# Localized and Distributed Beyond Diagonal Reconfigurable Intelligent
Surfaces with Lossy Interconnections: Modeling and Optimization

Matteo Nerini, , Golsa Ghiaasi, , 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).G. Ghiaasi is with Silicon Austria Labs (SAL), Graz
A-8010, Austria (e-mail: golsa.ghiaasi@silicon-austria.com).

###### Abstract

Reconfigurable intelligent surface (RIS) is a key technology to control the
communication environment in future wireless networks. Recently, beyond
diagonal RIS (BD-RIS) emerged as a generalization of RIS achieving larger
coverage through additional tunable impedance components interconnecting the
RIS elements. However, conventional RIS and BD-RIS can effectively serve only
users in their proximity, resulting in limited coverage. To overcome this
limitation, in this paper, we investigate distributed RIS, whose elements are
distributed over a wide region, in opposition to localized RIS commonly
considered in the literature. The scaling laws of distributed BD-RIS reveal
that it offers significant gains over distributed conventional RIS and
localized BD-RIS, enabled by its interconnections allowing signal propagation
within the BD-RIS. To assess the practical performance of distributed BD-RIS,
we model and optimize BD-RIS with lossy interconnections through transmission
line theory. Our model accounts for phase changes and losses over the BD-RIS
interconnections arising when the interconnection lengths are not much smaller
than the wavelength. Numerical results show that the performance of localized
BD-RIS is only slightly impacted by losses, given the short interconnection
lengths. Besides, distributed BD-RIS can achieve orders of magnitude of gains
over conventional RIS, even in the presence of low losses.

###### Index Terms:

Beyond diagonal reconfigurable intelligent surface (BD-RIS), losses,
optimization, transmission line.

##  I Introduction

Reconfigurable intelligent surface (RIS) is a promising technology allowing
the control of the propagation environment in wireless networks [1, 2]. RIS is
a technology that involves deploying arrays consisting of numerous
individually controllable elements. The reflection coefficients of these
elements can be adjusted in real-time to manipulate electromagnetic waves,
allowing for dynamic control over the wireless communication environment and
the realization of smart radio environments [3]. RIS presents significant
appeal owing to its capacity to optimize the signal propagation in a cost-
effective manner and with minimal power consumption.

RIS has been used to improve wireless communication systems under several
aspects, such as to minimize the transmit power [4], to maximize the
achievable rate [5], to maximize the weighted sum-rate [6], in narrow-band as
well as wide-band systems [7]. RIS has been also applied to further improve
the data rate by modulating and encoding data into the reconfigurable elements
[8, 9]. In addition to communication systems, RISs have been also employed to
enhance the efficiency of wireless power transfer (WPT) systems [10],
simultaneous wireless information and power transfer (SWIPT) systems [11], and
localization systems [12]. The practical problem of channel estimation in RIS-
aided systems has been tackled in [13]. Besides, the performance of practical
RIS based on discrete reflection coefficients has been analyzed in [14].
Finally, prototypes of RIS have been presented in [15, 16, 17].

Conventionally, RIS has been realized by reconfiguring each RIS element
through a tunable load, leading to a diagonal phase shift matrix with limited
flexibility. To improve the limited flexibility of conventional RIS, beyond
diagonal RIS (BD-RIS) has been recently proposed [18]. The key novelty of BD-
RIS is the presence of tunable impedance components interconnecting the RIS
elements to each other. Depending on the topology of such interconnections,
multiple BD-RIS architectures have been proposed, such as group-/fully-
connected RIS [19, 20], forest-/tree-connected RIS [21], and non-diagonal RIS
based on dynamic interconnections [22, 23]. The improved flexibility of BD-RIS
enables higher performance gains than conventional RIS at the cost of a
moderate increase in the circuit complexity [24] and channel estimation
overhead [25]. Recent works on BD-RIS optimization have shown the superiority
of BD-RIS in multi-antenna [26, 27, 28], multi-user [29, 30], and mobile edge
computing (MEC) [31] systems. Furthermore, BD-RIS proved to be especially
beneficial in the presence of mutual coupling [32, 33].

