# Target-Mounted Intelligent Reflecting Surface for Secure Wireless Sensing

Xiaodan Shao, , and Rui Zhang X. Shao is with the School of Science and
Engineering, Chinese University of Hong Kong, Shenzhen, China 518172, (e-mail:
shaoxiaodan@zju.edu.cn).R. Zhang is with School of Science and Engineering,
Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong,
Shenzhen, Guangdong 518172, China. He is also with the Department of
Electrical and Computer Engineering, National University of Singapore,
Singapore 117583 (e-mail: elezhang@nus.edu.sg).

###### Abstract

In this paper, we consider a challenging secure wireless sensing scenario
where a legitimate radar station (LRS) intends to detect a target at unknown
location in the presence of an unauthorized radar station (URS). We aim to
enhance the sensing performance of the LRS and in the meanwhile prevent the
detection of the same target by the URS. Under this setup, conventional
stealth-based approaches such as wrapping the target with electromagnetic wave
absorbing materials are not applicable, since they will disable the target
detection by not only the URS, but the LRS as well. To tackle this challenge,
we propose in this paper a new target-mounted IRS approach, where intelligent
reflecting surface (IRS) is mounted on the outer/echo surface of the target
and by tuning the IRS reflection, the strength of its reflected radar signal
in any angle of departure (AoD) can be adjusted based on the signal’s angle of
arrival (AoA), thereby enhancing/suppressing the signal power towards the
LRS/URS, respectively. To this end, we propose a practical protocol for the
target-mounted IRS to estimate the LRS/URS channel and waveform parameters
based on its sensed signals and control the IRS reflection for/against the
LRS/URS accordingly. Specifically, we formulate new optimization problems to
design the reflecting phase shifts at IRS for maximizing the received signal
power at the LRS while keeping that at the URS below a certain level, for both
the cases of short-term and long-term IRS operations with different dynamic
reflection capabilities. To solve these non-convex problems, we apply the
penalty dual decomposition method to obtain high-quality suboptimal solutions
for them efficiently. Finally, simulation results are presented that verify
the effectiveness of the proposed protocol and algorithms for the target-
mounted IRS to achieve secure wireless sensing, as compared with various
benchmark schemes.

###### Index Terms:

Target-mounted IRS, wireless sensing, sensing security, passive beamforming.

##  I Introduction

The future sixth-generation (6G) wireless networks are envisioned as an
enabler for various emerging applications, such as extended reality, tactile
internet, intelligent transportation, massive Internet of Things (IoT), etc
[1]. These applications generally require extraordinarily high communication
rate and/or reliability, as well as highly accurate sensing. Moreover, with
the growing demand for wireless communication system capacity,
spectrum/hardware resources are becoming increasingly scarce. The above trends
have motivated a new research paradigm, called integrated sensing and
communications (ISAC), for enabling sensing and communication functions in the
same wireless network by sharing its platform and resources such as base
station (BS), antenna, spectrum, waveform, etc. and jointly designing their
operations to achieve balanced performance [2, 3, 4, 5]. It is thus
anticipated that wireless sensing, which aims to accurately and efficiently
detect, estimate, and extract useful physical information/features of
environmental targets by exploiting radio wave transmission, reflection,
diffraction, and scattering, will become a major service provided by 6G
wireless networks, in addition to communications.

However, existing wireless sensing systems such as cellular BS sensing still
face practical challenges. For example, the sensing range/accuracy of the
radar (BS) is practically limited due to the high round-trip radar signal
propagation loss between the radar and its sensing target as well as the
target’s finite radar cross section (RCS). Even under the line-of-sight (LoS)
condition, when the radar-target distance is long and/or the RCS of the target
is small, the echo signal received by the radar receiver is of low power,
which degrades the sensing performance (e.g., target detection probability)
against the background noise. Although this issue can be partially resolved by
increasing the radar signal power and/or employing multiple-input-multiple-
output (MIMO) radars, such conventional methods usually require substantially
more energy consumption and higher implementation cost.

In recent years, intelligent reflecting surface (IRS) has been proposed as a
promising new technology to achieve smart and reconfigurable signal
propagation for wireless communications [6, 7]. Typically, IRS is a planar
surface composed of a large number of low-cost passive reflecting elements,
and by jointly adjusting the amplitude and/or phases of IRS elements, the
direction and strength of the electromagnetic wave reflected by IRS can be
flexibly controlled (i.e., passive beamforming). Motivated by the significant
performance gains that IRS brings for wireless communications, recent studies
have also revealed the great potential of IRS for enhancing the performance of
wireless sensing [8, 9, 10, 11] as well as improving the communication-sensing
trade-off for ISAC systems [12, 13, 14]. In these works, IRS is employed as
additional anchor node in wireless network for improving the BS’s detection
accuracy, which, however, may not perform well in practice due to the
significant path loss of the radar signal that is reflected by both the target
(once) and the IRS (twice) before being received by the radar/BS receiver. To
overcome this issue, in [15] and [16], the authors proposed a new IRS active
sensing approach, where the IRS (instead of cellular BS) directly sends radar
signal, and sensors installed on the IRS receive the echo reflected by the
target for detection, thus avoiding the severe (double) path loss between the
sensing/ISAC BS and IRS in conventional IRS-aided cellular sensing systems.

On the other hand, security is another critical challenge for wireless sensing
due to the inherent broadcast nature of wireless signals. Since IRS has the
capability of enhancing wireless signals at desired receivers as well as
suppressing them at undesired receivers, it has been applied to improve the
communication security in IRS-aided ISAC systems [17, 18]. However, prior
works only focused on IRS reflection design to achieve enhanced
information/communication security in ISAC systems, but overlooked the
important sensing security threat. In practice, the physical characteristics
of sensing target (e.g., location and velocity) may be intercepted by
unauthorized radar stations (URSs), which thus brings a new sensing security
issue. There are two traditional methods for tackling this issue. One method
is by designing the target’s shape [19], such that it can reflect incident
signal to undetectable directions, while this method has limited practical
applications due to the specific target shape required. The other method is to
wrap the target with electromagnetic (EM) wave absorbing (i.e., EM stealthy)
materials [20], which can significantly reduce the reflected radar signal
power in all directions. Although the above methods can achieve sensing
security against URS, they inevitably render the target invisible to any
legitimate radar station (LRS) as well. In addition, other methods have been
recently proposed to enable the LRS to detect stealthy target such as
increasing the power-aperture product of the LRS, or the number of LRS radar
pulses in coherent processing [21]. However, these methods can also be applied
by the URS to enhance its target detection. Note that the active IRS sensing
approach [15, 16] aforementioned cannot solve the secure sensing problem too,
because it has no control over the reflected signal by the target.

Figure 1: System model of secure wireless sensing with target-mounted IRS.

In order to improve the target sensing performance while ensuring sensing
security at the same time, we propose in this paper a new approach by mounting
IRS on the outer surface of the target, namely target-mounted IRS.
Specifically, we consider a secure wireless sensing system as shown in Fig. 1,
where an LRS aims to detect an aerial target with its echo-surface covered by
an IRS, and a URS also intends to detect the same target. The target-mounted
IRS can help achieve secure sensing of the target for the LRS as well as
against the URS. Specifically, instead of using IRS as additional anchor node
in existing IRS-assisted sensing/ISAC systems, mounting IRS on the target can
directly control the reflected echo signal from the target. By tuning IRS
reflection, the echo signal towards the LRS/URS receiver can be greatly
enhanced/suppressed, thus achieving our secure wireless sensing goals. Note
that the above IRS reflection design generally replies on the knowledge of the
angles of arrival (AoAs) of the radar signals from the LRS and/or URS. To
achieve this end, we consider installing dedicated sensors along with the
reflecting elements at the IRS to enable its acquisition of such information
(see Fig. 1). The main contributions of this paper are summarized as follows.

  * •

First, we propose a practical operation protocol for target-mounted IRS to
achieve secure wireless sensing. The protocol consists of two steps. In the
first step, IRS sensors estimate the LRS/URS channel and waveform parameters
with all IRS reflecting elements switched off. Then, in the second step, based
on the estimated parameters, IRS reflection is designed to simultaneously
boost the received signal at the LRS receiver and suppress that at the URS
receiver, thus achieving secure detection of the target.

  * •

Next, to design the IRS reflection for/against the LRS/URS, we formulate new
optimization problems for both the cases of short-term and long-term IRS
operations with different dynamic reflection capabilities, aiming to maximize
the received signal power at the LRS while in the meanwhile keeping that at
the URS below a given threshold. However, the formulated optimization problems
are non-convex and thus difficult to be solved optimally. To overcome this
difficulty, we propose an efficient penalty dual decomposition (PDD)-based
algorithm to solve these problems, which yields high-quality suboptimal
solutions.

  * •

Finally, we evaluate the performance of our proposed designs via numerical
results. The results demonstrate that target-mounted IRS can greatly improve
the target’s detection accuracy by the LRS in the LRS-only scenario, or
degrade the detection performance by the URS in the URS-only scenario, as well
as simultaneously enhance/degrade the target sensing at LRS/URS when they are
both present. In addition, it is shown that the proposed algorithm achieves
comparable performance to the semi-definite relaxation (SDR)-based method, but
with much lower computational complexity required, thus is more suitable for
real-time implementation. Furthermore, it is shown that the target-mounted IRS
with proposed reflection optimization achieves more robust secure sensing
performance against IRS sensing errors and random LRS/URS locations, as
compared to benchmark systems without IRS or with random IRS reflection.

The rest of this paper is organized as follows. Section II presents the
channel and signal models of the proposed secure wireless sensing system with
target-mounted IRS. Section III describes the practical protocol for the
operation of target-mounted IRS. Section IV formulates the optimization
problems for IRS reflection design in both the short-term and long-term IRS
reflection cases, and proposes efficient algorithms for solving them. Section
V presents numerical results to evaluate the performance of the proposed
protocol and algorithms. Finally, Section VI concludes this paper.

_Notations_ : Boldface upper-case and lower-case letters denote matrices and
vectors, respectively, $(\cdot)^{*}$, $(\cdot)^{H}$, and $(\cdot)^{T}$
respectively denote conjugate, conjugate transpose, and transpose, $\otimes$
denotes the Kronecker product, $\odot$ denotes the Hadamard product,
$\mathbb{E}[\cdot]$ denotes the expected value of random variable,
$\left\|\cdot\right\|_{2}$ denotes the Euclidean norm, $|\cdot|$ denotes the
absolute value, $\Re(\cdot)$ denotes the real part of a complex number,
$\mathbf{I}_{m}$ denotes the $m$-dimensional identity matrix,
$\mathrm{diag}({\bf x})$ denotes a diagonal matrix with the diagonal entries
specified by a vector ${\bf x}$.

