Performance Prediction of Power Beacon-Aided Wireless Sensor-Powered Non-Orthogonal Multiple-Access Internet-of-Things Networks under Imperfect Channel State Information

Abstract

In this paper, we investigate a novel power beacon (PB)-aided wireless sensor-powered non-orthogonal multiple-access (NOMA) Internet-of-Things (IoT) network under imperfect channel state information (CSI). Furthermore, the exact expression outage probability (OP) of two IoT users is derived to analyze the performance of the considered network. To give further insight, the expression asymptotic OP and diversity order are also expressed when the transmit power at the PB goes to infinity. Furthermore, a deep neural network (DNN) framework is proposed to concurrently forecast IoT users’ OP in relation to real-time setups for IoT users. Additionally, when compared to the traditional analysis, our created DNN shows the shortest run-time prediction, and the outcomes predicted by the DNN model almost match those of the simulation. In addition, numerical results validate our analysis, simulation, and prediction through a Monte Carlo Simulation. Furthermore, the results show the impact of the main parameter on our proposed system. Finally, these findings show that NOMA performs better than the conventional orthogonal multiple-access (OMA) techniques.

1. Introduction

Recently, the Internet of Things (IoTs) has seen significant improvements in machine-to-machine (M2M) interactions, as well as a rising demand for IoT connections. IoT applications use intelligent devices to gather, process, and publish data on a regular basis. Billions of smart devices are connected to the Internet to enable a variety of applications, including smart building, industrial automation, smart health, and smart agriculture. Smart devices can access multimedia devices connected to the Internet of Things [1,2]. Several studies have supported the use of IoT wireless technology. However, it remains difficult to satisfy the demands of IoT applications in the future that will support hundreds of billions of linked devices [3,4]. Non-orthogonal multiple-access (NOMA) approaches are a well-studied strategy for offering significant spectral efficiency (SE) use in IoT networks [5,6,7], satellite networks [8,9,10], and V2V networks [11,12]. NOMA enables several users to use the same time and frequency block with varying transmit power allocations [13,14]. Furthermore, it employs superposition coding (SC) on the transmitter to accommodate signals from numerous users, and successive interference cancellation (SIC) at the receiver to separate and decrease interference [15,16]. NOMA technology is critical for future network capacity expansion. In particular, NOMA has been enhanced to increase cooperative communications, allowing users with superior channel conditions to act as a decode-and-forward (DF) or amplify-and-forward (AF) relay, improving information delivery and transmission reliability [17,18,19,20]. Moreover, the outage performance of a DF-based NOMA system under full-duplex (FD) and half-duplex (HD) situations was studied in [17]. Furthermore, over Nakagami-m fading, the outage probability (OP) and sum rate of the cooperative NOMA system with DF relay were investigated [18]. Additionally, the authors examined the downlink NOMA outage patterns using both DF and AF protocols [19], where it was possible to ascertain the decoding order of cell-edge users’ data based on incomplete channel status information. The two-way relay cooperative NOMA system’s OP and ergodic rate were studied in light of the possible performance boost provided by FD cooperative NOMA [20].

Wireless power transfer (WPT) is a potential technique for extending the life of energy-constrained wireless networks, such as wireless sensor networks and post-disaster emergency communication [21]. WPT-based networks are classified into two types: simultaneous wireless information and power transfer (SWIPT) [22,23,24] and wireless-powered communication networks (WPCN) [25,26,27]. SWIPT transports both information and energy via radio frequency (RF) signals, whereas WPCN harvests energy from a specialized power station and utilizes it for wireless information transmission (WIT) such as a power beacon (PB) or a hybrid access point (HAP). Both models have been thoroughly investigated in a variety of settings, focusing on the trade-off between data rate and energy harvesting (EH) amount with two techniques: the time-switching (TS) and power-splitting (PS) protocols.