Beyond achieving higher performance gains, BD-RIS working in the so-called
hybrid mode enables both the reflection and transmission of the incident
signal, allowing full-space coverage [34]. To further improve the performance
while preserving full-space coverage, multi-sector BD-RIS has been introduced,
where the elements are divided into multiple sectors, each covering a narrow
region of space [35]. BD-RIS with reflective and transmissive capabilities has
proved to improve the capacity and sensing precision in dual-function radar-
communication (DFRC) systems [36] and to improve the sum rate and enlarge the
coverage in rate splitting multiple access (RSMA) systems [37].

Although BD-RIS working in hybrid mode and multi-sector BD-RIS have shown
great coverage improvements [34]-[37], an RIS effectively enhances the link
between a transmitter and a receiver only when placed close to one of them
[2]. For this reason, an RIS can be particularly beneficial only to users in
its proximity, yielding reduced coverage capabilities. This reduced coverage
limitation arises from the fact that RIS has been commonly regarded as a
localized array of scattering elements, here denoted as localized RIS. In this
study, we overcome this limitation by investigating RIS made of a distributed
array of elements, in which the inter-element distance can be much longer than
the wavelength, referred to as distributed RIS. Our focus will extend
specifically to distributed BD-RIS, as they are expected to achieve
particularly enhanced performance due to the presence of tunable impedance
components interconnecting the RIS elements. These interconnections have the
potential to effectively guide the electromagnetic (EM) signal toward the
receiver by enabling its propagation within the RIS. Distributed BD-RIS could
be implemented by integrating the RIS elements into long cables, forming long
linear arrays. Specifically, distributed BD-RIS could drive the EM signal from
a fixed base station (BS) to the mobile user equipments (UEs) within a
coverage area proportional to their array length in highly obstructed indoor
environments, e.g., smart factories, as well as in urban outdoor settings.

The investigation of distributed BD-RIS requires modeling the interconnections
in the BD-RIS architecture by accounting for their losses, which is a
challenge overlooked in previous literature on BD-RIS. In previous works
[18]-[37], BD-RIS has been modeled through a lumped-element circuit model,
valid when the physical dimensions of the RIS reconfigurable impedance network
are much smaller than the wavelength. According to the lumped-element model,
voltages and currents are constant along the interconnections, which
simplifies the BD-RIS modeling and optimization. However, the length of the
interconnections between the RIS elements in BD-RIS can be a considerable
fraction of the wavelength in localized BD-RIS, or many wavelengths in
distributed BD-RIS. For example, even when the inter-element distance is half-
wavelength, the interconnections are at least half-wavelength long. In the
case the interconnections are not much smaller than the wavelength, two
critical effects take place. First, voltages and currents vary in phase along
the interconnections. Second, voltages and currents may vary in magnitude
along the interconnections due to losses. Thus, it is crucial to develop a BD-
RIS model accounting for these two effects, as they may lead to undesired
changes of phase and dissipation of power within the BD-RIS circuit. To this
end, in this study, we model and optimize localized and distributed BD-RIS by
characterizing its interconnections through transmission line theory. Our
contributions are summarized as follows.

First, we propose the concept of distributed RIS, in which the antenna
elements are not localized in a specific site but distributed over a wide
region. We analyze their scaling laws and quantify the gain of distributed RIS
over localized RIS, and of distributed BD-RIS over distributed conventional
RIS. The derived scaling laws show that lossless distributed BD-RIS can offer
substantial gains over distributed conventional RIS and localized BD-RIS, as
high as several orders of magnitude. These gains are enabled by the
interconnections present in BD-RIS, which allow the guided propagation of the
EM signal within the BD-RIS.

