# Energy-Efficient Wireless Communications with Distributed Reconfigurable
Intelligent Surfaces

Zhaohui Yang, , Mingzhe Chen, , Walid Saad, , Wei Xu, , Mohammad Shikh-Bahaei,
, H. Vincent Poor, , and Shuguang Cui Z. Yang and M. Shikh-Bahaei are with the
Centre for Telecommunications Research, Department of Engineering, King’s
College London, WC2R 2LS, UK, Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>Chen is with the Chinese University of Hong Kong,
Shenzhen, 518172, China, and also with the Department of Electrical
Engineering, Princeton University, Princeton, NJ, 08544, USA, Email:
<EMAIL_ADDRESS>Saad is with the Wireless@VT, Bradley Department of
Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA, 24060,
USA, Email<EMAIL_ADDRESS>W. Xu is with the National Mobile Communications
Research Laboratory, Southeast University, Nanjing 210096, China, Email:
<EMAIL_ADDRESS>Vincent Poor is with the Department of Electrical
Engineering, Princeton University, Princeton, NJ, 08544, USA, Email:
<EMAIL_ADDRESS>S. Cui is with the Shenzhen Research Institute of Big Data
and School of Science and Engineering, the Chinese University of Hong Kong,
Shenzhen, 518172, China, Email<EMAIL_ADDRESS>
###### Abstract

This paper investigates the problem of resource allocation for a wireless
communication network with distributed reconfigurable intelligent surfaces
(RISs). In this network, multiple RISs are spatially distributed to serve
wireless users and the energy efficiency of the network is maximized by
dynamically controlling the on-off status of each RIS as well as optimizing
the reflection coefficients matrix of the RISs. This problem is posed as a
joint optimization problem of transmit beamforming and RIS control, whose goal
is to maximize the energy efficiency under minimum rate constraints of the
users. To solve this problem, two iterative algorithms are proposed for the
single-user case and multi-user case. For the single-user case, the phase
optimization problem is solved by using a successive convex approximation
method, which admits a closed-form solution at each step. Moreover, the
optimal RIS on-off status is obtained by using the dual method. For the multi-
user case, a low-complexity greedy searching method is proposed to solve the
RIS on-off optimization problem. Simulation results show that the proposed
scheme achieves up to 33% and 68% gains in terms of the energy efficiency in
both single-user and multi-user cases compared to the conventional RIS scheme
and amplify-and-forward relay scheme, respectively.

###### Index Terms:

Energy efficiency, reconfigurable intelligent surface, phase shift
optimization, integer programming.

##  I Introduction

Driven by the rapid development of advanced multimedia applications, next-
generation wireless networks must support high spectral efficiency and massive
connectivity [1]. Due to high data rate demand and massive numbers of users,
energy consumption has become a challenging problem in the design of future
wireless networks [2]. In consequence, energy efficiency, defined as the ratio
of spectral efficiency over power consumption, has emerged as an important
performance index for deploying green and sustainable wireless networks [3, 4,
5, 6, 7].

Recently, reconfigurable intelligent surface (RIS)-assisted wireless
communication has been proposed as a potential solution for enhancing the
energy efficiency of wireless networks [8, 9, 10, 11, 12, 13]. An RIS is a
meta-surface equipped with low-cost and passive elements that can be
programmed to turn the wireless channel into a partially deterministic space.
In RIS-assisted wireless communication networks, a base station (BS) sends
control signals to an RIS controller so as to optimize the properties of
incident waves and improve the communication quality of users. The RIS acts as
a reflector and does not perform any digitalization operation. Hence, if
properly deployed, an RIS promises much lower energy consumption than
traditional amplify-and-forward (AF) relays [14, 15, 16]. However, the
effective deployment of energy-efficient RIS systems faces several challenges
ranging from performance characterization to network optimization [17].

A number of existing works such as in [18, 19, 20, 21, 22, 23, 24] has studied
the deployment of RISs in wireless networks. In [18], the downlink sum-rate of
an RIS assisted wireless communication system was characterized. An asymptotic
analysis of the uplink transmission rate in an RIS-based large antenna-array
system was presented in [19]. Then, in [20], the authors investigated the
asymptotic optimality of the achievable rate in a downlink RIS system.
Considering energy harvesting, an RIS was invoked for enhancing the sum-rate
performance of a simultaneous wireless information and power transfer aided
system [21]. Instead of considering the availability of instantaneous channel
state information (CSI), the authors in [22] proposed a two-time-scale
transmission protocol to maximize the achievable sum-rate for an RIS-assisted
multi-user system. Taking the secrecy into consideration, the work in [23]
investigated the problem of secrecy rate maximization of an RIS assisted
multi-antenna system. Further by considering imperfect CSI, the RIS was
considered to enhance the physical layer security of a wireless channel in
[24]. Beyond the above studies, the use of RISs for enhanced wireless energy
efficiency has been studied in [25]. In [25], the authors proposed a new
approach to maximize the energy efficiency of a multi-user multiple-input
single-output (MISO) system by jointly controlling the transmit power of the
BS and the phase shifts of the RIS. However, only a single RIS was considered
for simplicity in [25]. Deploying a number of low-cost power-efficient RISs in
future networks can cooperatively enhance the coverage of the networks. In
particular, deploying multiple RISs in wireless networks has several
advantages. First, distributed RISs can provide robust data-transmission since
different RISs can be deployed geometrically apart from each other. Meanwhile,
multiple RISs can provide multiple paths of received signals, which increases
the received signal strength. To our best knowledge, this is the first work
that optimizes the energy efficiency for a wireless network with multiple
RISs.

The main contribution of this paper is a novel energy efficient resource
allocation scheme for wireless communication networks with distributed RISs.
Our key contributions include:

  * •

We investigate a downlink wireless communication system with distributed RISs
that can be dynamically turned on or off depending on the network
requirements. To maximize the energy efficiency of the system, we jointly
optimize the phase shifts of all RISs, the transmit beamforming of the
transmitter, and the RIS on-off status vector. We formulate an optimization
problem with the objective of maximizing the energy efficiency under minimum
rate constraint of users, transmit power constraint, and unit-modulus
constraint of the RIS phase shifts.

  * •

To maximize the energy efficiency for a single user, a suboptimal solution is
obtained by using a low-complexity algorithm that iteratively solves two,
joint subproblems. For the joint phase and power optimization subproblem, a
suboptimal phase is obtained by using the successive convex approximation
(SCA) method with low complexity, and the optimal power is subsequently
obtained in closed form. For the RIS on-off optimization subproblem, the dual
method is used to obtain the optimal solution.

  * •

To maximize the energy efficiency for multiple users, an iterative algorithm
is proposed through solving the phase optimization subproblem, beamforming
optimization subproblem, and RIS on-off subproblem, iteratively. For the
subproblem of phase optimization or beamforming optimization, we use the SCA
method to obtain a suboptimal solution. For the RIS on-off optimization
subproblem, we propose a low-complexity search method that can determine the
on-off status of all RISs.

Simulation results show that our proposed approach can enhance the energy
efficiency by up to 33% and 68% gains compared to the conventional RIS scheme
and AF relay scheme, respectively.

The rest of this paper is organized as follows. Section II introduces the
system model and problem formulation. Section III and Section IV provide the
energy efficiency optimization with a single user and multiple users,
respectively. Simulation results are provided in Section V and conclusions are
given in Section VI.

Notations: In this paper, the imaginary unit of a complex number is denoted by
$j=\sqrt{-1}$. Matrices and vectors are denoted by boldface capital and lower-
case letters, respectively. Matrix $\text{diag}(x_{1},\cdots,x_{N})$ denotes a
diagonal matrix whose diagonal components are $x_{1},\cdots,x_{N}$. The real
and imaginary parts of a complex number $x$ are denoted by $\mathcal{R}(x)$
and $\mathcal{I}(\cdot)$, respectively. $\bm{x}^{*}$, $\bm{x}^{T}$, and
$\bm{x}^{H}$ respectively denote the conjugate, transpose, and conjugate
transpose of vector $\bm{x}$. $[\bm{x}]_{n}$ and $[\bm{X}]_{kn}$ denote the
$n$-th and $(k,n)$-th elements of the respective vector $\bm{x}$ and matrix
$\bm{X}$. $|\bm{x}|$ denotes the $\ell_{2}$-norm of vector $\bm{x}$. The
distribution of a circularly symmetric complex Gaussian variable with mean $x$
and covariance $\sigma$ is denoted by ${\mathcal{CN}}(x,\sigma)$.

##  II System Model and Problem Formulation

###  II-A Transmission Model

Figure 1: A downlink MISO system with multiple RISs.

Consider an RIS-assisted MISO downlink channel that consists of one BS, a set
$\mathcal{K}$ of $K$ users, and a set $\mathcal{L}$ of $L$ RISs, as shown in
Fig. 1. The number of transmit antennas at the BS is $M$, while each user is
equipped with one antenna. Such a setting has been used in many practical
scenarios such as in Internet-of-Things networks [18, 19, 20]. Each RIS,
$l\in\mathcal{L}$, has $N_{l}$ reflecting elements. The RISs are configured to
assist the communication between the BS and users. In particular, the RISs
will be installed on the walls of the surrounding high-rise buildings.

The transmitted signal at the BS is:

$\bm{s}=\sum_{k=1}^{K}\bm{w}_{k}s_{k},$ (1)

where $s_{k}$ is unit-power information symbol [25] and
$\bm{w}_{k}\in\mathbb{C}^{M}$ is the beamforming vector for user
$k\in\mathcal{K}$.

