# Hybrid Relay-Reflecting Intelligent Surface-Assisted Wireless Communication

Nhan Thanh Nguyen, Quang-Doanh Vu, Kyungchun Lee, , and Markku Juntti The
research has been supported in part by Academy of Finland under 6Genesis
Flagship (grant 318927) and EERA Project (grant 332362).N. T. Nguyen and
Markku Juntti are with Centre for Wireless Communications, University of Oulu,
P.O.Box 4500, FI-90014, Finland, (e-mail<EMAIL_ADDRESS>markku.juntti@oulu.fi)Quang-Doanh Vu was with the Centre for Wireless
Communications, University of Oulu. He is now with the Mobile Networks, Nokia,
90650 Oulu, Finland (e-mail: quang-doanh.vu@nokia.com)K. Lee are with the
Department of Electrical and Information Engineering, Seoul National
University of Science and Technology, Seoul 01811, Republic of Korea (e-mail:
kclee@seoultech.ac.kr).

###### Abstract

Reconfigurable intelligent surface (RIS) has emerged as a cost- and energy-
efficient solution to enhance the wireless communication capacity. However,
recent studies show that a very large surface is required for a RIS-assisted
communication system; otherwise, they may be outperformed by the conventional
relay. Furthermore, the performance gain of a RIS can be considerably degraded
by hardware impairments such as limited-resolution phase shifters. To overcome
those challenges, we propose a novel concept of hybrid relay-reflecting
intelligent surface (HR-RIS), in which a single or few elements are deployed
with power amplifiers (PAs) to serve as active relays, while the remaining
elements only reflect the incident signals. Two architectures are proposed,
including the fixed and dynamic HR-RIS. Their coefficient matrices are
obtained based on alternating optimization (AO) and power allocation
strategies, which enable understanding the fundamental performances of RIS and
relaying-based systems with a trade-off between the two. The simulation
results show that a significant improvement in both the spectral efficiency
(SE) and energy efficiency (EE) with respect to the conventional RIS-aided
system can be attained by the proposed schemes, especially, by the dynamic HR-
RIS. In particular, the favorable design and deployment of the HR-RIS are
analytically derived and numerically justified.

##  I Introduction

The requirement of enhanced wireless communication services has created the
demand to use higher frequency bands including the millimeter-wave (mmWave)
bands. The high free-space path loss in those bands can be compensated for by
using large antenna arrays at both ends of the link [1, 2, 3]. Therefore,
massive multiple-input multiple-output (MIMO) transmission is built-in into
the mmWave system design to attain a significant improvement in spectral
efficiency (SE) from the large bandwidth and high beamforming gains. However,
the mmWave massive MIMO system generally requires increased power consumption,
hardware cost, and computational complexity for large-array signal processing
[4]. Furthermore, most practical mmWave links require line-of-sight (LoS)
propagation, because otherwise, the received signal power is too low for
reliable communications. Recently, a new technology called _reconfigurable
intelligent surface_ (RIS), also known as intelligent reflecting surface (IRS)
[5] or large intelligent surface (LIS) [6] has emerged as a new promising
solution to overcome the challenge of mmWave massive MIMO systems [7, 8, 9,
10, 11]. It may enable LoS-type connectivity on the mmWave bands by steering a
narrow beam from the RIS to the receiver assuming that LoS links exist between
the transmitter and RIS as well as between the RIS and receiver. However, the
potential benefits of a RIS do not limit to the mmWave bands but are of
interest also for the sub-6 GHz frequencies.

###  I-A Related Works

RIS is often assumed to be designed as a planar meta-surface or realized as a
planar array of numerous _passive reflecting elements_ , which are connected
to a controller, allowing modifying the phases of the incident signals
independently in real-time. As a result, its phase shifts can be optimized to
make the wireless channel between the transmitter and receiver more favorable
for the communication [4, 5, 8, 9]. Several potential technologies for RIS
implementation exist. It can be constructed as an array of discrete phase
shifters, which control the impedance values of the elements. Thereby the
phase shifts can steer the reflected signal to the desired direction of the
receiver. Such RIS is called the _discrete RIS_ with no baseband processing
capability [8, 9, 11]. The phase shift control is typically assumed to be
designed in some active network element such as a base station (BS) or access
point (AP). The phase shift values are then transferred via some control link
from the BS/AP to the RIS. Furthermore, a RIS with active receive elements
introduced by Taha et al. in [12, 13] has also been considered to enable the
channel estimation at the RIS.

The initial studies on RIS in the literature mainly focus on the performance
and design aspects of the RIS-assisted communication systems. Specifically, Hu
et. al in [14, 15] show that the capacity per RIS area-unit converges to
$\frac{\bar{P}}{2\sigma^{2}}$ as the wavelength goes to zero, where $\bar{P}$
is the transmit power per area-unit, and $\sigma^{2}$ is the additive white
Gaussian noise (AWGN) power spectral density. In [16], an adaptive phase
shifter design based on hierarchical codebooks and limited feedback from the
mobile station (MS) is proposed for a RIS-aided mmWave MIMO system for both
accurate positioning and high data rate transmission. In [17], the asymptotic
achievable rate in a RIS-assisted downlink system is analyzed under practical
reflection coefficients, and a passive beamformer and a modulation scheme that
can be used in a RIS without interfering with existing users is proposed to
increase the achievable system sum-rate. Furthermore, in [18], the propagation
channel of point-to-point MIMO systems is enriched by using the RIS so that
more paths with different spatial angles are added. As a result, the rank of
the channel matrix is enhanced, and the multiplexing gain is achieved even
when the direct path has low rank. The gain of RIS is further investigated in
[19] and [20] for multiple-input-single-output (MISO) downlink channels
operated at sub-6 GHz and mmWave frequencies, respectively. It is shown that
the RIS of $N$ elements can achieve a total beamforming gain of $N^{2}$ [19]
and allows the received signal power to increase quadratically with $N$ [20].
Moreover, the achievable data rate of communication systems assisted by a
practical RIS with limited phase shifts and hardware impairments are
characterized in [21, 22, 23, 24, 25, 26].

Another important line of studies on RIS focuses on the capacity/data rate
maximization of various RIS-assisted communication systems [27]. Specifically,
in [28], the problem of maximizing the achievable rate of a RIS-enhanced
single-input-single-output (SISO) orthogonal frequency division multiplexing
(OFDM) system has been solved by jointly optimizing the transmit power
allocation and the RIS passive array reflection coefficients. In contrast, the
MISO systems have been considered in [29, 30, 31, 23, 32]. In particular,
while [29, 30, 31] focus on the joint transmit and passive beamforming problem
of RISs with continuous phase shifts to attain substantially increased
capacity/data rate, Han et al. in [23] consider the discrete phase shifts and
justify that only a two-bit quantizer is sufficient to guarantee a high
capacity, which agrees with the finding in [32]. The capacity/rate
optimization of RIS-assisted MIMO systems has been recently considered in [4,
33, 25]. In [4], an efficient alternating optimization (AO) method has been
proposed to find the locally optimal phase shifts of the RIS for both the
narrowband frequency-flat and broadband frequency-selective MIMO OFDM systems.
In particular, it is shown that with the proposed AO-based RIS design, the
channel total power, rank, and condition number can be significantly improved
for capacity enhancement. In [25], a RIS-enhanced full-duplex (FD) MIMO two-
way communication system is considered. The system sum-rate is maximized
through jointly optimizing the transmit beamforming and the RIS’s coefficient
matrix. Specifically, the non-convex optimization problem is decoupled into
three sub-problems, which are solved iteratively, and its efficiency is
justified by simulation results.

###  I-B Motivations and Contributions

In this paper, we propose a novel _semi-active RIS-aided beamforming_ concept
in which active beamforming (relaying) can be applied together with passive
beamforming (reflecting). The idea is to activate a few elements of the RIS by
connecting them with radio frequency (RF) chains and power amplifiers (PAs),
allowing them to modify not only the phases but also the amplitudes of the
incident signals. In this respect, these few activated elements become active
amplify-and-forward (AF) relaying elements, and the conventional RIS becomes a
_hybrid relay-reflecting intelligent surface_ (HR-RIS). We use this term
throughout the paper to refer to the proposed architecture.

The HR-RIS may potentially be realized by recently introduced low-power
complementary metal-oxide semiconductor (CMOS)-based technologies, such as the
reflection amplifier (RA) [34, 35], which can turn not only the phases but
also the amplitudes of the incident signals. Thereby a possible implementation
of the proposed architecture could produce a meta-surface with RA elements.
However, this method can be cost-inefficient because the number of elements in
the HR-RIS is large, while only a few active elements are required, as will be
proven in the sequel. Therefore, a more practical solution appears to be
upgrading the conventional RIS by replacing some elements with the RAs.
Obviously, compared to the RIS, HR-RIS requires additional complexity for
hardware implementation and signal processing of the active elements. However,
those are only required for a single or few active elements. Considering that
the total number of elements in the conventional RIS is very large, the
hardware and computational costs increase only moderately. However, the
detailed HR-RIS implementation is out of the scope of this paper. We focus on
proposing it and analyzing its potential for system performance improvement.

The HR-RIS is motivated by recent comparisons between the conventional relay
and RIS, the practical deployment of RIS with hardware impairments, and the
demand for channel estimation at the RIS. First, RIS can be thought in some
respect similar to an FD relay [36, 37], but it consists of passive elements
without the active power amplifier, enabling passive beamforming without
requiring any active power and introducing no thermal noise. However, a main
limitation of the RIS compared to the relays is the fact that the reflection
limits the degrees of freedom in the beamforming. Therefore, HR-RIS, a hybrid
architecture between the RIS and relay, can leverage the advantages while
mitigating the disadvantages of RIS and relay. Second, based on the
performance comparisons of relay and RIS in [19] and [36], we found that the
HR-RIS is promising to provide a remarkable performance improvement.
Specifically, both Björnson et al. [36] and Wu et al. [19] show that a very
large RIS is needed to outperform decode-and-forward (DF) relaying; otherwise,
it can be easily outperformed even by a half-duplex (HD) relay with few
elements. Furthermore, as the RIS becomes sufficiently large, the performance
improvement is not significant if the number of elements in the RIS increases.
This implies that if a few passive elements of the RIS are replaced by active
ones, the reduction in the passive beamforming gain is just marginal, while
the gain from active relaying can be significant. Third, it is shown in [22,
23, 24, 25] that RISs with limited-resolution phase shifts have considerable
performance loss with respect to those with sufficiently large resolution.
However, as the amplitudes of a few elements are adjustable, they can be
optimized to compensate for the performance loss due to the limited-resolution
phase shifts. The aforementioned aspects motivate a new architecture of the
intelligent meta-surface, allowing it to leverage the benefits of both the RIS
and relay, as the HR-RIS proposed in this paper.

We recognize that there are several practical open problems related to the
practical and efficient implementation of HR-RIS. What is more, assume perfect
channel state information (CSI) [19, 4, 22, 38]. However, with the proposed
HR-RIS architecture, the assumption is less restrictive than with a passive
RIS, because the active processing chains can be readily used for channel
estimation using their built-in active elements [12]. Thereby, our paper
proposes the concept and demonstrates its significant system-level potential.
This gives more understanding on the fundamental trade-offs between RIS and
relaying as part of a MIMO communication link.

The major contributions of the paper are summarized as follows:

  * •

We propose the novel HR-RIS concept, which enables a hybrid active-passive
beamforming scheme rather than fully-passive beamforming of the conventional
RIS. It requires activating only a single or few elements of the RIS to serve
as active relays to attain remarkable performance improvement. In particular,
HR-RIS exploits the benefits of relaying while mitigating the limitation of
passive reflecting.

  * •

We propose two HR-RIS architectures, namely, fixed and dynamic HR-RIS. In the
former, the active elements are fixed in manufacture; by contrast, those in
the latter can be adaptively configured to improve the performance and save
power consumption. The coefficient matrices of the HR-RIS are optimized in the
formulated SE maximization problem, which is solved by the AO and power
allocation schemes. Furthermore, the favorable deployment and performance
gains of the HR-RIS are derived analytically and numerically verified by
computer simulations.

  * •

The total power consumption and energy efficiency of the system with the
proposed HR-RIS is analyzed and compared to that with the conventional RIS. It
is shown that the former requires higher power consumption than the latter
because additional power is consumed for the active processing. However, with
a small number of active elements in the proposed HR-RIS, the improvement in
both the SE and energy efficiency (EE) is guaranteed, as numerically
demonstrated by the simulation results.

  * •

The paper bridges the theoretical gap between a passive RIS and active AF
relay aiding a MIMO communication link. This gives more insight into the
fundamental performance of each of the approaches and attainable SE and EE.

Structure: The rest of the paper are as follows. In Section II, we introduce
the concept of the HR-RIS and the system model of the HR-RIS-aided MIMO
system, and the problem of SE maximization is formulated. Its efficient
solution is found in Section III. Based on that, the coefficient matrices of
the fixed and dynamic HR-RIS architectures are derived in Section IV. Their
power consumption is investigated in Section V. Simulation results are shown
in Section VI, and, finally, conclusions are drawn in Section VII.