Second, we model BD-RIS (localized and distributed) with lossy
interconnections by using transmission line theory. Specifically, we model the
interconnections present in the BD-RIS circuit topology as tunable impedance
components in series with lossy transmission lines. Thus, we derive the
expressions of the elements of the BD-RIS admittance matrix as functions of
the tunable impedance components and the transmission line parameters.

Third, since it is difficult to derive engineering insights from the derived
BD-RIS model, we obtain three simplified models. First, by making specific
assumptions on the interconnection lengths, we show how losses impact the BD-
RIS admittance matrix elements. Second, by assuming lossless interconnections,
we illustrate how the presence of long interconnections affects the lossless
BD-RIS model. Third, by assuming lossless interconnections with specific
lengths, it is shown how the derived model boils down to the BD-RIS model
widely considered in previous literature.

Fourth, we optimize lossy BD-RIS based on the proposed models and assess its
performance, compared with conventional RIS. Numerical results show that the
performance of localized BD-RIS is only slightly impacted by losses given the
short interconnection lengths. Besides, the performance of distributed RIS,
even with losses, can achieve orders of magnitude of gains over conventional
RIS.

Organization: In Section II, we introduce the localized and distributed BD-
RIS-aided system model. In Section III, we derive and compare the scaling laws
of localized and distributed BD-RIS. In Section IV, we model BD-RIS with lossy
interconnections through transmission line theory. In Section V, we derive
three simplified models for lossy BD-RIS. In Section VI, we optimize BD-RIS
characterized with the proposed models. In Section VII, we present the
numerical results. Finally, Section VIII concludes this work.

Notation: Vectors and matrices are denoted with bold lower and bold upper
letters, respectively. Scalars are represented with letters not in bold font.
$\Re\\{a\\}$, $\Im\\{a\\}$, and $|a|$ refer to the real part, the imaginary
part, and the absolute value of a complex scalar $a$, respectively.
$[\mathbf{a}]_{i}$ and $\|\mathbf{a}\|$ refer to the $i$th element and
$l_{2}$-norm of a vector $\mathbf{a}$, respectively. $\mathbf{A}^{T}$,
$\mathbf{A}^{H}$, and $[\mathbf{A}]_{i,j}$ refer to the transpose, conjugate
transpose, and $(i,j)$th element of a matrix $\mathbf{A}$, respectively.
$\mathbb{R}$, $\mathbb{R}_{+}$, $\mathbb{R}_{*}$, and $\mathbb{C}$ denote the
real, positive real, non-zero real, and complex number sets, respectively.
$j=\sqrt{-1}$ denotes the imaginary unit. $\mathbf{0}$ and $\mathbf{I}$ denote
an all-zero matrix and an identity matrix with appropriate dimensions,
respectively. $\mathcal{CN}(\mathbf{0},\mathbf{I})$ 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$(a_{1},\ldots,a_{N})$ refers to a diagonal matrix with
diagonal elements being $a_{1},\ldots,a_{N}$, and diag$(\mathbf{a})$ refers to
a diagonal matrix with diagonal elements being the vector $\mathbf{a}$.

##  II Localized and Distributed BD-RIS-Aided System Model

Consider a multiple-input multiple-output (MIMO) system between an
$N_{T}$-antenna transmitter and an $N_{R}$-antenna receiver aided by an
$N$-element BD-RIS. The $N$ elements of the BD-RIS are connected to a $N$-port
reconfigurable impedance network, with scattering matrix
$\boldsymbol{\Theta}\in\mathbb{C}^{N\times N}$. Given the matrix
$\boldsymbol{\Theta}$, the wireless channel
$\mathbf{H}\in\mathbb{C}^{N_{R}\times N_{T}}$ writes as

$\mathbf{H}=\mathbf{H}_{RT}+\mathbf{H}_{R}\boldsymbol{\Theta}\mathbf{H}_{T},$
(1)

where $\mathbf{H}_{RT}\in\mathbb{C}^{N_{R}\times N_{T}}$,
$\mathbf{H}_{R}\in\mathbb{C}^{N_{R}\times N}$, and
$\mathbf{H}_{T}\in\mathbb{C}^{N\times N_{T}}$ refer to the channels from
transmitter to receiver, RIS to receiver, and transmitter to RIS, respectively
[19].

