MGF-Based Mutual Approximation of Hybrid Fading: Performance of Wireless/Power Line Relaying Communication for IoT

Wireless and power line communications (PLC) are important components of distribution network communication, and have a broad application prospect in the fields of intelligent power consumption and home Internet of Things (IoT). This study mainly analyzes the performance of a dual-hop wireless/power line hybrid fading system employing an amplify-and-forward (AF) relay in terms of outage probability and average bit error rate (BER). The Nakagami-m distribution captures the wireless channel fading; whereas the PLC channel gain is characterized by the Log-normal (LogN) distribution. Moreover, the Bernoulli-Gaussian noise model is used on the noise attached to the PLC channel. Owing to the similarity between LogN and Gamma distributions, the key parameters of probability density function (PDF) with approximate distribution are determined by using moment generating function (MGF) equations, joint optimization of s vectors, and approximation of LogN variable sum. The MGF of the harmonic mean of the dual Gamma distribution variables is derived to evaluate the system performance suitable for any fading parameter m value. Finally, Monte Carlo simulation is used to verify the versatility and accuracy of the proposed method, and the influence of the hybrid fading channel and multidimensional impulse noise parameters on system performance is analyzed.


Introduction
With the rapid development of 5G mobile communication, the power system pays more and more attention to communication technology. The application of cutting-edge technologies such as big data, cloud computing, Internet of Things (IoT), mobile Internet and physical information fusion system in power systems has also attracted widespread attention. With the development of power supply networks, the coverage area of power lines has progressed considerably. The optimization of existing power supply network resources and the transmission of reliable information on power lines have gradually attracted the attention of researchers [1]. The power line serves not only as a medium for transmitting power but also for transmitting data for communication. Power line communications (PLC) can reach anywhere power lines exist, especially where the wireless signal is weak, such as underground, under water and metal-walled rooms. In communication architectures, PLC [2] uses the existing power line infrastructure to transmit information with power and provides cost-effective and extensive coverage of smart grid solutions [3].
However, the power line was originally designed for power transmission rather than specifically for communication. Therefore, problems such as insufficient reliability and inaccessibility of the investigations, the present study uses the hybrid fading model of wireless Nakagami-m/power line LogN and power line impulse noise to analyze the system performance of the AF relay protocol for the dual-media cooperative communication system of wireless access and power line relay. A good research method has not been put forward in existing literature to analyze this type of complicated and non-intuitive system. The proposed model is different from the symmetrical dual-hop system considered in the literature and is also a scientific problem to be developed that can provide significant theoretical basis for relevant research. The main contributions of this study are as follows: (1) System performance is analyzed based on the mutual approximation of different channel fading.
Considering the similarity between LogN and Gamma distributions, we propose an s vectors joint optimization algorithm based on moment generating function (MGF) equations and the principle of minimum difference between PDFs to determine the key parameters of approximate fading PDF and achieve high approximate accuracy. (2) After the LogN distribution is approximated to the Gamma distribution (L2G), we derive the MGF of the harmonic mean of two Gamma distribution variables with the Appell function F1 form and use the MGF to obtain the BER, outage probability, and diversity order of the system. Compared with other studies, the MGF derived in this study must only calculate the Appell hypergeometric function once. This function has high accuracy, low complexity, and is suitable for any Nakagami-m fading parameter value. (3) This study proposes an algorithm similar to (1) for determining the optimal distribution parameters and optimal s values based on the approximation of the LogN variable sum. The Gamma distribution is approximated to the LogN distribution (G2L) based on the principle of minimizing the MGF difference. In addition, the Gauss-Hermite series, integral changes, Mehta, and other algorithms are comprehensively applied to analyze the performance indicators of the double LogN distribution system. (4) The versatility and accuracy of the two approximation algorithms are verified by using the Monte Carlo simulation, and the difference of the L2G and G2L algorithms is compared and analyzed in the calculated system performance for effectiveness and reliability. At the same time, the influence of the hybrid channel fading and multidimensional impulse noise on the system performance is analyzed. The simulation results show that both approximation algorithms have high accuracy, but the L2G algorithm can analyze system performance more accurately, and the G2L algorithm can be used to analyze other performances besides BER and outage since the PDF of terminal signal to noise ratio (SNR) is known.
