Physical Layer Security of the MIMO-NOMA Systems under Near-Field Scenario

Abstract

In this paper, we propose a secure transmission framework for near-field MIMO-NOMA systems. This architecture integrates beamforming mechanisms for both transmission and reception, allowing the base station to send encrypted information to authorized users, effectively countering eavesdropping attempts in a near-field environment. To optimize the secrecy communication capability in the near field, a two-phase alternating optimization algorithm is introduced. In the first phase, the semidefinite relaxation (SDR) method is used to relax constraints in the problem and convert it into a semidefinite programming (SDP) problem. In the second phase, the successive convex approximation (SCA) algorithm is employed to transform the original non-convex problem into a convex optimization problem, obtaining a locally optimal solution through multiple iterations. Simulation results validate that the proposed near-field communication strategy exhibits superior secrecy communication capabilities under various parameter settings compared to far-field communication strategies.

1. Introduction

In the wave of the commercialization of 5G, the telecommunications sector is undergoing unprecedented revolutionary changes. With the exploration of future 6G technologies, wireless communication technology is increasingly moving towards the adoption of Extremely Large Antenna Arrays (ELAA), and the development of millimeter-wave (mmWave) and terahertz (THz) frequencies [1,2]. This signifies that the communication field is advancing towards higher data rates, lower latency, and greater capacity. The widespread application of high-frequency technology provides technical support for achieving faster data transmission rates and more precise sensing capabilities. However, accompanying these advancements are a series of new challenges, particularly in the context of near-field propagation environments. Due to the utilization of high frequencies and extremely large antenna arrays, the electromagnetic wave propagation model has undergone significant changes, shifting from the far-field planar wave model to the near-field spherical wave model [3]. The introduction of the spherical wave propagation model integrates receiver directionality and distance information, allowing the radiation pattern of the antenna array to focus more accurately on specific areas in free space. This precise beam focusing provides new degrees of freedom for controlling the distribution of beam energy in the distance domain, resulting in higher spatial reuse gains [4].

To support the connection of a large number of devices and enhance spectral efficiency (SE), efficient next-generation multiple access technologies are crucial for 6G. In this context, non-orthogonal multiple access (NOMA) has emerged as a highly promising solution due to its superior compatibility and flexibility. NOMA technology is generally classified into two categories: power domain NOMA (PD-NOMA) and code domain NOMA (CD-NOMA). Currently, the primary research contributions in NOMA focus on the power domain, and this article specifically addresses PD-NOMA, hereafter referred to as NOMA. In stark contrast to orthogonal multiple access (OMA), the essence of NOMA lies in allowing multiple users to share the same resource block. This is achieved by introducing controllable multiple access interference (MAI) among users. At the receiver’s end, this interference is progressively eliminated through serial interference cancellation (SIC), enabling the decoding of user signals [5,6,7]. However, when multiple users share the same resource block, there is a potential risk to the confidentiality of information. In this context, physical layer security (PLS) serves as a powerful complement to traditional encryption techniques. PLS ensures communication secrecy by exploiting the physical characteristics of wireless channels, such as interference, fading, and noise, without relying on complex key management systems [8,9]. In recent years, most studies on the PLS of NOMA have mainly focused on far-field channel models, overlooking the fundamental impact of near-field effects on the propagation characteristics of electromagnetic waves [10,11,12]. Despite important PLS features being attainable due to the angular characteristics associated with the plane wave regime, particularly in large antenna systems, the growing interest in near-field communications has necessitated the development of appropriate channel models that take into account spherical wave propagation and all aspects of near-field communication. This is essential to fully characterize the PLS features of the extra-large antenna arrays envisioned for 6G. Fortunately, preliminary research has already utilized near-field beamforming to ensure the security of wireless communications. The authors of [13] considered the interference rejection characteristics of extra-large arrays in near-field communications. Due to the focusing effect in the near-field, the interference rejection properties in this region are far superior to those related to the far-field. In [14], the authors proposed optimized hybrid beamforming algorithms to increase the secrecy capacity in near-field MIMO downlink channels, highlighting the security advantages of communicating in this field zone. Ref. [15] explores a near-field legacy network, effectively providing services to additional NOMA users through the utilization of preconfigured beams, thereby enhancing the overall system throughput and connectivity. The literature [16] utilizes preconfigured spatial beams for near-field users, providing services to additional far-field users through the NOMA system, enabling coexistence between near-field and far-field communications. In a small-area scenario, the proposed near-field model in the literature [17] is capable of accurately distinguishing users with different channel conditions. Simultaneously, the near-field model outperforms traditional methods in ensuring rate fairness. However, dedicated secrecy beam focusing strategies for MIMO-NOMA systems are still under-researched. In such emerging communication environments, particularly when using precise antenna array models for beamforming, the study of physical layer security techniques becomes especially important, which is also the motivation for this work.

We propose a near-field secure transmission architecture. As illustrated in Figure 1, in a near-field scenario assumed to have eavesdroppers, the base station securely transmits information to two near-field users using beamforming technology. At the transmitter, beamforming enhances signal emission in a specific direction by adjusting the phase and amplitude of signals on different antennas. Similarly, at the receiver, the same technology is employed to maximize the gain of the desired signal while suppressing interference and noise. While ensuring a fixed transmission power budget at the base station and minimum communication rate for each user, the secrecy capacity is maximized by optimizing the beamforming vectors at both the transmitter and receiver. This paper develops a two-phase algorithm, initially employing SDR to replace and relax some variables and constraints in the original problem, followed by SCA to convert the non-convex problem into a convex one, obtaining a locally optimal solution through multiple iterations. The numerical results confirm the convergence of the proposed two-stage algorithm. It also reveals that (1) the superiority of near-field communication over far-field communication in terms of secrecy performance, as well as the advantages of NOMA over OMA; and (2) the secrecy performance of the near-field system depends on the distance from the user to the reference point of the BS, the maximum transmission power of the BS, and the number of antennas at the BS.

