Dual-Function Radar Communications: A Secure Optimization Approach Using Partial Group Successive Interference Cancellation

Abstract

As one of the promising technologies of 6G, dual-function radar communication (DFRC) integrates communication and radar sensing networks. However, with the application and deployment of DFRC, its security problem has become a significantly important issue. In this paper, we consider the physical layer security of a DFRC system where the base station communicates with multiple legitimate users and simultaneously detects the sensing target of interest. The sensing target is also a potential eavesdropper wiretapping the secure transmission. To this end, we proposed a secure design based on partial group successive interference cancellation through fully leveraging the split messages and partially decoding to improve the rate increment of legitimate users. In order to maximize the radar echo signal-to-noise ratio (SNR), we formulate an optimization problem of beamforming and consider introducing new variables and relaxing the problem to solve the non-convexity of the problem. Then, we propose a joint secure beamforming and rate optimization algorithm to solve the problem. Simulation results demonstrate the effectiveness of our design in improving the sensing and secrecy performance of the considered DFRC system.

1. Introduction

As the two of the most significant applications of radio frequency technologies, radar and wireless communication have been widely used in various fields, such as weather monitoring, localization, traffic management, autonomous vehicles, and unmanned systems [1,2,3,4,5]. Over the recent few decades, radar and communication systems have been always designed and developed independently, each catering to distinct functions and operating within their respective frequency bands [6,7]. However, the exponential proliferation of mobile internet devices has led to a heightened demand for a spectrum in wireless communications in recent years [8]. Due to the scarcity of spectrum, exploring the sharing of spectrum between wireless communication and radar has become an increasingly urgent research issue [9]. As a result, the radar and wireless communication sharing spectrum has attracted a lot of attention [10]. These include the coexistence of radar and wireless communication and dual-function radar communication (DFRC) [11,12]. Generally, the coexistence of radar and wireless communication can be implemented and developed independently according to their respective functions, so they will not interfere with each other [13]. However, with the advancement and growing demand for future networks, dual-function radar communication offers numerous advantages that are poised to establish it as a prominent trend in the field [14]. First, the integration of the radar and communication network can deal with the problem of spectrum scarcity more efficiently. In addition, dual-function radar communication meets the requirements of integrated communication and sensing in future networks [15,16]. Specifically, with the increasing complexity of the electromagnetic environment, future wireless communication networks need to integrate sensing data [17]. Moreover, radar not only enables sensing functions for communication networks but also enables communications to achieve more accurate channel estimation by allowing for better interference management enhanced signal processing capabilities, as well as optimized resource allocation in dynamic environments, which can improve the overall performance and reliability of wireless systems [18,19]. Therefore, the development of dual-function radar communication has received a lot of attention due to the need for radar and communications to meet the demands of spectrum efficiency, performance gain, and complex tasks [20].

A critical issue of dual-function radar communication is the security problem of the integrated system [21]. The main purpose of a radar system is to detect the sensing target of interest and the communication system needs to transmit the desired messages to the users. When integrating the radar system and the communication system into the dual-functional system, it is worth noting that the sensing target of interest could wiretap the communication between the dual-functional base station and the legitimate users through the dual-functional waveform [22,23]. In this case, the transmission messages are leaked to the sensing target, which leads to severe security challenges of the information transmission in the dual-functional systems [24]. Therefore, the security problem is significant in the dual-function radar communication system during the pursuit of communication and sensing performance [25]. Physical layer security technology exploits the difference in channel state information (CSI) between legitimate links and eavesdropping links, thereby reducing the risk of wiretap and enhancing confidentiality [26,27]. Compared with key-based cryptography technology, physical layer security has its unique advantages [28]. Key-based cryptography needs extra communication resources to share the key, and it is difficult to ensure information security after the key is cracked [29]. Unlike cryptography algorithms, physical layer security technology is based on the randomness and diversity of wireless channels to ensure the security of communication from the information theory [30]. Moreover, key-based cryptography technologies cause high computational costs, which may result in excessive latency. Considering that the dual-function radar and communication system need to support sensing tasks and a large number of heterogeneous communication devices simultaneously, the high computational cost and excessive latency will significantly affect the sensing and communication performance [17]. Therefore, it is of great significance to consider the physical layer security technology to improve the security performance while ensuring the radar sensing performance in the DFRC system.

1.1. Related Work

The joint design of integrating the radar system and the communication system into the DFRC system has been well studied in [31,32,33,34,35,36]. Beamforming techniques are proposed for the multi-user DFRC system, where two antenna deployments are considered for the radar and communication functions [31]. In the separated deployment, antennas of the dual-functional base station are divided into two groups, one for radar and the other for communication. The communication beamformer is optimized so that the obtained beampattern matches the radar beam pattern. In the shared deployment, all antennas transmit a joint waveform shared by radar and communication, and an appropriate detection beampattern is formulated while ensuring communication performance. Numerical results illustrate that the performance of shared deployment is significantly better than the separated deployment. A DFRC signal design framework for multiple-input and multiple-output (MIMO) systems is proposed in [32] that optimizes the signal-to-interference-plus-noise ratio (SINR) while addressing signal-dependent clutter through an innovative spectral position index and amplitude modulation method. The proposed iterative block enhancement approach effectively manages the non-convex optimization problem, ensuring compatibility with hardware constraints and communication demands that demonstrate its effectiveness in complex scenarios. In [33], the authors further consider the range sidelobe control in the MIMO DFRC system and propose a conventional waveform design. The studies in [31,32,33] demonstrate the feasibility of realizing the dual functions of communication and radar in the traditional MIMO communication system from the perspective of waveform level. On the basis of these studies, DFRC has been rapidly studied in a wide range of application scenarios. In the vehicular network, the joint radar sensing and communication function can assist vehicle localization and collision avoidance detection. In [37], two waveform design problems for the DFRC system are considered with the communication quality-of-service (QoS) requirements, and a strict energy constraint is designed to control the communication performance more accurately at every sampling moment. The authors in [34] introduce a novel beam tracking approach for millimeter wave (mmWave) communication systems and propose a tailored scheme based on DFRC. In order to realize the functions of sensing and communication at the roadside units, the authors in [35] propose a conventional framework for tracking and predicting the parameters of vehicles. Aided by radar’s sensing function, the overhead of the communication beam tracking can be greatly reduced while the proposed allocation scheme can ensure the total transmission rate and meliorate the sensing accuracy. The superiority of the proposed DFRC scheme in the considered vehicle-to-infrastructure network is demonstrated by the numerical results. In [36], a new watermarking framework DFRC waveform design is proposed and it shows the effectiveness in joint synthetic aperture radar imaging and communication. The above studies focus on specific beam tracking and information transmission design in DFRC systems for key application scenarios. However, these studies do not take into account security threats such as signal eavesdropping and interference that may exist in the application scenarios, which could seriously impact the task operation and performance of DFRC systems. Therefore, effectively addressing these security threats and ensuring the stability and security of DFRC systems has become one of the important research directions.

