Error Performance Analysis and PS Factor Optimization for SWIPT AF Relaying Systems over Rayleigh Fading Channels: Interpretation SWIPT AF Relay as Non-SWIPT AF Relay

Abstract

This paper presents an analytical study of the bit error rate (BER) and signal-to-noise ratio (SNR) performance in simultaneous wireless information and power transfer (SWIPT) amplify-and-forward (AF) relaying systems over Rayleigh fading channels. A power-splitting (PS) protocol is employed at the energy-constrained relay to divide the received signal for concurrent energy harvesting and information processing. Closed-form and asymptotic BER expressions are derived based on exact and bounded moment-generating functions (MGFs), offering insights into how the SNR balance between the source–relay (SR) and relay–destination (RD) links influences system performance. An asymptotic BER expression further reveals that a SWIPT AF relay system can be interpreted as a generalized AF relaying model, sharing the same diversity order as conventional AF systems. Based on this interpretation, an optimization method for the PS factor is proposed, effectively reducing the BER by reinforcing the weaker link. Simulation results confirm the tightness of the derived expressions and the effectiveness of the optimization strategy. Moreover, the analytical framework is extended to multiple SWIPT relaying systems, where multiple relays operate with individually optimized PS ratios. For such configurations, approximations for the system BER, outage probability, and channel capacity are derived and validated. Results demonstrate that increasing the number of relays significantly improves system performance, and the proposed analysis accurately captures these performance gains under varying channel conditions.

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. 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:

• 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.
• 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.
• 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.
• 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.
• 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.
• 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.

2. System Model

We consider a point-to-point wireless communication system in which a cooperative relay assists in transmitting data from the source to the destination. An AF relay operating in half-duplex mode is employed, and all nodes—including the source, relay, and destination—are assumed to be equipped with a single antenna. The relay node is considered energy-constrained and lacks a dedicated power supply. Instead, it operates by harvesting energy from the RF signal transmitted by the source. To enable simultaneous EH and IP at the relay, a PS receiver architecture is adopted. Specifically, the received signal power at the relay is split into two parts: one portion is used for EH, and the remaining portion is used for IP, as in [27,28].

2.1. Single SWIPT Relaying System Model

From this point onward, we describe the SWIPT AF relay system. Let $h_{\mathrm{sd}}$, $h_{\mathrm{sr}}$, and $h_{\mathrm{rd}}$ denote the complex channel gains of the source-to-destination (SD), source-to-relay (SR), and relay-to-destination (RD) links, respectively. These channels are assumed to undergo independent Rayleigh fading and are thus modeled as mutually independent complex Gaussian random variables with zero mean and variances given by ${\mathrm{\Omega }}_{\mathrm{sd}}\triangleq E\left[{\left|h_{\mathrm{sd}}\right|}^2\right]$, ${\mathrm{\Omega }}_{\mathrm{sr}}\triangleq E\left[{\left|h_{\mathrm{sr}}\right|}^2\right]$, and ${\mathrm{\Omega }}_{\mathrm{rd}}\triangleq E\left[{\left|h_{\mathrm{rd}}\right|}^2\right]$. Here, $E\left[·\right]$ denotes the stochastic expectation.

Due to the half-duplex constraint of the relay, communication occurs over two time slots. In the first slot, the source broadcasts the signal $x_s$, with $E\left[x_s\right]=0$ and $E\left[{\left|x_s\right|}^2\right]=P_s$, to both the relay and the destination. The received signals at the relay and the destination, denoted by $y_{\mathrm{sr}}$ and $y_{\mathrm{sd}}$, respectively, are given by

$$\begin{array}{cc}\hfill y_{\mathrm{sr}}& =h_{\mathrm{sr}}{x}_s+n_{\mathrm{sr}}\hfill \\ \hfill y_{\mathrm{sd}}& =h_{\mathrm{sd}}{x}_s+n_{\mathrm{sd}}\hfill \end{array}$$

(1)

where $n_{\mathrm{sr}}$ and $n_{\mathrm{sd}}$ represent the complex additive white Gaussian noise (AWGN) at the relay and destination, respectively. These are assumed to be zero-mean and mutually independent, with variances $E\left[{\left|n_{\mathrm{sr}}\right|}^2\right]=E\left[{\left|n_{\mathrm{sd}}\right|}^2\right]={\sigma }^2$.

The relay’s power splitter divides the received signal $y_{\mathrm{sr}}$ into two components, $y_{\mathrm{sr}}^E$ and $y_{\mathrm{sr}}^I$, for EH and IP, respectively. By introducing a PS factor $\rho$ with $0<\rho <1$ [28], we can express

$$\begin{array}{cc}\hfill y_{\mathrm{sr}}^E& =\sqrt{\rho }{y}_{\mathrm{sr}}=\sqrt{\rho }\left(h_{\mathrm{sr}}{x}_s+n_{\mathrm{sr}}\right).\hfill \end{array}$$

(2)

The total energy harvested at the relay during the first time slot is then given by

$$\begin{array}{cc}\hfill {\mathrm{E}}_h& =E\left[{\left|y_{\mathrm{sr}}^E\right|}^2\right]=\eta \rho P_s{\left|h_{\mathrm{sr}}\right|}^{2}T\hfill \end{array}$$

(3)

where $\eta \in \left(0,1\right]$ is the energy conversion efficiency, and T denotes the time duration of the time slot.

Since the AF relay uses an equal duration for receiving and forwarding data, the available power $P_r$ at the relay during the second slot is

$$\begin{array}{cc}\hfill P_r& =\eta \rho P_s{\left|h_{\mathrm{sr}}\right|}^2.\hfill \end{array}$$

(4)

Meanwhile, the information component $y_{\mathrm{sr}}^I$ is given by

$$\begin{array}{cc}\hfill y_{\mathrm{sr}}^I& =\sqrt{1-\rho }{y}_{\mathrm{sr}}+n_c=\sqrt{1-\rho }{h}_{\mathrm{sr}}{x}_s+n_r\hfill \end{array}$$

(5)

where $n_r=\sqrt{1-\rho }{n}_{\mathrm{sr}}+n_c$, and $n_c$ denotes the noise introduced during RF-to-baseband conversion. For simplicity, we assume $n_{\mathrm{sr}}$ and $n_c$ are mutually independent with zero mean, and variances $E\left[{\left|n_c\right|}^2\right]={\sigma }^2$ and ${\sigma }_r^2=E\left[{\left|n_r\right|}^2\right]=\left(2-\rho \right){\sigma }^2$.

During the second time slot, the relay forwards $y_{\mathrm{sr}}^I$ to the destination using the available power $P_r$. The transmitted signal from the relay is

$$\begin{array}{cc}\hfill x_r& =G{y}_{\mathrm{sr}}^I\hfill \end{array}$$

(6)

where $G=\sqrt{{\displaystyle \frac{\eta \rho P_s{\left|h_{\mathrm{sr}}\right|}^2}{\left(1-\rho \right){\left|h_{\mathrm{sr}}\right|}^2{P}_s+{\sigma }_r^2}}}$ is the amplification gain. Accordingly, the signal received at the destination from the relay is    

$$\begin{array}{cc}\hfill y_{\mathrm{rd}}& =h_{\mathrm{rd}}{x}_r+n_{\mathrm{rd}}\hfill \\ \hfill & =h_{\mathrm{rd}}G\sqrt{1-\rho }{h}_{\mathrm{sr}}{x}_s+h_{\mathrm{rd}}G\left(\sqrt{1-\rho }{n}_{\mathrm{sr}}+n_c\right)+n_{\mathrm{rd}}\hfill \end{array}$$

(7)

where $n_{\mathrm{rd}}$ is the AWGN at the destination in the second time slot, with $E\left[n_{\mathrm{rd}}\right]=0$ and $E\left[{\left|n_{\mathrm{rd}}\right|}^2\right]={\sigma }^2$. It is assumed that all noise terms $\left\{n_{\mathrm{sd}},n_{\mathrm{sr}},n_c,n_{\mathrm{rd}}\right\}$ are mutually independent.

2.2. Indirect Link SNR

From $y_{\mathrm{rd}}$ of (7), the instantaneous SNR of the indirect link is expressed as

$$\begin{array}{cc}\hfill {\gamma }_{\mathrm{id}}& ={\displaystyle \frac{P_s{G}^2\left(1-\rho \right){\left|h_{\mathrm{sr}}\right|}^2{\left|h_{\mathrm{rd}}\right|}^2}{G^2{\left|h_{\mathrm{rd}}\right|}^2{\sigma }_r^2+{\sigma }^2}}.\hfill \end{array}$$

(8)

At high SNR regimes (i.e., when $P_s\gg {\sigma }^2$) [28], the above expression can be approximated as

$$\begin{array}{cc}\hfill {\gamma }_{\mathrm{id}}& \simeq {\displaystyle \frac{{\gamma }_{\mathrm{sr}}{\beta }_{\mathrm{rd}}}{{\beta }_{\mathrm{rd}}+1}}\hfill \end{array}$$

(9)

where ${\gamma }_{\mathrm{sr}}$, ${\gamma }_{\mathrm{rd}}$, and ${\beta }_{\mathrm{rd}}$ represent the instantaneous SNRs of the SR and RD links and the normalized SNR of the RD link, respectively, defined as

$$\begin{array}{cc}\hfill {\gamma }_{\mathrm{sr}}& \triangleq {\displaystyle \frac{\left(1-\rho \right)P_s{\left|h_{\mathrm{sr}}\right|}^2}{{\sigma }_r^2}}\hfill \\ \hfill {\gamma }_{\mathrm{rd}}& \triangleq {\displaystyle \frac{\eta \rho P_s{\left|h_{\mathrm{rd}}\right|}^2}{{\sigma }^2}}\hfill \\ \hfill {\beta }_{\mathrm{rd}}& \triangleq {\displaystyle \frac{{\gamma }_{\mathrm{rd}}}{{\overline{\gamma }}_{\mathrm{sr}}/{\mathrm{\Omega }}_{\mathrm{sr}}}}.\hfill \end{array}$$

