Performance analysis of dual-media cooperative communication based on wireless and power line under hybrid fading

Abstract

Wireless communication and power line communication are extensively applied in various fields, such as household Internet of things. For cooperative communication of amplify–forward relay using wireless access and power line transmission, the hybrid model of universal Nakagami wireless fading and lognormal power line fading was used in this study. A comparative analysis of similarities between Gamma and lognormal distributions and lognormal distribution characteristics of relay link signal-to-noise ratio was also carried out. Given the deficiencies of lognormal variable add approximation method, lognormal distribution parameters combined computing method based on moment generation function was proposed. Theoretical formulas of outage probability and bit error rate after amplify–forward relay and maximal-ratio combining was derived on the basis of moment generation function of the total output signal-to-noise ratio of the system. Effectiveness and reliability of the algorithm and theoretical formula were verified through simulation, and influence rules of hybrid channel fading and power distribution on the system performance were analyzed.

Introduction

Power line communication (PLC) and wireless communication are widely applied in various fields, such as smart power grid and household Internet of things (IoT).^1–5 Although wireless communication can realize mobile access, a radio frequency signal of 2.4 GHz can be easily affected by obstacles with great signal attenuation. Taking household IoT as an example, in rooms far from the router, people often find that the signal strength of WLAN is relatively weak, thus a usage of PLC and wireless hybrid network may increase the coverage area and reliability of the access network. Power line and wireless cooperative communication technology^2–5 can integrate advantageous resources and save construction cost. Using diversity combining technologies, such as dual-media parallel communication,^2–4 maximal-ratio combining (MRC), ³ and dual-hop hybrid relay, ⁵ can improve the remote and reliable communication ability of the system.

The power line channel in the studies by Lee and Kim ² and Lai and Messier ³ was a deterministic compound fading channel coefficient without consideration of the influence of time-domain impulse noise during the diversity combining process. The power line channel was modeled into a fading model presenting lognormal (LogN) distribution in a recent study.^4–7 In the study by Chen et al., ⁴ the reliability of the indoor wireless and power line dual-media hybrid communication system was investigated after the use of amplify–forward (AF)/decode–forward (DF) relay and selective combining (SC). Influence rules of channel fading and power distribution on the system performance were also analyzed. However, this work was inapplicable to other fading scenes, such as outdoor wireless communication (e.g. Nakagami fading). In the study by Qian et al., ⁵ theoretical formulas, such as system capacity and outage probability, were obtained for double communication interface system models of power line and wireless with AF relay in the IoT scene. Dubey and Mallik ⁶ analyzed the performance of the PLC system when using AF relay under LogN fading and impulse noise conditions, but they ignored the cooperation of wireless communication. Studying theoretical expressions of Nakagami–LogN hybrid fading and dual-media cooperative communication system performance under impulse noise conditions is theoretically important in the universal wireless communication scene. Mathur et al.^7,8 studied the performance of wireless Nakagami fading and power line LogN fading two-hop relay system under DF relay and SC conditions. However, the influence of direct link was ignored in the studies by Mathur et al.,^7,8 and Bernoulli secondary noise model was adopted for the power line, which was not as comprehensive and objective as the Mid-A impulse noise model. ⁴

The AF protocol and MRC technology have favorable performance and are sensitive to channel parameter change. This study comprehensively analyzed the performance restriction mechanism of hybrid fading and impulse noise by investigating the performance of the AF relay and MRC combination-based dual-media cooperative relay system, which is different from DF relay and SC technology specified in the studies by Mathur et al.^7,8 The influence of wireless direct link was also considered. Some research contents will involve signal processing under LogN–LogN fading, but MRC will be used for signal combining, which is also different from the research method in the studies by Chen et al. ⁴ and Qian et al. ⁵

The innovative contributions of this study are summarized as follows.

1. Similarities between two probability density functions (PDFs) of Gamma to LogN and LogN distribution characteristics of relay link signal-to-noise ratio (SNR) were verified through physical and numerical simulation.

2. Two computing methods of system performance, namely, LogN variable add approximation method and moment generation function (MGF) equation set method, were proposed to address the lack of closed expression in the integration process of LogN hybrid fading.^5,6

3. The first method equivalently transformed Gamma distribution of wireless SNR into LogN distribution and transformed the system performance computation under Nakagami–LogN hybrid fading into LogN variable add approximation computation problem under LogN–LogN fading. As a result, a computing method of the system performance based on LogN distribution parameter of system total SNR was obtained.

4. The second method used the combined computing method based on MGF equation set because LogN variable add approximation method needed multiple approximation computations with low accuracy. LogN distribution parameters of relay link SNR were solved through MGF equation set without transforming Gamma distribution of wireless SNR into LogN distribution. This method provided an analytical idea for the system performance based on MGF.

System model and signal processing

Figure 1 shows the three-node two-hop relay system model under wireless and power line dual-media cooperative communication used in this study. Mobile end S realizes wireless communication with relay node R and target node D, whereas relay R conducts PLC with node D. The system is under half-duplex mode. ⁸ Specifically, end S sends signal X_S to nodes R and D at the transmitting power P_S in the first time slot. In the second time slot, relay R forwards the received signal using the AF protocol to obtain the relay signal X_R and then send it to node D at the power P_R through the power line channel.

Figure 1.

System model.