The security issues of dual-function radar communication systems have attracted widespread attention and research in recent years [38,39,40,41]. In [38], a novel secure pilot allocation method is proposed to protect the sensing and communication data by hiding the location information of the pilot. The sensing function of the DFCR system can provide secure communication by estimating the potential location of the eavesdropper. In [39], a sensing-aided physical layer security method is investigated where the dual-functional base station utilizes the combined Capon and approximate maximum likelihood technique to search the location of the eavesdropper. Then, a weighted optimization problem of the secrecy rate and the Cramér–Rao Bound (CRB) of sensing estimation. By meliorating the estimation accuracy, the sensing and secure function can benefit each other, which can lead to mutual performance improvement during the optimization process. Moreover, the spectral and energy efficiency of the DFRC system is considered to improve the physical layer security meeting the sensing requirement and the legitimate users’ minimum secrecy rate requirement [40]. Using the CRB as the sensing metric, a secure model is proposed and the trade-off between the sensing and communication performance under the physical layer security enhancement is characterized. The authors in [41] study the millimeter-wave (mmWave) DFRC system where the sensing target might wiretap the information between the dual-functional base station and the communication users. In order to maximize the SINR of the radar, the transmit waveform and receive beamforming are jointly optimized with security and power constraints.

However, the existing studies [38,39,40,41] mainly focus on optimizing the secure waveform from the sensing requirements and transmission rate requirements, while not taking the internal interference of the DFRC system into consideration for meliorating the secure performance. Motivated by the Han–Kobayashi scheme [42], dividing the transmitted messages into several layers enlarges the rate vector dimension and the receivers can partially decode the messages thereby. The achievable rate region and efficient rate allocation when partially cancelling the specific interference are investigated in [43]. In particular, the beampattern is optimized to approximate the desired radar beampattern to guarantee the radar sensing performance. However, it lacks the consideration of the secure performance and only considers the desired beampattern of target tracking without considering the echo signal SNR of the DFRC waveform. The authors in [44] study a new strategy in the DFRC system where the dual-functional transmitter employs rate splitting to split messages into common and private streams, and the receivers employ successive interference cancellation (SIC) to decode the messages. The proposed strategy achieves interference management between the communication users, interference management between the radar and communication function, and an improvement in sensing performance by utilizing the common stream SIC receivers. In [45], a conventional transmissive reconfigurable intelligent surface (TRIS) transceiver is proposed. The common stream is independently designed for the sensing task and an optimization problem is formulated to optimize the precoding matrix of the communication and sensing streams with the objective of sensing Quality of Service (QoS) criteria. In [46], the authors investigate a rate-splitting assisted DFRC beamforming design that uses the CRB as the radar performance metric in a multibeam satellite system and the numerical results show the superiority of interference management in the communication-sensing trade-off and target estimation performance. However, the investigation of [44,45,46] only splits the messages into common streams and private streams to manage the internal interference and lacks consideration of the receiver design and the secrecy performance of the system. The lack of existing research inspires us to consider whether the physical layer security of the DFRC system can be improved from the perspective of interference management.

1.2. Contribution and Paper Organization

Inspired by the above studies, we pay attention to the secrecy performance of the DFRC system from the perspective of internal interference rather than optimizing it only from the waveform perspective. In this paper, we investigate the physical layer security of the dual-function radar and communication system where the base station communicates with the users and detects the target of interest simultaneously. The sensing target wiretaps the communication between the base station and the users, which brings a security risk to the DFRC system. By taking interference management of the dual-functional system into consideration, we consider a secure approach that utilizes split messages at the dual-functional base station and the partial group successive interference cancellation (PGSIC) at the legitimate users to meliorate the transmission rate vector of the legitimate users, which can guarantee the physical layer security. Intending to maximize the radar sensing performance, we formulate an optimization problem of secure beamforming to maximize the echo signal-to-noise ratio (SNR) under the constraints of transmit power and secure transmission. Since the optimization problem is non-convex, we consider introducing the new variable and relaxing the problem. Then, we propose a joint secure beamforming and rate optimization algorithm that optimizes the secure beamforming design and the rate allocation. Simulation results verify the effectiveness of the proposed secure design in improving the sensing and secure transmission performance of the DFCR system. The main contributions of this paper are outlined as follows:

• Starting from the security of integrating radar and communication systems, we investigate a physical layer security issue in the dual-function radar communication scenario where the base station simultaneously performs radar target detection and communicates with multiple users. The target of interest has the potential to eavesdrop on the communication between the base station and legitimate users. By jointly considering the radar target echo SNR and increase in the legitimate transmission rate increment, we effectively optimize the radar target detection performance and secrecy performance of the system.
• Different from the existing security design and performance enhancement schemes, this paper fully takes into account the characteristics of the dual-function base station and legitimate users. By leveraging the split messages at the base station, we propose a secure design based on PGSIC that the legitimate user decodes the desired message in a partial and sequential way. Together with an optimized beamformer, our secure design can achieve further increments in the secure transmission rate of the DFRC system.
• In order to maximize the radar echo SNR, we formulate an optimization problem under the transmit power constraints and the legitimate rate increments and eavesdropping rate constraints, which guarantee the physical layer security requirement. To solve the non-convexity of the problem, we reformulate the optimization problem, introduce new variables, and relax the problem. Then, we propose a joint secure beamforming and rate increment optimization algorithm to iteratively optimize the rate increments of the legitimate users and the secure beamforming.

The remainder of this paper is organized as follows. Section 2 introduces the system model of the considered secure dual-functional radar and communication system. Section 3 provides the radar sensing and secure transmission design based on PGSIC. Section 4 presents the formulated problem and the joint secure beamforming and rate increment optimization algorithm. Section 5 provides numerical results and discussion. Section 6 concludes this paper.

2. System Model

Consider a dual-function radar communication system where a base station communicates with K legitimate users while detecting a target of interest simultaneously, as shown in Figure 1.

It is worth noting that with the complexity of the dual-function radar communication scenario, the detection target has the possibility of eavesdropping on the information between the base station and the legitimate users. Therefore, in this paper, the target of interest is assumed to be an eavesdropper who aims to wiretap secure communication between the base station and the legitimate users. The dual-functional base station is equipped with N antennas. The legitimate users and the eavesdropper are all equipped with a single antenna. The base station has knowledge of the channel state information of all nodes, which is a rational assumption for the dual-function radar communication system [47].

