# Integrating Intelligent Reflecting Surface into Base Station: Architecture,
Channel Model, and Passive Reflection Design

Yuwei Huang, Student Member, IEEE, Lipeng Zhu, Member, IEEE, and Rui Zhang,
Fellow, IEEE Y. Huang is with the Department of Electrical and Computer
Engineering, National University of Singapore, Singapore 117583, and also with
the NUS Graduate School, National University of Singapore, Singapore 119077
(e-mail: yuweihuang@u.nus.edu). L. Zhu is with the Department of Electrical
and Computer Engineering, National University of Singapore, Singapore 117583
(e-mail:zhulp@nus.edu.sg). R. Zhang is with The Chinese University of Hong
Kong, Shenzhen, and Shenzhen Research Institute of Big Data, Shenzhen, China
518172 (e-mail: rzhang@cuhk.edu.cn). He is also with the Department of
Electrical and Computer Engineering, National University of Singapore,
Singapore 117583 (e-mail: elezhang@nus.edu.sg).

###### Abstract

Intelligent reflecting surface (IRS) has emerged as a cost-efficient technique
to improve the wireless network’s capacity and performance. Existing works on
IRS have mainly considered IRS being deployed in the environment to
dynamically control the wireless channels between the base station (BS) and
its served users in favor of their communications. In contrast, we propose in
this paper a new integrated IRS-BS architecture by deploying IRSs inside the
BS’s antenna radome to directly reconfigure the signal radiation to/from the
BS’s antennas. In other words, the IRSs can be considered as auxiliary passive
arrays with real-time reconfigurability equipped at the BS to enhance its
communication performance cost-effectively. Since the distance between the
integrated IRSs and BS’s antenna array is practically small (in the order of
several to tens of wavelengths), the path loss among them is significantly
reduced as compared to conventional IRS deployed much farther away from the
BS, while the real-time control of the IRS’s reflection by the BS becomes
easier to implement. However, the resultant near-field channel model also
becomes drastically different from its far-field counterpart for conventional
far-away IRSs in the literature. Thus, we propose an element-wise channel
model for IRS to characterize the channel vector between each single-antenna
user and the antenna array of the BS, which includes the direct (without any
IRS’s reflection) as well as the single and double IRS-reflection channel
components. Based on this channel model, we formulate a problem to optimize
the reflection coefficients of all IRS reflecting elements for maximizing the
uplink sum-rate of the users. By considering two typical cases with/without
perfect channel state information (CSI) at the BS, the formulated problem is
solved efficiently by adopting the successive refinement method and iterative
random phase algorithm (IRPA), respectively. Numerical results validate the
substantial capacity gain of the integrated IRS-BS architecture over the
conventional multi-antenna BS without integrated IRS. Moreover, the proposed
algorithms significantly outperform other benchmark schemes in terms of sum-
rate, and the IRPA without CSI can approach the performance upper bound with
perfect CSI as the training overhead increases.

###### Index Terms:

Intelligent reflecting surface (IRS), integrated IRS-BS architecture, element-
wise channel model, passive reflection design.

##  I Introduction

With the development of digitally-controlled metamaterial technologies,
intelligent reflecting surface (IRS) has emerged as a cost-efficient technique
to achieve smart and reconfigurable radio environment for future wireless
communication systems. Specifically, IRS is a programmable metasurface
consisting of a large number of passive reflecting elements, whose amplitude
and/or phase shifts can be individually controlled in real time, thereby
enabling dynamic control over the wireless propagation channel for a variety
of purposes (e.g., passive beamforming, multi-path diversity, interference
nulling/cancellation, and so on) [1, 2, 3]. Moreover, IRS dispenses with
transmit and receive radio frequency (RF) chains and reflects ambient signals
only passively, which requires low hardware cost and energy consumption. Thus,
IRS has been extensively investigated in assorted wireless communication
systems, such as orthogonal frequency division multiplexing (OFDM)-based
wideband communications [4, 5], multi-antenna and/or multi-user communications
[6, 7], multi-cell networks [8, 9], non-orthogonal multiple access (NOMA)
systems [10, 11], relaying communications [12, 13], physical-layer security
[14, 15], unmanned aerial vehicle (UAV) communications [16, 17], mobile edge
computing (MEC) [18, 19], etc.

To fully reap their benefits, IRSs need to be properly deployed between the
base station (BS) (or access point (AP)) and its served users to decrease the
product-distance path loss of the BS-IRS-user cascaded channels [1]. Towards
this end, a promising solution is to deploy the IRS closer to the user
terminals, e.g., in hotpot area, cell edge, or vehicles. Following this
approach, the authors in [4] considered a single-user OFDM communication
system by deploying an IRS in the vicinity of a single-antenna user, where the
passive reflection of the IRS was optimized to maximize the achievable rate.
In [20], a user-side IRS was employed to assist in the communication between a
multi-antenna BS and a multi-antenna user, where the capacity of the
considered multiple-input multiple-output (MIMO) system was maximized by
alternately optimizing the passive reflection of the IRS and the covariance
matrix of the transmit signals. Besides, the authors in [21] employed an IRS
in a high-speed vehicle to enhance the communication performance between the
users on board and a remote BS, where an efficient two-stage transmission
protocol was proposed to conduct channel estimation and IRS reflection
optimization. In addition to the user-side deployment, IRSs can also be
deployed in the vicinity of the BS to assist in its communication with users
distributed at different locations. In this regard, the authors in [22]
proposed an IRS-empowered BS architecture by deploying multiple IRSs close to
the BS (e.g., in the range of several to tens of meters), where a novel user-
IRS association design was proposed to select a certain number of cascaded
user-IRS-BS channels for estimation under a pre-determined channel training
overhead, and then the reflection coefficients of all IRSs were jointly
optimized to maximize the minimum achievable rate among the users based on the
estimated channel state information (CSI). Furthermore, a hybrid deployment
strategy was proposed in [23] to combine the complementary advantages of the
BS- and user-side IRSs, which can yield higher design flexibility and provide
additional passive beamforming gain by exploiting the inter-IRS reflections.
In this context, the authors in [24] investigated a double-IRS system with one
IRS deployed near the multi-antenna BS and the other IRS deployed closer to
the multi-antenna user, where all the involved channels were assumed to have
dominant line-of-sight (LoS) paths, and the reflection coefficients of the two
IRSs were jointly optimized to maximize the capacity of the MIMO system.
Subsequently, the authors in [25] extended the double-IRS system to a more
practical scenario with multiple users, where an efficient algorithm by
exploiting alternating optimization was proposed for cooperative IRSs’
beamforming design to maximize the minimum signal-to-interference-plus-noise
ratio (SINR) among all users.

Note that both user-side and BS-side deployments in the aforementioned works
can be categorized into the standalone IRS, where the IRSs are deployed in the
environment and thus sufficiently far away (i.e., more than hundreds of
carrier wavelengths) from the users/BSs. In practice, such deployment
strategies still result in non-negligible product-distance path loss of the
BS-IRS-user cascaded channels, which may limit the performance gain provided
by IRS. Besides, to achieve full passive beamforming gain, the IRSs should be
connected to BSs via wireless/wired links to execute reflection control for
both channel estimation and data transmission, which requires a prohibitive
signaling overhead between the BS and IRSs, especially for the case of large-
size IRS and/or large number of IRSs.

Figure 1: Illustration of an integrated IRS-BS architecture.

To circumvent the above difficulties, in this paper we propose a new
integrated IRS-BS architecture, where four IRSs with different orientations
and an antenna array are deployed within the same cuboid antenna radome at the
BS, as shown in Fig. 1. Due to the ultra-short distance between the IRSs and
BS’s antenna array (i.e., in the order of several to tens of wavelengths), the
path loss of the reflection channels via IRSs can be significantly reduced. In
addition, besides the single-IRS reflections between the users and the antenna
array of the BS, the double reflections among the IRSs become much stronger
than their counterparts in the conventional double-IRS systems with IRSs
deployed much farther away from each other [25, 24], and thus their effect to
the system performance becomes significant and may even dominate over the
single-IRS reflections. Moreover, the signaling overhead between the BS and
IRSs is generally reduced because of their integrated architecture. Despite
the above mentioned advantages, new technical issues also arise, which need to
be investigated for our proposed integrated IRS-BS architecture. First, the
conventional uniform plane wave (UPW) assumption cannot be directly applied to
model the channels between the BS antenna array and each of IRSs as a whole
since the far-field condition is not satisfied between them [26, 27, 28],
which motivates us to develop a more practical and sophisticated channel model
for the proposed integrated IRS-BS system. Second, the signals from/to the
users experience single and double reflections by multiple integrated IRSs,
and the coupling effect of these channel components makes the optimization of
IRS passive reflection more intractable as compared to the conventional design
considering single-IRS reflections only. Third, the CSI becomes more
substantial and is harder to obtain in practice due to the additional inter-
IRS reflections involved, despite various channel estimation methods have been
proposed for single-IRS [29, 30, 31, 32] or double-IRS systems [33] (see [35,
34], and the reference therein). This is because our proposed architecture has
multiple IRSs integrated very close to the BS’s antenna array, and thus the
number of channel coefficients to be estimated is much larger than that in
conventional single-/double-IRS systems, which may require extremely high
pilot overhead for estimating all reflection channels via all IRSs. To tackle
the above challenges, we develop efficient algorithms for designing the IRS
reflection coefficients under the case with/without perfect CSI at the BS,
respectively. The main contributions of this paper are summarized as follows.

  * •

First, considering the ultra-short distance between the IRSs and BS’s antenna
array, we propose an element-wise channel model for IRS to characterize the
channel vector between each single-antenna user and the antenna array of the
BS, which consists of the direct (without any IRS’s reflection) as well as the
single and double IRS-reflection channel components. Specifically, the UPW
propagation is utilized to model the channels from each reflecting element
(instead of the whole surface of the IRS) to each antenna at the BS as well as
between the reflecting elements of different IRSs, where the reflection gain
of each IRS reflecting element considers the non-isotropic electromagnetic
response of signals incident from/reflected to different directions. In
addition, to integrate more reflecting elements into the antenna radome, we
generalize the integrated IRS-BS architecture by employing modular antenna
arrays, where the whole antenna array is divided into several antenna modules,
and each antenna module is surrounded by four smaller-size IRSs.

  * •

Next, we formulate an optimization problem to maximize the uplink sum-rate of
the users by designing the reflection coefficients of all IRSs, which is non-
convex and thus difficult to solve. To tackle this problem, we consider two
typical cases with/without perfect CSI at the BS and develop efficient
algorithms for solving the problems in both cases. For the case with perfect
CSI, we adopt the successive refinement method to alternately update the
reflection coefficient of each reflecting element by fixing those of other
reflecting elements. While for the case without CSI, we extend the
conventional random phase algorithm (RPA) to the iterative random phase
algorithm (IRPA) for optimizing the reflection coefficients of multiple IRSs
in an iterative manner. Specifically, we first decouple the reflection
coefficients of multiple IRSs, and then adopt the RPA to update the reflection
coefficients of each IRS by fixing those of other IRSs.

  * •

Finally, numerical results are provided to validate the efficacy of the
proposed integrated IRS-BS architecture over the conventional multi-antenna BS
without integrated IRS. The proposed algorithms for IRS passive reflection
design are shown to achieve considerable performance gain as compared to other
benchmark schemes in terms of sum-rate, and the proposed IRPA without perfect
CSI can approach the performance upper bound with perfect CSI as the training
overhead increases. Besides, the results demonstrate that the double-
reflection channel components have a more dominant impact on the achievable
sum-rate than the single-reflection channel components. Moreover, the effects
of key system parameters, including the total number of reflecting elements,
the suspension angle of the BS antenna radome, and the number of antenna
modules in the generalized IRS-BS architecture, are also evaluated. The
results show that the proposed integrated IRS-BS architecture is a cost-
efficient design for future-generation BSs/APs in wireless networks, since the
passive IRSs deployed in the near-field of the BS antenna array can serve as
auxiliary passive arrays that can be configured jointly with the antenna array
to significantly enhance the degrees of freedom (DoFs) for dynamically
controlling the signal radiation to/from the BS.

