# A Joint Precoding Framework for Wideband Reconfigurable Intelligent Surface-
Aided
Cell-Free Network

Zijian Zhang, , Linglong Dai All authors are with the Beijing National
Research Center for Information Science and Technology (BNRist) as well as the
Department of Electronic Engineering, Tsinghua University, Beijing 100084,
China (e-mails<EMAIL_ADDRESS>daill@tsinghua.edu.cn).This
work was supported by the National Science and Technology Major Project of
China under Grant 2018ZX03001004-003 and the National Natural Science
Foundation of China for Outstanding Young Scholars under Grant 61722109.

###### Abstract

Thanks to the strong ability against the inter-cell interference, cell-free
network has been considered as a promising technique to improve the network
capacity of future wireless systems. However, for further capacity
enhancement, it requires to deploy more base stations (BSs) with high cost and
power consumption. To address the issue, inspired by the recently proposed
technique called reconfigurable intelligent surface (RIS), we propose the
concept of RIS-aided cell-free network to improve the network capacity with
low cost and power consumption. The key idea is to replace some of the
required BSs by low-cost and energy-efficient RISs, and deploy more RISs in
the cell-free network for capacity enhancement. Then, for the proposed RIS-
aided cell-free network in the typical wideband scenario, we formulate the
joint precoding design problem at the BSs and RISs to maximize the network
capacity. Due to the non-convexity and high complexity of the formulated
problem, we develop an alternating optimization algorithm to solve this
challenging problem. In particular, we decouple this problem via Lagrangian
dual transform and fractional programming, and solve the subproblems
alternatively. Note that most of the considered scenarios in existing works
are special cases of the general scenario in this paper, and the proposed
joint precoding framework can also serve as a general solution to maximize the
capacity in most of existing RIS-aided scenarios. Finally, simulation results
verify that, compared with the conventional cell-free network, the network
capacity of the proposed scheme can be improved significantly.

###### Index Terms:

Cell-free network, reconfigurable intelligent surface (RIS), wideband, joint
precoding.

##  I Introduction

Network technique is the most essential technique to increase the capacity of
wireless communication systems [1]. Compared with 4G, the capacity of 5G
wireless network is expected to be increased by about 1000 times [2]. In the
currently deployed cellular networks, all users in a cell are mainly served by
one base station (BS), thus the users close to the cell boundary usually
suffer from the severe inter-cell interference, which is caused by the signals
from adjacent cells.

Ultra-dense network (UDN) has been proposed as a promising technique for 5G to
further enhance the network capacity [3, 4]. The key idea of UDN is to
increase the number of BSs and deploy small cells [5] in the cell-centric
cellular network. However, as the increase of the cell density, the inter-cell
interference grows larger and larger. The cooperation theory has determined
that [6], the upper limit of the network capacity will be bounded by the
inter-cell interference as long as the network-centric network is used. In
other words, inter-cell interference becomes the bottleneck for the network
capacity improvement of UDN. This problem is inherent to the cell-centric
network paradigm, and cannot be efficiently solved [7].

To address the issue, a novel user-centric network paradigm called cell-free
network has been recently proposed [7]. Unlike the classical cell-centric
design principle, the cell-free network utilizes the user-centric transmission
design, where all BSs in the network jointly serve all users cooperatively
without cell boundaries. Dut to the efficient cooperation among all
distributed BSs [8], the inter-cell interference can be effectively
alleviated, and thus the network capacity can be increased accordingly. This
promising technique has been considered as a potential candidate for future
communication system [9], and has attracted the increasing research interest
such as resource allocation [10], precoding/beamforming [11], channel
estimation [12] in recent years.

However, to improve the network capacity further, the deployment of more
distributed synchronized BSs requires high cost and power consumption in the
cell-free network. Fortunately, the emerging new technique called
reconfigurable intelligent surface (RIS) is able to provide an energy-
efficient alternative to enhance the network capacity. Equipped with a low-
cost, energy-efficient and high-gain metasurface, RIS is becoming a promising
smart radio technique for future 6G communications [13]. With a large number
of low-cost passive elements, RIS is able to reflect the elertromagentic
incident signals to any directions with extra high array gains by adjusting
the phase shifts of its elements [14]. Since the wireless environment can be
effectively manipulated with low cost and energy consumption [15], RIS can be
used to improve the channel capacity [16], reduce the transmit power [17],
enhance the transmission reliability [18], and enlarge the wireless coverage
[19].

###  I-A Prior works

The existing research works about RIS include antenna design [20], channel
estimation [21], joint precoding/beamforming [22], and etc. Particularly, the
prototype of RIS-based wireless communication has been recently developed in
[23] to demonstrate its function in real systems.

One key guarantee for RIS to improve the network capacity is the joint
precoding. Different from the conventional precoding at the BS only, the joint
precoding in RIS-based wireless systems refers to the joint design of the
beamforming vector at the BS and the phase shifts of the RIS elements.
Different RIS-based scenarios have been studied to maximize the capacity in
the literature. Specifically, the authors in [24] considered a scenario where
one BS and one RIS jointly serve a single user, which was an early attempt to
realize the capacity enhancement by using low-cost and energy-efficient RIS.
In [25], the authors considered a multi-user scenario and maximized the sum-
rate of all users. To obtain the cooperation gain, the multi-BS scenario was
considered in [19], while the multi-RIS case was investigated in [18],
respectively. Besides, to satisfy the requirements of wideband system, the
multi-carrier scenario has been discussed in [26].

Apart from the capacity maximization, the design goal for joint precoding can
be different. Specifically, to reduce the power consumption, some researchers
have proposed the methods to minimize the transmit power subject to the
constraints of the user requirements [17], and the authors in [15] have
developed a method to maximize the energy efficiency. Some researchers have
considered the fairness among users, and some methods have been proposed to
maximize the minimum signal-to-interference-plus-noise ratio (SINR) [22, 27].
In addition, minimizing the symbol error rate (SER) was considered in [28].

Moreover, RIS has been combined with different techniques in the literature.
For instance, the RIS-aided millimeter-wave (mmWave) system was considered in
[18] and [29], where the authors discussed the signal-to-noise ratio (SNR)
maximization problem and the joint hybrid precoding design, respectively. In
[26] and [30], the application of RIS in wideband orthogonal frequency
division multiplexing (OFDM) systems was discussed to maximize the sum-rate.
Moreover, in [19] and [27], the authors considered the cooperation of BSs and
RIS in a multi-cell system. How to exploit RIS to enhance the performance of
non-orthogonal multiple acces (NOMA) systems was discussed in [31]. In
addition, the secure transmission via RIS has been investigated in [32], and
RIS was also considered to assist the communication using unmanned aerial
vehicles (UAVs) [33].

###  I-B Our contributions

To address the challenge of cell-free network as mentioned above, in this
paper we consider to exploit RISs to realize the improvement of network
capacity with low cost and power consumption. Specifically, the contributions
of this paper can be summarized as follows.

  * •

We propose the concept of RIS-aided cell-free network to further improve the
network capacity of the cell-free network with low cost and power consumption.
The key idea is to replace some of the BSs in cell-free network by the energy-
efficient RISs and deploy more RISs in the system for capacity enhancement. In
the proposed RIS-aided cell-free network, all BSs and RISs are simultaneously
serving all users cooperatively. To the best of our knowledge, this is the
first attempt to exploit RIS in cell-free network to improve the network
capacity.

  * •

For the proposed RIS-aided cell-free network in the typical wideband scenario,
we formulate the joint precoding design problem at the BSs and RISs to
maximize the weighted sum-rate (WSR) of all users to optimize the network
capacity. Since the considered scenario is very general, i.e., multiple
antennas, multiple BSs, multiple RISs, multiple users, and multiple carriers,
most of the considered scenarios in existing works, such as single BS, single
RIS, single user, and single carrier, or some of them are mutiple, are all
special cases of the considered scenario in this paper.

  * •

Inspired by the methods introduced in [34] [35], we propose a joint precoding
framework to solve the formulated WSR maximization problem. Specifically, the
proposed framework is an alternating optimization algorithm, which can
approximate the local optimal solution gradually. We first decouple the active
precoding at the BSs and the passive precoding at the RISs via Lagrangian dual
reformulation and fractional programming. Then, the decoupled subproblems are
reformulated as the quadratically constrained quadratic program (QCQP)
problems, respectively. By solving the subproblems alternatively, the WSR will
converge to a local optimal solution. The simulation results show that RISs
can improve the cell-free network capacity significantly. Note that the
proposed joint precoding framework can also serve as a general solution to
maximize the WSR in most of the existing RIS-aided scenarios in the
literature.

###  I-C Organization and notation

Organization: The rest of the paper is organized as follows. The system model
of the proposed RIS-aided cell-free network and corresponding WSR maximization
problem formulation of joint precoding design are discussed in Section II. The
joint precoding framework to solve the formulated problem is proposed in
Section III. Numerical simulation results are provided in Section IV to
validate the performance of the proposed joint precoding framework for the
RIS-aided cell-free network. Finally, in Section V, conclusions are drawn and
future works are discussed.

Notations: $\mathbb{C}$, $\mathbb{R}$, and $\mathbb{R}^{+}$ denote the set of
complex, real, and positive real numbers, respectively; ${[\cdot]^{-1}}$,
${[\cdot]^{*}}$, ${[\cdot]^{T}}$, and ${[\cdot]^{H}}$ denote the inverse,
conjugate, transpose, and conjugate-transpose operations, respectively;
$\|\cdot\|$ denotes the Euclidean norm of its argument;
$\mathbb{E}\\{\cdot\\}$ is the expectation operator; ${\rm diag}(\cdot)$
denotes the diagonal operation; $\mathfrak{R}\\{\cdot\\}$ denotes the real
part of its argument; $\otimes$ denotes the Kronecker product; $\angle[\cdot]$
denotes the angle of its complex argument; $\mathbf{I}_{L}$ is an $L\times L$
identity matrix, and $\mathbf{0}_{L}$ is an $L\times L$ zero matrix; Finally,
$\mathbf{e}_{l}$ is an elementary vector with a one at the $l$-th position,
and $\mathbf{1}_{L}$ indicates an $L$-length vector with all elements are 1.

##  II System Model of The Proposed RIS-Aided Cell-Free Network

Figure 1: The proposed concept of RIS-aided cell-free network.

To increase the network capacity with low cost and power consumption, in this
paper we first propose the concept of RIS-aided cell-free network. Different
form the traditional cell-free system, we replace some BSs by RISs and add
more RISs in the network, where multiple BSs and multiple RISs are
simultaneously employed to serve multiple users cooperatively. In this
section, the architecture of the proposed RIS-aided cell-free network will be
introduced at first. Then, we will discuss the transmitters, channels, and
receivers, respectively. Finally, we will formulate the problem of capacity
maximization problem in the RIS-aided cell-free network.

###  II-A System architecture

In this paper, we consider a wideband RIS-aided cell-free network as shown in
Fig. 1, where multiple distributed BSs and RISs are deployed to cooperatively
serve all users. A central processing unit (CPU) is deployed for control and
planning, to which all BSs are connected by optical cables or wireless
backhaul [36]. All RISs are controlled by the CPU or BSs by wire or wireless
control. The considered network consists of $B$ BSs, $R$ RISs, and $K$ multi-
antenna users. The number of antennas at the $b$-th BS and that at the $k$-th
user are $M_{b}$ and $U_{k}$, respectively. The number of elements at the
$r$-th RIS is $N_{r}$. For simplicity but without loss of generality, we
assume $M_{b}$, $U_{b}$ and $N_{r}$ are equal to $M$, $U$ and $N$,
respectively. Finally, the multi-carrier transmission is considered and the
number of available subcarriers is $P$. Let ${\cal
M}=\left\\{{1,\cdots,M}\right\\}$, ${\cal U}=\left\\{{1,\cdots,U}\right\\}$,
${\cal N}=\left\\{{1,\cdots,N}\right\\}$, ${\cal
B}=\left\\{{1,\cdots,B}\right\\}$, ${\cal R}=\left\\{{1,\cdots,R}\right\\}$,
${\cal K}=\left\\{{1,\cdots,K}\right\\}$ and ${\cal
P}=\left\\{{1,\cdots,P}\right\\}$ denote the index sets of BS antennas, user
antennas, RIS elements, BSs, RISs, users, and subcarriers, respectively.