By employing the layered rate-splitting scheme in the dual-function radar communication system, the message transmitted to the k-th legitimate user from the dual-functional base station can be given as

$$x_k=\sqrt{{\displaystyle \frac{1}{L}}}\sum _{\ell =1}^L{x}_{k,\ell },\forall k\in \{1,\dots ,K\},$$

(1)

where L denotes the number of codebooks used for each message and $x_{k,\ell }$ denotes the unit-power input from the codebook $c_{k,\ell }$. The number of codebooks L can affect the freedom of interference cancellation, which is the essence of rate splitting. We define the set of all the codebooks for the k-th message is denoted by ${\mathcal{C}}_k\triangleq \{c_{k,1},\dots ,c_{k,L}\}$ with $c_{k,\ell }$ as a Gaussian codebook with rate $R_{k,\ell }$. Therefore, the rate of the k-th message is

$$R_k=\sum _{\ell =1}^L{R}_{k,\ell },\forall k\in \{1,\dots ,K\}.$$

(2)

The received signal at the k-th legitimate user can be given as

$$\begin{array}{cc}\hfill y_k& =\sum _{j=1}^K{\mathit{h}}_k^H{\mathit{w}}_j{x}_j+n_k\hfill \\ \hfill & =\sqrt{{\displaystyle \frac{1}{L}}}\sum _{j=1}^K{\mathit{h}}_k^H{\mathit{w}}_j\sum _{\ell =1}^L{x}_{k,l}+n_k,\forall k\in \{1,\dots ,K\},\hfill \end{array}$$

(3)

where ${\mathit{h}}_k\in {\mathbb{C}}^{N\times 1}$ denotes the channel between the base station and the k-th legitimate user, ${\mathit{w}}_j\in {\mathbb{C}}^{N\times 1}$ denotes the beamformer for the j-th message, and $n_k\sim \mathcal{CN}(0,{\sigma }_k^2)$ denotes the additive white Gaussian noise (AWGN) at the k-th legitimate user. The channel between the base station and the k-th legitimate user is detailed as

$${\mathit{h}}_k=\sqrt{\beta D_k^{-\lambda }}{\mathit{f}}_k,\forall k\in \{1,\dots ,K\},$$

(4)

where $\beta$ denotes the channel gain for the unit distance, $D_k$ denotes the distance between the base station and the k-th legitimate user, $\lambda$ denotes the path-loss exponent, and ${\mathit{f}}_k$ denotes the Non-Line-of-Sight (NLOS) component that is modelled as complex Gaussian distributed. Each element of ${\mathit{f}}_k$ follows the complex Gaussian distribution with $\mathcal{CN}(0,1)$. According to the calculation of the mean and variance of the complex Gaussian distribution, the distribution of ${\mathit{f}}_k$ is ${\mathit{f}}_k\sim \mathcal{CN}(\mathbf{0},\mathbf{I})$.

3. Radar Sensing and Secure Transmission Design

To enhance target tracking and ensure secrecy performance in the considered DFRC system, we consider the specific radar sensing requirement for detection performance and a secure design based on partial group successive interference cancellation to improve the secure rate increments of the legitimate users.

3.1. Radar Sensing Requirement

In pursuit of superior radar sensing performance and minimizing the unnecessary consumption of wireless resources during target detection, the minimum SNR of the echo signal is employed as the key performance metric for the considered dual-function radar communication system. To be specific, the echo signal SNR is detailed as

$${\zeta }_{echo}={\displaystyle \frac{{\sum }_{k=1}^K{\parallel \mathbf{H}\left({\theta }_0\right){\mathit{w}}_k\parallel }^2}{{\sigma }_b^2}},$$

(5)

where ${\sigma }_b^2$ represents the power of noise at the dual-functional base station. $\mathbf{H}\left({\theta }_0\right)$ is given by

$$\mathbf{H}\left({\theta }_0\right)=\varphi \beta D_E^{-\lambda }{\mathit{\alpha }}^{*}\left({\theta }_0\right){\mathit{\alpha }}^{†}\left({\theta }_0\right),$$

(6)

where $\varphi$ denotes the radar cross section, $D_E$ denotes the distance between the base station and the target (eavesdropper), and $\mathit{\alpha }\left({\theta }_0\right)\in {\mathbb{C}}^{N\times 1}$ denotes the steering vector. Specifically, $\mathit{\alpha }\left({\theta }_0\right)$ is expressed as

$$\mathit{\alpha }\left({\theta }_0\right)={[1,e^{j2\pi \eta sin\left({\theta }_0\right)},\dots ,e^{j2\pi (N-1)\eta sin\left({\theta }_0\right)}]}^T,$$

(7)

where ${\theta }_0$ denotes the angle of the target and $\eta$ denotes the interval normalized by the wavelength between adjoining antennas.

3.2. Eavesdropping Performance

Given that the detected target is regarded as a potential eavesdropper, the possibility of a wiretap by the target needs to be incorporated into the analysis of the secure performance of the dual-function radar communication system. Therefore, as the target plays the role of the eavesdropper, the eavesdropping SINR of the target towards the k-th legitimate user can be given as

$${\gamma }_e^k={\displaystyle \frac{|{\mathit{h}}_e^H{\mathit{w}}_k{|}^2}{{\sum }_{j=1,j\ne k}^K{\left|{\mathit{h}}_e^H{\mathit{w}}_j\right|}^2+{\sigma }_e^2}},\forall k\in \{1,\dots ,K\},$$

(8)

where ${\mathit{h}}_e\in {\mathbb{C}}^{N\times 1}$ denotes the channel vector between the base station and the eavesdropper (target) and ${\sigma }_e^2\sim \mathcal{CN}(0,{\sigma }_e^2)$ denotes AWGN at the eavesdropper. The channel vector ${\mathit{h}}_e$ is detailed as

$${\mathit{h}}_e=\sqrt{\beta D_E^{-\lambda }}\mathit{\alpha }\left({\theta }_0\right),$$

(9)

where $D_E$ denotes the distance between the base station and the eavesdropper (target).

