# Intelligent Omni Surfaces assisted Integrated Multi-Target Sensing and
Multi-User MIMO Communications

Ziheng Zhang, Wen Chen, ,
Qingqing Wu, , Zhendong Li,
Xusheng Zhu, and Jinhong Yuan, Z. Zhang, W. Chen, Q. Wu, Z. Li and X. Zhu are
with the Department of Electronic Engineering, Shanghai Jiao Tong University,
Shanghai 200240, China (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>qingqingwu@sjtu.edu.cn).Jinhong Yuan is with the School of Electrical
Engineering and Telecommunications, University of New South Wales, Sydney, NSW
2025, Australia (e-mail: j.yuan@unsw.edu.au).(_Corresponding author: Wen
Chen._)

###### Abstract

Drawing inspiration from the advantages of intelligent reflecting surfaces
(IRS) in wireless networks, this paper presents a novel design for intelligent
omni surface (IOS) enabled integrated sensing and communications (ISAC). By
harnessing the power of multi-antennas and a multitude of elements, the dual-
function base station (BS) and IOS collaborate to realize joint active and
passive beamforming, enabling seamless 360-degree ISAC coverage. The objective
is to maximize the minimum signal-to-interference-plus-noise ratio (SINR) of
multi-target sensing, while ensuring the multi-user multi-stream
communications. To achieve this, a comprehensive optimization approach is
employed, encompassing the design of radar receive vector, transmit
beamforming matrix, and IOS transmissive and reflective coefficients. Due to
the non-convex nature of the formulated problem, an auxiliary variable is
introduced to transform it into a more tractable form. Consequently, the
problem is decomposed into three sub-problems based on the block coordinate
descent algorithm. Semidefinite relaxation and successive convex approximation
methods are leveraged to convert the sub-problem into a convex problem, while
the iterative rank minimization algorithm and penalty function method ensure
the equivalence. Furthermore, the scenario is extended to mode switching and
time switching protocols. Simulation results validate the convergence and
superior performance of the proposed algorithm compared to other benchmark
algorithms.

###### Index Terms:

Intelligent omni surface, integrated sensing and communication, multi-stream
communication, multi-target sensing.

##  I Introduction

The next-generation wireless network is poised to enable intelligent and
diverse scenarios, necessitating the development of mobile network systems
with multidimensional capabilities. In addition to communications, sensing and
computation play crucial roles in future networks, with communication and
sensing systems sharing many commonalities in signal processing and radio
frequency (RF) units[1]. Specifically, higher frequency signals with wide
bandwidth enhance transmission rates and sensing resolution, while larger-
scale antenna arrays improve beam accuracy. Furthermore, wireless
communication and sensing systems exhibit similarities in waveform design,
signal processing and resource allocation [2]. Therefore, it is highly
meaningful to design a software and hardware resource-sharing system that
enables integrated sensing and communication (ISAC). This approach promises to
enhance spectral efficiency and reduce hardware costs in next-generation
wireless networks.

The essence of ISAC lies in striking a balance between communication and
sensing systems, and it is important to identify appropriate performance
metrics that reveal this fundamental trade-off. Communication rate and Fisher
information are widely recognized as classical performance metrics for
communication and radar systems, respectively [3]. To characterize this trade-
off, the notion of the Cramér-Rao bound-rate region has been introduced. The
points within this region portray the performance of the ISAC system, thereby
serving as a guiding framework for the design of the actual system [4].
However, designing a suitable unified waveform to balance the trade-off in
deterministic-random is challenging, as randomness can convey information but
may reduce sensing performance. A natural type of approach is embedding
information onto radar signals. Chirp signals, extensively employed in radar
systems, possess modulable parameters such as frequency, phase, and amplitude,
making them suitable for shift keying [5]. Another method is transmitting data
during the waiting time of pulsed radars, thereby achieving information
embedding in the time domain. This method reduces mutual interference between
the transmitted signal and the echo signal, enabling full duplex operation
[6]. However, the performance of this method can be affected by the distance
and number of clutters. To mitigate environmental interference, an alternative
method of conveying information involves embedding data into the spatial
domain. In this approach, the main beam of the radar system is employed for
sensing tasks, while the sidelobe level is harnessed to transmit the encoded
bitstream [7]. Additionally, improving the sensing ability of communication
signals is another type of design approach. Orthogonal frequency division
multiplexing-multiple input multiple out (OFDM-MIMO) signals are designed to
achieve multi-beam to sensing multi-target. Intercarrier interference can be
exploited to extract the target’s speed information [8]. Recently, an
orthogonal delay-Doppler division multiplexing (ODDM) or generally delay-
Doppler domain multicarrier modulation (DDMC) has been proposed as a potential
new waveform for future ISAC applications. Its design and performance in
practical ISAC systems are not yet reported [9, 10]. Although a fully unified
waveform improves resource utilization, its implementation requires more
complex signal processing and hardware support.

Non-overlapped waveform design is an additional method employed to achieve
ISAC, effectively mitigating interference between radar and communication
systems. Analogous to multiple access methods in communication, resource
allocation for ISAC can be realized in domains such as time, frequency,
spatial, and code. In [11], a time-division ISAC protocol was proposed for the
Internet of vehicles. The protocol adopts a subframe structure comprising both
communication and sensing symbols, enabling the efficient and timely sharing
of sensing data. Moreover, resource allocation can also be extended to the
frequency domain, as discussed in [12]. To further enhance the utilization of
wireless resources, Shi _et al._ proposed a successive interference
cancellation signal processing scheme, which enables code division in an OFDM
system. Notably, this scheme aims to improve the performance of the system by
mitigating interference. Additionally, the authors established a unified
channel model that comprehensively incorporates both communication and sensing
aspects. The model integrates essential parameters such as Doppler shift,
radar echo model, fading coefficients, average power, and delay to effectively
characterize the ISAC channel properties [13]. With the advancements in large-
scale antenna arrays, MIMO technology is increasingly employed in
communication and sensing systems. Consequently, spatial division to maximize
multiplexing gain has gained popularity in ISAC. Liu _et al._ minimized the
difference between the optimal sensing beam and the ISAC beam that satisfies
the SINR requirement of multi-user communication using manifold algorithms
[14]. To reduce the complexity of spatial division, designing the covariance
matrix of the transmitted signal can simplify the precoding problem, which
transforms the requirements for transmission power, sensing and communication
into convex constraints. To further reduce interference between multi-user,
this work in [15] introduced dirty paper coding. Simulation results show that
ISAC can achieve multi-user communication with only 1dB radar sensing beam
loss compared with traditional radar design. The ISAC research directions
mentioned above have mainly focused on scenarios with multiple-input single-
output (MISO) or single-target sensing. However, these approaches may not be
well-suited for current communication or sensing systems as they lack the
necessary adaptability. In addition, ISAC systems often exhibit a heavy
reliance on line-of-sight signals, which results in limited coverage of the
surrounding building structures, particularly in urban environments.
Therefore, further research is necessary to enhance the system’s performance
and improve model universality.

Intelligent reflecting surface (IRS) is a potential technology to break the
bottleneck in the development of ISAC. As a passive communication unit, IRS
enables channel reconstruction and beamforming to enhance signals and overcome
line-of-sight obstructions, and it has found extensive applications in
wireless communication [16]. Based on these characteristics, IRS is compatible
with radar systems. In recent studies [17, 18], passive IRS has been
investigated as an assistant to mono-static radar systems where the receiving
and transmitting antennas are colocated. In [17], IRS is employed to assist
distributed radar, improving the detection probability of targets compared to
traditional systems. Zhang _et al._ enhanced signal strength in multi-user
localization with an IRS-assisted system by optimizing the decision function
and IRS coefficients, establishing the relationship between system performance
and the number of elements [18]. However, the utilization of IRS in mono-
static radar systems may lead to an increase in path loss compared to
conventional systems. In order to tackle this challenge, a novel IRS-assisted
sensing architecture is proposed, capitalizing on the active IRS’s ability to
transmit and receive signals [19]. Similarly, in [20], the authors presented a
method that deploys a limited number of sensors within the IRS to reduce the
adverse impact caused by excessive reflections. It is important to emphasize
that the aforementioned architecture requires the IRS to possess signal
processing capabilities, leading to an inevitable increase in the deployment
cost of the IRS. To control the deployment cost of IRS and reduce path loss in
the sensing system, bi-static radar is a feasible design. Deploying multiple
antennas in a distributed manner can mitigate the effects of path loss caused
by reflection from the IRS, while also avoiding self interferences of radar
signals at BS [21].

IRS has gained significant attention in both communication and sensing
domains, leading to ongoing research on IRS-assisted ISAC [22, 23, 24, 25]. In
[22], an approach utilizing IRS is proposed to mitigate interference among
multi-user while maintaining similarity with sensing-only beams. The authors
further investigate the performance improvements and trade-offs in ISAC
compared to systems without IRS. Due to the ability of multiple IRS to achieve
more precise beamforming, He _et al._ proposed a double-IRS-assisted ISAC
system model where the communication and sensing systems operate with separate
transmission antennas [23]. Two IRSs are strategically positioned near the
transmitting and receiving antennas to eliminate interference between the two
systems. However, conventional IRS is limited to signal reflection[26] or
transmission [27] and only provides coverage in a half-space. To overcome this
limitation, intelligent omni surface (IOS), also known as simultaneously
transmitting and reflecting (STAR)-IRS, has emerged as a promising solution
[28, 29]. The application of IOS in wireless communication is explored in
[30], where Mu _et al._ compared the performance of IOS in various scenarios
with conventional IRSs and demonstrate that IOS exhibits superior performance
in both unicast and multicast communication. Moreover, the benefits of IOS can
also be extended to ISAC. Wang _et al._ proposed a scheme where the space is
divided into two halves: the transmitting space and the reflecting space,
dedicated to communication and sensing, respectively [31]. This approach
effectively reduces inter-user interference but imposes limitations on the
placement of communication users and sensing targets.

Motivated by the aforementioned backgrounds and challenges, our objective is
to achieve enhanced spatial division multiplexing gain in the ISAC system,
accomplish 360-degree coverage and ensure compatibility with existing
communication systems. To address these goals, we propose an ISAC system
empowered by IOS, wherein a dual-function base station (BS) transmits signals
for multi-user communication and detects multiple targets, while the IOS
reflects and simultaneously transmits incident signals. To mitigate path loss
resulting from multiple signal reflections and simplify the deployment of IOS-
assisted sensing systems, we introduce bistatic radar into our model. The main
focus of this paper is to maximize the minimum signal-to-interference-plus-
noise ratio (SINR) of multi-target sensing, while ensuring that the achievable
rates of all users surpass a predefined threshold. Since the formulated
problem is non-convex, it is imperative to devise an efficient algorithm to
solve it. The main contributions of this paper can be summarized as follows:

  * •

We propose an IOS-enabled ISAC system that leverages a dual-function BS and
IOS with multiple antennas and numerous elements. This system implements joint
active and passive beamforming to serve users and detect targets. In order to
enhance practicality, we consider the impact of echo interference caused by
multi-target, as well as multi-stream communication. To achieve 360-degree
ISAC, we formulate a maximization problem for minimum SINR of multi-target.
This problem involves joint optimization of the radar receive vectors,
transmit beamforming matrices, and IOS reflective and transmissive
coefficients. However, due to the high coupling of variables, both the
objective function and the communication achievable rate constraint are non-
convex. Consequently, solving this problem poses a significant challenge.

  * •

We propose a joint optimization algorithm to address the SINR maximization
problem in the context of energy splitting (ES) protocol. Specifically, we
decompose the non-convex problem into three sub-problems using the coordinate
descent (BCD) algorithm framework. The sub-problem is converted into a convex
problem through the utilization of semidefinite relaxation and the successive
convex approximation method. Additionally, the equivalence is guaranteed
through the use of iterative rank minimization algorithms and the penalty
function method. Furthermore, we extend the proposed algorithm to mode
switching (MS) and time switching (TS) protocols, allowing for more
flexibility and adaptability in different scenarios.

  * •

Numerical results demonstrate the convergence and effectiveness of the
proposed algorithm for the ES protocol. Additionally, our results reveal that
the number of transmit/receive antennas has a more significant impact on
performance compared with the number of IOS elements or maximum transmit power
does. The relative position of each target to the BS also makes a great
contribution to system performance, indicating the importance of BS location
selection.

The rest of this paper is as follows. In Section II, we introduce the system
model of IOS-assisted integrated multi-target sensing and multi-user MIMO
communications and formulate the optimization problem. Then, in Section III,
We elaborate on the joint optimization algorithm. Section IV reveals the
convergence and performance superiority of the proposed algorithm compared to
other benchmarks. Finally, Section V concludes this paper.

Notations: Matrices are represented by bold uppercase letters. Vectors are
denoted by bold lowercase letters. Scalars are represented by standard
lowercase letters. For a complex-valued scalar $x$, $\left|{x}\right|$ denotes
its absolute value. For a general matrix $\bf{A}$, $\text{rank}(\bf{A})$ and
${\bf{A}}^{H}$ denote its rank, conjugate transpose, respectively. For a
square matrix $\bf{X}$, $\text{rank}(\bf{X})$, ${\bf{X}}^{H}$, and
$\text{tr}(\bf{X})$ denote its rank, conjugate transpose, and trace.
${\bf{X}}\succeq 0$ denotes that $\bf{X}$ is a positive semidefinite matrix.
${\mathbb{C}^{M\times N}}$ represents the ${M\times N}$ dimensional complex
matrix space and $j$ is the imaginary unit.

##  II System Model And Problem Formulation

As shown in Fig. 1, the system model of IOS-assisted integrated multi-target
sensing and multi-user MIMO communications networks is first introduced. The
model consists of crucial components, namely a multi-antenna base station
enabling ISAC, an IOS that facilitates the transmission of ISAC signals in the
reflective and transmissive areas, $K$ multi-antenna users and $O$ sensed
targets. The direct links between BS and users or targets are blocked, and
this is a typical application scenario of the IRS. To mitigate self-
interference and minimize the attenuation effect of IOS on signals, the model
of a bistatic radar is considered. Specifically, receiving antennas are
implemented in both the reflective and the transmissive area to receive echo
signals. The dual function BS is equipped with ${{N}_{t}}$ transmitting
antennas, and the receiving arrays in two areas are quipped with ${{N}_{r}}$
receiving antennas.

Figure 1: IOS enables ISAC networks.

It is assumed that there are ${{K}_{R}}$ multi-antenna users in the coverage
range of the IOS reflective signal, indexed by
$k\in\left\\{1,\cdots,{{K}_{R}}\right\\}$ and ${{K}_{T}}$ multi-antenna users
in the coverage range of the IOS transmissive signal, indexed by
$k\in\left\\{{{K}_{r}}+1,\cdots,{{K}_{R}}+{{K}_{T}}\right\\}$, where
${{K}_{T}}+{{K}_{R}}=K$. The number of receiving antennas of user $k$ is
${{M}_{k}}$. Similarly, targets that need to be sensed are also located in the
reflective and transmissive areas. There are ${{O}_{R}}$ targets located in
the reflection area, indexed by $i\in\left\\{1,2,\cdots,{{O}_{R}}\right\\}$
and ${{O}_{T}}$ targets located in the transmissive area, indexed by
$i\in\left\\{{{O}_{R}}+1,{{O}_{R}}+2,\cdots,{{O}_{R}}+{{O}_{T}}\right\\}$,
where ${{O}_{R}}+{{O}_{T}}=O$.