###  II-B Transmitters

In the proposed RIS-aided cell-free network, all BSs are synchronized, which
is necessary to serve all users by coherent joint transmission [8]. We follow
the common assumption in the literature that all BSs transmit the same symbols
[37, 11, 38, 39]. Let
${{\bf{s}}_{p}}\triangleq{\left[{{s_{p,1}},\cdots,{s_{p,K}}}\right]^{T}}\in{{\mathbb{C}}^{K}}$,
where ${s}_{p,k}$ denotes the transmitted symbol to the $k$-th user on the
$p$-th subcarrier. We assume that the transmitted symbols have normalized
power, i.e.,
$\mathbb{E}\left\\{\mathbf{s}_{p}\mathbf{s}_{p}^{H}\right\\}=\mathbf{I}_{K},\forall
p\in\mathcal{P}$. In the downlink, the frequency-domain symbol ${s}_{p,k}$ is
firstly precoded by the precoding vector ${\bf
w}_{b,p,k}\in{{\mathbb{C}}^{M}}$ at the $b$-th BS, so the precoded symbol
${{\bf{x}}_{b,p}}$ at the $b$-th BS on the $p$-th subcarrier can be written as

${{\bf{x}}_{b,p}}=\sum\limits_{k=1}^{K}{{{{\bf{w}}_{b,p,k}}{s_{p,k}}}}.$ (1)

Then, by inverse discrete Fourier transform (IDFT), the frequency-domain
signal $\left\\{{{{\bf{x}}_{b,p}}}\right\\}_{p=1}^{P}$ on all $P$ subcarriers
at the $b$-th BS is converted to the time domain. After adding the cyclic
prefix (CP), the signal is up-converted to the radio frequency (RF) domain via
${M}$ RF chains of the $b$-th BS.111In this paper, the fully-digital precoding
is considered for the proposed RIS-aided cell-free network, since usually the
number of BS antennas is not very large. The extension to the hybrid precoding
[40] is left for future work.

###  II-C Channels

Figure 2: The downlink channels in the wideband RIS-aided cell-free network.

Thanks to the directional reflection supported by $R$ RISs, as shown in Fig.
2, the channel between each BS and each user in the proposed RIS-aided cell-
free network consists of two parts: the BS-user link and $R$ BS-RIS-user
links, where each BS-RIS-user link can be further divided into a BS-RIS link
and a RIS-user link. The signal reflection on the RISs can be modeled by
multiplying the incident signal with a phase shift matrix and forwarding the
phase shifted signal to the user [24]. Therefore, the equivalent channel
${{\bf{h}}_{b,k,p}^{H}}$ from the $b$-th BS to the $k$-th user on the $p$-th
subcarrier can be written as [41]

${{\bf{h}}_{b,k,p}^{H}}=\underbrace{{{\bf{H}}_{b,k,p}^{H}}}_{\text{BS-user
link}}+\underbrace{\sum\limits_{r=1}^{R}{{{\bf{F}}_{r,k,p}^{H}{\bf{\Theta}}_{r}^{H}{{\bf{G}}_{b,r,p}}}}}_{\text{BS-
RIS-user links}},$ (2)

where ${{\bf{H}}_{b,k,p}^{H}}\in\mathbb{C}^{U\times M}$,
${\bf{G}}_{b,r,p}\in\mathbb{C}^{N\times M}$, and
${\bf{F}}_{r,k,p}^{H}\in\mathbb{C}^{U\times N}$ denote the frequency-domain
channel on the subcarrier $p$ from the BS $b$ to the user $k$, from the BS $b$
to the RIS $r$, and from the RIS $r$ to the user $k$, respectively;
${\bf{\Theta}}_{r}\in\mathbb{C}^{N\times N}$ denotes the phase shift matrix at
the RIS $r$, which is written as

${{\bf{\Theta}}_{r}}\buildrel\Delta\over{=}{\mathop{\rm
diag}\nolimits}\left({{\theta_{r,1}},\cdots,{\theta_{r,N}}}\right),~{}\forall
r\in\mathcal{R},$ (3)

where ${{\theta_{r,n}}}\in{\cal F}$. Note that ${\cal F}$ is the feasible set
of the reflection coefficient (RC) at RIS. Here we assume ${\cal F}$ is the
ideal RIS case, i.e., both the amplitude and the phase of ${\theta_{r,n}}$
associated with the RIS element can be controlled independently and
continuously, i.e.,

${{\cal
F}}\triangleq\left\\{{\theta_{r,n}}{\Big{|}}\left|{{\theta_{r,n}}}\right|\leq
1\right\\},~{}\forall r\in\mathcal{R},~{}\forall n\in{{\cal N}}.$ (4)

Note that the more practical RIS reflection coefficients such as low-
resolution discrete phase shifts will be discussed in Subsection III-E later.

###  II-D Receivers

After passing through the equivalent channel ${{\bf{h}}_{b,k,p}^{H}}$ as shown
in (2), the signals will be received by the users. The time-domain signals
received by the users are down-converted to the baseband at first. After the
CP removal and the discrete Fourier transform (DFT), the frequency-domain
symbols can be finally recovered. Let ${{\bf y}_{b,k,p}}\in\mathbb{C}^{U}$
denote the baseband frequency-domain signal, which is and received by the user
$k$ on the subcarrier $p$ from the BS $b$. Then, according to the channel
model above and considering the additive white Gaussion noise (AWGN) ${\bf
v}_{b,k,p}$ at the receivers, ${{\bf y}_{b,k,p}}$ can be expressed by
combining (1) and (2) as

$\displaystyle{{\bf y}_{b,k,p}}={\bf{h}}_{b,k,p}^{H}{{\bf{x}}_{b,p}}+{\bf
v}_{b,k,p}$ (5)
$\displaystyle=\left({\bf{H}}_{b,k,p}^{H}+\sum\limits_{r=1}^{R}{{\bf{F}}_{r,k,p}^{H}{\bf{\Theta}}_{r}^{H}{{\bf{G}}_{b,r,p}}}\right)\sum\limits_{j=1}^{K}{{{\bf{w}}_{b,p,j}}{s_{p,j}}}+{\bf
v}_{b,k,p}.$

Since there are $B$ BSs serving $K$ users simultaneously, the received signal
at user $k$ is the superposition of the signals transmitted by $B$ BSs.
Therefore, let ${{\bf y}_{k,p}}\in\mathbb{C}^{U}$ denote the received signal
at the user $k$ on the subcarrier $p$, we have its expression as shown in (6)
at the bottom of this page,

$\begin{aligned} {{\bf y}_{k,p}}=&\sum\limits_{b=1}^{B}{{{\bf
y}_{b,k,p}}}+{\bf
z}_{k,p}=\sum\limits_{b=1}^{B}{\sum\limits_{j=1}^{K}{\left({\bf{H}}_{b,k,p}^{H}+\sum\limits_{r=1}^{R}{{\bf{F}}_{r,k,p}^{H}{\bf{\Theta}}_{r}^{H}{{\bf{G}}_{b,r,p}}}\right){{\bf{w}}_{b,p,j}}{s_{p,j}}}}+{\bf
z}_{k,p}\\\
=&\underbrace{\sum\limits_{b=1}^{B}{\left({\bf{H}}_{b,k,p}^{H}+\sum\limits_{r=1}^{R}{{\bf{F}}_{r,k,p}^{H}{\bf{\Theta}}_{r}^{H}{{\bf{G}}_{b,r,p}}}\right){{\bf{w}}_{b,p,k}}{s_{p,k}}}}_{\text{Desired
signal to user $k$}}+\underbrace{\sum\limits_{b=1}^{B}{\sum\limits_{j=1,j\neq
k}^{K}\left({\bf{H}}_{b,k,p}^{H}+\sum\limits_{r=1}^{R}{{\bf{F}}_{r,k,p}^{H}{\bf{\Theta}}_{r}^{H}{{\bf{G}}_{b,r,p}}}\right){{\bf{w}}_{b,p,j}}{s_{p,j}}}}_{\text{
Interference from other users}}+{{\bf z}_{k,p}}\end{aligned}.$ (6)

where ${\bf
z}_{k,p}\triangleq{\left[{z_{k,p,1}^{T},\cdots,z_{k,p,U}^{T}}\right]^{T}}$
denotes the AWGN with zero mean $\mathbf{0}_{U}$ and covariance
${\bf{\Xi}}_{k,p}=\sigma^{2}{{\bf{I}}_{U}}$. Note that the first term on the
right-hand side of (6) is the desired signal to user $k$, while the second
term denotes the interference from other users. As there are $P$ subcarriers
available in total, we can denote the received signal at user $k$ as
$\left\\{{{{\bf y}_{k,p}}}\right\\}_{p=1}^{P}$.

###  II-E Problem fomulation

Based on the system model above, we consider to maximize the WSR of the
proposed RIS-aided cell-free network in this subsection. At first, the
received signal ${{\bf y}_{b,k,p}}$ in (6) can be simplified as

$\displaystyle{{\bf{y}}_{k,p}}$
$\displaystyle=\sum\limits_{b=1}^{B}{\sum\limits_{j=1}^{K}{\left({{\bf{H}}_{b,k,p}^{H}+\sum\limits_{r=1}^{R}{{\bf{F}}_{r,k,p}^{H}{\bf{\Theta}}_{r}^{H}{{\bf{G}}_{b,r,p}}}}\right){{\bf{w}}_{b,p,j}}{s_{p,j}}}}+{{\bf{z}}_{k,p}}$
(7)
$\displaystyle\mathop{=}\limits^{(a)}\sum\limits_{b=1}^{B}{\sum\limits_{j=1}^{K}{\left({{\bf{H}}_{b,k,p}^{H}+{\bf{F}}_{k,p}^{H}{\bf{\Theta}}^{H}{{\bf{G}}_{r,p}}}\right){{\bf{w}}_{b,p,j}}{s_{p,j}}}}+{{\bf{z}}_{k,p}}$
$\displaystyle\mathop{=}\limits^{(b)}\sum\limits_{b=1}^{B}{\sum\limits_{j=1}^{K}{{\bf{h}}_{b,k,p}^{H}{{\bf{w}}_{b,p,j}}{s_{p,j}}}}+{{\bf{z}}_{k,p}}$
$\displaystyle\mathop{=}\limits^{(c)}\sum\limits_{j=1}^{K}{{\bf{h}}_{k,p}^{H}{{\bf{w}}_{p,j}}{s_{p,j}}}+{{\bf{z}}_{k,p}},$

where $(a)$ holds by defining ${\bf{\Theta}}={\rm
diag}\left({{\bf{\Theta}}_{1}},\cdots,{{\bf{\Theta}}_{R}}\right)$,
${{\bf{F}}_{k,p}}={\left[{{\bf{F}}_{1,k,p}^{T},\cdots,{\bf{F}}_{R,k,p}^{T}}\right]^{T}}$,
and
${{\bf{G}}_{r,p}}={\left[{{\bf{G}}_{1,r,p}^{T},\cdots,{\bf{G}}_{B,r,p}^{T}}\right]^{T}}$,
$(b)$ holds according to (2), and $(c)$ holds by defining
${\bf{h}}_{k,p}=\left[{\bf{h}}_{1,k,p}^{T},\cdots,{\bf{h}}_{B,k,p}^{T}\right]^{T}$
and
${\bf{w}}_{p,k}=\left[{\bf{w}}_{1,p,k}^{T},\cdots,{\bf{w}}_{B,p,k}^{T}\right]^{T}$.
Then, the SINR for the transmitted symbol $s_{p,k}$ at the user $k$ on the
subcarrier $p$ can be easily calculated as

$\displaystyle{\gamma_{k,p}}$ (8) $\displaystyle=$
$\displaystyle{\bf{w}}_{p,k}^{H}{{\bf{h}}_{k,p}}\\!{\left({\sum\limits_{j=1,j\neq
k}^{K}\\!\\!\\!{{\bf{h}}_{k,p}^{H}{{\bf{w}}_{p,j}}{{\left({{\bf{h}}_{k,p}^{H}{{\bf{w}}_{p,j}}}\right)}^{H}}}\\!\\!+\\!{{\bf{\Xi}}_{k,p}}}\right)^{-1}}\\!\\!{\bf{h}}_{k,p}^{H}{{\bf{w}}_{p,k}}.$

Then, the WSR $R_{\rm sum}$ of all $K$ users is given by

$\displaystyle R_{\rm sum}=$
$\displaystyle\sum\limits_{k=1}^{K}{{\eta_{k}}\sum\limits_{p=1}^{P}{{R_{k,p}}}}=\sum\limits_{k=1}^{K}{{\eta_{k}}\sum\limits_{p=1}^{P}{{{\log}_{2}}\left({1+{\gamma_{k,p}}}\right)}}$
(9) $\displaystyle=$
$\displaystyle\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{\eta_{k}}{{{\log}_{2}}\left({1+{\gamma_{k,p}}}\right)}},$

where $\eta_{k}\in\mathbb{R}^{+}$ represents the weight of the user $k$.

We assume that each BS has its own maximum transmit power constraint, which
can be written as

$\displaystyle\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{\left\|{{{\bf{w}}_{b,p,k}}}\right\|}}\leq{P_{b,\max}},~{}\forall
b\in\mathcal{B},$ (10)

where ${P_{b,\max}}$ denotes the maximum transmit power of the BS $b$.
Moreover, for each RIS, we have the phase shift constraint as shown in (4).