The rest of the paper is organized as follows. Section 2 discusses the wireless/power line hybrid fading system model of the AF relay protocol. Section 3 calculates the average BER and outage probability based on the proposed approximation algorithm to evaluate the performance of the system under consideration. Section 4 presents the analysis and simulation results. Finally, Section 5 draws the conclusions. Figure 1 shows the two-slot dual-media hybrid communication system considered in this work. In the first time slot, the source (S) uses the transmit power P S to communicate wirelessly with the relay (R). In the second time slot, R amplifies and forwards the received data and transmits these to the terminal (D) through the power line by power P R .

System Description
The source S in Figure 1 refers to power equipment or independent smart meters or sensors, such as smart meters in buildings and homes, wireless sensor nodes in underground substations, non-contact infrared temperature cameras in substations, mobile Radio Frequency Identification card readers, etc. This system model combines the advantages of wireless and PLC technologies to provide consumers with the last kilometer access scheme, thereby effectively solving the distance limitation problem of the traditional wireless communication system. It likewise provides the infrastructure for Sensors 2019, 19, 2460 4 of 18 many old buildings and residences that cannot achieve smart home conditions and effectively reduces cost. This system can also connect to the Internet in any room with weak wireless signals and expand network coverage. The system model uses mobile access and dual-hop relay technologies, which meet the practical requirements of flexible access and long-distance reliable transmission in smart grid communication.  Figure 1. System model for a dual-hop wireless/power line hybrid communication system with amplify-and-forward (AF) relay.

First Time Slot
In the first time slot, S transmits wireless information to R by transmitting power PS and R receives the following signals: where the noise nGR satisfies the normal distribution N(0,NW), and HGR is the wireless fading coefficient that satisfies the following Nakagami-m distribution.
where mR ≥ 0.5 is the Nakagami-m fading parameter, Г(x) is the gamma function, and ΩR = E( is the variance of the fading amplitude that is normalized to ensure that the fading does not change the average power of the received signal. Let ΩR = 1. According to the system model, let ΔGR = PS/NW denote the channel average SNR. Then, according to Equation (1), the instantaneous SNR GR γ of the wireless channel R can be obtained as follows: satisfies the Gamma distribution and is defined as follows: where the parameter relationship between the two distributions of Gamma and Nakagami-m satisfies αR = mR, βR = ΩR/mR. Based on the properties of the Gamma function under the same average SNR, when ΔGR is a constant,

Second Time Slot
In the second time slot, R uses the AF protocol to forward the received data to D by power PR. Let XR be the signal of R, then the signal LD y received by D is

First Time Slot
In the first time slot, S transmits wireless information to R by transmitting power P S and R receives the following signals: where the noise n GR satisfies the normal distribution N(0,N W ), and H GR is the wireless fading coefficient that satisfies the following Nakagami-m distribution.
where m R ≥ 0.5 is the Nakagami-m fading parameter, Γ(x) is the gamma function, and Ω R = E(|H GR | 2 ) is the variance of the fading amplitude that is normalized to ensure that the fading does not change the average power of the received signal. Let Ω R = 1. According to the system model, let ∆ GR = P S /N W denote the channel average SNR. Then, according to Equation (1), the instantaneous SNR γ GR of the wireless channel R can be obtained as follows: Let H 2,GI = H GI 2 , (I = R, D), H 2,GR satisfies the Gamma distribution and is defined as follows: where the parameter relationship between the two distributions of Gamma and Nakagami-m satisfies Based on the properties of the Gamma function under the same average SNR, when ∆ GR is a constant, H GR 2 ∆ GR ∼ G(γ GR ; m R , ∆ GR Ω R /m R ).