Notations: Throughout this paper, numbers, vectors, and matrices are denoted
by lower-case, bold-face lower-case, and bold-face upper-case letters,
respectively. $(\cdot)^{*}$ and $(\cdot)^{H}$ denote the conjugate of a
complex number and the conjugate transpose of a matrix or vector,
respectively. $\textbf{{I}}_{N}$ denotes the identity matrix of size $N\times
N$, and $\mathrm{diag}\\{a_{1},\ldots,a_{N}\\}$ represents a diagonal matrix
with diagonal entries $a_{1},\ldots,a_{N}$. Furthermore, $\left|\cdot\right|$
denotes either the absolute value of a scalar or determinant of a matrix, and
the expectation operator is denoted by $\mathbb{E}\left\\{\cdot\right\\}$.

##  II HR-RIS Architecture, System model, and SE Problem formulation

Fig. 1: Illustration of the fixed and dynamic HR-RIS architectures.

###  II-A Proposed HR-RIS Beamforming Architecture

The HR-RIS is equipped with $N$ elements, including $M$ passive reflecting and
$K$ active relay elements $(M+K=N)$. Here, the passive element can only shift
the phase while the active one can tune both the phase and amplitude of the
incident signal. As such, for $K=0$, the HR-RIS serves as the conventional
RIS; by contrast, for $K=N$, it becomes a conventional relay station. It is
noted that an active element would consume more power for processing compared
to the passive one due to the operation of the RF chain. Thus, for practical
deployment, we are interested in the case that $1\leq K\ll M$. Furthermore,
similar to the conventional RIS, we assume that each (active/passive) element
of the HR-RIS can re-scatter the signal independently (without cross-
interference) by an individual coefficient [4].

We now introduce some notations in order to mathematically model the HR-RIS.
In particular, let us denote by $\mathbb{A}$,
$\mathbb{A}\subset\\{1,2,\ldots,N\\}$, the index set of the positions of the
$K$ active elements. Let $\alpha_{n}$ denote the relay/reflection coefficient
used at the $n$th element, which is given as

$\displaystyle\alpha_{n}=\begin{cases}\left|\alpha_{n}\right|e^{j\theta_{n}},&\text{if
}n\in\mathbb{A}\\\ e^{j\theta_{n}},&\text{otherwise}\end{cases},$ (1)

where $\theta_{n}\in[0,2\pi)$ represent the phase shift. We note that
$\left|\alpha_{n}\right|=1$ for $n\notin\mathbb{A}$. We also define three
diagonal matrices constructed from $\\{\alpha_{n}\\}_{n=1,\ldots,N}$. Let
$\boldsymbol{\Upsilon}=\text{diag}\\{\alpha_{1},\ldots,\alpha_{N}\\}\in\mathbb{C}^{N\times
N}$ be the diagonal matrix of the coefficients. We define an additive
decomposition for it as
$\boldsymbol{\Upsilon}=\boldsymbol{\Phi}+\boldsymbol{\Psi}$, where
$\boldsymbol{\Phi}=\text{diag}\\{\phi_{1},\ldots,\phi_{N}\\}\in\mathbb{C}^{N\times
N}$,
$\boldsymbol{\Psi}=\text{diag}\\{\psi_{1},\ldots,\psi_{N}\\}\in\mathbb{C}^{N\times
N}$,

$\displaystyle\phi_{n}$ $\displaystyle=\begin{cases}0,&\text{if
}n\in\mathbb{A}\\\ \alpha_{n}=e^{j\theta_{n}},&\text{otherwise}\end{cases},$
$\displaystyle\psi_{n}$
$\displaystyle=\begin{cases}\alpha_{n}=\left|\alpha_{n}\right|e^{j\theta_{n}},&\text{if
}n\in\mathbb{A}\\\ 0,&\text{otherwise}\end{cases}.$

In other words, $\boldsymbol{\Phi}$ and $\boldsymbol{\Psi}$ contain the
passive reflecting and active relaying coefficients, respectively.

There can be two architectures for the HR-RIS, namely, the fixed and dynamic
HR-RIS. In the former, the number and positions of the active elements, i.e.,
set ${\mathbb{A}}$, are predefined and fixed as illustrated in Fig. 1(a) [13,
12]. By contrast, in the dynamic HR-RIS architecture, the number and positions
of active elements can be dynamically changed according to the propagation
condition. In other words, set ${\mathbb{A}}$ can be a design parameter in
this architecture. More specifically, there are a number of RF-PA chains each
can be turned on/off. An element of HR-RIS is active if it connects to an “ON”
RF-PA chain; otherwise, it serves as a passive reflecting element [13, 12,
39]. Such connection can be done via a switching (SW) network as illustrated
in Fig. 1(b). Furthermore, the signal processing in the active elements are
controlled by an HR-RIS controller and baseband unit [13, 12]. In this paper,
both the fixed and dynamic HR-RIS architectures are considered.

###  II-B System Model

Fig. 2: Illustration of the HR-RIS-aided communication system.

We consider a downlink transmission between a BS and an MS. Suppose that there
is no direct link between the BS and the MS due to, e.g., severe pathloss or
blockage [40], and the communications is aided by the HR-RIS as illustrated in
Fig. 2 [13, 12]. Let us denote by $N_{t}$ and $N_{r}$ the numbers of antennas
equipped at the BS and MS, respectively, where $N_{t}\geq 1$ and $N_{r}\geq
1$. Let $\textbf{{H}}_{t}\in\mathbb{C}^{N\times N_{t}}$ and
$\textbf{{H}}_{r}\in\mathbb{C}^{N_{r}\times N}$ denote the channel between the
BS and the HR-RIS, and between the HR-RIS and the MS, respectively. Let
$\textbf{{x}}\in\mathbb{C}^{N_{t}\times 1}$ be the transmitted signal vector,
where
$\mathbb{E}\left\\{\textbf{{x}}\textbf{{x}}^{H}\right\\}=P_{\mathrm{BS}}\textbf{{I}}_{N_{t}}$
and $P_{\mathrm{BS}}$ is the transmit power at the BS. With these newly
introduced notations, the received signals at the MS can be written as

y
$\displaystyle=\textbf{{H}}_{r}\boldsymbol{\Phi}\textbf{{H}}_{t}\textbf{{x}}+\textbf{{H}}_{r}\boldsymbol{\Psi}\textbf{{H}}_{t}\textbf{{x}}+\textbf{{H}}_{r}\boldsymbol{\Psi}\textbf{{n}}_{\mathrm{H}}+\textbf{{n}}_{\mathrm{MS}}$
$\displaystyle=\left(\textbf{{H}}_{r}\boldsymbol{\Phi}\textbf{{H}}_{t}+\textbf{{H}}_{r}\boldsymbol{\Psi}\textbf{{H}}_{t}\right)\textbf{{x}}+\textbf{{n}}$
$\displaystyle=\textbf{{H}}_{r}\boldsymbol{\Upsilon}\textbf{{H}}_{t}\textbf{{x}}+\textbf{{n}},$
(2)

where
$\textbf{{n}}_{\mathrm{H}}\sim\mathcal{CN}(0,\sigma_{\mathrm{H}}^{2}\textbf{{I}}_{K})$
and
$\textbf{{n}}_{\mathrm{MS}}\sim\mathcal{CN}(0,\sigma_{\mathrm{MS}}^{2}\textbf{{I}}_{N_{r}})$
are the complex additive white Gaussian noise (AWGN) vectors at the $K$ active
relay elements of the HR-RIS and at the MS, respectively; and
$\textbf{{n}}=\textbf{{H}}_{r}\boldsymbol{\Psi}\textbf{{n}}_{\mathrm{H}}+\textbf{{n}}_{\mathrm{MS}}$
represents the total effective noise at the MS. For notational simplicity, we
assume that $\sigma_{\mathrm{H}}^{2}=\sigma_{\mathrm{MS}}^{2}=\sigma^{2}$ and
$\textbf{{n}}\sim\mathcal{CN}\left(0,\sigma^{2}\left(\textbf{{I}}_{N_{r}}+\textbf{{H}}_{r}\boldsymbol{\Psi}\boldsymbol{\Psi}^{H}\textbf{{H}}_{r}^{H}\right)\right)$.

###  II-C Problem Formulation

Based on (2), the SE of an HR-RIS-aided MIMO system can be expressed as

$\displaystyle
f_{0}(\\{\alpha_{n}\\})=\log_{2}\left|\textbf{{I}}_{N_{r}}+\rho\textbf{{H}}_{r}\boldsymbol{\Upsilon}\textbf{{H}}_{t}\textbf{{H}}_{t}^{H}\boldsymbol{\Upsilon}^{H}\textbf{{H}}_{r}^{H}\textbf{{R}}^{-1}\right|,$

where $\\{\alpha_{n}\\}$ represents the set of all the relay/reflecting
coefficients on the main diagonal of $\boldsymbol{\Upsilon}$, i.e.,
$\\{\alpha_{n}\\}=\\{\alpha_{1},\alpha_{2},\ldots,\alpha_{N}\\}$. Furthermore,

$\displaystyle\textbf{{R}}=\textbf{{I}}_{N_{r}}+\textbf{{H}}_{r}\boldsymbol{\Psi}\boldsymbol{\Psi}^{H}\textbf{{H}}_{r}^{H}\in\mathbb{C}^{N_{r}\times
N_{r}}$ (3)

is the aggregate noise covariance matrix, and
$\rho=\frac{P_{\mathrm{BS}}}{\sigma^{2}}$. The transmit power of the active
elements of the HR-RIS is

$\displaystyle P_{\mathrm{a}}(\\{\alpha_{n}\\})$
$\displaystyle\triangleq\mathrm{trace}\left(\boldsymbol{\Psi}\left(\textbf{{H}}_{t}\mathbb{E}\left\\{\textbf{{x}}\textbf{{x}}^{H}\right\\}\textbf{{H}}_{t}^{H}+\sigma^{2}\textbf{{I}}_{N}\right)\boldsymbol{\Psi}^{H}\right)$
$\displaystyle=\mathrm{trace}\left(\boldsymbol{\Psi}\left(P_{\mathrm{BS}}\textbf{{H}}_{t}\textbf{{H}}_{t}^{H}+\sigma^{2}\textbf{{I}}_{N}\right)\boldsymbol{\Psi}^{H}\right).$
(4)

The problem of designing relay/reflection coefficients of the HR-RIS for
maximizing the SE can be formulated as

$\displaystyle(\mathrm{P0})\quad\underset{\\{\alpha_{n}\\}}{\textrm{maximize}}\quad$
$\displaystyle f_{0}(\\{\alpha_{n}\\})$ (5a) subject to
$\displaystyle\left|\alpha_{n}\right|=1\text{ for }n\notin\mathbb{A}$ (5b)
$\displaystyle P_{\mathrm{a}}(\\{\alpha_{n}\\})\leq
P_{\mathrm{a}}^{\textrm{max}}$ (5c)

where $P_{\mathrm{a}}^{\textrm{max}}$ is the power budget of the active
elements. Function $f_{0}(\\{\alpha_{n}\\})$ is nonconvex with respect to
$\\{\alpha_{n}\\}$. In addition, the feasible set of $(\mathrm{P0})$ is
nonconvex due to the unit-modulus constraint (5b). Consequently, problem
$(\mathrm{P0})$ is intractable, and it is difficult to find an optimal
solution. In the following section, we develop an efficient solution to
$(\mathrm{P0})$.

##  III Efficient Solution to $(\mathrm{P0})$

###  III-A A Tractable Approximation of $(\mathrm{P0})$

As the first step of developing an efficient solution to $(\mathrm{P0})$, we
approximate the problem into a more tractable one. Specifically,
$f_{0}(\\{\alpha_{n}\\})$ is upper bounded by $f(\\{\alpha_{n}\\})$, as
follows:

$\displaystyle f_{0}(\\{\alpha_{n}\\})$
$\displaystyle=\log_{2}\left|\textbf{{I}}_{N_{r}}+\rho\textbf{{H}}_{r}\boldsymbol{\Upsilon}\textbf{{H}}_{t}\textbf{{H}}_{t}^{H}\boldsymbol{\Upsilon}^{H}\textbf{{H}}_{r}^{H}\textbf{{R}}^{-1}\right|$
$\displaystyle=\log_{2}\left|\textbf{{R}}+\rho\textbf{{H}}_{r}\boldsymbol{\Upsilon}\textbf{{H}}_{t}\textbf{{H}}_{t}^{H}\boldsymbol{\Upsilon}^{H}\textbf{{H}}_{r}^{H}\right|-\log_{2}\left|\textbf{{R}}\right|$
(6)
$\displaystyle\leq\log_{2}\left|\textbf{{I}}_{N_{r}}+\textbf{{H}}_{r}\boldsymbol{\Psi}\boldsymbol{\Psi}^{H}\textbf{{H}}_{r}^{H}+\rho\textbf{{H}}_{r}\boldsymbol{\Upsilon}\textbf{{H}}_{t}\textbf{{H}}_{t}^{H}\boldsymbol{\Upsilon}^{H}\textbf{{H}}_{r}^{H}\right|$
(7) $\displaystyle\triangleq f(\\{\alpha_{n}\\}),$

where the inequality in (7) follows by substituting (3) to (6). Clearly, the
equality occurs when $\mathbb{A}=\emptyset$. In addition, the smaller the term
$\log_{2}\left|\textbf{{R}}\right|$ is, the tighter the bound is. We can also
observe that $\log_{2}\left|\textbf{{R}}\right|$ is small when the path loss
is large and the number of active elements is small (since $\boldsymbol{\Psi}$
becomes very sparse when only a single or few active elements are employed at
the HR-RIS). These observations motivate us to arrive at the following
approximation of $(\mathrm{P0})$

$\displaystyle\left(\mathrm{P}_{\mathrm{hybrid}}\right)\quad\underset{\\{\alpha_{n}\\}}{\textrm{maximize}}\quad$
$\displaystyle f(\\{\alpha_{n}\\})$ subject to
$\displaystyle\eqref{opt_hyb_con_unit_modul},\eqref{opt_hyb_con_relay_power}.$

Although $f(\\{\alpha_{n}\\})$ is still nonconvex with respect to
$\\{\alpha_{n}\\}$, its structure allows us to develop an efficient solution
via the AO method. The details are presented in the following subsection.

