Beamforming Design for Joint Target Sensing and Proactive Eavesdropping^††thanks: Manuscript received.

Integrated sensing and communication (ISAC) technology can share spectrum, hardware platforms, and even baseband waveforms and signal processing between communication and sensing to improve the spectral efficiency, energy efficiency, and hardware efficiency of the system for integration gain. Further, the mutual assistance and mutual gain of the two functions can also be used to improve each other’s performance and obtain coordination gain [1]-[4].

Recently, the performance for various ISAC systems was investigated to achieve concurrent sensing and communication, which were classified into the radar-centric systems [5]-[9], the communication-centric systems [10]-[13], and the joint-design systems [14]-[15]. The authors in [5] investigated beamforming designs for the ISAC systems with multiple communication users (CUs) for scenarios with point or extended targets. In particular, the Cramér-Rao bound (CRB) of the azimuth angle or the response matrix of the target was minimized by designing the dual-functional beamforming matrix. The authors in [6] proposed an integrated sensing, communication, and computation over-the-air framework to enable simultaneous communication, target sensing, and over-the-air computation (AirComp). The shared and separated schemes were proposed, and the accuracy of AirComp was optimized by jointly designing the data transmission beamformer and the aggregation beamformer or the data beamformer and the radar sensing beamformer. In [7], a new angular waveform similarity metric was proposed and combined with the integrated main-lobe-to-sidelobe ratio (IMSR) of the transmit beampattern to evaluate the waveform ambiguity properties. The complementary waveform and linear precoder matrix were jointly designed to minimize the weighted sum of IMSR while considering a predefined signal-to-interference-plus-noise ratio (SINR), power, and peak-to-average-power ratio (PAPR) constraints. In [8], the authors investigated the performance of the ISAC systems with multiple targets and CUs. Both the scenarios without and with prior target knowledge at the BS were considered, and the multi-target response matrix and the reflection coefficients/angles were estimated, respectively. The CRB of multi-target estimation was minimized while satisfying the constraints of a minimum multicast communication rate and a maximum transmit power. The authors in [9] proposed a new robust beamforming scheme for ISAC systems with different point targets. The maximum CRB of the estimated direction of arrival (DOA) was minimized by designing the beamforming matrix while ensuring the quality-of-service (QoS) for all the CUs. In [10], the authors proposed a new waveform design based on constructive interference for the ISAC system with multiple CUs. Their results show that the proposed scheme can reduce the transmit power for the given constraint on estimation accuracy or increase the communication signal-to-noise ratio (SNR) with a given power budget. In [11], the joint active and passive beamforming design problem for the reconfigurable intelligent surface (RIS)-assisted full-duplex (FD) ISAC systems with multiple CUs. The sum rate was maximized by jointly optimizing the transmit beamformer, the linear postprocessing filters, the users’ uplink transmission power, and the reflected phase shift of RIS. In [12], two optimization problems were formulated for the FD ISAC systems. The power consumption was minimized and the sum rate was maximized by jointly optimizing the downlink dual-functional transmit signal, the uplink receive beamformers at the BS, and the transmit power at the uplink users. The authors in [13] investigated the SNR-constrained and CRB-constrained joint beamforming and reflection design problems for the RIS-aided ISAC systems. The beamforming matrix and RIS reflection matrix were jointly optimized to maximize the sum of CUs’ communication rate while considering the SNR/CRB, the BS’s transmission power, and the RIS’s unit-modulus constraints. Due to the possibility of mutual interference between the multiplexed signals, there is a trade-off between target sensing and communication when multi-beam transmitted signals were reused for both sensing and communications. In [14], the angle and the reflection coefficient of the target and the CRB of the angle estimation were utilized as the performance metric of the point target sensing scenario. The authors adopted three performance metrics, the trace, the maximum eigenvalue, and the determinant of the CRB matrix, respectively, to estimate the complete target response matrix with the extended target sensing scenario. The Pareto boundary of the achievable CRB-rate region was given. The energy efficiency (EE) of the ISAC with an extended target and a multi-antenna CU was investigated in [15].  The energy consumption at the transmitter with the on-off scheme was minimized by jointly designing the transmit covariance matrix and the time allocation while taking the QoS for the CU and the CRB constraints for the target into account.

