Short-Packet Communications in Wireless Energy Transfer Full-Duplex IoT Networks with Deep Learning Design

I. INTRODUCTION

SHORT-packet communication (SPC) has gained significant attention due to its potential applications in critical scenarios with stringent latency requirements such as autonomous driving, remote surgery, and factory automation [2], [3]. For the Internet-of-things (IoT) systems and wireless networks, SPC was considered to achieve low transmission latency, and high reliability with a small block error rate (BLER) [4], [5]. Recent studies in SPC with finite blocklength codes have primarily focused on the analysis of IoT systems with half-duplex (HD) transmissions, where strict requirements of 99.999% reliability and 1 ms latency demand utmost consideration [6].

Deep learning (DL) has recently emerged as a powerful solution to tackle a variety of practical problems in contemporary wireless IoT systems, including resource allocation, queue management, and congestion control [7]. By precisely estimating functions with high nonlinearity at a minimum of complexity, DL has been increasingly applied in wireless networks to improve various aspects of systems including resource allocation, throughput prediction, and channel estimation [8]–[10]. Furthermore, employing deep learning for performance prediction in IoT networks can expedite realtime configurations. DL-based models have the capability to precisely predict desired performance measurements from intricate datasets with high dimensionality, even in complex network scenarios and highly dynamic environments where mathematical derivations may not be practical.

A. Literature Survey

Recently, the studies of SPCs have been explored in nonorthogonal multiple access (NOMA) systems incorporating simultaneous transmitting and reflecting (STAR) reconfigurable intelligent surfaces (RIS) [11]. The average BLER, throughput, goodput, latency, reliability, and age of information of SPCbased unmanned aerial vehicle (UAV)-based NOMA systems were analyzed in [12], taking into account imperfect channel state information and successive interference cancellation. In order to achieve a good balance performance between cellcenter users and cell-edge users while satisfying quality-ofservice requirements, a hybrid long-and short-packet communication approach was investigated in [13]. In [14], the partial relay selection was proposed for dual-hop cooperative relaying networks with short packet communications. However, most of these recent works have just focused on half-duplex transmission in SPC environments. To improve the BLER and throughput of full-duplex IoT networks, the energy harvesting with hybrid power-time splitting protocol was designed for IoT devices with infinite and finite blocklength codes conditions [15]. A hybrid power-time splitting strategy with a nonlinear energy harvesting model was also investigated in fullduplex cooperative IoT networks [16], where the reliability, goodput, and energy efficiency performances were analyzed. In [17], a performance analysis was conducted to compare full-duplex relaying (FDR) and half-duplex relaying under SPC constraints. The finding results indicated that FDR was a favorable option for systems with lower transmit power, less strict block-error rate requirements, and effective mitigation of loop interference. However, different residual interference models have not yet been fully explored in these works.

The application of deep learning techniques to address complex optimization problems has been explored in the studies [8], [18]. Particularly, Hendra et al. in [18] proposed a deep learning approach to maximize spectral efficiency in multiple-input multiple-output (MIMO)-NOMA systems. On the other hand, Nguyen et al. [8] designed a new deep convolutional neural network (CNN) architecture to obtain the optimal solution for maximizing the achievable rate in multihop IoT systems. In [2], a DNN design for a relay selection scheme was shown to be more effective than the state-ofthe- art machine learning models in predicting the throughput of cognitive IoT networks with a non-linear EH model. In [10], a novel CNN architecture was designed with feature enhancement-collection blocks to simultaneously predict the BLER and throughput of multi-hop SPC IoT systems with high accuracy and low execution time as compared to conventional CNN. In [19], the CNN was used for achieving solutions of downlink beamforming design of multiple-input singleoutput under the per-antenna power constraints. The practical testbed experiments of deep learning using software-defined radio were also built to demonstrate the effectiveness of the proposed system. However, these aforementioned works have not considered full-duplex transmission in conjunction with SPCs and deep learning designs in these IoT systems.