###  III-B Efficient Solution to $\left(\mathrm{P}_{\mathrm{hybrid}}\right)$:
an AO Approach

Generally, the proposed solution is a sequential procedure: in each iteration,
a specific coefficient of HR-RIS is updated when the others are fixed. We
first further expand $f(\\{\alpha_{n}\\})$ to extract the role of variable
$\alpha_{n}$, for any $n\in\\{1,...,N\\}$. Specifically, let
$\textbf{{r}}_{n}\in\mathbb{C}^{N_{r}\times 1}$ be the $n$th column of
$\textbf{{H}}_{r}$, and $\textbf{{t}}_{n}^{H}\in\mathbb{C}^{1\times N_{t}}$ be
the $n$th row of $\textbf{{H}}_{t}$, i.e.,
$\textbf{{H}}_{r}=[\textbf{{r}}_{1},\ldots,\textbf{{r}}_{N}]$ and
$\textbf{{H}}_{t}=[\textbf{{t}}_{1},\ldots,\textbf{{t}}_{N}]^{H}$. Since
$\boldsymbol{\Upsilon}$ and $\boldsymbol{\Psi}$ are diagonal matrices, we have
$\textbf{{H}}_{r}\boldsymbol{\Upsilon}\textbf{{H}}_{t}=\sum_{n=1}^{N}\alpha_{n}\textbf{{r}}_{n}\textbf{{t}}_{n}^{H}$
and
$\textbf{{H}}_{r}\boldsymbol{\Psi}=\sum_{n\in\mathbb{A}}\alpha_{n}\textbf{{r}}_{n}$.
Consequently, we can express $f(\\{\alpha_{n}\\})$ as follows

$\displaystyle
f(\\{\alpha_{n}\\})=\log_{2}\left|\textbf{{A}}_{n}+\left|\alpha_{n}\right|^{2}\textbf{{B}}_{n}+\alpha_{n}\textbf{{C}}_{n}+\alpha_{n}^{*}\textbf{{C}}_{n}^{H}\right|,$
(8)

where matrices $\textbf{{A}}_{n}$, $\textbf{{B}}_{n}$, and $\textbf{{C}}_{n}$
are given by

$\displaystyle\textbf{{A}}_{n}$
$\displaystyle\triangleq\textbf{{I}}_{N_{r}}+\left(\sum_{i\in\mathbb{A},i\neq
n}\alpha_{i}\textbf{{r}}_{i}\right)\left(\sum_{i\in\mathbb{A},i\neq
n}\alpha_{i}^{*}\textbf{{r}}_{i}^{H}\right)$
$\displaystyle\quad+\rho\left(\sum_{i=1,i\neq
n}^{N}\alpha_{i}\textbf{{r}}_{i}\textbf{{t}}_{i}^{H}\right)\left(\sum_{i=1,i\neq
n}^{N}\alpha_{i}^{*}\textbf{{t}}_{i}\textbf{{r}}_{i}^{H}\right),$ (9)
$\displaystyle\textbf{{B}}_{n}$
$\displaystyle\triangleq\textbf{{r}}_{n}\textbf{{r}}_{n}^{H}+\rho\textbf{{r}}_{n}\textbf{{t}}_{n}^{H}\textbf{{t}}_{n}\textbf{{r}}_{n}^{H},$
(10) $\displaystyle\textbf{{C}}_{n}$
$\displaystyle\triangleq\textbf{{r}}_{n}\left(\sum_{i\in\mathbb{A},i\neq
n}\alpha_{i}^{*}\textbf{{r}}_{i}^{H}\right)+\rho\textbf{{r}}_{n}\textbf{{t}}_{n}^{H}\left(\sum_{i=1,i\neq
n}^{N}\alpha_{i}^{*}\textbf{{t}}_{i}\textbf{{r}}_{i}^{H}\right)$ (11)

for $n\in\mathbb{A}$, and

$\displaystyle\textbf{{A}}_{n}$
$\displaystyle\triangleq\textbf{{I}}_{N_{r}}+\rho\left(\sum_{i=1,i\neq
n}^{N}\alpha_{i}\textbf{{r}}_{i}\textbf{{t}}_{i}^{H}\right)\left(\sum_{i=1,i\neq
n}^{N}\alpha_{i}^{*}\textbf{{t}}_{i}\textbf{{r}}_{i}^{H}\right),$ (12)
$\displaystyle\textbf{{B}}_{n}$
$\displaystyle\triangleq\rho\textbf{{r}}_{n}\textbf{{t}}_{n}^{H}\textbf{{t}}_{n}\textbf{{r}}_{n}^{H},$
(13) $\displaystyle\textbf{{C}}_{n}$
$\displaystyle\triangleq\rho\textbf{{r}}_{n}\textbf{{t}}_{n}^{H}\left(\sum_{i=1,i\neq
n}^{N}\alpha_{i}^{*}\textbf{{t}}_{i}\textbf{{r}}_{i}^{H}\right)$ (14)

for $n\notin\mathbb{A}$. We note that matrices
$\textbf{{A}}_{n},\textbf{{B}}_{n},$ and $\textbf{{C}}_{n}$ do not contain
variable $\alpha_{n}$. This means that if all variables
$\\{\alpha_{i}\\}_{i=1,i\neq n}^{N}$ are fixed, the three matrices are
determined.

Similarly, we also extract the role of variable $\alpha_{n}$ in
$P_{\mathrm{a}}(\\{\alpha_{n}\\})$ by rewriting

$\displaystyle P_{\mathrm{a}}(\\{\alpha_{n}\\})$
$\displaystyle=\mathrm{trace}\left(P_{\mathrm{BS}}\boldsymbol{\Psi}\textbf{{H}}_{t}\textbf{{H}}_{t}^{H}\boldsymbol{\Psi}^{H}+\sigma^{2}\boldsymbol{\Psi}\boldsymbol{\Psi}^{H}\right)$
$\displaystyle=\sigma^{2}\mathrm{trace}\left(\boldsymbol{\Psi}\boldsymbol{\Psi}^{H}\right)+P_{\mathrm{BS}}\mathrm{trace}\left(\boldsymbol{\Psi}\textbf{{H}}_{t}\textbf{{H}}_{t}^{H}\boldsymbol{\Psi}^{H}\right)$
$\displaystyle=\sigma^{2}\sum_{n\in\mathbb{A}}\left|\alpha_{n}\right|^{2}+P_{\mathrm{BS}}\sum_{n\in\mathbb{A}}\left|\alpha_{n}\right|^{2}\left\lVert\textbf{{t}}_{n}\right\rVert^{2}$
$\displaystyle=\sum_{n\in\mathbb{A}}\left|\alpha_{n}\right|^{2}\xi_{n},$ (15)

where
$\xi_{n}\triangleq\sigma^{2}+P_{\mathrm{BS}}\left\lVert\textbf{{t}}_{n}\right\rVert^{2}$.
Let $\tilde{P_{\mathrm{a}}}\triangleq\sum_{i\in\mathbb{A},i\neq
n}\left|\alpha_{i}\right|^{2}\xi_{i}$, $n\in\mathbb{A}$. We can see that
$\tilde{P_{\mathrm{a}}}$ is a constant if all variables
$\\{\alpha_{i}\\}_{i=1,i\neq n}^{N}$ are fixed. Therefore, the role of
$\alpha_{n}$ in $P_{\mathrm{a}}(\\{\alpha_{n}\\})$ can be seen in the
following expression of $P_{\mathrm{a}}(\\{\alpha_{n}\\})$:

$\displaystyle
P_{\mathrm{a}}(\\{\alpha_{n}\\})=\left|\alpha_{n}\right|^{2}\xi_{n}+\tilde{P_{\mathrm{a}}}.$
(16)

####  III-B1 Problem of Updating $\alpha_{n}$

In the AO approach, in an iteration of updating $\alpha_{n}$,
$\\{\alpha_{i}\\}_{i=1,i\neq n}^{N}$ are fixed. Therefore, the objective
function $f(\\{\alpha_{n}\\})$ in (8) can be rewritten in the explicit form

$\displaystyle
f_{n}(\alpha_{n})=\log_{2}\left|\textbf{{A}}_{n}+\left|\alpha_{n}\right|^{2}\textbf{{B}}_{n}+\alpha_{n}\textbf{{C}}_{n}+\alpha_{n}^{*}\textbf{{C}}_{n}^{H}\right|,$
(17)

which has only one variable, i.e., $\alpha_{n}$. Moreover, because
$\textbf{{A}}_{n}$ is of full-rank ($\textrm{rank}(\textbf{{A}}_{n})=N_{r}$)
and invertible, we can rewrite $f_{n}(\alpha_{n})$ as

$\displaystyle
f_{n}(\alpha_{n})=\log_{2}\left|\textbf{{A}}_{n}\right|+g_{n}(\alpha_{n}),$
(18)

where

$\displaystyle
g_{n}(\alpha_{n})\triangleq\log_{2}\left|\textbf{{I}}_{N_{r}}+\left|\alpha_{n}\right|^{2}\textbf{{A}}_{n}^{-1}\textbf{{B}}_{n}+\alpha_{n}\textbf{{A}}_{n}^{-1}\textbf{{C}}_{n}+\alpha_{n}^{*}\textbf{{A}}_{n}^{-1}\textbf{{C}}_{n}^{H}\right|.$
(19)

In (18), $\log_{2}\left|\textbf{{A}}_{n}\right|$ is a constant with
$\alpha_{n}$. Therefore, the problem of updating $\alpha_{n}$, denoted by
$\left(\mathrm{P_{update}}\right)$, is given by

$\displaystyle\begin{cases}\underset{\alpha_{n}}{\textrm{maximize}}\;g_{n}(\alpha_{n})\;\textrm{subject
to}\;\left|\alpha_{n}\right|=1,&\textrm{for}\;n\notin\mathbb{A}\\\
\underset{\alpha_{n}}{\textrm{maximize}}\;g_{n}(\alpha_{n})\;\textrm{subject
to}\;\left|\alpha_{n}\right|^{2}\leq\frac{P_{\mathrm{a}}^{\mathrm{max}}-\tilde{P_{\mathrm{a}}}}{\xi_{n}},&\textrm{for}\;n\in\mathbb{A}\end{cases}$
(20)

####  III-B2 Solution to $\left(\mathrm{P_{update}}\right)$

$\left(\mathrm{P_{update}}\right)$ in (20) admits a closed-form solution, and,
thus, it is efficient for practical implementation. To derive it, we rewrite
$g_{n}(\alpha_{n})$ as

$\displaystyle g_{n}(\alpha_{n})$
$\displaystyle=\log_{2}\left|\textbf{{D}}_{n}+\alpha_{n}\textbf{{A}}_{n}^{-1}\textbf{{C}}_{n}+\alpha_{n}^{*}\textbf{{A}}_{n}^{-1}\textbf{{C}}_{n}^{H}\right|$
$\displaystyle=\log_{2}\left|\textbf{{D}}_{n}\right|+\log_{2}\left|\textbf{{I}}_{N_{r}}+\alpha_{n}\textbf{{E}}_{n}^{-1}\textbf{{C}}_{n}+\alpha_{n}^{*}\textbf{{E}}_{n}^{-1}\textbf{{C}}_{n}^{H}\right|,$
(21)

where
$\textbf{{D}}_{n}=\textbf{{I}}_{N_{r}}+\left|\alpha_{n}\right|^{2}\textbf{{A}}_{n}^{-1}\textbf{{B}}_{n}$,
and $\textbf{{E}}_{n}=\textbf{{A}}_{n}\textbf{{D}}_{n}$. We start
investigating the objective function $g_{n}(\alpha_{n})$ by considering the
first term in (21), i.e., $\log_{2}\left|\textbf{{D}}_{n}\right|$.
Specifically, we note that in $\textbf{{D}}_{n}$,
$\mathrm{rank}(\textbf{{A}}_{n}^{-1}\textbf{{B}}_{n})\leq\mathrm{rank}(\textbf{{B}}_{n})=1$.
In addition, the probability that
$\mathrm{rank}(\textbf{{A}}_{n}^{-1}\textbf{{B}}_{n})=0$ is almost zero (it
only happens when $\textbf{{A}}_{n}^{-1}\textbf{{B}}_{n}=\textbf{0}$).
Consequently, we generally have
$\mathrm{rank}(\textbf{{A}}_{n}^{-1}\textbf{{B}}_{n})=1$. Also, we observe
that $\textbf{{A}}_{n}^{-1}\textbf{{B}}_{n}$ is non-diagonalizable iff
$\mathrm{trace}(\textbf{{A}}_{n}^{-1}\textbf{{B}}_{n})=0$ [4], which rarely
happens in general. Thus, we almost surely have
$\mathrm{trace}(\textbf{{A}}_{n}^{-1}\textbf{{B}}_{n})\neq 0$ and
$\textbf{{A}}_{n}^{-1}\textbf{{B}}_{n}$ is diagonalizable. As a result, we can
write as
$\textbf{{A}}_{n}^{-1}\textbf{{B}}_{n}=\textbf{{Q}}_{n}\boldsymbol{\Gamma}_{n}\textbf{{Q}}_{n}^{-1}$
based on the eigenvalue decomposition (EVD) where
$\textbf{{Q}}_{n}\in\mathbb{C}^{N_{r}\times N_{r}}$,
$\boldsymbol{\Gamma}_{n}=\mathrm{diag}\left\\{\gamma_{n},0,\ldots,0\right\\}$,
and $\gamma_{n}$ is the sole non-zero eigenvalue of
$\textbf{{A}}_{n}^{-1}\textbf{{B}}_{n}$. Finally, since $\textbf{{A}}_{n}$ and
$\textbf{{B}}_{n}$ are both positive semi-definite, $\gamma_{n}$ is non-
negative and real. Thus, we can write

