# Cooperative Beamforming Design for Multiple RIS-Assisted Communication
Systems

Xiaoyan Ma, Yuguang Fang, Haixia Zhang,
Shuaishuai Guo, and Dongfeng Yuan This work was supported in part by the
Project of International Cooperation and Exchanges NSFC under Grant No.
61860206005, in part by the National Natural Science Foundation of China under
Grant No. 62171262 and in part by the Major Scientific and Technological
Innovation Project of Shandong Province 2020CXGC010108 (_Corresponding author:
Haixia Zhang_). X. Ma and D. Yuan are with Shandong Key Laboratory of Wireless
Communication Technologies, Jinan 250061, China, and School of Information
Science and Engineering, Shandong University, Qingdao 266237, China (email:
<EMAIL_ADDRESS>dfyuan@sdu.edu.cn).H. Zhang and S. Guo are with
Shandong Key Laboratory of Wireless Communication Technologies, Jinan 250061,
China, and School of Control Science and Engineering, Shandong University,
Jinan 250061, China (email<EMAIL_ADDRESS>haixia.zhang@sdu.edu.cn).Y. Fang is with the Computer, Electrical and
Mathematical Science and Engineering Division, Department of Electrical and
Computer Engineering University of Florida, Gainesville, FL, USA (email:
fang@ece.ufl.edu).

###### Abstract

Reconfigurable intelligent surface (RIS) provides a promising way to build
programmable wireless transmission environments. Owing to the massive number
of controllable reflecting elements on the surface, RIS is capable of
providing considerable passive beamforming gains. At present, most related
works mainly consider the modeling, design, performance analysis and
optimization of single-RIS-assisted systems. Although there are a few of works
that investigate multiple RISs individually serving their associated users,
the cooperation among multiple RISs is not well considered as yet. To fill the
gap, this paper studies a cooperative beamforming design for multi-RIS-
assisted communication systems, where multiple RISs are deployed to assist the
downlink communications from a base station to its users. To do so, we first
model the general channel from the base station to the users for arbitrary
number of reflection links. Then, we formulate an optimization problem to
maximize the sum rate of all users. Analysis shows that the formulated problem
is difficult to solve due to its non-convexity and the interactions among the
decision variables. To solve it effectively, we first decouple the problem
into three disjoint subproblems. Then, by introducing appropriate auxiliary
variables, we derive the closed-form expressions for the decision variables
and propose a low-complexity cooperative beamforming algorithm. Simulation
results have verified the effectiveness of the proposed algorithm through
comparison with various baseline methods. Furthermore, these results also
unveil that, for the sum rate maximization, distributing the reflecting
elements among multiple RISs is superior to deploying them at one single RIS.

###### Index Terms:

Reconfigurable intelligent surface (RIS), cooperative beamforming design,
multi-hop transmission

##  I Introduction

Reconfigurable intelligent surface (RIS) has emerged as a promising tool in
improving the transmission environments for the wireless communications [1, 2,
3]. Specifically, RIS can be a planar surface, which comprises a large number
of low-cost passive reflecting elements. Each element can be digitally
controlled and induce an independent amplitude change and/or phase shift to
the incident signals, thereby collaboratively altering the wireless channels
[4, 5, 6]. Besides, from the implementation perspective, RISs possess
appealing features such as low profile and lightweight, thus can be easily
mounted on walls or ceilings without significantly damaging the aesthetic
beauty of the environment [7]. Since RISs are complementary devices, deploying
them in the existing wireless systems only need necessary modifications of the
communication protocols, and no other changes on standards and hardware are
required [8]. Furthermore, RISs are usually much cheaper than active small
base stations (SBSs)/relays, therefore can be easily deployed rapidly [9].

Due to such great advantages, RISs have attracted intensive attentions lately
and a lot of related research works have been done, including transmission
protocol designs [10, 11, 12], system capacity analysis [13, 14, 15, 16, 17,
18], energy/spectral efficiency [19, 20, 21, 22, 23, 24], physical layer
security [25, 26, 27, 28], modulation/coding schemes [29, 30, 31], and so on.
And a comprehensive introduction to RIS can be found in [32]. Existing works
mainly consider one or multiple distributed RISs, where each RIS individually
reflects the signals to users, and there is no signal cooperation among
multiple RISs. This simplified approach generally contribute to sub-optimal
performance [33], since the multi-hop links that can provide high cooperation
beamforming gains are not well utilized. In addition, multi-hop reflections
established through inter-RIS links can provide higher channel diversity,
therefore can bypass the dense obstacles and provide blockage-free links with
higher transmission quality.

Inspired by the advantages of cooperative transmission in multi-RIS-assisted
systems, Han et al. investigated a double-RIS aided single-user wireless
communication system. In [34], it is demonstrated that by cooperatively
designing the phase shifts, a double-reflection link can provide passive
beamforming gain that increases quartically with the total number of RISs’
reflecting elements, which is significantly higher than the quadratic growth
of the passive beamforming gain of single-RIS link with the same number of
total reflecting elements. This work has been further extended to multi-user
setups [33, 35], where the superiority of deploying double-RIS is further
confirmed compared with single-RIS models. Specifically, Zheng, You and Zhang
[33] have investigated a double-RIS-assisted multi-user uplink communication
system. By jointly designing the receive beamforming at the BS and the
cooperative reflection at these two RISs, the minimum signal-to-interference-
plus-noise ratio (SINR) is maximized. With similar system setups made in [33],
a double-RIS-assisted system is designed to enhance the secrecy performance
[35]. The above works only consider double RISs, Huang et. al [36] then
extended to more general multi-RIS scenarios. In [36], only the multi-hop link
established through multiple RISs is utilized to bypass the severe obstacles
between the transmitter and the receivers, and their results also prove that
the multiplicative beamforming gain obtained through inter-RIS channels can
significantly improve the system capacity.

Except the aforementioned works, general multi-RIS-assisted multi-user
communication systems have not been fully investigated. Under the general
setups with more available RISs, users at different locations can be served
via various rich paths established through multi-hop reflections as shown in
Fig. 1. And this gives rise to a new cooperative beamforming design problem,
where multiple RISs serving multi-user through cooperative transmissions. In
this work, we have been devoted to solve this problem to maximize the system
sum rate. The main contributions are summarized as follows.

  * •

We establish a multi-RIS-assisted multi-user communication system, where the
general channel model for arbitrary reflection links is formulated. Different
from the traditional multiple RIS-assisted systems, where only one multi-hop
link that established through all the RISs is utilized. We address the
possibility of distributed RISs in providing higher access flexibility. With
this general setups, users at different locations can be served via various
rich paths established through multi-hop reflections, which significantly
improve the system capacity.

  * •

Based on the proposed channel model, an optimization problem is formulated to
maximize the sum rate of all users, which is multivariate and non-convex. To
solve it effectively, we decompose the original problem into three disjoint
sub-problems by utilizing closed-form fractional programming techniques.
Specifically, we first decouple the sum-of-logarithms-of-ratio problem into
sum-of-logarithms and sum-of-ratio problems by introducing auxiliary
variables, then utilize quadratic transform to further simplify the sum-of-
ratio part.

  * •

The formulated problem is a computationally intensive task due to the large
number of RISs’ reflecting elements and the interaction among different RISs.
To facilitate the cooperative beamforming design, we reformulate the user’s
equivalent channel, so that the coupled phase shift matrix of each RIS can be
extracted and iteratively updated. With the reformulated equivalent channel,
we are also able to derive the closed-form optimal expression for each RISs’
phase shift matrix, thereby significantly reduce the computational complexity.

  * •

To further simplify the overall computational complexity for cooperative
beamforming design, we derive the closed-form optimal expressions for all the
decision variables, and design a low-complexity cooperative beamforming
algorithm to maximize the sum rate of the system. Analysis shows that the
complexity of the proposed algorithm linearly increases with the number of
RISs’ reflecting elements. Extensive simulations have been conducted and
demonstrated that the proposed algorithm achieves significantly higher sum
rate than the baseline schemes. Furthermore, the results also unveil that
distributing the reflecting elements among multiple RISs is preferable to
deploying them at one single RIS in achieving higher sum rate.

Figure 1: The proposed multi-RIS-assisted cooperative transmission system,
where users at different locations can access the BS through proper RIS-
assisted links.

The remainder of this paper is organized as follows. The system model and
problem formulation are introduced in Section II. In Section III, we introduce
the proposed cooperative beamforming algorithm for multi-RIS-assisted multi-
user communication systems. To verify the superiority of the proposed method,
simulation results are presented in Section V. Finally, we conclude this paper
in Section VI.

The notations used in this paper are listed as follows. Bold symbols in
capital letter and small letter denote matrices and vectors, respectively.
$\mathcal{CN}(\mu,\sigma^{2})$ denotes the circularly symmetric complex
Gaussian (CSCG) distribution with mean $\mu$ and variance $\sigma^{2}$.
$||\mathbf{w}||$ denotes the Euclidean norm while $||\mathbf{W}||_{F}$ denotes
the Frobenius norm. $\mathbf{I}_{M}$ represents the $M\times M$ identity
matrix. $\mathbf{G}^{T}$ and $\mathbf{G}^{H}$ denote the transpose and
conjugate transpose of matrix $\mathbf{G}$, respectively. $\mathbf{G}^{-1}$
represents the inverse of $\mathbf{G}$. $\otimes$ is the Kronecker product.
$\mathbb{E}\\{\cdot\\}$ denotes the statistical expectation. For any real
number, $\sqrt{a}$ denotes the square root of $a$. For any complex variable
$x$, $|x|$ denotes the absolute value, $\angle x$ represents its phase angle,
$x^{*}$ denotes its conjugate, $\Re\\{x\\}$ and $\Im\\{x\\}$ represent its
real part and imaginary part, respectively. $A_{i,j}$ represents the element
at row $i$, column $j$ of matrix $\mathbf{A}$ and $a_{n}$ represents the
$n$-th elements of vector $\bm{a}$.
$\mathbf{A}=\mathop{\mathrm{diag}}(\bm{a})$ means that $\mathbf{A}$ is the
diagonal matrix of the vector $\bm{a}$. $\mathbf{0}$ is the zero matrix.

##  II System Model and Problem Formulation

###  II-A System and Channel Model

We consider a cooperative multi-RIS-assisted multi-user communication system
as shown in Fig. 1, in which $L$ distributed RISs are deployed to assist the
downlink transmission from the $M$ antenna BS to $K$ single-antenna users. It
is assumed that the direct links between the BS and users are severely blocked
by the obstacles, and $L$ RISs are deployed between the BS and user cluster to
assist the downlink transmissions. Without loss of generality, RIS $1$ is
deployed very close to the BS to guarantee line-of-sight (LoS) channel, i.e.,
directly above the BS, other RISs (i.e., RIS $2,3,...,L$) are deployed
sequentially to form multi-hop links. The $l$-th RIS, $l=1,2,...,L$, is
composed of $N_{l}$ reflecting elements, and each RIS is connected to a
controller that adjusts its phase shifts. To be specific, we consider the
general case where there exist arbitrary reflection links in the communication
systems. We let $\mathbf{G}_{1}\in\mathbb{C}^{N_{1}\times M}$,
$\mathbf{H}_{l-1,l}\in\mathbb{C}^{N_{l}\times N_{l-1}}$ and
$\bm{g}_{l,k}\in\mathbb{C}^{1\times N_{l}}$ denote the channels from the BS to
RIS $1$, from RIS $l-1$ to RIS $l$ and from RIS $l$ to user $k$, respectively.
The reflection coefficient matrix of RIS $l$ can be expressed as
$\mathbf{\Phi}_{l}\in\mathbb{C}^{N_{l}\times N_{l}}$, which can be further
expressed as
$\mathbf{\Phi}_{l}=\mathop{\mathrm{diag}}(\boldsymbol{\phi}_{l})$.
$\boldsymbol{\phi}_{l}=[\phi_{l,1},\phi_{l,2},...,\phi_{l,N_{l}}]^{T}$ is the
corresponding reflection vector of the $l$-th RIS, where
$\phi_{l,n}=\beta_{l,n}e^{j\theta_{l,n}}$, $n=1,2,...,N_{l}$. Moreover,
$\theta_{l,n}\in[0,2\pi)$ and $\beta_{l,n}\in[0,1]$ represent the phase and
amplitude change brought by the $n$-th reflecting element of RIS $l$ to the
incident signals. Without loss of generality, we set $\beta_{l,n}=1$, which
implies that all the elements of the RISs are switched on to fully reflect the
incident signals.