Finally, the original WSR maximization optimization problem ${\cal P}^{\rm o}$
to maximize the WSR (9) with the active precoding design of ${\bf{W}}$ at the
BSs and the passive precoding design of $\bf{\Theta}$ at the RISs, subject to
the BS power constraint (10) and RIS phase shift constraint (4) can be
originally formulated as

$\displaystyle{\cal P}^{\rm o}:~{}$
$\displaystyle\mathop{{\rm{max}}}\limits_{{\bf{\Theta}},{\bf{W}}}~{}f({\bf{\Theta}},{\bf{W}}){\rm{=}}\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{{\eta_{k}}{{\log}_{2}}\left({1+{\gamma_{k,p}}}\right)}}$
(11a)
$\displaystyle\,{\rm{s.t.}}~{}~{}~{}C_{1}:\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{\left\|{{{\bf{w}}_{b,p,k}}}\right\|}}\leq{P_{b,\max}},~{}\forall
b\in\mathcal{B},$ (11b)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\,C_{2}:{{\theta_{r,n}}}\in{\cal
F},~{}\forall r\in{{\cal R}},\forall n\in{{\cal N}},$ (11c)

where we have defined ${\bf{W}}$ as follows for expression simplicity:

$\displaystyle{\bf{W}}\\!=\\!{\left[{\bf{w}}_{1,1}^{T},{\bf{w}}_{1,2}^{T},\cdots,{\bf{w}}_{1,K}^{T},{\bf{w}}_{2,1}^{T},{\bf{w}}_{2,2}^{T},\cdots,{\bf{w}}_{P,K}^{T}\right]^{T}}.$
(12)

However, due to the non-convex complex objective function (11a), the joint
optimization of the phase shift matrix $\bf{\Theta}$ and the precoding vector
${\bf{W}}$ is very challenging. Fortunately, inspired by the fractional
programming (FP) methods designed for solving the non-convex problems [34,
35], we propose a joint precoding framework to find an approximate solution to
the problem ${\cal P}^{\rm o}$ in (11) in the following Section III.

##  III Proposed Joint Precoding Framework

In this section, we present the proposed joint precoding framework to solve
the WSR optimization problem ${\cal P}^{\rm o}$ in (11). Specifically, the
section is summarized as follows. An overview of the proposed framework is
provided in Subsection III-A, where the problem ${\cal P}^{\rm o}$ in (11) is
divided into three subproblems. Next, the detailed algorithms to solve these
three subproblems are given in Subsections III-B, III-C, and III-D,
respectively. Then, the discussions of the practical RIS reflection
coefficients are given in Subsection III-E. Finally, the convergence and
complexity of the proposed framework are provided in Subsection III-F.

###  III-A Overview of the proposed joint precoding framework

Firstly, to deal with the complexity of sum-logarithms in the WSR maximization
problem ${\cal P}^{\rm o}$ in (11), based on the multidimensional complex
Lagrangian dual reformulation, a method has been proposed in [34] to decouple
the logarithms, based on which we have the following Proposition 1.

###### Proposition 1

By introducing an auxiliary variable ${\bm{\rho}}\in\mathbb{R}^{PK}$ with
${\bm{\rho}}={[{\rho_{1,1}},\rho_{1,2},\cdots,\rho_{1,K},\rho_{2,1},\rho_{2,2},\cdots,\rho_{P,K}]^{T}}$,
the original problem ${{\cal P}^{\rm o}}$ in (11) is equivalent to

$\displaystyle{\bar{\cal P}}:{}{}$
$\displaystyle\mathop{{\rm{max}}}\limits_{{\bf{\Theta}},{\bf{W}},{\bm{\rho}}}~{}~{}~{}f({\bf{\Theta}},{\bf{W}},{\bm{\rho}})$
(13)
$\displaystyle\,{\rm{s.t.}}~{}~{}~{}C_{1}:\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{\left\|{{{\bf{w}}_{b,p,k}}}\right\|}}\leq{P_{b,\max}},~{}\forall
b\in\mathcal{B},$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\,C_{2}:{{\theta_{r,n}}}\in{\cal
F},~{}\forall r\in{{\cal R}},\forall n\in{{\cal N}},$

where the new objective function $f({\bf{\Theta}},{\bf{W}},{\bm{\rho}})$ is

$\displaystyle f({\bf{\Theta}},{\bf{W}},{\bm{\rho}})=$
$\displaystyle\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{{\eta_{k}}{{\log}_{2}}\left({1+{\rho_{k,p}}}\right)}}-\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{{\eta_{k}}{\rho_{k,p}}}}$
(14)
$\displaystyle+\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{{\eta_{k}}(1+{\rho_{k,p}}){f_{k,p}}({\bf{\Theta}},{\bf{W}})}},$

where the function $f_{k,p}({\bf{\Theta}},{\bf{W}})$ is denoted by

$\displaystyle{f_{k,p}}({\bf{\Theta}},{\bf{W}})=$ (15)
$\displaystyle{\bf{w}}_{p,k}^{H}{{\bf{h}}_{k,p}}{\left({\sum\limits_{j=1}^{K}{{\bf{h}}_{k,p}^{H}{{\bf{w}}_{p,j}}{{\left({{\bf{h}}_{k,p}^{H}{{\bf{w}}_{p,j}}}\right)}^{H}}}+{{\bf{\Xi}}_{k,p}}}\right)^{-1}}{\bf{h}}_{k,p}^{H}{{\bf{w}}_{p,k}}.$

###### Proof:

Detailed proof of this reformulation method is given in [35]. ∎

Then, we propose the joint active and passive precoding framework to optimize
the variables ${\bm{\rho}}$, ${\bf{W}}$, and ${\bf{\Theta}}$ in (13)
iteratively. The key idea of the framework is to fix the other two variables
and optimize the remained one, e.g., fix $({\bm{\rho}},{\bf{\Theta}})$ and
optimize ${\bf{W}}$. After introducing two auxiliary variables ${\bm{\xi}}$
and $\bm{\varpi}$, the proposed joint precoding framework to maximize the WSR
is summarized in Algorithm 1, where the optimal solution of $\bf X$ is denoted
by ${\bf X}^{\rm opt}$. In this framework, the variables ${\bm{\rho}}$,
${\bm{\xi}}$, ${\bf W}$, ${\bm{\varpi}}$, and ${\bm{\Theta}}$ are updated
iteratively, until the convergence of the objective function is achieved.

The optimal solutions to these variables in each step will be introduced in
the following three subsections. Specifically, the solution to
${\bm{\rho}}^{\rm opt}$ is firstly present in Subsection III-B. Then, the
solutions to ${\bm{\xi}}^{\rm opt}$ and ${\bf W}^{\rm opt}$ for the active
precoding design are provided in Subsection III-C. After that, the solutions
to ${\bm{\varpi}}^{\rm opt}$ and ${\bm{\Theta}}^{\rm opt}$ for the passive
precoding design are finally discussed in Subsection III-D.

Algorithm 1 Proposed Joint Precoding Framework.

0: The channels ${{\bf{H}}_{b,k,p}},{\bf{G}}_{b,r,p}$ and ${\bf{F}}_{r,k,p}$
where $\forall b\in{\cal B},k\in{\cal K},p\in{\cal P}$; user weights
$\eta_{k},~{}\forall k\in{\cal K}$.

0: Optimized active precoding vector $\bm{W}$; Optimized passive precoding
matrix $\bm{\Theta}$; Weighted sum-rate $R_{{\rm sum}}$.

1: Initialize $\bm{\Theta}$ and $\bm{W}$;

2: while no convergence of $R_{{\rm sum}}$ do

3: Update ${\bm{\rho}}^{\rm opt}$ by (16);

4: Update ${\bm{\xi}}^{\rm opt}$ by (20);

5: Update ${\bf W}^{\rm opt}$ by solving (25);

6: Update ${\bm{\varpi}}^{\rm opt}$ by (31);

7: Update ${\bm{\Theta}}^{\rm opt}$ by solving (37);

8: Update $R_{{\rm sum}}$ by (9);

9: end while

10: return ${\bm{\Theta}}$, $\bf{W}$ and $R_{{\rm sum}}$.

###  III-B Fix $({\bf{\Theta}},{\bf{W}})$ and solve ${\bm{\rho}}^{\rm opt}$

Given fixed $({\bf{\Theta}^{\star}},{\bf{W}^{\star}})$, the optimal
$\bm{\rho}$ in (14) can be obtained by solving $\partial
f/\partial\rho_{k,p}=0$ for $\forall k\in\mathcal{K},\forall p\in\mathcal{P}$.
The solution can be written as

$\displaystyle\rho_{k,p}^{\rm opt}={\gamma_{k,p}},~{}~{}\forall
k\in\mathcal{K},\forall p\in\mathcal{P}.$ (16)

By substituting $\rho_{k,p}^{\rm opt}$ in (16) back in $f$ in (14), one can
notice that, only the last term in (14) is associated with the variables
${\bf{W}}$ and ${\bf{\Theta}}$. Hence, the objective function $f$ in (14) can
be further simplified and solved as shown in the following two subsections.

###  III-C Active precoding: fix $({\bf{\Theta}},{\bm{\rho}})$ and solve ${\bf
W}^{\rm opt}$

In the case of given $({\bf{\Theta}^{\star}},{\bm{\rho}^{\star}})$, the
equivalent WSR maximization problem $\bar{\cal P}$ in (13) can be reformulated
as the following subproblem ${{\cal P}_{\rm active}}$ for the active precoding
design at BSs:

$\displaystyle{{\cal P}_{\rm active}:}{}$
$\displaystyle\mathop{{\rm{max}}}\limits_{\bf{W}}~{}g_{1}({\bf{W}})=\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{\mu_{k,p}{f_{k,p}}({\bf{\Theta}^{\star}},{\bf{W}})}}$
(17)
$\displaystyle\,{\rm{s.t.}}~{}C_{1}:\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{\left\|{{{\bf{w}}_{b,p,k}}}\right\|}}\leq{P_{b,\max}},~{}\forall
b\in\mathcal{B},$

where $\mu_{k,p}={\eta_{k}}(1+{\rho_{k,p}^{\star}})$ holds. Notice that the
reformulated subproblem ${\cal P}_{\rm active}$ in (17) is still too difficult
to solve due to the complexity of matrix inversion in $f_{k,p}$ in (15).
Actually, this subproblem is a multi-variable sum-of-ratios problem, which has
been discussed in [34]. Inspired by the methods proposed in [34], here we can
utilize the multidimensional complex quadratic transform to reformulate this
subproblem, and then we obtain Proposition 2 as below.

###### Proposition 2

With multidimensional complex quadratic transform, by introducing auxiliary
variables ${\bm{\xi}}_{p,k}\in\mathbb{C}^{U}$ and
${\bm{\xi}}={[{\bm{\xi}_{1,1}},\bm{\xi}_{1,2},\cdots,\bm{\xi}_{1,K},\bm{\xi}_{2,1},\bm{\xi}_{2,2},\cdots,\bm{\xi}_{P,K}]}$,
the subproblem ${{\cal P}_{\rm active}}$ in (17) can be further reformulated
as

$\displaystyle{\bar{\cal P}_{\rm active}:}{}$
$\displaystyle\mathop{{\rm{max}}}\limits_{\bf{W},{\bm{\xi}}}~{}~{}~{}g_{2}({\bf{W},{\bm{\xi}}})$
(18)
$\displaystyle\,{\rm{s.t.}}~{}C_{1}:\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{\left\|{{{\bf{w}}_{b,p,k}}}\right\|}}\leq{P_{b,\max}},~{}\forall
b\in\mathcal{B},$

where

$\displaystyle{{g}_{2}}({\bf{W}},{\bm{\xi}})=\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{2\sqrt{{\mu}_{k,p}}\,\,{\mathfrak{R}}\left\\{{\bm{\xi}}_{k,p}^{H}{\bf{h}}_{k,p}^{H}{{\bf{w}}_{p,k}}\right\\}}}$
(19)
$\displaystyle-\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{{\bm{\xi}}_{k,p}^{H}\left(\sum\limits_{j=1}^{K}{{\bf{h}}_{k,p}^{H}{{\bf{w}}_{p,j}}{{\left({{\bf{h}}_{k,p}^{H}{{\bf{w}}_{p,j}}}\right)}^{H}}}+{{\bf{\Xi}}_{k,p}}\right){{\bm{\xi}}_{k,p}}}}.$

###### Proof:

Detailed proof of this transform method is given in [35]. ∎

Therefore, we can optimize the variables ${\bf{W}}$ and ${\bm{\xi}}$ in (18)
alternatively. The reformulated subproblem ${\bar{\cal P}_{\rm active}}$ in
(18) can be further divided into two subproblems and respectively solved as
follows.