2. System Model and Problem Formulation

2.1. System Model

As shown in Figure 1, we consider a near-field (NF) MIMO-NOMA downlink communication system, consisting of a base station (BS), two NOMA users, ${\mathrm{U}}_1$ and ${\mathrm{U}}_2$, and a potential external eavesdropper (Eve). Considering the more general scenario, we assume that there are direct links between BS and ${\mathrm{U}}_1$, ${\mathrm{U}}_2$, and Eve. Additionally, both the BS, ${\mathrm{U}}_1$, and ${\mathrm{U}}_2$ employ uniform linear arrays (ULA), where the BS is equipped with $\mathrm{M}$ antennas, ${\mathrm{U}}_1$ is equipped with ${\mathrm{M}}_1$ antennas, ${\mathrm{U}}_2$ is equipped with ${\mathrm{M}}_2$ antennas. In order to reduce costs, decrease complexity, and enhance operational discretion, consideration is given to equipping Eve with a single antenna. We further assume that the direct links BS to ${\mathrm{U}}_1$, BS to ${\mathrm{U}}_2$, and BS to Eve experience free-space path loss. The BS operates in the 28 GHz millimeter-wave frequency band and attempts to transmit secure signals to ${\mathrm{U}}_1$ and ${\mathrm{U}}_2$ in the presence of external eavesdropper Eve. Both ${\mathrm{U}}_1$, ${\mathrm{U}}_2$ and Eve are located in the near-field region.

Assuming the distance between BS and users is shorter than the Rayleigh distance $d_R=\frac{2{\left(D_1+D_2\right)}^2}{\lambda }$ (where $\lambda$ is the wavelength, $D_1$ is the antenna aperture of BS, $D_2$ is the antenna aperture of users). Thus, the transmitted wavefronts follow spherical propagation. To prevent confidential information from being intercepted by Eve, the BS utilizes transmit-receive beamforming technology to enhance the signal strength for users, while suppressing information leakage to Eve.

Considering that M data streams are sent by each user, the transmit signal of ${\mathrm{U}}_i(i=1,2)$ can be expressed as $\mathrm{x}\in {\mathbb{C}}^{M\times 1}$, which is the sum of symbols from two users multiplied by their respective transmission beamforming vectors (also referred to as spatial features or linear precoders in the literature) according to the literature

$$\mathbf{x}={\mathbf{f}}_1{\mathrm{s}}_1+{\mathbf{f}}_2{\mathrm{s}}_2,$$

(1)

where $s_i\in \mathbb{C}(i=1,2)$ is the information signal with satisfying $\mathbb{E}|s_i{|}^2=1$, respectively, while ${\mathbf{f}}_i\in {\mathbb{C}}^{M\times 1}(i=1,2)$ denote the beamforming vector. Assuming that BS has a maximum transmit power budget $P_{\max }$, we have ${∥{\mathbf{f}}_1∥}^2+{∥{\mathbf{f}}_2∥}^2\le P_{\max }$.

Let ${\mathbf{H}}_1\in {\mathbb{C}}^{M\times M_1}$, ${\mathbf{H}}_2\in {\mathbb{C}}^{M\times M_2}$ and ${\mathbf{h}}_3\in {\mathbb{C}}^{M\times 1}$ represent the channel from BS to ${\mathrm{U}}_1$, that from BS to ${\mathrm{U}}_2$ and that from BS to eavesdropper Eve, respectively. Without loss of generality, it is assumed that ${\mathrm{U}}_1$ and ${\mathrm{U}}_2$ are arranged in descending order according to the simplified effective channel. To figure out the theoretical security capability, we assume that the channel state information (CSI) is known to all users in the case when Eves also play the role of legitimate receivers but are untrusted by the BS. Thus, while the Eves are not supposed to receive the confidential information from users, it is a rational consideration that the Eves engage in cooperation with legitimate nodes for CSI exchange. After receiving the signal vector, the BS applies linear receive beamforming to obtain the signal. Therefore, the received signal vector for ${\mathrm{U}}_i$ and Eve can be transformed as follows

$${\mathrm{y}}_i={\mathbf{H}}_i({\mathbf{f}}_1{\mathrm{s}}_1+{\mathbf{f}}_2{\mathrm{s}}_2)+{\mathrm{n}}_i,i\in \left\{1,2\right\},$$

(2)

$${\mathrm{y}}_3={\mathbf{h}}_3({\mathbf{f}}_1{\mathrm{s}}_1+{\mathbf{f}}_2{\mathrm{s}}_2)+{\mathrm{n}}_3,$$

(3)

where $n_i\sim \mathcal{CN}\left(0,{\sigma }_i^2\right)(i=1, 2, 3)$ is the complex additive white Gaussian noise (AWGN). When users receive signals, they decode the user with the highest power coefficient first using SIC technology. ${\mathrm{U}}_1$ decodes $s_2$ first, then subtracts the signal from ${\mathrm{U}}_2$ in the current superimposed signal, and then decodes its own signal $s_1$ from the remaining signal. In this situation, the achievable rates in bits/second/Hertz (bps/Hz) for the legitimate ${\mathrm{U}}_1$ is given by $R_1=\min \left\{R_{1,s_1},R_{1,s_2}\right\}$

$$R_{1,s_1}={\log }_2\left(1+\frac{{\left|{\mathbf{w}}_1^H{\mathbf{H}}_1{\mathbf{f}}_1\right|}^2}{{\sigma }_1^2}\right),$$