Like other wireless communication systems, ISAC systems also face security issues, and the confident signals are accessible to be wiretapped [4]. Since the transmit signals were reused for both sensing and communications, the security issue becomes imperative, especially when the target is a potential eavesdropper (PE) [16]-[21]. In [16], the radar target was assumed to be a PE, and the SNR at the PE was minimized by designing the radar beam pattern while ensuring the SINR requirement at legitimate users (LUs). Both the scenarios with perfect/imperfect target angle and the channel state information (CSI) were considered, respectively. In [17], destructive interference was utilized to suppress the PE in the ISAC systems, and the SINR of the radar signals at the BS was maximized by jointly designing the transmit and receive beamforming matrices. Assuming the estimation error of the azimuth angle of the PE obeys the Gaussian distribution, the authors in [18] proposed a new CSI error model. Both the scenarios with the perfect eavesdropping CSI/a single LU and with imperfect eavesdropping CSI/multiple LUs were considered. The sum secrecy rate of the considered ISAC system was maximized by optimizing a beamforming vector while considering the CRB and QoS constraints. In [19], the scenarios with perfect and imperfect CSI for the ISAC with multiple CUs and PE were considered. The weighted sum of the mean square error (MSE) between the designed and desired beampatterns and the mean-squared cross-correlation pattern was minimized by jointly optimizing the covariance matrix of the transmitted signals. Ref. [20] investigated the joint secure transmit beamforming designs for ISAC systems with a single LU and multiple PEs. The weighted sum of beampattern matching MSE and cross-correlation patterns was minimized by jointly optimizing the information and the sensing covariance matrix. The scenario with bounded CSI errors and Gaussian CSI errors were considered, and the worst-case secrecy rate and the secrecy outage constraints were considered, respectively. In [21], the authors proposed a novel sensing-aided secure scheme for the ISAC systems. The PEs’ direction was estimated using the combined capon and approximate maximum likelihood technique. Then, the dual-functional beamforming and the artificial noise (AN) matrix were jointly optimized to maximize the normalized weighted sum of the secrecy rate and estimated CRB under the estimated angle errors and the system power budget constraints.

In ISAC systems, the coexisted strong radar signals can be utilized as inherent jamming signals to fight the external eavesdroppers and improve the transmission security [4]. The authors in [22] investigated the joint designs of secure transmit beamforming for ISAC systems with multiple eavesdroppers. For the scenario with the eavesdropping CSI, the maximum eavesdropping SINR of multiple eavesdroppers was minimized by jointly designing the transmit beamformers of communication and radar systems while satisfying the QoS of LUs, the SINR constraint of radar target echo, and the transmit power budget. For the scenario without the eavesdropping CSI, an AN-aided transmit beamforming scheme was proposed. The total transmit power of the BS and radar was minimized by jointly designing the transmit beamformer and the covariance matrices of the AN vector at the BS and radar under the SINR constraints of LUs communication and radar target echoes. In [23], an unmanned aerial vehicle (UAV) was utilized as the RIS-aided ISAC base station (BS) and a secure transmission scheme was proposed to maximize the average achievable rate or the EE by jointly designing the transmit power allocation, the scheduling of users and targets, the phase shifts at RIS, as well as the trajectory and velocity of the UAV.

The above works implement communication and sensing functions within the ISAC system. To the best of the author’s knowledge, no open literature addresses the joint target sensing and proactive eavesdropping (JTSAPE) system. In this work, we try to answer the following questions: How to design the beamforming in the JTSAPE system? How is the effect of joint proactive eavesdropping and target sensing? A trade-off between parameter estimation CRB and eavesdropping SNR is achieved by a weighted design.

For clarity, the contributions of this work are listed as follows.

1. 1.

   We consider the scenario in which a multi-antenna JTSAPE BS actively wiretaps the illegal links while sensing the illegitimate transmitter. The beam transmitted by the BS to sense the target is also utilized as an AN to interfere with illegal links and enable proactive eavesdropping. We optimize the transmitting beam of the BS to maximize the eavesdropping SNR or minimize the CRB, respectively. The semidefinite relaxation (SDR) method was employed to solve the proactive eavesdropping subproblem and target sensing subproblem after dropping the rank one constraint.

2. 2.

   To realize the trade-off between proactive eavesdropping and target sensing, the joint optimization of proactive eavesdropping and target sensing is investigated, and the normalized weighted optimization problem is formulated. To address the complexity of the original problem, the formulated problem is decomposed into two subproblems: proactive eavesdropping and target sensing, which are solved by the SDR technique.

3. 3.

   Relative to [5] and [9] in which the CRB is solely considered as an optimization objective to maximize sensing performance, a trade-off between sensing CRB and eavesdropping SNR is achieved in this work. Relative to [16], [21], and [22] in which the AN is considered, the radar waves are regarded as AN in this work.

