# Reconfigurable Intelligent Surfaces Based on Single, Group, and Fully
Connected Discrete-Value Impedance Networks

Matteo Nerini, Bruno Clerckx M. Nerini and B. Clerckx are with the Department
of Electrical and Electronic Engineering, Imperial College London, London SW7
2AZ, U.K. (e-mail: m.nerini20, b.clerckx@imperial.ac.uk).

###### Abstract

Reconfigurable Intelligent Surfaces (RISs) allow to control the propagation
environment in wireless networks by properly tuning multiple reflecting
elements. Traditionally, RISs have been realized through single connected
reconfigurable impedance networks, in which each RIS element is independently
controlled by an impedance connected to ground. In a recent work, this
architecture has been extended by realizing more efficient RISs with group and
fully connected reconfigurable impedance networks. However, impedance networks
tunable with arbitrary precision are hard to realize in practice. In this
paper, we propose a practical RIS design strategy based on reconfigurable
impedance networks with discrete values. Besides, we address the problem of
how to group the RIS elements in group connected architectures. We optimize
single, group, and fully connected architectures considering finite-resolution
elements, and we compare them in terms of received signal power. Through Monte
Carlo simulations, supported by theoretical justifications, we show that only
a few resolution bits per reconfigurable impedance are sufficient to achieve
the performance upper bound. In particular, while four resolution bits are
needed to reach the upper bound in single connected architectures, only a
single resolution bit is sufficient in fully connected ones, simplifying
significantly the future development of these promising RIS architectures.

###### Index Terms:

Reconfigurable intelligent surface, Discrete-value impedance network, Pattern
search, $k$-means clustering, Alternating optimization.

##  I Introduction

Reconfigurable Intelligent Surfaces (RISs), or intelligent reflecting
surfaces, are an emerging technology that will enhance the performance of
future wireless communications [1, 2, 3]. This technology relies on large
planar surfaces comprising multiple reflecting elements, each of them able to
induce a certain amplitude and phase change to the incident electromagnetic
wave. Thus, an RIS can steer the reflected signal towards the intended
direction by smartly coordinating the reflection coefficients of its elements.
RIS-aided communication systems benefit from three main advantages. Firstly,
the received signal power is increased by means of the reconfigurable
propagation environment created by the RIS. Secondly, less inter-user
interference is experienced in multi-user systems and, consequently, a larger
capacity region is achieved. Finally, the outage probability is decreased
since RISs can better serve users in dead zones and at the cell edge. Beyond
these conceptual benefits, RISs are a cost-effective solution, since they are
composed solely of passive reflective elements, with no expensive Radio
Frequency (RF) chains.

In order to avoid difficult optimization problems, the vast majority of
studies on RISs do not pose limitations on the allowed reflection coefficient
values. However, in practical implementation, they are selected from a finite
number of discrete values. Indeed, reflection coefficients tunable with finer
resolution require a more complex hardware design, which can be prohibitive
when the number of RIS elements is high [4]. To enable the practical
development of RISs, research on RISs with discrete amplitude and phase shifts
is crucial. Considering a single-user RIS-aided Single-Input Single-Output
(SISO) system, the effects of discrete phase shifts on the diversity order [5,
6] and on the ergodic capacity [7] have been investigated. In addition,
several discrete optimization strategies have been recently presented with
different objectives.

RISs with discrete reflection coefficients have been proposed to enhance the
performance of Multiple-Input Multiple-Output (MIMO) communications by solving
rate maximization problems in single-user systems [8, 9, 10, 11] and sum-rate
maximization problems in multi-user systems [12, 13, 14, 15, 16]. In [10],
besides the discrete reflection coefficients assumption, a practical model
capturing the phase-dependent amplitude variation in the reflection
coefficients is considered while maximizing the user rate. In [16], only
statistical Channel State Information (CSI) is assumed to design the discrete
RIS reflection coefficients. To guarantee user fairness, the minimum user rate
maximization problem has been solved in [17], considering an RIS-aided Non-
Orthogonal Multiple Access (NOMA) system. RISs with discrete reflection
coefficients have been applied also with the objective of enhancing resource
utilization. In [18, 19], the transmit power is minimized by jointly
optimizing the continuous transmit precoding and the discrete phase shifts at
the RIS, subject to a given set of minimum Signal-to-Interference-plus-Noise
Ratio (SINR) constraints at the receivers. Orthogonal Multiple Access (OMA)
and NOMA schemes in RIS-aided communications have been compared in terms of
minimum transmit power required with given user rates in [20]. For RIS-aided
downlink communications, works have been conducted to maximize the energy
efficiency [21], and the spectral efficiency in presence of statistical CSI
[22]. In addition, RISs with discrete reflection coefficients have been
investigated for Terahertz (THz) MIMO communications, where the sparsity
characteristic of THz channels has been exploited to simplify the RIS design
[23, 24, 25, 26], and for Simultaneous Wireless Information and Power Transfer
(SWIPT) [27, 28].

In the aforementioned literature, it is always assumed that each RIS element
is independently controlled by a tunable impedance connected to ground [29].
Recently, a more general RIS architecture has been proposed in [30]. More
precisely, the authors have generalized the widely adopted single connected
architecture by connecting all or a subset of RIS elements through a
reconfigurable impedance network, resulting in the fully connected and group
connected architecture, respectively. These architectures lead to non-diagonal
scattering matrices, in contrast with the conventional diagonal scattering
matrix of the single connected architecture. Because of the additional degrees
of freedom provided by the more complex architecture, it has been proved that
the fully connected architecture enables the best performance gain with
respect to all the other RIS models proposed to date [30]. However, the
promising achievements of the group and fully connected architectures have
been studied with no practical constraints on the reflecting elements. Driven
by the success of these novel RIS architectures, in this paper we propose a
practical design strategy for group and fully connected RISs. In this way, we
generalize the well-established approach based on discrete phase shifts valid
for the single connected architecture. Furthermore, in the case of group
connected architectures, we investigate how the RIS elements should be grouped
to maximize the performance with reduced hardware complexity. The
contributions of this paper are summarized as follows.

First, we propose a practical RIS design strategy based on discrete-value
reconfigurable impedance networks. Previous literature on the single connected
architecture with discrete phase shifts is extended to the group and fully
connected cases. More precisely, firstly we propose an RIS design based on
scalar quantization, assigning a finite number of resolution bits to each
reconfigurable impedance. Then, a second discretization strategy based on
vector quantization is derived, in which groups of discrete-value
reconfigurable impedances are jointly optimized. RISs designed with these two
strategies are denoted as scalar-discrete RISs and vector-discrete RISs,
respectively. Vector-discrete RISs achieve higher performance than scalar-
discrete RISs at the cost of increased optimization computational complexity.

Second, we address the problem of grouping the RIS elements in the group
connected architecture, namely the grouping strategy. We describe two
approaches to design the grouping strategy: one based on the channel
statistics, and the other based on the channel realization. When the grouping
strategy is based on the channel statistics, it is fixed in time. In this
case, we propose two practical families of grouping strategies. The first,
called correlated grouping, groups together RIS elements that are close to
each other, experiencing correlated fading. The second, called uncorrelated
grouping, groups together RIS elements experiencing uncorrelated fading. Among
these two families, uncorrelated grouping is optimal to maximize the received
signal power in RISs with no practical constraints on the reflection
coefficients and in scalar-discrete RISs. However, correlated grouping becomes
beneficial when vector-discrete RISs are considered with low levels of
resolutions. When the grouping strategy is based on the channel realization,
it is continuously optimized at run-time for each channel realization. To this
end, we propose an iterative algorithm since the exhaustive search has
prohibitive complexity.

Third, we assess the performance in terms of received signal power of single
connected, group connected, and fully connected RISs employing different
numbers of resolution bits. Two scenarios including independent and
identically distributed (i.i.d.) Rayleigh fading and correlated fading
channels have been considered in our comparison. For each RIS architecture, we
analyze how many resolution bits are necessary to achieve the performance
upper bound, given by the RIS optimized with no resolution constraints.
Besides the simulation results, we provide theoretical reasons to justify the
obtained performance. We verify that the more connections are present in the
reconfigurable impedance network, the less resolution is needed to achieve the
optimality. Because of this property, in the fully connected architecture, a
single resolution bit allocated to each reconfigurable impedance is sufficient
to achieve the performance upper bound. In contrast, in the single connected
architecture, four resolution bits are needed to reach the upper bound. In a
practical development, this simplifies significantly the hardware complexity
of the fully connected RIS.

Fourth, we compare the single connected, group connected, and fully connected
architectures when the number of total resolution bits is limited. On the one
hand, the single connected is the simplest architecture having the lowest
number of impedances to be optimized. On the other hand, the fully connected
architecture includes a larger reconfigurable impedance network, but it
allows, at the same time, better performance with lower resolution per
reconfigurable impedance. Between these two extremes, a trade-off is given by
the group connected architecture. The fully connected architecture with one
resolution bit per reconfigurable impedance provides the best performance.
However, we show how single and group connected architectures should be used
to maximize the received signal power under a constrained number of total
resolution bits.

Organization: In Section II, we introduce the system model and the properties
of the three RIS architectures. In Section III, we present our practical RIS
design strategy based on discrete-value reconfigurable impedance networks. In
Section IV, we analyze the problem of grouping the RIS elements in the case of
group connected architecture, proposing two approaches to design the grouping
strategy. In Section V, we assess the obtained performance in terms of
received signal power. Finally, Section VI contains the concluding remarks.

Notation: Vectors and matrices are denoted with bold lower and bold upper
letters, respectively. Scalars are represented with letters not in bold font.
$\left|a\right|$, and $\arg\left(a\right)$ refer to the modulus and phase of a
complex scalar $a$, respectively. $\left[\mathrm{\mathbf{a}}\right]_{i}$ and
$\left\|\mathbf{a}\right\|$ refer to the $i$-th element and $l_{2}$-norm of
vector $\mathrm{\mathbf{a}}$, respectively. $\mathrm{\mathbf{A}}^{T}$,
$\mathrm{\mathbf{A}}^{H}$, and $\left[\mathrm{\mathbf{A}}\right]_{i,j}$ refer
to the transpose, conjugate transpose, and $\left(i,j\right)$-th element a
matrix $\mathrm{\mathbf{A}}$, respectively. $\mathbb{R}$ and $\mathbb{C}$
denote real and complex number set, respectively. $j=\sqrt{-1}$ denotes
imaginary unit. $\mathbf{0}$ and $\mathbf{I}$ denote an all-zero matrix and an
identity matrix, respectively, with appropriate dimensions.
$\mathcal{CN}\left(\mathbf{0},\mathbf{I}\right)$ denotes the distribution of a
circularly symmetric complex Gaussian random vector with mean vector
$\mathbf{0}$ and covariance matrix $\mathbf{I}$ and $\sim$ stands for
“distributed as”. diag$\left(a_{1},\ldots,a_{N}\right)$ refers to a diagonal
matrix with diagonal elements being $a_{1},\ldots,a_{N}$.
diag$\left(\mathrm{\mathbf{A}}_{1},\ldots,\mathrm{\mathbf{A}}_{N}\right)$
refers to a block diagonal matrix with blocks being
$\mathrm{\mathbf{A}}_{1},\ldots,\mathrm{\mathbf{A}}_{N}$.

