Enhancing wireless communication systems through large intelligent surfaces: a performance analysis of LIS-assisted NOMA networks

Abstract

The integration of large intelligent surfaces (LIS) with non-orthogonal multiple access (NOMA) networks has emerged as a promising solution to enhance the capacity and coverage of wireless communication systems. In this study, we analyse the performance of a NOMA network with the assistance of LIS. We propose a system model where a base station (BS) equipped with a LIS serves multiple users. The LIS consists of many passive elements that can influence the wireless channel by adjusting the reflection coefficients. We consider a downlink scenario where the BS transmits to multiple users simultaneously using NOMA, and the LIS helps to improve the signal quality and coverage. We additionally evaluate the efficiency of the suggested LIS-assisted NOMA network. In addition, we evaluate the efficiency of the LIS-assisted NOMA network in comparison to conventional NOMA systems that do not utilize LISs. The findings indicate that the LIS has a notable impact on enhancing the system's performance in terms of diversity gain, probability of error, and pairwise error probability (PEP). Moreover, the suggested LIS-assisted NOMA network is shown to be superior to conventional NOMA systems through comparison. These findings offer useful insights into the performance analysis of LIS-assisted NOMA networks. They also serve as inspiration and motivation for future research and development in this new subject, with the potential to revolutionize wireless communication systems.

Introduction

Implementing fifth-generation (5G) mobile networks aims to establish a linked society by enabling many forms of communication, including interactions between computers, humans, and other entities. However, these emerging models have disadvantages, such as increased energy consumption and reduced delay, when establishing connections between several data-intensive heterogeneous devices with varying requirements.

To overcome these challenges, it is imperative to have cutting-edge technologies that can effectively manage high data rates, numerous connections, and minimal latency. Non-orthogonal multiple access (NOMA) has emerged as a vital technology for 5G networks because of its advantages, such as reduced latency, improved connection reliability, and remarkable energy and spectrum economy. However, wireless channels are very unpredictable and susceptible to such problems as signal loss, obstruction, and absorption, leading to inevitable instances of multipath fading. While MIMO approaches can address these problems, they do result in reduced energy efficiency, which poses a significant challenge for the development of 5G and future wireless networks. In recent times, LIS technology has become increasingly popular because of its energy and spectrally-efficient nature. It enables users to have precise control over the wireless environment according to their preferences. LIS can address these constraints by employing reflecting elements (REs), which control incoming electromagnetic waves by reflection, refraction, absorption, steering, focusing, and polarization.

The integration of Localized Information Sharing (LIS) into Non-Orthogonal Multiple Access (NOMA) systems has produced promising outcomes. This integration enables the effective implementation of NOMA systems and the ability to tailor the propagation environment to prioritize individual users. Furthermore, LIS can modify user priority based on system requirements rather than solely dependent on the unpredictable wireless propagation environment. The integration of the LIS has been observed by the research community, as it can enhance NOMA systems. 6G communications are currently in the nascent phase of development, with no established standards or specifications as of yet. Nevertheless, specialists predict that 6G will provide accelerated data speeds, increased bandwidth, reduced latency, and enhanced dependability as compared to 5G. The implementation of 6G technology would enable operation in the terahertz (THz) frequency bands, resulting in improved data rates. Additionally, it would leverage AI-driven communication to optimize network efficiency. Furthermore, 6G would explore the potential of quantum communication for establishing secure channels. Moreover, it would utilize holographic technology to provide more immersive experiences. Furthermore, 6G would focus on improving energy efficiency and facilitating multiple connections across various communication channels. To summarize, 6G communication is an upcoming wireless technology that is currently being developed with the goal of delivering superior performance in comparison to 5G. The incorporation of Local Information Sharing (LIS) into Non-Orthogonal Multiple Access (NOMA) systems and the investigation of prospective functionalities for 6G communication signify notable progress in wireless networks.

Related work

The integration of LIS with NOMA networks is a rapidly growing field of study that combines wireless communication and intelligent surface technologies. In recent years, researchers have examined many elements of NOMA networks and intelligent surface applications. This previous work has provided the foundation for the inquiry reported in this article.