(4)

$$R_{1,s_2}={\log }_2\left(1+\frac{{\left|{\mathbf{w}}_1^H{\mathbf{H}}_1{\mathbf{f}}_2\right|}^2}{{\left|{\mathbf{w}}_1^H{\mathbf{H}}_1{\mathbf{f}}_1\right|}^2+{\sigma }_1^2}\right),$$

(5)

according to the principles of downlink NOMA, decodes the composite signal $s_2$ directly, the achievable rates of legitimate ${\mathrm{U}}_2$ is given by $R_2=R_{2,s_2}$

$$R_{2,s_2}={\log }_2\left(1+\frac{{\left|{\mathbf{w}}_2^H{\mathbf{H}}_2{\mathbf{f}}_2\right|}^2}{{\left|{\mathbf{w}}_2^H{\mathbf{H}}_2{\mathbf{f}}_1\right|}^2+{\sigma }_2^2}\right),$$

(6)

the achievable rates of Eve eavesdropping on ${\mathrm{U}}_1$ and ${\mathrm{U}}_2$ is given by

$$R_{3,s_1}={\log }_2\left(1+\frac{{\left|{\mathbf{h}}_3^H{\mathbf{f}}_1\right|}^2}{{\left|{\mathbf{h}}_3^H{\mathbf{f}}_2\right|}^2+{\sigma }_3^2}\right),$$

(7)

$$R_{3,s_2}={\log }_2\left(1+\frac{{\left|{\mathbf{h}}_3^H{\mathbf{f}}_2\right|}^2}{{\left|{\mathbf{h}}_3^H{\mathbf{f}}_1\right|}^2+{\sigma }_3^2}\right).$$

(8)

In the framework of physical layer security in information theory, the measure of confidentiality performance is often described using the secrecy capacity. The secrecy capacity is defined as the positive difference between the legitimate achievable rate and the eavesdropping achievable rate, that is, $R_{sec,1}={\left[R_1-R_{3,s_1}\right]}^{+}$ and $R_{sec,2}={\left[R_2-R_{3,s_2}\right]}^{+}$, where ${[x]}^{+}=\max \{x,0\}$ [12].

2.2. Near-Field Channel Model

In the near-field channel model shown in Figure 2, we assume that the coordinates of the midpoint of the BS antennas is $(0,0)$. The m-th antenna of the BS can be denoted as $(0,\tilde{m}d)$, where $\tilde{m}=m-\frac{M-1}{2}$ with m = 0, 1,…, $M-1$ and d denotes the spacing between antennas. Similarly, the coordinates of the m-th antenna of ${\mathrm{U}}_1$ and ${\mathrm{U}}_2$ can be represented as $\left(0,{\tilde{m}}_{1}d\right)$ and $\left(0,{\tilde{m}}_{2}d\right)$, respectively, where ${\tilde{m}}_1=m_1-\frac{M_1-1}{2}$ and ${\tilde{m}}_2=m_2-\frac{M_2-1}{2}$ [18].

Accordingly, the line-of-sight (LoS) near-field channel between the BS and ${\mathrm{U}}_1$, as well as ${\mathrm{U}}_2$ can be modeled as [11]

$${\mathbf{H}}_i(d,\theta )={\left[{\mathbf{h}}_{i,1},\cdots ,{\mathbf{h}}_{i,m_i},\cdots ,{\mathbf{h}}_{i,M_i}\right]}^T,i\in \left\{1,2\right\},$$

(9)

where ${\mathbf{h}}_{1,m_1}=(1/\sqrt{M})[g_{m_1,1}{e}^{-j\frac{2\pi f}{c}\left(d_{m_1,1}-d_{m_1}\right)},\cdots$$g_{m_1,m}{e}^{-j\frac{2\pi f}{c}\left(d_{m_1,m}-d_{m_1}\right)}\cdots ,g_{m_1,M}$

$e^{-j\frac{2\pi f}{c}\left(d_{m_1,M}-d_{m_1}\right)}{]}^T$, ${\mathbf{h}}_{2,m_2}=(1/\sqrt{M})[g_{m_2,1}{e}^{-j\frac{2\pi f}{c}\left(d_{m_2,1}-d_{m_2}\right)},\cdots g_{m_2,m}$$e^{-j\frac{2\pi f}{c}\left(d_{m_2,m}-d_{m_2}\right)}\cdots ,$$g_{m_2,M}{e}^{-j\frac{2\pi f}{c}\left(d_{m_2,M}-d_{m_2}\right)}{]}^T$. Note that $\left|g_{1,m}\right|=\frac{c}{4\pi f{d}_{1,m}}$ and $\left|g_{2,m}\right|=\frac{c}{4\pi f{d}_{2,m}}$ denotes the free-space large-scale path loss between the m-th antenna of the BS and the $m_i$-th (i = 1, 2) antenna of ${\mathrm{U}}_i(i=1,2)$. Here, $d_{m_1}$ and $d_{m_2}$ denote the reference distance from $(0,0)$ to $\left(x_1,y_1+{\tilde{m}}_{1}d\right)$ and $\left(x_2,y_2+{\tilde{m}}_{2}d\right)$, and the distance between the m-th antenna of the BS and the $m_i$-th antenna of ${\mathrm{U}}_i$ are derived using the cosine theorem

$$d_{m_i,m}={\left(d_{m_i}^2+{\left(\tilde{m}d\right)}^2-2d_{m_i}\tilde{m}dsin{\theta }_{m_i}\right)}^{1/2},i\in \left\{1,2\right\}.$$