##  II System Model

Let us consider a single-user SISO scenario in which the communication is
aided by an RIS, as represented in Figure 1111Though a SISO system is
considered through this paper, the results and the observations are extendable
and applicable also to MIMO systems.. We denote with $N_{T}=1$ the number of
antennas at the transmitter, $N_{R}=1$ the number of antennas at the receiver,
and $N_{I}$ the number of antennas at the RIS. The $N_{I}$ antennas of the RIS
are connected to a $N_{I}$-port reconfigurable impedance network, with
scattering matrix $\boldsymbol{\Theta}\in\mathbb{C}^{N_{I}\times N_{I}}$.
Defining $x\in\mathbb{C}$ as the transmitted signal and $y\in\mathbb{C}$ as
the received signal, we have $y=hx+n$, where $n$ is the Additive White
Gaussian Noise (AWGN) at the receiver. To characterize the channel $h$, this
scenario can be treated as an $N$-port network, where $N=N_{T}+N_{R}+N_{I}$,
and analyzed using scattering parameter network theory, as discussed in [30].
The general communication model resulting from the network analysis does not
allow to write $h$ as a function of $\boldsymbol{\Theta}$. Thus, it is
difficult to know the role played by $\boldsymbol{\Theta}$ and to optimize the
reconfigurable impedance network. For this reason, we simplify the general
model by making the following two assumptions. Firstly, we assume that there
is no impedance mismatching at the transmitter, receiver and RIS by
considering all the antennas perfectly matched to the characteristic
impedance, set as $Z_{0}=50\>\Omega$. Secondly, we assume no mutual coupling
between the antennas, which can be achieved in practice by setting the antenna
spacing larger than half-wavelength. With these two assumptions, the channel
$h$ can be written as

$h=h_{RT}+\mathbf{h}_{RI}\boldsymbol{\Theta}\mathbf{h}_{IT},$ (1)

where $h_{RT}\in\mathbb{C}$, $\mathbf{h}_{RI}\in\mathbb{C}^{1\times N_{I}}$,
and $\mathbf{h}_{IT}\in\mathbb{C}^{N_{I}\times 1}$ refer to the channels from
the transmitter to receiver, from the RIS to the receiver, and from the
transmitter to the RIS, respectively.

Figure 1: RIS-aided communication system model.

According to network theory [31], denoting with
$\mathbf{Z}_{I}\in\mathbb{C}^{N_{I}\times N_{I}}$ the impedance matrix of the
$N_{I}$-port reconfigurable impedance network, $\boldsymbol{\Theta}$ can be
expressed as

$\boldsymbol{\Theta}=\left(\mathbf{Z}_{I}+Z_{0}\mathbf{I}\right)^{-1}\left(\mathbf{Z}_{I}-Z_{0}\mathbf{I}\right).$
(2)

The $N_{I}$-port reconfigurable impedance network is constructed with passive
elements which can be adapted to the channel, to properly reflect the incident
signal. To maximize the power reflected by the RIS, we consider
$\mathbf{Z}_{I}$ purely reactive and we can write
$\mathbf{Z}_{I}=j\mathbf{X}_{I}$, where
$\mathbf{X}_{I}\in\mathbb{R}^{N_{I}\times N_{I}}$ denotes the reactance matrix
of the $N_{I}$-port reconfigurable impedance network. Hence,
$\boldsymbol{\Theta}$ is given by

$\boldsymbol{\Theta}=\left(j\mathbf{X}_{I}+Z_{0}\mathbf{I}\right)^{-1}\left(j\mathbf{X}_{I}-Z_{0}\mathbf{I}\right).$
(3)

Furthermore, the reconfigurable impedance network is also reciprocal so that
we have $\mathbf{X}_{I}=\mathbf{X}_{I}^{T}$ and
$\boldsymbol{\Theta}=\boldsymbol{\Theta}^{T}$. Depending on the connections
between the RIS elements, three different RIS architectures have been
identified by Shen et al., which are described in the following [30].

###  II-A Single Connected Reconfigurable Impedance Network

This is the conventional architecture traditionally employed in the
literature. In this architecture, each port of the RIS is connected to ground
with a reconfigurable impedance and is not connected to the other ports. We
denote with $Z_{n_{I}}=jX_{n_{I}}$ the impedance connecting the $n_{I}$-th
port to ground, for $n_{I}=1,\ldots,N_{I}$. Thus, the reactance matrix
$\mathbf{X}_{I}$ is a diagonal matrix given by

$\mathbf{X}_{I}=\mathrm{diag}\left(X_{1},X_{2},\ldots,X_{N_{I}}\right).$ (4)

According to (3), $\boldsymbol{\Theta}$ is also a diagonal matrix whose
$\left(n_{I},n_{I}\right)$-th element is the reflection coefficient of the
reconfigurable impedance $Z_{n_{I}}$, and is given by
$\frac{jX_{n_{I}}-Z_{0}}{jX_{n_{I}}+Z_{0}}=e^{j\theta_{n_{I}}}$. Hence, the
scattering matrix $\boldsymbol{\Theta}$ in the single connected case can be
written as

$\boldsymbol{\Theta}=\mathrm{diag}\left(e^{j\theta_{1}},e^{j\theta_{2}},\ldots,e^{j\theta_{N_{I}}}\right).$
(5)

###  II-B Fully Connected Reconfigurable Impedance Network

In a more general perspective, the fully connected reconfigurable network is
obtained by connecting every port of the network to all the other ports.
Therefore, the reactance matrix $\mathbf{X}_{I}$ can be an arbitrary full
matrix depending on the impedances chosen to build the reconfigurable
impedance network, assumed purely reactive [30]. For the reciprocity of the
$N_{I}$-port, the only constraint on $\mathbf{X}_{I}$ in the fully connected
architecture is

$\mathbf{X}_{I}=\mathbf{X}_{I}^{T}.$ (6)

According to network theory [31], and (3), $\boldsymbol{\Theta}$ is a complex
symmetric unitary matrix

$\boldsymbol{\Theta}=\boldsymbol{\Theta}^{T},\>\boldsymbol{\Theta}^{H}\boldsymbol{\Theta}=\boldsymbol{\mathrm{I}}.$
(7)

###  II-C Group Connected Reconfigurable Impedance Network

The group connected architecture has been proposed to reach a trade-off
between performance and complexity. On the one hand, the single connected
architecture is the simplest one, but also the most limiting since
$\boldsymbol{\Theta}$ is a diagonal matrix. On the other hand, the fully
connected architecture offers the highest degree of freedom but at the cost of
a complex circuit topology. In the group connected architecture, the $N_{I}$
elements are divided into $G$ groups, each having $N_{G}=\frac{N_{I}}{G}$
elements. Each element of the $N_{I}$-port is connected to all the other
elements in its group, while there is no connection inter-group. In other
words, for the $g$-th group, a $N_{G}$-port fully connected reconfigurable
impedance network with reactance matrix of
$\mathbf{X}_{I,g}\in\mathbb{R}^{N_{G}\times N_{G}}$ is used. Thus,
$\mathbf{X}_{I}$ is a block diagonal matrix given by

$\mathbf{X}_{I}=\mathrm{diag}\left(\mathbf{X}_{I,1},\mathbf{X}_{I,2},\ldots,\mathbf{X}_{I,G}\right),\>\mathbf{X}_{I,g}=\mathbf{X}_{I,g}^{T},\>\forall
g.$ (8)

According to (3), the following constraints can be found for the scattering
matrix in the group connected architecture

$\boldsymbol{\Theta}=\mathrm{diag}\left(\boldsymbol{\Theta}_{1},\boldsymbol{\Theta}_{2},\ldots,\boldsymbol{\Theta}_{G}\right),\>\boldsymbol{\Theta}_{g}=\boldsymbol{\Theta}_{g}^{T},\>\boldsymbol{\Theta}_{g}^{H}\boldsymbol{\Theta}_{g}=\boldsymbol{\mathrm{I}},\>\forall
g,$ (9)

which show that $\boldsymbol{\Theta}$ is a block diagonal matrix with each
block $\boldsymbol{\Theta}_{g}$ being a complex symmetric unitary matrix, for
$g=1,\ldots,G$.

##  III Discrete RIS Design

In this section, we formulate the received signal power maximization problem
in a single-user RIS-aided SISO system and we propose a novel practical
solution. Our ultimate goal is to design $\boldsymbol{\Theta}$ to maximize the
received signal power, which is given by
$P_{R}=P_{T}\left|h_{RT}+\mathbf{h}_{RI}\boldsymbol{\Theta}\mathbf{h}_{IT}\right|^{2}$,
where $P_{T}=\mathrm{E}\left[\left|x\right|^{2}\right]$ is the transmitted
signal power. Maximizing the received signal power is equivalent to maximize
$\left|\mathbf{h}_{RI}\boldsymbol{\Theta}\mathbf{h}_{IT}\right|^{2}$ since
$h_{RT}$ can be always combined in phase with
$\mathbf{h}_{RI}\boldsymbol{\Theta}\mathbf{h}_{IT}$. Depending on the
considered RIS architecture, the scattering matrix $\boldsymbol{\Theta}$ is
subject to different constraints, and different optimization problems are
formulated.