####  III-C1 Fix $\bf W$ and solve ${\bm{\xi}}^{\rm opt}$

While fixing $\bf W$ in $\bar{\cal P}_{\rm active}$ in (18), by setting
$\partial g_{2}/\partial{\bm{\xi}}_{k,p}$ to zero, the optimal ${\bm{\xi}}$
can be obtained by

$\displaystyle{\bm{\xi}}_{k,p}^{\rm opt}=\sqrt{{\mu}_{k,p}}$
$\displaystyle{\left(\sum\limits_{j=1}^{K}{{\bf{h}}_{k,p}^{H}{{\bf{w}}_{p,j}}{{\left({{\bf{h}}_{k,p}^{H}{{\bf{w}}_{p,j}}}\right)}^{H}}}\\!\\!+{{\bf{\Xi}}_{k,p}}\right)^{-1}}\\!\\!{\bf{h}}_{k,p}^{H}{{\bf{w}}_{p,j}},$
(20) $\displaystyle\forall k\in\mathcal{K}~{},\forall p\in\mathcal{P}.$

####  III-C2 Fix ${\bm{\xi}}$ and solve ${\bf W}^{\rm opt}$

While fixing ${\bm{\xi}}$ in $\bar{\cal P}_{\rm active}$ in (18), for
simplification and clarity of (18), we can first define

$\displaystyle{{\bf{a}}_{p}}=\sum\limits_{k=1}^{K}{{\bf{h}}_{k,p}{{\bm{\xi}}_{k,p}}{\bm{\xi}}_{k,p}^{H}{{\bf{h}}_{k,p}^{H}}},$
(21a) $\displaystyle{{\bf{A}}_{p}}={\bf I}_{K}\otimes{{\bf{a}}_{p}},$ (21b)
$\displaystyle
v_{k,p}={\bm{\xi}}_{k,p}^{H}{\bf{h}}_{k,p}^{H}{{\bf{w}}_{p,k}}.$ (21c)

Then, by substituting (21) into $g_{2}$ in (19), we can rewritten ${{g}_{2}}$
as

${{g}_{3}}\left({\bf{W}}\right)=-{{\bf{W}}^{H}}{\bf{AW}}+{\mathfrak{R}}\left\\{2{{\bf{V}}^{H}}{\bf{W}}\right\\}-Y,$
(22)

where

$\displaystyle{\bf{A}}=\left[{\begin{array}[]{*{20}{c}}{{{\bf{A}}_{1}}}&{}\hfil&{}\hfil\\\
{}\hfil&\ddots&{}\hfil\\\
{}\hfil&{}\hfil&{{{\bf{A}}_{P}}}\end{array}}\right],$ (23d)
$\displaystyle{\bf{V}}={[{v_{1,1}},v_{1,2},\cdots,v_{1,K},v_{2,1},v_{2,2},\cdots,v_{P,K}]^{T}},$
(23e) $\displaystyle
Y=\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{{\bm{\xi}}_{k,p}^{H}{{\bf{\Xi}}_{k,p}}{{\bm{\xi}}_{k,p}}}}.$
(23f)

Next, For further clarity, we consider to simplify the constraint $C_{1}$ in
(18), and we can obtain the following Lemma 1 as follows.

###### Lemma 1

The constraint $C_{1}$ in (18) can be rewritten as

${\bf{W}}^{H}{{\bf D}_{b}}{\bf{W}}\leq{P_{b,\max}},~{}\forall
b\in\mathcal{B},$ (24)

where ${{\bf
D}_{b}}={{\bf{I}}_{PK}}\otimes\left\\{\left({{\bf{e}}_{b}}{\bf{e}}_{b}^{H}\right)\otimes{{\bf{I}}_{M}}\right\\}$
and ${\bf{e}}_{b}\in\mathbb{R}^{B}$.

###### Proof:

See Appendix A. ∎

Therefore, by using (22) and (24), the active precoding problem ${\bar{\cal
P}_{\rm active}}$ in (18) can be further simplified as

$\displaystyle{\hat{\cal P}_{\rm active}:}{}{}$
$\displaystyle\mathop{{\rm{max}}}\limits_{\bf{W}}~{}g_{3}({\bf{W}})=-{{\bf{W}}^{H}}{\bf{AW}}+{\mathfrak{R}}\left\\{2{{\bf{V}}^{H}}{\bf{W}}\right\\}-Y$
(25) $\displaystyle\,{\rm{s.t.}}~{}~{}~{}~{}C_{1}:{\bf{W}}^{H}{{\bf
D}_{b}}{\bf{W}}\leq{P_{b,\max}},~{}\forall b\in\mathcal{B}.$

Finally, notice that since the matrices $\bf A$ and ${\bf D}_{b},\forall
b\in\mathcal{B}$ are all positive semidefinite, the simplified subproblem
${\hat{\cal P}_{\rm active}}$ in (25) is a standard QCQP problem, which can be
directly solved by the standard convex tools such as CVX [42]. We also give a
feasible solution to obtain ${\bf W}^{\rm opt}$ in Appendix B.

###  III-D Passive precoding: fix $({\bm{\rho}},{\bf{W}})$ and solve
${\bf\Theta}^{\rm opt}$

Based on the given $({\bm{\rho}}^{\star},{\bf{W}}^{\star})$, for the
equivalent WSR maximization problem ${\bar{\cal P}}$ in (13), the subproblem
of the passive precoding design at RISs can be equivalently written as

$\displaystyle{{\cal P}_{\rm passive}:}{}{}{}$
$\displaystyle\mathop{{\rm{max}}}\limits_{\bf{\Theta}}~{}~{}g_{4}({\bf{\Theta}})=\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{\mu_{k,p}{f_{k,p}}({\bf{\Theta}},{\bf{W}^{\star}})}}$
(26) $\displaystyle~{}{\rm{s.t.}}~{}~{}~{}C_{2}:{{\theta_{r,n}}}\in{\cal
F},~{}\forall r\in{{\cal R}},\forall n\in{{\cal N}},$

where $\mu_{k,p}={\eta_{k}}(1+{\rho_{k,p}^{\star}})$. Similiarly, to reduce
the complexity, we wish to simplify the expression of $g_{4}$ in (26).
Firstly, by defining a new auxiliary function with respect to ${\bf{\Theta}}$
as

$\displaystyle{g_{4}}({\bf{\Theta}})=\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{\sqrt{{\mu_{k,p}}}{\bf{Q}}_{k,p,k}^{H}\left({\bf{\Theta}}\right){{\left({\sum\limits_{j=1}^{K}{{{\bf{Q}}_{k,p,j}}\left({\bf{\Theta}}\right){\bf{Q}}_{k,p,k}^{H}\left({\bf{\Theta}}\right)}+{{\bf{\Xi}}_{k,p}}}\right)}^{-1}}{{\bf{Q}}_{k,p,k}}\left({\bf{\Theta}}\right)}}.$
(27)

${{\bf{Q}}_{k,p,j}}\left({\bf{\Theta}}\right)=\sum\limits_{b=1}^{B}{\left({{\bf{H}}_{b,k,p}^{H}+{\bf{F}}_{k,p}^{H}{{\bf{\Theta}}^{H}}{{\bf{G}}_{b,p}}}\right){{\bf{w}}_{b,p,j}}},$
(28)

we can rewrite $g_{4}$ in (26) as (27) at the bottom of this page. However,
this subproblem is still hard to solve due to matrix inversion in $f_{k,p}$ in
(15). Again, we consider to exploit the multidimensional complex quadratic
transform [34] to address this issue by using the following Proposition 3.

###### Proposition 3

With multidimensional complex quadratic transform, by introducing an auxiliary
variable ${\bm{\varpi}}_{p,k}\in\mathbb{C}^{U}$ and
${\bm{\varpi}}={[{\bm{\varpi}_{1,1}},\bm{\varpi}_{1,2},\cdots,\bm{\varpi}_{1,K},\bm{\varpi}_{2,1},\bm{\varpi}_{2,2},\cdots,\bm{\varpi}_{P,K}]}$,
the passive precoding subproblem ${{\cal P}_{\rm passive}}$ in (26) can be
reformulated as

$\displaystyle{\bar{\cal P}_{\rm passive}:}~{}~{}$
$\displaystyle\mathop{{\rm{max}}}\limits_{\bf{\Theta}}~{}~{}{{g}_{5}}\left({\bf{\Theta}},{\bm{\varpi}}\right)=\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{{g_{k,p}}\left({\bf{\Theta}},{\bm{\varpi}}\right)}}$
(29a) $\displaystyle~{}{\rm{s.t.}}~{}~{}C_{2}:{{\theta_{r,n}}}\in{\cal
F},~{}\forall r\in{{\cal R}},\forall n\in{{\cal N}},$ (29b)

where

$\displaystyle{{g}_{k,p}}({\bf{\Theta}},{\bm{\varpi}})=2\sqrt{{\mu_{k,p}}}\,{\mathfrak{R}}\left\\{{{\bm{\varpi}}_{k,p}^{H}{{\bf{Q}}_{k,p,k}}\left({\bf{\Theta}}\right)}\right\\}$
(30)
$\displaystyle-{\bm{\varpi}}_{k,p}^{H}\left(\sum\limits_{j=1}^{K}{{{\bf{Q}}_{k,p,j}}\left({\bf{\Theta}}\right){\bf{Q}}_{k,p,j}^{H}\left({\bf{\Theta}}\right)}+{{\bf{\Xi}}_{k,p}}\right){{\bm{\varpi}}_{k,p}}.$

###### Proof:

Detailed constructive proof of this reformulation method is given in [35]. ∎

Next, similar to the previous processing of the subproblem ${\bar{\cal P}_{\rm
active}}$ in (18), we consider to optimize two variables ${\bf{\Theta}}$ and
${\bm{\varpi}}$ in (29) alternatively. The reformulated subproblem ${\bar{\cal
P}_{\rm passive}}$ in (29) can be further divided into two subproblems and
respectively solved as follows.

####  III-D1 Fix $\bm{\Theta}$ and solve ${\bm{\varpi}}^{\rm opt}$

For given fixed $\bm{\Theta}^{\star}$ in $\bar{\cal P}_{\rm passive}$ in (29),
by solving $\partial g_{5}/\partial{\bm{\varpi}}_{k,p}=0$ for $\forall
k\in\mathcal{K}$ and $\forall p\in\mathcal{P}$, we can obtain the optimal
${\bm{\varpi}}$ by

$\displaystyle{\bm{\varpi}}_{k,p}^{\rm opt}=$ (31)
$\displaystyle\sqrt{{\mu_{k,p}}}{\left(\sum\limits_{j=1}^{K}{{{\bf{Q}}_{k,p,j}}\left({{{\bf{\Theta}}^{\rm{\star}}}}\right){\bf{Q}}_{k,p,j}^{H}\left({{{\bf{\Theta}}^{\rm{\star}}}}\right)}+{{\bf{\Xi}}_{k,p}}\right)^{-1}}\\!\\!{{\bf{Q}}_{k,p,j}}\left({\bf{\Theta}^{\star}}\right),$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\quad\quad\quad~{}~{}\forall
k\in\mathcal{K}~{},\forall p\in\mathcal{P}.$

####  III-D2 Fix $\bm{\varpi}$ and solve ${\bf\Theta}^{\rm opt}$

While fixing $\bm{\varpi}$ in $g_{5}$ in (29), due to the complexity of
$\bar{\cal P}_{\rm passive}$ in (29), we first consider to simplify the
expression of $g_{5}$ by using the new auxiliary function
${{\bf{Q}}_{k,p,j}}\left({\bf{\Theta}}\right)$ with respect to ${\bf{\Theta}}$
in (28) as follows:

$\displaystyle{\bm{\varpi}}_{k,p}^{H}{{\bf{Q}}_{k,p,j}}\left({\bf{\Theta}}\right)$
(32)
$\displaystyle\mathop{=}\limits^{(a)}\sum\limits_{b=1}^{B}{\left({{\bm{\varpi}}_{k,p}^{H}{\bf{H}}_{b,k,p}^{H}{{\bf{w}}_{b,p,j}}+{\bm{\varpi}}_{k,p}^{H}{\bf{F}}_{k,p}^{H}{{\bf{\Theta}}^{H}}{{\bf{G}}_{b,p}}{{\bf{w}}_{b,p,j}}}\right)}$
$\displaystyle\mathop{=}\limits^{(b)}\sum\limits_{b=1}^{B}{{\bm{\varpi}}_{k,p}^{H}{\bf{H}}_{b,k,p}^{H}{{\bf{w}}_{b,p,j}}}\\!+\\!{{\bm{\theta}}^{H}}\\!\sum\limits_{b=1}^{B}{{\rm
diag}\left({{\bm{\varpi}}_{k,p}^{H}{\bf{F}}_{k,p}^{H}}\right){{\bf{G}}_{b,p}}{{\bf{w}}_{b,p,j}}}$
$\displaystyle\mathop{=}\limits^{(c)}{c_{k,p,j}}+{{\bm{\theta}}^{H}}{{\bf{g}}_{k,p,j}},$

where $(a)$ holds according to (27), $(b)$ is obtained by defining
${\bm{\theta}}={\bm{\Theta}}{\bf 1}_{RN}$, and $(c)$ is achieved by defining