$\displaystyle\log_{2}\left|\textbf{{D}}_{n}\right|$
$\displaystyle=\log_{2}\left|\textbf{{I}}_{N_{r}}+\left|\alpha_{n}\right|^{2}\textbf{{Q}}_{n}\boldsymbol{\Gamma_{n}}\textbf{{Q}}_{n}^{-1}\right|$
$\displaystyle=\log_{2}\left|\textbf{{Q}}_{n}\left(\textbf{{I}}_{N_{r}}+\left|\alpha_{n}\right|^{2}\boldsymbol{\Gamma_{n}}\right)\textbf{{Q}}_{n}^{-1}\right|$
$\displaystyle=\log_{2}\left|\textbf{{I}}_{N_{r}}+\left|\alpha_{n}\right|^{2}\boldsymbol{\Gamma_{n}}\right|$
$\displaystyle=\log_{2}\left(1+\left|\alpha_{n}\right|^{2}\gamma_{n}\right).$
(22)

We now focus on the second term in (21). Following the similar arguments for
the first term, we also almost surely have
$\textbf{{E}}_{n}^{-1}\textbf{{C}}_{n}$ diagonalizable. Thus, we have
$\textbf{{E}}_{n}^{-1}\textbf{{C}}_{n}=\textbf{{U}}_{n}\boldsymbol{\Lambda}_{n}\textbf{{U}}_{n}^{-1}$
based on the EVD, where $\textbf{{U}}_{n}\in\mathbb{C}^{N_{r}\times N_{r}}$,
$\boldsymbol{\Lambda}_{n}=\mathrm{diag}\left\\{\lambda_{n},0,\ldots,0\right\\}$,
and $\lambda_{n}$ is the sole non-zero eigenvalue of
$\textbf{{E}}_{n}^{-1}\textbf{{C}}_{n}$. Now, let
$\textbf{{V}}_{n}\triangleq\textbf{{U}}_{n}^{H}\textbf{{E}}_{n}\textbf{{U}}_{n}$,
$v_{n}$ be the first element of the first column of $\textbf{{V}}_{n}^{-1}$,
and $v_{n}^{\prime}$ be the first element of the first row of
$\textbf{{V}}_{n}$. As a result, we can write [4]

$\displaystyle\log_{2}\left|\textbf{{I}}_{N_{r}}+\alpha_{n}\textbf{{E}}_{n}^{-1}\textbf{{C}}_{n}+\alpha_{n}^{*}\textbf{{E}}_{n}^{-1}\textbf{{C}}_{n}^{H}\right|$
$\displaystyle=\log_{2}\left(1+\left|\alpha_{n}\right|^{2}\left|\lambda_{n}\right|^{2}+2\mathfrak{R}{\left(\alpha_{n}\lambda_{n}\right)}-v_{n}^{\prime}v_{n}\left|\lambda_{n}\right|^{2}\right),$
(23)

where $\mathfrak{R}{\left(\cdot\right)}$ denotes the real part of a complex
number. The detailed derivation of (23) can be found in [4]. We note that,
compared to [4], the additional coefficient $\left|\alpha_{n}\right|^{2}$ is
due to the active relaying coefficient in the HR-RIS, which does not exist for
the conventional RIS [4].

In summary, based on (22) and (23), we have

$\displaystyle
g_{n}(\alpha_{n})=\log_{2}\left(1+\left|\alpha_{n}\right|^{2}\gamma_{n}\right)$
$\displaystyle+\log_{2}\left(1+\left|\alpha_{n}\right|^{2}\left|\lambda_{n}\right|^{2}+2\mathfrak{R}{\left(\alpha_{n}\lambda_{n}\right)}-v_{n}^{\prime}v_{n}\left|\lambda_{n}\right|^{2}\right).$
(24)

It is observed that an optimal solution to $\left(\mathrm{P_{update}}\right)$,
denoted by $\alpha_{n}^{\star}$, admits the form

$\displaystyle\alpha_{n}^{\star}=\begin{cases}\left|\alpha_{n}^{\star}\right|e^{-j\arg\\{\lambda_{n}\\}},&n\in\mathbb{A}\\\
e^{-j\arg\\{\lambda_{n}\\}},&\text{otherwise}\end{cases}.$ (25)

It is observed that the amplitude $\left|\alpha_{n}^{\star}\right|$ is not
available at this stage because it requires a determined $\mathbb{A}$. To this
end, it is natural to consider two scenarios: $\mathbb{A}$ is predetermined
and fixed in the manufacture, and $\mathbb{A}$ is dynamic and optimized based
on the propagation condition, associated with the fixed and dynamic HR-RIS
architectures illustrated in Figs. 1(a) and 1(b), respectively. The solution
$\\{\alpha_{n}^{\star}\\}$ dedicated to these architectures will be derived in
the next section.

##  IV Beamforming Coefficient Matrix of Fixed and Dynamic HR-RIS

###  IV-A Fixed HR-RIS Architecture

In the fixed HR-RIS, $\mathbb{A}$ is available to determine
$\\{\left|\alpha_{n}^{\star}\right|\\}_{n\in\mathbb{A}}$. Specifically,
$\left|\alpha_{n}^{\star}\right|$, $n\in\mathbb{A}$, can be determined based
on the fact that, given the optimal form in (25), $g_{n}(\alpha_{n}^{\star})$
monotonically increases with $\left|\alpha_{n}^{\star}\right|$. Therefore,
from (20), we obtain

$\displaystyle\left|\alpha_{n}^{\star}\right|=\sqrt{\frac{P_{\mathrm{a}}^{\mathrm{max}}-\tilde{P_{\mathrm{a}}}}{\xi_{n}}},n\in\mathbb{A}.$
(26)

As a result, the optimal solution to $\left(\mathrm{P_{update}}\right)$ is
given as

$\displaystyle\alpha_{n}^{\star}=\begin{cases}\sqrt{\frac{P_{\mathrm{a}}^{\mathrm{max}}-\tilde{P_{\mathrm{a}}}}{\xi_{n}}}e^{-j\arg\\{\lambda_{n}\\}},&n\in\mathbb{A}\\\
e^{-j\arg\\{\lambda_{n}\\}},&\text{otherwise}\end{cases}.$ (27)

The solution (27) and the given $\mathbb{A}$ are readily used to obtain the
coefficient matrix of the fixed HR-RIS scheme, as outlined in Algorithm 1.
Specifically, at the initial stage, coefficients $\\{\alpha_{n}\\}$ are
randomly generated to have arbitrary phases and amplitudes satisfying
$\left|\alpha_{n}\right|=1,n\notin\mathbb{A}$ and
$\sum_{n\in\mathbb{A}}\left|\alpha_{n}\right|^{2}\xi_{n}=P_{\mathrm{a}}^{\mathrm{max}}$,
based on (5c), (15), and (26). During steps 2–13, the coefficients are
alternatively updated based on (27). It terminates when a convergence criteria
on the objective $f(\\{\alpha_{n}\\})$ is reached. The iteration procedure in
Algorithm 1 guarantees the convergence. To see this, we note that objective
function $f(\\{\alpha_{n}\\})$ is upper bounded. In addition,
$\left(\mathrm{P_{update}}\right)$ is solved optimally by (27). Thus, the
resultant sequence of $f(\\{\alpha_{n}\\})$ is non-decreasing over iterations
[4]. We make two remarks on the design and deployment of the fixed HR-RIS.

###### Remark 1

Inserting $\tilde{P_{\mathrm{a}}}\triangleq\sum_{i\in\mathbb{A},i\neq
n}\left|\alpha_{i}\right|^{2}\xi_{i}$, $n\in\mathbb{A}$ to (26), it is
observed that a larger $K$ results in a smaller
$\left|\alpha_{n}^{\star}\right|$. Therefore, increasing $K$ does not always
guarantee the SE improvement of the fixed HR-RIS with respect to the
conventional RIS. In particular, with a limited power budget
$P_{\mathrm{a}}^{\mathrm{max}}$, the HR-RIS can have
$\left|\alpha_{n}^{\star}\right|<1$, causing signal attenuation, and hence
degrades the SE. In this case, the fixed HR-RIS with a small $K$ is easier to
attain SE gains than that with numerous active elements. Furthermore, in the
fixed HR-RIS, we have
$\sum_{n\in\mathbb{A}}\left|\alpha_{n}\right|^{2}\xi_{n}=P_{\mathrm{a}}^{\mathrm{max}}$,
implying that $P_{\mathrm{a}}^{\mathrm{max}}>\sum_{n\in\mathbb{A}}\xi_{n}$ is
necessary for $\left|\alpha_{n}\right|>1,\forall n\in\mathbb{A}$.

###### Remark 2

By inserting
$\xi_{n}\triangleq\sigma^{2}+P_{\mathrm{BS}}\left\lVert\textbf{{t}}_{n}\right\rVert^{2}$
to (26), it is observed that with a fixed $P_{\mathrm{a}}^{\mathrm{max}}$, a
lower transmit power $P_{\mathrm{BS}}$ and/or smaller channel gain
$\left\lVert\textbf{{t}}_{n}\right\rVert^{2}$ from the transmitter results in
a higher power amplifier coefficients $\left|\alpha_{n}^{\star}\right|$, and
thus, a more significant performance improvement can be achieved.

Remarks 1 and 2 will be further numerically justified and discussed in Section
VI. The fixed number and positions of active elements in the fixed HR-RIS
limit the beamforming gain of the HR-RIS. In the next section, we propose a
more robust architecture, named the dynamic HR-RIS, which overcomes the
drawbacks of the fixed HR-RIS to guarantee the SE improvement.

Algorithm 1 Find coefficients of the fixed HR-RIS

0: $\textbf{{H}}_{t},\textbf{{H}}_{r},\mathbb{A}$.

0: $\\{\alpha_{1}^{\star},\ldots,\alpha_{N}^{\star}\\}$.

1: Randomly generate $\\{\alpha_{n}\\}$ with $\left|\alpha_{n}\right|=1$,
$n\notin\mathbb{A}$, and
$\sum_{n\in\mathbb{A}}\left|\alpha_{n}\right|^{2}\xi_{n}=P_{\mathrm{a}}^{\mathrm{max}}$.

2: while objective value does not converge do

3: for $n=1\rightarrow N$ do

4: if $n\in\mathbb{A}$ then

5: Compute $\textbf{{A}}_{n},\textbf{{B}}_{n}$, and $\textbf{{C}}_{n}$ based
on (9)-(11).

6:
$\textbf{{D}}_{n}=\textbf{{I}}_{N_{r}}+\left|\alpha_{n}\right|^{2}\textbf{{A}}_{n}^{-1}\textbf{{B}}_{n}$,
$\textbf{{E}}_{n}=\textbf{{A}}_{n}\textbf{{D}}_{n}$.

7: else

8: Compute $\textbf{{A}}_{n},\textbf{{B}}_{n}$, and $\textbf{{C}}_{n}$ based
on (12)-(14).

9:
$\textbf{{D}}_{n}=\textbf{{I}}_{N_{r}}+\textbf{{A}}_{n}^{-1}\textbf{{B}}_{n}$,
$\textbf{{E}}_{n}=\textbf{{A}}_{n}\textbf{{D}}_{n}$.

10: end if

11: Find $\lambda_{n}$ as the sole non-zero eigenvalue of
$\textbf{{E}}_{n}^{-1}\textbf{{C}}_{n}$.

12: Update $\alpha_{n}^{\star}$ as (27).

13: end for

14: end while

###  IV-B Dynamic HR-RIS Architecture

Unlike the fixed HR-RIS architecture, in the dynamic HR-RIS illustrated in
Fig. 1(b), the number and positions of the active elements can be cast as
design parameters. In particular, the problem of SE maximization for the
dynamic HR-RIS can be formulated as

$\displaystyle(\mathrm{P1})\quad\underset{\\{\alpha_{n}\\},\mathbb{A}}{\textrm{maximize}}\quad$
$\displaystyle f_{0}(\\{\alpha_{n}\\})$ (28a) subject to
$\displaystyle\eqref{opt_hyb_con_unit_modul}-\eqref{opt_hyb_con_relay_power},$
(28b) $\displaystyle\left|\mathbb{A}\right|\leq
K,\mathbb{A}\subset\\{1,2,\ldots,N\\},$ (28c)

where the constraint (28c) means that the number of active elements in the HR-
RIS does not exceed the maximum number of RF-PA chains. Problem
$(\mathrm{P1})$ does not only inherit the numerical challenge of
$(\mathrm{P0})$ but also includes the difficulty from cardinality constraint
(28c). We develop an efficient solution to $(\mathrm{P1})$ for practical use
below.