###  III-A Discrete Single Connected RISs

In the single connected case, $\boldsymbol{\Theta}$ is characterized as in
(5). Thus, it is completely described by the phases
$\theta_{n_{I}}\in\left[0,2\pi\right)$, $\forall n_{I}=1,\ldots,N_{I}$.
Assuming no restrictions on the phases $\theta_{n_{I}}$, the power
maximization problem can be formulated as follows

$\displaystyle\underset{\theta_{n_{I}}}{\mathsf{\mathrm{max}}}\;\;$
$\displaystyle\left|\mathbf{h}_{RI}\boldsymbol{\Theta}\mathbf{h}_{IT}\right|^{2}$
(10) $\displaystyle\mathsf{\mathrm{s.t.}}\;\;\;$
$\displaystyle\boldsymbol{\Theta}=\mathrm{diag}\left(e^{j\theta_{1}},e^{j\theta_{2}},\ldots,e^{j\theta_{N_{I}}}\right),$
(11) $\displaystyle\theta_{n_{I}}\in\left[0,2\pi\right),\>\forall n_{I}.$ (12)

However, the assumption of continuous phase shifts is difficult to realize in
practice since complex hardware is needed to tune with high precision the
reconfigurable impedances.

To design practical RISs, recent works agree on a well-established solution to
discretize the phase shifts of $\boldsymbol{\Theta}$. They are discretized
uniformly in the interval $\left[0,2\pi\right)$ by defining $2^{B}$
quantization levels, where $B$ is the number of resolution bits. In other
words, the phase shifts of each diagonal element of $\boldsymbol{\Theta}$ are
chosen in the codebook

$\boldsymbol{\Omega}=\left\\{0,\delta,\ldots,\left(2^{B}-1\right)\delta\right\\},$
(13)

where $\delta=\frac{2\pi}{2^{B}}$ is the quantization step. Considering this
constraint, we can reformulate the received signal power maximization problem
(10)-(12) in the discrete case as

$\displaystyle\underset{\theta_{n_{I}}}{\mathsf{\mathrm{max}}}\;\;$
$\displaystyle\left|\mathbf{h}_{RI}\boldsymbol{\Theta}\mathbf{h}_{IT}\right|^{2}$
(14) $\displaystyle\mathsf{\mathrm{s.t.}}\;\;\;$
$\displaystyle\boldsymbol{\Theta}=\mathrm{diag}\left(e^{j\theta_{1}},e^{j\theta_{2}},\ldots,e^{j\theta_{N_{I}}}\right),$
(15) $\displaystyle\theta_{n_{I}}\in\boldsymbol{\Omega},\>\forall n_{I}.$ (16)

To solve this problem, an exhaustive search could be employed since the phases
can assume a finite number of values. However, this is prohibitive in practice
because the RIS system would be required to search among $2^{N_{I}B}$ possible
$\boldsymbol{\Theta}$ matrices, a number exponentially increasing with $N_{I}$
and $B$. To reduce the search space, we solve this problem through alternating
optimization, also known as successive refinement, or local search, widely
applied in the literature to optimize RIS with discrete phases [9, 10, 18]. In
this sub-optimal method, each of the $N_{I}$ phases is optimized individually
in an iterative manner by fixing the other $N_{I}-1$ phases. This procedure is
repeated for all the $N_{I}$ elements for $I$ iterations until the convergence
is reached. The total number of search times in a complete iteration is
decreased to $N_{I}2^{B}$.

###  III-B Scalar-Discrete Group Connected RISs

Let us now consider the group connected architecture. In the following, the
fully connected architecture is not explicitly mentioned since it is
considered a special case of the group connected, i.e., when $G=1$ and
$N_{G}=N_{I}$. The constraints (9) indicate that the scattering matrix
$\boldsymbol{\Theta}$ is a block diagonal matrix with each block being a
complex symmetric unitary matrix, which complicates the optimization. Thus,
the relationship between $\boldsymbol{\Theta}$ and the reactance matrix
$\mathbf{X}_{I}$ given by (3) is exploited to write the maximization problem
as

$\displaystyle\underset{\mathbf{X}_{I,g}}{\mathsf{\mathrm{max}}}\;\;$
$\displaystyle\left|\mathbf{h}_{RI}\boldsymbol{\Theta}\mathbf{h}_{IT}\right|^{2}$
(17) $\displaystyle\mathsf{\mathrm{s.t.}}\;\;\;$
$\displaystyle\boldsymbol{\Theta}=\mathrm{diag}\left(\boldsymbol{\Theta}_{1},\boldsymbol{\Theta}_{2},\ldots,\boldsymbol{\Theta}_{G}\right),$
(18)
$\displaystyle\boldsymbol{\Theta}_{g}=\left(j\mathbf{X}_{I,g}+Z_{0}\mathbf{I}\right)^{-1}\left(j\mathbf{X}_{I,g}-Z_{0}\mathbf{I}\right),\>\forall
g,$ (19) $\displaystyle\mathbf{X}_{I,g}=\mathbf{X}_{I,g}^{T},\>\forall g.$
(20)

which is unconstrained since $\mathbf{X}_{I,g}$ can be an arbitrary
$N_{G}\times N_{G}$ real symmetric matrix. This problem has been solved in
[30] by using the quasi-Newton method to find the optimal upper triangular
part of each $\mathbf{X}_{I,g}$ (having $\frac{N_{G}\left(N_{G}+1\right)}{2}$
entries) without any constraints. Although this approach has been proved to
achieve the theoretical upper bound [30], the entries of the reactance matrix
are allowed to assume any real value, which is hard to realize in practice.

Figure 2: Design of scalar-discrete group connected RISs.

Since the entries of $\mathbf{X}_{I,g}$ are not distributed in a finite
interval, to define the best discrete codebook is not as straightforward as in
the single connected case. For this reason, when considering the optimization
of discrete group connected RISs, a two-fold problem arises: i) the definition
of a suitable discrete codebook for the elements of $\mathbf{X}_{I,g}$, and
ii) the optimization of $\mathbf{X}_{I,g}$ over the possible values given by
the codebook. In Figure 2, we highlight the two stages in our RIS design
strategy that aim to solve these two problems. In our solution, we assume that
the optimal codebook $\boldsymbol{\Psi}$ is symmetric around zero

$\boldsymbol{\Psi}=\left\\{\pm\psi_{1},\pm\psi_{2},\ldots,\pm\psi_{2^{B-1}}\right\\}.$
(21)

Furthermore, the two cases in which $B=1$ and $B>1$ are treated differently.

####  III-B1 Case $B=1$

When the number of resolution bits per reconfigurable reactance is $B=1$, the
entries of the reactance matrix $\mathbf{X}_{I}$ can assume only the two
values in $\boldsymbol{\Psi}=\left\\{-\psi_{1},+\psi_{1}\right\\}$, where
$\psi_{1}$ is a positive real number. Thus, the following optimization problem
can be considered

$\displaystyle\underset{\mathbf{X}_{I,g}}{\mathsf{\mathrm{max}}}\;\;$
$\displaystyle\left|\mathbf{h}_{RI}\boldsymbol{\Theta}\mathbf{h}_{IT}\right|^{2}$
(22) $\displaystyle\mathsf{\mathrm{s.t.}}\;\;\;$
$\displaystyle\boldsymbol{\Theta}=\mathrm{diag}\left(\boldsymbol{\Theta}_{1},\boldsymbol{\Theta}_{2},\ldots,\boldsymbol{\Theta}_{G}\right),$
(23)
$\displaystyle\boldsymbol{\Theta}_{g}=\left(j\mathbf{X}_{I,g}+Z_{0}\mathbf{I}\right)^{-1}\left(j\mathbf{X}_{I,g}-Z_{0}\mathbf{I}\right),\>\forall
g,$ (24) $\displaystyle\mathbf{X}_{I,g}=\mathbf{X}_{I,g}^{T},\>\forall g,$
(25)
$\displaystyle\left[\mathbf{X}_{I,g}\right]_{i,j}\in\left\\{-\psi_{1},+\psi_{1}\right\\},\>\forall
g.$ (26)

Similarly to the problem (14)-(16), we can apply alternating optimization to
find the reactance matrix $\mathbf{X}_{I}$ which maximize the objective.
However, the optimal $\mathbf{X}_{I}$ is function of $\psi_{1}$, which also
needs to be optimized. To this end, we introduce a second optimization problem
as follows

$\displaystyle\underset{\psi_{1}}{\mathsf{\mathrm{max}}}\;\;$
$\displaystyle\left|\mathbf{h}_{RI}\boldsymbol{\Theta}\mathbf{h}_{IT}\right|^{2}$
(27) $\displaystyle\mathsf{\mathrm{s.t.}}\;\;\;$
$\displaystyle\boldsymbol{\Theta}=\mathrm{diag}\left(\boldsymbol{\Theta}_{1},\boldsymbol{\Theta}_{2},\ldots,\boldsymbol{\Theta}_{G}\right),$
(28)
$\displaystyle\boldsymbol{\Theta}_{g}=\left(j\mathbf{X}_{I,g}+Z_{0}\mathbf{I}\right)^{-1}\left(j\mathbf{X}_{I,g}-Z_{0}\mathbf{I}\right),\>\forall
g,$ (29) $\displaystyle\mathbf{X}_{I,g}\text{ solves \eqref{eq:P-GC-D-
obj1}-\eqref{eq:P-GC-D-con1-4}},\>\forall g.$ (30)

These two nested optimization problems can be formalized as a bilevel
programming problem, in which (27)-(30) is the upper-level problem, and
(22)-(26) is the lower-level problem. In general, the upper-level problem
(27)-(30) is difficult to solve since we do not have an expression of the
constraint (30), and we cannot write explicitly $\mathbf{X}_{I,g}$ as function
of $\psi_{1}$. However, we are able to evaluate $\mathbf{X}_{I,g}$ as function
of $\psi_{1}$ by solving (22)-(26). Thus, the upper-level problem is solvable
with pattern search, an optimization algorithm that only requires the
evaluation of the objective function in a series of points until the
convergence is reached, without using the derivatives. In a practical
development, we aim to adjust the reconfigurable impedance network choosing
among the values of a fixed codebook. For this reason, we propose to optimize
$\psi_{1}$ offline, on the basis of a training set of channel realizations,
and to solve the lower-level problem (22)-(26) online, depending on the actual
channels $\mathbf{h}_{RI}$ and $\mathbf{h}_{IT}$.

####  III-B2 Case $B>1$

Pattern search can be used to solve the upper-level problem (27)-(30) only
when the optimization involves a $1$-dimensional search, that is only when
$B=1$. When more resolution bits are considered, pattern search becomes highly
suboptimal since it is likely to converge to a local maximum in a multi-
dimensional search space. For this reason, we rely on another method to define
the codebook $\boldsymbol{\Psi}$ when $B>1$.

