# Energy-Efficient Active Element Selection in RIS-aided Massive MIMO Systems

Wilson Souza Jr, Taufik Abrão, José Carlos Marinello J. C. Marinello is with
the Electrical Engineering Department, Federal University of Technology PR,
Cornélio Procópio, PR, Brazil. (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>jcmarinello@utfpr.edu.br).Wilson Souza Jr and T.
Abrao are with the Department of Electrical Engineering Londrina State
University, Parana Brazil (e-mail<EMAIL_ADDRESS>
###### Abstract

This paper delves into the critical aspects of optimizing energy efficiency
(EE) in active reconfigurable intelligent surface (RIS)-assisted massive MIMO
(M-MIMO) wireless communication systems. We develop a comprehensive and
unified theoretical framework to analyze the boundaries of EE within M-MIMO
systems integrated with active RIS while adhering to practical constraints.
Our research focuses on a formulated EE optimization problem aiming to
maximize the EE for active RIS-assisted M-MIMO communication systems. Our goal
is strategically finding the number of active RIS elements for outperforming
the EE attainable by an entirely passive RIS. Besides, the proposed novel
solution has been specifically tailored to the innovative problem considered.
The formulation and solution design take into account analytical optimization
techniques, such as lagrangian dual transform (LDT) and fractional programming
(FP) optimization, facilitating the effective implementation of RIS-aided
M-MIMO applications in real-world settings. In particular, our results show
that the proposed algorithm is able to provide up to 120$\%$ higher EE in
comparison to the entirely passive RIS. Besides, we found that the active RIS
can operate with lower than half of reflecting elements with respect to the
entirely passive RIS. Finally, in view of active RIS achieving the complete
utilization of amplification power available, it should be equipped with a
reasonable number of reflecting elements, above $N=49$.

###### Index Terms:

RIS element selection; Energy efficiency (EE); Massive MIMO (M-MIMO); Spectral
efficiency; Reconfigurable intelligent surface (RIS); Heuristic evolutionary
optimization; Convex optimization.

##  I Introduction

The proliferation of the fifth generation (5G) wireless network has catalyzed
worldwide incentive and development efforts in research, resulting in an
intensified exploration of possibilities offered by the sixth generation (6G)
wireless network. Projected to be characterized and recognized by exceptional
data-driven capabilities and instantaneous connectivity global ubiquity, 6G is
expected to provide a wide range of applications and services that will
revolutionize communication systems. These innovations span across diverse
domains, such as extended reality (XR), holographic communications, etc.,
promising to redefine the technological scenario. The advent of 6G will bring
forth plenty of new services and withal elevate existing ones to higher
levels. However, in view of reaching this aim, the integration of cutting-edge
technologies becomes essential to facilitate the deployment of these services
and ensure their optimal functioning in the technological scenario. In
essence, the commercialization of 6G will necessitate a proactive approach in
adopting state-of-the-art technologies, marking a paradigm shift in how we
envision wireless communication networks’ potential.

Among all emerging technologies, RIS stands out as one of the most promising
recent enabling techniques poised to revolutionize future communication
systems, extending beyond the 5G and 6G eras. RIS has the potential to
significantly enhance transmission capabilities by establishing a “virtual”
re-configurable/re-programmable communication link. Essentially the RIS, is a
thin planar surface consisting of a massive number of low-cost reflecting
elements composed of meta-materials, to reconfigure and redirect the impinging
signal, by changing its electromagnetic properties (phase and/or magnitude)
[1], so that, the reflected signal can be focused toward a specific direction
with a specific gain. The electromagnetic properties of the impinging waves
are precisely controlled by utilizing electronic components such as PIN diodes
or radio frequency (RF) switches integrated into the RIS panel. For instance,
changing the impedance of PIN diodes or tuning the RF switch facilitates the
manipulation of the reflection with respect to its direction and amplitude,
respectively. This dynamic adjustment serves to enhance specific target
metrics. Therefore, RIS is promising due to its capacity to control in a
software-defined manner the wireless environment and its potential to mitigate
certain intrinsic effects. The specific advantages of the RIS arrays can be
summarized as follows [2]:
1) Easiness in implementation;
2) spectral efficiency (SE) and EE enhancement;
3) Sustainability;
4) Compatibility.

Due to the aforementioned characteristics, RIS s are recognized as a potential
solution for solving a wide range of challenges in commercial and civilian
applications. Besides, in contrast to conventional relaying methods such as
Amplify-and-Forward (AF) and Decode-and-Forward (DF), RIS presents many
advantages. It offers a cost-effective yet efficient solution that optimizes
both system’s EE and SE. On this matter, like any emerging technology, new
applications and a range of problems arise as a result. Besides, from the
perspective of communication and signal processing, open problems such as
phase-shift optimization of scattering elements, and channel acquisition of
additional links imposed by deploying the RIS are extremely necessary to be
addressed and efficiently solved.

Although RIS presents a range of advantages, there are some critical issues
that are found in real-world applications. Consequently, various architectural
variants for the RIS have been introduced to address these challenges and
enhance its overall effectiveness. These variations aim to significantly
enhance performance, offering innovative solutions to overcome challenges and
elevate the effectiveness of RIS-supported communication systems. Among them,
we can remark the active RIS structure, which is an alternative for providing
better conditions for the double-fading attenuation link. In active RIS, each
reflecting element is supported by a set of active-load impedances that enable
the active RIS to operate as an active reflector reflecting and applying power
amplification on the impinging signal. An interesting advantage of the active
RIS elements is their capability to estimate the channel between the
transmitter-RIS in downlink (DL) and uplink (UL) at the RIS side [3]. Besides,
since the active RIS has the ability to amplify the incident signal, fewer
reflecting elements are required to achieve the same signal-to-noise ratio
(SNR) at the receiver when compared with the conventional entirely passive RIS
[4]. However, evaluating the EE for the entirely passive RIS versus active RIS
is an important problem that should be further investigated.

Conversely, the proliferation of M-MIMO technology, characterized by the
deployment of an extensive array of transmit/receiver antennas at base station
(BS), plays a pivotal role in enhancing the SE of the system. This technology
is instrumental in efficiently accommodating multiples users’ equipment (UEs)
by leveraging the same physical resources. This strategic utilization enhances
the SE and optimizes the resource utilization, exemplifying the versatility
and efficiency of M-MIMO in contemporary wireless communication systems.

Besides the aforementioned technologies, a promising technology to be present
in future communications networks is the millimeter-wave (mmWave) technology.
It emerges as a strategic pivotal solution in addressing the evolving
connectivity demands. Operating within the high-frequency spectrum, typically
spanning from 30 to 100 GHz, mmWave technology presents a promising solution
capable of supplying characteristics such as the utilization of higher
bandwidth for contemporary communication networks. Furthermore, with the large
arrays, the near-field emerges, enabling the beam focusing instead of the
traditional beam steering for far-field communications [5, 6, 7].

However, the implementation of these higher-frequency systems, which often
entails the use of large-scale arrays at the BS or the RIS, introduces
inherent complexities in both passive and active beamforming, demanding low-
complexity, approximate yet efficient solutions for accurate channel
estimation and RIS configuration. The joint operation of RIS, M-MIMO, and
mmWave technologies emerges as key enablers for future communication systems.
Expected to play central roles in the upcoming generation of wireless
networks, these technologies are designed to support high data rates and
accommodate a substantial number of connected UEs, operating with expanded
array sizes. In response to this evolution, there is a fundamental growth to
prioritize the system’s EE, measured in bits-per-Joule111Alternatively, Joule-
per-bit., as a crucial performance metric. The focus on EE emphasizes the
matter of developing sustainable communication systems, especially as
efficient energy utilization becomes a global concern across various fields,
including telecommunications. In function of alarming levels of climate
change, there is an urgent need to limit global warming to 1.5 degrees above
the pre-industrial mark [8]. Within this context, the development of the next-
generation telecommunication standard, often referred to as Beyond-5G (B5G) or
6G, assumes particular importance. It aims to establish a highly energy-
efficient system capable of meeting expected requirements, such as a peak
throughput of 1 [Tbps] and latency in the order of microseconds [9]. This
forward-looking approach aligns with the broader goal of enabling
technological advancements that not only meet the requirements necessary in
terms of performance but actively contribute to the sustainability challenges
of our time.

In light of these challenges, the development of pioneering strategies that
not only enhance EE but also facilitate the intricate processes of channel
estimation and configuration for large-scale array systems is of paramount
importance. In this paper, our focus stays dedicated to addressing these
challenges with innovative approaches and practical solutions tailored to the
modern network’s infrastructures.

###  I-A Contributions of the paper

The contributions of this paper lie in analyzing the RIS-assisted M-MIMO
scenario, which will become essential components of future communication
systems. Specifically, the aim is to devise a solution that effectively
optimizes the signal amplification of active RIS concerning the EE, envisaging
the determination of the number of active RIS elements that outperforms
entirely passive RIS. Elaborate further, in this paper, we systematically:

  * •

Propose a comprehensive optimization methodology for system EE maximization,
focusing on the deployment of specific optimization tools, including convex
optimization procedures in view of reducing the computational complexity
aiming the application of the algorithm in a real-world scenario;

  * •

Optimize the phase and amplitude of the active RIS in a sustainable
perspective, $i.e.$, the phase and amplitude that results in the power
consumption necessary to obtain high-performance gains sufficient to
outperform the totally passive RIS;

  * •

Investigate the number of RIS active elements to be selected that surpasses
the EE performance of the entirely passive RIS within the system EE
optimization framework and different configurations/scenarios parameters.

###  I-B Organization

The remainder of this paper is organized into seven sections. These sections
are meticulously structured to provide a comprehensive exploration of the
various aspects of the RIS-assisted M-MIMO system. In Section II, we summarize
the related works; while in Section III, we present the groundwork by
formulating a general system model that effectively captures the key aspects
associated with the integration of the active RIS into the M-MIMO systems.
Also, we explore the SE and EE definitions and present the coupling power
consumption model for RIS-assisted M-MIMO systems. In Section IV, we review
the definitions, features, and concepts of some fundamental techniques for
addressing our intended problem, such as FP techniques and sum-of-ratio
techniques. Moving forward, in Section V, we meticulously present the defined
problem formulation, emphasizing its relevance and significance in
contemporary real-world applications. The motivations and justifications for
adopting this particular problem are thoroughly expounded, shedding light on
the specific challenges and opportunities that emphasize its importance in
modern communication systems. Section VI provides a complete and detailed
description of our proposed solution for the EE problem in active RIS-assisted
M-MIMO systems. Section VII presents numerical results, accompanied by an
extensive and in-depth discussion on the effectiveness, efficiency, and
potential of the proposed technique. The section also includes a relevant
discussion on the advantages/drawbacks of the active RIS structure in contrast
to the entirely passive RIS within the context of EE.