The rest of this paper is organized as follows. Section II presents our
proposed system architecture and channel model. Section III presents the
problem formulation and IRS passive reflection design with/without perfect
CSI. Section IV provides numerical results to verify the efficacy of our
proposed integrated IRS-BS architecture and algorithms for IRS passive
reflection design. Finally, Section V concludes this paper.

Notation: In this paper, scalars, vectors, and matrices are denoted by italic,
bold-face lower-case, and bold-face upper-case letters, respectively. For a
matrix $A$, its transpose, conjugate transpose, and determinant are denoted as
$\mbox{\boldmath{$A$}}^{T}$, $\mbox{\boldmath{$A$}}^{H}$, and
$\det(\mbox{\boldmath{$A$}})$, respectively. $\mbox{\boldmath{$I$}}_{M}$
denotes the identity matrix of size $M$. $\mathbb{C}^{x\times y}$ denotes the
set of $x\times y$-dimensional complex-valued matrices, and
$\mathbb{R}^{x\times y}$ denotes the set of $x\times y$-dimensional real-
valued matrices. For a complex number $s$, $s^{*}$ and $|s|$ denote its
conjugate and amplitude, respectively. $s\sim\mathcal{CN}(0,\sigma^{2})$ means
that $s$ is a circularly symmetric complex Gaussian (CSCG) random variable
with mean zero and variance $\sigma^{2}$. $i$ denotes the imaginary unit,
i.e., $i=\sqrt{-1}$. For a vector $a$, $\text{diag}(\mbox{\boldmath{$a$}})$
denotes a diagonal matrix whose diagonal elements are specified by $a$, and
$\lVert\mbox{\boldmath{$a$}}\rVert$ denotes its 2-norm. Notation
$\log_{2}(\cdot)$ denotes the logarithm function with base $2$; $\otimes$
represents the Kronecker product; and $\mathcal{O}(\cdot)$ denotes the
Landau’s symbol to describe the order of complexity.

##  II System Architecture and Channel Model

###  II-A System Architecture

Figure 2: System model of the integrated IRS-BS and its served users.

As shown in Fig. 1, we consider an integrated IRS-BS/AP architecture, where an
antenna array and $J=4$ IRSs are deployed within a cuboid antenna radome, with
$\mathcal{J}\triangleq\\{1,2,3,4\\}$ denoting the set of all IRSs.
Specifically, the BS’s antenna array is deployed at the center of the back
surface of the antenna radome, while IRSs 1–4 are respectively deployed at its
left, right, top, and bottom surfaces perpendicular to the antenna array. We
consider a three-dimensional (3D) local coordinate system (LCS) here, where
the center of the antenna array is set as the origin $O$, and the antenna
array is located on the $x$-$O$-$z$ plane. The antenna array is assumed to be
a uniform planar array (UPA) with the size of $M=M_{x}\times M_{z}$, where
$M_{x}$ and $M_{z}$ are the number of antennas along axes $x$ and $z$,
respectively. The size of each IRS $j$ is denoted by $N_{j}=N_{j,1}\times
N_{j,2}$, $j\in\mathcal{J}$, with $N_{j,1}$ and $N_{j,2}$ representing the
number of reflecting elements along axes $y$ and $z$ (or $x$),
respectively111In practice, since the antenna radome of cellular BSs or WiFi
APs has a limited size (e.g., in the order of several to tens of carrier
wavelengths), the number of IRS reflecting elements that can be integrated
into the antenna radome is limited. However, the distance between the IRSs and
the antenna array of the BS is small, which can significantly reduce the path
loss of the reflected signals by IRSs as compared to conventional IRSs
deployed much farther away from the BS/AP [22, 23, 36].. The sets of elements
for the BS UPA and IRS $j$, $j\in\mathcal{J}$, are denoted by
$\mathcal{M}\triangleq\\{1,2,\cdots,M\\}$ and
$\mathcal{N}_{j}\triangleq\\{1,2,\cdots,N_{j}\\}$, respectively. In the
considered LCS, the location of the $m$-th antenna is denoted as
$\mbox{\boldmath{$s$}}_{m}=[s_{m,x},s_{m,y},s_{m,z}]^{T}\in\mathbb{R}^{3\times
1},~{}m\in\mathcal{M}$, and that of the $n_{j}$-th reflecting element of IRS
$j$ is denoted as
$\mbox{\boldmath{$w$}}_{j,n_{j}}=[w_{j,n_{j},x},w_{j,n_{j},y},w_{j,n_{j},z}]^{T}\in\mathbb{R}^{3\times
1},~{}j\in\mathcal{J},~{}n_{j}\in\mathcal{N}_{j}$. Since the IRS can only
reflect signals to/from its front half-space, each IRS should be deployed to
face the antenna array and all other IRSs so as to efficiently reflect signals
to/from them. Denote $\mbox{\boldmath{$\delta$}}_{j}$ as the normal vector of
the plane which IRS $j$ is located in, with
$\|\mbox{\boldmath{$\delta$}}_{j}\|=1,~{}j\in\mathcal{J}$. According to the
orientation of the IRSs, we have
$\mbox{\boldmath{$\delta$}}_{1}=[-1,0,0]^{T}$,
$\mbox{\boldmath{$\delta$}}_{2}=[1,0,0]^{T}$,
$\mbox{\boldmath{$\delta$}}_{3}=[0,0,-1]^{T}$, and
$\mbox{\boldmath{$\delta$}}_{4}=[0,0,1]^{T}$. Moreover, as shown in Fig. 2,
the antenna radome is deployed at altitude $H_{AR}$ with a suspension angle of
$\theta_{tilt}\in[0,\pi/2]$ with respect to (w.r.t.) the vertical direction
(which is perpendicular to the ground). Note that the suspension angle
$\theta_{tilt}$ can be adjusted according to the requirement of specific
communication applications, such as $\theta_{tilt}\in[0,\pi/12]$ for cellular
BSs [37] (see Fig. 3(a)) and $\theta_{tilt}=\pi/2$ for WiFi APs mounted on the
ceiling (see Fig. 3 (b)). In this paper, we focus on the uplink transmission
from $K$ users to the BS, where the set of users is denoted by
$\mathcal{K}\triangleq\\{1,2,\cdots,K\\}$.

Figure 3: Examples of integrated IRS-BS/AP.

###  II-B Channel Model

Let $\vartheta_{j,n_{j}}$ denote the unit-modulus reflection coefficient for
the $n_{j}$-th element of IRS $j$ with
$|\vartheta_{j,n_{j}}|=1,~{}j\in\mathcal{J},n_{j}\in\mathcal{N}_{j}$, and
$\mbox{\boldmath{$\Theta$}}\triangleq\\{\vartheta_{j,n_{j}},~{}j\in\mathcal{J},n_{j}\in\mathcal{N}_{j}\\}$.
As shown in Fig. 2, the effective channel between each user and the BS’s
antenna array (i.e., UPA) in our proposed integrated IRS-BS architecture is a
superposition of the direct channel component from the user to the antenna
array of the BS without any IRS’s reflection and the reflection channel
components via IRSs. Due to the high path loss induced by multiple reflections
among the IRSs, the power of the reflection channels is dominated by the
single-reflection and double-reflection channel components. Thus, the
effective channel from user $k$ to the antenna array of the BS is given by

$\displaystyle\mbox{\boldmath{$h$}}_{k}(\mbox{\boldmath{$\Theta$}})=\tilde{\mbox{\boldmath{$h$}}}_{k}+\underbrace{\sum_{j=1}^{J}\sum_{n_{j}=1}^{N_{j}}\mbox{\boldmath{$f$}}_{k,j}^{n_{j}}\vartheta_{j,n_{j}}}_{\text{single-
reflection}}+\underbrace{\sum_{j=1}^{J}\sum_{q=1,q\neq
j}^{J}\sum_{n_{j}=1}^{N_{j}}\sum_{n_{q}=1}^{N_{q}}\mbox{\boldmath{$g$}}_{k,j,q}^{n_{j},n_{q}}\vartheta_{j,n_{j}}\vartheta_{q,n_{q}}}_{\text{double-
reflection}},$ (1)

where
$\tilde{\mbox{\boldmath{$h$}}}_{k}=[\tilde{h}_{k,1},\cdots,\tilde{h}_{k,M}]^{T}\in\mathbb{C}^{M\times
1}$ represents the direct channel component from user $k$ to the antenna array
of the BS;
$\mbox{\boldmath{$f$}}_{k,j,m}^{n_{j}}=[f_{k,j,1}^{n_{j}},\cdots,f_{k,j,M}^{n_{j}}]^{T}\in\mathbb{C}^{M\times
1}$ denotes the channel component from user $k$ to the antenna array of the BS
via a single reflection by the $n_{j}$-th element of IRS $j$; and
$\mbox{\boldmath{$g$}}_{k,j,q}^{n_{j},n_{q}}=[g_{k,j,q,1}^{n_{j},n_{q}},\cdots,g_{k,j,q,M}^{n_{j},n_{q}}]^{T}\in\mathbb{C}^{M\times
1}$ is the channel component from user $k$ to the antenna array of the BS via
double reflections by the $n_{j}$-th element of IRS $j$ and subsequently the
$n_{q}$-th element of IRS $q$. Next, we present the models of the direct,
single-reflection, and double-reflection channel components, respectively.

####  II-B1 Direct Channel Component

Figure 4: Illustration of the elevation and azimuth angles w.r.t. the BS’s
antenna array and IRSs.

For the direct channel component from each user to the antenna array of the
BS, the far-field condition is practically satisfied due to the much longer
signal propagation distance between the user and BS w.r.t. the size of the
BS’s UPA. Thus, the UPW model can be used to characterize the channel
response, where the one-dimensional (1D) array response vector can be defined
as [21]

$\displaystyle\mbox{\boldmath{$e$}}(\phi,\bar{M})=[1,e^{i\pi\phi},\cdots,e^{i(\bar{M}-1)\pi\phi}]^{T},$
(2)

with $\phi$ denoting the phase difference (normalized to $\pi$) of the
receive/transmit signals between two adjacent antenna elements, and $\bar{M}$
denoting the number of elements of interest in the 1D array. Thus, the direct
channel component between user $k$ and the antenna array of the BS is given by

$\displaystyle\tilde{\mbox{\boldmath{$h$}}}_{k}=\sum_{l=1}^{L_{k}}\tilde{a}_{k,l}\sqrt{\tilde{G}(\tilde{\theta}_{k,l},\tilde{\phi}_{k,l})}\mbox{\boldmath{$e$}}\left(\frac{2d_{A}}{\lambda}\sin\tilde{\theta}_{k,l}\cos\tilde{\phi}_{k,l},M_{z}\right)\otimes\mbox{\boldmath{$e$}}\left(\frac{2d_{A}}{\lambda}\sin\tilde{\theta}_{k,l}\sin\tilde{\phi}_{k,l},M_{x}\right),$
(3)

where $L_{k}$ is the number of (significant) signal propagation paths for user
$k$,222In this paper, we consider the narrow-band channel model, while the
channel dispersion effect is assumed to be negligible by applying multi-
carrier modulation techniques, such as OFDM. with
$\mathcal{L}_{k}\triangleq\\{1,2,\cdots,L_{k}\\}$ denoting the set of all
paths; $\tilde{a}_{k,l}\in\mathbb{C}$ denotes the complex gain of the $l$-th
path from user $k$ to the BS; $\tilde{\theta}_{k,l}\in[0,\pi/2]$ and
$\tilde{\phi}_{k,l}\in[0,2\pi)$ represent the elevation and azimuth AoAs of
the $l$-th path w.r.t. the antenna array of the BS as shown in Fig. 4; $d_{A}$
is the spacing between two adjacent antenna elements along axes $x$ and $z$;
$\lambda$ denotes the carrier wavelength; and
$\tilde{G}(\tilde{\theta}_{k,l},\tilde{\phi}_{k,l})$ is the antenna gain
corresponding to the $l$-th path of user $k$, which is determined by the
elevation and azimuth AoAs, $\tilde{\theta}_{k,l}$ and $\tilde{\phi}_{k,l}$,
in general for any given antenna radiation pattern at the BS333Various types
of BS/AP antenna radiation patterns can be used here, such as the isotropic
antenna pattern, cosine-function based pattern, 3GPP antenna radiation
patterns [38], etc..