We assume that all the channels follow Rician fading. Similar to [37], we
further assume that the antenna elements form a half-wavelength uniform linear
array (ULA) configuration at the BS, and the reflecting elements form a
uniform planar array (UPA) configuration at RISs. RIS $l$ consists of a sub-
wavelength UPA with $N_{l}=N_{lx}N_{ly}$ passive reflecting elements, where
$N_{lx}$ and $N_{ly}$ denote the number of the reflecting elements along the
x- and y-axis as shown in Fig. 2, the adjacent elements are separated by
$d_{x}$ and $d_{y}$, respectively. We use $\mathbf{G}_{1}$ and $\bm{g}_{1,k}$
as examples to illustrate the adopted channel model in detail. Specifically,
$\mathbf{G}_{1}$ can be modeled as

Figure 2: The channel model for multi-RIS-assisted system.

$\mathbf{G}_{1}\\!=\\!\sqrt{\beta_{\mathbf{G}_{1}}}\left(\sqrt{\frac{\mathcal{F}_{\mathbf{G}_{1}}}{\mathcal{F}_{\mathbf{G}_{1}}+1}}\mathbf{G}_{1,LoS}\\!+\\!\sqrt{\frac{1}{\mathcal{F}_{\mathbf{G}_{1}}+1}}\mathbf{G}_{1,NLoS}\right),$
(1)

where $\beta_{\mathbf{G}_{1}}$ denotes the free-space path loss;
$\mathcal{F}_{\mathbf{G}_{1}}$ represents the corresponding Rician factor; the
matrices $\mathbf{G}_{1,LoS}$ and $\mathbf{G}_{1,NLoS}$ represent the line-of-
sight (LoS) and non-LoS (NLoS) components of the channel $\mathbf{G}_{1}$,
respectively. The LoS component $\mathbf{G}_{1,LoS}$ can be expressed as

$\displaystyle\mathbf{G}_{1,LoS}=\mathbf{a}_{1,R}(\eta_{1,R},\vartheta_{1,R})\mathbf{a}_{T}^{H}(\vartheta_{1,R}),$
(2)

where $\mathbf{a}_{1,R}(\eta_{1,R},\vartheta_{1,R})\in\mathbb{C}^{N_{1}\times
1}$ represents the receiving array response of RIS 1 and
$\mathbf{a}_{T}(\vartheta_{1,R})\in\mathbb{C}^{M\times 1}$ denotes the
transmit array response of the BS. Let $\lambda$ denote the signal wavelength,
by defining $\bar{d}_{x}=d_{x}/\lambda$ and $\bar{d}_{y}=d_{y}/\lambda$, we
express the response vector $\mathbf{a}_{1,R}(\eta_{1,R},\vartheta_{1,R})$ as

$\displaystyle\mathbf{a}_{1,R}(\eta_{1,R},\vartheta_{1,R})=$
$\displaystyle[1,...,e^{j2\pi(N_{1x}-1)\bar{d}_{x}\Theta_{1,R}}]^{T}$ (3)
$\displaystyle\otimes[1,...,e^{j2\pi(N_{1y}-1)\bar{d}_{y}\Omega_{1,R}}]^{T},$

where $\Theta_{1,R}=\sin(\vartheta_{1,R})\cos(\eta_{1,R})$ and
$\Omega_{1,R}=\sin(\vartheta_{1,R})\sin(\eta_{1,R})$ are the angle parameters
with $\vartheta_{1,R}$ representing the zenith angle of arrival (AoA) and
$\eta_{1,R}$ being the azimuth AoA at RIS 1 as shown in Fig. 2. The transmit
array response $\mathbf{a}_{T}(\vartheta_{1,R})$ can be similarly obtained as
$\mathbf{a}_{T}(\vartheta_{1,R})=[1,e^{j\pi\sin\vartheta_{1,R}},...,e^{j\pi(M-1)\sin\vartheta_{1,R}}]^{T}$
with $\vartheta_{1,R}$ representing the angle of departure (AoD) from the BS
to RIS 1. The matrix $\mathbf{G}_{1,NLoS}$ represents the NLoS part, whose
elements are chosen from $\mathcal{CN}(0,1)$. Following the same processes, we
obtain the channel $\bm{g}_{1,k}$ from RIS 1 to user $k$ as

$\displaystyle\bm{g}_{1,k}\\!\\!=\\!\\!\sqrt{\beta_{\bm{g}_{1,k}}}\\!\\!\left(\sqrt{\frac{\mathcal{F}_{\bm{g}_{1,k}}}{\mathcal{F}_{\bm{g}_{1,k}}+1}}\bm{g}_{1,k,LoS}\\!\\!+\\!\\!\sqrt{\frac{1}{\mathcal{F}_{\bm{g}_{1,k}}+1}}\bm{g}_{1,k,NLoS}\right),$
(4)

where $\beta_{\bm{g}_{1,k}}$ represents the free-space path loss;
$\mathcal{F}_{\bm{g}_{1,k}}$ is the corresponding Rician factor; the vectors
$\bm{g}_{1,k,LoS}$ and $\bm{g}_{1,k,NLoS}$ are the LoS and NLoS components of
the channel, respectively. The LoS component $\bm{g}_{1,k,LoS}$ can be
represented by
$\bm{g}_{1,k,LoS}=\mathbf{a}_{1,T}^{T}(\eta_{1,T},\vartheta_{1,T})$ with
$\mathbf{a}_{1,T}(\eta_{1,T},\vartheta_{1,T})$ denoting the transmit array
response of RIS 1. Similarly, by defining the angle parameters
$\Theta_{1,T}=\sin(\vartheta_{1,T})\cos(\eta_{1,T})$ and
$\Omega_{1,T}=\sin(\vartheta_{1,T})\sin(\eta_{1,T})$ with AoDs
$\vartheta_{1,T}$ and $\eta_{1,T}$, we express
$\mathbf{a}_{1,T}(\eta_{1,T},\vartheta_{1,T})$ as

$\displaystyle\mathbf{a}_{1,T}(\eta_{1,T},\vartheta_{1,T})=$
$\displaystyle[1,...,e^{j2\pi(N_{1x}-1)\bar{d}_{x}\Theta_{1,T}}]^{T}$ (5)
$\displaystyle\otimes[1,...,e^{j2\pi(N_{1y}-1)\bar{d}_{y}\Omega_{1,T}}]^{T}.$

The vector $\bm{g}_{1,k,NLoS}$ in (4) is the NLoS component, whose elements
are chosen from $\mathcal{CN}(0,1)$. The other channels can also be generated
by following the same processes, we will not go into details here. The channel
state information (CSI) for the BS-RIS $1$ link and the inter-RIS links can be
considered to be known in advance, since once they are properly deployed,
high-quality LoS channels can be established between them and those CSI can be
directly utilized. For the RIS-user channel information, there are two types
of approaches to acquire CSI in the RIS-assisted systems, depending on whether
the RIS’s reflecting elements have the capability of sensing or not. For the
case that RIS has only passive elements, since the RISs are usually properly
deployed at the places that can establish LoS links to the users, such as the
facade of the tall buildings, the location based channel estimation schemes
can be used to get the CSI to assist the beamforming design [38, 39]. Besides,
some other advanced channel estimation methods were also proposed to perform
separate channel estimation in RIS-assisted systems. For example, Wei et al
[40] proposed a parallel factor decomposition based method to unfold the
cascaded channel model, which can also be adopted. For the second case, where
there is semi-passive RIS architecture, a small portion of the RIS’s
reflecting elements are able to process the received signal for facilitating
the channel estimation. Based on the pilot signals received by these semi-
passive elements, the RIS can estimate the channels between itself and the
user through various existing estimation methods [41]. With the CSI, the
cooperative beamforming can be done. In the following, we will focus on the
cooperative beamforming design for the multi-RIS-assisted communication
system.

###  II-B Problem Formulation

Let $x_{k}$ denote the transmitted signal from the BS to user $k$ with
$\mathbb{E}\\{|x_{k}|^{2}\\}=1,k=1,2,\cdots,K$. Then, the transmitted signals
for all users at the BS can be expressed as

$\displaystyle\begin{aligned}
\mathbf{x}=\sum_{k=1}^{K}\bm{w}_{k}x_{k},\end{aligned}$ (6)

where $\bm{w}_{k}\in\mathbb{C}^{M\times 1}$ is the corresponding transmit
beamforming vector for user $k$. The equivalent channel from the BS to user
$k$ can be represented as

$\displaystyle\begin{aligned}
\bm{h}_{k}^{H}=\bm{g}_{1,k}\mathbf{\Phi}_{1}\mathbf{G}_{1}+\sum_{l=2}^{L}\bm{g}_{l,k}\\!\\!\\!\\!\\!\\!\prod_{i=l,l-1,...,2}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathbf{\Phi}_{i}\mathbf{H}_{i-1,i}\mathbf{\Phi}_{1}\mathbf{G}_{1},\end{aligned}$
(7)

the first term $\bm{g}_{1,k}\mathbf{\Phi}_{1}\mathbf{G}_{1}$ in (7) represents
the single-reflection channel from the RIS $1$ to the user $k$, the second
term in (7) is the multi-hop links established through the multiple RISs.
Considering that the obstacle may very close to the BS, which can block the
direct link from BS to the users and the other RISs excerpt for RIS $1$.
Thereby in the proposed channel model, we let BS generate narrow beam with the
total transmit power directly towards RIS $1$. With the equivalent channel
$\bm{h}_{k}^{H}$, the received signal at user $k$ can be expressed as

$\displaystyle\begin{aligned}
y_{k}=\bm{h}_{k}^{H}\mathbf{x}+n_{k},\end{aligned}$ (8)

where $n_{k}\sim\mathcal{CN}(0,\sigma_{0}^{2})$ denotes the complex additive
white Gaussian noise (AWGN) at the $k$-th user. The $k$-th user treats all the
signals for other users as interference. Thus, the corresponding SINR at user
$k$ can be expressed as

$\begin{split}&\gamma_{k}=\frac{|\bm{h}_{k}^{H}\bm{w}_{k}|^{2}}{\sum_{i=1,i\neq
k}^{K}|\bm{h}_{k}^{H}\bm{w}_{i}|^{2}+\sigma_{0}^{2}},\end{split}$ (9)

with $\sum_{k=1}^{K}||\bm{w}_{k}||^{2}\leq P_{T}$ being the power constraint
and $P_{T}$ is the maximum transmit power at the BS. When the BS utilizes all
the power to transmit signals for user $k$, i.e., $||\bm{w}_{k}||^{2}=P_{T}$,
the maximum SINR $\gamma_{k}^{max}=P_{T}||\bm{h}_{k}^{H}||^{2}/\sigma_{0}^{2}$
can be achieved, therefore we have $0\leq\gamma_{k}\leq\gamma_{k}^{max}$.

Let $\mathbf{W}=[\bm{w}_{1},\bm{w}_{2},...,\bm{w}_{K}]\in\mathbb{C}^{M\times
K}$ denote the overall transmit beamforming matrix. In this paper, we aim to
maximize the sum rate of all users at the downlink transmission by jointly
designing the active beamforming $\mathbf{W}$ at the BS and the passive
beamforming $\boldsymbol{\phi}_{l}$, $l=1,2,...,L$ at each RIS, subject to the
BS’s transmit power constraint and RISs’ phase shift constraints.
Mathematically, the optimization problem can be formulated as

$\displaystyle(P1)\quad$
$\displaystyle\max_{\mathbf{W},\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L}}\quad
f(\mathbf{W},\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L})=\sum_{k=1}^{K}\log(1+\gamma_{k}),$
(10)
$\displaystyle\quad\quad\text{s.t.}\quad\quad|\phi_{l,n}|=1,~{}l=1,...,L,~{}n=1,...,N_{l},$
(11)
$\displaystyle\quad\quad\quad\quad\quad\sum_{k=1}^{K}||\bm{w}_{k}||^{2}\leq
P_{T}.$ (12)

In ($P1$), the objective variables that need to be obtained are $\mathbf{W}$
and $\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L}$, which are the active
beamforming at the base station and the passive beamforming for RIS $l$,
$l=1,2,...,L$. By cooperatively designing the transmit beamforming matrix
$\mathbf{W}$ and the passive beamforming vectors $\boldsymbol{\phi}_{l}$, the
inter-user interference at the receiving sides can be mitigated, leading to
the improvement of the sum rate. However, solving ($P1$) is extremely
challenging since the objective function
$f(\mathbf{W},\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L})$ and the phase
shift constraint (11) are non-convex. Moreover, the optimization variables
$\mathbf{W}$ and $\boldsymbol{\phi}_{l}$ are coupled with each other in the
expression of the sum rate. All these bring great difficulties to find the
global optimal solution. As an alternative, we attempt to find a low-
complexity sub-optimal solution to ($P1$).