Multiple studies have examined NOMA's advantages in wireless communication systems, highlighting its ability to improve spectral efficiency and support concurrent usage by multiple users. NOMA enables the sharing of time–frequency resources across numerous users, utilizing the power domain for user multiplexing. Research in NOMA has focused on diverse aspects, including resource allocation strategies, power control mechanisms, and decoding techniques, to optimize the overall system performance^1,2. The work in³ addresses challenges in network communications due to the evolving mobile Internet and Internet of Things. It introduces a 5G-oriented NOMA technology, exploring user pairing and power allocation algorithms. Simulation results reveal that the traversal search pairing scheme exhibits slightly better throughput than grouping, and specific power allocation algorithms outperform the flower pollination algorithm in various scenarios. The paper⁴ proposes a NOMA-based cooperative cellular system with a relay, showcasing its comparable performance to conventional multiple access in terms of diversity order and sum rate through the outage and ergodic sum rate analyses. Numerical simulations highlight NOMA's potential for increased spectral efficiency and user fairness by serving more users simultaneously. Research in⁵ demonstrates the application of NOMA in 6G networks, investigating the benefits and limitations of Successive Interference Cancellation (SIC) based on quality-of-service and channel state information. Additionally, it investigates how NOMA, particularly in large-scale interconnections, can fulfil the performance requirements of 6G networks. Unlike orthogonal multiple access (OMA), non-terrestrial NOMA networks supported by RIS exhibit greater outage resilience. The work in⁶ proposes using discrete wavelet transform (DWT) for NOMA pulse shaping instead of the traditional FFT-based method, showing that wavelet-based NOMA (WNOMA) has a lower bit error rate (BER) in additive white Gaussian noise. The study in⁷ introduces a downlink NOMA-based coordinated direct and relay system with one cell-center user and multiple cell-edge users using a decode-and-forward relay in full-duplex (FD) or half-duplex (HD) modes. It finds that FD outperforms HD at low SNR, while HD is better at high SNR, with mutual interference having a significant impact on performance. The research in⁸ investigates a UAV-enabled massive MIMO NOMA full-duplex two-way relay (TWR) system with low-resolution ADCs/DACs for multi-pair ground users, deriving expressions for sum spectral and energy efficiency (SE/EE) under imperfect conditions. Results show that increasing antenna scale and proper power scaling enhance performance, UAV height adjustment benefits SE, and the SE/EE trade-off is optimized by selecting the correct number of ADC/DAC quantization bits.

However, the use of reconfigurable intelligent surfaces (RIS) has gained significant attention because of its ability to modify and control wireless propagation conditions⁹. The work in¹⁰ offers a detailed summary of the latest research progress in RIS, highlighting its ability to enhance the performance of future wireless communication networks by adjusting to changing propagation conditions. This analysis focuses on critical issues that affect the profitability of future deployments. It includes topics such as practical hardware design, artificial intelligence techniques, system models, use cases, and strategies for improving the physical layer. These surfaces consist of several passive parts and can dynamically change reflection coefficients, which can effectively impact wireless channels^11,12. Prior research has investigated the utilization of LIS in many communication settings, such as improving coverage, signal quality, and overall system capacity in large MIMO systems and millimeter-wave communication. The study in¹³ is a comprehensive examination of the functioning, enhancement, and evaluation of reconfigurable intelligent surfaces. It delves into many optimization frameworks that aim to achieve objectives like energy efficiency, sum rate, secrecy rate, and coverage. RIS is emerging as a promising candidate for supporting 6G wireless networks¹⁴. The research in¹⁵ delves into the forefront of 6G wireless communication, highlighting innovative materials, radio-frequency architectures, and transformative communication paradigms. RISs are identified as pioneering 6G technology, offering programmable artificial structures capable of manipulating electromagnetic fields for diverse networking goals. The paper categorizes advancements in RIS hardware, unit element modeling, and signal propagation, with a focus on channel estimation for optimized RIS integration. Additionally, it explores the relevance of RIS in current wireless standards, shedding light on ongoing and future standardization efforts for RIS technology and its empowered networking approaches. Further emphasizing the potential of reconfigurable intelligent surfaces (RISs) to enhance wireless network performance, Hassouna et al.¹⁶ covers principles, performance analysis, and challenges in integrating passive components efficiently. The study compares channel estimation for different RIS types and deployment scenarios, concluding with proposed future research areas for RIS-aided wireless communication systems. To address challenges posed by internet-dependent 5G technology, Bariah et al.¹⁷ proposes implementing Internet of Things (IoT)-based 6G technology. It develops a closed-form formula that accurately determines the coverage probability for nearby and distant users in a NOMA system with RIS support. The work in¹⁸ explores the role of RISs in wireless communication, comparing RIS technology with the SISO case and evaluating performance in terms of data rate and energy efficiency. The study investigates the impact on wireless sensor networks, showcasing spectral efficiency gains with sufficiently large RIS sizes. Key open issues for maximizing RIS benefits in wireless communications and networks are discussed.