3.3. Secure Design

Since the target of interest has the potential of eavesdropping the communication between the base station and the legitimate users in the considered scenario, it is important to maintain the radar detection performance and secure transmission performance of the dual-function radar and communication system. Considering that the messages transmitted from the base station are split into several layers, it allows for partial decoding to enhance the transmission rate for legitimate users. Since the sensing target is not a legitimate communication user in the system, the sensing target cannot implement the same decoding process as the legitimate users. In this way, the secure transmission rates can be improved. Therefore, we consider deploying a more advanced interference management, partial group successive interference cancellation (PGSIC) scheme, at the receiver of the legitimate users in the secure dual-function radar and communication system to improve the secrecy performance in the presence of the eavesdropping target. Specifically, denote $\mathcal{A}\left(k\right)$ as the set of the desired messages of the k-th legitimate user. The k-th legitimate user in the system performs the partial group successive interference cancellation scheme to obtain the messages in $\mathcal{A}\left(k\right)$. For a given set of beamformer $\left\{{\mathit{w}}_j\right\}$ and rate vector ${\mathit{R}}^0=\left\{R_{j,\ell }^0\right\}$, it first executes PGSIC(OP) in the Appendix A to obtain the optimal partition $\underline{\mathcal{Q}}\left(k\right)\triangleq \{{\mathcal{Q}}_1\left(k\right),\dots ,{\mathcal{Q}}_{p_k}\left(k\right),{\mathcal{Q}}_{p_k+1}\left(k\right)\}$, where ${\mathcal{Q}}_1\left(k\right)$ denotes the set of layers to be jointly decoded during the first stage and ${\mathcal{Q}}_{p_k}\left(k\right)$ denotes the set of layers to be jointly decoded during the $p_k$-th stage of successive interference cancellation process. Once the $\underline{\mathcal{Q}}\left(k\right)$ and the $\mathit{R}$ are obtained, the k-th legitimate user can execute the PGSIC(GRI) and PGSIC(IRI) to obtain the rate increments. The search process of determining the best set of sub-messages to be decoded and the derivation of the achievable rate region can be further referenced in [43]. The detailed process can be briefly described as follows.

First, the optimal partition needs to be computed. For the k-th legitimate user, we denote ${\mathit{R}}_k^0=\left\{R_{j,\ell }^0\right|(j,\ell )\in \mathcal{A}\left(k\right)\}$. The channel vector between the base station and the k-th legitimate user is ${\mathit{h}}_k$. The k-th legitimate user employs the partial group successive interference cancellation to obtain the optimal partition $\underline{\mathcal{Q}}\left(k\right)\triangleq \{{\mathcal{Q}}_1\left(k\right),\dots ,{\mathcal{Q}}_{p_k}\left(k\right),{\mathcal{Q}}_{p_k+1}\left(k\right)\}$. The maximum group size of layers that can be jointly decoded at each stage is $\mu$. To be specific, the k-th legitimate user decodes the layers in the first $p_k$ stages of the $\underline{\mathcal{Q}}\left(k\right)$ successively. In the j-th stage, $1\le j\le p_k$, the layers in ${\mathcal{Q}}_j\left(k\right)$ are jointly decoded with $\{{\mathcal{Q}}_{j+1}\left(k\right),\cdots ,{\mathcal{Q}}_{p_k+1}\}$ treated as noise. After the layers in ${\mathcal{Q}}_j\left(k\right)$ have been successfully decoded, the received signal will subtract the layers in ${\mathcal{Q}}_j\left(k\right)$ for further interference cancellation. Then, the k-th legitimate user decodes the layers in ${\mathcal{Q}}_{j+1}$ while treating the layers in $\{{\mathcal{Q}}_{j+2}\left(k\right),\dots ,{\mathcal{Q}}_{p_k+1}\}$ as noise. The same operation is performed until the $p_k$ stage has been completed. The optimal partition and the decoding order can be obtained by PGSIC(OP) as detailed in the Appendix A. Then, the dual-functional base station allocates the rates with the feedback. One way is global rate increment assignment (PGSIC(GRI)) where the base station calculates the minimum value of all legitimate users. The other way is iterative rate increment assignment (PGSIC(IRI)) where the base station calculates a specific scalar value of all users to allocate the rate increments.

With the optimal partition of the interference cancellation order obtained by PGSIC(OP), each legitimate user implements PGSIC(GRI) and PGSIC(IRI) to compute the rate increments of the DFRC system. Denote all indices of all layered sub-messages as $\mathcal{M}$. Specifically, for any two disjoint subset $\mathcal{W}$, $\mathcal{Y}\subseteq \mathcal{M}$, denote the achievable rate region of the k-th legitimate user as ${\mathcal{C}}_k(\mathcal{W},\mathcal{Y})$. Specifically, it represents that the k-th legitimate user decodes the sub-messages in the set $\mathcal{W}$ while taking the consideration of the set $\mathcal{Y}$ as noise. ${\mathcal{C}}_k(\mathcal{W},\mathcal{Y})$ can be expressed as

$$\begin{array}{c}\hfill {\mathcal{C}}_k(\mathcal{W},\mathcal{Y})=\{{\mathit{R}}_{\mathcal{W}}\in {\mathbb{R}}_{+}^{\left|\mathcal{W}\right|}:\forall \mathcal{B}\subseteq \mathcal{W},\mathcal{B}\ne Ø,\\ \hfill \sum _{(j,\ell )\in \mathcal{B}}{R}_{j,\ell }\le {\mathcal{R}}_k(\mathcal{B},\mathcal{Y})\}.\end{array}$$

(10)

where $\mathcal{B}$ denotes any non-empty subset of $\mathcal{W}$. In particular, ${\mathcal{R}}_k(\mathcal{B},\mathcal{Y})$ can be given as

$${\mathcal{R}}_k(\mathcal{B},\mathcal{Y})\triangleq {log}_2\left(1+{\displaystyle \frac{{\sum }_{(j,\ell )\in \mathcal{B}}\frac{1}{L}{\left|{\mathit{h}}_k^H{\mathit{w}}_j\right|}^2}{{\sum }_{(j,\ell )\in \mathcal{Y}}\frac{1}{L}{\left|{\mathit{h}}_k^H{\mathit{w}}_j\right|}^2+{\sigma }_k^2}}\right).$$

(11)

The achievable region of the PGSIC in the considered secure dual-function radar communication system can be expressed as ${\cap }_{k=1}^K{\cap }_{i=1}^{p_k}{\mathcal{C}}_k\left({\mathcal{Q}}_i\left(k\right)\right)$. The messages in each partition stage are decoded sequentially by the k-th legitimate user. Denote $\overline{{\mathcal{Q}}_i\left(k\right)}={\cup }_{m=i+1}^{p_k+1}{\mathcal{Q}}_m\left(k\right)$. Therefore, the secure transmission rate requirement of the k-th legitimate user can be expressed as

$$\begin{array}{c}\hfill {log}_2\left(1+{\displaystyle \frac{{\sum }_{(j,\ell )\in {\mathcal{U}}_i}\frac{1}{L}{\left|{\mathit{h}}_k^H{\mathit{w}}_j\right|}^2}{{\sum }_{(j,\ell )\in \overline{{\mathcal{Q}}_i\left(k\right)}}\frac{1}{L}{\left|{\mathit{h}}_k^H{\mathit{w}}_j\right|}^2+{\sigma }_k^2}}\right)\ge \sum _{(j,\ell )\in {\mathcal{U}}_i}{R}_{j,\ell },\\ \hfill \forall i\in \{1,\dots ,p_k\},\forall {\mathcal{U}}_i\subseteq {\mathcal{Q}}_i\left(k\right),{\mathcal{U}}_i\ne Ø,\forall k\in \{1,\dots ,K\}.\end{array}$$

(12)

4. Problem Formulation and Optimization Algorithm

In this paper, we aim to improve the radar detection performance in the DFRC scenario. Considering that the detection target may pose a security threat to the transmission, a secure design based on the partial group successive interference cancellation scheme for legitimate users is proposed. Based on this, in this section, we formulate the radar echo SNR maximization problem constrained by the power and physical layer security requirements of transmission rates. Since the problem is non-convex, we introduce new variables and relax the problem. Then, a joint secure beamforming and rate optimization problem is proposed.