##  II System Model

In this paper, we consider a secure wireless sensing system aided by the
target-mounted IRS as illustrated in Fig. 1, where an LRS aims to detect a
target at unknown location, while a URS also intends to detect the same target
by sending/intercepting radar signals. Specifically, we consider that an IRS
is equipped on the echo surface of the target to help boost/suppress its
reflected signal towards the LRS/URS, which are both assumed to be a mono-
static radar (i.e., with the radar transmitter and receiver co-located). In
addition, we assume that the remaining echo surface of the target, which is
uncovered by the IRS, is wrapped with EM-stealth material [20], such that the
IRS can completely control the reflected radar signal from the target111Note
that the target is assumed to be passive and thus it cannot emit signals to
the LRS for facilitating its detection, due to various practical
considerations (e.g., information/sensing security, energy/hardware
constraint)..

###  II-A Channel Model

As shown in Fig. 2, we consider that the LRS/URS is equipped with a uniform
planar array (UPA) parallel to the $y-z$ plane. The LRS comprises
$M=M_{{y}}\times M_{{z}}$ transmit/receive antennas, with $M_{{y}}$ and
$M_{{z}}$ denoting the numbers of transmit/receive antennas along the $y$\-
and $z$-axes, respectively. The URS consists of $D=D_{{y}}\times D_{{z}}$
transmit/receive antennas, with $D_{{y}}$ and $D_{{z}}$ being the
corresponding numbers of antennas along the $y$\- and $z$-axes. The IRS is
assumed to be another UPA with $N=N_{y}\times N_{x}$ reflecting elements,
where $N_{y}$ and $N_{x}$ are the respective numbers of reflecting elements
along the $y$\- and $x$-axes, and a controller is attached to the IRS for
performing signal processing and adjusting its reflection over time. At the
IRS, each reflecting element re-scatters the received signal with an
independently tunable reflection coefficient. Specifically, we denote the IRS
reflection coefficients as
$\bm{\theta}=[\beta_{1}e^{j\omega_{1}},\cdots,\beta_{N}e^{j\omega_{N}}]^{T}$,
where each reflection coefficient comprises a phase shift
$\omega_{n}\in[0,2\pi)$ and an on/off amplitude parameter
$\beta_{n}\in\\{0,1\\},n=1,2,\cdots,N$. Note that $\beta_{n}=1$ means that the
$n$-th reflecting element fully reflects the incident signal without amplitude
change, while $\beta_{n}=0$ implies that the incident signal is fully absorbed
by this reflecting element (similar to EM wave absorbing material). Moreover,
to enable its sensing function, we assume that the IRS is equipped with
$L=L_{y}+L_{x}-1$ sensors (for receiving signals only) with $L_{y}$ and
$L_{x}$ denoting the number of sensors along the $y$\- and $x$-axes,
respectively (see Fig. 2), which is also termed as “semi-passive” IRS [6].

Figure 2: Channel model.

Due to the high altitude of aerial targets, we assume that the channels
between the LRS/URS and the target-mounted IRS follow the LoS model
(otherwise, reliable sensing cannot be achieved). As the URS/LRS is assumed to
be mono-static radar where the transmit and receive arrays are placed at the
same location, the AoD from its transmitter to the IRS is equivalent to the
AoA from the IRS to its receiver. As depicted in Fig. 2, we select the bottom-
left elements of the UPAs at the LRS/URS transmitter, LRS/URS receiver, and
IRS as their respective reference elements for modeling the LoS channels
between them (to be specified later). Furthermore, let $\phi_{U}$ and
$\eta_{U}$ denote the URS’s elevation and azimuth AoD/AoA with respect to
(w.r.t.) the IRS, and ${\phi}_{{L}}$ and ${\eta}_{{L}}$ denote the elevation
and azimuth AoD/AoA at the LRS w.r.t. the IRS, respectively. Note that due to
the far-field propagation between the LRS/URS and IRS (or target), the above
angles apply to all antenna/reflecting/sensor elements on their corresponding
UPAs.

Let $\mathbf{d}(B,\zeta)$ denote the one-dimensional (1D) steering vector
function of a given 1D array, which is defined as

$\displaystyle\mathbf{d}(B,\zeta)$
$\displaystyle=\left[1,e^{j\frac{2\pi{d}}{\lambda}\zeta},\cdots,e^{j\frac{2\pi{d}}{\lambda}(B-1)\zeta}\right]^{T},$
(1)

where $B$ denotes the array size, $\zeta$ denotes the steering angle,
$\lambda$ denotes the signal wavelength, and ${d}$ denotes the distance
between any two adjacent antenna/reflecting/sensor elements. Accordingly,
assuming that the IRS’s UPA is parallel to the $x$-$y$ plane, the two-
dimensional (2D) steering vectors at the IRS w.r.t. LRS for the incident and
reflected signals can be expressed as

$\displaystyle\mathbf{a}_{{l}}({\phi}_{{L}},{\eta}_{{L}})$
$\displaystyle=\mathbf{d}(N_{x},\zeta_{l}^{a})\otimes\mathbf{d}(N_{y},\zeta_{l}^{e}),$
(2) $\displaystyle\mathbf{a}_{{l}}(\pi-{\phi}_{{L}},\pi+{\eta}_{{L}})$
$\displaystyle=\mathbf{d}(N_{x},\zeta_{l}^{\bar{a}})\otimes\mathbf{d}(N_{y},\zeta_{l}^{\bar{e}}),$
(3)

respectively, where
$\zeta_{l}^{a}\triangleq\sin({\phi}_{{L}})\cos({\eta}_{{L}})$,
$\zeta_{l}^{e}\triangleq\sin({\phi}_{{L}})\sin({\eta}_{{L}})$,
$\zeta_{l}^{\bar{a}}\triangleq\sin(\pi-{\phi}_{{L}})\cos(\pi+{\eta}_{{L}})\triangleq-\sin({\phi}_{{L}})\cos({\eta}_{{L}})$,
and
$\zeta_{l}^{\bar{e}}\triangleq\sin(\pi-{\phi}_{{L}})\sin(\pi+{\eta}_{{L}})\triangleq-\sin({\phi}_{{L}})\sin({\eta}_{{L}})$.
Similarly, the 2D steering vectors for the incident and reflected signals at
the IRS w.r.t. URS are respectively given by

$\displaystyle\mathbf{a}_{{u}}({\phi}_{{U}},{\eta}_{{U}})$
$\displaystyle=\mathbf{d}(N_{x},\zeta_{u}^{a})\otimes\mathbf{d}(N_{y},\zeta_{u}^{e}),$
(4) $\displaystyle\mathbf{a}_{{u}}(\pi-{\phi}_{{U}},\pi+{\eta}_{{U}})$
$\displaystyle=\mathbf{d}(N_{x},\zeta_{u}^{\bar{a}})\otimes\mathbf{d}(N_{y},\zeta_{u}^{\bar{e}}),$
(5)

where $\zeta_{u}^{a}\triangleq\sin({\phi}_{{U}})\cos({\eta}_{{U}})$,
$\zeta_{u}^{e}\triangleq\sin({\phi}_{{U}})\sin({\eta}_{{U}})$,
$\zeta_{u}^{\bar{a}}\triangleq\sin(\pi-{\phi}_{{U}})\cos(\pi+{\eta}_{{U}})\triangleq-\sin({\phi}_{{U}})\cos({\eta}_{{U}})$,
and
$\zeta_{u}^{\bar{e}}\triangleq\sin(\pi-{\phi}_{{U}})\sin(\pi-{\eta}_{{U}})\triangleq-\sin({\phi}_{{U}})\sin({\eta}_{{U}})$.

In addition, for the LRS’s UPA assumed to be parallel to the $y$-$z$ plane,
its transmit and receive 2D steering vectors w.r.t. the IRS are respectively
given by

$\displaystyle\mathbf{b}({\phi}_{{L}},{\eta}_{{L}})$
$\displaystyle=\mathbf{d}(M_{{y}},\varphi_{t}^{a})\otimes\mathbf{d}(M_{{z}},\varphi_{t}^{e}),$
(6) $\displaystyle\mathbf{b}(\pi-{\phi}_{{L}},\pi+{\eta}_{{L}})$
$\displaystyle=\mathbf{d}(M_{{y}},\varphi_{r}^{a})\otimes\mathbf{d}(M_{{z}},\varphi_{r}^{e}),$
(7)

where $\varphi_{t}^{a}\triangleq\sin({\phi}_{{L}})\sin({\eta}_{{L}})$,
$\varphi_{t}^{e}\triangleq\cos({\phi}_{{L}})$,
$\varphi_{r}^{a}\triangleq\sin(\pi-{\phi}_{{L}})\sin(\pi+{\eta}_{{L}})\triangleq-\sin({\phi}_{{L}})\sin({\eta}_{{L}})$,
and
$\varphi_{r}^{e}\triangleq\cos(\pi-{\phi}_{{L}})\triangleq-\cos({\phi}_{{L}})$.
Similarly, assuming that the URS is parallel to the $y$-$z$ plane, its 2D
transmit and receive steering vectors are respectively expressed as

$\displaystyle\mathbf{c}({\phi}_{{U}},{\eta}_{{U}})$
$\displaystyle=\mathbf{d}(D_{{y}},\upsilon_{t}^{a})\otimes\mathbf{d}(D_{{z}},\upsilon_{t}^{e}),$
(8) $\displaystyle\mathbf{c}(\pi-{\phi}_{{U}},\pi+{\eta}_{{U}})$
$\displaystyle=\mathbf{d}(D_{{y}},\upsilon_{r}^{a})\otimes\mathbf{d}(D_{{z}},\upsilon_{r}^{e}),$
(9)

where $\upsilon_{t}^{a}\triangleq\sin({\phi}_{{U}})\sin({\eta}_{{U}})$,
$\upsilon_{t}^{e}\triangleq\cos({\phi}_{{U}})$,
$\upsilon_{r}^{a}\triangleq\sin(\pi-{\phi}_{{U}})\sin(\pi+{\eta}_{{U}})\triangleq-\sin({\phi}_{{U}})\sin({\eta}_{{U}})$,
and
$\upsilon_{r}^{e}\triangleq\cos(\pi-{\phi}_{{U}})\triangleq-\cos({\phi}_{{U}})$.