Figure 1 corresponds to the typical application scenario; that is, PLC cannot access wireless communications, high-band radio wave penetration ability is limited, and the fading is serious. Thus, the smart meter or the sensor (S) and the gateway D use wireless access (S→R) and PLC relay (R→D) hybrid cooperation to achieve wireless access and long-distance communication. PLC relay can overcome the influence of obstacles, such as walls, doors, and windows, make up for the defects of high-frequency signal fading in wireless communication.

In Figure 1, node D carries on a signal combination from the wireless and PLC link. The channels and standards of the links are different. For example, The PLC adopts IEEE 1901 standard and the wireless link uses IEEE 802.11n. The frequency bands of two links are unequable; however, the physical layer of both standards adopt orthogonal frequency division multiplexing (OFDM) modulation. After a radio frequency signal downconversion for wireless link, signal combining and noise reduction processing can be performed on the same baseband or intermediate frequency signal processing framework, thereby making dual-medium hybrid cooperation possible. ⁹ In order to investigate a performance analysis method over hybrid fading, we consider a similar modulation parameter and frame structure.

In the system model, three communication branches are affected by multiplicative fading and additive noise, where;

1. In wireless branches SR and SD in the first time slot, noises n_WR and n_WD satisfy normal distribution N (0, N_W), wireless channel fading coefficient H_WI satisfies Nakagami distribution, and I∈{D, R}. m_I ≥ 0.5 is the Nakagami shape factor, and ${\Omega }_{\mathrm{I}}=E(H_{\mathrm{WI}}^2)$ is the mean value of fading amplitude square. ¹⁰ $H_{\mathrm{WI}}^2$ satisfies Gamma distribution G(m_I, Ω_I/m_I). Ω_I is set as 1 to ensure that fading will not change the average power of the received signal. Average SNR of the channel is defined as Δ_W = P_S/N_W, and receiving-end SNR ${\gamma }_{\mathrm{WI}}$ of the wireless channel is expressed as

   $${\gamma }_{\mathrm{WI}}=H_{\mathrm{WI}}^2{\mathrm{\Delta }}_{\mathrm{W}}$$

   (1)

2. Power line relay link RD in the second time slot and power line fading coefficient H_PlD satisfy LogN distribution LogN(µ_PlD, ${\sigma }_{\mathrm{PlD}}^2$ ), where µ_PlD and σ_PlD are the mean value and standard deviation of lnH_PlD, respectively. Fading energy is normalized, namely, $E(H_{\mathrm{PlD}}^2)=\exp$ $(2{\mu }_{\mathrm{PlD}}+2{\sigma }_{\mathrm{PlD}}^2)=1$ is set. Accordingly, ${\mu }_{\mathrm{PlD}}=-{\sigma }_{\mathrm{PlD}}^2$ can be obtained.

Power line channel noise n_PlD is impulse noise, and Mid-A impulse noise model is adopted. The model consists of Gaussian background noise N_G and impulse noise N_I, and PDF of impulse noise amplitude z is

$$P\left(z\right)=\sum _{k=0}^{\infty }\frac{e^{-A}{A}^k}{k!}\frac{1}{\sqrt{2\pi N_k}}\exp -\frac{z^2}{N_k}$$

(2)

where N_k = N₀(k/A+T)/(1+T) is the transient total noise power of the power line, T = N_G/N_I is the ratio of background noise power to impulse noise power, and N₀ = N_G+N_I is the average total noise power. At concrete sampling time, impulse noise of the Mid-A model is formed through superposition of k Gaussian noises, and each noise model satisfies N (0, N_I/A). The number k of pulses complies with Poisson distribution with the mean value denoted as A.

Average SNR of the power line branch is defined as Δ_Pl,k = P_R/N_k. Under high SNR value (Δ_Pl,k >> 1, Δ_W >> 1), transient SNR ${\mathrm{\gamma }}_{\mathrm{PlD}}$ of node D in the relay link RD can be expressed as ¹¹

$${\gamma }_{\mathrm{PlD}}\approx {\mathrm{\Gamma }}_{\mathrm{k}}={\left(\frac{1}{H_{\mathrm{WR}}^2{\mathrm{\Delta }}_{\mathrm{W}}}+\frac{1}{H_{\mathrm{PlD}}^2{\mathrm{\Delta }}_{\mathrm{Pl},\mathrm{k}}}\right)}^{-1}$$

(3)

After relay link RD and direct link SD are combined using MRC at node D, the total output SNR ${\mathrm{\gamma }}^{\mathrm{AF}}$ of the system is

$${\gamma }^{\mathrm{AF}}={\gamma }_{\mathrm{WD}}+{\gamma }_{\mathrm{PlD}}$$

(4)

The transient mutual information of the system is

$$I^{\mathrm{AF}}=\frac{1}{2}\log (1+{\gamma }^{\mathrm{AF}})$$

(5)

Relay system performance analysis under dual-media cooperation

Theoretical formulas of the system under hybrid fading and impulse noise conditions were determined by analyzing the probability density distribution characteristics of transient SNRs of different branches and system total output SNR. Similarities between Gamma and LogN distributions were verified on the basis of PDF of various branches and system total SNRs (section “Characteristic analysis of probability density of SNR in the relay end-to-end link”). The receiving-end SNR of actual AF relay link complied with LogN distribution as obtained by the Monte Carlo simulation method.

A traditional performance analysis method, which was abbreviated as LogN variable add approximation algorithm, was proposed (section “System performance computation based on LogN variable add approximation method”). In this algorithm, Gamma distribution of wireless link SNR was approximated as LogN distribution, and the system performance analysis under Nakagami–LogN hybrid fading was transformed into analysis problem under LogN–LogN. Moreover, sum of LogN variables was used in computing system performance, such as outage probability, which also satisfied LogN distribution. An approximate calculation method for solving MGF equation set was proposed to further improve theoretical formulas of the system (section “MGFs of dual-variable harmonic means based on Gamma–LogN distribution and system performance computation”). MGF of receiving-end SNR was solved through reducing approximation times to calculate theoretical formulas of the system.

