# Joint Secure Transmit Beamforming Designs for Integrated Sensing and
Communication Systems ††thanks: Part of this paper has been presented in the
IEEE Wireless Communications and Networking Conference (WCNC), 2022 [1].
††thanks: J. Chu, R. Liu, M. Li, and Y. Liu are with the School of Information
and Communication Engineering, Dalian University of Technology, Dalian 116024,
China (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>yangliu_613@dlut.edu.cn). ††thanks: Q. Liu is with the School
of Computer Science and Technology, Dalian University of Technology, Dalian
116024, China (e-mail: qianliu@dlut.edu.cn).

Jinjin Chu, Rang Liu, Ming Li,
Yang Liu, and Qian Liu

###### Abstract

Integrated sensing and communication (ISAC), which allows individual radar and
communication systems to share the same spectrum bands, is an emerging and
promising technique for alleviating spectrum congestion problems. In this
paper, we investigate how to exploit the inherent interference from strong
radar signals to ensure the physical layer security (PLS) for the considered
multi-user multi-input single-output (MU-MISO) communication and colocated
multi-input multi-output (MIMO) radar coexistence system. In particular, with
known eavesdroppers’ channel state information (CSI), we propose to jointly
design the transmit beamformers of communication and radar systems to minimize
the maximum eavesdropping signal-to-interference-plus-noise ratio (SINR) on
multiple legitimate users, while guaranteeing the communication quality-of-
service (QoS) of legitimate transmissions, the requirement of radar detection
performance, and the transmit power constraints of both radar and
communication systems. When eavesdroppers’ CSI is unavailable, we develop a
joint artificial noise (AN)-aided transmit beamforming design scheme, which
utilizes residual available power to generate AN for disrupting malicious
receptions as well as satisfying the requirements of both legitimate
transmissions and radar target detection. Extensive simulations verify the
advantages of the proposed joint beamforming designs for ISAC systems on
secure transmissions and the effectiveness of the developed algorithms.

###### Index Terms:

Integrated sensing and communication (ISAC), physical layer security (PLS),
multi-user multi-input single-output (MU-MISO), artificial noise (AN),
interference exploitation.

##  I Introduction

With the explosive growth of wireless devices, exponentially increased
bandwidth is required to support a variety of high data-rate services.
Consequently, spectrum resources have been increasingly scarce, which
motivates the development of advanced spectrum sharing technologies [2]. Since
large portions of the spectrum are available at radar frequency bands,
spectrum sharing between radar and communication systems has led to
substantial research interest [3]-[6]. This line of research is referred to as
integrated sensing and communication (ISAC), which is also known as joint
radar-communication (JRC), joint communication and radar (JCR), joint
communication and radar sensing (JCAS), etc. ISAC allows radar and
communication systems to share the same spectrum bands, which can
significantly improve the spectrum efficiency and thus alleviates the spectrum
congestion problem. It is foreseeable that ISAC will become a critical
enabling technology for future wireless networks, supporting various vital
applications, including vehicular networks [7], Internet of Things (IoT) [8],
etc.

Research on ISAC can be generally categorized into two main directions: Dual-
functional radar-communication (DFRC) and radar and communication coexistence
(RCC). The former focuses on using the same signals transmitted from one
fully-shared hardware platform to simultaneously perform communication and
radar sensing functionalities [9]-[12]. Although it has the benefits of low
power consumption and smaller-size platform, the trade-off between the radar
and communication functionalities requires sophisticated optimizations on the
unified dual-functional waveform. Moreover, the resulting hardware complexity
significantly hinders practical applications. On the contrary, RCC enables
separately deployed communication and radar platforms to cooperatively perform
their respective functions using independent transmitted signals. Therefore,
RCC is more suitable for many existing scenarios such as sharing spectrum
between air-borne early warning radars and 3.5 GHz time-division duplex long-
term evolution (TDD-LTE) [13], [14], between battlefield/ground surveillance
and vessel traffic service (VTS) radars and WLAN networks [15], [16], etc.

In RCC systems, the interference management between the non-colocated base
station (BS) and radar transmitter is vital for achieving good communication
and radar sensing performance. Therefore, the cooperative design of these two
systems is necessary for practical applications [17]-[28]. Meanwhile, the
multi-input multi-output (MIMO) architecture has been widely deployed in both
radar and communication systems to provide additional spatial degrees of
freedom (DoFs) for pursuing more considerable beamforming gains.

Various signal processing techniques have been proposed to design transmit
beamformers for multi-antenna BS and MIMO radar to realize efficient
interference management. A null space projection (NSP) method was proposed in
[29], [30], where the radar transmit beamforming is designed to project the
radar signals onto the null space of the effective interference channels to
eliminate the interference at the communication receivers. However, the target
echo may fall into the row space of the interference channels, resulting in
degraded radar performance. As a step further, the authors of [31] expanded
the projection space to include the subspace corresponding to the small non-
zero singular values under a specified threshold, expecting to control the
interference to communication systems and achieve different trade-offs between
radar and communication performance. However, these NSP-based beamforming
designs significantly reduce the DoFs for optimization and cannot guarantee to
meet specific radar sensing requirements. In order to overcome these
disadvantages, investigations on the joint design of the transmit beamformers
for both radar and communication systems have been proposed in [32], [33],
where different performance metrics of these two systems are satisfied with
controllable constraints. In addition to the above scenarios with perfectly
known channel state information (CSI), a robust beamforming design was
proposed in [34] under the assumption of imperfect CSI.

As mentioned above, the existing literature on beamforming designs for RCC
systems essentially aims at suppressing the interference between radar and
communication systems. While radar interference is usually deemed as the most
significant harmful component to the communication system, from another
perspective, it can be utilized to disrupt illegal receivers to safeguard
confidential information against eavesdropping. Utilizing interfering/jamming
signals to disrupt potential eavesdroppers has been widely considered in the
literature on physical layer security (PLS) [35]-[44]. In [35], constructive
interference was leveraged to implement secure beamforming using a symbol-
level precoding approach. In [36], a multi-antenna cooperative jammer was
employed to assist the secure communications. In order to confuse the
eavesdropper, the authors in [37] utilized the idea of destructive
interference to push the received symbols at the eavesdropper towards the
destructive region where the wrong symbol will be detected. The authors in
[38] proposed a cooperative secure transmission scheme, in which the
legitimate information and interference signals lie in different subspaces at
the destination of the confidential transmission, but are aligned along the
same subspace at the eavesdropper.

Noting that the above secure beamformer designs require the knowledge of the
eavesdropper’s CSI, which is not always available in practical applications.
In such cases, artificial noise (AN) technology was introduced to realize PLS
[39], which uses a large amount of additional energy to generate interfering
signals for disrupting potential malicious receptions. The existing
communication literature generally forces the AN to be uniformly distributed
onto the null space of the confidential transmission channels to disturb the
eavesdropper’s reception but not harm legitimate users [40]. Inspired by this
concept, in addition to realizing the respective performance requirements of
communication and radar systems, the residual power can be used to generate
AN. However, this null space projection design can only exploit limited
spatial DoFs to generate AN, especially for a system with many users. In
recent works [41], [42], the AN and transmit beamforming were jointly
optimized to implement secure DFRC transmissions, making full use of available
DoFs for generating AN. In addition, the authors in [43] proposed to enhance
secure performance by deploying an RIS in DFRC systems. However, how to ensure
the physical layer security in RCC systems remains an open problem. Instead of
consuming additional power to generate AN, the authors in [44] proposed to
exploit the inherent multi-user interference as a helpful resource by
converting it to act like AN or distributed friendly jammers to improve the
security performance. Inspired by the concept of interference exploitation,
the radar signal, which usually has very strong signal power, is an up-and-
coming candidate as the jamming signal to enhance the physical layer security
performance for the considered RCC system.

Motivated by the above findings, we investigate the PLS problem for multi-user
multi-input single-output (MU-MISO) communication and colocated MIMO radar
coexistence systems in this paper. In particular, the considered RCC system
includes a multi-antenna BS serving multiple single-antenna users in the
presence of multiple eavesdroppers and a colocated MIMO radar attempting to
detect a point-like target. We aim to exploit the coexisted strong radar
signals as inherent jamming signals to disrupt the eavesdroppers’
reception111Since the radar signal that is only used to detect the target does
not contain any confidential information, it will not cause security/privacy
problem to the considered RCC system. . The transmit beamformers of
communication and radar systems and radar receive filter are jointly designed
to ensure security performance and satisfy the requirements of legitimate
transmissions and radar target detection. The main contributions of this paper
can be summarized as follows:

  * •

We consider the physical layer security in an ISAC system and innovatively
propose to exploit the coexisted strong radar signals as inherent
jamming/interfering signals to weaken the reception of potential eavesdroppers
and enhance the transmission security. Joint secure transmit beamforming and
radar receive filter designs are investigated to achieve this goal.

  * •

With the knowledge of eavesdroppers’ CSI, we jointly design the transmit
beamforming and radar receive filter to minimize the maximum eavesdropping
signal-to-interference-plus-noise ratio (SINR) on multiple legitimate users,
while satisfying the quality-of-service (QoS) requirements of the legitimate
users, the radar output SINR constraint, and the transmit power budgets of
radar and communication systems. An efficient algorithm based on the block
coordinate descent (BCD), fractional programming (FP), and semi-definite
relaxation (SDR) methods is developed to solve the resulting non-convex
optimization problem.

  * •

When eavesdroppers’ CSI is unavailable, we propose to jointly design the AN-
aided transmit beamforming and radar receive filter to maximize the power of
AN under the same constraints. In this case, all available transmit power of
the BS and radar should be utilized to generate AN as much as possible to
destruct eavesdroppers’ reception. A double-loop BCD and SDR based algorithm
is employed to convert the resulting complicated non-convex optimization
problem into two more tractable sub-problems that can be alternatively solved.

  * •

Extensive simulation results show that the eavesdropping SINR is generally
several orders of magnitude smaller than the SINR of legitimate communication
users, which verify the significant advancement of utilizing radar signals as
inherent jamming/interference signals to enhance the secure transmissions for
ISAC systems and the effectiveness of the proposed joint secure transmit
beamforming design algorithms.

The rest of this paper is organized as follows. Section ii@ introduces the
system model of the ISAC system in the presence of eavesdroppers and develops
a joint transmit beamforming and radar receive filter design with known
eavesdroppers’ CSI. Section iii@ investigates the joint AN-aided transmit
beamforming and radar receive filter design without eavesdroppers’ CSI.
Simulation results are demonstrated in Section iv@, and finally, conclusions
are provided in Section<EMAIL_ADDRESS>
Notations: Boldface lower-case and upper-case letters indicate column vectors
and matrices, respectively. $(\cdot)^{H}$ and $(\cdot)^{-1}$ denote the
transpose-conjugate and inverse operations, respectively. $\mathbb{C}$ denotes
the set of all complex numbers. $|a|$, $\|\mathbf{a}\|$, and
$\|\mathbf{A}\|_{F}$ are the magnitude of a scalar $a$, the norm of a vector
$\mathbf{a}$, and the Frobenius norm of a matrix $\mathbf{A}$, respectively.
$\mathbb{E}\\{\cdot\\}$ represents statistical expectation.
$\text{Tr}\\{\mathbf{A}\\}$ and $\text{Rank}\\{\mathbf{A}\\}$ are the trace
and rank of matrix $\mathbf{A}$, respectively.

