# Environment Sensing Considering the Occlusion Effect: A Multi-View Approach

Xin Tong, Zhaoyang Zhang, Yihan Zhang,
Zhaohui Yang, Chongwen Huang, Kai-Kit Wong, and Mérouane Debbah X. Tong, Z.
Zhang (Corresponding Author), Y. Zhang, Z. Yang and C. Huang (e-mails: {tongx,
ning_ming, zhangyihan, yang_zhaohui<EMAIL_ADDRESS>are with the
College of Information Science and Electronic Engineering, Zhejiang
University, Hangzhou 310027, China, and the Key Laboratory of Information
Processing, Communication and Networking of Zhejiang Province (IPCAN),
Hangzhou 310027, China, the Key Laboratory of Collaborative Sensing and
Autonomous Unmanned Systems of Zhejiang Province, Hangzhou 310015, China, and
the International Joint Innovation Center, Zhejiang University, Haining
314400, China. Z. Yang is also with Zhejiang Lab, Hangzhou 311121, China.K.
Wong (e-mails<EMAIL_ADDRESS>is with the Department of Electronic
and Electrical Engineering, University College London, UK.M. Debbah is with
the Technology Innovation Institute, 9639 Masdar City, Abu Dhabi, United Arab
Emirates (email<EMAIL_ADDRESS>and also with CentraleSupelec,
University Paris-Saclay, 91192 Gif-sur-Yvette, France.

###### Abstract

In this paper, we consider the problem of sensing the environment within a
wireless cellular framework. Specifically, multiple user equipments (UEs) send
sounding signals to one or multiple base stations (BSs) and then a centralized
processor retrieves the environmental information from all the channel
information obtained at the BS(s). Taking into account the occlusion effect
that is common in the wireless context, we make full use of the different
views of the environment from different users and/or BS(s), and propose an
effective sensing algorithm called GAMP-MVSVR (generalized-approximate-
message-passing-based multi-view sparse vector reconstruction). In the
proposed algorithm, a multi-layer factor graph is constructed to iteratively
estimate the scattering coefficients of the cloud points and their occlusion
relationship. In each iteration, the occlusion relationship between the cloud
points of the sparse environment is recalculated according to a simple
occlusion detection rule, and in turn, used to estimate the scattering
coefficients of the cloud points. Our proposed algorithm can achieve improved
sensing performance with multi-BS collaboration in addition to the multi-views
from the UEs. The simulation results verify its convergence and effectiveness.

###### Index Terms:

Integrated sensing and communication (ISAC), environment sensing, occlusion
effect, multi-view sensing.

##  I Introduction

###  I-A Motivation

The emergence of innovative wireless communication technologies, such as
ultra-massive multiple input multiple output (MIMO) technology, intelligent
reflective surface (IRS), and wireless artificial intelligence (AI) [1, 2, 3],
etc., provides more possibilities for the development of future wireless
communication systems design. In the foreseeable future wireless communication
application scenarios, autonomous driving [4], intelligent robot localization
[5], and unmanned aerial vehicle (UAV) control [6] require not only wireless
broadband connections, but also accurate environmental information, including
the location, shape, and electromagnetic (EM) characteristics, etc. of objects
in the environment. Therefore, integrated sensing and communication (ISAC), as
a research hotspot for the next generation wireless communication system, aims
to use the wireless communication equipment and infrastructure to achieve
environment sensing.

Such kind of environment sensing has long been accomplished by traditional
radar technology and its later evolution of joint radar and communication [7].
The latter usually aims to achieve the sharing of software and hardware
equipment and time-frequency resources between radar equipment and
communication equipment. Compared with traditional radar or early joint radar
and communication systems, efficient use of communication signals in the
existing communication systems for environment sensing, and efficient
exploitation of the environment sensing results to enhance communication, are
two major design goals of the ISAC systems. For the former goal, one
straightforward realization is to fully exploit the environmental information
embedded in the received communication data, due to the simple fact that the
distribution of environmental scatterers dramatically affects the wireless
multipath channels. For the latter, direct channel prediction (reconstruction)
can be conducted based on the sensed distribution and characteristics of
scatterers within the environment thus to enhance the performance of
communication [8, 9, 10]. In this paper, we mainly focus on the first part and
study how to perform efficient and accurate environment sensing based on the
uplink data received from multiple users, in a way compatible with existing
communication systems thus to achieve a smooth integration of communication
and sensing.

One major challenge to do this is the large number of unknown variables
embedded in the generally complicated environment, including the unknown
location and EM characteristics of the environmental scatterers themselves,
and the very common occlusion effect within and between scatterers, which
blocks the propagation of EM waves and makes portions of some scatterers or
even all of them invisible to the receivers. In such a case, it is hard for a
single user or base station (BS) to fully sense the entire environment and the
different views of the environment from different users and/or BS(s) should be
exploited. However, a challenge still remains since in each separate sensing
(transmitting and receiving), we generally cannot know which scatterers or
which part of them participates in the final generation of the received signal
in an environment yet to know, and it differs from user to user and BS to BS.
This motivates us to develop a practical and efficient multi-view sensing
algorithm to deal with the above problem.

###  I-B Related Works

So far, the design of ISAC system has raised extensive research effort. In
[11], the author comprehensively discussed the novel applications, key
performance requirements, challenges, and future research directions of ISAC
system designs. In [12], the author compared the difference between the radar
sensing scheme and the ISAC scheme. The author showcased that a common signal
wave should be jointly designed to realize the integration of sensing and
communication, and the advantages and disadvantages of various design
methodologies are also discussed.

Rapid progresses have been witnessed in the practical ISAC system and
algorithm design recently. [13] designed an ISAC system based on the wireless
cellular networks. The author jointly designed signal waveforms suitable for
sensing and communication based on 5G NR (New Radio) and its frame structure.
Also under the 5G framework, [14] proposed a multi-beam based sensing and
communication scheme. In addition to the beams used for communication, some
other ones are used to sense the environment around the BS. Based on the 4G
and 5G frameworks, [15] studied the processing principles, implementation
challenges, and performance of orthogonal frequency division multiplexing
(OFDM)-based radars, and achieved excellent sensing performance by using high
channel bandwidth and configurable sub-carriers. In a Wi-Fi communication
context, based on the Doppler frequency shift embedded in the channel state
information (CSI), [16] achieved the sensing of human body posture. In the
regime of internet of vehicles, [17] designed a fast beam alignment and
tracking algorithm based on a hardware test platform to achieve vehicle
tracking while performing millimeter wave communication with low latency and
high data rate. As the research interest on ISAC keeps on increasing, more and
more potential technologies are also introduced into this field, such as
channel modeling [18], joint beam optimization [19] and machine learning [20],
etc.

As for the environment sensing system design, some works have made full use of
the sparsity of environmental information and achieved better and lower-
complexity solutions based on the compressed sensing (CS) [21, 22] framework.
In the field of microwave computational imaging, the utilization of the
intrinsic sparsity of scatterers within the environment is key to their
effective detection. This type of algorithm usually treats the scatterer as a
sparsely distributed point cloud, thereby convert the imaging problem into a
CS reconstruction problem, which is solved by orthogonal matching pursuit
(OMP) [23], approximate message passing (AMP) [24], generalized approximate
message passing (GAMP) [25] and other widely used methods.

Recently, in [26] and [27], the authors innovatively used BS to sense
actively, and based on the block sparse Bayesian learning (BSBL) algorithm,
proposed an IRS-assisted microwave computing imaging method. Although the
above-mentioned methods effectively exploit the sparsity of the environment
but ignore the occlusion effect caused by the scatterers. They ideally believe
that EM signals can be freely transmitted to any location in the environment,
and there still lacks sufficient consideration of the occlusion effect in the
ISAC system design.

###  I-C Main Ideas and Contributions

In this paper, based on the existing wireless communication framework, we
design a multi-view environment sensing scheme taking into full consideration
the occlusion effect in an outdoor scenario.

Different from the above-mentioned application scenarios, we have achieved
ISAC by noninvasively making use of the multi-user uplink communication data.
At the same time, we have designed an effective sensing algorithm called GAMP-
MVSVR which refers to the generalized-approximate-message-passing-based multi-
view sparse vector reconstruction. In the uplink communication scenario, we
consider an environment sensing scenario with single or multiple BSs, where
users and BSs perform multi-view and collaborative sensing, and a centralized
processor retrieving the environmental information from all the channel
information obtained at the BSs. As a special case of the multi-BS scheme, a
single BS itself can also complete all the sensing tasks, and we compare the
performance differences between the two schemes.

Our design is depicted as follows. In one or more time slots, the BS estimates
the multi-user uplink channel after detecting user pilot symbols (sounding
signals). We propose an occlusion detection model based on the geometric
location of the cloud points and convert the environment sensing problem into
a CS reconstruction problem with the occlusion effect. Then we propose a
probabilistic reasoning model based on the factor graph by analyzing the
relationship among multipath channels, environmental information, and
occlusion effects. Based on the well-known AMP method, the proposed GAMP-MVSVR
algorithm achieves environment sensing by iteratively estimating the
scattering coefficients of the cloud points and their occlusion relationship.
In each iteration, the occlusion relationship between the cloud points of the
sparse environment is recalculated according to the proposed occlusion
detection rule, and in turn, used to estimate the scattering coefficients of
the cloud points. Our proposed algorithm achieves improved sensing performance
with multi-BS collaboration in addition to the multi-views from the users. To
the best of our knowledge, there has been little work on the design of such an
ISAC system in the literature.

Figure 1: The environment sensing scenario with the occlusion effect.

The main contributions of this paper are summarized as follows:

  * •

We propose an ISAC scheme, which makes full use of the different views of the
environment from different users and/or BS(s) to achieve improved sensing
performance with multi-BS collaboration in addition to the multi-views from
the UEs.

  * •

We propose an effective environment sensing algorithm called GAMP-MVSVR that
considers the occlusion effect, where a multi-layer factor graph is
constructed to iteratively estimate the scattering coefficients of the cloud
points and derive their occlusion relationship based on a simple occlusion
detection rule.

  * •

We analyze the impact of the multi-view schemes on environment sensing range
and system performance. Extensive simulation results verify the convergence
and effectiveness of the proposed algorithm.

The rest of this paper is organized as follows. Section II presents the
environment setting and system model in the uplink communication scenario.
Section III proposes the environment sensing problem formulation. Section IV
proposes the GAMP-MVSVR algorithm under the occlusion effect. In section V, we
analyze the impact of the multi-view schemes on environment sensing range and
system performance. Finally, section VII presents the numerical results, and
section VIII concludes the paper.

Notations: Fonts $a$, $\bm{a}$ and $\mathbf{A}$ represent scalars, vectors and
matrices, respectively. Notations $\mathbf{A}^{\rm{T}}$ and
$\|\mathbf{A}\|_{F}$ denote transpose and Frobenius norm of $\mathbf{A}$,
respectively. $\mathbf{A}_{i,:}$ and $\mathbf{A}_{:,j}$ represent the $i$-th
row and $j$-th column of matrix $\mathbf{A}$, respectively. $\odot$ represents
the Hadamard product between two matrices. Finally, notation ${\rm
diag}(\bm{a})$ represents a diagonal matrix with the entries of $\bm{a}$ on
its main diagonal, and $\delta(\cdot)$ is the Dirac delta function.

##  II Environment Setting and System Model

###  II-A Environment Setting

As shown in Fig. 1, we consider that in the outdoor wireless communication
scenario, multiple BSs are deployed and there are multiple active user
equipments (UEs). There are some buildings (target objects, serve as
scatterers) in the outdoor environment that affect the wireless communication
channel. We assume w.l.o.g. that the channel is a quasi-static channel where
one block of transmission time $T_{\rm b}$ is much shorter than the channel
coherence time $T_{\rm c}$, $T_{\rm b}<<T_{\rm c}$, and the UE is stationary
within the coherence time of the multipath channel estimation by UE pilots. In
the uplink communication scenario, multiple single-antenna UEs send uplink
communication data to multi-antenna BSs. The transmitted signal is scattered
by the scatterer, and received by BSs through multipath channel. Therefore,
the received signal of the BS contains environmental information.

We consider two schemes: a single-BS scheme and a multi-BS joint scheme to
achieve multi-view environment sensing. The specific analysis is as follows.

  * •

As shown in Fig. 1, the single-BS environment sensing scheme is treated as a
special case of the multi-BS joint scheme. When multiple UEs send uplink data
to BS 1, the uplink communication channel is affected by the target scatterer.
In the scenario, due to the occlusion effect, the multipath channel of UE 1 is
only affected by the scattering of target scatterers 4 and 5, and the
multipath channel of UE 2 is only affected by the scattering of target
scatterers 5. Therefore, not all scatterers in the environment will affect the
multipath channel of the same user, and the occlusion effect makes it hard for
a single user to fully sensed the entire environment. The different views of
the environment from different users should be exploited to deal with the
occlusion effect.

  * •

Fig. 1 shows the scenario where multiple BSs are used for joint environment
sensing, where the multi-BS scheme is performed. Sensing by multiple BSs will
effectively expand the sensing range. In addition to the same user multi-view
method as in the single-BS scheme, the different views of the environment from
different BSs should be exploited. For example, in Fig. 1, BS 2 cannot receive
the scattered signal from the target scatterer 3, and BS 1 and 3 also have
scatterers that cannot be observed. Therefore, a multi-BS joint scheme should
be designed to achieve the sensing of the entire environment.

###  II-B System Model

For the receiving antennas deployed on the BS, the single-BS scheme and the
multi-BS joint scheme have the same system model. Let the number of users in
the environment be $N_{\rm{u}}$, and the total number of receiving antennas of
all BSs is $N_{\rm{R}}$. In the uplink communication scenario, the channel
from the user to the BS is mainly composed of two parts: the line-of-sight
channel ${\bf{H}}^{{\rm{LOS}}}$ from the user to the BS, and the multipath
channel ${\bf{H}}^{{\rm{S}}}$ caused by the scatterer.

Figure 2: The discretized targeted environment object.

We discretize environmental information and treat the environmental
information in the entire environment as point clouds. Each point in the point
cloud represents the environmental information of the small cube with length
$l_{\rm{s}}$, width $w_{\rm{s}}$, and height $h_{\rm{s}}$ around it, and these
small cubes are called pixels. The inside of each pixel may be empty, or there
may be scatterers. The length, width and height of the environment to be
sensed are $L_{\rm{s}}$, $W_{\rm{s}}$ and $H_{\rm{s}}$ respectively, then the
number of point clouds is
${N_{\rm{s}}}=\left({{{L_{\rm{s}}}}\mathord{\left/{\vphantom{{{L_{\rm{s}}}}{{l_{\rm{s}}}}}}\right.\kern-1.2pt}{{l_{\rm{s}}}}}\right)\times\left({{{W_{\rm{s}}}}\mathord{\left/{\vphantom{{{W_{\rm{s}}}}{{w_{\rm{s}}}}}}\right.\kern-1.2pt}{{w_{\rm{s}}}}}\right)\times\left({{{H_{\rm{s}}}}\mathord{\left/{\vphantom{{{H_{\rm{s}}}}{{h_{\rm{s}}}}}}\right.\kern-1.2pt}{{h_{\rm{s}}}}}\right)$.
Fig. 2 shows the method of discretizing environmental information into point
clouds. We use the scattering coefficient ${x_{{n_{\rm{s}}}}}$ to represent
the scattering coefficient of the small cube where the $n_{\rm{s}}$-th cloud
point is located. If the small cube is empty, then ${x_{{n_{\rm{s}}}}}=0$.
Otherwise, ${x_{{n_{\rm{s}}}}}>0$. Therefore, the environmental information to
be sensed is expressed as
${\bm{x}}={\left[{{x_{1}},{x_{2}},\cdots,{x_{{N_{\rm{s}}}}}}\right]^{\rm{T}}}.$

Multiple users in the space share time-frequency resources. On a single
subcarrier, the antennas of all BSs are analyzed independently. The signal
received at the $n_{\rm{R}}$-th receiving antenna can be expressed as

$\small{\bf{Y}}\left({{n_{\rm{R}}}}\right)={\bf{S}}\left({{{\bf{H}}^{\rm{S}}}\left({{n_{\rm{R}}}}\right)+{{\bf{H}}^{{\rm{LOS}}}}\left({{n_{\rm{R}}}}\right)}\right)+{\bf{W}},$
(1)

where ${\bf{S}}\in\mathbb{C}^{N_{\rm{T}}\times N_{\rm{u}}}$ represents
codewords with the length of $N_{\rm{T}}$ sent by $N_{\rm{u}}$ users,
${\bf{Y}}\left({{n_{\rm{R}}}}\right)\in\mathbb{C}^{N_{\rm{T}}\times 1}$
represents the received signal of the $n_{\rm{R}}$-th BS antenna, and $\bf{W}$
represents noise. The term
${{\bf{H}}^{{\rm{LOS}}}}\left({{n_{\rm{R}}}}\right)\in\mathbb{C}^{N_{\rm{u}}\times
1}$ represents the channel coefficient of the line-of-sight path from the user
to the receiving antenna, where the free space channel is decomposed into
amplitude attenuation and phase shift,
$h^{{\rm{LOS}}}_{n_{\rm{u}}}\left({{n_{\rm{R}}}}\right)=\alpha^{\rm{LOS}}\cdot
e^{j\varphi^{\rm{LOS}}}$, $\alpha^{\rm{LOS}}$ is the line-of-sight channel
amplitude attenuation, and $\varphi^{\rm{LOS}}$ is the line-of-sight channel
phase shift. The line-of-sight channel occlusion can be detected through beam
scanning, received power, etc. The term
${{\bf{H}}^{{\rm{S}}}}\left({{n_{\rm{R}}}}\right)\in\mathbb{C}^{N_{\rm{u}}\times
1}$ represents the channel coefficient of the multipath channel from the user
to the receiving antenna.