Furthermore, the RF-EH protocols for a dual-hop relaying communication system are modeled after the TS and PS SWIPT protocols for non-relay point-to-point networks. In particular, these techniques, called time switching-based relaying (TSR) and power splitting-based relaying (PSR), collect energy from the source information stream [28]. The authors of [29] investigate multiple DF energy-constrained relays utilizing best relay selection and a hybrid TSR/PSR SWIPT RF-EH protocol, calculating OP and ergodic capacity with the extreme value theorem. In [30], the authors investigate a dual-hop cooperative relaying system with numerous energy-constrained relays utilizing the TSR SWIPT RF-EH protocol. The system’s OP is calculated using Nakagami-m fading in tight lower limits. In [31], the authors proposed a dual-hop cooperative relaying system with an energy-constrained DF relay that harvests energy from the source information stream. The exact closed-form OP equations are generated using special functions.

Recently, NOMA and cooperative relaying NOMA (C-NOMA) combined with EH have been hotly studied [32,33,34,35,36,37,38]. In [32], the authors investigated spatial modulation-based CNOMA for the BS downlink using a full-duplex near user that might collect energy and transmit data to a far user. Ref. [33] investigated collaborative user pairing and resource allocation for WPT-based CNOMA with EH near users. In [34], EH is carried out with CNOMA, and it is taken into consideration by choosing the optimal (decode-and-forward) DF relay. Only a two-user downlink NOMA and the dual-hop scenario are taken into account in the system model and analysis. When there are many EH-based amplify-and-forward (AF) relays between the base station and the far user, the best one can be chosen for data forwarding [35]. In [36], a two-user CNOMA network is explored, with the source node’s power consumption lowered to meet the needed transmission rates using both TS relaying and PS relaying strategies. The Lyapunov optimization methodology is utilized by the authors in [37] in order to maximize throughput or restrict transmit power levels. For a wireless-enabled IoT network, the author in [38] proposes a fairness-aware NOMA-based scheduling system. Additionally, they suggest boosting network through throughput-aware NOMA-based scheduling.

Recent research has demonstrated that deep learning (DL) methodologies can help with a variety of real-world challenges in wireless communication networks, such as resource allocation, congestion control, and queue management [39]. In [40], a deep neural network (DNN) was utilized to boost a cell-edge user’s productivity under both perfect SIC and imperfect SIC in wireless-powered CR-NOMA-based IoT relay networks. In [41], a DNN was developed to tackle classification and regression issues in cognitive two-way relay networks for a relay. The authors of [42] investigated a DNN incorporated into a NOMA system and achieved outstanding performance in terms of channel encoding, decoding, and detection. Ref. [43] described a unique way to improve service distribution in IoT networks that employed DL to discover the best distribution strategy.

Motivation and Contribution

Due to the benefit of sharing SE and achieving EE through the combination of NOMA and RF-EH, evaluating the performance of integrated systems of IoT networks has become more difficult and has garnered attention in several recent studies. In [44], the authors investigated the optimal sum throughput (STP) of SWIPT IoT relay NOMA systems in relation to the TS factor. The maximum throughput for the PB-assisted EH NOMA multiuser relaying systems across Nakagami-m channels was examined in [45] in order to enhance the NOMA IoT-based system performance. In the meanwhile, with faulty CSI and hardware impairment, the throughput and OP of cooperative PB-assisted EH were studied in [46]. In a recent study, ref. [47] looked into the cooperative multiple PBs’ OP for uplink NOMA systems in wirelessly powered IoT applications. However, these works do not consider DNN methods to be able to predict with a low latency and high accuracy, towards practical real-time configurations of C-NOMA IoT networks.

Based on these advantages, we propose a wireless sensor-powered NOMA IoT network under imperfect channel state information (CSI). Furthermore, the PB transmits energy to both the BS and relay to transmit the information. Subsequently, the exact OP is derived to evaluate the outage performance of the proposed network. In particular, we employ the DNN model to predict the performance of the considered system to reduce the complexity of the simulation and calculation. The key differences between our study and previous research are summarized in

Table 1. Comparison of this work with the reference works.