Figure 1: An ISAC system at the presence of eavesdroppers.

##  II Joint Transmit Beamforming Design with Known Eavesdroppers’ CSI

###  II-A System Model and Problem Formulation

We consider an ISAC system in which an MU-MISO communication system coexists
with a colocated MIMO radar system operating on the same frequency band, as
shown in Fig. 1. In particular, a BS equipped with $N$ antennas in a uniform
linear array (ULA) serves $K$ single-antenna users in the presence of $I$
eavesdroppers who attempt to intercept the confidential information
transmissions between the BS and the legitimate users. Meanwhile, a colocated
MIMO radar with $M$ transmit/receive antennas in a ULA attempts to detect a
point-like target. For convenience, the BS, the legitimate communication
users, and the eavesdroppers are referred to as Alice, Bobs, and Eves,
respectively. In this paper, we aim to exploit the coexisted strong radar
signals as inherent jamming signals to disrupt eavesdroppers’ reception under
different assumptions about the availability of Eves’ CSI.

We first assume that the CSI of Eves is perfectly known. This assumption is
valid, for example, Eves may be legitimate users who intend to overhear other
users’ confidential information. With known Eves’ CSI, we jointly design the
transmit beamformers of the BS and the radar to ensure secure transmissions
and satisfactory radar target detection performance for the ISAC system.

The received signal of the $k$-th Bob can be written as

$\displaystyle
y_{k}=\mathbf{h}_{k}^{H}\mathbf{W}\mathbf{s}+\mathbf{g}_{k}^{H}\mathbf{F}\mathbf{c}+n_{k},$
(1)

where $\mathbf{h}_{k}\in\mathbb{C}^{N}$ is the channel vector between the BS
and the $k$-th Bob,
$\mathbf{W}\triangleq\left[\mathbf{w}_{1},\mathbf{w}_{2},\ldots,\mathbf{w}_{K}\right]\in\mathbb{C}^{N\times
K}$ is the beamforming matrix with $\mathbf{w}_{k}$ representing the
beamforming vector of the $k$-th Bob, $\mathbf{s}\in\mathbb{C}^{K}$ is the
symbol vector with $\mathbb{E}\\{\mathbf{s}\mathbf{s}^{H}\\}=\mathbf{I}_{K}$,
$\mathbf{g}_{k}\in\mathbb{C}^{M}$ is the channel vector between the radar and
the $k$-th Bob, $\mathbf{F}\in\mathbb{C}^{M\times M}$ is the radar beamforming
matrix, $\mathbf{c}\in\mathbb{C}^{M}$ is the radar transmit waveform with
$\mathbb{E}\\{\mathbf{c}\mathbf{c}^{H}\\}=\mathbf{I}_{M}$, and $n_{k}$ is the
additive white Gaussian noise (AWGN) with
$n_{k}\sim\mathcal{CN}(0,\sigma_{k}^{2})$. We assume that the information
symbol vector $\mathbf{s}$ is statistically independent with the radar
waveform $\mathbf{c}$. Thus, the SINR of the $k$-th Bob can be calculated as

$\displaystyle\mathrm{SINR}_{k}$
$\displaystyle=\frac{|\mathbf{h}_{k}^{H}\mathbf{w}_{k}|^{2}}{\sum\limits_{j\neq
k}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{j}|^{2}+\|\mathbf{g}_{k}^{H}\mathbf{F}\|^{2}+\sigma_{k}^{2}}$
(2)
$\displaystyle=\frac{\mathbf{h}_{k}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{h}_{k}}{\sum\limits_{j\neq
k}^{K}\mathbf{h}_{k}^{H}\mathbf{w}_{j}\mathbf{w}_{j}^{H}\mathbf{h}_{k}+\mathbf{g}_{k}^{H}\mathbf{F}\mathbf{F}^{H}\mathbf{g}_{k}+\sigma_{k}^{2}}.$

Similarly, the received signal of the $i$-th Eve can be expressed as

$\displaystyle
y_{\mathrm{e},i}=\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{W}\mathbf{s}+\mathbf{g}^{H}_{\mathrm{e},i}\mathbf{F}\mathbf{c}+n_{\mathrm{e},i},$
(3)

where $\mathbf{h}_{\mathrm{e},i}\in\mathbb{C}^{N}$ is the channel vector
between the BS and the $i$-th Eve,
$\mathbf{g}_{\mathrm{e},i}\in\mathbb{C}^{M}$ is the channel vector between the
radar and the $i$-th Eve, and $n_{\mathrm{e},i}$ is the AWGN with
$n_{\mathrm{e},i}\sim\mathcal{CN}(0,\sigma_{\mathrm{e},i}^{2})$. The
eavesdropping SINR of the $i$-th Eve on the $k$-th Bob is thus given by

$\displaystyle\mathrm{SINR}_{i,k}^{\mathrm{e}}$
$\displaystyle=\frac{|\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{k}|^{2}}{\sum\limits_{j\neq
k}^{K}|\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{j}|^{2}+\|\mathbf{g}^{H}_{\mathrm{e},i}\mathbf{F}\|^{2}+\sigma_{\mathrm{e},i}^{2}}$
(4)
$\displaystyle=\frac{\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{h}_{\mathrm{e},i}}{\sum\limits_{j\neq
k}^{K}\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{j}\mathbf{w}_{j}^{H}\mathbf{h}_{\mathrm{e},i}+\mathbf{g}^{H}_{\mathrm{e},i}\mathbf{F}\mathbf{F}^{H}\mathbf{g}_{\mathrm{e},i}+\sigma_{\mathrm{e},i}^{2}}.$

To ensure secure communications, we aim to minimize the maximum eavesdropping
SINR on the $K$ communication users while guaranteeing the communication QoS
of legitimate transmissions.

In the radar system, the received signal, which includes the echo signal from
the target, interference from the BS, and noise, can be written as

$\displaystyle\mathbf{y}_{\mathrm{r}}=\alpha\mathbf{a}_{\mathrm{r}}(\theta_{0})\mathbf{a}_{\mathrm{t}}^{H}(\theta_{0})\mathbf{F}\mathbf{c}+\mathbf{Q}^{H}\mathbf{W}\mathbf{s}+\mathbf{n}_{\mathrm{r}},$
(5)

where $\alpha$ is the complex target amplitude with
$\mathbb{E}\\{|\alpha|^{2}\\}=\sigma_{0}^{2}$. The vectors
$\mathbf{a}_{\mathrm{t}}(\theta_{0})\in\mathbb{C}^{M}$ and
$\mathbf{a}_{\mathrm{r}}(\theta_{0})\in\mathbb{C}^{M}$ denote the transmit and
receive steering vectors of the radar antenna array,

$\displaystyle\mathbf{a}_{\mathrm{t}}(\theta_{0})=\mathbf{a}_{\mathrm{r}}(\theta_{0})\triangleq[1,e^{j\frac{2\pi}{\lambda}\Delta\mathrm{sin}(\theta_{0})},\ldots,e^{j\frac{2\pi}{\lambda}(M-1)\Delta\mathrm{sin}(\theta_{0})}]^{T},$
(6)

where $\theta_{0}$ represents the azimuth angle of the target222In radar
related literature, the direction of the target is typically known to the
transmitter since it can be readily estimated [45]-[47] at previous
observations, or given by the center of angular sector-of-interest. , $\Delta$
denotes the antenna spacing and $\lambda$ the wavelength. The matrix
$\mathbf{Q}\in\mathbb{C}^{N\times M}$ denotes the interfering channel between
the BS and the radar receiver, and $\mathbf{n}_{\mathrm{r}}\in\mathbb{C}^{M}$
is the AWGN with
$\mathbf{n}_{\mathrm{r}}\sim\mathcal{CN}(0,\sigma_{\mathrm{r}}^{2}\mathbf{I})$.
By defining
$\mathbf{A}\triangleq\mathbf{a}_{\mathrm{r}}(\theta_{0})\mathbf{a}_{\mathrm{t}}^{H}(\theta_{0})$,
the received signal at the radar can be concisely re-written as

$\mathbf{y}_{\mathrm{r}}=\alpha\mathbf{A}\mathbf{F}\mathbf{c}+\mathbf{Q}^{H}\mathbf{W}\mathbf{s}+\mathbf{n}_{\mathrm{r}}.$
(7)

To achieve better radar sensing performance, the radar utilizes a receive
filter $\mathbf{u}\in\mathbb{C}^{M}$ to suppress the interference from the BS
and the noise. The signal after filtering is

$\displaystyle
y_{\mathrm{r}}=\alpha\mathbf{u}^{H}\mathbf{A}\mathbf{F}\mathbf{c}+\mathbf{u}^{H}\mathbf{Q}^{H}\mathbf{W}\mathbf{s}+\mathbf{u}^{H}\mathbf{n}_{\mathrm{r}},$
(8)

and radar output SINR can thus be written as

$\displaystyle\mathrm{SINR}_{\mathrm{r}}$
$\displaystyle=\frac{\|\alpha\mathbf{u}^{H}\mathbf{A}\mathbf{F}\|^{2}}{\sum\limits_{k=1}^{K}|\mathbf{u}^{H}\mathbf{Q}^{H}\mathbf{w}_{k}|^{2}+\mathbb{E}\left\\{|\mathbf{u}^{H}\mathbf{n}_{\mathrm{r}}|^{2}\right\\}}$
(9)
$\displaystyle=\frac{\sigma_{\mathrm{0}}^{2}\mathbf{u}^{H}\mathbf{A}\mathbf{F}\mathbf{F}^{H}\mathbf{A}^{H}\mathbf{u}}{\mathbf{u}^{H}\Big{(}\sum\limits_{k=1}^{K}\mathbf{Q}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\Big{)}\mathbf{u}}.$

For radar systems, the radar SINR is widely utilized as the metric to evaluate
radar sensing performance. Thus, we attempt to design the transmit beamforming
and the radar receive filter to guarantee that the radar SINR is no less than
a pre-defined threshold for achieving satisfactory radar target detection
performance.

