## 1 Introduction  
In some practical scenarios, the traditional energy-limited communication system has a limited operational life. To maintain the continuity and connectivity of energy-limited wireless relay networks, nodes are usually equipped with fixed energy. Traditional battery-powered systems need to be periodically replaced or charged to maintain network connectivity [1; 2]. However, in certain situations (such as in vivo sensors), battery charging is inconvenient or impossible, and the life cycle of the wireless network is limited. To solve this problem, proposed energy acquisition schemes are proposed. However, the energy collected from the external environment is random and susceptible to many factors, such as weather and geography, and a continuous, stable energy supply is difficult to provide to the wireless network, making normal communication more difficult. In recent years, because radio-frequency (RF) signals can simultaneously carry information and energy, technology involving the use of RF signals for energy collection has received extensive attention. This technology has the advantages of controllability and measurability, which can be used in wireless systems compared with traditional energy harvesting technologies, thereby providing an effective solution to solve the energy supply problem of energy-constrained nodes in harsh environments. According to the interaction between wireless energy transmission and wireless information transmission, two main architectures for implementing wireless energy information [3; 4] are proposed, namely, power splitting (PS) and time switching (TS). For the TS protocol [5; 6], the energy and information signals are transmitted through different time slots. The data source first collects energy from the power signal and then uses the collected energy to transmit the information signal [7]. For the PS architecture [8; 9], the receiver divides the received power between energy harvesting and information processing. According to the time (TS) receiver architecture, we propose a closed expression of system throughput based on an adaptive TS protocol, as seen in [10]. In [11], for PS optimization of decode-and-forward (DF) and amplify-and-forward (AF) relays, an efficient algorithm was proposed by using the PS strategy in a simultaneously (SWIPT), multi-relay assisted, dual-hop relay system. In order to make full use of the collected energy, this study analyzes important performance indicators [12; 13; 14; 15; 16; 17] such as energy efficiency and outage probability. The analytic expressions of ergodic capacity and outage probability based on adaptive TS protocol were deduced in [17]. However, these tasks were only considered with regard to HD mode. Because HD relay transmits and receives signals in different frequency bands, it causes significant loss of spectral efficiency. In order to reduce the loss of spectral efficiency, FD architecture was proposed in [18]. With the latest advances in antenna and signal processing technology, FD relays are considered to be possible [19; 20; 21; 22]. Since the working mode of the relay node changes from HD to FD, the hardware structure also changes, therefore, previous power allocation schemes are not applicable; we expect new resource allocation schemes and transmission protocols to become the main directions in future research. In [23], the authors studied wireless-powered FD relay systems based on a TS protocol, where the number of antennas of relay nodes in the energy harvesting phase was considered and the analytical expressions of throughput maximization in three different communication modes (instantaneous transmission, delay-constrained transmission, and delay-tolerant transmission) were derived by optimizing the TS factor. Moreover, the throughput of the wireless-powered multi-antenna FD relay system was analyzed in [24]. In [25], the author proposed a new optimal power allocation scheme for bidirectional FD relay.