The power consumption of an RIS depends on both the type and the resolution of
the reflecting elements that effectively perform phase shifting on the
impinging signal [25, 26, 27]. Considering the power consumption of RISs due
to controlling the phase shift values of the reflecting elements [25], it is
often not energy efficient to turn on all the RISs. We now introduce a binary
variable $x_{l}\in\\{0,1\\}$, where $x_{l}=1$ indicates that RIS $l$ is on.
When $x_{l}=1$, the phase shift matrix of RIS $l$ can be optimized through a
diagonal matrix
$\bm{\Theta}_{l}=\text{diag}(\text{e}^{j\theta_{l1}},\cdots,\text{e}^{j\theta_{lN_{l}}})\in\mathbb{C}^{N_{l}\times
N_{l}}$ with $\theta_{ln}\in[0,2\pi]$, $l\in\mathcal{L}$, and
$n\in\mathcal{N}_{l}=\\{1,\cdots,N_{l}\\}$, where $\bm{\Theta}_{l}$ captures
the effective phase shifts applied by all reflecting elements of RIS $l$. In
contrast, when $x_{l}=0$, RIS $l$ is off and does not consume any power. Then,
with the multiple RISs, the received signal at user $k$ can be given by:

$y_{k}=\left(\bm{g}_{k}^{H}+\sum_{l=1}^{L}x_{l}\bm{h}_{kl}^{H}\bm{\Theta}_{l}\bm{G}_{l}\right)\bm{s}+n_{k},$
(2)

where $\bm{g}_{k}\in\mathbb{C}^{M}$, $\bm{G}_{l}\in\mathbb{C}^{N_{l}\times
M}$, and $\bm{h}_{kl}\in\mathbb{C}^{N_{l}}$, respectively, denote the channel
responses from the BS to user $k$, from the BS to RIS $l$, and from RIS $l$ to
user $k$, and $n_{k}\sim\mathcal{CN}(0,\sigma^{2})$ is the additive white
Gaussian noise.

Based on (1) and (2), the received signal-to-interference-plus-noise ratio
(SINR) at user $k$ is:

$\gamma_{k}=\frac{\left|\left(\bm{g}_{k}^{H}+\sum_{l=1}^{L}x_{l}\bm{h}_{kl}^{H}\bm{\Theta}_{l}\bm{G}_{l}\right)\bm{w}_{k}\right|^{2}}{\sum_{i=1,i\neq
k}^{K}\left|\left(\bm{g}_{k}^{H}+\sum_{l=1}^{L}x_{l}\bm{h}_{kl}^{H}\bm{\Theta}_{l}\bm{G}_{l}\right)\bm{w}_{i}\right|^{2}+\sigma^{2}}.$
(3)

As a result, the sum-rate of all users is:

$R_{\text{t}}=B\sum_{k=1}^{K}\log_{2}(1+\gamma_{k}),$ (4)

where $B$ is the bandwidth of the channel.

###  II-B Power Consumption Model

The total power consumption of the considered RIS-assisted system includes the
transmit power of the BS, the circuit power consumption of both the BS and all
users, and the power consumption of all RISs. Consequently, the total power of
the system will be given by:

$P_{\text{t}}=\underbrace{\sum_{k=1}^{K}\mu\bm{w}_{k}^{H}\bm{w}_{k}}_{\text{transmit
power of the BS}}+\underbrace{P_{\text{B}}}_{\text{circuit power of the
BS}}+\underbrace{\sum_{k=1}^{K}P_{k}}_{\text{circuit power of all
users}}+\underbrace{\sum_{l=1}^{L}x_{l}N_{l}P_{\text{R}}}_{\text{power
consumption of all RISs}},$ (5)

where $\mu=\nu^{-1}$ with $\nu$ being the power amplifier efficiency of the
BS, $P_{\text{B}}$ is the circuit power consumption of the BS, $P_{k}$ is the
circuit power consumption of user $k$, and $P_{\text{R}}$ is the power
consumption of each reflecting element in the RIS. In (5),
$x_{l}N_{l}P_{\text{R}}$ is the power consumption of RIS $l$.

###  II-C Problem Formulation

Given the considered system model, our objective is to jointly optimize the
reflection coefficients matrix, beamforming vector, and RIS on-off vector so
as to maximize the energy efficiency under the minimum rate requirements and
total power constraint. Mathematically, the problem for the distributed RISs
can be given by:

$\displaystyle\mathop{\max}_{\bm{\theta},\bm{w},\bm{x}}\quad$
$\displaystyle\frac{R_{\text{t}}}{P_{\text{t}}}=\frac{B\sum_{k=1}^{K}\log_{2}\left(1+\frac{\left|\left(\bm{g}_{k}^{H}+\sum_{l=1}^{L}x_{l}\bm{h}_{kl}^{H}\bm{\Theta}_{l}\bm{G}_{l}\right)\bm{w}_{k}\right|^{2}}{\sum_{i=1,i\neq
k}^{K}\left|\left(\bm{g}_{k}^{H}+\sum_{l=1}^{L}x_{l}\bm{h}_{kl}^{H}\bm{\Theta}_{l}\bm{G}_{l}\right)\bm{w}_{i}\right|^{2}+\sigma^{2}}\right)}{\mu\bm{w}^{H}\bm{w}+\sum_{k=1}^{K}P_{k}+P_{\text{B}}+\sum_{l=1}^{L}x_{l}N_{l}P_{\text{R}}}$
(6) s.t.
$\displaystyle{B\log_{2}\left(1+\frac{\left|\left(\bm{g}_{k}^{H}+\sum_{l=1}^{L}x_{l}\bm{h}_{kl}^{H}\bm{\Theta}_{l}\bm{G}_{l}\right)\bm{w}_{k}\right|^{2}}{\sum_{i=1,i\neq
k}^{K}\left|\left(\bm{g}_{k}^{H}+\sum_{l=1}^{L}x_{l}\bm{h}_{kl}^{H}\bm{\Theta}_{l}\bm{G}_{l}\right)\bm{w}_{i}\right|^{2}+\sigma^{2}}\right)}\geq
R_{k},\quad\forall k\in\mathcal{K},$ (6a) $\displaystyle\bm{w}^{H}\bm{w}\leq
P_{\max},$ (6b) $\displaystyle\theta_{ln}\in[0,2\pi],\quad\forall
l\in\mathcal{L},n\in\mathcal{N}_{l},$ (6c) $\displaystyle
x_{l}\in\\{0,1\\},\quad\forall l\in\mathcal{L},$ (6d)

where
$\bm{\theta}=[\theta_{11},\cdots,\theta_{1N_{1}},\cdots,\theta_{LN_{L}}]^{T}$,
$\bm{w}=[\bm{w}_{1};\cdots;\bm{w}_{K}]$, $\bm{x}=[x_{1},\cdots,x_{L}]^{T}$,
$R_{k}$ is the minimum data rate requirement of user $k$, and $P_{\max}$ is
the maximum transmit power of the BS. The minimum rate constraint for each
user is given in (6a) and (6b) represents the total power constraint. The
phase shift constraint for each reflecting element is provided in (6c), which
can also be seen as the unit-modulus constraint since
$|\text{e}^{j\theta_{ln}}|=1$. The problem in (6) is a mixed-integer nonlinear
program (MINLP) even for the single-user case with $K=1$. It is generally
difficult to obtain the globally optimal solution of the MINLP problem in (6).
In the following, we propose two iterative algorithms to obtain suboptimal
solutions of problem (6) for the single-user case and the multi-user case,
respectively.

##  III Energy Efficiency Optimization with A Single User

In this section, we consider the single-user case, i.e., $K=1$. For $K=1$,
problem (6) becomes:

$\displaystyle\mathop{\max}_{\bm{\theta},\bm{w}_{1},\bm{x}}\quad$
$\displaystyle\frac{B\log_{2}\left(1+\frac{\left|\left(\bm{g}_{1}^{H}+\sum_{l=1}^{L}x_{l}\bm{h}_{1l}^{H}\bm{\Theta}_{l}\bm{G}_{l}\right)\bm{w}_{1}\right|^{2}}{\sigma^{2}}\right)}{\mu\bm{w}_{1}^{H}\bm{w}_{1}+P_{1}+P_{\text{B}}+\sum_{l=1}^{L}x_{l}N_{l}P_{\text{R}}}$
(7) s.t.
$\displaystyle{B\log_{2}\left(1+\frac{\left|\left(\bm{g}_{1}^{H}+\sum_{l=1}^{L}x_{l}\bm{h}_{1l}^{H}\bm{\Theta}_{l}\bm{G}_{l}\right)\bm{w}_{1}\right|^{2}}{\sigma^{2}}\right)}\geq
R_{1},$ (7a) $\displaystyle\bm{w}_{1}^{H}\bm{w}_{1}\leq P_{\max},$ (7b)
$\displaystyle\theta_{ln}\in[0,2\pi],\quad\forall
l\in\mathcal{L},n\in\mathcal{N}_{l},$ (7c) $\displaystyle
x_{l}\in\\{0,1\\},\quad\forall l\in\mathcal{L}.$ (7d)

Since there is no multi-user interference, it is well-known that beaming as
the the maximum ratio transmission (MRT) at the BS is optimal [28]. That is:

$\bm{w}_{1}=\sqrt{p_{1}}\frac{\bm{g}_{1}+\sum_{l=1}^{L}x_{l}\bm{G}_{l}^{H}\bm{\Theta}_{l}^{H}\bm{h}_{1l}}{|\bm{g}_{1}+\sum_{l=1}^{L}x_{l}\bm{G}_{l}^{H}\bm{\Theta}_{l}^{H}\bm{h}_{1l}|},$
(8)

where $p_{1}\leq P_{\max}$ helps satisfy the transmit power constraint at the
BS. Substituting the optimal beamforming in (8) into problem (7), it becomes:

$\displaystyle\mathop{\max}_{\bm{\theta},p_{1},\bm{x}}\quad$
$\displaystyle\frac{B\log_{2}\left(1+\frac{p_{1}\left|\bm{g}_{1}^{H}+\sum_{l=1}^{L}x_{l}\bm{h}_{1l}^{H}\bm{\Theta}_{l}\bm{G}_{l}\right|^{2}}{\sigma^{2}}\right)}{\mu
p_{1}+P_{1}+P_{\text{B}}+\sum_{l=1}^{L}x_{l}N_{l}P_{\text{R}}}$ (9) s.t.
$\displaystyle{B\log_{2}\left(1+\frac{p_{1}\left|\bm{g}_{1}^{H}+\sum_{l=1}^{L}x_{l}\bm{h}_{1l}^{H}\bm{\Theta}_{l}\bm{G}_{l}\right|^{2}}{\sigma^{2}}\right)}\geq
R_{1},$ (9a) $\displaystyle 0\leq p_{1}\leq P_{\max},$ (9b)
$\displaystyle\theta_{ln}\in[0,2\pi],\quad\forall
l\in\mathcal{L},n\in\mathcal{N}_{l},$ (9c) $\displaystyle
x_{l}\in\\{0,1\\},\quad\forall l\in\mathcal{L}.$ (9d)

Due to the involvement of integer variable $\bm{x}$, it is difficult to obtain
the globally optimal solution of (9). As such, we propose an iterative
algorithm to solve problem (9) sub-optimally with low complexity. The proposed
iterative algorithm contains two major steps. In the first step, we jointly
optimize phase and power $(\bm{\theta},p_{1})$ with given $\bm{x}$. Then, in
the second step, we update RIS on-off vector $\bm{x}$ with the optimized
$(\bm{\theta},p_{1})$ in the previous step.