In the considered MU-MISO communication and MIMO radar coexisted system, the
CSI $\mathbf{h}_{k}$, $\mathbf{g}_{k}$, $\forall k$ and $\mathbf{Q}$ are
assumed to be perfectly known by appropriate channel estimation approaches333
The CSI $\mathbf{h}_{k}$, $\forall k$ can be obtained by conventional uplink
training, i.e., users send orthogonal pilot sequences and the BS estimates CSI
by classic channel estimation algorithms. Utilizing the same pilot signals
sent by users, the CSI $\mathbf{g}_{k}$, $\forall k$ can also be acquired by
the radar without additional signaling overhead. The channel $\mathbf{Q}$
between the BS and the radar requires specific pilot signaling for channel
estimation. Fortunately, since the geometric locations of the BS and the radar
are fixed, channel $\mathbf{Q}$ is generally quasi-static and requires less
estimation, which allows acceptable signaling overhead. . Our objective, in
this case, is to jointly design the BS transmit beamformer $\mathbf{W}$, the
radar transmit beamformer $\mathbf{F}$ and the radar receive filter
$\mathbf{u}$ to minimize the maximum eavesdropping SINR of the $I$
eavesdroppers on the $K$ communication users. Meanwhile, the communication QoS
requirements of the legitimate users, the radar target detection performance
constraint and the power constraints of both communication and radar systems
are satisfied. Therefore, the optimization problem can be formulated as

$\displaystyle\underset{\mathbf{W},\mathbf{F},\mathbf{u}}{\min}\quad$
$\displaystyle\underset{i,k}{\max}\quad\mathrm{SINR}_{i,k}^{\mathrm{e}}$ (10a)
s.t. $\displaystyle\mathrm{SINR}_{k}\geq\Gamma_{k},~{}\forall k,$ (10b)
$\displaystyle\mathrm{SINR}_{\mathrm{r}}\geq\Gamma_{\mathrm{r}},$ (10c)
$\displaystyle\left\|\mathbf{W}\right\|_{F}^{2}\leq P_{\mathrm{c}},$ (10d)
$\displaystyle\left\|\mathbf{F}\right\|_{F}^{2}\leq P_{\mathrm{r}},$ (10e)

where $\Gamma_{k}$ is the SINR requirement of the $k$-th Bob,
$\Gamma_{\mathrm{r}}$ is the pre-defined threshold for achieving required
target detection performance, $P_{\mathrm{c}}$ and $P_{\mathrm{r}}$ denote the
total power budgets of the BS and the radar, respectively. Due to the
quadratic fractional objective (10a), the quadratic fractional constraints
(10b) and (10c), and the coupled variables, problem (10) is a complicated non-
convex problem that cannot be directly solved. In order to tackle these
difficulties, in the next subsection, we first utilize the BCD method to
convert the original problem into two sub-problems, and then employ efficient
algorithms based on FP and SDR methods to iteratively solve them.

###  II-B Proposed Joint Transmit Beamforming Design

In order to decouple the transmit beamformers $\mathbf{W}$ and $\mathbf{F}$
and the radar receive filter $\mathbf{u}$ in the non-convex constraint (10c),
we adopt the BCD method to iteratively solve them, which are described in
detail as follows.

Update $\mathbf{W}$ and $\mathbf{F}$: With fixed $\mathbf{u}$, the sub-problem
for updating $\mathbf{W}$ and $\mathbf{F}$ is re-arranged as

$\displaystyle\underset{\mathbf{w}_{k},\forall
k,\mathbf{F}}{\min}~{}~{}\underset{i,k}{\max}~{}~{}\frac{\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{h}_{\mathrm{e},i}}{\sum\limits_{j\neq
k}^{K}\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{j}\mathbf{w}_{j}^{H}\mathbf{h}_{\mathrm{e},i}+\mathbf{g}^{H}_{\mathrm{e},i}\mathbf{F}\mathbf{F}^{H}\mathbf{g}_{\mathrm{e},i}+\sigma_{\mathrm{e},i}^{2}}$
(11a)
$\displaystyle\text{s.t.}~{}~{}~{}\frac{\mathbf{h}_{k}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{h}_{k}}{\sum\limits_{j\neq
k}^{K}\mathbf{h}_{k}^{H}\mathbf{w}_{j}\mathbf{w}_{j}^{H}\mathbf{h}_{k}+\mathbf{g}_{k}^{H}\mathbf{F}\mathbf{F}^{H}\mathbf{g}_{k}+\sigma_{k}^{2}}\geq\Gamma_{k},\forall
k,$ (11b)
$\displaystyle~{}~{}~{}~{}~{}~{}\frac{\sigma_{\mathrm{0}}^{2}\mathbf{u}^{H}\mathbf{A}\mathbf{F}\mathbf{F}^{H}\mathbf{A}^{H}\mathbf{u}}{\mathbf{u}^{H}\Big{(}\sum\limits_{k=1}^{K}\mathbf{Q}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\Big{)}\mathbf{u}}\geq\Gamma_{\mathrm{r}},$
(11c)
$\displaystyle~{}~{}~{}~{}~{}~{}\sum\limits_{k=1}^{K}\left\|\mathbf{w}_{k}\right\|^{2}\leq
P_{\mathrm{c}},$ (11d)
$\displaystyle~{}~{}~{}~{}~{}~{}\left\|\mathbf{F}\right\|_{F}^{2}\leq
P_{\mathrm{r}},$ (11e)

which is a complicated min-max problem. To tackle the min-max problem, we
introduce an auxiliary variable $z$ to re-formulate it into a more favorable
form as

$\displaystyle\underset{\mathbf{w}_{k},\forall k,\mathbf{F},z}{\min}\quad z$
(12a)
$\displaystyle\text{s.t.}~{}~{}\frac{\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{h}_{\mathrm{e},i}}{\sum\limits_{j\neq
k}^{K}\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{j}\mathbf{w}_{j}^{H}\mathbf{h}_{\mathrm{e},i}+\mathbf{g}^{H}_{\mathrm{e},i}\mathbf{F}\mathbf{F}^{H}\mathbf{g}_{\mathrm{e},i}+\sigma_{\mathrm{e},i}^{2}}\leq
z,~{}\forall i,k,$ (12b) $\displaystyle\qquad(\ref{eq:known update W and
F}\text{b})-(\ref{eq:known update W and F}\text{e}),$ (12c)

which is a minimization problem but still difficult to solve due to the
fractional constraint (12b) and the non-convex constraints (11b) and (11c).
Noting that problem (12) has a similar form as the max-min-ratio fractional
programming problems [48], Dinkelbach’s transform can be applied to convert it
into a more tractable form [49]. Specifically, the fractional constraint (12b)
can be converted into a polynomial expression (13) as shown on the top of this
page

$\displaystyle\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{h}_{\mathrm{e},i}-c_{i,k}\big{(}\sum\limits_{j\neq
k}^{K}\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{j}\mathbf{w}_{j}^{H}\mathbf{h}_{\mathrm{e},i}+\mathbf{g}^{H}_{\mathrm{e},i}\mathbf{F}\mathbf{F}^{H}\mathbf{g}_{\mathrm{e},i}+\sigma_{\mathrm{e},i}^{2}\big{)}\leq
z,~{}\forall i,k,$ (13)

by introducing an auxiliary variable $c_{i,k}$, which essentially represents
the eavesdropping SINR of the $i$-th Eve on the $k$-th Bob and is
alternatively updated with the transmit beamformers $\mathbf{w}_{k},\forall
k$, and $\mathbf{F}$.

With given $\mathbf{w}_{k},\forall k$, and $\mathbf{F}$, the optimal
$c_{i,k}^{\star}$ can be easily obtained by

$\displaystyle
c_{i,k}^{\star}=\frac{\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{h}_{\mathrm{e},i}}{\sum\nolimits_{j\neq
k}^{K}\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{j}\mathbf{w}_{j}^{H}\mathbf{h}_{\mathrm{e},i}+\mathbf{g}^{H}_{\mathrm{e},i}\mathbf{F}\mathbf{F}^{H}\mathbf{g}_{\mathrm{e},i}+\sigma_{\mathrm{e},i}^{2}},~{}\forall
i,k.$ (14)

Then, the optimization problem for updating $\mathbf{w}_{k}$, $\forall k$, and
$\mathbf{F}$ can be formulated as

$\displaystyle\underset{\mathbf{w}_{k},\forall k,\mathbf{F},z}{\min}\quad z$
(15a) $\displaystyle\text{s.t.}~{}~{}\eqref{eq:c}$
$\displaystyle~{}~{}~{}~{}~{}~{}\frac{\mathbf{h}_{k}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{h}_{k}}{\sum\limits_{j\neq
k}^{K}\mathbf{h}_{k}^{H}\mathbf{w}_{j}\mathbf{w}_{j}^{H}\mathbf{h}_{k}+\mathbf{g}_{k}^{H}\mathbf{F}\mathbf{F}^{H}\mathbf{g}_{k}+\sigma_{k}^{2}}\geq\Gamma_{k},\forall
k,$ (15b)
$\displaystyle~{}~{}~{}~{}~{}~{}\frac{\sigma_{\mathrm{0}}^{2}\mathbf{u}^{H}\mathbf{A}\mathbf{F}\mathbf{F}^{H}\mathbf{A}^{H}\mathbf{u}}{\mathbf{u}^{H}\Big{(}\sum\limits_{k=1}^{K}\mathbf{Q}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\Big{)}\mathbf{u}}\geq\Gamma_{\mathrm{r}},$
(15c)
$\displaystyle~{}~{}~{}~{}~{}~{}\sum\limits_{k=1}^{K}\left\|\mathbf{w}_{k}\right\|^{2}\leq
P_{\mathrm{c}},$ (15d)
$\displaystyle~{}~{}~{}~{}~{}~{}\left\|\mathbf{F}\right\|_{F}^{2}\leq
P_{\mathrm{r}}.$ (15e)

It is easy to see that the constraints (13), (15b), and (15c) are non-convex
with respect to variables $\mathbf{w}_{k},\forall k$, and $\mathbf{F}$ and are
hard to tackle. Therefore, we apply the SDR method to convert them into
primary variables for an easier solution. Specifically, by defining

$\displaystyle\mathbf{W}_{k}\triangleq\mathbf{w}_{k}\mathbf{w}_{k}^{H},~{}\forall
k,$ (16)
$\displaystyle\mathbf{R}_{\mathrm{F}}\triangleq\mathbf{F}\mathbf{F}^{H},$

the quadratic terms $\mathbf{w}_{k}\mathbf{w}_{k}^{H}$ and
$\mathbf{F}\mathbf{F}^{H}$ are converted into the primary variables
$\mathbf{W}_{k}$ and $\mathbf{R}_{\mathrm{F}}$, respectively. In the meantime,
the rank-one Hermitian positive semidefinite matrices $\mathbf{W}_{k}$,
$\forall k$, and the Hermitian positive semidefinite matrix
$\mathbf{R}_{\mathrm{F}}$ should satisfy

$\displaystyle\mathbf{W}_{k}=\mathbf{W}_{k}^{H},~{}\mathbf{W}_{k}\succeq
0,~{}\forall k,$ (17a) $\displaystyle\mathrm{Rank}(\mathbf{W}_{k})=1,$ (17b)
$\displaystyle\mathbf{R}_{\mathrm{F}}=\mathbf{R}_{\mathrm{F}}^{H},~{}\mathbf{R}_{\mathrm{F}}\succeq
0.$ (17c)

For simplicity, we define the set of all $N\times N$-dimensional Hermitian
positive semidefinite matrices as
$\mathbb{S}_{N}\triangleq\\{\mathbf{S}|\mathbf{S}=\mathbf{S}^{H},~{}\mathbf{S}\succeq
0\\}$. Afterwards, problem (15) is transformed into