Similar to problem $(\mathrm{P0})$, firstly, we employ the upper bound of
$f_{0}(\\{\alpha_{n}\\})$ (see (7)) to obtain an approximate but more
tractable problem of $(\mathrm{P1})$, which is given as

$\displaystyle\left(\mathrm{P}_{\mathrm{dyn}}\right)\quad\underset{\\{\alpha_{n}\\},\mathbb{A}}{\textrm{maximize}}\quad$
$\displaystyle f(\\{\alpha_{n}\\})$ (29a) subject to
$\displaystyle\eqref{opt_hyb_con_unit_modul}-\eqref{opt_hyb_con_relay_power},$
(29b) $\displaystyle\left|\mathbb{A}\right|\leq
K,\mathbb{A}\subset\\{1,2,\ldots,N\\}$ (29c)

At this point, a naive approach to solve
$\left(\mathrm{P}_{\mathrm{dyn}}\right)$ would be based on an exhaustive
search, i.e., using the result presented in Section III to determine the
coefficients $\\{\alpha_{n}\\}$ corresponding to each valid set $\mathbb{A}$,
then choosing the one providing the best performance. Clearly, such an
approach requires high computational cost due to excessively large numbers of
combinations of $\mathbb{A}$.

We now present an efficient solution to
$\left(\mathrm{P}_{\mathrm{dyn}}\right)$. Recall that we can write
$\alpha_{n}=\left|\alpha_{n}\right|e^{j\theta_{n}}$. Based on this, let us
introduce two matrices as
$\boldsymbol{\Theta}=\mathrm{diag}\left\\{e^{j\\{\theta_{1}\\}},\ldots,e^{j\\{\theta_{N}\\}}\right\\}$
and
$\hat{\boldsymbol{\Upsilon}}=\mathrm{diag}\left\\{\left|\alpha_{1}\right|,\ldots,\left|\alpha_{N}\right|\right\\}$
with $\left|\alpha_{n}\right|=1,n\notin\mathbb{A}$. Then we can write
$\boldsymbol{\Upsilon}=\boldsymbol{\Theta}\hat{\boldsymbol{\Upsilon}}$. This
motivates us develop an alternating procedure applied on three sets of
variables $\\{\boldsymbol{\Theta},\hat{\boldsymbol{\Upsilon}},\mathbb{A}\\}$.
Next, we describe the approach determining each of
$\\{\boldsymbol{\Theta},\hat{\boldsymbol{\Upsilon}},\mathbb{A}\\}$ when the
others are fixed, which are the key steps in our proposed solution.

####  IV-B1 Determine $\\{\theta_{n}\\}$

For this step, we suppose that $\left|\alpha_{n}\right|=1,\forall n$, which
corresponds to the conventional RIS where all elements are passive. Then, we
determine $\\{\theta_{n}\\}$ via solving the following problem extracted from
$\left(\mathrm{P}_{\mathrm{dyn}}\right)$

$\displaystyle\left(\mathrm{P1}_{\mathrm{dyn}}\right)\quad\underset{\\{\alpha_{n}\\}}{\textrm{maximize}}\quad$
$\displaystyle f(\\{\alpha_{n}\\})$ subject to
$\displaystyle\left|\alpha_{n}\right|=1,\forall n.$

A solution to $\left(\mathrm{P1}_{\mathrm{dyn}}\right)$ can be obtained by
using the same approach presented in Section III-B (i.e., see (25)) for the
case $n\notin\mathbb{A}$.

####  IV-B2  Determine $\mathbb{A}$ and $\\{\left|\alpha_{n}\right|\\}$

We recall that the passive reflecting elements have unit modulus, i.e.,
$\left|\alpha_{n}\right|=1,\forall n\notin\mathbb{A}$. Thus, we only need to
determine $\left|\alpha_{n}\right|,\forall n\in\mathbb{A}$. Furthermore, an
active element only results in performance gain with respect to a passive
element if $\left|\alpha_{n}\right|>1$; otherwise, it causes performance loss
due to signal attenuation. Therefore, to minimize the loss, in the dynamic HR-
RIS architecture, the RF-PA chain connected to the $n$th element should be
turned off if $\left|\alpha_{n}\right|<1$. As a result, in the dynamic HR-RIS
scheme, we have $\left|\alpha_{n}\right|>1,\forall n\in\mathbb{A}$, and the
solution in (25) becomes

$\displaystyle\alpha_{n}^{\star}=\begin{cases}\left|\alpha_{n}^{\star}\right|e^{-j\arg\\{\lambda_{n}\\}},&\left|\alpha_{n}^{\star}\right|>1,n\in\mathbb{A}\\\
e^{-j\arg\\{\lambda_{n}\\}},&\text{otherwise}\end{cases},$

with $\\{\arg\\{\lambda_{n}\\}\\}$ being determined in the previous step. With
this solution, it is obvious that
$\mathfrak{R}{\left(\alpha_{n}^{\star}\lambda_{n}\right)}=\left|\alpha_{n}^{\star}\right|\left|\lambda_{n}\right|$,
and (24) can be rewritten as

$\displaystyle g_{n}(\alpha_{n}^{\star})$
$\displaystyle=\log_{2}\left[\left(1+\left|\alpha_{n}^{\star}\right|^{2}\gamma_{n}\right)\right.$
$\displaystyle\left.\quad\quad\times\left(1+\left|\alpha_{n}^{\star}\right|^{2}\left|\lambda_{n}\right|^{2}+2\left|\alpha_{n}^{\star}\right|\left|\lambda_{n}\right|-v_{n}^{\prime}v_{n}\left|\lambda_{n}\right|^{2}\right)\right]$
$\displaystyle=\log_{2}\left[1+\zeta_{n}^{2}\left|\alpha_{n}^{\star}\right|^{4}+o\left(\left|\alpha_{n}^{\star}\right|^{3}\right)\right],$
(30)

where $\zeta_{n}\triangleq\left|\lambda_{n}\right|\sqrt{\gamma_{n}}$, and
$o\left(\left|\alpha_{n}^{\star}\right|^{3}\right)$ represents the sum of the
remaining terms associated with $\left|\alpha_{n}^{\star}\right|^{c},c\leq 3$.
Because $\left|\alpha_{n}\right|>1,n\in\mathbb{A}$, we can ignore
$o\left(\left|\alpha_{n}^{\star}\right|^{3}\right)$ in (30) and approximately
maximize $g_{n}(\alpha_{n})$ via maximizing
$\hat{g}_{n}(\alpha_{n}^{\star})\triangleq\log_{2}\left(1+\zeta_{n}^{2}\left|\alpha_{n}^{\star}\right|^{4}\right)$.
Moreover, because $\hat{g}_{n}(\alpha_{n}^{\star})$ monotonically increases
with $\left|\alpha_{n}^{\star}\right|$, maximizing it is equivalent to
maximizing
$\hat{\hat{g}}_{n}(\alpha_{n}^{\star})\triangleq\log_{2}\left(1+\zeta_{n}\left|\alpha_{n}^{\star}\right|^{2}\right)$.
The observation implies that to find $\left|\alpha_{n}^{\star}\right|$ that
maximizes $g_{n}(\alpha_{n}^{\star}),\forall n\in\mathbb{A}$, we can consider
the following problem

$\displaystyle\underset{\mathbb{A},\\{\left|\alpha_{n}\right|^{2}\\}}{\textrm{maximize}}\quad$
$\displaystyle\sum_{n\in\mathbb{A}}\log_{2}\left(1+\zeta_{n}\left|\alpha_{n}\right|^{2}\right)$
(31a) subject to
$\displaystyle\sum_{n\in\mathbb{A}}\left|\alpha_{n}\right|^{2}\xi_{n}\leq
P_{\mathrm{a}}^{\mathrm{max}}\text{ and }\eqref{con_cardinality},$ (31b)

where the first constraint in (31b) is the transmit power constraint of the
active elements. Let $p_{n}\triangleq\left|\alpha_{n}\right|^{2}\xi_{n}$
denote the power allocated to the $n$th active element, $n\in\mathbb{A}$. The
problem in (31) can be rewritten as

$\displaystyle\underset{\mathbb{A},\\{p_{n}\\}}{\textrm{maximize}}\quad$
$\displaystyle\sum_{n\in\mathbb{A}}\log_{2}\left(1+\frac{\zeta_{n}}{\xi_{n}}p_{n}\right)$
(32a) subject to $\displaystyle\sum_{n\in\mathbb{A}}p_{n}\leq
P_{\mathrm{a}}^{\mathrm{max}}\text{ and }\eqref{con_cardinality},$ (32b)

which allows $\mathbb{A}$ and $\\{p_{n}\\}$ to be obtained. Specifically, it
is clear from (32a) that the optimal set of active elements, i.e.,
$\mathbb{A}^{\star}$, is the set of indices of the $K$ largest elements in
$\left\\{\frac{\zeta_{1}}{\xi_{1}},\ldots,\frac{\zeta_{N}}{\xi_{N}}\right\\}$,
where $\frac{\zeta_{n}}{\xi_{n}},\forall n$ are obtained during determining
$\\{\theta_{n}\\}$ in the previous step, i.e., in Section IV-B1.

Once $\mathbb{A}^{\star}$ is determined, $\\{p_{n}\\}$ can be solved by
performing power allocation to the active elements in $\mathbb{A}^{\star}$.
The optimal solution is the well-know water-filling solution
$p_{n}^{\star}=\max\left\\{\frac{1}{\mu}-\frac{\xi_{n}}{\zeta_{n}},0\right\\}$,
with $\mu$ satisfying (32b). As a result, we obtain
$\left|\alpha_{n}^{\star}\right|^{2}=\frac{p_{n}^{\star}}{\xi_{n}}=\max\left\\{\frac{1}{\mu\xi_{n}}-\frac{1}{\zeta_{n}},0\right\\}$,
$n\in\mathbb{A}^{\star}$. Consequently, the amplitudes of elements in the
dynamic HR-RIS scheme is given as

$\displaystyle\left|\alpha_{n}^{\star}\right|=\begin{cases}\max\left\\{\sqrt{\frac{1}{\mu\xi_{n}}-\frac{1}{\zeta_{n}}},1\right\\},&n\in\mathbb{A}^{\star}\\\
1,&\text{otherwise}\end{cases}.$ (33)

The proposed procedure finding efficient solution to the problem of SE
maximization in dynamic HR-RIS architecture is outlined in Algorithm 2. It
starts with randomly generating $\\{\alpha_{n}\\}$ with unit modules. Then
steps 2–9 are executed to determine $\\{\theta_{n}^{\star}\\}$. After that,
$\mathbb{A}^{\star}$ is determined in steps 10 and 11, allowing
$\\{\left|\alpha_{n}^{\star}\right|\\}$ to be obtained in step 12. Finally,
the coefficients of the dynamic HR-RIS are derived in step 13 based on (25).

Algorithm 2 Find coefficients of the dynamic HR-RIS

0: $\textbf{{H}}_{t},\textbf{{H}}_{r}$.

0: $\mathbb{A}^{\star},\\{{\alpha_{n}^{\star}}\\}$.

1: Randomly generate
$\boldsymbol{\Theta}=\mathrm{diag}\left\\{e^{j\theta_{1}},\ldots,e^{j\theta_{N}}\right\\}$
and assign $\boldsymbol{\Upsilon}=\boldsymbol{\Theta}$.

2: while objective value does not converge do

3: for $n=1\rightarrow N$ do

4: Compute $\textbf{{A}}_{n},\textbf{{B}}_{n}$, and $\textbf{{C}}_{n}$ based
on (12)-(14).

5:
$\textbf{{D}}_{n}=\textbf{{I}}_{N_{r}}+\textbf{{A}}_{n}^{-1}\textbf{{B}}_{n}$,
$\textbf{{E}}_{n}=\textbf{{A}}_{n}\textbf{{D}}_{n}$.

6: Find $\lambda_{n}$ and $\gamma_{n}$ as the sole non-zero eigenvalue of
$\textbf{{E}}_{n}^{-1}\textbf{{C}}_{n}$ and
$\textbf{{A}}_{n}^{-1}\textbf{{B}}_{n}$, respectively.

7: $\theta_{n}^{\star}=-\arg\\{\lambda_{n}\\}$,
$\alpha_{n}^{\star}=e^{-j\theta_{n}^{\star}}$.

8: end for

9: end while

10: Compute $\frac{\zeta_{n}}{\xi_{n}}$, $n=1,\ldots,N$, where
$\zeta_{n}\triangleq\left|\lambda_{n}\right|\sqrt{\gamma_{n}}$ and
$\xi_{n}\triangleq\sigma^{2}+P_{\mathrm{BS}}\left\lVert\textbf{{t}}_{n}\right\rVert^{2}$.

11: Set $\mathbb{A}^{\star}$ to the indices of the $K$ largest value in
$\left\\{\frac{\zeta_{1}}{\xi_{1}},\ldots,\frac{\zeta_{N}}{\xi_{N}}\right\\}$.

12: Obtain $\\{\left|\alpha_{n}^{\star}\right|\\}$ given in (33).

13: Obtain $\\{\alpha_{n}^{\star}\\}$ based on (25).