(10)

where ${\theta }_{m_1}$ and ${\theta }_{m_2}$ denote the azimuth angle of the $m_i$-th antenna of ${\mathrm{U}}_i$ with respect to $(0,0)$. In the same way, the near-field eavesdropping channel ${\mathbf{h}}_3$ can be derived as ${\mathbf{h}}_3=$$(1/\sqrt{M}){\left[g_{3,1}{e}^{-j\frac{2\pi f}{c}{d}_{3,1}},\cdots g_{3,m}{e}^{-j\frac{2\pi f}{c}{d}_{3,m}}\cdots ,g_{3,M}{e}^{-j\frac{2\pi f}{c}{d}_{3,M}}\right]}^H$. The path loss and distance for near-field eavesdropping are respectively given by $\left|g_{3,m}\right|=\frac{c}{4\pi f{d}_{3,m}}$ and $d_{3,m}={\left(d_3^2+{\left(\tilde{m}d\right)}^2-2d_3\tilde{m}dsin{\theta }_3\right)}^{1/2}$. For simplicity, in the following text, we omit the $(d,\theta )$ in ${\mathbf{H}}_1(d,\theta )$ and ${\mathbf{H}}_2(d,\theta )$, and refer to them simply as ${\mathbf{H}}_1$ and ${\mathbf{H}}_2$. Note that in contrast to existing works on far-field secure communications, the spherical-wave channels in the near-field communications contain the extra distance information, which helps to secures the legitimate transmission.

2.3. Problem Formulation

In this paper, our primary objective is to maximize the secrecy capacity through the optimization of beamforming vectors at both the transmitter and receiver. We must ensure that, while maintaining a fixed transmission power budget at the BS, each user achieves the minimum communication rate to meet the practical constraints of the wireless communication system. The specific formulation of the problem is given by

$$\begin{array}{cc}\hfill \underset{\begin{array}{c}{\mathbf{w}}_1,{\mathbf{w}}_2\\ {\mathbf{f}}_1,{\mathbf{f}}_2\end{array}}{\max }& R_{\sec ,1}+R_{\sec ,2}\hfill \end{array}$$

(11a)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& {∥{\mathbf{w}}_1∥}^2={∥{\mathbf{w}}_2∥}^2=1,\hfill \end{array}$$

(11b)

$$\begin{array}{cc}\hfill & {∥{\mathbf{f}}_1∥}^2+{∥{\mathbf{f}}_2∥}^2\le P_{\max },\hfill \end{array}$$

(11c)

$$\begin{array}{cc}\hfill & R_1\ge R_{\min ,1},\hfill \end{array}$$

(11d)

$$\begin{array}{cc}\hfill & R_2\ge R_{\min ,2},\hfill \end{array}$$

(11e)

$$\begin{array}{cc}\hfill & {\left|{\mathbf{w}}_1{\mathbf{H}}_1{\mathbf{f}}_1\right|}^2\ge {\left|{\mathbf{w}}_2{\mathbf{H}}_2{\mathbf{f}}_2\right|}^2.\hfill \end{array}$$

(11f)

where (11b) denotes the normalized receive beamforming vector at ${\mathrm{U}}_i(i=1,2)$. The constraint (11c) is the transmit power constraint of the base station’s maximum transmission power $P_{\max }$, (11d) and (11e) are the minimum rate requirements for ${\mathrm{U}}_1$ and ${\mathrm{U}}_2$, and (11f) assumes, without loss of generality, that ${\mathrm{U}}_1$ and ${\mathrm{U}}_2$ are further arranged in descending order according to the simplified effective channel.

It is evident that the optimization problem in (11) is highly challenging. Firstly, the objective function in (11a) is inherently a non-convex optimization problem, involving the coupling of various optimization variables. Secondly, the presence of power constraints and the minimum quality of service (QoS) requirements for legitimate users [13] further complicates the optimization process. However, we find that the problem can be efficiently solved when one of ${\mathbf{f}}_i(i=1,2)$ and ${\mathbf{w}}_i(i=1,2)$ is fixed. This inspires us to propose an algorithm based on alternating optimization to solve the optimization problem in a suboptimal manner, by optimizing one of ${\mathbf{f}}_i$ and ${\mathbf{w}}_i$ in each iteration while keeping the other fixed, until convergence is reached.

3. Algorithm Design

In this section, the paper employs the alternating optimization (AO) method [19] as a strategic approach to tackle the intricate coupling between the base station’s transmit beamforming vectors and the users’ receive beamforming vectors in problem (11). The AO method is a powerful optimization technique that facilitates the resolution of complex problems by iteratively updating and optimizing subsets of variables while keeping the others fixed. This iterative process allows the algorithm to converge toward an optimal solution. Specifically, within the framework of the AO method applied here, the optimization process involves fixing one set of variables while alternately updating the other set. In the context of problem (11), this means that either the base station’s transmit beamforming vectors or the users’ receive beamforming vectors are held constant during each iteration, while the other set is adjusted to enhance the overall system performance. This alternating update scheme continues iteratively until a convergence criterion is met, signifying that the optimization process has reached a stable and satisfactory solution.

Sub1: Fix ${\mathbf{f}}_1$ and ${\mathbf{f}}_2$, optimize ${\mathbf{w}}_1$ and ${\mathbf{w}}_2$