The remaining organization of this paper is summarized as follows: Sect. II introduces the system model of the joint proactive eavesdropping and target sensing communication. In Sect. III, an efficient tradeoff algorithm is proposed to solve the formulated problem based on SDR and sequential rank-one constraint relaxation (SROCR) techniques. In Sect. IV, the simulation results of the algorithm are analyzed. Finally, Sect. V summarizes the work of the paper.

Notations: Vectors and matrices are represented in lowercase and uppercase boldface letters, respectively. $\bf I$ denotes an identity matrix with appropriate dimensions. ${\left(\cdot\right)^{-1}}$ and ${\left(\cdot\right)^{T}}$ represents the inverse and transpose of the matrix, respectively. ${\left(\cdot\right)^{*}}$ and ${\left(\cdot\right)^{H}}$ represent the conjugate and conjugate transpose of the matrix, respectively. $tr\left(\cdot\right)$ represents the trace of a matrix. $\left(\cdot\right)_{\left(i,j\right)}$denoted the $\left(i,j\right)$-th element of the matrix. $\odot$ denotes Hadamard product. ${\mathop{\rm Re}\nolimits}\left(\cdot\right)$ and ${\mathop{\rm Im}\nolimits}\left(\cdot\right)$ signify the real and the imaginary part of a complex-valued matrix, respectively.

The transmitted signal, ${{\bf x}_{E}\in\mathbb{C}^{N_{t}\times 1}}$, at $E$ is expressed as

where ${\bf w}\in\mathbb{C}^{N_{t}\times 1}$ is the transmit beamformer to be designed, ${s}$ denotes the symbols with unit power. It assumes that the data symbols are approximately orthogonal to each other, which denotes ${\mathbb{E}}\left\{{{{\left|s\right|}^{2}}}\right\}=1$.

Like [12, 22], since the radar host generally has a strong signal separation capability, and a variety of methods can be used for filtering. Then, $E$ is assumeded that can completely separate the signal and eliminate the noise. When there are $K$ targets, the echo received by the radar is expressed as

where ${{\bf{A}}_{r}}=\left[{{{\bf{a}}_{r}}\left({{\theta_{1}}}\right),...,{{\bf{a}}_{r}}\left({{\theta_{K}}}\right)}\right]$, ${{\bf{A}}_{t}}=\left[{{{\bf{a}}_{t}}\left({{\theta_{1}}}\right),...,{{\bf{a}}_{t}}\left({{\theta_{K}}}\right)}\right]$, ${\bm{\beta}}=diag\left({{{\left[{{\beta_{1}},...,{\beta_{K}}}\right]}^{T}}}\right)$, ${\bm{\theta}}=\left[{{\theta_{1}},...,{\theta_{K}}}\right]^{T}$, ${\beta_{i}}={b_{{R_{i}}}}+j{b_{{I_{i}}}}$, and ${\bf Z}\sim CN\left(0,\sigma^{2}{\bf{I}}\right)$, ${{\bf a}_{t}}\left(\theta\right)=[1,{e^{j\frac{{2\pi d}}{{\lambda}}\sin\left(\theta\right)}},...,{e^{j\frac{{2\pi d}}{{\lambda}}\left({{N_{t}}-1}\right)\sin\left(\theta\right)}}]^{T}$ and ${{\bf a}_{r}}\left(\theta\right)=[1,{e^{j\frac{{2\pi d}}{{\lambda}}\sin\left(\theta\right)}},...,{e^{j\frac{{2\pi d}}{{\lambda}}\left({{N_{r}}-1}\right)\sin\left(\theta\right)}}]^{T}$ are the steering vector of the transmitting and receiving arrays, $d$ is the antenna spacing, $\lambda$ denotes the wavelength, $\theta$ is the angle of the target when the first antenna is utilized as the reference point, $\sigma^{2}$ is noise power, the normalized parameter $\beta={\beta_{R}}+j{\beta_{I}}$ depends on the complex amplitude of the target reflection and path loss coefficient of the $S-D$ link. According to (2), ${\bf Y}\sim CN\left({{{\bf{A}}_{r}}\left(\bm{\theta}\right)\bm{\beta}{\bf{A}}_{t}^{H}\left(\bm{\theta}\right){{\bf x}_{E}},\sigma^{2}{\bf{I}}}\right)$. The Fisher information matrix (FIM) of a vector $\left({\bm{\xi}}\right)$ is expressed as

where ${{\bf{C}}\left({\bm{\xi}}\right)}$ and ${{\bm{\mu}}\left({\bm{\xi}}\right)}$ denote the covariance matrix and mean vector [27]. In this work, the target direction $\bm{\theta}$, the real and imaginary parts of the coefficient $\bm{\beta}$ are estimated. Since $\sigma^{2}{\bf{I}}$ is independent of the estimator [8, 21, 30], therefore, the FIM for ${\bm{\zeta}}=\left[{{{\bm{\theta}}},{\bm{\beta}_{R}},{{{\bm{\beta}}}_{I}}}\right]$ is given in the following Lemma.