B. Motivation and Contributions

DL-based performance prediction in wireless energy transfer (WET) full-duplex IoT networks (WFINs) with different residual self-interference (RSI) models considering shortpacket transmissions over Nakagami-m channels, which induces the intricacy of theoretical analysis, has not investigated in these previous works. This hinders a comprehensive investigation of the effects of RSI and finite block length under SPC studies. Moreover, when IoT networks are on a large-scale deployment, where the number of relays, and the number of antennas at the power beacon and destination are increasing exponentially with complex distributions. The traditional mathematical analysis of these systems may be intricate and these methods are not suitable for real-time applications in modern wireless communications systems. If we analyze the average BLER of the system with the RSI-II model, the resulting expression will have integral forms with several special functions (e.g., exponential integral function, gamma function, and Bessel functions), which cannot be further analyzed since they involved many complex functions. This motivates us to build a new framework relying on CNN design for performance evaluation. The DL framework allows us to estimate quickly the system performance, such as average BLER and throughput, whenever any changes in the network settings. The training process will be performed offline to obtain the trained model for online prediction, which helps to reduce the deployment time for network designing, planning, and monitoring in practice. We fill this gap by investigating full-duplex IoT networks consisting of multiple FD relays powered by a multi-antenna power beacon (PB) with different RSI models and evaluated by a new deep CNN architecture design. This work also emphasizes demonstrating the feasibility of a DL framework for efficiently evaluating WFINs wireless networks in which conventional mathematical modelbased approaches become unfeasible. The main contributions of this paper can be summarized as follows:

· We propose a full relay selection (FRS) scheme in WFINs, where two RSI models are considered at the IoT devices under SPC constraints. We also derive new closed-form expressions for the average BLER and throughput of the FRS scheme over Nakagami-m fading channels with the RSI-I model under the considered system setup.

· We perform the approximation analysis of the average BLER and throughput of FRS scheme with the RSI-I model to show the effects of fading severity and locations of relay and power beacon on the system performance. The approximation approach has not been studied in recent works since the complexity of Nakagami-m fading channels.

· We develop a new deep convolutional neural network architecture to estimate the BLER and throughput of the FRS scheme with the RSI-II model, where a multioutput regression problem is designed to train network parameters for high-accuracy performance prediction.

· Simulation results reveal that the new CNN design provides the smallest root-mean-square error (RMSE) among the traditional designs such as the deep neural network (DNN) and conventional CNN approaches, emerging as an outstanding performance estimator in IoT networks.

· It is also revealed that the CNN-based scheme attains the same BLER and throughput as the FRS one, but with significantly reduced complexity. Additionally, the FRS scheme with RSI-I outperforms HD one in terms of BLER and throughput, which corroborates the efficiency of FD transmission.

The paper is organized as follows. Section II introduces the system model with two RSI scenarios and the relay selection strategy. The system performance analysis is provided in Section III focusing on average BLER and throughput. Section IV provides the new CNN design for BLER and throughput prediction of the FRS scheme. Section V provides simulations to confirm the theoretic results of Sections III and CNN design in IV. Section VI concludes the paper. The appendix provides proof of Lemma 1 with derivations of the theoretical results.

Mathematical Notations: Boldface represents vector and [TeX:] $$\mathbb{E}\{\cdot\}$$ are the expectation operator.

II. SYSTEM MODEL

It is considered a wireless energy transfer full-duplex IoT system, where an IoT source (S) equipped with a singleantenna transmits short packets to a robot destination (D) with M-antennas through SPC channels with the assistance of K FD relays [TeX:] $$\mathcal{R}=\left\{\mathrm{R}_k\right\}_{k=1}^K.$$ At each relay, two isolated antennas are installed for transmission and reception, employing selfinterference cancellation (SIC) to mitigate loop interference. However, RF impairments can prevent the complete elimination of residual self-interference [20], [21]. Being energyconstrained IoT devices, the source and relays use the harvested energy from an N-antenna power beacon to send their packets under the assumption of perfect CSI.

We denote by [TeX:] $$g_{n, \mathrm{S}}, g_{{n, \mathrm{R}_k}},$$ with [TeX:] $$n \in\{1, \cdots, N\}, h_{\mathrm{S}, \mathrm{R}_k}, \text{ and } h_{\mathrm{R}_k, \mathrm{D}_m}, \text{ with } m \in \{1, \cdots, M\},$$ the channel coefficients in the system from the n-antenna at power beacon to [TeX:] $$\mathrm{S} \text{ and } \mathrm{R}_k,$$ from [TeX:] $$\mathrm{S} \text{ to } \mathrm{R}_k,$$ and from [TeX:] $$\mathrm{R}_k \text{ to } \mathrm{D}_m,$$ respectively. Each channel coefficient is modeled as [TeX:] $$f=\sqrt{G_f} \tilde{f}$$ with [TeX:] $$\tilde{f} \in\left\{g_{n, \mathrm{~S}}, g_{n, \mathrm{R}_k}, h_{\mathrm{S}, \mathrm{R}_k}, h_{\mathrm{R}_k, \mathrm{D}_m}\right\},$$ where [TeX:] $$G_f$$ represents the large-scale path loss and [TeX:] $$\tilde{f}$$ represents the small-scale fading following Nakagami-m fading model. Thus, the channel gain [TeX:] $$|f|^2$$ follows the gamma distribution with parameter [TeX:] $$\lambda_f$$. The large-scale path loss can be presented as [TeX:] $$G_f=\sigma^{P L}\left(d / d_0\right)^{-P L},$$ where [TeX:] $$d, \mathrm{PL}, d_0 \text {, and } \sigma^A$$ are the distance between two nodes, the path loss exponent, the reference distance, and the power attenuation at [TeX:] $$d_0,$$ respectively.