As a result, the IRS$\rightarrow$LRS channel
$\mathbf{H}_{LI}\in\mathbb{C}^{M\times N}$, LRS$\rightarrow$IRS channel
$\mathbf{H}_{IL}\in\mathbb{C}^{N\times{M}}$, IRS$\rightarrow$URS channel
$\mathbf{H}_{UI}\in\mathbb{C}^{D\times{N}}$, and URS$\rightarrow$IRS channel
$\mathbf{H}_{IU}\in\mathbb{C}^{N\times{D}}$ can be written as

$\displaystyle\mathbf{H}_{LI}$
$\displaystyle=\alpha_{l}\mathbf{b}(\pi-{\phi}_{{L}},\pi+{\eta}_{{L}})\mathbf{a}_{{l}}^{H}(\pi-{\phi}_{{L}},\pi+{\eta}_{{L}}),$
(10) $\displaystyle\mathbf{H}_{IL}$
$\displaystyle=\alpha_{l}\mathbf{a}_{{l}}({\phi}_{{L}},{\eta}_{{L}})\mathbf{b}^{H}({\phi}_{{L}},{\eta}_{{L}}),$
(11) $\displaystyle\mathbf{H}_{UI}$
$\displaystyle=\alpha_{u}\mathbf{c}(\pi-{\phi}_{{U}},\pi+{\eta}_{{U}})\mathbf{a}_{{u}}^{H}(\pi-{\phi}_{{U}},\pi+{\eta}_{{U}}),$
(12) $\displaystyle\mathbf{H}_{IU}$
$\displaystyle=\alpha_{u}\mathbf{a}_{{u}}({\phi}_{{U}},{\eta}_{{U}})\mathbf{c}^{H}({\phi}_{{U}},{\eta}_{{U}}),$
(13)

respectively, where $\alpha_{l}=e^{j\nu_{l}}\bar{\alpha}_{l}$ and
$\alpha_{u}=e^{j\nu_{u}}\bar{\alpha}_{u}$ with $\bar{\alpha}_{l}$ and
$\bar{\alpha}_{u}$ denoting the real-valued path gain of the LRS-IRS and URS-
IRS LoS channels, respectively, and $\nu_{l}=\frac{2\pi d_{LI}}{\lambda}$ and
$\nu_{u}=\frac{2\pi d_{UI}}{\lambda}$ being the reference phase of the LRS-IRS
and URS-IRS channels, respectively. In the above, $d_{LI}$ and $d_{UI}$
represent the distance between the LRS and target and that between the URS and
target, respectively, w.r.t. their reference elements. To characterize the
effect of small variations of the LRS-/URS- target distance (i.e., $d_{LI}$ or
$d_{UI}$) due to target position local perturbation on the signal phases
$\nu_{l}$ and $\nu_{u}$, we model $\nu_{l}$ and $\nu_{u}$ as independent and
uniformly distributed random variables in $[0,2\pi)$ [22].

###  II-B Signal Model

Figure 3: Illustration of received LRS/URS signals at IRS.

As shown in Fig. 3, we assume that the LRS transmits one coherent burst of
$K_{L}$ non-consecutive radar pulses with a constant pulse repetition interval
(PRI), denoted as $T_{L}$. The duration over which all these signals are
reflected by the target and received by the LRS is called the coherent-
processing interval (CPI), denoted by $T_{\mathrm{CPI}}$, which is equal to
$K_{L}\times T_{L}$. The pulse durations of LRS and URS are denoted as $t_{L}$
and $t_{U}$, respectively, with $t_{L}<T_{L}$ and $t_{U}<T_{U}$, where the PRI
of the URS is denoted as $T_{U}$. For simplicity, we assume that
$T_{L}=T_{U}=T$ in this paper222The proposed design in this paper can be
extended to the general case of $T_{L}\neq T_{U}$, which is omitted due to
space limitations.. Furthermore, we assume that the target location as well as
the PRIs and pulse durations of both LRS and URS remain unchanged during each
LRS CPI, but they may change from one CPI to another. In addition, we express
the radar pulse waveforms of LRS and URS during each PRI respectively as

$\displaystyle x(t)=\begin{cases}\sqrt{P_{L}}p(t),&0\leq t\leq t_{L}\\\
0,&t_{L}<t\leq T\\\ \end{cases}$ (14)

and

$\displaystyle s(t)=\begin{cases}\sqrt{P_{U}}q(t),&0\leq t\leq t_{U}\\\
0,&t_{U}<t\leq T\\\ \end{cases}$ (15)

where $P_{L}$ and $P_{U}$ represent the transmit signal power of LRS and URS,
respectively; $p(t)$ and $q(t)$ are the corresponding radar pulses with
normalized power, i.e., $\frac{1}{t_{L}}\int_{0}^{t_{L}}|p(t)|^{2}dt=1$ and
$\frac{1}{t_{U}}\int_{0}^{t_{U}}|q(t)|^{2}dt=1$.

As shown in Fig. 3, the signal received at the IRS during each LRS CPI may
fall into the following three cases depending on whether the received signals
from LRS and URS are overlapped or not, i.e., Case 1 (LRS-signal only case):
the received signal is from LRS only; Case 2 (URS-signal only case): the
received signal is from URS only; and Case 3 (overlapped LRS and URS signal):
the received signal consists of overlapped signals from both LRS and URS. For
convenience, we use $t_{o}$ to denote the overlapped signal duration in Case
3, with $0\leq t_{o}\leq\min\\{t_{L},t_{U}\\}$. Note that if $t_{o}=0$, then
the signals from LRS and URS are completely non-overlapped at the IRS. Without
loss of generality, assuming that the first LRS signal arrives at IRS before
that of URS during each LRS CPI, we present the signal models for the above
three cases in the following, respectively.

First, for Case 1 (i.e., $0\leq t\leq t_{L}-t_{o}$) with only the LRS signal
reflected by the IRS, the signals received by the LRS receiver through the
link ①$\rightarrow$② in Fig. 1 (i.e.,
LRS$\rightarrow$target/IRS$\rightarrow$LRS) and the URS receiver via the link
①$\rightarrow$④ in Fig. 1 (i.e., LRS$\rightarrow$target/IRS$\rightarrow$URS)
at time $t$ can be respectively expressed as

$\displaystyle y_{LL}(t)$
$\displaystyle=\mathbf{w}_{{L}}^{T}\mathbf{H}_{LI}\text{diag}(\bm{\theta})\mathbf{H}_{IL}\mathbf{w}_{{L}}x(t)+z_{L}(t),$
(16) $\displaystyle y_{LU}(t)$
$\displaystyle=\mathbf{w}_{{U}}^{T}\mathbf{H}_{UI}\text{diag}(\bm{\theta})\mathbf{H}_{IL}\mathbf{w}_{{L}}x(t)+z_{U}(t),$
(17)

where $z_{L}(t)$ and $z_{U}(t)$ denote the additive white Gaussian noise
(AWGN) with zero mean and average power $\sigma_{L}^{2}$ and $\sigma_{U}^{2}$,
respectively, $\mathbf{w}_{{L}}\in\mathbb{C}^{M\times 1}$ and
$\mathbf{w}_{{U}}\in\mathbb{C}^{D\times 1}$ are the transmit beamformers at
LRS and URS, respectively, and $\mathbf{w}_{{L}}^{T}\in\mathbb{C}^{1\times M}$
and $\mathbf{w}_{{U}}^{T}\in\mathbb{C}^{1\times D}$ denote their matching
receive beamformers at LRS and URS, respectively.

Second, for Case 2 (i.e., $t_{L}<t\leq t_{L}+t_{U}-t_{o}$) with only the URS
signal reflected by the IRS, the signals received by the LRS receiver via the
link ③$\rightarrow$② in Fig. 1 (i.e.,
URS$\rightarrow$target/IRS$\rightarrow$LRS) and the URS receiver through the
link ③$\rightarrow$④ in Fig. 1 (i.e.,
URS$\rightarrow$target/IRS$\rightarrow$URS) are respectively given by

$\displaystyle y_{UL}(t)$
$\displaystyle=\mathbf{w}_{{L}}^{T}\mathbf{H}_{LI}\text{diag}(\bm{\theta})\mathbf{H}_{IU}\mathbf{w}_{{U}}s(t)+z_{L}(t),$
(18) $\displaystyle y_{UU}(t)$
$\displaystyle=\mathbf{w}_{{U}}^{T}\mathbf{H}_{UI}\text{diag}(\bm{\theta})\mathbf{H}_{IU}\mathbf{w}_{{U}}s(t)+z_{U}(t).$
(19)

Third, for Case 3 (i.e., $t_{L}-t_{o}<t\leq t_{L}$) with the overlapped signal
reflected by the IRS, the signals received by the LRS receiver through both
links ①$\rightarrow$② and ③$\rightarrow$② in Fig. 1 and the URS receiver via
both links ①$\rightarrow$④ and ③$\rightarrow$④ in Fig. 1 are respectively
expressed as

$\displaystyle y_{OL}(t)$
$\displaystyle=\mathbf{w}_{{L}}^{T}\mathbf{H}_{LI}\text{diag}(\bm{\theta})\mathbf{H}_{IL}\mathbf{w}_{{L}}x(t)$
$\displaystyle+\mathbf{w}_{{L}}^{T}\mathbf{H}_{LI}\text{diag}(\bm{\theta})\mathbf{H}_{IU}\mathbf{w}_{{U}}s(t)+z_{L}(t),$
(20) $\displaystyle y_{OU}(t)$
$\displaystyle=\mathbf{w}_{{U}}^{T}\mathbf{H}_{UI}\text{diag}(\bm{\theta})\mathbf{H}_{IU}\mathbf{w}_{{U}}s(t)$
$\displaystyle+\mathbf{w}_{{U}}^{T}\mathbf{H}_{UI}\text{diag}(\bm{\theta})\mathbf{H}_{IL}\mathbf{w}_{{L}}x(t)+z_{U}(t).$
(21)

In practice, the performance of target detection/estimation (e.g., detecting
the presence of a target or estimating a target’s AoA) improves with the
increase of received signal power [23]. This is intuitively expected, since
larger signal power results in higher signal-to-noise ratio (SNR) of the
received echo signal, thus leading to lower detection/estimation error.
Therefore, we use the received signal powers at the LRS/URS as the performance
metric for their target detection/estimation, and derive them for the above
three cases, respectively, as follows.