##  V Power Consumption Analysis

###  V-A Fixed HR-RIS Architecture

In the fixed HR-RIS, the number of active and passive elements are fixed to
$K$ and $M$, respectively. Therefore, the total power consumption of the whole
MIMO communications system aided by the fixed HR-RIS can be modeled as [41,
42, 36]

$\displaystyle
P_{{\mathrm{H}}}^{\mathrm{fix.}}=\frac{P_{\mathrm{BS}}}{\tau_{\mathrm{BS}}}+\frac{P_{\mathrm{a}}}{\tau_{\mathrm{a}}}+P_{\mathrm{c,H}}^{\mathrm{fix.}}+MP_{\mathrm{p}},$
(34)

where $\tau_{\mathrm{BS}},\tau_{\mathrm{a}}\in(0,1]$ are the power amplifier
efficiencies of the BS and active elements at the HR-RIS, respectively,
$P_{\mathrm{a}}$ is given in (16), and $P_{\mathrm{p}}$ is the power required
for a passive reflecting element [36]. Furthermore,
$P_{\mathrm{c,H}}^{\mathrm{fix.}}$ is the total circuit power consumption of
the fixed HR-RIS-aided MIMO system and can be computed as [41]

$\displaystyle
P_{\mathrm{c,H}}^{\mathrm{fix.}}=N_{t}P_{\mathrm{BS,dynamic}}+KP_{\mathrm{a,dynamic}}+P_{\mathrm{BS,static}}+P_{\mathrm{a,static}},$

with $P_{\mathrm{BS,dynamic}}$ denoting the dynamic power consumption of each
RF chain of the BS, and $P_{\mathrm{BS,static}}$ denoting the static power
overhead of the BS, including baseband processing, power suply, and cooling
power consumption. Similarly, $P_{\mathrm{a,dynamic}}$ and
$P_{\mathrm{a,static}}$ are the dynamic and static power consumption of the
active relay elements in the HR-RIS.

###  V-B Dynamic HR-RIS Architecture

In the dynamic HR-RIS architecture, the number of active and passive elements
vary depending on $P_{\mathrm{a}}^{\mathrm{max}}$ and $\xi_{n}$ according to
(33). Let $\left|\mathbb{A}^{\star}\right|$ denote the number of actual active
elements whose amplitudes are larger than one, where $\mathbb{A}^{\star}$ is
obtained in Algorithm 2. As a result, the number of passive elements is given
as $N-\left|\mathbb{A}^{\star}\right|$, and the total power consumption of a
dynamic HR-RIS-assisted MIMO system is given by

$\displaystyle
P_{{\mathrm{H}}}^{\mathrm{dyn.}}=\frac{P_{\mathrm{BS}}}{\tau_{\mathrm{BS}}}+\frac{P_{\mathrm{a}}}{\tau_{\mathrm{a}}}+P_{\mathrm{c,H}}^{\mathrm{dyn.}}+(N-\left|\mathbb{A}^{\star}\right|)P_{\mathrm{p}}+NP_{\mathrm{SW}},$
(35)

where the last term accounts for the total power consumed by the $N$ switches
in Fig. 1(b), each requires a power of $P_{\mathrm{SW}}$, and
$P_{\mathrm{c,H}}^{\mathrm{dyn.}}$ is the total circuit power consumption of
the dynamic HR-RIS-aided MIMO system, given as [41]

$\displaystyle P_{\mathrm{c,H}}^{\mathrm{dyn.}}$
$\displaystyle=N_{t}P_{\mathrm{BS,dynamic}}+\left|\mathbb{A}^{\star}\right|P_{\mathrm{a,dynamic}}$
$\displaystyle\hskip 85.35826pt+P_{\mathrm{BS,static}}+P_{\mathrm{a,static}}.$

###  V-C Conventional RIS Architecture

We compare the total power consumption of the proposed HR-RIS-assisted system
to that of the conventional RIS-assisted system. We note that in the latter,
there is no active relay element, but $N=M+K$ passive reflecting elements are
used at the RIS. Therefore, the total power consumption of the RIS-aided
system can be expressed as

$\displaystyle
P_{\mathrm{RIS}}=\frac{P_{\mathrm{BS}}}{\tau_{\mathrm{BS}}}+P_{\mathrm{c,RIS}}+NP_{\mathrm{p}},$
(36)

where $P_{\mathrm{c,RIS}}=N_{t}P_{\mathrm{BS,dynamic}}+P_{\mathrm{BS,static}}$
is the total circuit power consumption of the dynamic HR-RIS-aided MIMO system

By comparing (34) and (35) to (36), with the note that
$P_{\mathrm{passive}}\ll P_{\mathrm{a,dynamic}}$, the increased total power
consumption of the fixed and dynamic HR-RIS-aided systems with respect to the
RIS-assisted system can be respectively given as

$\displaystyle\Delta
P^{\mathrm{fix.}}_{\mathrm{H}}=\frac{P_{\mathrm{a}}}{\tau_{\mathrm{a}}}+K\left(P_{\mathrm{a,dynamic}}-P_{\mathrm{p}}\right)+P_{\mathrm{a,static}},$
$\displaystyle\Delta
P^{\mathrm{dyn.}}_{\mathrm{H}}=\frac{P_{\mathrm{a}}}{\tau_{\mathrm{a}}}+\left|\mathbb{A}^{\star}\right|\left(P_{\mathrm{a,dynamic}}-P_{\mathrm{p}}\right)+P_{\mathrm{a,static}},$

which linearly increases with the number of active elements, i.e., $K$ in the
fixed HR-RIS and $\left|\mathbb{A}^{\star}\right|$ in the dynamic HR-RIS. The
numerical comparison of these architectures will be presented in the next
section.

##  VI Simulation Results

Fig. 3: Horizontal locations of the BS, MS, and HR-RIS.

In this section, numerical results are provided to validate the proposed HR-
RIS schemes. We assume that uniform linear arrays (ULAs) are deployed at the
BS and MS, respectively. By contrast, an uniform planar array (UPA) of $N$
elements is employed for the HR-RIS. Furthermore, half-wavelength distancing
between array elements are assumed for the BS, MS, and HR-RIS. As shown in
Fig. 3, in a two-dimensional coordinate, we assume that the BS is deployed at
a fixed location $(0,0)$. By contrast, the HR-RIS and MS are located at
$(x_{\mathrm{H}},0)$ and $(x_{\mathrm{MS}},y_{\mathrm{MS}})$, respectively,
where $x_{\mathrm{H}}$ and $x_{\mathrm{MS}}$ can vary. As a result, the
distance between the BS and MS is given by $d_{t}=x_{\mathrm{H}}$, while that
between the HR-RIS and the MS is
$d_{r}=\sqrt{(x_{\mathrm{H}}-x_{\mathrm{MS}})^{2}+y_{\mathrm{MS}}^{2}}$. The
path loss of a link distance $d$ is given by [19, 4]
$\beta(d)=\beta_{0}\left(\frac{d}{1\mathrm{m}}\right)^{\epsilon}$, where
$\beta_{0}$ is the path loss at the reference distance of $1$ meter (m), and
$\epsilon$ is the path loss exponent.

For small-scale fading, we assume the Rician fading channel model [19, 4]. As
a result, the small-scale fading channel between the BS and the HR-RIS is
modeled by [19, 4]

$\displaystyle\tilde{\textbf{{H}}}_{t}=\left(\sqrt{\frac{\kappa_{t}}{1+\kappa_{t}}}\textbf{{H}}_{t}^{\mathrm{LoS}}+\sqrt{\frac{1}{1+\kappa_{t}}}\textbf{{H}}_{t}^{\mathrm{NLoS}}\right),$
(37)

where $\textbf{{H}}_{t}^{\mathrm{LoS}}$ and $\textbf{{H}}_{t}^{\mathrm{NLoS}}$
represent the deterministic LoS and non-LoS (NLoS) components, respectively.
The NLoS channel is modeled by the Rayleigh fading, with the entry on the
$i$th row and $j$th column of $\textbf{{H}}_{t}^{\mathrm{NLoS}}$ being given
as $h_{t,ij}^{\mathrm{NLoS}}\sim\mathcal{CN}(0,1)$. The LoS component for each
channel is modeled as a deterministic component, i.e.,
$\textbf{{H}}_{t}^{\mathrm{LoS}}=\textbf{{a}}_{\mathrm{H}}(\theta_{\mathrm{H}},\phi_{\mathrm{H}})\textbf{{a}}_{\mathrm{BS}}^{H}(\theta_{\mathrm{BS}})$,
where $\textbf{{a}}_{\mathrm{BS}}(\theta_{\mathrm{BS}})$ and
$\textbf{{a}}_{\mathrm{H}}(\theta_{\mathrm{H}},\phi_{\mathrm{H}})$ are the
array response vectors at the BS and HR-RIS, respectively. Here, the $n$th
element of $\textbf{{a}}_{\mathrm{BS}}(\theta_{\mathrm{BS}})$ and
$\textbf{{a}}_{\mathrm{H}}(\theta_{\mathrm{H}},\phi_{\mathrm{H}})$ are given
as
$a_{\mathrm{BS},n}(\theta_{\mathrm{BS}})=e^{j\pi(n-1)\sin\theta_{\mathrm{BS}}},n=1,\ldots,N_{t}$
and
$a_{\mathrm{H},n}(\theta_{\mathrm{H}},\phi_{\mathrm{H}})=e^{j\pi\left(\lfloor\frac{n}{N_{x}}\rfloor\sin\theta_{\mathrm{H}}\sin\phi_{\mathrm{H}}+\left(n-\lfloor\frac{n}{N_{x}}\rfloor
N_{x}\right)\sin\theta_{\mathrm{H}}\cos\phi_{\mathrm{H}}\right)}$,
$n=1,\ldots,N$ [4]. $\theta_{\mathrm{BS}}$, $\theta_{\mathrm{H}}\in[0,2\pi)$
denote the angle-of-departure (AoD) at the BS and the azimuth angle-of-arrival
(AoA) at the HR-RIS, respectively, and $\phi_{\mathrm{H}}\in[-\pi/2,\pi/2)$
denotes the elevation AoA at the HR-RIS. Furthermore, in (37), $\kappa_{t}$ is
the Rician factor. With $\kappa_{t}\rightarrow 0$, $\tilde{\textbf{{H}}}_{t}$
approaches the Rayleigh fading channel, and with
$\kappa_{t}\rightarrow\infty$, $\tilde{\textbf{{H}}}_{t}$ becomes the LoS
channel. The channel matrix between the BS and the HR-RIS, i.e.,
$\textbf{{H}}_{t}$, is obtained by multiplying $\tilde{\textbf{{H}}}_{t}$ by
the square root of the corresponding path loss $\beta(d_{t})$, i.e.,
$\textbf{{H}}_{t}=\sqrt{\beta(d_{t})}\tilde{\textbf{{H}}}_{t}$. The channel
matrix between the HR-RIS and MS is modeled similarly.

In this work, unless otherwise stated, we set $\beta_{0}=-30$ dB,
$\\{\epsilon_{t},\epsilon_{r}\\}=\\{2.2,2.8\\}$, $\sigma^{2}=-80$ dBm,
$\\{x_{\mathrm{MS}},y_{\mathrm{MS}},x_{\mathrm{H}}\\}=\\{40,2,51\\}$ m, and
$\\{\kappa_{t},\kappa_{r}\\}=\\{\infty,0\\}$ [19]. Furthermore, the resolution
of the phase shifts is set to $b=2$ bits, i.e., the phase shifts are selected
from the set $\mathcal{F}=\left\\{\frac{2\pi(q-1)}{Q}\right\\}$,
$q=1,\ldots,Q$, where $Q=2^{b}$. All the numerical results are averaged over
$100$ independent channel realizations.

###  VI-A Improvement in SE of the Proposed HR-RIS

Fig. 4: SEs of the proposed HR-RIS schemes compared to those obtained by the
exhaustive search for $N_{t}=4$, $N_{r}=2$, $N=4$, $K=1$, $M=N-K=3$, and
$P_{\mathrm{a}}^{\mathrm{max}}=\\{-10,0,10\\}$ dBm. Fig. 5: SEs of the
proposed HR-RIS schemes compared to those of the conventional RIS schemes for
$N_{t}=32$, $N_{r}=2$, $N=50$, $K=1$, $M=N-K=49$, and
$P_{\mathrm{a}}^{\mathrm{max}}=\\{-10,0,10\\}$ dBm.

We at first consider in Fig. 4 a $4\times 2$ MIMO system, where a small HR-RIS
with $N=4$ and $K=1$ is deployed, to show that the proposed schemes can
perform near-optimal. With $K=1$, the HR-RIS has only a single active element,
whose transmit power is set to $P_{\mathrm{a}}^{\mathrm{max}}=\\{-10,0,10\\}$
dBm. To obtain the optimal performance, we perform the exhaustive search over
all the possible solutions. Specifically, for the fixed HR-RIS, the index of
the active element is set to one, and the optimal solution is searched over
all combinations of $N$ phase values, each is in $\mathcal{F}$. For the
dynamic HR-RIS, the exhaustive search scheme searches for the optimal
combination of $N$ phase values and the index of the active element. It is
observed from Fig. 4 that the proposed schemes perform very close to the
optimal exhaustive search counterparts for all the considered values of
$P_{\mathrm{a}}^{\mathrm{max}}$. In the subsequent simulations, we consider
larger systems and omit the results for exhaustive search because they require
excessively high computational and time complexity.

In Fig. 5, we show the performance improvement of the proposed HR-RIS
architectures in terms of SE. Specifically, a $32\times 2$ MIMO system aided
by an intelligent surface of $N=50$ elements is considered. The SEs of the
proposed fixed and dynamic HR-RIS schemes are shown for $K=1$, $M=N-K=49$, and
$P_{\mathrm{a}}^{\mathrm{max}}=\\{-10,0,10\\}$ dBm. For comparison, we also
show the SEs of the conventional RIS with $N$ fully passive reflecting
elements, whose phases are either randomly generated or optimized using the AO
method in [4]. Furthermore, in all the simulation results for the fixed HR-RIS
scheme, $\mathbb{A}$ is fixed to $\\{1,\ldots,K\\}$. It is noted that the
coefficients of the HR-RIS and AO-based RIS [4] are obtained based on the same
optimization method, i.e., the AO approach.