The main idea is to learn the discrete codebook from the distribution of the
optimal continuous elements of $\mathbf{X}_{I}$ which solve (17)-(20). This
can be realized with $k$-means clustering, an unsupervised machine learning
technique able to group the elements of a distribution into $k$ clusters. More
precisely, since the codebook is assumed symmetric around zero, we need to
obtain the $2^{B-1}$ values of the positive half of the codebook. To this end,
a training set is formed with the modulus of the optimal continuous
$\mathbf{X}_{I}$ elements. Thus, $\psi_{1},\psi_{2},\ldots,\psi_{2^{B-1}}$ are
given by the $k$ cluster centers obtained by clustering the training set, with
$k=2^{B-1}$. Once the codebook has been learned offline, the online
optimization process consists of applying alternating optimization to solve

$\displaystyle\underset{\mathbf{X}_{I,g}}{\mathsf{\mathrm{max}}}\;\;$
$\displaystyle\left|\mathbf{h}_{RI}\boldsymbol{\Theta}\mathbf{h}_{IT}\right|^{2}$
(31) $\displaystyle\mathsf{\mathrm{s.t.}}\;\;\;$
$\displaystyle\boldsymbol{\Theta}=\mathrm{diag}\left(\boldsymbol{\Theta}_{1},\boldsymbol{\Theta}_{2},\ldots,\boldsymbol{\Theta}_{G}\right),$
(32)
$\displaystyle\boldsymbol{\Theta}_{g}=\left(j\mathbf{X}_{I,g}+Z_{0}\mathbf{I}\right)^{-1}\left(j\mathbf{X}_{I,g}-Z_{0}\mathbf{I}\right),\>\forall
g,$ (33) $\displaystyle\mathbf{X}_{I,g}=\mathbf{X}_{I,g}^{T},\>\forall g,$
(34)
$\displaystyle\left[\mathbf{X}_{I,g}\right]_{i,j}\in\left\\{\pm\psi_{1},\pm\psi_{2},\ldots,\pm\psi_{2^{B-1}}\right\\},\>\forall
g.$ (35)

In the group connected architecture, the total number of search times in a
complete iteration of the alternating optimization algorithm is
$G\frac{N_{G}\left(N_{G}+1\right)}{2}2^{B}$, which boils down to
$\frac{N_{I}\left(N_{I}+1\right)}{2}2^{B}$ in the fully connected case. The
entire process leading to the design of scalar-discrete group connected RISs
is summarized in Figure 2. The two main stages are emphasized: firstly, an
offline learning has the objective of building the codebook
$\boldsymbol{\Psi}$; secondly, during the online deployment, alternating
optimization is used to find the discrete $\mathbf{X}_{I}$ which maximizes the
received signal power.

###  III-C Vector-Discrete Group Connected RISs

In the previous section, the scalar codebook $\boldsymbol{\Psi}$ has been
introduced to define the possible values that each element in $\mathbf{X}_{I}$
can assume. This concept can be extended by introducing a second
discretization strategy based on vector quantization. For a group connected
architecture with group size $N_{G}$, let us introduce a vector codebook
$\boldsymbol{\Psi}_{V}$ as

$\boldsymbol{\Psi}_{V}=\left\\{\boldsymbol{\psi}_{1},\boldsymbol{\psi}_{2},\ldots,\boldsymbol{\psi}_{2^{B_{V}}}\right\\}.$
(36)

where the $b$-th element
$\boldsymbol{\psi}_{b}\in\mathbb{R}^{\frac{N_{G}\left(N_{G}+1\right)}{2}}$ is
a possible value assumed by each block $\mathbf{X}_{I,g}$, and $B_{V}$ is the
number of resolution bits assigned to each $\mathbf{X}_{I,g}$. Similarly to
the scalar discretization case, $\boldsymbol{\Psi}_{V}$ can be obtained with
$k$-means clustering through an offline learning phase, with the difference
that a multi-dimensional feature space with dimension
$\frac{N_{G}\left(N_{G}+1\right)}{2}$ is now considered, and $k=2^{B_{V}}$.
Thus, the online optimization problem is formulated as

$\displaystyle\underset{\mathbf{X}_{I,g}}{\mathsf{\mathrm{max}}}\;\;$
$\displaystyle\left|\mathbf{h}_{RI}\boldsymbol{\Theta}\mathbf{h}_{IT}\right|^{2}$
(37) $\displaystyle\mathsf{\mathrm{s.t.}}\;\;\;$
$\displaystyle\boldsymbol{\Theta}=\mathrm{diag}\left(\boldsymbol{\Theta}_{1},\boldsymbol{\Theta}_{2},\ldots,\boldsymbol{\Theta}_{G}\right),$
(38)
$\displaystyle\boldsymbol{\Theta}_{g}=\left(j\mathbf{X}_{I,g}+Z_{0}\mathbf{I}\right)^{-1}\left(j\mathbf{X}_{I,g}-Z_{0}\mathbf{I}\right),\>\forall
g,$ (39) $\displaystyle\mathbf{X}_{I,g}=\mathbf{X}_{I,g}^{T},\>\forall g,$
(40) $\displaystyle\mathbf{X}_{I,g}\in\boldsymbol{\Psi}_{V},\>\forall g.$ (41)

Alternating optimization is the selected strategy to solve this problem, in
which the $G$ blocks $\mathbf{X}_{I,g}$ are iteratively optimized by searching
among their possible values contained in $\boldsymbol{\Psi}_{V}$.

The cost of this second strategy relies in its complexity. For example, if $B$
bits are allocated to each reactance element, the number of clusters needed to
quantize a vector of $\frac{N_{G}\left(N_{G}+1\right)}{2}$ elements is
$k=2^{B\frac{N_{G}\left(N_{G}+1\right)}{2}}$, growing exponentially with the
square of $N_{G}$. Consequently, the number of search times in a complete
iteration of the alternating optimization algorithms becomes
$G2^{B\frac{N_{G}\left(N_{G}+1\right)}{2}}$. However, this discretization
strategy based on vector quantization brings two benefits. First, for the same
number of total resolution bits, vector quantization is inherently more
efficient than scalar quantization. And second, the number of total resolution
bits can be chosen with more degrees of freedom, since it is no longer limited
to $B$ bits for each reactance element.

##  IV Grouping Strategy

In this section, we address the problem of grouping the RIS elements in the
case of a group connected architecture. We refer to a particular partition of
the $N_{I}$ RIS elements into $G$ groups, each containing
$N_{G}=\frac{N_{I}}{G}$ elements, as a grouping strategy. The role played by
the grouping strategy is still unexplored in the literature. For this reason,
we firstly show the importance of selecting the correct grouping strategy
through a toy example. Secondly, we define the characteristics of the grouping
strategy which maximizes the upper bound of the received signal power.
Finally, we present two approaches to design the grouping strategy: one based
on the channel statistics, and the other based on the channel realization.

###  IV-A Importance of the Grouping Strategy

In order to obtain insights into the impact of the grouping strategy, we
quantify the received signal power for two different grouping strategies in a
toy example. Let us consider an RIS-aided SISO system, as introduced in
Section III. For simplicity, we assume unitary transmitted signal power and
neglect the direct channel $h_{RT}$. Thus, the received signal power can be
expressed as
$P_{R}=\left|\mathbf{h}_{RI}\boldsymbol{\Theta}\mathbf{h}_{IT}\right|^{2}$. In
this toy example, we consider $N_{I}=4$ RIS antennas arranged in a Uniform
Linear Array (ULA). Furthermore, we assume that
$\left[\mathbf{h}_{RI}\right]_{1}=\left[\mathbf{h}_{RI}\right]_{2}$,
$\left[\mathbf{h}_{IT}\right]_{1}=\left[\mathbf{h}_{IT}\right]_{2}$, and that
$\left[\mathbf{h}_{RI}\right]_{3}=\left[\mathbf{h}_{RI}\right]_{4}$,
$\left[\mathbf{h}_{IT}\right]_{3}=\left[\mathbf{h}_{IT}\right]_{4}$. In other
words, we assume that the fading effects experienced in the first two antennas
of the ULA are identical, as well as the fading effects experienced in the
last two antennas. According to [30], the received signal power achieved with
a single connected and fully connected RIS is upper bounded by

$\displaystyle P_{R}^{\mathrm{Single}}$
$\displaystyle=\left(\sum_{n_{I}=1}^{N_{I}}\left|\left[\mathbf{h}_{RI}\right]_{n_{I}}\left[\mathbf{h}_{IT}\right]_{n_{I}}\right|\right)^{2},$
(42) $\displaystyle P_{R}^{\mathrm{Fully}}$
$\displaystyle=\left\|\mathbf{h}_{RI}\right\|^{2}\left\|\mathbf{h}_{IT}\right\|^{2},$
(43)

respectively. Thus, in our toy example, these upper bounds can be expressed as

$\displaystyle P_{R}^{\mathrm{Single}}$
$\displaystyle=\left(2\left|\left[\mathbf{h}_{RI}\right]_{1}\left[\mathbf{h}_{IT}\right]_{1}\right|+2\left|\left[\mathbf{h}_{RI}\right]_{3}\left[\mathbf{h}_{IT}\right]_{3}\right|\right)^{2},$
(44) $\displaystyle P_{R}^{\mathrm{Fully}}$
$\displaystyle=\left(2{\left|\left[\mathbf{h}_{RI}\right]_{1}\right|}^{2}+2{\left|\left[\mathbf{h}_{RI}\right]_{3}\right|}^{2}\right)\left(2{\left|\left[\mathbf{h}_{IT}\right]_{1}\right|}^{2}+2{\left|\left[\mathbf{h}_{IT}\right]_{3}\right|}^{2}\right).$
(45)

Furthermore, using the Cauchy-Schwarz inequality, we have
$P_{R}^{\mathrm{Single}}\leq P_{R}^{\mathrm{Fully}}$, and the equality holds
if and only if

$\frac{\left|\left[\mathbf{h}_{RI}\right]_{1}\right|}{\left|\left[\mathbf{h}_{RI}\right]_{3}\right|}=\frac{\left|\left[\mathbf{h}_{IT}\right]_{1}\right|}{\left|\left[\mathbf{h}_{IT}\right]_{3}\right|}.$
(46)

Let us now consider the group connected case, which is the trade-off between
the single and the fully connected in terms of hardware complexity. According
to [30], the received signal power in a group connected architecture is upper
bounded by