Considering that the communication scenario is downlink, the BS is required to
transmit multi stream data with dimension ${{D}_{k}}$ to user $k$
simultaneously. In addition, in order to ensure the performance of the radar,
additional signal streams need to be introduced. By introducing an additional
sensing stream equal to the number of transmit antennas, the full rank of the
autocorrelation matrix of the transmitted signal can be ensured [32]. The
signal transmitted by the BS can be represented as

$\displaystyle{\bf{x}}=\sum\limits_{k=1}^{K}{{{\bf{W}}_{k}}{{\bf{d}}_{k}}}+\sum\limits_{t=1}^{{N_{T}}}{{{\bf{m}}_{t}}{s_{t}}},$
(1)

where ${{\mathbf{d}}_{k}}\in{{\mathbb{C}}^{{{D}_{k}}\times 1}}$ represents the
multi-stream data sent by the BS to the user $k$. In order to ensure that the
data is well received, the transmitted data needs to meet
${{D}_{k}}\leq\min\left\\{{{N}_{t}},{{M}_{k}}\right\\}$. It is assumed that
the data ${{\mathbf{d}}_{k}}$ between different users is independent and
follows a Gaussian distribution
${{\mathbf{d}}_{k}}\sim\mathcal{C}\mathcal{N}\left(0,{\mathbf{I}_{{{D}_{K}}}}\right)$.
${{\bf{W}}_{k}}\in{{\mathbb{C}}^{{{N}_{T}}\times{{D}_{k}}}}$ represents the
communication beamforming matrix of the user $k$. ${{s}_{t}}$ denotes the
orthogonal base-band radar signal transmitted by the $t$-th antenna and
${{\bf{m}}_{t}}$ represents the beamforming vector of the $t$-th sensing
stream. The power of the transmitted signal can then be expressed as

$\displaystyle{P_{t}}=E\left({\left\|{\bf{x}}\right\|_{2}^{2}}\right)={\rm{tr}}\left({\sum\limits_{k=1}^{K}{{{\bf{W}}_{k}}{\bf{W}}_{k}^{H}}+\sum\limits_{t=1}^{{N_{T}}}{{{\bf{m}}_{t}}{\bf{m}}_{t}^{H}}}\right).$
(2)

###  II-A IOS Model

The IOS system is composed of $N$ elements for ISAC, indexed by
$n\in\left\\{1,\cdots,N\right\\}$ and its reflective and transmissive
coefficient matrices are respectively represented as

$\displaystyle{{\bf{\Theta}}_{R}}={\text{diag}}\left({{{\bf{\theta}}_{R}}}\right)={\text{diag}}\left({\left[{\sqrt{{\alpha_{1,R}}}{e^{j{\beta_{1,R}}}},\sqrt{{\alpha_{2,R}}}{e^{j{\beta_{2,R}}}},\cdots\sqrt{{\alpha_{N,R}}}{e^{j{\beta_{N,R}}}}}\right]}\right),$
(3)
$\displaystyle{{\bf{\Theta}}_{T}}={\text{diag}}\left({{{\bf{\theta}}_{T}}}\right)={\text{diag}}\left({\left[{\sqrt{{\alpha_{1,T}}}{e^{j{\beta_{1,T}}}},\sqrt{{\alpha_{1,T}}}{e^{j{\beta_{2,T}}}},\cdots\sqrt{{\alpha_{1,T}}}{e^{j{\beta_{N,T}}}}}\right]}\right),$
(4)

where $\sqrt{{{\alpha}_{n,R}}}{{e}^{j{{\beta}_{n,R}}}}$ and
$\sqrt{{{\alpha}_{n,R}}}{{e}^{j{{\beta}_{n,R}}}}$ represent the coefficients
of reflective and transmissive for the $n$-th element respectively, and
${{\alpha}_{n,R}},{{\alpha}_{n,T}}\in\left[0,1\right],{{e}^{j{{\beta}_{n,R}}}},{{e}^{j{{\beta}_{n,T}}}}\in\left[0,2\pi\right)$.
According to power constraints, the amplitude of the coefficients must satisfy
the following condition

$\displaystyle{{\alpha}_{n,R}}+{{\alpha}_{n,T}}=1,\forall n.$ (5)

There are three common protocols utilized in IOS-assisted wireless
communication: energy splitting (ES), mode switching (MS), and time switching
(TS). In the upcoming section, we will initially develop and address the
problem based on the ES protocol. Subsequently, in sections III-$E$ and
III-$F$, we will explore the modifications introduced by the MS and TS
protocols, respectively, along with their respective solutions.

(1) ES: In the ES protocol, each IOS element operates in both reflection and
transmission modes, dividing the incident energy into reflective and
transmissive signals, with an energy ratio denoted as
${{\alpha}_{n,R}}:{{\alpha}_{n,T}}$, where
${{\alpha}_{n,R}},{{\alpha}_{n,T}}\in\left[0,1\right],\forall n$. The ES
protocol offers flexibility in design but introduces significant information
exchange overhead.

(2) MS: In the MS protocol, each IOS element of IOS operates either in
reflective mode or transmissive mode, but not both simultaneously. The $N$ IOS
elements are divided into two groups: ${{N}_{R}}$ elements operate in the
reflective mode and ${{N}_{T}}$ elements operate in the transmissive mode. It
is important to note that ${{N}_{R}}+{{N}_{T}}=N$ and
${{\alpha}_{n,R}},{{\alpha}_{n,T}}\in\left\\{0,1\right\\},\forall n$. Compared
with the ES protocol, the MS protocol cannot achieve the same beam gain, but
it is relatively easier to implement.

(3) TS: In the TS protocol, all elements of the IOS work in the same mode
periodically. The time proportion of IOS elements working in the transmission
mode is denoted as $0\leq{{\lambda}_{R}}\leq 1$, while the time proportion in
the reflection mode is denoted as $0\leq{{\lambda}_{T}}\leq 1$, where
${{\lambda}_{R}}+{{\lambda}_{T}}=1$. Under this protocol, the coefficient
design for each IOS element is relatively simple as the transmission and
reflection coefficients are not coupled. However, precise synchronization in
the time domain is essential for accurate implementation of this protocol.

###  II-B Channel Model

To represent the downlink model, we adopt the Rician channel modeling,
assuming all channels as quasi-static fading. The channel from the BS to the
IOS can be represented as

$\displaystyle{{\bf{H}}_{{\rm{RB}}}}=\sqrt{\beta{{\left({\frac{{{d_{{\rm{RB}}}}}}{{{d_{0}}}}}\right)}^{-\alpha}}}\left({\sqrt{\frac{\kappa}{{\kappa+1}}}{{\bf{H}}_{{\rm{RB}},{\rm{LoS}}}}+\sqrt{\frac{1}{{\kappa+1}}}{{\bf{H}}_{{\rm{RB}},{\rm{NLoS}}}}}\right),$
(6)

where $\alpha$ represents the path loss exponent, $\beta$ represents the
channel loss exponent when the reference distance ${{d}_{0}}=1$ meter (m), and
${{d}_{\text{RB}}}$ represents the distance between BS and IOS.
${{\mathbf{H}}_{\text{RB,NLoS}}}\sim\mathcal{C}\mathcal{N}\left(0,{{\mathbf{I}}_{N\times{{N}_{t}}}}\right)$
and ${{\bf{H}}_{{\rm{RB,LoS}}}}$ represent the LoS component and NLoS
component respectively. Here, we assume that the IOS is a uniform planar
array, i.e., $N={{N}_{x}}\times{{N}_{z}}$, ${{N}_{x}}$ and ${{N}_{z}}$
represent the number of elements in the horizontal and vertical directions of
the IOS respectively. The transmitting antenna at the BS is a uniform linear
array, and ${{\mathbf{H}}_{\text{RB,LoS}}}$ can be expressed as

$\displaystyle{{\mathbf{H}}_{\text{RB,LoS}}}=\mathbf{u}\left({{\theta}_{\text{RB,h}}},{{\theta}_{\text{RB,v}}}\right)\cdot{{\mathbf{l}}^{T}}\left({{\theta}_{\text{RB,h}}}\right)\in{{\mathbb{C}}^{N\times{{N}_{t}}}},$
(7)

where
${\bf{l}}\left({{\theta_{{\rm{RB,h}}}}}\right)={\bf{a}}\left({{\varphi_{{\rm{BS}}}},{N_{t}}}\right)$
denotes the transmit vector from the BS to IOS,
$\mathbf{u}\left({{\theta}_{\text{RB,h}}},{{\theta}_{\text{RB,v}}}\right)=\mathbf{a}\left({{\varphi}_{\text{IOS,h}}},{{N}_{x}}\right)\otimes\mathbf{a}\left({{\varphi}_{\text{IOS,v}}},{{N}_{z}}\right)$
denotes the receive vector from the BS to IOS,
${\bf{a}}\left({\tilde{\theta},\tilde{N}}\right)$ represents the steering
vector where $\tilde{\theta}$ represents the phase difference between two
adjacent elements or antennas, and $\tilde{N}$ represents the number of
antennas transmitted or received, which can be expressed as

$\displaystyle\mathbf{a}\left(\tilde{\theta},\tilde{N}\right)={{\left[1,{{e}^{j2\pi\tilde{\theta}}},\cdots{{e}^{j2\pi\left(\tilde{N}-1\right)\tilde{\theta}}}\right]}^{T}}\in{{\mathbb{C}}^{\tilde{N}\times
1}}.$ (8)

and
${{\varphi}_{\text{RB,h}}}={{{f}_{c}}{{d}_{\text{IOS,x}}}\sin\left({{\theta}_{\text{RB,h}}}\right)\sin\left({{\theta}_{\text{RB,v}}}\right)}/{c}$,
${{\varphi}_{\text{RB,v}}}={{{f}_{c}}{{d}_{\text{IOS,z}}}\sin\left({{\theta}_{\text{RB,h}}}\right)\cos\left({{\theta}_{\text{RB,v}}}\right)}/{c}$
and
${{\varphi}_{\text{BS}}}={{{f}_{c}}{{d}_{\text{BS}}}\sin\left({{\theta}_{\text{RB,h}}}\right)}/{c}$.
${{d}_{\text{{IOS,x}}}}$ and ${{d}_{\text{IOS,z}}}$ represent horizontal and
vertical distances of IOS unit respectively. ${{\theta}_{\text{{RB,h}}}}$ and
${{\theta}_{\text{RB,v}}}$ represent respectively the physical azimuth and
vertical angles-of-arrival (AoA) at the IOS respectively. ${{d}_{\text{BS}}}$
represents the distance between adjacent transmitting antennas, ${{f}_{c}}$
represents carrier frequency. Similarly, the channel from IOS to the $k$-th
user can be represented as

$\displaystyle{{\mathbf{H}}_{k,\text{UR}}}=\sqrt{\beta{{\left(\frac{{{d}_{k\text{,UR}}}}{{{d}_{0}}}\right)}^{-\alpha}}}\left(\sqrt{\frac{\kappa}{\kappa+1}}{{\mathbf{H}}_{k\text{,UR,LoS}}}+\sqrt{\frac{1}{\kappa+1}}{{\mathbf{H}}_{k\text{,UR,NLoS}}}\right),\forall
k.$ (9)

The definition of physical quantities in Eq. (9) is similar to that in Eq. (6)

###  II-C Communication Model

For the $k$-th user, the received signal can be represented as

$\begin{split}{{\mathbf{y}}_{k}}=\left({{\mathbf{H}}_{k\text{,UR}}}{{\mathbf{\Theta}}_{{{S}_{K}}\left(k\right)}}{{\mathbf{H}}_{\text{RB}}}\right)\mathbf{x}+{{\mathbf{n}}_{k}},\forall
k,\end{split}$ (10)

where ${\mathbf{y}_{k}}\in{{\mathbb{C}}^{{{M}_{k}}\times 1}}$ represents the
signal received by the $k$-th user,
$\mathbf{x}\in{{\mathbb{C}}^{{{N}_{t}}\times 1}}$ represents the signal
transmitted by the dual function BS, ${\mathbf{n}_{k}}$ represents additive
white Gaussian noise (AWGN) introduced at the $k$-th user’s receiving antenna
and
${\mathbf{n}_{k}}\sim\mathcal{C}\mathcal{N}\left(0,\sigma_{n}^{2}{\mathbf{I}_{{{M}_{k}}}}\right)$.
${{S}_{K}}\left(k\right)$ is a character function that indicates the area
where the $k$-th user is located, which can be expressed as

${{S}_{K}}\left(k\right)=\left\\{\begin{aligned} &R,1\leq k\leq{{K}_{R}},\\\
&T,{{K}_{R}}+1\leq k\leq{{K}_{R}}+{{K}_{T}},\\\ \end{aligned}\right.$ (11)

For MIMO communications, the achievable rate (bps/Hz) of $k$-th user can then
be expressed as

$\displaystyle{{R}_{k}}=\log\det\left({{\mathbf{I}}_{{{D}_{k}}}}+\mathbf{W}_{k}^{H}\mathbf{H}_{k}^{H}\mathbf{J}_{k}^{-1}{{\mathbf{H}}_{k}}{{\mathbf{W}}_{k}}\right),\forall
k,$ (12)

where
${{\mathbf{H}}_{k}}={{\mathbf{H}}_{k\text{,UR}}}{{\mathbf{\Theta}}_{{{S}_{K}}\left(k\right)}}{{\mathbf{H}}_{\text{RB}}}$
and
${{\mathbf{J}}_{k}}=\sigma_{k}^{2}{{\mathbf{I}}_{{{M}_{k}}}}+\sum\limits_{l=1,l\neq
k}^{K}{{{\mathbf{H}}_{k}}{{\mathbf{W}}_{l}}\mathbf{W}_{l}^{H}\mathbf{H}_{k}^{H}}+\sum\limits_{t=1}^{{{N}_{T}}}{{{\mathbf{H}}_{k}}{{\mathbf{m}}_{t}}\mathbf{m}_{t}^{H}\mathbf{H}_{k}^{H}}$.
The communications quality-of-service (QoS) requirements for all users must be
satisfied, meaning that the achievable rate for each user ${{R}_{k}}$ should
be greater than or equal to a certain threshold, ${{R}_{\text{th}}}$, i.e.,
${{R}_{k}}\geq{{R}_{\text{th}}},\forall k$.