In related literature, an RIS has been regarded as an antenna array with an
inter-element distance comparable with the wavelength. We refer to this type
of RIS as localized RIS, as its elements are all localized in a specific site.
With a localized RIS, the channels $\mathbf{H}_{R}$ and $\mathbf{H}_{T}$ have
been commonly modeled as
$\mathbf{H}_{R}^{\text{Loc}}=\sqrt{\rho_{R}}\widetilde{\mathbf{H}}_{R}$ and
$\mathbf{H}_{T}^{\text{Loc}}=\sqrt{\rho_{T}}\widetilde{\mathbf{H}}_{T}$, where
$\rho_{R}$ and $\rho_{T}$ are the path-gains while
$\widetilde{\mathbf{H}}_{R}\in\mathbb{C}^{N_{R}\times N}$ and
$\widetilde{\mathbf{H}}_{T}\in\mathbb{C}^{N\times N_{T}}$ are the small-scale
fading effects. Opposed to localized RISs, we investigate in this study the
performance of distributed RISs, i.e., RISs whose elements are distributed
over a wide region, with an inter-element distance that can be much longer
than the wavelength. In the presence of a distributed RIS, the channel
$\mathbf{H}_{R}$ is expressed as
$\mathbf{H}_{R}^{\text{Dis}}=\widetilde{\mathbf{H}}_{R}\mathbf{R}_{R}^{1/2}$,
where $\mathbf{R}_{R}=\text{diag}(\boldsymbol{\rho}_{R})$, with
$\boldsymbol{\rho}_{R}\in\mathbb{R}^{N\times 1}$ introduced such that
$[\boldsymbol{\rho}_{R}]_{n}$ is the path-gain from the $n$th RIS element to
the receiver. Similarly, we have
$\mathbf{H}_{T}^{\text{Dis}}=\mathbf{R}_{T}^{1/2}\widetilde{\mathbf{H}}_{T}$,
where $\mathbf{R}_{T}=\text{diag}(\boldsymbol{\rho}_{T})$, with
$\boldsymbol{\rho}_{T}\in\mathbb{R}^{N\times 1}$ introduced such that
$[\boldsymbol{\rho}_{T}]_{n}$ is the path-gain from the transmitter to the
$n$th RIS element.

In this BD-RIS-aided system, we denote the transmitted signal as
$\mathbf{x}=\mathbf{w}s\in\mathbb{C}^{N_{T}\times 1}$, where
$\mathbf{w}\in\mathbb{C}^{N_{T}\times 1}$ is the precoding vector subject to
$\|\mathbf{w}\|=1$, and $s\in\mathbb{C}$ is the transmitted symbol with
average power $P_{T}=\text{E}[|s|^{2}]$. Denoting the received signal as
$\mathbf{y}\in\mathbb{C}^{N_{R}\times 1}$, we have
$\mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{n}$, where
$\mathbf{n}\in\mathbb{C}^{N_{R}\times 1}$ is the additive white Gaussian noise
(AWGN) at the receiver with power $\sigma_{n}^{2}$. By using a combining
vector $\mathbf{g}\in\mathbb{C}^{1\times N_{R}}$ subject to
$\|\mathbf{g}\|=1$, the signal used for detection
$\hat{s}=\mathbf{g}\mathbf{y}$ can be expressed as