$\displaystyle{c_{k,p,j}}=\sum\limits_{b=1}^{B}{{\bm{\varpi}}_{k,p}^{H}{\bf{H}}_{b,k,p}^{H}{{\bf{w}}_{b,p,j}}},$
(33a) $\displaystyle{{\bf{g}}_{k,p,j}}=\sum\limits_{b=1}^{B}{{\rm
diag}\left({{\bm{\varpi}}_{k,p}^{H}{\bf{F}}_{k,p}^{H}}\right){{\bf{G}}_{b,p}}{{\bf{w}}_{b,p,j}}}.$
(33b)

By substituting (32) into (30), we can obtain:

$\displaystyle{{{g}}_{k,p}}({\bf{\Theta}})=2\sqrt{{{\mu}_{k,p}}}\,{\mathfrak{R}}\left\\{{{c_{k,p,k}}+{{\bm{\theta}}^{H}}{{\bf{g}}_{k,p,k}}}\right\\}$
(34)
$\displaystyle-\\!\sum\limits_{j=1}^{K}{\left({c_{k,p,j}}+{{\bm{\theta}}^{H}}{{\bf{g}}_{k,p,j}}\right)\left(c_{k,p,j}^{*}+{{\bf{g}}^{H}_{k,p,j}}{\bm{\theta}}\right)}\\!-\\!{\bm{\varpi}}_{k,p}^{H}{{\bf{\Xi}}_{k,p}}{{\bm{\varpi}}_{k,p}}.$

Then, we can further substitute (34) into (29a), so that $g_{5}$ in (29a) can
be rewritten as

$\displaystyle{{g}_{6}}({\bm{\Theta}})=-{{\bm{\theta}}^{H}}{\bf{\Lambda}}{\bm{\theta}}+{\mathfrak{R}}\left\\{2{{\bm{\theta}}^{H}}{\bm{\nu}}\right\\}-\zeta,$
(35)

where

$\displaystyle{\bm{\Lambda}}=$
$\displaystyle\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{\sum\limits_{j=1}^{K}{{{\bf{g}}_{k,p,j}}{\bf{g}}_{k,p,j}^{H}}}},$
(36a) $\displaystyle{\bm{\nu}}=$
$\displaystyle\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{\sqrt{{\mu_{k,p}}}{{\bf{g}}_{k,p,k}}}}\\!-\\!\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{\sum\limits_{j=1}^{K}{c_{k,p,j}^{*}{{\bf{g}}_{k,p,j}}}}},$
(36b) $\displaystyle{\zeta}=$
$\displaystyle\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{\sum\limits_{j=1}^{K}{{{\left|{{c_{k,p,j}}}\right|}^{2}}}}}+\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{{\bm{\varpi}}_{k,p}^{H}{{\bf{\Xi}}_{k,p}}{{\bm{\varpi}}_{k,p}}}}$
(36c)
$\displaystyle-2\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{\sqrt{{\mu_{k,p}}}}~{}{\mathfrak{R}}\left\\{{c_{k,p,k}}\right\\}}.$
(36d)

Therefore, the reformulated passive precoding subproblem ${\bar{\cal P}_{\rm
passive}}$ in (29) can be further simplified as

$\displaystyle{\hat{\cal P}_{\rm passive}:}{}{}$
$\displaystyle\mathop{{\rm{max}}}\limits_{\bf{\Theta}}~{}~{}{{g}_{6}}({\bm{\Theta}})=-{{\bm{\theta}}^{H}}{\bf{\Lambda}}{\bm{\theta}}+{\mathfrak{R}}\left\\{2{{\bm{\theta}}^{H}}{\bm{\nu}}\right\\}-\zeta$
(37) $\displaystyle\,{\rm{s.t.}}~{}~{}~{}C_{2}:{{\theta_{r,j}}}\in{\cal
F},~{}\forall r\in{{\cal R}},\forall j\in{{\cal N}}.$

This simplified subproblem ${\hat{\cal P}_{\rm passive}}$ is similar to those
in [43, 41, 44]. Obviously, since the matrix ${\bm{\Lambda}}$ is positive
semidefinite, the objective function is convex. Besides, since we have ${{\cal
F}}\triangleq\\{{\theta_{r,j}}{\Big{|}}\left|{{\theta_{r,j}}}\right|\leq 1\\}$
according to (4), the constraint $C_{2}$ is also convex. Therefore,
${\bf\Theta}^{\rm opt}$ can be directly obtained by the standard convex tools
such as CVX [42]. Again, we also give a feasible solution to ${\hat{\cal
P}_{\rm passive}}$ in Appendix C.

###  III-E Extension to non-ideal RIS cases

Up to now, we have provided a complete joint precoding framework for the
proposed RIS-aided cell-free network, where ideal RIS case as mentioned in
Subsection II-C is considered. In this subsection, we will extend this
framework to the more practical non-ideal RIS cases.

####  III-E1 Non-ideal RIS cases

According to (4) in Subsection II-C, ${\cal F}$ is defined as the ideal RIS
case, where both the amplitude and phase of ${\theta_{r,j}}$ associated with
the RIS element can be controlled independently and continuously. However,
limited by the hardware implementation of metamaterials, the RISs in practice
are usually non-ideal. For consistent discussion, here we redefine the ideal
RIS case ${\cal F}$ as ${\cal F}_{1}$. We also define ${\cal F}_{2}$ as the
general case where only the phase of ${\theta_{r,j}}$ can be controlled
continuously, and ${\cal F}_{3}$ as the practical case where the low-
resolution phase of ${\theta_{r,n}}$ is discrete [23], i.e.,

$\displaystyle{{\cal
F}_{1}}\triangleq\left\\{{\theta_{r,n}}{\Big{|}}\left|{{\theta_{r,n}}}\right|\leq
1\right\\},$ (38a) $\displaystyle{{\cal
F}_{2}}\triangleq\\{{\theta_{r,n}}{\Big{|}}\left|{{\theta_{r,n}}}\right|=1\\},$
(38b) $\displaystyle{{\cal
F}_{3}}\triangleq\left\\{{\theta_{r,n}}{\Big{|}}{\theta_{r,n}}\in\left\\{1,{e^{j\frac{{2\pi}}{L}}},\cdots,{e^{j\frac{{2\pi(L-1)}}{L}}}\right\\}\right\\},$
(38c)

where $L$ indicates that ${{\cal F}_{3}}$ contains $L$ discrete phase shifts.

####  III-E2 Approximation in non-ideal RIS cases

According to the simplified subproblem $\hat{\cal P}_{\rm passive}$ in (37),
for the passive precoding design at the RISs, when the phase shift constraint
${\cal F}$ is ${\cal F}_{1}$ as discussed above, the subproblem ${\hat{\cal
P}_{\rm passive}}$ can be directly solved due to the convexity of ${\cal
F}_{1}$.

However, when the constraint ${\cal F}$ in (37) is ${\cal F}_{2}$ or ${\cal
F}_{3}$, the simplified subproblem ${\hat{\cal P}_{\rm passive}}$ in (37)
becomes non-convex. The common solution to address such problem is to relax
the constraint of the phase shift matrix in ${\cal F}_{2}$ or ${\cal F}_{3}$
[43, 41]. Hence, by introducing an auxiliary variable $\bf{\tilde{\Theta}}$,
the simplified subproblem ${\hat{\cal P}_{\rm passive}}$ in (37) can be
relaxed as

$\displaystyle{\hat{\cal P}_{\text{ passive-relaxed}}:}{}$
$\displaystyle\mathop{{\rm{max}}}\limits_{\bf{\tilde{\Theta}}}~{}{{g}_{7}}({\bm{\tilde{\Theta}}})=-{{\bm{\tilde{\theta}}}^{H}}{\bf{\Lambda}}{\bm{\tilde{\theta}}}+{\mathfrak{R}}\left\\{2{{\bm{\tilde{\theta}}}^{H}}{\bm{\nu}}\right\\}-\zeta$
(39)
$\displaystyle~{}{\rm{s.t.}}~{}~{}~{}C_{2}:{{\tilde{\theta}_{r,j}}}\in{\cal
F}_{1},\forall r\in{{\cal R}},\forall j\in{{\cal N}}.$

This relaxed subproblem ${\hat{\cal P}_{\text{ passive-relaxed}}}$ is convex,
which is now similar to the subproblem $\hat{\cal P}_{\rm passive}$ in (37).
Thus, ${\bm{\tilde{\Theta}}}^{\rm opt}$ can be directly solved by the standard
convex tools such as CVX [42] as well. Then, based on the proximity principle,
we can map the solved ${\bm{\tilde{\Theta}}}^{\rm opt}$ to the elements in
${\cal F}_{2}$ or ${\cal F}_{3}$ by a quantization operation, which is common
in existing works [43, 22, 26], and can be written as

$\displaystyle\\!\\!\left\\{\begin{array}[]{l}\left|{\theta_{r,j}^{\rm
sub}}\right|=1,\\\ \angle\theta_{r,j}^{\rm
sub}{\rm{=}}\mathop{{\rm{argmin}}}\limits_{\phi\in{\cal F}_{2}~{}{\rm
or}~{}{\cal F}_{3}}\left|{\angle\tilde{\theta}_{r,j}^{\rm
opt}-\angle\phi}\right|,\end{array}\right.~{}~{}\forall r\in{{\cal R}},\forall
j\in{{\cal N}},$ (40)

where $\theta_{r,j}^{\rm sub}$ denotes the approximated sub-optimal solution
to $\theta_{r,j}$. Thus, we can finally obtain ${\bm{\Theta}}^{\rm sub}$ by
using the approximation (40) above.

With the approximation in (40), the proposed joint precoding framework can be
also used in non-ideal RIS cases ${\cal F}_{2}$ and ${\cal F}_{3}$ as well. It
should be pointed out that, since the loss caused by the approximation might
be large and hard to estimate, some other discrete optimization methods such
as adaptive cross-entropy (ACE) algorithm [45] might be used to alleviate the
approximation loss in the non-ideal RIS cases, which is left for future work.

###  III-F Convergence and complexity

####  III-F1 Convergence

In the ideal case when ${\cal F}={\cal F}_{1}$, the proposed joint precoding
framework has strict convergence, since each step of the iteration, i.e.,
(16), (20), (25), (31) and (37), can be easily proved to be monotonous.
However, In the non-ideal cases when ${\cal F}={\cal F}_{2}$ and ${\cal
F}={\cal F}_{3}$, the convergence of the proposed framework can not be proved
strictly, since the update of ${\bm{\Theta}}$ has no guarantee of monotony in
some cases due to the approximation operation in (40). Fortunately, the rest
of iterative steps are all monotonous, so the loss caused by the approximation
operation has little adverse effect on the global convergence, which will be
verified by simulation results in the next Section IV.

####  III-F2 Complexity

The overall complexity of the proposed joint precoding framework is mainly
introduced by the update of the variables ${\bm{\rho}}$, ${\bm{\xi}}$, ${\bf
W}$, ${\bm{\varpi}}$, and ${\bm{\Theta}}$, as shown in (16), (20), (25), (31)
and (37), respectively. Notice that the optimal solutions to ${\bm{\rho}}$,
${\bm{\xi}}$, and ${\bm{\varpi}}$ are all in closed-form, which has lower
complexity than those to ${\bf W}$ and ${\bm{\Theta}}$, since the optimal
${\bf W}$ and ${\bm{\Theta}}$ have to be obtained iteratively. Specifically,
the computational complexities of ${\bm{\rho}}$, ${\bm{\xi}}$, and
${\bm{\varpi}}$ are mainly caused by the matrix inversion in (15), (20), and
(31), respectively. The dimensions of these three matrices are all $U$, i.e.,
the number of user antennas, so the complexity is ${\cal O}(U^{3})$. The
computational complexities of ${\bf W}$ and ${\bm{\Theta}}$ mainly comes from
the subproblems in (25) and (37), and they are strictly related to the
dimensions of the variables to be optimized. The dimension of ${\bf W}$ is
$BMPK$, while that of ${\bm{\Theta}}$ is $RN$. Thanks to the sparsity of the
matrices ${\bf A}$ and ${\bm{\Lambda}}$, and since the subproblems $\hat{\cal
P}_{\rm active}$ in (25) and $\hat{\cal P}_{\rm passive}$ in (37) are standard
QCQP problems, the complexities of solving them are acceptable.

###### Remark 1

Different from the prior works, in which only some of the system parameters
such as the number of RISs are multiple, our proposed RIS-aided cell-free
network is the first one to simultaneously consider multiple antennas,
multiple BSs, multiple users, multiple RISs and multiple subcarriers. It
indicates that most of the prior discussed scenarios [41, 15, 43, 18, 19, 26],
are our special cases. Moreover, by simply setting $M$, $U$, $N$, $B$, $R$,
$K$, $P$ to the required values in these special cases, our proposed joint
precoding framework can serve as a general solution to maximize the WSR in
these special cases. For instance, by setting $B=1$, $P=1$, and $R=1$, the
proposed framework can be used to solve the WSR maximization problem in a
single-carrier scenario where one BS assisted by one RIS is deployed to serve
multiple users as discussed in [43].