##  III Low Complexity Beamforming Design for Multi-RIS-Assisted Multi-user
Systems

In this section, the cooperative beamforming at both the BS and RISs are
carefully designed to mitigate the inter-user interference and maximize the
sum rate. In the original objective function (10),
$\sum_{k=1}^{K}\log(1+\gamma_{k})$ is a sum-of-log-of-ratio term, and the
objective variables $\mathbf{W}$ and $\boldsymbol{\phi}_{l}$ are coupled
together. In order to solve the optimization problem efficiently, we first
introduce several auxiliary variables to transform ($P1$) into a problem of
much low-complex by applying the closed-form fractional programming (FP)
techniques [42, 43]. The closed-form FP techniques is proposed to deal with
the sum-of-log-of-ratio problem as follows

$\max_{\bm{x}}\sum_{k=1}^{K}\log\left(1+\frac{|A_{k}(\bm{x})|^{2}}{B_{k}(\bm{x})-|A_{k}(\bm{x})|^{2}}\right),$

where $B_{k}(\bm{x})\geq|A_{k}(\bm{x})|^{2}$ for all $k$. Conceptually, the
closed-form FP approach has two key steps:

_a) Lagrangian Dual Transform._ By introducing an auxiliary variable
$\alpha_{k}$, the logarithm function can be tackled based on the following
equation

$\log(1+\gamma_{k})=\max_{\alpha_{k}\geq
0}\left(\log(1+\alpha_{k})-\alpha_{k}+\frac{(1+\alpha_{k})\gamma_{k}}{1+\gamma_{k}}\right).$

And the equivalency is guaranteed when $\alpha_{k}=\gamma_{k}$. Then, the
original problem is equivalently transformed to

$\displaystyle\max_{\bm{x},\bm{\alpha}}\quad\sum_{k=1}^{K}\left(\log(1+\alpha_{k})-\alpha_{k}+(1+\alpha_{k})\frac{|A_{k}(\bm{x})|^{2}}{B_{k}(\bm{x})}\right),$
$\displaystyle~{}\text{s.t.}\quad\quad\alpha_{k}\geq
0,\quad\forall\,k=1,2,...,K,$

where $\bm{\alpha}=[\alpha_{1},\alpha_{2},...,\alpha_{K}]^{T}$ is the
auxiliary variable vector.

_b) Quadratic Transform._ Given $\bm{\alpha}$, we then focus on the following
sum-of-ratios problem

$\max_{\bm{x}}\sum_{k=1}^{K}\frac{|A_{k}(\bm{x})|^{2}}{B_{k}(\bm{x})}.$

The key idea is introducing auxiliary variables
$\bm{\beta}=[\beta_{1},\beta_{2},...,\beta_{k}]^{T}$, and then the above
problem is equivalently translated to

$\max_{\bm{x},\bm{\beta}}\sum_{k=1}^{K}\left(2\Re\\{\beta_{k}^{*}A_{k}(\bm{x})\\}-|\beta_{k}|^{2}B_{k}(\bm{x})\right).$

The equivalence can be verified by substituting
$\beta_{k}=\frac{A_{k}(\bm{x})}{B_{k}(\bm{x})}$ into the above equation.

Based on the reformulated problem, we aim to derive the closed-form optimal
solutions for each decision variable. Specifically, the reformulated problem
can be written as

$\displaystyle\begin{aligned}
(P2)&\max_{\boldsymbol{\alpha},\mathbf{W},\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L}}\quad
f_{2}(\boldsymbol{\alpha},\mathbf{W},\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L}),\\\
&\quad\quad\text{s.t.}\quad\quad\quad|\phi_{l,n}|\\!\\!=\\!\\!1,~{}l=1,2,...,L,\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad~{}n=1,2,...,N_{l},\\\
&\quad\quad\quad\quad\quad\quad\sum_{k=1}^{K}||\bm{w}_{k}||^{2}\leq P_{T},\\\
&\quad\quad\quad\quad\quad\quad\alpha_{k}\geq 0,k=1,2,...,K,\end{aligned}$
(13)

where

$\displaystyle\begin{aligned}
f_{2}(\boldsymbol{\alpha},\mathbf{W},\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L})=&\sum_{k=1}^{K}\log(1+\alpha_{k})-\sum_{k=1}^{K}\alpha_{k}\\\
&+\sum_{k=1}^{K}\frac{(1+\alpha_{k})\gamma_{k}}{1+\gamma_{k}}.\end{aligned}$
(14)

In $(P2)$, the objective variables that need to obtain are
$\boldsymbol{\alpha}$, $\mathbf{W}$ and
$\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L}$.
$\boldsymbol{\alpha}=[\alpha_{1},\alpha_{2},...,\alpha_{K}]^{T}$ is the
auxiliary variable vector, each element $\alpha_{k}$ in it is an auxiliary
variable that introduced for user $k$. By doing so, we decouple the sum-of-
log-of-ratio term $\sum_{k=1}^{K}\log(1+\gamma_{k})$ into the sum-of-log term
$\sum_{k=1}^{K}\log(1+\alpha_{k})$ and the sum-of-ratio term
$\sum_{k=1}^{K}\frac{(1+\alpha_{k})\gamma_{k}}{1+\gamma_{k}}$ as shown in
(14), which greatly reduces the difficulty in finding the optimal solutions.
In the following, we further decouple $(P2)$ into three disjoint subproblems,
and target to find the closed-form optimal solutions for each subproblem.

_1) The closed-form optimal solution for $\boldsymbol{\alpha}$._ First, we aim
to find the optimal closed-form expression of $\boldsymbol{\alpha}$ for given
$\mathbf{W}$ and $\boldsymbol{\phi}_{l}$, $l=1,2,...,L$. By assuming the other
parameters are fixed, the objective function in $(P2)$ can be seen as an
optimization problem only with respect to $\boldsymbol{\alpha}$, and we have
the following proposition regarding its solution.

######  Proposition 1.

When $\mathbf{W}$ and $\boldsymbol{\phi}_{1}$, …, $\boldsymbol{\phi}_{L}$ are
fixed, the optimal $\alpha_{k}$ can be expressed as

$\displaystyle\begin{aligned} \alpha_{k}^{op}=\gamma_{k}.\end{aligned}$ (15)

###### Proof:

The partial derivative of
$f_{2}(\boldsymbol{\alpha},\\!\mathbf{W},\\!\boldsymbol{\phi}_{1}\\!,...,\\!\boldsymbol{\phi}_{L})$
with respect to $\alpha_{k}$ can be written as

$\displaystyle\begin{aligned} \frac{\partial
f_{2}(\boldsymbol{\alpha},\mathbf{W},\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L})}{\partial\alpha_{k}}&=\frac{\gamma_{k}-\alpha_{k}}{(1+\gamma_{k})(1+\alpha_{k})}.\end{aligned}$
(16)

By setting $\frac{\partial
f_{2}(\boldsymbol{\alpha},\mathbf{W},\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L})}{\partial\alpha_{k}}=0$,
it is easy to find that $\alpha_{k}^{op}=\gamma_{k}$ maximizes
$f_{2}(\boldsymbol{\alpha},\mathbf{W},\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L})$,
since $\partial
f_{2}(\boldsymbol{\alpha},\mathbf{W},\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L})/\partial\alpha_{k}>0$
when $\alpha_{k}<\gamma_{k}$ and $\partial
f_{2}(\boldsymbol{\alpha},\mathbf{W},\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L})/\partial\alpha_{k}<0$
when $\alpha_{k}>\gamma_{k}$. ∎

Besides, by defining
$\boldsymbol{\alpha}^{op}=[\alpha_{1}^{op},\alpha_{2}^{op},\cdots,\alpha_{K}^{op}]^{T}$
and substituting $\alpha_{k}^{op}=\gamma_{k}$ into (14), we have

$\displaystyle\begin{aligned}
f_{2}(\boldsymbol{\alpha}^{op},\mathbf{W},\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L})\\!=\\!\sum_{k=1}^{K}\log(1+\gamma_{k})\\!=\\!f(\mathbf{W},\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L}),\end{aligned}$

which guarantees the equivalence between the original problem $(P1)$ and the
reformulated problem $(P2)$.

_2) The closed-form optimal solution for $\mathbf{W}$._ When
$\boldsymbol{\alpha}$ is fixed, the variable left for optimization in $(P2)$
is $\gamma_{k}$ only. Then the corresponding optimization problem can be
rewritten as

$\begin{split}(P2.1)\max_{\mathbf{W},\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L}}&\quad\sum_{k=1}^{K}\frac{\tilde{\alpha}_{k}\gamma_{k}}{1+\gamma_{k}},\\\
\text{s.t.}\quad&\quad|\phi_{l,n}|=1,~{}l=1,2,...,L,~{}n=1,2,...,N_{l},\\\
&\quad\sum_{k=1}^{K}||\bm{w}_{k}||^{2}\leq P_{T},\end{split}$ (17)

where $\tilde{\alpha}_{k}=1+\alpha_{k}$ for $k=1,2,...,K$. Substituting the
expression (LABEL:SINR_Ori) of SINR into (17), we can reformulate the
objective function in $(P2.1)$ as

$\displaystyle\begin{aligned}
\sum_{k=1}^{K}\frac{\tilde{\alpha}_{k}\gamma_{k}}{1+\gamma_{k}}=\sum_{k=1}^{K}\frac{\tilde{\alpha}_{k}|\mathbf{h}_{k}^{H}\bm{w}_{k}|^{2}}{\sum_{i=1}^{K}|\mathbf{h}_{k}^{H}\bm{w}_{i}|^{2}+\sigma_{0}^{2}}.\end{aligned}$
(18)

Then, with given $\boldsymbol{\alpha}$ and
$\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L}$, optimizing $\mathbf{W}$
becomes

$\displaystyle\begin{aligned} (P2.1a)\quad&\max_{\mathbf{W}}\quad
f_{2.1a}(\mathbf{W})\\!=\\!\sum_{k=1}^{K}\frac{\tilde{\alpha}_{k}|\mathbf{h}_{k}^{H}\bm{w}_{k}|^{2}}{\sum_{i=1}^{K}|\mathbf{h}_{k}^{H}\bm{w}_{i}|^{2}\\!+\\!\sigma_{0}^{2}},\\\
&~{}\text{s.t.}\quad\quad\sum_{k=1}^{K}||\bm{w}_{k}||^{2}\leq
P_{T}.\end{aligned}$ (19)

In (19), $(P2.1a)$ is a multiple-ratio fractional programming problem. Using
the quadratic transform proposed in [42, 43], $(P2.1a)$ can be reformulated as
a biconvex optimization problem with a new objective function as

$\displaystyle\begin{aligned}
f_{2.1b}(\mathbf{W},\boldsymbol{\xi})=&\sum_{k=1}^{K}\sqrt{\tilde{\alpha}_{k}}\left(\xi_{k}^{*}\mathbf{h}_{k}^{H}\bm{w}_{k}+\bm{w}_{k}^{H}\mathbf{h}_{k}\xi_{k}\right)\\\
&-\sum_{k=1}^{K}|\xi_{k}|^{2}\left(\sum_{i=1}^{K}|\mathbf{h}_{k}^{H}\bm{w}_{i}|^{2}+\sigma_{0}^{2}\right),\end{aligned}$
(20)

where $\xi_{1},\xi_{2},...,\xi_{K}$ are the newly introduced complex auxiliary
variables and $\boldsymbol{\xi}=[\xi_{1},\xi_{2},...,\xi_{K}]^{T}$ is the
auxiliary variable vector. Then, as proved in [42] and [43], solving $(P3.1a)$
with respect to $\mathbf{W}$ is equivalent to solving the following biconvex
problem with respect to $\mathbf{W}$ and $\boldsymbol{\xi}$, which can be
further represented as

$\displaystyle\begin{aligned}
(P2.1b)\quad&\max_{\mathbf{W},\boldsymbol{\xi}}\quad
f_{2.1b}(\mathbf{W},\boldsymbol{\xi}),\\\
&~{}\text{s.t.}\quad\sum_{k=1}^{K}||\bm{w}_{k}||^{2}\leq P_{T}.\end{aligned}$
(21)

By analyzing $(P2.1b)$, we have the following proposition regarding to its
solution.