It can be observed from Fig. 5 that the RIS with random phases has poor
performance compared to the optimized RIS. By contrast, the HR-RIS schemes
achieve significant improvement in the SE with respect to the conventional
RIS, especially at low $P_{\mathrm{BS}}$. This numerically justifies the
observation in Remark 2, which states that a higher performance gain can be
obtained by the HR-RIS for low $P_{\mathrm{BS}}$. Furthermore, it is seen that
the HR-RIS only requires a small $P_{\mathrm{a}}^{\mathrm{max}}$ to achieve
remarkable SE improvement at low $P_{\mathrm{BS}}$, and with
$P_{\mathrm{a}}^{\mathrm{max}}=\\{0,10\\}$ dBm, sustainable SE gains of the
HR-RIS are seen for the whole considered range of $P_{\mathrm{BS}}$. For
example, the fixed and dynamic HR-RIS only requires
$P_{\mathrm{a}}^{\mathrm{max}}=0$ dBm to save more than $10$ dBm and $15$ dBm
for the transmit power at the BS, respectively. However, with a small power
budget at the HR-RIS, but a large transmit power at the BS, e.g., with
$P_{\mathrm{a}}^{\mathrm{max}}=-10$ dBm and $P_{\mathrm{BS}}\geq 35$ dBm, the
performance gain of the HR-RIS decreases, and it performs comparably with the
conventional RIS. This observation agrees with the discussion in Remarks 1 and
2. The SE gains of the proposed schemes for low $P_{\mathrm{BS}}$ have been
clearly shown in this figure. Therefore, in the following results on SE, we
consider high values of $P_{\mathrm{BS}}$.

Fig. 6: SE improvement for different positions of the HR-RIS with $N_{t}=32$,
$N_{r}=2$, $K=1$, $P_{\mathrm{BS}}=30$ dBm, and
$P_{\mathrm{a}}^{\mathrm{max}}=0$ dBm.

In Fig. 6, we investigate the SEs of the HR-RIS for different geographical
deployments of the RIS/HR-RIS by varying $x_{\mathrm{H}}$. We note that for
comparison, the RIS and HR-RIS are set to be placed at the same position. In
this simulation, we set $N_{t}=32$, $N_{r}=2$, $K=4$, $P_{\mathrm{BS}}=30$
dBm, and $P_{\mathrm{a}}^{\mathrm{max}}=0$ dBm. The BS is placed at the
coordinate $(0,0)$, while two positions are considered for the MS, including
$\\{(40,2),(100,2)\\}$. The HR-RIS/RIS is placed at $(0,x_{\mathrm{H}})$,
where $x_{\mathrm{H}}\in[10,100]$ m, implying that the HR-RIS/RIS moves along
the $x-$axis in Fig. 3. An observation from Fig. 6, which agrees with the
finding in [36], is that both the RIS and HR-RIS have decreased SEs when they
move far away from either the BS or the MS, and the highest SEs are attained
when they are closest to the MS. However, in this figure, we focus more on the
SE gains of the proposed HR-RIS versus $x_{\mathrm{H}}$. Specifically, it is
seen that as $x_{\mathrm{H}}$ grows, path loss $\beta(d_{t})$ increases,
leading to a smaller $\xi_{n}$ in (26) and (33). As result, more significant
active beamforming gains are attained, as discussed in Remark 2. Furthermore,
the dynamic HR-RIS exhibits large gains over the others in the worst
scenarios, i.e., when the MS is far from both the HR-RIS/RIS and BS. It can be
concluded from Fig. 6 that the proposed HR-RIS is robust over geographical
deployment.

Fig. 7: SEs of the proposed HR-RIS schemes compared to those of the
conventional RIS schemes and the relay with $K$ active elements for
$N_{t}=32$, $N_{r}=2$, $N=50$, $K=\\{1,2,\ldots,N\\}$, $P_{\mathrm{BS}}=30$
dBm, and $P_{\mathrm{a}}^{\mathrm{max}}=\\{-10,0,10\\}$ dBm.

Furthermore, we investigate the SE improvement of the proposed schemes for
different values of $K$ in Fig. 7. In this figure, the proposed schemes are
compared to not only the conventional RIS with random phases and AO-based RIS,
which are equipped with $N$ elements, but also to the RIS and the fully-active
relay equipped with only $K$ independent elements. We consider
$P_{\mathrm{BS}}=30$ dBm, $P_{\mathrm{a}}^{\mathrm{max}}=\\{-10,0,10\\}$ dBm,
and the other simulation parameters are the same as those in Fig. 5. The
following observations are noted:

  * •

Clearly, the SEs of the conventional fully-passive RIS schemes are constant
with $K$. Considering the RIS and relay with only $K$ elements, their SEs
monotonically increase with $K$. Furthermore, it is seen that as $K$
increases, the RIS performs closer to the relay with the same number of
elements. In particular, a sufficiently large RIS can outperform the relay
with a limited power budget, as seen in Fig. 7(a). These observations agree
with the findings in [36] and [19]. However, for a small $K$, the relay
performs far better than the RIS, even with a limited power budget, which has
motivated the proposal of the HR-RIS in this work.

  * •

Comparing the SEs of the proposed HR-RIS and the conventional RIS with $N$
elements, in all the considered scenarios, the former only requires a single
active element to outperform the latter. However, as we discussed in Remark 1,
it is clear for the fixed HR-RIS that increasing $K$ does not guarantee the SE
improvement, especially for low $P_{\mathrm{a}}^{\mathrm{max}}$ in Fig. 7(a).
More specifically, as $K$ increases, the SEs of both the fixed and dynamic HR-
RIS quickly increases for small $K$, then they gradually reach their peaks at
a small or moderate $K$ before degradation. Therefore, we can conclude that a
small number of active elements is sufficient for the HR-RIS to achieve
significant improvement in SE with respect to the conventional RIS.

  * •

The advantage of the dynamic HR-RIS in terms of SE compared to the fixed HR-
RIS can be clearly seen in Fig. 7. Specifically, unlike the fixed HR-RIS, in
all the considered scenarios, dynamic HR-RIS achieves higher or the same SE
compared to the AO-based RIS. A higher $P_{\mathrm{a}}^{\mathrm{max}}$
provides it with a higher gain, but small $P_{\mathrm{a}}^{\mathrm{max}}$ does
not cause any performance loss because the active elements can be deactivated
to serve as the conventional reflecting elements if they cannot amplify the
signal. In general, it can be concluded that the dynamic HR-RIS is more robust
than the fixed one.

Fig. 8: SEs of the proposed HR-RIS schemes compared to those of the
conventional RIS schemes for $N_{t}=32$, $N_{r}=2$, $N\in[20,200]$, $K=1$,
$P_{\mathrm{BS}}=30$ dBm, and $P_{\mathrm{a}}^{\mathrm{max}}=\\{-10,0,10\\}$
dBm

In Fig. 8, the SEs of the proposed HR-RIS and the conventional RIS schemes are
shown for $N_{t}=32$, $N_{r}=2$, $K=1$, $P_{\mathrm{BS}}=30$ dBm,
$P_{\mathrm{a}}^{\mathrm{max}}=\\{-10,0,10\\}$ dBm, and $N\in[20,200]$. It is
seen that the HR-RIS schemes attain the improvement in SE for all the
considered values of $N$ with respect to the conventional RIS schemes. In
particular, it is observed that the performance improvement becomes less
significant as $N$ increases. Nevertheless, the gain of the proposed schemes
is still considerable at large $N$, especially with
$P_{\mathrm{a}}^{\mathrm{max}}=10$ dBm. With a larger
$P_{\mathrm{a}}^{\mathrm{max}}$, the dynamic HR-RIS achieves higher SE gains
with respect to the fixed one.

###  VI-B Power Consumption and EE of the HR-RIS Schemes

In this subsection, the power consumption and EE of the proposed HR-RIS
architectures are investigated and compared to those of the conventional RIS.
Specifically, the total power consumption of the MIMO system aided by the HR-
RIS schemes and RIS, i.e., $P_{{\mathrm{H}}}^{\mathrm{fix.}}$,
$P_{{\mathrm{H}}}^{\mathrm{dyn.}}$, and $P_{\mathrm{RIS}}$, are computed based
on (34), (35) and (36), respectively. The component power consumption is
assumed as follows: $P_{\mathrm{BS,dynamic}}=40$ dBm,
$P_{\mathrm{a,dynamic}}=35$ dBm, $P_{\mathrm{BS,static}}=35$ dBm,
$P_{\mathrm{a,static}}=30$ dBm, $P_{\mathrm{p}}=P_{\mathrm{SW}}=5$ mW, and
$\tau_{\mathrm{a}}=\tau_{\mathrm{BS}}=0.5$ [42, 36, 43]. Then, the EE of a
scheme is given as $\mathrm{EE}=\frac{W\mathrm{SE}}{P}$ [bps/W], where $W$
denotes the system bandwidth, and
$P\in\\{P_{{\mathrm{H}}}^{\mathrm{fix.}},P_{{\mathrm{H}}}^{\mathrm{dyn.}},P_{\mathrm{RIS}}\\}$
is the total power consumption of the HR-RIS or RIS-assisted MIMO system. We
consider the system bandwidth of $10$ MHz [4].

Fig. 9: EEs of the proposed HR-RIS schemes compared to those of the
conventional RIS schemes for $N_{t}=32$, $N_{r}=2$, $N=50$, $K=1$, $M=N-K=49$,
and $P_{\mathrm{a}}^{\mathrm{max}}=\\{-10,0,10\\}$ dBm.

In Fig. 9, the EEs corresponding to the SEs in Fig. 5 are shown. The proposed
HR-RIS schemes achieve much higher EEs compared to those of the conventional
RIS schemes for low and moderate $P_{\mathrm{BS}}$. At high $P_{\mathrm{BS}}$,
HR-RIS and RIS have comparable EEs. The EE improvement of the HR-RIS schemes
shown in this figure is similar to the SE improvement shown in Fig. 5. In the
following results on the EE, we focus on comparing the power consumption and
EEs of the HR-RIS and RIS for high $P_{\mathrm{BS}}$ because that for low
$P_{\mathrm{BS}}$ is clear from this figure. In particular, the dynamic HR-RIS
will be shown to perform robustly and can save power to ensure high EEs with
low $P_{\mathrm{a}}^{\mathrm{max}}$.

Fig. 10: Total power consumption of the proposed HR-RIS schemes compared to
those of the conventional RIS schemes for $N_{t}=32$, $N_{r}=2$, $N=50$,
$K=\\{1,2,\ldots,N\\}$, $P_{\mathrm{BS}}=30$ dBm, and
$P_{\mathrm{a}}^{\mathrm{max}}=\\{-10,0,10\\}$ dBm.

Assuming the same simulation parameters as those in Fig. 7, we show the power
consumption of the considered schemes for $K=\\{1,2,\ldots,N\\}$ in Fig. 10.
As analyzed in Section V, it is clear that the proposed HR-RIS architectures
require higher power consumption than the RIS. However, for small
$P_{\mathrm{a}}^{\mathrm{max}}$, as $K$ increases, the changing properties of
$P_{{\mathrm{H}}}^{\mathrm{fix.}}$ and $P_{{\mathrm{H}}}^{\mathrm{dyn.}}$ are
of much different. Specifically, for all the considered
$P_{\mathrm{a}}^{\mathrm{max}}$, $P_{{\mathrm{H}}}^{\mathrm{fix.}}$ linearly
increases with $K$, which is, however, only correct for the dynamic HR-RIS
with $P_{\mathrm{a}}^{\mathrm{max}}=\\{0,10\\}$ dBm. In Fig. 10(a),
$P_{{\mathrm{H}}}^{\mathrm{dyn.}}$ almost linearly increases at first, then
quickly drops to a power value that is as low as $P_{\mathrm{RIS}}$. The
reason is that, with numerous active elements,
$P_{\mathrm{a}}^{\mathrm{max}}=-10$ dBm is not sufficient to share among all
the active elements for signal amplifying. Therefore, the dynamic HR-RIS
deactivates them to save power and also to preserve the SE, as justified in
Fig. 7(a). Focusing on the results for small $K$, e.g.,
$K=\\{1,2,\ldots,4\\}$, it is observed from Figs. 5, 7, and 10 that the HR-RIS
requires a reasonably additional power consumption to attain significant
improvement in the SE.

Fig. 11: EEs of the proposed HR-RIS schemes compared to those of the
conventional RIS schemes for $N_{t}=32$, $N_{r}=2$, $N=50$,
$K=\\{1,2,\ldots,N\\}$, $P_{\mathrm{BS}}=30$ dBm, and
$P_{\mathrm{a}}^{\mathrm{max}}=\\{-10,0,10\\}$ dBm.