As mentioned in Section I-B, we consider that in ISAC system design, the
utilization of the intrinsic sparsity of objects or scatterers within an
environment is key to the effective detection. We separate the free space
channel (microwave background field) and scatterers in the multipath channel
model to obtain sparse environmental information. In this way, the occlusion
relationship between the scatterer and the antenna can be modeled by the
existence of the free space channel between them. The multipath channel from
the user to the receiving antenna
${{\bf{H}}^{{\rm{S}}}}\left({{n_{\rm{R}}}}\right)$ is expressed as

$\displaystyle{{\bf{H}}^{{\rm{S}}}}\left({{n_{\rm{R}}}}\right)={\bf{\tilde{H}}}\left({{n_{\rm{R}}}}\right){\bm{x}}$
(2a) $\displaystyle=$
$\displaystyle\left[{{\bf{H}}\left({{n_{\rm{R}}}}\right)\odot{\bf{V}}\left({{n_{\rm{R}}}}\right)}\right]{\bm{x}}$
(2b) $\displaystyle=$ $\displaystyle\left({{{\bf{H}}^{{\rm{U\rightarrow
s}}}}\odot{{\bf{V}}^{{\rm{U\rightarrow
s}}}}}\right){\rm{diag}}\left({{{\bf{H}}^{{\rm{s\rightarrow
B}}}}\left({{n_{\rm{R}}}}\right)\odot{{\bf{V}}^{{\rm{s\rightarrow
B}}}}\left({{n_{\rm{R}}}}\right)}\right){\bm{x}},$ (2c)

where the free space channel coefficient
${\bf{\tilde{H}}}\left({{n_{\rm{R}}}}\right)$ with occlusion is expressed as
the Hadamard product of the free space channel coefficient
${\bf{H}}\left({{n_{\rm{R}}}}\right)\in\mathbb{C}^{N_{\rm{u}}\times
N_{\rm{s}}}$ and the occlusion matrix
${\bf{V}}\left({{n_{\rm{R}}}}\right)\in\left\\{0,1\right\\}^{N_{\rm{u}}\times
N_{\rm{s}}}$. The element 0 in the occlusion matrix
${\bf{V}}\left({{n_{\rm{R}}}}\right)$ indicates that the path from
$n_{\rm{u}}$-th user to $n_{\rm{R}}$-th receiving antenna is blocked, and the
element 1 indicates that the path from $n_{\rm{u}}$-th user to $n_{\rm{R}}$-th
receiving antenna is not blocked. ${{\bf{H}}^{{\rm{U\rightarrow
s}}}}\in\mathbb{C}^{N_{\rm{u}}\times N_{\rm{s}}}$ represents the channel
coefficient from the user to the point cloud position, where
$h^{{\rm{U\rightarrow s}}}_{n_{\rm{u}},n_{\rm{s}}}=\alpha^{\rm{U\rightarrow
s}}\cdot e^{j\varphi^{\rm{U\rightarrow s}}}$, $\alpha^{\rm{U\rightarrow s}}$
represents the channel amplitude attenuation from the user to the point cloud
position, $\varphi^{\rm{U\rightarrow s}}$ represents the channel phase shift
from the user to the point cloud position. ${{\bf{H}}^{{\rm{s\rightarrow
B}}}}\left({{n_{\rm{R}}}}\right)\in\mathbb{C}^{N_{\rm{s}}\times 1}$ represents
the channel coefficient from the point cloud position to the BS receiving
antenna, where $h^{{\rm{s\rightarrow
B}}}_{n_{\rm{s}}}\left({{n_{\rm{R}}}}\right)=\alpha^{\rm{s\rightarrow B}}\cdot
e^{j\varphi^{\rm{s\rightarrow B}}}$, $\alpha^{\rm{s\rightarrow B}}$ represents
the channel amplitude attenuation from the point cloud position to the BS
receiving antenna, $\varphi^{\rm{s\rightarrow B}}$ represents the channel
phase shift from the point cloud position to the BS receiving antenna,
${\bf{H}}\left({{n_{\rm{R}}}}\right)={{\bf{H}}^{{\rm{U\rightarrow
s}}}}{\rm{diag}}\left({{{\bf{H}}^{{\rm{s\rightarrow
B}}}}\left({{n_{\rm{R}}}}\right)}\right)$. ${{\bf{V}}^{{\rm{U\rightarrow
s}}}}\in\left\\{0,1\right\\}^{N_{\rm{u}}\times N_{\rm{s}}}$ represents the
occlusion matrix from the user to the point cloud position, and its
distribution reflects the different views from different users. When
${{\bf{V}}^{{\rm{U\rightarrow s}}}}$ contains an all-zero column, it means
that all users cannot sense the corresponding pixel, and the corresponding
pixel is out of the sensing range. ${{\bf{V}}^{{\rm{s\rightarrow
B}}}}\left({{n_{\rm{R}}}}\right)\in\left\\{0,1\right\\}^{N_{\rm{s}}\times 1}$
represents the occlusion matrix from the point cloud position to the receiving
antenna, and its distribution reflects the different views from different BSs.
When ${{\bf{V}}^{{\rm{s\rightarrow B}}}}\left({{n_{\rm{R}}}}\right)$ contains
a zero element, it means that the receiving antenna cannot sense the
corresponding pixel, and the corresponding pixel is also out of the sensing
range, ${\bf{V}}\left({{n_{\rm{R}}}}\right)={{\bf{V}}^{{\rm{U\rightarrow
s}}}}{\rm{diag}}\left({{{\bf{V}}^{{\rm{s\rightarrow
B}}}}\left({{n_{\rm{R}}}}\right)}\right)$.

In (2), the model is derived as follows. First, in (2a), we decompose the
multipath channel ${{\bf{H}}^{\rm{S}}}\left({{n_{\rm{R}}}}\right)$ in (1) into
free space channel with occlusion ${\bf\tilde{H}}\left({{n_{\rm{R}}}}\right)$
and environmental information $\bm{x}$. Then, in (2b), we decompose free space
channel with occlusion ${\bf\tilde{H}}\left({{n_{\rm{R}}}}\right)$ into free
space channel ${\bf H}\left({{n_{\rm{R}}}}\right)$ and occlusion matrix ${\bf
V}\left({{n_{\rm{R}}}}\right)$. Finally, in (2c), we further decompose the
free space channel with occlusion ${\bf\tilde{H}}\left({{n_{\rm{R}}}}\right)$
into the channels from the user, the base station to the point cloud position
${{\bf{H}}^{{\rm{U\rightarrow s}}}}$, ${{\bf{H}}^{{\rm{s\rightarrow
B}}}}\left({{n_{\rm{R}}}}\right)$ and their corresponding occlusion matrices
${{\bf{V}}^{{\rm{U\rightarrow s}}}}$, ${{\bf{V}}^{{\rm{s\rightarrow
B}}}}\left({{n_{\rm{R}}}}\right)$.

Figure 3: An illustration of the environmental scatterer occlusion detection.

##  III Problem Formulation

In this section, we analyze the detection method of the occlusion effect in
environment sensing, and based on this, we convert the environment sensing
problem into a CS reconstruction problem with occlusion effect.

###  III-A Occlusion Detection Model

We detect the occlusion effect between scatterers based on the geometric
positions between pixels. As shown in Fig. 3, there is a pixel C with a
scattering coefficient greater than the threshold $\eta$ between pixels A and
B. We believe that pixel C will cause an occlusion effect. Let the position of
pixel A be the origin of coordinates, then the position vectors of pixel B and
C are expressed as $\bm{b}$ and $\bm{c}$ respectively. According to the
following conditions, it is considered that pixel C blocks the line-of-sight
path between pixels A and B.

  * •

The distance $d$ between pixel C and the line-of-sight path between pixels A
and B is less than the threshold $l$,

$\small
d=\frac{{\left\lvert{{\bm{c}}\times{\bm{b}}}\right\rvert}}{{\left\lvert{\bm{b}}\right\rvert}}<l,$
(3)

where the detection threshold $l$ should be selected according to the side
length or diagonal length of the small cube, that is,
$l\in\left[\min\left\\{l_{\rm{s}},w_{\rm{s}},h_{\rm{s}}\right\\},\left(l_{\rm{s}}^{2}+w_{\rm{s}}^{2}+h_{\rm{s}}^{2}\right)^{1/2}\right]$.

  * •

The angle between vector $\bm{b}$ and vector $\bm{c}$ is an acute angle, which
means that pixel C is located in the same direction of the line-of-sight path
from pixel A to pixel B, but not pixel A is located between the pixel B and
the pixel C. It is expressed as

$\small{\bm{b}}\cdot{\bm{c}}>0.$ (4)

  * •

Pixel C is located between pixel A and pixel B, which is expressed as

$\small\left\lvert{{\bm{c}}\cdot{\bm{b}}}\right\rvert<\left\lvert{\bm{b}}\right\rvert^{2}.$
(5)

According to the above occlusion detection method, we design the environment
sensing algorithm. However, the proposed occlusion detection method is not the
only method, and the proposed environment sensing algorithm does not depend on
the uniqueness of the occlusion detection method.

###  III-B Environment Compressed Sensing Model

As mentioned in Section I, since the distribution of scatterers in the
environment is sparse, according to the CS theory, the optimization problem of
solving environmental information is expressed as

$\small{\bm{\hat{x}}}=\arg\mathop{\min}\limits_{\bm{x}}{\left\|{\bm{x}}\right\|_{1}}\quad{\rm{s}}{\rm{.t}}{\rm{.}}\quad{\left\|{{\bf{Y}}-{\bf{S}}\left({{{\bf{H}}^{\rm{S}}}+{{\bf{H}}^{{\rm{LOS}}}}}\right)}\right\|_{2}}\leq\varepsilon,$
(6)

where $\varepsilon$ is a slack variable. In the system model of Section II-B,
We can calculate the free space channel coefficients
${{\bf{H}}^{{\rm{LOS}}}}\left({{n_{\rm{R}}}}\right)$ and
${{\bf{H}}}\left({{n_{\rm{R}}}}\right)$ through the basic free space path
propagation model. In addition, some empirical models, simplified models, or
interpolation methods are also suitable, and the appropriate model needs to be
selected according to the practical environment. The BS estimates the
multipath channel ${{\bf{H}}^{{\rm{S}}}}\left({{n_{\rm{R}}}}\right)$ by
detecting the UE pilot and calculating the relationship with the received
signal ${\bf{Y}}\left({{n_{\rm{R}}}}\right)$. Therefore, in general, in order
to achieve accurate channel estimation (e.g. least-squares channel
estimation), the pilot length should be greater than the number of users, that
is, $N_{\rm T}>N_{\rm u}$. The proposed algorithm does not rely much on the
method of channel estimation. Note that the channel estimation is only the
first step of the overall integrated sensing and communication process. The
subsequent environment sensing problem studied here is always an
underdetermined CS problem with given channel matrix. Due to the large
propagation range of wireless signals, according to different precision
requirements, the number of pixels $N_{\rm s}$ will be greater than $N_{\rm
u}N_{\rm R}$. At the same time, based on the sparsity of the environmental
information $\bm{x}$, the optimization problem (6) is expressed as the CS
reconstruction problem equation

$\displaystyle{\left[{\begin{array}[]{*{20}{c}}{{{\bf{H}}^{\rm{S}}}\left(1\right)}\\\
\vdots\\\ {{{\bf{H}}^{\rm{S}}}\left({{n_{\rm{R}}}}\right)}\\\ \vdots\\\
{{{\bf{H}}^{\rm{S}}}\left({{N_{\rm{R}}}}\right)}\end{array}}\right]_{{N_{\rm{u}}}{N_{\rm{R}}}\times
1}}$
$\displaystyle={\left[{\begin{array}[]{*{20}{c}}{{\bf{\tilde{H}}}\left(1\right)}\\\
\vdots\\\ {{\bf{\tilde{H}}}\left({{n_{\rm{R}}}}\right)}\\\ \vdots\\\
{{\bf{\tilde{H}}}\left({{N_{\rm{R}}}}\right)}\end{array}}\right]_{{N_{\rm{u}}}{N_{\rm{R}}}\times{N_{\rm{s}}}}}{\left[{\bm{x}}\right]_{{N_{\rm{s}}}\times
1}},$ (17) $\displaystyle\Rightarrow{{\bf{H}}^{\rm{S}}}$
$\displaystyle={\bf{\tilde{H}}}{\bm{x}},$ (18)

where ${\bf{\tilde{H}}}$ is formed by splicing the multipath channel matrix
${{\bf{\tilde{H}}}\left({{n_{\rm{R}}}}\right)}$ between $N_{\rm{R}}$ receiving
antennas and $N_{\rm{u}}$ users. In the multi-BS joint environment sensing
scheme, $N_{\rm{R}}$ receiving antennas include the number of all antennas on
multiple BSs.

##  IV Environment Sensing Algorithm under The Occlusion Effect

In this section, we propose an environment sensing algorithm called GAMP-MVSVR
under the occlusion effect to solve the environment information $\bm{x}$ in
the (18). According to (2), (18) is expressed as

$\small{{\bf{H}}^{\rm{S}}}={\bf{\tilde{H}}}{\bm{x}}=\left({\bf{H}}\odot{\bf{V}}\right){\bm{x}},$
(19)

where channel matrix ${\bf{H}}$ is formed by splicing channel matrix
${\bf{H}}\left(n_{\rm{R}}\right)$ of $N_{\rm{R}}$ receiving antennas, and
occlusion matrix ${\bf{V}}$ is formed by splicing occlusion matrix
${\bf{V}}\left(n_{\rm{R}}\right)$ of $N_{\rm{R}}$ receiving antennas.

Figure 4: A sketch of the proposed GAMP-MVSVR algorithm. Figure 5: The factor
graph of joint posterior distribution.

###  IV-A Overview of the Proposed GAMP-MVSVR Algorithm

Different from the traditional CS reconstruction problem, solving the
environment sensing problem with occlusion effect requires not only solving
the environment information ${\bm{x}}$ but also needs to consider the unknown
occlusion matrix ${\bf{V}}$. Traditional CS reconstruction algorithms, such as
GAMP algorithm [25], can solve environmental information in an ideal (ignoring
occlusion, ${\bf{\tilde{H}}}={\bf{H}}$) situation. In this way, since the
accurate measurement matrix ${\bf{\tilde{H}}}$ cannot be obtained, the
environmental information ${\bm{x}}$ cannot be accurately solved. Compared
with the GAMP algorithm, the Bilinear GAMP algorithm [28] can solve the
bilinear problem, that is, simultaneously solve the occlusion matrix
${\bf{V}}$ and the environmental information ${\bm{x}}$. However, it does not
consider the relationship between the occlusion effect and the environmental
information, the Bilinear GAMP algorithm cannot constrain the uniqueness and
accuracy of the occlusion matrix ${\bf{V}}$ and the environmental information
${\bm{x}}$.