Second Time Slot
In the second time slot, R uses the AF protocol to forward the received data to D by power P R . Let X R be the signal of R, then the signal y LD received by D is where H LD is the power line fading coefficient that satisfies the following LogN(µ D , σ 2 D ) distribution: where µ D and σ D are the mean and mean square deviation of ln H LD , respectively. Let E( The envelope energy of the channel fading can be normalized to ensure that the channel fading does not change the average power of the signal. On the basis of the properties of the LogN distribution, H LD 2 satisfies the LogN(2µ D , 4σ 2 D ) distribution. To accurately characterize the PLC channel, the noise is assumed to consist of background and impulsive noise components [19]. These noise types are modeled using the two-term Bernoulli-Gaussian noise model [20]. Therefore, the PDF of the total noise can be simply expressed as where n g is considered the Gaussian with zero mean and variance σ 2 g , while the impulsive part n i is modeled as the Bernoulli-Gaussian random process. As n LD is the total noise n i = bA, A is the white Gaussian noise with mean zero, and b is the Bernoulli process with the probability mass function.
Therefore, the PDF of the total noise can be simply expressed as where N(0, σ 2 ) refers to the Gaussian distribution with mean zero and σ 2 as variance, and p is the arrival probability of the impulsive component of the Bernoulli-Gaussian noise. The noise power on the PLC link is σ 2 g in the presence of only the background noise, while the total noise power is σ 2 g (1 + T) in the presence of background and impulsive noises, where T = σ 2 i /σ 2 g is the impulsive noise index. Let ∆ LD = P R /N P , and then the instantaneous SNR of the power line receiver can be expressed as Therefore, the total SNR γ GL of the dual-hop AF relay protocol communication system is When a high SNR (P S /N W , P R /N P 1) exists, Equation (11) can be approximated as

Mutual Approximation Algorithm Based on the PDF and MGF Equations
Determining the PDF of each branch SNR is necessary to obtain the system's outage probability and BER. However, the PDF of the destination node D of the hybrid fading system is difficult to solve, making it difficult, in turn, to analyze the system performance. At the same time, it is mathematically difficult to deal with because the LogN variable integration does not have a closed expressionin the process of analyzing the performance of the system BER. Therefore, the H LD 2 of LogN(2µ D , 4σ 2 D ) distribution is approximated to G(H 2,GD ; α D , β D ) based on the similarity between two distributions [21] for further analysis of the system.
The approximation accuracy is poor because the PDF approximation is obtained in the same way as the mean and variance [22]. Therefore, we propose the PDF approximation algorithm based on the MGF equations in this work, and the obtained approximation function has high precision.
H LD 2 satisfies the LogN(2µ D , 4σ 2 D ) distribution, and the MGF is presented as where s i (i = 1, 2) is an adjustable variable. N is the Hermite integration order, and w n and a n are the weights of the Gauss-Hermite formula and their abscissas [23], respectively. Similarly, the MGF equation of G(H 2,GD ; α D , β D ) is satisfied with the following equation: Then the L2G of H LD 2 using the MGF equations The approximated parameters α D and β D can be solved by using the fsolve function in MATLAB.
The values of the independent variables s 1 and s 2 in the MGF equations are directly related to the determination of the approximate parameters α D and β D and the approximate accuracy of the PDF. This study proposes a mathematical optimization model based on the s vectors optimization algorithm with the minimum difference of the PDF curves, the key parameters of the approximate fading PDF are obtained by solving a set of optimal s 1 and s 2 combinations. For the known power line fading parameters µ D and σ D , we use PDF Equations (4) and (6) to solve the wireless fading parameters α D and β D with s 1 and s 2 as variables. Hence, the following mathematical model is established: subject to: where H i denote the sample values of the PDF of SNR with fading, and N is the total number of sample values of the PDF (N = 100 is selected in this study). The mathematical model considers the matching degree of PDF curves of the LogN and Gamma distributions as the optimization objective and performs the weighting calculation on the square of the difference of the PDF values to the corresponding fading sampling values H i . Specifically, 2 , which is called the objective function in this study. When the selected s value minimizes the objective function, the simultaneous equations can obtain the PDF key parameters of approximated Gamma distribution by the known LogN distribution. The optimal s value can be solved by an intelligent optimization algorithm, such as the genetic algorithm.
Therefore, the numerical calculation and analysis are conducted in this study under channel fading normalization (E( H LD 2 ) = 1). Figure  The mathematical model considers the matching degree of PDF curves of the LogN and Gamma distributions as the optimization objective and performs the weighting calculation on the square of the difference of the PDF values to the corresponding fading sampling values Hi. Specifically, , which is called the objective function in this study.