(10)

Furthermore, the average SNRs of each link are given by

$$\begin{array}{cc}\hfill {\overline{\gamma }}_{\mathrm{sr}}& ={\displaystyle \frac{\left(1-\rho \right)P_s{\mathrm{\Omega }}_{\mathrm{sr}}}{{\sigma }_r^2}}\hfill \\ \hfill {\overline{\gamma }}_{\mathrm{rd}}& ={\displaystyle \frac{\eta \rho P_s{\mathrm{\Omega }}_{\mathrm{rd}}}{{\sigma }^2}}\hfill \\ \hfill {\overline{\beta }}_{\mathrm{rd}}& ={\displaystyle \frac{{\overline{\gamma }}_{\mathrm{rd}}}{{\overline{\gamma }}_{\mathrm{sr}}/{\mathrm{\Omega }}_{\mathrm{sr}}}}.\hfill \end{array}$$

(11)

3. Performance Analysis for Single SWIPT Relaying System

In this section, we first derive the average SNR and the moment-generating function (MGF) of ${\gamma }_{\mathrm{id}}$. Based on the derived MGF, the average BER for BPSK modulation is subsequently obtained. Moreover, asymptotic expressions for the BER are presented to offer further analytical insights.

3.1. Average SNR Derivation for the Indirect Link

The joint probability density function (PDF) of ${\gamma }_{\mathrm{sr}}$ and ${\beta }_{\mathrm{rd}}$ can be written as

$$\begin{array}{cc}\hfill f_{{\gamma }_{\mathrm{sr}},{\beta }_{\mathrm{rd}}}\left(x,y\right)& ={\displaystyle \frac{1}{{\overline{\gamma }}_{\mathrm{sr}}}}exp\left(-{\displaystyle \frac{x}{{\overline{\gamma }}_{\mathrm{sr}}}}\right){\displaystyle \frac{1}{{\overline{\beta }}_{\mathrm{rd}}}}exp\left(-{\displaystyle \frac{y}{{\overline{\beta }}_{\mathrm{rd}}}}\right)u\left(x\right)u\left(y\right)\hfill \end{array}$$

(12)

where $u\left(·\right)$ denotes the unit step function [40]. By employing (9) together with (12), the average SNR of the indirect link, ${\overline{\gamma }}_{\mathrm{id}}$, can be obtained after some manipulations:   

$$\begin{array}{cc}\hfill {\overline{\gamma }}_{\mathrm{id}}=E\left[{\displaystyle \frac{{\gamma }_{\mathrm{sr}}{\beta }_{\mathrm{rd}}}{{\beta }_{\mathrm{rd}}+1}}\right]& ={\overline{\gamma }}_{\mathrm{sr}}-{\overline{\gamma }}_{\mathrm{sr}}{\displaystyle \frac{exp\left(1/{\overline{\beta }}_{\mathrm{rd}}\right)}{{\overline{\beta }}_{\mathrm{rd}}}}{E}_1\left({\displaystyle \frac{1}{{\overline{\beta }}_{\mathrm{rd}}}}\right)\hfill \end{array}$$

(13)

where the exponential integral function is defined as $E_1\left({\displaystyle \frac{1}{\tau }}\right)={\int }_{1/\tau }^{\infty }{\displaystyle \frac{e^{-y}}{y}}dy$ [41,42].

3.2. MGF Expression for the Indirect Link

The MGF of ${\gamma }_{\mathrm{id}}$ is derived as

$$\begin{array}{cc}\hfill M_{{\gamma }_{\mathrm{id}}}\left(\mathsf{s}\right)& ={\int }_{y=0}^{\infty }\left[{\int }_{x=0}^{\infty }{\displaystyle \frac{1}{{\overline{\gamma }}_{\mathrm{sr}}}}{e}^{-{\displaystyle \frac{xy}{y+1}}\mathsf{s}}{e}^{-{\displaystyle \frac{x}{{\overline{\gamma }}_{\mathrm{sr}}}}}dx\right]{\displaystyle \frac{1}{{\overline{\beta }}_{\mathrm{rd}}}}{e}^{-{\displaystyle \frac{y}{{\overline{\beta }}_{\mathrm{rd}}}}}dy\hfill \\ \hfill & ={\displaystyle \frac{1}{{\overline{\gamma }}_{\mathrm{sr}}\mathsf{s}+1}}+{\displaystyle \frac{{\overline{\gamma }}_{\mathrm{sr}}\mathsf{s}}{{\overline{\gamma }}_{\mathrm{sr}}\mathsf{s}+1}}{\displaystyle \frac{exp\left(1/\alpha \right)}{{\overline{\gamma }}_{\mathrm{sr}}{\overline{\beta }}_{\mathrm{rd}}\mathsf{s}+{\overline{\beta }}_{\mathrm{rd}}}}{E}_1\left({\displaystyle \frac{1}{\alpha }}\right)\hfill \end{array}$$

(14)

with $\alpha =a{\overline{\beta }}_{\mathrm{rd}}$ and $a={\overline{\gamma }}_{\mathrm{sr}}\mathsf{s}+1$ [41,42,43]. Furthermore, an upper bound for the MGF is given by [42]

$$\begin{array}{cc}\hfill M_{{\gamma }_{\mathrm{id}}}\left(\mathsf{s}\right)& <{\displaystyle \frac{1}{a}}+{\displaystyle \frac{a-1}{a}}{\displaystyle \frac{ln\left(1+\alpha \right)}{\alpha }}=M_{{\gamma }_{\mathrm{id}}}^{\mathrm{App}}\left(\mathsf{s}\right).\hfill \end{array}$$

(15)

3.3. Average Error Rate Expressions

By applying maximum ratio combining to $y_{\mathrm{sd}}$ in (1) and $y_{\mathrm{rd}}$ in (7) at the destination,

$$\begin{array}{cc}\hfill {\gamma }_{\mathrm{sd}}& \triangleq {\displaystyle \frac{P_s{\left|h_{\mathrm{sd}}\right|}^2}{{\sigma }^2}}\hfill \end{array}$$

(16)

represents the direct link SNR, whose PDF is given by $f_{{\gamma }_{\mathrm{sd}}}\left(x\right)={\displaystyle \frac{1}{{\overline{\gamma }}_{\mathrm{sd}}}}exp\left(-{\displaystyle \frac{x}{{\overline{\gamma }}_{\mathrm{sd}}}}\right)u\left(x\right)$, with ${\overline{\gamma }}_{\mathrm{sd}}={\displaystyle \frac{P_s{\mathrm{\Omega }}_{\mathrm{sd}}}{{\sigma }^2}}$ and MGF $M_{{\gamma }_{\mathrm{sd}}}\left(\mathsf{s}\right)={\displaystyle \frac{1}{{\overline{\gamma }}_{\mathrm{sd}}\mathsf{s}+1}}$. The average BER for BPSK modulation over the SD, indirect, and combined links can be expressed as

$$\begin{array}{cc}\hfill P_{B,\mathrm{sd}}& ={\displaystyle \frac{1}{\pi }}{\int }_0^{\pi /2}{M}_{{\gamma }_{\mathrm{sd}}}\left(\mathsf{s}\right)d\theta \hfill \\ \hfill P_{B,\mathrm{id}}& ={\displaystyle \frac{1}{\pi }}{\int }_0^{\pi /2}{M}_{{\gamma }_{\mathrm{id}}}\left(\mathsf{s}\right)d\theta \hfill \\ \hfill P_{B,\mathrm{cb}}& ={\displaystyle \frac{1}{\pi }}{\int }_0^{\pi /2}{M}_{{\gamma }_{\mathrm{sd}}}\left(\mathsf{s}\right)M_{{\gamma }_{\mathrm{id}}}\left(\mathsf{s}\right)d\theta \hfill \end{array}$$

(17)

where $\mathsf{s}=g_{\mathrm{PSK}}/{sin}^2\left(\theta \right)$ and $g_{\mathrm{PSK}}={sin}^2(\pi /2)$ [43].

By substituting $M_{{\gamma }_{\mathrm{id}}}\left(\mathsf{s}\right)$ in (17) with its approximation $M_{{\gamma }_{\mathrm{id}}}^{\mathrm{App}}\left(\mathsf{s}\right)$ from (15), the approximated BER bounds for the indirect and combined links can be derived as

$$\begin{array}{cc}\hfill P_{B,\mathrm{id}}<P_{B,\mathrm{id}}^{\mathrm{App}}& ={\displaystyle \frac{1}{\pi }}{\int }_0^{\pi /2}{M}_{{\gamma }_{\mathrm{id}}}^{\mathrm{App}}\left(\mathsf{s}\right)d\theta \hfill \\ \hfill P_{B,\mathrm{cb}}<P_{B,\mathrm{cb}}^{\mathrm{App}}& ={\displaystyle \frac{1}{\pi }}{\int }_0^{\pi /2}{M}_{{\gamma }_{\mathrm{sd}}}\left(\mathsf{s}\right)M_{{\gamma }_{\mathrm{id}}}^{\mathrm{App}}\left(\mathsf{s}\right)d\theta .\hfill \end{array}$$

(18)

For the indirect link, the BER bound $P_{B,\mathrm{id}}$ in (17) can be further approximated asymptotically under the high SNR assumption as

$$\begin{array}{cc}\hfill P_{B,\mathrm{id}}& \approx P_{B,\mathrm{id}}^{\mathrm{Asym}}\hfill \\ \hfill & ={\displaystyle \frac{1}{4{\overline{\gamma }}_{\mathrm{sr}}+1}}+{\displaystyle \frac{1}{4{\overline{\gamma }}_{\mathrm{rd}}{\mathrm{\Omega }}_{\mathrm{sr}}/ln\left(4{\overline{\gamma }}_{\mathrm{rd}}{\mathrm{\Omega }}_{\mathrm{sr}}+1\right)}}.\hfill \end{array}$$

(19)

The detailed derivation for $P_{B,\mathrm{id}}^{\mathrm{Asym}}$ is provided in Appendix A.