Characteristic analysis of probability density of SNR in the relay end-to-end link

Gamma and specific LogN distributions have high similarities under different parameter conditions. ¹² Figure 2 shows the comparison curve between Gamma distribution corresponding to actual $H_{\mathrm{WI}}^2$ and PDF of LogN distribution of other different parameters when Nakagami distribution parameter $m_{\mathrm{R}}$ of wireless fading coefficient H_WI(I∈{D, R}) is 3.2. In the traditional performance analysis process, ¹¹ PDF parameter after the approximation can be calculated by matching secondary moment of PDFs of LogN and Gamma. Mean value and variance of PDFs are set as equal before and after the approximation. LogN distribution parameters can be obtained as

$${\sigma }_{\mathrm{R}}^2=\ln (1+\frac{1}{m_{\mathrm{R}}})$$

(6)

$${\mu }_{\mathrm{R}}=\ln \left(\frac{{\mathrm{\Omega }}_{\mathrm{R}}}{\sqrt{1+\frac{1}{m_{\mathrm{R}}}}}\right)$$

(7)

Figure 2.

Comparison chart of PDFs before and after the approximation (solid line: actual Gamma distribution; dotted line: approximated LogN distribution).

As shown in Figure 2, parameters obtained through secondary moment approximation are µ_R = –0.14 and ${\sigma }_{\mathrm{R}}=0.52$ , and LogN distribution curve slightly deviates from the original Gamma distribution curve. LogN distribution curves of different parameters are selected for comparison with actual Gamma distribution. Some LogN distribution parameters of high goodness of fit exist, such as µ_R = –0.02 and ${\sigma }_{\mathrm{R}}=0.57$ , which accords with the original Gamma distribution. Therefore, the follow-up computational accuracy of the system performance can be improved from aspects of PDF parameter solving and reduction in approximation times.

Sample data (10⁶) of transient SNR ${\mathrm{\Gamma }}_{\mathrm{k}}$ of the relay link are acquired using Monte Carlo simulation, and a comparative analysis of probability density is conducted. As shown in Figure 3, the solid line is the PDF curve simulation SNR ${\mathrm{\Gamma }}_{\mathrm{k}}$ , the mean value of samples after the statistics is 0.0365, and the mean square error is 0.0715. The hybrid line is a theoretical LogN distribution curve with the same mean and variance as ${\mathrm{\Gamma }}_{\mathrm{k}}$ . The dotted line is a theoretical gamma distribution curve with the same mean and variance as ${\mathrm{\Gamma }}_{\mathrm{k}}$ . Comparison results show that a certain similarity exists between actual PDF curve and LogN theoretical value of samples. Specifically, transient SNR ${\mathrm{\Gamma }}_{\mathrm{k}}$ of the relay link under Nakagami–LogN hybrid fading has LogN distribution characteristics, thereby providing a simulation basis for the follow-up system performance computation.

Figure 3.

Comparison chart of PDFs of receiving-end SNR in the AF relay link (solid line: actual PDF curve of the SNR of the receiver; hybrid line: similar LogN distribution curve; dotted line: similar Gamma distribution curve).

System performance computation based on LogN variable add approximation method

In general, SNR ${\mathrm{\Gamma }}_{\mathrm{k}}$ approximately conforms to LogN distribution, and the key of system performance analysis is a computing problem of distribution parameters. Considering that ${\mathrm{\gamma }}^{\mathrm{AF}}$ is the sum of wireless link SNR ${\mathrm{\gamma }}_{\mathrm{WD}}$ and power line link SNR ${\mathrm{\gamma }}_{\mathrm{PlD}}$ , the steps of LogN variable add approximation method are summarized as follows.

1. Secondary moment approximation method is used to approximate $\mathrm{G}(m_{\mathrm{I}},{\mathrm{\Omega }}_{\mathrm{I}}/m_{\mathrm{I}})$ of wireless fading $H_{\mathrm{WI}}^2$ into LogN(µ_I, ${\sigma }_{\mathrm{I}}^2$ ) and I∈{D, R}.

2. The LogN distribution properties indicate that SNRs ${\gamma }_{\mathrm{WI}}$ and $H_{\mathrm{PlD}}^2{\mathrm{\Delta }}_{\mathrm{Pl},\mathrm{k}}$ of all branches conform to LogN distribution, and reciprocals of LogN variables still conform to LogN distribution. We have

   $$\begin{array}{c}{H}_{\mathrm{WD}}^2{\mathrm{\Delta }}_{\mathrm{W}}-\mathrm{LogN}(\ln {\mathrm{\Delta }}_{\mathrm{W}}+{\mu }_{\mathrm{D}};{\sigma }_{\mathrm{D}}^2)\hfill \\ \frac{1}{\left(H_{\mathrm{PlD}}^2{\mathrm{\Delta }}_{\mathrm{Pl},\mathrm{k}}\right)-\mathrm{LogN}(2{\sigma }_{\mathrm{PlD}}^2-\ln {\mathrm{\Delta }}_{\mathrm{Pl},\mathrm{k}};4{\sigma }_{\mathrm{PlD}}^2)}\hfill \\ \frac{1}{\left(H_{\mathrm{WR}}^2{\mathrm{\Delta }}_{\mathrm{W}}\right)-\mathrm{LogN}(-\ln {\mathrm{\Delta }}_W-{\mu }_R;{\sigma }_{\mathrm{R}}^2)}\hfill \end{array}$$