Different from long packet communications, which are usually based on the asymptotic Shannon theory, short packets with new coding rates require a different performance analysis approach. Short packets are subject to finite-blocklength effects, which means that the capacity rate and error probability depend on the blocklength and the channel conditions. In particular, the capacity rate expression using SPCs is a very complicated function of the transmission power and channel uses [8], making the design and evaluation of SPCs under the full-duplex scenarios with WET much more computationally challenging [22].

The system operation is divided into the WET and packet transmission (PT) phases. The period for the WET is [TeX:] $$c_E T$$ while that for PT is [TeX:] $$\left(c_T-c_E\right) T,$$ where [TeX:] $$c_E, c_T \text {, and } T$$ denote the number of channel uses (CUs) for the WET phase, the total number of CUs, and duration of each CU, respectively. In the WET phase, each IoT device will wirelessly harvest energy from the power beacon for the data transmission. We denote by [TeX:] $$\eta \in(0,1) \text { and } P \text {, }$$ the energy conversion efficiency and the transmit power of each antenna at the PB, respectively. The energy harvested at each IoT device X, with [TeX:] $$\mathrm{X} \in\left\{\mathrm{S}, \mathrm{R}_k\right\},$$ from PB can be expressed as

where [TeX:] $$g_{n, \mathrm{X}}$$ is the channel coefficient between PB and X. After performing WET as in (1) during the period of [TeX:] $$c_E T,$$ the transmit power of [TeX:] $$\mathrm{R}_k$$ over the PT period of [TeX:] $$\left(c_T-c_E\right) T$$ can be calculated as [TeX:] $$P_{\mathrm{X}}=\frac{E_{\mathrm{X}}}{\left(c_T-c_E\right) T},$$ which yields

By using the FD transmission, the self-interference at [TeX:] $$\mathrm{R}_k$$ will include not only the additive white Gaussian noise, denoted by [TeX:] $$n_{\mathrm{R}_k},$$ but also the RSI signal caused by the incomplete cancellation of the undesired interference, denoted by [TeX:] $$u_{\mathrm{RSI}} \text {, }$$ which can be written as follows:

where [TeX:] $$u_{\mathrm{RSI}}$$ is the RSI signal at [TeX:] $$\mathrm{R}_k, x_{\mathrm{S}} \text { and } x_{\mathrm{R}_k}$$ denote the signals at the [TeX:] $$\mathrm{S} \text { and } \mathrm{R}_k \text {, }$$ respectively.

The RSI in (3) can affect the quality and reliability of the desired signal and thus needs to be modeled and mitigated properly. Therefore, we consider [TeX:] $$u_{\mathrm{RSI}}$$ with two models such as RSI-I and RSI-II as follows:

· (I) RSI-I model: [TeX:] $$u_{\mathrm{RSI}}$$ in (3) is zero-mean, additive and Gaussian following [TeX:] $$\mathcal{C N}\left(0, l \sigma_k^2\right),$$ where l reflects the quality of the SIC technique and [TeX:] $$\sigma_k^2$$ is the RSI variance [20], [23]. The smaller value of l indicates a better selfinterference cancellation quality. The RSI-I model has zero-mean, which implies that there is no bias or offset in the RSI model since its average value is zero. It is Gaussian means that the RSI-I model has a normal distribution with zero-mean and variance of [TeX:] $$l \sigma_k^2$$. If the interference is not Gaussian, the RSI-I model can be seen as the worst-case scenario in terms of achievable rate [24]. The Gaussian assumption might hold in reality due to the various sources of imperfections in the cancellation process (i.e., due to the central limit theorem). Moreover, it is additive indicating that it is added to the desired signal as the external noise at the receiver. This implies that the RSI can be treated as an independent noise source and its effect can be analyzed using standard techniques such as signal-to-interference-plus-noise ratio (SINR) or error probability [25].