Fig. 4 shows a sketch of the proposed GAMP-MVSVR algorithm. In Section IV-B,
we propose a factor graph based on the decomposition result of the joint
posterior probability. Based on the sum-product algorithm (SPA) algorithm, we
calculate the message passed in the factor graph, and define the prior
probability and likelihood function. In Section IV-C, we approximate the
message from factor node to variable node. Through the approximate process, we
use the result in the previous iteration to calculate its corresponding
multipath channel. We compare it with the channel
${{\bf{\hat{H}}}^{{\rm{S}}}}$ estimated by the pilot and calculate the
residual to guide this iteration. In Section IV-D, we approximate the message
from variable node to factor node. Through the approximate process, according
to the occlusion effect, we estimate ${\bf{\tilde{H}}}$ and ${\bm{x}}$ based
on the residual. In Section IV-E, We update the intermediate variables in
Section IV-C and IV-D, and achieve the iterative execution of the estimation
methods in Section IV-C and IV-D.

###  IV-B Factor Graph and Passed Messages

As mentioned in Section III-B, let ${{\bf{\hat{H}}}^{\rm{S}}}$ be the
multipath channel estimated by the pilot, then
${{\bf{\hat{H}}}^{\rm{S}}}={{\bf{H}}^{\rm{S}}}+{\bf{W}_{\rm e}}$, where
${\bf{W}_{\rm e}}$ represents the estimation error. The estimation error
${\bf{W}_{\rm e}}$ is caused by two main reasons, including the multipath
channel estimation error caused by pilot estimation and the free space channel
estimation error caused by modeling estimation. In order to simplify the
analysis, we assume that the noise ${\bf W}_{\rm e}$ obeys a Gaussian
distribution. The joint posterior distribution of the occluded multipath
channel ${\bf{\tilde{H}}}$ and the environmental information ${\bm{x}}$ is
expressed as

$\small\begin{split}&{p_{{\bm{x}},{\bf{\tilde{H}}}|{{\bf{\hat{H}}}^{\rm{S}}}}}\left({{\bm{x}},{\bf{\tilde{H}}}|{{\bf{\hat{H}}}^{\rm{S}}}}\right)\\\
=&p_{{{\bf{\hat{H}}}^{\rm{S}}}|{\bm{x}},{\bf{\tilde{H}}}}\left({{{\bf{\hat{H}}}^{\rm{S}}}|{\bm{x}},{\bf{\tilde{H}}}}\right){p_{\bm{x}}}\left({\bm{x}}\right){p_{{\bf{\tilde{H}}}|{\bm{x}}}}\left({{\bf{\tilde{H}}}|{\bm{x}}}\right)/{{p_{{\bf{\hat{H}}}^{\rm{S}}}}\left({{\bf{\hat{H}}}^{\rm{S}}}\right)}\\\
\propto&p_{{{\bf{\hat{H}}}^{\rm{S}}}|{\bf{\tilde{H}}}{\bm{x}}}\left({{{\bf{\hat{H}}}^{\rm{S}}}|{\bf{\tilde{H}}}{\bm{x}}}\right){p_{\bm{x}}}\left({\bm{x}}\right){p_{{\bf{\tilde{H}}}|{\bm{x}}}}\left({{\bf{\tilde{H}}}|{\bm{x}}}\right)\\\
=&p_{{{\bf{\hat{H}}}^{\rm{S}}}|{{\bf{H}}^{\rm{S}}}}\left({{{\bf{\hat{H}}}^{\rm{S}}}|{{\bf{H}}^{\rm{S}}}}\right){p_{\bm{x}}}\left({\bm{x}}\right){p_{{\bf{\tilde{H}}}|{\bm{x}}}}\left({{\bf{\tilde{H}}}|{\bm{x}}}\right),\end{split}$
(20)

where ${{p_{{\bf{H}}^{\rm{S}}}}\left({{\bf{H}}^{\rm{S}}}\right)}$ is a
constant. To simplify the expression, let
$N_{\rm{H}}={N_{\rm{u}}}{N_{\rm{R}}}$ denote the length of the multipath
channel vector ${{\bf{\hat{H}}}^{\rm{S}}}$. As shown in Fig. 5, according to
the (20), the posterior distribution is expressed in the form of a factor
graph.

As shown in Fig. 5, the circles represent the variable nodes $\bf{\tilde{H}}$
and $\bm{x}$, the squares represent the factor node
$p_{{{\bf{\hat{H}}}^{\rm{S}}}|{{\bf{H}}^{\rm{S}}}}({{{\bf{\hat{H}}}^{\rm{S}}}|{{\bf{H}}^{\rm{S}}}})$,
${p_{\bm{x}}}\left({\bm{x}}\right)$ and
${p_{{\bf{\tilde{H}}}|{\bm{x}}}}({{\bf{\tilde{H}}}|{\bm{x}}})$. The lines
between the nodes represent that the variables are included in the function of
the factor nodes. Specifically, the environmental information prior
probability function ${p_{\bm{x}}}\left({\bm{x}}\right)$ contains the
environmental information variable $\bm{x}$. The relationship between the
environmental information variable $\bm{x}$ and the channel with occlusion
$\bf{\tilde{H}}$ conforms to the occlusion detection model function
${p_{{\bf{\tilde{H}}}|{\bm{x}}}}({{\bf{\tilde{H}}}|{\bm{x}}})$. The multipath
channel observation function
$p_{{{\bf{\hat{H}}}^{\rm{S}}}|{{\bf{H}}^{\rm{S}}}}({{{\bf{\hat{H}}}^{\rm{S}}}|{{\bf{H}}^{\rm{S}}}})$
contains the environmental information variable $\bm{x}$ and the free space
channel $\bf{\tilde{H}}$. The specific function is expressed as follows:
${{\bf{\hat{H}}}^{\rm{S}}}$ can be considered as the observation of
${{\bf{H}}^{\rm{S}}}$ under the interference of additive white Gaussian noise,
i.e.,

$\small
p_{{\hat{h}^{\rm{S}}_{n_{\rm{H}}}}|{h_{n_{\rm{H}}}^{\rm{S}}}}\left({{\hat{h}_{n_{\rm{H}}}^{\rm{S}}}|{h_{n_{\rm{H}}}^{\rm{S}}}}\right)={\mathcal{N}}\left({\hat{h}_{n_{\rm{H}}}^{\rm{S}};{h_{n_{\rm{H}}}^{\rm{S}}},{\sigma^{\rm{w}}}}\right),$
(21)

where ${\sigma^{\rm{w}}}$ is the variance of Gaussian white noise. Due to the
sparseness of environmental information $\bm{x}$, we assume that the $\bm{x}$
obeys a Bernoulli-Gaussian distribution,

$\small{p_{\bm{x}}}\left({\bm{x}}\right)=\left({1-\lambda}\right)\delta\left(x\right)+\lambda{\cal
N}\left({x;{\theta^{\rm{x}}},{\sigma^{\rm{x}}}}\right),$ (22)

where $\lambda$ is the sparsity of the environment, $\theta^{\rm{x}}$ and
$\sigma^{\rm{x}}$ represent the mean and variance of the scattering
coefficient of the environmental scatterer.
${p_{{\bf{\tilde{H}}}|{\bm{x}}}}\left({{\bf{\tilde{H}}}|{\bm{x}}}\right)$
represents the occlusion effect of environmental information $\bm{x}$ on the
channel, i.e.,

$\small\begin{split}&{p_{{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}|{\bm{x}}}}\left({{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}|{\bm{x}}}\right)=\\\
&\left\\{\begin{array}[]{l}\delta\left({{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}-{h_{{n_{\rm{H}}},{n_{\rm{s}}}}}}\right)\quad{v_{{n_{\rm{H}}},{n_{\rm{s}}}}}=f_{\rm{occ}}\left({\bm{x}}\right)=1\\\
\delta\left({{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}}\right)\quad\quad\quad\quad\;\;\;{v_{{n_{\rm{H}}},{n_{\rm{s}}}}}=f_{\rm{occ}}\left({\bm{x}}\right)=0\end{array}\right.,\end{split}$
(23)

where $f_{\rm{occ}}\left(\cdot\right)$ represents the occlusion detection
function, the specific method is as described in the Section III-A, and the
occlusion detection threshold is set to $\eta={\theta^{\rm{x}}}/4$.

Based on the sparse prior information and occlusion relationship of
environmental information, we aim to obtain the minimum mean square error
(MMSE) estimate of environmental information

$\small\hat{x}_{{n_{\rm{s}}}}^{{\rm{MMSE}}}=\arg\mathop{\min}\limits_{{{\hat{x}}_{{n_{\rm{s}}}}}}{\mathbb{E}_{\bm{x}}}\left[{{{\left({{x_{{n_{\rm{s}}}}}-{{\hat{x}}_{{n_{\rm{s}}}}}}\right)}^{2}}}\right].$
(24)

According to the loopy belief propagation (LBP) theory [29], based on the
factor graph model, we calculate the MMSE estimated value of the environment
information through the SPA [30]. As explained in Appendix A, the method of
SPA message passing is as follows. As shown in Fig. 5, the output message of
the variable node along the edge (solid line) is obtained by calculating the
product of all its input messages. The output message of the factor node is
calculated by multiplying and integrating the function of the factor node and
its input message. The marginal distribution of the variable is calculated
from the product of all messages input to the node. Since it is difficult to
accurately calculate the passed message , we approximate the message in the
following sections.

###  IV-C Approximated Factor-to-Variable Messages

In this section, we approximate the message from the factor node to the
variable node according to the central limit theorem (CLT) and Gaussian
approximation based on the idea of approximate message passing [24, 25, 28].
Based on CLT, when $N_{\rm{s}}\rightarrow\infty$, $N_{\rm{H}}/N_{\rm{s}}$
remains constant, we drop some high-order infinitesimal terms. In the $t$
iteration, the sparsity $\lambda$ in the prior distribution of environmental
information ${p_{\bm{x}}}\left({\bm{x}}\right)$ remains constant. As mentioned
in Section III-A and Section V, the number of zero elements in the occlusion
matrix $\bf{V}$ in the variable $\bf{\tilde{H}}$ remains fixed, and the
position of the zero elements is determined by ${\hat{x}_{{n_{\rm{s}}}}}$.
Therefore, we consider that variable ${\hat{x}_{{n_{\rm{s}}}}}$ is independent
of variable ${\hat{h}_{{n_{\rm{H}}},{n_{\rm{s}}}}}$ in the $t$-th iteration.
As explained in Appendix B, the passed message in (77) is approximated as

$\displaystyle{\mu_{\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}\to{x_{{n_{\rm{s}}}}}}}\left({t,{x_{{n_{\rm{s}}}}}}\right)=\log\int_{h_{{n_{\rm{H}}}}^{\rm{S}}}{{p_{\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}|h_{{n_{\rm{H}}}}^{\rm{S}}}}\left({\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}\left|{h_{{n_{\rm{H}}}}^{\rm{S}}}\right.}\right)}$
$\displaystyle\times{\cal
N}\left({h_{{n_{\rm{H}}}}^{\rm{S}};{{\bar{h}}_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}}\left(t\right){x_{{n_{\rm{s}}}}}+{{\bar{p}}_{{n_{\rm{s}}},{n_{\rm{H}}}}}\left(t\right),}\right.$
$\displaystyle\quad\quad\quad\left.{\sigma_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{h}}\left(t\right)x_{{n_{\rm{s}}}}^{2}+\sigma_{{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{p}}\left(t\right)}\right)+c$
$\displaystyle={F_{{n_{\rm{H}}}}}\left({{{\bar{h}}_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}}\left(t\right){x_{{n_{\rm{s}}}}}+{{\bar{p}}_{{n_{\rm{s}}},{n_{\rm{H}}}}}\left(t\right),}\right.$
$\displaystyle\quad\quad\quad\left.{\sigma_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{h}}\left(t\right)x_{{n_{\rm{s}}}}^{2}+\sigma_{{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{p}}\left(t\right);\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}}\right)+c,$
(25)

where

$\small{F_{{n_{\rm{H}}}}}\left({{\bar{p}},{\sigma^{\rm{p}}};{{\hat{h}}^{\rm{S}}}}\right)=\log\int_{{h^{\rm{S}}}}{{p_{{{\hat{h}}^{\rm{S}}}|{h^{\rm{S}}}}}\left({{{\hat{h}}^{\rm{S}}}|{h^{\rm{S}}}}\right){\cal
N}\left({{{\hat{h}}^{\rm{S}}};{\bar{p}},{\sigma^{\rm{p}}}}\right)}.$ (26)

Adding the $n_{\rm{s}}$-th term to the summation in
${\bar{p}_{{n_{\rm{s}}},{n_{\rm{H}}}}}\left(t\right)$ in (86), then
${\bar{p}_{{n_{\rm{H}}}}}\left(t\right)$ and
$\sigma_{{n_{\rm{H}}}}^{\rm{p}}\left(t\right)$ are considered as an estimate
of ${h_{{n_{\rm{H}}}}^{\rm{S}}}$, i.e.,

$\small{\bar{p}_{{n_{\rm{H}}}}}\left(t\right)=\sum_{k=1}^{N_{\rm{s}}}{{{\bar{h}}_{{n_{\rm{H}}}k,{n_{\rm{H}}}}}\left(t\right){{\bar{x}}_{k,{n_{\rm{H}}}}}\left(t\right)},$
(27)
$\small\begin{split}\sigma_{{n_{\rm{H}}}}^{\rm{p}}&\left(t\right)=\sum_{k=1}^{N_{\rm{s}}}{\left({\bar{h}_{{n_{\rm{H}}}k,{n_{\rm{H}}}}^{2}\left(t\right)\sigma_{k,{n_{\rm{H}}}}^{\rm{x}}\left(t\right)}\right.}\\\
&{\left.{+\sigma_{{n_{\rm{H}}}k,{n_{\rm{H}}}}^{\rm{h}}\left(t\right)\bar{x}_{k,{n_{\rm{H}}}}^{2}\left(t\right)+\sigma_{{n_{\rm{H}}}k,{n_{\rm{H}}}}^{\rm{h}}\left(t\right)\sigma_{k,{n_{\rm{H}}}}^{\rm{x}}\left(t\right)}\right)}.\end{split}$
(28)

We define a residual variable whose mean value is,

$\small{\bar{s}_{n_{\rm{H}}}}\left(t\right)={F^{\prime}_{{n_{\rm{H}}},1}}\left({{{\bar{p}}_{{n_{\rm{H}}}}}\left(t\right),\sigma_{{n_{\rm{H}}}}^{\rm{p}}\left(t\right);\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}}\right),$
(29)

due to the Gaussian property of ${F_{{n_{\rm{H}}}}}$, the variance of the
residual is expressed as,

$\small\sigma_{n_{\rm{H}}}^{\rm{s}}\left(t\right)=-{F^{\prime\prime}_{{n_{\rm{H}}},1}}\left({{{\bar{p}}_{{n_{\rm{H}}}}}\left(t\right),\sigma_{{n_{\rm{H}}}}^{\rm{p}}\left(t\right);\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}}\right).$
(30)

where ${F^{\prime}_{{n_{\rm{H}}},1}}$ and
${F^{\prime\prime}_{{n_{\rm{H}}},1}}$ represent the first and second
derivatives of the first argument of ${F_{{n_{\rm{H}}}}}$, and
${F^{\prime}_{{n_{\rm{H}}},2}}$ represents the first derivative of the second
argument of ${F_{{n_{\rm{H}}}}}$.