When the selected s value minimizes the objective function, the simultaneous equations can obtain the PDF key parameters of approximated Gamma distribution by the known LogN distribution. The optimal s value can be solved by an intelligent optimization algorithm, such as the genetic algorithm. Therefore, the numerical calculation and analysis are conducted in this study under channel fading normalization (E(  (1) The choice of s value that will affect the prediction of the PDF curve coincidence is verified.
(2) Although the MGF equation approximation lacks analytical expressions, it can be optimized through s1 and s2 with increased accuracy.
For the sake of clarity, Figure 3 presents the PDF curve comparison of the Gamma distribution in the first time slot with the approximated LogN distribution under normalized conditions (E( 2 GR H ) = 1) for different combinations of the parameters s1, s2,and mR.
Using the above method, Table 1 presents the approximate optimization table of the MGF equation for reference in practical applications, where s1 and s2 are the optimal values of different σD, and mD and D are the PDF parameters that are optimal approximated to the Gamma distribution by the usual σD, s1 and s2.  (1) (The choice of s value that will affect the prediction of the PDF curve coincidence is verified. (2) Although the MGF equation approximation lacks analytical expressions, it can be optimized through s 1 and s 2 with increased accuracy.
For the sake of clarity, Figure  Similarly, Table 2 shows the optimization results of the LogN distribution approximated by the parameters of the Gamma distribution of the first time slot in the system.  Using the above method, Table 1 presents the approximate optimization table of  and β D are the PDF parameters that are optimal approximated to the Gamma distribution by the usual σ D , s 1 and s 2 . Similarly, Table 2 shows the optimization results of the LogN distribution approximated by the parameters of the Gamma distribution of the first time slot in the system. As the fading parameters of the channel after the approximation no longer satisfy the normalization condition, it would be cumbersome to use above numerical experimental method to determine the appropriate s 1 and s 2 values. In addition, the accuracy of s 1 and s 2 determined by the experiments in a large numerical range will not be guaranteed because of the lack of help from mathematical theory tools. Thus, this study first approximates the normalized PDF of channel fading and then calculates the equivalent parameters of the instantaneous SNR based on the average SNR of the first hop. According to γ LD = H 2 LD ∆ LD and its properties, the key parameters of γ LD approximating into G γ GD ; α D , β D distribution are   Similarly, in the approximation process of the Gamma distribution to the LogN distribution of Figure 5, according to γ GR = H 2 GR ∆ GR , the key parameters after the variable γ GR is approximated to LogN(µ R , σ 2 R ) are σ 2 R = σ 2 R and µ R = ln ∆ GR + µ R .

Performance Analysis of MGF Based on Harmonic Mean of the Double Gamma Distribution System
The performance analysis under hybrid fading is transformed into the performance analysis under the double Gamma distribution after L2G in the second time slot R-D link. Many studies have investigated the BER and outage performance of the AF relaying system under Nakagami-m fading. However, these studies have the following limitations: (1) The use of the approximate function for end-to-end SNR, such as γGRγGD/(γGR + γGD + 1) ≈ min(γGR,γGD) [24], leads to imprecise results. (2) Numerous performance formulas contain multiple series [25], Whittaker functions [26], or Appell F4 functions [27], thereby causing high computation complexity. (3) Some methods can only evaluate the system outage probability [28]. (4) Some precise conclusions are only suitable for a special fading parameter m (integer [24,25] or integer plus one-half [25]). To reduce the computational complexity of the harmonic mean, some studies use the least function approximation calculation. Some basic frame works are also proposed but with increased complexity, where in two Appell functions need to be calculated [16]. Motivated by all of the above, we present an effective performance evaluation approach using the basic theory of probability. The MGF for terminal-to-terminal instantaneous SNR is derived first in the form of the Appell function F1, which involves a single integral that can be computed by the general software MATLAB. Suitable for the arbitrary Nakagami-m fading parameter m, the MGF is then used to derive the system performance of the diversity order, BER, and outage probability.
