## 1 Introduction
Reference [13] studies the dual-hop relay forwarding model of the power line based on an energy collection device, wherein the relay node can effectively improve the energy efficiency of the power line relay system via energy harvesting and effective power allocation strategy. In terms of wireless and PLC cooperation technologies, a system structure of parallel communication of wireless and power line is proposed in [14] and the bit error rate (BER) performance is analyzed under impulse noise. However, the fading coefficient of the power line is set to a fixed constant. Reference [15] adopts the indoor wireless and power line channel model of LogN to study the reliability of an indoor dual-media cooperative communication system based on AF and DF protocols and analyze the rules of LogN fading and power affecting system performance. In this study, the Gauss and Middleton-A noise models are used to model the two transmission media, and the fading models are all LogN distributions. The closed expression of the system performance can be obtained by using an approximate algorithm. [PERSON] et al. [16] analyze the average BER performance of the power line/wireless hybrid cooperative communication system when using dual-hop parallel communication and decoding and forwarding protocols. Similarly, reference [17] verifies the complementarity between PLC and wireless communication through measured data and obtains significant diversity gain by using MRC and selective combining. These two studies on hybrid fading systems are both focused on the DF protocol, hard decisions can result in the loss of useful information and error code, and its performance is significantly worse than that of the AF protocol. Study [18] also proposes a hybrid architecture of wireless and PLC networks installed in rural and sparsely populated environments to provide broadband and smart grid services. Most of the works in the literature are focused on the performance of a dual-hop wireless or PLC system. In such a system, performance analysis is simplified owing to the symmetry of the system model as both links are subjected to the same channel and noise. Compared with existinginvestigations, 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.