Finally, in Section VIII, we provide a relevant and comprehensive discussion
that encapsulates our perspective into the problem-solving approach adopted
throughout this paper as well as the obtained insights. Our analysis and
conclusions are framed within the broader context of the current state of the
art, aiming to provide valuable insights and potential directions for future
research.

##  II Related Works

An extensive number of works including [13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26, 27, 28, 29, 30] have studied the RIS deployment impact in
M-MIMO systems under different objectives, such as: maximizing the weigthed
sum-rate (WSR) [13, 14], maximizing the sum-rate (SR) [15], maximizing the EE
[16, 17, 18], RIS-assisted sensing [19, 20], minimizing the total transmit
power [21], maximizing the minimum rate [22, 23], for interference nulling
[24, 25], and active RIS [26, 27, 28, 29, 30].

The above-cited studies demonstrate the immense potential and versatility of
RIS within wireless systems. The work in [13] investigates the weighted SR
maximization. It uses a convex optimization approach to design the BS
precoding matrix and the RIS-phase shift for the passive RIS. The paper
proposes two different techniques for the analyzed problem. The proposed
optimization method utilizes FP techniques aiming at the problem
”convexation”. Besides, an block coordinate descent (BCD) method is utilized
to optimize sequentially the BS precoding matrix and the RIS phase shift,
where analytical methods and Riemannian conjugate gradient have been utilized.
The proposed method is available under two different scenarios: perfect and
imperfect channel estimation. Numerical simulations confirm the effectiveness
and efficiency of the proposed method.

The work in [14] suggests that by attempting to reduce inter-cell
interference, the weighted SR in RIS-aided DL multicell could be maximized.
Hence, to achieve this aim, [14] jointly optimizes the active precoding
matrices at BS s and the phase shifts at the RIS subject to each BS ’s power
constraint and unit modulus constraint, respectively. In this case, the
optimization problem does not consider any Quality of Service (QoS)
restriction, $e.g.$, a minimum rate requirement for the users. As a result,
this solution tends to maximize the rate of users with better channel
conditions.

In [15], the authors investigated the ergodic rate of an RIS-assisted M-MIMO
system. Applying random matrix theory, the authors derive a closed-form lower
bound expression for the system ergodic rate. The derived bound reveals
tightness over several system and channel parameters. Besides, it is promising
since the statistical channel state information (CSI) optimization can be
deployed based on this bound, being useful to alleviate the channel estimation
burden since it matches very well with the expected results. Besides, they
provided a gradient-ascent method to optimize the RIS angle phase shift in
view of maximizing the system SR.

The EE maximization problem is addressed in [16] by considering UL multi-user
single-antenna devices and Multiple-input single-output (MISO) RIS-aided
systems. The joint optimization involves the transmit power at users, phase
shift at the RIS, and combining matrix at BS receiver. The proposed BCD
iterative procedure can handle one variable during each iteration while the
others are held fixed. Numerical results indicate that the proposed scheme
outperforms baselines that jointly optimize two of three variables.

In [17], the joint number of antennas, power allocation, and passive RIS
reflecting coefficients optimization for EE maximization with statistical CSI
is studied. Analytical techniques have been applied in view of maximizing the
EE with low overhead for channel estimation. EE problem has been maximized
with a low-complexity algorithm, where the power allocation, RIS phase shift,
and the number of active antennas at BS have been optimized based on closed-
form expressions, utilizing BCD method. The method was demonstrated to be
promising, outperforming the gradient ascend methodology for passive RIS
reflecting coefficients optimization with statistical CSI.

In [18], the authors also analyzed the EE in RIS-assisted systems. In this
study, the EE maximization has been analyzed under joint optimization of power
consumption for both RIS and BS. The authors proposed an innovative power
consumption model on the RIS by considering the ON and OFF state of RIS unit
reflecting element, matching further precisely with the real-world behavior of
physical components. The paper also proposes low-complexity and effective
algorithms, which have been proven by the simulation results to be promising.

In [19], the authors studied the integrated sensing and communication (ISAC)
system for a double RIS-assisted system. The research primarily focuses on
addressing the challenge of mutual interference between the radar and
communication functionalities by leveraging a joint optimization approach for
beamforming and radar operations. To manage the complexity of the solution,
the authors introduce a penalty dual decomposition (PDD)-based approach and a
BCD algorithm. These methods effectively reduce computational complexity,
offering practical and efficient solutions for real-world implementation.
Moreover, the work presented in [20] addresses the challenges of a sensing-
assisted multi-user mmWave system. The study focuses on optimizing the channel
sensing duration for each UE to enhance the overall system SR. By effectively
managing the sensing duration, the research aims to improve the performance of
the mmWave system, offering potential advancements for future wireless
communication technologies.

In [21], the phase shift and BS beamforming optimization techniques have been
proposed in view of minimizing the UL transmit power in an RIS-aided Internet
of Things (IoT) network. The proposed technique was demonstrated to be
promising since it can reduce to about half of the UL transmit power over the
conventional scheme without RIS. The methodology exploits the product
Riemannian manifold structure of the sets of unit-modulus phase shifts and
unit-norm beamforming vectors. Besides, the manifold method converts the non-
convex UL transmit power minimization problem into an unconstrained problem
and then finds the optimal solution over the product Riemannian manifold.

In [22], the focus was on the study of the maximum-minimum rate problem. The
research delved into the joint optimization of transmit and passive
beamforming, considering the uncertainty of UE s locations. To address this
challenge, the authors proposed an alternating optimization approach combined
with sucessive convex approximation (SCA) technique and a penalty algorithm
employing a double loop.

In the research paper [23], the authors demonstrated that the non-convex
$\mathrm{max}-\mathrm{min}$ rate problem in RIS-assisted multi-user
communication systems can be reformulated into a second order conic
programming (SOCP) problem. They also explored the application of semi-defined
relaxation (SDR) and SCA techniques to optimize the passive and active
beamforming for the system under consideration.

In [24], the RIS structure is deployed aiming at interference nulling, where
an alternating projection method is utilized. This method is able to converge
for a solution that completely eliminates the interference. However, the
number of elements for achieving this goal can vary according to some
parameters, such as the number of served UEs. Besides, the authors also
explored the max-min rate problem. The numerical results are revealed to be
promising since the proposed procedure achieves high performance with low
complexity. Adhering to a similar approach, the authors in [25] proposed a
method to determine the angles for two distinct passive RISs, aiming to
achieve complete orthogonalization of channels between UEs and thereby nulling
interference. The investigation delves specifically into both diagonal and
beyond-diagonal RISs, employing a Manifold-based algorithm to find the angles
for both RISs, adhering to the unitary matrix constraint essential for
achieving complete orthogonalization of channels between the UEs. Notably, the
study highlights that signal amplification is unnecessary for achieving
orthogonalization. Furthermore, the paper introduces an efficient channel
estimation method to complement the proposed approach.

Our previous related works and contributions on the resource efficiency (RE)
optimization, EE and SE in M-MIMO systems aided or not by RIS include: [17,
31, 32, 33, 34, 35, 36, 37, 38, 39]. In this paper, we propose to deploy the
FP methodology to solve the phase-shift and amplification coefficients of an
active RIS aiming to EE maximization problem considering the context of the DL
RIS-aided multiuser M-MIMO systems.

##  III General System Model for RIS-aided M-MIMO

We start the section by describing the general model for M-MIMO systems
assisted by an RIS setup considered. This system model is conceived to cover a
diverse range of practical scenarios. Subsequently, we present the models for
the DL received baseband signals. Built such model, we finish the section by
formulating expressions for SE and EE within the specified RIS-aided M-MIMO
scenarios. The notation deployed in this paper is entirely outlined in Table
I.

TABLE I: List of notation and symbols for the RIS-aided M-MIMO Scenario  Symbol | Description
---|---
$M\in\mathbb{Z}_{+}$ | Number of BS antennas
$K\in\mathbb{Z}_{+}$ | Number of UEs
$N\in\mathbb{Z}_{+}$ | Number of total RIS elements
$\mathcal{K}=\\{1,2,\dots,K\\}$ |  UEs set
$\mathcal{N}=\\{1,2,\dots,N\\}$ |  RISs elements set
$\lambda>0$ | Carrier wavelength
$\beta_{1}\in\mathbb{R}$ | Large-scale fading coefficient of the channel from BS to RIS
$\beta_{2,k}\in\mathbb{R}$ | Large-scale fading coefficient of the channel from RIS to $k$-th UE
$\bm{H}\in\mathbb{C}^{M\times K}$ | Cascaded channel matrix of the BS and UEs through RIS
$\bm{H}_{1}\in\mathbb{C}^{M\times N}$ | Channel between the BS and RIS
$\bar{\bm{H}}_{1}\in\mathbb{C}^{M\times N}$ |  Line-of-sight (LoS) component of the BS and RIS channel
$\tilde{\bm{H}}_{1}\in\mathbb{C}^{M\times N}$ | Multipath component of the BS and RIS channel
$\bm{H}_{2}\in\mathbb{C}^{N\times K}$ | Channel between the RIS and UE s
$\bar{\bm{H}}_{2}\in\mathbb{C}^{N\times K}$ |  LoS component of the RIS and UE s channel
$\tilde{\bm{H}}_{2}\in\mathbb{C}^{N\times K}$ | Multipath component of the RIS and UE s channel
$\bm{v}\in\mathbb{C}^{N}$ |  RIS amplitude/phase-shift vector
$\bm{\Phi}\in\mathbb{C}^{N\times N}$ |  RIS amplitude/phase-shift diagonal matrix
$\kappa_{1}\in[0,\infty)$ | Rician $\kappa$-factor for RIS and BS channel
$\kappa_{2,k}\in[0,\infty)$ | Rician $\kappa$-factor for RIS and $k$-th UE
$p_{k}$ $\in$ $\mathbb{R}^{+}$ | Power designated to the $k$-th UE
$\bm{W}$ $\in$ $\mathbb{C}^{M\times K}$ | Precoding matrix
$s_{k}$ $\in$ $\mathbb{C}$ | Information symbol intended to the $k$-th UE
$n_{k}$ $\in$ $\mathbb{C}$ |  additive white gaussian noise (AWGN) sample at the $k$-th UE
$\bm{z}$ $\in$ $\mathbb{C}^{N\times 1}$ |  AWGN sample vector at the RIS
$\alpha_{n}$ $\in\mathbb{R}_{+}$ | Amplitude value of $n$-th element of RIS
$\theta_{n}$ $\in[0,2\pi)$ | Phase-shift value of $n$-th element of RIS
$\alpha_{\max}$ $\in\mathbb{R}_{+}^{*}$ | Maximum amplitude provided by the RIS
$P_{\max}^{\rm RIS}$ $\in\mathbb{R}_{+}$ | Maximum amplification power of RIS