First, according to (16) and (17), the received signal powers at LRS and URS
in Case 1 are respectively given by

$\displaystyle{Q}_{LL}$
$\displaystyle={|\mathbf{w}_{{L}}^{T}\mathbf{H}_{LI}\text{diag}(\bm{\theta})\mathbf{H}_{IL}\mathbf{w}_{{L}}x(t)|^{2}}$
$\displaystyle={|\bar{\alpha}_{l}\mathbf{w}_{{L}}^{T}\mathbf{b}(\pi-{\phi}_{{L}},\pi+{\eta}_{{L}}){\mathbf{u}}^{H}\bm{\theta}\bar{\alpha}_{l}\mathbf{b}^{H}({\phi}_{{L}},{\eta}_{{L}})\mathbf{w}_{{L}}x(t)|^{2}}$
$\displaystyle={\frac{Q_{LS}^{2}}{P_{L}}\times|{\mathbf{u}}^{H}\bm{\theta}|^{2}},$
(22)

and

$\displaystyle{Q}_{LU}$
$\displaystyle={|\mathbf{w}_{{U}}^{T}\mathbf{H}_{UI}\text{diag}(\bm{\theta})\mathbf{H}_{IL}\mathbf{w}_{{L}}x(t)|^{2}}$
$\displaystyle={|\bar{\alpha}_{u}\mathbf{w}_{{U}}^{T}\mathbf{c}(\pi-{\phi}_{{U}},\pi+{\eta}_{{U}}){\mathbf{v}}^{H}\bm{\theta}\bar{\alpha}_{l}\mathbf{b}^{H}({\phi}_{{L}},{\eta}_{{L}})\mathbf{w}_{{L}}x(t)|^{2}}$
$\displaystyle={\frac{Q_{LS}Q_{US}}{P_{U}}\times|{\mathbf{v}}^{H}\bm{\theta}|^{2}},$
(23)

with

$\displaystyle{\mathbf{u}}$
$\displaystyle=\mathbf{a}_{l}(\pi-{\phi}_{{L}},\pi+{\eta}_{{L}})\odot\mathbf{a}_{l}^{*}({\phi}_{{L}},{\eta}_{{L}})$
$\displaystyle\triangleq\mathbf{d}(N_{x},\zeta_{l}^{\bar{a}}-\zeta_{l}^{a})\otimes\mathbf{d}(N_{y},\zeta_{l}^{\bar{e}}-\zeta_{l}^{e})\in\mathbb{C}^{N\times
1},$ (24) $\displaystyle{\mathbf{v}}$
$\displaystyle=\mathbf{a}_{u}(\pi-{\phi}_{{U}},\pi+{\eta}_{{U}})\odot\mathbf{a}_{l}({\phi}_{{L}},{\eta}_{{L}})$
$\displaystyle\triangleq\mathbf{d}(N_{x},\zeta_{u}^{\bar{a}}-\zeta_{l}^{a})\otimes\mathbf{d}(N_{y},\zeta_{u}^{\bar{e}}-\zeta_{l}^{e})\in\mathbb{C}^{N\times
1},$ (25) $\displaystyle Q_{LS}$
$\displaystyle=|\bar{\alpha}_{l}\mathbf{b}^{H}({\phi}_{{L}},{\eta}_{{L}})\mathbf{w}_{{L}}x(t)|^{2},$
(26) $\displaystyle Q_{US}$
$\displaystyle=|\bar{\alpha}_{u}\mathbf{w}_{{U}}^{T}\mathbf{c}(\pi-{\phi}_{{U}},\pi+{\eta}_{{U}})s(t)|^{2},$
(27)

where $|{\mathbf{u}}^{H}\bm{\theta}|^{2}$ and
$|{\mathbf{v}}^{H}\bm{\theta}|^{2}$ depend on the IRS reflection, i.e.,
$\bm{\theta}$, and $Q_{LS}$ and $Q_{US}$ represent the received signal powers
at IRS from LRS and URS, respectively.

Similarly, based on (18) and (19), the received signal powers at LRS and URS
in Case 2 are respectively

$\displaystyle{Q}_{UL}$
$\displaystyle={|\mathbf{w}_{{L}}^{T}\mathbf{H}_{LI}\text{diag}(\bm{\theta})\mathbf{H}_{IU}\mathbf{w}_{{U}}s(t)|^{2}}$
$\displaystyle={|\bar{\alpha}_{l}\mathbf{w}_{{L}}^{T}\mathbf{b}(\pi-{\phi}_{{L}},\pi+{\eta}_{{L}}){\mathbf{r}}^{H}\bm{\theta}\bar{\alpha}_{u}\mathbf{c}^{H}({\phi}_{{U}},{\eta}_{{U}})\mathbf{w}_{{U}}s(t)|^{2}}$
$\displaystyle={\frac{Q_{LS}Q_{US}}{P_{L}}\times|{\mathbf{r}}^{H}\bm{\theta}|^{2}},$
(28)

and

$\displaystyle{Q}_{UU}$
$\displaystyle={|\mathbf{w}_{{U}}^{T}\mathbf{H}_{UI}\text{diag}(\bm{\theta})\mathbf{H}_{IU}\mathbf{w}_{{U}}s(t)|^{2}}$
$\displaystyle={|\bar{\alpha}_{u}\mathbf{w}_{{U}}^{T}\mathbf{c}(\pi-{\phi}_{{U}},\pi+{\eta}_{{U}}){\mathbf{g}}^{H}\bm{\theta}\bar{\alpha}_{u}\mathbf{c}^{H}({\phi}_{{U}},{\eta}_{{U}})\mathbf{w}_{{U}}s(t)|^{2}}$
$\displaystyle={\frac{Q_{US}^{2}}{P_{U}}\times|{\mathbf{g}}^{H}\bm{\theta}|^{2}},$
(29)

with

$\displaystyle{\mathbf{r}}$
$\displaystyle=\mathbf{a}_{l}(\pi-{\phi}_{{L}},\pi+{\eta}_{{L}})\odot\mathbf{a}_{u}^{*}({\phi}_{{U}},{\eta}_{{U}})$
$\displaystyle\triangleq\mathbf{d}(N_{x},\zeta_{l}^{\bar{a}}-\zeta_{u}^{a})\otimes\mathbf{d}(N_{y},\zeta_{l}^{\bar{e}}-\zeta_{u}^{e})\in\mathbb{C}^{N\times
1},$ (30) $\displaystyle{\mathbf{g}}$
$\displaystyle=\mathbf{a}_{u}(\pi-{\phi}_{{U}},\pi+{\eta}_{{U}})\odot\mathbf{a}_{u}^{*}({\phi}_{{U}},{\eta}_{{U}})$
$\displaystyle\triangleq\mathbf{d}(N_{x},\zeta_{u}^{\bar{a}}-\zeta_{u}^{a})\otimes\mathbf{d}(N_{y},\zeta_{u}^{\bar{e}}-\zeta_{u}^{e})\in\mathbb{C}^{N\times
1},$ (31)

where $|{\mathbf{r}}^{H}\bm{\theta}|^{2}$ and
$|{\mathbf{g}}^{H}\bm{\theta}|^{2}$ depend on the IRS reflection, i.e.,
$\bm{\theta}$.

Furthermore, from (20) and (21), the average received signal powers at LRS and
URS in Case 3 are respectively obtained as

$\displaystyle{Q}_{OL}$
$\displaystyle=\mathbb{E}\left[|\mathbf{w}_{{L}}^{T}\mathbf{H}_{LI}\text{diag}(\bm{\theta})\mathbf{H}_{IL}\mathbf{w}_{{L}}x(t)\right.$
$\displaystyle\left.+\mathbf{w}_{{L}}^{T}\mathbf{H}_{LI}\text{diag}(\bm{\theta})\mathbf{H}_{IU}\mathbf{w}_{{U}}s(t)|^{2}\right]$
(32)
$\displaystyle={|\mathbf{f}|^{2}}+{|\bar{\mathbf{f}}|^{2}}+\mathbb{E}[e^{j\nu_{l}}]\mathbb{E}[e^{-j\nu_{u}}]\mathbf{f}\bar{\mathbf{f}}^{*}$
$\displaystyle+\mathbb{E}[e^{-j\nu_{l}}]\mathbb{E}[e^{j\nu_{u}}]\mathbf{f}^{*}\bar{\mathbf{f}}$
(33)
$\displaystyle={\frac{Q_{LS}^{2}}{P_{L}}\times|{\mathbf{u}}^{H}\bm{\theta}|^{2}}+{\frac{Q_{LS}Q_{US}}{P_{L}}\times|{\mathbf{r}}^{H}\bm{\theta}|^{2}},$
(34)

and

$\displaystyle{Q}_{OU}$
$\displaystyle=\mathbb{E}\left[|\mathbf{w}_{{U}}^{T}\mathbf{H}_{UI}\text{diag}(\bm{\theta})\mathbf{H}_{IL}\mathbf{w}_{{L}}x(t)\right.$
$\displaystyle\left.+\mathbf{w}_{{U}}^{T}\mathbf{H}_{UI}\text{diag}(\bm{\theta})\mathbf{H}_{IU}\mathbf{w}_{{U}}s(t)|^{2}\right]$
(35)
$\displaystyle={|\mathbf{e}|^{2}}+{|\bar{\mathbf{e}}|^{2}}+\mathbb{E}[e^{j\nu_{l}}]\mathbb{E}[e^{-j\nu_{u}}]\mathbf{e}\bar{\mathbf{e}}^{*}$
$\displaystyle+\mathbb{E}[e^{-j\nu_{l}}]\mathbb{E}[e^{j\nu_{u}}]\mathbf{e}^{*}\bar{\mathbf{e}}$
(36)
$\displaystyle={\frac{Q_{LS}Q_{US}}{P_{U}}\times|{\mathbf{v}}^{H}\bm{\theta}|^{2}}+{\frac{Q_{US}^{2}}{P_{U}}\times|{\mathbf{g}}^{H}\bm{\theta}|^{2}},$
(37)

where
$\mathbf{f}=e^{2j\nu_{l}}\bar{\alpha}_{l}^{2}\mathbf{w}_{{L}}^{T}\mathbf{b}(\pi-{\phi}_{{L}},\pi+{\eta}_{{L}}){\mathbf{u}}^{H}\bm{\theta}\mathbf{b}^{H}({\phi}_{{L}},{\eta}_{{L}})\mathbf{w}_{{L}}x(t)$,
$\bar{\mathbf{f}}=e^{j\nu_{l}}e^{j\nu_{u}}\bar{\alpha}_{l}\bar{\alpha}_{u}\mathbf{w}_{{L}}^{T}\mathbf{b}(\pi-{\phi}_{{L}},\pi+{\eta}_{{L}}){\mathbf{r}}^{H}\bm{\theta}\mathbf{c}^{H}({\phi}_{{U}},{\eta}_{{U}})\mathbf{w}_{{U}}s(t)$,
$\mathbf{e}=e^{j\nu_{l}}e^{j\nu_{u}}\bar{\alpha}_{u}\bar{\alpha}_{l}\mathbf{w}_{{U}}^{T}\mathbf{c}(\pi-{\phi}_{{U}},\pi+{\eta}_{{U}}){\mathbf{v}}^{H}\bm{\theta}\mathbf{b}^{H}({\phi}_{{L}},{\eta}_{{L}})\mathbf{w}_{{L}}x(t)$,
and
$\bar{\mathbf{e}}=e^{2j\nu_{u}}\bar{\alpha}_{u}^{2}\mathbf{w}_{{U}}^{T}\mathbf{c}(\pi-{\phi}_{{U}},\pi+{\eta}_{{U}}){\mathbf{g}}^{H}\bm{\theta}\mathbf{c}^{H}({\phi}_{{U}},{\eta}_{{U}})\mathbf{w}_{{U}}s(t)$.
In the above, the expectations are taken w.r.t. the independent random phases
$\nu_{l}$ and $\nu_{u}$, while (33) and (36) hold since $\mathbb{E}[e^{\pm
j\nu_{l}}]=0$ and $\mathbb{E}[e^{\pm j\nu_{u}}]=0$.