##  IV Simulation Results

In this section, we provided extensive simulation results under different
conditions to validate the performance of the proposed concept of RIS-aided
cell-free network and the corresponding joint precoding framework.

###  IV-A Simulation setup

For the simulation of the proposed RIS-aided cell-free network, the number of
antennas at each BS is set as $M=8$ and that at each user is set as $U=2$. The
number of RIS elements is set as $N=32$. The number of subcarriers is set as
$P=6$, and the noise power is set as ${\sigma^{2}}=-120$ dBm.

For the channel model, firstly we consider a large-scale fading model
according to [46], where the BS-RIS-user link suffers the “double-fading
effect” discussed in [47]. Therefore, let $d_{\rm Bu}$, $d_{\rm BR}$, $d_{\rm
Ru}$ denote the distance between BS and user, BS and RIS, RIS and user,
respectively, the distance-dependent path loss of the BS-user link can be
written as

$\displaystyle{f_{\rm d}}({d_{\rm Bu}})={C_{\rm d}}{G_{\rm B}}{G_{\rm
u}}{d_{\rm Bu}}^{-{\kappa_{\rm Bu}}},$ (41)

and that of the BS-RIS-user link can be written as

$\displaystyle{f_{\rm r}}({d_{\rm BR}},{d_{\rm Ru}})=C_{\rm r}{G_{\rm
B}}{G_{\rm u}}d_{\rm BR}^{-{\kappa_{\rm BR}}}d_{\rm Ru}^{-{\kappa_{\rm Ru}}},$
(42)

where $C_{\rm d}$ and $C_{\rm r}$ are the channel fading variables related to
the wavelength, channel status, and angle of departure (AOD), and $C_{\rm r}$
is also related to the angle of arrival (AOA), the material and size of RIS
element [46]; $G_{\rm B}$ and $G_{\rm u}$ denote the antenna gain of the BSs
and users, respectively, and here we assume ${C_{\rm d}}{G_{\rm B}}{G_{\rm
u}}=-30$ dB and $C_{\rm r}{G_{\rm B}}{G_{\rm u}}=-40$ dB [43]; $\kappa$
denotes the path loss exponent, and the path loss exponents of the BS-RIS
link, RIS-user link, and BS-user link are set as ${\kappa_{\rm BR}}=2$,
${\kappa_{\rm Ru}}=2$, and ${\kappa_{\rm Bu}}=3$ [46], respectively.

Figure 3: The simulated scenario where two BSs assisted by two RISs serve four
users.

Then, to account for the small-scale fading, we further consider a Rician
fading channel model, thus the BS-user channel ${\bf H}$ is obtained by

$\displaystyle{\bf
H}=\sqrt{\frac{\beta_{\mathrm{Bu}}}{1+\beta_{\mathrm{Bu}}}}{\bf
H}^{\mathrm{LoS}}+\sqrt{\frac{1}{1+\beta_{\mathrm{Bu}}}}{\bf
H}^{\mathrm{NLoS}},$ (43)

where $\beta_{\mathrm{Bu}}$ denotes the Rician factor, and ${\bf
H}^{\mathrm{LoS}}$ and ${\bf H}^{\mathrm{NLoS}}$ denote the LoS and Rayleigh
fading components, respectively. Note that ${\bf H}$ is equivalent to a LoS
channel when $\beta_{\mathrm{Bu}}\rightarrow\infty$, and a Rayleigh fading
channel when $\beta_{\mathrm{Bu}}=0$. Then, ${\bf H}$ is multiplied by the
square root of the distance-dependent path loss ${f_{\rm d}}({d_{\rm Bu}})$ in
(41). Similarly, the BS-RIS and RIS-user channels can also be generated by the
above procedure, and let $\beta_{\mathrm{BR}}$ and $\beta_{\mathrm{Ru}}$
denote the Rician factors of them, respectively. Here we assume
$\beta_{\mathrm{BR}}\rightarrow\infty$, $\beta_{\mathrm{Bu}}=0$ and,
$\beta_{\mathrm{Ru}}=0$ [41].

At last, for the joint precoding framework, we set the weights of users as
$\eta_{k}=1$, and we assume that the channel state information (CSI) is
perfectly known222In this paper, the CSI is assumed to be perfectly known,
while the challenging problem of channel estimation [21] is left for future
work.. As an alternating algorithm, $\bm{\Theta}$ is initialized by random
values in ${\cal F}$, and $\bf W$ is initialized by identical power and random
phases.

###  IV-B Weighted sum-rate performance of the RIS-aided cell-free network

The WSR of the proposed RIS-aided cell-free network is evaluated in this
subsection. For simplicity but without loss of generality, we consider a RIS-
aided cell-free system with the topology as shown in Fig. 3, where two BSs and
two RISs are located at $(0,-20\,{\rm m})$, $(80\,{\rm m},-20\,{\rm m})$,
$(30\,{\rm m},3\,{\rm m})$ and $(50\,{\rm m},3\,{\rm m})$, respectively, and
four users are randomly distributed in a circle centered at $(L,0)$ with
radius $\rm 1\,m$, where $L$ denotes the horizontal distance between the BS 1
and the circle centre. The maximum transmit power at the BS is set as
$P_{b,\max}=0$ dB.

Figure 4: Weighted sum-rate against the distance $L$.

Fig. 4 shows the WSR against the distance $L$, where the conventional cell-
free network without RIS (which is denote by “No RIS”) is considered as
benchmark to be compared with the proposed RIS-aided cell-free network.
Specifically, the “Ideal RIS case” means that both the amplitude and phase of
RIS elements can be controlled continuously and independently, i.e., ${\cal
F}={\cal F}_{1}$, while “Continuous phase shift” denotes the non-ideal case
${\cal F}={\cal F}_{2}$. More discussions about the non-ideal RIS cases will
be provided in Fig. 5 and Fig. 10 later. For the conventional cell-free
network without RIS, the WSR maximization problem can be rewritten as

$\displaystyle{\cal P}_{\text{no-RIS}}^{\rm o}:{}{}$
$\displaystyle\mathop{{\rm{max}}}\limits_{{\bf{W}}}~{}~{}f_{0}({\bf{W}}){\rm{=}}\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{{\eta_{k}}{{\log}_{2}}\left({1+{\gamma_{k,p}^{\rm
o}}}\right)}}$ (44)
$\displaystyle~{}{\rm{s.t.}}~{}C_{1}:\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{\left\|{{{\bf{w}}_{b,p,k}}}\right\|}}\leq{P_{b,\max}},~{}\forall
b\in\mathcal{B},$

where

$\displaystyle{\gamma}_{k,p}^{\rm o}=$ (45)
$\displaystyle{\bf{w}}_{p,k}^{H}{{\bf{H}}_{k,p}}\\!{\left(\sum\limits_{j=1,j\neq
k}^{K}\\!\\!\\!{{\bf{H}}_{k,p}^{H}{{\bf{w}}_{p,j}}{{\left({{\bf{H}}_{k,p}^{H}{{\bf{w}}_{p,j}}}\right)}^{H}}}\\!\\!\\!\\!+\\!{{\bf{\Xi}}_{k,p}}\right)^{-1}}\\!\\!\\!{\bf{H}}_{k,p}^{H}{{\bf{w}}_{p,k}}.$

Obviously, the problem ${\cal P}_{\text{no-RIS}}^{\rm o}$ in (44) is a special
case of our problem ${\cal P}^{\rm o}$ in (11) for the RIS-aided cell-free
network. However, to the best of our knowledge, the WSR maximization problem
${\cal P}_{\text{no-RIS}}^{\rm o}$, which is subject to the power constraints
in the wideband cell-free system, has not been investigated in the literature.
Thus, the performance curve named “No RIS” is achieved by the proposed joing
precoding framework by simply setting the number of RISs as $R=0$.

We have two observations from Fig. 4. First, as the users move away from the
BSs, the WSR of the users decreases rapidly due to the signal attenuation in
the conventional cell-free network. However, for the proposed RIS-aided cell-
free network, we can see two obvious peaks at $L=30\,{\rm m}$ and $L=50\,{\rm
m}$, which indicates that the WSR rises when the users approach one of the two
RISs, since the users can receive strong signals reflected from the RISs. For
the conventional scheme without RIS, these two peaks will not appear. Thus, we
can conclude that the network capacity can be substantially increased by
deploying RISs in the network, and the signal coverage can be accordingly
extended. Second, we can find that the curves “Ideal RIS case” and “Continuous
phase shift” are very close to each other. It indicates that, for practical
hardware implementation, there is no need to design amplitude-controllable
RISs, since the continuous phase shift can approach the best performance in
the ideal case.

###  IV-C Convergence of the joint precoding framework

Figure 5: Weighted sum-rate against the number of iterations.

We present the convergence of the proposed algorithm by plotting the WSR
against the number of iterations in Fig. 5. The simulation parameters and the
network topology are same as those used in IV-B, and we fix the distance $L$
as $L=25\,{\rm m}$. To show the convergence in non-ideal low-resolution phase
shift cases, we add the curves “1-bit phase shift” and “2-bit phase shift” to
denote the case ${\cal F}={\cal F}_{3}$, where the discrete available phase
shift sets are $\\{0,\pi\\}$ and $\\{0,\pi/2,\pi,3\pi/2\\}$, respectively. The
results in Fig. 5 clearly illustrate that, when the convergence error is no
more than $2\%$, the proposed joint precoding framework can converge within 15
iterations. To be more specific, the ideal RIS case and continuous phase shift
case can converge within 15 iterations. The discrete phase shift case can
converge within 10 iterations. Since the conventional cell-free network
without RIS does not need to address the passive precoding design at RISs, the
WSR in the case “No RIS” converges within 8 iterations. The results indicate
that, although the approximation operation in (40) causes the uncertainty of
convergence, the proposed framework still enjoys the global convergence and a
fast convergence speed. Besides, we can also notice that, the low-resolution
phase shifts only suffer from an acceptable performance loss. For example,
after 15 iterations, the WSR in “Ideal RIS case” is about $82\,{\rm
bit/s/Hz}$, while that of the cases “2-bit phase shift” is about $75\,{\rm
bit/s/Hz}$, which suffers a loss about $8\%$ of the ideal case. Considering
the hardware implementation of RIS [23], since the low-resolution phase shift
is much easier for implementation than the continuous one, this result is very
encouraging for the massive deployment of RIS in the RIS-aided cell-free
network.

###  IV-D The impact of key system parameters

To reveal more design insights of the proposed RIS-aided cell-free network
with different system parameters, we consider a new simulation setup as shown
in Fig. 6 in this subsection, where eight RISs and two users are uniformly
distributed along a circle of radius $15\,{\rm m}$. Two BSs are located at
$(-10\,{\rm m},0)$ and $(10\,{\rm m},0)$, respectively. The BS maximum
transmit power is set as $P_{b,\max}=-10$ dB.

Figure 6: The simulation scenario where two BSs and eight RISs serve four
users.

####  IV-D1 Weighted sum-rate against BS transmit power

To analyze the effect of BS transmit power on the proposed RIS-aided cell-free
network, we present the average weighted sum-rate against the BS transmit
power in Fig. 7. It can be seen that, with the increase of the BS transmit
power, the weighted sum-rate rises rapidly in all cases, which indicates the
intuitive conclusion that increasing the transmit power can be a
straightforward way to improve the WSR in the proposed network. However, we
have a more important finding that, the performance gain brought by RISs is
significant only when the BS transmit power is moderate (e.g., form -20 dB to
0 dB), while the performance gain becomes negligible when the BS transmit
power is too low (e.g. -40 dB) or too high (e.g. 10 dB). The reason is that,
the reflected signal by RIS is too weak when the BS transmit power is too low,
while the received signal at the user is dominated by the direct BS-user link
instead of the indirect BS-RIS-user link when the BS transmit power is too
high due to the “double-fading effect” of the RIS. Particularly, with the
increase of the BS transmit power, the link gain difference between the direct
and indirect links becomes larger, thus the BSs tend to directly orient the
beam towards the users, which makes the role of RISs less obvious. That is to
say, when the BS transmit power is high enough, simply increasing it will
weaken the benefits brought by RISs. Therefore, to improve the performance of
the proposed RIS-aided cell-free network more cost and power effective, we
should choose the appropriate BS transmit power333In this paper, we only
consider the transmit power, while the hardware power consumption and the
corresponding energy efficiency optimization for the proposed RIS-aided cell-
free network are left for future work..

Figure 7: Weighted sum-rate against the BS transmit power.

####  IV-D2 Weighted sum-rate against number of BS/user antennas

Figure 8: Weighted sum-rate against the number of BS antennas. Figure 9:
Weighted sum-rate against the number of user antennas.