Consider a DL RIS-assisted M-MIMO scenario where the direct link between the
BS and UEs is completely obstructed, thus, $K$ single antenna UEs are served
by the BS with $M$ elements through an RIS with $N$ elements. Besides, we
assume a direct connection between the RIS and the BS is made by means of a
dedicated link via a programmable controller, enabling the BS to manage both
phase shifts and amplitude coefficients of the RIS. Furthermore, we assume
that the time necessary for RIS amplitude/phase configuration is shorter than
the channel coherence time, $i.e.$, at the start of each time slot, the BS
should configure the RIS appropriately for its operation. Let us denote
$\bm{H}_{1}$ as the channel from the BS to the RIS and $\bm{H}_{2}$ as the
channel from the RIS to the UEs. The cascaded channel between the BS and UEs
through the RIS can be given as
$\bm{H}=\left[\bm{h}_{1},\bm{h}_{2},\dots,\bm{h}_{K}\right]$; more
specifically, denoted as

$\bm{h}_{k}=\bm{H}_{1}\bm{\Phi}\bm{h}_{2,k},\quad\forall k\in\mathcal{K},$ (1)

where $\bm{\Phi}={\rm diag}(\bm{v})$ $\in$ $\mathbb{C}^{N\times N}$, with
$\bm{v}=[v_{1}^{*},v_{2}^{*},\dots,v_{N}^{*}]^{H}$ $\in$ $\mathbb{C}^{N\times
1}$, being $v_{n}=\alpha_{n}e^{j\theta_{n}}$ $\in$ $\mathbb{C}$ the phase-
shift and amplitude applied by the $n$-th element of the RIS, with
$\alpha_{n}$ $\in[0,\alpha_{\max}]$ and $\theta_{n}$ $\in[0,2\pi)$. The BS
uses a precoding matrix
$\bm{W}=\left[\bm{w}_{1},\bm{w}_{2},\dots,\bm{w}_{K}\right]$ $\in$
$\mathbb{C}^{M\times K}$, with $||\bm{w}_{k}||^{2}=p_{k}$ $\in$
$\mathbb{R}^{+}$, in view of beamforming the signal towards the UEs, where
$p_{k}$ is the power allocated for the $k$-th UE served. Here, for simplicity,
we adopt maximum ratio (MR) precoding with equal power allocation (EPA).

Utilizing the system model explained above, the DL baseband transmitted signal
$\bm{x}$ $\in\mathbb{C}^{M\times 1}$ can be written as

$\bm{x}=\sum_{k=1}^{K}\bm{w}_{k}s_{k},$ (2)

where $s_{k}$ is the symbol intended to the $k$-th UE, with
$\mathbb{E}[|s_{k}|^{2}]=1$. While the received signal by the $k$-th UE is
given as

$y_{k}=\bm{w}_{k}^{H}\bm{h}_{k}s_{k}+\sum_{j=1,j\neq
K}^{K}\bm{w}_{j}^{H}\bm{h}_{k}s_{k}+\bm{h}_{2,k}^{H}\bm{\Phi}\bm{z}+n_{k},$
(3)

in which $n_{k}$ $\in\mathbb{C}$ is the AWGN sample at the $k$-th UE,
following $n_{k}\sim\mathcal{CN}(0,\sigma^{2})$, $\forall k\in\mathcal{K}$,
and $\mathbf{z}$ $\in\mathbb{C}^{N\times 1}$ is the AWGN at the RIS, with
$\mathbf{z}\sim\mathcal{CN}(\mathbf{0}_{N},\sigma_{{\rm
RIS}}^{2}\mathbf{I}_{N})$ .

###  III-A RIS-aided M-MIMO Channel Modelling

The RIS takes the form of a thin uniform squared planar array (USPA),
featuring $N=N_{h}\times N_{v}$ total elements, where $N_{h}=N_{v}=\sqrt{N}$
elements in the horizontal and vertical directions, with $\sqrt{N}$ being a
positive integer ($\sqrt{N}\in\mathbb{Z}_{+}$). Furthermore, the elements are
vertically and horizontally equally spaced with spacing $d_{\rm R}$. Besides,
the BS is assumed to be equipped with a uniform linear array (ULA) of $M$
total elements disposed in the $x$ axis, $i.e.$, parallel to the RIS.

In the channel model for RIS-aided M-MIMO, we make the assumption of far-field
wavefront propagation regime between the BS and the RIS panels, as well as
between the RIS and the UE s, $i.e.$, the planar wavefront is assumed in both
links.

BS-RIS link: to be more specific, we establish the channel between the BS and
the RIS based on the Rician-fading model, given by [40, 41, 42]:

$\bm{H}_{1}=\sqrt{\frac{\beta_{1}\kappa_{1}}{\kappa_{1}+1}}\bar{\bm{H}}_{1}+\sqrt{\frac{\beta_{1}}{\kappa_{1}+1}}\tilde{\bm{H}}_{1},$
(4)

where $\bar{\bm{H}}_{1}$ $\in\mathbb{C}^{M\times N}$ corresponds to the
deterministic LoS channel component, $\tilde{\bm{H}}_{1}$
$\in\mathbb{C}^{M\times N}$ to the Rayleigh-fading multipath component,
$\kappa_{1}\geq 0$ denotes the Rician $\kappa$-factor, and $\beta_{1}$ is the
large-scale fading coefficient for this link specifically. The large-scale
fading coefficient is given by

$\beta_{1}=\frac{\beta_{0}}{d_{\rm BR}^{\lambda_{\rm BR}}},$ (5)

in which $\beta_{0}$ is the path loss at a reference distance of 1 meter,
$d_{\rm BR}$, and $\lambda_{\rm BR}$ is the distance and the path-loss
exponent between the BS and RIS, respectively. Particularly, the deterministic
LoS component can be given through the steering vector of a two-dimensional
uniform planar array (UPA). Let us consider the RIS centered in the origin of
the Cartesian coordinates; therefore, each column of LoS channel matrix
$\bar{\bm{H}}_{1}$ is given as

$\bar{\bm{H}}_{1}=\bm{a}_{\rm B}(\vartheta_{\rm B})\bm{a}_{\rm
R}(\vartheta_{\rm R},\varphi_{\rm R})^{H},$ (6)

where $\vartheta_{\rm B}$ is the azimuthal angle of departure (AoD) of the
signal departing from the BS towards the RIS, and $\vartheta_{\rm R}$ and
$\varphi_{\rm R}$ are the azimuthal and elevation angle of arrival (AoA) at
the RIS from the BS. The array response vector of the BS ULA and of the RIS
UPA, are defined, respectively as

$\bm{a}_{\rm B}(\vartheta_{\rm B})=\left[1,e^{j\pi\frac{2d_{\rm
B}}{\lambda}\sin\vartheta_{\rm B}},\dots,e^{j(M-1)\pi\frac{2d_{\rm
B}}{\lambda}\sin\vartheta_{\rm B}}\right]^{H},$ (7) $\displaystyle\bm{a}_{\rm
R}(\vartheta_{\rm R},\varphi_{\rm R})$ $\displaystyle=$
$\displaystyle\left[1,e^{j\pi\frac{2d_{\rm R}}{\lambda}\sin\vartheta_{\rm
R}\cos\varphi_{\rm R}},\dots,e^{j(\sqrt{N}-1)\pi\frac{2d_{\rm
R}}{\lambda}\sin\vartheta_{\rm R}\cos\varphi_{\rm R}}\right]^{H}$
$\displaystyle\hskip 56.9055pt\otimes\left[1,e^{j\pi\frac{2d_{\rm
R}}{\lambda}\sin\varphi_{\rm R}},\dots,e^{j(\sqrt{N}-1)\pi\frac{2d_{\rm
R}}{\lambda}\sin\varphi_{\rm R}}\right]^{H},$

with $d_{\rm B}$ being the distance between the antenna elements at the BS.
Moreover, the multipath component follows an uncorrelated Rayleigh
distribution such that
$\mathbf{\tilde{h}}_{1,i}\sim\mathcal{CN}(\mathbf{0}_{N},\mathbf{I}_{N})$,
$\forall i\in\\{1,\dots,M\\}$, with $\tilde{\mathbf{h}}_{1,i}$ being the
$i$-th column of matrix $\tilde{\mathbf{H}}_{1}$.

RIS-UE link: the link between the RIS and UEs,
$\bm{H}_{2}=[\bm{h}_{2,1}\,\bm{h}_{2,2}\,\dots\,\bm{h}_{2,K}]$, is also
assumed to be Rician fading and given as

$\bm{h}_{2,k}=\sqrt{\frac{\beta_{2,k}{\kappa}_{2,k}}{{\kappa}_{2,k}+1}}\bar{\bm{h}}_{2,k}+\sqrt{\frac{\beta_{2,k}}{{\kappa}_{2,k}+1}}\tilde{\bm{h}}_{{2,k}},\quad\forall
k\in\mathcal{K}$ (9)

where $\bar{\mathbf{h}}_{{2,k}}$ $\in\mathbb{C}^{N\times 1}$ corresponds to
the deterministic LoS channel component between the RIS and the $k$-th UE,
$\tilde{\bm{h}}_{2,k}$ $\in\mathbb{C}^{N\times 1}$ to the Rayleigh-fading
multipath component between RIS and $k$-th UE, and $\kappa_{2,k}$ denotes the
Rician $\kappa$-factor for this same link, with $\kappa_{2,k}\geq 0$ $\forall
k$ $\in\mathcal{K}$. Besides, $\beta_{2,k}$ corresponds to the large-scale
fading from the RIS to the $k$-th UE, given as

$\beta_{2,k}=\frac{\beta_{0}}{d_{{\rm RU},k}^{\lambda_{\rm RU}}},$ (10)

with $d_{{\rm RU},k}$ being the distance between the RIS and $k$-th UE, and
$\lambda_{\rm RU}$ the path loss exponent between the RIS and UEs.

