Analyzing Physical-Layer Security of PLC Systems Using DCSK: A Copula-Based Approach

This study analyzes the physical layer security (PLS) performance of a differential chaos shift keying (DCSK) modulation-based Power Line Communication (PLC) system by exploiting the novel Farlie-Gumbel-Morgenstern (FGM) Copula approach. A power line Wyner’s wiretap channel model is investigated, where the main channel and the wiretap channel are assumed to be correlated and Log-normally distributed. The Gamma approximation to the Log-normal distribution is employed to simplify the computation. Concurrently, the PLC channel noise is modeled as a Bernoulli-Gaussian random process. Utilizing a Copula based approach to model the dependence among the correlated PLC channels, the PLS performance of the PLC system is evaluated in terms of the secure outage probability (SOP) and the strictly positive secrecy capacity (SPSC). It is revealed through the asymptotic SOP analysis that the secrecy diversity order depends on the shaping parameter <inline-formula> <tex-math notation="LaTeX">$(m_{\gamma _{M}})$ </tex-math></inline-formula> of the main channel. We also propose an algorithm to maximize the secrecy throughput under SOP constraints. Based on the insights from this analysis, it has been seen that the SOP performance degrades when the value of the dependence parameter <inline-formula> <tex-math notation="LaTeX">$(\theta)$ </tex-math></inline-formula> increases. Also, the secrecy throughput performance improves with a lower optimal threshold value of the signal-to-noise ratio (SNR), <inline-formula> <tex-math notation="LaTeX">$\gamma _{\textrm {th}}$ </tex-math></inline-formula>. Furthermore, some other insightful observations are presented by studying the impact of different parameters such as spreading factor <inline-formula> <tex-math notation="LaTeX">$(\beta)$ </tex-math></inline-formula>, impulsive noise occurrence probability <inline-formula> <tex-math notation="LaTeX">$(p)$ </tex-math></inline-formula>, transmitted power <inline-formula> <tex-math notation="LaTeX">$(P_{T})$ </tex-math></inline-formula>, and impulsive noise index <inline-formula> <tex-math notation="LaTeX">$(K)$ </tex-math></inline-formula>.


I. INTRODUCTION
O VER the past years, the fourth industrial revolution, also named Industry 4.0, has amalgamated modern information and communication technologies (ICTs) to design a complete, efficient, reliable, robust, optimized, and smart industrial system. Industry 4.0 mainly consists of cyber-physical systems and the Internet of Things (IoT) [1], where multiple devices, machines, vehicles, and buildings connect to each other via the integration of various ICTs. In Industry 4.0, ICTs are required to offer a very good quality-of-service (QoS) for the last mile access ("to the home") and the last inch access ("in the home") [2]. QoS covers various key performance indicators (KPIs) such as better network connectivity, high data rates, low latency, high network coverage, low maintenance and installation cost, cybersecurity, etc. In this context, Power Line Communication (PLC) is a suitable contender for fulfilling the above mentioned KPIs in Industry 4.0. PLC is a wireline technology that allows an exchange of digital data over the existing electric wires or power distribution networks. The notable advantage of PLC lies in its use of the existing ubiquitous wired infrastructure of the power grid, making it a low-cost technology and offering high network coverage. High data rates can be supported by PLC as demonstrated in HomePlug Powerline Alliance (HomePlug) and IEEE P1901 standards for home networking [3]. Moreover, PLC can play a considerable role in IoT applications including smart buildings, smart vehicles, smart grids, and smart industries [3]. Consequently, due to its appealing characteristics PLC technologies play a significant role in Industry 4.0 and IoT networks.
have analyzed and compared the PLS performance of a single-input single-output (SISO) PLC system and a wireless system. It was found that the secrecy rate is lower in PLC systems than in wireless communication systems. In [20], the PLS of a multi-input multi-output (MIMO) PLC system has been evaluated. Here, it was established that MIMO PLC is more secure than SISO PLC systems. The secrecy performance of a cooperative PLC system was investigated over correlated Log-normal communication channels in the presence of a passive eavesdropper in [21]. Similarly, the authors of [22] analyzed the PLS performance of PLC systems in terms of average secrecy capacity (ASC), secure outage probability (SOP), and strictly positive secrecy capacity (SPSC) for independent and correlated channels.