First, introduce new variables $\mathbf{W}$ and $\mathbf{T}$ such that ${\mathbf{W}}_1={\mathbf{w}}_1{\mathbf{w}}_1^H$, ${\mathbf{W}}_2={\mathbf{w}}_2{\mathbf{w}}_2^H$. It is given that $rank\left({\mathbf{W}}_1\right)=rank\left({\mathbf{W}}_2\right)=1$, ${\mathbf{W}}_1\ge 0$, ${\mathbf{W}}_2\ge 0$. Regarding $\mathbf{T}$, we have ${\mathbf{T}}_{1,1}=$${\mathbf{H}}_1{\mathbf{f}}_1{\mathbf{f}}_1^H{\mathbf{H}}_1^H,{\mathbf{T}}_{1,2}={\mathbf{H}}_1{\mathbf{f}}_2{\mathbf{f}}_2^H{\mathbf{H}}_1^H,{\mathbf{T}}_{2,2}={\mathbf{H}}_2{\mathbf{f}}_2{\mathbf{f}}_2^H{\mathbf{H}}_2^H,{\mathbf{T}}_{2,1}={\mathbf{H}}_2{\mathbf{f}}_1{\mathbf{f}}_1^H{\mathbf{H}}_2^H$. Secondly, for computational convenience, set the inverse of the covariance matrices for ${\mathrm{U}}_i$ noise as ${\gamma }_i=1/{\sigma }_i^2(i=1,2)$. The achievable rates of the users in Equations (4)–(6) can be re-expressed as

$$\begin{array}{cc}\hfill R_{1,s_1}& ={\log }_2\left(1+{\gamma }_1\mathrm{Tr}\left({\mathbf{T}}_{1,1}{\mathbf{W}}_1\right)\right.,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill R_{1,s_2}& ={\log }_2\left(1+\frac{{\gamma }_1\mathrm{Tr}\left({\mathbf{T}}_{1,2}{\mathbf{W}}_1\right)}{{\gamma }_1\mathrm{Tr}\left({\mathbf{T}}_{1,1}{\mathbf{W}}_1\right)+1}\right),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill R_{2,s_2}& ={\log }_2\left(1+\frac{{\gamma }_2\mathrm{Tr}\left({\mathbf{T}}_{2,2}{\mathbf{W}}_2\right)}{{\gamma }_2\mathrm{Tr}\left({\mathbf{T}}_{2,1}{\mathbf{W}}_2\right)+1}\right),\hfill \end{array}$$

(14)

where $\mathrm{Tr}\left(\mathrm{X}\right)$ represents the trace of matrix $\mathrm{X}$. Let the variable ${\mathrm{t}}_1=\min \left\{R_{1,s_2},R_{2,s_2}\right\}\Rightarrow \left\{\begin{array}{c}{R}_{1,s_2}\ge {\mathrm{t}}_1\hfill \\ R_{2,s_2}\ge {\mathrm{t}}_1\hfill \end{array}\right.$ At this point, $R_{3,s_1}$ and $R_{3,s_2}$ are constants, and after removing the constant values, the optimization function is expressed as

$$\begin{array}{cc}\hfill \underset{{\mathbf{W}}_1,{\mathbf{W}}_2,{\mathrm{t}}_1}{\max }& {\log }_2\left(1+{\gamma }_1\mathrm{Tr}\left({\mathbf{T}}_{1,1}{\mathbf{W}}_1\right)\right)+t_1\hfill \end{array}$$

(15a)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& {\mathbf{W}}_1\ge 0,{\mathbf{W}}_2\ge 0,\hfill \end{array}$$

(15b)

$$\begin{array}{cc}\hfill & \mathrm{Tr}\left({\mathbf{W}}_1\right)=\mathrm{Tr}\left({\mathbf{W}}_2\right)=1,\hfill \end{array}$$

(15c)

$$\begin{array}{cc}\hfill & \mathrm{Tr}\left({\mathbf{T}}_{1,1}{\mathbf{W}}_1\right)\ge \mathrm{Tr}\left({\mathbf{T}}_{2,2}{\mathbf{W}}_2\right),\hfill \end{array}$$

(15d)

$$\begin{array}{cc}\hfill & {\tilde{R}}_{1,s_2}\ge {\mathrm{t}}_1,{\tilde{R}}_{2,s_2}\ge {\mathrm{t}}_1.\hfill \end{array}$$

(15e)

where (15b) indicates that ${\mathbf{W}}_1$ and ${\mathbf{W}}_2$ are symmetric positive semi-definite matrices, and (15c) imposes the rank constraint of 1 on ${\mathbf{W}}_1$ and ${\mathbf{W}}_2$. Equation (15e) can be addressed using SCA to handle these non-convex constraints [20]. Specifically, by employing a first-order Taylor approximation, approximately equivalent ${\tilde{R}}_{1,s_2}$ and ${\tilde{R}}_{2,s_2}$ can be obtained.

During this stage, it is noteworthy that Equation (15) conforms to the structure of second-order cone programming (SOCP), a mathematical optimization framework. This particular type of problem can be effectively solved using the CVX tool. The utilization of CVX streamlines the optimization process, making it more efficient and numerically robust. Subsequently, the algorithm proceeds by iteratively updating the beamforming matrices $\tilde{\mathbf{W}}1$ and $\tilde{\mathbf{W}}2$ to obtain an approximate solution for Sub1. Algorithm 1 delineates the step-by-step procedure involved in this iterative optimization process. The detailed content of the algorithm section is presented in Appendix A.

Algorithm 1 Alternating Optimization for Solving Sub 1

Input:  ${\mathbf{H}}_1,{\mathbf{H}}_2,R_{\min ,1},R_{\min ,2},{\sigma }_1^2,{\sigma }_2^2$. Output:  ${\mathbf{w}}_1,{\mathbf{w}}_2.$ 1: Initialize $i_1=0$, ${\tilde{\mathbf{W}}}_1^{\left(0\right)}$, ${\tilde{\mathbf{W}}}_2^{\left(0\right)}$, $\epsilon$ denotes accuracy. 2: Set ${\tilde{\mathbf{w}}}_1={\mathbf{w}}_1{\mathbf{w}}_1^H$ and ${\tilde{\mathbf{w}}}_2={\mathbf{w}}_2{\mathbf{w}}_2^H$. 3: repeat 4:     Based on Equation (15), compute ${\mathbf{W}}_1^{\left(i_1\right)}$, ${\mathbf{W}}_2^{\left(i_1\right)}$, and $i_1$ using CVX. 5:     Update ${\tilde{\mathbf{W}}}_1^{(i_1+1)}={\mathbf{W}}_1^{\left(i_1\right)}$, ${\tilde{\mathbf{W}}}_2^{(i_1+1)}={\mathbf{W}}_2^{\left(i_1\right)}$. 6:     Update $i_1=i_1+1$ 7: until$\frac{\left|R_{\sec ,1}^{(i_1+1)}+R_{\sec ,2}^{(i_1+1)}-R_{\sec ,1}^{\left(i_1\right)}-R_{\sec ,2}^{\left(i_1\right)}\right|}{R_{\sec ,1}^{\left(i_1\right)}+R_{\sec ,2}^{\left(i_1\right)}}\le \epsilon$, loop termination. 8: Recover ${\mathbf{w}}_1$ and ${\mathbf{w}}_2$ via Gaussian Randomization, respectively.