As explained in Appendix C, we get the final approximation result of the
passed message
${\mu_{\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}\to{x_{{n_{\rm{s}}}}}}}\left({t,{x_{{n_{\rm{s}}}}}}\right)$
in (77)

$\displaystyle{\mu_{\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}\to{x_{{n_{\rm{s}}}}}}}\left({t,{x_{{n_{\rm{s}}}}}}\right)$
$\displaystyle\approx\left[{{{\bar{s}}_{{n_{\rm{H}}}}}\left(t\right){{\bar{h}}_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}}\left(t\right)+\sigma_{{n_{\rm{H}}}}^{\rm{s}}\left(t\right)\hat{h}_{{n_{\rm{H}}},{n_{\rm{s}}}}^{2}\left(t\right){{\hat{x}}_{{n_{\rm{s}}}}}\left(t\right)}\right]{x_{{n_{\rm{s}}}}}$
$\displaystyle-\frac{1}{2}\left[{\sigma_{{n_{\rm{H}}}}^{\rm{s}}\left(t\right)\hat{h}_{{n_{\rm{H}}},{n_{\rm{s}}}}^{2}\left(t\right)-\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{h}}\left(t\right)}\right.$
$\displaystyle\times\left.{\left({\bar{s}_{{n_{\rm{H}}}}^{2}\left(t\right)-\sigma_{{n_{\rm{H}}}}^{\rm{s}}\left(t\right)}\right)}\right]x_{{n_{\rm{s}}}}^{2}+c,$
(31)

According to (27), ${\bar{p}_{{n_{\rm{H}}}}}$ is considered to be an estimate
of ${{\bf{H}}^{\rm{S}}}$, we can express the posterior probability of
${{\bf{H}}^{\rm{S}}}$ as the product of the prior probability ${\cal
N}\left({h_{{n_{\rm{H}}}}^{\rm{S}};{{\bar{p}}_{{n_{\rm{H}}}}}\left(t\right),\sigma_{{n_{\rm{H}}}}^{\rm{p}}\left(t\right)}\right)$
and the likelihood
${p_{\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}|h_{{n_{\rm{H}}}}^{\rm{S}}}}({\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}|h_{{n_{\rm{H}}}}^{\rm{S}}})$
as follows

$\small\begin{split}&{p_{h_{{n_{\rm{H}}}}^{\rm{S}}|{p_{{n_{\rm{H}}}}}}}\left({h_{{n_{\rm{H}}}}^{\rm{S}}|{{\bar{p}}_{{n_{\rm{H}}}}}\left(t\right);\sigma_{{n_{\rm{H}}}}^{\rm{p}}\left(t\right)}\right)\\\
&=\frac{1}{C}{p_{\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}|h_{{n_{\rm{H}}}}^{\rm{S}}}}\left({\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}|h_{{n_{\rm{H}}}}^{\rm{S}}}\right){\cal
N}\left({h_{{n_{\rm{H}}}}^{\rm{S}};{{\bar{p}}_{{n_{\rm{H}}}}}\left(t\right),\sigma_{{n_{\rm{H}}}}^{\rm{p}}\left(t\right)}\right),\end{split}$
(32)

where
$C=\int_{h^{\rm{S}}}{p_{h_{{n_{\rm{H}}}}^{\rm{S}}|{p_{{n_{\rm{H}}}}}}}\left({h^{\rm{S}}|{{\bar{p}}}\left(t\right);\sigma^{\rm{p}}\left(t\right)}\right)$
is a normalization constant. Then the mean and variance of (32) are the mean
and variance of calculated ${{\bf{H}}^{\rm{S}}}$ as follows

$\small\bar{h}_{{n_{\rm{H}}}}^{\rm{S}}\left(t\right)={\mathbb{E}}\left\\{{h_{{n_{\rm{H}}}}^{\rm{S}}|{{\bar{p}}_{{n_{\rm{H}}}}}\left(t\right);\sigma_{{n_{\rm{H}}}}^{\rm{p}}\left(t\right)}\right\\},$
(33)
$\small\sigma_{{n_{\rm{H}}}}^{{\rm{Hs}}}\left(t\right)={\mathbb{D}}\left\\{{h_{{n_{\rm{H}}}}^{\rm{S}}|{{\bar{p}}_{{n_{\rm{H}}}}}\left(t\right);\sigma_{{n_{\rm{H}}}}^{\rm{p}}\left(t\right)}\right\\},$
(34)

where $\mathbb{E}(X)=\int{xf(x){\rm d}x}$ represents the mean of $x$ and
$\mathbb{D}(X)=\int{\left[x-\mathbb{E}(X)\right]^{2}f(x){\rm d}x}$ represents
the variance of $x$.

Pluging (32), (33) and (34) into (29) and (30). Based on the likelihood in
(21), according to the derivation in [25, 28], the mean and variance of the
residual are expressed as

$\small{\bar{s}_{{n_{\rm{H}}}}}\left(t\right)=\frac{{\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}-{{\bar{p}}_{{n_{\rm{H}}}}\left(t\right)}}}{{\sigma_{{n_{\rm{H}}}}^{\rm{p}}\left(t\right)+{\sigma^{\rm{w}}}}},$
(35) $\small\sigma_{{n_{\rm{H}}}}^{\rm
s}\left(t\right)=\frac{1}{{\sigma_{{n_{\rm{H}}}}^{\rm{p}}\left(t\right)+{\sigma^{\rm{w}}}}}.$
(36)

As mentioned in this section, in the $t$-th iteration, we consider that
variable $\bm{x}$ is independent of variable $\bf{\tilde{H}}$, so the same as
the derivation of (31), we approximate the message
${\mu_{\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}\to{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}}}({t,{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}})$
in (79) as

$\displaystyle{\mu_{\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}\to{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}}}\left({t,{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}}\right)\approx$
$\displaystyle\left[{{{\bar{s}}_{{n_{\rm{H}}}}}\left(t\right){{\bar{x}}_{{n_{\rm{s}}},{n_{\rm{H}}}}}\left(t\right)+\sigma_{{n_{\rm{H}}}}^{\rm{s}}\left(t\right)\hat{x}_{{n_{\rm{s}}}}^{2}\left(t\right){{\hat{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right)}\right]{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}$
$\displaystyle-\frac{1}{2}\left[{\sigma_{{n_{\rm{H}}}}^{\rm{s}}\left(t\right)\hat{x}_{{n_{\rm{s}}}}^{2}\left(t\right)-\sigma_{{n_{\rm{s}}}}^{\rm{x}}\left(t\right)}\right.$
$\displaystyle\times\left.{\left({\bar{s}_{{n_{\rm{H}}}}^{2}\left(t\right)-\sigma_{{n_{\rm{H}}}}^{\rm{s}}\left(t\right)}\right)}\right]\tilde{h}_{{n_{\rm{H}}},{n_{\rm{s}}}}^{2}+c.$
(37)

In summary, in this section we approximate the message from factor node to
variable node as follows. We calculate
${{{\bar{p}}_{{n_{\rm{H}}}}}\left(t\right)}$ (as shown in (27)) to obtain the
calculated channel ${\bar{h}_{{n_{\rm{H}}}}^{\rm{S}}\left(t\right)}$ (as shown
in (33)). Then we calculate the residual
${\bar{s}_{{n_{\rm{H}}}}}\left(t\right)$ according to
${\bar{h}_{{n_{\rm{H}}}}^{\rm{S}}\left(t\right)}$ and the channel estimation
value using the pilot ${\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}}$ (as shown in (35)).
Finally, we use the residual and the approximate mean and variance of the
passed message to represent the message from the factor node to the variable
node (as shown in (31) and (37)).

###  IV-D Approximated Variable-to-Factor Messages

In this section, we approximate the message from the factor node to the
variable node. We plug (31) in (78), the message
${{\mu_{{x_{r}}\to\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}}}\left({t+1,{x_{r}}}\right)}$
from variable node $\bm{x}$ to factor node
$p_{{{\bf{\hat{H}}}^{\rm{S}}}|{{\bf{H}}^{\rm{S}}}}({{{\bf{\hat{H}}}^{\rm{S}}}|{{\bf{H}}^{\rm{S}}}})$
is approximated as

$\small\begin{split}&{\mu_{{x_{{n_{\rm{s}}}}}\to\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}}}\left({t+1,{x_{{n_{\rm{s}}}}}}\right)\\\
&=\log{p_{\bm{x}}}\left({{x_{{n_{\rm{s}}}}}}\right)-\frac{1}{{2\sigma_{{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{r}}\left(t\right)}}{\left({{x_{{n_{\rm{s}}}}}-{{\bar{r}}_{{n_{\rm{s}}},{n_{\rm{H}}}}}\left(t\right)}\right)^{2}}+c\\\
&=\log\left({{p_{\bm{x}}}\left({{x_{{n_{\rm{s}}}}}}\right){\cal
N}\left({{x_{{n_{\rm{s}}}}};{{\bar{r}}_{{n_{\rm{s}}},{n_{\rm{H}}}}}\left(t\right),\sigma_{{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{r}}\left(t\right)}\right)}\right)+c,\end{split}$
(38)

where

$\displaystyle{{\bar{r}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right)$
$\displaystyle={{\hat{x}}_{{n_{\rm{s}}}}}\left(t\right)\left({1+\sigma_{{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{r}}\left(t\right)\sum\limits_{k\neq{n_{\rm{H}}}}{\sigma_{k,{n_{\rm{s}}}}^{\rm{h}}\left(t\right)\left[{\bar{s}_{k}^{2}\left(t\right)-\sigma_{k}^{\rm{s}}\left(t\right)}\right]}}\right)$
$\displaystyle+\sigma_{{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{r}}\left(t\right)\sum\limits_{k\neq
n_{\rm{H}}}{{{\hat{h}}_{k,{n_{\rm{s}}}}}\left(t\right){{\bar{s}}_{k}}\left(t\right)},$
(39)

$\small\begin{split}&\sigma_{{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{r}}\left(t\right)\\\
&={\left({\sum\limits_{k\neq{n_{\rm{H}}}}{{{\hat{h}}_{k,{n_{\rm{s}}}}}\left(t\right)\sigma_{k}^{\rm{s}}\left(t\right)-\sigma_{k,{n_{\rm{s}}}}^{\rm{h}}\left(t\right)\left({\bar{s}_{k}^{2}\left(t\right)-\sigma_{k}^{\rm{s}}\left(t\right)}\right)}}\right)^{-1}}.\end{split}$
(40)

Then the mean ${\bar{x}_{{n_{\rm{s}}},{n_{\rm{H}}}}}\left({t+1}\right)$ and
variance $\sigma_{{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{x}}\left({t+1}\right)$ of
the passed message in (38) are expressed as,

$\small\begin{split}&{{\bar{x}}_{{n_{\rm{s}}},{n_{\rm{H}}}}}\left({t+1}\right)\\\
&=\frac{1}{C}\int_{{x_{{n_{\rm{s}}}}}}{{x_{{n_{\rm{s}}}}}{p_{\bm{x}}}\left({{x_{{n_{\rm{s}}}}}}\right){\cal
N}\left({{x_{{n_{\rm{s}}}}};{{\bar{r}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right),\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{r}}\left(t\right)}\right)}\\\
&={G_{\bm{x}}}\left({{{\bar{r}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right),\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{r}}\left(t\right)}\right),\end{split}$
(41)
$\small\sigma_{{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{x}}\left({t+1}\right)=\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{r}}\left(t\right){G^{\prime}_{\bm{x}}}\left({{{\bar{r}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right),\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{r}}\left(t\right)}\right),$
(42)

where
$C=\int_{{x_{{n_{\rm{s}}}}}}{{p_{\bm{x}}}\left({{x_{{n_{\rm{s}}}}}}\right){\cal
N}\left({{x_{{n_{\rm{s}}}}};{{\bar{r}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right),\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{r}}\left(t\right)}\right)}$
is a normalization constant, and ${G^{\prime}_{\bm{x}}}$ represents the first
derivative of the first argument of the ${{G}_{\bm{x}}}$. The specific
derivation of the relationship between (41) and (42) was shown in [25].

Adding the $n_{\rm{H}}$-th term to the summation in
${{\bar{r}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right)$, then
${{\bar{r}}_{{n_{\rm{s}}}}}\left(t\right)$ and
$\sigma_{{n_{\rm{s}}}}^{\rm{r}}\left(t\right)$ are expressed as

$\displaystyle{{\bar{r}}_{{n_{\rm{s}}}}}\left(t\right)$
$\displaystyle={{\hat{x}}_{{n_{\rm{s}}}}}\left(t\right)\left({1+\sigma_{{n_{\rm{s}}}}^{\rm{r}}\left(t\right)\sum_{k=1}^{N_{\rm{H}}}{\sigma_{k,{n_{\rm{s}}}}^{\rm{h}}\left(t\right)\left[{\bar{s}_{k}^{2}\left(t\right)-\sigma_{k}^{\rm{s}}\left(t\right)}\right]}}\right)$
$\displaystyle+\sigma_{{n_{\rm{s}}}}^{\rm{r}}\left(t\right)\sum\limits_{k=1}^{N_{\rm{H}}}{{{\hat{h}}_{k,{n_{\rm{s}}}}}\left(t\right){{\bar{s}}_{k}}\left(t\right)},$
(43)

$\displaystyle\sigma_{{n_{\rm{s}}}}^{\rm{r}}\left(t\right)$
$\displaystyle={\left({\sum_{k=1}^{N_{\rm{H}}}{{{\hat{h}}_{k,{n_{\rm{s}}}}}\left(t\right)\sigma_{k}^{\rm{s}}\left(t\right)}}\right.}{\left.{{-\sigma_{k,{n_{\rm{s}}}}^{\rm{h}}\left(t\right)\left({\bar{s}_{k}^{2}\left(t\right)-\sigma_{k}^{\rm{s}}\left(t\right)}\right)}}\right)^{-1}},$
(44)

Based on AMP [24, 25, 28], ${{\bar{r}}_{{n_{\rm{s}}}}}\left(t\right)$ and
$\sigma_{{n_{\rm{s}}}}^{\rm{r}}\left(t\right)$ are considered as an estimate
of $x_{n_{\rm{s}}}$ as follows

$\small{\hat{x}}_{n_{\rm{s}}}\left(t+1\right)={G_{\bm{x}}}\left({{{\bar{r}}_{{n_{\rm{s}}}}}\left(t\right),\sigma_{{n_{\rm{s}}}}^{\rm{r}}\left(t\right)}\right),$
(45)
$\small\sigma_{{n_{\rm{s}}}}^{\rm{x}}\left(t+1\right)=\sigma_{{n_{\rm{s}}}}^{\rm{r}}\left(t\right){G^{\prime}_{\bm{x}}}\left({{{\bar{r}}_{{n_{\rm{s}}}}}\left(t\right),\sigma_{{n_{\rm{s}}}}^{\rm{r}}\left(t\right)}\right),$
(46)

where ${G^{\prime}_{\bm{x}}}$ represent the first derivative of the first
argument of ${G_{\bm{x}}}$. Finally, we calculate the relationship between the
mean of the passed message
${\bar{x}_{{n_{\rm{s}}},{n_{\rm{H}}}}}\left({t+1}\right)$ and the mean of the
marginal distribution ${\hat{x}_{{n_{\rm{s}}}}}\left({t+1}\right)$ as follows

$\small\begin{split}&{{\bar{x}}_{{n_{\rm{s}}},{n_{\rm{H}}}}}\left({t+1}\right)\\\
&\approx{G_{\bm{x}}}\left({{{\bar{r}}_{{n_{\rm{s}}}}}\left(t\right)-\sigma_{{n_{\rm{s}}}}^{\rm{r}}\left(t\right){{\hat{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right){{\bar{s}}_{{n_{\rm{H}}}}}\left(t\right),\sigma_{{n_{\rm{s}}}}^{\rm{r}}\left(t\right)}\right)\\\
&\approx{G_{\bm{x}}}\left({{{\bar{r}}_{{n_{\rm{s}}}}}\left(t\right),\sigma_{{n_{\rm{s}}}}^{\rm{r}}\left(t\right)}\right)\\\
&-\sigma_{{n_{\rm{s}}}}^{\rm{r}}\left(t\right){{\hat{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right){{\bar{s}}_{{n_{\rm{H}}}}}\left(t\right){{G^{\prime}}_{\bm{x}}}\left({{{\bar{r}}_{{n_{\rm{s}}}}}\left(t\right),\sigma_{{n_{\rm{s}}}}^{\rm{r}}\left(t\right)}\right)\\\
&={{\hat{x}}_{{n_{\rm{s}}}}}\left({t+1}\right)-{{\hat{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right){{\bar{s}}_{{n_{\rm{H}}}}}\left(t\right)\sigma_{n}^{\rm{x}}\left({t+1}\right),\end{split}$
(47)

in which we drop infinitesimal terms such as
$\sigma_{{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{r}}\left(t\right)-{\sigma_{{n_{\rm{s}}}}^{\rm{r}}\left(t\right)}$
in the second step of approximation, and we perform Taylor series expansion on
the first argument of the function at point
${{\bar{r}}_{{n_{\rm{s}}}}}\left(t\right)$ and drop the infinitesimal term in
the second step of approximation.