LoS Component: similarly to the BS and RIS channel model, the deterministic
LoS component $\bar{\bm{h}}_{2,k}$ can be given by the steering vector:

$\displaystyle\bar{\bm{h}}_{2,k}$ $\displaystyle=$ $\displaystyle\bm{a}_{{\rm
R},k}(\vartheta_{{\rm R},k},\varphi_{{\rm R},k})$ $\displaystyle=$
$\displaystyle\left[1,e^{j\pi\frac{2d_{\rm R}}{\lambda}\sin\vartheta_{\rm
R}\cos\varphi_{\rm R}},\dots,e^{j(\sqrt{N}-1)\pi\frac{2d_{\rm
R}}{\lambda}\sin\vartheta_{\rm R}\cos\varphi_{\rm
R}}\right]^{H}\otimes\left[1,e^{j\pi\frac{2d_{\rm R}}{\lambda}\sin\varphi_{\rm
R}},\dots,e^{j(\sqrt{N}-1)\pi\frac{2d_{\rm R}}{\lambda}\sin\varphi_{\rm
R}}\right]^{H}.$

###  III-B SE and EE Formulation in RIS-Assisted M-MIMO

In this subsection, we introduce two critical and crucial performance metrics
for analyzing the performance of communication systems. We initiate the
discussion with the concept of SE and then expand into EE.

####  III-B1 Spectral Efficiency

To assess the wireless system performance, it is crucial to evaluate the
capacity for transmitting bits per second per Hertz, namely as SE, a
fundamental metric that measures the efficiency of information transmission
through a channel. The SE reflects the average number of bits transmitted per
channel use across varying fading conditions, serving as a common metric for
evaluating communication system capabilities. Since for the RIS-assisted
systems, the channel is a generalization of the conventional M-MIMO systems,
the SE can be equally represented as the following

${\rm SE}_{k}=\log_{2}(1+{\rm SINR}_{k}),\qquad\left[\frac{\rm bit}{\rm s\cdot
Hz}\right]$ (12)

where the signal-to-interference-plus-noise ratio (SINR) of the $k$-th UE,
${\rm SINR}_{k}$, can be expressed as

${\rm SINR}_{k}=\frac{|\bm{w}_{k}^{H}\mathbf{h}_{k}|^{2}}{\sum_{j=1,j\neq
K}^{K}|\bm{w}_{j}\mathbf{h}_{k}|^{2}+|\bm{z}^{H}\bm{\Phi}\bm{h}_{2,k}|^{2}+|n_{k}|^{2}}.$
(13)

Finally, we can define the system’s normalized SR as the following

$R=\sum_{k=1}^{K}\log_{2}\left(1+\frac{|\bm{w}_{k}^{H}\mathbf{h}_{k}|^{2}}{\sum_{j=1,j\neq
K}^{K}|\bm{w}_{j}\mathbf{h}_{k}|^{2}+|\bm{z}^{H}\bm{\Phi}\bm{h}_{2,k}|^{2}+|n_{k}|^{2}}\right).$
(14)

####  III-B2 Energy Efficiency

The metric for evaluating a system’s sustainable capacity is referred to as
EE. This metric essentially quantifies the number of bits that can be reliably
transmitted per unit of energy, expressed as:

$\mathrm{EE}=\frac{\mathrm{Normalized~{}Sum~{}Rate~{}[bit/s/Hz]}}{\mathrm{{System}~{}Power~{}Consumption~{}[W]}}=\frac{\sum_{k}{\rm
SE}_{k}}{P_{{\rm total}}}\qquad{\left[\frac{\rm bit}{\rm Joule\cdot
Hz}\right]}$ (15)

where $P_{\rm total}$ is the total power consumed by the communication system
as detailed in Remark III-B2 and formulated in Subsection III-B3 in the next.

Remark 1 The EE optimization as a function of the amount of transmit RF power
in (15) involves a ratio of a convex function in the numerator and an affine
function in the denominator. Hence, one can formulate the EE optimization
problem as a single-ratio problem, briefly revisited in subsection IV-B.

Remark 2 The power consumption for RIS-assisted M-MIMO systems requires more
detailed consideration than traditional M-MIMO systems since although the RIS
technology achieves low power consumption, its consumption is not entirely
negligible and should be thoroughly examined. Therefore, in the next
subsection III-B3 we further detail the adopted system power consumption
model.

####  III-B3 System’s Power Consumption Model

Different from most prior works, which mostly consider entirely passive RIS,
here, we adopt a more practical, realistic energy consumption model by taking
into account two primary components: the power consumption of the BS and the
power consumption related to the active RIS. Generically, the total power
consumption model can be written as

$P_{{\rm total}}=P_{{\rm BS}}+P_{{\rm RIS}}.$ (16)

The power consumption at the BS is related directly to three main factors: the
constant power required for BS operation, $i.e.$, site-cooling, control
signaling, load-independent power of backhaul infrastructure, baseband
processor, etc; the power necessary to run the circuit components attached to
each antenna, such as converters, mixers, filters, etc; and the RF transmit
power allocated to the UEs [43], resulting in

$P_{{\rm BS}}=P_{0,{\rm BS}}+MP_{{\rm
M}}+\sum_{k=1}^{K}\varrho||\bm{w}_{k}||^{2},$ (17)

where $\varrho$ denotes the inefficiency of the transmit power amplifier.

On the other hand, the RIS needs a controller222The controller includes a
field programmable gate array (FPGA) and drive circuits., which is required
for receiving the external signals, processing data, and programming, as well
as configuring the RIS unit cells. Overall, the total power consumption
dissipated to operate the RIS consists of three parts; a) one is the static
power consumption generated by the FPGA control board and drive circuits,
namely $P_{\rm CB}$ [44, 45]; b) the power consumption by the RIS unit cells,
being, the power necessary for controlling the impedance of each element via
amplifier and phase turner, in order to configure the phase/amplitude of
reflection; and c) the active power for RF signal amplification. Accordingly,
the power consumed at the active RIS can be expressed as

$P_{{\rm RIS}}=P_{{\rm CB}}+NP_{{\rm N}}+P_{{\rm RIS}}^{{\rm out}}-P_{{\rm
RIS}}^{{\rm in}}.$ (18)

where $P_{{\rm RIS}}^{{\rm out}}$ and $P_{{\rm RIS}}^{{\rm in}}$ is the signal
RF power that arrives at the RIS and departs from the RIS, respectively, given
by [46]

$P_{{\rm RIS}}^{{\rm
in}}=||\bm{z}||^{2}+\sum_{k=1}^{K}||\bm{w}_{k}^{H}\bm{H}_{1}||^{2}=\textrm{tr}(\bm{Z}\bm{Z}^{H})+\sum_{k=1}^{K}\textrm{tr}(\bm{U}_{k}\bm{U}_{k}^{H}),$
(19) $P_{{\rm RIS}}^{{\rm
out}}=||\bm{\Phi}\bm{z}||^{2}+\sum_{k=1}^{K}||\bm{w}_{k}^{H}\bm{H}_{1}\bm{\Phi}||^{2}=\textrm{tr}\left(\bm{Z}\bm{v}\bm{v}^{H}\bm{Z}^{H}\right)+\sum_{k=1}^{K}\textrm{tr}(\bm{U}_{k}\bm{v}\bm{v}^{H}\bm{U}_{k}^{H}),$
(20)

where $\bm{Z}\triangleq\textrm{diag}(\bm{z})$ and
$\bm{U}_{k}\triangleq\textrm{diag}(\bm{H}_{1}^{H}\bm{w}_{k})$. Thereby, the
total RF power amplification at the active RIS can be expressed as

$\displaystyle P_{{\rm RIS}}^{{\rm out}}-P_{{\rm RIS}}^{{\rm in}}$
$\displaystyle=$
$\displaystyle\textrm{tr}\left(\bm{Z}\left(\bm{v}\bm{v}^{H}-\mathbf{I}_{N}\right)\bm{Z}^{H}\right)+\sum_{k=1}^{K}\textrm{tr}\left(\bm{U}_{k}\left(\bm{v}\bm{v}^{H}-\mathbf{I}_{N}\right)\bm{U}_{k}^{H}\right)$
$\displaystyle=$
$\displaystyle\textrm{tr}\left(\left(\bm{v}\bm{v}^{H}-\mathbf{I}_{N}\right)\bm{Z}^{H}\bm{Z}\right)+\textrm{tr}\left(\left(\bm{v}\bm{v}^{H}-\mathbf{I}_{N}\right)\sum_{k=1}^{K}\bm{U}_{k}^{H}\bm{U}_{k}\right)$
$\displaystyle=$
$\displaystyle\textrm{tr}\left(\left(\bm{v}\bm{v}^{H}-\mathbf{I}_{K}\right)\bm{Q}\right),$
$\displaystyle=$ $\displaystyle\bm{v}^{H}\bm{Q}\bm{v}-\textrm{tr}(\bm{Q})$

where

$\bm{Q}\triangleq\bm{Z}^{H}\bm{Z}+\sum_{k=1}^{K}\bm{U}_{k}^{H}\bm{U}_{k}$ (22)

Therefore, Eq. (18) also can be written as a function of $\bm{v}$ as
following333Herein, we assume that the RIS is unable to absorb the incident
power, although it is possible by means of energy harvesting circuits embedded
at the RIS. Therefore, $P_{\textrm{RIS}}^{\rm out}\geq P_{\textrm{RIS}}^{\rm
in}$ always hold, with the equality when $|v_{n}|=1$, $\forall n$
$\in\mathcal{N}$.