In order to improve the security of communication systems, researchers have also explored different secure modulation schemes. Chaos-based modulation techniques have been widely studied for their impactful advantages over other traditional modulation schemes [23], [24], [25]. Chaotic signals are easier to generate, non-periodic, wide-band, and possess good auto/cross-correlation properties. Furthermore, chaotic modulation schemes provide robustness to multipath fading environments [26], jamming resistance along with a low probability of interception (LPI) [27], and secure communications [28]. Among chaos-based schemes, coherent chaos shift keying (CSK) and non-coherent differential CSK (DCSK) are popular schemes that have been broadly researched in recent years due to the advantages mentioned above. In the literature, the first study of chaos-based modulation schemes was reported in [29], where permutationbased DCSK and multiple-access DCSK systems were analyzed to enhance data security. In [30], [31], [32], the authors have used the concept of eigen beamforming, spacetime coding, and multi-carrier to enhance security of CSK and DCSK schemes. In [28], the anti-synchronization CSK (ACSK) scheme was realized and analyzed using the generalized Lorenz system's anti-synchronization properties. Moreover, it was shown that the resulting ACSK communication scheme has the potential of providing a high degree of security with a low receiver complexity. Furthermore, the PLS performance of CSK and DCSK were evaluated over additive white Gaussian noise and Rayleigh fading channels in [33]. Moreover, generalized correlation delay shift keying (GCDSK) is proposed in [34], where the authors compared the BER performance of the GCDSK with the CDSK scheme and DCSK scheme.
In the past few years, PLC systems have been designed considering many traditional modulation schemes. Meanwhile, the DCSK modulation scheme has been introduced in PLC systems for its robustness. In [25], the performance of DCSK systems over the multipath power line channel model was investigated in the presence of background, impulsive, and phase noise, where the authors concluded that the DCSK schemes perform better than direct sequence differential phase shift keying (DS-DPSK) and the direct sequence code division multiple access (DS-CDMA) schemes. Similarly, the authors of [35] analyzed the performance of M-ary DCSK (M-DCSK) with background, impulsive, and phase noises, over narrowband PLC, where it was shown that M-DCSK outperforms direct sequence M-ary differential phase-shift keying (DS-MDPSK). Further, the authors of [35] extended their work in [36] where the replica piecewise (RP)-MDCSK system was designed and studied for PLC with asynchronous impulsive noise. Moreover, the authors demonstrated that RP-MDCSK has a significantly lower bit error rate (BER) than DCSK and DS-MDPSK.
Copulas are novel mathematical functions that provide an easy, flexible, and alternate approach to model correlated multivariate distribution functions in terms of their marginal distributions and mutual dependencies. Copulas have been utilized in statistics, economics, survival analysis, image processing, machine learning, and a wide range of engineering applications [37], [38]. Recently, Copula theory has been explored by researchers in the field of communication channel modeling [39]. Copula-based approaches offer excellent flexibility in mathematical modeling and simplification in computational load compared to the non-Copula-based approaches [40]. In [41], the authors utilized the Copula method to obtain the capacity of a correlated ergodic MIMO channel over Nakagami-m fading, where three types of Copulas were used, i.e., normal, Clayton, and t Copula for channel parameter estimation. In [39], the authors computed a closed-form expression for the outage probability under cross-correlated Weibull fading using selection combining reception technique by exploiting the survival-Gumbel Copula approach. Similarly, the authors in [42] computed the average symbol error rate and average BER for PLC systems using a quadrature phase shift keying (QPSK) modulation scheme over Nakagami-m distributed additive background noise, wherein the Copula approach was used to analyze dependence among decision variables. Furthermore, the authors in [43] utilized the Farlie-Gumbel-Morgenstern (FGM) Copula method to establish the closed-form expressions for the outage probability and coverage region over correlated Rayleigh fading multiple access channels. In [44], the authors analyzed the PLS performance of a correlated Rayleigh distributed wiretap fading channel, where closedform expressions for the ASC, SOP, and secrecy coverage region (SCR) were derived using the FGM Copula approach.