###  II-D Sensing Model

When the receiving arrays process the echo signal of the $i$-th target, the
echo signals of the other targets can be considered as interference. Thus, the
received signal can be represented as

$\begin{split}&{{\mathbf{p}}_{i}}={{\alpha}_{i}}\mathbf{A}\left({{\theta}_{i,\text{OR,h}}},{{\theta}_{i,\text{OR,v}}},{{\theta}_{i,\text{RO,h}}}\right){{\mathbf{\Theta}}_{{{S}_{o}}\left(i\right)}}{{\mathbf{H}}_{\text{RB}}}\mathbf{x}\\\
&+\sum\limits_{o=1+\delta\left({i-{O_{R}}}\right)\cdot{O_{R}},o\neq
i}^{{{O}_{R}+\delta\left({i-{O_{R}}}\right)\cdot{O_{T}}}}{{{\alpha}_{o}}\mathbf{A}\left({{\theta}_{o,\text{OR,h}}},{{\theta}_{o,\text{OR,v}}},{{\theta}_{o,\text{RO,h}}}\right){{\mathbf{\Theta}}_{{{S}_{o}}\left(i\right)}}{{\mathbf{H}}_{\text{RB}}}\mathbf{x}}+{\mathbf{z}_{i}},\forall
i,\end{split}$ (13)

where
$\mathbf{A}\left({{\theta}_{i,\text{OR,h}}},{{\theta}_{i,\text{OR,v}}},{{\theta}_{i,\text{RO,h}}}\right)=\mathbf{l}\left({{\theta}_{i,\text{RO,h}}}\right)\cdot{{\mathbf{u}}^{T}}\left({{\theta}_{i,\text{OR,h}}},{{\theta}_{i,\text{OR,v}}}\right)\in{{\mathbb{C}}^{{{N}_{r}}\times{{N}_{t}}}}$,
${{\theta}_{i,\text{OR,h}}}$ and ${{\theta}_{i,\text{OR,v}}}$ denote the
azimuth and vertical AoAs from the IOS to target, ${{\theta}_{i,\text{RO,h}}}$
denote the azimuth angles-of-departure (AoDs) from the targe to receiving
antenna. ${\mathbf{z}_{i}}$ represents Gaussian white noise at the receiving
antenna and
${{\bf{z}}_{i}}\sim\mathcal{C}\mathcal{N}\left({0,\sigma_{i}^{2}{{\bf{I}}_{{N_{r}}}}}\right)$.
$\delta\left(x\right)=1,x>0;\delta\left(x\right)=0,x\leq 0$.
${{S}_{O}}\left(i\right)$ is a character function that indicates the area
where the $i$-th target is located, which can be expressed as

${{S}_{O}}\left(i\right)=\left\\{\begin{aligned} &R,1\leq i\leq{{O}_{R}},\\\
&T,{{O}_{R}}+1\leq i\leq{{O}_{R}}+{{O}_{T}},\\\ \end{aligned}\right.$ (14)

The BS utilizes multiple antennas to achieve receiving beamforming. We use
${\bf{g}_{i}}$ to represent the filter when receiving the echo from the $i$-th
target and the SINR of the echo signal of the $i$-th target can be expressed
as

$\displaystyle{\text{SIN}}{{\text{R}}_{i}}=\frac{{\alpha_{i}^{2}{{\left|{{\bf{g}}_{i}^{H}{{\bf{S}}_{i}}{\bf{x}}}\right|}^{2}}}}{{{\bf{g}}_{i}^{H}\left({\sum\limits_{o=1+\delta\left({i-{O_{R}}}\right)\cdot{O_{R}},o\neq
i}^{{{O}_{R}+\delta\left({i-{O_{R}}}\right)\cdot{O_{T}}}}{\alpha_{o}^{2}{{\bf{S}}_{o}}{\bf{x}}{{\bf{x}}^{H}}{\bf{S}}_{o}^{H}}+\sigma_{i}^{2}{\bf{I}}_{{{N}_{r}}}}\right){{\bf{g}}_{i}}}},\forall
i.$ (15)

where
${{\mathbf{S}}_{i}}=\mathbf{A}\left({{\theta}_{i,\text{OR,h}}},{{\theta}_{i,\text{OR,v}}},{{\theta}_{i,\text{RO,h}}}\right){{\mathbf{\Theta}}_{{{S}_{o}}\left(i\right)}}{{\mathbf{H}}_{\text{RB}}}$
and it represents the channel from the BS to the radar receive array for the
echo signal from the $i$-th target.

###  II-E Problem Formulation

In multi-target detection, it is necessary to ensure the echo signal of all
targets can be effectively detected. Because the detection performance of the
radar is correlated with the SINR of the perceived signal, the goal of the
system design is to maximize the SINR of the multi-target. Therefore, the
design problem of the IOS-assisted ISAC system can be expressed as

$\displaystyle\text{(P0)}:\quad$
$\displaystyle\underset{\begin{smallmatrix}{{\mathbf{W}}_{k}},{{\mathbf{m}}_{t}},{{\mathbf{g}}_{i}}\\\
{{\theta}_{R}},{{\theta}_{T}}\end{smallmatrix}}{\mathop{\max}}\,\underset{i}{\mathop{\min}}\quad\text{SIN}{{\text{R}}_{i}},$
(16a) $\displaystyle\rm{s.t.}\quad$
$\displaystyle\text{tr}\left(\sum\limits_{k=1}^{K}{{{\mathbf{W}}_{k}}\mathbf{W}_{k}^{H}}+\sum\limits_{t=1}^{{{N}_{T}}}{{{\mathbf{m}}_{t}}\mathbf{m}_{t}^{H}}\right)\leq{{P}_{\max}},$
(16b) $\displaystyle{{R}_{k}}\geq{{R}_{\text{th}}},\forall k,$ (16c)
$\displaystyle{{\alpha}_{n,R}},{{\alpha}_{n,T}}\in\left[0,1\right],{{\alpha}_{n,R}}+{{\alpha}_{n,T}}=1,\forall
n,$ (16d)
$\displaystyle{{\beta}_{n,R}},{{\beta}_{n,T}}\in\left[0,2\pi\right),\forall
n.$ (16e)

where $P_{\max}$ indicates the maximum transmission power of the BS. (16b)
denotes the BS transmit power limitation. (16c) guarantees the performance of
multi -user multi-stream communications. (16d) and (16e) represent the
constraints on the design of IOS amplitude and phase coefficients
respectively. Because the objective function and constraints in problem (P0)
are non-convex and the variables therein are highly coupled, it is difficult
to directly solve the primal problem and we propose a joint active and passive
beamforming design algorithm in the following.

##  III Joint Active And Passive Beamforming Design For IOS-Assisted ISAC
Network

We divide the problem (P0) into three sub-problems to solve based on the BCD
algorithm. Specifically, the sub-problem 1 is to optimize radar receive
beamforming vectors ${\mathbf{g}_{i}}$ for fixed IOS reflective and
transmissive coefficient ${{\mathbf{\theta}}_{R}},{{\mathbf{\theta}}_{T}}$ and
beamforming matrices ${\mathbf{W}_{k}},{\mathbf{m}_{t}}$ . In the sub-problem
2, the radar receive beamforming vectors ${\mathbf{g}_{i}}$ and IOS reflective
and transmissive coefficient ${{\mathbf{\theta}}_{R}},{{\mathbf{\theta}}_{T}}$
are fixed, beamforming matrices ${\mathbf{w}_{k}},{\mathbf{m}_{t}}$ are
jointly optimized. In the sub-problem 3, the radar receive beamforming vectors
${\mathbf{g}_{i}}$ and beamforming matrices
${\mathbf{w}_{k}},{\mathbf{m}_{t}}$ are given and IOS reflective and
transmissive coefficient ${{\mathbf{\theta}}_{R}},{{\mathbf{\theta}}_{T}}$ are
jointly optimized. Finally, the three sub-problems are optimized alternately
until convergence is achieved.

###  III-A Sub-Problem 1: Radar Receive Vectors Design

The optimal design for ${\mathbf{g}_{i}}$ should maximize the SINR of the echo
signal of the $i$-th target. Without losing generality, the received beam
energy is normalized here, and the sub-problem 1 can be expressed as

$\displaystyle\text{(P1)}:\quad$
$\displaystyle\mathop{\max}\limits_{{{\bf{g}}_{i}}}{\rm{}}\frac{{\alpha_{i}^{2}{\bf{g}}_{i}^{H}{{\bf{S}}_{i}}{\bf{x}}{{\bf{x}}^{H}}{\bf{S}}_{i}^{H}{{\bf{g}}_{i}}}}{{{\bf{g}}_{i}^{H}\left({\sum\limits_{o=1+\delta\left({i-{O_{R}}}\right)\cdot{O_{R}},o\neq
i}^{{{O}_{R}+\delta\left({i-{O_{R}}}\right)\cdot{O_{T}}}}{\alpha_{o}^{2}{{\bf{S}}_{o}}{\bf{x}}{{\bf{x}}^{H}}{\bf{S}}_{o}^{H}}+\sigma_{i}^{2}{\mathbf{I}_{{{N}_{r}}}}}\right){{\bf{g}}_{i}}}},\forall
i,$ (17a) $\displaystyle\rm{s.t.}\quad$
$\displaystyle{{\bf{g}}_{i}}{\bf{g}}_{i}^{H}=1,\forall i.$ (17b)

By defining
${{\mathbf{Q}}_{i}}={{\mathbf{S}}_{i}}\mathbf{x}{{\mathbf{x}}^{H}}\mathbf{S}_{i}^{H}$
and
${{\mathbf{T}}_{i}}=\sum\limits_{o=1+\delta\left({i-{O_{R}}}\right)\cdot{O_{R}},o\neq
i}^{{{O}_{R}+\delta\left({i-{O_{R}}}\right)\cdot{O_{T}}}}{\alpha_{o}^{2}{{\mathbf{S}}_{o}}\mathbf{x}{{\mathbf{x}}^{H}}\mathbf{S}_{o}^{H}}+\sigma_{i}^{2}{\mathbf{I}_{{{N}_{r}}}}$,
the optimal ${\mathbf{g}_{i}}$ can be expressed as

$\displaystyle\mathbf{g}_{i}^{*}=\underset{{{\mathbf{g}}_{i}}}{\mathop{\arg\max}}\,\frac{\mathbf{g}_{i}^{H}{{\mathbf{Q}}_{i}}{{\mathbf{g}}_{i}}}{\mathbf{g}_{i}^{H}{{\mathbf{T}}_{i}}{{\mathbf{g}}_{i}}},\forall
i.$ (18)

It’s obvious that ${{\mathbf{Q}}_{i}}$ is a Hermite matrix, and
${{\mathbf{T}}_{i}}$ is a positive definite Hermite matrix. Based on
eigenvalue decomposition, ${{\mathbf{T}}_{i}}$ can be represented as

$\displaystyle{{\bf{T}}_{i}}={{\bf{U}}_{i}}{\text{diag}}\left({{\lambda_{i,1}},{\lambda_{i,2}},\cdots{\lambda_{i,{N_{r}}}}}\right){\bf{U}}_{i}^{H}={\left({{{\bf{U}}_{i}}{\text{diag}}\left({\sqrt{{\lambda_{i,1}}},\sqrt{{\lambda_{i,2}}},\cdots\sqrt{{\lambda_{i,{N_{r}}}}}}\right){\bf{U}}_{i}^{H}}\right)^{2}},\forall
i,$ (19)

where ${{\mathbf{U}}_{i}}$ is a unitary matrix,
${{\lambda}_{i,1}},{{\lambda}_{i,2}},\cdots{{\lambda}_{i,{N_{r}}}}$ are
${N_{r}}$ positive eigenvalues of ${{\mathbf{U}}_{i}}$. By defining
${{\mathbf{B}}_{i}}=\left({{\mathbf{U}}_{i}}\text{diag}\left(\sqrt{{{\lambda}_{i,1}}},\sqrt{{{\lambda}_{i,2}}},\cdots\sqrt{{{\lambda}_{{i,{N}_{r}}}}}\right)\mathbf{U}_{i}^{H}\right)$,
which is also a definite Hermite matrix, its inverse matrix is still a Hermite
matrix. Then, by defining
${{\mathbf{f}}_{i}}={{\mathbf{B}}_{i}}{{\mathbf{g}}_{i}}$, the problem of
finding optimal ${{\mathbf{g}}_{i}}$ can be equivalent to finding optimal
${{\mathbf{f}}_{i}}$ and the problem (P1) can be represent as

$\displaystyle\text{(P1.1)}:\qquad$
$\displaystyle\underset{{{\mathbf{f}}_{i}}}{\mathop{\max}}\,\text{
}\mathbf{f}_{i}^{H}\mathbf{B}_{i}^{-1}{{\mathbf{Q}}_{i}}\mathbf{B}_{i}^{-1}{{\mathbf{f}}_{i}},\forall
i,$ (20a) $\displaystyle\rm{s.t.}\qquad$
$\displaystyle\mathbf{f}_{i}^{H}{{\mathbf{f}}_{i}}={{r}_{i}},\forall i,$ (20b)

where ${{r}_{i}}$ is a constant whose value does not affect the solution of
the problem. By constructing Lagrange functions
$L\left({{\mathbf{f}}_{i}},\lambda\right)=\mathbf{f}_{i}^{H}\mathbf{B}_{i}^{-1}{{\mathbf{Q}}_{i}}\mathbf{B}_{i}^{-1}{{\mathbf{f}}_{i}}+\lambda\left(\mathbf{f}_{i}^{H}{{\mathbf{f}}_{i}}-{{r}_{i}}\right)$,
the optimal ${{\mathbf{f}}_{i}}$ can be obtained by calculating the stagnation
point method, i.e., ${\partial
L\left({{\mathbf{f}}_{i}},\lambda\right)}/{\partial{{\mathbf{f}}_{i}}}\;=0\Rightarrow\mathbf{B}_{i}^{-1}{{\mathbf{Q}}_{i}}\mathbf{B}_{i}^{-1}{{\mathbf{f}}_{i}}=\lambda{{\mathbf{f}}_{i}}$,
which means the optimal ${{\mathbf{f}}_{i}}$ is the eigenvector corresponding
to the maximum eigenvalue of
${\bf{B}}_{i}^{-1}{{\bf{Q}}_{i}}{\bf{B}}_{i}^{-1}$ and
$\mathbf{g}_{i}^{*}=\mathbf{B}_{i}^{-1}\mathbf{f}_{i}^{*}$. Therefore the
optimal minimum SINR of multi-target in the $m$-th after solving problem (P1)
can be expressed as

$\displaystyle
q_{m,1}^{*}=\underset{i}{\mathop{\min}}\,\frac{{{\left(\mathbf{g}_{i}^{*}\right)}^{H}}{{\mathbf{Q}}_{i}}\mathbf{g}_{i}^{*}}{{{\left(\mathbf{g}_{i}^{*}\right)}^{H}}{{\mathbf{T}}_{i}}\mathbf{g}_{i}^{*}}.$
(21)

###  III-B Sub-Problem 2: Beamforming Design of Transmitting Antenna

In this subsection, given the radar receive beamforming vectors
${{\mathbf{g}}_{i}}$ and IOS reflective and transmissive coefficient
${{\mathbf{\theta}}_{R}},{{\mathbf{\theta}}_{T}}$, beamforming matrices
${{\mathbf{W}}_{k}}$ and ${{\mathbf{m}}_{t}}$ are jointly optimized. then the
problem (P0) can be written as the problem (P2), which is expressed as