3. The sum of LogN variables also conforms to LogN distribution, that is, ${\mathrm{\Gamma }}_{\mathrm{k}}=((1/H_{\mathrm{WR}}^2{\mathrm{\Delta }}_{\mathrm{W}})+(1/H_{\mathrm{PlD}}^2{\mathrm{\Delta }}_{\mathrm{Pl},\mathrm{k}}){)}^{-1}$ satisfies LogN(µ_k, ${\sigma }_{\mathrm{k}}^2$ ). Parameters µ_k and ${\sigma }_{\mathrm{k}}^2$ of LogN distribution of ${\mathrm{\Gamma }}_{\mathrm{k}}$ can be obtained using traditional methods, such as Fenton–Wilkinson. ¹³ It computes the mean (or variance) of $G=(1/H_{\mathrm{WR}}^2{\mathrm{\Delta }}_{\mathrm{W}})+(1/H_{\mathrm{PlD}}^2{\mathrm{\Delta }}_{\mathrm{Pl},\mathrm{k}})$ by an addition of the mean (or variance) of $1/H_{\mathrm{WR}}^2{\mathrm{\Delta }}_{\mathrm{W}}$ and $1/H_{\mathrm{PlD}}^2{\mathrm{\Delta }}_{\mathrm{Pl},\mathrm{k}}$ . Finally, we get parameters µ_k and ${\sigma }_{\mathrm{k}}^2$ by

   $${\mu }_{\mathrm{k}}=-\ln \left(\frac{\mathrm{E}\left[G\right]}{{\left(\frac{\mathrm{Var}\left[G\right]}{{\left(\mathrm{E}\left[G\right]\right)}^2+1}\right)}^{1/2}}\right)$$

   (8)
   $${\sigma }_{\mathrm{k}}^2=\ln \left(\frac{\mathrm{Var}\left[G\right]}{{\left(\mathrm{E}\left[G\right]\right)}^2+1}\right)$$

   (9)

   where $\mathrm{E}[G]$ and $\mathrm{Var}[G]$ represent the expectation and variance of G.

4. Considering that ${\gamma }_{\mathrm{WD}}$ and ${\mathrm{\Gamma }}_{\mathrm{k}}$ satisfy LogN distribution, $\mathrm{LogN}({\mu }_{\mathrm{AF}},{\sigma }_{\mathrm{AF}}^2)$ distribution parameters of the system total output SNR ${\mathrm{\gamma }}^{\mathrm{AF}}$ can be obtained in a similar way. $\mathrm{LogN}({\mu }_{\mathrm{AF}},{\sigma }_{\mathrm{AF}}^2)$ is used to obtain theoretical formulas of the system, such as outage probability and bit error rate (BER), which can be expressed as series forms based on Q function. For outage probability

   $$\begin{array}{c}{P}_{\mathrm{AF}}^{\mathrm{out}}=\mathrm{Pr}(I^{\mathrm{AF}}<R_{\mathrm{th}})=\sum _{k=0}^{\infty }\frac{e^{-A}{A}^k}{k}\mathrm{Pr}({\mathrm{\gamma }}_k^{\mathrm{AF}}<{\mathrm{\gamma }}_{\mathrm{th}})\hfill \\ =\sum _{k=0}^{\infty }\frac{e^{-A}{A}^k}{k}\underset{-\infty }{\overset{{\mathrm{\gamma }}_{\mathrm{th}}}{\int }}\frac{1}{\mathrm{\gamma }\sqrt{2\pi {\sigma }_{\mathrm{AF},\mathrm{k}}}}\exp \left(\frac{-{(\ln \mathrm{\gamma }-{\mu }_{\mathrm{AF},\mathrm{k}})}^2}{2{\left({\sigma }_{\mathrm{AF},\mathrm{k}}\right)}^2}\right)d\mathrm{\gamma }\hfill \end{array}$$

We set $t=(\ln {\gamma }_{\mathrm{th}}-{\mu }_{\mathrm{AF},\mathrm{k}})/{\sigma }_{\mathrm{AF},\mathrm{k}}$ and carry on an integral transformation, then

$$P_{\mathrm{AF}}^{\mathrm{out}}=\sum _{k=0}^{\infty }\frac{e^{-A}{A}^k}{k}Q\left(\frac{{\mu }_{\mathrm{AF},\mathrm{k}}-\ln {\gamma }_{\mathrm{th}}}{{\sigma }_{\mathrm{AF},\mathrm{k}}}\right)$$

(10)

where $R_{\mathrm{th}}$ is the threshold value, ${\gamma }_{\mathrm{th}}=\exp (2R_{\mathrm{th}})-1$ is the SNR threshold of system outage.

Notably, the first method mainly provides the computing idea of LogN distribution parameters of the system total SNR. In accordance with the main algorithm steps, approximation from Gamma to LogN distribution and approximation of sum of LogN variables to LogN distribution should be carried out twice when this method is used to compute the system performance. The follow-up simulation indicates that this algorithm has low accuracy. Therefore, in consideration of influence factors on the accuracy, such as PDF parameter solving and approximation times, a system performance analysis algorithm featuring first-order approximate calculation based on MGF equation set was further proposed as the main contribution of this study.

