 https://ror.org/033vjfk17 grid.49470.3e 0000 0001 2331 6153 School of Electrical Engineering and Automation, Wuhan University, Wuhan, 430072 China
2 https://ror.org/04gsp2c11 grid.1011.1 0000 0004 0474 1797 College of Science and Engineering, James Cook University, Smithfield, QLD 4878 Australia
3 https://ror.org/05jhnwe22 grid.1038.a 0000 0004 0389 4302 School of Science, Edith Cowan University, Joondalup, WA 6027 Australia
9 4 2024
9 4 2024
2024
14 833521 11 2023
2 4 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by/4.0/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Enhancing information security has become increasingly significant in the digital age. This paper investigates the concept of physical layer security (PLS) within a relay-aided power line communication (PLC) system operating over a multiple-input multiple-output (MIMO) channel based on MK model. Specifically, we examine the transmission of confidential signals between a source and a distant destination while accounting for the presence of multiple eavesdroppers, both colluding and non-colluding. We propose a two-phase jamming scheme that leverages a full-duplex (FD) amplify-and-forward (AF) relay to address this challenge. Our primary objective is to maximize the secrecy rate, which necessitates the optimization of the jamming precoding and transmitting precoding matrices at both the source and the relay while adhering to transmit power constraints. We present a formulation of this problem and demonstrate that it can be efficiently solved using an effective block coordinate descent (BCD) algorithm. Simulation results are conducted to validate the convergence and performance of the proposed algorithm. These findings confirm the effectiveness of our approach. Furthermore, the numerical analysis reveals that our proposed algorithm surpasses traditional schemes that lack jamming to achieve higher secrecy rates. As a result, the proposed algorithm offers the benefit of guaranteeing secure communications in a realistic channel model, even in scenarios involving colluding eavesdroppers.

Subject terms

Information technology
Electrical and electronic engineering
http://dx.doi.org/10.13039/501100001809 National Natural Science Foundation of China 62173256 62373290 Kong Zhengmin Ding Li issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

The power line channel has gained significant attention in the realm of communication networks due to its utilization of existing power line infrastructure, thus avoiding the need for additional infrastructure deployment1. Power line communication (PLC) has established itself as a mature technology, finding applications in various domains such as indoor, outdoor, and in-vehicle communication systems2,3. Meanwhile, PLC system in the home automation4, smart grid5–7, smart city8 and remote sensing9 and other fields, PLC is a strong competitor of wireless communication system. In addition, the application of PLC is not limited to the above fields, in the vehicle10,11, aviation equipment12, ship13,14 and train15 communication environment, PLC has the value of application. However, the transmission of high-frequency signals over power lines, which were not originally designed for communication purposes, poses challenges in terms of considerable attenuation over long distances. In addition to attenuation, other detrimental factors such as multipath effects resulting from impedance mismatching, distortion caused by impulsive noise16–18, and coupling loss (when PLC is connected to the electric power grid) further degrade the quality of data transmission in power line system19.

In addition to the aforementioned challenges faced by power line communication (PLC), the strict limitations on transmit power spectral density imposed by electromagnetic compatibility regulations further hinder its coverage capabilities. These limitations overlook the negative factors affecting PLC, thereby constraining its potential for reliable and high-capacity communication over long distances20. As a result, achieving robust and efficient PLC under unpredictable channel conditions becomes even more challenging. To overcome these limitations and enhance the reliability and reach of PLC networks, researchers have explored the application of relay-aided communication21. By leveraging relay nodes, which serve as intermediaries between the source and destination, relay-aided PLC offers promising opportunities for long-distance transmission in the high-frequency band22.