The channel coefficients HGR and HGD of the S-R and R-D links are subject to Nakagami-m fading with the shape parameters mR and mD and satisfy E[|HGR| 2 ] = ΩR = 1 and E[|HGD| 2 ] = ΩD. Then, Figure 5. Comparison of PDF for instantaneous SNR with the Gamma and approximated LogN distributions using the best s value optimization method.

Performance Analysis of MGF Based on Harmonic Mean of the Double Gamma Distribution System
The performance analysis under hybrid fading is transformed into the performance analysis under the double Gamma distribution after L2G in the second time slot R-D link. Many studies have investigated the BER and outage performance of the AF relaying system under Nakagami-m fading. However, these studies have the following limitations: (1) The use of the approximate function for end-to-end SNR, such as γ GR γ GD /(γ GR + γ GD + 1) ≈ min(γ GR ,γ GD ) [24], leads to imprecise results.
(2) Numerous performance formulas contain multiple series [25], Whittaker functions [26], or Appell F4 functions [27], thereby causing high computation complexity. (3) Some methods can only evaluate the system outage probability [28]. (4) Some precise conclusions are only suitable for a special fading parameter m (integer [24,25] or integer plus one-half [25]). To reduce the computational complexity of the harmonic mean, some studies use the least function approximation calculation. Some basic frame works are also proposed but with increased complexity, where in two Appell functions need to be calculated [16]. Motivated by all of the above, we present an effective performance evaluation approach using the basic theory of probability. The MGF for terminal-to-terminal instantaneous SNR is derived first in the form of the Appell function F1, which involves a single integral that can be computed by the general software MATLAB. Suitable for the arbitrary Nakagami-m fading parameter m, the MGF is then used to derive the system performance of the diversity order, BER, and outage probability.
The channel coefficients H GR and H GD of the S-R and R-D links are subject to Nakagami-m fading with the shape parameters m R and m D and satisfy E[|H GR | 2 ] = Ω R = 1 and E[|H GD | 2 ] = Ω D . Then, the instantaneous SNR of the links are γ GR = |H GR | 2 ∆ GR and γ GD = |H GD | 2 ∆ GD , where ∆ GR and ∆ GD are the average SNRs. Consequently, each γ GI (I = R, D) satisfies the Gamma distribution denoted as γ GI~G (γ GI ; α I , β I ), where α I = m I and β I = ∆ GI Ω I /m I . The equivalent end-to-end SNR γ GG at terminal D is similar to that of Equation (11). F 1 (·) represents the first type of Appell hypergeometric function, and the MGF of the harmonic mean of the two variables with the Gamma distribution satisfies the following lemma. Lemma 1. If X I (I = R,D) satisfies G(X I; α I , β I ), then the MGF of X I 's harmonic mean Z = X R × X D /(X R + X D ) has the following expression: Proof. We mainly derive the PDF and MGF for γ GI . Then, by using the PDF of γ GI , which is denoted as f γ GI (x), we can obtain the PDF of 1/γ GI , Y = 1/γ GR + 1/γ GD and Z = 1/Y, which are presented as follows: where U(z) = 1 for z ≥ 0 and U(z) = 0 for z < 0. Then, by substituting the PDF of γ GI , which is G(X I; α I , β I ) into Equation (20-c), we derive the following: According to the MGF definition M Z (s) = ∞ 0 e −sz f Z (z)dz, the following expression is obtained: [23], the MGF of Equation (22) can be rewritten as where B(α R , α D ) is the beta function. Finally, according to the definition of the Appell function F1 [23], a concise form of the MGF is obtained as Hence, the final MGF only involves a single integral, which can be solved by quad and other integral functions in MATLAB or Appell F1 function in Mathematics.
In particular, when the PDF parameters of X R and X D are the same, that is, α R = α D = α and β R = β D = β, Equation (19) can be defined as Finally, through integral transformation, the above equationcan be simplified to where 2 F 1 (·) represents the hypergeometric function [29]. The function 2 F 1 (·) can also be expressed as the following simple integral: For the coherently M-PSK modulation, the system BER is defined as With the help of the MGF derived in Equation (22), we can then compute the BER according to [27] or use the double integral function in MATLAB directly, where g PSK = sin 2 (π/M). If Θ = (M−1)π M , Equation (28) can be further simplified to Similarly, the outage probability P out is defined as the probability value when the instantaneous total SNR is lower than the fixed threshold (R th ). The outage probability can also be calculated by substituting Equation (22) into the following equation [25] after some necessary alterations: where β 0 = 2 and β n = 1(n is a positive integer), and the values of A, N and Q determine the accuracy of the numerical calculation, which is estimated by the overall truncation error term E(A, N, Q).