$P_{R}^{\mathrm{Group}}=\left(\sum_{g=1}^{G}\left\|\mathbf{h}_{RI,g}\right\|\left\|\mathbf{h}_{IT,g}\right\|\right)^{2},$
(47)

where the vectors $\mathbf{h}_{RI,g}\in\mathbb{C}^{1\times N_{G}}$ and
$\mathbf{h}_{IT,g}\in\mathbb{C}^{N_{G}\times 1}$ contain the $N_{G}$ elements
of $\mathbf{h}_{RI}$ and $\mathbf{h}_{IT}$, respectively, which are grouped
into the $g$-th group. In our toy example, we consider the only possible group
size, that is $N_{G}=2$, and compare the impact of two different grouping
strategies. The first grouping strategy groups together adjacent antennas in
the ULA. Thus, one group is composed of antennas $1$ and $2$, and the other
group is composed of antennas $3$ and $4$. In this case, applying (47), the
received signal power is upper bounded by

$P_{R_{A}}^{\mathrm{Group}}=\left(\sqrt{2}\left|\left[\mathbf{h}_{RI}\right]_{1}\right|\sqrt{2}\left|\left[\mathbf{h}_{IT}\right]_{1}\right|+\sqrt{2}\left|\left[\mathbf{h}_{RI}\right]_{3}\right|\sqrt{2}\left|\left[\mathbf{h}_{IT}\right]_{3}\right|\right)^{2}.$
(48)

Comparing (48) with (44), we notice that
$P_{R_{A}}^{\mathrm{Group}}=P_{R}^{\mathrm{Single}}$. Hence, there is no
additional benefit in using the group connected architecture with this
grouping strategy with respect to the single connected architecture. The
second considered grouping strategy maximizes the minimum distance between
antenna elements grouped together. Thus, one group is given by antennas $1$
and $3$, and the other is given by antennas $2$ and $4$. In this second case,
using (47), the received signal power is upper bounded by

$P_{R_{B}}^{\mathrm{Group}}=\left(\sqrt{{\left|\left[\mathbf{h}_{RI}\right]_{1}\right|}^{2}+{\left|\left[\mathbf{h}_{RI}\right]_{3}\right|}^{2}}\sqrt{{\left|\left[\mathbf{h}_{IT}\right]_{1}\right|}^{2}+{\left|\left[\mathbf{h}_{IT}\right]_{3}\right|}^{2}}+\sqrt{{\left|\left[\mathbf{h}_{RI}\right]_{1}\right|}^{2}+{\left|\left[\mathbf{h}_{RI}\right]_{3}\right|}^{2}}\sqrt{{\left|\left[\mathbf{h}_{IT}\right]_{1}\right|}^{2}+{\left|\left[\mathbf{h}_{IT}\right]_{3}\right|}^{2}}\right)^{2}.$
(49)

Comparing (49) with (45), it can be easily verified that
$P_{R_{B}}^{\mathrm{Group}}=P_{R}^{\mathrm{Fully}}$. Hence, this grouping
strategy is beneficial since it allows to achieve the same performance as the
fully connected architecture, but with reduced hardware complexity.

This toy example showed that, with fixed group size $N_{G}$, the performance
of the group connected architecture is highly dependent on the grouping
strategy. In particular, the received signal power can be as low as in the
single connected architecture (with consequent waste of hardware complexity),
or as high as in the fully connected one. For this reason, it is crucial to
characterize the best grouping strategy which allows to approach the
performance of the fully connected architecture.

Let us consider the upper bounds of the received signal power in the case of
fully and group connected architecture, which are given by (43) and (47),
respectively. Using the Cauchy-Schwarz inequality, we can deduce that
$P_{R}^{\mathrm{Group}}\leq P_{R}^{\mathrm{Fully}}$, and the equality is
achieved if and only if

$\left\|\mathbf{h}_{RI,g}\right\|=\alpha\left\|\mathbf{h}_{IT,g}\right\|,\>\forall
g,$ (50)

where $\alpha$ can assume any positive real value. This condition is
achievable only for particular realizations of the channels $\mathbf{h}_{RI}$
and $\mathbf{h}_{IT}$. Since a grouping strategy satisfying (50) does not
exist for every channel realizations, we reformulate this condition to
characterize the best grouping strategy. We introduce the vectors
$\mathbf{m}_{RI}=\left[\left\|\mathbf{h}_{RI,1}\right\|,\left\|\mathbf{h}_{RI,2}\right\|,\ldots,\left\|\mathbf{h}_{RI,G}\right\|\right]$
with $\mathbf{m}_{RI}\in\mathbb{R}^{1\times G}$, and
$\mathbf{m}_{IT}=\left[\left\|\mathbf{h}_{IT,1}\right\|,\left\|\mathbf{h}_{IT,2}\right\|,\ldots,\left\|\mathbf{h}_{IT,G}\right\|\right]^{T}$
with $\mathbf{m}_{IT}\in\mathbb{R}^{G\times 1}$. Thus, fulfilling condition
(50) is equivalent to making the vectors $\mathbf{m}_{RI}$ and
$\mathbf{m}_{IT}$ linearly dependent. In this way, the quality of a specific
grouping strategy can be measured with the cosine similarity between
$\mathbf{m}_{RI}$ and $\mathbf{m}_{IT}$, defined as

$\rho=\frac{\mathbf{m}_{RI}\mathbf{m}_{IT}}{\left\|\mathbf{h}_{RI}\right\|\left\|\mathbf{h}_{IT}\right\|}.$
(51)

Given specific realizations of $\mathbf{h}_{RI}$ and $\mathbf{h}_{IT}$, the
best grouping strategy is defined as the one that maximizes this cosine
similarity $\rho$. In the following, we propose two practical strategies to
design the grouping strategy: the former, based on the channel statistics,
maintain the grouping strategy fixed in time; the latter, based on the channel
realization, adapt the grouping strategy on a per channel realization basis.

###  IV-B Grouping Based on the Channel Statistics: Correlated Grouping and
Uncorrelated Grouping

Figure 3: Correlated grouping (on the left) and uncorrelated grouping (on the
right) performed on uni-polarized RIS (on top) and dual-polarized RIS (on the
bottom), with $N_{I}=16$ and $N_{G}=4$.

Let us consider an RIS realized through a ULA composed of $N_{I}$ antenna
elements equally spaced. These elements can be either all vertically polarized
(uni-polarized RIS), or $\pm 45^{\circ}$ polarized (dual-polarized RIS). With
a focus on these categories of RIS, we propose two opposite families of
grouping strategies based on the channel statistics, namely correlated
grouping and uncorrelated grouping. On the one hand, correlated grouping
groups together adjacent antenna elements, which experience spatially
correlated fading effects. On the other hand, uncorrelated grouping maximizes
the minimum distance between antenna elements grouped together. In the case of
dual-polarized RIS, uncorrelated grouping maximizes also the minimum distance
between antenna elements grouped together having the same polarization. In
this way, antenna elements within the same group are expected to experience
uncorrelated fading effects. A graphical representation of these two families
of grouping strategies is shown in Figure 3, where different colors correspond
to different groups.

In the case of RISs built with continuous-value impedance networks,
uncorrelated grouping is expected to outperform correlated grouping. This is
because uncorrelated grouping averages out the small scale effects within each
element of the vectors $\mathbf{m}_{RI}$ and $\mathbf{m}_{IT}$. Thus, the
elements of $\mathbf{m}_{RI}$ tend to take on similar values to each other, as
well as the elements of $\mathbf{m}_{IT}$. As a result, $\mathbf{m}_{RI}$ and
$\mathbf{m}_{IT}$ tend to be aligned, satisfying more easily condition (50).
Conversely, when correlated grouping is employed, the elements in each group
experience highly correlated fading effects. Thus, the $g$-th element of
$\mathbf{m}_{RI}$ and $\mathbf{m}_{IT}$ depends on the fading experienced
locally in the $g$-th group. Since all the elements of $\mathbf{m}_{RI}$ are
likely to be different, and independent from the elements of
$\mathbf{m}_{IT}$, condition (50) is more hardly satisfied. The benefit of
adopting a grouping strategy based on the channel statistics is that it does
not change at run-time, resulting in simple hardware specifications.

###  IV-C Grouping Based on the Channel Realization: Optimal Grouping

Input: $N_{I},N_{G},\mathbf{h}_{RI},\mathbf{h}_{IT}$

Output: The optimal grouping strategy $s^{*}$

1 $i\leftarrow 0$;

2 $s_{0}$ is given by uncorrelated grouping;

3 repeat

4 $i\leftarrow i+1$;

5 From $s_{i-1}$, generate the set $\mathcal{S}$ containing all the
$\frac{N_{I}N_{G}\left(G-1\right)}{2}$ grouping strategies obtainable by
swapping two elements;

6 $s_{i}$ is given by the best grouping strategy among the ones in
$\mathcal{S}\cup s_{i-1}$;

7

8until  _$s_{i}=s_{i-1}$_ ;

return _$s^{*}\leftarrow s_{i}$_ ;

Algorithm 1 Grouping strategy optimization based on the channel realization.

Now, our goal is to find the grouping strategy that maximizes the cosine
similarity in (51) for every channel realization. In an RIS with $N_{I}$
elements grouped into $G$ groups, each having $N_{G}=\frac{N_{I}}{G}$
elements, the number of possible grouping strategies is given by

$N_{S}=\frac{1}{G!}\binom{N_{I}}{N_{G}}\binom{N_{I}-N_{G}}{N_{G}}\cdots\binom{N_{G}}{N_{G}}=\frac{1}{G!}\frac{N_{I}!}{{\left(N_{G}!\right)}^{G}}.$
(52)

Due to the high value of $N_{S}$, an exhaustive search among all the possible
$N_{S}$ grouping strategies would be too computationally expensive for medium-
high values of $N_{I}$. For instance, considering $N_{I}=32$, we have
$N_{S}>{10}^{8}$ for every $N_{G}$ non-trivial divisor of $N_{I}$. Thus, to
simplify this problem, we propose the iterative process shown in Algorithm 1.
The initial grouping strategy, denoted as $s_{0}$, is given by uncorrelated
grouping. At each iteration, we generate all the possible grouping strategies
obtainable by swapping two elements in $s_{i-1}$. Here, the operation of
swapping two elements consists in selecting two RIS elements belonging to two
different groups and assigning each of them to the group of the other. The
resulting set is denoted as $\mathcal{S}$ and its cardinality is
$\frac{N_{I}N_{G}\left(G-1\right)}{2}$. Then, $s_{i}$ is given by the grouping
strategy which maximizes (51) among the ones in $\mathcal{S}\cup s_{i-1}$.
Finally, the convergence is considered to be reached when $s_{i}=s_{i-1}$. We
refer to the grouping strategy returned by Algorithm 1 $s^{*}$ as the optimal
grouping.

