Sum Secrecy Rate Maximization for Full Duplex ISAC Systems

The work in [13] utilizes a sequence of snapshots for robust and secure resource allocation in an ISAC system consisting of a multi-antenna DFRC BS serving multiple single-antenna legitimate users, while sensing potential single-antenna eavesdroppers. Similarly, the authors in [14] consider a downlink (DL) secure ISAC system with a multi-antenna BS transmitting confidential messages to single-antenna CUs while simultaneously sensing targets. Furthermore, [15] explores the sensing-aided secure ISAC system, where a dual-functioning base station (BS) emits omnidirectional waveform in search of potential eavesdroppers and simultaneously sends confidential messages to CUs. The work intends to maximize the secrecy rate, while minimizing the eavesdroppers’ Cramér-Rao Bound (CRB). Moreover, [16] aims at designing an ISAC beamformer that ensures secure communication rates, while generating a beampattern sufficiently close to the desired one, under the given transmit power budget constraint. Meanwhile, [17] investigates a secure cell-free ISAC system, where distributed access points collaboratively serve CUs and perform target sensing in the presence of multiple eavesdroppers. For that, the authors propose to jointly optimize the communication and sensing beamforming vectors, with the goal of maximizing both the secrecy rate for all CUs and the sensing signal-to-noise ratio (SNR) of the target. Also, the work in [18] studies outage beamforming strategies for dual-functional radar and communication (DFRC) ISAC systems, but does not touch upon the security vulnerabilities that are inherent to ISAC systems.

Recently, full duplex (FD) communications have emerged as a key enabler for ISAC applications, thanks to their ability to support simultaneous DL and uplink (UL) transmissions across the entire frequency band. One of the main challenges in establishing in-band monostatic FD communication is the detrimental effect of self interference (SI). It constitutes a significant part of the received signal, drains high unnecessary power (up to $100\operatorname{dB}$ higher than those of the signals of interest [7] [19]), yet does not contain any useful information. There are three main components of SI: leakage SI, direct SI or spillover, and reflected SI [19]. The major portion of SI must be canceled in the analog domain, i.e. before it reaches analog to digital conversion, after which the residual SI is treated in the digital domain. From FD-ISAC perspective, the work in [20] proposes an FD waveform design for a monostatic ISAC setup, wherein a single ISAC node aims at simultaneously sensing a radar target while communicating with a communication receiver. The design can enhance the communication and the probability of target detection, assuming effective suppression of SI, by time-multiplexing the classic pulsed radar waveform with SAC signals. Furthermore, [21] considers an FD-ISAC system, in which the BS simultaneously executes target detection and communicates with multiple DL and UL users by reusing the same time and frequency resources, where the aim is to maximize the sum rate while minimizing power consumption. The authors in [22] consider the joint active and passive beamforming design problem in RIS-assisted FD UL communication systems. The study in [23] proposes that FD-ISAC can address the “echo-miss problem”, a phenomenon where radar returns are suppressed by strong residual self-interference (SI) from the transmitter in a monostatic ISAC receiver. The solution involves jointly designing the beamformers, UL transmit precoder, and DL receive combiner to maximize communication rates, enhancing radar beampattern power, and suppressing residual SI. The authors in [24] present a novel system termed integrated sensing and backscatter communication (ISABC), which consists of an FD BS that extracts environmental information from its transmitted signal, a backscatter tag that reflects the signal, and a user who receives the data. The authors discover that, while the power allocation at the BS plays a crucial role in influencing user and sensing rates, it does not affect the backscatter rate. The work in [25] emphasizes the design challenges in integrating communication and sensing/radar systems and investigates possible solutions, such as simultaneous transmission and reception, multibeam beamforming, and joint waveform optimization. In addition, [26] investigates an FD-ISAC system where the BS is tasked with target detection while sharing the same resources for simultaneous DL/UL communications, where the focus is on tackling the robust energy efficiency maximization problem for a sustainable FD-ISAC architecture. Also, [27] focuses on near-field FD-ISAC beamforming for multi-target detection. Our previous work in [10] addresses the problem of satisfying UL and DL secrecy rates, under integrated sidelobe to mainlobe ratio (ISMR) constraints, while minimizing the total power consumption of the FD system, in the presence of eavesdropping targets.