Innovatively, the research in¹⁹ proposes a reconfigurable reflective meta surface with integrated sensing capabilities, aiming to enhance wireless communication and power transfer by providing the reflective surface with prior knowledge of the propagation environment. This technology, by modifying tunable meta-atoms, can sample incident waves and detect properties such as the angle of arrival, potentially reducing the number of required sensors through tunable multiplexing. The proposed technology holds promise for applications in wireless communications, wireless power transfer, RF sensing, and intelligent sensors. Addressing the challenges of massive connections and green communication, the study in²⁰ combines RIS and NOMA, focusing on a downlink RIS-aided NOMA system. The evaluation of system performance through adequate capacity (EC) for real-time services highlights NOMA's superiority due to reduced transmission time. In two-way communication scenarios²¹, users assisted by an RIS in Rayleigh fading channels are investigated, considering both reciprocal and non-reciprocal channels. The study in²² introduces a system for serving power-domain NOMA users by optimizing passive beamforming at RISs, demonstrating superior performance over its orthogonal counterpart. In the context of NOMA networks, the performance of imperfect and perfect SIC is investigated in²³ by exploring the IRS application. Numerical results affirm the enhanced performance of IRS-assisted NOMA networks in both ergodic rate and energy efficiency over conventional cooperative communications. The work in²⁴ actively shapes incident signals through passive beamforming to enhance wireless system performance, focusing on an IRS-assisted uplink NOMA system. The proposed NOMA-based scheme outperforms OMA, emphasizing the influence of the reflecting element count on the sum rate.

Recent attention has shifted towards simultaneously transmitting and reflecting Reconfigurable Intelligent Surface (STAR-RIS) assisted NOMA due to its high secrecy ability and near-optimal performance with low complexity²⁵. The study in²⁶ explores the rate performance of a STAR-RIS-aided NOMA system, while²⁷ characterizes the coverage region of STAR-RIS-aided two-user downlink communication systems. Furthermore, Guo et al.²⁸ investigates secrecy energy efficiency maximization in uplink NOMA systems using STAR-RISs mounted on unmanned aerial vehicles (UAVs), and²⁹ proposes joint caching and simultaneous wireless information and power transfer (SWIPT) in STAR-RIS-empowered NOMA systems. Finally, Wen et al.³⁰ examines a STAR-RIS-aided NOMA system with unbalanced users, suggesting strategies for short-range applications and specific power scenarios.

The existing literature sets the stage for our investigation, highlighting the individual merits of NOMA and LIS. Our study extends this knowledge by exploring the synergies between these technologies and uncovering the unique benefits of incorporating LIS in NOMA networks. The insights gained from this research contribute to the evolving landscape of intelligent surface-assisted communication systems, guiding future developments and innovations in this promising field.

Contributions

This research study greatly enhances wireless communication systems by examining the incorporation of LIS with NOMA networks and doing a thorough analysis of its effects on system performance. This study presents the following significant contributions:

• Our suggested comprehensive system architecture has a base station (BS) integrated with a LIS. The BS uses NOMA to serve numerous users simultaneously in a downlink communication situation. The LIS employs a variety of passive devices that may be adjusted to control the reflection coefficients, improving the signal's quality and coverage. This unique system model establishes the basis for examining the interactions between LIS and NOMA in a real-world wireless communication setting.