$P_{\rm RIS}=P_{{\rm CB}}+NP_{{\rm
N}}+\bm{v}^{H}\bm{Q}\bm{v}-\textrm{tr}\left(\bm{Q}\right).$ (23)

A detailed discussion on the static power consumption generated by the FPGA
board and drive circuits $P_{\rm CB}$ is found in [44, 45].

##  IV Optimization Techniques

In order to fully clarify the proposed optimization strategy for tackling the
EE active RIS-aided M-MIMO optimization problem, it is crucial to delve into
some key optimization techniques that act in our proposed solution. These
techniques play essential roles in our approach to addressing the investigated
problem, and an entire comprehension of their concepts is essential for a
careful understanding of the proposed solution. In the following, we provide a
detailed review of these techniques, including LDT and fractional single and
multiple-ratio problems.

###  IV-A Lagrangian Dual Transform

In addressing the challenge posed by the sum-of-logarithms problem, specific
techniques prove to be valuable in easing the management of the optimization
problem, thereby enabling the possibility of finding a closed-form, low-
complexity solution. Given that communication system-related problems
frequently involve logarithmic function summation, finding any form of
solution can be exceedingly challenging as the presence of logarithmic
functions hinders the analytical tractability and, consequently, optimization.
As a result, an optimization technique introduced by [11], namely LDT
leverages the Lagrangian duality to externalize the argument of logarithm
functions to the outside, eliminating the dependence of the variable of
interest with the logarithmic functions, resulting in easier management and
analytical tractability of the problem. To further understand the LDT, let us
define the following target maximization problem:

$\displaystyle\underset{\mathbf{x}}{\mathrm{maximize}}$
$\displaystyle\qquad\sum_{\ell=1}^{L}\log\left(1+\frac{f_{\ell}(\mathbf{x})}{g_{\ell}(\mathbf{x})}\right),$
(24) $\displaystyle\mathrm{subject\,to}$ $\displaystyle\qquad
h_{i}(\mathbf{x})\leq 0,\quad i\in\\{1,\dots,I\\}$ (25a)

where $f_{\ell}(\mathbf{x}):$ $\mathbb{C}^{N}\rightarrow\mathbb{R}_{+}$ is a
non-negative function, and $g_{\ell}(\mathbf{x}):$
$\mathbb{C}^{N}\rightarrow\mathbb{R}_{+}^{*}$ is a positive function,
$\forall\ell$ $\in\\{1,\dots,L\\}$. It is important to observe that the ratio
$f_{\ell}(\mathbf{x})/g_{\ell}(\mathbf{x})$ can have a physical interpretation
as SINR in our context. The problem (24) is assumed to be a non-convex
problem, with a non-convex constraint (25a). According to [12], the problem
(24) can be transformed in the following equivalent problem

$\displaystyle\underset{\mathbf{x},\,\bm{\alpha}}{\mathrm{maximize}}$
$\displaystyle\qquad\sum_{\ell=1}^{L}\log(1+\gamma_{\ell})-\sum_{\ell=1}^{L}\gamma_{\ell}+\sum_{\ell=1}^{L}(1+\gamma_{\ell})\frac{f_{\ell}(\mathbf{x})}{f_{\ell}(\mathbf{x})+g_{\ell}(\mathbf{x})},$
(26) $\displaystyle\mathrm{subject\,to}$ $\displaystyle\qquad
h_{i}(\mathbf{x})\leq 0,\qquad\forall i\in\\{1,\dots,I\\}$ (28a)
$\displaystyle\qquad\gamma_{\ell}\geq 0,\qquad\forall\ell\in\\{1,\dots,L\\}.$

where
$\bm{\gamma}^{(t)}=[\gamma_{1}^{(t)},\gamma_{2}^{(t)},\dots,\gamma_{L}^{(t)}]^{T}$
is an auxiliary variable, which is iteratively updated, introduced for each
ratio term $f_{\ell}(\mathbf{x}^{(t-1)})/g_{\ell}(\mathbf{x}^{(t-1)})$. It’s
important to note that the two problems, (24) and (26), are equivalent. In
other words, at the convergence, the solution $\mathbf{x}$ to (24) is
identical to the solution to (26), and their respective optimal objective
values are also equal [11, III-B]. Furthermore, it is worth highlighting that,
according to the Karush–Kuhn–Tucker (KKT) conditions, we can obtain the
following

$\displaystyle\frac{\partial\left(\displaystyle\sum_{\ell=1}^{L}\log(1+\gamma_{\ell}^{(t+1)})-\displaystyle\sum_{\ell=1}^{L}\gamma_{\ell}^{(t+1)}+\displaystyle\sum_{\ell=1}^{L}\dfrac{(1+\gamma_{\ell}^{(t+1)})f_{\ell}(\mathbf{x}^{(t)})}{f_{\ell}(\mathbf{x}^{(t)})+g_{\ell}(\mathbf{x}^{(t)})}\right)}{\partial\gamma_{\ell}}=0$
(29)
$\displaystyle\gamma_{\ell}^{\star(t+1)}=\frac{f_{\ell}(\mathbf{x}^{(t)})}{g_{\ell}(\mathbf{x}^{(t)})}$
(30)

This equivalence allows for a seamless transition between the two problem
formulations while preserving the optimality.

###  IV-B Fractional Programming

Fractional programming theory is the branch of optimization theory concerned
with the properties and optimization of fractional functions, _i.e._ , a ratio
of two functions that are generally nonlinear. They can be found in several
areas and are generically subdivided into two classes: single-ratio problems
and multiple-ratio problems. Below, we review techniques designed to address
each of these classes and their respective solution methodologies.

#### Single-Ratio Problems

There are many transforms that deal consistently with this type of problem;
the most classical is named Dinkelbach’s Transform [47], [12], [11]. For a
better understanding, let us define the following optimization problem.

$\displaystyle\underset{\mathbf{x}}{\mathrm{maximize}}$
$\displaystyle\quad\frac{f(\mathbf{x})}{g(\mathbf{x})},\vspace{0.2cm}$ (31)
$\displaystyle\mathrm{subject\,to}$ $\displaystyle\quad h_{i}(\mathbf{x})\leq
0,\qquad\forall i\in\\{1,\dots,I\\}$ (32a)

where $f(\mathbf{x}):\mathbb{C}^{N}\rightarrow\mathbb{R}_{+}$ and
$g(\mathbf{x}):\mathbb{C}^{N}\rightarrow\mathbb{R}^{*}_{+}$ are non-negative
function and positive function, respectively. The conventional approach for
dealing with this FP is decoupling the numerator and denominator and treating
it jointly. The Dinkelbach’s transform reformulates the single-ratio problem
(31) as the following

$\displaystyle\underset{\mathbf{x},\eta}{\mathrm{maximize}}$
$\displaystyle\qquad f(\mathbf{x})-\eta g(\mathbf{x}),$ (33)
$\displaystyle\mathrm{subject\,to}$ $\displaystyle\qquad h_{i}(\mathbf{x})\leq
0,\qquad\forall i\in\\{1,\dots,I\\}$ (34a)

where $\eta$ is an auxiliary variable that is iteratively updated by

$\eta^{(t)}=\frac{f(\mathbf{x}^{(t-1)})}{g(\mathbf{x}^{(t-1)})},$ (35)

where $t$ is the iteration index.

#### Multiple-Ratio Problems

Although the classic Dinkelbach’s transform [47] works well for single-ratio
problems, they cannot be easily extended to the multiple-ratio problems, since
for single-ratio problems, the optimal solution is the same for the original
FP and the transformed problem, but not the value of the objective function of
both. To solve the multi-ratio problem, several different techniques have been
proposed, such as Quadratic Transform [11], and [48]. Herein, we focus on the
technique developed by [48]. Let us define the following multi-ratio problem:

$\displaystyle\underset{\mathbf{x}}{\mathrm{maximize}}$
$\displaystyle\qquad\sum_{\ell=1}^{L}\frac{f_{\ell}(\mathbf{x})}{g_{\ell}(\mathbf{x})},$
(36) $\displaystyle\mathrm{subject\,to}$ $\displaystyle\qquad
h_{i}(\mathbf{x})\leq 0,\qquad i\in\\{1,\dots,I\\}$ (37a)

where $f_{\ell}(\mathbf{x}):\mathbb{C}^{N}\rightarrow\mathbb{R}_{+}$ and
$g_{\ell}(\mathbf{x}):\mathbb{C}^{N}\rightarrow\mathbb{R}_{+}^{*}$,
$\forall\ell$ $\in\\{1,\dots,{L}\\}$. In [48], the authors proposed an
iterative solution in order to solve problem (36), where the equivalent
problem can be given as

$\displaystyle\underset{\mathbf{x},\mathbf{u},\bm{\beta}}{\mathrm{maximize}}$
$\displaystyle\qquad\sum_{\ell=1}^{L}u_{\ell}\left(f_{\ell}(\mathbf{x})-\beta_{\ell}g_{\ell}(\mathbf{x})\right),$
(38) $\displaystyle\mathrm{subject\,to}$ $\displaystyle\qquad
h{{}_{i}}(\mathbf{x})\leq 0,\qquad i\in\\{1,\dots,I\\}$ (39a)

where $\mathbf{u}$ and $\bm{\beta}$ are auxiliary variables which are
iteratively updated as

$u_{\ell}^{(t+1)}=\frac{1}{g{{}_{\ell}}(\mathbf{x}^{(t)})},\quad\forall\ell=1,\dots,L,$
(40)
$\beta_{\ell}^{(t+1)}=\frac{f{{}_{\ell}}(\mathbf{x}^{(t)})}{g_{\ell}(\mathbf{x}^{(t)})},\quad\forall\ell=1,\dots,L.$
(41)

##  V Problem Formulation

In view of addressing all the aforementioned arguments, with regard to EE
exposed in Section I, we formulate an optimization problem whose main
objective is optimizing the reflecting coefficients of an active RIS in order
to assess the necessary number of RIS elements that should be selected to
outperform the entirely passive RIS in terms of EE. The idea is to reduce the
number of operating elements in the RIS in view of reducing their power
consumption while increasing the total SR by amplifying the incoming signal
intended for the UEs. Hence, such an objective can be formulated as the
following optimization problem, denoted as $\mathcal{P}_{0}$:

$\displaystyle\mathcal{P}_{0}:\quad$
$\displaystyle\underset{\bm{v}}{\mathrm{maximize}}$
$\displaystyle\qquad\frac{\displaystyle\sum_{k=1}^{K}\log_{2}\left(1+\dfrac{|\bm{w}_{k}^{H}\bm{h}_{k}|^{2}}{\sum_{j=1,j\neq
K}^{K}|\bm{w}_{j}\bm{h}_{k}|^{2}+|\bm{z}^{H}\bm{\Phi}\bm{h}_{2,k}|^{2}+|n_{k}|^{2}}\right)}{P_{0,{\rm
BS}}+P_{\rm CB}+MP_{{\rm M}}+\sum_{k=1}^{K}\varrho||\bm{w}_{k}||^{2}+NP_{{\rm
N}}+\bm{v}^{H}\bm{Q}\bm{v}-\textrm{tr}\left(\bm{Q}\right)},\vspace{0.2cm}$
(42) $\displaystyle\mathrm{subject\,to}$ $\displaystyle\hskip
22.76228pt\textrm{tr}(\bm{Q})\leq\bm{v}^{H}\bm{Q}\bm{v}\leq\mathrm{tr}(\bm{Q})+P_{\max}^{{\rm
RIS}},$ (44a) $\displaystyle\hskip
22.76228pt|v_{n}|\leq\alpha_{\max},\qquad\forall n\in\mathcal{N}.$

Constraint (44a) considers the maximum amplification of power provided by the
active RIS, while constraint (44a) considers the maximum amplitude imposed by
each element of the RIS. One should notice that problem $\mathcal{P}_{0}$ is
more challenging than the EE optimization with entirely passive RIS since the
RIS configuration vector also appears in the denominator of the objective
function (42).

##  VI Proposed Solution

In Subsection VI-A, we introduce the proposed algorithm namely FP-based
Amplitude/Phase Beamforming Design, which utilizes the FP optimization
technique. Specifically, to enhance the convexity of the optimization problem,
this algorithm incorporates auxiliary variables derived from Dilkelbach’s
Transform, LDT, and from the methodology outlined in [48]. This integration
strategically molds the objective function into a convex form, facilitating
effective optimization.

Subsequently, in Subsection VI-B, we conduct a comprehensive analysis of the
algorithm’s complexity. This examination provides valuable insights,
emphasizing the algorithm’s efficiency and suitability for real-world
applications.

Finally, in Section VII, we present a comparative evaluation of the proposed
algorithm’s performance within typical RIS-aided M-MIMO channel and system
scenarios. This comparative analysis serves to validate and illustrate the
algorithm’s efficacy across a range of practical and diverse operating
conditions.

###  VI-A FP-Based Solution Method

Before delving directly into the problem at hand, it is useful to establish a
more concise nomenclature. Therefore, firstly, for the sake of tractability,
we shall rewrite Equation (13) as follows

$\textrm{SINR}_{k}=\frac{|\bm{w}_{k}^{H}\bm{h}_{k}|^{2}}{\sum_{j=1,j\neq
K}^{K}|\bm{w}_{j}\bm{h}_{k}|^{2}+|\bm{z}^{H}\bm{\Phi}\bm{h}_{2,k}|^{2}+|n_{k}|^{2}}=\frac{\bm{v}^{H}\bm{A}_{k}\bm{v}}{\bm{v}^{H}\bm{B}_{k}\bm{v}+|n_{k}|^{2}},$
(45)

where with few analytical manipulations, we obtain that

$\displaystyle\bm{A}_{k}$ $\displaystyle\triangleq$
$\displaystyle\bm{H}_{2,k}^{H}\bm{H}_{1}^{H}\bm{w}_{k}\bm{w}_{k}^{H}\bm{H}_{1}\bm{H}_{2,k}\,,$
(46) $\displaystyle\bm{B}_{k}$ $\displaystyle\triangleq$
$\displaystyle\bm{H}_{2,k}^{H}\bm{z}\bm{z}^{H}\bm{H}_{2,k}+\sum_{j=1,j\neq
k}^{K}\bm{H}_{2,k}^{H}\bm{H}_{1}^{H}\bm{w}_{j}\bm{w}_{j}\bm{H}_{1}\bm{H}_{2,k}\,,$
(47)

with $\mathbf{H}_{2,k}=\textrm{diag}(\mathbf{h}_{2,k})$. This manipulation
enables us to effortlessly address our problem, thereby facilitating the
derivation of subsequent equations. Rewriting problem (42), we have the
following

$\displaystyle\mathcal{P}_{0}:\quad$
$\displaystyle\underset{\bm{v}}{\mathrm{maximize}}$
$\displaystyle\qquad\frac{\displaystyle\sum_{k=1}^{K}\log_{2}\left(1+\frac{\bm{v}^{H}\bm{A}_{k}\bm{v}}{\bm{v}^{H}\bm{B}_{k}\bm{v}+|n_{k}|^{2}}\right)}{\hat{P}+\bm{v}^{H}\bm{Q}\bm{v}},\vspace{0.2cm}$
$\displaystyle\mathrm{subject\,to}$
$\displaystyle\qquad\eqref{eq:c1},~{}\eqref{eq:c2},$

where we define
$\hat{P}\triangleq\sum_{k=1}^{K}\varrho||\bm{w}_{k}||^{2}+P_{0,\rm{BS}}+P_{\rm
CB}+MP_{\rm{M}}+NP_{{\rm N}}-\rm{tr}(\mathbf{Q})$. One can see that Problem
(VI-A) is a non-convex problem, with a non-convex constraint (44a).
Furthermore, we also should notice that (VI-A) is a single ratio problem.
Therefore, prior to solving the optimization process, a pre-processing step
should be applied to the objective function due to its intractable fractional
form. To decouple the numerator and denominator of (VI-A), we employ the
classic Dinkelbach’s methodology [47]. Specifically, by introducing an
auxiliary variable $\eta$, according to subsection IV-B, the original problem
can be equivalently reformulated as

$\displaystyle\mathcal{P}_{1}:\quad$
$\displaystyle\underset{\bm{v},\eta}{\mathrm{maximize}}$
$\displaystyle\qquad\displaystyle\sum_{k=1}^{K}\log_{2}\left(1+\frac{\bm{v}^{H}\bm{A}_{k}\bm{v}}{\bm{v}^{H}\bm{B}_{k}\bm{v}+|n_{k}|^{2}}\right)-\eta\left(\hat{P}+\bm{v}^{H}\bm{Q}\bm{v}\right),\vspace{0.2cm}$
$\displaystyle\mathrm{subject\,to}$
$\displaystyle\qquad\eqref{eq:c1},~{}\eqref{eq:c2},$

where the optimal $\eta^{*}$ in the $\ell$-th iteration can be directly
computed as

$\eta^{\star{(\ell)}}=\dfrac{\displaystyle\sum_{k=1}^{K}\log_{2}\left(1+\dfrac{\bm{v}^{{(\ell-1)}H}\bm{A}_{k}\bm{v}^{{(\ell-1)}}}{\bm{v}^{{(\ell-1)}H}\bm{B}_{k}\bm{v}^{{(\ell-1)}}+\sigma^{2}}\right)}{{\hat{P}+\bm{v}^{{(\ell-1)}H}\bm{Q}\bm{v}^{{(\ell-1)}}}}.$
(50)

Thus, we should seek to solve $\mathcal{P}_{1}$, for fixed $\eta$ (at the
$\ell$-th iteration). In this way, we should see that the objective function
of $\mathcal{P}_{1}$ is still non-convex. Moreover, it is composed of a scaled
version of a multiple-ratio function; therefore, applying the LDT technique,
we can introduce the auxiliary variable $\bm{\gamma}$, and rewrite
$\mathcal{P}_{1}$ in an equivalent way, obtaining the following optimization
problem, which we denote as $\mathcal{P}_{2}$

$\displaystyle\mathcal{P}_{2}:\quad$
$\displaystyle\underset{\bm{v},\bm{\gamma}}{\mathrm{maximize}}$
$\displaystyle\quad\displaystyle\sum_{k=1}^{K}\log_{2}\left(1+\gamma_{k}\right)-\gamma_{k}+\vphantom{\sum}\frac{(1+\gamma_{k})\bm{v}^{H}\bm{A}_{k}\bm{v}}{\left(\bm{v}^{H}\left(\bm{A}_{k}+\bm{B}_{k}\right)\bm{v}+|n_{k}|^{2}\right)}-\eta\left(\hat{P}+\bm{v}^{H}\bm{Q}\bm{v}\right),$
$\displaystyle\mathrm{subject\,to}$
$\displaystyle\qquad\eqref{eq:c1},\eqref{eq:c2}.$

The $\gamma$ that maximizes $\mathcal{P}_{2}$ at the $\ell$-th iteration is
given by

$\gamma_{k}^{\star{(\ell)}}=\frac{\bm{v}^{{(\ell-1)}H}\bm{A}_{k}\bm{v}^{{(\ell-1)}}}{\bm{v}^{{(\ell-1)}H}\bm{B}_{k}\bm{v}^{{(\ell-1)}}+\sigma^{2}},\qquad\forall
k\in\mathcal{K}.$ (51)

By proceeding, following methodology of [48], and introducing the following
auxiliary variables, $\bm{\beta}$, and $\bm{u}$, we can rewrite
$\mathcal{P}_{2}$ as

$\displaystyle\mathcal{P}_{2}^{{}^{\prime}}:\quad$
$\displaystyle\underset{\bm{v},\bm{u},\bm{\beta}}{\mathrm{maximize}}$
$\displaystyle\quad\displaystyle\sum_{k=1}^{K}\log_{2}\left(1+\gamma_{k}\right)-\gamma_{k}+u_{k}\left[\vphantom{\sum}{(1+\gamma_{k})\bm{v}^{H}\bm{A}_{k}\bm{v}}-\beta_{k}\left(\bm{v}^{H}\left(\bm{A}_{k}+\bm{B}_{k}\right)\bm{v}+|n_{k}|^{2}\right)\right]$
$\displaystyle\hskip
142.26378pt-\eta\left(\hat{P}+\bm{v}^{H}\bm{Q}\bm{v}\right),$
$\displaystyle\mathrm{subject\,to}$
$\displaystyle\qquad\eqref{eq:c1},\eqref{eq:c2}.$