The analytical BER expression in (19) introduces a new analytical regime for evaluating SWIPT-based relaying systems. It allows the SWIPT relay to be interpreted as an equivalent conventional AF relay with a dedicated power source, where the SNRs of the SR and RD links are given by ${\overline{\gamma }}_{\mathrm{sr}}$ and

$$\begin{array}{c}\hfill {\overline{\gamma }}_{\mathrm{rd}}^{\prime }={\displaystyle \frac{{\overline{\gamma }}_{\mathrm{rd}}{\mathrm{\Omega }}_{\mathrm{sr}}}{ln\left(4{\overline{\gamma }}_{\mathrm{rd}}{\mathrm{\Omega }}_{\mathrm{sr}}+1\right)}},\end{array}$$

(20)

respectively. For example, it is observed that the BER performance of the SWIPT relay system is primarily determined by the weaker link between the SR link SNR, ${\overline{\gamma }}_{\mathrm{sr}}$, and the modified RD link SNR, ${\overline{\gamma }}_{\mathrm{rd}}^{\prime }$. Moreover, it is evident from (19) that the SWIPT relay does not experience any diversity loss compared to a conventional AF relay powered by its own energy source.

By applying the result (14-4-18) in [44] with appropriate manipulations, the BER expression for the combined link, $P_{B,\mathrm{cb}}$ in (17), can be asymptotically approximated as [24]

$$\begin{array}{c}\hfill P_{B,\mathrm{cb}}\approx P_{B,\mathrm{cb}}^{\mathrm{Asym}}={\displaystyle \frac{3}{4{\overline{\gamma }}_{\mathrm{sd}}+1}}{P}_{B,\mathrm{id}}^{\mathrm{Asym}}.\end{array}$$

(21)

3.4. Optimization for the PS Factor $\rho$

Note that $P_{B,\mathrm{id}}^{\mathrm{Asym}}$ in (19) depends on $\rho$, $\eta$, and the channel conditions. To optimize the PS factor $\rho$, we define

$$\begin{array}{cc}\hfill {\rho }_{\mathrm{opt}}& =arg\underset{0<\rho <1}{min}{P}_{B,\mathrm{id}}^{\mathrm{Asym}}\left(\rho \right)\hfill \end{array}$$

(22)

which can be efficiently determined via numerical methods using the asymptotic BER expression in (19).

3.5. Approximation SWIPT Relay to AF Relay

As mentioned earlier, the BER expression in (19) offers new insight into the performance characteristics of SWIPT relaying systems. In this subsection, we demonstrate that the BER expression for the SWIPT relay can also be derived from that of a conventional non-SWIPT AF relay by replacing the SR and RD link SNRs with ${\overline{\gamma }}_{\mathrm{sr}}$ and ${\overline{\gamma }}_{\mathrm{rd}}^{\prime }$, respectively. Accordingly, the indirect link SNR can be approximated as

$$\begin{array}{c}\hfill {\displaystyle \frac{{\gamma }_{\mathrm{sr}}{\gamma }_{\mathrm{rd}}^{\prime }}{{\gamma }_{\mathrm{sr}}+{\gamma }_{\mathrm{rd}}^{\prime }}}\simeq min\left\{{\gamma }_{\mathrm{sr}},{\gamma }_{\mathrm{rd}}^{\prime }\right\}=\gamma \end{array}$$

(23)

where the PDF of $\gamma$ is given by $f_{\gamma }\left(x\right)={\displaystyle \frac{1}{\overline{\gamma }}}exp\left(-{\displaystyle \frac{x}{\overline{\gamma }}}\right)u\left(x\right)$ with $\overline{\gamma }={\left(1/{\overline{\gamma }}_{\mathrm{sr}}+1/{\overline{\gamma }}_{\mathrm{rd}}^{\prime }\right)}^{-1}$ [16,23]. The average BERs for the indirect and combined links can be expressed, respectively, as

$$\begin{array}{cc}\hfill P_{B,\mathrm{id}}^{\mathrm{AF}-\mathrm{App}}& ={\displaystyle \frac{1}{\pi }}{\int }_0^{\pi /2}{\displaystyle \frac{1}{\overline{\gamma }\mathsf{s}+1}}d\theta \hfill \\ \hfill P_{B,\mathrm{cb}}^{\mathrm{AF}-\mathrm{App}}& ={\displaystyle \frac{1}{\pi }}{\int }_0^{\pi /2}{M}_{{\gamma }_{\mathrm{sd}}}\left(\mathsf{s}\right){\displaystyle \frac{1}{\overline{\gamma }\mathsf{s}+1}}d\theta .\hfill \end{array}$$

(24)

It is worth noting that by the same argument as Appendix A, $P_{B,\mathrm{id}}^{\mathrm{AF}-\mathrm{App}}$ of (24) can be asymptotically approximated as the form of (19).

4. Extension to Multiple SWIPT AF Relaying Systems: Performance Analysis

To validate the effectiveness of the proposed approximation treating SWIPT AF relays as their non-SWIPT counterparts, we extend the analysis to systems employing multiple SWIPT AF relays and perform a comprehensive performance evaluation.

For a system with R SWIPT relays, the Equations from (1) to (11) can be generalized to the rth relay by adding the subscript $r\in \{1,2,\cdots ,R\}$. Specifically, for the rth relay, the SR link SNR (denoted as ${\gamma }_r={\gamma }_{{\mathrm{sr}}_r}$), the RD link SNR, and the normalized RD link SNR are, respectively, presented as

$$\begin{array}{cc}\hfill {\gamma }_r& ={\displaystyle \frac{\left(1-{\rho }_r\right)P_s{\left|h_{{\mathrm{sr}}_r}\right|}^2}{{\sigma }_r^2}}={\gamma }_{{\mathrm{sr}}_r}\hfill \\ \hfill {\gamma }_{{\mathrm{rd}}_{\mathrm{r}}}& ={\displaystyle \frac{\eta {\rho }_r{P}_s{\left|h_{{\mathrm{rd}}_r}\right|}^2}{{\sigma }^2}}\hfill \\ \hfill {\beta }_r& ={\gamma }_{{\mathrm{rd}}_r}{\displaystyle \frac{{\mathrm{\Omega }}_{{\mathrm{sr}}_r}}{{\overline{\gamma }}_{{\mathrm{sr}}_r}}}={\beta }_{{\mathrm{sr}}_r}\hfill \end{array}$$

(25)

where

$$\begin{array}{cc}\hfill {\overline{\gamma }}_r& =\left(1-{\rho }_r\right)P_s{\mathrm{\Omega }}_{{\mathrm{sr}}_r}/{\sigma }_r^2={\overline{\gamma }}_{{\mathrm{sr}}_r}\hfill \\ \hfill {\overline{\gamma }}_{{\mathrm{rd}}_r}& =\eta {\rho }_r{P}_s{\mathrm{\Omega }}_{{\mathrm{rd}}_r}/{\sigma }^2\hfill \\ \hfill {\overline{\beta }}_r& ={\overline{\gamma }}_{{\mathrm{rd}}_r}{\mathrm{\Omega }}_{{\mathrm{sr}}_r}/{\overline{\gamma }}_{{\mathrm{sr}}_r},\hfill \end{array}$$

(26)

${\mathrm{\Omega }}_{{\mathrm{sr}}_r}=E\left[{\left|h_{{\mathrm{sr}}_r}\right|}^2\right]$, and ${\mathrm{\Omega }}_{{\mathrm{rd}}_r}=E\left[{\left|h_{{\mathrm{rd}}_r}\right|}^2\right]$.