Motivation: It is important to develop robust, reliable, and trustworthy PLC communication networks for Industry 4.0 and IoT applications. Since PLC technology is broadcast in nature, it becomes crucial to analyze the security of PLC systems for reliable deployment in Industry 4.0 and IoT networks. Contrary to the aforementioned research works [18], [19], [20], [21], [22], where the investigation of the PLS performance of PLC systems is limited to the use of traditional modulation schemes and conventional statistical methods; in this work, we propose a DCSK modulation scheme to investigate the secrecy performance of PLC systems. Furthermore, the correlation between the main PLC channel and the eavesdropper PLC channel is mathematical modeled by exploiting a novel FGM Copula approach. Different from [27], [28], and [32], where it is demonstrated that chaos-based modulation schemes have the ability to provide secure communications against the eavesdropper in wireless communication scenarios; in this work, we employ DCSK modulation in PLC scenarios to analyze the secrecy performance of PLC systems. In contrast to [25], [35], [36], where DCSK modulation was utilized to improve the BER performance of PLC system; in this research, DCSK modulation is utilized to enhance the secrecy performance of PLC systems. Further, contrary to [39], [41], [42], [43], [44], we analyze the PLS performance of a DCSK-based PLC system for correlated Log-normally distributed channels corrupted by additive Bernoulli-Gaussian noise, where the FGM Copula-based approach facilitates the mathematical modeling of PLS performance metrics. We utilize Copula-based methods as they have been empirically proven to provide an easy mathematical modeling and help reduce the computation of jointly distributed multivariate functions [39], [40]. Thus, motivated by the aforementioned works in the literature, we perform the secrecy analysis of a DCSK-based PLC system by exploiting FGM Copula approach.
Contributions: In this work, we investigate the PLS performance of a DCSK-based PLC system by utilizing a novel FGM Copula approach, where a power line Wyners wiretap channel model is analyzed to compute the secrecy between the main channel and the wiretap channel. Both PLC channels are considered to be Log-normally distributed and correlated, and the Bernoulli-Gaussian model characterizes the PLC channel noise. Further, the PLS performance is numerically expressed in terms of SOP and SPSC. To obtain useful insight into the PLC system, the asymptotic SOP analysis is conducted. Furthermore, an algorithm is presented to maximize the secrecy throughput under SOP constraints.
To the best of our knowledge, no previous works have considered Copula theory to investigate the PLS performance of DCSK-based PLC systems. The main contributions of this work are summarized as follows: • We propose combining the DCSK modulation scheme and FGM Copula approach to investigate the PLS performance of the considered PLC system. The DCSK modulation scheme is utilized to enhance the security of the considered system. Moreover, the FGM Copula approach eases the numerical analysis and shows statistical dependence between the correlated PLC channels. • To get deep insights into the SOP analysis, the asymptotic behavior of the SOP is investigated, where useful insights, such as diversity order and coding gain of the considered PLC system are revealed. It is shown that the secrecy diversity order depends on the shaping parameter (m γ M ) of the main channel. • In order to comprehensively investigate the secrecy performance, we formulate an optimization problem to maximize the secrecy throughput under SOP constraints. We also propose an algorithm to obtain the optimal threshold value of the signal-to-noise ratio (SNR) for SOP. • We present a comprehensive investigation of the effect of different parameters, such as the spreading factor (β), transmitted power (P T ), impulsive noise occurrence probability (p), and dependence parameter (θ ) on the PLS performance metrics of the considered PLC system. We also compare the SOP performance of the proposed system with GCDSK, DS-DPSK, and CDMA. It is found that the proposed system performs better than GCDSK and CDMA for certain values of β. Organization of the Paper: The rest of the paper is arranged as follows: Section II presents the considered PLC system model, where the DCSK system, received power, PLC channel and noise models, and the characterization of the instantaneous SNR are described. After reviewing Copula theory, the detailed secrecy analysis of the proposed PLC system in terms of SOP and SPSC is presented in Section III. Further, the asymptotic analysis of the SOP is conducted. We also carry out the secrecy throughput optimization analysis in Section III. In Section IV, we demonstrate and discuss the numerical results, while conclusions are drawn in Section V.