##  III Proposed Protocol for Target-Mounted IRS

Figure 4: Proposed protocol for secure wireless sensing with the target-
mounted IRS.

In this section, we propose a practical protocol for the operation of the
target-mounted IRS for secure wireless sensing, as illustrated in Fig. 4. The
protocol consists of two steps in each LRS CPI. In the first step (Step I),
IRS reflecting elements are switched off and IRS sensors estimate the required
LRS/URS channel and waveform parameters. In the second step (Step II), based
on the estimated information, IRS controller first designs the phase shifts of
IRS reflecting elements, and then they are switched on/off with their phase
shifts set accordingly. In the following, we elaborate the above two steps in
detail.

In Step I, with all IRS reflecting elements switched off, i.e.,
$\beta_{n}=0,\forall n=1,...,N$, IRS uses its sensors to receive the radar
signals from LRS and URS for estimating their respective AoAs, i.e.,
$\\{\phi_{L},\eta_{L}\\}$ and $\\{\phi_{U},\eta_{U}\\}$, which are required
for the IRS reflection design in Step II. Specifically, these AoAs can be
estimated by applying high-resolution angle estimation methods such as
multiple signal classification (MUSIC)[24]. In addition, the LRS/URS PRI $T$,
their pulse durations, i.e., $t_{L}$ and $t_{U}$, and overlapped signal
duration $t_{o}$ are also needed for the subsequent IRS reflection design,
which, if not a priori known, can be estimated based on e.g, the
deinterleaving approach [25, 26]. Specifically, the autocorrelation-based
method and specialized finite impulse response filter can be utilized to
extract different PRIs [25] and pulse durations [27, 28], respectively, and
then these estimated parameters are compared with their known values stored
for LRS to identify those corresponding to URS, which also helps identify the
previously estimated AoAs belonging to LRS or URS. Finally, the received
signal powers at IRS from LRS/URS, i.e., $Q_{LS}$ and $Q_{US}$, are also
required for the design of IRS reflection, which can be measured from the
received signals at IRS sensors after resolving the LRS/URS AoAs by e.g., the
power gain estimation method [29], even when there is signal overlap at the
IRS from both LRS and URS (i.e., Case 3)[30].

With the above estimated parameters, IRS controller then designs the phase
shifts of its reflecting elements in Step II for simultaneously boosting the
received signal at the LRS receiver and suppressing that at the URS receiver
(see Section IV for details). In particular, we consider two IRS reflection
design approaches, namely short-term or long-term IRS reflection, which offer
different trade-offs between performance and complexity. Specifically, in the
short-term IRS reflection case, as shown in Fig. 4(a), the IRS reflecting
elements are switched on (i.e., $\beta_{n}=1,\forall n=1,...N$) in Step II
during Cases 1, 2, and 3, with the corresponding reflection phase shifts given
by $\bm{\theta}_{1}$, $\bm{\theta}_{2}$, and $\bm{\theta}_{3}$, respectively;
otherwise, the IRS reflecting elements are switched off (i.e.,
$\beta_{n}=0,\forall n=1,...N$) to save power. In contrast, with long-term IRS
reflection, as shown in Fig. 4(b), the IRS reflecting elements are switched on
(i.e., $\beta_{n}=1,\forall n=1,...N$) during the entire Step II, with fixed
IRS reflection phase shifts given by $\bm{\theta}_{0}$.

Comparing the short-term and long-term IRS reflections, we can observe that
short-term reflection offers more flexible IRS passive beamforming design for
different cases and thus is expected to achieve better performance (in terms
of enhanced/suppressed signal power at LRS/URS) than long-term reflection.
However, it also incurs higher computational and implementation complexities
due to more design variables, and more frequent IRS phase shift adjustment.

##  IV IRS Reflection Design

In this section, we consider IRS reflection design in Step II of the proposed
protocol for short-term and long-term IRS reflections, respectively.

###  IV-A Short-Term IRS Reflection Design

For short-term IRS reflection, we need to optimize the IRS reflection
separately for three cases within each LRS CPI, namely, LRS-signal only, URS-
signal only, and overlapped LRS and URS signal. Specifically, for the LRS-
signal-only case, we aim to maximize the power of the IRS-reflected LRS signal
over the LRS$\rightarrow$IRS$\rightarrow$LRS link, while keeping that over the
LRS$\rightarrow$IRS$\rightarrow$URS link below a certain level, by optimizing
the IRS reflection phase shifts. Consequently, based on (II-B) and (II-B), the
optimization problem can be formulated as

$\displaystyle(\mathcal{P}_{1}):\mathop{\max}\limits_{\bm{\theta}_{1}}~{}$
$\displaystyle~{}\underbrace{Q_{LS}^{2}\times|{\mathbf{u}}^{H}\bm{\theta}_{1}|^{2}}_{\text{LRS}\rightarrow\text{IRS}\rightarrow\text{LRS}}$
(38a) s.t.
$\displaystyle~{}\underbrace{{\frac{Q_{LS}Q_{US}}{P_{U,\min}}\times|{\mathbf{v}}^{H}\bm{\theta}_{1}|^{2}}}_{\text{LRS}\rightarrow\text{IRS}\rightarrow\text{URS}}\leq\gamma,$
(38b) $\displaystyle~{}|{\theta}_{1,n}|=1,n=1,\cdots,N,$ (38c)

where $P_{U,\min}$ denotes the minimum value of $P_{U}$ which is assumed to be
known, and $\gamma$ denotes the maximum received signal power threshold at the
URS, below which the URS cannot achieve its desired target
detection/estimation performance.

Next, for the URS-signal-only case, our goal is to maximize the power of IRS-
reflected URS signal over the URS$\rightarrow$IRS$\rightarrow$LRS link for
exploiting the URS’s radar signal for the LRS’s target detection, subject to
that over the URS$\rightarrow$IRS$\rightarrow$URS link being below the given
threshold $\gamma$. According to (II-B) and (II-B), the IRS reflection phase
shifts can be optimized by solving the following problem,

$\displaystyle(\mathcal{P}_{2}):\mathop{\max}\limits_{\bm{\theta}_{2}}~{}$
$\displaystyle~{}\underbrace{Q_{LS}Q_{US}\times|{\mathbf{r}}^{H}\bm{\theta}_{2}|^{2}}_{\text{URS}\rightarrow\text{IRS}\rightarrow\text{LRS}}$
(39a) s.t.
$\displaystyle~{}\underbrace{{\frac{Q_{US}^{2}}{P_{U,\min}}\times|{\mathbf{g}}^{H}\bm{\theta}_{2}|^{2}}}_{\text{URS}\rightarrow\text{IRS}\rightarrow\text{URS}}\leq\gamma,$
(39b) $\displaystyle~{}|{\theta}_{2,n}|=1,n=1,\cdots,N.$ (39c)

Last, for the case of overlapped LRS and URS signal, we aim to maximize the
average power of IRS-reflected overlapped LRS and URS signal at the LRS
receiver, while keeping that at the URS receiver below $\gamma$. To achieve
this, we can formulate the following IRS reflection phase shifts optimization
problem based on (34) and (37),

$\displaystyle\\!\\!\\!\\!(\mathcal{P}_{3}):\mathop{\max}\limits_{\bm{\theta}_{3}}~{}$
$\displaystyle~{}\underbrace{{{Q_{LS}^{2}}\times|{\mathbf{u}}^{H}\bm{\theta}_{3}|^{2}}}_{\text{LRS}\rightarrow\text{IRS}\rightarrow\text{LRS}}+\underbrace{{{Q_{LS}Q_{US}}\times|{\mathbf{r}}^{H}\bm{\theta}_{3}|^{2}}}_{\text{URS}\rightarrow\text{IRS}\rightarrow\text{LRS}}$
(40a) s.t.
$\displaystyle~{}\underbrace{{\frac{Q_{US}^{2}}{P_{U,\min}}\times|{\mathbf{g}}^{H}\bm{\theta}_{3}|^{2}}}_{\text{URS}\rightarrow\text{IRS}\rightarrow\text{URS}}+\underbrace{{\frac{Q_{LS}Q_{US}}{P_{U,\min}}\times|{\mathbf{v}}^{H}\bm{\theta}_{3}|^{2}}}_{\text{LRS}\rightarrow\text{IRS}\rightarrow\text{URS}}\leq\gamma,$
(40b) $\displaystyle~{}|{\theta}_{3,n}|=1,n=1,\cdots,N.$ (40c)

_Remark 1_ : The estimates of LRS/URS AoAs $\\{\phi_{L},\eta_{L}\\}$ and
$\\{\phi_{U},\eta_{U}\\}$ in Step I of the proposed protocol are required for
constructing the vectors $\mathbf{u}$, $\mathbf{v}$, $\mathbf{r}$, and
$\mathbf{g}$ involved in problems $(\mathcal{P}_{1})$ to $(\mathcal{P}_{3})$.
In addition, the estimates of received signal powers from LRS and URS at IRS,
i.e., $Q_{LS}$ and $Q_{US}$ in Step I are also needed for formulating the
objective functions and constraints in $(\mathcal{P}_{1})$ to
$(\mathcal{P}_{3})$.