• We conduct a comprehensive analysis of the proposed network's performance that uses LIS to support NOMA. We analyze essential performance indicators such as diversity gain, likelihood of mistake, and pairwise error probability (PEP). By conducting thorough analysis and simulation experiments, we present a complete understanding of how the LIS impacts the NOMA network's overall performance. This research provides valuable insights for scholars and practitioners in this sector.

• We do a comparative analysis by evaluating the performance of the LIS-assisted NOMA network in comparison to traditional NOMA systems that do not include LIS integration. This study compares the benefits brought by the LIS in terms of increased system capacity, expanded coverage, and improved signal quality. The findings of this comparison demonstrate that the suggested LIS-assisted NOMA network outperforms the alternatives. They contribute significant information to the ongoing discourse on intelligent surface-enhanced communication systems.

• Our research offers valuable insights into the unexplored capabilities and difficulties of LIS-assisted NOMA networks. Through the identification of areas that require improvement and potential optimizations, we provide the groundwork for future research and development in this expanding subject. Our study generates knowledge that encourages researchers to delve deeper into the intricacies of LIS-NOMA integration and its broader implications for the evolution of wireless communication systems, catalysing further exploration in this emerging field.

In summary, this work advances the current understanding of wireless communication systems by investigating the novel fusion of LIS and NOMA networks. The suggested system model, performance analysis, and comparative study add to what is already known in this area. They give useful information and will help guide future research that aims to utilize intelligent surface technologies in wireless communication fully.

The subsequent sections of this paper are structured as follows: "System model" section introduces the system model for the LIS-assisted NOMA network, presenting the associated mathematical expressions. In "Asymptotic diversity order" section, we discuss the pairwise error probability, while "Simulation analysis" section explores the asymptotic diversity order of the LIS-assisted NOMA. In Sect. 5, we substantiate the efficacy of the LIS-assisted NOMA through a detailed simulation analysis. Lastly, Sect. 6 offers the conclusion of this paper, summarizing the significant findings derived from our investigation.

System model

This study focuses on a single LIS that supports a downlink NOMA system and has N stacks of REs, where each stack contains K REs as indicated in Fig. 1¹⁷. Thus, the total number of REs equipped on the LIS is N × K. Data transfer between BS and N NOMA User Equipments (UEs) UE − 1, UE − 2,…, UE − N is the intended usage of this technology. The LIS is distributed into N stacks denoted as S₁, S₂,…, S_N to simplify operations. In this configuration, it is up to each stack to direct the signal to the appropriate user. The BS transmits a composite message comprising N independent messages for N users per the NOMA principles. The power domain multiplexes these communications. Since we assume large-scale fading dominates the system, the distances between users and the LIS mainly influence the channel properties. We focus on the first user, UE − 1, as the weakest link since the user is far from the LIS. On the other hand, the Nth user, UE − N, is closest to the LIS and has the most robust channel. The relationship between the LIS's distances from each user is represented as d₁ > d₂ > … d_n > … > d_N, where d_n is the LIS's distance from the nth user. Users are given power coefficients based on their locations to optimize the power distribution. UEs at a greater distance receive higher power levels than those closer. Remembering that our analytical paradigm may address both near-field and far-field scenarios is crucial. However, in our study, we mainly take the far-field case into account for simplicity. We also assumed that the UE is not in the vicinity of the BS to evaluate the performance of the LIS-assisted NOMA in the absence of the direct link between BS and UE. The distances between the REs in each stack and the UEs who belong to them are also assumed to be the same. The baseband signal received by the nth UE is expressed as:

1

$$r_n=\frac{1}{\sqrt{d_B^{\alpha }{d}_n^{\alpha }}}\sum _{i=1}^N\left[\sum _{k=1}^K{g}_{S_n,k}{h}_{k,n}^{S_n}{a}_{S_n,k}{e}^{j{\theta }_{k,n}}\right]\sqrt{P_i}{x}_i+w_n$$ where, d_B is the distance separation between the BS and LIS and α is the path loss exponent. The small-scale fading channel coefficient between the BS and the kth RE in the nth stack is denoted by $g_{S_n,k}$ , and the fading channel between the kth RE in the nth stack and the nth UE is denoted by $h_{k,n}^{S_n}$ . The stack index of the fading coefficients is discarded for notational convenience throughout the remainder of the paper. Thus, the coefficients $g_{S_n,k}$ and $h_{k,n}^{S_n}$ are denoted as g_k and h_k,n, respectively. We have also assumed no direct line-of-sight (LoS) communication between BS-LIS and LIS–UE pairs. Hence, the fading coefficients g_k and h_k,n are assumed to be circularly symmetric complex Gaussian (CSCG) distributed with zero mean and variance σ². The amplitude reflection coefficient $a_{S_n,k}$ used in the Eq. (1) is normalized to unity. The phase shift of the kth RE in the nth stack is chosen to be θ_k,n. Assuming complete knowledge of the channel phases ${\theta }_{g_k}$ and ${\theta }_{h_{k,n}}$ of g_k and h_k,n, respectively, the phase shift θ_k,n is computed from ${\theta }_{k,n}=-\left({\theta }_{g_k}+{\theta }_{h_{k,n}}\right)$ . Further, P_i and x_i are the power level and transmitting symbol of the ith UE respectively. The power level P_i is equal to ρ_iP_T, where ρ_i is the power coefficient of ith UE and P_T is the total power of the BS, which is normalized to unity. The additive white noise w_n ~ CN(0, W₀) is assumed to be Gaussian distributed and has zero mean and W₀ power spectral density. Thus, the signal received by the nth UE is expressed as:

2

$$r_n=q_n\sum _{i=1}^N\sqrt{P_i}{x}_i+w_n$$ where, the end-to-end channel coefficient, q_n, is represented by:

3

$$q_n=\frac{1}{\sqrt{d_B^{\alpha }{d}_n^{\alpha }}}\sum _{k=1}^K\left|g_k\right|\left|h_{k,n}\right|$$

Figure 1 [Images not available. See PDF.]

LIS-integrated NOMA: modeling with N UEs.

Lower power signals, such as x_n+1, x_n+2,…, x_N, are considered additive noise, whereas higher power signals, such x₁, x₂,…, x_n−1, are cancelled from multiuser interference at the nth UE by employing SIC. As a result, the output signal of the nth SIC receiver is written as:

4

$$r_n^{\prime }=q_n\left(\sqrt{P_n}{x}_n+X_n\right)+w_n$$ with

5

$$X_n=\sum _{i=1}^{n-1}\sqrt{P_i}{\delta }_i+\sum _{j=n+1}^N\sqrt{P_j}{x}_j$$ where, ${\widehat{x}}_i$ represents the detected symbol of ith SIC iteration and ${\delta }_i=x_i-{\widehat{x}}_i$ the output of the ith SIC iteration. Be mindful that δ_i can have either a successful or unsuccessful SIC because δ_i can be equal to 0 or not equal to 0.

Decoding of NOMA system

The process of NOMA decoding is done as follows:

• Decoding Order: The decoding order is determined based on the users' channel conditions, with stronger users decoding and subtracting the signals of weaker users before decoding their own signals. This technique adheres to the concepts of successive interference cancellation (SIC) employed in NOMA systems.

• SIC: This technique involves the more powerful user decoding the message meant for the less powerful user by considering its own signal as noise. After decoding and removing the signal from the weaker user, the stronger user proceeds to decode its own signal.

• The role of LIS in decoding: We elucidate the impact of the LIS on the decoding process by augmenting the signal-to-interference-plus-noise ratio (SINR) for both users. The enhanced reflection coefficients of the LIS enhance the quality of the received signal, hence enabling more efficient successive interference cancellation (SIC).

For instance, in a scenario where User-1 (a user with more strength) and User-2 (a user with lesser strength) both receive overlapping signals from the base station, User–1 would prioritize decoding User-2's message by considering its own signal as interference. Upon deciphering the message from User-2, User-1 proceeds to subtract it from the received signal and subsequently decodes its own message. User-2 autonomously deciphers its own message by utilizing the enhanced Signal-to-Interference-plus-Noise Ratio (SINR) resulting from the transmission supported by the Large Intelligent Surface (LIS).