$\displaystyle\text{(P2)}:\quad$
$\displaystyle\underset{\begin{smallmatrix}{{\mathbf{W}}_{k}},{{\mathbf{m}}_{t}}\\\
\end{smallmatrix}}{\mathop{\max}}\,\underset{i}{\mathop{\min}}\,\text{
SIN}{{\text{R}}_{i}},$ (22a) $\displaystyle\rm{s.t.}\quad$
$\displaystyle\text{
tr}\left(\sum\limits_{k=1}^{K}{{{\mathbf{W}}_{k}}\mathbf{W}_{k}^{H}}+\sum\limits_{t=1}^{{{N}_{T}}}{{{\mathbf{m}}_{t}}\mathbf{m}_{t}^{H}}\right)\leq{{P}_{\max}},$
(22b) $\displaystyle{{R}_{k}}\geq{{R}_{\text{th}}},\forall k.$ (22c)

Considering that problem (P2) is an optimization problem with respect to
${{\mathbf{W}}_{k}},{{\mathbf{m}}_{t}}$ and
${{\mathbf{W}}_{k}},{{\mathbf{m}}_{t}}$ in the objective function appears in
quadratic form, we use SDR to solve. Specifically, by defining
${{\mathbf{\tilde{W}}}_{k}}={{\mathbf{W}}_{k}}\mathbf{W}_{k}^{H}$ and
${{\mathbf{\tilde{M}}}_{t}}={{\mathbf{m}}_{t}}\mathbf{m}_{t}^{H}$, quadratic
optimization problems can be transformed into linear problems. To ensure that
the solved ${{\mathbf{W}}_{k}}$ and ${{\mathbf{m}}_{t}}$ can be decomposed
into the product of the corresponding dimensional beamforming matrix,
additional rank constraints need to be introduced, i.e.,
$\text{rank}\left({{\mathbf{\tilde{W}}}_{k}}\right)\leq{{D}_{k}}$ and
$\text{rank}\left({{\mathbf{\tilde{M}}}_{t}}\right)=1$. Because the objective
function is in the max-min form and includes fractional programming,
${{q}_{m,x}}$ is introduced as an auxiliary variable, which represents the
minimum SINR of multi-target when solving problem (P$x$) in the $m$-th
iteration and problem (P2) can be represented as

$\displaystyle\text{(P2.1)}:\quad$
$\displaystyle\underset{{{{{\bf{\tilde{W}}}}_{k}}},{{{{\bf{\tilde{M}}}}_{t}}},{{q}_{m,2.1}}}{\mathop{\max}}\,{{q}_{m,2.1}},$
(23a) $\displaystyle\rm{s.t.}\quad$
$\displaystyle{\text{tr}}\left(\sum\limits_{k=1}^{K}{{{{\mathbf{\tilde{W}}}}_{k}}}+\sum\limits_{t=1}^{{{N}_{T}}}{{{{\mathbf{\tilde{M}}}}_{t}}}\right)\leq{{P}_{\max}},$
(23b) $\displaystyle\text{ }{{R}_{k}}\geq{{R}_{\text{th}}},\forall k,$ (23c)
$\displaystyle\text{
tr}\left(\left({{{\tilde{\mathbf{S}}}}_{i,i}}-{{q}_{m,2.1}}\sum\limits_{o=1+\delta\left({i-{O_{R}}}\right)\cdot{O_{R}},o\neq
i}^{{{O}_{R}+\delta\left({i-{O_{R}}}\right)\cdot{O_{T}}}}{{{{\tilde{\mathbf{S}}}}_{i,o}}}\right)\left(\sum\limits_{k=1}^{K}{{{{\mathbf{\tilde{W}}}}_{k}}}+\sum\limits_{t=1}^{{{N}_{T}}}{{{{\mathbf{\tilde{M}}}}_{t}}}\right)\right)\geq{{q}_{m,2.1}}{{c}_{i}},\forall
i,$ (23d) $\displaystyle\text{
rank}\left({{{\mathbf{\tilde{W}}}}_{k}}\right)\leq{{D}_{k}},\forall k,$ (23e)
$\displaystyle\text{ rank}\left({{{\mathbf{\tilde{M}}}}_{t}}\right)=1,\forall
t,$ (23f)

where
${{\mathbf{\tilde{S}}}_{i,i}}=\alpha_{i}^{2}\mathbf{S}_{i}^{H}{\mathbf{g}_{i}}\mathbf{g}_{i}^{H}{{\mathbf{S}}_{i}},{{\mathbf{\tilde{\mathbf{S}}}}_{i,o}}=\alpha_{o}^{2}\mathbf{S}_{o}^{H}{\mathbf{g}_{i}}\mathbf{g}_{i}^{H}{{\mathbf{S}}_{o}},{{c}_{i}}=\sigma_{i}^{2}\text{tr}\left({{\mathbf{g}}_{i}}\mathbf{g}_{i}^{H}{{\mathbf{I}}_{{{N}_{r}}}}\right)$.
Because constraint (23d) is joint nonconvex with respect to
${{\mathbf{\tilde{W}}}_{k}},{{\mathbf{\tilde{M}}}_{t}},{{q}_{m,2.1}}$, we
approximate ${{q}_{m,2.1}}$ on the left side of the inequality with a fixed
point $q_{m,1}^{*}$, which can be calculated according to Eq. (21) [33]. The
problem (P2.1) can be approximated as problem (P2.2) as follows

$\displaystyle\text{(P2.2)}:\quad$
$\displaystyle\underset{{{{\mathbf{\tilde{W}}}}_{k}},{{{\mathbf{\tilde{M}}}}_{t}},{{q}_{m,2.2}}}{\mathop{\max}}\,\text{
}{{q}_{m,2.2}},$ (24a) $\displaystyle\rm{s.t.}\quad$ $\displaystyle\text{
tr}\left(\left({{{\mathbf{\tilde{S}}}}_{i,i}}-{{q}_{m,1}}\sum\limits_{o=1+\delta\left({i-{O_{R}}}\right)\cdot{O_{R}},o\neq
i}^{{{O}_{R}+\delta\left({i-{O_{R}}}\right)\cdot{O_{T}}}}{{{{\mathbf{\tilde{S}}}}_{i,o}}}\right)\left(\sum\limits_{k=1}^{K}{{{{\mathbf{\tilde{W}}}}_{k}}}+\sum\limits_{t=1}^{{{N}_{T}}}{{{{\mathbf{\tilde{M}}}}_{t}}}\right)\right)\geq{{q}_{m,2.2}}{{c}_{i}},\forall
i,$ (24b)
$\displaystyle\text{(\ref{p2.1}b)},\text{(\ref{p2.1}c)},\text{(\ref{p2.1}e)},\text{(\ref{p2.1}c)},$
(24c)

where constraint (24b) is joint convex with respect to
${{\mathbf{\tilde{W}}}_{k}},{{\mathbf{\tilde{M}}}_{t}},{{q}_{m,2.2}}$.

Proposition 1: After solving problem (P2.2) in the $m$-th iteration,
$\mathbf{\tilde{W}}_{k}^{\left(m,2.2\right)},\mathbf{\tilde{M}}_{t}^{\left(m,2.2\right)}$
can be obtained,
${{q}_{n,2.1}}\left(\mathbf{\tilde{W}}_{k}^{\left(m,2.2\right)},\mathbf{\tilde{M}}_{t}^{\left(m,2.2\right)}\right)\geq{{q}_{m,1}}$
always holds which ensures the convergence of the algorithm.

_Proof_ : The proof is given in Appendix A. $\hfill\blacksquare$

Next, (22c) is nonconvex with respect to
${{\mathbf{\tilde{W}}}_{k}},{{\mathbf{\tilde{M}}}_{t}}$ and needs to be
approximated. According to the properties of the determinant, the achievable
rate of user $k$ can be expressed as

$\displaystyle\log\det\left(\sigma_{k}^{2}{{\mathbf{I}}_{{{M}_{k}}}}+\sum\limits_{k=1}^{K}{{{\mathbf{H}}_{k}}{{{\mathbf{\tilde{W}}}}_{k}}\mathbf{H}_{k}^{H}}+\sum\limits_{t=1}^{{{N}_{T}}}{{{\mathbf{H}}_{k}}{{{\mathbf{\tilde{M}}}}_{t}}\mathbf{H}_{k}^{H}}\right)-\log\det\left({{\mathbf{J}}_{k}}\right),\forall
k,$ (25)

where the first term is already a concave function and the second term is a
concave function. We can apply the first-order Taylor expansion to the convex
function to get its lower bound. Specially,
$\log\det\left({{\mathbf{J}}_{k}}\right)$ can be approximated as follows

$\begin{split}&\log\det\left({{{\bf{J}}_{k}}}\right)\leq\log\det\left({\mathop{{{\bf{J}}_{k}}}\limits^{\left(m\right)}}\right)\\\
&+{\rm{tr}}\left({\mathop{{\bf{J}}_{k}^{-1}}\limits^{\left(m\right)}\left({\sum\limits_{l=1,l\neq
k}^{K}{{{\bf{H}}_{k}}{{{\bf{\tilde{W}}}}_{l}}{\bf{H}}_{k}^{H}}+\sum\limits_{t=1}^{{N_{T}}}{{{\bf{H}}_{k}}{{{\bf{\tilde{M}}}}_{t}}{\bf{H}}_{k}^{H}}}\right)}\right)\\\
&-{\rm{tr}}\left({\mathop{{\bf{J}}_{k}^{-1}}\limits^{\left(m\right)}\left({\sum\limits_{l=1,l\neq
k}^{K}{{{\bf{H}}_{k}}{{\mathop{{\bf{\tilde{W}}}}\limits^{\left(m\right)}}_{l}}{\bf{H}}_{k}^{H}}+\sum\limits_{t=1}^{{N_{T}}}{{{\bf{H}}_{k}}{{\mathop{{\bf{\tilde{M}}}}\limits^{\left(m\right)}}_{t}}{\bf{H}}_{k}^{H}}}\right)}\right)\triangleq{{\left(\log\det\left(\overset{\left(m\right)}{\mathop{{{\mathbf{J}}_{k}}}}\,\right)\right)}^{ub}},\forall
k,\end{split}$ (26)

where
$\left(\overset{\left(m\right)}{\mathop{{{\mathbf{J}}_{k}}}}\,,\overset{\left(m\right)}{\mathop{\mathbf{J}_{k}^{-1}}}\,,\overset{\left(m\right)}{\mathop{{{{\mathbf{\tilde{W}}}}_{k}}}}\,,\overset{\left(m\right)}{\mathop{{{{\mathbf{\tilde{M}}}}_{t}}}}\,\right)$
represents fixed points at the $m$-th iteration. Constraint (22c) can be
represent as

$\log\det\left(\sigma_{k}^{2}{{\mathbf{I}}_{{{M}_{k}}}}+\sum\limits_{k=1}^{K}{{{\mathbf{H}}_{k}}{{{\mathbf{\tilde{W}}}}_{k}}\mathbf{H}_{k}^{H}}+\sum\limits_{t=1}^{{{N}_{T}}}{{{\mathbf{H}}_{k}}{{{\mathbf{\tilde{M}}}}_{t}}\mathbf{H}_{k}^{H}}\right)-{{\left(\log\det\left(\overset{\left(m\right)}{\mathop{{{\mathbf{J}}_{k}}}}\,\right)\right)}^{ub}},\forall
k.$ (27)

Constraint (27) is a convex constraint. Next, because rank constraints (23e)
and (23d) are nonconvex, we first use the following Proposition 2 [34] to
handle the rank-$n$ constraint.

Proposition 2: As for a matrix ${\bf{X}}\in S_{+}^{n}$, the eigenvalues of
which are ${{\lambda}_{1}},{{\lambda}_{2}},\cdots,{{\lambda}_{n}}$ and
$0\leq{{\lambda}_{1}}\leq{{\lambda}_{2}}\leq\cdots{{\lambda}_{n}}$. The
$(r+1)$-th largest eigenvalue ${{\lambda}_{n-r}}$ of matrix ${\bf{X}}$ is no
greater than if and only if
$e{{\bf{I}}_{n-r}}-{{\bf{V}}^{H}}{\bf{X}}{\bf{V}}\geq 0$, where
${{\bf{I}}_{n-r}}$ is the identity matrix with dimension $n-r$ and
${\bf{V}}\in{{\mathbb{C}}^{n\times n-r}}$ are the eigenvectors corresponding
to the $n-r$ smallest eigenvalues of ${\bf{X}}$.

_Proof_ : Assuming that the eigenvalues of ${\bf{X}}$ are sorted in descending
order, $e{{\bf{I}}_{n-r}}-{{\bf{V}}^{H}}{\bf{X}}{\bf{V}}$ is a diagonal matrix
whose diagonal elements are
$\left[e-{{\lambda}_{n-r}},e-{{\lambda}_{n-r-1}},\cdots
e-{{\lambda}_{1}}\right]$.
$e{{\bf{I}}_{n-r}}-{{\bf{V}}^{H}}{\bf{X}}{\bf{V}}$is a semi-positive definite
matrix if and only if $0\leq e-{{\lambda}_{n-r}}\leq e-{{\lambda}_{n-r-1}}\leq
e-{{\lambda}_{1}}$, which indicates ${{\lambda}_{n-r}}$ is no greater than
$e$.