######  Proposition 2.

The optimal beamforming vector $\bm{w}_{k}$ can be updated by

$\displaystyle\begin{aligned}
\bm{w}_{k}^{op}=\sqrt{\tilde{\alpha}_{k}}\xi_{k}^{op}\left(\sum_{i=1}^{K}|\xi_{i}^{op}|^{2}\mathbf{h}_{i}\mathbf{h}_{i}^{H}+\lambda^{op}\mathbf{I}_{M}\right)^{-1}\mathbf{h}_{k},\end{aligned}$
(22)

where

$\displaystyle\begin{aligned}
\xi_{k}^{op}=\frac{\sqrt{\tilde{\alpha}_{k}}\mathbf{h}_{k}^{H}\bm{w}_{k}}{\sum_{i=1}^{K}|\mathbf{h}_{k}^{H}\bm{w}_{i}|^{2}+\sigma_{0}^{2}}.\end{aligned}$
(23)

In (22), $\lambda^{op}$ is the Lagrangian dual variable introduced for the
power constraint $\sum_{k=1}^{K}||\bm{w}_{k}^{op}||^{2}=P_{T}$, which can be
further expressed as

$\displaystyle\begin{aligned} \lambda^{op}=\min\left\\{\lambda^{op}\geq
0:\sum_{k=1}^{K}||\bm{w}_{k}^{op}||^{2}=P_{T}\right\\}.\end{aligned}$ (24)

###### Proof:

See Appendix A. ∎

_3) The closed-form optimal solution for phase shift vectors
$\boldsymbol{\phi}_{1},\boldsymbol{\phi}_{2},...,\boldsymbol{\phi}_{L}$._ For
the optimized $\boldsymbol{\alpha}$, the objective function in $(P2)$ is
reduced to $(P2.1)$ in (17) as discussed above. By assuming that the
beamforming matrix $\mathbf{W}$ at the BS side is fixed, we will optimize the
reflection vector of each RIS iteratively, i.e, optimizing the reflection
vector $\boldsymbol{\phi}_{l}$, $l\in\\{1,2,..,L\\}$, of RIS $l$ by assuming
the rest are fixed. In the original expression of the equivalent channel for
user $k$ in (7), we can see that the phase shift matrix of each RIS is coupled
together due to the multi-hop transmission, which makes it impossible to solve
them out. In order to overcome this, we perform channel reformulation as shown
in the following, with which we can extract the phase shift matrix of each RIS
as an individually objective variable. This will help us derive out the
closed-form expression and thus they can be updated one by one with quite low
complexity. Specifically, the reformulated channel model is represented as

$\mathbf{h}_{k}^{H}=\bar{\boldsymbol{\phi}_{l}}\mathbf{A}_{l}+\mathbf{b}_{l}.$
(25)

In (25), $\bar{\boldsymbol{\phi}_{l}}$ is a row vector that is composed of
$L-l+1$ reflecting vectors $\boldsymbol{\phi}_{l}^{H}$ of RIS $l$, i.e.,
$\bar{\boldsymbol{\phi}_{l}}=[\boldsymbol{\phi}_{l}^{H},...,\boldsymbol{\phi}_{l}^{H}]\in\mathbb{C}^{1\times(L-l+1)N_{l}}$.
The matrix $\mathbf{A}_{l}\in\mathbb{C}^{(L-l+1)N_{l}\times M}$ is represented
as

$\begin{split}\mathbf{A}_{l}=\left[\mathop{\mathrm{diag}}(\mathbf{g}_{l,k})\mathbf{R}_{l},\mathop{\mathrm{diag}}(\mathbf{u}_{l+1})\mathbf{R}_{l},...,\mathop{\mathrm{diag}}(\mathbf{u}_{L})\mathbf{R}_{l}\right]^{T},\end{split}$
(26)

where

$\mathbf{R}_{l}=\left\\{\begin{aligned}
&\quad\mathbf{G}_{1},\quad\quad\quad\quad\quad\quad\quad\quad l=1,\\\
&\\!\\!\prod_{i=l,l-1,...,2}\\!\\!\\!\\!\\!\\!\\!\mathbf{H}_{i-1,i}\mathbf{\Phi}_{i-1}\mathbf{G}_{1},\quad
l=2,...,L,\end{aligned}\right.$ (27)

and

$\mathbf{u}_{i}=\mathbf{g}_{i,k}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\prod_{j=i,i-1,...,l+1}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathbf{\Phi}_{j}\mathbf{H}_{j-1,j},\quad\quad
i=l+1,l+2,...,L.$ (28)

In (25), $\mathbf{b}_{l}\in\mathbb{C}^{1\times M}$ is represented as

$\mathbf{b}_{l}=\left\\{\begin{aligned}
\\!\\!\mathbf{0},\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad&l=1,\\\
\\!\\!\\!\mathbf{g}_{1,k}\mathbf{\Phi}_{1}\mathbf{G}_{1},\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad&l=2,\\\
\\!\\!\mathbf{g}_{1,k}\mathbf{\Phi}_{1}\mathbf{G}_{1}\\!\\!+\\!\\!\sum_{i=2}^{l-1}\mathbf{g}_{i,k}\\!\\!\\!\\!\\!\\!\\!\\!\\!\prod_{j=i,i-1,...,2}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathbf{\Phi}_{j}\mathbf{H}_{j-1,j}\mathbf{\Phi}_{1}\mathbf{G}_{1},\quad&l=3,...,L.\end{aligned}\right.$
(29)

With the reformulated $\bm{h}_{k}^{H}$, we can rewrite the objective function
in (17) with respect to $\boldsymbol{\phi}_{l}$ as

$\begin{split}f_{3.1a}(\boldsymbol{\phi}_{l})&=\sum_{k=1}^{K}\frac{\tilde{\alpha}_{k}|(\bar{\boldsymbol{\phi}_{l}}\mathbf{A}_{l}+\bm{b}_{l})\bm{w}_{k}|^{2}}{\sum_{i=1}^{K}|(\bar{\boldsymbol{\phi}_{l}}\mathbf{A}_{l}+\bm{b}_{l})\bm{w}_{i}|^{2}+\sigma_{0}^{2}}\\\
&=\sum_{k=1}^{K}\frac{\tilde{\alpha}_{k}|\bar{\boldsymbol{\phi}_{l}}\bm{z}_{l,k}+c_{l,k}|^{2}}{\sum_{i=1}^{K}|\bar{\boldsymbol{\phi}_{l}}\bm{z}_{l,i}+c_{l,i}|^{2}+\sigma_{0}^{2}}.\end{split}$
(30)

In (30), the vector $\bm{z}_{l,i}\in\mathbb{C}^{(L-l+1)N_{l}\times
1}=\mathbf{A}_{l}\bm{w}_{i}$ and the complex number
$c_{l,i}=\bm{b}_{l}\bm{w}_{i}$. According to the quadratic transform proposed
in [42], we further transform (30) into

$\begin{split}f_{3.2b}(\boldsymbol{\phi}_{l},\boldsymbol{\varepsilon})\\!\\!=\\!\\!&\sum_{k=1}^{K}\sqrt{\tilde{\alpha}_{k}}\left(\varepsilon_{k}^{*}(\bar{\boldsymbol{\phi}_{l}}\bm{z}_{l,k}+c_{l,k})\\!+\\!\\!(\bm{z}_{l,k}^{H}\bar{\boldsymbol{\phi}_{l}}^{H}\\!+\\!c_{l,k}^{*})\varepsilon_{k}\right)\\\
&-\sum_{k=1}^{K}|\varepsilon_{k}|^{2}\left(\sum_{i=1}^{K}|\bar{\boldsymbol{\phi}_{l}}\bm{z}_{l,i}+c_{l,i}|^{2}+\sigma_{0}^{2}\right),\end{split}$
(31)

where $\varepsilon_{1},\varepsilon_{2},...,\varepsilon_{K}$ are the newly
introduced complex auxiliary variables and
$\boldsymbol{\varepsilon}=[\varepsilon_{1},\varepsilon_{2},...,\varepsilon_{K}]^{T}$
is the auxiliary variable vector. Then, as proved in [42] and [43], optimizing
$f_{3.2b}(\boldsymbol{\phi}_{l},\boldsymbol{\varepsilon})$ with respect to
$\boldsymbol{\phi}_{l}$, $l=1,2,...L$, is equivalent to solving the following
problem with respect to $\boldsymbol{\phi}_{l}$ and
$\boldsymbol{\varepsilon}$, and we have the following objective function

$\displaystyle\begin{aligned}
(P3.2b)\quad&\max_{\boldsymbol{\phi}_{l},\boldsymbol{\varepsilon}}\quad
f_{3.2b}(\boldsymbol{\phi}_{l},\boldsymbol{\varepsilon}),\\\
&~{}\text{s.t.}\quad\quad|\phi_{l,n}|=1,\quad n=1,2,...,N_{l}.\end{aligned}$
(32)

By analyzing $(P3.2b)$, we have the following conclusion regarding to the
optimal solution for $\bm{\phi}_{l}$.

######  Proposition 3.

The optimal expression $\phi_{l,n},n=1,2,...,N_{l}$, of the reflecting vector
$\boldsymbol{\phi}_{l}$, $l=1,2,...,L$, is updated by

$\displaystyle\begin{aligned} \phi_{l,n}^{op}=e^{-j\angle\eta_{n}},\quad
n=1,2,...,N_{l},\end{aligned}$ (33)

where

$\displaystyle\begin{aligned} \eta_{n}=d_{l,n}-\sum_{j=1,j\neq
n}^{(L-l+1)N_{l}}p_{n,j}\bar{\phi}^{*}_{l,j},\end{aligned}$ (34)

with $d_{l,n}$ being the $n$-th element of vector

$\displaystyle\begin{aligned}
\bm{d}_{l}=\sum_{k=1}^{K}\sqrt{\tilde{\alpha}_{k}}(\varepsilon_{k}^{op})^{*}\bm{z}_{l,k}-\sum_{k=1}^{K}|\varepsilon_{k}^{op}|^{2}\sum_{i=1}^{K}\bm{z}_{l,i}c_{l,i}^{*}.\end{aligned}$
(35)

Among the denotations, $p_{n,j}$ represents the element at row $n$ and column
$j$ of matrix
$\mathbf{P}=\sum_{k=1}^{K}|\varepsilon_{k}^{op}|^{2}\sum_{i=1}^{K}\bm{z}_{l,i}\bm{z}_{l,i}^{H}$,
and $\bar{\phi}_{l,j}$ represents the $j$-th element of the vector
$\bar{\boldsymbol{\phi}_{l}}$. Finally, $\varepsilon_{k}^{op}$ is proved to be

$\displaystyle\begin{aligned}
\varepsilon_{k}^{op}=\frac{\sqrt{\tilde{\alpha}_{k}}(\bar{\boldsymbol{\phi}_{l}}\bm{z}_{l,k}+c_{l,k})}{\sum_{i=1}^{K}|\bar{\boldsymbol{\phi}_{l}}\bm{z}_{l,i}+c_{l,i}|^{2}+\sigma_{0}^{2}}.\end{aligned}$
(36)

###### Proof:

See Appendix B. ∎

Algorithm 1 :Cooperative beamforming design for multi-RIS-assisted multi-user
systems.