###  IV-B Long-Term IRS Reflection Design

The objective of long-term IRS reflection is to maximize the total energy of
IRS-reflected LRS signal with duration $t_{L}$ and URS signal with duration
$t_{U}$ at the LRS receiver, while keeping their overlapped signal peak power
(in Case 3 by assuming the worst case of $t_{o}>0$) at the URS receiver below
the given threshold $\gamma$. Similar to ($\mathcal{P}_{3}$), the IRS
reflection phase shifts during Step II can be optimized by the following
problem,

$\displaystyle(\mathcal{P}_{4}):\mathop{\max}\limits_{\bm{\theta}_{0}}~{}$
$\displaystyle~{}\underbrace{t_{L}{{Q_{LS}^{2}}\times|{\mathbf{u}}^{H}\bm{\theta}_{0}|^{2}}}_{\text{LRS}\rightarrow\text{IRS}\rightarrow\text{LRS}}+\underbrace{t_{U}{{Q_{LS}Q_{US}}\times|{\mathbf{r}}^{H}\bm{\theta}_{0}|^{2}}}_{\text{URS}\rightarrow\text{IRS}\rightarrow\text{LRS}}$
(41a) s.t.
$\displaystyle~{}\underbrace{{\frac{Q_{US}^{2}}{P_{U,\min}}\times|{\mathbf{g}}^{H}\bm{\theta}_{0}|^{2}}}_{\text{URS}\rightarrow\text{IRS}\rightarrow\text{URS}}+\underbrace{{\frac{Q_{LS}Q_{US}}{P_{U,\min}}\times|{\mathbf{v}}^{H}\bm{\theta}_{0}|^{2}}}_{\text{LRS}\rightarrow\text{IRS}\rightarrow\text{URS}}\leq\gamma,$
(41b) $\displaystyle~{}|{\theta}_{0,n}|=1,n=1,\cdots,N.$ (41c)

_Remark 2_ : The estimation of AoA values $\\{\phi_{L},\eta_{L}\\}$ and
$\\{\phi_{U},\eta_{U}\\}$, as well as received signal powers $Q_{LS}$ and
$Q_{US}$ in Step I of the proposed protocol is required for formulating
problem $(\mathcal{P}_{4})$. In addition, the LRS and URS pulse durations,
i.e., $t_{L}$ and $t_{U}$, respectively, are explicitly used for formulating
the objective function of $(\mathcal{P}_{4})$, which, if not a priori known,
also need to be estimated in Step I of the proposed protocol.

Note that if the constraint (41b) in problem $(\mathcal{P}_{4})$ is satisfied,
then constraints (38b) in problem $(\mathcal{P}_{1})$, (39b) in problem
$(\mathcal{P}_{2})$, and (40b) in problem $(\mathcal{P}_{3})$ are also
satisfied. Thus, the optimal solution $\bm{\theta}_{0}^{o}$ to problem
$(\mathcal{P}_{4})$ is in general only a feasible solution to problems
$(\mathcal{P}_{1})$–$(\mathcal{P}_{3})$. Consequently, we have the following
inequality,

$\displaystyle
t_{L}\times{{Q_{LS}^{2}}|{\mathbf{u}}^{H}\bm{\theta}_{0}^{o}|^{2}}+t_{U}\times{{Q_{LS}Q_{US}}|{\mathbf{r}}^{H}\bm{\theta}_{0}^{o}|^{2}}$
$\displaystyle\leq(t_{L}-t_{o})\times{{Q_{LS}^{2}}|{\mathbf{u}}^{H}\bm{\theta}_{1}^{o}|^{2}}+(t_{U}-t_{o})\times{{Q_{LS}Q_{US}}|{\mathbf{r}}^{H}\bm{\theta}_{2}^{o}|^{2}}$
$\displaystyle+t_{o}\times({{Q_{LS}^{2}}|{\mathbf{u}}^{H}\bm{\theta}_{3}^{o}|^{2}}+{{Q_{LS}Q_{US}}|{\mathbf{r}}^{H}\bm{\theta}_{3}^{o}|^{2}}),$
(42)

where $\bm{\theta}_{1}^{o}$, $\bm{\theta}_{2}^{o}$, and $\bm{\theta}_{3}^{o}$
denote the optimal solutions to problems
($\mathcal{P}_{1}$)–($\mathcal{P}_{3}$), respectively. The above inequality
indicates that the maximum LRS received signal energy with long-term IRS
reflection is upper-bounded by that with short-term IRS reflection. In
particular, when the LRS and URS waveforms have the same duration and they are
completely overlapped, i.e., $t_{o}=t_{L}=t_{U}$, (IV-B) holds with equality
since $t_{L}-t_{o}=0$, $t_{U}-t_{o}=0$, and
$\bm{\theta}_{0}^{o}=\bm{\theta}_{3}^{o}$. As a result, the long-term and
short-term IRS reflections yield the same LRS received signal energy.

The above formulated optimization problems, namely
$(\mathcal{P}_{1})$–$(\mathcal{P}_{4})$, share the same structure and can be
all expressed by the following general problem,

$\displaystyle(\mathcal{P}_{5}):\mathop{\max}\limits_{\bm{\theta}}~{}$
$\displaystyle~{}\sum_{i=1}^{2}|\mathbf{q}_{i}^{H}\bm{\theta}|^{2}$ (43a) s.t.
$\displaystyle~{}\sum_{i=1}^{2}|{\mathbf{h}}_{i}^{H}\bm{\theta}|^{2}\leq\gamma,$
(43b) $\displaystyle~{}|{\theta}_{n}|=1,n=1,\cdots,N,$ (43c)

with $\mathbf{q}_{1}=\sqrt{{{Q_{LS}^{2}}}}\mathbf{u}$,
$\mathbf{q}_{2}=\mathbf{0}$,
$\mathbf{h}_{1}=\sqrt{\frac{Q_{LS}Q_{US}}{P_{U,\min}}}\mathbf{v}$, and
$\mathbf{h}_{2}=\mathbf{0}$ for problem $(\mathcal{P}_{1})$;
$\mathbf{q}_{1}=\sqrt{{{Q_{LS}Q_{US}}}}\mathbf{r}$,
$\mathbf{q}_{2}=\mathbf{0}$,
$\mathbf{h}_{1}=\sqrt{\frac{Q_{US}^{2}}{P_{U,\min}}}\mathbf{g}$, and
$\mathbf{h}_{2}=\mathbf{0}$ for problem $(\mathcal{P}_{2})$;
$\mathbf{q}_{1}=\sqrt{{{Q_{LS}^{2}}}}\mathbf{u}$,
$\mathbf{q}_{2}=\sqrt{{{Q_{LS}Q_{US}}}}\mathbf{r}$,
$\mathbf{h}_{1}=\sqrt{\frac{Q_{US}^{2}}{P_{U,\min}}}\mathbf{g}$, and
$\mathbf{h}_{2}=\sqrt{\frac{Q_{LS}Q_{US}}{P_{U,\min}}}\mathbf{v}$ for problem
$(\mathcal{P}_{3})$; and
$\mathbf{q}_{1}=\sqrt{t_{L}{{Q_{LS}^{2}}}}\mathbf{u}$,
$\mathbf{q}_{2}=\sqrt{t_{U}{{Q_{LS}Q_{US}}}}\mathbf{r}$,
$\mathbf{h}_{1}=\sqrt{\frac{Q_{US}^{2}}{P_{U,\min}}}\mathbf{g}$, and
$\mathbf{h}_{2}=\sqrt{\frac{Q_{LS}Q_{US}}{P_{U,\min}}}\mathbf{v}$ for problem
$(\mathcal{P}_{4})$. As such, it suffices to solve problem
$(\mathcal{P}_{5})$. Note that $(\mathcal{P}_{5})$ can be reformulated such
that the SDR method can be used to solve it sub-optimally [31]. However, the
SDR-based method is computationally prohibitive, especially when the number of
IRS reflecting elements is large. To reduce the computational complexity, we
propose in the following an alternative PDD-based algorithm for solving
$(\mathcal{P}_{5})$ more efficiently.

###  IV-C PDD-based Algorithm

In this subsection, we present a computationally efficient algorithm to solve
($\mathcal{P}_{5}$). It is noted that ($\mathcal{P}_{5}$) is a non-convex
optimization problem because the objective in (43a) is non-concave and the
constraints in (43c) are non-convex. Hence, we adopt the PDD method [32],
which is based on the augmented Lagrangian and can separate the coupled
optimization variables by penalizing the auxiliary equality constraints via
dual decomposition. To apply the PDD method, we first decouple the constraints
(43b) and (43c) by introducing auxiliary variables
$\bm{\vartheta}=[\vartheta_{1},\vartheta_{2},\cdots,\vartheta_{N}]^{T}\in\mathbb{C}^{N\times
1}$, which satisfy $\bm{\vartheta}=\bm{\theta}$. As a result, problem
($\mathcal{P}_{5}$) can be equivalently recast as

$\displaystyle(\mathcal{P}_{6}):\mathop{\max}\limits_{\bm{\theta}}~{}$
$\displaystyle~{}\sum_{i=1}^{2}|\mathbf{q}_{i}^{H}\bm{\theta}|^{2}$ (44a) s.t.
$\displaystyle~{}\sum_{i=1}^{2}|{\mathbf{h}}_{i}^{H}\bm{\theta}|^{2}\leq\gamma,$
(44b) $\displaystyle~{}\bm{\theta}=\bm{\vartheta},$ (44c)
$\displaystyle~{}|{\vartheta}_{n}|=1,n=1,\cdots,N.$ (44d)

Then, by penalizing the equality constraint in (44c), the dual optimization
problem of ($\mathcal{P}_{6}$) can be written as

$\displaystyle(\mathcal{P}_{7}):\mathop{\min}\limits_{\bm{\theta},\bm{\vartheta},\bm{\lambda}}~{}$
$\displaystyle~{}-\sum_{i=1}^{2}|\mathbf{q}_{i}^{H}\bm{\theta}|^{2}+\frac{1}{2\rho}\|\bm{\theta}-\bm{\vartheta}+\rho\bm{\lambda}\|_{2}^{2}$
(45a) s.t.
$\displaystyle~{}\sum_{i=1}^{2}|{\mathbf{h}}_{i}^{H}\bm{\theta}|^{2}\leq\gamma,$
(45b) $\displaystyle~{}|\theta_{n}|\leq 1,n=1,\cdots,N,$ (45c)
$\displaystyle~{}|{\vartheta}_{n}|=1,n=1,\cdots,N,$ (45d)

where $\rho$ is the penalty parameter and $\bm{\lambda}$ is the dual variable
corresponding to the equality constraint $\bm{\theta}=\bm{\vartheta}$. To
solve the problem ($\mathcal{P}_{7}$), we use inner iterations to update
$\bm{\theta}$ and $\bm{\vartheta}$, and outer iterations to update the dual
variables $\bm{\lambda}$ with the details given below.

For the inner loop, we fix $\bm{\lambda}$ as a constant. Then, we partition
the remaining optimization variables in $(\mathcal{P}_{7})$ into two blocks,
i.e., $\bm{\vartheta}$ and $\bm{\theta}$, and optimize them iteratively as
follows. Specifically, when $\bm{\vartheta}$ is given, $\bm{\theta}$ can be
updated by solving the following problem.