###  III-A Joint Phase and Power Optimization

For a fixed integer variable $\bm{x}$, problem (9) becomes:

$\displaystyle\mathop{\max}_{\bm{\theta},p_{1}}\quad$
$\displaystyle\frac{B\log_{2}\left(1+\frac{p_{1}\left|\bm{g}_{1}^{H}+\sum_{l=1}^{L}x_{l}\bm{h}_{1l}^{H}\bm{\Theta}_{l}\bm{G}_{l}\right|^{2}}{\sigma^{2}}\right)}{\mu
p_{1}+P_{1}+P_{\text{B}}+\sum_{l=1}^{L}x_{l}N_{l}P_{\text{R}}}$ (10) s.t.
$\displaystyle{B\log_{2}\left(1+\frac{p_{1}\left|\bm{g}_{1}^{H}+\sum_{l=1}^{L}x_{l}\bm{h}_{1l}^{H}\bm{\Theta}_{l}\bm{G}_{l}\right|^{2}}{\sigma^{2}}\right)}\geq
R_{1},$ (10a) $\displaystyle 0\leq p_{1}\leq P_{\max},$ (10b)
$\displaystyle\theta_{ln}\in[0,2\pi],\quad\forall
l\in\mathcal{L},n\in\mathcal{N}_{l}.$ (10c)

From the objective function (10) and the constraint in (10a), we observe that
the optimal $\bm{\theta}$ is the one that maximizes the channel gain, i.e.,
$\left|\left(\bm{g}_{1}^{H}+\sum_{l=1}^{L}x_{l}\bm{h}_{1l}^{H}\bm{\Theta}_{l}\bm{G}_{l}\right)\right|^{2}$.
With this in mind, the optimal solution of problem (10) can be obtained in two
stages, i.e., obtain the value of $\bm{\theta}$ that maximizes
$\left|\left(\bm{g}_{1}^{H}+\sum_{l=1}^{L}x_{l}\bm{h}_{1l}^{H}\bm{\Theta}_{l}\bm{G}_{l}\right)\right|^{2}$
in the first stage and, then, calculate the optimal $p_{1}$ in the second
stage with the obtained $\bm{\theta}$ in the first stage.

####  III-A1 First stage

We first optimize the phase shift vector $\bm{\theta}$ of problem (10). Before
optimizing $\bm{\theta}$, we show that
$\bm{h}_{1l}^{H}\bm{\Theta}_{l}\bm{G}_{l}=\bm{\theta}_{l}^{T}\bm{U}_{1l}$,
where
$\bm{U}_{1l}=\text{diag}(\bm{h}_{1l}^{H})\bm{G}_{l}\in\mathbb{C}^{N_{l}\times
M}$ and
$\bm{\theta}_{l}=[\text{e}^{j\theta_{l1}},\cdots,\text{e}^{j\theta_{lN_{l}}}]^{T}$.
According to problem (10), the optimal $\bm{\theta}$ can be calculated by
solving the following problem:

$\displaystyle\mathop{\max}_{\bm{\theta}}\quad$
$\displaystyle\left|\bm{g}_{1}^{H}+\sum_{l=1}^{L}x_{l}\bm{\theta}_{l}^{T}\bm{U}_{1l}\right|^{2}$
(11) s.t. $\displaystyle\theta_{ln}\in[0,2\pi],\quad\forall
l\in\mathcal{L},n\in\mathcal{N}_{l}.$ (11a)

Let $\bm{\theta}_{l}^{*}$ be the conjugate vector of $\bm{\theta}_{l}$. The
total number of elements for all RISs is denoted by $Q=\sum_{l=1}^{L}N_{l}$.
Denote
$\bm{v}=[\bm{\theta}_{1}^{*};\cdots;\bm{\theta}_{L}^{*}]\in\mathbb{C}^{Q}$ and
$\bm{U}_{1}=[x_{1}\bm{U}_{1l};\cdots;x_{L}\bm{U}_{1L}]\in\mathbb{C}^{Q\times
M}$. Problem (11) can be rewritten as:

$\displaystyle\mathop{\max}_{{\bm{v}}}\quad$
$\displaystyle\left|{\bm{g}}_{1}+{\bm{U}}_{1}^{H}{\bm{v}}\right|^{2}$ (12)
s.t. $\displaystyle|v_{q}|=1,\quad\forall q\in\mathcal{Q},$ (12a)

where $\mathcal{Q}=\\{1,\cdots,Q\\}$.

To solve the optimization problem in (12), various methods were proposed by
techniques such as semidefinite relaxation (SDR) technique [29] and successive
refinement (SR) algorithm [30]. However, the SDR method imposes high
complexity to obtain a rank-one solution and the SR algorithm requires a large
number of iterations due to the need for updating the phase shifts in a one-
by-one manner. To reduce the computational complexity, we propose the SCA
method to solve the phase shift optimization problem (12).

To handle the nonconvexity of objective function (12), we adopt the SCA method
and, consequently, objective function (12) can be approximated by:

$\displaystyle\quad
2\mathcal{R}(({\bm{g}}_{1}+{\bm{U}}_{1}^{H}{\bm{v}}^{(n-1)})^{H}{\bm{U}}_{1}^{H}{\bm{v}})-\left|{\bm{g}}_{1}+{\bm{U}}_{1}^{H}{\bm{v}}^{(n-1)}\right|^{2},$
(13)

which is the first-order Taylor series of
$\left|{\bm{g}}_{1}+{\bm{U}}_{1}^{H}{\bm{v}}\right|^{2}$ and the superscript
$(n-1)$ represents the value of the variable at the $(n-1)$-th iteration.

With the above approximation (13), the nonconvex problem (12) can be
approximated by the following convex problem:

$\displaystyle\mathop{\max}_{{\bm{v}}}\quad$ $\displaystyle
2\mathcal{R}(({\bm{g}}_{1}+{\bm{U}}_{1}^{H}{\bm{v}}^{(n-1)})^{H}{\bm{U}}_{1}^{H}{\bm{v}})-\left|{\bm{g}}_{1}+{\bm{U}}_{1}^{H}{\bm{v}}^{(n-1)}\right|^{2}$
(14) s.t. $\displaystyle|v_{q}|\leq 1,\quad\forall q\in\mathcal{Q},$ (14a)

where constraint (12a) is temporarily relaxed as (14a). In the following
lemma, we show that (14a) always holds with equality for the optimal solution
of problem (14).

###### Lemma 1

The optimal solution of problem (14) is:

$\bm{v}=\text{e}^{-j\angle({\bm{U}}_{1}({\bm{g}}_{1}+{\bm{U}}_{1}^{H}{\bm{v}}^{(n-1)}))},$
(15)

where $\angle(\cdot)$ represents the angle vector of a vector, i.e,
$[\angle(\bm{y})]_{q}=\arctan\frac{\mathcal{I}([\bm{y}]_{q})}{\mathcal{R}([\bm{y}]_{q})}$.

Proof: To maximize
$({\bm{g}}_{1}+{\bm{U}}_{1}^{H}{\bm{v}}^{(n-1)})^{H}{\bm{U}}_{1}^{H}{\bm{v}}$
in (14), the optimal $\bm{v}$ should be chosen such that
$[({\bm{g}}_{1}+{\bm{U}}_{1}^{H}{\bm{v}}^{(n-1)})^{H}{\bm{U}}_{1}^{H}]_{q}[{\bm{v}}]_{q}$
is a real number and $|[{\bm{v}}]_{q}|=1$ for any $q$, i.e., the optimal
$\bm{v}$ should be given as (15). $\Box$

From (15) and Lemma 1, we can see that the optimal phase vector $\bm{v}$
should be adjusted such that the signal that goes through all RISs is aligned
to be a signal vector with equal phase at each element. We can also see that
the optimal phase vector $\bm{v}$ is independent of the amplitude of the
channel ${\bm{U}}_{1}({\bm{g}}_{1}+{\bm{U}}_{1}^{H}{\bm{v}}^{(n-1)})$.

Algorithm 1 SCA Method for Phase Optimization

1: Initialize $\bm{v}^{(0)}$. Set iteration number $n=1$.

2: repeat

3: Set
$\bm{v}^{(n)}=\text{e}^{-j\angle({\bm{U}}_{1}({\bm{g}}_{1}+{\bm{U}}_{1}^{H}{\bm{v}}^{(n-1)}))},$
and $n=n+1$.

4: until the objective value (12) converges.

5: Output $\bm{\theta}=(\bm{v}^{(n)})^{*}$.

The SCA algorithm for solving problem (12) is summarized in Algorithm 1. The
convergence of Algorithm 1 is guaranteed by the following lemma that follows
directly from [31, Proposition 3]:

###### Lemma 2

The objective value (12) obtained in Algorithm 1 is monotonically non-
decreasing and the sequence $\bm{v}^{(n)}$ converges to a point fulfilling the
Karush-Kuhn-Tucker (KKT) optimal conditions of the original nonconvex problem
(12).

Lemma 2 shows that Algorithm 1 will always converge to a locally optimal
solution of problem (12).

