## 1 Introduction  
In next-generation wireless communication systems, energy supply has emerged as one of the most critical challenges [1]. Among various energy supply and harvesting techniques, utilizing received radio frequency (RF) signals that simultaneously carry information and energy has gained significant attention. To exploit the far-field energy of RF signals, energy harvesting (EH)-enabled communication has been extensively studied over the past decade. In particular, two prominent models have been widely investigated: the simultaneous wireless information and power transfer (SWIPT) model, where a transmitter concurrently delivers information and energy signals to both information and EH receivers [1, 2, 3], and the wireless powered communication network (WPCN) model, where all transmitting nodes are powered by RF energy broadcast from a dedicated energy beacon [4; 5; 6; 7; 8]. These models offer practical and scalable solutions for sustainable wireless networks [9; 10]. Meanwhile, relay-based communication schemes have been extensively studied due to their simple operation, ability to extend cell coverage, and capability to enhance communication reliability [11; 12; 13; 14]. Early research on relay systems typically assumed that relays were equipped with dedicated power supplies [15; 16; 17; 18; 19; 20; 21; 22; 23; 24]. However, recent efforts have shifted toward the development of SWIPT-enabled relays, where relays forward received information signals using energy harvested from incoming RF signals. This approach can significantly prolong relay battery lifespan and enables the deployment of relays without built-in power sources. Furthermore, recent advancements in the practical implementation of EH hardware have enhanced the feasibility and applicability of SWIPT-based systems [25; 26]. Depending on how the information decoding and EH processes are separated, SWIPT relays can adopt one of two practical receiver architectures: power splitting (PS) and time switching (TS) [1; 27; 28]. In the TS architecture, dedicated time slots are allocated separately for EH and information decoding. In contrast, the PS architecture divides the received signal power into two streams--one for EH and the other for information processing (IP). Among these, the PS receiver is more commonly employed in SWIPT relay systems, as TS requires accurate time synchronization and may cause interruptions in subsequent data transmission [5; 6; 8]. An effective approach to understanding a communication system is through analytical performance evaluation using mathematical models. For conventional non-SWIPT relaying systems, extensive analytical results have been reported in the literature [15; 16; 17; 18; 19; 20; 21; 23; 24]. In [27], the authors analyzed the performance of a SWIPT amplify-and-forward (AF) relaying scheme with a PS receiver, focusing on the outage probability of the achievable information rate. This work was later extended to more generalized scenarios, including the presence of a direct link between the source and destination [28], and the case of correlated Nakagami-\(m\) fading using copula theory [29]. Additionally, the performance of flipped SWIPT relay systems, where an energy-constrained source is powered by nearby relays, was studied in [30]. The bit error rate (BER) performance of WPCN-like AF relaying systems, where both the source and the relay are powered by an external energy beacon, was investigated in [31]. In [32; 33], exact closed-form expressions for the BER were derived for SWIPT AF relaying networks employing differential modulation over Rayleigh fading channels. The work in [34] analyzed the average BER and outage probability performance of SWIPT cooperative AF relaying systems with both binary phase shift keying (BPSK) and differential BPSK under composite Nakagami-lognormal fading conditions. In [35], the average BER performance of dual-hop SWIPT AF and decode-and-forward (DF) relaying systems over Nakagami-\(m\) fading channels was investigated.