The $\beta_{k}$ and $u_{k}$ that maximize $\mathcal{P}_{2}^{{}^{\prime}}$ at
the $\ell$-th iteration can be obtained by the KKT conditions which are given
as:

$\displaystyle\beta_{k}^{(\ell)\star}$ $\displaystyle=$
$\displaystyle\frac{(1+\gamma_{k}^{(\ell)})\bm{v}^{{(\ell-1)}H}\bm{A}_{k}\bm{v}^{{(\ell-1)}}}{\bm{v}^{{(\ell-1)}H}\left(\bm{A}_{k}+\bm{B}_{k}\right)\bm{v}^{{(\ell-1)}}+|n_{k}|^{2}},\qquad\forall
k\in\mathcal{K},$ (53) $\displaystyle\quad u_{k}^{(\ell)\star}$
$\displaystyle=$
$\displaystyle\frac{1}{\bm{v}^{{(\ell-1)}H}\left(\bm{A}_{k}+\bm{B}_{k}\right)\bm{v}^{{(\ell-1)}}+|n_{k}|^{2}},\qquad\forall
k\in\mathcal{K}.$ (54)

Rewriting $\mathcal{P}_{2}^{{}^{\prime}}$ in a more compact form, we arrive at
the following equivalent problem

$\displaystyle\mathcal{P}_{3}:\quad$
$\displaystyle\underset{\bm{v}}{\mathrm{maximize}}$
$\displaystyle\qquad\bm{v}^{H}\bm{C}\bm{v},\vspace{0.2cm}$
$\displaystyle\mathrm{subject\,to}$
$\displaystyle\qquad\eqref{eq:c1},~{}\eqref{eq:c2},$

where

$\bm{C}^{(\ell)}=-\eta^{(\ell)}\bm{Q}+\sum_{k=1}^{K}u_{k}^{(\ell)}(1+\gamma_{k}^{(\ell)})\bm{A}_{k}-u_{k}^{(\ell)}\beta_{k}^{(\ell)}\left(\bm{A}_{k}+\bm{B}_{k}\right).$
(56)

The problem $\mathcal{P}_{3}$ can be rewritten in an alternative manner as
follows:

$\displaystyle\mathcal{P}_{4}:\quad$
$\displaystyle\underset{\bm{V}}{\mathrm{maximize}}$ $\displaystyle\qquad{\rm
tr}\left(\bm{C}\bm{V}\right),\vspace{0.2cm}$ (57)
$\displaystyle\mathrm{subject\,to}$
$\displaystyle\qquad\textrm{tr}\left(\bm{C}\bm{V}\right)\leq\mathrm{tr}(\bm{Q})+P_{\max}^{{\rm
RIS}},$ (61a)
$\displaystyle\qquad\textrm{tr}\left(\bm{C}\bm{V}\right)\geq\textrm{tr}\left(\bm{Q}\right)$
$\displaystyle\qquad[\bm{V}]_{n,n}\leq\alpha_{\max}^{2},\quad\forall
n\in\mathcal{N}$ $\displaystyle\qquad{\rm rank}(\bm{V})=1,$

where the $\bm{v}^{H}\bm{C}\bm{v}=\textrm{tr}(\bm{C}\bm{V})$ property is
utilized, with $\bm{V}=\bm{v}\bm{v}^{H}$. In particular, both the objective
function and constraints in $\mathcal{P}_{4}$, given by Eq. (57), are linear
in the matrix $\bm{V}$ [49].

Notice that the constraint (61a) is a non-convex constraint; however, we can
relax it in order to obtain a sub-optimal solution for problem
$\mathcal{P}_{4}$. In summary, the optimization problem $\mathcal{P}_{4}$ can
be globally solved by standard fractional programming tools, such as CVX [50].
If the obtained solution of $\bm{V}$ has a unit rank, then $\bm{V}$ is also
feasible for the original problem. Otherwise, a feasible solution can be
obtained by randomization techniques [49, 51].

Based on the above derivations, the overall FP-based Amplitude/Phase
Beamforming design algorithm is summarized in Algorithm 1. By appropriately
initializing the variable $\bm{v}$ and the BS transmit power $\bm{p}$, the RIS
reflection/amplification matrix $\bm{\Phi}={\rm diag}(\bm{v})$ is iteratively
updated, until the EE of the RIS-assisted M-MIMO system converges.

Algorithm 1 FP-based Amplitude/Phase Beamforming Design

Input: ${\bm{p}}$, $\mathbf{A}_{k}$, $\mathbf{B}_{k}$, $\forall k$
$\in\\{1,\dots,K\\}$

Set $\ell=1$;

Set feasible value for ${\bm{v}^{(\ell-1)}}$;

Set $\eta^{(\ell-1)}=0$;

repeat

Step 1: Compute $\eta^{\star(\ell)}$ by (50);

Step 2: Update $\bm{\gamma}^{\star(\ell)}$ by (51);

Step 3: Compute $\bm{\beta}^{\star(\ell)}$ by (53);

Step 4: Update $\bm{u}^{\star(\ell)}$ by (54);

Step 5: Obtain $\bm{v}^{(\ell)}$ by solving $\mathcal{P}_{4}$;

Step 6: $\ell=\ell+1$;

until $|\eta^{\star(\ell)}-\eta^{\star(\ell-1)}|<\epsilon$

Output:
$\bm{v}^{\star}=[\alpha_{1}^{\star}e^{-j\theta_{1}^{\star}},\alpha_{2}^{\star}e^{-j\theta_{2}^{\star}},\dots,\alpha_{N}^{\star}e^{-j\theta_{N}^{{}^{\star}}}]^{H}$

###  VI-B Complexity

So far, we have completed the design of active RIS phase shift matrix
$\bm{\Phi}$. For clarity of this scheme procedure, we summarize the main ideas
of the entire proposed FP-based amplitude/phase beamforming design algorithm
as follows. First, by fixing $\eta$, we can obtain $\bm{\gamma}$,
$\bm{\beta}$, and $\bm{u}$, with closed-form expressions, and the RIS phase-
shift matrix $\bm{v}$ by solving $\mathcal{P}_{4}$, as illustrated on
Algorithm 1. The alternating iterating procedure is repeated among four
variables until the termination condition is reached. Herein, we describe the
complexity of the proposed iterative algorithm. It is known that, the
complexity to update $\bm{\gamma}$, $\bm{\beta}$, $\bm{u}$, and solving
$\mathcal{P}_{4}$ are $\mathcal{O}(2KN^{2})$, $\mathcal{O}(2KN^{2})$,
$\mathcal{O}(KN^{2})$, and $\mathcal{O}(N^{2})$ [49] respectively. Besides,
the $\eta$ updating requires $\mathcal{O}(3KN^{2})$. Therefore, the overall
complexity is given by
$\mathcal{O}\left(8LKN^{2}\ln(\frac{1}{\epsilon})\right)$ Floating Point
Operations (FLOPs), where $L$ denotes the number of loop iterations. In
addition, it is worthy to notice that the complexity of the Gaussian
randomization method is of $\mathcal{O}(N^{2})$ per draw [52].

##  VII Numerical Results

Herein, in this section we illustrate simulation results in view to
demonstrate the effectiveness of the proposed FP-based amplitude/phase
beamforming design algorithm, where the active RIS amplitude/phase vector is
obtained by running Algorithm 1, where we solve $\mathcal{P}_{4}$ via convex
optimization with CVX solver, in MATLAB. For the simulation setup, we assume
that the UEs are positioned in a circle with a radius of $10~{}m$ from the
center $(30~{}m,30~{}m)$, the BS is located at $(-10~{}m,5~{}m)$ and the RIS
is located at the origin $(0~{}m,0~{}m)$. For the BS RIS link, we defined
$\beta_{0}^{{\rm BR}}=1e^{-3}$ and $\lambda^{\rm BR}=2$, while for the RIS-UEs
link, we assume $\beta_{0}^{{\rm RU}}=1e^{-3}$ and $\lambda^{\rm RU}=2.5$.
Unless specified otherwise, the remainder of the parameters are listed in
Table II. Aiming for a fair comparison, we assume the same power budget for
entirely passive RIS and active RIS, where the maximum power for RIS
amplification, $P_{\max}^{{\rm RIS}}$, is given by $P_{\max}^{{\rm
RIS}}=\tau_{{\rm RIS}}P_{{\rm TX}}$, where $\tau_{\rm RIS}$ $\in[0,1]$.

TABLE II: List of simulation parameters.  Parameter | Description
---|---
$P_{\textsc{tx}}=[10:5:50]$ dBm | Maximum power budget at BS
$\sigma_{k}^{2}=-95$ dBm | Noise variance at UEs, $\forall k$ $\in\mathcal{K}$
$\sigma_{\rm\lx@glossaries@gls@link{acronym}{ris}{\leavevmode RIS}}^{2}=-80$ dBm | Noise variance at RIS
$K=5$ | Numbers of UE s
$M=32$ | Number of antennas at BS
$N=64$ | Number of reflecting meta-surfaces elements
$\varrho$ = 1.2 | Power amplifier inefficiency
$P_{0,{\rm BS}}$ = 9 dBW | Fixed power consumption at BS
$P_{\rm{M}}$ = 1 W | Power consumed by a BS transceiver chain
$P_{\rm{CB}}$ = 4.8 W | Power consumed by control board
$P_{\rm{N}}$ = 10 dBm | Power of RIS element configuration
$\alpha_{\max}=10$ | Maximum amplitude gain
$\kappa_{1}=0.1$ | Rician coefficient for BS RIS link
$\kappa_{2,k}=0.01$ | Rician coefficient for RIS UEs links, $\forall k$ $\in\mathcal{K}$
$\tau_{\rm RIS}=0.15$ | Coefficient for RIS amplification
$\mathcal{T}={500}$ | Realizations Monte-Carlo Simulation (MCs)

###  VII-A Energy Efficiency vs. Transmit Power Budget

Initially, in Fig. 1, we assess and compare the EE of both RIS architecture as
a function of transmit power $P_{\rm TX}$. For each point in the curve, the
proposed algorithm is employed to optimize the power of reflection of the
active RIS array, while for the passive RIS we just optimize the phase shift
angles. We denote as “Passive RIS” the entirely passive RIS, “Active RIS” the
active RIS, “Random RIS” as the entirely passive RIS with random precoding and
random RIS reflecting coefficients matrix.