$\displaystyle\underset{\mathbf{W}_{k},\forall
k,\mathbf{R}_{\mathrm{F}},z}{\min}\quad z$ (18a)
$\displaystyle\text{s.t.}~{}~{}\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{W}_{k}\mathbf{h}_{\mathrm{e},i}$
(18b) $\displaystyle\quad~{}~{}~{}-c_{i,k}\big{(}\sum\limits_{j\neq
k}^{K}\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{W}_{j}\mathbf{h}_{\mathrm{e},i}+\mathbf{g}^{H}_{\mathrm{e},i}\mathbf{R}_{\mathrm{F}}\mathbf{g}_{\mathrm{e},i}+\sigma_{\mathrm{e},i}^{2}\big{)}\leq
z,~{}\forall i,k,$
$\displaystyle\quad~{}~{}~{}\frac{\mathbf{h}_{k}^{H}\mathbf{W}_{k}\mathbf{h}_{k}}{\sum\limits_{j\neq
k}^{K}\mathbf{h}_{k}^{H}\mathbf{W}_{j}\mathbf{h}_{k}+\mathbf{g}_{k}^{H}\mathbf{R}_{\mathrm{F}}\mathbf{g}_{k}+\sigma_{k}^{2}}\geq\Gamma_{k},~{}\forall
k,$ (18c)
$\displaystyle\quad~{}~{}~{}\frac{\sigma_{\mathrm{0}}^{2}\mathbf{u}^{H}\mathbf{A}\mathbf{R}_{\mathrm{F}}\mathbf{A}^{H}\mathbf{u}}{\mathbf{u}^{H}\Big{(}\sum\limits_{k=1}^{K}\mathbf{Q}^{H}\mathbf{W}_{k}\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\Big{)}\mathbf{u}}\geq\Gamma_{\mathrm{r}},$
(18d)
$\displaystyle\quad~{}~{}~{}\sum_{k=1}^{K}\mathrm{Tr}(\mathbf{W}_{k})\leq
P_{\mathrm{c}},$ (18e)
$\displaystyle\quad~{}~{}~{}\mathrm{Tr}(\mathbf{R}_{\mathrm{F}})\leq
P_{\mathrm{r}},$ (18f)
$\displaystyle\quad~{}~{}~{}\mathbf{W}_{k}\in\mathbb{S}_{N},~{}\forall
k,~{}\mathbf{R}_{\mathrm{F}}\in\mathbb{S}_{M},$ (18g)
$\displaystyle\quad~{}~{}~{}\mathrm{Rank}(\mathbf{W}_{k})=1,~{}\forall k.$
(18h)

It is evident that the rank-one constraint (18h) tremendously hinders finding
a straightforward solution. Thus, we apply the SDR algorithm by dropping the
rank-one constraint (18h) and relaxing the problem (18) as

$\displaystyle\underset{\mathbf{W}_{k},\forall
k,\mathbf{R}_{\mathrm{F}},z}{\min}~{}~{}$ $\displaystyle z$ (19a) s.t.
$\displaystyle(\ref{eq:SDR}\text{b})-(\ref{eq:SDR}\text{g}),$ (19b)

which is a semi-definite programming (SDP) problem that can be efficiently
solved by various existing algorithms and toolboxes such as CVX. Since the
rank-one constraint is temporarily neglected, the optimal objective value of
problem (19) only serves as a lower bound. After obtaining
$\mathbf{W}_{k},\forall k$, the eigenvalue decomposition (EVD) is usually used
to obtain the optimal solution $\mathbf{w}_{k}$ if the resulting
$\mathbf{W}_{k}$ satisfies the rank-one constraint. Otherwise, Gaussian
randomization is required to convert the high-rank solution to a feasible
rank-one solution to the problem (11). In our considered case, the rank-1
solution can be guaranteed, whose proof is given in Appendix A. On the other
hand, with resulting optimal $\mathbf{R}_{\text{F}}$, the radar beamforming
matrix $\mathbf{F}$ can be obtained by utilizing Cholesky decomposition.

Algorithm 1 Joint transmit beamforming design algorithm for solving problem
(10)

0: $\mathbf{h}_{\mathrm{e},i}$, $\mathbf{g}_{\mathrm{e},i}$,
$\sigma_{\mathrm{e},i}^{2}$, $\forall i$, $\mathbf{h}_{k}$, $\mathbf{g}_{k}$,
$\sigma_{k}^{2}$, $\Gamma_{k}$, $\forall k$, $\mathbf{A}$, $\mathbf{Q}$,
$\sigma_{\text{0}}^{2}$, $\sigma_{\text{r}}^{2}$,
$~{}~{}~{}~{}~{}\Gamma_{\mathrm{r}}$, $P_{\mathrm{c}}$, $P_{\mathrm{r}}$.

0: $\mathbf{w}_{k}^{\star}$, $\forall k$, $\mathbf{F}^{\star}$, and
$\mathbf{u}^{\star}$.

1: Initialize $\mathbf{F}$, $\mathbf{u}$, and $c_{i,k}$, $\forall i,k$.

2: while the objective value (10a) does not converge do

3: while no convergence do

4: Calculate $\mathbf{W}_{k},\forall k$, and $\mathbf{R}_{\mathrm{F}}$ by
solving (19).

5: Update $\mathbf{w}_{k}$ from $\mathbf{W}_{k},\forall k$, by EVD.

6: Update $\mathbf{F}$ from $\mathbf{R}_{\mathrm{F}}$ by Cholesky
decomposition.

7: Update $c_{i,k}$, $\forall i,k$, by (14).

8: end while

9: Update $\mathbf{u}$ by solving (20).

10: end while

11: Return $\mathbf{w}_{k}^{\star},\forall k$, $\mathbf{F}^{\star}$, and
$\mathbf{u}^{\star}$.

Update $\mathbf{u}$: It can be seen that the variable $\mathbf{u}$ only exists
in the constraint (10c) of the problem (10). Therefore, with given
$\mathbf{W}$ and $\mathbf{F}$, problem (10) is transformed into a feasibility-
check problem. For the sake of leaving enough freedoms for solving
$\mathbf{W}$ and $\mathbf{F}$ in the subsequent optimization process and
accelerating the convergence, we propose to optimize $\mathbf{u}$ to maximize
the radar output SINR. The optimization problem is formulated as

$\mathbf{u}^{\mathrm{opt}}=\arg\mathop{\max}\limits_{\mathbf{u}}\frac{\sigma_{\mathrm{0}}^{2}\mathbf{u}^{H}\mathbf{A}\mathbf{F}\mathbf{F}^{H}\mathbf{A}^{H}\mathbf{u}}{\mathbf{u}^{H}\Big{(}\sum\limits_{k=1}^{K}\mathbf{Q}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\Big{)}\mathbf{u}}.$
(20)

We observe that problem (20) is a typical generalized Rayleigh quotient, whose
optimal solution can be easily obtained as the generalized eigenvector
corresponding to the largest eigenvalue of the matrix
$\sigma_{\mathrm{0}}^{2}\big{(}\sum\nolimits_{k=1}^{K}\mathbf{Q}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\big{)}^{-1}\mathbf{A}\mathbf{F}\mathbf{F}^{H}\mathbf{A}^{H}$.

###  II-C Summary, Initialization, Convergence, and Complexity Analysis

####  II-C.1 Summary

Based on the above derivations, the proposed joint transmit beamforming and
radar receive filter design algorithm is straightforward and summarized in
Algorithm 1. With appropriate initialization, problems (11) and (20) are
iteratively solved to respectively update $\mathbf{w}_{k}$, $\forall k$,
$\mathbf{F}$, and $\mathbf{u}$ until the objective value (10a) converges. For
updating $\mathbf{w}_{k}$, $\forall k$, and $\mathbf{F}$, we iteratively
update the communication beamforming $\mathbf{w}_{k}$, $\forall k$, and the
radar beamforming $\mathbf{F}$ by solving the SDP problem (19) and then
recovering feasible solutions using EVD and Cholesky decomposition, and the
auxiliary variables $c_{i,k},\forall i,k$, until the convergence of problem
(11) is found.

####  II-C.2 Initialization

In order to solve sub-problem (15) for updating $\mathbf{w}_{k}$, $\forall k$,
and $\mathbf{F}$, we need to initialize $\mathbf{u}$ and the auxiliary
variables $c_{i,k},~{}\forall i,k$. The initial value of the radar receive
filter is selected as $\mathbf{u}=\mathbf{a}_{\mathrm{r}}(\theta_{0})$ via a
phase alignment operation for better radar detection performance. Since
$c_{i,k}$ represents the eavesdropping SINR, which is generally several orders
of magnitude smaller than the communication SINR, initializing $c_{i,k}$ with
the pre-defined threshold of communication SINR $\Gamma_{k}$ can guarantee the
feasibility. The obtained $\mathbf{w}_{k}$, $\forall k$, and $\mathbf{F}$ by
solving (15) can be set as the initial value for solving sub-problem (20).

####  II-C.3 Convergence

We will briefly prove the convergence of the proposed algorithm as follows.
Denote $\eta\big{(}\mathbf{W},\mathbf{F},\mathbf{u}\big{)}$ as the objective
value of the original problem (10). First, in the transmit beamformer design,
we apply Dinkelbach’s transform to convert it into a more tractable form and
transform it into the problem (15). According to [50], it is easy to prove the
convergence of the algorithm given the non-increasing property of the
auxiliary variable $c_{i,k}$. Since the optimal solution of problem (11) is
obtained with given $\mathbf{u}^{t}$, we have

$\eta\big{(}\mathbf{W}^{t},\mathbf{F}^{t},\mathbf{u}^{t}\big{)}\geq\eta\big{(}\mathbf{W}^{t+1},\mathbf{F}^{t+1},\mathbf{u}^{t}\big{)},$
(21)

where the superscript $t$ denotes the index of iterations. Second, with fixed
$\left\\{\mathbf{W},\mathbf{F}\right\\}$, the sub-problem for updating
$\mathbf{u}$ is a feasibility-check problem. After solving the problem (20), a
better radar output SINR than the original requirement is achieved with the
obtained radar receive filter $\mathbf{u}$ in the current iteration, i.e., the
feasible domain of the original problem (10) is expanded while the objective
value is fixed. In other words, with given
$\left\\{\mathbf{W}^{t+1},\mathbf{F}^{t+1}\right\\}$, we have

$\eta\big{(}\mathbf{W}^{t+1},\mathbf{F}^{t+1},\mathbf{u}^{t}\big{)}=\eta\big{(}\mathbf{W}^{t+1},\mathbf{F}^{t+1},\mathbf{u}^{t+1}\big{)}.$
(22)

Based on the above analysis, we have the relationship of the objective values
between iterations as

$\eta\big{(}\mathbf{W}^{t},\mathbf{F}^{t},\mathbf{u}^{t}\big{)}\geq\eta\big{(}\mathbf{W}^{t+1},\mathbf{F}^{t+1},\mathbf{u}^{t+1}\big{)},$
(23)

which indicates that the objective value of problem (10) is non-increasing
during the iterations of Algorithm 1. Since the objective value of problem
(10) is greater than zero, the proposed Algorithm 1 can converge to a local
optimum point.