II. SYSTEM MODEL
Let us consider a discrete memoryless PLC wiretap channel model illustrated in Fig. 1, where Alice (A) transmits a message to an authorized receiver Bob (M) over the main channel. At the same time, a passive eavesdropper (E) wants to listen and attempts to decode the message of A via a wiretap power line channel.

A. DCSK MODULATION
DCSK modulation was recently proposed for PLC systems in [25]. In this modulation, a chaos signal is generated by the chaos generator using a chaotic map. Each signal period is divided into two parts: the first part is assigned to the reference samples and the second part is assigned to the data samples. If the data samples are the same as the reference samples, bit 1 is transmitted, whereas if the data samples are inverted versions of the reference samples, a bit 0 is transmitted. Here, the modulated bits are denoted by b i ∈ {+1, −1}. Let β denote the spreading factor. For the i th bit interval, the modulated output of the DCSK transmitter represented by I, is given as where s l indicates the chaotic signal used as reference samples and s l−β is the delayed version of it. Based on the Gaussian approximation, for large values of β, following assumption can be made [45] E

B. RECEIVED POWER
The received signals at M and E are given by where δ k = e −(a 0 +a 1 ·f c )d characterizes the attenuation of a power line link and a 0 , a 1 , and c are the attenuation parameters at a frequency f and path length d. h M and h E are the correlated complex channel gains of the main channel and wiretap channel, respectively, which are assumed to undergo flat fading. This is a valid assumption as the considered PLC network has a small number of branches with short lengths (0-300 m), and therefore the signal bandwidth is less than the channel's coherence bandwidth. Consequently, the PLC channels experience flat fading [46], [47]. In Eq. (3), n M and n E represent the channel noise on the main channel and eavesdropper channel, respectively, and I l denotes the single carrier DCSK modulated symbol transmitted by A.

C. PLC CHANNEL MODEL
For the considered PLC system, we assume that both the main and wiretap channels are Log-normally distributed. The probability density function (PDF) of the PLC channel gain is expressed where μ k and σ 2 k are the mean and variance of the random variable (RV) ln |h k |, respectively. Moreover, the channel state information (CSI) of the main and wiretap channels are assumed to be known. In our scenario, it is observed that carrying out the whole analysis with the Log-normal distribution in (4) is quite intractable to obtain the closedform solution for the double integral expressions. Therefore, we utilize the Gamma approximation of the Log-normal distribution [48, eq. ( In (5), the parameters m h k and h k hold the following relations with σ k and μ k : where k ∈ {M, E}, m h k denotes the shaping parameter, and h k is the mean power.

D. NOISE MODELING
PLC channels generally yield a mixture of background noise and impulsive noise [49], [50]. Background noise occurs due to the addition of all low power noise sources, which can cause disturbance in frequency. Impulsive noise is due to switching transients, which occur all over a power supply network at irregular intervals [51]. The additive noise in a PLC system is usually modeled by the Bernoulli-Gaussian random process [50]. Hence, the total noise can be described by where G n k and G n k are Gaussian RVs corresponding to the background noise and impulsive noise component of the Bernoulli-Gaussian PLC noise having zero mean and variances σ 2 B and σ 2 I , respectively. I n k denotes the impulsive noise, where the Bernoulli RV (P B ) denotes the arrival of impulsive noise component of the PLC noise with the random amplitude of G n k . Here, G n k , G n k , and P B are assumed to be mutually independent.