According to Proposition 2, when $e=0$, $\text{rank}\left({\bf{X}}\right)\leq
r$ if and only if $e{{\bf{I}}_{n-r}}-{{\bf{V}}^{H}}{\bf{X}}{\bf{V}}\geq 0$.
Therefore, constraint (23e) can be equivalently represented as
${{e}_{k}}{{\bf{I}}_{{{N}_{t}}-{{D}_{k}}}}-{{\bf{V}}_{k}^{H}}{{\tilde{\bf{W}}}_{k}}{\bf{V}}_{k}\geq
0,\forall k$. However, because it’s impossible to obtain eigenvector of
${{\tilde{\bf{W}}}_{k}}$ before solving problem (P2.2), we satisfy (23e)
approximately through iterative rank minimization methods. Specifically, the
${\bf{V}}_{k}$ obtained in $m-1$-th iteration is used to approximate the in
$m$-th iteration and the problem can be expressed as

$\displaystyle\text{(P2.3)}:\quad$
$\displaystyle\underset{{{{\tilde{\bf{W}}}}_{k}},{{{\tilde{\bf{M}}}}_{t}},{{q}_{m,2.2}},{{e}_{k}}}{\mathop{\max}}\,{{q}_{n,2.2}}-\sum\limits_{k=1}^{K}{{{\gamma}_{k}}{{\left(\gamma\right)}^{m}}{{e}_{k}}},$
(28a) $\displaystyle\rm{s.t.}\quad$
$\displaystyle{{e}_{k}}{{\bf{I}}_{{{N}_{t}}-{{D}_{k}}}}-\overset{\left(m\right)}{\mathop{{{\bf{V}}_{k}^{H}}}}\,{{{\tilde{\bf{W}}}}_{k}}\overset{\left(m\right)}{\mathop{{\bf{V}}_{k}}}\,\geq
0,\forall k,$ (28b)
$\displaystyle{{e}_{k}}\leq\overset{\left(m\right)}{\mathop{{{e}_{k}}}},\forall
k,$ (28c)
$\displaystyle\text{(\ref{p2.1}b)},\text{(\ref{p2.1}f)},\text{(\ref{p2.2}b)},\text{(\ref{rkconvex})},$
(28d)

where ${{\gamma}_{0}}$ represents initial penalty factor,
$\gamma\left(\gamma>1\right)$ represents the exponential explosion factor,
$\overset{\left(m\right)}{\mathop{\bf{V}}_{k}}\,$ represents is eigenvectors
corresponding to the smallest ${{N}_{t}}-{{D}_{k}}$ eigenvalues of given
$\overset{\left(m\right)}{\mathop{{{{\tilde{\bf{W}}}}_{k}}}}\,$. Constraint
(28c) ensures the monotonicity of ${{e}_{k}}$ after each iteration, in other
words, minimize the value of ${{e}_{k}}$ as much as possible through penalty
terms and constraints, so that ${{e}_{k}}$ gradually converges to 0. When
${{e}_{k}}$ approaches 0, that the $n-r+1$-th largest eigenvalue of a also
approaches 0, making
$\text{rank}\left({{{\tilde{\bf{W}}}}_{k}}\right)\leq{{D}_{k}}$ hold.
Constraint (24c) is a rank-one constraint, which can be solved through the
difference of convex algorithm [30] and we omit the specific algorithm here.
After completing the problem (P2.3) solution, ${{\mathbf{\tilde{W}}}_{k}}$ can
be decomposed as

$\displaystyle{{\tilde{\bf{W}}}_{k}}={{\bf{U}}_{k}}{{{\bf{\Sigma}}}_{{{r}_{k}}}}{\bf{U}}_{k}^{H},$
(29)

where ${{r}_{k}}$ is the rank of ${{\tilde{\bf{W}}}_{k}}$,
${{{\bf{\Sigma}}}_{{{r}_{k}}}}\in{{\mathbb{R}}^{{{r}_{k}}\times{{r}_{k}}}}$ is
a diagonal matrix, the diagonal elements of which are ${{r}_{k}}$ positive
eigenvalues of ${{\tilde{W}}_{k}}$ and
${{\bf{U}}_{k}}\in{{\mathbb{R}}^{{{N}_{t}}\times{{r}_{k}}}},{\bf{U}}_{k}^{H}{{\bf{U}}_{k}}={{\bf{I}}_{{{r}_{k}}}}$.
If ${{r}_{k}}={{D}_{k}}$, the beamforming matrix for the multi-stream
communication of user $k$ can be designed as
${{\bf{W}}_{k}}={\bf{U}}{\bf{\Sigma}}_{{{r}_{k}}}^{{1}/{2}\;}$. If
${{r}_{k}}<{{D}_{k}}$, select some column vector from
${\bf{U}}\Sigma_{{{r}_{k}}}^{{1}/{2}\;}$ for splitting in amplitude. The split
column vector replaces the original column vector to expand the number of
matrix columns. The total energy of split vectors should be equal to the
energy of the original vector. For example, if ${{r}_{k}}+1={{D}_{k}}$,
${\bf{U}}{\bf{\Sigma}}_{{{r}_{k}}}^{{1}/{2}\;}=\left[{{v}_{1}},{{v}_{2}},\cdots{{v}_{{{r}_{k}}}}\right]\in{{\mathbb{C}}^{{{N}_{t}}\times{{r}_{k}}}}$,
a feasible design method for ${{\bf{W}}_{k}}$ is
${{\bf{W}}_{k}}=\left[{{v}_{1}},{{v}_{2}},\cdots{{{v}_{{{r}_{k}}}}}/{\sqrt{2}}\;,{{{v}_{{{r}_{k}}}}}/{\sqrt{2}}\;\right]\in{{\mathbb{C}}^{{{N}_{t}}\times{{D}_{k}}}}$.

###  III-C Sub-Problem 3: Design of Reflective and Transmissive Coefficients
for IOS

In this subsection, given the radar receive beamforming vectors
${\mathbf{g}_{i}}$ and, beamforming matrices
${\mathbf{W}_{k}},{\mathbf{m}_{t}}$,
${\mathbf{\theta}_{R}},{\mathbf{\theta}_{T}}$ are jointly optimized. then the
problem (P0) can be written as the problem (P3), which is expressed as

$\displaystyle\text{(P3)}:\quad$
$\displaystyle\underset{{{{\bf{\Theta}}}_{r}},{{{\bf{\Theta}}}_{t}}}{\mathop{\max}}\,\underset{i}{\mathop{\min}}\quad\text{SIN}{{\text{R}}_{i}},$
(30a) $\displaystyle\rm{s.t.}\quad$
$\displaystyle{{R}_{k}}\geq{{R}_{\text{th}}},\forall k,$ (30b)
$\displaystyle\alpha_{n}^{t},\alpha_{n}^{r}\in\left[0,1\right],\alpha_{n}^{t}+\alpha_{n}^{r}=1,\forall
n,$ (30c)
$\displaystyle\beta_{n}^{t},\beta_{n}^{r}\in\left[0,2\pi\right),\forall n.$
(30d)

The problem (P3) is challenging for the following reasons. First, constraint
(30b) is non-convex with respect to
${\mathbf{\theta}_{T}},{\mathbf{\theta}_{R}}$. Then, the traditional IOS
coefficient optimization problem based on MISO scenario, where IOS coefficient
matrix can be expressed as vector through the product exchange of vector and
diagonal matrix. However, in MIMO communication, the expressions for reachable
rates are all matrix products, which makes it difficult to apply the SDR
algorithm to process quadratic forms of IOS coefficient directly. Therefore,
we use the weighted MMSE approach [35] to represent constraint (30b).

Proposition 3: The constraint (30b) to ensure the achievable rate of each user
is equivalent to the weighted MSE constraint when the weight matrix satisfies
${\mathbf{V}_{k}}=\mathbf{E}_{k}^{-1}$, which can be expressed as

$\displaystyle{{f}_{k}}=\text{tr}\left({\mathbf{V}_{k}}{\mathbf{E}_{k}}\right)-\log\det\left({\mathbf{V}_{k}}\right)\leq{{D}_{k}}-{{R}_{\text{th}}},\forall
k,$ (31)

with

$\begin{split}&{\mathbf{E}_{k}}=E\left[\left({\mathbf{d}_{k}}-{{{\hat{\mathbf{d}}}}_{k}}\right){{\left({\mathbf{d}_{k}}-{{{\hat{\mathbf{d}}}}_{k}}\right)}^{H}}\right]\\\
&=E\left[\left(\left({\mathbf{d}_{k}}-{\mathbf{G}_{k}}{\mathbf{H}_{k}}{\mathbf{W}_{k}}{\mathbf{d}_{k}}\right)-{\mathbf{G}_{k}}{\mathbf{H}_{k}}\left(\sum\limits_{i=1,i\neq
k}^{K}{{\mathbf{W}_{i}}{\mathbf{d}_{i}}}+\sum\limits_{t=1}^{{{N}_{T}}}{{\mathbf{m}_{t}}{{s}_{t}}}\right)+{\mathbf{G}_{k}}{\mathbf{n}_{k}}\right)\right.\\\
&\left.{{\left(\left({\mathbf{d}_{k}}-{\mathbf{G}_{k}}{\mathbf{H}_{k}}{\mathbf{W}_{k}}{\mathbf{W}_{k}}\right)-{\mathbf{G}_{k}}{\mathbf{H}_{k}}\left(\sum\limits_{i=1,i\neq
k}^{K}{{\mathbf{W}_{i}}{\mathbf{d}_{i}}}+\sum\limits_{t=1}^{{{N}_{T}}}{{\mathbf{m}_{t}}{{s}_{t}}}\right)+{\mathbf{G}_{k}}{\mathbf{n}_{k}}\right)}^{H}}\right],\forall
k,\end{split}$ (32)

where ${\mathbf{V}_{k}}$ represents the MSE weight matrix and
${\mathbf{E}_{k}}$ is the MSE matrix of user $k$. ${\mathbf{G}_{k}}$
represents the MIMO communication receiving beamforming matrix.

_Proof_ : The proof is given in Appendix B. $\hfill\blacksquare$

However, The entire optimization problem introduces two additional variables
${\mathbf{G}_{k}}$ and ${\mathbf{V}_{k}}$ through weighted MMSE approach.
Therefore, we use alternating optimization methods to update these three
variables. The optimal $\mathbf{G}_{k}^{*}$ and $\mathbf{V}_{k}^{*}$ can be
obtained by calculating $\partial{{{f}_{k}}}/{\partial}\mathbf{G}_{k}^{*}=0$
and $\partial{{{f}_{k}}}/{\partial}\mathbf{V}_{k}^{*}=0$ respectively. i.e.,

$\displaystyle\mathbf{G}_{k}^{*}=\frac{\mathbf{W}_{k}^{H}\mathbf{H}_{k}^{H}}{\sum\limits_{i=1}^{K}{{\mathbf{H}_{k}}{\mathbf{W}_{i}}\mathbf{W}_{i}^{H}\mathbf{H}_{k}^{H}}+\sum\limits_{t=1}^{{{N}_{T}}}{{\mathbf{H}_{k}}{\mathbf{m}_{t}}\mathbf{m}_{t}^{H}\mathbf{H}_{k}^{H}}+\sigma_{k}^{2}\mathbf{I}_{{M}_{k}}},\mathbf{V}_{k}^{*}=\mathbf{E}_{k}^{-1}.$
(33)

Then, we use SDR to handle the quadratic forms of ${{\theta}_{T}}$ and
${{\theta}_{R}}$. Utilizing the properties of trace,
$\text{tr}\left({\mathbf{V}_{k}}{\mathbf{E}_{k}}\right)$ can be expressed as

$\begin{split}&\text{tr}\left({{\mathbf{V}}_{k}}{{\mathbf{E}}_{k}}\right)=\text{tr}\left({{\mathbf{V}}_{k}}+E\left[\sum\limits_{i=1}^{K}{\mathbf{d}_{i}^{H}\mathbf{W}_{i}^{H}\mathbf{H}_{k}^{H}\mathbf{G}_{k}^{H}{{\mathbf{V}}_{k}}{{\mathbf{G}}_{k}}{{\mathbf{H}}_{k}}{{\mathbf{W}}_{i}}{{\mathbf{d}}_{i}}}-\mathbf{d}_{k}^{H}\mathbf{W}_{k}^{H}\mathbf{H}_{k}^{H}\mathbf{G}_{k}^{H}{{\mathbf{V}}_{k}}{{\mathbf{d}}_{k}}\right.\right)\\\
&\left.-\left.\mathbf{d}_{k}^{H}{{\mathbf{V}}_{k}}{{\mathbf{G}}_{k}}{{\mathbf{H}}_{k}}{{\mathbf{W}}_{k}}{{\mathbf{d}}_{k}}+\sum\limits_{t=1}^{{{N}_{T}}}{s_{t}^{H}\mathbf{m}_{t}^{H}\mathbf{H}_{k}^{H}\mathbf{G}_{k}^{H}{{\mathbf{V}}_{k}}{{\mathbf{G}}_{k}}{{\mathbf{H}}_{k}}{\mathbf{m}_{t}}{{s}_{t}}}\right]+\sigma_{k}^{2}{{\mathbf{V}}_{k}}{{\mathbf{G}}_{k}}\mathbf{G}_{k}^{H}\right).\end{split}$
(34)

By applying the change of variables
$\mathbf{Z}_{k,k}^{H}=\text{diag}\left(\tilde{\mathbf{d}}_{k}^{H}\right)\mathbf{H}_{k,\text{UR}}^{H}\mathbf{G}_{k}^{H}$,
$\mathbf{Z}_{k,i}^{H}=\text{diag}\left(\tilde{\mathbf{d}}_{i}^{H}\right)\mathbf{H}_{k,\text{UR}}^{H}\mathbf{G}_{k}^{H}$,
$\tilde{\mathbf{s}}_{t}^{H}=s_{t}^{H}\mathbf{m}_{t}^{H}\mathbf{H}_{\text{RB}}^{H}$
and
$\tilde{\mathbf{d}}_{k}^{H}\mathbf{\Theta}_{{S_{K}}\left(k\right)}^{H}=\mathbf{\theta}_{{S_{K}}\left(k\right)}^{H}\text{diag}\left(\tilde{\mathbf{d}}_{k}^{H}\right)$,
where
$\tilde{\mathbf{d}}_{k}^{H}=\mathbf{d}_{k}^{H}\mathbf{W}_{k}^{H}\mathbf{H}_{\text{RB}}^{H}$
and
$\tilde{\mathbf{s}}_{t}^{H}=s_{t}^{H}\mathbf{m}_{t}^{H}\mathbf{H}_{\text{RB}}^{H}$.
Thus, ${{f}_{k}}$ is equivalent to

$\displaystyle\text{tr}\left({\mathbf{V}_{k}}+\sigma_{k}^{2}{\mathbf{V}_{k}}{\mathbf{G}_{k}}\mathbf{G}_{k}^{H}\right)-\log\det\left({\mathbf{V}_{k}}\right)+\text{tr}\left(\sum\limits_{i=1}^{K}{\mathbf{\theta}_{{S_{K}}\left(k\right)}^{H}\mathbf{Z}_{k,i}^{H}{\mathbf{V}_{k}}{\mathbf{Z}_{k,i}}{{\mathbf{\theta}}_{{S_{K}}\left(k\right)}}}\right.$
(35)
$\displaystyle+\left.\sum\limits_{t=1}^{{{N}_{T}}}{\theta_{{S_{K}}\left(k\right)}^{H}\mathbf{Z}_{k,t}^{H}{\mathbf{V}_{k}}{\mathbf{Z}_{k,t}}}{\mathbf{\theta}_{{S_{K}}\left(k\right)}}-\mathbf{\theta}_{{S_{K}}\left(k\right)}^{H}\mathbf{Z}_{k,k}^{H}{\mathbf{V}_{k}}{\mathbf{d}_{k}}-\mathbf{d}_{k}^{H}{\mathbf{V}_{k}}{\mathbf{Z}_{k,k}}{\mathbf{\theta}_{{S_{K}}\left(k\right)}}\right).$

Eq. (35) is a non-convex quadratic function, which can be reformulated as a
homogeneous quadratic function by introducing an auxiliary variable $t$.
Specifically, Eq. (35) can be represented as
$\text{tr}\left({\mathbf{V}_{k}}+\sigma_{k}^{2}{\mathbf{v}_{k}}{\mathbf{G}_{k}}\mathbf{G}_{k}^{H}+E\left[{\mathbf{A}_{1,k}}\right]{{{\tilde{\mathbf{\Theta}}}}_{k}}\right)$,
where