####  II-C.4 Complexity Analysis

In this subsection, the computational complexity of Algorithm 1 is analyzed as
follows. We first analyze the computational complexity of solving for
$\mathbf{w}_{k}$, $\forall k$, and $\mathbf{F}$. Problem (19) is a convex
problem with $K$ $N\times N$-dimensional and an $M\times M$-dimensional
variable to be optimized, $(I+1)K+1$ second-order cone (SOC) constraints and
$K+1$ linear matrix inequality (LMI) constraints. Using the CVX solver, the
computational complexity is of order
$\mathcal{O}\\{\text{ln}(1/\xi)2\sqrt{5}IK^{1.5}M^{6}\\}$, where $\xi$ is the
convergence threshold. The computational complexity of updating $c_{i,k}$,
$\forall i,k$, is of order $\mathcal{O}\\{M^{3}\\}$. Other calculations have
much lower complexities. For example, updating $\mathbf{w}_{k}$, $\forall k$,
has negligible computational of order $\mathcal{O}\\{N^{3}\\}$. Thus, the
total complexity to obtain $\mathbf{w}_{k},\forall k$, and $\mathbf{F}$ is of
order $\mathcal{O}\\{N_{\text{FP}}\text{ln}(1/\xi)2\sqrt{5}IK^{1.5}M^{6}\\}$,
where $N_{\text{FP}}$ is the number of iterations of the inner loop. The
computational complexity of updating $\mathbf{u}$ is of order
$\mathcal{O}\\{M^{3}\\}$. Therefore, the total computational complexity of the
proposed BCD-FP-SDR algorithm is of order
$\mathcal{O}\\{N_{\text{tot}}N_{\text{FP}}\text{ln}(1/\xi)2\sqrt{5}IK^{1.5}M^{6}\\}$,
where $N_{\text{tot}}$ is the number of iterations of the outer loop.

##  III Joint AN-Aided Transmit Beamforming Design without Eavesdroppers’ CSI

###  III-A System Model and Problem Formulation

When Eves are pure passive eavesdroppers, Alice is unaware of Eves’ CSI or
even their existence. In the sequel, the proposed design in the previous
section cannot be adopted to ensure security performance. In such cases, AN is
a very effective method to improve the physical layer security by disrupting
Eves’ reception. Specifically, in addition to transmitting confidential
information or probing signals, the BS and the radar also use the available
transmit power to generate AN for disturbing potential eavesdroppers as much
as possible. Therefore, the transmit beamforming and AN of both the BS and the
radar are jointly designed to guarantee good communication and radar sensing
performance while safeguarding the communication system against potential
malicious eavesdroppers.

Based on the above descriptions, the received signal of the $k$-th Bob can be
written as

$\displaystyle
y_{k}=\mathbf{h}_{k}^{H}(\mathbf{W}\mathbf{s}+\mathbf{z})+\mathbf{g}_{k}^{H}(\mathbf{F}\mathbf{c}+\mathbf{v})+n_{k},$
(24)

where $\mathbf{z}\sim\mathcal{CN}(0,\mathbf{R}_{\mathrm{z}})$ and
$\mathbf{v}\sim\mathcal{CN}(0,\mathbf{R}_{\mathrm{v}})$ are AN vectors
generated by the BS and the radar, respectively. We assume that the
information symbol vector $\mathbf{s}$, the AN vector $\mathbf{z}$ generated
by the BS, the radar waveform $\mathbf{c}$, and the AN vector $\mathbf{v}$
generated by the radar are statistically independent of each other. Thus, the
SINR of the $k$-th Bob can be calculated as

$\displaystyle\mathrm{SINR}_{k}$ (25)
$\displaystyle=\frac{|\mathbf{h}_{k}^{H}\mathbf{w}_{k}|^{2}}{\sum\limits_{j\neq
k}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{j}|^{2}+\mathbb{E}\left\\{|\mathbf{h}_{k}^{H}\mathbf{z}|^{2}\right\\}+\|\mathbf{g}_{k}^{H}\mathbf{F}\|^{2}+\mathbb{E}\left\\{|\mathbf{g}_{k}^{H}\mathbf{v}|^{2}\right\\}+\sigma_{k}^{2}}$
$\displaystyle=\frac{\mathbf{h}_{k}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{h}_{k}}{\mathbf{h}_{k}^{H}(\sum\limits_{j\neq
k}^{K}\mathbf{w}_{j}\mathbf{w}_{j}^{H}+\mathbf{R}_{\mathrm{z}})\mathbf{h}_{k}+\mathbf{g}_{k}^{H}(\mathbf{F}\mathbf{F}^{H}+\mathbf{R}_{\mathrm{v}})\mathbf{g}_{k}+\sigma_{k}^{2}}.$

From the communication perspective, the proposed joint AN-aided transmit
beamforming design aims to guarantee the communication QoS requirements of
legitimate transmission while interfering with Eves as much as possible.

On the radar side, the echo wave received by the radar is expressed as

$\mathbf{y}_{\mathrm{r}}=\alpha\mathbf{A}(\mathbf{F}\mathbf{c}+\mathbf{v})+\mathbf{Q}^{H}(\mathbf{W}\mathbf{s}+\mathbf{z})+\mathbf{n}_{\mathrm{r}}.$
(26)

After passing through the receive filter $\mathbf{u}$, the radar output is

$y_{\mathrm{r}}=\alpha\mathbf{u}^{H}\mathbf{A}(\mathbf{F}\mathbf{c}+\mathbf{v})+\mathbf{u}^{H}\mathbf{Q}^{H}(\mathbf{W}\mathbf{s}+\mathbf{z})+\mathbf{u}^{H}\mathbf{n}_{\mathrm{r}}.$
(27)

The radar output SINR is thus given by

$\displaystyle\mathrm{SINR}_{\mathrm{r}}$ (28)
$\displaystyle=\frac{\|\alpha\mathbf{u}^{H}\mathbf{A}\mathbf{F}\|^{2}}{\mathbb{E}\left\\{|\alpha\mathbf{u}^{H}\mathbf{A}\mathbf{v}|^{2}\right\\}+\sum\limits_{k=1}^{K}|\mathbf{u}^{H}\mathbf{Q}^{H}\mathbf{w}_{k}|^{2}+\mathbb{E}\left\\{|\mathbf{u}^{H}\mathbf{Q}^{H}\mathbf{z}|^{2}+|\mathbf{u}^{H}\mathbf{n}_{\mathrm{r}}|^{2}\right\\}}$
$\displaystyle=\frac{\sigma_{\mathrm{0}}^{2}\mathbf{u}^{H}\mathbf{A}\mathbf{F}\mathbf{F}^{H}\mathbf{A}^{H}\mathbf{u}}{\mathbf{u}^{H}\Big{(}\sigma_{\mathrm{0}}^{2}\mathbf{A}\mathbf{R}_{\mathrm{v}}\mathbf{A}^{H}+\mathbf{Q}^{H}(\sum\limits_{k=1}^{K}\mathbf{w}_{k}\mathbf{w}_{k}^{H}+\mathbf{R}_{\mathrm{z}})\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\Big{)}\mathbf{u}}.$

From the radar perspective, the radar output SINR is guaranteed to be no less
than a pre-defined threshold for achieving satisfactory target detection
performance. Meanwhile, as much power as possible is used to generate AN for
interfering with Eves.

It is intuitive that higher transmission power of the confidential information
poses a higher risk of being intercepted, since Eves’ eavesdropping SINR is
directly proportional to the transmission power. Thus, the BS should minimize
the transmit power to satisfy Bobs’ QoS and use the residual power to generate
AN signals. Similarly, for the radar system, the minimum required power is
allocated to generate directional signals, whose main lobe points to the
direction of the target for achieving satisfactory detection performance. The
huge residual power is used to generate omni-directional AN signals, which
will bring excellent security performance in the presence of potential
eavesdroppers. Therefore, our objective is to jointly design the BS transmit
beamformer $\mathbf{W}$, the covariance $\mathbf{R}_{\mathrm{z}}$ of the AN
vector $\mathbf{z}$ generated by the BS, the radar transmit beamformer
$\mathbf{F}$, the covariance $\mathbf{R}_{\mathrm{v}}$ of the AN vector
$\mathbf{v}$ generated by the radar, and the radar receive filter $\mathbf{u}$
to minimize the total transmit power used by the BS and radar beamformers.
Meanwhile, the communication QoS requirements of the legitimate users, the
radar output SINR constraint, and the power constraints of both communication
and radar systems are satisfied. Therefore, the optimization problem is
formulated as

$\displaystyle\underset{\mathbf{W},\mathbf{R}_{\mathrm{z}},\mathbf{F},\mathbf{R}_{\mathrm{v}},\mathbf{u}}{\min}\quad$
$\displaystyle\left\|\mathbf{W}\right\|_{F}^{2}+\left\|\mathbf{F}\right\|_{F}^{2}$
(29a) s.t. $\displaystyle\mathrm{SINR}_{k}\geq\Gamma_{k},~{}\forall k,$ (29b)
$\displaystyle\mathrm{SINR}_{\mathrm{r}}\geq\Gamma_{\mathrm{r}},$ (29c)
$\displaystyle\left\|\mathbf{W}\right\|_{F}^{2}+\mathrm{Tr}(\mathbf{R}_{\mathrm{z}})=P_{\mathrm{c}},$
(29d)
$\displaystyle\left\|\mathbf{F}\right\|_{F}^{2}+\mathrm{Tr}(\mathbf{R}_{\mathrm{v}})=P_{\mathrm{r}},$
(29e)
$\displaystyle\mathbf{R}_{\mathrm{z}}\in\mathbb{S}_{N},~{}\mathbf{R}_{\mathrm{v}}\in\mathbb{S}_{M}.$
(29f)

We observe that problem (29) is a non-convex problem that is difficult to
solve for the following two reasons. First, the variables are intricately
coupled in the constraints (29b) and (29c). Second, these two constraints are
quadratic and fractional. In order to efficiently solve this problem, we
employ the BCD and SDR algorithms to convert it into two more tractable sub-
problems and then alternately solve each of them until convergence is
achieved.

###  III-B Proposed AN-aided Joint Transmit Beamforming Design

In this subsection, we propose an efficient double-loop BCD-SDR algorithm to
tackle the non-convex problem (29). It can be seen that problem (29) is very
complicated due to the coupling variables in the SINR constraints (29b) and
(29c). To this end, we divide this problem into two sub-problems with respect
to the radar and communication systems, and utilize a two-block BCD algorithm
to iteratively solve them.