####  II-B2 Single-Reflection Channel Component

The single-reflection channel component is comprised of the link from the user
to each IRS and the reflection link from that IRS to the antenna array of the
BS. In our proposed integrated IRS-BS architecture, the far-field condition
between the users and IRSs is satisfied due to the long signal propagation
distance between them, where the UPW-based channel model can be applied,
similar to the direct channel from each user to the BS’s UPA. However, the
distance between the IRSs and the antenna array of the BS is much shorter due
to the limited size of the antenna radome and can be even smaller than the
size of the UPA/IRSs; as a result, the UPW assumption between each IRS and the
BS UPA does not hold. In this regard, we adopt the uniform spherical wave
(USW) to model the element-wise channel from each IRS’s reflecting element to
the BS UPA (or equivalently, the UPW model for the channel from each IRS’s
reflecting element to each element of the BS’s UPA) [26, 27]. Notice that the
single-reflection channel component from a user to the antenna array of the BS
is determined by the channels between the user and all IRSs’ reflecting
elements, the reflection gain provided by all IRSs’ reflecting elements, the
channels between all IRSs’ reflecting elements and the BS antennas, and the
antenna gain at the BS. Next, we present the models of the above elements in
detail.

Without loss of generality, we define the $n_{j}$-th element of IRS $j$ as the
one located at row $n_{j,1}$ ($1\leq n_{j,1}\leq N_{j,1}$) and column
$n_{j,2}$ ($1\leq n_{j,2}\leq N_{j,2}$), where we define $n_{j}$ in terms of
$(n_{j,1},n_{j,2})$ as $n_{j}=n_{j,1}+n_{j,2}(N_{j,1}-1)$. According to the
UPW model, we can obtain the array response vector of the $l$-th path between
user $k$ and IRS $j$ as
$\mbox{\boldmath{$\alpha$}}_{k,l,j}\in\mathbb{C}^{N_{j}\times 1}$, with the
$n_{j}$-th entry given by

$\displaystyle\alpha_{k,l,j}^{n_{j}}(\hat{\theta}_{k,l,j},\hat{\phi}_{k,l,j})=e^{i\pi\frac{2d_{I}}{\lambda}\left[(n_{j,1}-1)\sin\hat{\theta}_{k,l,j}\cos\hat{\phi}_{k,l,j}+(n_{j,2}-1)\sin\hat{\theta}_{k,l,j}\sin\hat{\phi}_{k,l,j}\right]+i\varphi_{j}},$
(4)

where $\varphi_{j}$ denotes the initial phase of the signals received at IRS
$j$ relative to the origin $O$; $d_{I}$ is the spacing between any two
adjacent IRS reflecting elements; and $\hat{\theta}_{k,l,j}\in[0,\pi)$ and
$\hat{\phi}_{k,l,j}\in[0,2\pi)$ are the elevation and azimuth AoAs of the
$l$-th path for user $k$ w.r.t. IRS $j$ as shown in Fig. 4. According to
geometry, we obtain the relation between the AoAs w.r.t. the antenna array and
those w.r.t. the IRSs as

$\displaystyle\sin\hat{\theta}_{k,l,j}\cos\hat{\phi}_{k,l,j}=\cos\tilde{\theta}_{k,l},~{}\sin\hat{\theta}_{k,l,j}\sin\hat{\phi}_{k,l,j}=\sin\tilde{\theta}_{k,l}\cos\tilde{\phi}_{k,l},~{}j=1,2,$
(5)
$\displaystyle\sin\hat{\theta}_{k,l,j}\cos\hat{\phi}_{k,l,j}=\cos\tilde{\theta}_{k,l},~{}\sin\hat{\theta}_{k,l,j}\sin\hat{\phi}_{k,l,j}=\sin\tilde{\theta}_{k,l}\sin\tilde{\phi}_{k,l},~{}j=3,4,$
(6)
$\displaystyle\cos\hat{\theta}_{k,l,1}=-\sin\tilde{\theta}_{k,l}\sin\tilde{\phi}_{k,l},~{}\cos\hat{\theta}_{k,l,2}=\sin\tilde{\theta}_{k,l}\sin\tilde{\phi}_{k,l},$
(7)
$\displaystyle\cos\hat{\theta}_{k,l,3}=-\sin\tilde{\theta}_{k,l}\cos\tilde{\phi}_{k,l},~{}\cos\hat{\theta}_{k,l,4}=\sin\tilde{\theta}_{k,l}\cos\tilde{\phi}_{k,l}.$
(8)

Note that the electromagnetic response for each reflecting element of IRS is
not isotropic for different incident/reflected angles in general. Thus, we
define the reflection gain of each IRS element as [39]

$\displaystyle
G(\hat{\theta}^{AoA},\hat{\theta}^{AoD})=\begin{cases}2\times\frac{A\cos\hat{\theta}^{AoA}}{\lambda^{2}/4\pi}\times\frac{A\cos\hat{\theta}^{AoD}}{\lambda^{2}/4\pi},~{}&\text{if}~{}\hat{\theta}^{AoA}\in[0,\frac{\pi}{2})~{}\text{and}~{}\hat{\theta}^{AoD}\in[0,\frac{\pi}{2}),\\\
0,~{}&\text{otherwise},\end{cases}$ (9)

where $\hat{\theta}^{AoA}$ and $\hat{\theta}^{AoD}$ denote the elevation AoA
of the incident signals and the elevation AoD of the reflected signals w.r.t.
the IRS reflecting element, respectively, factor $2$ is due to the half-space
reflection of the IRS [40], and $A$ is the area of each reflecting element. As
can be observed from (9), each IRS element can only provide reflection gain
when the signals are incident from and reflected to its front half-space,
i.e., $\hat{\theta}^{AoA}\in[0,\frac{\pi}{2})$ and
$\hat{\theta}^{AoD}\in[0,\frac{\pi}{2})$; otherwise, the reflection gain is
set as zero.

Let $\xi_{j,m}^{n_{j}}$ denote the complex channel gain between the $n_{j}$-th
reflecting element of IRS $j$ and the $m$-th antenna at the BS. Since the
sizes of each IRS reflecting element and antenna element are of sub-
wavelength, the UPW assumption still holds for the LoS channel between any two
IRS/antenna elements, despite their short distance [41]. As such, we have

$\displaystyle\xi_{j,m}^{n_{j}}=\frac{\lambda}{4\pi||\mbox{\boldmath{$s$}}_{m}-\mbox{\boldmath{$w$}}_{j,n_{j}}||}e^{-i\frac{2\pi}{\lambda}||\mbox{\boldmath{$s$}}_{m}-\mbox{\boldmath{$w$}}_{j,n_{j}}||}.$
(10)

Next, the single-reflection channel component from user $k$ to the $m$-th
antenna of the BS via the $n_{j}$-th reflecting element of IRS $j$,
$f_{k,j,m}^{n_{j}},~{}k\in\mathcal{K},j\in\mathcal{J},n_{j}\in\mathcal{N}_{j},m\in\mathcal{M}$,
can be expressed as the summation of the single-reflection responses for the
$L_{k}$ paths from user $k$, where the single-reflection response of each path
is equal to the product of the channel gain between user $k$ and the
$n_{j}$-th reflecting element of IRS $j$, the reflection gain provided by the
$n_{j}$-th reflecting element of IRS $j$, the channel gain between the
$n_{j}$-th reflecting element of IRS $j$ and the $m$-th antenna, and the
antenna gain of the $m$-th antenna, i.e.,

$\displaystyle
f_{k,j,m}^{n_{j}}=\sum_{l=1}^{L_{k}}\tilde{a}_{k,l}\alpha_{k,l,j}^{n_{j}}(\hat{\theta}_{k,l,j},\hat{\phi}_{k,l,j})\sqrt{G(\hat{\theta}_{k,l,j},\hat{\theta}_{j,m}^{n_{j}})}\xi_{j,m}^{n_{j}}\sqrt{\tilde{G}(\tilde{\theta}_{m,j}^{n_{j}},\tilde{\phi}_{m,j}^{n_{j}})},$
(11)

where $\tilde{a}_{k,l}\in\mathbb{C}$ is the complex gain of the $l$-th path
from user $k$ to the BS given in (3);
$G(\hat{\theta}_{k,l,j},\hat{\theta}_{j,m}^{n_{j}})$ denotes the reflection
gain provided by the $n_{j}$-th reflecting element of IRS $j$ as defined in
(9), with
$\hat{\theta}_{j,w_{m}}^{n_{j}}=\arccos\frac{(\mbox{\boldmath{$s$}}_{m}-\mbox{\boldmath{$w$}}_{j,n_{j}})^{T}\mbox{\boldmath{$\delta$}}_{j}}{||\mbox{\boldmath{$s$}}_{m}-\mbox{\boldmath{$w$}}_{j,n_{j}}||}$
being the elevation angle of the direction of the $m$-th antenna w.r.t. the
$n_{j}$-th reflecting element of IRS $j$; and
$\tilde{G}(\tilde{\theta}_{m,j}^{n_{j}},\tilde{\phi}_{m,j}^{n_{j}})$ denotes
the antenna gain provided by the $m$-th antenna as defined in (3), with
$\tilde{\theta}_{m,j}^{n_{j}}=\arccos\frac{w_{j,n_{j},y}-s_{m,y}}{||\mbox{\boldmath{$w$}}_{j,n_{j}}-\mbox{\boldmath{$s$}}_{m}||}$
and
$\tilde{\phi}_{m,j}^{n_{j}}=\arctan\frac{w_{j,n_{j},x}-s_{m,x}}{w_{j,n_{j},z}-s_{m,z}}$
being the elevation and azimuth angles of the direction of the $n_{j}$-th
reflecting element of IRS $j$ w.r.t. the $m$-th antenna, respectively.