$\hat{s}=\mathbf{g}\left(\mathbf{H}_{RT}+\mathbf{H}_{R}\boldsymbol{\Theta}\mathbf{H}_{T}\right)\mathbf{w}s+\tilde{n},$
(2)

where $\tilde{n}=\mathbf{g}\mathbf{n}$ is the AWGN with power
$\sigma_{n}^{2}$. Thus, when reconfiguring the BD-RIS, $\boldsymbol{\Theta}$
is optimized jointly with $\mathbf{g}$ and $\mathbf{w}$ to maximize the
received signal power, given by

$P_{R}=P_{T}\left|\mathbf{g}\left(\mathbf{H}_{RT}+\mathbf{H}_{R}\boldsymbol{\Theta}\mathbf{H}_{T}\right)\mathbf{w}\right|^{2}.$
(3)

In the following, we analyze and compare the received signal power scaling
laws of localized and distributed BD-RIS.

##  III Scaling Laws

To compare the fundamental limits of localized and distributed BD-RIS, we
derive their received signal power scaling laws. For simplicity, we consider a
single-input single-output (SISO) system, i.e., $N_{R}=1$ and $N_{T}=1$, with
obstructed direct channel, i.e., $\mathbf{H}_{RT}=\mathbf{0}$, and lossless
BD-RIS, i.e., with unitary $\boldsymbol{\Theta}$. With no loss of generality,
we assume transmit power $P_{T}=1$, such that the received signal power writes
as $P_{R}=|\mathbf{h}_{R}\boldsymbol{\Theta}\mathbf{h}_{T}|^{2}$. We model the
path-gain of the channels $\mathbf{h}_{R}$ and $\mathbf{h}_{T}$ through the
distance-dependent model and their small-scale fading as independent and
identically distributed (i.i.d.) Rayleigh distributed. Specifically, for
localized BD-RIS, we have
$\mathbf{h}_{i}^{\text{Loc}}\sim\mathcal{CN}(\mathbf{0},\rho_{i}\mathbf{I})$,
with $\rho_{i}=C_{0}d_{i}^{-a}$, for $i\in\\{R,T\\}$, where $C_{0}$ refers to
the path-gain at the reference distance 1 m, $a$ is the path-loss exponent,
and $d_{R}$ (resp. $d_{T}$) is the distance between the RIS and the receiver
(resp. the transmitter). Similarly, for distributed BD-RIS, we have
$\mathbf{h}_{i}^{\text{Dis}}\sim\mathcal{CN}(\mathbf{0},\mathbf{R}_{i})$, with
$\mathbf{R}_{i}=\text{diag}(\boldsymbol{\rho}_{i})$,
$[\boldsymbol{\rho}_{i}]_{n}=C_{0}[\mathbf{d}_{i}]_{n}^{-a}$, for
$i\in\\{R,T\\}$, where $[\mathbf{d}_{R}]_{n}$ (resp. $[\mathbf{d}_{T}]_{n}$)
is the distance between the $n$th RIS element and the receiver (resp. the
transmitter).

In the following, we study the localized and distributed versions of single-
and fully-connected RIS, which are the least and the most complex BD-RIS
architectures, respectively. Single-connected RIS is the conventional RIS
characterized by a unitary and diagonal scattering matrix, i.e.,
$\boldsymbol{\Theta}=\text{diag}(e^{j\theta_{1}},\ldots,e^{j\theta_{N}})$,
with $\theta_{n}\in[0,2\pi)$, for $n=1,\ldots,N$. Besides, fully-connected RIS
is the BD-RIS architecture with the highest flexibility as it is characterized
by an arbitrary symmetric and unitary scattering matrix [19].