1: Initialization: Initialize
$\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L}$ that satisfy (11).
Initialize $\mathbf{W}$ that satisfy (12). Set the threshold value $\ell$ and
the maximum number of iterations $I_{m}$. Iteration time $i=1$.

2: Step 1: Update $\alpha_{k}$ according to (15).

3: Step 2: Update $\bm{w}_{k}$, $\xi_{k}$ and $\lambda$ according to (22),
(23) and (24), accordingly.

4: Step 3: Update $\phi_{l,n}$, $l=1,2,...L$, $n=1,2,...,N_{l}$, and
$\varepsilon_{k}$ according to (33) and (36), respectively.

5: $i=i+1$.UNTILThe objective function in (14) satisfies
$f_{2}(\boldsymbol{\alpha}^{i},\mathbf{W}^{i},\boldsymbol{\phi}_{1}^{i},...,\boldsymbol{\phi}_{L}^{i})-f_{2}(\boldsymbol{\alpha}^{i-1},\mathbf{W}^{i-1},\boldsymbol{\phi}_{1}^{i-1},...,\boldsymbol{\phi}_{L}^{i-1})\leq\ell$
or the iteration time $i=I_{m}$.

Following the above approach, the phase shift of each RIS can be updated
iteratively until the objective function converges. The overall algorithm to
optimize the multi-RIS-assisted multi-user communication system is summarized
in Algorithm 1, where we optimize the active beamforming matrix $\mathbf{W}$
and the phase shift vectors $\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L}$
iteratively to maximize the sum rate. Algorithm 1 is guaranteed to converge
(c.f. Proposition 4) and its computational complexity mainly comes from the
process of optimizing the active beamforming and updating the phase shifts. In
the cooperative beamforming design processes, we first update $\alpha_{k}$,
$k=1,2,...,K$, according to (15), whose computational complexity is
$\mathcal{O}(K)$. To update the active beamforming matrix
$\mathbf{W}=[\bm{w}_{1},\bm{w}_{2},...,\bm{w}_{K}]$ at the BS side, we derive
the closed-form optimal solution for $k$-th user’s beamforming vector
$\bm{w}_{k}$ as shown in (22) with the required auxiliary variable
$\xi_{k}^{op}$ shown in (23). The complexity for updating $\xi_{k}^{op}$ is
$\mathcal{O}(K)$ and the complexity for calculating $\mathbf{W}$ is
$\mathcal{O}(M^{3}K)$, thereby the overall computational complexity for
updating $\mathbf{W}$ is $\mathcal{O}(M^{3}K+K)$. Finally, for the phase shift
design at each RIS, we iteratively update each RIS’s phase shift vector
$\boldsymbol{\phi}_{l}$. The optimal expression $\phi_{l,n},n=1,2,...,N_{l}$,
of the reflecting vector $\boldsymbol{\phi}_{l}$, $l=1,2,...,L$, is updated
according to (33) with an auxiliary variable $\varepsilon_{k}^{op}$ calculated
according to (36). The complexity for updating $\varepsilon_{k}^{op}$,
$k=1,2,...,K$, is $\mathcal{O}(K)$. And we iteratively update each RIS’s phase
shift vector $\boldsymbol{\phi}_{l}$ according (33) and its computational
complexity is $\mathcal{O}(N_{l})$. Thus, the aggregated complexity for
getting $\boldsymbol{\phi}_{l}$ is $\mathcal{O}\left(N_{l}+K\right)$. Since we
deploy $L$ distributed RISs in the system, the overall complexity for updating
RISs’ phase shift is represented as $\mathcal{O}(KL+\sum_{l=1}^{L}N_{l})$.
Summarizing all above, the complexity of the proposed algorithm can be
expressed as
$\mathcal{O}\left(\left((M^{3}+L+2)K+\sum_{l=1}^{L}N_{l}\right)I_{1}\right)$,
with $I_{1}$ denoting the required number of outer iterations.

######  Proposition 4.

Algorithm 1 is guaranteed to converge.

###### Proof:

See Appendix C. ∎

##  IV Performance Evaluation

In this section, performance evaluations are done to show the effectiveness of
the proposed algorithm. We first investigate the convergence property of the
proposed cooperative beamforming algorithm, and then simulate its performance
in terms of the complexity and the sum rate. For comparison purpose, we also
adopt some existing cooperative beamforming algorithms for baselines. We
investigate the impacts of the number of the reflecting elements and the
channel estimation error on the sum rate. In our simulations, we assume that
there are $L$ RISs (RIS 1, RIS 2,…, RIS $L$) deployed to assist the downlink
communications, and RIS $l$ has $N_{l}$ reflecting elements. Since the RIS-
reflected channel follows the product-distance path loss model, thereby we
always deploy RIS 1 near the BS and RIS $L$ near the users to produce smaller
multiplicative poss loss. Other RISs are deployed sequentially between RIS 1
and RIS $L$ to form multi-hop transmission links. Under the 3D Cartesian
coordinate, the BS is deployed at
$(0\,\mathrm{m},0\,\mathrm{m},0\,\mathrm{m})$, the deployment location of RIS
$l$, $l=1,2,...,L$, is represented by
$(x_{l}\,\mathrm{m},y_{l}\,\mathrm{m},z_{l}\,\mathrm{m})$. The users are
located within a target circular area $\mathcal{A}$, where $\mathcal{A}$ is
centered at $A=(D\,\mathrm{m},0\,\mathrm{m},0\,\mathrm{m})$ with the radius of
$r$ $\mathrm{m}$. At the BS and user sides, we consider the widely used half-
wavelength dipole antennas with corresponding maximum effective area
$0.13\lambda^{2}$ [44], and the size of each RIS element is
$\lambda/5\times\lambda/5$ [45]. By considering the physical area of the RIS
element as the effective area, we can calculate the corresponding free-space
path loss of each channel according to Friis transmission formula [46]. Other
system parameters are summarized in Table I according to [37]. All presented
results are obtained by averaging $1000$ independent realizations of the
users’ locations and the wireless channels.

TABLE I: Simulation Setups The central location $A$ of the user cluster |  $(10\,\mathrm{m},0\,\mathrm{m},0\,\mathrm{m})$
---|---
The radius $r$ of the target area |  $8$ $\mathrm{m}$
The transmit power $P_{T}$ at the BS |  10 dBW
The Rician factor for all the channels |  3
The noise power $\sigma_{0}^{2}$ |  -110 dBm
The carrier frequency |  2.4 GHz

###  IV-A The Convergence Property of the Proposed Cooperative Beamforming
Algorithm

Figure 3: The convergence property of the proposed beamforming algorithm.
There are $L=2$ RISs (RIS 1 and RIS 2). RIS 1 is deployed at
$(1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$ and RIS 2 is deployed at
$(9\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$.
Figure 4: The convergence property of the proposed beamforming algorithm.
There are $L=3$ RISs (RIS 1, RIS 2 and RIS 3). RIS 1 is deployed at
$(1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$, RIS 2 is deployed at
$(5\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$ and RIS 3 is deployed at
$(9\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$.

The convergence property of the proposed cooperative beamforming algorithm is
first investigated under various system settings. The results are illustrated
in Figs. 3 and 4. Specifically, Fig. 3 shows the convergence property of the
proposed beamforming algorithm for $L=2$ RISs (RIS 1 and RIS 2), where RIS 1
is deployed at $(1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$ and RIS 2 is
deployed at $(9\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$. In the simulations,
the numbers of the reflecting elements at these two RISs are set to be equal,
i.e., $N_{1}=N_{2}$. It can be observed in Fig. 3 that the proposed
beamforming algorithm converges within several iterations in cases with
$N_{1}=N_{2}=9$, $N_{1}=N_{2}=16$ and $N_{1}=N_{2}=25$, respectively. This
verifies Proposition 4 and the fast convergence implies a relatively low
computational complexity of the proposed cooperative beamforming algorithm.

Fig. 4 shows the convergence behavior of the proposed beamforming algorithm
for $L=3$ RISs, where RIS 1 is deployed at
$(1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$, RIS 2 is deployed at
$(5\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$ and RIS 3 is deployed at
$(9\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$. In the simulations, the numbers
of the reflecting elements at these three RISs are also set to be the same,
i.e., $N_{1}=N_{2}=N_{3}$. Similarly, for the 3 RISs case, the proposed
algorithm also converges within several iterations. Comparing with Fig. 3, we
observe that although the number of the deployed RISs increases, our proposed
algorithm can still converge quickly. This implies that the proposed
beamforming algorithm does have a good expandability.

###  IV-B Comparisons with the Existing Cooperative Beamforming Algorithms

For comparison, we choose the semidefinite relaxation (SDR)-based and
alternating direction method of multipliers (ADMM)-based algorithms as
benchmarks. We choose these two algorithms because they are two different
methods handling non-convex rank-one constraint optimization problems, which
is also the constraint in our design. The SDR-based method can directly relax
the rank-one constraint, while ADMM-based algorithm transfers the non-convex
constraint to the objective function by adding a penalty factor. Specifically,

  * •

The SDR-based algorithm [47] can relax the non-convex rank-one constraint of
the RIS’s phase shifts, and transform the original objective function into a
convex semidefinite program (SDP) to make such problems easily be solved
through existing convex optimization methods.

  * •

The ADMM-based algorithm [48] solves such problem by introducing an auxiliary
vector $\bm{q}_{l}$, then adding a penalty term into the objective function.
By doing so, the objective function can be reformulated into Lagrange dual
problem and solved by gradient descent method.

The computational complexity of these algorithms are listed in Table II. We
can observe from Table II that the complexity of the proposed algorithm
linearly increases with the number of the RIS’s reflecting elements, which
indicates that the proposed algorithm can be easily extended to systems with
more RISs. On the contrary, the SDR- and ADMM-based algorithms cannot be
directly applied to systems with large-scale RISs because of the prohibitively
high computational complexity. For comparsion purpose, we show the convergence
property of these algorithms in Fig. 5. It is observed that the convergence
speed of the proposed algorithm is obviously faster than the SDR- and ADMM-
based algorithms. The reason is that in each iteration, our proposed algorithm
is updated with the closed-form optimal solutions, which not only improves the
sum rate, but also reduces the computational complexity.

TABLE II: Computational Complexity for Different Cooperative Beamforming
Algorithms

Algorithm |  Computational complexity for $L=2$ RISs
---|---
Proposed algorithm |  $\mathcal{O}\left((M^{3}K+N_{1}+N_{2}+4K)I_{1}\right)$
ADMM-based algorithm |  $\mathcal{O}\left((M^{3}K+N_{1}^{3}+N_{2}^{3}+2K)I_{2}\right)$
SDR-based algorithm |  $\mathcal{O}\left((M^{3}K+N_{1}^{6}+N_{2}^{6}+2K)I_{3}\right)$

  * *

$I_{n}$ with $n=1,2,3$ is the number of outer iterations when the algorithm
converges.

Figure 5: The convergence properties of different algorithms.There are $L=2$
RISs (RIS 1 and RIS 2). RIS 1 is deployed at
$(1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$ and RIS 2 is deployed at
$(9\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$. $M=10$ and $K=3$.
Figure 6: The sum rate comparison for different SNR values. There are $L=2$
RISs (RIS 1 and RIS 2).
RIS 1 is deployed at $(1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$ and RIS 2
is deployed at $(9\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$. $M=10$ and
$K=3$.

Fig. 6 shows the sum rate performance of these cooperative beamforming
algorithms for different SNR values. It is observed that as the SNR increases,
the proposed algorithm can always achieve the highest sum rate. This is owing
to that the closed-form optimal solutions of the individual variables are
obtained in each iteration. Based on the above observations in Table II, Fig.
5 and Fig. 6, we claim that the proposed algorithm can provide higher sum rate
with much lower computational complexity.

Figure 7: The sum rate performance with different numbers of quantization bits
$b$. There are $L=2$ RISs (RIS 1 and RIS 2).
RIS 1 is deployed at $(1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$ and RIS 2
is deployed at $(9\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$,
$N_{1}=N_{2}=25$, $M=10$ and $K=3$.