The works in [28], [29], [30], and [31] all investigate the application of reconfigurable intelligent surface (RIS) in physical layer security (PLS) of ISAC systems. The authors in [29] use high-power radar signal as the interference to disturb the eavesdropper’s interception and RIS is used to expand the coverage area. The work in [30] considers RIS-based backscatter systems, where an ISAC BS attempts to communicate with multiple IoT devices in the presence of an aerial eavesdropper. Finally, the work in [31] examines a RIS-based terahertz (THz) ISAC system with delay alignment modulation. Also, [32] utilizes RIS for secure transmissions within unmanned aerial vehicle (UAV) ISAC systems, whereby a methodology is proposed taking into account the velocity and positions of the UAV.

This work centers on a secure FD design, where the DFRC BS transmits ISAC signals optimized for SAC, with the goal of preventing multiple eavesdropping targets to decode these signals in FD-ISAC mode. To that purpose, we have summarized our contributions as follows.

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  Maximum Secrecy Optimization for Secure FD ISAC. We first introduce our FD-ISAC scenario that includes DL and UL CUs associated with a a DFRC BS in the presence of multiple eavesdropping targets, where the DFRC BS would like to sense these eavesdropping targets. The injected artificial noise (AN) signal is not only used for securing the signal, but also aids in the sensing tasks through proper ISMR regulations. We then formulate an optimization framework that aims at maximizing the FD sum secrecy rate, which includes both UL and DL sum secrecy rates. For sensing, we impose an ISMR constraint which also has the ability of revealing minimal communication information, while maximizing the radar backscattered signal for sensing purposes.

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  Solution via Taylor series approximations and block cyclic coordinate descent. As the proposed maximum FD secrecy optimization problem is non-convex and highly non-linear, we prove a series of lemmas, which are crucial as they allow us to transform the problem into a more tractable form. Subsequently, we derive a method termed iterative joint Taylor-BCCD (IJTB), which cycles between two sub-problems, one of which produces the UL beamformers in closed-form, while the other approximate problem, generates the UL power allocation vector, the AN covariance generator and the DL beamforming vectors, thanks to a series of Taylor approximations applied to ”convexify” the problem. The proposed IJTB algorithm for maximum-secrecy FD-ISAC iterates between the two problems and converges to a stable solution providing the optimal beamformers, artificial noise statistics, and power allocation vector for the UL CUs, in addition to the UL beamformers.

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  Extensive simulation results. We present extensive simulation results that highlight the superiority, as well as the potential of the proposed design and algorithm with respect to many benchmarks, in terms of UL and DL sum secrecy rate, achieved ISMR, in addition to algorithmic convergence.

Unlike our previous work [10], this work essentially focuses on the maximization of the sum secrecy rate of both UL and DL users, under sensing and various power budget constraints. In particular, we are interested in studying the maximum sum secrecy rates for both UL and DL, under sensing and power constraints, which serves a complementary study to our previous work in [10], that mainly focuses on power efficient ways to satisfy secure and ISAC constraints. Moreover, we unveil some important insights, i.e.

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  The IJTB method consistently outperforms all benchmarks in terms of uplink (UL) sum secrecy rate, regardless of the distance of the eavesdropper or the UL radial distance. More specifically, as the UL radial distance increases, the UL sum secrecy rate decreases for all methods. Meanwhile, IJTB can adapt to different eavesdropper distances, maintaining a higher UL sum secrecy rate even as the eavesdropper gets closer. For instance, all benchmarks show a significant drop in the UL sum secrecy rate, nearing zero when the eavesdropper distance reaches $17\operatorname{m}$, highlighting their vulnerability when compared to IJTB.

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  Regarding AN exploitation, IJTB adapts well to varying eavesdropper distances, maintaining its performance consistently and outperforming all other methods. For example, when the UL CU is situated at $50\operatorname{m}$, IJTB achieves a sum secrecy rate of about $21.4\operatorname{bits/sec/Hz}$, compared to $18.86\operatorname{bits/sec/Hz}$ by the isotropic no-AN benchmark. This shows that IJTB can explore the extra degrees-of-freedom (DoF) offered by AN, even when performing sensing tasks, i.e. under ISMR constraints.