####  III-B.1 The Sub-problem for Radar System

Given the variables $\mathbf{w}_{k}$, $\forall k$, and
$\mathbf{R}_{\mathrm{z}}$ of the communication system, the transmit beamformer
$\mathbf{F}$, the covariance $\mathbf{R}_{\mathrm{v}}$ of the AN vector
$\mathbf{v}$, and the receive filter $\mathbf{u}$ of the radar system are
jointly optimized. The problem for updating $\mathbf{F}$,
$\mathbf{R}_{\mathrm{v}}$, and $\mathbf{u}$ can be formulated as

$\displaystyle\underset{\mathbf{F},\mathbf{R}_{\mathrm{v}},\mathbf{u}}{\min}~{}~{}\left\|\mathbf{F}\right\|_{F}^{2}$
(30a)
$\displaystyle\text{s.t.}~{}~{}\frac{\mathbf{h}_{k}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{h}_{k}}{\mathbf{h}_{k}^{H}(\sum\limits_{j\neq
k}^{K}\mathbf{w}_{j}\mathbf{w}_{j}^{H}+\mathbf{R}_{\mathrm{z}})\mathbf{h}_{k}+\mathbf{g}_{k}^{H}(\mathbf{F}\mathbf{F}^{H}+\mathbf{R}_{\mathrm{v}})\mathbf{g}_{k}+\sigma_{k}^{2}}\geq\Gamma_{k},\forall
k,$ (30b)
$\displaystyle~{}~{}~{}~{}~{}\frac{\sigma_{\mathrm{0}}^{2}\mathbf{u}^{H}\mathbf{A}\mathbf{F}\mathbf{F}^{H}\mathbf{A}^{H}\mathbf{u}}{\mathbf{u}^{H}\Big{(}\sigma_{\mathrm{0}}^{2}\mathbf{A}\mathbf{R}_{\mathrm{v}}\mathbf{A}^{H}+\mathbf{Q}^{H}(\sum\limits_{k=1}^{K}\mathbf{w}_{k}\mathbf{w}_{k}^{H}+\mathbf{R}_{\mathrm{z}})\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\Big{)}\mathbf{u}}\geq\Gamma_{\mathrm{r}},$
(30c)
$\displaystyle~{}~{}~{}~{}~{}\left\|\mathbf{F}\right\|_{F}^{2}+\mathrm{Tr}(\mathbf{R}_{\mathrm{v}})=P_{\mathrm{r}},$
(30d) $\displaystyle~{}~{}~{}~{}~{}\mathbf{R}_{\mathrm{v}}\in\mathbb{S}_{M}.$
(30e)

Since the variables $\mathbf{F}$, $\mathbf{R}_{\mathrm{v}}$, and $\mathbf{u}$
are highly coupled in constraint (30c), which makes problem (30) very
difficult to solve, we adopt a two-block BCD scheme to iteratively solve them
as follows.

Update $\mathbf{F}$ and $\mathbf{R}_{\mathrm{v}}$: With fixed $\mathbf{u}$,
the sub-problem for updating $\mathbf{F}$ and $\mathbf{R}_{\mathrm{v}}$ is re-
arranged as

$\displaystyle\underset{\mathbf{F},\mathbf{R}_{\mathrm{v}}}{\min}~{}~{}$
$\displaystyle\left\|\mathbf{F}\right\|_{F}^{2}$ (31a) s.t.
$\displaystyle(\ref{eq:bcd radar}\text{b})-(\ref{eq:bcd radar}\text{e}).$
(31b)

Considering that the constraints (30b) and (30c) are still non-convex due to
the quadratic terms with respect to $\mathbf{F}$, we transform problem (31)
into

$\displaystyle\underset{\mathbf{R}_{\mathrm{F}},\mathbf{R}_{\mathrm{v}}}{\min}~{}~{}\mathrm{Tr}(\mathbf{R}_{\mathrm{F}})$
(32a)
$\displaystyle\text{s.t.}~{}\frac{\mathbf{h}_{k}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{h}_{k}}{\mathbf{h}_{k}^{H}(\sum\limits_{j\neq
k}^{K}\mathbf{w}_{j}\mathbf{w}_{j}^{H}+\mathbf{R}_{\mathrm{z}})\mathbf{h}_{k}+\mathbf{g}_{k}^{H}(\mathbf{R}_{\mathrm{F}}+\mathbf{R}_{\mathrm{v}})\mathbf{g}_{k}+\sigma_{k}^{2}}\geq\Gamma_{k},\forall
k,$ (32b)
$\displaystyle\frac{\sigma_{\mathrm{0}}^{2}\mathbf{u}^{H}\mathbf{A}\mathbf{R}_{\mathrm{F}}\mathbf{A}^{H}\mathbf{u}}{\mathbf{u}^{H}\Big{(}\sigma_{\mathrm{0}}^{2}\mathbf{A}\mathbf{R}_{\mathrm{v}}\mathbf{A}^{H}+\mathbf{Q}^{H}(\sum\limits_{k=1}^{K}\mathbf{w}_{k}\mathbf{w}_{k}^{H}+\mathbf{R}_{\mathrm{z}})\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\Big{)}\mathbf{u}}\geq\Gamma_{\mathrm{r}},$
(32c)
$\displaystyle\mathrm{Tr}(\mathbf{R}_{\mathrm{F}})+\mathrm{Tr}(\mathbf{R}_{\mathrm{v}})=P_{\mathrm{r}},$
(32d)
$\displaystyle\mathbf{R}_{\mathrm{F}}\in\mathbb{S}_{M},~{}\mathbf{R}_{\mathrm{v}}\in\mathbb{S}_{M}.$
(32e)

Obviously, problem (32) is an SDP problem and can be solved by convex tools,
e.g., CVX. After solving $\mathbf{R}_{\text{F}}$, the radar beamforming matrix
$\mathbf{F}$ can be obtained by utilizing Cholesky decomposition.

Update $\mathbf{u}$: Notice that the variable $\mathbf{u}$ only exists in the
constraint (30c) of the problem (30). Therefore, with given $\mathbf{F}$ and
$\mathbf{R}_{\mathrm{v}}$, problem (30) is transformed into a feasibility-
check problem. In order to leave more freedoms for minimizing
$\left\|\mathbf{F}\right\|_{F}^{2}$ in the next iteration, we propose to
optimize $\mathbf{u}$ to maximize the radar output SINR. The optimization
problem is formulated as

$\displaystyle\mathbf{u}^{\mathrm{opt}}=$ (33)
$\displaystyle\arg\mathop{\max}\limits_{\mathbf{u}}\frac{\sigma_{\mathrm{0}}^{2}\mathbf{u}^{H}\mathbf{A}\mathbf{F}\mathbf{F}^{H}\mathbf{A}^{H}\mathbf{u}}{\mathbf{u}^{H}\Big{(}\sigma_{\mathrm{0}}^{2}\mathbf{A}\mathbf{R}_{\mathrm{v}}\mathbf{A}^{H}+\mathbf{Q}^{H}(\sum\limits_{k=1}^{K}\mathbf{w}_{k}\mathbf{w}_{k}^{H}+\mathbf{R}_{\mathrm{z}})\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\Big{)}\mathbf{u}},$

whose optimal solution can be easily obtained as the generalized eigenvector
corresponding to the largest eigenvalue of matrix
$\sigma_{\mathrm{0}}^{2}\big{[}\sigma_{\mathrm{0}}^{2}\mathbf{A}\mathbf{R}_{\mathrm{v}}\mathbf{A}^{H}\\!+\\!\mathbf{Q}^{H}(\sum\limits_{k=1}^{K}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\\!+\\!\mathbf{R}_{\mathrm{z}})\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\big{]}^{-1}\mathbf{A}\mathbf{F}\mathbf{F}^{H}\mathbf{A}^{H}$.

Algorithm 2 Joint AN-aided transmit beamforming design algorithm for solving
problem (29)

0: $\mathbf{h}_{k}$, $\mathbf{g}_{k}$, $\Gamma_{k}$, $\sigma_{k}^{2}$,
$\forall k$, $\mathbf{A}$, $\mathbf{Q}$, $\sigma_{\mathrm{0}}^{2}$,
$\sigma_{\text{r}}^{2}$, $\Gamma_{\mathrm{r}}$, $P_{\mathrm{c}}$,
$P_{\mathrm{r}}$.

0: $\mathbf{w}_{k}^{\star}$, $\forall k$, $\mathbf{R}_{\mathrm{z}}^{\star}$,
$\mathbf{F}^{\star}$, $\mathbf{R}_{\mathrm{v}}^{\star}$, and
$\mathbf{u}^{\star}$.

1: Initialize $\mathbf{w}_{k}$, $\forall k$, $\mathbf{R}_{\mathrm{z}}$, and
$\mathbf{u}$.

2: while the objective value (29a) does not converge do

3: while the objective value (30a) does not converge do

4: Calculate $\mathbf{R}_{\mathrm{F}}$ and update $\mathbf{R}_{\mathrm{v}}$ by
solving (32).

5: Update $\mathbf{F}$ from $\mathbf{R}_{\mathrm{F}}$ by Cholesky
decomposition.

6: Update $\mathbf{u}$ by (LABEL:eq:update_u).

7: end while

8: Calculate $\mathbf{W}_{k}$, $\forall k$, and update
$\mathbf{R}_{\mathrm{z}}$ by solving (35).

9: Update $\mathbf{w}_{k}$ from $\mathbf{W}_{k}$, $\forall k$, by EVD.

10: end while

11: Return $\mathbf{w}_{k}^{\star}$, $\forall k$,
$\mathbf{R}_{\mathrm{z}}^{\star}$, $\mathbf{F}^{\star}$,
$\mathbf{R}_{\mathrm{v}}^{\star}$, and $\mathbf{u}^{\star}$.

####  III-B.2 The Sub-problem for Communication System

With fixed $\mathbf{F}$, $\mathbf{R}_{\mathrm{v}}$, and $\mathbf{u}$, the sub-
problem for updating the transmit beamformer $\mathbf{W}$ and the covariance
$\mathbf{R}_{\mathrm{z}}$ of the AN vector $\mathbf{z}$ is re-arranged as

$\displaystyle\underset{\mathbf{w}_{k},\forall
k,\mathbf{R}_{\mathrm{z}}}{\min}~{}~{}$
$\displaystyle\sum_{k=1}^{K}\mathrm{Tr}(\mathbf{w}_{k}\mathbf{w}_{k}^{H})$
(34a) s.t. $\displaystyle(\ref{eq:bcd radar}\text{b}),~{}(\ref{eq:bcd
radar}\text{c})$ (34b)
$\displaystyle\sum_{k=1}^{K}\mathrm{Tr}(\mathbf{w}_{k}\mathbf{w}_{k}^{H})+\mathrm{Tr}(\mathbf{R}_{\mathrm{z}})=P_{\mathrm{c}},$
(34c) $\displaystyle\mathbf{R}_{\mathrm{z}}\in\mathbb{S}_{N}.$ (34d)

Similarly, we utilize the SDR algorithm to convert this problem into an SDP
problem by using the definitions in (16) and dropping the rank-one constraint
(17b) as

$\displaystyle\underset{\mathbf{W}_{k},\forall
k,\mathbf{R}_{\mathrm{z}}}{\min}~{}~{}\sum_{k=1}^{K}\mathrm{Tr}(\mathbf{W}_{k})$
(35a) s.t.
$\displaystyle\frac{\mathbf{h}_{k}^{H}\mathbf{W}_{k}\mathbf{h}_{k}}{\mathbf{h}_{k}^{H}(\sum\limits_{j\neq
k}^{K}\mathbf{W}_{j}+\mathbf{R}_{\mathrm{z}})\mathbf{h}_{k}+\mathbf{g}_{k}^{H}(\mathbf{F}\mathbf{F}^{H}+\mathbf{R}_{\mathrm{v}})\mathbf{g}_{k}+\sigma_{k}^{2}}\geq\Gamma_{k},\forall
k,$ (35b)
$\displaystyle\frac{\sigma_{\mathrm{0}}^{2}\mathbf{u}^{H}\mathbf{A}\mathbf{F}\mathbf{F}^{H}\mathbf{A}^{H}\mathbf{u}}{\mathbf{u}^{H}\Big{(}\sigma_{\mathrm{0}}^{2}\mathbf{A}\mathbf{R}_{\mathrm{v}}\mathbf{A}^{H}+\mathbf{Q}^{H}(\sum\limits_{k=1}^{K}\mathbf{W}_{k}+\mathbf{R}_{\mathrm{z}})\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\Big{)}\mathbf{u}}\geq\Gamma_{\mathrm{r}},$
(35c)
$\displaystyle\sum_{k=1}^{K}\mathrm{Tr}(\mathbf{W}_{k})+\mathrm{Tr}(\mathbf{R}_{\mathrm{z}})=P_{\mathrm{c}},$
(35d) $\displaystyle(\ref{eq:Wk constraint}\text{a}),~{}(\ref{eq:bcd
communication}\text{d}),$ (35e)

and then solve the resulting SDP problem by various existing algorithms or
toolboxes, e.g., CVX. The obtained $\mathbf{W}_{k}$, $\forall k$ can also be
guaranteed to satisfy the rank-1 constraint in this case. The proof is given
in Appendix A. We use the same method as in the previous section to obtain
$\mathbf{w}_{k}$, $\forall k$.