####  II-B3 Double-Reflection Channel Component

Similar to the single-reflection channel component, the double-reflection
channel component from user $k$ to the $m$-th antenna of the BS via the
$n_{j}$-th reflecting element of IRS $j$ and then the $n_{q}$-th reflecting
element of IRS $q$, $g_{k,j,q,m}^{n_{j},n_{q}},~{}k\in\mathcal{K},j\neq
q\in\mathcal{J},n_{j}\in\mathcal{N}_{j},n_{q}\in\mathcal{N}_{q},m\in\mathcal{M}$,
is the summation of the double-reflection responses for the $L_{k}$ paths from
user $k$. The double-reflection response of each path is given by the product
of the channel gain between user $k$ and the $n_{j}$-th reflecting element of
IRS $j$, the reflection gain provided by the $n_{j}$-th reflecting element of
IRS $j$, the channel gain between the $n_{j}$-th reflecting element of IRS $j$
and the $n_{q}$-th reflecting element of IRS $q$, the reflection gain provided
by the $n_{q}$-th reflecting element of IRS $q$, the channel gain between the
$n_{q}$-th reflecting element of IRS $q$ and the $m$-th antenna, and the
antenna gain of the $m$-th antenna, i.e.,

$\displaystyle
g_{k,j,q,m}^{n_{j},n_{q}}=\sum_{l=1}^{L_{k}}\tilde{a}_{k,l}\alpha_{k,l,j}^{n_{j}}(\hat{\theta}_{k,l,j},\hat{\phi}_{k,l,j})\sqrt{G(\hat{\theta}_{k,l,j},\hat{\theta}_{j,q}^{n_{j},n_{q}})}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\times\zeta_{j,q}^{n_{j},n_{q}}\sqrt{G(\hat{\theta}_{q,j}^{n_{q},n_{j}},\hat{\theta}_{q,m}^{n_{q}})}\xi_{q,m}^{n_{q}}\sqrt{\tilde{G}(\tilde{\theta}_{m,q}^{n_{q}},\tilde{\phi}_{m,q}^{n_{q}})},$
(12)

where

$\displaystyle\zeta_{j,q}^{n_{j},n_{q}}=\frac{\lambda}{4\pi||\mbox{\boldmath{$w$}}_{q,n_{q}}-\mbox{\boldmath{$w$}}_{j,n_{j}}||}e^{-i\frac{2\pi}{\lambda}||\mbox{\boldmath{$w$}}_{q,n_{q}}-\mbox{\boldmath{$w$}}_{j,n_{j}}||}$
(13)

denotes the complex channel gain for the LoS path between the $n_{j}$-th
reflecting element of IRS $j$ and the $n_{q}$-th reflecting element of IRS $q$
due to their short distance;
$G(\hat{\theta}_{k,l,j},\hat{\theta}_{j,q}^{n_{j},n_{q}})$ is the reflection
gain provided by the $n_{j}$-th reflecting element of IRS $j$, with
$\hat{\theta}_{j,q}^{n_{j},n_{q}}=\arccos\frac{(\mbox{\boldmath{$w$}}_{q,n_{q}}-\mbox{\boldmath{$w$}}_{j,n_{j}})^{T}\mbox{\boldmath{$\delta$}}_{j}}{||\mbox{\boldmath{$w$}}_{q,n_{q}}-\mbox{\boldmath{$w$}}_{j,n_{j}}||}$
denoting the elevation angle of the direction of the $n_{q}$-th reflecting
element of IRS $q$ w.r.t. the $n_{j}$-th reflecting element of IRS $j$;
$G(\hat{\theta}_{q,j}^{n_{q},n_{j}},\hat{\theta}_{q,m}^{n_{q}})$ is the
reflection gain provided by the $n_{q}$-th reflecting element of IRS $q$, with
$\hat{\theta}_{q,j}^{n_{q},n_{j}}=\arccos\frac{(\mbox{\boldmath{$w$}}_{j,n_{j}}-\mbox{\boldmath{$w$}}_{q,n_{q}})^{T}\mbox{\boldmath{$\delta$}}_{q}}{||\mbox{\boldmath{$w$}}_{j,n_{j}}-\mbox{\boldmath{$w$}}_{q,n_{q}}||}$
and
$\hat{\theta}_{q,m}^{n_{q}}=\arccos\frac{(\mbox{\boldmath{$s$}}_{m}-\mbox{\boldmath{$w$}}_{q,n_{q}})^{T}\mbox{\boldmath{$\delta$}}_{q}}{||\mbox{\boldmath{$s$}}_{m}-\mbox{\boldmath{$w$}}_{q,n_{q}}||}$
denoting the elevation angles of the directions of the $n_{j}$-th reflecting
element of IRS $j$ and the $m$-th antenna w.r.t. the $n_{q}$-th reflecting
element of IRS $q$; and
$\tilde{G}(\tilde{\theta}_{m,q}^{n_{q}},\tilde{\phi}_{m,q}^{n_{q}})$ is the
antenna gain provided by $m$-th antenna, with
$\tilde{\theta}_{m,q}^{n_{q}}=\arccos\frac{w_{q,n_{q},y}-s_{m,y}}{||\mbox{\boldmath{$w$}}_{q,n_{q}}-\mbox{\boldmath{$s$}}_{m}||}$
and
$\tilde{\phi}_{m,q}^{n_{q}}=\arctan\frac{w_{j,n_{j},x}-s_{m,x}}{w_{j,n_{j},z}-s_{m,z}}$
denoting the elevation and azimuth angles of the direction of the $n_{q}$-th
reflecting element of IRS $q$ w.r.t. the $m$-th antenna, respectively. For
ease of reading, the main symbol notations used in this paper and their
physical meanings are summarized in Table I.

TABLE I: Symbols and Physical Meanings Symbol |  Physical meaning |  Symbol |  Physical meaning
---|---|---|---
$M$ |  Number of BS antennas |  $J$ |  Number of IRSs
$N_{j}$ |  Number of reflecting elements of IRS $j$ |  $N$ |  Total number of reflecting elements of all IRSs
$M_{x}$/$M_{z}$ |  Number of antennas along axis $x$ or axis $z$ |  $N_{j,1}$/$N_{j,2}$ |  Number of reflecting elements along axis $y$ or axis $z$($x$) for IRS $j$
$K$ |  Number of users |  $\mbox{\boldmath{$s$}}_{m}$ |  Coordinates of the $m$-th antenna
$\mbox{\boldmath{$w$}}_{j,n_{j}}$ |  Coordinates of the $n_{j}$-th reflecting element of IRS $j$ |  $\mbox{\boldmath{$\delta$}}_{j}$ |  Normal vector of IRS $j$
$\vartheta_{j,n_{j}}$ |  Reflection coefficient of the $n_{j}$-th reflecting element of IRS $j$ |  $H_{AR}$ |  Altitude of the antenna radome
$\theta_{tilt}$ |  Suspension angle of the BS antenna radome |  $A$ |  Area of each reflecting element
$\lambda$ |  Carrier wavelength |  $d_{A}$/$d_{I}$ |  Spacing between two adjacent elements of the antenna array/IRSs
$\tilde{\theta}_{m,j}^{n_{j}}$/$\tilde{\phi}_{m,j}^{n_{j}}$ |  Elevation/azimuth angle of the direction of the $n_{j}$-th reflecting element of IRS $j$ w.r.t. the $m$-th BS antenna |  $\hat{\theta}_{j,m}^{n_{j}}$/$\hat{\theta}_{j,q}^{n_{j},n_{q}}$ |  Elevation angle of the direction of the $m$-th antenna/the $n_{q}$-th reflecting element of IRS $q$ w.r.t. the $n_{j}$-th reflecting element of IRS $j$
$\tilde{G}(\cdot)$ |  Antenna gain of the BS antennas |  $G(\cdot)$ |  Reflection gain of the IRS element
$\mbox{\boldmath{$f$}}_{k,j}^{n_{j}}$ |  Channel component from user $k$ to the antenna array of the BS via a single reflection at the $n_{j}$-th element of IRS $j$ |  $\mbox{\boldmath{$g$}}_{k,j,q}^{n_{j},n_{q}}$ |  Channel component from user $k$ to the antenna array of the BS via double reflections at the $n_{j}$-th element of IRS $j$ and then the $n_{q}$-th reflecting element of IRS $q$
$\xi_{k,j,m}^{n_{j}}$ |  Complex channel gain between the $n_{j}$-th reflecting element of IRS $j$ and the $m$-th BS antenna |  $\zeta_{j,q}^{n_{j},n_{q}}$ |  Complex channel gain between the $n_{j}$-th reflecting element of IRS $j$ and the $n_{q}$-th reflecting element of IRS $q$
$\varphi_{j}$ |  Initial phase of the received signals at IRS $j$ relative to the origin $O$ |  $L_{k}$ |  Number of paths from user $k$
$\tilde{a}_{k,l}$ |  Complex path gain of the $l$-th path from user $k$ |  $\alpha_{k,l,j}^{n_{j}}(\hat{\theta}_{k,l,j},\hat{\phi}_{k,l,j})$ |  Array response of the $n_{j}$-th reflecting element of IRS $j$ for the $l$-th path from user $k$
$\tilde{\theta}_{k,l}$/$\tilde{\phi}_{k,l}$ |  Elevation/azimuth AoA of the $l$-th path from user $k$ w.r.t. the BS antenna array |  $\hat{\theta}_{k,l,j}$/$\hat{\phi}_{k,l,j}$ |  Elevation/azimuth AoA of $l$-th path from user $k$ w.r.t. IRS $j$
$P$ |  Transmit power of each user |  $\sigma^{2}$ |  Noise power per BS antenna
$\eta$ |  Number of antenna modules in generalized IRS-BS architecture with modular antenna array |  |

###  II-C Generalized IRS-BS Architecture With Modular Antenna Arrays

In this subsection, we generalize the proposed integrated IRS-BS architecture
by employing modular antenna arrays, so that more reflecting elements can be
accommodated inside the antenna radome. Specifically, we divide the antenna
array into $\eta\geq 1$ modules, where each module consisting of $M/\eta$
antennas (assumed to be an integer) is surrounded by four smaller-size IRSs
parallel to the left, right, top, and bottom surfaces. Note that the
integrated IRS-BS architecture proposed in Section II-A can be considered as a
special case of the generalized IRS-BS architecture by setting $\eta=1$.
However, the integrated IRS-BS architecture proposed in Section II-A does not
need to change the structure of existing antenna arrays and thus has lower
implementation cost compared to the generalized IRS-BS architecture with
$\eta>1$. Fig. 5 shows an example of the generalized IRS-BS architecture with
modular antenna arrays, where the number of BS antennas is set as $M=16$ with
$M_{x}=M_{z}=4$. It can be seen that the total number of reflecting elements
increases as $\eta$ increases, e.g., $2N$ elements for $\eta=4$ and $4N$
elements for $\eta=16$. However, the number of reflecting elements surrounding
each antenna module decreases as $\eta$ increases, e.g., $N/2$ elements for
$\eta=4$ and $N/4$ for $\eta=16$. Note that the channels between the users and
each antenna module for $\eta>1$ can be modelled in a similar way to that for
$\eta=1$ as shown in the previous subsection, while we assume that there are
no signal reflections among the reflecting elements surrounding different
antenna modules. In particular, as the number of antenna modules (i.e.,
$\eta$) increases, the number of single-reflection and double-reflection
channel components between any user and each antenna element decreases, while
the path loss of such reflection channel components is also reduced due to
shorter signal propagation distance among the reflecting elements surrounding
the same antenna module. The performance comparison between the generalized
IRS-BS architectures under different numbers of antenna modules will be
provided in Section IV via simulations.

Figure 5: Illustration of the generalized integrated IRS-BS architecture with
modular antenna arrays, where the yellow circles represent antennas and the
red lines indicate the surfaces to deploy IRS reflecting elements.

##  III Problem Formulation and Passive Reflection Design

In this section, we formulate and solve the problem for optimizing the
reflection coefficients of all IRSs in the integrated IRS-BS architecture
presented in Section II-A, while the results are also applicable to the
generalized IRS-BS architecture with modular antenna arrays given in Section
II-C.