	Our Work	[44]	[45]	[46]	[47]	[48]	[49]
NOMA	✓	✓	✓	✓	✓	✓	✓
Energy harvesting	✓	✓	✓	✓	✓	✓	✓
Imperfect CSI	✓			✓
Outage probability analysis	✓	✓	✓	✓	✓	✓	✓
Diversity order	✓			✓
DNN method	✓

. The main contribution of our paper can be summarized as follows:

• We investigate a novel wireless sensor-powered NOMA IoT network, where a PB adopts the TS protocol to transmit the energy to both BS and relaying to transmit information. In particular, the imperfect CSI is considered in the information transmission phase to estimate the complex channel intricacies.
• We analyze the performance in terms of outage probability (OP). To obtain more insight into the considered network, the asymptotic expression OP and diversity order are given. Based on the asymptotic expression OP, we show the main parameters that affect the considered network in high SNR.
• We design the DNN frame to reduce the complexity of the Monte Carlo simulation and the traditional analysis towards practical real-time configurations. The DNN models are demonstrated to be efficient in predicting performance in complicated network scenarios using a low-latency inference technique.

The remainder of this work is organized as follows. Section 2 describes the proposed system model. The performance of the proposed system is given in Section 3. The numerical findings of our suggested system are presented in Section 5. Finally, the conclusion is summarized in Section 6.

2. System Model

This paper considered the power beacon (PB)-aided wireless sensor-powered NOMA IoT networks as in Figure 1, where a base station $\left(S\right)$ harvests energy from a PB to transmit the information to two users (an IoT far user (FU) and an IoT near user (NU)) through a DF relay (R). In addition, we assumed that every node had a single antenna and that every node had complete knowledge of the CSI. $g_{PS}$ and $g_{PR}$ are the channel coefficients from PB to S and R, respectively; $g_{SR}$, $g_{RN}$, and $g_{RF}$ are the channel coefficients from S to R, R to NU and FU, respectively. Furthermore, all channels follow Rayleigh fading. Thus, $|g_j{|}^2,j\in \{PS,PR,SR,RN,RF\}$ is an exponential random variable with the probability density function (PDF) and the cumulative distribution function (CDF) expressed, respectively, as

$$\begin{array}{cc}\hfill f_{\left|g_j\right|}\left(x\right)& =\frac{1}{{\Omega }_j}{e}^{-\frac{x}{{\Omega }_j}}\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill F_{\left|g_j\right|}\left(x\right)& =1-e^{-\frac{x}{{\Omega }_j}}\hfill \end{array}$$

(2)

where ${\Omega }_j=d_j^{\ell }$ denotes the average channel gain, $d_j$ denotes the distance, and ℓ denotes the path-loss exponent.

Furthermore, we considered the TS protocol for our proposed network as depicted in Figure 2. In particular, the PB transmits the energy to S and R during the period time of $\beta T$ in the EH phase, where $\beta$ denotes the TS factor, and T denotes the coherence time block. For the information transmission phase, the remaining time, $(1-\beta )T$, is divided evenly into two hops of transmission, where S and R employ the transmit power derived from the gathered energy for transmitting and relaying operations. Moreover, in the information phase, the channel cannot be perfectly estimated due to the complex channel intricacies. Thus, the channel for the information phase can be modeled as [50]

$$\begin{array}{c}\hfill g_z=\epsilon {\widehat{g}}_z+\sqrt{1-{\epsilon }^2}{\tilde{g}}_z\end{array}$$

(3)

where $z\in \{SR,RN,RF\}$, ${\tilde{h}}_z$ denotes a circular symmetric complex Gaussian random variable with zero mean and the same variance of $h_z$, and $\epsilon$ denotes the channel correlation factor that models the accuracy of the channel estimate.

2.1. Energy-Harvesting Phase

In the EH phase, S and R harvest energy from PB during the time period of $\beta T$. The total harvested energy at S and R are expressed, respectively, as [48]

$$\begin{array}{cc}\hfill E_S& =\eta \beta T{P}_{PB}{\left|g_{PS}\right|}^2\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill E_R& =\eta \beta T{P}_{PB}{\left|g_{PR}\right|}^2\hfill \end{array}$$