##  V Numerical Results

In this section, we evaluate the performance of an RIS-aided SISO system with
different discrete RIS configurations, each characterized by its number of
elements $N_{I}$, group size $N_{G}$, grouping strategy, and number of
resolution bits. The performance is measured in terms of received signal
power. $P_{T}$ is assumed unitary and the effect of $h_{RT}$ is neglected
since it does not depend on the RIS configuration. Thus, the received signal
power is
$P_{R}=\left|\mathbf{h}_{RI}\boldsymbol{\Theta}\mathbf{h}_{IT}\right|^{2}$.

###  V-A Dataset Description

We deploy discrete RISs in two different scenarios to analyze the effect of
uncorrelated and correlated channels. In the former scenario, we assume i.i.d.
Rayleigh fading channels. Thus, we have
$\mathbf{h}_{RI}\sim\mathcal{CN}\left(\boldsymbol{0},\mathbf{I}\right)$ and
$\mathbf{h}_{IT}\sim\mathcal{CN}\left(\boldsymbol{0},\mathbf{I}\right)$. In
the latter scenario, the channels have been generated with QuaDRiGa version
2.4, a Matlab based statistical ray-tracing channel simulator [32]. To this
end, we consider a three-dimensional coordinate system $(x,y,z)$, in which the
$z$-axis represents the height above the ground in meters (m). The RIS is a
ULA composed of $N_{I}$ elements arranged along the $y$-axis. The center of
the array is located at $(0,0,20)$ and the antenna spacing is half wavelength,
with a carrier frequency $f_{c}=2.5$ GHz. The single-antenna transmitter is
located at $(50,-100,25)$ and the single-antenna receiver is randomly placed
in a circular area parallel to the $xy$ plane with radius 20 m and centered in
$(40,0,1.5)$. An urban macrocell propagation environment is simulated, with
Line-of-Sight (LoS) between the transmitter and the RIS, and with Non-Line-of-
Sight (NLoS) between the RIS and the receiver. The transmitter antenna and the
receiver antenna are omnidirectional vertically polarized. The performance of
both uni-polarized and dual-polarized RIS is investigated. In the former case,
the antenna elements of the RIS are all vertically polarized. In the latter
case, the elements are $\pm 45^{\circ}$ polarized. The channels generated with
Quadriga can be expressed as
$\widetilde{\mathbf{h}}_{RI}=\Lambda_{RI}^{-1/2}\mathbf{h}_{RI}$ and
$\widetilde{\mathbf{h}}_{IT}=\Lambda_{IT}^{-1/2}\mathbf{h}_{IT}$, where the
scalars $\Lambda_{RI}^{-1/2}$ and $\Lambda_{IT}^{-1/2}$ contain the path loss
and the shadowing, while $\mathbf{h}_{RI}$ and $\mathbf{h}_{IT}$ are the
fading effects. In order to obtain a received signal power purely dependent on
the RIS configuration, the dataset has been normalized according to

$\mathbf{h}_{RI}\leftarrow\Lambda_{RI}^{1/2}\widetilde{\mathbf{h}}_{RI},\>\mathbf{h}_{IT}\leftarrow\Lambda_{IT}^{1/2}\widetilde{\mathbf{h}}_{IT},$
(53)

such that $\left\|\mathbf{h}_{RI}\right\|^{2}=N_{I}$ and
$\left\|\mathbf{h}_{IT}\right\|^{2}=N_{I}$ for every channel realization.

For both scenarios, offline learning stages have been carried out to define
the optimal codebooks. In the case of scalar-discrete RISs with $B=1$,
$\psi_{1}$ has been obtained by averaging the outcomes of pattern search over
20 channel realizations. Otherwise, $k$-means clustering has been applied to a
training dataset composed of 100 channel realizations for scalar-discrete
RISs, and 300 channel realizations for vector-discrete RISs. Before applying
$k$-means clustering, the distributions of the optimal $\mathbf{X}_{I}$
elements deriving from the training datasets have been cleaned by removing
possible outliers. We define outliers as elements more than four scaled Median
Absolute Deviation (MAD) from the median value, with scaling factor set to
$c={\left(\sqrt{2}\text{erfc}^{-1}(1/2)\right)}^{-1}$, where
$\text{erfc}\left(x\right)$ refers to the complementary error function. With
this scaling factor, when a Gaussian distribution is cleaned, the scaled MAD
approximates its standard deviation. After obtaining the suitable discrete
codebooks for each RIS configuration, they have been used in the online
optimization processes. Finally, the average received signal power with each
RIS configuration has been computed using the Monte Carlo method.

###  V-B Impact of the Grouping Strategy

Figure 4: Average power gains of the group and fully connected architectures
over the single connected architectures in the cases of uni-polarized “UP”
RISs (on the left) and dual-polarized “DP” RISs (on the right). Correlated
grouping “CG”, uncorrelated grouping “UG”, and optimal grouping “OG” are
compared.

First of all, we investigate the impact of the grouping strategy on the
received signal power in the case of RISs based on continuous-value impedance
networks. To this end, we consider the correlated fading scenario, i.e., the
channels generated with QuaDRiGa. To quantify the increase in the received
signal power in the presence of group and fully connected RISs compared to
single connected RISs, we introduce the power gains of the group and fully
connected architectures respectively as
$\mathcal{G}^{\mathrm{Group}}=P_{R}^{\mathrm{Group}}/P_{R}^{\mathrm{Single}}$
and
$\mathcal{G}^{\mathrm{Fully}}=P_{R}^{\mathrm{Fully}}/P_{R}^{\mathrm{Single}}$.
Here, $P_{R}^{\mathrm{Single}}$, $P_{R}^{\mathrm{Group}}$, and
$P_{R}^{\mathrm{Fully}}$ are the upper bounds respectively given in (42),
(47), and (43), coinciding with the the performance of optimized RISs based on
continuous-value impedance networks [30]. The average power gains
$\mathcal{G}^{\mathrm{Group}}$ (for group size $N_{G}\in\\{2,4\\}$) and
$\mathcal{G}^{\mathrm{Fully}}$ are shown in Figure 4, where the three grouping
strategies proposed are compared. As expected, we observe that uncorrelated
grouping outperforms correlated grouping, and that optimal grouping is the
best strategy among the three proposed. More precisely, optimal grouping with
group size $N_{G}=4$ achieves the same performance as the fully connected
architecture, making clear the benefit of optimizing the grouping strategy on
a per channel realization basis.

###  V-C Scalar-Discrete RISs

Figure 5: Average received signal power versus the number of RIS elements in
the single connected architecture.

We now evaluate the received signal power in the case of scalar-discrete RISs.
In the correlated fading scenario, uni-polarized RISs are considered unless
otherwise specified. We start by analyzing the performance of the traditional
single connected architecture, i.e., with group size $N_{G}=1$. Figure 5 shows
the received signal power for different values of $N_{I}$, using from one to
four resolution bits per phase shift. In the case of i.i.d. Rayleigh fading
channels, the performance of the discrete RISs is compared with the
theoretical upper bound, defined as

$\mathrm{E}\left[P_{R}^{\mathrm{Single}}\right]=N_{I}+N_{I}\left(N_{I}-1\right)\Gamma\left(1.5\right)^{4},$
(54)

where $\Gamma\left(x\right)$ refers to the gamma function [30]. This upper
bound coincides with the performance of a single connected RIS with optimized
continuous phase shifts [30]. In the case of correlated fading channels, we
report the empirical upper bound given by continuous phase shift RISs,
optimized with the quasi-Newton method. In both scenarios, the upper bound is
approximately achieved by quantizing the phase shifts with only four
resolution bits. Thus, a further increase in the resolution only adds more
hardware complexity to the RIS, with no performance improvement. Furthermore,
we notice that with correlated fading the received signal power is higher than
with i.i.d. Rayleigh fading channels. In fact, the single connected
architecture achieves a higher power in LoS channels, as described by the
scaling laws presented in [30].

Figure 6: Average received signal power versus the number of RIS elements in
group connected architectures. In the case of correlated fading, correlated
grouping “CG”, uncorrelated grouping “UG”, and optimal grouping “OG” are
compared.

For the group and fully connected architectures, our novel approach presented
in Section III is used to maximize the received signal power. The 12 plots in
Figure 6 represent the received signal power when group connected
architectures (with group size $N_{G}\in\\{2,4,8\\}$) are used in presence of
i.i.d. Rayleigh fading channels and correlated channels. In the case of
correlated channels, the three grouping strategies proposed are compared.
Figure 7 shows the received signal power achieved with fully connected
architectures. In Figures 6 and 7, the theoretical upper bound reported in the
i.i.d. Rayleigh fading scenario is

$\mathrm{E}\left[P_{R}^{\mathrm{Group}}\right]=N_{I}N_{G}+\frac{N_{I}}{N_{G}}\left(\frac{N_{I}}{N_{G}}-1\right)\left(\frac{\Gamma\left(N_{G}+\frac{1}{2}\right)}{\Gamma\left(N_{G}\right)}\right)^{4},$
(55)

which boils down to $\mathrm{E}\left[P_{R}^{\mathrm{Fully}}\right]=N_{I}^{2}$
in the fully connected case [30]. In the correlated fading scenario, we report
the performance of the optimized continuous-value RISs found with the quasi-
Newton method.

Figure 7: Average received signal power versus the number of RIS elements in
the fully connected architecture.

Let us first compare the performance of the three grouping strategies in the
case of group connected RISs. As expected, uncorrelated grouping achieves a
higher received signal power than correlated grouping, and optimal grouping is
the best strategy among the three proposed. This holds for all the three group
sizes $N_{G}\in\\{2,4,8\\}$, and for every level of resolution considered.