· (II) RSI-II model: [TeX:] $$u_{\mathrm{RSI}}$$ is an independent Nakagami-m random variable and can be presented as [TeX:] $$u_{\mathrm{RSI}}=\sqrt{P_{\mathrm{R}}} h_{\mathrm{RR}_k},$$ where [TeX:] $$h_{\mathrm{RR}_k},$$ is the SI channel at [TeX:] $$\mathrm{R}_k.$$ Thus, [TeX:] $$\left|h_{\mathrm{RR}_k}\right|^2$$ follows gamma distribution with spread parameter [TeX:] $$\lambda_R$$ and shape parameter [TeX:] $$m_3.$$ The shape parameter influences the shape of the distribution, while the spread parameter controls the scale of the distribution. Modeling RSI as a Nakagami-m fading channel can capture various fading scenarios, but it also introduces high mathematical complexity due to the involvement of Gamma functions and incomplete Gamma functions. Therefore, several existing works used approximations or bounds to simplify the analysis, but these approaches may lose accuracy [26].

From (3), the instantaneous SNR at relays is expressed as

where [TeX:] $$\kappa \triangleq \eta c_E /\left(c_T-c_E\right), \psi=P / \sigma_n^2, z_{\mathrm{RSI}}=\sigma_k^2 / \sigma_n^2$$ for RSI-I model, and [TeX:] $$z_{\mathrm{RSI}}=P_{\mathrm{R}}\left|h_{\mathrm{RR}_k}\right|^2 / \sigma_n^2$$ for RSI-II model. The maximal ratio combining is used at the robot destination, and the instantaneous SNR at [TeX:] $$\mathrm{D}_m$$ can be expressed as

By using the decode-and-forward protocol, an FD relay is chosen to achieve the highest end-to-end (e2e) SNR, i.e.,

The criterion (6) will improve reliable packet communication in WFINs under short blocklength transmission and imperfection of the SI channels. The e2e SNR of the considered network can be expressed as

Based on the e2e SNR, the average BLER, throughput, reliability, and latency will be analyzed in the following section.

III. PERFORMANCE ANALYSIS

1) BLER Analysis: Assuming finite blocklength transmission through channel [TeX:] $$i \in\left\{\mathrm{SR}_k, \mathrm{R}_k \mathrm{D}_m\right\},$$ the S transmits τ information bits (message size) encoded into a packet over the blocklength [TeX:] $$t \gt 100.$$ The received SNR and the channel coding rate are denoted as [TeX:] $$\gamma_i \text { and } r_i \triangleq \tau / t,$$ respectively, the average BLER can be calculated as

where [TeX:] $$Q^{-1}(.)$$ denotes the inverse Gaussian Q-function, [TeX:] $$V(x) \triangleq\left(1-(1+x)^{-2}\right)\left(\log _2 e\right)^2$$ is the channel dispersion, and [TeX:] $$C(x) \triangleq \log _2(1+x)$$ is the Shannon capacity. The involvement of a complex Gaussian Q-function in (8) makes the exact closed-form expression for the system BLER quite complex. A tight approximation method for Q-function will be used to solve integral in (8), where the Q-function is approximated as [TeX:] $$Q\left(\frac{C\left(\gamma_i\right)-r_i}{\sqrt{V\left(\gamma_i\right) / t}}\right) \approx \Xi\left(\gamma_i\right) \text {, with } \Xi\left(\gamma_i\right)$$ being expressed as [8], [16], [17]

where [TeX:] $$\delta_i=\left[2 \pi\left(2^{2 r_i}-1\right)\right]^{-1 / 2}, v_i=\zeta_i-1 /\left(2 \delta_i \sqrt{t}\right),$$ [TeX:] $$\zeta_i \triangleq 2^{r_i}-1 \text {, and } u_i=\zeta_i+1 /\left(2 \delta_i \sqrt{t}\right) \text {. }$$ Using this feasible approximation, [TeX:] $$\varepsilon_i$$ can be calculated as

where [TeX:] $$F_{\gamma_i}(.)$$ is the cumulative distribution function (CDF) of [TeX:] $$\gamma_i$$ and step (a) is due to the integration by parts. Making Ruse of the first order Riemann integral approximation, i.e., [TeX:] $$\int_x^y f(z) d z=(y-x) f\left(\frac{x+y}{2}\right),$$ for (10), which yields

Next, we find the CDF of [TeX:] $$\gamma_i$$ of the considered FD system based on the criterion (6).

Theorem 1. The CDFs of [TeX:] $$\gamma_{\mathrm{SR}_k} \text { and } \gamma_{\mathrm{R}_k \mathrm{D}_m}$$ over Nakagami-m fading channels with RSI-I model are expressed, respectively, as

Proof: Please see Appendix A.