In the $t$-th iteration, we consider that variable $\bm{x}$ is independent of
variable $\bf{\tilde{H}}$ and recalculate the relationship between variables
$\bf{\tilde{H}}$ and $\bm{x}$ after each iteration. Therefore, we approximate
the passed message from variable node $\bf{\tilde{H}}$ to factor node
$p_{{{\bf{\hat{H}}}^{\rm{S}}}|{{\bf{H}}^{\rm{S}}}}({{{\bf{\hat{H}}}^{\rm{S}}}|{{\bf{H}}^{\rm{S}}}})$
the same way as (38), and calculate the mean
${\hat{h}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left({t+1}\right)$ and the variance
$\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{h}}\left({t+1}\right)$ of the
marginal distribution of variable $\bf{\tilde{H}}$ the same way as (45) and
(46), i.e.,

$\small{\hat{h}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left({t+1}\right)={G_{\bf{H}}}\left({{{\bar{q}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right),\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{q}}\left(t\right)}\right),$
(48)
$\small\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{h}}\left({t+1}\right)=\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{q}}\left(t\right){G^{\prime}_{\bf{H}}}\left({{{\bar{q}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right),\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{q}}\left(t\right)}\right),$
(49)
$\small\begin{split}&{G_{\bf{H}}}\left({{{\bar{q}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right),\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{q}}\left(t\right)}\right)\\\
&\quad\quad\quad\quad\quad=\frac{1}{C}\int_{\tilde{h}}{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}{p_{{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}|{\bm{x}}}}\left({{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}|{\bm{x}}}\right)\\\
&\quad\quad\quad\quad\quad\times{\cal
N}\left({{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}};{{\bar{q}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right),\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{q}}\left(t\right)}\right),\end{split}$
(50)

where
$C=\int_{\tilde{h}}{{p_{{{\tilde{h}}}|{\bm{x}}}}({{{\tilde{h}}}|{\bm{x}}}){\cal
N}}({{{\tilde{h}}};{{\bar{q}}}\left(t\right),\sigma^{\rm{q}}\left(t\right)})$
is a normalization constant, ${G^{\prime}_{\bf{H}}}$ represent the first
derivative of the first argument of ${G_{\bf{H}}}$,
${\bar{q}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right)$ and
$\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{q}}\left(t\right)$ are considered as
an estimate of ${{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}$ as follows

$\displaystyle{\bar{q}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right)={\hat{h}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right)\left({1+\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{q}}\left(t\right)\sigma_{{n_{\rm{s}}}}^{\rm{x}}\left(t\right)}\right.$
$\displaystyle\left.{\left({\bar{s}_{{n_{\rm{H}}}}^{2}\left(t\right)-\sigma_{{n_{\rm{H}}}}^{\rm{s}}\left(t\right)}\right)+\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{q}}\left(t\right){{\bar{x}}_{{n_{\rm{s}}},{n_{H}}}}\left(t\right){{\bar{s}}_{{n_{H}}}}\left(t\right)}\right),$
(51)
$\small\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{q}}\left(t\right)={\left({\hat{x}_{{n_{\rm{s}}}}^{2}\left(t\right)\sigma_{{n_{\rm{H}}}}^{\rm{s}}\left(t\right)-\sigma_{{n_{\rm{s}}}}^{\rm{x}}\left(t\right)\left({\bar{s}_{{n_{\rm{H}}}}^{2}\left(t\right)-\sigma_{{n_{\rm{H}}}}^{\rm{s}}\left(t\right)}\right)}\right)^{-1}}.$
(52)

Similar to the approximation process of (47), we calculate the relationship
between the mean of the passed message
${\bar{h}_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}}\left({t+1}\right)$ and the
mean of the marginal distribution
${\hat{h}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left({t+1}\right)$ as follows

$\small{\bar{h}_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}}\left({t+1}\right)={\hat{h}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left({t+1}\right)-{\hat{x}_{{n_{\rm{s}}}}}\left(t\right){\bar{s}_{{n_{\rm{H}}}}}\left(t\right)\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{h}}\left(t\right).$
(53)

Since the environmental information $\bm{x}$ is an unknown variable, in the
$t+1$-th iteration, we use ${\bm{\hat{x}}}\left(t\right)$ instead of $\bm{x}$
for occlusion detection, (50) is expressed as

$\displaystyle{G_{\bf{H}}}\left({{{\bar{q}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right),\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{q}}\left(t\right)}\right)=\frac{1}{C}\int_{\tilde{h}}{{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}}$
$\displaystyle{{p_{{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}|{{\bm{\hat{x}}}\left(t\right)}}}\left({{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}|{{\bm{\hat{x}}}\left(t\right)}}\right){\cal
N}}\left({{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}};{{\bar{q}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right),\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{q}}\left(t\right)}\right).$
(54)

In summary, in this section we approximate the message from variable node to
factor node as follows. Firstly, we calculate
${{\bar{r}}_{{n_{\rm{s}}}}}\left(t\right)$ and
${\bar{q}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right)$ based on the residual
${{{\bar{s}}_{{n_{H}}}}\left(t\right)}$ (as shown in (43) and (51)), and
calculate the new estimated values
${\hat{x}_{{n_{\rm{s}}}}}\left({t+1}\right)$ and
${\hat{h}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left({t+1}\right)$ (as shown in (45)
and (48)). Secondly, since the message from the variable node to the factor
node is similar to the posterior probability, we approximate the relationship
between the estimated value ${\hat{x}_{{n_{\rm{s}}}}}\left({t+1}\right)$,
${\hat{h}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left({t+1}\right)$ and the passed
message ${\bar{x}_{{n_{\rm{s}}},{n_{\rm{H}}}}}\left({t+1}\right)$,
${\bar{h}_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}}\left({t+1}\right)$ (as
shown in (47) and (53)).

Algorithm 1 Proposed GAMP-MVSVR Algorithm

0: Given the free space channel coefficient $\bf{H}$ and the estimated value
of the uplink multipath channel ${{\bf{\hat{H}}}^{\rm{S}}}$.

1: Initialization: For each ${n_{\rm{H}}}$ and ${n_{\rm{s}}}$, set the prior
probability ${p_{\bm{x}}}\left({\bm{x}}\right)$ of environmental information
$\bm{x}$ in (22), and choose ${\hat{x}_{{n_{\rm{s}}}}\left(1\right)}$ and
${\sigma_{{n_{\rm{s}}}}^{\rm{x}}\left(1\right)}$. Set the likelihood
$p_{{\hat{h}^{\rm{S}}_{n_{\rm{H}}}}|{h_{n_{\rm{H}}}^{\rm{S}}}}({{\hat{h}_{n_{\rm{H}}}^{\rm{S}}}|{h_{n_{\rm{H}}}^{\rm{S}}}})$
in (21). Set ${\bar{s}_{{n_{\rm{H}}}}}\left(0\right)=0$.

2: For each ${n_{\rm{H}}}$ and ${n_{\rm{s}}}$, plug
${\hat{x}_{{n_{\rm{s}}}}\left(1\right)}$ into (23), calculate the free space
channel coefficient distribution
${p_{{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}|{\bm{x}}\left(1\right)}}({{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}|{\bm{x}}\left(1\right)})$
with occlusion, and choose
${{{\hat{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right)}$ and
$\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{h}}\left(t\right)$.

3: for $t=1,2,\ldots,T_{\rm{max}}$ do

4: For each ${n_{\rm{H}}}$, plug ${\hat{x}_{{n_{\rm{s}}}}\left(t\right)}$,
${\sigma_{{n_{\rm{s}}}}^{\rm{x}}\left(t\right)}$,
${{{\hat{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right)}$, and
$\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{h}}\left(t\right)$ into (55) and (56)
to obtain ${\bar{p}_{{n_{\rm{H}}}}}\left(t\right)$ and
$\sigma_{n_{\rm{H}}}^{\rm{p}}\left(t\right)$.

5: For each ${n_{\rm{H}}}$, plug ${\bar{p}_{{n_{\rm{H}}}}}\left(t\right)$,
$\sigma_{n_{\rm{H}}}^{\rm{p}}\left(t\right)$, and
$p_{{\hat{h}^{\rm{S}}_{n_{\rm{H}}}}|{h_{n_{\rm{H}}}^{\rm{S}}}}({{\hat{h}_{n_{\rm{H}}}^{\rm{S}}}|{h_{n_{\rm{H}}}^{\rm{S}}}})$
into (33) and (34) to obtain $\bar{h}_{{n_{\rm{H}}}}^{\rm{S}}\left(t\right)$
and $\sigma_{{n_{\rm{H}}}}^{{\rm{Hs}}}\left(t\right)$.

6: For each ${n_{\rm{H}}}$, plug $\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}$,
${{\bar{p}}_{{n_{\rm{H}}}}\left(t\right)}$, and
$\sigma_{{n_{\rm{H}}}}^{\rm{p}}\left(t\right)$ into (35) and (36) to obtain
${\bar{s}_{{n_{\rm{H}}}}}\left(t\right)$ and
$\sigma_{{n_{\rm{H}}}}^{\rm{s}}\left(t\right)$.

7: For each ${n_{\rm{s}}}$, plug ${\hat{x}_{{n_{\rm{s}}}}}\left(t\right)$,
${\sigma_{{n_{\rm{s}}}}^{\rm{x}}\left(t\right)}$,
${{{\hat{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right)}$,
${\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{h}}\left(t\right)}$,
${{{\bar{s}}_{n_{\rm{H}}}}\left(t\right)}$, and
${\sigma_{n_{\rm{H}}}^{\rm{s}}\left(t\right)}$ into (57) and (58) to obtain
${\bar{r}_{{n_{\rm{s}}}}}\left(t\right)$ and
$\sigma_{{n_{\rm{s}}}}^{\rm{r}}\left(t\right)$.

8: For each ${n_{\rm{H}}}$ and ${n_{\rm{s}}}$, plug
${\hat{x}_{{n_{\rm{s}}}}}\left(t\right)$,
${\sigma_{{n_{\rm{s}}}}^{\rm{x}}\left(t\right)}$,
${{{\hat{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right)}$,
${\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{h}}\left(t\right)}$,
${{{\bar{s}}_{n_{\rm{H}}}}\left(t\right)}$, and
${\sigma_{n_{\rm{H}}}^{\rm{s}}\left(t\right)}$ into (59) and (60) to obtain
${\bar{q}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right)$ and
$\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{q}}\left(t\right)$.

9: For each ${n_{\rm{s}}}$, plug ${\bar{r}_{{n_{\rm{s}}}}}\left(t\right)$ and
$\sigma_{{n_{\rm{s}}}}^{\rm{r}}\left(t\right)$ into (45) and (46) to obtain
${\hat{x}}_{n_{\rm{s}}}\left(t+1\right)$ and
$\sigma_{{n_{\rm{s}}}}^{\rm{x}}\left(t+1\right)$.

10: For each ${n_{\rm{H}}}$ and ${n_{\rm{s}}}$, plug
${\bar{q}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right)$,
$\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{q}}\left(t\right)$, and
${p_{{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}|{\bm{x}}\left(t\right)}}({{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}|{\bm{x}}\left(t\right)})$
into (48) and (49) to obtain
${\hat{h}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left({t+1}\right)$ and
$\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{h}}\left({t+1}\right)$.

11: For each ${n_{\rm{H}}}$ and ${n_{\rm{s}}}$, plug
${\hat{x}_{{n_{\rm{s}}}}\left(t+1\right)}$ into (23) to obtain the free space
channel coefficient distribution with occlusion
${p_{{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}|{\bm{\hat{x}}}\left(t+1\right)}}({{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}|{\bm{\hat{x}}}\left(t+1\right)})$.

12: If
$\sum_{{n_{\rm{H}}}}{\left|{\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}}-{\bar{h}_{{n_{\rm{H}}}}^{\rm{S}}}\left({t}\right)\right|}>\varepsilon_{\rm{t}}$,
where $\varepsilon_{\rm{t}}$ is a given error tolerance value, stop the
iteration.

13: end for

13: Estimated sparse vector ${\hat{x}}_{n_{\rm{s}}}\left(t\right)$ and
$\sigma_{{n_{\rm{s}}}}^{\rm{x}}\left(t\right)$.

###  IV-E Algorithm Iterative Execution

In Section IV-C and Section IV-D, we have obtained the estimation methods for
variables ${\hat{x}_{{n_{\rm{s}}}}}\left({t+1}\right)$ and
${\hat{h}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left({t+1}\right)$, but the
intermediate variables ${\bar{p}_{{n_{\rm{H}}}}}\left(t\right)$,
${{\bar{r}}_{{n_{\rm{s}}}}}\left(t\right)$, and
${\bar{q}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right)$ contain the passed
messages ${\bar{x}_{{n_{\rm{s}}},{n_{\rm{H}}}}}\left({t+1}\right)$ and
${\bar{h}_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}}\left({t+1}\right)$. We need
to replace the passed message with the current estimated variable
${\hat{x}_{{n_{\rm{s}}}}}\left({t+1}\right)$ and
${\hat{h}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left({t+1}\right)$ to calculate these
intermediate variables again to achieve the iteration of the algorithm.

Pluging (47) and (53) into (27), we can get

$\displaystyle{\bar{p}_{{n_{\rm{H}}}}}\left(t\right)\approx\sum\limits_{k=1}^{{N_{\rm{s}}}}{{{\hat{h}}_{{n_{\rm{H}}},k}}\left(t\right){{\hat{x}}_{k}}\left(t\right)}-{\bar{s}_{{n_{\rm{H}}}}}\left({t-1}\right)$
$\displaystyle\quad\times\sum\limits_{k=1}^{{N_{\rm{s}}}}{\left({\sigma_{{n_{\rm{H}}},k}^{\rm{h}}\left(t\right)\hat{x}_{k}^{2}\left(t\right)+\hat{h}_{{n_{\rm{H}}},k}^{2}\left(t\right)\sigma_{k}^{\rm{x}}\left(t\right)}\right)},$
(55)

where we replace ${\hat{x}_{k}\left(t\right)}{\hat{x}_{k}\left(t-1\right)}$
with ${\hat{x}_{k}^{2}\left(t\right)}$, replace
${\hat{h}_{{n_{\rm{H}}},k}\left(t\right)}{\hat{h}_{{n_{\rm{H}}},k}\left(t-1\right)}$
with ${\hat{h}_{{n_{\rm{H}}},k}^{2}\left(t\right)}$, and drop the
infinitesimal term under the CLT condition. In most AMP algorithms [25, 28],
(55) can be interpreted as the Onsager correction to
${{{\hat{h}}_{{n_{\rm{H}}},k}}\left(t\right){{\hat{x}}_{k}}\left(t\right)}$,
and (28) is expressed as

$\displaystyle\sigma_{n_{\rm{H}}}^{\rm{p}}\left(t\right)\approx$
$\displaystyle\sum\limits_{k=1}^{{N_{\rm{s}}}}{\left({\sigma_{{n_{\rm{H}}},k}^{\rm{h}}\left(t\right)\hat{x}_{k}^{2}\left(t\right)}\right.}$
$\displaystyle{\left.{+\hat{h}_{{n_{\rm{H}}},k}^{2}\left(t\right)\sigma_{k}^{\rm{x}}\left(t\right)+\sigma_{{n_{\rm{H}}},k}^{\rm{h}}\left(t\right)\sigma_{k}^{\rm{x}}\left(t\right)}\right)}.$
(56)

Similar to the approximate method in the (55), pluging (47) and (53) into (43)
and (44), it yields

$\displaystyle{\bar{r}_{{n_{\rm{s}}}}}\left(t\right)$
$\displaystyle\approx{\hat{x}_{{n_{\rm{s}}}}}\left(t\right)\left({1-\sigma_{{n_{\rm{s}}}}^{\rm{r}}\left(t\right)\sum\limits_{k=1}^{{N_{\rm{H}}}}{\sigma_{k,{n_{\rm{s}}}}^{\rm{h}}\left(t\right)\sigma_{k}^{\rm{s}}\left(t\right)}}\right)$
$\displaystyle+\sigma_{{n_{\rm{s}}}}^{\rm{r}}\left(t\right)\sum\limits_{k=1}^{{N_{\rm{H}}}}{\left({{{\hat{h}}_{k,{n_{\rm{s}}}}}\left(t\right){{\bar{s}}_{k}}\left(t\right)}\right)},$
(57)
$\small\sigma_{{n_{\rm{s}}}}^{\rm{r}}\left(t\right)\approx{\left({\sum\limits_{k=1}^{{N_{\rm{H}}}}{\hat{h}_{k,{n_{\rm{s}}}}^{2}\left(t\right)\sigma_{{n_{\rm{s}}}}^{\rm{s}}\left(t\right)}}\right)^{-1}}.$
(58)

And similar to the approximate method in the (55), pluging (47) and (53) into
(51) and (52), it yields

$\displaystyle{\bar{q}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right)$
$\displaystyle\approx{\hat{h}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left(t\right)\left({1-\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{q}}\left(t\right)\sigma_{{n_{\rm{s}}}}^{\rm{x}}\left(t\right)\sigma_{{n_{\rm{H}}}}^{\rm{s}}\left(t\right)}\right)$
$\displaystyle+\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{q}}\left(t\right){\hat{x}_{{n_{\rm{s}}}}}\left(t\right){\bar{s}_{{n_{\rm{H}}}}}\left(t\right),$
(59)
$\small\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{q}}\left(t\right)\approx{\left({\hat{x}_{{n_{\rm{s}}}}^{2}\left(t\right)\sigma_{{n_{\rm{H}}}}^{\rm{s}}\left(t\right)}\right)^{-1}}.$
(60)

We summarize the proposed GAMP-MVSVR algorithm in Algorithm 1. During the
execution process of the GAMP-MVSVR algorithm, the approximate message passing
and the occlusion detection method execute repeatedly and converge to the
estimated value of environmental information. The computational complexity of
the GAMP-MVSVR algorithm consists of two parts: approximate message passing
and occlusion detection method. The computational complexity is expressed as
$\mathcal{O}\left(N_{\rm{H}}N_{\rm{s}}+N_{\rm{H}}N_{\rm{s}}\left\lVert\bm{x}\right\rVert_{0}\right)$,
where $\left\lVert\bm{x}\right\rVert_{0}$ is the number of the environmental
scatterers, the number of pixels $N_{\rm{s}}$ has a major influence on the
computational complexity. The setting of $T_{\rm max}$ is to stop the
iteration in time when the system performance is poor. Due to the small order
of magnitude of iterations, the number of algorithm iterations will hardly
reach $T_{\rm max}$, and we omit the term $T_{\rm max}$ when calculating the
computational complexity.