Pairwise error probability analysis

This section presents a definitive equation for the PEP that offers valuable information about the error rate of the situation being analyzed. The PEP quantifies the probability of an error occurring in a certain scenario, including transmission and detection. Furthermore, the calculation of the PEP serves as the primary element for establishing an upper limit on the BER. The likelihood of incorrectly decoding the symbol ${\overline{x}}_n$ provided that symbol $x_n$ was communicated, where ${\overline{x}}_n\ne x_n,\forall n$ is known as the PEP of the nth UE. The conditional PEP of the nth UE may be stated as follows using the maximum-likelihood (ML) rule.

6

$$Pr\left(Pr{x}_n\to {\overline{x}}_n|q_n\right)=Pr\left({\left|r_n^{\prime }-q_n\sqrt{P_n}{\overline{x}}_n\right|}^2\le {\left|r_n^{\prime }-q_n\sqrt{P_n}{x}_n\right|}^2\right)$$

It should be noted that the PEP expression in Eq. (6) depends on the fading coefficient q_n and the broadcast and detected symbols of every UE. After certain mathematical simplifications and the insertion of (4) into (6), the PEP can be rewritten as follows:

7

$$Pr\left(x_n\to {\overline{x}}_n|q_n\right)=Pr\left({\left|q_n\left(\sqrt{P_n}{\overline{\mathrm{\Delta }}}_n+X_n\right)+w_n\right|}^2\right)\left(\le {\left|q_n{X}_n+w_n\right|}^2\right)$$ where ${\overline{\mathrm{\Delta }}}_n=x_n-{\overline{x}}_n$ .

8

$$Pr\left(x_n\to {\overline{x}}_n|q_n\right)=Pr\left(\underset{W_n}{\underbrace{2\sqrt{P_n}\Re \left\{{\overline{\mathrm{\Delta }}}_{n}w_n^{\ast }\right\}}}\le -q_n\left({\left|\sqrt{P_n}{\overline{\mathrm{\Delta }}}_n+X_n\right|}^2-{\left|X_n\right|}^2\right)\right)$$

The conditional PEP can be represented as in Eq. (8), which is shown at the bottom of the page after extending Eq. (7). The noise term W_n in Eq. (8) is treated as a Gaussian random variable with zero mean and variance ${\sigma }_W^2=2P_n{\left|{\overline{\mathrm{\Delta }}}_n\right|}^2{W}_0$ depending on the q_n value.

9

$$Pr\left(x_n\to {\overline{x}}_n|q_n=q\right)=Q\left(\frac{q{\vartheta }_n}{{\lambda }_n}\right)$$

When the Gaussian Q-function is represented by Q(∙). Also

10

$${\vartheta }_n=\sqrt{P_n}{\left|{\overline{\mathrm{\Delta }}}_n\right|}^2+2\Re \left\{{\overline{\mathrm{\Delta }}}_{n}X_n^{\ast }\right\}$$ and

11

$${\lambda }_n=\left|{\overline{\mathrm{\Delta }}}_n\right|\sqrt{2W_0}$$

The conditional PEP can be constrained by applying the Chernoff bound on the Q-function in Eq. (9), which results in:

12

$$Pr\left(x_n\to {\overline{x}}_n|q_n=q\right)\le exp\left(-\frac{q^2{\vartheta }_n^2}{2{\lambda }_n^2}\right)$$

We take the average of the PDF of the whole fading coefficient, q_n, where q_n ≥ 0, to derive the unconditional PEP.

13

$$Pr\left(x_n\to {\overline{x}}_n\right)\approx \frac{exp\left(-\frac{{\overline{\mu }}_n^2}{2{\overline{\sigma }}_n^2}\right)}{\sqrt{2\pi {\overline{\sigma }}_n^2}}\times \underset{0}{\overset{\infty }{\int }}exp\left(-\frac{q^2({\vartheta }_n^2{\overline{\sigma }}_n^2+{\lambda }_n^2)}{2{\overline{\sigma }}_n^2{\lambda }_n^2}+\frac{{\overline{\mu }}_{n}q}{{\overline{\sigma }}_n^2}\right)dq$$

Asymptotic diversity order

The asymptotic diversity order, defined as the PEP's slope at a high signal-to-noise ratio (SNR) regime, has been efficiently quantified using the PEP, which has gained widespread adoption as a measurement tool. The nth UE's asymptotic diversity order can be calculated as follows¹⁷:

14

$$d_s=-\underset{\overline{\gamma }\to \infty }{lim}\frac{log\left(Pr\left(x_n\to {\overline{x}}_n\right)\right)}{log\overline{\gamma }}$$ where the nth UE's average transmit SNR is $\overline{\gamma }$ . We initially take advantage of Meijer's G-function's asymptotic expansion when in order to obtain the asymptotic diversity order. Keep in mind that Meijer's G-function argument approaches infinity as $\overline{\gamma }\to \infty$ .

Simulation analysis

This section provides analytical and Monte Carlo simulation results to substantiate the proposed mathematical framework and draw significant conclusions regarding the performance of LIS-assisted NOMA systems. In this section, we also describe the simulation setup and channel modelling used to evaluate the performance of the LIS-assisted NOMA network. The simulation involves multiple users, a base station (BS), and an LIS consisting of many passive reflecting elements. The BS is assumed to be located at the origin (0,0) of a 2D plane and NOMA users are randomly distributed within a radius of 6 m from the BS. The LIS is placed at a fixed position with coordinates (x_LIS, y_LIS). The channel model consists of three main components namely direct link, reflected link and composite channel gain. The direct link is the channel from the BS to the user and the reflected Link is the channel from the BS to the LIS and then from the LIS to the users. The composite channel gain is the combination of direct and reflected links. Table 1 presents the remaining simulation parameters for reference.

Table 1. Simulation parameters.

Parameters	Value
Modulation	BPSK
Carrier frequency	2.5 GHz
Bandwidth	20 MHz
Noise power spectral density	− 174 dBm/Hz
Transmit power of BS	40 dBm
Path-loss exponent (α)	3
Number of UEs (N)	2
Number of REs (K)	Variable
Far user distance from REs	6 m
Near user distance from REs	2.5 m
Distance between the BS and the LIS	1 m
The power coefficient of UE-1(P1)	0.8
The power coefficient of UE-2(P2)	0.2

Channel correlation

Each LIS component can receive and broadcast signals. On the other hand, the antennae are used to connect with the LIS and are commonly found on mobile devices or base stations. The relationship between the channels that the antennas and the LIS elements detected is described by the channel correlation matrix. The correlation coefficient quantifies the linear relationship between two variables, ranging from − 1 to + 1. A value of 1 indicates a perfect positive correlation, − 1 represents a perfect negative correlation, and 0 signifies no correlation between the variables. A high correlation coefficient means a strong connection between the signals received at one antenna and the signals sent out by another, as shown in Fig. 2.

Figure 2 [Images not available. See PDF.]

Channel correlation matrix.

Impulse response

The function known as the channel impulse response defines how the channel reacts to a brief energy pulse. The signal's amplitude and latency as it travels through the channel are commonly displayed on the graph of the channel impulse response. The signal's amplitude indicates its power at a given instant, and the latency means how long it takes for the signal to travel a certain distance via the channel. Remarkably, increasing multipath fading and interference might lead to a more complicated channel impulse response when there are more LIS elements and users. The channel coefficient, which depicts how the LIS affects the signal, can affect the channel impulse response's form as shown in Fig. 3. Generally speaking, a more significant channel coefficient means that the LIS enhances the signal more vigorously, leading to better signal quality, channel capacity, and energy efficiency.

Figure 3 [Images not available. See PDF.]

LIS-channel impulse response.

Covariance matrix

The degree to which two variables vary is indicated by their covariance, which measures their combined variability. When two variables have a zero covariance, they have no linear relationship. When two variables have a positive covariance, they tend to rise or fall together. They tend to move in opposite ways when they have a negative covariance. In LIS, the covariance matrix is typically used to assess the joint variability between the signals at the received antennas and the signals that the LIS elements are transmitting, as depicted in Fig. 4. A sizeable positive covariance means that the signals emitted by a certain LIS element tend to rise together with the signals received at a specific receive antenna. This makes it possible for the system to concentrate the sent signals on the highly correlated antennas, enhancing the signal quality and lowering interference. This can be helpful for beamforming and precoding.