Sub2: Fix ${\mathbf{w}}_1$ and ${\mathbf{w}}_2$, optimize ${\mathbf{f}}_1$ and ${\mathbf{f}}_2$

To address the non-convex unit modulus constraints of ${\mathbf{f}}_1$ and ${\mathbf{f}}_2$, this paper adopts the SDR technique to update ${\mathbf{f}}_1$ and ${\mathbf{f}}_2$. Specifically, by defining a new variable $\mathbf{F}$, such that ${\mathbf{F}}_1={\mathbf{f}}_1{\mathbf{f}}_1^H,{\mathbf{F}}_2={\mathbf{f}}_2{\mathbf{f}}_2^H$, this paper ensures that $rank\left({\mathbf{F}}_1\right)=rank\left({\mathbf{F}}_2\right)=1,{\mathbf{F}}_1\ge 0,{\mathbf{F}}_2\ge 0$. A new auxiliary variable $\mathbf{S}$ is defined, where ${\mathbf{S}}_1={\mathbf{H}}_1{\mathbf{w}}_1{\mathbf{w}}_1^H{\mathbf{H}}_1^H,{\mathbf{S}}_2={\mathbf{H}}_2{\mathbf{w}}_2{\mathbf{w}}_2^H{\mathbf{H}}_2^H,{\mathbf{S}}_3={\mathbf{h}}_3{\mathbf{h}}_3^H$. Then, set the inverse of the eavesdropper’s noise covariance matrix as ${\gamma }_i=1/{\sigma }_i^2(i=1,2,3)$. The achievable rates of the legitimate users in (4–6) can be re-expressed as

$$\begin{array}{cc}\hfill R_{1,s_1}& ={\log }_2\left(1+{\gamma }_1\mathrm{Tr}\left({\mathbf{S}}_1{\mathbf{F}}_1\right)\right),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill R_{1,S_2}& ={\log }_2\left(1+\frac{{\gamma }_1\mathrm{Tr}\left({\mathbf{S}}_1{\mathbf{F}}_2\right)}{{\gamma }_1\mathrm{Tr}\left({\mathbf{S}}_1{\mathbf{F}}_1\right)+1}\right),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill R_{2,s_2}& ={\log }_2\left(1+\frac{{\gamma }_2\mathrm{Tr}\left({\mathbf{S}}_2{\mathbf{F}}_2\right)}{{\gamma }_2\mathrm{Tr}\left({\mathbf{S}}_2{\mathbf{F}}_1\right)+1}\right).\hfill \end{array}$$

(18)

The achievable rates of eavesdropping from (7–8) can be re-expressed as

$$\begin{array}{cc}\hfill R_{3,S_1}& ={\log }_2\left(1+\frac{{\gamma }_3\mathrm{Tr}\left({\mathrm{S}}_3{\mathrm{F}}_1\right)}{{\gamma }_3\mathrm{Tr}\left({\mathrm{S}}_3{\mathrm{F}}_2\right)+1}\right),\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill R_{3,S_2}& ={\log }_2\left(1+\frac{{\gamma }_3\mathrm{Tr}\left({\mathrm{S}}_3{\mathrm{F}}_2\right)}{{\gamma }_3\mathrm{Tr}\left({\mathrm{S}}_3{\mathrm{F}}_1\right)+1}\right).\hfill \end{array}$$

(20)

Let the variable ${\mathrm{t}}_2=\min \left\{R_{1,s_2},R_{2,s_2}\right\}\Rightarrow \left\{\begin{array}{c}{R}_{1,s_2}\ge {\mathrm{t}}_2\hfill \\ R_{2,s_2}\ge {\mathrm{t}}_2\hfill \end{array}\right.$ The optimization function is expressed as

$$\begin{array}{cc}\hfill \underset{{\mathbf{F}}_1,{\mathbf{F}}_2,{\mathrm{t}}_2}{\max }& {\log }_2\left(1+{\gamma }_1\mathrm{Tr}\left({\mathbf{S}}_1{\mathbf{F}}_1\right)\right)+{\mathrm{t}}_2-{\tilde{R}}_{3,s_1}-{\tilde{R}}_{3,s_2}\hfill \end{array}$$

(21a)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& {\mathbf{F}}_1\ge 0,{\mathbf{F}}_2⪰0,\hfill \end{array}$$

(21b)

$$\begin{array}{cc}\hfill & \mathrm{Tr}\left({\mathbf{F}}_1+{\mathbf{F}}_2\right)\le P_{\max },\hfill \end{array}$$

(21c)

$$\begin{array}{cc}\hfill & \mathrm{Tr}\left({\mathbf{S}}_1{\mathbf{F}}_1\right)\ge \mathrm{Tr}\left({\mathbf{S}}_2{\mathbf{F}}_2\right),\hfill \end{array}$$

