## 1 Introduction
The average error rate of SWIPT DF relaying systems was further studied in [36]. However, the resulting error expressions in these studies are mathematically complex and lack analytical tractability, making it difficult to evaluate cooperative diversity performance based on the derived BER. In the context of multi-antenna cooperative cognitive radio networks supporting Device-to-Device (D2D) communications, a comparative performance analysis of DF and AF relaying in SWIPT-enabled systems was conducted in [37]. This study derived closed-form outage probability expressions and demonstrated that DF relaying achieves better energy efficiency at low transmit power, while AF relaying outperforms DF at high transmit power. Moreover, ref. [38] analyzed the outage performance of a full-duplex two-way DF relay network, enabled by SWIPT, under the combined impact of residual hardware impairments and residual self-interference, revealing a system performance ceiling effect beyond which reliable communication is no longer feasible regardless of the signal-to-noise ratio (SNR). Additionally, ref. [29] studied the impact of fading channel correlation on dual-hop DF SWIPT relay networks using a power splitting-based relaying protocol. The authors derived closed-form expressions for ergodic capacity and outage probability under correlated Nakagami-_m_ fading using copula theory, demonstrating that positive dependence between the source-relay and relay-destination links significantly enhances system performance compared to negative or independent fading scenarios. Furthermore, the AF-SWIPT source power minimization problem for multi-hop cooperative relaying was formulated in [39]. Despite these advancements, to the best of our knowledge, a tractable analytical framework that interprets the SWIPT-AF relay system as an equivalent non-SWIPT (i.e., power-supplied) AF relay system has not yet been proposed. It is also noteworthy that most prior studies on power-splitting factor (PSF) optimization primarily focused on outage probability or transmit power minimization, rather than BER performance [28; 29; 37; 38; 39]. In this paper, we provide a comprehensive analytical framework for evaluating the performance of SWIPT AF relaying systems, including the extension to multiple relays. The main contributions are as follows:  
1. _Exact and Upper-Bound BER Expressions_: We derive exact and upper-bound BER expressions for SWIPT AF relaying systems under Rayleigh fading using the MGF-based approach. 2. _Closed-Form Asymptotic BER and SNR Interpretation:_ An intuitive asymptotic BER expression is presented, revealing the harmonic-mean interpretation of the end-to-end SNR, similar to conventional non-SWIPT AF systems. 3. _PS Factor Optimization:_ Based on the derived asymptotic BER, we propose an efficient PS optimization method that enhances BER performance while significantly reducing search complexity. 4. _Extension to Multiple SWIPT Relays_: The proposed framework is extended to a multiple-relay SWIPT system, and its performance is analyzed in terms of BER, outage probability, and channel capacity under different relay configurations and channel conditions. 5. _Unified Analytical Bound:_ For the multiple-relay scenario, we derive approximated but tractable analytical expressions that capture the overall performance trend and validate their accuracy through simulation. 6. _Simulation Validation:_ Extensive simulations confirm the validity of the proposed analysis and demonstrate the impact of the number of relays and optimized PS factors on system performance. The rest of this paper is organized as follows. Section 2 describes the system model of the SWIPT AF relaying system. Section 3 derives the analytical BER expressions and PS optimization. In Section 4, we extend the analysis to the multiple SWIPT relay scenario and derive performance bounds. Section 5 presents simulation results, and concluding remarks are given in Section 6.