Previous research efforts have made significant strides in the development of long-distance transmission techniques for power line communication (PLC) systems employing relay nodes23–26. However, it is important to note that the majority of these studies primarily concentrate on the decode-and-forward (DF) relaying protocol25,26, with only a limited number exploring amplify-and-forward (AF) relay-aided PLC systems24. In comparison to DF relaying, AF approaches utilize simpler relay nodes that do not necessitate additional time for decoding, quantization, or digital signal processing. Through the straightforward process of amplifying and forwarding the received signals, AF relays can reduce the overall end-to-end transmission delay. Consequently, the performance of PLC systems based on AF relaying schemes has become an intriguing area of investigation, warranting further exploration.

The establishment of a reliable power line communication system necessitates the improvement of coverage and the guarantee of data rates, both of which are challenging due to power and bandwidth limitations27. In addition to the introduction of relay nodes, one practical approach to enhance data rates within a given channel quality is to improve spectral efficiency. In this regard, in-band full-duplex (IBFD) technology has showed up as a viable solution, originally explored in the realm of wireless communication28–31. IBFD enables simultaneous transmission and reception of signals within the same frequency band32. Furthermore, the application of IBFD can enhance the overall relaying capacity of a multi-hop network by enabling full-duplex relaying. This approach mitigates the repeating delays associated with each relay node, effectively doubling the data throughput33. IBFD has recently garnered significant attention in the context of PLC34. As a result, the development of IBFD technology holds great promise in enhancing the performance of long-distance PLC systems35.

Another significant issue in communication is to enhance security against potential negative factors36,37. The cheap and ubiquitous power line system is not perfect. How to enhance the security of the communication system based on power line is a serious problem with the electromagnetic radiation in the power line and the existence of malicious wired users38,39. Despite the incorporation of relays and in-band full duplex (IBFD) communication, long-distance power line communication (PLC) systems are inherently more vulnerable to security risks compared to conventional wireless communication. These vulnerabilities arise from factors such as impedance mismatch and non-Gaussian noise, which are characteristic of power line channels. Furthermore, in a relay-aided PLC system, the presence of malicious users who can potentially eavesdrop on messages transmitted between the source and relay nodes poses significant security threats. The security of such systems is further compromised by imperfect channel state information (CSI), which can deteriorate the overall security posture. It is important to note that power line channels and wireless channels share similarities, including frequency selectivity, frequency-dependent attenuation, and an open nature that allows any wireless or PLC device to intercept the exchanged messages. Consequently, both types of communication channels are susceptible to exploitation by malicious users40,41. In light of these shared vulnerabilities, it becomes possible and necessary to leverage well-investigated wireless communication technologies and security mechanisms in the context of PLC. Existing research has already demonstrated the feasibility and importance of applying established wireless communication technologies to enhance the security of PLC systems24,42.

The security of power line communication (PLC) can be achieved through two primary approaches: cryptographic protocols and physical layer security (PLS). While cryptographic protocols are effective in securing data transmission, PLS leverages the quality of the channel to protect against eavesdropping attacks. Compared to cryptography-based methods, PLS techniques have lower complexity and have attracted recent research interests43. The concept of PLS was initially introduced in the 1970s44–46 and has since undergone extensive research. Initially, studies focused on the degraded wiretap channel44, followed by investigations into the nondegraded wiretap channel45,46. More recently, research has delved into the analysis of fading wiretap channels47,48 and multiple-input–multiple-output (MIMO) wiretap channels49–51. PLS has also found applications in other communication scenarios, such as fiber optical networks52. In the context of PLC, research efforts have explored the application of PLS in different system configurations. Studies initially concentrated on PLS for single-input, single-output (SISO) PLC systems51, followed by investigations into PLS techniques for MIMO-based PLC systems53.