###  III-A Problem Formulation

Let $P$ denote the maximum transmit power of each user. By employing the
minimum mean square error (MMSE) combining and successive interference
cancellation (SIC) techniques at the BS, the sum-rate for decoding the signals
from all the $K$ users in bits per second per Hertz (bps/Hz) is given by [42]

$\displaystyle
R(\mbox{\boldmath{$\Theta$}})=\log_{2}\text{det}\left(\mbox{\boldmath{$I$}}_{M}+\sum_{k=1}^{K}\frac{P}{\sigma^{2}}\mbox{\boldmath{$h$}}_{k}(\mbox{\boldmath{$\Theta$}})\mbox{\boldmath{$h$}}_{k}(\mbox{\boldmath{$\Theta$}})^{H}\right),$
(14)

where $\sigma^{2}$ denotes the independent CSCG noise power at each of the
receive antennas at the BS.

In this paper, we aim to maximize the sum-rate of the $K$ users by optimizing
the passive reflection coefficients of all the IRSs, i.e., $\Theta$. The
corresponding problem is formulated as

$\displaystyle\text{(P1):}~{}\max_{\mbox{\boldmath{$\Theta$}}}~{}$
$\displaystyle R(\mbox{\boldmath{$\Theta$}})$ s.t.
$\displaystyle|\vartheta_{j,n_{j}}|=1,~{}\forall
j\in\mathcal{J},n_{j}\in\mathcal{N}_{j}.$ (15)

Problem (P1) is difficult to solve for the following reasons. On one hand, the
CSI acquisition is a non-trivial task in practice because of much unknown
channel parameters in $\\{\tilde{\mbox{\boldmath{$h$}}}_{k}\\}$,
$\\{\mbox{\boldmath{$f$}}_{k,j}^{n_{j}}\\}$, and
$\\{\mbox{\boldmath{$g$}}_{k,j,q}^{n_{j},n_{q}}\\}$, which may result in high
overhead for channel estimation. On the other hand, problem (P1) is a non-
convex optimization problem due to the non-concave objective function and
unit-modulus constraints in (15). Besides, the passive reflection coefficients
of all reflecting elements at all IRSs are coupled intricately in the
objective function due to both the single- and double-reflection signals. In
the following, we consider two cases with/without perfect CSI at the BS to
solve (P1), respectively.

###  III-B Passive Reflection Design With Perfect CSI

In this subsection, we assume that the perfect CSI for
$\\{\tilde{\mbox{\boldmath{$h$}}}_{k}\\}$,
$\\{\mbox{\boldmath{$f$}}_{k,j}^{n_{j}}\\}$, and
$\\{\mbox{\boldmath{$g$}}_{k,j,q}^{n_{j},n_{q}}\\}$ is available at the BS by
exploiting the channel estimation methods proposed in IRS-aided communications
[34, 29, 30, 31, 32, 33], where the obtained solution provides an upper bound
on the achievable sum-rate of the users. To tackle the non-convex problem
(P1), in the following, we propose an efficient algorithm by exploiting the
successive refinement method [20], where we optimize each of the $N$
reflection coefficients with the other $(N-1)$ reflection coefficients being
fixed in an iterative manner over the $N$ reflection coefficients.

For the given $\vartheta_{q,n_{q}},~{}q\neq
j\in\mathcal{J},n_{q}\in\mathcal{N}_{q}$, and
$\vartheta_{j,\hat{n}_{j}},~{}\hat{n}_{j}\neq n_{j}\in\mathcal{N}_{j}$, the
objective function of (P1) w.r.t. $\vartheta_{j,n_{j}}$ can be re-written as

$\displaystyle
R_{j,n_{j}}(\vartheta_{j,n_{j}})\triangleq\log_{2}\text{det}\left(\mbox{\boldmath{$A$}}_{j,n_{j}}+\vartheta_{j,n_{j}}\mbox{\boldmath{$B$}}_{j,n_{j}}+\vartheta_{j,n_{j}}^{*}\mbox{\boldmath{$B$}}_{j,n_{j}}^{H}\right),~{}j\in\mathcal{J},n_{j}\in\mathcal{N}_{j},$
(16)

where $\mbox{\boldmath{$A$}}_{j,n_{j}}$ and $\mbox{\boldmath{$B$}}_{j,n_{j}}$
are constants independent to $\vartheta_{j,n_{j}}$ and respectively given by

$\displaystyle\mbox{\boldmath{$A$}}_{j,n_{j}}=\mbox{\boldmath{$I$}}_{M}+\frac{P}{\sigma^{2}}\sum_{k=1}^{K}\Big{(}\Big{(}\mbox{\boldmath{$h$}}_{k}+\sum_{\hat{n}_{j}\neq
n_{j}}^{N_{j}}\mbox{\boldmath{$f$}}_{k,j}^{\hat{n}_{j}}\vartheta_{j,\hat{n}_{j}}+\sum_{q\neq
j}\sum_{n_{q}=1}^{N_{q}}\mbox{\boldmath{$f$}}_{k,q}^{n_{q}}\vartheta_{q,n_{q}}+\sum_{\hat{n}_{j}\neq
n_{j}}^{N_{j}}\sum_{q\neq
j}\sum_{n_{q}=1}^{N_{q}}\Big{(}\mbox{\boldmath{$g$}}_{k,j,q}^{\hat{n}_{j},n_{q}}$
$\displaystyle+\mbox{\boldmath{$g$}}_{k,q,j}^{n_{q},\hat{n}_{j}}\Big{)}\vartheta_{j,\hat{n}_{j}}\vartheta_{q,n_{q}}+\sum_{q\neq
j}\sum_{r\neq q,r\neq
j}\sum_{n_{q}=1}^{N_{q}}\sum_{n_{r}=1}^{N_{r}}\mbox{\boldmath{$g$}}_{k,q,r}^{n_{q},n_{r}}\vartheta_{q,n_{q}}\vartheta_{r,n_{r}}\Big{)}\times\Big{(}\mbox{\boldmath{$h$}}_{k}+\sum_{\hat{n}_{j}\neq
n_{j}}^{N_{j}}\mbox{\boldmath{$f$}}_{k,j}^{\hat{n}_{j}}\vartheta_{j,\hat{n}_{j}}$
$\displaystyle+\sum_{q\neq
j}\sum_{n_{q}=1}^{N_{q}}\mbox{\boldmath{$f$}}_{k,q}^{n_{q}}\vartheta_{q,n_{q}}+\sum_{\hat{n}_{j}\neq
n_{j}}^{N_{j}}\sum_{q\neq
j}\sum_{n_{q}=1}^{N_{q}}\Big{(}\mbox{\boldmath{$g$}}_{k,j,q}^{\hat{n}_{j},n_{q}}+\mbox{\boldmath{$g$}}_{k,q,j}^{n_{q},\hat{n}_{j}}\Big{)}\vartheta_{j,\hat{n}_{j}}\vartheta_{q,n_{q}}$
$\displaystyle+\sum_{q\neq j}\sum_{r\neq q,r\neq
j}\sum_{n_{q}=1}^{N_{q}}\sum_{n_{r}=1}^{N_{r}}\mbox{\boldmath{$g$}}_{k,q,r}^{n_{q},n_{r}}\vartheta_{q,n_{q}}\vartheta_{r,n_{r}}\Big{)}^{H}+\Big{(}\mbox{\boldmath{$f$}}_{k,j}^{n_{j}}+\sum_{q\neq
j}\sum_{n_{q}=1}^{N_{q}}\Big{(}\mbox{\boldmath{$g$}}_{k,j,q}^{n_{j},n_{q}}+\mbox{\boldmath{$g$}}_{k,q,j}^{n_{q},n_{j}}\Big{)}\vartheta_{q,n_{q}}\Big{)}$
$\displaystyle\times\Big{(}\mbox{\boldmath{$f$}}_{k,j}^{n_{j}}+\sum_{q\neq
j}\sum_{n_{q}=1}^{N_{q}}(\mbox{\boldmath{$g$}}_{k,j,q}^{n_{j},n_{q}}+\mbox{\boldmath{$g$}}_{k,q,j}^{n_{q},n_{j}})\vartheta_{q,n_{q}}\Big{)}^{H}\Big{)},~{}j\in\mathcal{J},n_{j}\in\mathcal{N}_{j},$
(17)

and

$\displaystyle\mbox{\boldmath{$B$}}_{j,n_{j}}=\frac{P}{\sigma^{2}}\sum_{k=1}^{K}\Big{(}\Big{(}\mbox{\boldmath{$f$}}_{k,j}^{n_{j}}+\sum_{q\neq
j}\sum_{n_{q}=1}^{N_{q}}\Big{(}\mbox{\boldmath{$g$}}_{k,j,q}^{n_{j},n_{q}}+\mbox{\boldmath{$g$}}_{k,q,j}^{n_{q},n_{j}}\Big{)}\vartheta_{q,n_{q}}\Big{)}$
$\displaystyle\times\Big{(}\mbox{\boldmath{$h$}}_{k}+\sum_{\hat{n}_{j}\neq
n_{j}}^{N_{j}}\mbox{\boldmath{$f$}}_{k,j}^{\hat{n}_{j}}\vartheta_{j,\hat{n}_{j}}+\sum_{q\neq
j}\sum_{n_{q}=1}^{N_{q}}\mbox{\boldmath{$f$}}_{k,q}^{n_{q}}\vartheta_{q,n_{q}}+\sum_{\hat{n}_{j}\neq
n_{j}}^{N_{j}}\sum_{q\neq
j}\sum_{n_{q}=1}^{N_{q}}\Big{(}\mbox{\boldmath{$g$}}_{k,j,q}^{\hat{n}_{j},n_{q}}+\mbox{\boldmath{$g$}}_{k,q,j}^{n_{q},\hat{n}_{j}}\Big{)}\vartheta_{j,\hat{n}_{j}}\vartheta_{q,n_{q}}$
$\displaystyle+\sum_{q\neq j}\sum_{r\neq q,r\neq
j}\sum_{n_{q}=1}^{N_{q}}\sum_{n_{r}=1}^{N_{r}}\mbox{\boldmath{$g$}}_{k,q,r}^{n_{q},n_{r}}\vartheta_{q,n_{q}}\vartheta_{r,n_{r}}\Big{)}^{H}\Big{)},~{}\forall
j\in\mathcal{J},n\in\mathcal{N}_{j}.$ (18)

Therefore, the subproblem for optimizing $\vartheta_{j,n_{j}}$ can be
expressed as

$\displaystyle\text{(P1-$j$-$n_{j}$):}~{}\max_{\vartheta_{j,n_{j}}}~{}$
$\displaystyle R_{j,n_{j}}(\vartheta_{j,n_{j}})$ s.t.
$\displaystyle|\vartheta_{j,n_{j}}|=1.$ (19)

Although the objective function of (P1-$j$-$n_{j}$) is concave w.r.t.
$\vartheta_{j,n_{j}}$, it is still non-convex due to the unit-modulus
constraint in (19). To address this issue, we relax it as
$|\vartheta_{j,n_{j}}|\leq 1$ and denote the relaxed problem as
(P2-$j$-$n_{j}$), which is convex and thus can be solved by standard convex
optimization methods, such as the interior point method [43]. Denote
$\vartheta_{j,n_{j}}^{\star}$ as the optimal solution for (P2-$j$-$n_{j}$),
which may not be feasible to (P1-$j$-$n_{j}$) due to the unit-modulus
constraint in (19). As such, the obtained solution should be projected to the
feasible region of (P1-$j$-$n_{j}$) as follows [44]

$\displaystyle\vartheta^{\star\star}_{j,n_{j}}=\vartheta_{j,n_{j}}^{\star}/|\vartheta_{j,n_{j}}^{\star}|.$
(20)

With the solution for (P1-$j$-$n_{j}$) derived in (20), we are ready to
complete our proposed solution to solve (P1) with perfect CSI. Specifically,
we first randomly generate $T\geq 1$ solutions of
$\\{\vartheta_{j,n_{j}},~{}j\in\mathcal{J},n_{j}\in\mathcal{N}_{j}\\}$
satisfying $|\vartheta_{j,n_{j}}|=1$, where the phase shifts of
$\vartheta_{j,n_{j}}$’s follow the uniform distribution in $[0,2\pi)$. Then,
we select the solution achieving the maximum sum-rate as the initial point.
Subsequently, we update the reflection coefficients of all reflecting elements
based on (20) sequentially from $j=1$ and $n_{j}=1$ to $j=J$ and
$n_{j}=N_{J}$. After the reflection coefficients of all reflecting elements
are updated, we check whether the increment of the objective function in (P1)
is smaller than a given small positive value $\epsilon_{1}$ or the maximum
number of iterations $I_{\max}$ is reached. If this is the case, the algorithm
terminates; otherwise, the update for the reflection coefficients of all
reflecting elements is repeated as above. The overall solution to solve
problem (P1) with perfect CSI is summarized in Algorithm 1. Denote $I$ as the
number of iterations required to update the reflection coefficients of all
IRSs in Algorithm 1 (i.e., line 3–line 11) with $I\leq I_{\max}$. Notice that
the reflection coefficient of each reflecting element is updated by applying
the standard interior-point method to solve problem (P2-$j$-$n_{j}$), which
incurs the complexity of $\mathcal{O}(3K^{3}+2K^{2}M)$ [45], and the
calculation for sum-rate has the complexity of $\mathcal{O}(M^{2})$. As a
result, the overall computational complexity of Algorithm 1 is
$\mathcal{O}(TM^{2}+IN(3K^{3}+2K^{2}M)+IM^{2})$.