4.1. Average SNR for the rth Indirect Link

From (9), (25), and (26), the SNR of the rth indirect link can be approximated as

$$\begin{array}{c}\hfill {\gamma }_{{\mathrm{id}}_r}\simeq {\displaystyle \frac{{\gamma }_r{\beta }_r}{{\beta }_r+1}}\end{array}$$

(27)

and its average SNR is expressed, following (13), as 

$$\begin{array}{cc}\hfill {\overline{\gamma }}_{{\mathrm{id}}_r}& =E\left[{\displaystyle \frac{{\gamma }_r{\beta }_r}{{\beta }_r+1}}\right]={\overline{\gamma }}_r-{\overline{\gamma }}_r{\displaystyle \frac{1}{{\overline{\beta }}_r}}exp\left({\displaystyle \frac{1}{{\overline{\beta }}_r}}\right)E_1\left({\displaystyle \frac{1}{{\overline{\beta }}_r}}\right).\hfill \end{array}$$

(28)

4.2. MGF Expression for the rth Indirect Link

From (14), (25), and (26), the MGF of ${\gamma }_{{\mathrm{id}}_r}$ can be expressed as

$$\begin{array}{cc}\hfill M_{{\gamma }_{{\mathrm{id}}_r}}\left(\mathsf{s}\right)& ={\displaystyle \frac{1}{{\overline{\gamma }}_r\mathsf{s}+1}}+{\displaystyle \frac{{\overline{\gamma }}_r\mathsf{s}}{{\overline{\gamma }}_r\mathsf{s}+1}}{\displaystyle \frac{e^{1/\left({\overline{\gamma }}_r{\overline{\beta }}_r\mathsf{s}+{\overline{\beta }}_r\right)}}{{\overline{\gamma }}_r{\overline{\beta }}_r\mathsf{s}+{\overline{\beta }}_r}}{E}_1\left({\displaystyle \frac{1}{\alpha }}\right)\hfill \end{array}$$

(29)

where $\alpha ={\overline{\beta }}_{r}a$ and $a={\overline{\gamma }}_r\mathsf{s}+1$ [41,42,43]. Furthermore, the MGF in (29) can be upper bounded as [42]

$$\begin{array}{cc}\hfill M_{{\gamma }_{{\mathrm{id}}_r}}\left(\mathsf{s}\right)& <{\displaystyle \frac{1}{a}}+{\displaystyle \frac{a-1}{a}}{\displaystyle \frac{ln\left(1+\alpha \right)}{\alpha }}=M_{{\gamma }_{{\mathrm{id}}_r}}^{\mathrm{Up}}\left(\mathsf{s}\right).\hfill \end{array}$$

(30)

4.3. Average Error Rate Expressions

Applying maximum ratio combining to the direct link signal $y_{\mathrm{sd}}$ in (1) and the R indirect link signals $y_{{\mathrm{id}}_r}$ in (7) at the destination, the combined SNR is given by    

$$\begin{array}{c}\hfill {\gamma }_{\mathrm{cb}}={\gamma }_{\mathrm{sd}}+{\gamma }_{\mathrm{id}}={\gamma }_0+\sum _{r=1}^R{\gamma }_{{\mathrm{id}}_r}\end{array}$$

(31)

where ${\gamma }_0={\gamma }_{\mathrm{sd}}=P_s{\left|h_{\mathrm{sd}}\right|}^2/{\sigma }^2$ represents the direct link SNR with the PDF of $f_{{\gamma }_0}\left(x\right)={\displaystyle \frac{1}{{\overline{\gamma }}_0}}exp\left(-{\displaystyle \frac{x}{{\overline{\gamma }}_0}}\right)u\left(x\right)$, ${\overline{\gamma }}_0={\overline{\gamma }}_{\mathrm{sd}}$, and the MGF of $M_{{\gamma }_0}\left(\mathsf{s}\right)=1/\left({\overline{\gamma }}_0\mathsf{s}+1\right)$. Assuming independence between ${\gamma }_0$ and ${\gamma }_{{\mathrm{id}}_r}$, the average symbol error rates (SERs) for the indirect links and the combined link are expressed as

$$\begin{array}{cc}\hfill P_{S,\mathrm{id}}& ={\displaystyle \frac{1}{\pi }}{\int }_0^{(M-1)\pi /M}\prod _{r=1}^R{M}_{{\gamma }_{{\mathrm{id}}_r}}\left(\mathsf{s}\right)d\theta \hfill \\ \hfill P_{S,\mathrm{cb}}& ={\displaystyle \frac{1}{\pi }}{\int }_0^{(M-1)\pi /M}{M}_{{\gamma }_{\mathrm{sd}}}\left(\mathsf{s}\right)\prod _{r=1}^R{M}_{{\gamma }_{{\mathrm{id}}_r}}\left(\mathsf{s}\right)d\theta \hfill \end{array}$$

(32)

where $\mathsf{s}=g_{\mathrm{PSK}}/{sin}^2\left(\theta \right)$ and $g_{\mathrm{PSK}}={sin}^2(\pi /M)$ [43].

By replacing $M_{{\gamma }_{{\mathrm{id}}_r}}\left(\mathsf{s}\right)$ in (32) with the upper bound $M_{{\gamma }_{{\mathrm{id}}_r}}^{\mathrm{Up}}\left(\mathsf{s}\right)$ from (30), the approximated SER bounds for the indirect and the combined links are given as

$$\begin{array}{cc}\hfill P_{S,\mathrm{id}}<P_{S,\mathrm{id}}^{\mathrm{Up}}& ={\displaystyle \frac{1}{\pi }}{\int }_0^{(M-1)\pi /M}\prod _{r=1}^R{M}_{{\gamma }_{{\mathrm{id}}_r}}^{\mathrm{Up}}\left(\mathsf{s}\right)d\theta \hfill \\ \hfill P_{S,\mathrm{cb}}<P_{S,\mathrm{cb}}^{\mathrm{Up}}& ={\displaystyle \frac{1}{\pi }}{\int }_0^{(M-1)\pi /M}{M}_{{\gamma }_{\mathrm{sd}}}\left(\mathsf{s}\right)\prod _{r=1}^R{M}_{{\gamma }_{{\mathrm{id}}_r}}^{\mathrm{Up}}\left(\mathsf{s}\right)d\theta .\hfill \end{array}$$

(33)

From (19) and (26), the asymptotic BER for BPSK for the rth indirect link is approximated as

$$\begin{array}{cc}\hfill P_{B,{\mathrm{id}}_r}& \approx P_{B,{\mathrm{id}}_r}^{\mathrm{Asym}}\hfill \\ \hfill & ={\displaystyle \frac{1}{4{\overline{\gamma }}_{{\mathrm{sr}}_r}+1}}+{\displaystyle \frac{1}{4{\overline{\gamma }}_{{\mathrm{rd}}_r}{\mathrm{\Omega }}_{{\mathrm{sr}}_r}/ln\left(4{\overline{\gamma }}_{{\mathrm{rd}}_r}{\mathrm{\Omega }}_{{\mathrm{sr}}_r}+1\right)}}.\hfill \end{array}$$

(34)

From (34), the help of (14-4-18) in [44], and after some algebraic manipulations, the asymptotic approximations for $P_{S,\mathrm{id}}^{\mathrm{Up}}$ and $P_{S,\mathrm{cb}}^{\mathrm{Up}}$ in (33) can be expressed for BPSK as [24]

$$\begin{array}{cc}\hfill P_{B,\mathrm{id}}<P_{B,\mathrm{id}}^{\mathrm{Asym}}& =\left({\displaystyle \genfrac{}{}{0pt}{}{2R-1}{R}}\right)\prod _{r=1}^R{P}_{B,{\mathrm{id}}_r}^{\mathrm{Asym}}\hfill \\ \hfill P_{B,\mathrm{cb}}<P_{B,\mathrm{cb}}^{\mathrm{Asym}}& ={\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2R+1}{R+1}\right)}{4{\overline{\gamma }}_{\mathrm{sd}}+1}}\prod _{r=1}^R{P}_{B,{\mathrm{id}}_r}^{\mathrm{Asym}}.\hfill \end{array}$$

(35)

4.4. Optimization for the PS Factor ${\rho }_r$

Notice that $P_{B,{\mathrm{id}}_r}^{\mathrm{Asym}}$ in (34) is a function of ${\rho }_r$, $\eta$, and the channel conditions. To design the PS factor ${\rho }_r$, we determine the optimal ${\rho }_r$ as

$$\begin{array}{cc}\hfill {\rho }_r^{\mathrm{opt}}& =arg\underset{0<{\rho }_r<1}{min}{P}_{B,{\mathrm{id}}_r}^{\mathrm{Asym}}\left({\rho }_r\right)\hfill \end{array}$$

(36)

which can be obtained numerically using $P_{B,{\mathrm{id}}_r}^{\mathrm{Asym}}$ given in (34).

4.5. Approximation SWIPT Relay to AF Relay

With careful inspection of (34), we gain insight into the SWIPT relaying system such that the asymptotic BER in (34) can also be obtained for a general AF relay system with the SR link average SNR of ${\overline{\gamma }}_{{\mathrm{sr}}_r}$ and the modified RD link average SNR of ${\overline{\gamma }}_{{\mathrm{rd}}_r}^{\prime }$. At the general AF relay system, the rth indirect link SNR can be approximated as    

$$\begin{array}{c}\hfill {\gamma }_{{\mathrm{id}}_r}={\displaystyle \frac{{\gamma }_r{\beta }_r}{{\beta }_r+1}}\approx {\displaystyle \frac{{\gamma }_{{\mathrm{sr}}_r}{\gamma }_{{\mathrm{rd}}_r}^{\prime }}{{\gamma }_{{\mathrm{sr}}_r}+{\gamma }_{{\mathrm{rd}}_r}^{\prime }}}\simeq min\left\{{\gamma }_{{\mathrm{sr}}_r},{\gamma }_{{\mathrm{rd}}_r}^{\prime }\right\}={\gamma }_{{\mathrm{id}}_r}^{\mathrm{AF}}\end{array}$$

(37)

where ${\gamma }_{{\mathrm{sr}}_r}$ and ${\gamma }_{{\mathrm{rd}}_r}^{\prime }$ are the rth SR and the modified RD link SNRs, respectively [16,23]. Then, the PDF of ${\gamma }_{{\mathrm{id}}_r}^{\mathrm{AF}}$ is given by $f_{{\gamma }_{{\mathrm{id}}_r}^{\mathrm{AF}}}\left(x\right)={\displaystyle \frac{1}{{\overline{\gamma }}_r^{min}}}exp\left(-{\displaystyle \frac{x}{{\overline{\gamma }}_r^{min}}}\right)u\left(x\right)$ with ${\left.{\overline{\gamma }}_r^{min}\right|}_{r=1}^R={\left(1/{\overline{\gamma }}_{{\mathrm{sr}}_r}+1/{\overline{\gamma }}_{{\mathrm{rd}}_r}^{\prime }\right)}^{-1}$ and ${\overline{\gamma }}_{{\mathrm{rd}}_r}^{\prime }={\overline{\gamma }}_{{\mathrm{rd}}_r}{\mathrm{\Omega }}_{{\mathrm{sr}}_r}/ln\left(4{\overline{\gamma }}_{{\mathrm{rd}}_r}{\mathrm{\Omega }}_{{\mathrm{sr}}_r}+1\right)$.