Lemma 2.
If X I (I = R,D) satisfies G(X I; α I , β I ), the diversity order of the AF relay system is min(α R , α D ). See Appendix A for the specific certification process.

Performance Analysis of the Double LogN Distribution System Based on the Approximation of the LogN Variable Sum
In this section, we will similarly operate G2L in the first time slot of the system to analyze the performance. In particular, the system performance analysis under the hybrid fading condition is converted into the double LogN distribution condition. Accordingly, the best approximation scheme can be further selected by comparing the accuracy, reliability, and complexity of the two approximation algorithms with those in Sub-Section 3.2. In addition, the method directly calculates the BER and outage probability of the system by using the SNR of the system terminal D, which is beyond the limitation of the MGF and can easily analyze various performance indexes of the system.
We assume that the process of approximating the Nakagami-m distribution to the LogN distribution in the first slot has been completed because of the similarity of the approximation process (Section 3.1). . The instantaneous SNR of the link is γ LI = |H LI | 2 ∆ LI , where ∆ LR = P S /N W and ∆ LD = P R /N P are the average SNRs of the channel. Then, the total SNR γ LL of the AF relay protocol communication system can be directly written as The instantaneous mutual information I of the system is According to the properties of the LogN distribution, P S |H LR | 2 /N W and P R |H LD | 2 /N P satisfy the LogN distribution when P S /N W and P R /N P are constants. As the reciprocal of the LogN variable still satisfies the LogN distribution, 1/(P S | H LR | 2 /N W ) and 1/(P R |H LD | 2 /N P ) also satisfy the LogN distribution. Hence, the reciprocal of the sum of the LogN distribution variables, namely, the SNR γ LL of the system, also satisfies the LogN distribution.
If G = 1/(P S H LR 2 /N W ) + 1/(P R H LD 2 /N P ) , then γ LL ∼ Log(µ G , σ 2 G ). When the known parameters µ I , σ 2 I and the Mehta algorithm are used, the distribution parameters µ G and σ G of variable G satisfy the following relations: where w n and a n refer to literature [22], . The two equations in the simultaneous Equation (33) can solve µ G and σ G by using the fsolve function of MATLAB. The Mehta algorithm has high precision, but the parameters of µ G and σ G obtained by equation solution lacks analytical expression.
Similar to the modeling method of Equation (16), joint parameter optimization based on the principle of minimum difference of the MGF curve is conducted to fit the LogN distribution parameters of variable G using an objective function If the threshold is R th and γ = exp(2R th )−1, then the P LL out is By substituting the cumulative distribution function F(λ) = Q(−(ln γ + µ G ))/σ G ) of γ LL into Equation (35), we obtain where p 0 = 1 − p, and p 1 = p, which represent the probability of whether impulse noise exists in the power line channel. µ Gj and σ Gj are the mean and variance of the attenuation distribution for the Gauss channel and the channel with impulse noise ( j = 0, 1), respectively. The BER P LL BER of the system is obtained by using F(λ) in the following equation: To simplify the calculation, let t = (−µ Gj − ln λ)/σ Gj , the integral transformation of the above equation is defined as follows: Then, exp(− exp(x)) is approximated to the form of Gaussian function weighting, and the BER P LL BER of the system is expressed as follows [11]: which includes

Numerical Results and Discussion
In this section, Monte Carlo computer simulation experiments are performed using MATLAB software to verify the reliability and accuracy of the theoretical formulas and illustrate the effects of various system parameters on the BER and outage probability of the system under binary phase shift keying modulation. In all simulations, the simulation process uses the following default settings unless mentioned otherwise: (1) The total system power is 2, P S = P R = 1. (2) To highlight the impact of channel fading and noise on the performance, we assume that the average SNR of the system channel is ∆, N W = N P = 1/∆, that is, ∆ GR = ∆ LD = ∆.  