Considering the deployment cost of RIS, is it possible to consider discrete
values for the RIS’ phase shifts in the practical implementations. In discrete
phase case, we refer to $b$ as the number of quantization bits and $B=2^{b}$
as the discrete phase shift level. Thus the set of discrete phase-shift values
at each element is given by
$\mathcal{F}=\\{0,\Delta\theta,...,(B-1)\Delta\theta\\},$ where
$\Delta\theta=2\pi/B$. To investigate the impact of the discrete phase shift
level on the sum rate performance, we depicts the sum rate of all users versus
the number of quantization bits $b$ in Fig. 7. As the number of quantization
bits increases, the sum rate obtained with discrete phase shifts approaches
that in the continuous case and our proposed algorithm can always achieve the
best sum rate performance compared with the benchmark schemes.

###  IV-C Impacts of the Number of Reflecting Elements and the Channel
Estimation Errors

Figure 8: The sum rate with different reflecting element numbers for $L=3$
RISs (RIS 1, RIS 2 and RIS 3).
RIS 1 is deployed at $(1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$, RIS 2 is
deployed at $(5\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$ and RIS 3 is
deployed at $(9\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$.

The performance of the RIS-assisted communication systems greatly depends on
the number of the reflecting elements. To investigate the dependency between
them, we simulate the sum rate performance of the proposed algorithm for
different numbers of the RISs’ reflecting elements. In the simulations, $L=3$
RISs (RIS 1, RIS 2 and RIS 3) are deployed, and these three RISs have the same
number of reflecting elements. The SDR- and ADMM-based cooperative beamforming
algorithms are also simulated for comparison. For $M=10$, $K=3$, the
simulation results are shown in Fig. 8. It is observed that as the number of
the RISs’ reflecting elements increases, the sum rates of all the three
algorithms increase and the proposed algorithm always achieves the highest sum
rate compared with the chosen baseline algorithms. We can also observe from
Fig. 8 that the improvement of the sum rate is limited by the inter-user
interference. Since the RISs do not have any signal processing ability and
they reflect the overall aggregated signals, thereby as the reflecting element
number continuously increases, the sum rate performance tends to converge to
ceiling.

For the beamforming algorithms, imperfect CSI will affect the final sum rate
performance. To show how the channel estimation ambiguity impacts the system
performance when applying our proposed algorithm, we conduct simulations to
evaluate the sum rate under different channel estimation errors. Specifically,
we model all the estimated channel vectors or matrices, i.e., BS-RIS 1
channel, inter-RIS channels and RIS-user channels, as
$\hat{\bm{r}}=\bm{r}+\bm{r}_{e}$ or
$\hat{\bm{\Upsilon}}=\bm{\Upsilon}+\bm{\Upsilon}_{e}$ with $\bm{r}$ and
$\bm{\Upsilon}$ representing the actual channel, $\bm{r}_{e}$ and
$\bm{\Upsilon}_{e}$ denoting the channel estimation error. Each element in
$\bm{r}_{e}$ and $\bm{\Upsilon}_{e}$ is chosen from
$\mathcal{CN}(0,\sigma_{e}^{2})$. And $\sigma_{e}^{2}$ can reflect the level
of the channel estimation ambiguity. when $\sigma_{e}^{2}$ is bigger, it
indicates that the channel estimation error is larger, and vice versa
indicates that the channel estimation error is small. It is observed from Fig.
9 that as the channel estimation error increases, the sum rate obtained by all
the three algorithms decrease and our proposed algorithm always achieves the
best sum rate. It is also shown that when the channel estimation error is
small, e.g., $\sigma_{e}^{2}=10^{-4}$, the sum rate achieved by our proposed
algorithm is even better than that of the ADMM- and SDR-based algorithms with
perfect CSI.

###  IV-D Comparisons with Existing Design Strategies for Equal Total
Reflecting Elements

Figure 9: The sum rate comparison for different channel estimation errors.
There are $L=2$ RISs (RIS 1 and RIS 2).
RIS 1 is deployed at $(1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$ and RIS 2
is deployed at $(9\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$.
$N_{1}=N_{2}=16$, $M=10$ and $K=3$.
Figure 10: The sum rate comparison for the proposed channel model and existing
baseline schemes with equal total reflecting elements. In double-RIS systems,
RIS 1 is deployed at $(1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$, RIS 2 is
deployed at $(9\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$. In single-RIS
systems, the RIS is deploying at three different locations, i.e.,
$(1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$,
$(5\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$ and
$(9\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$.

To show how much sum rate gains can be obtained by the proposed transmission
strategy, we also compare the sum rate for double- and single-RIS-assisted
systems with equal total reflecting elements, the results are shown in Fig.
10. Specifically, for the double-RIS systems, we compare the proposed multi-
reflection transmission model with the baseline scheme that only studies the
single reflection links, i.e., only BS-RIS 1-user links and BS-RIS 2-user
links. And in these two double-RIS scenarios, we assume RIS 1 has $N_{1}$
reflecting elements and deployed at
$(1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$, RIS 2 has $N_{2}$ reflecting
elements and deployed at $(9\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$. We
also compare the proposed cooperative transmission strategy with single-RIS
scheme for equal total reflecting elements. In single-RIS scenario, the RIS
has $N=N_{1}+N_{2}$ reflecting elements and for thorough comparison, the
performance is simulated for the $N$ elements in different locations,
$(1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$,
$(5\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$ and
$(9\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$. We can observe that it is
beneficial to adopt the proposed cooperative transmission scheme in most
cases, since the multi-reflection link established through RIS 1 and RIS 2 can
bring significant multiplicative beamforming gains compared with other
benchmark schemes. We also observe that the number of the reflecting elements
at RIS 1 and RIS 2 have great effect on the sum rate, and the peak sum rate is
obtained when $N_{1}=40$ and $N_{2}=20$ for the simulated scenario. In
particular, for the baseline scheme that only considers the single-reflection
links of these two distributed RISs, due to the smaller number of reflecting
elements at each RIS and the lack of multiplicative beamforming gains, its
performance is even worse than single-RIS schemes.

Figure 11: The sum rate comparison for the proposed channel model and existing
baseline schemes when $D$ increasing. In double-RIS scenario, RIS 1 is
deployed at $(1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$, RIS 2 is deployed
at $(D-1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$. In single-RIS scenario,
the RIS is deployed at three different locations, i.e.,
$(1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$,
$(D/2\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$ and
$(D-1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$.

To further verify the superiority of the proposed multi-RIS cooperative
transmission scheme, we study the sum rate performance when the distance $D$
between the center of the user cluster and the BS varies. In Fig. 11, we show
the sum rate achieved by double- and single-RIS-assisted systems. In double-
RIS scenarios, we also compare the proposed multi-reflection transmission
model with the baseline scheme that only studies the single reflection links,
i.e., only BS-RIS 1-user links and BS-RIS 2-user links. And for these two
schemes, we use the same systems setup, i.e., RIS 1 has $N_{1}$ reflecting
elements and is deployed at $(1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$, RIS
2 has $N_{2}$ reflecting elements and is deployed at
$(D-1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$. For the comparative single-
RIS case, we consider equal total reflecting elements, the RIS has
$N=N_{1}+N_{2}$ reflecting elements, and again the system is simulated when
the RIS is depoyed at different locations, i.e.,
$(1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$,
$(D/2\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$ and
$(D-1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$. It can be observed from Fig.
11 that as $D$ increases, the sum rate of both the single- and double-RIS
assisted systems decrease due to the increasing transmit distance. However,
the proposed cooperative multi-RIS transmission scheme can always achieve
higher sum rate with the help of the multiplicative beamforming gains provided
by the multi-reflection links.

###  IV-E The Impact of the Number of Distributed RISs and the Deployment
Location on Sum Rate

Figure 12: The sum rate performance under different RISs’ deployment
locations. In double-RIS scenario, RIS 1 is deployed at
$(x_{1}\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$ while RIS 2 is deployed at
$\left((20-x_{1})\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m}\right)$. In single-
RIS scenario, the RIS is deployed at
$(2\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$,
$(10\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$ and
$(18\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$, respectively.

To gain some useful insights for the optimal deployment locations of the RISs,
we simulate the sum rate performance under different RISs’ deployment
locations in Fig. 12 with $L=2$ distributed RISs. In doing so, we consider a
double-RIS setup. At the beginning, RIS $1$ and RIS $2$ are closed to the base
station and the user cluster, respectively, then they gradually move closer to
each other. The results in Fig. 12 show that the placement of the RISs have
great influences on the sum rate performance. For the double-RIS scenario, the
best placement for RIS 1 and RIS 2 are the points of
$(2\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$ and
$(18\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$, since the RIS-reflected
channel follows the product-distance path loss model, it is better to make
some channels have longer transmission distance while some channels have
shorter transmission distance. As RIS 1 and RIS 2 are getting closer to each
other, the sum rate performance first decreases then gradually increases. This
is because when the two RISs getting closer, the transmission distance between
the BS and RIS 1 becomes longer, resulting in heavier channel pass loss, which
severely restricts the sum rate performance. However, when RIS 1 and RIS 2 are
getting even closer, the inter-path between these two RISs can provide
considerable multiplicative beamforming gain, thereby improves the sum rate.
For single-RIS case, when the RIS is deployed close to the base station, the
sum rate performance will be better, but when the RIS is deployed in the
middle between the BS and the user cluster, the sum rate will be worse due to
the heavy product-distance path loss.

Figure 13: The sum rate performance under different distributed-RIS number $L$
with fixed total available reflecting elements $N=60$, each RIS has
$N_{l}=N/L$ reflecting elements.

To investigate the impact of the distributed-RIS number on the sum rate
performance, we simulated the sum rate performance under different
distributed-RIS numbers in Fig. 13. Specifically, with fixed total available
reflecting elements $N=60$, we change the number $L$ of the distributed RISs,
and each RIS has $N_{l}=N/L$ reflecting elements. For L-RIS scenario, when
$L=1$, the RIS is deployed at $(1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$;
when $L=2$, these two RISs are deployed at
$(1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$ and
$(9\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$, respectively, and each RIS has
$N_{1}=N_{2}=30$ reflecting elements. When $L=3$, RIS 1, RIS 2 and RIS 3 are
deployed at $(1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$,
$(5\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$ and
$(9\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$, each RIS has
$N_{1}=N_{2}=N_{3}=20$ reflecting elements. When $L=4$, these four RISs are
deployed at $(1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$,
$(3\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$,
$(6\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$ and
$(9\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$, each RIS has
$N_{1}=N_{2}=N_{3}=N_{4}=15$ reflecting elements. When $L=5$, these RISs are
deployed at $(1\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$,
$(3\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$,
$(5\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$,
$(7\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$ and
$(9\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$, each RIS has
$N_{1}=N_{2}=N_{3}=N_{4}=N_{5}=12$ reflecting elements. When $L=6$, these RISs
are deployed at $(0\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$,
$(2\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$,
$(4\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$,
$(6\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$,
$(8\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$ and
$(10\,\mathrm{m},0\,\mathrm{m},3\,\mathrm{m})$, each RIS has
$N_{1}=N_{2}=N_{3}=N_{4}=N_{5}=N_{6}=10$ reflecting elements. From Fig. 13, we
observe that when the total number of the available reflecting elements is
fixed, the number of distributed RISs has a significant impact on sum rate
performance. When $L$ increasing, the sum rate first increases due to the
multiplicative beamforming gains obtained through multi-reflection channels,
and then decreases. This is because with fixed total available reflecting
elements, the continuous growth of $L$ will lead to a decrease in the number
of available reflecting elements per RIS, resulting in a smaller passive
beamforming gain at each RIS and finally a decrease of the sum rate
performance.

##  V Conclusions

In this paper, we have investigated the cooperative beamforming design for
multi-RIS-assisted multi-user communication systems. We have formulated a
downlink sum rate optimization problem for arbitrary reflection links. The
formulated problem is diffcult to solve due to the non-convexity and the
interactions among the decision variables, we have decoupled the original
problem into three disjoint subproblems. By introducing appropriate auxiliary
variables in each subproblem, we have derived the closed-form expressions for
the optimization decision variables. Based on these closed-form expressions,
we have designed a low-complexity cooperative beamformig algorithm. We have
carried out extensive simulation studies and shown that the proposed algorithm
has linear complexity with the number of RISs’ reflecting elements. We have
also verified that the proposed algorithm achieves higher sum rate with lower
complexity compared with other commonly used alternative approaches. The
results have also unveiled that appropriately distributing the reflecting
elements among multiple RISs performs better than deploying them at one single
RIS in terms of achievable sum rate.