Thus, the PDF of the PLC channel noise on the main channel (A to M) and the eavesdropper channel (A to E) is given by [52] where p 1 = 1 − p, p 2 = p, p is the probability of arrival of the impulsive noise, σ 2 k 1 = σ 2 B k is the background noise power, and σ 2 is the noise power of both background and impulsive noises. K = σ 2 I k /σ 2 B k denotes the impulsive noise to the background noise power ratio.

E. CHARACTERIZATION OF THE INSTANTANEOUS SNR
The instantaneous SNR of the received DCSK signals at M and E can be written as [52] and γ k =γ k /(K + 1). Further, the PDF of γ M j and γ E j can be written as where Here, (9), the cumulative distribution function (CDF) of γ M j and γ E j can be expressed as Using the chain rule in (11), the corresponding joint PDF can be written as where C f (F(α 1 ), F(α 2 ), . . . , F(α d )) represents the Copula density function and f (α d 1 ) denotes the marginal PDFs. Various kinds of Copula-based approaches exist, where FGM Copula is particularly popular in many practical applications due to its ability to provide easy and fast computation and to describe the actual dependence between the RVs [40]. In our considered system, the FGM Copula is utilized to analyze the PLS performance metrics. According to the FGM Copula method the joint PDF of γ M j and γ E j can be written as Further, the FGM Copula function for γ M j and γ E j is defined as follows: where θ ∈ [−1, 1] quantifies the dependency between the RVs γ M j and γ E j . A negative dependence between the two RVs is corresponds to θ ∈ [−1, 0), θ ∈ (0, 1] represents positive dependence between the two RVs, while θ = 0 denotes independence. Taking the partial derivative of (14), the FGM Copula density function can be calculated as Finally, by substituting (15) into (13) and after some mathematical simplifications, the joint PDF of γ M j and γ E j can be expressed as

B. SOP ANALYSIS
The SOP is the probability that the instantaneous secrecy capacity falls below a predefined secrecy rate (R th ) [54]. By following the same assumption that was mentioned in the work of [55], [56], [57], [58], [59], [60], we have assumed the upper bound on the channel capacity with full knowledge of the noise state available at the transmitter and receivers. Thus, the SOP of the considered system can be expressed as [21], [22], [56], [59], [60] where where C s represents the instantaneous secrecy capacity and λ = exp(R th ). Further, the exact SOP analysis is intractable; due to this, obtaining the closed-form solution of the double integral expressions following (18) becomes quite difficult. Therefore, we proceed with computing the lower bound on SOP. In this context, assuming λ(γ E j +1)−1 ≈ λγ E j , we get Substituting (16) into (19), we have where In Theorems 2-5, we derive the series-based expressions for (21)- (24).
Theorem 2: The series-based expressions of P 1,j is given as where χ = ( Proof: Refer to Appendix A for the proof. Theorem 3: The series-based expressions of P 2,j is expressed as Proof: Refer to Appendix B for the proof. Theorem 4: The series-based expressions of P 3,j is derived as Proof: Refer to Appendix C for the proof. Theorem 5: The series-based expressions of P 4,j is given as Proof: Refer to Appendix D for the proof.