The BER decreases as the system SNR increases, and the resulting curve matches well with the Monte Carlo simulations, thereby implying the correctness of the derived analytical expressions and the validity of the theoretical derivation of the two approximation algorithms. The coincidence degree between the theory and simulation of the L2G algorithm is likewise higher than that of the G2L algorithm in terms of algorithm comparison, especially at high SNR, which is caused by multiple approximations in the theoretical calculation of the latter algorithm. The L2G algorithm is highly accurate and suitable for arbitrary m values rather than special parameters, which are reflected in the BER curves of the three sets of parameters in the figure. However, the simulation process of the L2G algorithm is sensitive to the selection of parameters, and such sensitivity is related to the high complexity of the algorithm and the large proportion of integrals involved. Although the G2L algorithm is not accurate enough, the theoretical calculation is simply for end-to-end SNR and multiple approximations, which can be used to analyze many performance indicators and solve a wide range of problems. Figure 7 shows the simulation and theoretical comparison of the system outage probability and the average SNR of the two approximated algorithms under different thresholds R th . The setting of the three sets of fading parameters is the same as that in Figure 6, and the Bernoulli-Gaussian noise parameter is the default setting. The three sets of curves in the figure show that the outage probability of the different parameters increases continuously as the value increases from 0.2 to 0.4, and the theoretical curves of the two algorithms are extremely close to the simulation results, thus verifying the correctness of our analysis. Furthermore, the curve of the L2G algorithm almost coincides with the simulation curve and the accuracy is higher than that of the G2L algorithm, however, the coincidence degree is slightly reduced at high SNR. Figure 7 shows that the algorithm can obtain sufficient computational accuracy in a certain range of SNR based on the consumption complexity.
terms of algorithm comparison, especially at high SNR, which is caused by multiple approximations in the theoretical calculation of the latter algorithm. The L2G algorithm is highly accurate and suitable for arbitrary m values rather than special parameters, which are reflected in the BER curves of the three sets of parameters in the figure. However, the simulation process of the L2G algorithm is sensitive to the selection of parameters, and such sensitivity is related to the high complexity of the algorithm and the large proportion of integrals involved. Although the G2L algorithm is not accurate enough, the theoretical calculation is simply for end-to-end SNR and multiple approximations, which can be used to analyze many performance indicators and solve a wide range of problems.  Figure 7 shows the simulation and theoretical comparison of the system outage probability and the average SNR of the two approximated algorithms under different thresholds Rth. The setting of the three sets of fading parameters is the same as that in Figure 6, and the Bernoulli-Gaussian noise parameter is the default setting. The three sets of curves in the figure show that the outage probability of the different parameters increases continuously as the value increases from 0.2 to 0.4, and the theoretical curves of the two algorithms are extremely close to the simulation results, thus verifying the correctness of our analysis. Furthermore, the curve of the L2G algorithm almost coincides with the simulation curve and the accuracy is higher than that of the G2L algorithm, however, the coincidence degree is slightly reduced at high SNR. Figure7 shows that the algorithm can obtain sufficient computational accuracy in a certain range of SNR based on the consumption complexity. Figure 8 illustrates the performance of BER in the two approximation algorithms with the different approximation accuracies caused by the value of the s vectors. The fading parameters are , and the other parameters are set by default. Figure 8a,b shows that the coincidence between the theoretical algorithms and the simulation of the two approximation theoretical algorithms with the best s values is much higher than that with the random selection of s values. This outcome proves that the selection of the best s values is more conducive to obtaining the fading parameters of the approximate distribution accurately in the approximation process based on the PDF and MGF equations, wherein the system performance can be analyzed more accurately.     Figure 8a,b shows that the coincidence between the theoretical algorithms and the simulation of the two approximation theoretical algorithms with the best s values is much higher than that with the random selection of s values. This outcome proves that the selection of the best s values is more conducive to obtaining the fading parameters of the approximate distribution accurately in the approximation process based on the PDF and MGF equations, wherein the system performance can be analyzed more accurately.     