$\displaystyle{{\tilde{\mathbf{\Theta}}}_{k}}={{\tilde{\mathbf{\theta}}}_{k}}\tilde{\mathbf{\theta}}_{k}^{H},{{\tilde{\mathbf{\theta}}}_{k}}=\left[{{\mathbf{\theta}}_{{S_{K}}\left(k\right)}},t\right],{\mathbf{A}_{1,k}}=\left[\begin{matrix}\sum\limits_{i=1}^{K}{\mathbf{Z}_{k,i}^{H}{\mathbf{V}_{k}}{\mathbf{Z}_{k,i}}}+\sum\limits_{t=1}^{{{N}_{T}}}{\mathbf{Z}_{k,t}^{H}{\mathbf{V}_{k}}{\mathbf{Z}_{k,t}}}&-\mathbf{Z}_{k,k}^{H}{\mathbf{V}_{k}}{\mathbf{d}_{k}}\\\
-\mathbf{d}_{k}^{H}{\mathbf{V}_{k}}{\mathbf{Z}_{k,k}}&0\end{matrix}\right].$
(36)

In addition, the amplitude of the linear term needs to remain constant i.e.,
$\left|t\right|=1$. Similarly, we use SDR to express the performance of radar
system. By defining

$\displaystyle\mathbf{\tilde{g}}_{i,i}^{H}=\mathbf{g}_{i}^{H}\mathbf{A}\left({{\theta}_{i,\text{OR,h}}},{{\theta}_{i,\text{OR,v}}},{{\theta}_{i,\text{RO,h}}}\right),\mathbf{\tilde{g}}_{i,o}^{H}=\mathbf{g}_{i}^{H}\mathbf{A}\left({{\theta}_{o,\text{OR,h}}},{{\theta}_{o,\text{OR,v}}},{{\theta}_{o,\text{RO,h}}}\right),$
(37)
$\displaystyle{{\mathbf{Y}}_{i,i}}=\alpha_{i}^{2}\text{diag}\left(\mathbf{\tilde{g}}_{i,i}^{H}\right){{\mathbf{H}}_{RB}}\left(\sum\limits_{k=1}^{K}{{{{\mathbf{\tilde{W}}}}_{k}}}+\sum\limits_{t=1}^{{{N}_{T}}}{{{{\mathbf{\tilde{M}}}}_{t}}}\right)\mathbf{H}_{RB}^{H}\text{diag}\left({{{\mathbf{\tilde{g}}}}_{i,i}}\right),$
(38) $\displaystyle{{\mathbf{Y}}_{i,o}}=\sum\limits_{o=1,o\neq
i}^{{{O}_{r}}}{\alpha_{o}^{2}\text{diag}\left(\mathbf{\tilde{g}}_{i,o}^{H}\right){{\mathbf{H}}_{RB}}\left(\sum\limits_{k=1}^{K}{{{{\mathbf{\tilde{W}}}}_{k}}}+\sum\limits_{t=1}^{{{N}_{T}}}{{{{\mathbf{\tilde{M}}}}_{t}}}\right)}\mathbf{H}_{RB}^{H}\text{diag}\left({{{\mathbf{\tilde{g}}}}_{i,o}}\right),$
(39)

and leveraging
$\tilde{\mathbf{g}}_{i,i}^{H}\mathbf{\Theta}_{{{S}_{O}}\left(i\right)}^{H}=\mathbf{\theta}_{{{S}_{O}}\left(i\right)}^{H}\text{diag}\left(\mathbf{g}_{i,i}^{H}\right)$,
$\tilde{\mathbf{g}}_{i,o}^{H}\mathbf{\Theta}_{{{S}_{O}}\left(i\right)}^{H}=\mathbf{\theta}_{{{S}_{O}}\left(i\right)}^{H}\text{diag}\left(\mathbf{g}_{i,o}^{H}\right)$,
the SINR of the $i$-th target echo signal can be expressed as

$\displaystyle\text{SIN}{{\text{R}}_{i}}=\frac{\text{tr}\left({\mathbf{Y}_{i,i}}{{\mathbf{\theta}}_{{{S}_{O}}\left(i\right)}}\mathbf{\theta}_{{{S}_{O}}\left(i\right)}^{H}\right)}{\text{tr}\left({\mathbf{Y}_{i,o}}{{\mathbf{\theta}}_{{{S}_{O}}\left(i\right)}}\mathbf{\theta}_{{{S}_{O}}\left(i\right)}^{H}\right)},\forall
i.$ (40)

To ensure that the dimensions of the optimization variables are the same as
those in communication constraints, we define
${{\tilde{\mathbf{\Theta}}}_{i}}$, ${{\tilde{\mathbf{\theta}}}_{i}}$,
${\mathbf{A}_{2,i}}$ and ${\mathbf{A}_{2,i}}$, which are

$\displaystyle{{\tilde{\mathbf{\Theta}}}_{i}}={{\tilde{\mathbf{\theta}}}_{i}}\tilde{\mathbf{\theta}}_{i}^{H},{{\tilde{\mathbf{\theta}}}_{i}}=\left[{{\mathbf{\theta}}_{{S_{O}}\left(i\right)}},t\right],{\mathbf{A}_{2,i}}=\left[\begin{matrix}{\mathbf{Y}_{i,i}}&0\\\
0&0\\\
\end{matrix}\right],{\mathbf{A}_{3,i}}=\left[\begin{matrix}{\mathbf{Y}_{i,o}}&0\\\
0&0\\\ \end{matrix}\right],\forall i.$ (41)

Therefore, $\text{SIN}{{\text{R}}_{i}}$ can be represented as
${\text{tr}\left({\mathbf{A}_{2,i}}{{\tilde{\mathbf{\Theta}}}_{i}}\right)}/{\text{tr}\left({\mathbf{A}_{3,i}}{{\tilde{\mathbf{\Theta}}}_{i}}\right)}$.
Finally, we use the same approximation method as sub-problem 2 to handle max-
min problem in objective function and problem (P3) can be expressed as

$\displaystyle\text{(P3.1)}:\qquad$
$\displaystyle\underset{{{{\tilde{\mathbf{\Theta}}}}_{i}},{{{\tilde{\mathbf{\Theta}}}}_{k}},{{q}_{m,3.1}}}{\mathop{\max}}\,{{q}_{m,3.1}},$
(42a) $\displaystyle\rm{s.t.}\qquad$
$\displaystyle\text{tr}\left({\mathbf{V}_{k}}+\sigma_{k}^{2}{\mathbf{V}_{k}}{\mathbf{G}_{k}}\mathbf{G}_{k}^{H}+E\left[{\mathbf{A}_{1,k}}\right]{{{\tilde{\mathbf{\Theta}}}}_{k}}\right)-\log\det\left({\mathbf{V}_{k}}\right)\leq{{D}_{k}}-{{R}_{\text{th}}},\forall
k,$ (42b)
$\displaystyle\text{tr}\left(\left({\mathbf{A}_{2,i}}-{{q}_{m,2.1}}{\mathbf{A}_{3,i}}\right)\tilde{\Theta}_{i}\right)\geq{{q}_{m,3.1}}{{c}_{i}},\forall
i,$ (42c)
$\displaystyle{{{\tilde{\mathbf{\Theta}}}}_{k}}\left[N+1,N+1\right]={{{\tilde{\mathbf{\Theta}}}}_{i}}\left[N+1,N+1\right]=1,$
(42d) $\displaystyle\text{(\ref{p3}c)},\text{(\ref{p3}d)},$ (42e)

where ${{q}_{m,2.1}}$ denotes the minimum SINR which has been obtained after
solving problem (P2.2) in the $m$-th iteration.
${{\tilde{\mathbf{\Theta}}}_{k}}\left[N+1,N+1\right]$ and
${{\tilde{\mathbf{\Theta}}}_{i}}\left[N+1,N+1\right]$ represent the element in
$N+1$-th row and $N+1$-th column of ${{\tilde{\mathbf{\Theta}}}_{k}}$ and
${{\tilde{\mathbf{\Theta}}}_{i}}$ respectively.
${{q}_{m,1}}\leq{{q}_{m,2.1}}\leq{{q}_{m,3.1}}\leq{{q}_{m+1,1}}$ always hold,
which ensures the convergence of the entire algorithm and will be proved in
Appendix A. After the above equivalence transformation and approximation,
problem (P3.1) is a standard SDP problem and can be solved by using CVX
toolbox.

###  III-D The Overall Joint Active And Passive Beamforming Design For IOS-
Assisted ISAC Network

In this subsection, we propose the overall joint radar receive vectors,
transmit beamforming matrices, and IOS reflective and transmissive
coefficients optimization algorithm and summarize it in Algorithm 1. We can
respectively solve the problem (P1.1), (P2.3) and (P3.1) to obtain radar
receive vectors, transmit beamforming matrices, and IOS reflective and
transmissive coefficients. Finally, the three sub-problems are optimized
alternately until the entire problem converges.

Algorithm 1 Joint Active And Passive Beamforming Design For IOS-Assisted ISAC
Network

1:Input: Initialize feasible points
$\overset{\left(1\right)}{\mathop{{{\mathbf{g}}_{i}}}},\overset{\left(1\right)}{\mathop{{{{\mathbf{\tilde{W}}}}_{k}}}},\overset{\left(1\right)}{\mathop{{{{\mathbf{\tilde{M}}}}_{t}}}},\overset{\left(1\right)}{\mathop{{{\mathbf{\Theta}}_{R}}}},\overset{\left(1\right)}{\mathop{{{\mathbf{\Theta}}_{T}}}}$,
the penalty factor ${{\gamma}_{0}}$ and exponential explosion factor
${\gamma}$.

2:repeat: outer loop

3: Set iteration index $m=1$ for inner loop.

4: repeat: inner loop

5: Solve the problem (P1.1) to obtain radar receive vectors
${{\mathbf{g}}_{i}}$ and update
$\overset{\left(m+1\right)}{\mathop{{{\mathbf{g}}_{i}}}}$.

6: Solve the problem (P2.3) to obtain transmit beamforming matrices
${{\mathbf{\tilde{W}}}_{k}},{{\mathbf{\tilde{M}}}_{t}}$ and updata
$\overset{\left(m+1\right)}{\mathop{{{{\mathbf{\tilde{W}}}}_{k}}}},\overset{\left(m+1\right)}{\mathop{{{{\mathbf{\tilde{M}}}}_{t}}}}$.

7: Solve the problem (P3.1) to obtain IOS reflective and transmissive
coefficients ${{\mathbf{\Theta}}_{R}},{{\mathbf{\Theta}}_{T}}$ and updata
$\overset{\left(m+1\right)}{\mathop{{{\mathbf{\Theta}}_{R}}}},\overset{\left(m+1\right)}{\mathop{{{\mathbf{\Theta}}_{T}}}},m=m+1$.

8: until The fractional decrease of the objective function value is below a
predefined threshold.

9: Updata ${{\gamma}_{0}}={{\gamma}_{0}}\cdot\gamma$ and feasible points
$\overset{\left(1\right)}{\mathop{{{\mathbf{g}}_{i}}}},\overset{\left(1\right)}{\mathop{{{{\mathbf{\tilde{W}}}}_{k}}}},\overset{\left(1\right)}{\mathop{{{{\mathbf{\tilde{M}}}}_{t}}}},\overset{\left(1\right)}{\mathop{{{\mathbf{\Theta}}_{R}}}},\overset{\left(1\right)}{\mathop{{{\mathbf{\Theta}}_{T}}}}$
with current solutions.

10:until The constraint violation is below a predefined threshold.

11:Output:
$\mathbf{g}_{i}^{*},\mathbf{\tilde{W}}_{k}^{*},\mathbf{\tilde{M}}_{t}^{*},\mathbf{\Theta}_{R}^{*},\mathbf{\Theta}_{T}^{*}$.

###  III-E Proposed Solution for MS

Compared with the ES protocol, the algorithm design of MS protocol only
changes the design way of the IOS coefficient without changing that of radar
receive beamforming and transit beamforming. Specifically, the amplitude of
the reflective and transmissive coefficients of each element should be 0 or 1,
which is a discrete binary constraint and can be represented as

$\displaystyle\alpha_{n,R}^{2}-{{\alpha}_{n,R}}=0,\alpha_{n,T}^{2}-{{\alpha}_{n,T}}=0.$
(43)

The coefficient amplitude obtained by solving problem (P2) is between 0 and 1,
leading the value of $\alpha_{n,R}^{2}-{{\alpha}_{n,R}}$ and
$\alpha_{n,T}^{2}-{{\alpha}_{n,T}}$ is always less than 0. To satisfy Eq.
(43), a penalty term can be introduced to make the coefficient amplitude
approaches 0 or 1. Since the penalty term is nonconvex, it needs to be
converted into convex function through first-order Taylor expansion, i.e.,

$\displaystyle\alpha_{n,R}^{2}-{{\alpha}_{n,R}}\geq\left(\overset{\left(m\right)}{\mathop{2{{\alpha}_{n,R}}}}\,-1\right){{\alpha}_{n,R}}-\overset{\left(m\right)}{\mathop{\alpha_{n,R}^{2}}}\,\triangleq\overset{\left(m\right)}{\mathop{{{\gamma}_{n,R}}}},$
(44)
$\displaystyle\alpha_{n,T}^{2}-{{\alpha}_{n,T}}\geq\left(\overset{\left(m\right)}{\mathop{2{{\alpha}_{n,T}}}}\,-1\right){{\alpha}_{n,T}}-\overset{\left(m\right)}{\mathop{\alpha_{n,T}^{2}}}\,\triangleq\overset{\left(m\right)}{\mathop{{{\gamma}_{n,T}}}}.$
(45)

where $\overset{\left(m\right)}{\mathop{{{\alpha}_{n,R}}}}$ and
$\overset{\left(m\right)}{\mathop{{{\alpha}_{n,T}}}}$ are the given point in
the $m$-th iteration. Therefore, the optimization problem can be represented
as

$\displaystyle\text{(P4)}:$
$\displaystyle\underset{{{\tilde{\mathbf{\Theta}}}_{k}},{{\tilde{\mathbf{\Theta}}}_{i}},{{q}_{m,3.1}}}{\mathop{\max}}\,{{q}_{m,3.1}}-{{\nu}_{0}}{{\left(\nu\right)}^{m}}\sum\limits_{n=1}^{N}{\left(\overset{\left(m\right)}{\mathop{{{\gamma}_{n,R}}}}\,+\overset{\left(m\right)}{\mathop{{{\gamma}_{n,T}}}}\,\right)},$
(46) $\displaystyle\rm{s.t.}$
$\displaystyle\quad\text{(\ref{p3}c)},\text{(\ref{p3}d)},\text{(\ref{p3.1}b)},\text{(\ref{p3.1}c)},\text{(\ref{p3.1}d)},$
(47)

where ${{\nu}_{0}}$ represents the initial penalty factor and
$\nu\left(\nu>1\right)$ represents the exponential explosion factor. Then,
problem (P4) is a standard convex optimization problem that can be solved by
using CVX toolbox.