4.5.1. PDF and CDF of ${\gamma }_{\mathrm{id}}^{\mathrm{AF}}$ and ${\gamma }_{\mathrm{cb}}^{\mathrm{AF}}$

By approximating the SWIPT relay as a general AF relay, the indirect and combined link SNRs can be approximately expressed, based on (31) and (37), as 

$$\begin{array}{cc}\hfill {\gamma }_{\mathrm{id}}=\sum _{r=1}^R{\gamma }_{{\mathrm{id}}_r}& \simeq \sum _{r=1}^R{\gamma }_{{\mathrm{id}}_r}^{\mathrm{AF}}={\gamma }_{\mathrm{id}}^{\mathrm{AF}}\hfill \\ \hfill {\gamma }_{\mathrm{cb}}={\gamma }_0+{\gamma }_{\mathrm{id}}& \simeq {\gamma }_0+{\gamma }_{\mathrm{id}}^{\mathrm{AF}}={\gamma }_{\mathrm{cb}}^{\mathrm{AF}}.\hfill \end{array}$$

(38)

The PDFs of ${\gamma }_{\mathrm{id}}^{\mathrm{AF}}$ and ${\gamma }_{\mathrm{cb}}^{\mathrm{AF}}$ can then be expressed as

$$\begin{array}{cc}\hfill f_{{\gamma }_{\mathrm{id}}^{\mathrm{AF}}}\left(x\right)& =\sum _{r=1}^R{\displaystyle \frac{{\pi }_{{\mathrm{id}}_r}}{{\overline{\gamma }}_r^{min}}}exp\left(-{\displaystyle \frac{x}{{\overline{\gamma }}_r^{min}}}\right)u\left(x\right)\hfill \\ \hfill f_{{\gamma }_{\mathrm{cb}}^{\mathrm{AF}}}\left(x\right)& =\sum _{r=0}^R{\displaystyle \frac{{\pi }_{{\mathrm{cb}}_r}}{{\overline{\gamma }}_r^{min}}}exp\left(-{\displaystyle \frac{x}{{\overline{\gamma }}_r^{min}}}\right)u\left(x\right)\hfill \end{array}$$

(39)

where ${\pi }_{{\mathrm{id}}_r}={\prod }_{i=1,i\ne r}^R{\displaystyle \frac{{\overline{\gamma }}_r^{min}}{{\overline{\gamma }}_r^{min}-{\overline{\gamma }}_i^{min}}}$, ${\pi }_{{\mathrm{cb}}_r}={\prod }_{i=0,i\ne r}^R{\displaystyle \frac{{\overline{\gamma }}_r^{min}}{{\overline{\gamma }}_r^{min}-{\overline{\gamma }}_i^{min}}}$, and ${\overline{\gamma }}_0^{min}={\overline{\gamma }}_0$. Then, the cumulative distribution functions (CDFs) of ${\gamma }_{\mathrm{id}}^{\mathrm{AF}}$ and ${\gamma }_{\mathrm{cb}}^{\mathrm{AF}}$ can be obtained as

$$\begin{array}{cc}\hfill F_{{\gamma }_{\mathrm{id}}^{\mathrm{AF}}}\left(\gamma \right)={\int }_0^{\gamma }{f}_{{\gamma }_{\mathrm{id}}^{\mathrm{AF}}}\left(x\right)dx& =\sum _{r=1}^R{\pi }_{{\mathrm{id}}_r}\left(1-exp\left(-{\displaystyle \frac{\gamma }{{\overline{\gamma }}_r^{min}}}\right)\right)\hfill \\ \hfill F_{{\gamma }_{\mathrm{cb}}^{\mathrm{AF}}}\left(\gamma \right)={\int }_0^{\gamma }{f}_{{\gamma }_{\mathrm{cb}}^{\mathrm{AF}}}\left(x\right)dx& =\sum _{r=0}^R{\pi }_{{\mathrm{cb}}_r}\left(1-exp\left(-{\displaystyle \frac{\gamma }{{\overline{\gamma }}_r^{min}}}\right)\right).\hfill \end{array}$$

(40)

4.5.2. Average SER and BER Expressions

The average BERs for the indirect and the combined links can be approximated, respectively, as 

$$\begin{array}{cc}\hfill P_{S,\mathrm{id}}^{\mathrm{AF}-\mathrm{App}}& ={\displaystyle \frac{1}{\pi }}{\int }_0^{(M-1)\pi /M}\prod _{r=1}^R{\displaystyle \frac{{\pi }_{{\mathrm{id}}_r}}{{\overline{\gamma }}_r^{min}\mathsf{s}+1}}d\theta \hfill \\ \hfill P_{S,\mathrm{cb}}^{\mathrm{AF}-\mathrm{App}}& ={\displaystyle \frac{1}{\pi }}{\int }_0^{(M-1)\pi /M}\prod _{r=0}^R{\displaystyle \frac{{\pi }_{{\mathrm{cb}}_r}}{{\overline{\gamma }}_r^{min}\mathsf{s}+1}}d\theta \hfill \end{array}$$

(41)

with $\mathsf{s}=g_{\mathrm{PSK}}/{sin}^2\left(\theta \right)$ and $g_{\mathrm{PSK}}={sin}^2(\pi /M)$ [43]. It is noteworthy that (41) with $M=2$ yields the asymptotic BERs in (35). For the specific case of BPSK ($M=2$), the average BERs can also be expressed in a more tractable closed-form as

$$\begin{array}{cc}\hfill P_{B,\mathrm{id}}^{\mathrm{AF}-\mathrm{App}}& =\sum _{r=1}^R{\displaystyle \frac{{\pi }_{{\mathrm{id}}_r}}{2}}\left[1-\sqrt{{\displaystyle \frac{{\overline{\gamma }}_r^{min}}{{\overline{\gamma }}_r^{min}+1}}}\right]\hfill \\ \hfill P_{B,\mathrm{cb}}^{\mathrm{AF}-\mathrm{App}}& =\sum _{r=0}^R{\displaystyle \frac{{\pi }_{{\mathrm{cb}}_r}}{2}}\left[1-\sqrt{{\displaystyle \frac{{\overline{\gamma }}_r^{min}}{{\overline{\gamma }}_r^{min}+1}}}\right].\hfill \end{array}$$

(42)

4.5.3. Approximated Outage Probability

The outage probability $P_{\mathrm{out}}$ is defined as the probability that the channel mutual information I falls below the required transmission rate $T_R$. For the considered relaying system, the mutual information is expressed as $I={\displaystyle \frac{1}{R+1}}{log}_2(1+{\gamma }_{\mathrm{tot}})$, where ${\gamma }_{\mathrm{tot}}$ can be either ${\gamma }_{\mathrm{id}}$ or ${\gamma }_{\mathrm{cb}}$. The scaling factor $1/\left(R+1\right)$ accounts for the use of $(R+1)$ orthogonal time slots or channels required to transmit data from the source to the destination through R relays. Accordingly, the outage probability can be approximated as:

$$\begin{array}{cc}\hfill P_{\mathrm{out},\mathrm{id}}& =\mathrm{Pr}\left({\displaystyle \frac{1}{R+1}}{log}_2(1+{\gamma }_{\mathrm{id}})\le T_R\right)\hfill \\ \hfill & =\mathrm{Pr}\left({\gamma }_{\mathrm{id}}\le 2^{(R+1)T_R}-1\right)\hfill \\ \hfill & \simeq \mathrm{Pr}\left({\gamma }_{\mathrm{id}}^{\mathrm{AF}}\le 2^{(R+1)T_R}-1\right)=P_{\mathrm{out},\mathrm{id}}^{\mathrm{AF}-\mathrm{App}}.\hfill \end{array}$$

(43)

Based on the CDF expressions $F_{{\gamma }_{\mathrm{id}}^{\mathrm{AF}}}\left(\gamma \right)$ and $F_{{\gamma }_{\mathrm{cb}}^{\mathrm{AF}}}\left(\gamma \right)$ given in (40), the approximated outage probabilities are given by

$$\begin{array}{cc}\hfill P_{\mathrm{out},\mathrm{id}}^{\mathrm{AF}-\mathrm{App}}& =F_{{\gamma }_{\mathrm{id}}^{\mathrm{AF}}}\left(2^{(R+1)T_R}-1\right)\hfill \\ \hfill P_{\mathrm{out},\mathrm{cb}}^{\mathrm{AF}-\mathrm{App}}& =F_{{\gamma }_{\mathrm{cb}}^{\mathrm{AF}}}\left(2^{(R+1)T_R}-1\right).\hfill \end{array}$$

(44)

In addition, the outage probability of the direct SD link (i.e., without relaying) is expressed as

$$\begin{array}{cc}\hfill P_{\mathrm{out},\mathrm{sd}}& =F_{{\gamma }_0}\left(2^{T_R}-1\right)\hfill \end{array}$$

(45)

where the CDF of ${\gamma }_0$ is given by $F_{{\gamma }_0}\left(x\right)=\left(1-exp\left(-{\displaystyle \frac{x}{{\overline{\gamma }}_0}}\right)\right)u\left(x\right)$.