C. CONVERGENCE ANALYSIS
On substituting (25)- (28) into (17), the series-based expressions of the SOP lower bound is obtained. It is observed from (26)-(28) that the series-based expressions for P 2,j , P 3,j , and P 4,j consist of infinite summations. Let us present the convergence analysis of the infinite series here. In order to verify the convergence of (17), we calculate the absolute error, defined as Absolute error = |Exact value Approximated value|, (29) where the exact value is obtained from (17) and the approximated value is computed from (20) using (25)- (28). Fig. 2 shows the plot of the absolute error against the number of summation terms (S t ) for various values of p, when β = 32, θ = 0.1, K = 100, and the transmitted power on the A to M and A to E links is fixed at 20 dB and 10 dB, respectively. It can be noticed from the figure that as the value of S t increases, the absolute error tends to zero. This demonstrates the convergence of the summation terms in the infinite series of (17). Therefore, in what follows, we consider S t = 33 for our analysis.
The infinite summation vanishes in (26)-(28) because, at high average SNR, only the terms corresponding to φ = ψ = 0 dominate. Hence, the terms P 1,j , P 2,j , P 3,j , and P 4,j , can be approximated as shown in (30)-(33): where In (34b), (34c), and (34d Finally, substituting (30)- (33) into (17), the series-based expressions of the asymptotic SOP can be computed. It is known that the asymptotic analysis of any system is characterized by two parameters: coding gain (Cg) and diversity order (Do). The coding gain shows the relative horizontal shift of the SOP versus SNR plot on a log-log scale. The diversity order specifies the asymptotic slope of the curve at high SNR values. The relation between coding gain and diversity order for the considered PLC system is computed from (30)-(33) as Furthermore, from (35), the overall diversity order is obtained as Do = min{Do 1,j , Do 2,j , Do 3,j , Do 4,j } = min{Do 1,j , Do 3,j }. Thus, (35) can be re-written as where Do 1,j = Do 3,j = m γ M j .

E. SPSC ANALYSIS
In order to guarantee perfect secrecy over the communication channel, the secrecy capacity must be a non-zero quantity, which is also interpreted as SPSC. The SPSC for the considered PLC system is given by where Here, the integral in (38) can be solved using (16) as where the series-based expressions for P 1,j , P 2,j , P 3,j , and P 4,j can be calculated similar to (25)-(28) by substituting λ = 0 in (25)-(28) as follows:

F. SECRECY THROUGHPUT OPTIMIZATION ANALYSIS
In this section, we consider the secrecy throughput optimization for the considered PLC system. Our objective is to maximize the secrecy throughput under the SOP constraints; an algorithm is proposed for the same in this subsection. Let us assume the Wyner's wiretap encoding scheme for the considered system model, where A transmits information at rate R M to M by maintaining a predefined secrecy rate R th against E. The net rate difference R D = R M − R th estimates the cost of securing data transmission. For perfect secrecy, the main channel (A to M link) must follow the condition C M > R th . Otherwise, perfect secrecy fails, i.e., C E > R D , where C M and C E are the instantaneous channel capacities of M and E, respectively. Thus, the conditional SOP is formulated as [61] P csop j Pr C E j > R M j − R th | data transmission . (44) Using (44), the overall conditional SOP of the considered system can be written as We aim to maximize the secrecy throughput for the considered PLC system. The secrecy throughput is defined as where P tx denotes the transmission probability. Hence, the optimization problem can be formulated as where ρ ∈ [0, 1] and ν ∈ [0, 1] represent the constraints for secure and reliable transmission, respectively. In addition, we assume that the data transmission is possible whenever the value of γ M is greater than a certain threshold SNR, γ th , with γ th ≥ exp(R th ) − 1. Therefore, the transmission probability can be expressed as Let us evaluate the conditional