(5)

where $\eta$ denotes the energy efficiency, and $P_{PB}$ denotes the transmit power of PB. Furthermore, the transmit power of S and R that was attained in the remaining time $(1-\beta )T/2$ can be given, respectively, as

$$\begin{array}{cc}\hfill P_S& =\frac{E_S}{\frac{\left(1-\beta \right)T}{2}}=\frac{2\eta \beta P_{PB}{\left|g_{PS}\right|}^2}{\left(1-\beta \right)}=\vartheta P_{PB}{\left|g_{PS}\right|}^2\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill P_R& =\frac{E_R}{\frac{\left(1-\beta \right)T}{2}}=\frac{2\eta \beta P_{PB}{\left|g_{PR}\right|}^2}{\left(1-\beta \right)}=\vartheta P_{PB}{\left|g_{PR}\right|}^2\hfill \end{array}$$

(7)

where $\vartheta =\frac{2\eta \beta }{\left(1-\beta \right)}$.

2.2. Information Transmission Phase

In the first information transmission phase, by applying the NOMA protocol, S transmits the superimposed signal $\sqrt{{\alpha }_N}{x}_N+\sqrt{{\alpha }_F}{x}_F$ to R, where $x_N$ and $x_F$ denotes the signal of NU and FU, respectively, with $\mathbb{E}[{\left|x_N\right|}^2]\hspace{0.17em}=\mathbb{E}[|x_F{|}^2]=1$; ${\alpha }_N$ and ${\alpha }_F$ are the power allocation with ${\alpha }_N<{\alpha }_F$ and ${\alpha }_N+{\alpha }_F=1$. Thus, the received signal at R is given by

$$\begin{array}{c}\hfill y_R=g_{SR}\left(\sqrt{{\alpha }_N}{x}_N+\sqrt{{\alpha }_F}{x}_F\right)\sqrt{P_S}+n_R\end{array}$$

(8)

where $n_R$ denotes the additive white Gaussian noise with $\mathcal{CN}(0,N_0)$. The signal-to-interference-plus-noise ratios (SINRs) at R when detecting $x_F$ and $x_N$ are given, respectively, as

$$\begin{array}{cc}\hfill {\gamma }_{R\to x_F}& =\frac{P_S{\epsilon }^2{\alpha }_F{\left|{\widehat{g}}_{SR}\right|}^2}{P_S{\alpha }_N{\epsilon }^2{\left|{\widehat{g}}_{SR}\right|}^2+P_S\left(1-{\epsilon }^2\right){\Omega }_{SR}+N_0}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\frac{\vartheta {\epsilon }^2{\alpha }_F{\left|{\widehat{g}}_{SR}\right|}^2{\left|g_{PS}\right|}^2{\rho }_{PB}}{{\alpha }_N{\epsilon }^2{\left|{\widehat{g}}_{SR}\right|}^2{\rho }_{PB}+\vartheta \left(1-{\epsilon }^2\right){\Omega }_{SR}{\left|g_{PS}\right|}^2{\rho }_{PB}+N_0}\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\gamma }_{R\to x_N}& =\frac{\vartheta {\alpha }_N{\epsilon }^2{\left|{\widehat{h}}_{SR}\right|}^2{\left|h_{PS}\right|}^2{\rho }_{PB}}{\vartheta \left(1-{\epsilon }^2\right){\Omega }_{SR}{\left|h_{PS}\right|}^2{\rho }_{PB}+1}\hfill \end{array}$$

(10)

where ${\rho }_{PB}=\frac{P_{PB}}{N_0}$ denotes the average signal-to-noise ratio (SNR).

In the second information transmission phase, R decodes the signal from S and transmits a superimposed signal to two users NU and FU. The received signal at NU and FU are given, respectively, by

$$\begin{array}{cc}\hfill y_{NU}& =g_{RN}\left(\sqrt{{\alpha }_N}{x}_N+\sqrt{{\alpha }_F}{x}_F\right)\sqrt{P_S}+n_{NU}\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill y_{NF}& =g_{RF}\left(\sqrt{{\alpha }_N}{x}_N+\sqrt{{\alpha }_F}{x}_F\right)\sqrt{P_S}+n_{FU}\hfill \end{array}$$