At present, there are some related fields of research as showed in Table 1. For example, a comprehensive study examines the ergodic secrecy achievable rate of an in-home system in the presence of an adjacent malicious wireless device54. Subsequently, a separate investigation analyzes the effective secrecy throughput utilizing an experimental dataset55. In order to enhance security, a scheme is proposed that incorporates artificial noise to improve the hybrid channel56. Furthermore, the research explores the PLS of cooperative relaying PLC systems under the presence of an eavesdropper57. To address channel noise, a scheme based on full-duplex communication with artificial noise injection is introduced58. Additionally, the research delves into the analysis of secrecy rates for a MIMO system, employing the IBFD jamming technique to secure the transmitted data59. The effective secrecy throughput is studied under the scenario of passive colluding wireless eavesdroppers, considering both in-home and broadband PLC systems60. Furthermore, PLS in the presence of passive eavesdropping is investigated, taking into account the impact of Bernoulli–Gaussian impulsive noise61. Finally, the study provides a thorough analysis of secrecy rate, secrecy outage probability, and secrecy capacity of a broadband system62.Table 1 Overview of the existing literature.

Contributions	This work	54	55	56	57	58	60	61	59	62	
Log-normal channel model	MK model	✓	✓	✓	✓	✓	✓	✓		✓	
Bernoulli–Gaussian impulsive noise	✓							✓		✓	
Imperfect CSI	✓		✓	✓	✓			✓	✓	✓	
Multi-hop system	✓				✓			✓			
Relay	AF				AF			AF			
Multiple Eves	✓			✓		✓	✓				
Proposing scheme	✓			✓	✓	✓	✓		✓	✓	

To the best of our knowledge, there is currently a research gap regarding relay-aided PLS in the presence of multiple eavesdroppers. Motivated by previous studies on cooperative precoding to enhance channel quality for legitimate users63–65 and cooperative jamming to impair channel quality for unauthorized users in wireless communication66–68, we propose a novel cooperative jamming and precoding PLS scheme for IBFD DF relay-aided PLC systems. Unlike other studies assuming perfect channel information69,70, our proposed approach takes into account imperfect channel information and aims to design the precoding matrices at the legitimate nodes. The objective is to maximize the secrecy rate of AF relay-aided PLC systems with imperfect channel information in the presence of multiple eavesdroppers. To achieve this, we utilize an efficient BCD algorithm to iteratively optimize the precoding matrices. By jointly optimizing these matrices, we aim to enhance the secrecy rate by leveraging the cooperative capabilities of the relay and introducing intentional jamming to disrupt the eavesdroppers’ reception. The proposed scheme addresses the challenges posed by multiple eavesdroppers and imperfect channel information, which are critical considerations in practical PLC systems.

To ensure the proposed scheme accurately models real-world power line channels, this paper conducts a characterization of the statistical MIMO PLC channel based on an analysis of a set of experimental field measurements71. The analysis takes into account various factors that impact data transfer, including fading effects, multipath propagation, and signal frequency. In addition, the noise in PLC is modeled as Bernoulli–Gaussian impulsive noise72. Previous research on PLS has considered different scenarios involving imperfect knowledge of CSI, ranging from passive eavesdroppers with unknown CSI55–57,61,62 to those with estimation errors59,73. We consider a system with globally imperfect channels to provide a more comprehensive and realistic approach. In this scenario, all CSIs are partially known by the legitimate nodes in the PLC system due to channel estimation errors. Furthermore, we extend the study from a single-eavesdropper scenario to a multiple-eavesdropper scenario, considering two types of eavesdropping scenarios: non-colluding and colluding. In the non-colluding scenario, the eavesdroppers operate independently and do not share information, while in the colluding scenario, all eavesdroppers collaborate to intercept the legitimate transmission. Specifically, we investigate the severe colluding case to gain deeper insights into the security performance of the proposed scheme.

The subsequent sections of this paper are organized as follows. “System model” presents a detailed description of the system model. In “Simulation and results”, the proposed optimization problem is proved to be solvable by a series of transformations. net section showcases the numerical results obtained from the proposed scheme. Finally, “Conclusion” concludes the findings and provides insights for future research directions.

Notations: To simplify the formulation, we denote AAHasAK and the vec(A) denotes the vectorization of a matrix A.