4.5.4. Approximated Channel Capacity

The channel capacity in Shannon’s sense is a fundamental performance metric, representing the maximum achievable data rate under which reliable communication is possible. For the direct SD link without relaying, the normalized channel capacity is given by

$$\begin{array}{cc}\hfill C_{\mathrm{sd}}=E\left[{log}_2\left(1+{\gamma }_0\right)\right]& ={\int }_0^{\infty }{log}_2\left(1+\gamma \right)f_{{\gamma }_0}\left(\gamma \right)d\gamma \hfill \\ \hfill & ={\displaystyle \frac{1}{ln\left(2\right)}}exp\left({\displaystyle \frac{1}{{\overline{\gamma }}_0}}\right)E_1\left({\displaystyle \frac{1}{{\overline{\gamma }}_0}}\right).\hfill \end{array}$$

(46)

For ${\gamma }_{tot}\in \left\{{\gamma }_{\mathrm{id}},{\gamma }_{\mathrm{cb}}\right\}$, the average channel capacity for the relaying system is expressed as

$$\begin{array}{cc}\hfill C& ={\displaystyle \frac{1}{R+1}}E\left[{log}_2\left(1+{\gamma }_{tot}\right)\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{R+1}}{\int }_0^{\infty }{log}_2\left(1+\gamma \right)f_{{\gamma }_{tot}}\left(\gamma \right)d\gamma \hfill \end{array}$$

(47)

where the factor ${\displaystyle \frac{1}{R+1}}$ accounts for the use of $(R+1)$ orthogonal time slots or subchannels required for cooperative transmission. The approximated average channel capacity of the indirect link, using the PDF from (39), can be expressed as    

$$\begin{array}{cc}\hfill C_{\mathrm{id}}& ={\displaystyle \frac{1}{R+1}}E\left[{log}_2\left(1+{\gamma }_{\mathrm{id}}\right)\right]\hfill \\ \hfill & \approx {\displaystyle \frac{1}{R+1}}{\int }_0^{\infty }{log}_2\left(1+\gamma \right)f_{{\gamma }_{\mathrm{id}}^{\mathrm{AF}}}\left(\gamma \right)d\gamma \hfill \\ \hfill & ={\displaystyle \frac{1}{(R+1)ln\left(2\right)}}\sum _{r=1}^R{\pi }_{{\mathrm{id}}_r}exp\left({\displaystyle \frac{1}{{\overline{\gamma }}_r^{min}}}\right)E_1\left({\displaystyle \frac{1}{{\overline{\gamma }}_r^{min}}}\right)=C_{\mathrm{id}}^{\mathrm{AF}-\mathrm{App}}.\hfill \end{array}$$

(48)

Similarly, the approximated average capacity of the combined link is given, from (39) and (47), by 

$$\begin{array}{cc}\hfill C_{\mathrm{cb}}& ={\displaystyle \frac{1}{R+1}}E\left[{log}_2\left(1+{\gamma }_{\mathrm{cb}}\right)\right]\hfill \\ \hfill & \approx {\displaystyle \frac{1}{R+1}}{\int }_0^{\infty }{log}_2\left(1+\gamma \right)f_{{\gamma }_{\mathrm{cb}}^{\mathrm{AF}}}\left(\gamma \right)d\gamma \hfill \\ \hfill & ={\displaystyle \frac{1}{(R+1)ln\left(2\right)}}\sum _{r=0}^R{\pi }_{{\mathrm{cb}}_r}exp\left({\displaystyle \frac{1}{{\overline{\gamma }}_r^{min}}}\right)E_1\left({\displaystyle \frac{1}{{\overline{\gamma }}_r^{min}}}\right)=C_{\mathrm{cb}}^{\mathrm{AF}-\mathrm{App}}.\hfill \end{array}$$

(49)

The above expressions for $C_{\mathrm{id}}^{\mathrm{AF}-\mathrm{App}}$ and $C_{\mathrm{cb}}^{\mathrm{AF}-\mathrm{App}}$ provide tractable estimates of the average achievable rates under multiple SWIPT AF relaying. The accuracy of these approximations will be validated in Section 5.2.3.

5. Numerical and Simulation Results

In this section, we present numerical evaluations of the derived performance metrics and validate their accuracy through comparison with Monte Carlo simulation results. Simulations are conducted for both the single-relay SWIPT AF system described in Section 2 and the multi-relay scenario presented in Section 4. All simulations are carried out using MATLAB software (version R2024a) under the following assumptions. For the single-relay system, BPSK modulation (i.e., $M=2$) is assumed, and all wireless channels $\left\{h_{\mathrm{sd}},h_{\mathrm{sr}},h_{\mathrm{rd}}\right\}$ are modeled as mutually independent Rayleigh fading. Using MATLAB, the channel gains $\left\{h_{\mathrm{sd}},h_{\mathrm{sr}},h_{\mathrm{rd}}\right\}$ are independently generated ${10}^8$ times. AWGN with variance ${\sigma }^2$ is assumed at all receivers. Using MATLAB, the noise terms $\left\{n_{\mathrm{sd}},n_{\mathrm{sr}},n_c,n_{\mathrm{rd}}\right\}$ are also independently generated ${10}^8$ times. These assumptions are similarly extended to the multi-relay case. In addition, the average SNR is set to $\mathrm{SNR}={\overline{\gamma }}_{\mathrm{sd}}$. In all figures presented in this section, the blue and red lines represent the derived numerical results for the indirect and combined links, respectively.

5.1. Single SWIPT Relay System

Table 1. Channel power gains due to scenario for single SWIPT relay system ($d_{\mathrm{sd}}=d_{\mathrm{sr}}$, ${\mathrm{\Omega }}_{\mathrm{sd}}={\mathrm{\Omega }}_{\mathrm{sr}}$).

$\left[{\mathbf{\Omega }}_{\mathbf{sd}},{\mathbf{\Omega }}_{\mathbf{sr}},{\mathbf{\Omega }}_{\mathbf{rd}}\right]$	Distance	Relationship	Weak Link with $\mathit{\rho }=0.1$	Weak Link with $\mathit{\rho }={\mathit{\rho }}_{\mathbf{opt}}$	At
$\left[1,1,4\right]$	$d_{\mathrm{rd}}=d_{\mathrm{sr}}/2$	$4{\mathrm{\Omega }}_{\mathrm{sr}}={\mathrm{\Omega }}_{\mathrm{rd}}$	RD	SR	Figure 2
$\left[1,1,1\right]$	$d_{\mathrm{rd}}=d_{\mathrm{sr}}$	${\mathrm{\Omega }}_{\mathrm{sr}}={\mathrm{\Omega }}_{\mathrm{rd}}$	RD	SR	Figure 3
$\left[1,1,1/2\right]$	$d_{\mathrm{rd}}=\sqrt{2}{d}_{\mathrm{sr}}$	${\mathrm{\Omega }}_{\mathrm{sr}}=2{\mathrm{\Omega }}_{\mathrm{rd}}$	RD	SR	Figure 4
$\left[1,1,1/4\right]$	$d_{\mathrm{rd}}=2d_{\mathrm{sr}}$	${\mathrm{\Omega }}_{\mathrm{sr}}=4{\mathrm{\Omega }}_{\mathrm{rd}}$	RD	SR ≈ RD	Figure 5

lists the average channel power gains used for the single-relay SWIPT system. In the figures presented for this scenario:

• The label ‘Theory’ corresponds to the numerical results derived from (17) using the exact MGF in (14).
• ‘Theory, App’ represents the results based on the approximated ASER expression in (18) and the MGF bound in (15).
• ‘Theory, Asym’ denotes the high-SNR asymptotic expressions obtained from (19) and (21).
• ‘Theory, AF-App’ refers to the approximated AF-based BER results given in (24).

Additionally:

• ‘Theory, ${\overline{\gamma }}_{\mathrm{sr}}$’, ‘Theory, ${\overline{\gamma }}_{\mathrm{rd}}$’, and ’Theory, ${\overline{\gamma }}_{\mathrm{id}}$’ correspond to the analytical expressions for ${\overline{\gamma }}_{\mathrm{sr}}$ and ${\overline{\gamma }}_{\mathrm{rd}}$ from (11) and ${\overline{\gamma }}_{\mathrm{id}}$ from (13), respectively.

The simulation results labeled as ‘Simulation’ are obtained by directly using the amplification gain G as defined in (6).

Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5 illustrate the BER performance as a function of the direct link SNR for various system parameters, including the energy conversion efficiency $\eta$, power-splitting ratio $\rho$, and channel power gains $\left[{\mathrm{\Omega }}_{\mathrm{sd}},{\mathrm{\Omega }}_{\mathrm{sr}},{\mathrm{\Omega }}_{\mathrm{rd}}\right]$.

From Figure 1a, it can be observed that the indirect link exhibits inferior performance compared to the direct link, mainly due to the lack of a dedicated power source at the relay. However, when the direct and indirect links are combined, the SWIPT relay system outperforms the direct link alone across the entire SNR range. Furthermore, the numerical results labeled as ‘Theory’, ‘Theory, App’, and ’Theory, AF-App’ show a strong agreement with the simulation results for all SNR values. In particular, ‘Theory, Asym’ accurately reflects the asymptotic BER behavior in the high SNR regime. In Figure 1b, it is shown that the theoretical link SNRs closely match the simulation results, with only minor discrepancies caused by the approximation used in (9), especially in the low SNR region. Additionally, it is confirmed that the SNR of the indirect link, ${\overline{\gamma }}_{\mathrm{id}}$, is limited by the weaker of the SR and RD links, which is consistent with typical behavior in conventional AF relaying systems.