(12)

where $n_{NU}$ and $n_{FU}$ are the AWGN with $\mathcal{CN}(0,N_0)$. The SINR at FU when detecting its own signal $x_F$ is given by

$$\begin{array}{cc}\hfill {\gamma }_{FU\to x_F}& =\frac{P_R{\epsilon }^2{\alpha }_F{\left|{\widehat{g}}_{FU}\right|}^2}{P_R{\alpha }_N{\epsilon }^2{\left|{\widehat{g}}_{FU}\right|}^2+P_R\left(1-{\epsilon }^2\right){\Omega }_{FU}+1}\hfill \\ \hfill & =\frac{\vartheta {\epsilon }^2{\alpha }_F{\left|g_{PR}\right|}^2{\left|{\widehat{g}}_{FU}\right|}^2{\rho }_{PB}}{\vartheta {\alpha }_N{\epsilon }^2{\left|g_{PR}\right|}^2{\left|{\widehat{g}}_{FU}\right|}^2{\rho }_{PB}+\vartheta \left(1-{\epsilon }^2\right){\Omega }_{FU}{\left|g_{PR}\right|}^2{\rho }_{PB}+1}\hfill \end{array}$$

(13)

By adopting the SIC process, NU first detects the signal of FU, i.e., $x_F$, then detects its own signal, i.e., $x_n$. Thus, the SINRs at NU when detecting signals $x_F$ and $x_N$ are given, respectively, by

$$\begin{array}{cc}\hfill {\gamma }_{NU\to x_F}& =\frac{\vartheta {\epsilon }^2{\alpha }_F{\left|g_{PR}\right|}^2{\left|{\widehat{g}}_{NU}\right|}^2{\rho }_{PB}}{\vartheta {\alpha }_N{\epsilon }^2{\left|g_{PR}\right|}^2{\left|{\widehat{g}}_{NU}\right|}^2{\rho }_{PB}+\vartheta \left(1-{\epsilon }^2\right){\Omega }_{NU}{\left|g_{PR}\right|}^2{\rho }_{PB}+1}\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {\gamma }_{NU\to x_U}& =\frac{\vartheta {\alpha }_N{\epsilon }^2{\left|g_{PR}\right|}^2{\left|{\widehat{g}}_{NU}\right|}^2{\rho }_{PB}}{\vartheta \left(1-{\epsilon }^2\right){\Omega }_{NU}{\left|g_{PR}\right|}^2{\rho }_{PB}+1}\hfill \end{array}$$

(15)

3. Outage Performance Analysis

3.1. Outage Probability Analysis

The outage event takes place when R and FU cannot properly decode the $x_F$ signal. Therefore, the OP of FU can be determined as

$$\begin{array}{c}\hfill P_{FU}=Pr\left(min\left({\gamma }_{R\to x_F},{\gamma }_{FU\to x_F}\right)<{\gamma }_{FU}\right)\end{array}$$

(16)

where ${\gamma }_{FU}=2^{\frac{2R_F}{1-\beta }}-1$ denotes the threshold, and $R_F$ denotes the target rate.

Theorem 1.

The exact expression OP of FU is expressed as

$$\begin{array}{c}\hfill P_{FU}=1-\frac{4{\theta }_{FU}{e}^{-2{\theta }_{FU}\vartheta \left(1-{\epsilon }^2\right){\rho }_{PB}}}{\sqrt{{\Omega }_{SR}{\Omega }_{PS}{\Omega }_{FU}{\Omega }_{PR}}}{K}_1\left(2\sqrt{\frac{{\theta }_{FU}}{{\Omega }_{SR}{\Omega }_{PS}}}\right)K_1\left(2\sqrt{\frac{{\theta }_{FU}}{{\Omega }_{FU}{\Omega }_{PR}}}\right)\end{array}$$

(17)

Proof of Theorem 1.

It can be seen in Appendix A. □

The outage event of NU takes place when R and NU cannot properly decode the $x_N$ signal. Thus, the OP of NU can be given by

$$\begin{array}{c}\hfill P_{NU}=Pr\left(min\left({\gamma }_{R\to x_N},{\gamma }_{NU\to x_N}\right)<{\gamma }_{NU}\right)\end{array}$$

(18)

Theorem 2.