System model

We consider a secure transmission system as shown in Figs. 1 and 2, where a source tries to transmit confidential information to legitimate users via an FD relay in the presence of multiple eavesdroppers eavesdropping in different time phases. More specifically, we assume all the eavesdroppers can be divided into two sets by their eavesdropping time. The first one can only eavesdrop on messages in the first time phase and the second one can only eavesdrop messages in the second time phase. Because the relay runs in the full-duplex model, self-interference should be involved. In addition, considering the huge attenuation of signals over long distances in the PLC system, the direct link from the source to the legitimate users can be ignored. In the system, the source(S), the relay(R) and the users(D) are involved NS, NR and ND ports, respectively. Two sets of eavesdroppers are equivalent to two multiple-ports eavesdroppers (E1 and E2). Here, both sets of eavesdroppers are equipped with NE ports.

In the system, channels are described by channel transfer function (CTF) Hij,k as the matrix of coefficients, where i, j and k denote the transmitter, receiver and transmission time phases, respectively. Note that HRR,1 refers to the self-interference matrix , and Hij,k stays constant in the transmission process. In this paper, considering the multipath effect, frequency-selective effect, and time delay of power lines, this paper adopts the MK model71 to model the channel, with channel noise characterized as a Bernoulli–Gaussian pulse noise. Due to the imprecision of channel estimation/feedback and the stealthiness of eavesdroppers, it is challenging for the transmitter to obtain accurate channel state information between the receiver and the transmitter. Therefore, we consider all channels to be imperfect channels.

The imperfect channels are described by the deterministic uncertainty model:1 Hij,k∈Hij,k=Hij,k|Hij,k=H¯ij,k+Δij,k,Δij,k≤δij,k,

where Δij,k and H¯ij,k denote the CTF error and the mean CTF, respectively.Figure 1 Phase 1 in PLC system.

The PLC system is operated over two-time phases with the relay in the AF model. In the two time slots, confidential information is transmitted via a source and a relay. While one of the legitimate nodes propagates the information forward, the other sends jamming signal to deal with possible eavesdroppers. By considering the possible worst-case scenario, the model introduces two groups of eavesdroppers who eavesdrop on two time slots respectively and considers the effect of self-interference at the relay.

In the first time phase, the source broadcasts confidential messages to the relay and the messages are inevitably eavesdropped by E1. More accurately, the confidential messages are modeled as symbols S∈CN(0,1) mapped by vector W∈CNS×1:2 XS=WS,

Next, we consider the messages emitted by the relay which only emits jamming:3 XR=VZ,

where V∈CNR×1 and Z∈CN(0,1) denote jamming vector and symbol.

With self-interference, the messages received by the relay can be formulated:4 YR1=HSR,1WS+HRR,1VZ+nR1,

where nR1 is actually PLC noise based on the Bernoulli–Gaussian noise model at the relay.

Meanwhile, E1 receives the messages from both the source and the relay:5 YE1=HSE,1WS+HRE,1VZ+nE1,

where nE1 is Bernoulli–Gaussian noise at E1.Figure 2 Phase 2 in the PLC system.

In the second time phase, the source emits the jamming precoded from the relay:6 XS2=AZ,

Meanwhile, the relay is working in the AF mode so it amplifies and forwards the messages it received to both the users and E2:7 XR2=GYR1=G(HSR,1WS+HRR,1VZ+nR1),

where G∈CNR×NR denotes the amplifying matrix.

Because of the distance, the jamming from the source will not interrupt users but E2, i.e.8 YE2=HSE,2AZ+HRE,2G(HSR,1WS+HRR,1VZ+nR1)+nE2

9 YD=HRD,2XR2+nD=HRD,2G(HSR,1WS+HRR,1VZ+nR1)+nD,

where nD is Bernouil–Gaussia noise at the users and nE2 is Bernouil–Gaussia noise at E2.