4.1. Problem Formulation

In order to maximize the radar echo SNR while guaranteeing the physical layer security requirements, we formulate an optimization problem of transmit beamforming design with the considered secure design. Specifically, for a given valid partition of the k-th legitimate user obtained by the PGSIC $\underline{\mathcal{Q}}\left(k\right)=\{{\mathcal{Q}}_1\left(k\right),{\mathcal{Q}}_2\left(k\right),\dots ,{\mathcal{Q}}_{p_k+1}\left(k\right)\}$, the optimization problem can be formulated as

$$\begin{array}{cc}\mathrm{(13a)}& \hfill \underset{\mathit{w}}{max}& {\displaystyle \frac{{\sum }_{k=1}^K{\parallel \mathbf{H}\left({\theta }_0\right){\mathit{w}}_k\parallel }^2}{{\sigma }_b^2}}\hfill \hfill \mathrm{s}.\mathrm{t}.& {log}_2\left(1+{\displaystyle \frac{{\sum }_{(j,\ell )\in {\mathcal{U}}_i}\frac{1}{L}{\left|{\mathit{h}}_k^H{\mathit{w}}_j\right|}^2}{{\sum }_{(j,\ell )\in \overline{{\mathcal{Q}}_i\left(k\right)}}\frac{1}{L}{\left|{\mathit{h}}_k^H{\mathit{w}}_j\right|}^2+{\sigma }_k^2}}\right)\ge \sum _{(j,\ell )\in {\mathcal{U}}_i}{R}_{j,\ell },\hfill \\ \mathrm{(13b)}& \hfill & \forall i\in \{1,\dots ,p_k\},\forall {\mathcal{U}}_i\subseteq {\mathcal{Q}}_i\left(k\right),{\mathcal{U}}_i\ne Ø,\forall k\in \{1,\dots ,K\},\hfill \mathrm{(13c)}& \hfill & {\displaystyle \frac{|{\mathit{h}}_e^H{\mathit{w}}_k{|}^2}{{\sum }_{j=1,j\ne k}^K{\left|{\mathit{h}}_e^H{\mathit{w}}_j\right|}^2+{\sigma }_e^2}}\le {\mu }_{\mathrm{e},\mathrm{th}}^k,\forall k\in \{1,\cdots ,K\},\hfill \mathrm{(13d)}& \hfill & \sum _{k=1}^K{\parallel {\mathit{w}}_k\parallel }^2\le P,\hfill \end{array}$$

where ${\mu }_{\mathrm{e},\mathrm{th}}^k$ denotes the SINR threshold of the eavesdropping rate towards the k-th legitimate user and P denotes the total transmit power of the dual-functional base station of the system. Noted that the Constraints (13b) and (13c) guarantee the physical layer security of the DFRC system [48]. However, the optimization problem (13) is non-convex and is difficult to solve.

4.2. Problem Relaxation

To solve the non-convex optimization problem, we introduce new variables, rewrite the optimization problem, and relax the problem. First, the Constraint (13b) is difficult to solve in the expression of logarithm operations. Therefore, we consider writing the rate $R_{j,\ell }$ on the right side of (13b) as the logarithm of the corresponding SINR, so as to remove the logarithmic operations of the constraint. Denote

$$R_{j,\ell }\triangleq {log}_2({\mu }_{j,\ell }+1),\forall j\in \{1,\dots ,K\},\ell \in \{1,\dots ,L\},$$

(14)

where ${\mu }_{j,\ell }$ represents the corresponding SINR in the logarithm operations of $R_{j,\ell }$. Therefore, the right-hand side of (13b) can be expressed as

$$\sum _{(j,\ell )\in {\mathcal{U}}_i}{R}_{j,\ell }=\sum _{(j,\ell )\in {\mathcal{U}}_i}{log}_2({\mu }_{j,\ell }+1)={log}_2\left(\prod _{(j,\ell )\in {\mathcal{U}}_i}({\mu }_{j,\ell }+1)\right).$$

(15)

For the ease of expression of (13b), we can define

$$f\left({\mu }_{{\mathcal{U}}_i}\right)\triangleq \prod _{(j,\ell )\in {\mathcal{U}}_i}\left({\mu }_{j,\ell }+1\right)-1.$$

(16)

Consequently, the formulated problem (13) can be expressed as

$$\begin{array}{cc}\mathrm{(17a)}& \hfill \underset{\mathit{w}}{max}& {\displaystyle \frac{{\sum }_{k=1}^K{\parallel \mathbf{H}\left({\theta }_0\right){\mathit{w}}_k\parallel }^2}{{\sigma }_b^2}}\hfill \hfill \mathrm{s}.\mathrm{t}.& {\displaystyle \frac{{\sum }_{(j,\ell )\in {\mathcal{U}}_i}\frac{1}{L}{\left|{\mathit{h}}_k^H{\mathbf{w}}_j\right|}^2}{{\sum }_{(j,\ell )\in \overline{{\mathcal{Q}}_i\left(k\right)}}\frac{1}{L}{\left|{\mathit{h}}_k^H{\mathbf{w}}_j\right|}^2+{\sigma }_k^2}}\ge f\left({\mu }_{{\mathcal{U}}_i}\right)\hfill \\ \mathrm{(17b)}& \hfill & \forall i\in \{1,\dots ,p_k\},\forall {\mathcal{U}}_i\subseteq {\mathcal{Q}}_i\left(k\right),{\mathcal{U}}_i\ne Ø,\forall k\in \{1,\dots ,K\},\hfill \mathrm{(17c)}& \hfill & {\displaystyle \frac{|{\mathit{h}}_e^H{\mathit{w}}_k{|}^2}{{\sum }_{j=1,j\ne k}^K{\left|{\mathit{h}}_e^H{\mathit{w}}_j\right|}^2+{\sigma }_e^2}}\le {\mu }_{\mathrm{e},\mathrm{th}}^k,\forall k\in \{1,\dots ,K\},\hfill \mathrm{(17d)}& \hfill & \sum _{k=1}^K{\parallel {\mathit{w}}_k\parallel }^2\le P,\hfill \end{array}$$