####  III-A2 Second stage

We now obtain the optimal power allocation $p_{1}$. With the obtained
${\bm{\theta}}$ in Algorithm 1 and defining
$\bar{g}_{1}=\frac{\left|\bm{g}_{1}^{H}+\sum_{l=1}^{L}x_{l}\bm{h}_{1l}^{H}\bm{\Theta}_{l}\bm{G}_{l}\right|^{2}}{\sigma^{2}}$,
problem (10) reduces to:

$\displaystyle\mathop{\max}_{p_{1}}\quad$
$\displaystyle\frac{B\log_{2}\left(1+\bar{g}_{1}p_{1}\right)}{\mu
p_{1}+P_{0}}$ (16) s.t. $\displaystyle P_{\min}\leq p_{1}\leq P_{\max},$ (16a)

where $P_{0}=P_{1}+P_{\text{B }}+\sum_{l=1}^{L}x_{l}N_{l}P_{\text{R}}$, and
$P_{\min}=\frac{\left(2^{\frac{R_{1}}{B}}-1\right)}{\bar{g}_{1}}$. In (16a),
$P_{\min}$ is used to guarantee the minimum rate requirement for user 1.

For the energy efficiency optimization problem (16), the Dinkelbach method
from [32] can be used. The Dinkelbach method involves solving a series of
convex subproblems, which increases the computational complexity. However, the
optimal solution of (16) can be derived in closed form using the following
theorem.

###### Theorem 1

The optimal transmit power of the energy efficiency maximization problem in
(16) is:

$p_{1}=\left[\frac{\bar{g}_{1}P_{0}-\mu}{\mu\bar{g}_{1}W\left(\frac{(\bar{g}_{1}P_{0}-\mu)}{\mu\text{e}}\right)}-\frac{1}{\bar{g}_{1}}\right]_{P_{\min}}^{P_{\max}},$
(17)

where $W(\cdot)$ is the Lambert-W function and
$[a]_{b}^{c}=\min\\{\max\\{a,b\\},c\\}$.

Proof: The first-order derivative of the objective function (16) with respect
to power $p_{1}$ is:

$\frac{\partial\frac{B\log_{2}(1+\bar{g}_{1}p_{1})}{\mu p_{1}+P_{0}}}{\partial
p_{1}}=\frac{B({\bar{g}_{1}}(\mu
p_{1}+P_{0})-\mu(1+\bar{g}_{1}p_{1})\ln(1+\bar{g}_{1}p_{1}))}{(1+\bar{g}_{1}p_{1})(\mu
p_{1}+P_{0})^{2}\ln 2}.$ (18)

To show the increasing trend of the objective function (16), we further
denote:

$f_{1}(p_{1})=\bar{g}_{1}(\mu
p_{1}+P_{0})-\mu(1+\bar{g}_{1}p_{1})\ln(1+\bar{g}_{1}p_{1}),\quad\forall
p_{1}>0.$ (19)

The first-order derivative of function $f_{1}(p_{1})$ is:

$f_{1}^{\prime}(p_{1})=-\mu\bar{g}_{1}\ln(1+\bar{g}_{1}p_{1})<0,$ (20)

which indicates that $f_{1}(p_{1})$ is a monotonically decreasing function.
Since $f_{1}(0)=\bar{g}_{1}P_{0}>0$ and
$\lim_{p_{1}\rightarrow\infty}f_{1}(p_{1})<0$, there must exist a unique
$\bar{p}_{1}$ such that $f_{1}(\bar{p}_{1})=0$, where

$\bar{p}_{1}=\frac{\bar{g}_{1}P_{0}-\mu}{\mu\bar{g}_{1}W\left(\frac{(\bar{g}_{1}P_{0}-\mu)}{\mu\text{e}}\right)}-\frac{1}{\bar{g}_{1}}.$
(21)

Hence, the objective function (16) first increases in interval
$(0,\bar{p}_{1}]$ and then decreases in interval $(\bar{p}_{1},\infty)$, which
indicates that the optimal solution can be presented as in (17). $\Box$

From Theorem 1, the optimal power control of problem (16) is obtained in
closed-form, as shown in (17). According to (17), it is shown that the optimal
power control lies in one of three values, i.e., the minimum transmit power,
the power with zero first-order derivative, and the maximum transmit power.

###  III-B RIS On-Off Optimization

Substituting the phase and power variables $(\bm{\theta},p_{1})$ obtained in
the previous section, problem (9) becomes:

$\displaystyle\mathop{\max}_{\bm{x}}\quad$
$\displaystyle\frac{B\log_{2}\left(1+\frac{p_{1}\left|\bm{g}_{1}^{H}+\sum_{l=1}^{L}x_{l}\bm{h}_{1l}^{H}\bm{\Theta}_{l}\bm{G}_{l}\right|^{2}}{\sigma^{2}}\right)}{\mu
p_{1}+P_{1}+P_{\text{B}}+\sum_{l=1}^{L}x_{l}N_{l}P_{\text{R}}}$ (22) s.t.
$\displaystyle{B\log_{2}\left(1+\frac{p_{1}\left|\bm{g}_{1}^{H}+\sum_{l=1}^{L}x_{l}\bm{h}_{1l}^{H}\bm{\Theta}_{l}\bm{G}_{l}\right|^{2}}{\sigma^{2}}\right)}\geq
R_{1},$ (22a) $\displaystyle x_{l}\in\\{0,1\\},\quad\forall l\in\mathcal{L}.$
(22b)

We introduce an auxiliary variable $y$ and problem (22) is equivalent to:

$\displaystyle\mathop{\max}_{\bm{x},y}\quad$
$\displaystyle\frac{B\log_{2}\left(1+\frac{p_{1}y}{\sigma^{2}}\right)}{\mu
p_{1}+P_{1}+P_{\text{B}}+\sum_{l=1}^{L}x_{l}N_{l}P_{\text{R}}}$ (23) s.t.
$\displaystyle
y\leq{\left|\bm{g}_{1}^{H}+\sum_{l=1}^{L}x_{l}\bm{h}_{1l}^{H}\bm{\Theta}_{l}\bm{G}_{l}\right|^{2}},$
(23a) $\displaystyle
y\geq\left(2^{\frac{R_{1}}{B}}-1\right)\frac{\sigma^{2}}{p_{1}},$ (23b)
$\displaystyle x_{l}\in\\{0,1\\},\quad\forall l\in\mathcal{L},$ (23c)

where constraint (23b) is used to ensure the minimum rate demand. For the
optimal solution of problem (23), constraint (23a) will always hold with
equality since the objective function monotonically increases with $y$.
Although problem (23) has a simplifier form compared to (22), it is still a
nonconvex MINLP. There are two difficulties in solving problem (23). The first
difficulty is that objective function (23) has a fractional form, which is
difficult to solve. The second difficulty is that constraint (23a) is
nonconvex.

To handle the first difficulty, we use the parametric approach in [32] and
consider the following problem:

$H(\lambda)=\mathop{\max}_{(\bm{x},y)\in\mathcal{F}}{B\log_{2}\left(1+\frac{p_{1}y}{\sigma^{2}}\right)}-\lambda\left({\mu
p_{1}+P_{1}+P_{\text{B}}+\sum_{l=1}^{L}x_{l}N_{l}P_{\text{R}}}\right),$ (24)

where $\mathcal{F}$ is the feasible set of $(\bm{x},y)$ satisfying constraints
(23a)-(23c). It was proved in [32] that solving (23) is equivalent to finding
the root of the nonlinear function $H(\lambda)$, which can be obtained by
using the Dinkelbach method. By introducing parameter $\lambda$, the objective
function of problem (23) can be simplified, as shown in (24).

To handle the second difficulty, due to the fact that $x_{l}\in\\{0,1\\}$, we
can rewrite the right hand side of constraint (23a) as:

$\displaystyle\left|\bm{g}_{1}^{H}+\sum_{l=1}^{L}x_{l}\bm{h}_{1l}^{H}\bm{\Theta}_{l}\bm{G}_{l}\right|^{2}=D_{0}+\sum_{l=1}^{L}D_{l}x_{l}+\sum_{l=2}^{L}\sum_{m=1}^{l-1}D_{lm}x_{l}x_{m},$
(25)

where $D_{0}=\bm{g}_{1}^{H}\bm{g}_{1}$,
$D_{l}=\bm{h}_{1l}^{H}\bm{\Theta}_{l}\bm{G}_{l}\bm{G}_{l}^{H}\bm{\Theta}_{l}^{H}\bm{h}_{1l}+\bm{g}_{1}^{H}\bm{G}_{l}^{H}\bm{\Theta}_{l}^{H}\bm{h}_{1l}+\bm{h}_{1l}^{H}\bm{\Theta}_{l}\bm{G}_{l}\bm{g}_{1}$,
and
$D_{lm}=\bm{h}_{1l}^{H}\bm{\Theta}_{l}\bm{G}_{l}\bm{G}_{m}^{H}\bm{\Theta}_{m}^{H}\bm{h}_{1m}+\bm{h}_{1m}^{H}\bm{\Theta}_{m}\bm{G}_{m}\bm{G}_{l}^{H}\bm{\Theta}_{l}^{H}\bm{h}_{1l}$.
To solve problem (23), we introduce new variable $z_{lm}=x_{l}x_{m}$. Since
$x_{l}\in\\{0,1\\}$, constraint $z_{lm}=x_{l}x_{m}$ is equivalent to:

$z_{lm}\geq x_{l}+x_{m}-1,\quad 0\leq z_{lm}\leq 1,$ (26) $z_{lm}\leq
x_{l},\quad z_{lm}\leq x_{m},$ (27)

for all $l=2,\cdots,L$, $m=1,\cdots,l-1$. According to (24)-(27), problem (23)
can be reformulated as:

$\displaystyle\mathop{\max}_{\bm{x},y,\bm{z}}\quad$
$\displaystyle{B\log_{2}\left(1+\frac{p_{1}y}{\sigma^{2}}\right)}-\lambda\left({\mu
p_{1}+P_{1}+P_{\text{B}}+\sum_{l=1}^{L}x_{l}N_{l}P_{\text{R}}}\right)$ (28)
s.t. $\displaystyle y\leq
D_{0}+\sum_{l=1}^{L}D_{l}x_{l}+\sum_{l=2}^{L}\sum_{m=1}^{l-1}D_{lm}z_{lm},$
(28a) $\displaystyle z_{lm}\geq x_{l}+x_{m}-1,z_{lm}\leq x_{l},z_{lm}\leq
x_{m},\quad\forall l=2,\cdots,L,m=1,\cdots,l-1,$ (28b) $\displaystyle
y\geq\left(2^{\frac{R_{1}}{B}}-1\right)\frac{\sigma^{2}}{p_{1}},$ (28c)
$\displaystyle 0\leq z_{lm}\leq 1,\quad\forall l=2,\cdots,L,m=1,\cdots,l-1,$
(28d) $\displaystyle x_{l}\in\\{0,1\\},\quad\forall l\in\mathcal{L},$ (28e)

where $\bm{z}=[z_{21},z_{31},z_{32},\cdots,z_{L(L-1)}]^{T}$.