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  From sensing perspective for ISAC, when ISMR constraint is stringent, the sum secrecy rate of IJTB drops below that of isotropic benchmarks, due to power allocation strategies that focus on minimizing sidelobe power. Despite this ”drop”, IJTB still achieves competitive sum secrecy rates, especially as the ISMR constraint becomes less restrictive, reflecting a favorable increase for sensing performance. Putting things in context, when ISMR is tuned to $-20\operatorname{dB}$ (i.e. suggesting very pointy radar beams), IJTB can still achieve a total sum secrecy rate of about $8.84\operatorname{bits/sec/Hz}$, which is comparable with other benchmarks.

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  IJTB demonstrates rapid convergence, stabilizing in about $10$ iterations.

The detailed structure of the following paper is given as follows: The full-duplex integrated sensing and communication system model along with its key performance indicators is presented in Section II. The formulation of the proposed optimization framework bespoke for the secure FD-ISAC design problem is outlined in Section III. Consequently, its solution in a form of an algorithm is described in Section IV, while Section V describes the simulation findings. Lastly, the conclusion of the paper can be found in Section VI.

Notation: A vector is denoted by a lower-case boldface letter, e.g. $\boldsymbol{x}$, while its $\ell_{2}$ norm is denoted as $\|\boldsymbol{x}\|$. The zero-vector is denoted by $\boldsymbol{0}$, while the all-ones vector of appropriate dimensions is denoted by $\boldsymbol{1}$. A matrix is denoted by an upper-case boldface letter and its transpose, conjugate, and transpose-conjugate are denoted by $(.)^{T}$, $(.)^{*}$, and $(.)^{H}$, respectively. A vector with all non-negative entries is denoted by $\boldsymbol{x}\succeq\boldsymbol{0}$, while a positive semi-definite matrix is denoted by $\boldsymbol{A}\succeq\boldsymbol{0}$. The $N\times N$ identity matrix is $\boldsymbol{I}_{N}$. When it comes to matrix indexing, $[\boldsymbol{A}]_{i,j}$ denotes the $(i,j)^{th}$ entry of matrix $\boldsymbol{A}$ and $[\boldsymbol{A}]_{:,j}$ denotes its $j^{th}$ column. The operator $\operatorname{EVD}$ stands for eigenvalue decomposition. The magnitude and the angle of any complex number $z\in\mathbb{C}$ are denoted by $|z|$ and $\angle z$, respectively. $\mathbb{E}\{\}$ denotes the statistical expectation.

We assume an FD-ISAC scenario, in which there is a threat of malicious eavesdroppers intercepting data, as depicted in Fig.1. The DFRC BS considered is equipped with $N_{T}$ transmitting antennas and $N_{R}$ receiving antennas displaced via a mono-static radar setting with no isolation between both arrays. The following $N_{T}\times 1$ secure ISAC baseband signal vector is transmitted by the DFRC BS \useshortskip

where $\boldsymbol{V}\in\mathbf{C}^{N_{T}\times L}$ is the DFRC beamforming matrix with its $\ell^{th}$ column loaded with the beamforming vector designated towards the $\ell^{th}$ CU; and $L$ denotes the total number of DL CUs. Also, $\boldsymbol{s}$ is a vector of transmit data symbols, whose $\ell^{th}$ entry is the data symbol intended to the $\ell^{th}$ DL CU. We assume independent symbols with unit variance, i.e $\mathbb{E}(\boldsymbol{s}\boldsymbol{s}^{H})=\boldsymbol{I}_{L}$. Moreover, a superimposed ISAC AN vector $\boldsymbol{w}$, assumed to be independent from the signal $\boldsymbol{s}$, is transmitted based on $\boldsymbol{w}\sim\mathcal{N}(\boldsymbol{0},\boldsymbol{W})$, where $\boldsymbol{W}\succeq\boldsymbol{0}$ stands for the spatial ISAC AN covariance matrix, which is unknown, and has to be optimized for. Likewise, $\boldsymbol{w}$ and $\boldsymbol{s}$ are assumed to be independent of one another, in which case, $\operatorname{tr}(\boldsymbol{W})$ would reflect the total transmit power invested on AN. The beamformer $\boldsymbol{V}$ is designed for DFRC ISAC beamforming to maintain good SAC properties. As a consequence, although the secure ISAC signal $\boldsymbol{x}$ is intended for the DL CUs, it is also designed to function as a radar signal through both $\boldsymbol{V}$ and $\boldsymbol{W}$.