Figure 1: Average EEvs the transmit power budget [dBm] at the BS
($P_{\rm{TX}}$). Performance evaluation of the proposed algorithm for the
active RISvs entirely passive RIS and random phase shift/precoding, with
$N=25$.

We can first observe the considerable impact that both passive and active RIS
can exert on the system performance, particularly in terms of EE. This
emphasizes the critical importance of optimizing both precoding and the matrix
of reflecting coefficients of the RIS within the RIS-aided M-MIMO scenario
since it can deal with the interference, improving the system SE, and
consequently the system EE. Furthermore, delving deeper into our findings, the
result reveals that the proposed active RIS optimization method significantly
improves the EE of the RIS-assisted M-MIMO system compared to the entirely
passive RIS configuration. Specifically, our result indicates that the active
RIS can outperform the performance of its entirely passive counterpart,
achieving an impressive improvement of up to 120$\%$ in the peak value at
$\approx 36$ dBm. This emphasizes the efficacy of deploying active RIS in
maximizing the EE, demonstrating its ability to deal with the double-fading
attenuation drawback significantly.

###  VII-B Percentage of Amplification Power Utilized

Figure 2 illustrates the percentage of utilized amplification power as a
function of the number of reflecting elements for two distinct strategies: 1)
SR maximization, and 2) maximization of EE. One can see through this result
that amplification power $P_{\max}^{\rm RIS}$ increases as the number of
reflecting elements increases. A reason for such a behavior is that when $N$
is low, utilizing fully $P_{\max}^{\rm RIS}$ can result in higher
interference, and consequently poor EE since the denominator of (42) also
increases with $P_{\max}^{\rm RIS}$. Besides, by increasing the number of
reflecting elements, the RIS can easily achieve equality on the right side of
constraint (44a), showing that as higher is $N$, better the RIS can manage the
interference. Furthermore, one can see that the utilized power amplification
for the SR maximization strategy is also limited for scenarios when $N$
assumes low values, confirming that this power management is necessary for the
optimization of both metrics.

Figure 2: Percentage of power amplification utilized vs the number of
reflecting elements for the proposed algorithm. In this setup, we set $P_{\rm
TX}=35$ dBm.

In Figure 3, we plot the cumulative density function (CDF) of the average
amplitude gain of the active RIS,
$\bar{\alpha}=\frac{1}{N}\sum_{n=1}^{N}\alpha_{n}$, for some values of
$N\in[9;64]$ RIS elements. One can observe that the probability of
$\bar{\alpha}$ to assume a low value, is higher for $N=9$, than
$N=[16,25,36,49,64]$. This coincides with the result shown in Fig. 2,
justifying the fact that low $N$ values ($N=\\{9,16\\}$) utilize lower
amplification power. However, for $N=64$, the algorithm also applies low
values of $\bar{\alpha}$. This is because, within scenarios where $N$ assumes
high values, the maximum available amplification power is easily achieved. We
can see that by increasing $N$ from $N=49$, the $\bar{\alpha}$ value starts to
decrease.

Figure 3: CDF of the average value of amplitude obtained with the proposed
algorithm. Here we set $P_{\rm TX}=35$ dBm.

###  VII-C Number of RIS Reflecting Elements

In the sequel, Fig. 4 illustrates the system EE as a function of the number of
reflecting elements ($N$) of RIS. Additionally, the performance of an entirely
passive RIS, with a fixed number of elements $N=64$, is depicted in the plot.
We can see that the active RIS exhibits comparable performance to the passive
RIS when equipped with $\approx 23$ reflecting elements. In other words, an
active RIS with $N=25$ ($N_{v}=N_{h}=5$) can surpass, in terms of EE, the
entirely passive RIS-aided M-MIMO with $N=64$ elements. Furthermore, we can
see when we increase the number of UEs $K$, the number of RIS elements is
slightly affected. This again demonstrates the potential of amplifying the
incoming signal at the RIS, enabling the possibility of reducing the number of
elements on the RIS without compromising the system EE performance.
Additionally, it shows the potential of active RIS to reduce the channel
estimation overhead, keeping the same performance as entirely passive RIS,
which can be very useful for practical systems since the channel estimation
overhead is one of the most critical drawbacks associated with the RIS
technology.

This unveils an interesting aspect of the active RIS, that enables it to
achieve the same EE performance as passive RIS while reducing the size of the
array physically. Let’s assume elements with a size of $d=\lambda$ and an
adjacent spacing between elements of $d_{\rm R}=\frac{\lambda}{2}$. For a
frequency operation of $f=5$ GHz, where the wavelength is $\lambda=60$ mm, the
length of RIS can be calculated using the formula $\mathcal{L}=\lambda
N_{h}+\frac{\lambda(N_{h}-1)}{2}$. With $N=64$, the length is $\mathcal{L}=69$
cm, while for active RIS with $N=25$ the length is 42 cm. This demonstrates
that the active RIS can achieve comparable performance with a physically
smaller array, providing potential deployment advantages.

Figure 4: Average EEvs number of RIS reflecting units. Performance evaluation
of the proposed algorithm for the active RISvs entirely passive RIS with fixed
number of elements $N=64$, for $P_{\rm TX}=35$ dBm and two different number of
UEs, $K=5$ and $K=10$.

Moreover, envisioning a scenario where system priorities may shift between SE
or EE metrics, a dynamic approach to controlling the number of active RIS
elements becomes paramount. For instance, when the SE is the priority,
employing a higher quantity of operating elements is a natural choice.
However, when prioritizing EE, a conscious reduction in the number of acting
elements becomes crucial. This flexibility allows an opportunistic and
consistent selection of RIS elements conditioned to the system’s specific
needs in different operational contexts.

###  VII-D Convergence Performance of Proposed Algorithm

Fig. 5 illustrates the convergence behavior of the proposed algorithm under
various conditions, showcasing their effectiveness for each one. The monotonic
convergence patterns are examined across three distinct values of $P_{\rm TX}$
with constant $N$, and for the same $P_{\rm TX}$ under two different values of
$N$. It is clear that the EE initially experiences an upward trend and
subsequently stabilizes with increasing iterations across all modes,
highlighting the efficiency of the proposed algorithm. In addition, a
noteworthy trend can be observed: as the number of reflective elements $N$
increases, so does the achieved EE. Moreover, we can see that the proposed
algorithm demonstrates rapid convergence (number of iterations) since it does
not need many iterations (6 to 9 iterations) to converge, $e.g.$, for the case
when $P_{\rm TX}=35$, and $N=64$, the number of iteration necessary to
converge is 6, while for $P_{\rm TX}=50$, and $N=25$, the number of iteration
necessary to converge is 9. This result shows that the power level has a more
significant impact on convergence than the number of reflective elements,
providing valuable insights into the algorithm’s efficiency under varying
operational conditions.

Figure 5: Average EEvs number of iterations for the proposed algorithm.
Performance evaluation of the proposed algorithm for the active RIS with fixed
number of elements $N=25$, for $P_{\rm TX}=\\{10,35,50\\}$ dBm and for fixed
transmit power budget $P_{\rm TX}=35$ dBm and $N=\\{49,64\\}$.

##  VIII Conclusions

We have examined the profound implications of optimizing the EE for active RIS
scenarios, seeking to understand and find the number of elements that need to
be selected for acting in active RIS given outperforming its entirely passive
RIS counterpart. By exploring various optimization techniques, we aim to
provide a low-complexity solution and ensure its proximity to the optimal
solution. This approach allows us to assess the effectiveness of our
analytical methodology to handle complex optimization challenges.

The numerical results show that the active RIS can easily outperform the
entirely passive RIS in terms of system EE, achieving a gain of about 120$\%$.
Furthermore, for typical system and channel scenarios, active RIS can operate
with less than half of elements compared with the passive RIS in view of
achieving about the same performance, enabling an operational cost reduction
for physical arrays and lower channel estimation overhead for practical
implementations. However, to achieve better performance in signal
amplification, the active RIS needs to be equipped with a reasonable number of
reflective elements, mainly in scenarios where the available power for signal
amplification is high. Finally, we show how the proposed algorithm can be
promising since it has required a few iterations for convergence.

We delved into the optimization of energy efficiency (EE) in RIS-assisted
M-MIMO systems. While significant progress has been made, several crucial
aspects pose open research challenges, offering promising avenues for
exploration. Our initial focus was on optimizing the RIS reflection
coefficients within the framework of MR precoding with EPA. However, it is
noteworthy that the precoding strategy at the BS also presents an opportunity
for optimization. Considering the joint optimization of RIS and BS precoding
can potentially unveil enhanced EE performance.

Moreover, our system model has assumed a far-field propagation channel.
Extending and investigating the analysis of near-field propagation channels is
a very relevant topic for future works. This extension is particularly
pertinent due to the prevalence of large aperture arrays aimed at mitigating
the multiplicative path-loss effect, and higher frequency operations for
increased bandwidth, which inherently increase the Rayleigh distance. Notably,
this has extended the range where near-field propagation dominates.
Furthermore, new opportunities arise in the near-field context; for instance,
the beam-focusing effect becomes a prominent phenomenon. This effect not only
allows adjustment of the direction of the reflected beam but also offers the
intriguing capability to control the distance within a specific angle.
Developing strategies to improve the EE of RIS-assisted M-MIMO networks
leveraging the beam-focusing effect for near-field scenarios is thus of
significant relevance.

Additionally, the current investigation centered on a single active RIS.
Extending the system model to accommodate a scenario with multiple active RIS
units is a valuable avenue for future research, offering insights into the
complexities and opportunities associated with such configurations.

Finally, the optimization scope can be broadened by considering the joint
optimization of BS and RIS antenna elements. By optimizing both components
simultaneously, the overall EE of the system can be finely tuned for improved
performance and resource utilization. These avenues highlight the richness of
potential research directions in advancing the understanding and optimization
of active RIS-assisted M-MIMO systems.

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