##  Appendix A Proof of Proposition 2

The optimization problem is

$\displaystyle\begin{aligned}
(P2.1b)\quad&\max_{\mathbf{W},\boldsymbol{\xi}}\quad
f_{2.1b}(\mathbf{W},\boldsymbol{\xi}),\\\
&~{}\text{s.t.}\quad\sum_{k=1}^{K}||\bm{w}_{k}||^{2}\leq P_{T}.\end{aligned}$
(37)

with

$\displaystyle\begin{aligned}
f_{2.1b}(\mathbf{W},\boldsymbol{\xi})=&\sum_{k=1}^{K}\sqrt{\tilde{\alpha}_{k}}\left(\xi_{k}^{*}\mathbf{h}_{k}^{H}\bm{w}_{k}+\bm{w}_{k}^{H}\mathbf{h}_{k}\xi_{k}\right)\\\
&-\sum_{k=1}^{K}|\xi_{k}|^{2}\left(\sum_{i=1}^{K}|\mathbf{h}_{k}^{H}\bm{w}_{i}|^{2}+\sigma_{0}^{2}\right).\end{aligned}$
(38)

First, when $\mathbf{W}$ is fixed, the partial gradient of
$f_{2.1b}(\mathbf{W},\boldsymbol{\xi})$ with respect to $\xi_{k}$ can be
expressed as

$\displaystyle\begin{aligned} \frac{\partial
f_{2.1b}(\mathbf{W},\boldsymbol{\xi})}{\partial\xi_{k}}\\!=\\!2\sqrt{\tilde{\alpha}_{k}}\mathbf{h}_{k}^{H}\mathbf{w}_{k}\\!-\\!2\xi_{k}\sum_{i=1}^{K}|\mathbf{h}_{k}^{H}\bm{w}_{i}|^{2}\\!-\\!2\xi_{k}\sigma_{0}^{2}.\end{aligned}$
(39)

By solving $\partial f_{2.1b}(\mathbf{W},\bm{\beta})/\partial\xi_{k}=0$, the
optimal $\xi_{k}^{op}$ can be derived as shown in (23). Next, with fixed
$\xi_{k}^{op}$, $f_{2.1b}(\mathbf{W},\boldsymbol{\xi})$ can be rewritten as

$\displaystyle\begin{aligned}
f_{2.1b}(\mathbf{W},\boldsymbol{\xi}^{op})=&\sum_{k=1}^{K}\sqrt{\tilde{\alpha}_{k}}\left(\xi_{k}^{op^{*}}\mathbf{h}_{k}^{H}\bm{w}_{k}+\bm{w}_{k}^{H}\mathbf{h}_{k}\xi_{k}^{op}\right)\\\
&-\sum_{k=1}^{K}|\xi_{k}^{op}|^{2}\sum_{i=1}^{K}\bm{w}_{i}^{H}\mathbf{h}_{k}\mathbf{h}_{k}^{H}\bm{w}_{i}.\end{aligned}$
(40)

Considering the power constraint $\sum_{k=1}^{K}||\bm{w}_{k}||^{2}=P_{T}$, we
write the Lagrangian function of $f_{2.1b}(\mathbf{W},\boldsymbol{\xi}^{op})$
as

$\displaystyle\begin{aligned}
&f_{2.1b}^{(L)}(\lambda,\mathbf{W},\boldsymbol{\xi}^{op})\\!\\!=\\!\\!\sum_{k=1}^{K}\sqrt{\tilde{\alpha}_{k}}\left(\xi_{k}^{op^{*}}\mathbf{h}_{k}^{H}\bm{w}_{k}\\!+\\!\bm{w}_{k}^{H}\mathbf{h}_{k}\xi_{k}^{op}\right)\\\
&-\sum_{k=1}^{K}|\xi_{k}^{op}|^{2}\sum_{i=1}^{K}\bm{w}_{i}^{H}\mathbf{h}_{k}\mathbf{h}_{k}^{H}\bm{w}_{i}\\!-\\!\lambda\left(\sum_{k=1}^{K}\bm{w}_{k}^{H}\bm{w}_{k}\\!-\\!P_{T}\right),\end{aligned}$
(41)

where $\lambda$ is the Lagrangian dual variable. Taking the partial derivative
of $f_{2.1b}^{(L)}(\lambda,\mathbf{W},\boldsymbol{\xi}^{op})$ with respect to
$\bm{w}_{k}$, we obtain

$\displaystyle\begin{aligned} &\frac{\partial
f_{2.1b}^{(L)}(\lambda,\mathbf{W},\boldsymbol{\xi}^{op})}{\partial\bm{w}_{k}}=\\\
&2\sqrt{\tilde{\alpha}_{k}}\mathbf{h}_{k}\xi_{k}^{op}\\!-\\!2\sum_{k=1}^{K}|\xi_{k}^{op}|^{2}\sum_{i=1}^{K}\mathbf{h}_{k}\mathbf{h}_{k}^{H}\bm{w}_{i}\\!-\\!2\lambda\mathbf{I}_{M}\bm{w}_{i}.\end{aligned}$
(42)

Then, by solving $\partial
f_{2.1a}^{(L)}(\lambda,\mathbf{W},\boldsymbol{\xi}^{op})/\partial\bm{w}_{k}=0$,
we can get the optimal expression of $\bm{w}_{k}^{op}$ in (22). For
$\lambda^{op}$, taking the partial derivative of
$f_{2.1b}^{(L)}(\lambda,\mathbf{W},\boldsymbol{\xi}^{op})$ with respect to
$\lambda$, we have

$\displaystyle\begin{aligned} &\frac{\partial
f_{2.1b}^{(L)}(\lambda,\mathbf{W},\boldsymbol{\xi}^{op})}{\partial\lambda}=-\sum_{k=1}^{K}\bm{w}_{k}^{H}\bm{w}_{k}\\!+\\!P_{T}.\end{aligned}$
(43)

By solving the equation ${\partial
f_{2.1b}^{(L)}(\lambda,\mathbf{W},\boldsymbol{\xi}^{op})}/{\partial\lambda}=0$,
i.e., $\sum_{k=1}^{K}||\bm{w}_{k}^{op}||^{2}=P_{T}$, we can obtain all the
feasible solutions for $\lambda^{op}$. Among all the solutions, the smallest
one is chosen since $\lambda\geq 0$,
$\sum_{k=1}^{K}\bm{w}_{k}^{H}\bm{w}_{k}-P_{T}\leq 0$, and the dual function of
$f_{2.1b}^{(L)}(\lambda,\mathbf{W},\boldsymbol{\xi}^{op})$, i.e.,
$\min\limits_{\lambda}\mathcal{G}(\lambda)=\min\max\limits_{\mathbf{W},\boldsymbol{\xi}^{op}}f_{2.1b}^{(L)}(\lambda,\mathbf{W},\boldsymbol{\xi}^{op})$
increases with $\lambda$. Among all the feasible solutions, the smallest one
is selected since minimizing $\mathcal{G}(\lambda)$ equals to maximize
$f_{2.1b}^{(L)}(\lambda,\mathbf{W},\boldsymbol{\xi}^{op})$. Finally, we can
get $\lambda^{op}$ in (24). $\blacksquare$

##  Appendix B Proof of Proposition 3

The optimization problem is

$\displaystyle\begin{aligned}
(P3.2b)\quad&\max_{\boldsymbol{\phi}_{l},\boldsymbol{\varepsilon}}\quad
f_{3.2b}(\boldsymbol{\phi}_{l},\boldsymbol{\varepsilon}),\\\
&~{}\text{s.t.}\quad\quad|\phi_{l,n}|=1,\quad n=1,2,...,N_{l},\end{aligned}$
(44)

with

$\begin{split}f_{3.2b}(\boldsymbol{\phi}_{l},\boldsymbol{\varepsilon})\\!\\!=\\!\\!&\sum_{k=1}^{K}\sqrt{\tilde{\alpha}_{k}}\left(\varepsilon_{k}^{*}(\bar{\boldsymbol{\phi}_{l}}\bm{z}_{l,k}+c_{l,k})\\!+\\!\\!(\bm{z}_{l,k}^{H}\bar{\boldsymbol{\phi}_{l}}^{H}\\!+\\!c_{l,k}^{*})\varepsilon_{k}\right)\\\
&-\sum_{k=1}^{K}|\varepsilon_{k}|^{2}\left(\sum_{i=1}^{K}|\bar{\boldsymbol{\phi}_{l}}\bm{z}_{l,i}+c_{l,i}|^{2}+\sigma_{0}^{2}\right).\end{split}$
(45)

In $(P3.2b)$, taking partial derivative of
$f_{3.2b}(\boldsymbol{\phi}_{l},\boldsymbol{\varepsilon})$ with respect to
$\varepsilon_{k}$, we obtain

$\displaystyle\begin{aligned} \frac{\partial
f_{3.2b}(\boldsymbol{\phi}_{l},\boldsymbol{\varepsilon})}{\partial\varepsilon_{k}}=&2\sqrt{\tilde{\alpha}_{k}}(\bar{\boldsymbol{\phi}_{l}}\bm{z}_{l,k}+c_{l,k})\\\
&-2\varepsilon_{k}\left(\sum_{i=1}^{K}|\bar{\boldsymbol{\phi}_{l}}\bm{z}_{l,i}+c_{l,i}|^{2}+\sigma_{0}^{2}\right).\end{aligned}$
(46)

By setting $\partial
f_{3.2b}(\boldsymbol{\phi}_{l},\boldsymbol{\varepsilon})/\partial\varepsilon_{k}=0$,
we can get the optimal $\varepsilon_{k}^{op}$ in (36) as

$\displaystyle\begin{aligned}
\varepsilon_{k}^{op}=\frac{\sqrt{\tilde{\alpha}_{k}}(\bar{\boldsymbol{\phi}_{l}}\bm{z}_{l,k}+c_{l,k})}{\sum_{i=1}^{K}|\bar{\boldsymbol{\phi}_{l}}\bm{z}_{l,i}+c_{l,i}|^{2}+\sigma_{0}^{2}}.\end{aligned}$
(47)

Substituting $\varepsilon_{k}^{op}$ into
$f_{3.2b}(\boldsymbol{\phi}_{l},\boldsymbol{\varepsilon})$, we have

$\displaystyle\begin{aligned}
&f_{3.2b}(\boldsymbol{\phi}_{l},\boldsymbol{\varepsilon}^{op})\\\
&=\sum_{k=1}^{K}\sqrt{\tilde{\alpha}_{k}}\left((\varepsilon_{k}^{op})^{*}(\bar{\boldsymbol{\phi}_{l}}\bm{z}_{l,k}\\!+\\!c_{l,k})\\!+\\!(\bm{z}_{l,k}\bar{\boldsymbol{\phi}_{l}}^{H}\\!\\!+\\!\\!c_{l,k}^{*})\varepsilon_{k}^{op}\right)\\\
&\quad\quad-\sum_{k=1}^{K}|\varepsilon_{k}^{op}|^{2}\left(\sum_{i=1}^{K}|\bar{\boldsymbol{\phi}_{l}}\bm{z}_{l,i}+c_{l,i}|^{2}+\sigma_{0}^{2}\right)\\\
&=-\bar{\boldsymbol{\phi}_{l}}\mathbf{P}\bar{\boldsymbol{\phi}_{l}}^{H}+2\Re\\{\bar{\boldsymbol{\phi}_{l}}\bm{d}_{l}+q_{1}\\}-c_{1}.\end{aligned}$
(48)

where

$\displaystyle\begin{aligned}
\mathbf{P}=\sum_{k=1}^{K}|\varepsilon_{k}^{op}|^{2}\sum_{i=1}^{K}\bm{z}_{l,i}\bm{z}_{l,i}^{H},\end{aligned}$
(49)

and

$\begin{split}\bm{d}_{l}=&\sum_{k=1}^{K}\sqrt{\tilde{\alpha}_{k}}(\varepsilon_{k}^{op})^{*}\bm{z}_{l,k}-\sum_{k=1}^{K}|\varepsilon_{k}^{op}|^{2}\sum_{i=1}^{K}\bm{z}_{l,i}c_{l,i}^{*}.\end{split}$
(50)