Due to constraints (28e), it is difficult to handle problem (28). By relaxing
the integer constraints (28e) with $x_{l}\in[0,1]$, problem (28) becomes a
convex problem. For problem (28) with relaxed constraints, the optimal
solution can be obtained through the dual method [33]. We show that the dual
method obtains the integer solution, which guarantees both optimality and
feasibility of the original problem. To obtain the optimal solution of problem
(28), we have the following theorem.

###### Theorem 2

For problem (28), the optimal RIS on-off vector $\bm{x}$ and auxiliary
variables $(y,\bm{z})$ can be respectively expressed as:

$x_{l}=\left\\{\begin{array}[]{ll}\\!\\!1,&\text{if}\;C_{l}>0,\\\
\\!\\!0,&\text{otherwise},\end{array}\right.$ (29)
$y=\left.\left(\frac{B}{(\ln
2)\alpha}-\frac{\sigma^{2}}{p_{1}}\right)\right|_{\left(2^{\frac{R_{1}}{B}}-1\right)\frac{\sigma^{2}}{p_{1}}},$
(30)

and

$z_{lm}=\left\\{\begin{array}[]{ll}\\!\\!1,&\text{if}\;\alpha
D_{lm}+\kappa_{1lm}-\kappa_{2lm}-\kappa_{3lm}>0,\\\
\\!\\!0,&\text{otherwise},\end{array}\right.$ (31)

where

$C_{l}\\!=\\!\left\\{\begin{array}[]{l}\\!\\!\\!-\lambda
N_{1}P_{\text{R}}+\alpha
D_{1}+\sum_{m=2}^{L}(\kappa_{3ml}-\kappa_{1ml}),\quad\text{if}\;l=1,\\\
\\!\\!\\!-\lambda N_{l}P_{\text{R}}+\alpha
D_{l}+\sum_{m=1}^{l-1}(\kappa_{2lm}-\kappa_{1lm})+\sum_{m=l+1}^{L}(\kappa_{3ml}-\kappa_{1ml}),\quad\text{if}\;2\leq
l\leq L-1,\\\ \\!\\!\\!-\lambda N_{L}P_{\text{R}}+\alpha
D_{L}+\sum_{m=1}^{L-1}(\kappa_{2lm}-\kappa_{1lm}),\quad\text{if}\;l=L,\end{array}\right.$
(32)

$\\{\alpha,\kappa_{1lm},\kappa_{2lm},\kappa_{3lm}\\}$ are the Lagrange
multipliers associated with corresponding constraints of problem (28), and
$a_{b}=\max\\{a,b\\}$.

Proof: The dual problem of problem (28) with relaxed constraints is:

$\mathop{\min}_{\alpha,\bm{\kappa}}\quad\mathcal{D}(\alpha,\bm{\kappa}),$ (33)

where

$\mathcal{D}(\alpha,\bm{\kappa})=\left\\{\begin{array}[]{ll}\\!\\!\\!\mathop{\max}\limits_{\bm{x},y,\bm{z}}&\mathcal{L}(\bm{x},y,\bm{z},\alpha,\bm{\kappa})\\\
\\!\\!\rm{s.t.}&y\geq\left(2^{\frac{R_{1}}{B}}-1\right)\frac{\sigma^{2}}{p_{1}},\\\
&0\leq z_{lm}\leq 1,\quad\forall l=2,\cdots,L,m=1,\cdots,l-1,\\\ &0\leq
x_{l}\leq 1,\quad\forall l\in\mathcal{L},\end{array}\right.$ (34)
$\displaystyle\mathcal{L}(\bm{x},y,$
$\displaystyle\bm{z},\alpha,\bm{\kappa})={B\log_{2}\left(1+\frac{p_{1}y}{\sigma^{2}}\right)}-\lambda\left({\mu
p_{1}+P_{1}+P_{\text{B}}+\sum_{l=1}^{L}x_{l}N_{l}P_{\text{R}}}\right)$
$\displaystyle+\alpha\left(D_{0}+\sum_{l=1}^{L}D_{l}x_{l}+\sum_{l=2}^{L}\sum_{m=1}^{l-1}D_{lm}z_{lm}-y\right)$
$\displaystyle+\sum_{l=2}^{L}\sum_{m=1}^{l-1}\left[\kappa_{1lm}(z_{lm}-x_{l}-x_{m}+1)+\kappa_{2lm}(x_{l}-z_{lm})+\kappa_{3lm}(x_{m}-z_{lm})\right],$
(35)

and
$\bm{\kappa}=\\{\kappa_{1lm},\kappa_{2lm},\kappa_{3lm}\\}_{l=2,\cdots,L,m=1,\cdots,l-1}$.

To maximize the objective function in (34), which is a linear combination of
$x_{l}$ and $z_{lm}$, we must let the positive coefficients corresponding to
the $x_{l}$ and $z_{lm}$ be 1. Therefore, the optimal $x_{l}$ and $z_{lm}$ are
thus given as (29) and (31), respectively.

To optimize $y$ from (34), we set the first derivative of objective function
to zero, i.e.,

$\frac{\partial\mathcal{L}(\bm{x},y,\bm{z},\alpha,\bm{\kappa})}{\partial
y}=\frac{Bp_{1}}{(p_{1}y+\sigma^{2})\ln 2}-\alpha=0,$ (36)

which yields $y=\frac{B}{(\ln 2)\alpha}-\frac{\sigma^{2}}{p_{1}}$. Considering
constraint (28c), we obtain the optimal solution to problem (28) as (30).
$\Box$

Theorem 2 states that RIS $l$ that has a positive coefficient $C_{l}$ should
be on. According to the expression of coefficient $C_{l}$ in (32), the
negative term $-\lambda N_{l}P_{\text{R}}$, is the effect of introducing
additional power consumption if RIS $l$ is on, while the remaining term
represents the benefit of increasing the transmit rate by keeping RIS $l$ in
operation. When $C_{l}>0$, the benefit of increasing the transmit rate is
larger than the effect of introducing additional power consumption, which
means that the energy efficiency can be improved if RIS $l$ is on.

The values of $(\alpha,\bm{\kappa})$ are determined by the sub-gradient method
[34, 35, 36, 37], which can be given by

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\alpha=\left[\alpha-\phi\left(D_{0}+\sum_{l=1}^{L}D_{l}x_{l}+\sum_{l=2}^{L}\sum_{m=1}^{l-1}D_{lm}z_{lm}-y\right)\right]^{+},$
(37)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\kappa_{1lm}=[\kappa_{1lm}-\phi(z_{lm}-x_{l}-x_{m}+1)]^{+},$
(38)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\kappa_{2lm}=\left[\kappa_{2lm}-\phi(x_{l}-z_{lm})\right]^{+},$
(39)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\kappa_{3lm}=\left[\kappa_{3lm}-\phi(x_{m}-z_{lm})\right]^{+},$
(40)

where $\phi>0$ is a dynamically chosen step-size sequence and
$[a]^{+}=\max(a,0)$.

Algorithm 2 Dual Method for Problem (23)

1: Initialize parameter $\lambda$ and set the accuracy $\epsilon$.

2: repeat

3: Initialize dual variables $(\alpha,\bm{\kappa})$.

4: repeat

5: Update the RIS on-off vector $\bm{x}$ and the auxiliary variables
$(y,\bm{z})$ according to (29)-(31).

6: Update dual variables $(\alpha,\bm{\kappa})$ based on (37)-(40).

7: until the objective value (28) converges

8: Denote the objective value (28) by $H(\lambda)$.

9: Update
$\lambda=\frac{B\log_{2}\left(1+\frac{p_{1}y}{\sigma^{2}}\right)}{\mu
p_{1}+P_{1}+P_{\text{B}}+\sum_{l=1}^{L}x_{l}N_{l}P_{\text{R}}}$.

10: until $H(\lambda)<\epsilon$.

By iteratively optimizing primal variables $(\bm{x},y,\bm{z})$ and dual
variables $(\alpha,\bm{\kappa})$, the optimal RIS on-off vector is obtained.
The dual method for solving problem (28) and the Dinkelbach method to update
parameter $\lambda$ are given in Algorithm 2. Notice that the optimal $x_{l}$
is either 0 or 1 according to (29), even though we relax $x_{l}$ as (34).
Consequently, the optimal solution to problem (28) is obtained by using the
dual method, i.e., $H(\lambda)$ in (24) is obtained for given $\lambda$. Using
the Dinkelbach method, we can obtain the root of $H(\lambda)=0$, which
indicates that the optimal solution of fractional energy efficiency
optimization problem (23) is obtained.

###  III-C Complexity Analysis

Algorithm 3 Iterative Optimization for Problem (9)

1: Initialize $(\bm{\theta}^{(0)},p_{1}^{(0)},\bm{x}^{(0)})$. Set iteration
number $n=1$.

2: repeat

3: Given $\bm{x}^{(n-1)}$, solve the phase optimization problem (12) by using
Algorithm 1 and the solution is denoted by $\bm{\theta}^{(n)}$.

4: Given $\bm{x}^{(n-1)}$ and the optimized $\bm{\theta}^{(n)}$, solve the
power control problem (16) according to Theorem 1 and the optimal power is
denoted by $p_{1}^{(n)}$.

5: Given $(\bm{\theta}^{(n)},p_{1}^{(n)})$, solve the RIS on-off optimization
problem (23) by using Algorithm 2 and the solution is denoted by
$\bm{x}^{(n)}$.

6: Set $n=n+1$.

7: until the objective value (9) converges.