Figure 4 [Images not available. See PDF.]

Covariance matrix of LIS channel.

LIS channel ratios

This graph only represents the SNR or channel quality between the LIS and the users in a wireless communication system, as depicted in Fig. 5. It displays the proportion of received signal power to received noise power. The channel ratio may be used to assess the effectiveness of various signal processing methods, modulation systems, and power allocation techniques in a sizable, intelligent, surface-aided wireless communication system. The location and configuration of the LIS elements can be adjusted by utilizing the channel ratio, hence enhancing the functionality and efficiency of the wireless network. An indication of a strong and reliable signal is typically observed when the channel ratio of the graph is near 1. The channel ratio is a measure of the signal-to-noise ratio (SNR) in a wireless communication system. It quantifies the ratio between the strength of the received signal and the power of the noise. When the signal-to-noise ratio (SNR) is high, the strength of the received signal surpasses that of the background noise, facilitating easy recognition and decoding. By implementing this approach, the components of the LIS can effectively enhance both the quality and intensity of the signal, leading to a higher channel ratio.

Figure 5 [Images not available. See PDF.]

LIS channel ratios between UE–n and 1.

Figure 6 displays the asymptotic diversity order of the initial user in a NOMA system based on LIS for three different values of K = 2, 8, and 14. To provide clarity, the figure does not display the findings of the second user, even though they share the same diversity order. These conclusions on the asymptotic diversity order in the system are supported by Monte Carlo simulations. The findings show that LIS has excellent potential for improving NOMA systems' error rate performance, with the number of REs, K, being a critical factor in defining the asymptotic diversity order of the underlying system model.

Figure 6 [Images not available. See PDF.]

Diversity order of the two UEs for various values of K.

Figure 7 depicts the BER union bound for K = 2 and K = 5 for the LIS-assisted NOMA system with two users. The outcomes support the benefits of applying LIS to improve NOMA's error rate performance. The noteworthy enhancement may significantly influence the integration of LIS into NOMA systems in error rate performance and its extended coverage. For instance, Fig. 7 shows that at 10⁻³ error rate, the first and second users' performances improve by around 11 and 9.5 dB, respectively, when K increases from 2 to 5. Additionally, Fig. 7 shows that both users encounter the same diversity order.

Figure 7 [Images not available. See PDF.]

BER comparisons among NOMA and LIS-NOMA for different numbers of REs.

Figure 8 depicts the first user's PEP performance, showcasing both analytical and simulation results for K = 2, 5, and 10. The solid lines represent the derived bounds/approximations, while the dashed lines represent the actual (simulated) PEP. Notably, the simulation results demonstrate that the analytical results provide an accurate upper limit on the PEP performance across the entire SNR range, affirming the precision of the theoretical expressions generated. Moreover, the fact that both analytical and simulation findings exhibit the same diversity order reinforces the validity of the mathematical framework established.

Figure 8 [Images not available. See PDF.]

Pairwise Error Probability of the first NOMA with different REs numbers.

Conclusions

In conclusion, the performance analysis of LIS-assisted NOMA networks has shown promising results in enhancing system capacity, improving spectral efficiency, and enabling better coverage. The wireless communication system can be significantly enhanced by employing LIS, which consists of many passive reflecting elements. The error rate performance of a NOMA system supported by an LIS is examined in this study using a comprehensive mathematical framework. The framework for NOMA users includes deriving the PEP expression and the PDF of the end-to-end fading channel. For the situations of a single RE and many REs, we obtain innovative PEP expressions using the central limit theorem (CLT) in the latter case. The BER tight union bound is then evaluated using the resulting PEP equation. Additionally, we assess the asymptotic PEP and employ it to calculate the system's asymptotic diversity order. The outcomes also demonstrate the benefits of LIS over traditional amplify-and-forward relaying, which can only support a diversity order of unity in terms of enhancing the diversity order of NOMA users.

Acknowledgements

The KSU authors acknowledge the funding from Researchers Supporting Project Number (RSP2024R355), King Saud University, Riyadh, Saudi Arabia.

Author contributions

All the authors have contributed equally to this article.

Data availability

The data used to support the findings of this study are included in the article.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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