Among the denotations, $q_{1}$ and $c_{1}$ are complex numbers, which are
given by
$q_{1}=\sum_{k=1}^{K}\sqrt{\tilde{\alpha}_{k}}(\varepsilon_{k}^{op})^{*}c_{l,k}$
and
$c_{1}=\sum_{k=1}^{K}|\varepsilon_{k}^{op}|^{2}\sum_{i=1}^{K}c_{l,i}c_{l,i}^{*}+\sum_{k=1}^{K}|\varepsilon_{k}^{op}|^{2}\sigma_{0}^{2}$,
respectively. In (LABEL:phi3), the first term
$\bar{\boldsymbol{\phi}_{l}}\mathbf{P}\bar{\boldsymbol{\phi}_{l}}^{H}$ can be
further expanded as

$\begin{split}\bar{\boldsymbol{\phi}_{l}}\mathbf{P}\bar{\boldsymbol{\phi}_{l}}^{H}=&\sum_{i=1}^{(L-l+1)N_{l}}\sum_{j=1}^{(L-l+1)N_{l}}\bar{\phi}_{l,i}p_{i,j}\bar{\phi}^{*}_{l,j}\\\
=&\bar{\phi}_{l,n}p_{n,n}\bar{\phi}^{*}_{l,n}+\sum_{j=1,j\neq
n}^{(L-l+1)N_{l}}\bar{\phi}_{l,n}p_{n,j}\bar{\phi}^{*}_{l,i}\\\
&+\\!\\!\\!\\!\\!\\!\\!\sum_{i=1,i\neq
n}^{(L-l+1)N_{l}}\\!\\!\\!\\!\\!\bar{\phi}_{l,i}p_{i,n}\bar{\phi}^{*}_{l,n}\\!\\!+\\!\\!\\!\\!\\!\\!\sum_{i=1,i\neq
n}^{(L-l+1)N_{l}}\sum_{j=1,j\neq
n}^{(L-l+1)N_{l}}\\!\\!\\!\\!\\!\\!\bar{\phi}_{l,i}^{*}p_{i,j}\bar{\phi}_{l,j}\\\
\overset{(a)}{=}&\bar{\phi}_{l,n}p_{n,n}\bar{\phi}_{l,n}^{*}+2\Re\left\\{\sum_{j=1,j\neq
n}^{(L-l+1)N_{l}}\bar{\phi}_{l,n}p_{n,j}\bar{\phi}_{l,i}^{*}\right\\}\\\
&+\sum_{i=1,i\neq n}^{(L-l+1)N_{l}}\sum_{j=1,j\neq
n}^{(L-l+1)N_{l}}\bar{\phi}_{l,i}p_{i,j}\bar{\phi}^{*}_{l,j}.\end{split}$ (51)

In (51), $(a)$ holds since
$\mathbf{P}=\sum_{k=1}^{K}|\varepsilon_{k}^{op}|^{2}\sum_{i=1}^{K}\bm{z}_{l,i}\bm{z}_{l,i}^{H}$
is a Hermitian matrix and satisfies $p_{i,j}=p_{j,i}^{*}$.

The second term $\bar{\boldsymbol{\phi}_{l}}\bm{d}_{l}$ of (LABEL:phi3) can be
further expanded as

$\begin{split}\bar{\boldsymbol{\phi}_{l}}\bm{d}_{l}=\sum_{i=1}^{(L-l+1)N_{l}}\bar{\phi}_{l,i}d_{l,i}=\bar{\phi}_{l,n}d_{l,n}+\sum_{i=1,i\neq
n}^{(L-l+1)N_{l}}\bar{\phi}_{l,i}d_{l,i}.\end{split}$ (52)

Substituting (52) and (51) into (LABEL:phi3) and ignoring the constant term
$c_{1}$, $f_{3.2b}(\boldsymbol{\phi}_{l},\boldsymbol{\varepsilon}^{op})$ can
be rewritten as

$\begin{split}f_{3.2c}(\bar{\phi}_{l,n},\boldsymbol{\varepsilon}^{op})=&-\bar{\phi}_{l,n}p_{n,n}\bar{\phi}^{*}_{l,n}\\\
&+\\!2\Re\left\\{\bar{\phi}_{l,n}(d_{l,n}\\!-\\!\\!\\!\\!\\!\sum_{j=1,j\neq
n}^{(L-l+1)N_{l}}p_{n,j}\bar{\phi}^{*}_{l,j})\right\\}\\!+\\!c_{2}\\\
\overset{(b)}{=}&-p_{n,n}+2\Re\\{\bar{\phi}_{l,n}\eta_{n}\\}+c_{2},\end{split}$
(53)

where $(b)$ holds since $\bar{\phi}_{l,n}\bar{\phi}_{l,n}^{*}=1$ always holds
and $\eta_{n}=d_{l,n}-\sum_{j=1,j\neq
n}^{(L-l+1)N_{l}}p_{n,j}\bar{\phi}^{*}_{l,j}$, the constant term $c_{2}$ is
represented as

$\begin{split}c_{2}=&-\sum_{i=1,i\neq n}^{(L-l+1)N_{l}}\sum_{j=1,j\neq
n}^{(L-l+1)N_{l}}\bar{\phi}_{l,i}p_{i,j}\bar{\phi}^{*}_{l,j}\\\
&+2\Re\\{\sum_{i=1,i\neq
n}^{(L-l+1)N_{l}}\bar{\phi}_{l,i}d_{l,i}+q_{1}\\},\end{split}$ (54)

which is independent of $\bar{\phi}_{l,n}$.

From (53), we obtain that maximizing
$f_{3.2c}(\bar{\phi}_{l,n},\boldsymbol{\varepsilon}^{op})$ is equivalent to
maximizing $2\Re\\{\bar{\phi}_{l,n}\eta_{n}\\}$, which gives rise to the
solution in (33). $\blacksquare$

##  Appendix C Convergence analysis of Algorithm 1

In Algorithm 1, Proposition 1 ensures that the update of $\boldsymbol{\alpha}$
always produces an improving value, thus the objective function in (14) always
satisfies
$f_{2}(\boldsymbol{\alpha}^{i},\mathbf{W}^{i-1},\boldsymbol{\phi}_{1}^{i-1},...,\boldsymbol{\phi}_{L}^{i-1})\geq
f_{2}(\boldsymbol{\alpha}^{i-1},\mathbf{W}^{i-1},\boldsymbol{\phi}_{1}^{i-1},...,\boldsymbol{\phi}_{L}^{i-1})$
in the $i$-th iteration.

For the BS’s transmit beamforming optimization, when $\boldsymbol{\alpha}$ and
$\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L}$ are fixed, the function in
(20) is a biconvex problem with respect $\bm{w}_{k}$ and $\xi_{k}$. For fixed
$\mathbf{W}$, the optimal $\xi_{k}^{op}$ can be obtained by setting $\partial
f_{2.1b}(\mathbf{W},\boldsymbol{\xi})/\partial\xi_{k}=0$. For fixed $\xi_{k}$,
the optimization problem to get the optimal $\mathbf{W}$ can be expressed as

$\displaystyle\begin{aligned} \max_{\mathbf{W}}\quad
f_{2.1b}(\mathbf{W})=&\sum_{k=1}^{K}\sqrt{\tilde{\alpha}_{k}}\left(\xi_{k}^{*}\mathbf{h}_{k}^{H}\bm{w}_{k}+\bm{w}_{k}^{H}\mathbf{h}_{k}\xi_{k}\right)\\\
&-\sum_{k=1}^{K}|\xi_{k}|^{2}\sum_{i=1}^{K}\bm{w}_{i}^{H}\mathbf{h}_{k}\mathbf{h}_{k}^{H}\bm{w}_{i},\\\
\text{s.t.}\quad\quad\quad\quad\quad\quad&\sum_{k=1}^{K}||\bm{w}_{k}||^{2}\leq
P_{T}.\end{aligned}$ (55)

The corresponding Lagrangian function is given by

$\displaystyle\begin{aligned}
&f_{2.1b}^{(L)}(\mathbf{W})=-\sum_{k=1}^{K}|\xi_{k}|^{2}\sum_{i=1}^{K}\bm{w}_{i}^{H}\mathbf{h}_{k}\mathbf{h}_{k}^{H}\bm{w}_{i}\\\
&\\!\\!+\\!\\!\sum_{k=1}^{K}\sqrt{\tilde{\alpha}_{k}}\left(\xi_{k}^{*}\mathbf{h}_{k}^{H}\bm{w}_{k}\\!\\!+\\!\\!\bm{w}_{k}^{H}\mathbf{h}_{k}\xi_{k}\right)\\!\\!-\\!\\!\lambda\left(\sum_{k=1}^{K}\bm{w}_{k}^{H}\bm{w}_{k}\\!\\!-\\!\\!P_{T}\right).\end{aligned}$
(56)

By setting $\partial f_{2.1b}^{(L)}(\mathbf{W})/\partial\bm{w}_{k}=0$, the
optimal expression $\bm{w}_{k}$ in (22) can be obtained. Thus,
$f_{2}(\boldsymbol{\alpha}^{i},\mathbf{W}^{i},\boldsymbol{\phi}_{1}^{i-1},...,\boldsymbol{\phi}_{L}^{i-1})\geq
f_{3}(\boldsymbol{\alpha}^{i-1},\mathbf{W}^{i-1},\boldsymbol{\phi}_{1}^{i-1},...,\boldsymbol{\phi}_{L}^{i-1})$
is guaranteed.

With fixed $\mathbf{W}$, the objective functions
$f_{3.2b}(\boldsymbol{\phi}_{l},\boldsymbol{\varepsilon})$ in (31) is solved
by deriving the optimal solution for $\boldsymbol{\phi}_{l}$, $l=1,2,...,L$
and $\boldsymbol{\varepsilon}$, respectively. Specifically, we get the optimal
$\varepsilon_{k}^{op}$ by setting $\partial
f_{3.2b}(\boldsymbol{\phi}_{2},\boldsymbol{\varepsilon})/\partial\varepsilon_{k}=0$.
Thus, we can optimize $\boldsymbol{\phi}_{l}$ sequentially according to (33).
Based on the above analysis, we have
$f_{3}(\boldsymbol{\alpha}^{i},\mathbf{W}^{i},\boldsymbol{\phi}_{1}^{i},...,\boldsymbol{\phi}_{L}^{i})\geq
f_{3}(\boldsymbol{\alpha}^{i-1},\mathbf{W}^{i-1},\boldsymbol{\phi}_{1}^{i-1},...,\boldsymbol{\phi}_{L}^{i-1})$
after the $i$-th iteration. Thus, the algorithm generates a non-decreasing
sequence.

To prove the convergence of Algorithm 1, we also need to show that the
sequence generated from Algorithm 1 is upper bounded. As shown in (16), for
$\alpha_{k}\geq 0$ and
$\boldsymbol{\alpha}=[\alpha_{1},\alpha_{2},...,\alpha_{K}]^{T}$,
$f_{2}(\boldsymbol{\alpha},\mathbf{W},\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L})$
is an increasing function with respect to $\boldsymbol{\alpha}$. The maximum
value of
$f_{2}(\boldsymbol{\alpha},\mathbf{W},\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L})$
with respect to $\boldsymbol{\alpha}$ is achieved by setting
$\alpha_{k}^{op}=\gamma_{k}$, leading to
$f_{2}(\boldsymbol{\alpha},\mathbf{W},\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L})=f(\mathbf{W},\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L})$.
This shows that
$f_{2}(\bm{\alpha},\mathbf{W},\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L})$
is upper-bounded by the original objective function
$f(\mathbf{W},\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L})$. Since
$\gamma_{k}$ ranges from $0$ to $\gamma_{k}^{max}$,
$f(\mathbf{W},\boldsymbol{\phi}_{1},...,\boldsymbol{\phi}_{L})$ must have a
finite upper bound. Thus, Algorithm 1 generates a bounded non-decreasing
sequence and hence converges. $\blacksquare$

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