SOP as follows: Considering the lower bound in (49), i.e., λ(γ E j + 1) − 1 ≈ λγ E j , we obtain where = ( 1+γ th λ ) − 1. Here, the double integral in (50) can be divided into two parts as follows: By substituting (17) into (38), we get I 1,j ≈ I 11,j − I 12,j − I 13,j + I 14,j and I 2,j ≈ I 21,j − I 22,j − I 23,j + I 24,j . The seriesbased expressions for I 1i,j , i = {1, 2, 3, 4} can be obtained following a similar approach to (21)-(24) as Likewise, the series-based expressions for I 2i,j , i = {1, 2, 3, 4} is obtained as Algorithm 1: Algorithm for Finding the Optimal γ th to Maximize the Secrecy Throughput 1 Initialization: R th , ρ, and ν; 2 Optimization problem given in (47) and obtain CSOP and P tx given in (45) and (48), respectively; 3 To maximize τ , we aim to find the optimal value of γ th for the considered design problem given in (60a) to (60c); 4 do 5 Calculate γ th by (48) 6 while P tx (γ th ) ≥ ν, γ th ≥ exp(R th ) − 1, and R th > 0; 7 Find optimal value of γ th ; 8 do 9 Calculate γ th using expression CSOP(γ th , R th ) = ρ 10 while CSOP(γ th , R th ) ≤ ρ; 11 Finally, a feasible range of optimal γ th is given by (61); (45), the series-based expressions of CSOP can be obtained. Now, to maximize the secrecy throughput, we find the optimal value of γ th for the considered optimization problem as follows: where P tx and CSOP are the function of γ th and R th . Therefore, using (48) the possible range of γ th is obtained by following P tx (γ th ) ≥ ν. Thus, γ th ∈ [ exp(R th ) − 1, γ th,u ], where γ th,u is the upper limit of γ th computed from (48). Herein, to maximize the secrecy throughput the smallest value of γ th is considered whilst satisfying security constraints. Since, (45) is a decreasing function of γ th , the optimal γ th can be obtained using CSOP(γ th , R th ) = ρ, by following CSOP(γ th , R th ) ≤ ρ, as where γ th,u represents the upper limit of γ th . It should be noted that the optimal value of γ th given by (45) and (48) is calculated using MATLAB software for mathematical tractability. The algorithm to compute the optimal value of γ th that maximizes the secrecy throughput is summarized as the algorithm 1.

IV. NUMERICAL RESULTS AND DISCUSSION
In this section, numerical results and discussion are presented to analyze the performance of the proposed system. The analytical plots are obtained using the derived theoretical results. We assume that the frequency is 1 MHz and the distance from A to M and from A to E is set to 300 m. Moreover, a 0 = 8.40 × 10 −3 m −1 , a 1 = 3 × 10 −9 s/m, and c = 1. In Fig. 3, we plot the coding gain as a function of the diversity order for various values of p, when θ = 0.1, β = 32, K = 100, and the transmitted power on the A to M and the A to E links are 20 dB and 10 dB, respectively. It can be observed from the figure that as the secrecy diversity order m γ M j increases, the overall C g,j decreases. Moreover, the C g,j improves as the value of p is reduced from 0.7 to 0.01. Fig. 4 illustrates the SOP versusγ M curves for various values of p, when θ = 0.1, β = 32, K = 100, and transmitted power on the A to E link is fixed at 20 dB. It is explicitly shown in the figure that the SOP performance is better for lower values of p and increasing p degrades the SOP. This is because the frequent occurrence of impulsive noise causes, the PLC system's overall noise power to increase, which negatively affects the SOP performance of the considered system. Moreover, it is observed that the lower bound SOP curves are very close to the exact SOP curves, which is the accuracy of the derived SOP lower bound. Additionally, the asymptotic SOP as a function of the average SNR is shown in Fig. 4. Here, the SOP is 5.97×10 −5 and 5.97×10 −6 at 70 dB and 80 dB, respectively, for p = 0.01, θ = 0.1, β = 32, and  the transmitted power on the A to E link is fixed at 20 dB. Therefore, the asymptotic slope is computed as log 10 (5.97 × 10 −5 ) − log 10 (5.97 × 10 −6 ) = 1 = min{m γ M j , 2m γ M j }, which is in line with the asymptotic slope theoretically calculated in (35).