Then, we investigate the effect of the number of BS and user antennas. We plot
the average weighted sum-rate against the number of BS antennas in Fig. 8, and
that against the number of user antennas in Fig. 9, respectively. The general
trend of the curves is consistent with the expectation, since we can see that
the performance will be increased with the number of BS/user antennas.
However, since the dimension of $\bf W$ is $BMPK$, the complexity of solving
(25) grows rapidly with the number of BS antennas number $M$, and the increase
of user antennas also improve the complexity of matrix inversion in (15).
Besides, too many antennas also makes the channel estimation more challenging.
Therefore, in practical systems, we should choose the reasonable number of
antennas to enhance the network capacity.

####  IV-D3 Weighted sum-rate against number of RIS elements

Figure 10: Weighted sum-rate against the number of RIS elements.

At last, the average weighted sum-rate against the number of RIS elements is
shown in Fig. 10. We can see that the WSR performance gain of the RIS-aided
cell-free network increases as the number of RIS elements rises, because more
RIS elements provide higher beamforming gain. As the RIS is low-cost and
energy-efficient, we can come to the conclusion that increasing the number of
RIS elements can be a cost-effective way to improve the performance. More
importantly, we can find an interesting observation that, with the increase of
RIS elements, the approximation loss of the low-resolution phase shift becomes
larger. For instance, when $N=32$, the approximation loss in the case “2-bit
phase shift” is about $5\%$ of the ideal case, and this loss grows to $9\%$
when $N=64$. The observation implies that, when the number of RIS is large, is
is necessary to use more accurate phase shifts for passive precoding, so that
the signals reflected from RISs can reach the users more precisely. In
addition, since the dimension of $\bf\Theta$ is $RN$, the complexity of
solving (37) grows with the number of RIS elements $N$, and too more RIS
elements will also make the channel estimation and hardware implementation
more challenging. Thus, it is essential to choose the number of RIS elements
reasonably.

##  V Conclusions and Future Works

In this paper, we first propose the concept of RIS-aided cell-free network,
which aims to improve the network capacity with low cost and power
consumption. Then, for the proposed RIS-aided cell-free network, in the
typical wideband scenario, we formulate the joint precoding design problem to
maximize the weighted sum-rate of all users to optimize the network capacity,
subject to the transmit power constraint of BS and the phase shift constraint
of RIS. Finally, we propose a joint precoding framework to solve this problem.
Since most of the considered scenarios in existing works are special cases of
the general scenario in this paper, the proposed joint precoding framework can
also serve as a general solution to maximize the capacity in most of existing
RIS-aided scenarios. The simulation results demonstrate that, with the assist
of low-cost and energy-efficient RISs, the proposed RIS-aided cell-free
network can realize higher capacity than the conventional cell-free network.

For the proposed concept of RIS-aided cell-free network, only the weighted
sum-rate maximization is considered in this paper, while some other important
performance metrics such as the energy efficiency [15], BS transmit power
[17], user SINR [22, 27], and secrecy rate [32] are left for future works.
Moreover, since the perfectly known CSI is assumed in this paper, the
challenging problem of channel estimation [21] should be studied. In addition,
the fully digital precoding at the BSs is considered in this paper, while the
more energy-efficient hybrid precoding [40] can be investigated in future
works.

##  Appendix A Proof of Lemma 1

From ${\bf{w}}_{p,k}=[{\bf{w}}_{1,k,p}^{T},\cdots,{\bf{w}}_{B,k,p}^{T}]^{T}$,
firstly we can set $b$ to 1 as an instance, and then obtain:

$\begin{aligned}
{{\bf{w}}_{1,p,k}}={\left[{\begin{array}[]{*{20}{c}}{{{\bf{I}}_{M}}}&\cdots&{{{\bf{0}}_{M}}}\\\
\vdots&\ddots&\vdots\\\
{{{\bf{0}}_{M}}}&\cdots&{{{\bf{0}}_{M}}}\end{array}}\right]}{{\bf{w}}_{p,k}}=\left\\{{\left({{{\bf{e}}_{1}}{\bf{e}}_{1}^{H}}\right)\otimes{{\bf{I}}_{M}}}\right\\}{{\bf{w}}_{p,k}}\end{aligned},$
(46)

where ${{\bf{e}}_{1}}\in\mathbb{R}^{B}$ holds. This operation is essentially
to take out the elements correlated with the first BS in ${\bf{w}}_{p,k}$.
Since $({{{\bf{e}}_{1}}{\bf{e}}_{1}^{H}})\otimes{{\bf{I}}_{M}}$ is a diagonal
matrix and all non-zero elements are 1, we have

$\displaystyle{\left\\{{\left({{{\bf{e}}_{1}}{\bf{e}}_{1}^{H}}\right)\otimes{{\bf{I}}_{M}}}\right\\}^{H}}\left\\{{\left({{{\bf{e}}_{1}}{\bf{e}}_{1}^{H}}\right)\otimes{{\bf{I}}_{M}}}\right\\}=\left({{{\bf{e}}_{1}}{\bf{e}}_{1}^{H}}\right)\otimes{{\bf{I}}_{M}}.$
(47)

Thus the power of ${{\bf{w}}_{1,p,k}}$ can be represented as

$\displaystyle\left\|{{{\bf{w}}_{1,p,k}}}\right\|$
$\displaystyle={\bf{w}}_{p,k}^{H}{\left\\{{\left({{{\bf{e}}_{1}}{\bf{e}}_{1}^{H}}\right)\otimes{{\bf{I}}_{M}}}\right\\}^{H}}\left\\{{\left({{{\bf{e}}_{1}}{\bf{e}}_{1}^{H}}\right)\otimes{{\bf{I}}_{M}}}\right\\}{{\bf{w}}_{p,k}}$
(48)
$\displaystyle={\bf{w}}_{p,k}^{H}\left\\{{\left({{{\bf{e}}_{1}}{\bf{e}}_{1}^{H}}\right)\otimes{{\bf{I}}_{M}}}\right\\}{{\bf{w}}_{p,k}}.$

Then, based on the definition
${\bf{W}}={[{\bf{w}}_{1,1}^{T},{\bf{w}}_{1,2}^{T},\cdots,{\bf{w}}_{1,K}^{T},{\bf{w}}_{2,1}^{T},{\bf{w}}_{2,2}^{T},\cdots,{\bf{w}}_{P,K}^{T}]^{T}}$,
we obtain

$\displaystyle\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{\left\|{{{\bf{w}}_{1,p,k}}}\right\|}}$
(49) $\displaystyle=$
$\displaystyle\sum\limits_{k=1}^{K}{\sum\limits_{p=1}^{P}{{\bf{w}}_{p,k}^{H}\left\\{{\left({{{\bf{e}}_{1}}{\bf{e}}_{1}^{H}}\right)\otimes{{\bf{I}}_{M}}}\right\\}{{\bf{w}}_{p,k}}}}$
$\displaystyle=$
$\displaystyle{{\bf{W}}^{H}}\left[{\begin{array}[]{*{20}{c}}{\left({{{\bf{e}}_{1}}{\bf{e}}_{1}^{H}}\right)\otimes{{\bf{I}}_{M}}}&{}\hfil&{}\hfil\\\
{}\hfil&\ddots&{}\hfil\\\
{}\hfil&{}\hfil&{\left({{{\bf{e}}_{1}}{\bf{e}}_{1}^{H}}\right)\otimes{{\bf{I}}_{M}}}\end{array}}\right]{\bf{W}}$
$\displaystyle=$
$\displaystyle{{\bf{W}}^{H}}\left\\{{{{\bf{I}}_{PK}}\otimes\left\\{{\left({{{\bf{e}}_{1}}{\bf{e}}_{1}^{H}}\right)\otimes{{\bf{I}}_{M}}}\right\\}}\right\\}{\bf{W}}$
$\displaystyle=$
$\displaystyle{{\bf{W}}^{H}}{{\bf{D}}_{1}}{\bf{W}}\leq{P_{1,\max}}.$

Then, by setting $b$ to other elements in $\cal B$, we can similarly deduce
all expressions of the power constraints by

${\bf{W}}^{H}{{\bf D}_{b}}{\bf{W}}\leq{P_{b,\max}},~{}\forall
b\in\mathcal{B},$ (50)

where ${{\bf
D}_{b}}={{\bf{I}}_{PK}}\otimes\left\\{\left({{\bf{e}}_{b}}{\bf{e}}_{b}^{H}\right)\otimes{{\bf{I}}_{M}}\right\\}$.

##  Appendix B Feasible solution to problem (25)

We consider to solve (25) via Lagrange dual decomposition. By introducing
${\bm{\lambda}}=\left[{{\lambda_{1}},\cdots,{\lambda_{B}}}\right]^{T}\in\mathbb{R}^{B}$,
we can equivalently transform (25) as

$\displaystyle\mathop{{\rm{min}}}\limits_{{\bm{\lambda}}}~{}~{}~{}\Pi_{1}({\bm{\lambda}})=\mathop{\max}\limits_{\bf{W}}\left\\{{\Gamma_{1}({\bf{W}},{\bm{\lambda}})}\right\\}$
(51) $\displaystyle\,{\rm{s.t.}}~{}~{}~{}{\lambda_{b}}\geq 0,~{}\forall
b\in\mathcal{B},$

where

$\displaystyle\Gamma_{1}({\bf{W}},{\bm{\lambda}})={g_{3}}({\bf{W}})-\sum\limits_{b=1}^{B}{{\lambda_{b}}\left({{\bf{W}}^{H}}{{\bf{D}}_{b}}{\bf{W}}-{P_{b,\max}}\right)}.$
(52)

Since the problem satisfies the Slater’s condition, the duality gap does not
exist strictly [48]. Then, by substituting $g_{3}$ into (42) and setting
$\partial\Gamma_{1}/\partial{\bf{W}}$ to zero, we obtain

$\displaystyle{{\bf{W}}^{\rm
opt}}={\left({\sum\limits_{b=1}^{B}{{\lambda_{b}}{{\bf{D}}_{b}}+{\bf{A}}}}\right)^{-1}}{\bf{V}}.$
(53)

Finally, the optimal dual variable vector ${\bm{\lambda}}^{\rm opt}$ can be
obtained via the ellipsoid method [48].

##  Appendix C Feasible solution to problem (37)

Similarly, we consider to solve (37) via Lagrange dual decomposition. To
simplify the expression, we use the following equivalent definition ${{\cal
F}}$ in (37) as

$\displaystyle{{\cal
F}}\triangleq\left\\{{\theta_{r,n}}{\Big{|}}\left|{{\theta_{r,n}}}\right|\leq
1\right\\}\triangleq\left\\{{\theta_{r,n}}{\Big{|}}\left|{{\theta_{r,n}}}\right|^{2}\leq
1\right\\}.$ (54)

Then, by introducing the dual variable vector
${\bm{\chi}}={\left[{{\chi_{1,1}},{\chi_{1,2}},\cdots,{\chi_{1,N}},{\chi_{2,1}},{\chi_{2,2}},\cdots,{\chi_{R,N}}}\right]^{T}}\in{\mathbb{R}^{RN}}$,
the problem (37) can be equivalently transformed as

$\displaystyle\mathop{{\rm{min}}}\limits_{{\bm{\chi}}}~{}~{}~{}\Pi_{2}({\bm{\chi}})=\mathop{\max}\limits_{\bf{\Theta}}\left\\{{\Gamma_{2}({\bf{\Theta}},{\bm{\chi}})}\right\\}$
(55) $\displaystyle\,{\rm{s.t.}}~{}~{}~{}{\chi_{r,n}}\geq 0,~{}\forall
r\in{{\cal R}},\forall n\in{{\cal N}},$

where

$\displaystyle{\Gamma_{2}({\bf{\Theta}},{\bm{\chi}})}$
$\displaystyle={{g}_{7}}({\bm{\Theta}})-\sum\limits_{r=1}^{R}{\sum\limits_{n=1}^{N}{\left({{{\left|{{{\theta}_{r,n}}}\right|}^{2}}-1}\right)}}$
(56)
$\displaystyle={{g}_{7}}({\bm{\Theta}})-\sum\limits_{r=1}^{R}{\sum\limits_{n=1}^{N}\left({{\chi_{r,n}}{{\bm{\theta}}^{H}}{\bf{e}}_{j}{\bf{e}}_{j}^{H}{\bm{\theta}}}-1\right)}.$

Note that $j=N(r-1)+n$ holds and ${\bf{e}}_{j}\in\mathbb{R}^{RN}$. Similarly,
the problem also satisfies the Slater’s condition and the duality gap doesn’t
exist [48]. Then, by setting $\partial\Gamma_{2}/\partial{\bm{\Theta}}$ to
zero, we have

$\displaystyle{{\bm{\theta}}^{\rm
opt}}={\left({\sum\limits_{r=1}^{R}{\sum\limits_{n=1}^{N}{\chi_{r,n}}{{{\bf{e}}_{j}}{\bf{e}}_{j}^{H}}}+{\bf{\Lambda}}}\right)^{-1}}{\bm{\nu}}.$
(57)

Finally, the optimal dual variable vector ${\bm{\chi}}^{\rm opt}$ can be
obtained via the ellipsoid method [48].