For localized single-connected RIS, it is possible to optimize the phase
shifts $\theta_{n}$ to obtain a received signal power given by

$P_{R}^{\text{Loc-
SC}}=\left(\sum_{n=1}^{N}\left|\left[\mathbf{h}_{R}^{\text{Loc}}\right]_{n}\right|\left|\left[\mathbf{h}_{T}^{\text{Loc}}\right]_{n}\right|\right)^{2}.$
(4)

Furthermore, the expected value of (4) can be derived by exploiting the i.i.d.
channels assumption, and that
$\text{E}[|[\mathbf{h}_{i}^{\text{Loc}}]_{n}|^{2}]=\rho_{i}$ and
$\text{E}[|[\mathbf{h}_{i}^{\text{Loc}}]_{n}|]=\frac{\sqrt{\pi}}{2}\sqrt{\rho_{i}}$,
for $i\in\\{R,T\\}$ and $n=1,\ldots,N$, because of the moments of the chi
distribution with $2$ degrees of freedom. Thus, it is possible to show that

$\displaystyle\text{E}\left[P_{R}^{\text{Loc-SC}}\right]$
$\displaystyle=\left(N+\frac{\pi^{2}}{16}N\left(N-1\right)\right)C_{0}^{2}d_{R}^{-a}d_{T}^{-a},$
(5)

giving the scaling law of localized single-connected RIS [19].

In the case of localized fully-connected RIS, it has been shown that it is
always possible to optimize the RIS to achieve a maximum received signal power
given by

$P_{R}^{\text{Loc-
FC}}=\left\|\mathbf{h}_{R}^{\text{Loc}}\right\|^{2}\left\|\mathbf{h}_{T}^{\text{Loc}}\right\|^{2},$
(6)

whose expected value writes as

$\displaystyle\text{E}\left[P_{R}^{\text{Loc-FC}}\right]$
$\displaystyle=N^{2}C_{0}^{2}d_{R}^{-a}d_{T}^{-a},$ (7)

following the i.i.d. channels assumption and the moments of the chi
distribution with $2$ degrees of freedom [19].

For distributed single-connected RIS, the achievable received signal power
writes as

$P_{R}^{\text{Dis-
SC}}=\left(\sum_{n=1}^{N}\left|\left[\mathbf{h}_{R}^{\text{Dis}}\right]_{n}\right|\left|\left[\mathbf{h}_{T}^{\text{Dis}}\right]_{n}\right|\right)^{2},$
(8)

similarly to (4). By exploiting the i.i.d. channels assumption, and that
$\text{E}[|[\mathbf{h}_{i}^{\text{Dis}}]_{n}|^{2}]=[\boldsymbol{\rho}_{i}]_{n}$
and
$\text{E}[|[\mathbf{h}_{i}^{\text{Dis}}]_{n}|]=\frac{\sqrt{\pi}}{2}\sqrt{[\boldsymbol{\rho}_{i}]_{n}}$,
for $i\in\\{R,T\\}$ and $n=1,\ldots,N$, it is possible to show that

$\text{E}\left[P_{R}^{\text{Dis-
SC}}\right]=C_{0}^{2}\left(\sum_{n=1}^{N}\left(\left[\mathbf{d}_{R}\right]_{n}\left[\mathbf{d}_{T}\right]_{n}\right)^{-a}\right.\\\
\left.+\frac{\pi^{2}}{16}\sum_{n\neq
m}\left(\left[\mathbf{d}_{R}\right]_{n}\left[\mathbf{d}_{T}\right]_{n}\left[\mathbf{d}_{R}\right]_{m}\left[\mathbf{d}_{T}\right]_{m}\right)^{-\frac{a}{2}}\right),$
(9)

giving the scaling law of distributed single-connected RIS.