Assuming a path-loss exponent of $\mu =2$ [18,23], the average channel power gains can be characterized in terms of the relative distances between nodes. Let $d_{\mathrm{sd}}$, $d_{\mathrm{sr}}$, and $d_{\mathrm{rd}}$ denote the distances of SD, SR, and RD links, respectively. Under this model, the ratio of the average power gains for the RD and SR links is given by ${\mathrm{\Omega }}_{\mathrm{rd}}/{\mathrm{\Omega }}_{\mathrm{sr}}=d_{\mathrm{sr}}^2/d_{\mathrm{rd}}^2$. Moreover, when $d_{\mathrm{sd}}=d_{\mathrm{sr}}$, it directly follows that ${\mathrm{\Omega }}_{\mathrm{sd}}={\mathrm{\Omega }}_{\mathrm{sr}}$. As summarized in Table 1, a change in the ratio ${\mathrm{\Omega }}_{\mathrm{rd}}/{\mathrm{\Omega }}_{\mathrm{sr}}$ from 4 to 1 corresponds to an increase in the RD link distance from one-half to twice the SR link distance. As a result, the modified RD link of (20) becomes increasingly weaker. Following this trend, we aim to investigate how the optimization of the power-splitting factor $\rho$ in the SWIPT relay system impacts the system BER performance. Figure 2, Figure 3, Figure 4 and Figure 5 illustrate the performance variations resulting from the optimization of the power-splitting ratio $\rho$. For the case of ‘Theory’, the optimal power-splitting (PS) factor was determined by performing an exhaustive search over the interval $[0,1]$, based on the analytical BER expression $P_{B,\mathrm{id}}^{\mathrm{Asym}}$ given in (19). In contrast, for the case of ‘Simulation’, we evaluated the numerical BER for all possible PS factors and selected the value that yielded the minimum BER.

Considering the case where $\left[{\mathrm{\Omega }}_{\mathrm{sd}},{\mathrm{\Omega }}_{\mathrm{sr}},{\mathrm{\Omega }}_{\mathrm{rd}}\right]=\left[1,1,4\right]$, $\rho =0.1$ (i.e., prior to $\rho$ optimization), and $\eta =1.0$, Figure 2b indicates that the RD link acts as the weak channel, while the SR link is dominant, leading to ${\overline{\gamma }}_{\mathrm{id}}<{\overline{\gamma }}_{\mathrm{rd}}$. Upon applying the optimized $\rho$ (i.e., ${\rho }_{\mathrm{opt}}<0.6$ in the high SNR region as seen in Figure 2c, the link roles are reversed, and the SNR gap increases, enhancing ${\overline{\gamma }}_{\mathrm{id}}$ and reducing the BER. Specifically, Figure 2a shows approximately 4 dB and 1.6 dB SNR gains at $\mathrm{BER}={10}^{-2}$ and $\mathrm{BER}={10}^{-4}$ for the indirect and combined links, respectively. Similar trends are observed for $\left[{\mathrm{\Omega }}_{\mathrm{sd}},{\mathrm{\Omega }}_{\mathrm{sr}},{\mathrm{\Omega }}_{\mathrm{rd}}\right]=\left[1,1,1\right]$. As shown in Figure 3, with $\rho =0.1$, the RD link remains the bottleneck. Optimization leads to ${\rho }_{\mathrm{opt}}\in (0.6,0.7)$ at high SNR in Figure 3c, improving ${\overline{\gamma }}_{\mathrm{id}}$ and reducing the BER. Corresponding SNR gains are approximately $7.5$ dB and $3.2$ dB at near $\mathrm{BER}={10}^{-2}$ and ${10}^{-4}$, respectively.

For the case of $\left[{\mathrm{\Omega }}_{\mathrm{sd}},{\mathrm{\Omega }}_{\mathrm{sr}},{\mathrm{\Omega }}_{\mathrm{rd}}\right]=\left[1,1,1/2\right]$, Figure 4 again highlights the RD link as the limiting factor. Optimization with ${\rho }_{\mathrm{opt}}\in (0.7,0.8)$ improves the RD link SNR and results in approximately 6 dB and 3 dB gains at near $\mathrm{BER}={10}^{-2}$ and ${10}^{-4}$, respectively. When $\left[{\mathrm{\Omega }}_{\mathrm{sd}},{\mathrm{\Omega }}_{\mathrm{sr}},{\mathrm{\Omega }}_{\mathrm{rd}}\right]=[1,1,1/4]$, the optimization process leads to a high power-splitting ratio, with ${\rho }_{\mathrm{opt}}>0.8$, resulting in a significant improvement in the RD link SNR (i.e., bringing it closer to that of the SR link). Nevertheless, improvements are still achieved, with about 2.5 dB gain in combined link performance at $\mathrm{BER}={10}^{-4}$, as shown in Figure 5a.

Across all scenarios, Figure 4a and Figure 5a confirm that the analytical expressions $P_{B,\mathrm{id}}^{\mathrm{AF}-\mathrm{App}}$ and $P_{B,\mathrm{Cb}}^{\mathrm{AF}-\mathrm{App}}$ in (24) provide a lower bound on the BER in the low SNR region, and an upper bound that closely approximates the asymptotic BER performance in the high SNR region. It is noteworthy that in the derivation of (19) from (A4), the second term associated with the RD link is obtained by applying an approximation process to the SR link SNR term. Without the optimization for the power-splitting ratio $\rho$, this approximation introduces inaccuracy in scenarios where the RD link is the dominant weak channel. As a result, discrepancies between the simulation and numerical results for both (19) and (21) are observed in Figure 2a, Figure 3a, Figure 4a and Figure 5a. On the other hand, when $\rho ={\rho }_{\mathrm{opt}}$ is employed as shown in Figure 2, Figure 3, Figure 4 and Figure 5, the SR link becomes the weaker channel. In this case, the asymptotic BER bounds given by (19) and (21) show excellent agreement with the simulation results in the high SNR region. Moreover, although minor mismatches exist, the approximations in (18), obtained via (22), demonstrate good consistency with simulations, affirming the effectiveness of the proposed $\rho$-optimization strategy. Consequently, from Figure 2, Figure 3, Figure 4 and Figure 5, it is confirmed that optimizing $\rho$ according to (22) effectively enhances the system performance by reinforcing the weaker link, particularly by increasing $\rho$ as the SNR increases and ${\mathrm{\Omega }}_{\mathrm{rd}}$ decreases (i.e., as $d_{\mathrm{rd}}$ increases).

Note that in the low SNR region, as shown in Figure 2, Figure 3, Figure 4 and Figure 5, significant discrepancies can be observed between the theoretical and simulation results. These discrepancies arise primarily because the theoretical predictions are based on asymptotic BER expressions, which are derived under high SNR assumptions and are thus not accurate in the low SNR regime.

5.2. Multiple SWIPT Relaying Systems

Table 2 shows the channel model used for the multiple SWIPT relaying system and the corresponding link channel power. In Table 2, Ch. Model $=\left[1,X,1\right]$ indicates the environment in which the RD link channel power can be different according to R. Also, Ch. Model $=\left[1,1,X\right]$ provides the SR link channel power-varying environment. Table 3 shows the relationship between the meaning, equation number, and symbols of the legend used in the resulting figures.

Table 2. Channel models for multiple SWIPT relaying system.

Ch. Model	Link	Ch. Power	R
$\left[1,1,X\right]$	SD	${\mathrm{\Omega }}_{\mathrm{sd}}=1$
	SR	${\mathrm{\Omega }}_{{\mathrm{sr}}_r}\in \left\{1\right\}$
	RD	${\mathrm{\Omega }}_{{\mathrm{rd}}_r}\in \left\{1\right\}$	1
		${\mathrm{\Omega }}_{{\mathrm{rd}}_r}\in \left\{1,2\right\}$	2
		${\mathrm{\Omega }}_{{\mathrm{rd}}_r}\in \left\{1/4,1,4\right\}$	3
		${\mathrm{\Omega }}_{{\mathrm{rd}}_r}\in \left\{1/2,1,2,4\right\}$	4
$\left[1,X,1\right]$	SD	${\mathrm{\Omega }}_{\mathrm{sd}}=1$
	SR	${\mathrm{\Omega }}_{{\mathrm{rd}}_r}\in \left\{1\right\}$	1
		${\mathrm{\Omega }}_{{\mathrm{sr}}_r}\in \left\{1,2\right\}$	2
		${\mathrm{\Omega }}_{{\mathrm{sr}}_r}\in \left\{1/4,1,4\right\}$	3
		${\mathrm{\Omega }}_{{\mathrm{sr}}_r}\in \left\{1/2,1,2,4\right\}$	4
	RD	${\mathrm{\Omega }}_{{\mathrm{rd}}_r}\in \left\{1\right\}$

Table 2. Channel models for multiple SWIPT relaying system.