Above all, we can calculate the signal-to-noise ratio (SNR) at all the receivers.10 γD=(HRD,2GHSR,1W)KQD-1,whereQD=(HRD,2GHRR,1V)K+σR2(HRD,2G)K+σD2I

11 γE1=(HSE,1W)KQE1-1,whereQE1=(HRE,1V)K+σE2I

12 γE2=(HRE,2GHSR,1W)KQE2-1,whereQE2=(HSE,2A)K+(HRE,2GHRR,1V)K+σR2(HRE,2G)K+σE2I

and σi is the amplitude of the corresponding Bernouil–Gaussia noise ni.

The achievable rate of the legitimate users is as follows:13 RD=logI+γD,

However, the situation for eavesdropping is more complicated, because the eavesdroppers can collude or not. In the colluding case, the eavesdroppers can utilize maximum ratio combining (MRC) to combine their received information. In this typical collusion strategy, the eavesdropping SNR is the sum of all the eavesdroppers. So the achievable rate of the eavesdroppers is as follows:14 RE=logI+γE1+γE2

Jamming precoding scheme

In this work, the goal is to maximize the secrecy rate of the system. With the transmit power constraint, the optimization problem can be formulated as follows.15 maxW,V,A,GminHij,k∈Hij,kRD-RE

16 s.t.W2⩽PS,A2⩽PS,V2⩽PR,tr((GHSR,1W)K+(GHRR,1V)K+σR2GK)⩽PR∀Hij,k∈Hij,k

Because of the high non-convexity of the function log·, the problem is hard to solve. To make the problem solvable, (15) is transformed into an equivalent form through WMMSE algorithm, which can be solved with the BCD algorithm. We first introduce WMMSE algorithm.

Lemma174: Define the MSE matrixM^≜(DH-I)K+DRDH

17 -RE=logQ-logQ+PK=logI+σE-2(HRE,1V)K00σE-2[(HSE,2A)K+(HRE,2GHRR,1V)K+σR2(HRE,2G)K]-logI+σE-2(HRE,1V)K00σE-2[(HSE,2A)K+(HRE,2GHRR,1V)K+σR2(HRE,2G)K]+σE-1P1σE-1P2K=logI+σE-2(HRE,1V)K⏟CE1+logI+σE-2[(HSE,2A)K+(HRE,2GHRR,1V)K+σR2(HRE,2G)K]⏟CE2+-logI+σE-2(HRE,1V)K00σE-2[(HSE,2A)K+(HRE,2GHRR,1V)K+σR2(HRE,2G)K]+σE-1P1σE-1P2K⏟ME3⏟CE3

where R≻0. Then we have18 -logM=maxS≻0logS-tr(SM)+tr(I)logI+R-1HK=maxS≻0,DlogS-tr(SM^)+tr(I)

Furthermore, auxiliary matrices Si,Mi,Di are introduced to reformulate the part of log· in the objective function in (15) as follows.19 RD=maxSD≻0,DDlogSD-tr(SDMD)+tr(I)

where20 MD=(DDHRD,2GHSR,1W-I)K+DDQDDDH

However, (14) is hard to explicit the Lemma1 directly. As a result, we need to transform (14) in a more compatible form.21 logI+γE1+γE2=logI+PKQ-1

whereQ=QE100QE2,P=P1P2=HSE,1WHRE,2GHSR,1W

Then we have (17), where RE are divided as CE1, CE2 and CE3 for subsequent transformation. Thus, in order to formulate the achievable rate of the eavesdroppers in the colluding case, CE1 and CE2 is equivalent to22 CE1=maxSE1≻0,DE1logSE1-tr(SE1ME1)+tr(I)

23 CE2=maxSE2≻0,DE2logSE2-tr(SE2ME2)+tr(I)

whereME1=(DE1HRE,1V-I)K+σE2DE1KME2=(DE21HSE,2AX+DE22HRE,2GHRR,1VX+σRDE23HRE,2G-I)K+σE2DE21K+DE22K+DE23K

24 f≜logSD-tr(SDMD)+logSE1-tr(SE1ME1)+logSE2-tr(SE2ME2)+logSE3-tr(SE3ME3)

25 g=ΔlogSD-βD+logSE1-βE1+logSE2-βE2+logSE3-βE3

Note the decomposition DE2=DE21DE22DE23 and X=10∈C1×NR.