$\displaystyle(\mathcal{P}_{8}):\mathop{\min}\limits_{\bm{\theta}}~{}$
$\displaystyle~{}-\sum_{i=1}^{2}|\mathbf{q}_{i}^{H}\bm{\theta}|^{2}+\frac{1}{2\rho}\|\bm{\theta}-\bm{\vartheta}+\rho\bm{\lambda}\|_{2}^{2}$
(46a) s.t.
$\displaystyle~{}\sum_{i=1}^{2}|{\mathbf{h}}_{i}^{H}\bm{\theta}|^{2}\leq\gamma,$
(46b) $\displaystyle~{}|\theta_{n}|\leq 1,n=1,\cdots,N.$ (46c)

Note that the objective function of ($\mathcal{P}_{8}$) is the difference of
two non-negative convex functions, which is non-convex in general. To
circumvent this difficulty, we approximate the convex function
$|\mathbf{q}_{i}^{H}\bm{\theta}|^{2}$ by its first-order Taylor expansion at a
given point ${\tilde{\bm{\theta}}}$, denoted by
$\underline{q}_{i}(\bm{\theta},{\tilde{\bm{\theta}}})$, i.e.,

$\displaystyle|\mathbf{q}_{i}^{H}\bm{\theta}|^{2}\geq\underline{q}_{i}(\bm{\theta},{\tilde{\bm{\theta}}})\triangleq
2\Re(\bm{\varepsilon}_{i}^{H}\bm{\theta})+\bar{\xi}_{i},$ (47)

where
$\bm{\varepsilon}_{i}\triangleq\mathbf{q}_{i}^{H}{\tilde{\bm{\theta}}}\mathbf{q}_{i}^{H}$
and $\bar{\xi}_{i}=-|\mathbf{q}_{i}^{H}{\tilde{\bm{\theta}}}|^{2}$. Therefore,
we obtain the following problem,

$\displaystyle(\mathcal{P}_{9}):\mathop{\min}\limits_{\bm{\theta}}$
$\displaystyle~{}-2\sum_{i=1}^{2}\Re(\bm{\varepsilon}_{i}^{H}\bm{\theta})-\sum_{i=1}^{2}\bar{\xi}_{i}+\frac{1}{2\rho}\|\bm{\theta}-\bm{\vartheta}+\rho\bm{\lambda}\|_{2}^{2}$
(48a) s.t.
$\displaystyle~{}\sum_{i=1}^{2}|{\mathbf{h}}_{i}^{H}\bm{\theta}|^{2}\leq\gamma,$
(48b) $\displaystyle~{}|\theta_{n}|\leq 1,n=1,\cdots,N.$ (48c)

Problem $(\mathcal{P}_{9})$ is a convex optimization problem, which can be
solved efficiently via existing software, e.g., CVX [33]. By repeatedly
solving $(\mathcal{P}_{9})$, and setting the point for Taylor expansion, i.e.,
${\tilde{\bm{\theta}}}$, in each iteration as the optimal solution obtained in
the previous iteration, the above procedure generates a sequence of solutions
that converge to a Karush-Kuhn-Tucker (KKT) solution of $(\mathcal{P}_{8})$
[34].

On the other hand, when $\bm{\theta}$ is given, $\bm{\vartheta}$ can be
updated by solving the following problem,

$\displaystyle(\mathcal{P}_{10}):\mathop{\min}\limits_{\bm{\vartheta}}$
$\displaystyle~{}\|\bm{\theta}-\bm{\vartheta}+\rho\bm{\lambda}\|_{2}^{2}$
(49a) s.t. $\displaystyle~{}|{\vartheta}_{n}|=1,n=1,\cdots,N,$ (49b)

for which the optimum is attained when the elements of $\bm{\vartheta}$
satisfy the following condition,

$\displaystyle{\vartheta}_{n}^{*}=\angle({\theta}_{n}+\rho{\lambda}_{n}).$
(50)

In summary, the inner loop alternatively updates $\bm{\vartheta}$ and
$\bm{\theta}$ until convergence is reached.

Next, we consider the outer loop, where the dual variable $\bm{\lambda}$ can
be updated by

$\displaystyle\bm{\lambda}:=\bm{\lambda}+\frac{1}{\rho}(\bm{\theta}-\bm{\vartheta}).$
(51)

The above PDD-based algorithm for solving ($\mathcal{P}_{5}$) is summarized in
Algorithm 1. In step 17 of Algorithm 1, $c<1$ is a positive constant scaling
factor that is used to enlarge the value of the penalty term involved in
problem ($\mathcal{P}_{7}$) in each outer iteration.

Finally, we compare the complexity of the proposed PDD-based method and SDR-
based method in [31]. By carefully checking the iterative process in Algorithm
1, its computational complexity is found to be dominated by that for solving
problem ($\mathcal{P}_{9}$), which has a complexity of order
$\mathcal{O}(N^{2})$. Therefore, the overall complexity of Algorithm 1 can be
shown in the order of $\mathcal{O}(J_{0}J_{1}J_{2}N^{2})$, where $J_{0}$ and
$J_{1}$ denote the maximum outer and inner iteration numbers, and $J_{2}$ is
the number of iterations for solving ($\mathcal{P}_{9}$). In contrast, the
computational complexity of the SDR-based method can be shown in the order of
$\mathcal{O}(N^{6.5})$ [31]. Thus, the complexity of the proposed Algorithm 1
is much lower than the SDR-based method, especially for large-size IRS with
large values of $N$.

##  V Numerical Results

In this section, we evaluate the performance of our proposed target-mounted
IRS enabled secure wireless sensing design via numerical results. In the
following simulations, the radar pulse waveforms of LRS and URS are given by
$x(t)=\exp(j\pi Bt^{2}/t_{L})$ and $s(t)=\exp(j\pi Bt^{2}/t_{U})$,
respectively, with the signal band-width $B=100$ MHz [2]. For simplicity, we
consider uniform linear array (ULA) for all the LRS, URS, and IRS, where the
numbers of antennas at the LRS and URS are $M=64$ and $D=64$, respectively,
the numbers of IRS reflecting elements and sensors are $N=64$ and $L=15$,
respectively, the wavelength is $\lambda=0.2$ m, the target is located at
$0^{\circ}$ horizontally from the IRS, and the transmit power of LRS and URS
are set as $P_{L}=P_{U}=30$ mW, if not specified otherwise. Additionally, the
LRS-target and the URS-target distances are set as $d_{\mathrm{LT}}=30$ m and
$d_{\mathrm{UT}}=20$ m, respectively; the LRS/URS PRI is $T=100$ us with the
pulse durations of $t_{L}=25$ us and $t_{U}=30$ us; the total physical size of
the IRS is $S=N{d}^{2}$; the target echo surface is set as $A=1.2\times S$
(i.e., the size of target echo surface is larger than that of the IRS); the
RCS of the target is set as $\kappa_{\mathrm{t}}=\frac{4\pi
A^{2}}{\lambda^{2}}$ in m2; and the IRS element separation is
${d}=\lambda/10$, unless otherwise stated. For the purpose of exposition, we
first assume that the LRS/URS channel and waveform parameters are perfectly
estimated by the IRS sensors, while the effects of their imperfect estimation
on the performance are then evaluated.

For comparison, we consider the following benchmark schemes:

  * 1)

Random Phase Shift: We assume that each IRS reflecting element takes a random
phase shift independently and uniformly distributed in $[0,2\pi)$, based on
which we obtain the received signal power at LRS or URS.

  * 2)

No Target-Mounted IRS: There is no IRS mounted on the target, as in the
traditional sensing system.

  * 3)

SDR-based Method: Problem ($\mathcal{P}_{5}$) is first transformed into an
semidefinite programming (SDP) problem by applying SDR and dropping the
nonconvex rank-one constraint, i.e.,
$\mathrm{rank}(\bm{\theta}\bm{\theta}^{H})=1$. The resulting convex SDP is
then optimally solved by CVX [31] with 10000 Gaussian randomizations generated
to find the best rank-1 feasible solution for $(\mathcal{P}_{5})$.

###  V-A LRS Present Only

First, we consider the special case where only the LRS is present in the
system to detect the target. In this case, there is no security concern and
our objective is to maximize the power of reflected echo signal from the
target/IRS towards the LRS receiver for enhancing its target sensing
performance, by optimizing the IRS reflection. This problem can be formulated
by setting $\mathbf{q}_{1}=\sqrt{{{Q_{LS}^{2}}}}\mathbf{u}$,
$\mathbf{q}_{2}=\mathbf{h}_{1}=\mathbf{h}_{2}=\mathbf{0}$ in problem
($\mathcal{P}_{5}$). It can be easily shown that the optimal solution to this
problem is given by $\bm{\theta}_{L}^{o}=\mathbf{u}$. This implies that the
optimal IRS reflection should align the reflected signal from IRS with its AoA
to IRS for maximizing the signal power at the LRS receiver to improve its
target detection/estimation accuracy.

Figure 5: Received signal power at LRS versus LRS transmit beam direction.
Figure 6: Received signal power at URS versus URS transmit beam direction.

Fig. 5 compares the received signal power at the LRS receiver under different
schemes versus the LRS transmit beam direction (i.e., by setting different
$\mathbf{w}_{L}$). In this simulation, we keep the location of the target
fixed, while the LRS transmitter scans different angles in the target space
via transmit beamforming based on the discrete Fourier transform (DFT) of size
$N$. First, it is shown that, despite that the IRS has a smaller size than the
target’s echo surface, the received signal power at LRS with target-mounted
IRS yields about 16.5 dB gain over the traditional sensing without IRS at the
target angle. This is because when the LRS beam hits the direction of the
target, the target-mounted IRS provides a strong passive beamforming gain for
the echo signal, which is not available for the target without IRS. Second,
with target-mounted IRS, the proposed reflection design achieves significantly
improved signal power at the LRS receiver over the scheme with random IRS
phase shifts, which is lack of IRS passive beamforming gain. Third, it is
observed that the scheme without target-mounted IRS performs better than the
scheme with random IRS phase shifts. This is because, although the reflected
signals in both schemes are not directional to the LRS, the size of the target
echo surface is larger than that of the IRS, thus resulting in more reflected
signal power at the LRS receiver on average.

###  V-B URS Present Only

Next, we consider another special case where only the URS is present in the
system to detect the same target. In this case, we aim to minimize the power
of echo signal from the target/IRS towards the URS receiver by optimizing the
IRS reflection. Accordingly, the optimization problem can be formulated as

$\displaystyle\mathop{\min}\limits_{\bm{\theta}}~{}$
$\displaystyle~{}\underbrace{|{\mathbf{g}}^{H}\bm{\theta}|^{2}}_{\text{URS}\rightarrow\text{IRS}\rightarrow\text{URS}}$
s.t. $\displaystyle~{}|{\theta}_{n}|=1,n=1,\cdots,N.$ (52)

By exploiting the structure of LoS-based channel vector $\mathbf{g}$ given in
(31), we show in the following proposition a closed-form optimal solution to
the above problem.

(a) $N=64$ (b) $N=128$

Figure 7: Received signal power at LRS versus received signal power threshold
$\gamma$ at URS.