The exact expression OP of NU is expressed as

$$\begin{array}{c}\hfill P_{NU}=1-\frac{4{\theta }_{NU}{e}^{-2{\theta }_{NU}\vartheta \left(1-{\epsilon }^2\right){\rho }_{PB}}}{\sqrt{{\Omega }_{SR}{\Omega }_{PS}{\Omega }_{NU}{\Omega }_{PR}}}{K}_1\left(2\sqrt{\frac{{\theta }_{NU}}{{\Omega }_{SR}{\Omega }_{PS}}}\right)K_1\left(2\sqrt{\frac{{\theta }_{NU}}{{\Omega }_{NU}{\Omega }_{PR}}}\right)\end{array}$$

(19)

Proof of Theorem 2.

With help from (10) and (15), the OP of NU is expressed as

$$\begin{array}{cc}\hfill P_{NU}& =1-Pr\left({\left|{\widehat{g}}_{SR}\right|}^2>\left(\vartheta \left(1-{\epsilon }^2\right){\Omega }_{SR}{\theta }_{NU}{\rho }_{PB}+\frac{{\theta }_{NU}}{{\left|g_{PS}\right|}^2}\right)\right)\hfill \\ \hfill & \times Pr\left({\left|{\widehat{g}}_{NU}\right|}^2>\left(\vartheta \left(1-{\epsilon }^2\right){\Omega }_{NU}{\theta }_{NU}{\rho }_{PB}+\frac{{\theta }_{NU}}{{\left|g_{PR}\right|}^2}\right)\right)\hfill \end{array}$$

(20)

where ${\theta }_{NU}=\frac{{\gamma }_{NU}}{\vartheta {\epsilon }^2{\rho }_{PB}{\alpha }_N}$. Then, the $P_{NU}$ in (20) can be calculated as

$$\begin{array}{cc}\hfill P_{NU}& =1-\underset{0}{\overset{\infty }{\int }}{f}_{{\left|g_{PS}\right|}^2}\left(x\right)\underset{0}{\overset{\infty }{\int }}{f}_{{\left|g_{PR}\right|}^2}\left(y\right)\hfill \\ \hfill & \times \hspace{0.17em}{F}_{{\left|{\widehat{g}}_{SR}\right|}^2}\left(\vartheta \left(1-{\epsilon }^2\right){\Omega }_{SR}{\theta }_{NU}{\rho }_{PB}+\frac{{\theta }_{NU}}{x}\right)dz\hfill \\ \hfill & \times \hspace{0.17em}{F}_{{\left|{\widehat{g}}_{NU}\right|}^2}\left(\vartheta \left(1-{\epsilon }^2\right){\Omega }_{NU}{\theta }_{NU}{\rho }_{PB}+\frac{{\theta }_{NU}}{y}\right)dy\hfill \end{array}$$

(21)

Similarly, in Appendix A, the exact expression OP of NU is given in (19). The proof is completed. □

3.2. Diversity Order Analysis

In this subsection, we express the diversity order to evaluate the OP in high SNRs (i.e., ${\rho }_{PB}\to \infty$) to obtain more insight into the proposed network. The diversity order is given by [27]

$$\begin{array}{c}\hfill d=-\underset{{\rho }_{PB}\to \infty }{lim}\frac{log\left(P^{\infty }\left({\rho }_{PB}\right)\right)}{log\left({\rho }_{PB}\right)}\end{array}$$

(22)

where $P^{\infty }\left({\rho }_{PB}\right)$ denotes the asymptotic OP of IoT users.

Corollary 1.

In high SNR (${\rho }_{PB}\to \infty$), we can approximate as $K_1\left(x\right)\approx \frac{1}{x}$. Thus, the expression asymptotic OP of FU is expressed as

$$\begin{array}{c}\hfill P_{FU}^{\infty }\approx 1-e^{-\frac{2{\gamma }_{FU}\left(1-{\epsilon }^2\right)}{{\epsilon }^2\left({\alpha }_F-{\gamma }_{FU}{\alpha }_N\right)}}\end{array}$$

(23)

Remark 1.