From (11), (12), and (13), the e2e BLER of the considered system can be expressed as

Based on Theorem 1, when the fading severity parameters of EH links and the number of antennas at the PB increase, their product inside the Bessel function is large, making the CDFs of [TeX:] $$\gamma_{\mathrm{SR}_k} \text { and } \gamma_{\mathrm{R}_k \mathrm{D}_m}$$ become small. However, these expressions are still in the Bessel function, and difficult to draw some insights into the system behavior directly. Therefore, we will perform the approximation analysis for the average BLER in the following Theorem.

Theorem 2. The approximated CDFs of [TeX:] $$\gamma_{\mathrm{SR}_k} \text { and } \gamma_{\mathrm{R}_k \mathrm{D}_m}$$ over Nakagami-m fading channels with RSI-I model are expressed, respectively, as

where

Proof: Please see Appendix B.

In Theorem 2, the CDFs of [TeX:] $$\gamma_{\mathrm{SR}_k} \text { and } \gamma_{\mathrm{R}_k \mathrm{D}_m}$$ are shown as exponential functions of fading severity parameters [TeX:] $$m_1, m_2,$$ and the channel uses for EH through parameters [TeX:] $$\Delta_1, \text{ and } \Delta_2.$$ The number of antennas at D and PB is proportional to the CDFs of [TeX:] $$\gamma_{\mathrm{SR}_k} \text { and } \gamma_{\mathrm{R}_k \mathrm{D}_m}$$. The positions of PB and FD relays expressed through parameters [TeX:] $$\lambda_{E, 1} \text { and } \lambda_{E, 2}$$ also considerably influence the CDFs of [TeX:] $$\gamma_{\mathrm{SR}_k} \text { and } \gamma_{\mathrm{R}_k \mathrm{D}_m}$$, which will be extensively evaluated in the numerical results Section. Similar to (11), the approximation of BLER is expressed as

The e2e BLER approximation of the considered system can be expressed as

Compared to the e2e BLER in (14), the product of two error probabilities, [TeX:] $$\varepsilon_{\mathrm{SR}_k}^{\mathrm{ap}}\left(\zeta_{\mathrm{SR}_k}\right) \text { and } \varepsilon_{\mathrm{R}_k \mathrm{D}_m}^{\mathrm{ap}}\left(\zeta_{\mathrm{R}_k \mathrm{D}_m}\right) \text {, }$$ is very small, approximating [TeX:] $$10^{-4}-10^{-6} \text {, }$$ when [TeX:] $$\psi \rightarrow \infty$$ in (22); thus, this product can be ignored.

2) Throughput Analysis: The system throughput [TeX:] $$\mathrm{R}_{th}$$ can be formulated by considering the delay-limited transmission with a fixed data transmission rate in bits per channel use (BPCU) as

where [TeX:] $$\varepsilon \in\left\{\varepsilon_{e 2 e}, \varepsilon_{e 2 e}^{\mathrm{ap}}\right\},$$ and [TeX:] $$\left(c_T-c_E\right) / c_T$$ presents the fraction of effective communication time and total transmission time.

Regarding the RSI-II model, the analysis of average BLER is quite intricate since it involves four random variables leading to the integral of the product of two Bessel functions. This challenge motivates us to build a new framework relying on deep learning design for performance evaluation. The new DL framework is projected to estimate the average BLER and throughput of the FRS scheme with the RSI-II model via a short-time inference process, which contributes to reducing the deployment time for network designing, planning, and monitoring in practice. The proposed DL framework will be presented in the next section.

IV. PROPOSED CNN DESIGN FOR BLER AND THROUGHPUT PREDICTION

We propose a deep CNN design to predict the average BLER and throughput of the considered system with the RSI-II scenario. The proposed CNN includes an input layer with a size of 1 × 11 for processing network parameters. The input variables consist of the number of relays (K), the number of antennas at PB (N), the number of antennas at D (M), the positions of relay [TeX:] $$\left(x_{\mathrm{R}} \text { and } y_{\mathrm{R}}\right) \text {, }$$ the position of [TeX:] $$\mathrm{PB~}\left(x_{\mathrm{PB}} \text { and } y_\mathrm{PB}\right) \text {, }$$ the transmit power of [TeX:] $$\mathrm{PB~}\left(P_{\mathrm{PB}}\right) \text {, }$$ the channel uses for EH [TeX:] $$\left(c_E\right),$$ the information bits τ, and the target data rate [TeX:] $$\left(R_{t h}\right).$$ We have chosen the above system parameters because they influence significantly the performance of the proposed WFINs. Over 160, 000 samples are generated for training and testing based on input variables in Table I.