## References

  * [1] D. López-Pérez, M. Ding, H. Claussen, and A. H. Jafari, “Towards 1 Gbps/UE in cellular systems: Understanding ultra-dense small cell deployments,” _IEEE Commun. Surveys Tuts._ , vol. 17, no. 4, pp. 2078–2101, Fourthquarter 2015.
  * [2] F. E. Idachaba, “5G networks: Open network architecture and densification strategies for beyond 1000x network capacity increase,” in _Proc. 2016 Future Technologies Conference (FTC’16)_ , Dec. 2016, pp. 1265–1269.
  * [3] W. Yu, H. Xu, H. Zhang, D. Griffith, and N. Golmie, “Ultra-dense networks: Survey of state of the art and future directions,” in _Proc. 2016 25th International Conference on Computer Communication and Networks (ICCCN’16)_ , Aug. 2016, pp. 1–10.
  * [4] M. Kamel, W. Hamouda, and A. Youssef, “Ultra-dense networks: A survey,” _IEEE Commun. Surveys Tuts._ , vol. 18, no. 4, pp. 2522–2545, Fourthquarter 2016.
  * [5] J. G. Andrews, X. Zhang, G. D. Durgin, and A. K. Gupta, “Are we approaching the fundamental limits of wireless network densification?” _IEEE Commun. Mag._ , vol. 54, no. 10, pp. 1558–1896, Oct. 2016.
  * [6] A. Lozano, R. W. Heath, and J. G. Andrews, “Fundamental limits of cooperation,” _IEEE Trans. Inf. Theory_ , vol. 59, no. 9, pp. 5213–5226, Sep. 2013.
  * [7] E. Nayebi, A. Ashikhmin, T. L. Marzetta, and H. Yang, “Cell-free massive MIMO systems,” in _Proc. 2015 49th Asilomar Conference on Signals, Systems and Computers (ACSSC’15)_ , Nov. 2015, pp. 695–699.
  * [8] G. Interdonato, E. Björnson, H. Q. Ngo, P. Frenger, and E. G. Larsson, “Ubiquitous cell-free massive MIMO communications,” _arXiv preprint arXiv:1804.03421_ , Apr. 2019.
  * [9] H. Q. Ngo, A. Ashikhmin, H. Yang, E. G. Larsson, and T. L. Marzetta, “Cell-free massive MIMO versus small cells,” _IEEE Trans. Wireless Commun._ , vol. 16, no. 3, pp. 1834–1850, Mar. 2017.
  * [10] S. Mosleh, H. Almosa, E. Perrins, and L. Liu, “Downlink resource allocation in cell-free massive MIMO systems,” in _Proc. 2019 International Conference on Computing, Networking and Communications (ICNC’19)_ , Feb. 2019, pp. 883–887.
  * [11] M. Attarifar, A. Abbasfar, and A. Lozano, “Modified conjugate beamforming for cell-free massive MIMO,” _IEEE Commun. Lett._ , vol. 8, no. 2, pp. 616–619, Apr. 2019.
  * [12] Y. Jin, J. Zhang, S. Jin, and B. Ai, “Channel estimation for cell-free mmwave massive MIMO through deep learning,” _IEEE Trans. Veh. Technol._ , vol. 68, no. 10, pp. 10 325–10 329, Oct. 2019.
  * [13] Y.-C. Liang, R. Long, Q. Zhang, J. Chen, H. V. Cheng, and H. Guo, “Large intelligent surface/antennas (LISA): Making reflective radios smart,” _arXiv preprint arXiv:1906.06578_ , Jun. 2019.
  * [14] E. Basar, M. D. Renzo, J. de Rosny, M. Debbah, M.-S. Alouini, and R. Zhang, “Wireless communications through reconfigurable intelligent surfaces,” _arXiv preprint arXiv:1906.09490_ , Jun. 2019.
  * [15] C. Huang, A. Zappone, G. C. Alexandropoulos, M. Debbah, and C. Yuen, “Reconfigurable intelligent surfaces for energy efficiency in wireless communication,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 8, pp. 4157–4170, Aug. 2019.
  * [16] K. Ntontin, M. D. Renzo, J. Song, F. Lazarakis, J. de Rosny, D. T. Phan-Huy, O. Simeone, R. Zhang, M. Debbah, G. Lerosey, M. Fink, S. Tretyakov, and S. Shamai, “Reconfigurable intelligent surfaces vs. relaying: Differences, similarities, and performance comparison,” _arXiv preprint arXiv:1908.08747_ , Aug. 2019.
  * [17] Q. Wu and R. Zhang, “Beamforming optimization for wireless network aided by intelligent reflecting surface with discrete phase shifts,” _IEEE Trans. Commun. (early access article)_ , 2019.
  * [18] P. Wang, J. Fang, X. Yuan, Z. Chen, H. Duan, and H. Li, “Intelligent reflecting surface-assisted millimeter wave communications: Joint active and passive precoding design,” _arXiv preprint arXiv:1908.10734_ , Aug. 2019\.
  * [19] C. Pan, H. Ren, K. Wang, W. Xu, M. Elkashlan, A. Nallanathan, and L. Hanzo, “Multicell MIMO communications relying on intelligent reflecting surface,” _arXiv preprint arXiv:1907.10864_ , Jul. 2019.
  * [20] C. Huang, C. Zhang, J. Yang, B. Sun, B. Zhao, and X. Luo, “Reconfigurable metasurface for multifunctional control of electromagnetic waves,” _Adv. Opt. Mater._ , vol. 5, no. 22, p. 1700485, Sep. 2017.
  * [21] C. Hu and L. Dai, “Two-timescale channel estimation for reconfigurable intelligent surface aided wireless communications,” _arXiv preprint arXiv:1912.07990_ , Dec. 2019.
  * [22] Q.-U.-A. Nadeem, A. Kammoun, A. Chaaban, M. Debbah, and M.-S. Alouini, “Asymptotic analysis of large intelligent surface assisted MIMO communication,” _arXiv preprint arXiv:1903.08127_ , Mar. 2019.
  * [23] L. Dai, B. Wang, M. Wang, X. Yang, J. Tan, S. Bi, S. Xu, F. Yang, Z. Chen, M. D. Renzo, and L. Hanzo, “Reconfigurable intelligent surface-based wireless communication: Antenna design, prototyping and experimental results,” _arXiv preprint arXiv:1912.03620_ , Dec. 2019.
  * [24] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless network: Joint active and passive beamforming design,” in _Proc. 2018 IEEE Global Communications Conference (GLOBECOM’18)_ , Dec. 2018, pp. 1–6.
  * [25] C. Huang, A. Zappone, M. Debbah, and C. Yuen, “Achievable rate maximization by passive intelligent mirrors,” in _Proc. 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP’18)_ , Apr. 2018, pp. 3714–3718.
  * [26] H. Li, R. Liu, M. Li, and Q. Liu, “IRS-enhanced wideband MU-MISO-OFDM communication systems,” _arXiv preprint arXiv:1909.11314_ , Sep. 2019.
  * [27] H. Xie, J. Xu, and Y.-F. Liu, “Max-min fairness in IRS-aided multi-cell MISO systems via joint transmit and reflective beamforming,” _arXiv preprint arXiv:1912.12827_ , Dec. 2019.
  * [28] J. Ye, S. Guo, and M.-S. Alouini, “Joint reflecting and precoding designs for SER minimization in reconfigurable intelligent surfaces assisted MIMO systems,” _arXiv preprint arXiv:1906.11466_ , Jun. 2019.
  * [29] B. Di, H. Zhang, L. Song, Y. Li, Z. Han, and H. V. Poor, “Hybrid beamforming for reconfigurable intelligent surface based multi-user communications: Achievable rates with limited discrete phase shifts,” _arXiv preprint arXiv:1910.14328_ , Oct. 2019.
  * [30] Y. Yang, S. Zhang, and R. Zhang, “IRS-enhanced OFDM: Power allocation and passive array optimization,” _arXiv preprint arXiv:1905.00604_ , May 2019\.
  * [31] X. Liu, Y. Liu, Y. Chen, and H. V. Poor, “RIS enhanced massive non-orthogonal multiple access networks: Deployment and passive beamforming design,” _arXiv preprint arXiv:2001.1036_ , Jan. 2020.
  * [32] X. Yu, D. Xu, and R. Schober, “Enabling secure wireless communications via intelligent reflecting surfaces,” _arXiv preprint arXiv:1904.09573_ , Apr. 2019.
  * [33] S. Li, B. Duo, X. Yuan, Y.-C. Liang, and M. D. Renzo, “Reconfigurable intelligent surface assisted UAV communication: Joint trajectory design and passive beamforming,” _arXiv preprint arXiv:1908.04082_ , Aug. 2019.
  * [34] K. Shen and W. Yu, “Fractional programming for communication systems—part I: Power control and beamforming,” _IEEE Trans. Signal Process._ , vol. 66, no. 10, pp. 2616–2630, May 2018.
  * [35] ——, “Fractional programming for communication systems—part II: Uplink scheduling via matching,” _IEEE Trans. Signal Process._ , vol. 66, no. 10, pp. 2631–2644, May 2018.
  * [36] U. Siddique, H. Tabassum, and E. Hossain, “Downlink spectrum allocation for in-band and out-band wireless backhauling of full-duplex small cells,” _IEEE Trans. Commun._ , vol. 65, no. 8, pp. 3538–3554, Aug. 2017.
  * [37] E. Nayebi, A. Ashikhmin, T. L. Marzetta, H. Yang, and B. D. Rao, “Precoding and power optimization in cell-free massive MIMO systems,” _IEEE Trans. Wireless Commun._ , vol. 16, no. 7, pp. 4445–4459, Jul. 2017\.
  * [38] X. Hu, C. Zhong, X. Chen, W. Xu, and Z. Zhang, “Rate analysis and ADC bits allocation for cell-free massive MIMO systems with low resolution ADCs,” in _Proc. 2018 IEEE Global Communications Conference (GLOBECOM’18)_ , Dec. 2018, pp. 1–6.
  * [39] S. Kusaladharma, W. P. Zhu, W. Ajib, and G. Amarasuriya, “Achievable rate analysis of NOMA in cell-free massive MIMO: A stochastic geometry approach,” in _Proc. 2019 IEEE International Conference on Communications (ICC’19)_ , May 2019, pp. 1–6.
  * [40] X. Gao, L. Dai, S. Han, C.-L. I, and R. W. Heath, “Energy-efficient hybrid analog and digital precoding for mmwave MIMO systems with large antenna arrays,” _IEEE J. Sel. Areas Commun._ , vol. 34, no. 4, pp. 998–1009, Apr. 2016.
  * [41] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 11, pp. 5394–5409, Nov. 2019.
  * [42] M. Grant and S. Boyd, “CVX: Matlab software for disciplined convex programming, version 2.1,” Mar. 2014.
  * [43] H. Guo, Y.-C. Liang, J. Chen, and E. G. Larsson, “Weighted sum-rate optimization for intelligent reflecting surface enhanced wireless networks,” _IEEE Trans. Wireless Commun. (early access article)_ , 2020.
  * [44] Q.-U.-A. Nadeem, A. Kammoun, A. Chaaban, M. Debbah, and M.-S. Alouini, “Asymptotic analysis of large intelligent surface assisted MIMO communication,” _arXiv preprint arXiv:1903.08127_ , Mar. 2019.
  * [45] X. Gao, L. Dai, Y. Sun, S. Han, and C.-L. I, “Machine learning inspired energy-efficient hybrid precoding for mmwave massive MIMO systems,” in _Proc. 2017 IEEE International Conference on Communications (ICC’17)_ , May 2017, pp. 1–6.
  * [46] Ö. Özdogan, E. Björnson, and E. G. Larsson, “Intelligent reflecting surfaces: Physics, propagation, and pathloss modeling,” _IEEE Wireless Commun. Lett. (early access article)_ , 2019.
  * [47] J. D. Griffin and G. D. Durgin, “Complete link budgets for backscatter-radio and RFID systems,” _IEEE Trans. Antennas Propag._ , vol. 51, no. 2, pp. 11–25, Apr. 2009.
  * [48] S. Boyd and L. Vandenberghe, “Convex optimization,” _Cambridge University Press_ , 2004.

  *[BS]: base station
  *[UDN]: Ultra-dense network
  *[RIS]: reconfigurable intelligent surface
  *[SINR]: signal-to-interference-plus-noise ratio
  *[SER]: symbol error rate
  *[mmWave]: millimeter-wave
  *[SNR]: signal-to-noise ratio
  *[OFDM]: orthogonal frequency division multiplexing
  *[NOMA]: non-orthogonal multiple acces
  *[UAVs]: unmanned aerial vehicles
  *[WSR]: weighted sum-rate
  *[QCQP]: quadratically constrained quadratic program
  *[IDFT]: inverse discrete Fourier transform
  *[CP]: cyclic prefix
  *[RC]: reflection coefficient
  *[DFT]: discrete Fourier transform
  *[AWGN]: additive white Gaussion noise
  *[ACE]: adaptive cross-entropy
  *[CSI]: channel state information