###  III-C Summary, Initialization, Convergence, and Complexity Analysis

####  III-C.1 Summary

Based on the above derivations, the proposed joint AN-aided transmit
beamforming design is straightforward and summarized in Algorithm 2. In the
inner loop, we iteratively solve problems (31) and (LABEL:eq:update_u) to
respectively update $\mathbf{F}$, $\mathbf{R}_{\mathrm{v}}$, and $\mathbf{u}$
until the objective value (30a) converges. In the outer loop, we iteratively
solve problems (30) and (34) for updating $\mathbf{w}_{k}$, $\forall k$, and
$\mathbf{R}_{\text{z}}$ until the objective value (29a) converges.

####  III-C.2 Initialization

In order to solve sub-problem (31) for updating $\mathbf{F}$ and
$\mathbf{R}_{\mathrm{v}}$, we investigate to properly initialize $\mathbf{W}$,
$\mathbf{R}_{\mathrm{z}}$, and $\mathbf{u}$. The concept of NSP is utilized to
initialize $\mathbf{W}$ and $\mathbf{R}_{\mathrm{z}}$. Specifically, we
jointly design the beamforming without AN aiming to guarantee the performance
requirements of communication and radar systems, and then project the AN of
the communication system onto the null space of the effective interference
channels between the BS and Bobs. Similarly, the initial value of the radar
receive filter is selected as $\mathbf{u}=\mathbf{a}_{\mathrm{r}}(\theta_{0})$
via a phase alignment operation. The obtained $\mathbf{F}$,
$\mathbf{R}_{\mathrm{v}}$, and $\mathbf{u}$ by solving (30) can be set as the
initial value for solving sub-problem (34).

####  III-C.3 Convergence

The convergence of the proposed algorithm will be briefly proven as follows.
Denote
$\eta\big{(}\mathbf{W},\mathbf{R}_{\mathrm{z}},\mathbf{F},\mathbf{R}_{\mathrm{v}},\mathbf{u}\big{)}$
as the objective value of the original problem (29). First, for the radar
system, the sub-problem for updating $\mathbf{F}$ and
$\mathbf{R}_{\mathrm{v}}$ is transformed into an SDP problem. It is obvious
that the transmit power is non-increasing between iterations, we have

$\eta\big{(}\mathbf{W}^{t},\mathbf{R}_{\mathrm{z}}^{t},\mathbf{F}^{t},\mathbf{R}_{\mathrm{v}}^{t},\mathbf{u}^{t}\big{)}\geq\eta\big{(}\mathbf{W}^{t},\mathbf{R}_{\mathrm{z}}^{t},\mathbf{F}^{t+1},\mathbf{R}_{\mathrm{v}}^{t+1},\mathbf{u}^{t}\big{)},$
(36)

with fixed
$\left\\{\mathbf{W}^{t},\mathbf{R}_{\mathrm{z}}^{t},\mathbf{u}^{t}\right\\}$.
Given
$\left\\{\mathbf{W}^{t},\mathbf{R}_{\mathrm{z}}^{t},\mathbf{F}^{t+1},\mathbf{R}_{\mathrm{v}}^{t+1}\right\\}$,
the sub-problem for updating $\mathbf{u}$ is a feasibility-check problem,
which follows that

$\eta\big{(}\mathbf{W}^{t},\mathbf{R}_{\mathrm{z}}^{t},\mathbf{F}^{t+1},\mathbf{R}_{\mathrm{v}}^{t+1},\mathbf{u}^{t}\big{)}=\eta\big{(}\mathbf{W}^{t},\mathbf{R}_{\mathrm{z}}^{t},\mathbf{F}^{t+1},\mathbf{R}_{\mathrm{v}}^{t+1},\mathbf{u}^{t+1}\big{)}.$
(37)

Second, the sub-problem for the communication system is also transformed into
an SDP problem. With given
$\left\\{\mathbf{F}^{t+1},\mathbf{R}_{\mathrm{v}}^{t+1},\mathbf{u}^{t+1}\right\\}$,
it follows that

$\displaystyle\eta\big{(}\mathbf{W}^{t},\mathbf{R}_{\mathrm{z}}^{t},\mathbf{F}^{t+1},\mathbf{R}_{\mathrm{v}}^{t+1},\mathbf{u}^{t+1}\big{)}$
(38)
$\displaystyle\geq\eta\big{(}\mathbf{W}^{t+1},\mathbf{R}_{\mathrm{z}}^{t+1},\mathbf{F}^{t+1},\mathbf{R}_{\mathrm{v}}^{t+1},\mathbf{u}^{t+1}\big{)}.$

Based on the above analysis, we have the relationship of the objective values
between iterations as

$\displaystyle\eta\big{(}\mathbf{W}^{t},\mathbf{R}_{\mathrm{z}}^{t},\mathbf{F}^{t},\mathbf{R}_{\mathrm{v}}^{t},\mathbf{u}^{t}\big{)}$
(39)
$\displaystyle\geq\eta\big{(}\mathbf{W}^{t+1},\mathbf{R}_{\mathrm{z}}^{t+1},\mathbf{F}^{t+1},\mathbf{R}_{\mathrm{v}}^{t+1},\mathbf{u}^{t+1}\big{)},$

which indicates that the objective value of problem (29) is non-increasing
during iterations of Algorithm 2. Since the objective value of problem (29) is
greater than zero, the proposed Algorithm 2 can converge to a local optimum
point.

####  III-C.4 Complexity Analysis

In the inner loop, problem (32) is a convex problem with two $M\times
M$-dimensional variables to be optimized, $K+1$ SOC constraints and an LMI
constraint. Using the CVX solver, the complexity for updating $\mathbf{F}$ and
$\mathbf{R}_{\mathrm{v}}$ is of order
$\mathcal{O}\\{\text{ln}(1/\xi)8\sqrt{2}K^{1.5}M^{6}\\}$, and solving problem
(LABEL:eq:update_u) for updating $\mathbf{u}$ has the complexity of order
$\mathcal{O}\\{M^{3}\\}$. Thus, the total complexity for obtaining
$\mathbf{F}$, $\mathbf{R}_{\mathrm{v}}$, and $\mathbf{u}$ is of order
$\mathcal{O}\\{N_{\text{inn}}\text{ln}(1/\xi)8\sqrt{2}K^{1.5}M^{6}\\}$, where
$N_{\text{inn}}$ is the number of iterations of the inner loop. Similarly,
problem (35) is a convex problem with $K+1$ $N\times N$-dimensional variables
to be optimized, $K+1$ SOC constraints, and an LMI constraint. The
computational complexity for updating $\mathbf{w}_{k}$, $\forall k$, and
$\mathbf{R}_{\text{z}}$ is of order
$\mathcal{O}\\{\text{ln}(1/\xi)\sqrt{2}K^{4.5}N^{6}\\}$. Other lower
complexity calculations are omitted. Thus, the total computational complexity
for solving the problem (29) is of order
$\mathcal{O}\\{N_{\text{out}}N_{\text{inn}}\text{ln}(1/\xi)8\sqrt{2}K^{1.5}M^{6}\\}$,
where $N_{\text{out}}$ is the number of iterations of the outer loop.

##  IV Simulation Results

In this section, extensive simulation results are provided to demonstrate the
performance of our proposed joint secure beamforming designs for the
considered ISAC systems under the assumption that Eves’ CSI is available or
not. Except for the radar-target link adopting the AoA model, the Rayleigh
fading channel model is adopted for all links, i.e., each entry of the channel
matrices is assumed to obey the standard complex Gaussian distribution. The
number of eavesdroppers is $I$ = 2. The antenna spacing of the radar is
$\Delta=\lambda/2$. The noise power at Eves and Bobs are set as
$\sigma_{\mathrm{e},i}^{2}=\sigma_{k}^{2}=10\mathrm{dBm},~{}\forall i,k$. The
total power budget of the BS is $P_{\mathrm{c}}$ = 10W. For simplicity, we
assume the communication QoS requirement of each Bob is the same and denoted
by $\Gamma_{\mathrm{c}}$. The target is located at the azimuth angle
$\theta_{0}=0^{\circ}$ and the radar cross section (RCS) is
$\sigma_{\mathrm{0}}^{2}=1$. The noise power to radar echoes is set as
$\sigma_{\mathrm{r}}^{2}=0\mathrm{dB}$444The radar echo passes through the
round-trip distance between the radar and the target, while the signal the
user receives only passes through the one-way distance from the BS to the
user. Thus, the noise power to radar echoes is usually a little larger than
that of the users.. Besides, the SINR requirement thresholds should be set
appropriately to prevent an infeasible case.

Figure 2: Convergence of the proposed algorithm. Figure 3: SINRs of Eve and
Bob versus the communication QoS requirement $\Gamma_{\mathrm{c}}$ ($M$ = 16,
$\Gamma_{\mathrm{r}}$ = 10dB, $P_{\mathrm{r}}$ = 500W). Figure 4: SINRs of Eve
and Bob versus the number of radar transmit/receive antennas $M$
($\Gamma_{\mathrm{c}}$ = 5dB, $\Gamma_{\mathrm{r}}$ = 10dB, $P_{\mathrm{r}}$ =
500W). Figure 5: SINRs of Eve and Bob versus the radar system transmit power
$P_{\mathrm{r}}$ ($M$ = 16, $\Gamma_{\mathrm{c}}$ = 5dB, $\Gamma_{\mathrm{r}}$
= 10dB).

###  IV-A Known Eavesdroppers’ CSI

In this subsection, we illustrate the performance of our proposed joint secure
beamforming design assuming perfect knowledge of Eves’ CSI. We assume that the
BS is equipped with $N=4$ antennas to serve $K=4$ Bobs. In all simulations,
the SINR of Eve denotes the maximum eavesdropping SINR and the SINR of Bob
denotes the minimum legitimate SINR, respectively.