##  V System Performance Analysis

In this section, we compare the performance of the single-BS environment
sensing scheme and the multi-BS joint environment sensing scheme from two
aspects: environment sensing range and environment sensing accuracy.

###  V-A Environment Sensing Range

In this section, we analyze the impact of the CS measurement matrix
${{\bf{\tilde{H}}}}$ on the system performance. According to the CS
reconstruction model in Section III-B, the environment sensing range is
determined by the CS measurement matrix ${{\bf{\tilde{H}}}}$. Specifically,
the number of 0 elements in the occlusion matrix ${{\bf{V}}}$,
${{\bf{V}}^{{\rm{U\rightarrow s}}}}$, and ${{\bf{V}}^{{\rm{s\rightarrow B}}}}$
is calculated to obtain the environment sensing range.

As mentioned in Section II-B, the number of pixels is $N_{\rm{s}}$, and the
position of ${n_{\rm{s}}}$-th pixel is
${\bm{a}}_{n_{\rm{s}}}^{\rm{s}}=\left[x_{n_{\rm{s}}}^{\rm{s}},y_{n_{\rm{s}}}^{\rm{s}},z_{n_{\rm{s}}}^{\rm{s}}\right]$.
The number of users in the environment is $N_{\rm{u}}$, and the position of
$n_{\rm{u}}$-th user is
${\bm{a}}_{n_{\rm{u}}}^{\rm{u}}=\left[x_{n_{\rm{u}}}^{\rm{u}},y_{n_{\rm{u}}}^{\rm{u}},z_{n_{\rm{u}}}^{\rm{u}}\right]$.
In the multi-BS joint environment sensing scheme, the number of BSs in the
environment is $N_{\rm{B}}$, and the position of $n_{\rm{B}}$-th BS is
${\bm{a}}_{n_{\rm{B}}}^{\rm{B}}=\left[x_{n_{\rm{B}}}^{\rm{B}},y_{n_{\rm{B}}}^{\rm{B}},z_{n_{\rm{B}}}^{\rm{B}}\right]$.
Under these definitions, the analysis of the occlusion matrix
${{\bf{V}}^{{\rm{U\rightarrow s}}}}$ and ${{\bf{V}}^{{\rm{s\rightarrow B}}}}$
is summarized in the following theorem.

###### Theorem 1

The probability that the element in the occlusion matrix
${{\bf{V}}^{{\rm{U\rightarrow s}}}}$ is 0 can be expressed as

$\small\begin{split}&p\left({{\bf{V}}_{{n_{\rm{u}}},{n_{\rm{s}}}}^{{\rm{U\rightarrow
s}}}=0}\right)=1-\frac{{{{\left\|{{{\bf{V}}^{{\rm{U\rightarrow
s}}}}}\right\|}_{1}}}}{{{N_{\rm{u}}}\times{N_{\rm{s}}}}}\\\
&\quad\quad\quad\quad\quad\quad=\int_{{\bm{a}}_{{n_{\rm{u}}}}^{\rm{u}},{\bm{a}}_{{n_{\rm{s}}}}^{\rm{s}}}{\frac{{{p_{\rm{u}}}\left({{\bm{a}}_{{n_{\rm{u}}}}^{\rm{u}}}\right)\lambda
l^{2}{{\left\|{{\bm{a}}_{{n_{\rm{u}}}}^{\rm{u}}-{\bm{a}}_{{n_{\rm{s}}}}^{\rm{s}}}\right\|}_{2}}}}{{{{\left({{L_{\rm{s}}}{W_{\rm{s}}}{H_{\rm{s}}}}\right)}^{2}}}}},\end{split}$
(61)

where ${{p_{\rm{u}}}\left({{\bm{a}}_{{n_{\rm{u}}}}^{\rm{u}}}\right)}$ is the
distribution function of $n_{\rm{u}}$-th user in the environment. For the
receiving antenna deployed on the $n_{\rm{B}}$-th BS, the probability that the
element in the occlusion matrix ${{\bf{V}}^{{\rm{s\rightarrow
B}}}}\left({{n_{\rm{R}}}}\right)$ is 0 can be expressed as

$\small\begin{split}&p\left({{\bf{V}}_{{n_{\rm{s}}}}^{{\rm{s\rightarrow
B}}}\left({{n_{\rm{R}}}}\right)=0}\right)=1-\frac{{{{\left\|{{{\bf{V}}^{{\rm{s\rightarrow
B}}}}}\right\|}_{1}}}}{{{N_{\rm{s}}}}}\\\
&\quad\quad\quad\quad\quad=\int_{{\bm{a}}_{{n_{\rm{B}}}}^{\rm{B}},{\bm{a}}_{{n_{\rm{s}}}}^{\rm{s}}}{\frac{{{p_{\rm{B}}}\left({{\bm{a}}_{{n_{\rm{B}}}}^{\rm{B}}}\right)\lambda
l^{2}{{\left\|{{\bm{a}}_{{n_{\rm{B}}}}^{\rm{B}}-{\bm{a}}_{{n_{\rm{s}}}}^{\rm{s}}}\right\|}_{2}}}}{{{{\left({{L_{\rm{s}}}{W_{\rm{s}}}{H_{\rm{s}}}}\right)}^{2}}}}},\end{split}$
(62)

where ${{p_{\rm{B}}}\left({{\bm{a}}_{{n_{\rm{B}}}}^{\rm{B}}}\right)}$ is the
distribution function of $n_{\rm{B}}$-th BS in the environment.

###### Proof:

According to the occlusion detection model in Section III-A, when there is a
scatterer in the space of size
$l^{2}{{\left\|{{\bm{a}}_{{n_{\rm{u}}}}^{\rm{u}}-{\bm{a}}_{{n_{\rm{s}}}}^{\rm{s}}}\right\|}_{2}}$
between the $n_{\rm{u}}$-th user and the ${n_{\rm{s}}}$-th pixel, the EM
propagation path is blocked. The location distribution function
${{p_{\rm{u}}}\left({{\bm{a}}_{{n_{\rm{u}}}}^{\rm{u}}}\right)}$ of each user
is independent, and the division of pixels is uniform. Therefore, in the space
of size ${{L_{\rm{s}}}{W_{\rm{s}}}{H_{\rm{s}}}}$, the probability of a
scatterer between the $n_{\rm{u}}$-th user and the ${n_{\rm{s}}}$-th pixel is
denoted as (61). In the occlusion detection model, the analysis methods of the
BS and the user are the same. Therefore, according to (61), replacing
${{p_{\rm{u}}}\left({{\bm{a}}_{{n_{\rm{u}}}}^{\rm{u}}}\right)}$ with
${{p_{\rm{B}}}\left({{\bm{a}}_{{n_{\rm{B}}}}^{\rm{B}}}\right)}$ and replacing
${\bm{a}}_{n_{\rm{s}}}^{\rm{s}}$ with ${\bm{a}}_{n_{\rm{B}}}^{\rm{B}}$, we
obtain (62). ∎

The occlusion matrix ${{\bf{V}}^{{\rm{U\rightarrow s}}}}$ reflects the
different views from different users. In the single-BS environment sensing
scheme and the multi-BS joint environment sensing scheme, the multi-views from
users are the same. As mentioned in Section II-B, when
${{\bf{V}}^{{\rm{U\rightarrow s}}}}$ contains an all-zero column, it means
that all users cannot sense the corresponding pixel, and the pixel is out of
the user sensing range. According to Theorem 1, when the number of users is
${N_{\rm{u}}}$, the probability that the pixels in the environment are outside
the user sensing range is

$\small p\left({{\bf{V}}_{:,{n_{\rm{s}}}}^{{\rm{U\rightarrow
s}}}=\bf{0}}\right)=p^{N_{\rm{u}}}\left({{\bf{V}}_{{n_{\rm{u}}},{n_{\rm{s}}}}^{{\rm{U\rightarrow
s}}}=0}\right).$ (63)

Let ${{\bf{V}}^{{\rm{s\rightarrow B}}}}=\left[{{\bf{V}}^{{\rm{s\rightarrow
B}}}}\left(1\right),{{\bf{V}}^{{\rm{s\rightarrow
B}}}}\left(2\right),\ldots,{{\bf{V}}^{{\rm{s\rightarrow
B}}}}\left({{N_{\rm{R}}}}\right)\right]\in\left\\{0,1\right\\}^{N_{\rm{s}}\times
N_{\rm{R}}}$ and the occlusion matrix ${{\bf{V}}^{{\rm{s\rightarrow B}}}}$
reflects the different views from different BSs. In the single-BS environment
sensing scheme, since the receiving antennas are deployed close to each other,
they have almost the same views to the scatterers in the environment.
Therefore, the probability that a pixel is outside the sensing range of a
single BS is $p\left({{\bf{V}}_{{n_{\rm{s}}}}^{{\rm{s\rightarrow
B}}}\left({{n_{\rm{R}}}}\right)=0}\right)$. In the multi-BS joint environment
sensing scheme, when ${{\bf{V}}^{{\rm{s\rightarrow B}}}}$ contains an all-zero
row, it means that all BSs cannot sense the corresponding pixel, and the pixel
is out of the BS sensing range. According to Theorem 1, since the location
distribution probability of each BS is independent, the probability that a
pixel in the environment is outside the sensing range of the BS is

$\small p\left({{\bf{V}}_{{n_{\rm{s}}},:}^{{\rm{s\rightarrow
B}}}=\bf{0}}\right)=p^{N_{\rm{B}}}\left({{\bf{V}}_{{n_{\rm{s}}}}^{{\rm{s\rightarrow
B}}}\left({{n_{\rm{R}}}}\right)=0}\right).$ (64)

According to (61), (62), (63), and (64), we calculate the mean value
$\bar{N}_{\rm{con}}$ of the number of pixels that cannot be sensed in the
single-BS environment sensing scheme and the mean value $\bar{N}_{\rm{dis}}$
of the number of pixels that cannot be sensed in the multi-BS joint
environment sensing scheme as

$\displaystyle\bar{N}_{\rm{con}}$
$\displaystyle=N_{\rm{s}}\left({p^{N_{\rm{u}}}\left({{\bf{V}}_{{n_{\rm{u}}},{n_{\rm{s}}}}^{{\rm{U\rightarrow
s}}}=0}\right)+p\left({{\bf{V}}_{{n_{\rm{s}}}}^{{\rm{s\rightarrow
B}}}\left({{n_{\rm{R}}}}\right)=0}\right)}\right.$
$\displaystyle\left.{-p^{N_{\rm{u}}}\left({{\bf{V}}_{{n_{\rm{u}}},{n_{\rm{s}}}}^{{\rm{U\rightarrow
s}}}=0}\right)p\left({{\bf{V}}_{{n_{\rm{s}}}}^{{\rm{s\rightarrow
B}}}\left({{n_{\rm{R}}}}\right)=0}\right)}\right),$ (65)
$\displaystyle\bar{N}_{\rm{dis}}$
$\displaystyle=N_{\rm{s}}\left({p^{N_{\rm{u}}}\left({{\bf{V}}_{{n_{\rm{u}}},{n_{\rm{s}}}}^{{\rm{U\rightarrow
s}}}=0}\right)+p^{N_{\rm{B}}}\left({{\bf{V}}_{{n_{\rm{s}}}}^{{\rm{s\rightarrow
B}}}\left({{n_{\rm{R}}}}\right)=0}\right)}\right.$ $\displaystyle-$
$\displaystyle
p^{N_{\rm{u}}}\left({{\bf{V}}_{{n_{\rm{u}}},{n_{\rm{s}}}}^{{\rm{U\rightarrow
s}}}=0}\right)p^{N_{\rm{B}}}\left({{\bf{V}}_{{n_{\rm{s}}}}^{{\rm{s\rightarrow
B}}}\left({{n_{\rm{R}}}}\right)=0}\right)\Big{)}.$ (66)

The multi-BS joint environment sensing has a larger sensing range than single-
BS environment sensing. In practical applications, a multi-BS scheme should be
implemented according to the BS location. When the multi-BS scheme is
difficult to implement, e.g., when the BSs are far away and users cannot
communicate with each BS, a single-BS scheme should be used. In addition, if
there is a failure of environment sensing for some pixels, a larger number of
base stations and users or users after moving locations should be used for
joint sensing.

###  V-B Environment Sensing Accuracy

In this section, we analyze the impact of environmental information $\bm{x}$
on system performance. The mean square error (MSE) between the estimated
environmental information $\bm{\hat{x}}$ and the real environmental
information $\bm{x}$ is used to describe the accuracy of environment sensing,
i.e.,

$\small{\rm{MSE}}=\frac{1}{N_{\rm{s}}}\left\|{{\bm{x}}-{\bm{\hat{x}}}}\right\|_{2}^{2}.$
(67)

According to the CS reconstruction model in Section III-B (6), the
reconstruction problem in (18) is expressed as

$\small{\bm{\hat{x}}}=\arg\mathop{\min}\limits_{\bm{x}}{\left\|{\bm{x}}\right\|_{1}}\quad\quad{\rm{s}}{\rm{.t}}{\rm{.}}\quad{\left\|{{\bf{H}}^{\rm{S}}}-{\bf{\tilde{H}}}{\bm{x}}\right\|_{2}}\leq\varepsilon,$
(68)

where $\varepsilon$ is a slack variable. We consider that the performance of
environment sensing methods for solving the occlusion problem should be as
close as possible to the upper limit in Theorem 2.

(a) The original environment scatterer distribution. (b) The result of the
GAMP algorithm. (c) The result of the Bilinear GAMP algorithm.

(d) The result of the proposed GAMP-MVSVR algorithm.

Figure 6: The original environment scatterer distribution and single-BS
environment sensing results.

(a) The original environment scatterer distribution. (b) The result of the
GAMP algorithm. (c) The result of the Bilinear GAMP algorithm.

(d) The result of the proposed GAMP-MVSVR algorithm.

Figure 7: The original environment scatterer distribution and multi-BS joint
environment sensing results.

###### Theorem 2

The upper bound of the environment sensing error when the CS measurement
matrix ${{\bf{\tilde{H}}}}={\bf{H}}$ under the ideal (no occlusion, the
occlusion detection is completely correct) situation is

$\small{\rm{MSE}}\leq\frac{m\cdot R^{2}\cdot\log{N_{\rm{s}}}}{N_{\rm{s}}\cdot
N_{\rm{u}}\cdot N_{\rm{R}}},$ (69)

where $m$ is a normalized constant, and

$\displaystyle
R=\left\\{\begin{array}[]{l}\\!R_{\rm{con}}=\left\lVert\bm{x}_{\rm{con}}\right\rVert_{1}=\lambda\theta^{\rm{x}}(N_{\rm{s}}-{\bar{N}}_{\rm{con}}),\\\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\text{for single-BS scheme},\\\
\\!R_{\rm{dis}}=\left\lVert\bm{x}_{\rm{dis}}\right\rVert_{1}=\lambda\theta^{\rm{x}}(N_{\rm{s}}-{\bar{N}}_{\rm{dis}}),\\\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\text{for multi-BS
scheme}.\end{array}\right.$ (74)

###### Proof:

According to the CS theory [21],

$\small\left\lVert{\bm{x}}-{\bm{\hat{x}}}\right\rVert_{2}\leq m\cdot
R\cdot\left(N_{\rm{H}}/\log{N_{\rm{s}}}\right)^{1/2-1/{p_{\rm{s}}}},$ (75)

where, $\left\lVert\bm{x}\right\rVert_{p_{\rm{s}}}\leq R$. Pluging (67) and
(68) into (75), we obtain (69). According to (68), let ${p_{\rm{s}}}=1$, we
get the values of $R_{\rm{con}}$ and $R_{\rm{dis}}$. ∎

###### Remark 1

When the number of BS antennas or the number of users increases, the upper
bound of environment sensing error MSE decreases. When the environment
sparsity $\lambda$ and other system parameters remain constant, the number of
pixels $N_{\rm{s}}$ increases (this means a wider sensing range or higher
sensing resolution), the upper bound of environment sensing error MSE
increases. Therefore, wider and more accurate sensing results require more
resource costs.