where (17b) represents the transmission requirement of the legitimate users, (17c) represents the eavesdropping requirement of the eavesdropper (target), and (17d) represents the transmit power requirement of the dual-function radar and communication system. Although the form of (17) is simplified compared to (13), (17) is still a non-convex problem and difficult to solve. This is because the objective function of (17) is convex and the convexity and concavity of the constraints (17b) and (17c) are difficult to determine subject to $\mathit{w}$. To this end, we consider that the variables of (17) are relaxed by introducing a matrix variable [49]. Specifically, we consider introducing the new variables ${\mathbf{W}}_k={\mathit{w}}_k{\mathit{w}}_k^H$, $\forall k\in \{1,\dots ,K\}$. Therefore, the optimization problem (17) can be rewritten as

$$\begin{array}{cc}\mathrm{(18a)}& \hfill \underset{\mathbf{W}}{max}& {\displaystyle \frac{{\sum }_{k=1}^K\mathrm{tr}\left(\overline{\mathbf{H}}\left({\theta }_0\right){\mathbf{W}}_k\right)}{{\sigma }_b^2}}\hfill \hfill \mathrm{s}.\mathrm{t}.& \sum _{(j,\ell )\in {\mathcal{U}}_i}\mathrm{tr}\left({\mathbf{H}}_k{\mathbf{W}}_j\right)-f\left({\mu }_{{\mathcal{U}}_i}\right)\sum _{(j,\ell )\in \overline{{\mathcal{Q}}_i\left(k\right)}}\mathrm{tr}\left({\mathbf{H}}_k{\mathbf{W}}_j\right)-f\left({\mu }_{{\mathcal{U}}_i}\right)L{\sigma }_k^2\ge 0,\hfill \\ \mathrm{(18b)}& \hfill & \forall i\in \{1,\dots ,p_k\},\forall {\mathcal{U}}_i\subseteq {\mathcal{Q}}_i\left(k\right),{\mathcal{U}}_i\ne Ø,\forall k\in \{1,\dots ,K\},\hfill \mathrm{(18c)}& \hfill & \mathrm{tr}\left({\mathbf{H}}_e{\mathbf{W}}_k\right)-{\mu }_{\mathrm{e},\mathrm{th}}^k\sum _{j=1,j\ne k}^K\mathrm{tr}\left({\mathbf{H}}_e{\mathbf{W}}_j\right)-{\mu }_{\mathrm{e},\mathrm{th}}^k{\sigma }_e^2\le 0,\forall k\in \{1,\cdots ,K\},\hfill \mathrm{(18d)}& \hfill & \sum _{k=1}^K\mathrm{tr}\left({\mathbf{W}}_k\right)\le P,\hfill \mathrm{(18e)}& \hfill & \mathrm{rank}\left({\mathbf{W}}_k\right)=1,\hfill \end{array}$$

where $\overline{\mathbf{H}}\left({\theta }_0\right)={\mathbf{H}}^H\left({\theta }_0\right)\mathbf{H}\left({\theta }_0\right)$, ${\mathbf{H}}_k={\mathit{h}}_k{\mathit{h}}_k^H$, ${\mathbf{H}}_e={\mathit{h}}_e{\mathit{h}}_e^H$. The objective function and constraints (18b) and (18c) are linear. However, due to the new constraint (18e) introduced during relaxation, problem (18) is still a non-convex problem. By considering the relaxation of the non-convex rank-one constraint (18e), the optimization problem can be expressed as

$$\begin{array}{cc}\mathrm{(19a)}& \hfill \underset{\mathbf{W}}{max}& {\displaystyle \frac{{\sum }_{k=1}^K\mathrm{tr}\left(\overline{\mathbf{H}}\left({\theta }_0\right){\mathbf{W}}_k\right)}{{\sigma }_b^2}}\hfill \hfill \mathrm{s}.\mathrm{t}.& \sum _{(j,\ell )\in {\mathcal{U}}_i}\mathrm{tr}\left({\mathbf{H}}_k{\mathbf{W}}_j\right)-f\left({\mu }_{{\mathcal{U}}_i}\right)\sum _{(j,\ell )\in \overline{{\mathcal{Q}}_i\left(k\right)}}\mathrm{tr}\left({\mathbf{H}}_k{\mathbf{W}}_j\right)-f\left({\mu }_{{\mathcal{U}}_i}\right)L{\sigma }_k^2\ge 0,\hfill \\ \mathrm{(19b)}& \hfill & \forall i\in \{1,\dots ,p_k\},\forall {\mathcal{U}}_i\subseteq {\mathcal{Q}}_i\left(k\right),{\mathcal{U}}_i\ne Ø,\forall k\in \{1,\dots ,K\},\hfill \mathrm{(19c)}& \hfill & \mathrm{tr}\left({\mathbf{H}}_e{\mathbf{W}}_k\right)-{\mu }_{\mathrm{e},\mathrm{th}}^k\sum _{j=1,j\ne k}^K\mathrm{tr}\left({\mathbf{H}}_e{\mathbf{W}}_j\right)-{\mu }_{\mathrm{e},\mathrm{th}}^k{\sigma }_e^2\le 0,\hfill \mathrm{(19d)}& \hfill & \sum _{k=1}^K\mathrm{tr}\left({\mathbf{W}}_k\right)\le P,\hfill \end{array}$$

which is a convex problem now. Problem (19) can be considered as the SDR of the problem (18). For the relaxed problem, we provide a joint optimization algorithm to solve it in the following.

4.3. Joint Secure Beamforming and Rate Optimization Algorithm

In the considered dual-function radar communication system, the dual-functional base station transmits messages to legitimate users while simultaneously detecting the target of interest. The target of interest wiretaps the secure communication between the base station and the legitimate users. In order to improve the secrecy performance while guaranteeing the detection performance of the system, we consider the echo signal SNR as the radar sensing requirement and provide a secure design based on PGSIC to improve the secure rate vector during the communication process. Specifically, the transmitted messages are split into several layers at the dual-functional base station. Legitimate users utilize the partial interference cancellation successively decoding the desired sub-messages to meliorate the secure rate increments. We formulate an optimization problem that aims to maximize the radar echo SNR while meliorating the secure communication performance of the DFRC system. Since the optimization problem is non-convex, we introduce new variables, transform the problem, and relax the problem to be convex. In the following, we provide a joint secure beamforming and rate optimization algorithm by optimizing the rate optimization achieved by the secure design based on PGSIC and secure beamforming design iteratively.