Proposition 1: The optimal solution to problem (V-B) is given by

$\displaystyle\bm{\theta}_{U}^{o}=\bm{\theta}_{x}^{o}\otimes\bm{\theta}_{y}^{o},$
(53)

with $\bm{\theta}_{x}^{o}=\mathbf{d}(N_{x},\varsigma_{x}^{o})$ and
$\bm{\theta}_{y}^{o}=\mathbf{d}(N_{y},\varsigma_{y}^{o})$, where
$\varsigma_{x}^{o}=(\zeta_{u}^{\bar{a}}-\zeta_{u}^{a})+\frac{\lambda}{N_{x}{d}}i_{x},~{}i_{x}=1,2,\cdots,N_{x}-1$
and
$\varsigma_{y}^{o}=(\zeta_{u}^{\bar{e}}-\zeta_{u}^{e})+\frac{\lambda}{N_{y}{d}}i_{y},~{}i_{y}=1,2,\cdots,N_{y}-1$.

###### Proof:

Please refer to Appendix A. ∎

Fig. 6 compares the received signal power at URS receiver under different
schemes versus the URS transmit beam direction (similarly assumed as that for
the LRS transmitter in Fig. 5). We observe that, at the target angle, the
received signal power at URS with target-mounted IRS has a significant
reduction compared to both traditional sensing without IRS and the scheme with
random IRS phase shifts, which thus validates the effectiveness of target-
mounted IRS for achieving secure wireless sensing against the URS. This is
because the target-mounted IRS can completely null the reflected URS signal
towards its receiver with the proposed IRS reflection design given in
Proposition 1.

###  V-C Both LRS and URS Present

Figure 8: Comparison of LRS received signal power with LRS and URS both
present versus with LRS present only.

Last, we consider the general case where both the LRS and URS are present in
the system. In this case, we assume that both the LRS and URS transmitters
scan the target space based on DFT codebook, and consider the most challenging
scenario when their beam directions both align with the target direction at
the same time (otherwise, the system reduces to the LRS-present only or URS-
present only as considered previously). Fig. 7 (a) plots the received signal
power of LRS versus the maximum received signal power threshold $\gamma$ at
the URS. It is observed that the received signal power at LRS is higher when
using short-term IRS reflection compared to long-term IRS reflection, which is
consistent with the result in (IV-B). In addition, Fig. 7 (a) shows that our
proposed PDD-based algorithm can achieve almost the same performance as the
SDR-based algorithm, thus being more practically appealing given its much
lower computational complexity. Moreover, the proposed short-term and long-
term IRS reflections both achieve significantly improved received signal power
at LRS over the benchmark schemes without target-mounted IRS or with random
IRS phase shifts. In Fig.7 (b), we increase the number of IRS reflecting
elements from $N=64$ to $N=128$ (by decreasing the IRS element separation
${d}$) with the physical size of the IRS (i.e., $S$) fixed. It is observed
that the performance gap between our proposed short-term/long-term IRS
reflection and the scheme without target-mounted IRS is enlarged as $N$
increases. This is because installing more reflecting elements at IRS provides
enhanced passive beamforming gain.

Next, we present a performance comparison of the proposed scheme with both LRS
and URS present ($\gamma=10^{-5}$ mW) versus the LRS-present-only scenario.
For both cases, the IRS reflection is optimized as above. As shown in Fig. 8,
the URS-target distance is kept constant, i.e., $d_{UT}=20$ m while the LRS
gradually moves away from the target. It is observed that the proposed scheme
shows better performance in terms of LRS received signal power when both LRS
and URS are present, as compared to the scenario where only LRS is present,
and their performance gap is more pronounced for a larger LRS-target distance
$d_{LT}$. This demonstrates the advantage of using URS radar signals for the
LRS target’s detection, especially when the LRS is located far away from the
target and its own radar signal is severely attenuated due to the increased
LRS-target distance.

Figure 9: The performance of the proposed scheme versus URS-target distance.
Figure 10: The performance of the proposed scheme versus the AoA difference
between URS and LRS.

In Fig. 9, we keep the AoAs from LRS and URS to target/IRS fixed, and plot the
received signal powers at LRS and URS with or without target-mounted IRS
versus the distance between the URS and target. For the case with target-
mounted IRS, the proposed PDD-based algorithm is applied with short-term IRS
reflection. It is observed that compared to the case without target-mounted
IRS, the proposed design with target-mounted IRS can significantly enhance the
received signal power at LRS and reduce that at URS at the same time, for a
large range of URS-target distance values.

In Fig. 10, we keep LRS-target distance and URS-target distance fixed, and
plot the received signal powers at LRS and URS with or without target-mounted
IRS versus the AoA difference between LRS and URS. It is shown that for the
scheme without target-mounted IRS, the received powers at LRS and URS are
constant due to fixed LRS-target and URS-target distances, regardless of their
AoAs to the target/IRS. In contrast, the proposed design with target-mounted
IRS can achieve greatly enhanced/suppressed power at LRS/URS receivers,
provided that the AoA difference between LRS and URS is sufficiently large,
say, larger than 0.1 radian (rad).

Figure 11: Received signal power at LRS versus the LRS/URS signal overlapping
ratio with short-term or long-term IRS reflection. Figure 12: Effect of
imperfect IRS sensor angle estimation to LRS/URS received signal power with
short-term or long-term IRS reflection.

Fig. 11 compares the performance of long-term versus short-term IRS reflection
for the proposed target-mounted IRS design under different ratios of the
LRS/URS overlapping signal duration to that of the LRS/URS pulse duration,
i.e., $t_{o}/t_{U}$ by assuming $t_{L}=t_{U}=30$ us. It is observed that the
received signal power at LRS with long-term IRS reflection is upper-bounded by
that with short-term IRS reflection, as expected. In addition, as the LRS/URS
signal overlapping ratio increases, the received signal power at LRS with
long-term IRS reflection increases, while that with short-term IRS reflection
decreases; and as the overlapping ratio approaches one, they become identical,
which is consistent with the discussion for (IV-B). Furthermore, it is
observed that more stringent security requirement (i.e., smaller $\gamma$) on
the URS received signal power also results in lower received signal power at
LRS. This is because the IRS reflection needs to further reduce the signal
power towards the URS, which has to sacrifice the passive beamforming gain
towards the LRS.

Finally, we evaluate the impact of imperfect LRS and URS AoA estimation by IRS
sensors on the received signal powers at LRS and URS for the proposed target-
mounted IRS design, as shown in Fig. 12. It is observed that the proposed
scheme with imperfect AoA estimation has less received signal power at LRS and
more received signal power at URS, as compared to the case with perfect AoA
estimation. Although inaccurate AoA estimation reduces the signal
enhancement/suppression gain of target-mounted IRS in practice, the
performance loss is observed to be quite small (less than 1.8%) if the angle
estimation error is below $1^{\circ}$.

##  VI Conclusions

In this paper, we proposed a new secure wireless sensing approach to
simultaneously enhance the target detection for LRS while preventing that
against the URS by utilizing the target-mounted IRS. We presented a practical
protocol for target-mounted IRS and designed the IRS reflection to maximize
the IRS reflected signal power to LRS, while keeping that at URS below a
certain threshold, for both short-term and long-term IRS reflection scenarios.
Simulation results showed the significant received signal power
improvement/reduction at LRS/URS achieved with our proposed target-mounted IRS
designs as compared to benchmark schemes, and also validated their robust
performance under different practical setups.

##  Appendix A The Proof of (53)

The IRS reflection gain $|{\mathbf{g}}^{H}\bm{\theta}|^{2}$ in (V-B) can be
decoupled as the product of the gains along the $x$\- and $y$-axes, i.e.,

$\displaystyle|{\mathbf{g}}^{H}\bm{\theta}|^{2}=|\mathbf{d}^{H}(N_{x},\zeta_{u}^{\bar{a}}-\zeta_{u}^{a})\bm{\theta}_{x}|^{2}\times|\mathbf{d}^{H}(N_{x},\zeta_{u}^{\bar{e}}-\zeta_{u}^{e})\bm{\theta}_{y}|^{2}.$
(54)

By setting the IRS reflection vector $\bm{\theta}$ along the $x$\- and
$y$-axes as $\bm{\theta}_{x}=\mathbf{d}(N_{x},\varsigma_{x})$ and
$\bm{\theta}_{y}=\mathbf{d}(N_{y},\varsigma_{y})$, respectively, with
$\varsigma_{x}$ and $\varsigma_{y}$ being the corresponding steering angles,
we have

$\displaystyle|\mathbf{d}^{H}(N_{x},\zeta_{u}^{\bar{a}}-\zeta_{u}^{a})\bm{\theta}_{x}|^{2}$
$\displaystyle=\left|\sum_{n=1}^{N_{x}}e^{j\frac{2\pi
d}{\lambda}(n-1)(\varsigma_{x}-(\zeta_{u}^{\bar{a}}-\zeta_{u}^{a}))}\right|^{2}$
$\displaystyle=\frac{\sin^{2}(\frac{\frac{2\pi
d}{\lambda}N_{x}\delta_{x}}{2})}{\sin^{2}(\frac{\frac{2\pi
d}{\lambda}\delta_{x}}{2})},$ (55)
$\displaystyle|\mathbf{d}^{H}(N_{y},\zeta_{u}^{\bar{e}}-\zeta_{u}^{e})\bm{\theta}_{y}|^{2}$
$\displaystyle=\left|\sum_{n=1}^{N_{y}}e^{j\frac{2\pi
d}{\lambda}(n-1)(\varsigma_{y}-(\zeta_{u}^{\bar{e}}-\zeta_{u}^{e}))}\right|^{2}$
$\displaystyle=\frac{\sin^{2}(\frac{\frac{2\pi
d}{\lambda}N_{y}\delta_{y}}{2})}{\sin^{2}(\frac{\frac{2\pi
d}{\lambda}\delta_{y}}{2})},$ (56)

where $\delta_{x}=\varsigma_{x}-(\zeta_{u}^{\bar{a}}-\zeta_{u}^{a})$ and
$\delta_{y}=\varsigma_{y}-(\zeta_{u}^{\bar{e}}-\zeta_{u}^{e})$. Based on (55)
and (56), when
$\delta_{x}=\frac{\lambda}{N_{x}d}i_{x},~{}i_{x}=1,2,\cdots,N_{x}-1$, it
follows that
$|\mathbf{d}^{H}(N_{x},\zeta_{u}^{\bar{a}}-\zeta_{u}^{a})\bm{\theta}_{x}|^{2}=0$.
Similarly, when
$\delta_{y}=\frac{\lambda}{N_{y}d}i_{y},~{}i_{y}=1,2,\cdots,N_{y}-1$, we have
$|\mathbf{d}^{H}(N_{y},\zeta_{u}^{\bar{e}}-\zeta_{u}^{e})\bm{\theta}_{y}|^{2}=0$.
Thus, the optimal solution to problem (V-B) is given by (53).

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