Then to solve CE3, we also apply Lemma1 :26 CE3=maxSE3≻0logSE3-tr(SE3ME3)+tr(I)

After substituting (22)–(26) into (15), the secrecy rate of the system with the colluding eavesdroppers can be rewritten as27 maxW,V,A,G,Si≻0,DiminHij,k∈Hij,kf(W,V,A,G,Si,Di),s.t.(16)

where f(W,V,A,G,Si,Di) is defined in (24).

The max-min problem and constrain tr((GHSR,1W)K+(GHRR,1V)K+σR2GK)≤PR can be transformed into an optimization problem by introducing constraints with slack variables βi as follows.28 tr(SiMi)≤βi,∀Hij,k∈Hij,k

The problem (27) can be further transformed as29 maxW,V,A,G,Si≻0,Dig(W,V,A,G,Si,Di),s.t.(16),(26)

where g(W,V,A,G,Si,Di) is defined in (25).

However, (25) is still convex because of the semi-infinite constraints (28). For i=D, tr(SDMD) can be rewritten as30 tr(SDMD)=vec(FD(DDHRD,2GHSR,1W-I))vec(FDDDHRD,2GHRR,1V)vec(σRFDDDHRD,2G)vec(σDFDDD)⏟ϕD2

where SD=FDHFD and the equality tr(AK)=vec(A)2 is applied.

Especially note that for i=E3, to obtain similiar form as (30), FE3 should be divided as31 FE3=FE31FE32,FE31,FE32∈C2NE×NE

So we have32 tr(SE3ME3)=vec(FE31)vec(FE32)vec(σE-1FE31HRE,1V)vec(σE-1FE32HSE,2A)vec(σE-1FE32HRE,2GHRR,1V)vec(σE-1σRFE32HRE,2G)vec(σE-1FE31HSE,1W)vec(σE-1FE32HRE,2GHSR,1W)2

Focusing on the uncertain CTF, (30) can be rewritten as follows.33 ϕD=ϕ¯D+∑jΩDjvec(Δj)⏟ΔD+∑kαkvec(Δk1)vecH(Δk2)⏟Δ~D

where the identity vec(ABC)=CT⊗AvecB is applied. ΔD and Δ~D is the linear part and the quadratic part of the CTF uncertainty, respectively. Actually the quadratic part is negligible. Then, we only consider asymptotic form of ϕD as34 ϕD=ϕ¯D+∑jΩDjvec(Δj)⏟ΔD

where35 ΩDSR,1=WT⊗FDDDH¯RD,2G000,ΩDSD,2=(GH¯SR,1W)T⊗FDDD(GH¯RR,1V)T⊗FDDDσRGT⊗FDDD0,ΩDRR,1=0VT⊗FDDDH¯RD,2G00

For other situations and for the constraint tr((GHSR,1W)K+(GHRR,1V)K+σR2GK)≤PR, similiar formulas can be obtained through the same method. While the power constraint does not involve any quadratic part of the CTF uncertainty, so the original problem constraints are not relaxed.

With (30) and (34) and by exploiting the Schur complement lemma75, (28) can be rewritten as matrix inequality .36 βDϕ¯DHϕ¯DI≻-0ΔDHΔD0

The constraint (36) still contains the uncertainty ΔD. The sign-definiteness lemma is applied to eliminate this uncertainty.