Generally, determining the feasibility of the partial partition in the considered secure sensing and communication system is difficult. Once the partition is obtained by the optimal partition algorithm PGSIC(OP), the feasibility of the transmission requirement can be guaranteed [43]. Based on the obtained optimal partition, the system coordinates the rate increments of all users globally and iteratively through PGSIC(GRI) and PGSIC(IRI). The difference between the two ways is whether all layers are assigned the same increment to improve the secrecy transmission rate of the system. With the obtained rate increments, the joint algorithm solves the formulated problem to optimize the secure beamforming design. It is worth noting that (19) is a convex problem and can be easily solved. However, the obtained optimal solution must be checked to see whether the rank-one constraint is satisfied. The desired optimal solution ${\mathit{w}}_k$ can be acquired directly by ${\mathbf{W}}_k={\mathit{w}}_k{\mathit{w}}_k^H$ if all matrices are rank-one. When not all matrices ${\mathbf{W}}_k$ are rank-one, $\mathrm{tr}\left({\mathbf{W}}_k\right)$ is a lower bound. If the rank of ${\mathbf{W}}_k$ is larger than 1, ${\mathit{w}}_k$ must be extracted from the matrices. We consider utilizing the Gaussian randomization to extract the rank-one solutions.

Once the secure beamformers are optimized with the given rate increments, the rate increments can be further optimized based on the obtained beamformers. Then, we can obtain the optimized rate increments. With the improved rate increments, the secure beamformers can be optimized again. It can be seen that the secure transmission and the radar detection performances can be further improved in this way. Therefore, the joint algorithm iteratively implements the secure beamforming optimization and the rate increment optimization of the partial group successive interference cancellation until convergence has been achieved. The whole optimization process is summarized in Algorithm 1.

Algorithm 1 Joint secure beamforming and rate optimization algorithm

1: Initialize all beamformers $\left\{{\mathit{w}}_k^0\right\}$. 2: Compute the initial rate increments $\left\{R_{j,\ell }^0\right\}$. 3: $\mathbf{Repeat}$ 4:  Run PGSIC(OP) to obtain the optimal partition $\left\{\underline{\mathcal{Q}}\left(k\right)\right\}$. 5:  Run PGSIC(GRI) or PGSIC(IRI) to allocate the rate increments $\mathit{R}=\left\{R_{j,\ell }\right\}$. 6:  Obtain $\left\{{\mathbf{W}}_k\right\}$ with the given rate increments by solving (19). 7:  $\mathbf{If}$ all matrices $\left\{{\mathbf{W}}_k\right\}$ satisfy rank-one requirement, obtain $\left\{{\mathit{w}}_k\right\}$ directly; 8:  $\mathbf{Else}$ utilize Gaussian Randomization for $\left\{{\mathit{w}}_k\right\}$. 9: Until The improvement of the radar echo SNR is smaller than a given threshold $ϵ$. 10: Output $\left\{R_{j,\ell }\right\}$ and $\left\{{\mathit{w}}_k\right\}$.

5. Results and Discussion

In this section, the numerical results verify the effectiveness of the proposed secure design in improving the sensing and security performance of the DFRC system. The dual-functional base station is equipped with $N=16$ antennas. The default number of legitimate users is $K=4$ and $\mu =1$. In the following results, GRI refers to the global rate increment assignment of the rate vector optimization (PGSIC(GRI)), and IRI refers to the iterative rate increment assignment of the rate vector optimization (PGSIC(IRI)). Set $\lambda =3$, $\beta =1\times {10}^{-4}$ [50], $D_k=30$ m, $D_E=80$ m, $\varphi =1$, and ${\theta }_0=0^{\circ }$, and ${\sigma }_k={\sigma }_e={\sigma }_b=-110$ dBm [51].

5.1. Radar Performance

Figure 2 shows the radar echo SNR under different transmit power of the base station achieved by the proposed secure design. First, as expected, all curves increase as the transmit power of the base station increases. When the transmit power $P=0.1$ W, the differences between the curves are small. As the increasing of the transmit power, the differences between the curves gradually become apparent. This is reasonable because the sensing and communication performance of the dual-functional system can be significantly improved with the increase in transmit power. Second, for the given transmit power and the same rate increment assignment way to obtain the rate vectors, the radar echo SNR increases with the number of layers increasing, which illustrates the sensing performance can be improved by the proposed secure approach. This is because splitting the desired messages and partially decoding the messages can improve the transmission rates of legitimate users, which can save more potential of the dual-function base station for detecting the target of interest. Third, with the given transmit power and the same number of split layers at the base station, IRI always achieves better radar sensing than GRI. Moreover, it can be observed that when $L=1$, the radar echo SNR achieved by the two methods is significantly different. With the increase in L, the radar echo SNR achieved by the two methods is improved, but the difference between the two ways becomes smaller.

5.2. System Level Secrecy Performance

Figure 3 and Figure 4 show the system-level secrecy performance achieved by our secure optimization approach with two rate increment assignments in the considered dual-functional radar and communication system. Specifically, the effects of layers L and group size $\mu$ are also compared in the two figures. We use the traditional single-user decoding scheme, minimum mean square error decoding scheme (MMSE), as the comparison to illustrate the effectiveness of our proposed secure design. Specifically, Figure 3 plots the sum secrecy rate of the DFRC system achieved by our secure optimization approach in GRI. It can be observed that the sum secrecy rates achieved by our proposed secure design with GRI are significantly better than the traditional MMSE condition, which demonstrates the effectiveness of the proposed secure design for the DFRC system. In Figure 3, the curves with GRI increase with the increasing transmit power of the base station. The sum secrecy rate of the curve $L=1,\mu =1$ increases slowly when the transmit power is low. This is reasonable since the small transmit power and the number of dividing layers limit the rate increments at the legitimate users. The other four curves increase significantly with the increase in transmit power, which shows the great potential of the proposed secure design in the DFRC system. By comparing the three curves $L=1,\mu =1$, $L=2,\mu =1$ and $L=3,\mu =1$, it can be seen that increasing the number of layers of the messages at the dual-functional base station can effectively improve the sum secrecy rate of the system when the group size $\mu$ at the legitimate is the same. It demonstrates that a larger L gives the receivers more freedom to decide which part of the interfering messages to decode and which to treat as noise. Moreover, the three curves $L=3,\mu =1$, $L=3,\mu =2$, and $L=3,\mu =3$ show that when the split layers are fixed at the transmitter, the increase in the decoding group size $\mu$ at the legitimate user can also improve the secrecy performance of the DFRC system.