An input sample to the deep network for BLER and throughput estimation can be expressed as follows:

The proposed CNN architecture is illustrated in Fig. 2, where the input size is specified as 1 × 11, representing the wireless system variables outlined in Table I. Following this, a convolutional (conv) layer with a kernel size of 1 × 1 is employed, succeeded by a batch normalization (bn) layer and a swish activation (swish) layer to aggregate features across three color channels. To capture more diversified features, we design an MRF block with three parallel connections, each of which is specified by a conv layer with the kernel size of 1 × 1, followed by a conv layer with different one-dimensional asymmetric kernels of sizes such as 1 × 3, 1 × 2, and 1 × 1. The bn and swish layers are used in each connection, where their outputs are subsequently collected along the depth dimension using a depth-wise concatenation (concat) layer to merge the outputs. To reduce computational expenses when the network goes deeper, a max-pooling layer (maxpool) with a stride of (1, 2) and a pool size of 1 × 2 is employed. This operation effectively reduces the spatial dimensions of the feature maps by half [27].

In order to highlight the significant features obtained from the previous layers, multiple processing P−Blocks are linked together in a serial manner to extract intrinsic features as the correlations between system variables, which are interleaved by a (maxpool) layer to decrease the dimension of feature maps. Each P−Block has three conv layers and an elementwise addition add layer. These layers are connected by following the structure of residual blocks of ResNet [28], which has been broadly applied to address the vanishing gradient problem in numerous learning tasks of wireless communications [29]. To further collect meaningful features, the outputs are then traveled to the next three P − Blocks, ended with a global average pooling (gap) layer, and followed by two fully connected (fc) layers to summarize spatial features in each map and estimate continuous values of BLER and throughput. The loss function of the regression problem measures the discrepancy between the predicted value, denoted by [TeX:] $$\mathbf{y}_{(t)},$$ and true target value, denoted by [TeX:] $$\tilde{\mathbf{y}}_{(t)},$$ what can be presented as [9]

where [TeX:] $$S_t$$ is the number of samples for training. The weights and biases of the CNN have been updated during the backpropagation process using the Adam optimizer.

Once the offline training process is finished, the resulting CNN model, comprising weights and biases, can be succinctly represented as a concise mapping function denoted as [TeX:] $$\mathcal{F}(.).$$ Typically, a well-trained CNN is capable of providing highly precise and real-time predictions. Thus, the obtained CNN model is used to predict the BLER and throughput values when any new data becomes available at the input. Particularly, each input sample, sorted into vector [TeX:] $$\mathbf{x},$$ at the input of CNN, the CNN will produce the respective output values of BLER and throughput, also arranged into vector [TeX:] $$\tilde{\mathbf{y}} \triangleq\left[\begin{array}{cc} \tilde{\varepsilon}_{e 2 e} & \tilde{\tau}_{e 2 e} \end{array}\right],$$ i.e.,

Through this short-time inference procedure, the CNN model can predict both the BLER and throughput within a short time.

V. NUMERICAL RESULTS AND CNN EVALUATION

In this section, we provide some illustrative examples for system performance evaluation. Monte Carlo simulations and CNN prediction results of the BLER and throughput are presented to verify our designed approach. The [TeX:] $$\mathrm{S}, \mathrm{D}, \mathrm{R}_k \text {, and } \mathrm{PB}$$ are located, respectively, at (0, 0), (15, 0), (10, 0), and (10, 3) on the Euclidean plane. We set [TeX:] $$d_0=1 \mathrm{~m}, \sigma^{P L}=-30 \mathrm{~dB} \text {, }$$ the shape parameters [TeX:] $$m_1=m_2=m_3=2, \mathrm{PL}=3.6$$ Unless otherwise specified, the remaining simulation parameters are set as follows. K = 3, M = 3, N = 3, [TeX:] $$c_T=900 \text {, }$$ the number of bits 600, [TeX:] $$c_E=400, \eta=0.8, l=0.01, I=12,$$ and the normalized noise variance [TeX:] $$\sigma^2=1.$$ The whole dataset is divided by 80% and 20% for training and testing, respectively. The CNN is trained for 60 epochs with random weights initialization, a gradient decay factor of 0.96, and the adaptive moment estimation optimizer. At the beginning of the training phase, the learning rate is initial with [TeX:] $$10^{-3}$$ and it is decreased by 90 % after 20 epochs.