MGFs of dual-variable harmonic means based on Gamma–LogN distribution and system performance computation

The approximation times were decreased, and the accuracy of the performance computation was improved by proposing a high-accuracy analysis algorithm based on first-order MGF parameter approximation by virtue of PDF distribution characteristics of the system total SNR in section “Characteristic analysis of probability density of SNR in the relay end-to-end link.” The main aim was to calculate MGFs of dual-variable harmonic means under Gamma and LogN distributions.

In general, $H_{\mathrm{WR}}^2$ conforms to Gamma distribution. Gamma distribution properties indicate that $H_{\mathrm{WR}}^2{\mathrm{\Delta }}_{\mathrm{W}}-G(m_{\mathrm{R}},{\mathrm{\Delta }}_{\mathrm{W}}{\mathrm{\Omega }}_{\mathrm{R}}/m_{\mathrm{R}})$ when Δ_W is a constant. Accordingly, MGF of $1/(H_{\mathrm{WR}}^2{\mathrm{\Delta }}_{\mathrm{W}})$ can be obtained

$$M_{\mathrm{R}}\left(s\right)=\frac{2}{\mathrm{\Gamma }\left(m_{\mathrm{R}}\right)}{\left(\frac{s·m_{\mathrm{R}}}{{\mathrm{\Omega }}_{\mathrm{R}}·{\mathrm{\Delta }}_{\mathrm{W}}}\right)}^{0.5m_{\mathrm{R}}}{K}_{m_{\mathrm{R}}}\left(2\sqrt{\frac{s·m_{\mathrm{R}}}{{\mathrm{\Omega }}_{\mathrm{R}}·{\mathrm{\Delta }}_{\mathrm{W}}}}\right)$$

(11)

where $\mathrm{\Gamma }*$ is the Gamma function and $K_{\mathrm{v}}(*)$ is the Class II modified Bessel function.

The analysis in section “System performance computation based on LogN variable add approximation method” indicates that $1/(H_{\mathrm{PlD}}^2{\mathrm{\Delta }}_{\mathrm{Pl},\mathrm{k}})$ satisfies LogN distribution. Integration processing of MGFs of LogN variables is carried out through the Gauss–Hermite series method to obtain MGF expressions of $1/(H_{\mathrm{PlD}}^2{\mathrm{\Delta }}_{\mathrm{Pl},\mathrm{k}})$ and ${\mathrm{\Gamma }}_{\mathrm{k}}$

$$\begin{array}{c}{M}_{\mathrm{PlD},\mathrm{k}}\left(s\right)\hfill \\ =\sum _{n=1}^N\frac{w_n}{\sqrt{\mathrm{\pi }}}\exp (-s·\exp (2\sqrt{2}{\sigma }_{\mathrm{PlD}}{a}_n+2{\sigma }_{\mathrm{PlD}}^2-\ln {\mathrm{\Delta }}_{\mathrm{Pl},\mathrm{k}}))\hfill \end{array}$$

(12)

$$M_{\mathrm{k}}\left(s\right)=\sum _{n=1}^N\frac{w_n}{\sqrt{\mathrm{\pi }}}\exp (-s·\exp (\sqrt{2}{\sigma }_{\mathrm{k}}{a}_n-{\mu }_{\mathrm{k}}))$$

(13)

where w_n and a_n are the weight and zero point of Gauss–Hermite formula.

MGF of the sum of two variables is equal to the product of MGFs of two variables. F(11) and F(12) show that MGF of ${\mathrm{\Gamma }}_{\mathrm{k}}$ is equal to

$$M_k\left(s\right)=\sum _{k=0}^{\infty }\frac{e^{-A}{A}^k}{k}{M}_{\mathrm{PlD},\mathrm{k}}\left(s\right)\times M_{\mathrm{R}}\left(s\right)$$

(14)

Substituting equation (13) into equation (14) yields

$$\begin{array}{c}\sum _{k=0}^{\infty }\frac{e^{-A}{A}^k}{k}{M}_{\mathrm{PlD},\mathrm{k}}\left(s\right)\times M_{\mathrm{R}}\left(s\right)\hfill \\ =\sum _{n=1}^N\frac{w_n}{\sqrt{\mathrm{\pi }}}\exp (-s·\exp (\sqrt{2}{\sigma }_{\mathrm{k}}{a}_n-{\mu }_{\mathrm{k}}))\hfill \end{array}$$

Two fixed values (s₁ and s₂) are selected. Under different channel fading conditions, selection of s value will be different. ¹³ When $m_{\mathrm{R}}⩽2$ , (s₁; s₂) = (1; 0.2) is selected; when $m_{\mathrm{R}}>2$ , (s₁; s₂) = (0.001; 0.005) is selected. Equation set of µ_k and ${\sigma }_{\mathrm{k}}^2$ can be acquired.

The value of s in the abovementioned conclusion is obtained through a numerical experiment. In the actual application, numerical modeling is conducted with the minimum difference of generation function curves at two sides of equation (14), and optimal s₁ and s₂ combination is solved

$$\underset{s_1{s}_2}{\min }\sum _{i=1}^M|M_{\mathrm{k}}\left(H_i\right)-M_{\mathrm{PlD},\mathrm{k}}\left(H_i\right)\times M_{\mathrm{R}}\left(H_i\right)|$$

$$\mathrm{s}.\mathrm{t}.\{\begin{array}{c}{M}_k\left(s_1\right)=\sum _{k=0}^{\mathrm{t}}\frac{e^{-A}{A}^k}{k}{M}_{\mathrm{PlD},k}\left(s_1\right)\times M_R\left(s_1\right)\hfill \\ \multicolumn{1}{c}{M_k\left(s_2\right)=\sum _{k=0}^{\mathrm{t}}\frac{e^{-A}{A}^k}{k}{M}_{\mathrm{PlD},k}\left(s_2\right)\times M_R\left(s_2\right)}\\ \multicolumn{1}{c}{H_{\mathrm{i}}=0.01+0.05(i-1)}\\ \multicolumn{1}{c}{s_1>0}\\ \multicolumn{1}{c}{s_2>0}\end{array}$$

where H_i is the sampling value of MGF, M is the total number of sampling points (M = 1000 in this study), and t is the maximum number of impulse noises (t = 100 in this study).