Lemma 276: Given a Hermitian matrix A and arbitrary matrices pair Pi,Qi,i∈1,2,…,N, the semi-infinite Linear Matrix Inequality (LMI)37 A≻∑iNPiHYiQi+QiHYiHPi,Yi≤δi

holds if and only if there exist nonnegative real numbers λ1,λ2,…,λN such that38 A-∑i=1NλiQiHQi-δ1P1H⋯-δNPNH-δ1P1δ1I⋯0⋮⋮⋱⋮-δNPN0⋯δNI≻0

39 h≜2logFD-βD+2logFE1-βE1+2logFE2-βE2+2logFE3-βE3

Appropriately choose the parameters as below40 AD=βDϕ¯DHϕ¯DI,QD1,QD2,QD3=[-1 mathvariant="bold">0],PD1=0ΩDSR,1H,PD2=0ΩDSD,2H,PD3=0ΩDRR,1H

Apply lemma 2 and insert (40) to (36) , and we have41 βD-λD1-λD2-λD3ϕ¯DHϕ¯DIθDHθDdiag(λD1I,λD2I,λD3I)≻0

where θD=-[δDSR,1PD1T,δDSD,2PD2T,δDRR,1PD3`T]T. Similarly, other inequalities tr(SiMi)≤βi can be transfered into the same form as following.42 βi-∑k=ljλkϕ¯iHϕ¯iIθiHθiλlI,⋯,λjI≻0

With all components, the problem is equivalent to43 maxW,V,A,G,Fi≻0,Di,λi≥0,βih(W,V,A,G,Fi,Di,λi,βi),s.t.(16),(40),(41)

where the function h(W,V,A,G,Fi,Di,λi,βi) is defined in (39).

Although (43) is still non-convex. However, it is convex with respective to Fi or any one in W,V,A,G,Di. As a result, (43) can be solved via BCD algorithm as below.

Algorithm 1 Jamming precoding scheme to solve problem (43).

The variables to be optimized in the optimization problem are divided into the following groups.44 Φ0={λi,βi},Φ1=Di,Φ0,Φ2=Fi,Φ0,Φ3=W,V,A,Φ0,,Φ4=G,Φ0

By optimizing the problem in order Φ1→Φ2→Φ3→Φ4, we denote yki as the optimal value optimized to Φk in the ith iteration. Since the variables to be optimized always meet the constraints in the optimization process, the optimal value will not decrease :45 yi=y4i≥y3i≥y2i≥y1i≥yi-1=y4i-1≥y3i-1≥y2i-1≥y1i-1⋯

Obviously with constraints (16), the secrecy rate is bounded and the objective function value increases in each iteration, which proves the convergence.

Simulation and results

In the simulation parts, the statistical MIMO PLC channels are generated by formula (1) in reference71, and the specific parameters are shown in Table 2. And the noises in PLC are also modeled as a Bernoulli–Gaussian impulsive noise.Table 2 Parameters of the PLC channels71.

Path loss parameter	Model	Parameters	
AdBS1,D1	N(μA,σA)	μA=-50.1dBσA=15.6dB	
AdBSm,Dnm∈[1,2,3]	AdBSm,Dn=AdBS1,D1+N(0,σSm,Dn)	σSm,Dn=05.13.82.95.75.26.67.86.94.65.95.1dB	
AdBS4,Dn	AdBS4,Dn=0.5×AdBS1,D1-30+N(0,σS4,Dn)		
a0	Eshift(μa0,δa0)	μa0=1.04×10-2δa0=-6.7×10-3	
a1	Constant	a1=4×10-10	
K	W(αK,βK)	αK=5.7×10-2βK=57.7	
Lmax	Constant	Lmax=800m	
Λ	Constant	Λ=0.2m-1	

In this section, numerical results are presented to prove the effectiveness of precoding jamming scheme in terms of average secrecy rate. In this part, without specific definition, we consider NS=NR=ND=NE=N=2. Besides, for the simplicity, the CTF uncertaninty bound δij,k are related to corresponding determinant of mean CTF with one certain coefficient, or δij,k=μHij,k¯. Apparently, it accords with the natural assumption that CTF with larger determinant tends to be more uncertain.Figure 3 Average secrecy rate versus numbers of iterations comparison of different ports number and CTF uncertainty.

Figure 4 Average secrecy rate versus numbers of iterations comparison of different ports number and CTF uncertainty.