###  III-F Proposed Solution for TS

When applying the TS protocol to the IOS system, the dual-function base
stations serve communication users and sensed targets in reflective and
transmissive areas periodically and The switching period is the same as that
of IOS. The communications volume is linearly affected by transmission time,
i.e., the achievable rate of $k$-th user will reduce to
${{\lambda}_{{{S}_{K}}\left(k\right)}}$ equivalently and can be expressed as

$\displaystyle
R_{k}^{\text{TS}}={{\lambda}_{{{S}_{K}}\left(k\right)}}\log\det\left({{\mathbf{I}}_{{{D}_{k}}}}+\mathbf{W}_{k}^{H}\mathbf{H}_{k}^{H}\mathbf{J}_{k}^{-1}{{\mathbf{H}}_{k}}{{\mathbf{W}}_{k}}\right),\forall
k.$ (48)

Similarly, the signal energy reflected by the perceived target is linearly
related to the accumulation time of the pulse, so the SINR of $i$-th target in
TS protocol can be represented as

$\displaystyle\text{SINR}_{i}^{\text{TS}}=\frac{{{\lambda}_{{{S}_{O}}\left(i\right)}}\alpha_{i}^{2}{{\left|\mathbf{g}_{i}^{H}{{\mathbf{S}}_{i}}\mathbf{x}\right|}^{2}}}{\mathbf{g}_{i}^{H}\left(\sum\limits_{o=1+\delta\left({i-{O_{R}}}\right)\cdot{O_{R}},o\neq
i}^{{{O}_{R}+\delta\left({i-{O_{R}}}\right)\cdot{O_{T}}}}{\alpha_{o}^{2}{{\mathbf{S}}_{o}}\mathbf{x}{{\mathbf{x}}^{H}}\mathbf{S}_{o}^{H}}+\sigma_{i}^{2}{{\mathbf{I}}_{{{N}_{r}}}}\right){{\mathbf{g}}_{i}}},\forall
i,$ (49)

where ${{\lambda}_{R}}$ and ${{\lambda}_{T}}$ are newly introduced
optimization variables compared with problem (P0). Therefore, we still divide
all optimization variables into three blocks which are
$\left\\{{\mathbf{g}_{i}}\right\\}$,
$\left\\{{\mathbf{W}_{k}},{\mathbf{m}_{t}}\right\\}$, and
$\left\\{{{\mathbf{\theta}}_{R}},{{\mathbf{\theta}}_{T}},{{\lambda}_{R}},{{\lambda}_{T}}\right\\}$
to optimizing alternately. The algorithm of optimize
$\left\\{{\mathbf{g}_{i}}\right\\}$ and
$\left\\{{\mathbf{W}_{k}},{\mathbf{m}_{t}}\right\\}$ are the same as that of
MS protocol and we consider optimizing $\left\\{{\mathbf{g}_{i}}\right\\}$ and
$\left\\{{\mathbf{W}_{k}},{\mathbf{m}_{t}}\right\\}$. Recalling Eq. (48). and
Eq. (49), the sub-problem can be expressed as

$\displaystyle\text{(P5)}:\quad$
$\displaystyle\underset{{{\tilde{\mathbf{\Theta}}}_{k}},{{\tilde{\mathbf{\Theta}}}_{i}},{{q}_{m,3.1}},{{\lambda}_{r}},{{\lambda}_{t}}}{\mathop{\max}}\,{{q}_{m,3.1}},$
(50a) $\displaystyle\rm{s.t.}\quad$
$\displaystyle\text{tr}\left({\mathbf{V}_{k}}+\sigma_{k}^{2}{\mathbf{V}_{k}}{\mathbf{G}_{k}}\mathbf{G}_{k}^{H}+E\left[{{A}_{1,k}}\right]{{\tilde{\mathbf{\Theta}}}_{k}}\right)-{{\lambda}_{{{S}_{K}}\left(k\right)}}\log\det\left({\mathbf{V}_{k}}\right)\leq{{D}_{k}}-{{R}_{\text{th}}},\forall
k,$ (50b)
$\displaystyle{{\lambda}_{{{S}_{O}}\left(i\right)}}\text{tr}\left(\left({\mathbf{A}_{2,i}}-{{q}_{m,2.1}}{\mathbf{A}_{3,i}}\right){{\tilde{\mathbf{\Theta}}}_{i}}\right)\geq{{q}_{m,3.1}}{{c}_{i}},\forall
i,$ (50c)
$\displaystyle{{\lambda}_{r}}+{{\lambda}_{t}}=1,0\leq{{\lambda}_{t}}\leq
1,0\leq{{\lambda}_{r}}\leq 1,$ (50d)
$\displaystyle\text{(\ref{p3}c)},\text{(\ref{p3}d)},\text{(\ref{p3.1}d)},$
(50e)

where (50b) is already a convex and (50c) is a bilinear constraint with
respect to ${{\lambda}_{{{S}_{O}}\left(i\right)}}$ and
${{\tilde{\mathbf{\Theta}}}_{i}}$ and is a non-convex constraint. Therefore,
we transform the left-hand side of (50c) into the following difference of
convex (DC) functions and find its lower bound by first-order Taylor
expansion. Specifically,
${{\lambda}_{{{S}_{O}}\left(i\right)}}\text{tr}\left(\left({{\mathbf{A}}_{2,i}}-{{q}_{m,2.1}}{{\mathbf{A}}_{3,i}}\right){{{\mathbf{\tilde{\Theta}}}}_{i}}\right)$
can be expressed as

$\begin{split}&\frac{{{\left({{\lambda}_{{{S}_{O}}\left(i\right)}}+\text{tr}\left(\left({{\mathbf{A}}_{2,i}}-{{q}_{m,2.1}}{{\mathbf{A}}_{3,i}}\right){{{\mathbf{\tilde{\Theta}}}}_{i}}\right)\right)}^{2}}}{4}-\frac{{{\left({{\lambda}_{{{S}_{O}}\left(i\right)}}-\text{tr}\left(\left({{\mathbf{A}}_{2,i}}-{{q}_{m,2.1}}{{\mathbf{A}}_{3,i}}\right){{{\mathbf{\tilde{\Theta}}}}_{i}}\right)\right)}^{2}}}{4}\\\
&\geq\frac{{{\left(\overset{\left(m\right)}{\mathop{{{\lambda}_{{{S}_{O}}\left(i\right)}}}}\,+\text{tr}\left(\left({{\mathbf{A}}_{2,i}}-{{q}_{m,2.1}}{{\mathbf{A}}_{3,i}}\right)\overset{\left(m\right)}{\mathop{{{{\mathbf{\tilde{\Theta}}}}_{i}}}}\,\right)\right)}^{2}}}{4}-\frac{{{\left(\overset{\left(m\right)}{\mathop{{{\lambda}_{{{S}_{O}}\left(i\right)}}}}\,-\text{tr}\left(\left({{\mathbf{A}}_{2,i}}-{{q}_{m,2.1}}{{\mathbf{A}}_{3,i}}\right){{{\mathbf{\tilde{\Theta}}}}_{i}}\right)\right)}^{2}}}{4}\\\
&+\frac{\left(\overset{\left(m\right)}{\mathop{{{\lambda}_{{{S}_{O}}\left(i\right)}}}}\,+\text{tr}\left(\left({{\mathbf{A}}_{2,i}}-{{q}_{m,2.1}}{{\mathbf{A}}_{3,i}}\right)\overset{\left(m\right)}{\mathop{{{{\mathbf{\tilde{\Theta}}}}_{i}}}}\,\right)\right)}{2}\text{tr}\left(\left({{\mathbf{A}}_{2,i}}-{{q}_{m,2.1}}{{\mathbf{A}}_{3,i}}\right)\left({{{\mathbf{\tilde{\Theta}}}}_{i}}-\overset{\left(m\right)}{\mathop{{{{\mathbf{\tilde{\Theta}}}}_{i}}}}\,\right)\right)\\\
&+\frac{\left(\overset{\left(m\right)}{\mathop{{{\lambda}_{{{S}_{O}}\left(i\right)}}}}\,+\text{tr}\left(\left({{\mathbf{A}}_{2,i}}-{{q}_{m,2.1}}{{\mathbf{A}}_{3,i}}\right)\overset{\left(m\right)}{\mathop{{{{\mathbf{\tilde{\Theta}}}}_{i}}}}\,\right)\right)}{2}\left({{\lambda}_{{{S}_{O}}\left(i\right)}}-\overset{\left(m\right)}{\mathop{{{\lambda}_{{{S}_{O}}\left(i\right)}}}}\,\right),\end{split}$
(51)

where ${\mathop{{\lambda}_{{{S}_{O}}\left(i\right)}}\limits^{\left(m\right)}}$
and ${\mathop{{{\tilde{\mathbf{\Theta}}}_{i}}}\limits^{\left(m\right)}}$ are
the given point in the $m$-th iteration. Therefore, problem (P5) can be
efficiently solved by using CVX.

##  IV Numerical Results

In this section, we verify the effectiveness of our proposed algorithm in IOS-
assisted ISAC system through numerical results. We consider a three-
dimensional coordinate system in this section. The IOS and BS are located at
(0, 0, 50m,) and (25m, 25m, 10m), respectively. ${{K}_{R}}=2$ user,
${{O}_{R}}=2$ sensed targets and are located randomly distributed in the
reflective area whose coordinate is $\left(-100\leq x\leq 100,-100\leq y\leq
0,z=0\right)$. ${{K}_{T}}=2$ users, ${{O}_{T}}=2$ sensed targets are located
randomly distributed in the transmissive area whose coordinate are
$\left(-100\leq x\leq 100,0\leq y\leq 100,z=0\right)$. Other main system
parameters are listed in Table I. We compare the proposed algorithm for ES,
MS, and TS protocols with the other two benchmarks:

Conventional IRS (C-IRS): In this scheme, $N$ elements are evenly divided into
two equal groups, which serve users and sensed targets in reflective and
transmissive areas respectively, i.e., for elements serving the reflective
area, the amplitude of their coefficients satisfies
${{\alpha}_{n,R}}=1,{{\alpha}_{n,T}}=0$ while for elements serving
transmissive area, the amplitude of their coefficients satisfies
${{\alpha}_{n,R}}=0,{{\alpha}_{n,T}}=1$.

Uniform Energy Allocation (UEA): In this scheme, for each element, the
amplitude of reflective and transmissive coefficients are equal. i.e.,
${{\alpha}_{n,R}}={{\alpha}_{n,T}}={1}/{2}\;,\forall n$.

TABLE I: Main Simulation Parameters Parameter | Value | Parameter | Value | Parameter | Value
---|---|---|---|---|---
${{N}_{t}}$ | $8$ | ${{N}_{r}}$ | $8$ | $N$ | $16$
${{D}_{k}}$ | $2$ | ${{M}_{k}}$ | $4$ | ${{R}_{\text{th}}}$ | $1$
${{P}_{max}}$ | $300w$ | $\sigma_{i}^{2}$ | $-60\text{dbm}$ | $\sigma_{k}^{2}$ | $-60\text{dbm}$

Figure 2: Convergence behavior of the proposed robust joint optimization
algorithm.

Figure 3: 3D beam pattern gain of the proposed algorithm.

First, the convergence of our algorithm is verified in IOS-enabled ISAC
networks. Fig. 2 shows the convergence behavior of the proposed algorithm for
ES protocol. It is obvious that the minimum SINR of the multi-target increases
until convergence as the number of iterations increases, which verifies our
proposed algorithm has good convergence. In addition, the number of
transmitting/receiving antennas and elements has an impact on the complexity
and convergence of the algorithm. Therefore, we set
${{N}_{t}}={{N}_{r}}=8,N=16$ as the benchmark, and change the value of
${N_{t}},{N_{r}},N$ and observe the robustness of the algorithm. It is
depicted in the Fig. 2 that our algorithm can always converge in 20 iterations
under different conditions.

Fig. 3 shows the beam pattern gain versus horizontal and vertical angel in
reflective area. We assume that the horizontal and vertical angles of the
three targets are ${{O}_{1}}=\left[-0.5,-0.3\right]$,
${{O}_{2}}=\left[0.5,-0.7\right]$ and ${{O}_{3}}=\left[0.9,-1.2\right]$
respectively. It’s obvious that our proposed beamforming scheme can make the
beam gain pointing towards targets greater than other directions including the
direction of the clutter, which indicates that our proposed algorithm can
achieve accurate beamforming while ensuring the communication quality of
users.

Figure 4: SINR versus the number of elements.

Figure 5: SINR versus the number of transmit/receive antennas.

Then, we discuss the impact of the number of IOS elements on system
performance. Fig. 4 shows the minimum SINR of multi-target versus the number
of elements. It can be observed that the minimum SINR increases as the number
of elements increases because more components will provide greater spatial
diversity gain. Specifically, ES and TS protocols are superior to the other
three protocols due to the differences in design flexibility. ES and TS
protocols exploit the ability of the whole elements that can reconstruct the
amplitude and phase of the incident signal in the highest measure. The TS
protocol can effectively eliminate interference in communication and improve
the pertinence of sensing beams at the cost of time. However, when the number
of elements is sufficient, the passive beamforming ability of IOS can
effectively achieve the above-mentioned functions without sacrificing time. As
a result, the ES protocol has the best performance when the number of elements
exceeds 25. In general, increasing the number of IOS elements can improve
system performance at a lower cost and has practical application value.

Next, Fig. 5 illustrates the variation of the minimum SINR of multi-target
with the number of transmit/receive antennas. Obviously, increasing the number
of antennas can greatly improve system performance without the need for
additional transmission power overhead. More antennas can enhance active
beamforming and increase beaming gain. UES and MS protocols cannot split the
energy of incident signals into reflective and transmissive areas, resulting
in a performance loss of approximately 4dB compared to the ES protocol. The
performance of C-IRS is the worst mainly because the beam gain achieved by IOS
is versus $N^{2}$ with increasing $N$. When IOS is divided into two groups
equally, the beam gain in each area is reduced to one-quarter of the original.
Although two groups of IRS serving different areas can reduce interference, it
cannot compensate for the performance loss caused by the decrease in the
number of IRS elements.

Figure 6: SINR versus the maximum transmit power.

Figure 7: SINR versus the number of targets.

Fig. 6 depicts the minimum SINR of multi-target sensing versus the maximum
transmit power of BS. It can be seen that greater transmission power can
achieve better system performance. However, the performance improvement
brought about by increasing transmit power is not as significant as increasing
the number of antennas and IRS elements. When the transmission power reaches a
certain threshold, the SINR of the target’s echo signal is mainly limited by
the interference brought by other targets and clutters. Increasing the number
of antennas or IOS elements can achieve more accurate beamforming and reduce
interference while increasing the transmission power can only reduce the
interference caused by noise significantly. Although the performance of TS
protocol is slightly better than ES protocol as the number of antennas
increases, the TS protocol requires synchronization of the switching cycles
between the BS and IRS, which will be a challenge in hardware implementation.