The iterative algorithm for solving problem (9) is given in Algorithm 3. From
Algorithm 3, the main complexity of solving problem (9) lies in solving the
phase optimization problem (12) and the RIS on-off optimization problem (23).

According to Algorithm 1, to solve the phase optimization problem (12), the
complexity lies in computing
$\bm{v}^{(n)}=\text{e}^{-j\angle({\bm{U}}_{1}({\bm{g}}_{1}+{\bm{U}}_{1}^{H}{\bm{v}}^{(n-1)}))}$
at each iteration, which involves the complexity of $\mathcal{O}(QM)$. Hence,
the total complexity of solving problem (12) with Algorithm 1 is
$\mathcal{O}(T_{1}QM)$, where $T_{1}$ is the total number of the iterations of
Algorithm 1.

According to Algorithm 2, the main complexity of solving problem (23) lies in
solving RIS on-off vector $\bm{x}$, which involves the complexity of
$\mathcal{O}(L^{2})$ based on (29) and (32). Hence, the complexity of solving
problem (23) with Algorithm 2 is $\mathcal{O}(T_{2}T_{3}L^{2})$, where $T_{2}$
is the number of inner iterations by updating primal variables and dual
variables and $T_{3}$ is the number of inner iterations by updating the
parameter $\lambda$.

As a result, the total complexity of solving problem (9) is
$\mathcal{O}(T_{0}T_{1}QM+T_{0}T_{2}T_{3}L^{2})$, where $T_{0}$ is the total
number of iterations for Algorithm 3. The complexity of the proposed Algorithm
3 grows quadratically with the number of all RISs and this complexity is lower
than that of the SDR-based algorithm in [29].

##  IV Energy Efficiency Optimization with Multiple Users

In this section, we consider a general case with multiple users. To solve the
energy efficiency optimization problem in (6), an iterative algorithm with low
complexity is proposed via alternatingly optimizing the phase vector,
beamforming vector, and RIS on-off vector.

###  IV-A Phase Optimization

Given beamforming vector $\bm{w}$ and RIS on-off vector $\bm{x}$, the total
power consumption of the system is fixed and the energy efficiency
maximization is equivalent to the sum-rate maximization. Thus, given
$(\bm{w},\bm{x})$, problem (6) reduces to:

$\displaystyle\mathop{\max}_{\bm{\theta},\bm{\eta}}\quad$
$\displaystyle\sum_{k=1}^{K}\log_{2}\left(1+\eta_{k}\right)$ (41) s.t.
$\displaystyle\eta_{k}\leq\frac{\left|\left(\bm{g}_{k}^{H}+\sum_{l=1}^{L}x_{l}\bm{h}_{kl}^{H}\bm{\Theta}_{l}\bm{G}_{l}\right)\bm{w}_{k}\right|^{2}}{\sum_{i=1,i\neq
k}^{K}\left|\left(\bm{g}_{k}^{H}+\sum_{l=1}^{L}x_{l}\bm{h}_{kl}^{H}\bm{\Theta}_{l}\bm{G}_{l}\right)\bm{w}_{i}\right|^{2}+\sigma^{2}},\quad\forall
k\in\mathcal{K},$ (41a) $\displaystyle\eta_{k}\geq
2^{\frac{R_{k}}{B}}-1,\quad\forall k\in\mathcal{K},$ (41b)
$\displaystyle\theta_{ln}\in[0,2\pi],\quad\forall
l\in\mathcal{L},n\in\mathcal{N}_{l},$ (41c)

where $\bm{\eta}=[\eta_{1},\cdots,\eta_{K}]^{T}$. In (41), $\bm{\eta}$ is a
slack vector, which ensures that constraint (41a) always holds with equality
for the optimal solution. Constraint (41b) is added to guarantee the minimum
rate requirement of each user.

Before optimizing $\bm{\theta}$, we denote
$s_{ln}=\text{e}^{j\theta_{ln}},\forall l\in\mathcal{L},n\in\mathcal{N}_{l}$,
$\bm{s}_{l}=[s_{l1},\cdots,s_{lN_{l}}]^{T}$, and
$\bm{s}=[s_{11},\cdots,s_{1N_{1}},\cdots,s_{LN_{L}}]^{T}$. With the help of
$\bm{s}_{l}$, we show
$\bm{h}_{kl}^{H}\bm{\Theta}_{l}\bm{G}_{l}\bm{w}_{i}=\bm{t}_{kli}^{H}\bm{s}_{l}$,
where
$\bm{t}_{kli}=(\text{diag}(\bm{h}_{kl}^{H})\bm{G}_{l}\bm{w}_{i})^{*}\in\mathbb{C}^{N_{l}}$.
Hence, constraint (41a) can be rewritten as:

$\eta_{k}\leq\frac{\left|{\bm{t}}_{kk}^{H}{\bm{s}}+\bar{g}_{kk}\right|^{2}}{\sum_{i=1,i\neq
k}^{K}\left|{\bm{t}}_{ki}^{H}{\bm{s}}+\bar{g}_{ki}\right|^{2}+\sigma^{2}},\quad\forall
k\in\mathcal{K},$ (42)

where ${\bm{t}}_{ki}=[{\bm{t}}_{kl1};\cdots;{\bm{t}}_{klL}]$,
$\bar{g}_{kk}={\bm{g}}_{k}^{H}\bm{w}_{k}$, and
$\bar{g}_{ki}={\bm{g}}_{k}^{H}\bm{w}_{i}$.

Problem (41) can be reformulated as:

$\displaystyle\mathop{\max}_{\bm{s},\bm{\eta}}\quad$
$\displaystyle\sum_{k=1}^{K}\log_{2}\left(1+\eta_{k}\right)$ (43) s.t.
$\displaystyle|s_{ln}|=1,\quad\forall l\in\mathcal{L},n\in\mathcal{N}_{l},$
(43a) $\displaystyle(\ref{eemumax1eq0}),(\ref{eemumax1}b).$ (43b)

To handle the nonconvexity constraint (43a), we use the penalty method and
problem (43) can be rewritten as:

$\displaystyle\mathop{\max}_{\bm{s},\bm{\eta}}\quad$
$\displaystyle\sum_{k=1}^{K}\log_{2}\left(1+\eta_{k}\right)+C\sum_{l=1}^{L}\sum_{n=1}^{N_{l}}(|s_{ln}|^{2}-1)$
(44) s.t. $\displaystyle|s_{ln}|\leq 1,\quad\forall
l\in\mathcal{L},n\in\mathcal{N}_{l},$ (44a)
$\displaystyle(\ref{eemumax1eq0}),(\ref{eemumax1}b),$ (44b)

where $C$ is a large positive constant. Note that the penalty part
$C\sum_{l=1}^{L}\sum_{n=1}^{N_{l}}(|s_{ln}|^{2}-1)$ enforces that
$|s_{ln}|^{2}-1=0$ for the optimal solution of (44). To solve the nonconvex
problem in (44), we utilize the SCA method. The objective function of (44) can
be approximated by:

$\sum_{k=1}^{K}\log_{2}\left(1+\eta_{k}\right)+2C\sum_{l=1}^{L}\sum_{n=1}^{N_{l}}s_{ln}^{(n-1)}(s_{ln}-s_{ln}^{(n-1)}),$
(45)

where the second part is the first-order Taylor series of
$C\sum_{l=1}^{L}\sum_{n=1}^{N_{l}}(|s_{ln}|^{2}-1)$ and the superscript
$(n-1)$ means the value of the variable at the $(n-1)$-th iteration. To handle
the nonconvexity of constraint (42), we introduce variable $\beta_{k}$ and
constraint (42) is equivalent to:

${\left|{\bm{t}}_{kk}^{H}{\bm{s}}+\bar{g}_{kk}\right|^{2}}\geq\beta_{k}\eta_{k}=\frac{1}{4}((\beta_{k}+\eta_{k})^{2}-(\beta_{k}-\eta_{k})^{2}),$
(46)

and

${\sum_{i=1,i\neq
k}^{K}\left|{\bm{t}}_{ki}^{H}{\bm{s}}+\bar{g}_{ki}\right|^{2}+\sigma^{2}}\leq\beta_{k},$
(47)

where (47) is convex, while it remains to handle the nonconvexity of (46). We
adopt an approximation of the difference of two convex functions (DC) and
constraint (46) can be approximated by:

$\displaystyle\quad
2\mathcal{R}(({\bm{t}}_{kk}^{H}{\bm{s}}^{(n-1)}+\bar{g}_{kk})^{H}{\bm{t}}_{kk}^{H}{\bm{s}})-{\left|{\bm{t}}_{kk}^{H}{\bm{s}}^{(n-1)}+\bar{g}_{kk}\right|^{2}}$
$\displaystyle\geq\frac{1}{4}((\beta_{k}+\eta_{k})^{2}-(\beta_{k}^{(n-1)}-\eta_{k}^{(n-1)})(\beta_{k}-\eta_{k})+(\beta_{k}^{(n-1)}-\eta_{k}^{(n-1)})^{2}),$
(48)

where the left hand side is the first-order Taylor expansion of
${\left|{\bm{t}}_{kk}^{H}{\bm{s}}+\bar{g}_{kk}\right|^{2}}$ with respect to
$\bm{s}$ at $\bm{s}=\bm{s}^{(n-1)}$.

With the above approximations, the nonconvex problem in (44) can be formulated
in the following approximated convex problem:

$\displaystyle\mathop{\max}_{{\bm{s}},\bm{\eta},\bm{\beta}}\quad$
$\displaystyle\sum_{k=1}^{K}\log_{2}\left(1+\eta_{k}\right)+2C\sum_{l=1}^{L}\sum_{n=1}^{N_{l}}s_{ln}^{(n-1)}(s_{ln}-s_{ln}^{(n-1)}),$
(49) s.t. $\displaystyle\beta_{k}\geq 0,\quad\forall k\in\mathcal{K}$ (49a)
$\displaystyle(\ref{eemumax1eq1_2}),(\ref{eemumax1eq1_3}),(\ref{eemumax01}a),$
(49b)

where $\bm{\beta}=[\beta_{1},\cdots,\beta_{K}]^{T}$. Problem (41) can be
solved by using the SCA method, where the approximated convex problem (49) is
solved at each iteration. The detailed process of using the SCA method to
solve problem (41) is analogous to Algorithm 1.