Corollary 1 gives some useful insights: (i) the main parameters that affect the OP of FU are the target rate of FU ${\gamma }_{FU}$, the power allocation (${\alpha }_F,{\alpha }_N$), and the channel correlation factor ε; ($ii$) upon substituting (23) into (22), the diversity order of FU is zero.

Corollary 2.

The expression asymptotic OP of NU is expressed as

$$\begin{array}{c}\hfill P_{NU}^{\infty }\approx 1-e^{-\frac{2{\gamma }_{NU}\left(1-{\epsilon }^2\right)}{{\epsilon }^2{\alpha }_N}}\end{array}$$

(24)

Remark 2.

Corollary 2 gives some useful insights: (i) the main parameters that affect the OP of NU are the target rate of NU ${\gamma }_{NU}$, the power allocation (${\alpha }_F,{\alpha }_N$), and the channel correlation factor ε; ($ii$) upon substituting (24) into (22), the diversity order of FU is zero.

4. Deep Neural Networks Analysis

In this section, we design and suggest a DNN that can predict the OP in situations where traditional analysis and Monte Carlo simulations are not practical, while having a fast run time and little computing cost. We describe our suggested strategy in more detail below.

4.1. Description of the DNN

Firstly, in order to solve a regression problem, we generated a DNN model. As shown in Figure 3, the DNN model was composed of an input layer, multiple hidden layers, and an output layer. Moreover, in an input layer, we had 15 neurons corresponding to 15 parameters as shown in

Table 2. The parameters of the DNN.

Input	Value	Input	Value
$\beta$	0.2	$d_{SR}$	[10,20] [m]
$\eta$	0.7	$d_{PS}$	[10,20] [m]
${\alpha }_F$	0.85	$d_{PR}$	[10,20] [m]
${\alpha }_N$	0.15	$d_{RF}$	[20,40] [m]
$R_F$	[0.01, 0.02]	$d_{RN}$	[15,30] [m]
$R_N$	[0.01, 0.02]	$N_0$	−94 [dBm]
$\epsilon$	0.95	$P_{PB}$	[−30,30] [dBm]
ℓ	2.7

. Next, each hidden layer $k=1,\dots ,D_{hiden}$ had $D_{neu}$ neurons. Furthermore, each hidden neuron employed the rectified linear unit (ReLU) as the nonlinear activation function, defined as [51]

$$\begin{array}{c}\hfill ReLU\left(x\right)=max(x,0)\end{array}$$

(25)

The output layer was composed of a single neuron that utilized the linear activation function to yield the anticipated OP value, $P^{Pre}\left[1\right]$. This was because the regression issue sought to estimate an output value without requiring any further conversions.

4.2. Setup for Dataset

In this subsection, each hidden layer k generated the row vector data set $\mathcal{M}$ with data set$\left[k\right]=\left[\mathbf{X}\left[k\right],P^{Sim}\right]$, where $\mathbf{X}\left[k\right]$ denotes the feature vector made up of all inputs from each of the parameters in Table 2. Next, the real-value OP sets from (16) and (18) were generated using each $\mathbf{X}\left[k\right]$ feature vector, which were then input into the simulation to obtain a distinct matching $P^{Sim}$. In summary, the data set was created by concatenating the ${10}^5$ samples and was then separated into three new data sets: 80% for training (${\mathcal{M}}_{train}$), 10% for validation (${\mathcal{M}}_{vali}$), and 10% for testing (${\mathcal{M}}_{test}$).

The mean squared error (MSE), defined as $MSE=\frac{1}{{\mathcal{M}}_{test}}{\sum }_{k=0}^{{\mathcal{M}}_{test}-1}(P^{Pre}-P^{sim})$, was used to assess how effectively the suggested DL approach performed. To calculate the difference between the natural and predicted OP values for the full test set, we use the root-mean-square error (RMSE) in the OP prediction, and it can be defined as $RMSE=\sqrt{MSE}$.