For the multiple output regression, the RMSE is used for the performance evaluation of the CNN design in predicting the BLER and throughput. RMSE measures the deviation between the true target and predicted values across the whole test set. By considering the samples in the test set, the RMSE can be calculated as

Fig. 3 illustrates the effects of the percentage of training data on different deep models. The first model is a DNN with five fully-connected layers, each of which consists of 256 neurons. The second model is a traditional CNN with a straightforward architecture involving six conv layers (utilizing 1 × 3 with the numbers of filters of 64, 64, 128, 128, 256, and 256), followed by activation ReLU layers and interleaved by max-pooling layers. As can be observed, the proposed CNN achieves the lowest RMSE while the deep neural network gets the highest RMSE when the percentage of training data increases. The reason is that the DNN model with a simple structure is not able to estimate a moderate-to-high-dimensional dataset, leading to the lowest performance. Although the traditional CNN performs BLER and throughput prediction more accurately than DNN, it is limited by multiple input correlations and vanishing gradients if compared with the proposed CNN model. Furthermore, it is realized that the RMSE of the CNN models decreases more significantly than that of the DNN ones as the amount of training data increases. DNNs are usually more prone to overfitting than CNNs, especially when dealing with large amounts of data, because DNNs have more parameters, thus leading to the model memorizing the training data rather than learning generalizable patterns. Besides the ability to capture local patterns, reduced dimensionality, and parameter efficiency, the proposed CNN model can map the original dataset into a higher dimensional space with complex patterns, resulting in error reduction and estimation accuracy improvement. Due to the lowest RMSE of the proposed CNN, it will be used for predicting the average BLER and throughput in the next simulations.

As shown in Fig. 4, the FD scheme with the RSI-I model delivers the best performance compared to the half-duplex scheme, highlighting the benefits of full-duplex transmission. Furthermore, the FRS scheme, which involves multiple relays in the relay selection process, achieves higher diversity and reliable transmissions than the random relay selection (RS) scheme, resulting in lower BLER. Furthermore, the Monte- Carlo simulations of the FRS scheme with the RSI-I model demonstrate excellent agreement with the analytical ones, affirming the correctness of the developed analysis. The approximation results are also tightly matched to the analysis one, validating our approximation approach. Moreover, the CNN prediction results of the RSI-II model closely align with the simulation ones, emphasizing the efficacy of CNN design in Section IV.

Fig. 5 reveals the effects of the number of antennas at D on the system BLER with different positions of FD relays. As can be observed, the number of antennas at D significantly affects the average BLER when the relay is closer to the source, but it does not affect the system BLER when the relay is farther from the source. The reason is that the e2e SNR is dominated by the instantaneous SNR at the relay, [TeX:] $$\gamma_{\mathrm{SR}_k},$$ in (6) when the distance between [TeX:] $$\mathrm{S} \text { and } \mathrm{R}_k$$ is large, and increasing M does not improve the e2e SNR. On the other hand, when the distance from source to relay is small, leads to the domination of instantaneous SNR at [TeX:] $$\mathrm{D}, \gamma_{{\mathrm{R}_k}{\mathrm{D}_m}},$$ in (6), and the e2e SNR is now a function of M.

Fig. 6 shows the significant effect of the number of antennas at the PB on the average BLER of FRS scheme with the RSI-I model. When the position of PB is close to the network, or when a greater number of antennas are deployed at the PB, the source and FD relays have more opportunities to harvest energy, which improves the average BLER. Moreover, when N is large, more degrees of freedom are contributed to the system. Thus, the improvement of the average BLER becomes significant when the PB is located near the network.

Fig. 7 shows the increase of throughput of all schemes as ψ is large. The [TeX:] $$\mathrm{S} \text { and } \mathrm{R}_k$$ will have more chances of harvesting sufficient energy from the PB when ψ is large, which results in the improvement of throughput. Again, the FRS scheme with the RSI-I model achieves the highest throughput, while the HD scheme yields the lowest performance. Furthermore, the CNN-based scheme achieves the same throughput as the FD RSI-II model, showing an exceptional CNN design approach.

Fig. 8 shows the effects of the number of FD relays and [TeX:] $$R_{t h}$$ on the throughput of FRS scheme. When [TeX:] $$R_{t h}$$ is increased, the average throughput of FRS scheme with RSI-I and RSI-II models considerably improves. When K is large, the system with the RSI-I model achieves almost the same throughput as its counterpart with the RSI-II one. It also shows that the CNN prediction of the RSI-II model is matched with the simulation one. This shows the correctness of our CNN design framework.

The reliability and latency of short-packet communications [6], [8], [10] can be calculated as

where [TeX:] $$T=3 \mu s$$[30].