In this work, it is assumed that there is one target. The singal target echo sensing signal received by $E$ is expressed as [19]

where ${\bf{n}}_{r}\sim CN\left(0,\sigma^{2}{\bf{I}}\right)$ is the additive white Gaussian noise (AWGN) matrix.

The CRB matrix is obtained as

It is assumed that a radar signal is used as AN to realize proactive eavesdropping, and the line-of-sight channel between $E$ and $D$ is expressed as [18]

where $\beta_{0}$ represents the channel gain at the reference distance, $d_{ED}$ represents the distance between $E$ and $D$, and $\alpha$ represents the path loss exponent. The channel between $S$ and $D$ is expressed as

where $d_{SD}$ represents the distance between $S$ and $D$. The radar signal transmitted from $E$ also acts as an AN singnals to reduce the illegal link transmission rate. Thus, the received signal at $D$ is expressed as

where $x_{S}$ denotes the signal sent by $S$, and ${n_{D}}\sim N\left({0,\sigma_{D}^{2}}\right)$ is the AWGN. The SNR of received signal at $D$ is expressed as

where $P_{S}$ is the transmit power of $S$, and $\mathcal{I}\left({{\theta_{D}}}\right)={\bf{a}}_{t}^{H}\left({{\theta_{D}}}\right){\bf{R}}{{\bf{a}}_{t}}\left({{\theta_{D}}}\right)$ signifies the beampattern from $E$.

The communication signals received at $E$ is expressed as

where ${\bf{n}}_{E}\sim N\left({0,\sigma_{E}^{2}{\bf{I}}}\right)$, ${{\bf h}_{SE}\in\mathbb{C}^{N_{r}\times 1}}$ is the $S-E$ channel. It is assumed that $E$ adopts the selection combining scheme, the SNR of $E$ is expressed as

where $h_{SE}=\sqrt{\frac{{{\beta_{0}}}}{{d_{SE}^{\alpha}}}}$, $d_{SE}$ represents the distance between $S$ and $E$.

In this work, it is assumed that there is an adaptive information transmission link between $S$ and $D$, $E$ can decode the received information without error, and the success of proactive eavesdropping needs to meet ${\gamma_{D}}\leq{\gamma_{E}}$ [24, 25, 26]. In this subsection, proactive eavesdropping is considered only, which indicates that the transmitting beam is designed to meet the eavesdropping demand. The problem of proactive eavesdropping is formulated as

where (16a) is the premise of successful eavesdropping, (16b) is the constraint on the total power, where $P_{0}$ denotes the power budget of $E$, (16c) indicates that the covariance matrix $\bf{R}$ is a positive semi-definite matrix, and (16d) indicates that the rank of $\bf{R}$ is one.

$\mathcal{P}_{1}$ is challengeable to solve because the convexity of (16a) is unknown and (16d) is non-convex. By substituting (13) and (15) into (16a), which is rewritten as

Based on (13), it can be found that maximing ${\gamma_{D}}$ equals to minimizing $\mathcal{I}\left({{\theta_{D}}}\right)$. Then, $\mathcal{P}_{1}$ is equivalent to

where $\mathcal{I}\left({{\theta_{D}}}\right)$ is non-negative. With the same method as [16], igoring (18d), $\mathcal{P}_{1.1}$ can be solved directly by the convex optimization toolbox.

CRB is the lower bound of the unbiased estimation variance, and therefore minimizing CRB is a strategy for improving the accuracy of the estimation. In this subsection, target sensing is considered only, which indicates that the transmitting beam is designed to minimize the CRB.

In this section, a normalized weighted optimization problem that reveals the performance tradeoff between proactive eavesdropping and target sensing is proposed. The design of a weighted optimization between the radar CRB and the eavesdropping rate presents the challenge that the two performance metrics have different units and potentially different magnitudes. Inspired by [21], the CRB and the eavesdropping rate are normalized with their respective bound obtained in $\mathcal{P}_{1.1}$ and $\mathcal{P}_{2.1}$, respectively.