Algorithm 1 Passive Reflection Design with Perfect CSI.

0: $\\{\tilde{\mbox{\boldmath{$h$}}}_{k}\\}$,
$\\{\mbox{\boldmath{$f$}}_{k,j}^{n_{j}}\\}$,
$\\{\mbox{\boldmath{$g$}}_{k,j,q}^{n_{j},n_{q}}\\}$, $P$, $\sigma^{2}$, and
$T$.

1: Randomly generate $T$ independent realizations of
$\\{\vartheta_{j,n_{j}},~{}j\in\mathcal{J},n_{j}\in\mathcal{N}_{j}\\}$, where
the phase shift of each $\vartheta_{j,n_{j}}$ is randomly chosen from interval
$[0,2\pi)$.

2: Select
$\\{\tilde{\vartheta}_{j,n_{j}}^{\star},~{}j\in\mathcal{J},n_{j}\in\mathcal{N}_{j}\\}$
as the realization yielding the largest objective value of (P1), and
initialize
$\vartheta_{j,n_{j}}=\tilde{\vartheta}_{j,n_{j}}^{\star},~{}j\in\mathcal{J},n_{j}\in\mathcal{N}_{j}$.

3: repeat

4: for $j=1\rightarrow J$ do

5: for $n_{j}=1\rightarrow N_{j}$ do

6: For the given $\vartheta_{q,n_{q}},~{}q\neq
j\in\mathcal{J},n_{q}\in\mathcal{N}_{q}$, and
$\vartheta_{j,\hat{n}_{j}},~{}\hat{n}_{j}\neq n_{j}\in\mathcal{N}_{j}$, obtain
$A_{j,n_{j}}$ and $B_{j,n_{j}}$ according to (17) and (18).

7: Obtain $\vartheta_{j,n_{j}}^{\star}$ via solving (P2-$j$-$n_{j}$) and
update $\vartheta^{\star\star}_{j,n_{j}}$ according to (20).

8: Set $\vartheta_{j,n_{j}}=\vartheta^{\star\star}_{j,n_{j}}$.

9: end for

10: end for

11: until the increment of the objective function in (P1) is smaller than
$\epsilon_{1}$ or the maximum number of iterations $I_{\max}$ is reached.

11: $\\{\vartheta_{j,n_{j}},~{}j\in\mathcal{J},n_{j}\in\mathcal{N}_{j}\\}$.

###  III-C Passive Reflection Design Without CSI

In practice, the CSI acquisition for the considered integrated IRS-BS
architecture is a challenging problem, because the direct channel components
$\\{\tilde{\mbox{\boldmath{$h$}}}_{k}\\}$, single-reflection channel
components $\\{\mbox{\boldmath{$f$}}_{k,j}^{n_{j}}\\}$, and double-reflection
channel components $\\{\mbox{\boldmath{$g$}}_{k,j,q}^{n_{j},n_{q}}\\}$ involve
$KM+KMN+KM\sum_{j=1}^{J}\sum_{q\neq j}N_{j}N_{q}$ unknown channel parameters
in total, which may result in extremely high training overhead if the number
of reflecting elements (i.e., $N$) is large, and thus a substantial
degradation of the achievable rates of the users for data transmission. To
address this issue, in this subsection, we propose practical methods for
designing the passive reflection coefficients of all IRSs without the CSI
estimated explicitly.

####  III-C1 Random Phase Algorithm (RPA)

The RPA has been widely used in the existing literature to optimize the
passive reflection coefficients of IRS (see, e.g., [20, 46, 4, 25]) without
assuming any knowledge of the CSI. This algorithm is also applicable to our
considered integrated IRS-BS system for solving (P1). Specifically, a given
number of feasible solutions of $\Theta$ are randomly generated at the BS,
where the phase shift of each reflecting element in $\Theta$ is chosen from
interval $[0,2\pi)$ uniformly and independently, and then the solution with
the maximum objective value of (P1) is selected as the final solution444In
practice, with each realization of $\Theta$, the BS needs to estimate the
effective channels from the users to the antenna array of the BS, i.e.,
$\\{\mbox{\boldmath{$h$}}_{k}(\mbox{\boldmath{$\Theta$}}),~{}k\in\mathcal{K}\\}$,
by applying the channel estimation schemes for the conventional multi-antenna
BS without integrated IRS.. Note that this algorithm is in fact used for the
initialization of Algorithm 1 assuming perfect CSI at the BS. The performance
of RPA is highly dependent on the phase control of the single-reflection and
double-reflection channel components with independent reflection coefficient
realizations. However, for the double-reflection channel components, due to
the coupled terms, i.e.,
$\mbox{\boldmath{$g$}}_{k,j,q}^{n_{j},n_{q}}\vartheta_{j,n_{j}}\vartheta_{q,n_{q}},~{}j\neq
q\in\mathcal{J},n_{j}\in\mathcal{N}_{j},n_{q}\in\mathcal{N}_{q}$ in (1),
independent phase control with randomly generated reflection coefficients for
all IRS simultaneously becomes impossible, which may result in the performance
loss of RPA. Thus, we use RPA as a benchmark algorithm and propose an improved
solution in the following.

####  III-C2 Iterative Random Phase Algorithm (IRPA)

To further improve the performance of RPA, we propose an IRPA to decouple the
design of reflection coefficients for multiple IRSs, where we apply the RPA to
optimize the reflection coefficients of one IRS each time only with those of
the other $(J-1)$ IRSs being fixed, which operates in an alternate manner over
the $J$ IRSs. We denote
$\mbox{\boldmath{$\tilde{\Theta}$}}_{j}\triangleq\\{\vartheta_{j,n_{j}},~{}n_{j}\in\mathcal{N}_{j}\\},~{}j\in\mathcal{J}$,
and rewrite the effective user-BS channels in (1) as
$\mbox{\boldmath{$h$}}_{k}(\\{\tilde{\mbox{\boldmath{$\Theta$}}}_{j},~{}j\in\mathcal{J}\\})$
and the sum-rate of all users in (14) as
$R(\\{\tilde{\mbox{\boldmath{$\Theta$}}}_{j},~{}j\in\mathcal{J}\\})$,
respectively.

For the given $\mbox{\boldmath{$\tilde{\Theta}$}}_{q},q\neq j\in\mathcal{J}$,
the effective user $k$-BS channel in (1) w.r.t. the reflection coefficients of
IRS $j$, i.e., $\mbox{\boldmath{$\tilde{\Theta}$}}_{j}$, is given by

$\displaystyle\mbox{\boldmath{$h$}}_{k,j}(\mbox{\boldmath{$\tilde{\Theta}$}}_{j})\triangleq\sum_{n_{j}=1}^{N_{j}}\mbox{\boldmath{$\gamma$}}_{k,j}^{n_{j}}\vartheta_{j,n_{j}}+\mbox{\boldmath{$\beta$}}_{k,j}^{n_{j}},$
(21)

where
$\mbox{\boldmath{$\gamma$}}_{k,j}^{n_{j}}=\mbox{\boldmath{$f$}}_{k,j}^{n_{j}}+\sum_{q\neq
j}\sum_{n_{q}=1}^{N_{q}}(\mbox{\boldmath{$g$}}_{k,j,q}^{n_{j},n_{q}}+\mbox{\boldmath{$g$}}_{k,q,j}^{n_{q},n_{j}})\vartheta_{q,n_{q}}$
and
$\mbox{\boldmath{$\beta$}}_{k,j}^{n_{j}}=\tilde{\mbox{\boldmath{$h$}}}_{k}+\sum_{q\neq
j}\sum_{n_{q}=1}^{N_{q}}\mbox{\boldmath{$f$}}_{k,q}^{n_{q}}\vartheta_{q,n_{q}}+\sum_{q\neq
j}\sum_{r\neq q,r\neq
j}\sum_{n_{q}=1}^{N_{q}}\sum_{n_{r}=1}^{N_{r}}\mbox{\boldmath{$g$}}_{k,q,r}^{n_{q},n_{r}}\vartheta_{q,n_{q}}\vartheta_{r,n_{r}}$
are constants independent to $\tilde{\mbox{\boldmath{$\Theta$}}}_{j}$. As can
be observed, the effective channel vector in (21) is linear to the reflection
coefficients of IRS $j$ and thus can be effectively controlled. As such, we
can employ the RPA to obtain the solution of
$\mbox{\boldmath{$\tilde{\Theta}$}}_{j}$ to maximize the objective value of
(P1) under given $\\{\mbox{\boldmath{$\tilde{\Theta}$}}_{q},~{}q\neq
j\in\mathcal{J}\\}$. In the following, we introduce the proposed IRPA for
designing the reflection coefficients of all IRSs without CSI in detail.