This numerical model takes goodness of fit of MGF curves as the optimization goal to calculate absolute value of difference value between two MGFs corresponding to sampling value H_i of each s for weighing processing. This numerical model can use multiple groups of s values to seek for optimal s₁ and s₂ combination for solving. First, we can randomly generate a sufficient number of pairs of s (10⁵) by simulation. Then, the pair of s with the minimum difference of MGF is selected and used to obtain the approximation parameters of LogN. The model can also be solved by a multiobjective optimization such as fgoalattain function of MATLAB. The aforementioned method can be used to establish a list of optimal s values under different SNRs and fading coefficients (Table 1).

Table 1.

Optimal s values under different channel conditions.

SNR/dB	2	10	20
m R = 0.75; σ PlD = 3	s 1 = 0.9629 s 2 = 2.9418	s 1 = 0.9876 s 2 = 2.9570	s 1 = 0.0089 s 2 = 0.0181
m R = 2; σ PlD = 2	s 1 = 0.9995 s 2 = 2.9175	s 1 = 0.0002 s 2 = 2.4940	s 1 = 0.0058 s 2 = 0.0076
m R = 3; σ PlD = 1	s 1 = 0.9669 s 2 = 2.9839	s 1 = 0.0091 s 2 = 2.5876	s 1 = 0.0134 s 2 = 0.0462

SNR: signal-to-noise ratio.

Evidently, this method is the only first-order PDF parameter approximate calculation in the second algorithm. SNR ${\mathrm{\Gamma }}_{\mathrm{k}}$ of the relay link RD finally satisfies $\mathrm{LogN}({\mu }_{\mathrm{k}},{\sigma }_{\mathrm{k}}^2)$ , and its MGF is

$$M_{{\mathrm{\Gamma }}_{\mathrm{k}}}\left(s\right)=\sum _{n=1}^N\frac{w_n}{\sqrt{\mathrm{\pi }}}\exp (-s·\exp (\sqrt{2}{\sigma }_{\mathrm{k}}{a}_n-{\mu }_{\mathrm{k}}))$$

(15)

In general, SNR of SD direct link is ${\gamma }_{\mathrm{WD}}=H_{\mathrm{WD}}^2{\mathrm{\Delta }}_{\mathrm{W}}$ , and $H_{\mathrm{WD}}^2$ satisfies Gamma distribution. The properties of Gamma distribution function indicate that ${\gamma }_{\mathrm{WD}}-G(m_{\mathrm{D}},{\mathrm{\Delta }}_{\mathrm{W}}{\mathrm{\Omega }}_{\mathrm{D}}/m_{\mathrm{D}})$ holds when Δ_W is a constant. Accordingly, the MGF form of ${\gamma }_{\mathrm{WD}}$ is shown as follows

$$M_{\mathrm{D}}\left(s\right)={(1+\frac{{\mathrm{\Delta }}_{\mathrm{W}}{\mathrm{\Omega }}_{\mathrm{D}}s}{m_{\mathrm{D}}})}^{-m_{\mathrm{D}}}$$

(16)

On the basis of MGFs of SNRs ${\gamma }_{\mathrm{WD}}$ and ${\mathrm{\Gamma }}_{\mathrm{k}}$ of receiving-end wireless and power line branches, statistical averaging of impulse noise parameter k is solved. As a result, the MGF of the system total SNR ${\mathrm{\gamma }}^{\mathrm{AF}}$ under MRC combining can be expressed as

$$M_{\mathrm{MRC}}\left(s\right)=\sum _{k=0}^{\infty }\frac{e^{-A}{A}^k}{k}{M}_{\mathrm{D}}\left(s\right)\times M_{{\mathrm{\Gamma }}_{\mathrm{k}}}\left(s\right)$$

(17)

The MGF of ${\mathrm{\gamma }}^{\mathrm{AF}}$ can be utilized to calculate BER of the system. Under multiple phase shift keying (M-PSK) modulation, the system BER $P_e$ can be expressed as

$$P_e=\frac{1}{\pi }\underset{0}{\overset{\frac{(M-1)\pi }{M}}{\int }}{M}_{\mathrm{MRC}}\left(\frac{g_{\mathrm{PSK}}}{\mathrm{si}{\mathrm{n}}^2\theta }\right)d\theta$$

(18)

where $g_{\mathrm{PSK}}=\mathrm{si}{\mathrm{n}}^2(\mathrm{\pi }/M)$ . Under binary phase shift keying (BPSK) modulation, $\Theta =(M-1)\mathrm{\pi }/M$ is defined, and equation (18) can be further simplified as ¹⁴

$$\begin{array}{c}{P}_e\approx (\frac{\mathrm{\Theta }}{2\mathrm{\pi }}-\frac{1}{6})M_{\mathrm{MRC}}\left(g_{\mathrm{PSK}}\right)+\frac{1}{4}{M}_{\mathrm{MRC}}\left(\frac{4}{3}{g}_{\mathrm{PSK}}\right)\hfill \\ +(\frac{\mathrm{\Theta }}{2\mathrm{\pi }}-\frac{1}{4})M_{\mathrm{MRC}}\left(\frac{g_{\mathrm{PSK}}}{\mathrm{si}{\mathrm{n}}^2\mathrm{\Theta }}\right)\hfill \end{array}$$