In Fig. 9, it can be observed that the message containing 256 bytes demonstrates greater reliability and lower latency compared to longer messages (i.e., those consisting of 512 or 1024 bytes). The message with the size of 512 byte can be encoded into packets over 6000 CUs to increase the reliability level to 90%, as shown in Fig. 9(b). However, its latency amounts to approximately 20 seconds, as indicated in Fig. 9(a), which surpasses the maximum allowed latency for URLLC services and applications [4]. On the other hand, the 1024-byte message exhibits a latency of 80 seconds and a reliability of less than 40% with the utilization of 6000 channel uses, failing to meet the stringent requirements for both latency and reliability. Consequently, it can be concluded that longer messages are unsuitable for supporting URLLCs in IoT networks and for facilitating low-delay communications in factory automation.

Finally, the running time of the throughput prediction is evaluated in Table II. To obtain the exact result at [TeX:] $$\psi=20 \mathrm{~dB},$$ each sample of Monte-Carlo is averaged over [TeX:] $$3 \times 10^6$$ trials. In all scenarios, when parameters K,N, M, and [TeX:] $$c_E$$ of the considered system increase, the CNN consistently predicts the BLER and throughput with a short period of time less than 6ms. In contrast, the running time of the simulation method grows rapidly with such parameters, taking over 24 s for the last case. Numerical results show the outstanding capability of the CNN-based scheme to effectively handle IoT networks with large-scale configurations.

VI. CONCLUSIONS

We studied the performance analysis of FEINs, where different RSI models for full-duplex IoT devices were considered. The full relay selection was proposed to achieve the highest e2e SNR of the system with the RSI-I model, based on which the closed-form expression for the average BLER and throughput were derived. The respective approximation analysis was also carried out to provide deeper insights into the system behavior. Aimed real-time configurations, a CNN architecture was designed for estimating the average BLER and throughput when mathematical analyses of the system with the RSI-II model became intricate. Simulation results demonstrated that the CNN design-based estimation scheme attained the exact average BLER and throughput values of the FRS RSI-II model. Our future works will investigate the FD-enabled multi-hop SPC incorporating new network paradigms such as reflecting surface and edge computing to enable distributed learning for future IoT systems.

APPENDIX A

PROOF OF THEOREM 1

We first consider (4) with RSI-I model, i.e., [TeX:] $$z_{\mathrm{RSI}}=\sigma_k^2 / \sigma_n^2,$$ let [TeX:] $$X=\sum_{n=1}^N\left|g_{n, \mathrm{~S}}\right|^2,$$ the CDF of [TeX:] $$\gamma_{\mathrm{SR}_k}$$ can be calculated as

where [TeX:] $$F_X(.) \text { and } f_{\left|h_{5, R_k}\right|^2}(.)$$ denote the CDF of X and PDF of [TeX:] $$\left|h_{\mathrm{S}, \mathrm{R}_k}\right|^2,$$ respectively. Over Nakagami-m fading channels, [TeX:] $$F_X(.) \text { and } f_{\left|h_{5, R_k}\right|^2}(.)$$ can be expressed, respectively, as

Plugging (31) and (32) into (30) and after some manipulations, we obtain as

By applying [31, Eq. (3.471.9)], the integral of (33), denoted by [TeX:] $$T_1$$, can be calculated as

By substituting (35) into (33) and after some manipulations, we obtain the CDF of [TeX:] $$\gamma_{\mathrm{SR}_k}$$ as shown in (12).

Next, we consider (5), let [TeX:] $$Y=\sum_{m=1}^M\left|h_{\mathrm{R}_k, \mathrm{D}_m}\right|^2$$ and [TeX:] $$Z=\sum_{n=1}^N\left|g_{n, \mathrm{R}_k}\right|^2,$$ the CDF of [TeX:] $$\gamma_{{\mathrm{R}_k} \mathrm{D}_m}$$ can be calculated as

The CDF of Z and PDF of Y in (37) can be expressed, respectively, as

Next, by plugging (38) and (39) into (37), we obtain as

Similar to steps to calculate (33), the integral in (40), denoted by [TeX:] $$T_2,$$ can be calculated by applying [31, Eq. (3.471.9)] as follows:

Plugging (42) into (40), and after some manipulations, the CDF of [TeX:] $$\gamma_{{\mathrm{R}_k} \mathrm{D}_m}$$ can be obtained as (13). The proof is complete.

APPENDIX B

PROOF OF THEOREM 2

By applying Lemma 1 in [8], [TeX:] $$K_{b-c}(x)$$ can be equivalently approximated as

where I presents the constrained upper limit of the summation, and it can be appropriately configured to meet the desired accuracy level. By applying (43), the CDFs in (12) and (13) of Theorem 1 can be approximated in compact forms as in (15) and (16) of Theorem 2, respectively.