Additionally, we observe that the higher the group size, the fewer resolution
bits are necessary to reach the upper bound. In particular, in the fully
connected architecture, a single resolution bit is sufficient to obtain
approximately the same performance as the optimized continuous-value RISs in
both fading scenarios, as shown in Figure 7. The rationale behind this
behavior is given by the necessary and sufficient condition for
$\boldsymbol{\Theta}$ to achieve the upper bound. More precisely, according to
[30], each block $\boldsymbol{\Theta}_{g}$ needs to be a complex symmetric
unitary matrix satisfying

$\frac{\mathbf{h}_{RI,g}^{H}}{\left\|\mathbf{h}_{RI,g}\right\|}=\boldsymbol{\Theta}_{g}\frac{\mathbf{h}_{IT,g}}{\left\|\mathbf{h}_{IT,g}\right\|},\forall
g,$ (56)

where the vectors $\mathbf{h}_{RI,g}\in\mathbb{C}^{1\times N_{G}}$ and
$\mathbf{h}_{IT,g}\in\mathbb{C}^{N_{G}\times 1}$ contain the $N_{G}$ elements
of $\mathbf{h}_{RI}$ and $\mathbf{h}_{IT}$, respectively, which are grouped
into the $g$-th group. This condition consists of an underdetermined system of
$G$ equations, where each equation has $\frac{N_{G}\left(N_{G}+1\right)}{2}$
unknowns. Thus, a higher group size $N_{G}$ implies more degrees of freedom in
(56), and fewer resolution bits are required to satisfy it. We want now to
verify that (56) is actually better satisfied when higher group sizes are
utilized. To this end, we consider the cosine similarity between
$\frac{\mathbf{h}_{RI}^{H}}{\left\|\mathbf{h}_{RI}\right\|}$ and
$\boldsymbol{\Theta}\frac{\mathbf{h}_{IT}}{\left\|\mathbf{h}_{IT}\right\|}$ to
measure to what extent these two complex vectors with unitary modulus are
aligned.

In Figure 8, we fix $N_{I}=64$ and we report the average values of this cosine
similarity for RISs with different group sizes, defined as

$\eta=\left|\left(\frac{\mathbf{h}_{RI}^{H}}{\left\|\mathbf{h}_{RI}\right\|}\right)^{H}\boldsymbol{\Theta}\frac{\mathbf{h}_{IT}}{\left\|\mathbf{h}_{IT}\right\|}\right|.$
(57)

For simplicity, we report only the cosine similarities corresponding to $B=1$
and $B=4$ resolution bits as the cases $B=2$ and $B=3$ lie between them. As
expected, higher values of $B$ correspond to higher cosine similarities.
Moreover, the curves approach the limit $\eta=1$ as the group size increases,
justifying the better performances reached with higher group sizes. In the
fully connected case ($N_{G}=64$ in Figure 8), (56) is completely satisfied
also with $B=1$. As a result, a single resolution bit per $\mathbf{X}_{I}$
element is sufficient to reach the performance upper bound in the fully
connected architecture.

Figure 8: Cosine similarity $\eta$ defined in (57), with $N_{I}=64$. In the
case of correlated fading, correlated grouping “CG”, uncorrelated grouping
“UG”, and optimal grouping “OG” are compared.

Let us now compare the performance obtained with different group sizes when
the number of RIS elements is fixed. To this end, we optimize RISs with
$N_{I}=64$ elements considering all the possible group sizes, spanning from
the single connected architecture ($N_{G}=1$) to the fully connected one
($N_{G}=64$). In Figure 9, the upper bound of the received signal power is
reported together with the power maximized with different resolutions, in the
two considered scenarios: i.i.d. Rayleigh fading channels and correlated
channels. In the case of correlated channels, we compare uni-polarized RISs
and dual-polarized RISs employing the three grouping strategies proposed.
Thus, to account for the effects of the polarization loss, the received signal
power reported in Figure 9 includes the path loss and the shadowing
contributions, and it is given by
$\widetilde{P}_{R}=\left|\widetilde{\mathbf{h}}_{RI}\boldsymbol{\Theta}\widetilde{\mathbf{h}}_{IT}\right|^{2}$.
For $N_{G}\geq 2$, we notice that the received signal power increases
monotonically with the group size, for every $B$. Furthermore, the difference
between the upper bound curve and the $B=1$ curve decreases with the group
size, becoming approximately zero in the fully connected architecture. In the
single connected case, the matrix $\boldsymbol{\Theta}$ is completely
described by its phase shifts $\theta_{n_{I}}\in\left[0,2\pi\right)$, $\forall
n_{I}=1,\ldots,N_{I}$, which can be directly quantized. Because of this
property, the curves in Figure 9 exhibit a different trend when $N_{G}=1$. In
fact, despite a four-bit resolution is not sufficient to achieve the upper
bound when $N_{G}=2$, it reaches the optimality when $N_{G}=1$. Finally, in
the presence of correlated fading, we notice that uni-polarized RISs achieve
higher received signal power than dual-polarized RISs. This is due to the
absence of polarization loss when uni-polarized RISs are employed with
vertically polarized antennas.

Figure 9: Average received signal power versus the group size. In the case of
correlated fading, uni-polarized “UP” and dual-polarized “DP” RISs with
correlated grouping “CG”, uncorrelated grouping “UG”, and optimal grouping
“OG” are compared.

Figure 10: Average received signal power versus the number of total resolution
bits. In the case of correlated fading, uni-polarized “UP” and dual-polarized
“DP” RISs with correlated grouping “CG”, uncorrelated grouping “UG”, and
optimal grouping “OG” are compared.

When $B$ bits are allocated to each $\mathbf{X}_{I}$ element, the number of
total bits employed scales with the number of $\mathbf{X}_{I}$ elements as
$B\frac{\left(N_{G}+1\right)}{2}N_{I}$. We fix now $N_{I}=64$, and we
investigate which are the RIS configurations, described by the pairs
$(N_{G},B)$, that maximize the received signal power when the number of total
bits is limited. In Figure 10, each point represents the received signal power
achievable with a specific RIS configuration, when
$B\frac{\left(N_{G}+1\right)}{2}\times 64$ total resolution bits are
available. Here, different colors represent different resolutions $B$, while
the numeric labels above the points identify the group size $N_{G}$. As
expected, in both i.i.d. Rayleigh fading and correlated fading scenarios, the
power is maximized by the fully connected architecture ($N_{G}=64$) with
$B=1$. However, to realize this RIS configuration, the number of total
resolution bits required is $32.5\times 64=2080$. Since a simpler architecture
may be needed for practical reasons, Figure 10 shows how the single connected
and group connected architectures with discrete reconfigurable impedances can
provide the compromise between performance and hardware complexity. In the
case of correlated fading, the RIS configurations designed with optimal
grouping achieve a higher received signal power than their corresponding
configurations using a grouping based on the channel statistics. Thus, we can
conclude that optimal grouping is the best grouping strategy when scalar-
discrete RISs are employed. Furthermore, among the two grouping strategies
based on the channel statistics considered, uncorrelated grouping is the
preferred one.

Figure 11: Average received signal power versus the number of RIS elements in
the vector-discrete group connected architecture. In the case of correlated
fading, correlated grouping “CG”, uncorrelated grouping “UG”, and optimal
grouping “OG” are compared.

###  V-D Vector-Discrete RISs

Let us analyze the performance of vector-discrete RISs, in comparison with
scalar-discrete RISs. To this end, we set the number of resolution bits
assigned to each block $\mathbf{X}_{I,g}$ to
$B_{V}=B\frac{N_{G}\left(N_{G}+1\right)}{2}$. This is equivalent, in terms of
total bits employed, to assigning $B$ bits to each reactance element, allowing
a comparison between scalar-discrete and vector-discrete RISs. Figure 11 shows
the performance achieved by the vector discretization strategy applied to
group connected architectures with group size $N_{G}=2$. In the correlated
fading scenario, uni-polarized RISs are considered. Comparing this figure with
the top of Figure 6, where scalar discretization was used to optimize group
connected RISs with $N_{G}=2$, we can notice a significant performance
improvement. In particular, in Figure 11, $B_{V}=9$ resolution bits per
impedance network block are sufficient to approach the performance upper
bound. In terms of total resolution bits employed, $B_{V}=9$ bits per
impedance network block are equivalent to $B=3$ bits per impedance element.
However, $B=3$ resolution bits per impedance element are not sufficient to
reach the upper bound when scalar discretization is considered, as shown in
Figure 6.

Finally, we compare the performance of RISs with different group sizes when
the same number of total bits is employed. To this end, vector discretization
has to be considered since it offers the needed flexibility in defining the
resolution. We fix the number of total bits used to $N_{I}$, which is the
minimum possible in the case of scalar-discrete RISs. Indeed, $N_{I}$ total
bits are required by the single connected architecture with $B=1$. In order to
use $N_{I}$ total bits, the feature space having
$\frac{N_{G}\left(N_{G}+1\right)}{2}$ dimensions is quantized with
$B_{V}=N_{G}$ resolution bits. Figure 12 shows the received signal power
achieved with different values of group size. Here, we can notice that higher
group sizes achieve better performance than lower group sizes. This is due to
the optimality condition (56), which has more degrees of freedom as the group
size increase. Thus, it can be more easily satisfied when higher group sizes
are considered. Interestingly, correlated grouping achieves higher received
signal power than uncorrelated grouping and optimal grouping. This is because
with correlated grouping there is a higher correlation between the optimal
reactance elements within each group $\mathbf{X}_{I,g}$. Vector-quantization
can benefit from this property, achieving better performance than in the
uncorrelated grouping and optimal grouping cases, when a low resolution is
used. In conclusion, differently from what observed for scalar-discrete RISs,
correlated grouping is the preferred grouping strategy in the case of vector-
discrete RISs used with a low number of total resolution bits.

Figure 12: Average received signal power versus the number of RIS elements for
different group sizes, with $N_{I}$ total resolution bits. In the case of
correlated fading, correlated grouping “CG”, uncorrelated grouping “UG”, and
optimal grouping “OG” are compared.

###  V-E Optimization Computational Complexity

To conclude our discussion, we compare RISs based on continuous-value
impedance networks with RISs based on discrete-value impedance networks in
terms of optimization computational complexity. On the one hand, when the
quasi-Newton method is used to optimize the continuous values of
$\mathbf{X}_{I}$, the computational complexity of each iteration is
$\mathcal{O}\left(\left(\frac{N_{I}\left(N_{G}+1\right)}{2}\right)^{2}\right)$
[30]. On the other hand, when alternating optimization is used to optimize the
discrete values of $\mathbf{X}_{I}$, we distinguish two cases: scalar-discrete
RISs and vector-discrete RISs. In the case of scalar-discrete RISs, the total
number of search times in a complete iteration of the alternating optimization
algorithm is $\frac{N_{I}\left(N_{G}+1\right)}{2}2^{B}$, as discussed in
Section III. Instead, in the case of vector-discrete RISs, the number of
search times per iteration becomes $\frac{N_{I}}{N_{G}}2^{B_{V}}$. The values
of these computational complexities are reported in Figure 13. In the case of
scalar-discrete RISs, we observe that our RIS design strategy based on
discrete-value impedance networks is beneficial also in terms of optimization
computational complexity. Even with $B=4$, the complexity is far less than the
complexity of the quasi-Newton method for medium-high numbers of RIS elements.