(19)

When the system information rate is smaller than the required minimum rate threshold γ₀, normal communication of the system will be interrupted. The system outage probability $P_{\mathrm{out}}$ can also be calculated using MGF as follows

$$P_{\mathrm{out}}={\mathrm{\zeta }}^{-1}\left(\frac{M_{\mathrm{MRC}}\left(s\right)}{s}\right)|{\mathrm{\gamma }}_0$$

(20)

where ${\mathrm{\zeta }}^{-1}$ (*) is the inverse Laplace transformation, and the following approximate expression of outage probability can be obtained through the necessary transformation ¹⁵

$$\begin{array}{c}{P}_{\mathrm{out}}=\frac{2^{-Q}\exp (0.5C)}{{\mathrm{\gamma }}_0}\sum _{q=0}^Q\left(\begin{array}{c}Q\hfill \\ \multicolumn{1}{c}{q}\end{array}\right)\sum _{n=0}^{N+q}\frac{{(-1)}^n}{{\beta }_n}\hfill \\ \times R\left\{\frac{M_{\mathrm{MRC}}\left(\frac{C+2\pi \mathrm{jn}}{2{\gamma }_0}\right)}{\frac{C+2\pi \mathrm{jn}}{2{\gamma }_0}}\right\}+\mathrm{E}(\mathrm{C},\mathrm{N},\mathrm{Q})\hfill \end{array}$$

(21)

where ${\beta }_1$  = 2 and ${\beta }_n$  = 1 (n is a positive integer). The accuracy of numerical estimation is decided by numerical values C, N, and Q and can be estimated through the overall truncation error $\mathrm{E}(\mathrm{C},\mathrm{N},\mathrm{Q})$ . Evidently, the second method adopts the computing idea of the system performance based on MGF of the system total SNR.

Simulation results

The accuracy of the theoretical formula was verified by Monte Carlo simulation experiment in MATLAB and by conducting a comparative analysis with theoretical expressions in the numerical simulation. Without loss of generality, parameters in the model shown in Figure 1 use the following default setting if no special instructions are necessary during the simulation and calculation process. After power normalization, the system total power is 2, P_S = 1, and P_R = 1. The influences of channel fading and noise distribution on the performance are highlighted under the assumption that average SNR of the system channel is SNR, namely, Δ_Pl,k = Δ_w = Δ. Parameters of impulse noise are set as A = 1 and T = 0.2, and the maximum value of k in the formula is Max_k = 100. Under different channel fading conditions, selection of s value will be slightly different. ¹³

Figures 4 and 5 show the PDF distribution diagram and cumulative distribution function (CDF) distribution diagram of simulation data of receiving-end SNR ${\mathrm{\Gamma }}_{\mathrm{k}}$ under different channel parameter conditions. The PDF and CDF curves determined by two methods, namely, LogN variable add approximation method and MGF equation set method, are compared. The figures show that, when different fading parameters are used, the proposed approximation method of MGF equation set has higher accuracy than the LogN variable add approximation method. This method also has a high goodness of fit with PDF and CDF curves of actual simulation data. Same as the conclusion in section “System performance computation based on LogN variable add approximation method” and the theoretical analysis results in section “MGFs of dual-variable harmonic means based on Gamma–LogN distribution and system performance computation,” the abovementioned simulation sufficiently indicates that receiving-end SNR ${\mathrm{\Gamma }}_{\mathrm{k}}$ satisfies LogN distribution.

Figure 4.

Comparison of PDF approximation effects under different parameters: (a) m_R = 2, σ_PlD = 2.8, and SNR = 8; (b) m_R = 2, σ_PlD = 2.8, and SNR = 18; (c) m_R = 3, σ_PlD = 2.1, and SNR = 8; and (d) m_R = 3, σ_PlD = 2.1, and SNR = 18.

Figure 5.

Comparison of CDF approximation effects under different parameters: (a) m_R = 1.7, σ_PlD = 2.8, and SNR = 8; (b) m_R = 1.7, σ_PlD = 2.8, and SNR = 18; (c) m_R = 3, σ_PlD = 2.1, and SNR = 8; and (d) m_R = 3, σ_PlD = 2.1, and SNR = 18.

The default parameter setting is adopted. Figures 6 and 7 compare outage probability and BER performances when MRC is used for the AF relay system under different channel fading parameters. m_D value is set as 1 and m_R value is greater than 1 in the simulation because direct link SD is not as good as channel conditions of close-range wireless link SR in actual application. The following conclusions can be obtained from the analysis: (1) MGF-based equation set approximate calculation has higher accuracy than LogN variable add approximation algorithm when approximation times are decreased and approximate calculation of MGF is adopted. The result is identical to the simulation performance. Therefore, the reliability of the theoretical formula is further verified. (2) m_R value increases (i.e. wireless channel fading degree decreases), system outage probability and BER decrease under fixed SNR and power line channel parameter ${\sigma }_{\mathrm{PlD}}$ . When SNR and wireless channel parameter m_R are the same, power line channel fading degree increases and the system reliability degrades as ${\sigma }_{\mathrm{PlD}}$ value increases.