According to (65) and (66), under the condition that the number of users is
the same and the total number of antennas of all BSs is the same, the number
of environmental scatterers sensed by multiple BSs is more than that of a
single BS, $R_{\rm{dis}}>R_{\rm{con}}$. Therefore, according to (69), for the
scatterers in the sensing range, the upper bound of MSE of multi-BS joint
environment sensing is higher than that of the single-BS scheme. However, in
practical applications, the total number of antennas deployed by multiple BSs
is usually more than that of a single BS. The increase in the number of
antennas $N_{\rm{R}}$ will significantly improve system performance.

We analyze the probability distribution of the occlusion matrix caused by the
environmental scatterers as shown in Corollary 1.

###### Corollary 1

The elements in the occlusion matrix $\bf V$ conform to the Bernoulli
distribution. The probability that the element in the occlusion matrix
${{\bf{V}}}_{{n_{\rm{u}}},{n_{\rm{s}}}}\left(N_{\rm R}\right)$ is 0 can be
expressed as

$\displaystyle p\left({{\bf{V}}_{{n_{\rm{u}}},{n_{\rm{s}}}}\left(N_{\rm
R}\right)=0}\right)$ $\displaystyle=$ $\displaystyle
p\left({{\bf{V}}_{{n_{\rm{u}}},{n_{\rm{s}}}}^{{\rm{U\rightarrow
s}}}=0}\right)+p\left({{\bf{V}}_{{n_{\rm{s}}}}^{{\rm{s\rightarrow
B}}}\left({{n_{\rm{R}}}}\right)=0}\right)$ $\displaystyle-$
$\displaystyle\frac{p\left({{\bf{V}}_{{n_{\rm{u}}},{n_{\rm{s}}}}^{{\rm{U\rightarrow
s}}}=0}\right)p\left({{\bf{V}}_{{n_{\rm{s}}}}^{{\rm{s\rightarrow
B}}}\left({{n_{\rm{R}}}}\right)=0}\right)}{N_{\rm s}}$ (76)

###### Proof:

According to ${\bf{V}}\left({{n_{\rm{R}}}}\right)={{\bf{V}}^{{\rm{U\rightarrow
s}}}}{\rm{diag}}\left({{{\bf{V}}^{{\rm{s\rightarrow
B}}}}\left({{n_{\rm{R}}}}\right)}\right)$, as shown in Theorem 1, the
probability that the element in the occlusion matrix
${{\bf{V}}}_{{n_{\rm{u}}},{n_{\rm{s}}}}\left(N_{\rm R}\right)$ is 0 can be
calculated by ${{\bf{V}}^{{\rm{U\rightarrow s}}}}$ and
${{{\bf{V}}^{{\rm{s\rightarrow B}}}}\left({{n_{\rm{R}}}}\right)}$. ∎

Therefore, the unknown occlusion matrix $\bf V$ is equivalent to causing a
multiplicative Bernoulli perturbation to the known free space channel matrix
$\bf H$ as shown in (19). At present, some works [31, 32] have analyzed the
compressed sensing reconstruction performance under the condition that there
is a specific perturbation in the compressed sensing measurement matrix. In
this paper, the proposed algorithm iteratively eliminates the perturbation to
the measurement matrix caused by the occlusion effect according to the results
of each iteration and obtains accurate imaging results. The system performance
analysis of this iterative detection process will be future work, and
extensive simulation results have verified the effectiveness and convergence
of the proposed algorithm.

##  VI Numerical Results

In this section, we simulate the performance of the algorithm. In order to
reduce the computational cost and reflect the environment sensing accuracy, we
set up a small-scale simulation sensing region, the environmental scatterers
are distributed in the sensing region of $5\rm{m}\times 5\rm{m}\times
5\rm{m}$, and the sensing region is evenly divided into $10\times 10\times 10$
pixels. Multiple BSs and users are randomly distributed in or around the
sensing region. The carrier frequency of the uplink signal is set to
$30\rm{GHz}$. The positions of scatterers distributed in sensing region are
random, and the scattering coefficient is set to
$x_{n_{\rm{s}}}\in\left[0,1\right]$. As shown in the Fig. 7 and Fig. 7, small
cubes are used to represent the position and scattering coefficient of pixels.
The lower the transparency of the small cubes, the higher the scattering
coefficient of the pixel.

In this section, as mentioned in Section IV-A, we choose GAMP and Bilinear
GAMP algorithm as the baseline algorithms to solve the environment sensing
problem under the occlusion effect. Compared with the two baseline algorithms,
in this paper, the proposed GAMP-MVSVR algorithm based on the occlusion effect
significantly improves the accuracy of environment sensing.

Figure 8: The relationship between the number of users and the MSE within the
sensing range of the single-BS scheme.

Fig. 7 shows the results of the single-BS environment sensing, where the gray
cubes represent scatterers that are out of the sensing range due to the
occlusion effect. The system parameters are set to the number of users
$N_{\rm{u}}=20$, a single BS is deployed with a $5\times 5$ array antenna, and
the signal-to-noise ratio (SNR) $E_{\rm{b}}/N_{\rm{0}}=20\rm{dB}$. Compared
with the baseline algorithms, it can be seen that the proposed algorithm
clearly and accurately obtains the shape of the original target.

Fig. 7 shows the results of the multi-BS joint environment sensing. In order
to make the total number of antennas deployed in the environment the same, we
reduce the number of antennas on each BS. The system parameters are set to the
number of users $N_{\rm{u}}=20$, 5 BSs are deployed with $5\times 1$ array
antennas, and the SNR $E_{\rm{b}}/N_{\rm{0}}=20\rm{dB}$. Compared with the
single-BS scheme, as analyzed in (65) and (66), multi-BS joint environment
sensing scheme effectively increases the environment sensing range. In Fig. 7,
there is no scatterer that cannot be sensed even though the occlusion effect
is exist. It can be seen that the sensing results of baseline algorithms are
more blurred than the results of the single-BS scheme. This is because, as
analyzed in Section V-B, when the total number of receiving antennas is the
same, multi-BS joint environment sensing has a larger environment sensing
range, which causes a small decrease in the accuracy of environment sensing
within the sensing range.

Figure 9: The relationship between the number of users and the MSE within the
sensing range of the multi-BS scheme.

Fig. 8 and Fig. 9 show the relationship between the single-BS environment
sensing performance, the multi-BS joint environment sensing performance and
the number of users $N_{\rm{u}}$, respectively. The MSE between the
environmental information within the sensing range and the environmental
information estimated by the algorithm reflects the accuracy of environment
sensing within the sensing range. The system parameters are set as follows:
the single BS in the scenario in Fig. 8 is deployed with a $5\times 5$ array
antenna, and 5 BSs in the scenario in Fig. 9 are deployed with $5\times 1$
array antennas, and the SNR $E_{\rm{b}}/N_{\rm{0}}=20\rm{dB}$.

###### Remark 2

As analyzed in (69), as the number of users increases or the number of
environmental scatterers decreases, the accuracy of environment sensing
increases. Both the proposed algorithm and the baseline algorithm perform
poorly when the number of users is small. But as the number of users
increases, the performance of the proposed algorithm improves significantly.

Figure 10: The relationship between the number of users and the MSE within the
full range of two sensing schemes. Figure 11: The relationship between the SNR
and the MSE within the sensing range of the single-BS scheme. Figure 12: The
relationship between the SNR and the MSE within the sensing range of the
multi-BS scheme. Figure 13: The relationship between the SNR and the MSE
within the full range of two sensing schemes. Figure 14: The convergence
performance of proposed algorithm.

Fig. 10 shows the relationship between the number of users and the MSE within
the full range of two sensing schemes. The MSE between the environmental
information within the full range and the environmental information estimated
by the algorithm reflects the size of the environment sensing range. The
system parameter settings of the two environment sensing schemes are the same
as those in Fig. 8 and Fig. 9, respectively.

###### Remark 3

As analyzed in (69), as the number of users increases or the number of
environmental scatterers decreases, the accuracy of environment sensing
increases. At the same time, as analyzed in (65) and (66), for the full range
of the environment, the multi-BS scheme has a lower sensing error, so it
effectively increases the environment sensing range.

Fig. 11 and Fig. 12 show the relationship between the single-BS environment
sensing performance, the multi-BS joint environment sensing performance and
the SNR $E_{\rm{b}}/N_{\rm{0}}$, respectively. The system parameters are set
as follows: the single BS in the scenario in Fig. 11 is deployed with a
$5\times 5$ array antenna, and 5 BSs in the scenario in Fig. 12 are deployed
with $5\times 1$ array antennas, and the number of users $N_{\rm{u}}=10$.

###### Remark 4

As the SNR increases, the estimated value ${{\bf{\hat{H}}}^{\rm{S}}}$ of the
multipath channel is more accurate, and the accuracy of environment sensing
increases. At the same time, the less sparse the environmental scatterer
distribution, the closer the performance of the proposed algorithm and the
baseline algorithm are. This is because the more environmental scatterers
cause more occlusion, which reduces the performance of all algorithms at the
same time.

Fig. 13 shows the relationship between the SNR and the MSE within the full
range of two sensing schemes. The system parameter settings of the two
environment sensing schemes are the same as those in Fig. 11 and Fig. 12,
respectively.

###### Remark 5

As the SNR increases, the accuracy of environment sensing increases. At the
same time, as analyzed in (65) and (66), for the full range of the
environment, the multi-BS scheme has a lower sensing error, so it effectively
increases the environment sensing range.

As shown in Fig. 14, we provide performance curves under various parameter
settings to verify the convergence of the proposed algorithm. The antenna
settings of the multi-base station scheme are the same as above.

###### Remark 6

As the number of iterations increases, the sensing error MSE gradually
decreases and converges. The sparser the environmental information, the faster
the algorithm converges and the better the system performs. Meanwhile, the
increase in the number of users and the signal-to-noise ratio accelerates
algorithm convergence and improves system performance.

##  VII Conclusion

In this paper, we have designed a multi-view environment sensing scheme taking
into full consideration the occlusion effect in an outdoor scenario. Based on
the occlusion effect, we have proposed a probabilistic reasoning model based
on the factor graph. According to the AMP theory, the GAMP-MVSVR algorithm is
proposed to achieve multi-view environment sensing. The proposed GAMP-MVSVR
algorithm achieves environment sensing by iteratively estimating the
scattering coefficients of the cloud points and their occlusion relationship.
In each iteration, the occlusion relationship between the cloud points of the
sparse environment is recalculated according to the proposed occlusion
detection rule, and in turn, used to estimate the scattering coefficients of
the cloud points. Our proposed algorithm achieves improved sensing performance
with multi-BS collaboration in addition to the multi-views from the users,
which will provide ideas for future ISAC system design.

## Appendix A

According to the factor graph shown in Fig. 5, the logarithmic form of the
passed message is calculated as follows. In the $t$-th iteration, the message
${\mu_{\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}\to{x_{{n_{\rm{s}}}}}}}\left({t,{x_{{n_{\rm{s}}}}}}\right)$
passed from the factor node
$p_{{{\bf{\hat{H}}}^{\rm{S}}}|{{\bf{H}}^{\rm{S}}}}({{{\bf{\hat{H}}}^{\rm{S}}}|{{\bf{H}}^{\rm{S}}}})$
to the variable node $\bm{x}$ is

$\displaystyle{\mu_{\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}\to{x_{{n_{\rm{s}}}}}}}\left({t,{x_{{n_{\rm{s}}}}}}\right)$
$\displaystyle=\log\int_{\left\\{{{{\tilde{h}}_{{n_{\rm{H}}},k}}}\right\\}_{k=1}^{{N_{\rm{s}}}},{{\left\\{{{x_{r}}}\right\\}}_{r\neq{n_{\rm{s}}}}}}{{p_{\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}|h_{{n_{\rm{H}}}}^{\rm{S}}}}\left({\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}\left|{\sum\limits_{k=1}^{N_{\rm{s}}}{{{\tilde{h}}_{{n_{\rm{H}}},k}}{x_{k}}}}\right.}\right)}$
$\displaystyle\times\prod\limits_{r\neq{n_{\rm{s}}}}{\exp\left({{\mu_{{x_{r}}\to\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}}}\left({t,{x_{r}}}\right)}\right)}$
$\displaystyle\times\prod\limits_{k=1}^{{N_{\rm{s}}}}{\exp\left({{\mu_{{{\tilde{h}}_{{n_{\rm{H}}},k}}\to\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}}}\left({t,{{\tilde{h}}_{{n_{\rm{H}}},k}}}\right)}\right)}+c,$
(77)

where the passed message $\mu$ is expressed in logarithmic form. $c$ is a
normalization constant, so that the integral of the exponential passed message
$\exp({\mu_{\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}\to{x_{{n_{\rm{s}}}}}}})$ is 1. In
the subsequent passed message and marginal distribution in this section, the
constant $c$ has the same meaning. In the $(t+1)$-th iteration, the message
${{\mu_{{x_{r}}\to\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}}}\left({t+1,{x_{r}}}\right)}$
passed from the variable node $\bm{x}$ to the factor node
$p_{{{\bf{\hat{H}}}^{\rm{S}}}|{{\bf{H}}^{\rm{S}}}}({{{\bf{\hat{H}}}^{\rm{S}}}|{{\bf{H}}^{\rm{S}}}})$
is

$\small\begin{split}{\mu_{{x_{{n_{\rm{s}}}}}\to\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}}}\left({t+1,{x_{{n_{\rm{s}}}}}}\right)&\\\
=\log{p_{x}}\left({{x_{{n_{\rm{s}}}}}}\right)&+\sum\limits_{k\neq{n_{\rm{H}}}}{{\mu_{\hat{h}_{k}^{\rm{S}}\to{x_{{n_{\rm{s}}}}}}}\left({t,{x_{{n_{\rm{s}}}}}}\right)}+c.\end{split}$
(78)

Let the mean and variance of the exponential form of the message passed in
(78) be ${\bar{x}_{{n_{\rm{s}}},{n_{\rm{H}}}}}\left({t+1}\right)$ and
$\sigma_{{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{x}}\left({t+1}\right)$, respectively.
In the $t$-th iteration, the message
${\mu_{\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}\to{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}}}({t,{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}})$
passed from the factor node
$p_{{{\bf{\hat{H}}}^{\rm{S}}}|{{\bf{H}}^{\rm{S}}}}({{{\bf{\hat{H}}}^{\rm{S}}}|{{\bf{H}}^{\rm{S}}}})$
to the variable node $\bf{\tilde{H}}$ is expressed as

$\displaystyle{\mu_{\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}\to{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}}}\left({t,{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}}\right)$
$\displaystyle=\log\int_{{{\left\\{{{{\tilde{h}}_{{n_{\rm{H}}},r}}}\right\\}}_{r\neq{n_{\rm{s}}}}},\left\\{{{x_{k}}}\right\\}_{k=1}^{{N_{\rm{s}}}}}{{p_{\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}|h_{{n_{\rm{H}}}}^{\rm{S}}}}\left({\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}\left|{\sum\limits_{k=1}^{{N_{\rm{s}}}}{{{\tilde{h}}_{{n_{\rm{H}}},k}}{x_{k}}}}\right.}\right)}$
$\displaystyle\times\prod\limits_{k=1}^{{N_{\rm{s}}}}{\exp\left({{\mu_{{x_{k}}\to\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}}}\left({t,{x_{k}}}\right)}\right)}$
$\displaystyle\times\prod\limits_{r\neq{n_{\rm{s}}}}{\exp\left({{\mu_{{{\tilde{h}}_{{n_{\rm{H}}},r}}\to\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}}}\left({t,{{\tilde{h}}_{{n_{\rm{H}}},r}}}\right)}\right)}+c.$
(79)

In the $(t+1)$-th iteration, similar to (78), the message
${\mu_{{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\to\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}}}({t+1,{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}})$
passed from the variable node $\bf{\tilde{H}}$ to the factor node
$p_{{{\bf{\hat{H}}}^{\rm{S}}}|{{\bf{H}}^{\rm{S}}}}({{{\bf{\hat{H}}}^{\rm{S}}}|{{\bf{H}}^{\rm{S}}}})$
is expressed as

$\displaystyle{\mu_{{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\to\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}}}\left({t+1,{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}}\right)=\log{p_{{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}|{\bm{x}}}}\left({{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}|{\bm{x}}}\right)$
$\displaystyle+\sum\limits_{k\neq{n_{\rm{H}}}}{{\mu_{\hat{h}_{k}^{\rm{S}}\to{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}}}\left({t,{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}}\right)}+c.$
(80)

Let the mean and variance of the exponential form of the message passed in
(80) be ${\bar{h}_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}}\left({t+1}\right)$
and
$\sigma_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{h}}\left({t+1}\right)$,
respectively.