Similar to Algorithm 1, we first randomly generate $T_{0}\geq 1$ solutions of
$\\{\tilde{\mbox{\boldmath{$\Theta$}}}_{j},~{}j\in\mathcal{J}\\}$ by selecting
the phase shift of each reflecting element from interval $[0,2\pi)$
independently, and then select the solution achieving the maximum sum-rate as
the initial point. With the initialized
$\\{\tilde{\mbox{\boldmath{$\Theta$}}}_{j},~{}j\in\mathcal{J}\\}$, we denote
its corresponding sum-rate of all users, i.e.,
$R(\\{\tilde{\mbox{\boldmath{$\Theta$}}}_{j},~{}j\in\mathcal{J}\\})$, as
$R_{o}$. Next, we apply the RPA to update the reflection coefficients of IRS
$j$ with those of the other IRSs $q\neq j\in\mathcal{J}$ being fixed
sequentially from $j=1$ to $j=J$.555For the generalized architecture with
$\eta>1$ modular antenna arrays, IRS $j,~{}j\in\mathcal{J}$ refers to four
IRSs surrounding the same antenna module. Specifically, when applying the RPA
to update the reflection coefficients of IRS $j$, we set the number of
randomly generated solutions of $\tilde{\mbox{\boldmath{$\Theta$}}}_{j}$ as
$T_{j}$ and fix the reflection coefficients of all other IRSs as
$\\{\tilde{\mbox{\boldmath{$\Theta$}}}_{q},q\neq j,~{}q\in\mathcal{J}\\}$. The
$t$-th generated solution is denoted as
$\tilde{\mbox{\boldmath{$\Theta$}}}_{j,t}$ with the phase shifts of its
reflecting elements randomly chosen from interval $[0,2\pi)$, and the
corresponding sum-rate of all users is given by
$R_{t}(\tilde{\mbox{\boldmath{$\Theta$}}}_{j,t},\\{\tilde{\mbox{\boldmath{$\Theta$}}}_{q},~{}q\neq
j\in\mathcal{J}\\})$. Then, the solution which achieves the maximum sum-rate
is selected, i.e.,
$\tilde{\mbox{\boldmath{$\Theta$}}}_{j}^{\star}=\tilde{\mbox{\boldmath{$\Theta$}}}_{j,t_{j}}$
with
$t_{j}=\arg\max_{t=1,2,\cdots,T_{j}}R_{t}(\tilde{\mbox{\boldmath{$\Theta$}}}_{j,t},\\{\tilde{\mbox{\boldmath{$\Theta$}}}_{q},~{}q\neq
j\in\mathcal{J}\\})$. If the objective value of (P1) is improved, i.e.,
$R(\mbox{\boldmath{$\tilde{\Theta}$}}_{j}^{\star},\\{\mbox{\boldmath{$\tilde{\Theta}$}}_{q},~{}q\neq
j\in\mathcal{J}\\})\geq R_{o}$, we update the reflection coefficients of IRS
$j$ as
$\tilde{\mbox{\boldmath{$\Theta$}}}_{j}=\tilde{\mbox{\boldmath{$\Theta$}}}_{j}^{\star}$
and
$R_{o}=R(\mbox{\boldmath{$\tilde{\Theta}$}}_{j}^{\star},\\{\mbox{\boldmath{$\tilde{\Theta}$}}_{q},~{}q\neq
j\in\mathcal{J}\\})$; otherwise, we keep the reflection coefficients of IRS
$j$ and $R_{o}$ unchanged. After the RPA has been applied to all IRSs, we
check whether the increment of the objective value in (P1) is smaller than a
given small positive value $\epsilon_{2}$ or the maximum number of iterations
$I_{r,\max}$ is reached. If this is the case, the IRPA terminates; otherwise,
we repeat the above procedure for updating the reflection coefficients of all
IRSs alternately via RPA. We summarize the proposed IRPA to solve (P1) without
CSI in Algorithm 2. Since we only update
$\tilde{\mbox{\boldmath{$\Theta$}}}_{j},~{}j\in\mathcal{J}$, when the sum-rate
of the users increases compared to the previous iteration, the objective
function in (P1) is non-decreasing and Algorithm 2 is guaranteed to converge.
Denote $I_{r}$ as the number of iterations in Algorithm 2 (i.e., line 6–line
17) with $I_{r}\leq I_{r,\max}$, and thus the total number of randomly
generated solutions in Algorithm 2 is given by
$T_{total}=T_{0}+I_{r}\sum_{j=1}^{J}T_{j}$, which is defined as the overall
training overhead required for implementing Algorithm 2 in practice.

Algorithm 2 Iterative Random Phase Algorithm (IRPA) Without CSI.

0: $P$, $\sigma^{2}$, $T_{0}$ and $\\{T_{j},~{}j\in\mathcal{J}\\}$.

1: for $t=1\rightarrow T_{0}$ do

2: Randomly generate the phase shift of each reflecting element in
$\\{\tilde{\mbox{\boldmath{$\Theta$}}}_{j},~{}j\in\mathcal{J}\\}$ from
interval $[0,2\pi)$, which is denoted as
$\\{\tilde{\mbox{\boldmath{$\Theta$}}}_{j,t},~{}j\in\mathcal{J}\\}$.

3: Based on the estimated effective channels from all users, i.e.,
$\\{\mbox{\boldmath{$h$}}_{k}(\\{\tilde{\mbox{\boldmath{$\Theta$}}}_{j,t},~{}j\in\mathcal{J}\\}),~{}k\in\mathcal{K}\\}$,
the BS calculates the sum-rate of all users as
$R_{t}(\\{\tilde{\mbox{\boldmath{$\Theta$}}}_{j,t},~{}j\in\mathcal{J}\\})$.

4: end for

5: Initialize
$\tilde{\mbox{\boldmath{$\Theta$}}}_{j}=\tilde{\mbox{\boldmath{$\Theta$}}}_{j,t_{0}},~{}j\in\mathcal{J}$,
where
$t_{0}=\text{arg}\max_{t=1,2,\cdots,T_{0}}R_{t}(\\{\tilde{\mbox{\boldmath{$\Theta$}}}_{j,t},~{}j\in\mathcal{J}\\}$
and the corresponding sum-rate of all users
$R(\\{\tilde{\mbox{\boldmath{$\Theta$}}}_{j},~{}j\in\mathcal{J}\\})$ is
denoted as $R_{o}$.

6: repeat

7: for $j=1\rightarrow J$ do

8: for $t=1\rightarrow T_{j}$ do

9: Randomly generate the phase shift of each reflecting element in
$\tilde{\mbox{\boldmath{$\Theta$}}}_{j}$ from interval $[0,2\pi)$, which is
denoted as $\tilde{\mbox{\boldmath{$\Theta$}}}_{j,t}$.

10: Based on the estimated effective channels from all users, i.e.,
$\\{\mbox{\boldmath{$h$}}_{k}(\tilde{\mbox{\boldmath{$\Theta$}}}_{j,t},\\{\tilde{\mbox{\boldmath{$\Theta$}}}_{q},~{}q\neq
j\in\mathcal{J}\\}),~{}k\in\mathcal{K}\\}$, the BS calculates the sum-rate of
all users as
$R_{t}(\tilde{\mbox{\boldmath{$\Theta$}}}_{j,t},\\{\tilde{\mbox{\boldmath{$\Theta$}}}_{q},~{}q\neq
j\in\mathcal{J}\\})$.

11: end for

12: Set
$\tilde{\mbox{\boldmath{$\Theta$}}}_{j}^{\star}=\tilde{\mbox{\boldmath{$\Theta$}}}_{j,t_{j}}$
with
$t_{j}=\text{arg}\max_{t=1,2,\cdots,T_{j}}R_{t}(\tilde{\mbox{\boldmath{$\Theta$}}}_{j,t},\\{\tilde{\mbox{\boldmath{$\Theta$}}}_{q},~{}q\neq
j\in\mathcal{J}\\})$.

13: if Objective value of (P1) is improved, i.e.,
$R(\tilde{\mbox{\boldmath{$\Theta$}}}_{j}^{\star},\\{\tilde{\mbox{\boldmath{$\Theta$}}}_{q},~{}q\neq
j\in\mathcal{J}\\})>R_{0}$ then

14: Update
$\tilde{\mbox{\boldmath{$\Theta$}}}_{j}=\tilde{\mbox{\boldmath{$\Theta$}}}_{j}^{\star}$
and
$R_{o}=R(\tilde{\mbox{\boldmath{$\Theta$}}}_{j}^{\star},\\{\tilde{\mbox{\boldmath{$\Theta$}}}_{q},~{}q\neq
j\in\mathcal{J}\\})$.

15: end if

16: end for

17: until the increment of the objective function in (P1) is smaller than
$\epsilon_{2}$ or the maximum number of iterations $I_{r,\max}$ is reached.

17: $\\{\tilde{\mbox{\boldmath{$\Theta$}}}_{j},~{}j\in\mathcal{J}\\}$.

##  IV Numerical Results

In this section, we provide numerical results to validate the performance of
the proposed integrated IRS-BS architecture and passive reflection designs.
Unless otherwise specified, the simulation parameters are set as follows. The
operating frequency is set as $f_{c}=6~{}\text{GHz}$, and thus the carrier
wavelength is $\lambda=0.05~{}\text{m}$. The size of the antenna array of the
BS is set as $M=16$ with $M_{x}=M_{z}=4$, where the antenna spacing is set as
$d_{a}=\lambda/2=0.025~{}\text{m}$. The number of reflecting elements in each
IRS $j\in\mathcal{J}$ is set to $N_{j}=8$ with $N_{j,1}=1$ and $N_{j,2}=8$,
and thus the total number of reflecting elements is
$N=\sum_{j=1}^{J}N_{j}=32$. The spacing between two adjacent reflecting
elements is set as $d_{I}=\lambda/2=0.025~{}\text{m}$, the area of each
reflecting element is set as $A=(\lambda/2)^{2}$, and the initial phase of the
received signals at IRS $j$ relative to the origin $O$ is set to
$\varphi_{j}=0,~{}j\in\mathcal{J}$, for simplicity. The suspension angle of
the antenna radome is set to $\theta_{tilt}=0$, the noise power is set as
$\sigma^{2}=-70~{}\text{dBm}$, and the transmit power at users is set as
$P=30~{}\text{dBm}$. For Algorithm 1, we set its stopping threshold as
$\epsilon_{1}=10^{-5}$, the maximum number of iterations as $I_{\max}=100$,
and the number of random initialization as $T=100$. The number of ground users
is set to $K=3$, which are assumed to have quasi-static/slow-fading channels
with the antenna array of the BS and all IRSs. For the considered geometric
multipath channels between users and the BS, we set $L_{k}=4$ and
$\tilde{a}_{k,l}\sim\mathcal{CN}(0,2\times
10^{-12}),~{}k\in\mathcal{K},l\in\mathcal{L}_{k}$, the elevation angle
$\tilde{\theta}_{k,l}$ is randomly generated in interval $[0,\pi/4)$, and the
azimuth angle $\tilde{\phi}_{k,l},$ is randomly generated in interval
$[0,2\pi)$. Furthermore, we adopt the 3GPP radiation pattern for the antenna
elements at the BS [38]. All results in the simulation are averaged over 100
independent realizations of random channels.

###  IV-A Convergence Behaviour of Algorithm 1

Figure 6: Convergence behaviour of Algorithm 1.

First, Fig. 6 shows the convergence behaviour of Algorithm 1 under the case of
one channel realization as well as that averaged over 100 independent channel
realizations. It is observed that the sum-rate monotonically increases for
both cases and converges after about 10 iterations. Moreover, for the case of
100 independent channel realizations, the objective value increases from 4.3
bps/Hz to 7.1 bps/Hz during the iterations, which reaps 65.1% increment on the
achievable sum-rate. The results verify the convergence and efficacy of
Algorithm 1 in improving the sum-rate performance of the considered integrated
IRS-BS architecture.

###  IV-B Performance Comparison between Proposed and Benchmark Algorithms

Next, we evaluate the efficacy of the proposed Algorithm 1 with perfect CSI
and Algorithm 2 (i.e., IRPA) without CSI by comparing them with the
conventional RPA introduced in Section III-C and another benchmark algorithm
called discrete Fourier transform (DFT)-based codebook search (DFT-CS), which
is described as follows.

  * •

DFT-CS: In this benchmark algorithm, we search the optimal codeword in a two-
dimensional DFT codebook for designing the IRS passive reflection
coefficients. For each IRS $j,~{}j\in\mathcal{J}$, the codebook is defined as
$\mathcal{W}_{j}=\\{\mbox{\boldmath{$w$}}_{j}|\mbox{\boldmath{$w$}}_{j}=\mbox{\boldmath{$w$}}_{j,1}\otimes\mbox{\boldmath{$w$}}_{j,2},\mbox{\boldmath{$w$}}_{j,1}\in\mathcal{W}_{j,1},\mbox{\boldmath{$w$}}_{j,2}\in\mathcal{W}_{j,2}\\}$,
where $\mathcal{W}_{j,1}$ and $\mathcal{W}_{j,2}$ represent the sets of
vectors including all the columns of DFT matrices of size $N_{j,1}$ and
$N_{j,2}$, respectively. Then, we jointly search the passive reflection
coefficients of all IRSs over their given codebooks for achieving the maximum
sum-rate $R(\mbox{\boldmath{$\Theta$}})$ in (14). Thus, the overall training
overhead required for implementing DFT-CS is given by
$\prod_{j=1}^{J}N_{j,1}N_{j,2}$.