Figure 6.

Outage probability performances under different channel parameters.

Figure 7.

BER performance under different channel parameters.

Figure 8 shows the relationships of the system outage probability performance with fading parameters m_R, σ_PlD, and SNR under default parameter setting. The figure shows that when m_R and σ_PlD are unchanged, the increase in the system average SNR evidently improves the system reliability. When m_R and SNR remain unchanged, fading of the power line branch becomes serious and the overall system performance becomes poor as power line fading coefficient σ_PlD increases. When σ_PlD and SNR are unchanged, wireless channel fading becomes negligible and the overall system performance enhances as the wireless fading coefficient m_R increases. When SNR = 10 dB and σ_PlD increases from 0.01 dB to 5 dB, system outage probability increases from 10^–4 to 10^–2. When m_R increases from 0.01 to 5 dB, system outage probability decreases from 10^–1 to 10^–3. Evidently, the system performance is highly sensitive to the change in parameter m_R because parameter m_R influences the fading degrees of two wireless branches SR and SD.

Figure 8.

Relationship between theoretical outage probability performance and fading parameter m_R/σ_PlD.

The channel fading energy is normalized to analyze the influence of unbalance of branch channels on the system performance. Figure 9 shows the schematic of the relationship of theoretical (Theo) and simulated (Simu) system performance with different relay transmitting powers P_R when the total power remains unchanged. In the figure, Oute is system outage probability, and BER is system BER. SNR = 8 dB, SNR = 18 dB, ${\sigma }_{\mathrm{PlD}}$  = 3, and $m_{\mathrm{R}}$  = 1.5 are set in the simulation, but the other parameters are identical to the default parameters. Figure 8 shows that (1) theoretical and simulated results of system performance are identical under different transmitting powers. (2) Under a high SNR (SNR = 18 dB), system outage probability changes evidently as the power increases. (3) The system performance is optimal because of the optimal power distribution factor in the system.

Figure 9.

Relationship between system performance and transmitting power.

In dual-media hybrid cooperative communication, the optimal transmitting power distribution problem with minimum system performance as the target can be described using the following numerical model

$$\{\begin{array}{c}\min P_e\left(\mathrm{or}{P}_{\mathrm{out}}\right)\hfill \\ \multicolumn{1}{c}{\mathrm{s}.\mathrm{t}.P_{\mathrm{S}}+P_{\mathrm{R}}=P}\end{array}$$

(22)

where $P_{\mathrm{S}}$ and $P_{\mathrm{R}}$ are transmitting powers of the source and relay nodes, respectively; P is the total transmitting power of the system and set as P = 2; and $P_e(\mathrm{or}{P}_{\mathrm{out}})$ is optimal system outage probability or BER performance.

As shown in Figure 9, system outage probability performance is a continuous smooth curve as the transmitting power P_R changes, and only an extreme point exists within the interval (0, 2). Therefore, outage probability and BER functions with transmitting power as the variable are continuously differentiable, and a global optimal solution exists. The solution obtained through numerical optimization algorithms, such as golden section, can be repeated. Accordingly, golden section and quadratic interpolation method are combined to solve the optimal power distribution coefficient. Golden section is used for global searching, whereas quadratic interpolation is used for refined searching. Tables 2 and 3 show the optimal power distribution parameters and corresponding outage probability and BER performance solved through golden section and quadratic interpolation methods. The proposed distribution algorithm can effectively optimize optimal power coefficient and has high accuracy.

Table 2.

Power distribution value corresponding to optimal outage probability.

σ PlD/dB	m R	m D	P R/W	P S/W	P out
3	1.5	1	0.7810	1.219	0.0089
3	2	1	0.7910	1.209	0.0079
3	3	1	0.7960	1.204	0.0071
1.5	3	1	0.8481	1.1519	0.0006
2	3	1	0.8193	1.1807	0.0018

Table 3.

Power distribution value corresponding to optimal bit error rate.

σ PlD/dB	m R	m D	P R/W	P S/W	P e
3	1.5	1	0.4815	1.5085	0.0096
3	2	1	0.5214	1.4786	0.0091
3	3	1	0.5570	1.4420	0.0086
1.5	3	1	0.6065	1.3935	0.0054
2	3	1	0.6012	1.3988	0.0062

Tables 2 and 3 show that when power line fading coefficient remains the same (e.g. σ_PlD = 3 dB), wireless channel fading weakens as m_R value increases. Thus, the power of PLC branch should be increased to obtain optimal system performance. When index m_R of the wireless link is unchanged, wireless communication power should be increased to optimize the system performance as σ_PlD value increases (i.e. channel condition becomes poor).

Conclusion

Power line and wireless dual-media cooperative relay can improve the remote and reliable communication ability of the system. For the AF relay system under dual-media cooperation, hybrid channel fading and multidimensional impulse noise model based on Nakagami and LogN distribution was used in this study. A system performance analysis framework based on relay link LogN distribution hypothesis, and MGF calculation was established. MGF equation set approximation algorithm was used to solve LogN distribution parameters and obtain theoretical system performance. The results provide necessary theoretical support for the application of dual-media cooperative communication.

Subsequent studies can be extended to the usage of mixture LogN approximation in analysis. For example, the MGF equation approximation together with a mixture LogN of two elements can be used to derive the PDF parameters of the SNR at the receiving end of the AF transmission system, and the outage probability and BER of the system can be further derived with an improved accuracy. Besides, we can investigate a more general analysis method for the other hybrid fading system except the group of Nakagami and LogN in this article.