According to the ISAC design, this model does not distinctly separate SAC signals, treating them as a unified signal instead. Consequently, the single-antenna DL CUs receive the following signal: \useshortskip

where the desired and interfering terms are elucidated, and the last term, i.e. $n_{\ell}$, is zero-mean additive white Gaussian noise (AWGN) noise with variance $\sigma_{\ell}^{2}$, i.e. $n_{\ell}\sim\mathcal{N}(0,\sigma_{\ell}^{2})$. Moreover, $\boldsymbol{h}_{r,\ell}$ denotes the mmWave channel between the $\ell^{th}$ DL CU and the DFRC BS modeled as [33] \useshortskip

where $\kappa_{\ell}$ is the Rician $K$-factor of the $\ell^{th}$ user. Further, $\boldsymbol{h}_{r,\ell}^{\mathrm{LoS}}$ is the LoS component between the $\ell^{th}$ user and the DFRC BS, and the non-line-of-sight (NLoS) component consists of $N_{\mathrm{cl}}$ clusters as \useshortskip$\boldsymbol{h}_{r,\ell}^{\mathrm{NLoS}}=\sqrt{1\Big{/}\sum_{q=1}^{N_{\mathrm{cl}}}N_{q}}\cdot\sum_{q=1}^{N_{\mathrm{cl}}}\sum_{r=1}^{N_{q}}u_{q,r}\boldsymbol{a}_{N_{T}}(\varphi_{q,r})$ [34], where $N_{q}$ is the number of propagation paths within the $q^{th}$ clusters. The steering vector departing from angle of departure (AoD) $\varphi$ is denoted by $\boldsymbol{a}_{N_{T}}(\varphi)\in\mathbb{C}^{N_{T}\times 1}$. Additionally, we define $u_{q,r}$ as the path attenuation and $\varphi_{q,r}$ as AoDs of the $r^{th}$ propagation path within the $q^{th}$ cluster. As the number of clusters grows, the path attenuation coefficients and angles between users and the DFRC BS become randomly distributed. Additionally, $q_{k,\ell}$ represents the channel coefficient between the $k^{th}$ UL CU and the $\ell^{th}$ DL user, which captures the UL-to-DL interference, and can be harmful when UL and DL users are in close proximity to each other. Furthermore, the $k^{th}$ UL symbol, formed by the $k^{th}$ user and intended for the DFRC BS, is denoted as $v_{k}\in\mathbb{C}$, with power level $\mathbb{E}(|v_{k}|^{2})=p_{k},\forall k$. Due to the presence of clutter, a portion of the DL signal scatters from the clutter towards the DL CUs, which is contained within the DL channels $\boldsymbol{h}_{r,\ell}$. The eavesdroppers are also intercepting DL signals, so the signal at the $p^{th}$ eavesdroppers is

where $g_{k,p}$ is the complex channel coefficient between the $k^{th}$ UL CU and the $p^{th}$ eavesdropper. The path-loss complex coefficient between the DFRC BS and the $p^{th}$ eavesdropper is denoted by $\alpha_{p}$. The steering vector arriving at angle of arrival (AoA) $\theta$ is denoted by $\boldsymbol{a}_{N_{R}}(\theta)\in\mathbb{C}^{N_{R}\times 1}$. The angles of arrival (AoAs) for the eavesdroppers are denoted as $\theta_{p}$. The AWGN at the $p^{th}$ eavesdropper is $z_{p}$, modeled as zero-mean with variance $\sigma_{p}^{2}$, i.e., $z_{p}\sim\mathcal{N}(0,\sigma_{p}^{2})$. We emphasize the relationship between the received signal at the $\ell^{th}$ legitimate DL user (as in (2)) and the signal at the eavesdropper (as in (3)). Both legitimate users and eavesdroppers receive the UL signals and the DFRC ISAC signal $\boldsymbol{x}$. However, the channel models differ between DFRC-DL (i.e., $\boldsymbol{h}_{r,\ell}$) and DFRC-eavesdropper (i.e., $\boldsymbol{a}(\theta_{p})$). Eavesdroppers are considered radar targets with a strong dominant line-of-sight (LoS) component (a Rician channel with a large $K$-factor) for localization and tracking applications. This model is particularly applicable in scenarios involving UAVs [35].