(21d)

$$\begin{array}{cc}\hfill & {\tilde{R}}_{1,s_2}\ge {\mathrm{t}}_2,{\tilde{R}}_{2,s_2}\ge {\mathrm{t}}_2.\hfill \end{array}$$

(21e)

where (21b) indicates that ${\mathbf{F}}_1$ and ${\mathbf{F}}_2$ are symmetric positive semidefinite matrices, (21c) is the transmit power constraint for the maximum transmission power $P_{\max }$ at the transmitter, and (21d) states that under general conditions, the trace of the equivalent channel of ${\mathrm{U}}_1$ is superior to that of ${\mathrm{U}}_2$. Equation (21e) are non-convex and are handled using the first-order Taylor approximation by the SCA algorithm. The maximum eavesdropping rates for the eavesdropper Eve are denoted by ${\tilde{R}}_{3,s_1}$ and ${\tilde{R}}_{3,s_2}$.

It is evident that the current problem (21) is convex and can be efficiently solved for a suboptimal solution using convex optimization tools (such as CVX). Then, by iteratively updating ${\tilde{\mathbf{F}}}_1$ and ${\tilde{\mathbf{F}}}_2$, an approximate solution for Sub2 is obtained. This process is elaborated in Algorithm 2. The detailed content of the algorithm section is presented in Appendix A.

Algorithm 2 Alternating Optimization for Solving Sub2

Input:  ${\mathbf{H}}_1,{\mathbf{H}}_2,{\mathbf{h}}_3,R_{\min ,1},R_{\min ,2},{\sigma }_1^2,{\sigma }_2^2,{\sigma }_3^2$. Output:  ${\mathbf{f}}_1,{\mathbf{f}}_2.$ 1: Initialize $i_2=0$, $\left\{{\tilde{\mathbf{F}}}_1^{\left(0\right)}\right\},\left\{{\tilde{\mathbf{F}}}_2^{\left(0\right)}\right\},\epsilon \mathrm{represents}\mathrm{precision}.$ 2: Set ${\tilde{\mathbf{F}}}_1={\mathbf{f}}_1{\mathbf{f}}_1^H$ and ${\tilde{\mathbf{F}}}_2={\mathbf{f}}_2{\mathbf{f}}_2^H$. 3: repeat 4:     Based on Equation (21), compute ${\mathbf{F}}_1^{\left(i_2\right)},{\mathbf{F}}_2^{\left(i_2\right)}$, and $i_2$ using CVX. 5:     Update ${\tilde{\mathbf{F}}}_1^{(i_1+1)}={\mathbf{F}}_1^{\left(i_1\right)}$, ${\tilde{\mathbf{F}}}_2^{(i_1+1)}={\mathbf{F}}_2^{\left(i_1\right)}$. 6:     Update $i_2=i_2+1$ 7: until$\frac{\left|R_{\sec ,1}^{(i_2+1)}+R_{\sec ,2}^{(i_2+1)}-R_{\sec ,1}^{\left(i_2\right)}-R_{\sec ,2}^{\left(i_2\right)}\right|}{R_{\sec ,1}^{\left(i_2\right)}+R_{\sec ,2}^{\left(i_2\right)}}\le \epsilon$, loop termination. 8: Recover ${\mathbf{f}}_1$ and ${\mathbf{f}}_2$ via Gaussian Randomization, respectively.

4. Numerical Results

In this section, we reveal numerical results of our proposed NF-NOMA scheme. We consider a 2D coordinate setup in meters (m), where the BS is located at (0, 0) m, while midpoints of the antennas of ${\mathrm{U}}_1$, ${\mathrm{U}}_2$ and Eve are respectively located 8 m, 20 m and 10 m from the coordinate (0, 0) m with the azimuth angle of 45°, 15°, 20°. All the ULA are positioned along the y-axis. Unless otherwise specified, the default parameters are set as $f=28\mathrm{GHz}$ [21], $d=\lambda /2$, $M=12$, $M_1=4$, $M_2=4$, ${\sigma }_i^2=-90$ dBm. The numerical results are averaged from 400 independent MonteCarlo experiments.

As a comparison, this section simulates NOMA and OMA schemes under near-field and far-field communications. Through simulation and comparison, we discuss the changes in system secrecy rate with variations in the distance ${\mathrm{d}}_1$ between the BS and ${\mathrm{U}}_1$, the maximum BS transmission power $P_{\max }$, and the number of BS antennas. The results validate that, compared to traditional far-field communication schemes, considering a near-field model introduces additional distance dimension information, leading to higher system secrecy rates.

In Figure 3, we illustrate the variation of the overall secrecy rate for four different communication strategies as the distance of user ${\mathrm{U}}_1$’s midpoint to the BS changes. The minimum achievable rate for two users is set to $R_{min}=1$ bps/Hz, and the BS transmission power is $P_{\max }=16$ dBm. As shown in Figure 3, it is evident that, with the increasing distance ${\mathrm{d}}_1$ from the BS to ${\mathrm{U}}_1$, the system secrecy rates for all four communication strategies are decreasing. This is because, as the distance becomes greater, the channel gain from the BS to ${\mathrm{U}}_1$ diminishes, leading to increased signal loss for legitimate users and, consequently, a decline in the system secrecy rate. Among them, NF-NOMA outperforms other methods at various distances, highlighting its advantage in secure transmission in near-field communication. In the near-field channel model, the radiation pattern of the antenna array integrates both directional and distance information from the receiver, allowing for a more precise distribution of energy in specific regions of free space. Therefore, compared to the far-field model, the near-field model provides additional distance-dimensional information that can be used to further optimize the security of the communication link. As the distance increases, signal path loss and attenuation increase, posing challenges faced by all wireless communication systems. However, due to the ability of near-field communication to more precisely focus and direct beams, it provides better secrecy rates over short distances. As the distance increases, the system gradually transitions to the far-field region, diminishing the advantages of beam focusing and resulting in a smaller difference in secrecy rates between near-field and far-field. Overall, these characteristics of near-field communication make it more effective in maintaining communication security over shorter distances, with this advantage gradually decreasing as the distance increases.