###  IV-B Beamforming Optimization

Given phase vector $\bm{\theta}$ and RIS on-off vector $\bm{x}$, problem (6)
becomes:

$\displaystyle\mathop{\max}_{\bm{w},\bm{\zeta}}\quad$
$\displaystyle\frac{B\sum_{k=1}^{K}\log_{2}\left(1+\zeta_{k}\right)}{\mu\bm{w}^{H}\bm{w}+\sum_{k=1}^{K}P_{k}+P_{\text{B}}+\sum_{l=1}^{L}x_{l}N_{l}P_{\text{R}}}$
(50) s.t.
$\displaystyle\zeta_{k}\leq\frac{\left|\tilde{\bm{g}}_{k}^{H}\bm{w}_{k}\right|^{2}}{\sum_{i=1,i\neq
k}^{K}\left|\tilde{\bm{g}}_{k}^{H}\bm{w}_{i}\right|^{2}+\sigma^{2}},\quad\forall
k\in\mathcal{K},$ (50a) $\displaystyle\bm{w}^{H}\bm{w}\leq P_{\max},$ (50b)
$\displaystyle\zeta_{k}\geq 2^{\frac{R_{k}}{B}}-1,\quad\forall
k\in\mathcal{K},$ (50c)

where $\bm{\zeta}=[\zeta_{1},\cdots,\zeta_{K}]^{T}$ and
$\tilde{\bm{g}}_{k}=\bm{g}_{k}+\sum_{l=1}^{L}x_{l}\bm{G}_{l}^{H}\bm{\Theta}_{l}^{H}\bm{h}_{kl}$.
In (50), $\bm{\zeta}$ is a slack variable, which ensures that constraint (50a)
always holds with equality for the optimal solution.

To handle the nonconvexity of constraint (50a), we introduce a slack variable
$\gamma_{k}>0$ and reformulate constraints (50a) into the following equivalent
form:

$|\tilde{\bm{g}}_{k}^{H}\bm{w}_{k}|^{2}\geq\gamma_{k}\zeta_{k},$ (51)

and

$\sum_{i=1,i\neq
k}^{K}|\tilde{\bm{g}}_{k}^{H}\bm{w}_{i}|^{2}+\sigma^{2}\leq\gamma_{k}.$ (52)

Without loss of generality, the term $\tilde{\bm{g}}_{k}^{H}\bm{w}_{k}$ in
constraint (51) can be expressed as a real number through an arbitrary
rotation to beamforming $\bm{w}_{k}$. As a result, constraint (51) can be
equivalent to
$\mathcal{R}(\tilde{\bm{g}}_{k}^{H}\bm{w}_{k})\geq\sqrt{\gamma_{k}\zeta_{k}}.$
Replacing concave function $\sqrt{\gamma_{k}\zeta_{k}}$ with the first-order
Taylor series, constraint (51) becomes:

$\mathcal{R}(\tilde{\bm{g}}_{k}^{H}\bm{w}_{k})\geq\sqrt{\gamma_{k}^{(n-1)}\zeta_{k}^{(n-1)}}+\frac{1}{2}\sqrt{\frac{\gamma_{k}^{(n-1)}}{\zeta_{k}^{(n-1)}}}(\zeta_{k}-\zeta_{k}^{(n-1)})+\frac{1}{2}\sqrt{\frac{\zeta_{k}^{(n-1)}}{\gamma_{k}^{(n-1)}}}(\gamma_{k}-\gamma_{k}^{(n-1)}).$
(53)

Using the above approximations, the nonconvex problem in (50) can be
formulated in the following approximated problem:

$\displaystyle\mathop{\max}_{\bm{w},\bm{\zeta},\bm{\gamma}}\quad$
$\displaystyle\frac{B\sum_{k=1}^{K}\log_{2}\left(1+\zeta\right)}{\mu\bm{w}^{H}\bm{w}+\sum_{k=1}^{K}P_{k}+P_{\text{B}}+\sum_{l=1}^{L}x_{l}N_{l}P_{\text{R}}}$
(54) s.t.
$\displaystyle(\ref{miso1min2eq2}),(\ref{miso1min2eq1_3}),(\ref{eemubeamax1}b),(\ref{eemubeamax1}c),$
(54a) $\displaystyle\gamma_{k}\geq 0,\quad\forall k\in\mathcal{K}.$ (54b)

Since the objective function is a concave function divided by a convex
function and the feasible set is convex, the optimal solution of problem (54)
can be obtained by the Dinkelbach method in [32]. As a result, the original
beamfoming optimization problem in (50) can be solved by using the SCA method,
where (54) is solved optimally by using the Dinkelbach method at each
iteration.

###  IV-C RIS On-Off Optimization

Algorithm 4 Greedy Method For RIS On-Off Optimization

1: Initialize $\mathcal{A}=\\{1,\cdots,L\\}$ and $x_{l}=1$, $\forall
l\in\mathcal{A}$.

2: Calculate the objective value (6), which is denoted by $E_{0}$.

3: while $\mathcal{A}\neq\emptyset$ do

4: for $l\in\mathcal{A}$ do

5: Turn off RIS $l$, i.e., construct a new RIS on-off solution, $x_{l}=0$,
$x_{m}=1$, $x_{n}=0$, $\forall m\in\mathcal{A}\setminus\\{l\\}$,
$n\in\mathcal{L}\setminus\mathcal{A}$.

6: If the new RIS on-off solution is feasible, calculate the objective value
(6), which is denoted by $E_{l}$.

7: If the new RIS on-off solution is infeasible, set $E_{l}=0$.

8: end for

9: Calculate $k=\arg\max_{j\in\mathcal{A}\cup\\{0\\}}E_{j}$.

10: if $k\neq 0$ then

11: Set $\mathcal{A}=\mathcal{A}\setminus\\{k\\}$ and $E_{0}=E_{k}$.

12: else

13: Break and jump to Step 16.

14: end if

15: end while

16: Output $x_{l}=1$, $x_{m}=0$, $\forall l\in\mathcal{A}$,
$m\in\mathcal{L}\setminus\mathcal{A}$.

Given phase vector $\bm{\theta}$ and beamforming vector $\bm{w}$, problem (6)
is a nonlinear integer optimization problem with respect to the RIS on-off
vector $\bm{x}$. Since the nonlinear integer optimization problem is NP-hard
in general, it is hard to obtain the globally optimal solution with polynomial
complexity. To tackle this difficulty, we use the greedy method to solve the
RIS on-off optimization problem.

To solve problem (6) with fixed $(\bm{\theta},\bm{w})$, we propose a greedy
method based algorithm to optimize the integer RIS on-off vector, which is
given in Algorithm 4. The basic idea of Algorithm 4 is that we try to turn off
one RIS at each time if the objective value (6) can be improved and the new
solution is feasible.

At step 1 of Algorithm 4, set $\mathcal{A}$ is the set of active RISs that are
serving users. At step 2, $E_{0}$ is the initial objective value. Step 5 means
that we construct a new solution by turning off one RIS from $\mathcal{A}$. If
the new solution is feasible, we calculate the objective value (6), as shown
at step 6. Otherwise, the new solution is infeasible, we denote the objective
value as zero at step 7. At step 9, we compare the energy efficiency values of
these new solutions and the initial solution. If $k\neq 0$, the index value
$k$ means that turning off one RIS can increase the energy efficiency and
turning off RIS $k$ leads to the highest energy efficiency compared to turning
off any other RIS $l\neq k$. In this case, it is energy efficient to turn off
RIS $k$, then we subtract index $k$ from the active set $\mathcal{A}$ and
update the initial energy efficiency value for the next iteration, as shown at
step 11. If $k=0$, it means that turning off any one RIS can lead to low
energy efficiency. In this case, turning off any one RIS cannot increase the
energy efficiency, which means that we need to terminate the loop and output
the active set.

Since the objective value of problem (6) is increasing at each iteration and
the objective value always has a finite upper bound, Algorithm 4 must
converge.

###  IV-D Complexity Analysis

In summary, the iterative algorithm for solving the general multi-user energy
efficiency maximization problem in (6) is given in Algorithm 5. From Algorithm
5, the complexity of solving problem (6) is dominated by the complexity of
solving the phase optimization problem in (41), beamforming optimization
problem (50), and optimizing the RIS on-off status.

The phase optimization problem (41) is solved by using the SCA method. Since
there are $3K$ constraints in problem (41), the number of iterations that are
required for SCA method is $\mathcal{O}(\sqrt{2K+Q}\log_{2}(1/\epsilon_{1}))$
[38], where $\epsilon_{1}$ is the accuracy of the SCA method for solving
problem (41). At each iteration, the complexity of solving problem (49) is
$\mathcal{O}(S_{1}^{2}S_{2})$, where $S_{1}=2K+Q$ is the total number of
variables and $S_{2}=3K+Q$ is the total number of constraints [39]. Thus, the
total complexity of the SCA method for solving problem (41) is
$\mathcal{O}(K^{3.5}\log_{2}(1/\epsilon_{1}))$. With similar analysis, the
total complexity of the SCA method for solving the beamforming optimization
problem (50) is $\mathcal{O}(TK^{3.5}\log_{2}(1/\epsilon_{2}))$, where $T$ is
the number of iterations for solving problem (54) with the Dinkelbach method
and $\epsilon_{2}$ is the accuracy of the SCA method for solving problem (50).
According to Algorithm 4, the main complexity of the RIS on-off optimization
lies in calculating the objective value (6), which involves the complexity of
$\mathcal{O}(QM)$. Thus, the total complexity of Algorithm 4 is
$\mathcal{O}(L^{2}QM)$, where $\mathcal{O}(L^{2})$ is the total number of
iterations for Algorithm 4.

As a result, the total complexity of Algorithm 5 for solving problem (6) is
$\mathcal{O}(S_{3}K^{3.5}\log_{2}(1/\epsilon_{1})+S_{3}TK^{3.5}\log_{2}(1/\epsilon_{2})+S_{3}L^{2}QM)$,
where $S_{3}$ is the number of iterations for Algorithm 5. The proposed
Algorithm 5 has a lower complexity than the SDR-based algorithm in [29].