Finally, we investigate the relationship between the minimum SINR of multi-
target sensing and the number of targets in Fig. Here, we assume
${{O}_{R}}=\left\lceil{O}/{2}\;\right\rceil,{{O}_{T}}=\left\lfloor{O}/{2}\;\right\rfloor$.
It is obvious that the minimum SINR of multi-target decreases as the increase
of the number of targets. The location of newly added targets will affect the
extent of system performance degradation. Specifically, when the location of a
new target is close to the location of an original target, the channel
correlation between these two targets is strong, which makes it difficult to
distinguish the echo signals from these two targets. Conversely, if the
channels of the new target and the original targets have good orthogonality,
radar signals can achieve good spatial multiplexing for multi-target sensing,
which makes system performance almost unchanged.

##  V Conclusions

In this paper, we investigate the minimum SINR of multi-target sensing
maximization problem for IOS-assisted ISAC system, where IOS enabled dual-
function BS to achieve 360-degree ISAC coverage. Specifically, radar receive
vectors, IOS reflective, and transmissive coefficient and beamforming matrices
are jointly designed while satisfying the requirements for reachable rates of
multi-user. Because of the high coupling between optimization variables, the
primal problem is decomposed into three sub-problems based on the introduction
of an auxiliary variable and BCD algorithm. To solve the sub-problem, SDR and
SCA methods are utilized to convert the non-convex objective functions and
constraints into convex functions. Then, the iterative rank minimization and
penalty function method are used to ensure the equivalence of the conversion.
In addition, We extend the algorithm design to MS and TS protocols to adapt to
different scenarios. Finally, the simulation results validate the
effectiveness of the proposed algorithm from multiple perspectives, including
performance improvement compared with other benchmark schemes under different
parameters, beam pattern gain and convergence.

##  Appendix A Proof of Proposition 1

${{q}_{m,x}}$ is a function with respect to
$\left({{\mathbf{g}}_{i}},{{\mathbf{W}}_{k}},{{\mathbf{M}}_{t}},{{\mathbf{\theta}}_{R}},{{\mathbf{\theta}}_{T}}\right)$.
Because the proposed algorithm is based on BCD, we only consider changes in
the corresponding optimization variables in each step of our proof.

(1)
${{q}_{m,2.1}}\left(\mathbf{\tilde{W}}_{k}^{\left(m,2.2\right)},\mathbf{\tilde{M}}_{t}^{\left(m,2.2\right)}\right)\geq{{q}_{m,1}^{*}}$

The proof is based on contradiction, i.e., we assume
${{q}_{m,1}^{*}}>{{q}_{m,2.1}}\left(\mathbf{\tilde{W}}_{k}^{\left(m,2.2\right)},\mathbf{\tilde{M}}_{t}^{\left(m,2.2\right)}\right)$.
When problem (P1.1) is solved in $m$-th, ${q_{m,1}}$ can be obtained and
$\left(\mathbf{\tilde{W}}_{k}^{\left(m,1\right)},\mathbf{\tilde{M}}_{t}^{\left(m,1\right)}\right)$
express a feasible solution. Therefore,
$\left(\mathbf{\tilde{W}}_{k}^{\left(m,1\right)},\mathbf{\tilde{M}}_{t}^{\left(m,1\right)}\right)$
can also be a feasible solution for problem (P2.2) and we can get

${{q}_{m,1}}={{q}_{m,2.2}}\left(\mathbf{\tilde{W}}_{k}^{\left(m,1\right)},\mathbf{\tilde{M}}_{t}^{\left(m,1\right)}\right).$
(52)

When problem (P2.2) is solved in $m$-th iteration, optimal solution
$\left(\mathbf{\tilde{W}}_{k}^{\left(m,2.2\right)},\mathbf{\tilde{M}}_{t}^{\left(m,2.2\right)}\right)$
and optimal value
${{q}_{m,2.2}}\left(\mathbf{\tilde{W}}_{k}^{\left(m,2.2\right)},\mathbf{\tilde{M}}_{t}^{\left(m,2.2\right)}\right)$
can be obtained. Moreover,
$\left(\mathbf{\tilde{W}}_{k}^{\left(m,2.2\right)},\mathbf{\tilde{M}}_{t}^{\left(m,2.2\right)}\right)$
can also be a feasible solution for problem (P2.1) and there must exist a
target $i^{\prime}$ that satisfies

$i^{\prime}=\underset{i}{\mathop{\arg\min}}\,q_{m,2.1}^{\left(i\right)}\left(\mathbf{\tilde{W}}_{k}^{\left(m,2.2\right)},\mathbf{\tilde{M}}_{t}^{\left(m,2.2\right)}\right),$
(53)

where $q_{m,2.1}^{\left(i\right)}$ represents the SINR of $i$-th target in
$\left(P2.1\right)$, which can be expressed as

$q_{m,2.1}^{\left(i\right)}=\frac{\text{tr}\left({{{\tilde{\mathbf{S}}}}_{i,i}}\left(\sum\limits_{k=1}^{K}{\tilde{\mathbf{W}}_{k}}+\sum\limits_{t=1}^{{{N}_{T}}}{\tilde{\mathbf{M}}_{t}}\right)\right)}{\text{tr}\left(\sum\limits_{o=1+\delta\left({i-{O_{R}}}\right)\cdot{O_{R}},o\neq
i}^{{{O}_{R}+\delta\left({i-{O_{R}}}\right)\cdot{O_{T}}}}{{{{\tilde{\mathbf{S}}}}_{i,o}}}\left(\sum\limits_{k=1}^{K}{\tilde{\mathbf{W}}_{k}}+\sum\limits_{t=1}^{{{N}_{T}}}{\tilde{\mathbf{M}}_{t}}\right)\right)+{{c}_{i}}}.$
(54)

Next, $q_{m,2.2}^{\left(i\right)}$ can be expressed as

${q_{m,2.2}^{\left(i\right)}}=\frac{\text{tr}\left(\left({{{\tilde{\mathbf{S}}}}_{i,i}}-{{q}_{m,1}}\sum\limits_{o=1+\delta\left({i-{O_{R}}}\right)\cdot{O_{R}},o\neq
i}^{{{O}_{R}+\delta\left({i-{O_{R}}}\right)\cdot{O_{T}}}}{{{{\tilde{\mathbf{S}}}}_{i,o}}}\right)\left(\sum\limits_{k=1}^{K}{\tilde{\mathbf{W}}_{k}}+\sum\limits_{t=1}^{{{N}_{T}}}{\tilde{\mathbf{M}}_{t}}\right)\right)}{{{c}_{i}}}.$
(55)

The numerical relationship between $q_{m,2.1}^{\left(i\right)}$ and
$q_{m,2.2}^{\left(i\right)}$ can be expressed as

$\begin{split}&q_{m,2.2}^{\left(i\right)}=q_{m,2.1}^{\left(i\right)}\\\
&+\frac{\text{tr}\left(\left(\sum\limits_{o=1+\delta\left({i-{O_{R}}}\right)\cdot{O_{R}},o\neq
i}^{{{O}_{R}+\delta\left({i-{O_{R}}}\right)\cdot{O_{T}}}}{{{{\tilde{\mathbf{S}}}}_{i,o}}}\right)\left(\sum\limits_{k=1}^{K}{\tilde{\mathbf{W}}_{k}}+\sum\limits_{t=1}^{{{N}_{T}}}{\tilde{\mathbf{M}}_{t}}\right)\right)}{{{c}_{i}}}\cdot\left(q_{m,2.1}^{\left(i\right)}-q_{m,1}\right),\end{split}$
(56)

It’s obvious that if
$q_{m,1}>q_{m,2.1}^{\left(i\right)}\left(\tilde{\mathbf{W}}_{k}^{\left(m,1\right)},\tilde{\mathbf{M}}_{t}^{\left(m,1\right)}\right)$,
$q_{m,2.1}^{\left(i\right)}>q_{m,2.2}^{\left(i\right)}$. Then we can get

$\begin{split}&{{q}_{m,1}}\overset{\left(a\right)}{\mathop{>}}\,q_{m,2.1}\left(\mathbf{\tilde{W}}_{k}^{\left(m,2.2\right)},\mathbf{\tilde{M}}_{t}^{\left(m,2.2\right)}\right)\overset{\left(b\right)}{\mathop{=}}\,{{q}_{m,2.1}^{\left(i^{\prime}\right)}}\left(\mathbf{\tilde{W}}_{k}^{\left(m,2.2\right)},\mathbf{\tilde{M}}_{t}^{\left(m,2.2\right)}\right)\\\
&\overset{\left(c\right)}{\mathop{>}}\,q_{m,2.2}^{\left(i^{\prime}\right)}\left(\mathbf{\tilde{W}}_{k}^{\left(m,2.2\right)},\mathbf{\tilde{M}}_{t}^{\left(m,2.2\right)}\right)\overset{\left(d\right)}{\mathop{\geq}}\,{{q}_{m,2.2}}\left(\mathbf{\tilde{W}}_{k}^{\left(m,2.2\right)},\mathbf{\tilde{M}}_{t}^{\left(m,2.2\right)}\right).\end{split}$
(57)

The procedure $(a)$ is due to assumptions in the contradiction, $(b)$ is due
to the definition of $i^{\prime}$, $(c)$ is due to Eq. (56) and $(d)$ is due
to the definition of ${{q}_{m,2.2}}$. Recalling Eq. (52), we have

${{q}_{m,1}}={{q}_{m,2.2}}\left(\mathbf{\tilde{W}}_{k}^{\left(m,1\right)},\mathbf{\tilde{M}}_{t}^{\left(m,1\right)}\right)>{{q}_{m,2.2}}\left(\mathbf{\tilde{W}}_{k}^{\left(m,2.2\right)},\mathbf{\tilde{M}}_{t}^{\left(m,2.2\right)}\right).$
(58)

However, problem (P2.2) is a convex problem while
$\left({{\bf{\tilde{W}}}_{k}^{\left({m,2.2}\right)},{\bf{\tilde{M}}}_{t}^{\left({m,2.2}\right)}}\right)$
is an optimal solution and
$\left({{\bf{\tilde{W}}}_{k}^{\left({m,1}\right)},{\bf{\tilde{M}}}_{t}^{\left({m,1}\right)}}\right)$
is a feasible solution, which is contradictory and the proof is completed.

(2)
${{q}_{m,3.1}}\left(\mathbf{\theta}_{R}^{\left(m,3.1\right)},\mathbf{\theta}_{T}^{\left(m,3.1\right)}\right)\geq{{q}_{m,2.1}}\left(\mathbf{\theta}_{R}^{\left(m,2.2\right)},\mathbf{\theta}_{T}^{\left(m,2.2\right)}\right)$

The proof process is similar to proof
${{q}_{m,2.1}}\left(\mathbf{W}_{k}^{\left(m,2.2\right)},\mathbf{M}_{t}^{\left(m,2.2\right)}\right)\geq
q_{m,1}^{*}$, and we omit this process here.

(3)
${{q}_{m+1,1}}\left(\mathbf{g}_{i}^{\left(m+1,1\right)}\right)\geq{{q}_{m,3.1}}\left(\mathbf{g}_{i}^{\left(m,1\right)}\right)$

The change of ${{\mathbf{g}}_{i}}$ will not affect the feasibility of the
original problem (P0). $\mathbf{g}_{i}^{\left(m+1,1\right)}$ is the optimal
solution for maximizing $\text{SIN}{{\text{R}}_{i}}$ given
$\left({{\mathbf{W}}_{k}},{{\mathbf{M}}_{t}},{{\mathbf{\theta}}_{R}},{{\mathbf{\theta}}_{T}}\right)$
after the completion of the $m$-th iteration. Therefore,

$q_{m+1,1}^{\left(i\right)}\left(\mathbf{g}_{i}^{\left(m+1,1\right)}\right)\geq
q_{m,3.1}^{\left(i\right)}\left(\mathbf{g}_{i}^{\left(m,1\right)}\right),\forall
i,$ (59)

Then we have

${{q}_{m+1,1}}=\underset{i}{\mathop{\min}}\,\left(q_{m+1,1}^{\left(i\right)}\left(\mathbf{g}_{i}^{\left(m+1,1\right)}\right)\right)\geq{{q}_{m,3.1}}=\underset{i}{\mathop{\min}}\,\left(q_{m,3.1}^{\left(i\right)}\left(\mathbf{g}_{i}^{\left(m,1\right)}\right)\right).$
(60)

and the proof is completed.

##  Appendix B Proof of Proposition 3

First, we prove $\log\det\left(\mathbf{V}_{k}^{-1}\right)={{R}_{k}}$,
${{\mathbf{E}}_{k}}$ can be expressed as

${{\bf{E}}_{k}}=\left({{{\bf{I}}_{{D_{k}}}}-{{\bf{G}}_{k}}{{\bf{H}}_{k}}{{\bf{W}}_{k}}}\right){\left({{{\bf{I}}_{{D_{k}}}}-{{\bf{G}}_{k}}{{\bf{H}}_{k}}{{\bf{W}}_{k}}}\right)^{H}}.$
(61)

By plugging back this optimum value of ${{{\bf{G}}_{k}}}$ in Eq. (33) and
using Woodbury matrix identity,

${{\mathbf{E}}_{k}}=\left({{\mathbf{I}}_{{{D}_{k}}}}-\mathbf{W}_{k}^{H}\mathbf{H}_{k}^{H}{{\left(\sigma_{k}^{2}{{\mathbf{I}}_{{{M}_{k}}}}+\sum\limits_{k=1}^{K}{{{\mathbf{H}}_{k}}{{\mathbf{W}}_{l}}\mathbf{W}_{l}^{H}\mathbf{H}_{k}^{H}}+\sum\limits_{t=1}^{{{N}_{T}}}{{{\mathbf{H}}_{k}}{{\mathbf{m}}_{t}}\mathbf{m}_{t}^{H}\mathbf{H}_{k}^{H}}\right)}^{-1}}{{\mathbf{H}}_{k}}{{\mathbf{W}}_{k}}\right).$
(62)

Then $\log\det\left(\mathbf{E}_{k}^{-1}\right)$ can be expressed as

$\log\det\left(\mathbf{E}_{k}^{-1}\right)=\log\det\left({{\mathbf{I}}_{{{D}_{k}}}}+\mathbf{W}_{k}^{H}\mathbf{H}_{k}^{H}\mathbf{J}_{k}^{-1}{{\mathbf{H}}_{k}}{{\mathbf{W}}_{k}}\right)={{R}_{k}}.$
(63)

Next,
$\text{tr}\left({{\mathbf{V}_{k}}{\mathbf{E}_{k}}}\right)-\log\det\left({{\mathbf{V}_{k}}}\right)$
and ${R_{k}}$ are convex function and concave function on
${{\mathbf{\Theta}}_{R}},{{\mathbf{\Theta}}_{T}}$ respectively. Therefore,
when ${{\mathbf{V}}_{k}}=\mathbf{E}_{k}^{-1}$,
$\text{tr}\left({{\mathbf{V}}_{k}}{{\mathbf{E}}_{k}}\right)={{D}_{k}}$ and
${{R}_{k}}\geq{{R}_{\text{th}}}$ is equivalent to
$\text{tr}\left({{\mathbf{V}}_{k}}{{\mathbf{E}}_{k}}\right)-\log\det\left({{\mathbf{V}}_{k}}\right)\leq{{D}_{k}}-{{R}_{\text{th}}}$.

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