The two graphs in Figure 9 compare the relationship of the system BER with the fading parameters m R and σ D under different channel parameter settings. We assume here that the fading parameters are 2.2 and 2.8 and that the average SNR ∆ values are 12 and 15 dB, respectively. To highlight the contrast, p and T have two additional values of 0.001 and 100 respectively in addition to the initial default values. For the horizontal and vertical analyses of Figure 9, the following conclusions can be drawn: (1) Figure 9a shows that the system BER decreases as the m R increases (the wireless link fading is reduced), while the other parameters remain unchanged. When m R is larger than 2.5, the BER curve gradually flattens and the influence on the BER performance decreases. Similarly, Figure 9b shows that the BER performance of the system increases with the increase of σ D (serious power line link fading) under the condition that other parameters are constant, and the change trend of the BER curve is more significant than that of Figure 9a. Therefore, we conclude that the BER performance of the system is more sensitive to the selection of σ D than that of m R . Therefore, for the dual-hop hybrid fading system, the LogN link occupies the dominant link, and the system performance can be optimized by significantly adjusting the environment variables of the link. (2) Figure 9a,b shows that the change of the channel noise parameters has a small impact on the BER. For example, when other parameters remain unchanged, the system performance improves slightly after p changes from 0.1 to 0.001, and the BER changes slightly after T changes from 10 to 100. (3) When the fading parameters p and T are the same, the BER performance of the system will significantly improve by increasing ∆, as reflected in both graphs. However, the improvement of the system will be different by increasing the same ∆ under different fading parameters. D constant, and the change trend of the BER curve is more significant than that of Figure 9a. Therefore, we conclude that the BER performance of the system is more sensitive to the selection of D σ than that of R m . Therefore, for the dual-hop hybrid fading system, the LogN link occupies the dominant link, and the system performance can be optimized by significantly adjusting the environment variables of the link. (2) Figure9a,b shows that the change of the channel noise parameters has a small impact on the BER. For example, when other parameters remain unchanged, the system performance improves slightly after p changes from 0.1 to 0.001, and the BER changes slightly after T changes from 10 to 100. (3) When the fading parameters p and T are the same, the BER performance of the system will significantly improve by increasing Δ, as reflected in both graphs. However, the improvement of the system will be different by increasing the same Δ under different fading parameters.

Conclusions
In this study, a novel type of dual-hop wireless power line hybrid communication system based on the AF relay protocol is analyzed. The proposed system uses the hybrid channel fading based on the Nakagami-m and LogN distributions, and the Bernoulli-Gaussian noise model is used for the mathematical modeling and theoretical deduction and analysis of the system. The difficulty in analyzing the system performance increases because the PDF at the end of the hybrid fading system is difficult to solve. Therefore, the MGF equation based on PDF is used to transform the performance analysis of the hybrid fading system into the performance analysis of the double Gamma fading system and further derive the precise closed-form expressions of the outage probability and BER of the system. At the same time, we execute the process of G2L, and the performance indicators of the double LogN distribution system are analyzed by using the approximation of the LogN variable sum, integral change, Mehta, and other algorithms. The numerical results show that the accuracy of the L2G algorithm is higher than that of the G2L algorithm. However, the L2G algorithm is also sensitive to parameter selection because of its high complexity. The G2L algorithm is beyond the limits of the MGF and can be used to analyze many performance indicators. The fading and noise parameters of each link can affect the system performance to varying degrees. Hence, the LogN distribution link is crucial in the entire system, and the combination of environmental variables of the link can be adjusted to optimize the system performance.  1 Ω(s, t)dt is transformed first using t = 1 + x/s. The right integral is then presented as with the aid of [30], Equation (A1) can be simplified as Similarly, by using t = x/s, the left integral ∞ 0 Ω(s, t)dt in the MGF can be transformed into As s→∞, |M γ (s; α R , α D , β R , β D )| = b|s| −d + o(|s| −d ), then the diversity order for the relaying system is d [31]. Based on Equations (A2) and (A3), we have M Z (s; α R , α D , β R , β D ) ≈ s −α D β D −α D + s −α R β R −α R (as s→∞), and the diversity order is min(α R , α D ) for the system model in this study.