Algorithm 5 Iterative Optimization for Problem (6)

1: Initialize $(\bm{\theta}^{(0)},\bm{w}^{(0)},\bm{x}^{(0)})$. Set iteration
number $n=1$.

2: repeat

3: Given $(\bm{w}^{(n-1)},\bm{x}^{(n-1)})$, solve the phase optimization
problem (41) by using the SCA method in Section IV-A and the solution is
denoted by $\bm{\theta}^{(n)}$.

4: Given $(\bm{\theta}^{(n)},\bm{x}^{(n-1)})$, solve the beamforming
optimization problem (50) by using the SCA method in Section IV-B and the
solution is denoted by $\bm{w}^{(n)}$.

5: Given $(\bm{\theta}^{(n)},\bm{w}^{(n)})$, optimize the RIS on-off vector by
using Algorithm 4 and the solution is denoted by $\bm{x}^{(n)}$.

6: Set $n=n+1$.

7: until the objective value (6) converges.

##  V Simulation Results and Analysis

There are $K$ users uniformly distributed in a square area of size $300$ m
$\times$ $300$ m with the BS located at its center. There are $L$ RISs and the
location of RIS $l$ is given by $(\cos(2l\pi/L),\sin(2l\pi/L))\times 100$ m.
The main system parameters are listed in Table I [40]. The relay is assumed to
transmit with the maximum power $P_{\text{T}}$. Unless specified otherwise, we
choose a maximum transmit power BS $P_{\max}=50$ dBm for the BS, a total of
$M=8$ BS transmit antennas, a total of $L=8$ RISs, a penalty factor
$C=10^{3}$, $K=1$, an equal number of reflecting elements
$N_{1}=\cdots=N_{L}=N=4$, and an equal rate demand $R_{1}=\cdots=R_{K}=R=1$
Mbps (i.e., 1 bps/Hz). We compare the proposed scheme using distributed RISs
(labeled ‘DRIS’) with the following schemes: the conventional scheme with the
central deployment of one RIS located at (100, 0) m in [25] (labeled ‘CRIS’)
and the conventional AF relay scheme [41] (labeled ‘AFR’). In particular, the
number of reflecting elements for one central RIS in CRIS is set as the total
number of reflecting elements for all RISs in DRIS. In AFR, we consider the
same deployment of DRIS, i.e., there are $L$ AF relays, where AF $l$ with $N$
antennas is located at $(\cos(2l\pi/L),\sin(2l\pi/L))\times 100$ m.

TABLE I: System Parameters Parameters | Values
---|---
Bandwidth of the BS $B$ | 1 MHz
Noise power $\sigma^{2}$ |  $-104$ dBm
Maximum transmit power of the AF relay $P_{\text{T}}$ | 30 dBm
Small scale fading model, $\forall k,l,m,n$ | $[\bm{g}_{k}]_{m},[\bm{h}_{kl}]_{m},[\bm{G}_{l}]_{mn}\sim\mathcal{CN}(0,1)$
Large scale fading model at distance $d$ | $\frac{10^{-3.53}}{d^{3.76}}$
Circuit power of the BS $P_{\text{B}}$ | 39 dBm
Power amplifier efficiency at the BS/ AF relay $\nu$ | 0.8
Circuit power of each user $P_{k}$ | 10 dBm
Circuit power of each RIS element $P_{\text{R}}$ | 10 dBm
Circuit power of each AF relay transmit-receive antenna $P_{\text{A}}$ | 10 dBm
Figure 2: Convergence behaviour of Algorithm 1 with different initial
solutions.

Fig. 2 illustrates the convergence of Algorithm 1 using different initial
solutions. In this figure, the normalized channel gain value means the channel
gain divided by the noise power, i.e.,
$\frac{\left|{\bm{g}}_{1}+{\bm{U}}_{1}^{H}{\bm{v}}\right|^{2}}{\sigma^{2}}$.
It can be seen that the proposed algorithm converges fast, and eight
iterations are sufficient to converge, which shows the effectiveness of the
proposed algorithm.

Figure 3: Energy efficiency versus the maximum transmit power $P_{\max}$ of
the BS.

Fig. 3 shows how the energy efficiency changes as the maximum transmit power
of the BS varies. In this figure, the EXH-DRIS scheme is an exhaustive search
method that can find a near optimal solution of problem (9). Hereinafter, the
EXH-DRIS scheme refers to the proposed DRIS algorithm with 1000 initial
starting points. In this simulation, EXH-DRIS can obtain 1000 solutions, and
the solution with the highest energy efficiency is treated as the near optimal
solution. It is shown that the energy efficiency of all schemes first
increases and then remains stable as the maximum transmit power of the BS
increases. This is because energy efficiency is not a monotonically increasing
function of the maximum transmit power, as shown in (17). When $P_{\max}\geq
25$ dBm, the exceed transmit power is not used since it will decrease the
energy efficiency. Fig. 3 also shows that the proposed DRIS scheme outperforms
the CRIS and AFR schemes. For high maximum transmit power of the BS, DRIS can
increase up to 27% and 68% energy efficiency compared to CRIS and AFR,
respectively. Moreover, the proposed DRIS scheme achieves almost the same
performance as the EXH-DRIS scheme, which indicates that the proposed DRIS can
achieve the near optimum solution.

Figure 4: Sum-rate versus the maximum transmit power $P_{\max}$ of the BS with
$R=0$.

Fig. 4 shows how the sum-rate changes as the maximum transmit power of the BS
varies. It is found that the sum-rate of all schemes linearly increases with
the logarithmic maximum transmit power of the BS. We can see that AFR achieves
the best performance. This is because the AF relay is an active terminal by
transmitting the received signal to the user, while the RIS is only a passive
reflecting structure. From Fig. 4, DRIS can increase up to 26% sum-rate
compared to CRIS. This is due to the benefits of distributed deployment.
Multiple RISs are spatially distributed in DRIS, which can provide more than
one path of received signal compared to CRIS with only one central RIS.

Figure 5: Energy efficiency versus the minimum rate demand $R$.

Fig. 5 shows the energy efficiency versus the minimum rate demand. From this
figure, DRIS achieves the best performance. In particular, DRIS can achieve up
to 33% and 67% gains in terms of energy efficiency compared to CRIS and AFR,
respectively. Both DRIS and CRIS always achieve a better performance than AFR.
This is due to the fact that the power consumption for the RIS is much lower
compared to the transmit power of the AF relay. It is also found that DRIS
achieves better performance than CRIS, which indicates the benefit of
distributed deployment of RISs. This is because DRIS exhibits a better
spectrum efficiency compared to CRIS, and CRIS is more sensitive to high
minimum rate demand than DRIS. From Fig. 5, we can observe that the energy
efficiency remains stable when minimum rate demand is low. However, for a high
minimum rate demand, the energy efficiency decreases rapidly for both CRIS and
DRIS. This is because a high minimum rate demand requires the BS to transmit
with high power, which consequently degrades the energy efficiency
performance. Fig. 5 also demonstrates that, as the minimum rate demand
increases, the energy efficiency of CRIS decreases faster than DRIS.

Figure 6: Energy efficiency versus the number of reflecting elements $N$ for
each RIS with $L=4$. Figure 7: Energy efficiency versus the number of RISs $L$
with $N=4$.

Figs. 6 and 7 show the energy efficiency versus the number of reflecting
elements for each RIS and the number of RISs, respectively. From these
figures, we can see that the energy efficiency of DRIS monotonically increases
with the number of reflecting elements and the number of RISs. This is because
large number of reflecting elements and RISs can lead to high spectral
efficiency and the power consumption of introducing additional reflecting
elements and RISs is low, which result in high energy efficiency of the
system. According to Figs. 6 and 7, it is also found that the energy
efficiency of DRIS increases faster with the number of RISs than that with the
number of reflecting elements for each RIS, which shows that it is more energy
efficient to deploy with multiple RISs. For the same total number of all
reflecting elements, i.e., the same $NL$, we consider the following two
configurations: (a) $N=12$, $L=4$, $P_{\max}=25$ dBm, and (b) $N=4$, $L=12$,
$P_{\max}=25$ dBm. From Figs. 6 and 7, the energy efficiency of configuration
(a) is 4.0 Mbits/Joule and the energy efficiency of configuration (b) is 4.3
Mbits/Joule, which shows that the energy efficiency of $L>N$ is better than
that of $L<N$.

Figure 8: Energy efficiency versus the number of transmit antennas $M$ at the
BS.

In Fig. 8, we show the energy efficiency versus the number of transmit
antennas at the BS with various maximum transmit power of the BS. Fig. 8
demonstrates that the energy efficiency increases rapidly for a small number
of transmit antennas at the BS, however, this increase becomes slower for a
larger number of transmit antennas at the BS. This is because a high number of
transmit antennas at the BS leads to high power consumption, which
consequently decreases the slope of increase of the energy efficiency. From
Fig. 8, we can also see that the energy efficiency increases with the maximum
transmit power of the BS for a given number of transmit antenna at the BS.
Moreover, for high value of the BS maximum transmit power, the energy
efficiency slowly increases with the maximum transmit power.

Figure 9: Energy efficiency versus the number of users.

The energy efficiency versus the number of users is shown in Fig. 9. From Fig.
9, we observe that the energy efficiency of AFR is always low due to the fact
that the transmit power of the AF relay is high. Clearly, the proposed DRIS is
always better than CRIS and AFR especially when the number of users is large.

##  VI Conclusion

In this paper, we have investigated the resource allocation problem for a
wireless communication network with distributed RISs. The RIS phase shifts, BS
transmit beamforming, and RIS on-off status were jointly optimized to maximize
the system energy efficiency while satisfying minimum rate demand, maximum
transmit power, and unit-modulus constraints. To solve this problem, we have
proposed two iterative algorithms with low complexity for the single-user case
and multi-user case, respectively. In particular, the phase optimization
problem was solved by using the SCA method, where the closed-form solution was
obtained at each step for the single-user case. Numerical results have shown
that the proposed scheme outperforms conventional schemes in terms of energy
efficiency, especially for small maximum transmit power and large number of
users.

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