Figure 4 plots the sum secrecy rate of the DFRC system achieved by our secure optimization approach in IRI. We can observe that the sum secrecy rates achieved by our proposed secure design with IRI are noticeably better than the traditional MMSE condition, which illustrates the efficiency of our design in improving security performance. Compared with Figure 3, it can be observed that IRI achieves higher secrecy rates under the same conditions, which illustrates that IRI can further improve the secrecy performance significantly by iteratively updating the rate increment according to different users. In addition, as expected, it can be observed that the sum secrecy rates in the curves with IRI increase with the increase in transmit power. Similar to Figure 3, when the decoding group size of the legitimate user $\mu$ is fixed, the sum secrecy rate increases with the increase in the number of split layers. When the number of message split layers of the dual-function transmitter is fixed, the sum secrecy rate is improved as the group size of the legitimate user increases. As we explained above in Figure 3, it demonstrates that our secure design can fully consider and utilize the characteristics of the transmitting end of the dual-functional base station and the receiving end of the legitimate user, so as to effectively improve the secrecy performance of the DFRC system. In addition, compared with Figure 3, it is worth noting that in Figure 4, the sum secrecy rate of the curve $L=1,\mu =1$ is still significantly improved with the increase in power. This illustrates that IRI has the potential to further meliorate the secrecy performance of the system compared with GRI when the transmit power is small. Comparing the curves $L=3,\mu =2$ and $L=3,\mu =3$ in Figure 3 and Figure 4, the difference between the curves in IRI is also larger than the difference in GRI. This indicates that IRI also has great potential to further enhance the system’s secrecy performance, even when the number of layers and the group size are large.

5.3. User Level Secrecy Performance

Figure 5 presents the transmission rates achieved by our proposed design using GRI and the traditional MMSE condition of each legitimate user and the eavesdropping rates for the corresponding legitimate user in the DFRC system with four legitimate users when $L=2,\mu =1,P=0.1$ W. It can be observed that the transmission rates achieved by GRI of all four users are greater than their respective eavesdropping rates, indicating that the proposed secure transmission design guarantees the physical layer security for all legitimate users. Compared with the traditional MMSE scheme, our proposed secure design achieves higher achievable rates than the traditional MMSE condition. Moreover, it can be seen that the secrecy performance achieved by MMSE barely guarantees physical layer security at some users, which is far less secure than the secrecy performance achieved by our design. Moreover, the eavesdropping rates of the four legitimate users remain at a relatively average level, and the transmission rate distribution achieved by GRI is also quite uniform. Consequently, the secrecy rate of User 3 is relatively low, while that for User 1 and User 4 is relatively high. It shows that GRI optimizes the rate increment uniformly across all users, leading to poorer security robustness in this scenario.

Figure 6 illustrates the transmission rates achieved by our proposed design using IRI and the traditional MMSE condition of each legitimate user and the eavesdropping rates for the corresponding legitimate user in the DFRC system with four legitimate users when $L=2,\mu =1,P=0.1$ W. We can observe that all users in Figure 6 also meet the requirements of the physical layer security for secure transmission by the proposed design. The transmission rates achieved by IRI of the four legitimate users are better than those achieved by GRI. Moreover, the transmission rates and secrecy performance achieved by IRI are significantly better than those achieved by the traditional MMSE scheme. Furthermore, the eavesdropping rates of the eavesdropper (target) in IRI show a greater disparity in eavesdropping rates across different users, which is different from the eavesdropping rate distribution in Figure 5. As a result, the variations in secrecy rates among all users in IRI are considerable. Specifically, while the secrecy rates of User 1 and User 2 are roughly comparable between the two assignments, the secrecy rates of User 3 and User 4 achieved by IRI are significantly better than those achieved by GRI. This is reasonable, as IRI optimizes the rate increments individually for each user. In addition, it can be observed that the minimum secrecy rate achieved by IRI is significantly improved compared to GRI in Figure 5, indicating that IRI can achieve better security robustness in the considered DFRC system.

5.4. Secrecy Performance with Different Numbers of Legitimate Users

Figure 7 depicts the sum secrecy rates achieved by the proposed secure approach and the traditional MMSE scheme with different numbers of legitimate users in the DFRC system with $L=1,\mu =2$. Compared with the traditional MMSE scheme, GRI and IRI both can achieve higher sum secrecy rates than MMSE under different numbers of legitimate users, which illustrates the superiority of our proposed design. Moreover, we have the following important observations. As the transmit power increases, the sum secrecy rate achieved by the proposed design of all curves is enhanced. The sum secrecy rates with three legitimate users in the system are greater than those with four and five legitimate users. This is because in a DFRC system with limited total power, when there are fewer legitimate communication users, the beamforming of the message expected by each user can allocate more power, which is more conducive to the improvement of the sum secrecy rate of the system. In addition, the two curves of three legitimate users show significant improvement under both methods as transmit power increases. It illustrates that the proposed secure design can notably enhance the system’s secrecy performance when the number of users is small. However, when the number of legitimate users is four and five, the sum secrecy rates are lower than in the case with three legitimate users, and the improvement with increasing power is not significant. It shows that when the DFRC system needs to meet more security transmission requirements of more users and ensure the radar sensing performance simultaneously, the sum secrecy rate of the system will deteriorate. Furthermore, comparing the four curves of four and five legitimate users, it can be seen that the sum secrecy rate achieved by IRI is higher than that of GRI. This suggests that in practical deployments of DFRC systems, when the number of legitimate users is high, using IRI of the proposed secure design can be more efficient than merely increasing transmission power to enhance the secrecy rate. Compared with the sum secrecy rates under four users in Figure 3 and Figure 4, it can be demonstrated that increasing the decoding size at legitimate communication users can also effectively enhance the system’s secrecy performance. Specifically, when the number of layers L is 1, the sum secrecy rates achieved by both ways with $\mu =2$ are higher than those with $\mu =1$, which also validates the effectiveness of the proposed secure design in fully utilizing the transmit and receiving ends to enhance the secrecy performance of the DFRC system.

5.5. Convergence Performance

Figure 8 plots the iterative process of the optimization under the condition of $L=3,\mu =3$, $P=0.4$ W. It can be observed that the sum secrecy rates of the DFRC system increase during the optimization process. After seven to eight iterations, the rates stabilized, demonstrating the convergence of the proposed joint optimization algorithm. Additionally, as shown in the figure, the sum secrecy rates achieved by IRI are always significantly higher than the secrecy rates achieved by GRI, which further validates the superiority of IRI in optimizing rate increments based on user-specific factors.

6. Conclusions

In this paper, we investigated a secure optimization approach to deal with the physical layer security problem of the dual-function radar communication system where the sensing target wiretaps the communication between the base station and the legitimate users. We considered a secure design based on partial group successive interference cancellation that the legitimate users can decode the split sub-messages in a sequential way, which can improve the legitimate rate increments. In order to maximize the radar echo SNR, we formulated an optimization problem of beamforming design constrained by the transmit power and physical layer security requirements. Since the optimization problem is non-convex and difficult to solve, we transformed it into a convex form by introducing new variables and relaxation. Then, a joint secure beamforming and rate optimization algorithm was considered to optimize the rate vector and the secure beamforming design. Numerical results demonstrate that our proposed secure scheme can significantly improve the sensing and secrecy performance of the DFRC system. In the future, we will expand our work into more emerging practical application scenarios of DFRC by taking the latency caused by PGSIC into consideration.