Figure 13: Optimization computational complexity versus the number of RIS
elements. The quasi-Newton method “QN” is compared with alternating
optimization for scalar-discrete RISs (on the left) and vector-discrete RISs
(on the right).

##  VI Conclusion

We propose a practical RIS design based on discrete-value reconfigurable
impedance networks. The proposed design strategy extends the previous research
on single connected RIS architectures with discrete phase shifts to group and
fully connected RIS architectures, recently introduced in [30]. Our strategy
is composed of two stages. Firstly, through the offline learning stage, we
build the codebook containing the possible values for the reconfigurable
impedances. Secondly, during the online deployment stage, we optimize the
reconfigurable impedances with alternating optimization, on the basis of the
designed codebook. Two different approaches are developed based on scalar
quantization and vector quantization.

We address the problem of grouping the RIS elements in the case of group
connected architectures. We describe two approaches to design the grouping
strategy: one based on the channel statistics, and the other, called optimal
grouping, based on the channel realization. Under the category of grouping
strategies based on the channel statistics, we propose two opposite solutions.
The former, correlated grouping, groups together RIS elements that are close
to each other. The latter, uncorrelated grouping, groups together RIS elements
experiencing uncorrelated fading.

We consider a single-user RIS-aided SISO system, and we assess the performance
in terms of received signal power of the single, group, and fully connected
architectures, discretized with different levels of resolution. Numerical
results show that the theoretical upper bound of the received signal power can
be reached also allocating a few resolution bits. In particular, a single
resolution bit per reconfigurable impedance is sufficient to achieve the
optimality in fully connected RISs. Furthermore, optimal grouping is the
grouping strategy that maximizes the received signal power in absence of
resolution constraints, and in the case of scalar-discrete RISs. Conversely,
correlated grouping is beneficial in the case of vector-discrete RISs, when
they are designed with low levels of resolution.

## References

  * [1] E. Basar, M. Di Renzo, J. De Rosny, M. Debbah, M.-S. Alouini, and R. Zhang, “Wireless communications through reconfigurable intelligent surfaces,” _IEEE access_ , vol. 7, pp. 116 753–116 773, 2019.
  * [2] Q. Wu and R. Zhang, “Towards smart and reconfigurable environment: Intelligent reflecting surface aided wireless network,” _IEEE Communications Magazine_ , vol. 58, no. 1, pp. 106–112, 2019.
  * [3] Y. Liu, X. Liu, X. Mu, T. Hou, J. Xu, M. Di Renzo, and N. Al-Dhahir, “Reconfigurable intelligent surfaces: Principles and opportunities,” _IEEE Communications Surveys & Tutorials_, 2021.
  * [4] Q. Wu, S. Zhang, B. Zheng, C. You, and R. Zhang, “Intelligent reflecting surface aided wireless communications: A tutorial,” _IEEE Transactions on Communications_ , 2021.
  * [5] M.-A. Badiu and J. P. Coon, “Communication through a large reflecting surface with phase errors,” _IEEE Wireless Communications Letters_ , vol. 9, no. 2, pp. 184–188, 2019.
  * [6] P. Xu, G. Chen, Z. Yang, and M. Di Renzo, “Reconfigurable intelligent surfaces-assisted communications with discrete phase shifts: How many quantization levels are required to achieve full diversity?” _IEEE Wireless Communications Letters_ , vol. 10, no. 2, pp. 358–362, 2020.
  * [7] D. Li, “Ergodic capacity of intelligent reflecting surface-assisted communication systems with phase errors,” _IEEE Communications Letters_ , vol. 24, no. 8, pp. 1646–1650, 2020.
  * [8] J. Xu, W. Xu, and A. L. Swindlehurst, “Discrete phase shift design for practical large intelligent surface communication,” in _2019 IEEE Pacific Rim Conference on Communications, Computers and Signal Processing (PACRIM)_. IEEE, 2019, pp. 1–5.
  * [9] C. You, B. Zheng, and R. Zhang, “Intelligent reflecting surface with discrete phase shifts: Channel estimation and passive beamforming,” in _ICC 2020-2020 IEEE International Conference on Communications (ICC)_. IEEE, 2020, pp. 1–6.
  * [10] S. Abeywickrama, R. Zhang, Q. Wu, and C. Yuen, “Intelligent reflecting surface: Practical phase shift model and beamforming optimization,” _IEEE Transactions on Communications_ , vol. 68, no. 9, pp. 5849–5863, 2020\.
  * [11] X. Qian, M. Di Renzo, J. Liu, A. Kammoun, and M.-S. Alouini, “Beamforming through reconfigurable intelligent surfaces in single-user MIMO systems: SNR distribution and scaling laws in the presence of channel fading and phase noise,” _IEEE wireless communications letters_ , vol. 10, no. 1, pp. 77–81, 2020.
  * [12] H. Guo, Y.-C. Liang, J. Chen, and E. G. Larsson, “Weighted sum-rate maximization for intelligent reflecting surface enhanced wireless networks,” in _2019 IEEE Global Communications Conference (GLOBECOM)_. IEEE, 2019, pp. 1–6.
  * [13] B. Di, H. Zhang, L. Song, Y. Li, Z. Han, and H. V. Poor, “Hybrid beamforming for reconfigurable intelligent surface based multi-user communications: Achievable rates with limited discrete phase shifts,” _IEEE Journal on Selected Areas in Communications_ , vol. 38, no. 8, pp. 1809–1822, 2020.
  * [14] X. Mu, Y. Liu, L. Guo, J. Lin, and N. Al-Dhahir, “Exploiting intelligent reflecting surfaces in NOMA networks: Joint beamforming optimization,” _IEEE Transactions on Wireless Communications_ , vol. 19, no. 10, pp. 6884–6898, 2020.
  * [15] M. Jung, W. Saad, M. Debbah, and C. S. Hong, “On the optimality of reconfigurable intelligent surfaces (RISs): Passive beamforming, modulation, and resource allocation,” _IEEE Transactions on Wireless Communications_ , 2021.
  * [16] M.-M. Zhao, Q. Wu, M.-J. Zhao, and R. Zhang, “Intelligent reflecting surface enhanced wireless network: Two-timescale beamforming optimization,” _IEEE Transactions on Wireless Communications_ , 2020.
  * [17] G. Yang, X. Xu, and Y.-C. Liang, “Intelligent reflecting surface assisted non-orthogonal multiple access,” in _2020 IEEE Wireless Communications and Networking Conference (WCNC)_. IEEE, 2020, pp. 1–6.
  * [18] Q. Wu and R. Zhang, “Beamforming optimization for wireless network aided by intelligent reflecting surface with discrete phase shifts,” _IEEE Transactions on Communications_ , vol. 68, no. 3, pp. 1838–1851, 2019.
  * [19] M. Fu, Y. Zhou, Y. Shi, and K. B. Letaief, “Reconfigurable intelligent surface empowered downlink non-orthogonal multiple access,” _IEEE Transactions on Communications_ , 2021.
  * [20] B. Zheng, Q. Wu, and R. Zhang, “Intelligent reflecting surface-assisted multiple access with user pairing: NOMA or OMA?” _IEEE Communications Letters_ , vol. 24, no. 4, pp. 753–757, 2020.
  * [21] C. Huang, G. C. Alexandropoulos, A. Zappone, M. Debbah, and C. Yuen, “Energy efficient multi-user MISO communication using low resolution large intelligent surfaces,” in _2018 IEEE Globecom Workshops (GC Wkshps)_. IEEE, 2018, pp. 1–6.
  * [22] Y. Han, W. Tang, S. Jin, C.-K. Wen, and X. Ma, “Large intelligent surface-assisted wireless communication exploiting statistical CSI,” _IEEE Transactions on Vehicular Technology_ , vol. 68, no. 8, pp. 8238–8242, 2019.
  * [23] X. Ma, Z. Chen, Y. Chi, W. Chen, L. Du, and Z. Li, “Channel Estimation for Intelligent Reflecting Surface Enabled Terahertz MIMO Systems,” in _2020 IEEE International Conference on Communications Workshops (ICC Workshops)_. IEEE, 2020, pp. 1–6.
  * [24] X. Ma, Z. Chen, W. Chen, Y. Chi, Z. Li, C. Han, and Q. Wen, “Intelligent reflecting surface enhanced indoor terahertz communication systems,” _Nano Communication Networks_ , vol. 24, p. 100284, 2020.
  * [25] W. Chen, X. Ma, Z. Li, and N. Kuang, “Sum-rate maximization for intelligent reflecting surface based terahertz communication systems,” in _2019 IEEE/CIC International Conference on Communications Workshops in China (ICCC Workshops)_. IEEE, 2019, pp. 153–157.
  * [26] Y. Lu and L. Dai, “Reconfigurable Intelligent Surface Based Hybrid Precoding for THz Communications,” _arXiv preprint arXiv:2012.06261_ , 2020.
  * [27] Q. Wu and R. Zhang, “Joint active and passive beamforming optimization for intelligent reflecting surface assisted SWIPT under QoS constraints,” _IEEE Journal on Selected Areas in Communications_ , vol. 38, no. 8, pp. 1735–1748, 2020.
  * [28] Y. Zhao, B. Clerckx, and Z. Feng, “Intelligent reflecting surface-aided swipt: Joint waveform, active and passive beamforming design,” _arXiv preprint arXiv:2012.05646_ , 2020.
  * [29] X. Yu, V. Jamali, D. Xu, D. W. K. Ng, and R. Schober, “Smart and Reconfigurable Wireless Communications: From IRS Modeling to Algorithm Design,” _arXiv preprint arXiv:2103.07046_ , 2021.
  * [30] S. Shen, B. Clerckx, and R. Murch, “Modeling and Architecture Design of Reconfigurable Intelligent Surfaces Using Scattering Parameter Network Analysis,” _IEEE Transactions on Wireless Communications_ , pp. 1–1, 2021\.
  * [31] D. M. Pozar, _Microwave engineering_. John wiley & sons, 2011.
  * [32] S. Jaeckel, L. Raschkowski, K. Börner, and L. Thiele, “QuaDRiGa: A 3-D multi-cell channel model with time evolution for enabling virtual field trials,” _IEEE Transactions on Antennas and Propagation_ , vol. 62, no. 6, pp. 3242–3256, 2014.