Figures 3 and 4 illustrate the relationship between the average secrecy rate and the number of iterations, assuming power constraints PS=PR=P=10dB. Notably, the average secrecy rate consistently stabilizes after approximately 40 iterations. This indicates that heightened CTF uncertainty detrimentally impacts the secrecy rate. Moreover, the proposed approach exhibits superior performance with an increased number of legitimate user ports and a reduced number of eavesdropper ports. This distinction becomes particularly pronounced in scenarios characterized by larger CTF uncertainty. In essence, the number of ports directly correlates with the capacity for receiving or intercepting information.Figure 5 Average secrecy rate versus power constraint comparison of different ports number and CTF uncertainty.

Figure 6 Average secrecy rate versus power constraint comparison of different ports number and CTF uncertainty.

We examine the characteristics of the proposed scheme under varying transmit power levels in Figs. 5 and 6. The analysis reveals an increase in the average secrecy rate as the transmit power is raised. However, beyond a transmission power of 10 dB, especially in scenarios with more eavesdropper ports and increased CTF uncertainty, the secrecy rate experiences only marginal improvement. This is attributed to the fact that elevating transmit power enhances not only the capacity of legitimate users but also that of eavesdroppers colluding to boost their eavesdropping rate. Consequently, this simultaneous enhancement leads to only a slight alteration in the overall secrecy rate. Additionally, when the numbers of legitimate user ports and eavesdropper ports both increase from 2 to 3, there is a notable decrease in the average secrecy rate. This observation suggests that in collusion scenarios, the expansion of the eavesdroppers’ port number has a more substantial impact on the PLC system than the growth in the number of legitimate users’ ports.Figure 7 Average secrecy rate versus power constraint comparison of different schemes.

We assess the influence of jamming by showcasing the numerical outcomes of our proposed schemes alongside a comparable one lacking jamming signals in Fig. 7. This figure shows improvements achieved by the proposed algorithm compared to the traditional one. As shown in Fig. 7, the proposed algorithm achieves higher secure rates than the traditional algorithm when μ=0.05 or μ=0.075, and the advantage increases with increasing power. Specifically, when μ=0.05 and the transmission power is 20 dB, the secure rate of the proposed algorithm can reach 0.85 bit/s/Hz, while the traditional algorithm achieves 0.61 bit/s/Hz. In contrast to the jamming-free scheme, our proposed approach demonstrates superior performance, particularly regarding the average secrecy rate, especially under conditions of lower CTF uncertainty and increased transmit power. This suggests that, to a certain degree, jamming has the capability to disrupt the interception efforts of eavesdroppers, even in scenarios characterized by higher CTF uncertainty.Figure 8 Average secrecy rate versus different eavesdropping ports number.

Figure 8 depicts how the ports of eavesdroppers affect the PLC system, where NS=NR=ND=2. It suggests the ability of eavesdroppers increases with the growth of their ports, especially in few ports.

Conclusion

The paper introduces a precoding jamming scheme aimed at bolstering the security of AF relay-aided PLC systems when faced with the challenge of multiple colluding eavesdroppers, while also considering CTF uncertainty. The numerical results unequivocally establish the superiority of our proposed scheme compared to a jamming-free alternative. Notably, the effectiveness of the proposed scheme is underscored, especially in scenarios characterized by elevated CTF uncertainty.

Author contributions

J.C. and Z.K. conceived study conception and design. J.C. and Z.K. conceived the experiments. J.C. and Z.K. and L.D. made draft manuscript preparation. J.C. and Z.K. conducted the experiments. L.D., T.H. and S.Y. analysed the results. All authors reviewed the manuscript.

Data availability

The datasets generated and/or analysed during the current study are available in the github repository, https://github.com/zilongmi/Jamming-Precoding-in-AF-Relay-aided-PLC-Systems-with-Multiple-Eavessdroppers.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

These authors contributed equally: Zhengmin Kong, Jiaxing Cui, Tao Huang and Shihao Yan.
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