Following the investigation of SEs and power consumption of the HR-RIS and RIS
in Figs. 7 and 10, with the same simulation parameters, we investigate their
EEs in Fig. 11. It is seen that the EE of the fixed HR-RIS rapidly decreases
as $K$ increases due to its increased power consumption, as shown in Fig. 10.
However, the EEs of the proposed HR-RIS architectures are the same or much
higher than those of the conventional AO-based RIS at small $K$. At large $K$,
the dynamic HR-RIS still exhibits the comparable EE with respect to the AO-
based RIS for $P_{\mathrm{a}}^{\mathrm{max}}=-10$ dBm. The results in this
figure and Fig. 7 further show that a spectral- and energy-efficient HR-RIS
should be equipped with a small number of active elements. However, if the
favorable conditions on $P_{\mathrm{BS}}$, $P_{\mathrm{a}}^{\mathrm{max}}$, or
$K$ are not guaranteed, the dynamic HR-RIS can still manage to ensure no or
small loss in EE.

##  VII Conclusion

We proposed a novel HR-RIS to assist MIMO communication systems with
significantly improved SE and EE compared to that assisted by the conventional
RIS. The HR-RIS is a semi-passive beamforming architecture, in which a few
elements are capable of adjusting the incident signals’ power. It has been
designed and optimized by the AO and power allocation strategies, resulting in
two different architectures, namely, the fixed and dynamic HR-RIS. In the
former, the active elements are fixed; by contrast, those in the latter can be
dynamically selected based on their capability of enhancing the SE, and are
powered based on the water-filling power allocation method. The analytical
results suggest the favorable design and deployment of the HR-RIS.
Specifically, it should be equipped with a small number of active elements
because employing numerous active elements does not guarantee SE or EE
improvement, while consuming high power. Furthermore, the HR-RIS can attain
significant SE and EE improvement when the transmit power at the transmitter
is small or moderate, and the improvement is more significant when the
distance between the transmitter and the HR-RIS is large. Finally, intensive
simulations have been performed to numerically justify the findings. The
results reveal that the HR-RIS has the potential to outperform both the
passive RIS and active AF relaying in certain communications channels.

Several important open problems for future work remain. The practical
implementation of the HR-RIS using efficient RAs needs more attention. In
particular, the realization of the FD type AF relay assumed has practical
challenges. Regardless, this paper provides theoretical insight in bridging
the gap between the RIS and active relaying concepts. The system optimization
from the EE perspective should be further addressed because we did not
directly optimize the EE metric. The CSI acquisition and HR-RIS performance
under imperfect CSI, hardware imperfections, and practical wireless channel
models need further investigation. Further benefits of HR-RIS can be expected
via channel estimation using the active components compared to the passive RIS
case. The active components were assumed to be ideal AF relaying with no
processing delays similar to the passive reflection. The practical delays and
use of DF relaying need to be addressed in future works.

## References

  * [1] A. Alkhateeb, O. El Ayach, G. Leus, and R. W. Heath, “Channel estimation and hybrid precoding for millimeter wave cellular systems,” _IEEE J. Sel. Topics Signal Process._ , vol. 8, no. 5, pp. 831–846, Oct 2014.
  * [2] J. He, T. Kim, H. Ghauch, K. Liu, and G. Wang, “Millimeter wave MIMO channel tracking systems,” in _Proc. IEEE GLOBECOM Workshop_ , Dec. 2014, pp. 416–421.
  * [3] R. W. Heath, N. González-Prelcic, S. Rangan, W. Roh, and A. M. Sayeed, “An overview of signal processing techniques for millimeter wave MIMO systems,” _IEEE J. Sel. Topics Signal Process._ , vol. 10, no. 3, pp. 436–453, April 2016.
  * [4] S. Zhang and R. Zhang, “Capacity characterization for intelligent reflecting surface aided mimo communication,” _IEEE J. Sel. Areas Commun._ , vol. 38, no. 8, pp. 1823–1838, 2020.
  * [5] Q. Wu and R. Zhang, “Beamforming optimization for wireless network aided by intelligent reflecting surface with discrete phase shifts,” _IEEE Trans. Commun._ , pp. 1–1, 2019.
  * [6] S. Hu, F. Rusek, and O. Edfors, “Beyond massive MIMO: The potential of positioning with large intelligent surfaces,” _IEEE Trans. Signal Process._ , vol. 66, no. 7, pp. 1761–1774, Apr. 2018.
  * [7] C. Liaskos, A. Tsioliaridou, A. Pitsillides, S. Ioannidis, and I. F. Akyildiz, “Using any surface to realize a new paradigm for wireless communications,” _Commun. ACM_ , vol. 61, pp. 30–33, 2018.
  * [8] E. Basar, M. Di Renzo, J. De Rosny, M. Debbah, M. Alouini, and R. Zhang, “Wireless communications through reconfigurable intelligent surfaces,” _IEEE Access_ , vol. 7, pp. 116 753–116 773, 2019.
  * [9] C. Huang, A. Zappone, G. C. Alexandropoulos, M. Debbah, and C. Yuen, “Reconfigurable intelligent surfaces for energy efficiency in wireless communication,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 8, pp. 4157–4170, 2019.
  * [10] M. D. Renzo, M. Debbah, D. T. P. Huy, A. Zappone, M. Alouini, C. Yuen, V. Sciancalepore, G. C. Alexandropoulos, J. Hoydis, H. Gacanin, J. de Rosny, A. Bounceur, G. Lerosey, and M. Fink, “Smart radio environments empowered by reconfigurable AI meta-surfaces: an idea whose time has come,” _EURASIP J. Wireless Comm. Network._ , vol. 2019, p. 129, 2019.
  * [11] J. He, H. Wymeersch, L. Kong, O. Silvén, and M. Juntti, “Large intelligent surface for positioning in millimeter wave MIMO systems,” in _Proc. of IEEE Veh. Tech. Conf. (VTC2020-Spring)_ , 2020, pp. 1–5.
  * [12] A. Taha, M. Alrabeiah, and A. Alkhateeb, “Enabling large intelligent surfaces with compressive sensing and deep learning,” _arXiv preprint arXiv:1904.10136_ , 2019.
  * [13] ——, “Deep learning for large intelligent surfaces in millimeter wave and massive MIMO systems,” in _IEEE Global Commun. Conf. (GLOBECOM)_ , 2019, pp. 1–6.
  * [14] S. Hu, F. Rusek, and O. Edfors, “The potential of using large antenna arrays on intelligent surfaces,” in _IEEE 85th Veh. Tech. Conf. (VTC Spring)_ , 2017, pp. 1–6.
  * [15] ——, “Beyond massive MIMO: The potential of data transmission with large intelligent surfaces,” _IEEE Trans. Signal Process._ , vol. 66, no. 10, pp. 2746–2758, 2018.
  * [16] J. He, H. Wymeersch, T. Sanguanpuak, O. Silvén, and M. Juntti, “Adaptive beamforming design for mmwave RIS-aided joint localization and communication,” in _IEEE Wireless Commun. Net. Conf. Work. (WCNCW)_ , 2020, pp. 1–6.
  * [17] M. Jung, W. Saad, M. Debbah, and C. S. Hong, “Asymptotic optimality of reconfigurable intelligent surfaces: Passive beamforming and achievable rate,” in _IEEE Int. Conf. Commun. (ICC)_ , 2020, pp. 1–6.
  * [18] Ö. Özdogan, E. Björnson, and E. G. Larsson, “Using intelligent reflecting surfaces for rank improvement in MIMO communications,” in _IEEE Int. Conf. Acoustics, Speech and Signal Process. (ICASSP)_ , 2020, pp. 9160–9164.
  * [19] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 11, pp. 5394–5409, 2019.
  * [20] P. Wang, J. Fang, X. Yuan, Z. Chen, and H. Li, “Intelligent reflecting surface-assisted millimeter wave communications: Joint active and passive precoding design,” _IEEE Trans. Veh. Tech._ , 2020.
  * [21] H. Zhang, B. Di, L. Song, and Z. Han, “Reconfigurable intelligent surfaces assisted communications with limited phase shifts: How many phase shifts are enough?” _IEEE Trans. Veh. Tech._ , vol. 69, no. 4, pp. 4498–4502, 2020\.
  * [22] H. Guo, Y.-C. Liang, J. Chen, and E. G. Larsson, “Weighted Sum-Rate Maximization for Intelligent Reflecting Surface Enhanced Wireless Networks,” in _IEEE Global Commun. Conf. (GLOBECOM)_. IEEE, 2019, pp. 1–6.
  * [23] Y. Han, W. Tang, S. Jin, C.-K. Wen, and X. Ma, “Large intelligent surface-assisted wireless communication exploiting statistical CSI,” _IEEE Trans. Veh. Tech._ , vol. 68, no. 8, pp. 8238–8242, 2019.
  * [24] G. Yang, X. Xu, and Y.-C. Liang, “Intelligent reflecting surface assisted non-orthogonal multiple access,” in _IEEE Wireless Commun. Network. Conf. (WCNC)_ , 2020, pp. 1–6.
  * [25] Y. Zhang, C. Zhong, Z. Zhang, and W. Lu, “Sum rate optimization for two way communications with intelligent reflecting surface,” _IEEE Commun. Lett._ , vol. 24, no. 5, pp. 1090–1094, 2020.
  * [26] J. V. Alegría and F. Rusek, “Achievable Rate with Correlated Hardware Impairments in Large Intelligent Surfaces,” in _IEEE Int. Workshop Computational Advances in Multi-Sensor Adaptive Process. (CAMSAP)_ , 2019, pp. 559–563.
  * [27] S. Gong, X. Lu, D. T. Hoang, D. Niyato, L. Shu, D. I. Kim, and Y.-C. Liang, “Towards smart wireless communications via intelligent reflecting surfaces: A contemporary survey,” _IEEE Commun. Surveys Tuts._ , 2020.
  * [28] Y. Yang, B. Zheng, S. Zhang, and R. Zhang, “Intelligent reflecting surface meets OFDM: Protocol design and rate maximization,” _IEEE Trans. Commun._ , 2020.
  * [29] X. Yu, D. Xu, and R. Schober, “MISO wireless communication systems via intelligent reflecting surfaces,” in _IEEE Int. Conf. Commun. (ICCC)_ , 2019, pp. 735–740.
  * [30] Y. Yang, S. Zhang, and R. Zhang, “IRS-enhanced OFDM: Power allocation and passive array optimization,” in _IEEE Global Commun. Conf. (GLOBECOM)_ , 2019, pp. 1–6.
  * [31] J. Yuan, Y.-C. Liang, J. Joung, G. Feng, and E. G. Larsson, “Intelligent reflecting surface-assisted cognitive radio system,” _IEEE Trans. Commun._ , 2020.
  * [32] B. Di, H. Zhang, L. Li, L. Song, Y. Li, and Z. Han, “Practical Hybrid Beamforming With Finite-Resolution Phase Shifters for Reconfigurable Intelligent Surface Based Multi-User Communications,” _IEEE Trans. Veh. Tech._ , vol. 69, no. 4, pp. 4565–4570, 2020.
  * [33] K. Ying, Z. Gao, S. Lyu, Y. Wu, H. Wang, and M.-S. Alouini, “GMD-based hybrid beamforming for large reconfigurable intelligent surface assisted millimeter-wave massive MIMO,” _IEEE Access_ , vol. 8, pp. 19 530–19 539, 2020.
  * [34] N. Landsberg and E. Socher, “Design and measurements of 100 GHz reflectarray and transmitarray active antenna cells,” _IEEE Trans. Antennas Propag._ , vol. 65, no. 12, pp. 6986–6997, 2017.
  * [35] ——, “A low-power 28-nm CMOS FD-SOI reflection amplifier for an active F-band reflectarray,” _IEEE Trans. Microw. Theory Techn._ , vol. 65, no. 10, pp. 3910–3921, 2017.
  * [36] E. Björnson, O. Özdogan, and E. G. Larsson, “Intelligent reflecting surface vs. decode-and-forward: How large surfaces are needed to beat relaying?” _IEEE Wireless Commun. Lett._ , pp. 1–1, 2019.
  * [37] M. Di Renzo, K. Ntontin, J. Song, F. H. Danufane, X. Qian, F. Lazarakis, J. De Rosny, D.-T. Phan-Huy, O. Simeone, R. Zhang _et al._ , “Reconfigurable intelligent surfaces vs. relaying: Differences, similarities, and performance comparison,” _IEEE Open J. Commun. Society_ , vol. 1, pp. 798–807, 2020.
  * [38] C. Pan, H. Ren, K. Wang, W. Xu, M. Elkashlan, A. Nallanathan, and L. Hanzo, “Multicell MIMO communications relying on intelligent reflecting surfaces,” _IEEE Trans. Wireless Commun._ , 2020.
  * [39] Q. Wu, S. Zhang, B. Zheng, C. You, and R. Zhang, “Intelligent reflecting surface aided wireless communications: A tutorial,” _arXiv preprint arXiv:2007.02759_ , 2020.
  * [40] Y. Han, S. Zhang, L. Duan, and R. Zhang, “Cooperative Double-IRS Aided Communication: Beamforming Design and Power Scaling,” _IEEE Wireless Commun. Lett._ , 2020.
  * [41] A. Zappone, P. Cao, and E. A. Jorswieck, “Energy efficiency optimization in relay-assisted MIMO systems with perfect and statistical CSI,” _IEEE Trans. Signal Process._ , vol. 62, no. 2, pp. 443–457, 2013.
  * [42] S. Gong, S. Wang, S. Chen, C. Xing, and L. Hanzo, “Robust energy efficiency optimization for amplify-and-forward mimo relaying systems,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 9, pp. 4326–4343, 2019.
  * [43] N. T. Nguyen and K. Lee, “Unequally sub-connected architecture for hybrid beamforming in massive MIMO systems,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 2, pp. 1127–1140, 2019.