Specifically, the DFRC aims to infer additional sensing information about eavesdropping targets, such as delay and Doppler, without exposing the communication information embedded within the transmitted ISAC signal $\boldsymbol{x}$. Meanwhile, a set of $K$ UL CUs associated with the DFRC base station (BS) coexist. It is important to note that while the DFRC BS transmits a secure signal vector $\boldsymbol{x}$, it also collects the UL signals along with other components clarified in this section. Let $\boldsymbol{h}_{t,k}$ be the UL channel vector between the $k^{th}$ UL CU and the DFRC BS. For Rayleigh fading, the UL channels $\boldsymbol{h}_{t,k}$ account for environmental clutter, where part of the signal propagates towards the clutter and back to the DFRC BS. Thus, the signal received by the DFRC BS from the single-antenna UL users is $\sum\nolimits_{k=1}^{K}\boldsymbol{h}_{t,k}v_{k}$.

We summarize the system model in Fig. (reference), which illustrates a group of UL and DL legitimate CUs associated with a DFRC BS. Additionally, a group of eavesdroppers intercepts the UL and DL signal exchanges. Note that the DL signal is an ISAC signal, and the DFRC BS directs beams towards the eavesdroppers to acquire physical sensing information about them. Furthermore, due to full-duplex (FD) operation, the simultaneous transmission of the signal vector $\boldsymbol{x}$ ”leaks” into the receiving unit of the DFRC BS, resulting in loop interference. This phenomenon, known as SI, is a crucial component when targeting FD behavior, especially when the transmitting and receiving units of the DFRC BS are colocated without isolation. The receiving unit experiences an SI component of $\sqrt{\beta}\boldsymbol{H}_{\mathrm{SI}}\in\mathbb{C}^{N_{R}\times N_{T}}$, where $\beta$ represents the residual SI power. Under the direct-coupling channel model for SI, the small-scale SI fading channel reads $[\boldsymbol{H}_{\mathrm{SI}}]_{i,j}=\exp\left(j2\pi\frac{d_{i,j}}{\lambda}\right)$, which contains the phases introduced between each pair of transmitting and receiving antennas. Also, $\lambda$ is the wavelength and $d_{i,j}$ is the distance between the $i$-th receive and $j^{th}$ transmit antennas. Additionally, the DFRC BS aims to direct sensing power towards $P$ eavesdroppers so that the backscattered echo returns experience good sensing SNR. To this end, the echo-plus-clutter returns can be written as \useshortskip

In (4), we denote $\gamma_{p}$ by the complex amplitude due to the $p^{th}$ eavesdropper and $\boldsymbol{c}\sim\mathcal{N}(\boldsymbol{0},\boldsymbol{R}_{c})$ is the clutter including the reflected signal from the UL/DL CUs. We assume the clutter covariance matrix $\boldsymbol{R}_{c}\in\mathbb{C}^{N_{R}\times N_{R}}$, capturing all environmental clutter effects, to be constant and that the eavesdropper AoAs are known. Due to the collocated transmit and receiving units, the AoAs and AoDs in equation (4) are identical. Lastly, the received signal at the DFRC BS is

In summary, the first part in (5) is the signal due to the $K$ UL CUs. The second part is the sensing echo-plus-clutter returns. The third part is the SI, and the AWGN follows $\boldsymbol{n}\sim\mathcal{N}(\boldsymbol{0},\sigma_{n}^{2}\boldsymbol{I}_{N_{R}})$.

Before tackling the secure FD-ISAC problem, it is important to define key performance indicators that will be critical for evaluating the system’s overall performance and for developing an appropriate optimization framework.