Fig. 7 shows the sum-rate of all users versus the overall training overhead
$T_{total}$. On one hand, for IRPA, we set its stopping threshold as
$\epsilon_{2}=10^{-5}$, the maximum number of iterations as $I_{r,\max}=10$,
$T_{0}=200$ and
$T_{j}=\frac{T_{total}-T_{0}}{4I_{r,\max}},~{}j\in\mathcal{J}$. On the other
hand, for the RPA, we set its overall training overhead (i.e., the total
number of randomly generated solutions for
$\\{\tilde{\mbox{\boldmath{$\Theta$}}}_{j},~{}j\in\mathcal{J}\\}$) as
$T_{total}$. It is observed that the sum-rates achieved by IRPA and RPA
increase with $T_{total}$ due to more randomly generated reflection
coefficient solutions. As expected, the IRPA outperforms the RPA because the
former decouples the design of reflection coefficients for different IRSs and
thus more effectively controls the double-reflection channel components.
Besides, the IRPA is observed to approach the performance upper-bound achieved
by Algorithm 1 with perfect CSI as $T_{total}$ increases, even without any CSI
initially. Furthermore, it is observed that the IRPA achieves better
performance than the DFT-CS despite the higher training overhead for DFT-CS,
i.e., $\prod_{j=1}^{J}N_{j,1}N_{j,2}=4096$ in this example. This indicates
that the conventional DFT-CS that performs well in the systems with the UPW
propagation adopted for the channels between the IRSs and the antenna array of
the BS, as well as only single-reflection channel components present [47], is
not applicable to our considered integrated IRS-BS architecture, since the
far-field condition is not satisfied between the IRSs and the antenna array of
the BS, and the more pronounced double-reflection channel components coexist
with the single-reflection channel components (see (1)) [48].

Figure 7: Sum-rate versus the overall training overhead, $T_{total}$.

###  IV-C Performance Comparison under Different Channel Setups

In addition, we aim to evaluate the impact of the single-reflection and
double-reflection channel components on the performance of the proposed
integrated IRS-BS architecture. Towards this end, we define two benchmark
channel setups for performance comparison, where only the single-reflection
channel components, i.e.,
$\mbox{\boldmath{$f$}}_{k,j}^{n_{j}},~{}k\in\mathcal{K},j\in\mathcal{J},n_{j}\in\mathcal{N}_{j}$,
are considered (termed as “single-reflection only”), and only the double-
reflection channel components, i.e.,
$\mbox{\boldmath{$g$}}_{k,j,q}^{n_{j},n_{q}},~{}k\in\mathcal{K},j\neq
q\in\mathcal{J},n_{j}\in\mathcal{N}_{j},n_{q}\in\mathcal{N}_{q}$, are
considered (termed as “double-reflection only”), respectively. Our proposed
channel setup in (1) is termed as “single- and double- reflection”. We also
consider the conventional multi-antenna BS without integrated IRS as a
benchmark scheme, which is termed as “no-IRS”. For all setups employing IRSs,
we adopt Algorithm 1 to design the reflection coefficients of all IRSs by
assuming that perfect CSI is available at the BS. Besides, we aim to study the
impact of the total number of reflecting elements $N$ on the performance of
all schemes with integrated IRSs. Thus, we increase $N_{j,1}$ from $1$ to $5$
and fix $N_{j,2}=8,~{}j\in\mathcal{J}$, and thus the total number of
reflecting elements $N$ is increased from $32$ to $160$.666To reduce the
computational complexity, we apply the IRS element grouping strategy [4, 29]
when $N\geq 32$ to group the reflecting elements in the same $y$-axis of each
IRS into a subsurface, which will employ a common reflection coefficient.
Under this strategy, the total number of subsurfaces under all considered $N$
values is fixed as $\tilde{N}=32$.

Figure 8: Sum-rate versus the total number of reflecting elements, $N$.

Fig. 8 shows the sum-rates achieved by all setups versus the total number of
reflecting elements, $N$. It is observed that the sum-rates achieved by the
setups with integrated IRSs monotonically increase with $N$ due to higher
passive array reflection gain. It is also observed that the “double-reflection
only” setup performs closely to “single- and double- reflection” setup, and
achieves a higher performance gain over the “single-reflection only” setup.
The performance gap between the “double-reflection only” and “single-
reflection only” setups increases with $N$. This is because in our proposed
integrated IRS-BS architecture, the path loss of the double-reflection channel
components is not severe due to the short distance between different IRSs.
Moreover, the passive array reflection gain provided by the double-reflection
channel components is larger than that provided by the single-reflection
channel components [25, 24], and the number of double-reflection components is
much larger than that of single-reflection components, i.e.,
$\sum_{j=1}^{J}\sum_{q\neq j}^{J}N_{j}N_{q}\gg N$. As a result, the double-
reflection channel components have a more dominant impact on the sum-rate
performance as compared to the single-reflection channel components.
Furthermore, despite a limited number of IRS reflecting elements, the “single-
and double- reflection” setup significantly outperforms the “no-IRS” benchmark
scheme employing the same number of antennas (i.e., $M=16$) or even more
antennas (i.e., $M=64$). Thus, the proposed integrated IRS-BS architecture is
an efficient solution for enhancing the communication rate performance or
reducing the number of antennas at the BSs for achieving the same rate
performance.

###  IV-D Effect of Suspension Angle, $\theta_{tilt}$

Figure 9: Sum-rates under different suspension angles, $\theta_{tilt}$.

Furthermore, we investigate the effect of suspension angle $\theta_{tilt}$ on
the system performance. Towards this end, we respectively set
$\theta_{tilt}=0$ (e.g., see Fig. 3(a)) and $\theta_{tilt}=\pi/2$ (e.g., see
Fig. 3(b)), where Algorithm 1 is adopted to design the passive reflection
coefficients of all IRSs by assuming perfect CSI at the BS. As can be observed
in Fig. 9, the BS can achieve better performance by setting its suspension
angle as $\theta_{tilt}=\pi/2$ than setting it as $\theta_{tilt}=0$. The
reason is that when $\theta_{tilt}=0$, all users are located outside the
reflection half-space of IRS 4, which is deployed at the bottom of the antenna
radome (see Fig. 1). This is more likely to lead to
$\hat{\theta}_{k,l,4}>\pi/2,~{}k\in\mathcal{K},l\in\mathcal{L}_{k}$, and thus
IRS 4 cannot provide reflection gains to the single-reflection and double-
reflection channel components from the users to it directly. Specifically, we
have $G(\hat{\theta}_{k,l,4},\hat{\theta}_{4,m}^{n_{4}})=0$ and
$G(\hat{\theta}_{k,l,4},\hat{\theta}_{4,j}^{n_{4},n_{j}})=0$, which result in
$f_{k,4,m}^{n_{4}}=0$ and
$g_{k,4,j,m}^{n_{4},n_{j}}=0,~{}m\in\mathcal{M},n_{4}\in\mathcal{N}_{4},j\neq
4\in\mathcal{J},n_{j}\in\mathcal{N}_{j}$. In contrast, if the suspension angle
is set as $\theta_{tilt}=\pi/2$, each IRS may provide reflection gains to all
users via all single-reflection and double-reflection channel components, thus
enhancing the system rate performance.

###  IV-E Performance Comparison under Different Numbers of Antenna Modules
for Generalized IRS-BS Architecture with Modular Antenna Arrays

Finally, we compare the performance of the generalized IRS-BS architecture
with modular antenna arrays under different numbers of antenna modules as
introduced in Section II-C, where Algorithm 1 is adopted to design the
reflection coefficients of all IRSs. Note that for this generalized IRS-BS
architecture, the LoS channels only exist between each antenna and its
surrounding IRSs (modelled as $\xi_{j,m}^{n_{j}}$ in (10)), and between two
reflecting elements surrounding the same antenna module (modelled as
$\zeta_{j,q}^{n_{j},n_{q}}$ in (13)).

Figure 10: Sum-rate versus the number of antenna modules, $\eta$.

Fig. 10 shows the sum-rate versus the number of antenna modules, i.e., $\eta$.
It is observed that the performance of the generalized IRS-BS architecture
with modular antenna arrays slightly increases with $\eta$ when $\eta\leq 4$
and reaches steady state when $\eta>4$. However, the total number of IRS
reflecting elements always increases with $\eta$, which results in an
increasing cost at the BS as well as more overhead for channel estimation.
Specifically, the sum-rates achieved by the “single- and double- reflection”
and “double-reflection only” setups first increase by 12% from $\eta=1$ to
$\eta=4$ and then become approximately static when $\eta>4$. The sum-rate
achieved by “single-reflection only” setup first increases by 10% from
$\eta=1$ to $\eta=4$ and then decreases to the same level as that for
$\eta=1$. This is because when $\eta\leq 4$, signal propagation distance of
the single-reflection and double-reflection channel components decreases as
$\eta$ increases, and thus the achievable sum-rate increases. In contrast,
when $\eta>4$, the number of single-reflection and double-reflection channel
components is reduced, which degrades the sum-rate performance, even though
their path loss still decreases as $\eta$ increases. The results in Fig. 10
provide useful guidelines for choosing proper integrated BS-IRS architecture
to balance the performance and cost trade-off.

##  V Conclusions

In this paper, we proposed a new integrated IRS-BS architecture to deploy IRSs
with different orientations and an antenna array within the same antenna
radome at the BS, and also extended it to a generalized IRS-BS architecture
with modular antenna arrays for accommodating more reflecting elements.
Considering the ultra-short propagation distance, we proposed an element-wise
channel model for IRS to characterize the direct (without any IRS’s
reflection) as well as the single and double IRS-reflection channel components
between each single-antenna user and the antenna array of the BS, where the
UPW was utilized to model the channels from each reflecting element (instead
of the whole surface of the IRS) to each antenna at the BS as well as between
the reflecting elements of different IRSs. Under the considered channel model,
we jointly optimized the reflection coefficients of all IRSs to maximize the
uplink sum-rate of the users. By considering two typical cases with/without
perfect CSI at the BS, the formulated problem was solved efficiently by
exploiting the successive refinement method and IRPA, respectively.

Numerical results demonstrated that the proposed integrated IRS-BS
architecture can achieve significant performance gain over the conventional
multi-antenna BS without integrated IRS. It was also shown that the developed
IRPA can outperform other benchmark algorithms in terms of sum-rate under the
case without perfect CSI, and approach the performance upper bound with
perfect CSI as the training overhead increases. Moreover, the double-
reflection channel components were shown to have a more dominant impact on the
achievable sum-rate than the single-reflection channel components due to the
short inter-IRS distance and the significantly larger number of double-
reflection channel components. Finally, the number of antenna modules in the
generalized IRS-BS architecture was shown to have a significant influence on
the system rate performance, which should be properly selected in practice to
balance the communication performance and implementation cost. It is
noteworthy that this paper is an initial study of integrating IRSs into BS/AP,
while the results can be extended to other promising directions in future
work, such as considering practical IRS phase-shift model [49], IRS passive
reflection optimization in frequency-selective wideband channels, joint
optimization of IRS passive reflection and user transmission scheduling, IRS
passive reflection optimization based on statistical CSI to reduce the
training overhead, and so on.

## References

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