According to the factor graph in Fig. 5, the logarithmic marginal distribution
of the variable $\bf{\tilde{H}}$ and $\bm{x}$ are calculated as follows. With
the iteration of the algorithm, the marginal distributions of variables
gradually converge to their MMSE estimates. The marginal distribution of the
variable $\bm{x}$ is

$\small\begin{split}{\mu_{{x_{{n_{\rm{s}}}}}}}\left({t+1,{x_{{n_{\rm{s}}}}}}\right)&\\\
=\log{p_{x}}&\left({{x_{{n_{\rm{s}}}}}}\right)+\sum\limits_{k=1}^{{N_{\rm{H}}}}{{\mu_{\hat{h}_{k}^{\rm{S}}\to{x_{{n_{\rm{s}}}}}}}\left({t,{x_{{n_{\rm{s}}}}}}\right)}+c.\end{split}$
(81)

Let the mean and variance of the exponential form of the marginal distribution
in (81) be ${\hat{x}_{{n_{\rm{s}}}}}\left({t+1}\right)$ and
$\sigma_{{n_{\rm{s}}}}^{\rm{x}}\left({t+1}\right)$, respectively. The marginal
distribution of the variable $\bf{\tilde{H}}$ is

$\small\begin{split}&{\mu_{{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}}}\left({t+1,{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}}\right)=\log{p_{{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}|{\bm{x}}}}\left({{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}|{\bm{x}}}\right)\\\
&+\sum\limits_{k=1}^{{N_{\rm{H}}}}{{\mu_{\hat{h}_{k}^{\rm{S}}\to{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}}}\left({t,{{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}}\right)}+c.\end{split}$
(82)

Let the mean and variance of the exponential form of the marginal distribution
in (82) be ${\hat{h}_{{n_{\rm{H}}},{n_{\rm{s}}}}}\left({t+1}\right)$ and
$\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{h}}\left({t+1}\right)$, respectively.

## Appendix B

Here we derive the approximate result (25). The Roman font is used to
represent the elements ${{\rm{h}}_{n_{\rm{H}}}^{\rm{S}}}$,
$\rm{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}$, and $\rm{x}_{n_{\rm{s}}}$ in the
matrix ${{\bf{H}}^{\rm{S}}}$, $\bf{\tilde{H}}$, and $\bm{x}$ in the case of
CLT. Let
$\Delta{h_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}}={\rm{\tilde{h}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}-{\bar{h}_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}}\left(t\right)$
and
$\Delta{x_{{n_{\rm{s}}},{n_{\rm{H}}}}}={{\rm{x}}_{{n_{\rm{s}}}}}-{\bar{x}_{{n_{\rm{s}}},{n_{\rm{H}}}}}\left(t\right)$,
then the mean is $0$, and the variance is
$\sigma_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{h}}\left({t}\right)$ and
$\sigma_{{n_{\rm{s}}}}^{\rm{x}}\left({t}\right)$.

In the $t$ iteration, the sparsity $\lambda$ in the prior distribution of
environmental information ${p_{\bm{x}}}\left({\bm{x}}\right)$ remains
constant. As mentioned in Section IV-C, we consider that variable $\bm{x}$ is
independent of variable $\bf{\tilde{H}}$ in the $t$-th iteration. Therefore,
the mean ${\hat{h}_{{n_{\rm{H}}},{n_{\rm{s}}}}}$ and the variance
$\sigma_{{n_{\rm{H}}},{n_{\rm{s}}}}^{\rm{h}}$ of variable $\bf{\tilde{H}}$ are
independent of variable $\bm{x}$. ${{\bf{H}}^{\rm{S}}}$ is expressed as

$\displaystyle\rm{h}_{{n_{\rm{H}}}}^{\rm{S}}$
$\displaystyle=\sum\limits_{k=1}^{N}{{{{\rm{\tilde{h}}}}_{{n_{\rm{H}}},{n_{\rm{s}}}}}{{\rm{x}}_{{n_{\rm{s}}}}}}$
$\displaystyle=\left({\Delta{{\hat{h}}_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}}+{{\bar{h}}_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}}\left(t\right)}\right){{\rm{x}}_{{n_{\rm{s}}}}}$
$\displaystyle+\sum\limits_{k\neq{n_{\rm{s}}}}\left({{{\bar{h}}_{{n_{\rm{H}}}k,{n_{\rm{H}}}}}\left(t\right){{\bar{x}}_{k,{n_{\rm{H}}}}}\left(t\right)+{{\bar{h}}_{{n_{\rm{H}}}k,{n_{\rm{H}}}}}\left(t\right)\Delta{x_{k,{n_{\rm{H}}}}}}\right.$
$\displaystyle+\left.{\Delta{{h}_{{n_{\rm{H}}}k,{n_{\rm{H}}}}}{{\bar{x}}_{k,{n_{\rm{H}}}}}\left(t\right)+\Delta{{h}_{{n_{\rm{H}}}k,{n_{\rm{H}}}}}\Delta{x_{k,{n_{\rm{H}}}}}}\right).$
(83)

Based on the CLT condition, ${{\rm{x}}_{{n_{\rm{s}}}}}={x_{{n_{\rm{s}}}}}$,
the mean and variance of $\rm{h}_{{n_{\rm{H}}}}^{\rm{S}}$ in (83) is written
as

$\small{\mathbb{E}}\left\\{{{\rm{h}}_{{n_{\rm{H}}}}^{\rm{S}}}\right\\}={\bar{h}_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}}\left(t\right){x_{{n_{\rm{s}}}}}+{\bar{p}_{{n_{\rm{s}}},{n_{\rm{H}}}}}\left(t\right),$
(84)
$\small{\mathbb{D}}\left\\{{{\rm{h}}_{{n_{\rm{H}}}}^{\rm{S}}}\right\\}=\sigma_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{h}}\left(t\right)x_{{n_{\rm{s}}}}^{2}+\sigma_{{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{p}}\left(t\right),$
(85)

where

$\small{\bar{p}_{{n_{\rm{s}}},{n_{\rm{H}}}}}\left(t\right)=\sum\limits_{k\neq{n_{\rm{s}}}}{{{\bar{h}}_{{n_{\rm{H}}}k,{n_{\rm{H}}}}}\left(t\right){{\bar{x}}_{k,{n_{\rm{H}}}}}\left(t\right)},$
(86)
$\small\begin{split}&\sigma_{{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{p}}\left(t\right)=\sum\limits_{k\neq{n_{\rm{s}}}}{\left({\bar{h}_{{n_{\rm{H}}}k,{n_{\rm{H}}}}^{2}\left(t\right)\sigma_{k,{n_{\rm{H}}}}^{\rm{x}}\left(t\right)}\right.}\\\
&{\left.{+\sigma_{{n_{\rm{H}}}k,{n_{\rm{H}}}}^{\rm{h}}\left(t\right)\bar{x}_{k,{n_{\rm{H}}}}^{2}\left(t\right)+\sigma_{{n_{\rm{H}}}k,{n_{\rm{H}}}}^{\rm{h}}\left(t\right)\sigma_{k,{n_{\rm{H}}}}^{\rm{x}}\left(t\right)}\right)}.\end{split}$
(87)

Under CLT conditions, we approximate the last two product terms of message
${\mu_{\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}\to{x_{{n_{\rm{s}}}}}}}\left({t,{x_{{n_{\rm{s}}}}}}\right)$
in (77) to the marginal distribution of variable
$\rm{h}_{{n_{\rm{H}}}}^{\rm{S}}$ in (83). Therefore, according to the
conditional Gaussian approximation, we obtain (25).

## Appendix C

Here we derive the approximate result (31). Pluging (27) and (28) into (25) we
obtain

$\displaystyle{F_{{n_{\rm{H}}}}}\left({{{\bar{h}}_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}}\left(t\right){x_{{n_{\rm{s}}}}}+{{\bar{p}}_{{n_{\rm{s}}},{n_{\rm{H}}}}}\left(t\right),}\right.$
$\displaystyle\quad\left.{\sigma_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{h}}\left(t\right)x_{{n_{\rm{s}}}}^{2}+\sigma_{{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{p}}\left(t\right);\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}}\right)+c$
$\displaystyle\approx{F_{{n_{\rm{H}}}}}\left({{{\bar{h}}_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}}\left(t\right)\left({{x_{{n_{\rm{s}}}}}-{{\hat{x}}_{{n_{\rm{s}}}}}\left(t\right)}\right)+{{\bar{p}}_{{n_{\rm{H}}}}}\left(t\right),}\right.$
$\displaystyle\quad\left.{\sigma_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{h}}\left(t\right)\left({x_{{n_{\rm{s}}}}^{2}-\hat{x}_{{n_{\rm{s}}}}^{2}\left(t\right)}\right)+\sigma_{{n_{\rm{H}}}}^{\rm{p}}\left(t\right);\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}}\right)+c,$
(88)

where, since $N_{\rm{s}}\rightarrow\infty$, we drop infinitesimal terms such
as
${{\bar{h}}_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}}\left(t\right)\left({{{\hat{x}}_{{n_{\rm{s}}}}}\left(t\right)-{{\bar{x}}_{{n_{\rm{s}}},{n_{\rm{H}}}}}\left(t\right)}\right)$
and so on.

As with most AMP algorithms [24, 25, 28], we make the same derivation. We
perform Taylor series expansion of variable ${x_{{n_{\rm{s}}}}}$ in (88) at
point ${{{\hat{x}}_{{n_{\rm{s}}}}}\left(t\right)}$, the passed message in (25)
is approximated as

$\displaystyle{\mu_{\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}\to{x_{{n_{\rm{s}}}}}}}\left({t,{x_{{n_{\rm{s}}}}}}\right)\approx
c+{F_{{n_{\rm{H}}}}}\left({{{\bar{p}}_{{n_{\rm{H}}}}}\left(t\right),\sigma_{{n_{\rm{H}}}}^{\rm{p}}\left(t\right);\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}}\right)$
$\displaystyle+{{\bar{h}}_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}}\left(t\right)\left({{x_{{n_{\rm{s}}}}}-{{\hat{x}}_{{n_{\rm{s}}}}}\left(t\right)}\right)$
$\displaystyle\quad\quad\quad\quad\times{{F^{\prime}}_{{n_{\rm{H}}},1}}\left({{{\bar{p}}_{{n_{\rm{H}}}}}\left(t\right),\sigma_{{n_{\rm{H}}}}^{\rm{p}}\left(t\right);\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}}\right)$
$\displaystyle+2\sigma_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{h}}\left(t\right){{\hat{x}}_{{n_{\rm{s}}}}}\left(t\right)\left({{x_{{n_{\rm{s}}}}}-{{\hat{x}}_{{n_{\rm{s}}}}}\left(t\right)}\right)$
$\displaystyle\quad\quad\quad\quad\times{{F^{\prime}}_{{n_{\rm{H}}},2}}\left({{{\bar{p}}_{{n_{\rm{H}}}}}\left(t\right),\sigma_{{n_{\rm{H}}}}^{\rm{p}}\left(t\right);\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}}\right)$
$\displaystyle+\frac{1}{2}\bar{h}_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}^{2}\left(t\right){\left({{x_{{n_{\rm{s}}}}}-{{\hat{x}}_{{n_{\rm{s}}}}}\left(t\right)}\right)^{2}}$
$\displaystyle\quad\quad\quad\quad\times{{F^{\prime\prime}}_{{n_{\rm{H}}},1}}\left({{{\bar{p}}_{{n_{\rm{H}}}}}\left(t\right),\sigma_{{n_{\rm{H}}}}^{\rm{p}}\left(t\right);\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}}\right)$
$\displaystyle+\sigma_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{h}}\left(t\right){\left({{x_{{n_{\rm{s}}}}}-{{\hat{x}}_{{n_{\rm{s}}}}}\left(t\right)}\right)^{2}}$
$\displaystyle\quad\quad\quad\quad\times{{F^{\prime}}_{{n_{\rm{H}}},2}}\left({{{\bar{p}}_{{n_{\rm{H}}}}}\left(t\right),\sigma_{{n_{\rm{H}}}}^{\rm{p}}\left(t\right);\hat{h}_{{n_{\rm{H}}}}^{\rm{S}}}\right).$
(89)

where ${F^{\prime}_{{n_{\rm{H}}},1}}$ and
${F^{\prime\prime}_{{n_{\rm{H}}},1}}$ represent the first and second
derivatives of the first argument of ${F_{{n_{\rm{H}}}}}$, and
${F^{\prime}_{{n_{\rm{H}}},2}}$ represents the first derivative of the second
argument of ${F_{{n_{\rm{H}}}}}$. We drop infinitesimal terms such as the
second derivative term of the second argument of ${F_{{n_{\rm{H}}}}}$.

According to the derivation in Appendix D that

$\small{F^{\prime}_{n_{\rm{H}},2}}=\frac{1}{2}\left[{{{\left({{F^{\prime}_{n_{\rm{H}},1}}}\right)}^{2}}+{F^{\prime\prime}_{n_{\rm{H}},1}}}\right].$
(90)

We plug (29), (30) and (90) into (89), and absorb the constant term into the
constant $c$, we obtain (31), where under the CLT condition, we use
${\hat{h}_{{n_{\rm{H}}},{n_{\rm{s}}}}^{2}\left(t\right)}$ instead of
$\bar{h}_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}^{2}\left(t\right)$, and
${\sigma_{{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{h}}\left(t\right)}$ instead of
$\sigma_{{n_{\rm{H}}}{n_{\rm{s}}},{n_{\rm{H}}}}^{\rm{h}}\left(t\right)$.

## Appendix D

Here we derive the equation (90). According to (26), we know the following

$\small\frac{{\partial\exp\left({{F_{{n_{\rm{H}}}}}}\right)}}{{\partial{\sigma^{p}}}}={F^{\prime}_{{n_{\rm{H}}},2}}\exp\left({{F_{{n_{\rm{H}}}}}}\right),$
(91)
$\small\frac{{{\partial^{2}}\exp\left({{F_{{n_{\rm{H}}}}}}\right)}}{{\partial{p^{2}}}}=\left[{{{\left({{F^{\prime}_{{n_{\rm{H}}},1}}}\right)}^{2}}+{F^{\prime\prime}_{{n_{\rm{H}}},1}}}\right]\exp\left({{F_{{n_{\rm{H}}}}}}\right).$
(92)

According to (26), we obtain

$\displaystyle\frac{{\partial\exp\left({{F_{{n_{\rm{H}}}}}}\right)}}{{\partial{\sigma^{\rm{p}}}}}$
$\displaystyle=\int_{{h^{\rm{S}}}}{{p_{{{\hat{h}}^{\rm{S}}}|{h^{\rm{S}}}}}\left({{{\hat{h}}^{\rm{S}}}|{h^{\rm{S}}}}\right)\frac{1}{{\sqrt{2\pi}}}\left[{-\frac{1}{2}{{\left({{\sigma^{\rm{p}}}}\right)}^{{\rm{-3/2}}}}}\right.}$
$\displaystyle{\left.{-{{\left({{\sigma^{\rm{p}}}}\right)}^{{\rm{-5/2}}}}\frac{{{{\left({{h^{\rm{S}}}-p}\right)}^{2}}}}{2}}\right]\exp\left({\frac{{{{\left({{h^{\rm{S}}}-p}\right)}^{2}}}}{{2{\sigma^{\rm{p}}}}}}\right)}$
$\displaystyle=\frac{1}{2}\frac{{{\partial^{2}}\exp\left({{F_{{n_{\rm{H}}}}}}\right)}}{{\partial{p^{2}}}}.$
(93)

Plugging (91) and (92) into (93), we obtain (90).

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