5. Numerical Result

In this section, we provide illustrative numerical results to evaluate the performance of the proposed network in terms of OP by using a Monte Carlo simulation. The main parameters were set as $\beta =0.2$, $d_{SR}=10$ m, $\eta =0.7$, $d_{PS}=10$ m, ${\alpha }_F=0.85$, $d_{PR}=10$ m, ${\alpha }_N=0.15$, $d_{RF}=20$ m, $d_{RN}=15$ m, $R_N=0.01$, $R_F=0.01$, $N_0=-94$ [dBm], $\epsilon =0.95$, and $\ell =2.7$, except for the special case. Furthermore, Ana., Sim., and Asymp. are the abbreviations for analytical, simulation, and asymptotic, respectively. For the DNN, we set five hidden layers with each layer having 128 neurons, which was implemented in Python 3.11.4 using Keras 2.8.0 and TensorFlow 2.8.0, with 100 epochs for training the DNN.

In Figure 4, we utilized the MSE of the training and the validation to evaluate the accuracy of the proposed DNN. As can be seen, the MSE converged to a value lower than ${10}^{-5}$ after 40 epochs. Figure 5 plots the OP of two users (FU and NU) versus $P_{PB}$ in dBm while varying the target rate $R_F=R_N$. It can be observed that over the whole range of $P_{PB}$ in dBm, the OP curves and analytical results agreed exceptionally well based on the Monte Carlo simulation. Furthermore, the asymptotic findings produced in (23) and (24) were extremely valid with the theoretical analysis ones at high $P_{PB}$, confirming the accuracy of our developed analysis approach. In addition, we can observe that the outage performance was improved when increasing the transmit power but did not change in high $P_{PB}$. The fact that the OP depended on more factors than only transmitting $P_{PB}$ can be shown in Remark 1 and 2. Another observation is that the excellent agreement between the simulation and DNN prediction outcomes validated our proposed methods.

Figure 6 depicts the OP of users versus $P_{PB}$ in dBm with varying $\epsilon$. As shown in Figure 6, we can observe that the OP increased when $\beta$ decreased and converge with a large $P_{PB}$. This is due to the OP depending on the interference channels in the SINR of two users when decoding the signal. In Figure 7, we plotted the OP versus ${\alpha }_F$ when varying $P_{PB}$ in dBm. First, we can observe that the OP of FU decreased when ${\alpha }_F$ increased. The reason was that when ${\alpha }_F$ increased, the transmit power of FU was greater than that of NU. Furthermore, the OP of NOMA was better that OMA in the range $0.5<\beta <0.8$. In addition, there existed an optimal point ${\alpha }_F$ for the OP of two users. Specifically, when $P_{PB}=-10$ dBm the optimal point for the OP of two users was ${\alpha }_F=0.64$, ${\alpha }_F=0.59$ for $P_{PB}=0$ dBm, and ${\alpha }_F=0.54$ for $P_{PB}=10$ dBm.

Figure 8 shows the OP of two users versus $\eta$ when varying $P_{PB}$. It can be observed that the OP was improved when increasing the energy efficiency $\eta$. This is due to the energy efficiency $\eta$ being larger, leading to a greater transmit power of PB in (4) and (5). Figure 9 illustrates the OP of two users versus $\beta$ when varying $P_{PB}$. The TS factor $\beta$ is important because it controls both the data transport and the amount of energy collected at S and R. We can observe that the best OP for two users for $P_{PB}=0$ dBm can be achieved when $0.4<\beta <0.5$, and for $P_{PB}=10$ dBm, the best OP can be achieved when $0.1<\beta <0.2$.

6. Conclusions

This paper investigated the PB-aided wireless sensor-powered NOMA IoT networks under imperfect CSI. The exact expression OP, expression asymptotic OP, and diversity order were derived to obtain more insight into the system characteristics. Based on the expression asymptotic OP, we showed the main parameter that affected the proposed network such as the channel correlation factor, the power allocation, the factor of TS, and the target rate of users. A DNN was developed to predict the performance in terms of OP with minimal complexity and high accuracy. In the numerical result, this paper found that using a DNN model for system performance evaluation, the OP outcomes closely resembled the Monte Carlo simulation and analysis results.