From Figure 4, we can observe the variation in the overall secrecy rate for four different communication strategies with respect to the maximum transmit power of the BS. The minimum achievable rate for two users is set to $R_{min}=1$ bps/Hz. As the maximum transmit power of the BS increases, there is an expected upward trend in the overall secrecy rates for all communication strategies. Firstly, the increase in transmit power enhances the signal propagation strength, effectively overcoming signal attenuation and propagation losses along the communication path. Secondly, higher transmit power also improves the SNR. With the signal’s intensity becoming more pronounced relative to the noise, the increase in SNR enhances the communication system’s ability to discern signals, consequently elevating the secrecy capacity of the communication link. Regarding the performance of communication strategies, NF-NOMA consistently provides the highest secrecy rates across all power settings. This suggests that within the considered power range, $\left|\right|{\mathbf{f}}_1{\left|\right|}^2+{\left|\right|\mathbf{f}2\left|\right|}^2\le P_{\max }$, NF-NOMA can finely adjust power allocation based on the specific location of the user and the characteristics of the near-field channel, maximizing secrecy rates. Taking these factors into account, NF-NOMA demonstrates superior secrecy performance as power increases, particularly in comparison to OMA technology, showcasing clear advantages in utilizing power and spatial resources.

In Figure 5, we can observe the impact of the number of antennas at the BS on the overall secrecy rate for four different communication strategies. The minimum achievable rate for two users is set to $R_{min}=1$ bps/Hz, and the BS transmission power is $P_{\max }=16$ dBm. It can be observed that the overall secrecy rate of the system gradually increases with the growth of the BS antenna count. Moreover, as M increases, the rate of growth in secrecy rate slows down. NF-NOMA still provides the highest overall secrecy rate across all antenna quantity settings, but the gain of the near-field strategy is far less pronounced than the far-field strategy. The reasons for this could be as follows:

• Near-field saturation effect: In near-field communication, due to the propagation of electromagnetic waves in the form of spherical waves, the beamforming capability of the antenna array has strong control over the directionality and energy concentration of the signal. Once the number of antennas reaches a certain quantity, the beam is concentrated enough, and adding more antennas may not significantly enhance signal quality or secrecy.
• Near-field distance limitation: The unique spatial constraints of the near-field may imply that beyond a certain point, additional antennas do not improve system performance. This is because in the near-field range, the distance from the base station to the user is more critical than the variation in distance between antennas.
• Limited ideal channel state information: The base station has perfect knowledge of the channel information for all users, and the beamforming algorithms designed for the near-field environment have already achieved near-optimal performance with a smaller number of antennas. In this scenario, adding more antennas may not have a substantial impact.
• Resource allocation: In near-field communication, precise adjustments in power and antenna resource allocation are required to accommodate the characteristics of the near-field channel. Increasing the number of antennas necessitates more complex resource allocation strategies, which may result in gains in secrecy transmission rates not being as significant as expected.

Furthermore, in all the simulated results, whether in near-field communication or far-field communication, NOMA demonstrates significant advantages over OMA in multiple aspects. Firstly, by allowing multiple users to communicate simultaneously within the same time and frequency resources, NOMA enhances spectral efficiency, effectively utilizing the limited spectrum resources. Secondly, NOMA supports simultaneous access for multiple users, thereby increasing the system’s access capacity, particularly suitable for high-density user scenarios. Lastly, NOMA’s dynamic power allocation and successive interference cancellation techniques enhance communication security, making it more robust.

5. Conclusions

In our simulation study, we examined how the overall secrecy rate under different communication strategies varies with the increasing distance between the user and the base station. The results indicate that, whether in the near-field or far-field, as the distance increases, the overall secrecy rate decreases for all strategies. Among them, NF-NOMA maintains the highest secrecy rate at all distance settings, demonstrating its clear advantage in near-field communication. This is primarily attributed to the beamforming and power allocation strategies in the near-field channel model, allowing more precise signal directionality and eavesdropping resistance. Furthermore, when considering the constraint of the base station’s maximum transmit power, the NF-NOMA strategy can more effectively utilize available power resources to enhance secrecy rates. Regarding the number of antennas, although increasing the number of base station antennas can improve the secrecy rates for all communication strategies, the gain for the near-field strategy is smaller compared to the far-field strategy. This may be due to the saturation of near-field effects and the complexity of antenna resource allocation. Due to the diversity of communication systems, exploring eavesdroppers with multiple antennas is a meaningful direction for future research. The use of multiple antennas enables eavesdroppers to simultaneously capture signals from different spatial directions, enhancing their signal capturing capability. In this scenario, the channel from the eavesdropper to the BS can be represented as ${\mathbf{H}}_3(d,\theta )={\left[{\mathbf{h}}_{3,1},\cdots ,{\mathbf{h}}_{3,m_3},\cdots ,{\mathbf{h}}_{3,M_3}\right]}^T$. If the eavesdropper has complete knowledge of the CSI, the pre-coding matrix can be estimated through near-field beamforming by adjusting the antenna’s direction and distance to enhance eavesdropping efficiency. However, if the eavesdropper lacks knowledge of the channel state information, the zero forcing (ZF) pre-coding method needs to be employed. This ensures that the multi-antenna receiving end is only influenced by the desired signal, minimizing interference between antennas. In the simulation results, as the number of antennas at the eavesdropper increases, the system’s secrecy performance may degrade due to the eavesdropper’s enhanced ability to more effectively capture communication signals.