Ch. Model	Link	Ch. Power	R
$\left[1,1,X\right]$	SD	${\mathrm{\Omega }}_{\mathrm{sd}}=1$
	SR	${\mathrm{\Omega }}_{{\mathrm{sr}}_r}\in \left\{1\right\}$
	RD	${\mathrm{\Omega }}_{{\mathrm{rd}}_r}\in \left\{1\right\}$	1
		${\mathrm{\Omega }}_{{\mathrm{rd}}_r}\in \left\{1,2\right\}$	2
		${\mathrm{\Omega }}_{{\mathrm{rd}}_r}\in \left\{1/4,1,4\right\}$	3
		${\mathrm{\Omega }}_{{\mathrm{rd}}_r}\in \left\{1/2,1,2,4\right\}$	4
$\left[1,X,1\right]$	SD	${\mathrm{\Omega }}_{\mathrm{sd}}=1$
	SR	${\mathrm{\Omega }}_{{\mathrm{rd}}_r}\in \left\{1\right\}$	1
		${\mathrm{\Omega }}_{{\mathrm{sr}}_r}\in \left\{1,2\right\}$	2
		${\mathrm{\Omega }}_{{\mathrm{sr}}_r}\in \left\{1/4,1,4\right\}$	3
		${\mathrm{\Omega }}_{{\mathrm{sr}}_r}\in \left\{1/2,1,2,4\right\}$	4
	RD	${\mathrm{\Omega }}_{{\mathrm{rd}}_r}\in \left\{1\right\}$

Table 3. Legend description for multiple SWIPT relaying system.

At	Legend	Symbol	Equation	Comment
Figure 6 and Figure 7	Theory, Exact	$P_{S,\mathrm{sd}}$, $P_{S,\mathrm{id}}$, $P_{S,\mathrm{cb}}$	(32)	Exact MGF
	Theory, Up	$P_{S,\mathrm{id}}^{\mathrm{Up}}$, $P_{S,\mathrm{cb}}^{\mathrm{Up}}$	(33)	Upper Bounded MGF
	Theory, Asym	$P_{B,\mathrm{id}}^{\mathrm{Asym}}$, $P_{B,\mathrm{cb}}^{\mathrm{Asym}}$	(35)	Asymptotic Bound
Figure 6 and Figure 7	Theory, AF-App	$P_{S,\mathrm{id}}^{\mathrm{AF}-\mathrm{App}}$, $P_{S,\mathrm{cb}}^{\mathrm{AF}-\mathrm{App}}$	(41)	AF-Approximation
		$P_{B,\mathrm{id}}^{\mathrm{AF}-\mathrm{App}}$, $P_{B,\mathrm{cb}}^{\mathrm{AF}-\mathrm{App}}$	(42)
Figure 8 and Figure 9	Theory, Exact	$P_{\mathrm{out},\mathrm{sd}}$	(45)	Exact only for SD
	Theory, AF-App	$P_{\mathrm{out},\mathrm{id}}^{\mathrm{AF}-\mathrm{App}}$, $P_{\mathrm{out},\mathrm{cb}}^{\mathrm{AF}-\mathrm{App}}$	(44)	AF-Approximation
Figure 10, Figure 11, Figure 12 and Figure 13	Theory, Exact	$C_{\mathrm{sd}}$	(46)	Exact only for SD
	Theory, AF-App	$C_{\mathrm{id}}^{\mathrm{AF}-\mathrm{App}}$, $C_{\mathrm{cb}}^{\mathrm{AF}-\mathrm{App}}$	(48) and (49)	AF-Approximation

Table 3. Legend description for multiple SWIPT relaying system.

At	Legend	Symbol	Equation	Comment
Figure 6 and Figure 7	Theory, Exact	$P_{S,\mathrm{sd}}$, $P_{S,\mathrm{id}}$, $P_{S,\mathrm{cb}}$	(32)	Exact MGF
	Theory, Up	$P_{S,\mathrm{id}}^{\mathrm{Up}}$, $P_{S,\mathrm{cb}}^{\mathrm{Up}}$	(33)	Upper Bounded MGF
	Theory, Asym	$P_{B,\mathrm{id}}^{\mathrm{Asym}}$, $P_{B,\mathrm{cb}}^{\mathrm{Asym}}$	(35)	Asymptotic Bound
Figure 6 and Figure 7	Theory, AF-App	$P_{S,\mathrm{id}}^{\mathrm{AF}-\mathrm{App}}$, $P_{S,\mathrm{cb}}^{\mathrm{AF}-\mathrm{App}}$	(41)	AF-Approximation
		$P_{B,\mathrm{id}}^{\mathrm{AF}-\mathrm{App}}$, $P_{B,\mathrm{cb}}^{\mathrm{AF}-\mathrm{App}}$	(42)
Figure 8 and Figure 9	Theory, Exact	$P_{\mathrm{out},\mathrm{sd}}$	(45)	Exact only for SD
	Theory, AF-App	$P_{\mathrm{out},\mathrm{id}}^{\mathrm{AF}-\mathrm{App}}$, $P_{\mathrm{out},\mathrm{cb}}^{\mathrm{AF}-\mathrm{App}}$	(44)	AF-Approximation
Figure 10, Figure 11, Figure 12 and Figure 13	Theory, Exact	$C_{\mathrm{sd}}$	(46)	Exact only for SD
	Theory, AF-App	$C_{\mathrm{id}}^{\mathrm{AF}-\mathrm{App}}$, $C_{\mathrm{cb}}^{\mathrm{AF}-\mathrm{App}}$	(48) and (49)	AF-Approximation

5.2.1. BER Comparison

Figure 6 and Figure 7 show the BER performance comparison with respect to the direct link SNR and different parameters of R and ${\rho }_r={\rho }_{\mathrm{opt}}$ for Ch. Model $=\left[1,1,X\right]$ and Ch. Model $=\left[1,X,1\right]$, respectively. From Figure 6 and Figure 7, it is shown that BER performance can be improved as R increases. Furthermore, it is verified that the numerical results for ‘Theory, Exact’, ‘Theory, Up’ of (33), and  ‘Theory, AF-App’ of (41) or (42) well match the simulation results regardless of SNR for both indirect and combined links. Although the larger R increases the discrepancy, we can additionally find that ‘Theory, Asym’ of (35) shows asymptotic BER bounds in good agreement in the high SNR region.

5.2.2. Outage Probability Comparison

In Figure 8 and Figure 9, the outage probability is compared with respect to the direct link SNR for varying values of R, with the power-splitting ratio set to the optimal value, ${\rho }_r={\rho }_{\mathrm{opt}}$, under Ch. Model $=\left[1,1,X\right]$ and Ch. Model $=\left[1,X,1\right]$, respectively. From Figure 8 and Figure 9, it can be observed that the outage probability performance improves as R increases. It is verified that the analytical approximation denoted as ‘Theory, AF-App’ provides an upper bound on the outage probability for both the indirect and combined links. Moreover, as R increases, the performance gap between the indirect and the combined links becomes smaller. As R increases, the outage performance difference between indirect and combined links decreases. Furthermore, the analytical result labeled as ‘Theory, Exact’ based on (45) shows an excellent match with the simulation results across all SNR values of the direct link. Although the gap between the simulation results and the ‘Theory, AF-App’ results derived from (44) becomes more pronounced as R increases, the approximate analysis still accurately captures the overall trend of the outage probability performance with increasing R.

5.2.3. Channel Capacity Comparison

Figure 10 and Figure 12 illustrate the channel capacity as a function of the direct link SNR for various values of R, with ${\rho }_r={\rho }_{\mathrm{opt}}$ under Ch. Model $=\left[1,1,X\right]$ and Ch. Model $=\left[1,X,1\right]$, respectively. In addition, Figure 11 and Figure 13 provide a comparison of channel capacity for $R\in \left\{1,2,3,4\right\}$, also under the respective channel models, while varying the SNR of the direct link. As observed in Figure 10, Figure 11, Figure 12 and Figure 13, as the channel capacity tends to decrease, R increases. The analytical results labeled as ‘Theory, AF-App’, derived from (48) and (49), serve as lower bounds on the channel capacity, particularly in the certain SNR region. For the indirect link, ‘Theory, AF-App’ shows a mismatch with the simulation results, and this mismatch is inversely proportional to R. On the other hand, ‘Theory, AF-App’ provides a well-aligned lower capacity bound for the combined link. Overall, it is confirmed that the approximately derived analytical results illustrate the channel capacity trend under various SNR and R conditions.

6. Conclusions

In this paper, we have derived the MGF and its approximated bound for SWIPT-enabled AF relaying systems operating without an external power supply at the relay. Based on these MGFs, closed-form BER expressions over quasi-static Rayleigh fading channels were obtained. Additionally, an asymptotic BER expression was derived to provide analytical insight into how the SNR relationship between the SR and RD links impacts the system performance. Using the derived asymptotic BER, we proposed a method to optimize the PS factor $\rho$, thereby enhancing the performance of the SWIPT relay system. Furthermore, we introduced an analytical approach that interprets the SWIPT relay as a generalized AF relay system, leading to an alternative approximated BER expression. The accuracy of the proposed BER expressions and the optimization framework was validated through simulations. The analytical expressions not only match the simulation results closely but also serve as simplified performance bounds, explicitly revealing the role of the SR and RD link SNRs in determining system performance.

Moreover, the analytical framework was successfully extended to the multiple SWIPT AF relaying system. For the multiple relay scenario, approximated performance metrics including BER, outage probability, and channel capacity were derived and verified. The results demonstrated that increasing the number of relays R improves the BER and outage performance, while the channel capacity tends to decrease with R. The derived asymptotic and approximate expressions provide useful bounds and accurately capture the system behavior across a range of SNRs and network configurations. This extension confirms the generality and robustness of the proposed analytical methods for practical multi-relay SWIPT systems.

Future work may extend this research to alternative SWIPT relaying protocols such as TS, multi-hop relaying, and WPCN configurations. Additionally, the development of energy-aware routing and power allocation algorithms for wireless-powered multi-hop networks remains a promising direction to further improve system efficiency and reliability.