In Fig. 5, the SOP is plotted as a function of the dependence parameter θ for p = 0.01 and β = 64 under different SNR conditions (γ M >γ E ,γ M =γ E , andγ M <γ E ). When γ M <γ E , the SOP performance deteriorates with increasing θ compared to the other two conditions. Receiving a high SNR at E increases its channel capacity, which implies higher chances of secrecy outage. Therefore, the PLC channel becomes less secure. Similarly, whenγ M >γ E the SOP performance is improved. This is because receiving a high SNR at M increases the capacity of the main PLC channel. As a result, there are less chance of an outage. Finally, for γ M =γ E the SOP performance is moderate. In all the three conditions, it is revealed that as θ increases to 1, the SOP also tends to 1. This observation is quite intuitive because high positive values of θ indicate higher dependency between the main channel and the eavesdropper channel, thereby making the PLC transmission less secure.   that the SPSC performance improves as the value of p is reduced from 0.4 to 0.05. Fig. 7 depicts the secrecy throughput versusγ M for different values of γ th , when p = 0.01, β = 32, K = 100, R th = 1, andγ E = 0 dB. Herein, γ th = 2.35 dB is obtained by γ th = exp(R th ) − 1. Likewise, γ th = 9.5 dB is computed assuming CSOP(γ th , R th ) = ρ for p = 0.01, β = 32, ρ = 0.2, and θ = 0.1, when the transmitted power on the A to M and the A to E links fixed at 20 dB and 8 dB, respectively. It is observed that secrecy throughput improves with a lower value of γ th because there are less chances of outage for lower values of γ th . Similarly, there is a high outage probability for high values of γ th , and therefore the secrecy throughput decreases with the increase of γ th . It is noted that, the computed values of γ th that maximize the secrecy throughput satisfy all the optimization conditions given in (60b) and (60c). For instance, it is observed from the figure that when γ th = 9.5 dB, CSOP = 0.134 and P tx = 0.987 for ρ = 0.2 and ν = 0.4, which satisfy the optimization conditions in (60b) and (60c).
The SOP of the considered PLC system as a function of transmitted power on the A to M link for different values of K is shown in Fig. 8. The curves are plotted for p = 0.1, θ = 0.1, β = 48, and the transmitted power of the A to E link is fixed at 20 dB. It is observed that the SOP performance degrades as the value of K increases. Moreover, it is inferred from the plot that the value of the impulsive noise index parameter severely affect the behavior of the proposed PLC communication system. Fig. 9 shows the comparison of SOP as a function of γ M for the different modulation schemes such as DCSK, DS-DPSK, GCDSK, and CDMA. It is found that the SOP performance of DCSK is superior to GCDSK for selected values of β and delay block (M). Additionally, the GCDSK system structure is complex and complicated. It requires an additional delay, correlator, and multiplier [35]. However, CDMA performs worse with a higher value of β and viceversa. Further, it is noticed that the SOP performance of DS-DPSK is better than DCSK. However, the DCSK system inherits chaos signal properties that make the interception and decoding of data more difficult for an eavesdropper.

V. CONCLUSION
In this paper, we investigated the secrecy performance of DCSK-based PLC systems under correlated Lognormally distributed PLC wiretap channel. The FGM Copula approach facilitated the mathematical modeling of the PLS performance metrics in terms of SOP and SPSC. Specifically, the FGM Copula explicitly shows the correlation between the main and the wiretap channels in terms of the dependence parameter. In addition, to further understand the PLC system performance the asymptotic analysis was carried out, where a relation was established between the secrecy coding gain and the secrecy diversity order. We also presented an algorithm to maximize the secrecy throughout under SOP constraints. The algorithm is used to find the optimal SNR for which secrecy throughput is maximum. Useful insights into the considered PLC system were ascertained by studying the impact of different parameters such as the spreading factor, impulsive noise occurrence probability, transmitted power, dependence parameter, and impulsive noise index.