We first illustrate the convergence performance of the proposed algorithm in
Fig. 2. It can be observed that our proposed algorithm converges very quickly
under different settings, which reveals the effectiveness of the proposed
algorithm and its great potential in practical applications.

The SINRs of Eve and Bob versus the communication QoS requirement
$\Gamma_{\mathrm{c}}$ are plotted in Fig. 3 to illustrate the performance of
secure communication. The “proposed” scheme denotes our proposed joint
transmit beamforming design for secure ISAC systems. For comparisons, we also
include the “no radar” scheme, which denotes that there is one single
communication system without radar interference, and the “separate” scheme,
which represents that the transmit beamformers of the communication system and
radar system are separately designed without considering the interference
between them. From Fig. 3, we can observe that in the scenario of “no radar”,
the required communication QoS of Bobs can be guaranteed, but the security of
the communication system is not very satisfactory since Eves’ SINR is not low
enough. When the radar joins the system without cooperative joint beamforming
design, Eves’ SINR is significantly reduced owing to the strong radar
interfering signals, which provides good confidentiality. However, at the same
time, legitimate transmissions are also severely damaged by radar
interference. Compared with these two scenarios, our proposed joint transmit
beamforming design significantly decreases Eves’ SINR to a minimum value,
while always maintaining Bobs’ SINR at a required level to satisfy the QoS
requirements of the legitimate transmissions.

In Fig. 4, we show the SINRs of Eve and Bob versus the number of radar
antennas $M$. It is natural that the number of radar antennas does not affect
the security and legitimate transmission in the “no radar” and “separate”
scenarios. With the proposed joint beamforming design algorithm, Eves’ SINR
decreases with increasing $M$ since additional spatial DoFs can be exploited
to provide stronger interference to Eve. This phenomenon verifies that a
higher DoF of MIMO radar is of great help for secure communication.

Next, we present the SINRs of Eve and Bob versus the radar system power budget
$P_{\mathrm{r}}$ in Fig. 5, where the same relationship between different
scenarios can be observed as that in Figs. 3 and 4. In addition, we observe
that the eavesdropping SINRs of both the “proposed” and “separate” schemes
decrease with a larger $P_{\mathrm{r}}$ due to stronger jamming signals.
Moreover, the proposed scheme achieves the least eavesdropping SINR thanks to
the cooperative joint beamforming design, which guarantees the most favorable
security performance.

###  IV-B Unknown Eavesdroppers’ CSI

Figure 6: Convergence of the proposed algorithm.

In this subsection, we demonstrate the performance of our proposed joint AN-
aided transmit beamforming design for the case that Eves’ CSI is unknown. The
BS is equipped with $N=8$ antennas. We first illustrate the convergence
performance of the proposed algorithm in Fig. 6. It can be observed that the
objective value (29a) monotonically converges within a limited number of
iterations under different settings, which verifies the effectiveness of the
proposed algorithm.

Next, we present the security performance of the proposed algorithm in Figs.
7-9. We also include a scheme that does not consider the physical layer
security. In other words, it only uses the least power to satisfy the
communication QoS requirements of the legitimate users and the radar output
SINR constraint (denoted as “No PLS”), and the scheme that utilizes the
tremendous residual power to generate AN in the null space of Bobs’ channels
(denoted as “Null space”). The security of the communication system is
evaluated in terms of the maximum eavesdropping SINR of the potential
eavesdroppers on the $K$ communication users.

Figure 7: SINR of Eves versus the communication QoS requirement
$\Gamma_{\mathrm{c}}$ ($M$ = 16, $\Gamma_{\mathrm{r}}$ = 10dB,
$P_{\mathrm{r}}$ = 1000W). Figure 8: SINR of Eves versus the number of radar
antennas $M$ ($\Gamma_{\mathrm{c}}$ = 5dB, $\Gamma_{\mathrm{r}}$ = 10dB,
$P_{\mathrm{r}}$ = 500W). Figure 9: SINR of Eves versus the radar system
transmit power $P_{\mathrm{r}}$ ($M$ = 16, $\Gamma_{\mathrm{c}}$ = 5dB,
$\Gamma_{\mathrm{r}}$ = 10dB).

In Fig. 7, we plot the maximum SINR of Eves versus the communication QoS
requirement $\Gamma_{\mathrm{c}}$. Not surprisingly, the maximum SINR of Eves
increases with increasing communication QoS requirements, because the stricter
communication QoS requirements require more power for the beamforming and
leave less power to generate AN, which brings higher risks of being
eavesdropped. In addition, we observe a considerable performance gap between
the “No PLS” scheme and the schemes with PLS design (“Null space” and
“Proposed”), since the generated AN greatly confuses the eavesdropper.
Moreover, it is seen that the proposed algorithm achieves notably better
security performance than the “Null space” scheme since more DoFs are
exploited in designing AN. These findings verify the advantages of the
proposed AN-aided transmit beamforming design in preventing potential
eavesdropping without the knowledge of Eves’ CSI.

Fig. 8 shows the maximum SINR of Eves versus the number of radar antennas $M$.
Since more antennas can exploit additional spatial DoFs, the transmit power
required to achieve the pre-set radar SINR becomes less. Therefore, for the
“No PLS” scheme, the jamming/interference from the radar to the eavesdropper
becomes weak and the security performance is degraded. On the other hand, for
the “Null space” scheme and the proposed scheme, more power is available for
generating AN, which can significantly improve the security performance. It is
noticed that, compared with Fig. 4, Eves’ SINR will not visibly decrease when
the number of radar antennas is greater than 16. This is because the spatial
DoFs of a radar system with 16 antennas are sufficient for providing enough
signal processing ability. More radar antennas would not significantly reduce
the transmit power for achieving the required radar SINR. Consequently, the
available power for generating AN will be relatively small with more radar
antenna than 16, and the Eves’ SINR will be maintained at almost the same
extremely low level, e.g. close to -44dB. These results can still verify that
the proposed AN-aided secure beamforming design can provide superior security
performance even when the MIMO radar has a moderate number of antennas.

Fig. 9 presents the maximum SINR of Eve versus the radar system power budget
$P_{\mathrm{r}}$. The same relationship between different design schemes can
be observed in Figs. 7 and 8. In addition, since more additional power can be
utilized for generating AN, the eavesdropping SINR decreases with the increase
of $P_{\mathrm{r}}$ for the “Null space” and the proposed schemes.

##  V Conclusions

This paper investigated joint secure transmit beamforming designs for ISAC
systems. When the eavesdroppers’ channels are available, the maximum
eavesdropping SINR was minimized under the communication QoS constraints, the
radar detection performance constraint, and the communication and radar power
constraints. An efficient BCD-FP-SDR-based algorithm was proposed to solve the
non-convex optimization problem. When the eavesdroppers’ channels are
unavailable, a joint AN-aided transmit beamforming design was developed to
disrupt the eavesdroppers’ reception while guaranteeing the legitimate
communication SINR and the radar output SINR requirements by utilizing the
available power of radar and communication systems to generate as much AN as
possible. We proposed a double-loop BCD-SDR-based algorithm to solve the
resulting non-convex optimization problem. Simulation results illustrated the
advantages of ISAC systems on secure transmissions and the effectiveness of
the proposed algorithms.

## Appendix A

Let $\widetilde{\mathbf{W}}_{k}$, $\forall k$, be an arbitrary global optimal
solution to problem (18). Then, we construct a new solution
$\widehat{\mathbf{W}}_{k}$, $\forall k$, based on
$\widetilde{\mathbf{W}}_{k}$, $\forall k$, as

$\widehat{\mathbf{W}}_{k}=\widetilde{\mathbf{W}}_{k},~{}\widehat{\mathbf{w}}_{k}=(\mathbf{h}_{k}^{H}\widetilde{\mathbf{W}}_{k}\mathbf{h}_{k})^{-1/2}\widetilde{\mathbf{W}}_{k}\mathbf{h}_{k},~{}\widehat{\mathbf{W}}_{k}=\widehat{\mathbf{w}}_{k}\widehat{\mathbf{w}}_{k}^{H}.$
(40)

Clearly, $\widehat{\mathbf{W}}_{k}$, $\forall k$, is rank-one and positive
semidefinite which means that constraints (18g) and (18h) hold. Next, we show
that $\widehat{\mathbf{W}}_{k}$ is also a global optimal solution to problem
(18). Based on (40), we have the following equality

$\sum_{k=1}^{K}\widetilde{\mathbf{W}}_{k}=\sum_{k=1}^{K}\widehat{\mathbf{W}}_{k},$
(41)

which implies that the constraints (18d) and (18e) hold. Moreover, the
constraint (18f) and the objective function hold since they do not contain the
variables $\widetilde{\mathbf{W}}_{k}$, $\forall k$.

In addition, since

$\mathbf{h}_{k}^{H}\widehat{\mathbf{W}}_{k}\mathbf{h}_{k}=\mathbf{h}_{k}^{H}\widehat{\mathbf{w}}_{k}\widehat{\mathbf{w}}_{k}^{H}\mathbf{h}_{k}=\mathbf{h}_{k}^{H}\widetilde{\mathbf{W}}_{k}\mathbf{h}_{k},$
(42)

the SINR constraint in (18c) can be re-written as

$\displaystyle\frac{(\Gamma_{k}+1)}{\Gamma_{k}}\mathbf{h}_{k}^{H}\widehat{\mathbf{W}}_{k}\mathbf{h}_{k}=\frac{(\Gamma_{k}+1)}{\Gamma_{k}}\mathbf{h}_{k}^{H}\widetilde{\mathbf{W}}_{k}\mathbf{h}_{k}$
(43)
$\displaystyle\geq\sum_{j=1}^{K}\mathbf{h}_{k}^{H}\widetilde{\mathbf{W}}_{j}\mathbf{h}_{k}+\mathbf{g}_{k}^{H}\mathbf{R}_{\mathrm{F}}\mathbf{g}_{k}+\sigma_{k}^{2}$
$\displaystyle=\sum_{j=1}^{K}\mathbf{h}_{k}^{H}\widehat{\mathbf{W}}_{j}\mathbf{h}_{k}+\mathbf{g}_{k}^{H}\mathbf{R}_{\mathrm{F}}\mathbf{g}_{k}+\sigma_{k}^{2},$

namely (18c) holds for $\widehat{\mathbf{W}}_{k}$, $\forall k$. Similarly, the
SINR constraint (18b) also holds for $\widehat{\mathbf{W}}_{k}$, $\forall k$.
Therefore, it is proven that the constructed $\widehat{\mathbf{W}}_{k}$,
$\forall k$ in (40) is a feasible and global optimal rank-1 solution to the
problem (18).

For problem (35), we also let $\widetilde{\mathbf{W}}_{k}$, $\forall k$, be an
arbitrary global optimal solution to problem (35). In the same way as (40), we
construct a new solution $\widehat{\mathbf{W}}_{k}$, $\forall k$, which
implies that (35e) holds. Similar to the previous proof, constraints
(35a)-(35d) hold. Thus, the constructed $\widehat{\mathbf{W}}_{k}$, $\forall
k$, in (40) being a feasible and global optimal solution to problem (35) is
verified.

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