Finally, in the case of distributed fully-connected RIS, the achievable
received signal power is

$P_{R}^{\text{Dis-
FC}}=\left\|\mathbf{h}_{R}^{\text{Dis}}\right\|^{2}\left\|\mathbf{h}_{T}^{\text{Dis}}\right\|^{2},$
(10)

with expected value

$\displaystyle\text{E}\left[P_{R}^{\text{Dis-FC}}\right]$
$\displaystyle=C_{0}^{2}\sum_{n=1}^{N}\left[\mathbf{d}_{R}\right]_{n}^{-a}\sum_{n=1}^{N}\left[\mathbf{d}_{T}\right]_{n}^{-a},$
(11)

following again the i.i.d. channels assumption and the moments of the chi
distribution with $2$ degrees of freedom 111Under line-of-sight (LoS) small-
scale fading, i.e.,
$[\mathbf{h}_{i}^{\text{Loc}}]_{n}=\sqrt{\rho_{i}}e^{j\phi_{i,n}}$ and
$[\mathbf{h}_{i}^{\text{Dis}}]_{n}=\sqrt{[\boldsymbol{\rho}_{i}]_{n}}e^{j\psi_{i,n}}$,
for $i\in\\{R,T\\}$ and $n=1,\ldots,N$, it is possible to prove that
$\text{E}[P_{R}^{\text{Loc-FC}}]$ and $\text{E}[P_{R}^{\text{Dis-FC}}]$ remain
as in (7) and (11), $\text{E}[P_{R}^{\text{Loc-
SC}}]=\text{E}[P_{R}^{\text{Loc-FC}}]$, and $\text{E}[P_{R}^{\text{Dis-
SC}}]=C_{0}^{2}(\sum_{n}([\mathbf{d}_{R}]_{n}[\mathbf{d}_{T}]_{n})^{-a/2})^{2}$.
Given the similarity between the scaling laws under Rayleigh and LoS fading,
the following insights on localized and distributed RIS obtained considering
Rayleigh fading are also valid under more general Rician fading..

In the following, we compare the four scaling laws derived in (5), (7), (9),
and (11) by analyzing four gains: i) the gain of fully- over single-connected
localized RIS, ii) the gain of fully- over single-connected distributed RIS,
iii) the gain of distributed over localized single-connected RIS, and iv) the
gain of distributed over localized fully-connected RIS. First, we carry out a
theoretical discussion valid for any values of the distances $d_{R}$, $d_{T}$,
$\mathbf{d}_{R}$, and $\mathbf{d}_{T}$. Second, to numerically quantify these
gains, we consider a three-dimensional coordinate system where the transmitter
and receiver are located at $(0,0,0)$ and $(20,0,0)$ in meters (m),
respectively. In the case of localized RIS, the RIS is located in $(20,0,2)$,
as shown in Fig. 1. In the case of distributed RIS, the RIS elements are
spread out over a long line. Specifically, the RIS is a uniform linear array
(ULA) with elements uniformly placed between $(0,0,2)$ and $(40,0,2)$, as
shown in Fig. 1.

Figure 1: (a) Localized and (b) distributed RIS-aided communication system.

###  III-A Gain of Fully- over Single-Connected Localized RIS

The gain of fully-connected over single-connected localized RIS, defined as
$\mathcal{G}^{\text{Loc}}=\text{E}[P_{R}^{\text{Loc-
FC}}]/\text{E}[P_{R}^{\text{Loc-SC}}]$, writes as

$\mathcal{G}^{\text{Loc}}=\frac{N}{1+\frac{\pi^{2}}{16}\left(N-1\right)},$
(12)

following (5) and (7), in agreement with [19]. Remarkably, we observe that
$\mathcal{G}^{\text{Loc}}$, which depends only on $N$, is lower bounded by
$\mathcal{G}^{\text{Loc}}\geq 1$ and upper bounded by
$\mathcal{G}^{\text{Loc}}<\frac{16}{\pi^{2}}\approx 1.62$. Thus, localized
fully-connected RIS is beneficial over localized single-connected RIS,
enabling a gain of at most $62\%$ in a SISO system [19].

###  III-B Gain of Fully- over Single-Connected Distributed RIS

We define the gain of fully-connected over single-connected distributed RIS as
$\mathcal{G}^{\text{Dis}}=\text{E}[P_{R}^{\text{Dis-
FC}}]/\text{E}[P_{R}^{\text{Dis-SC}}]$. By noticing that (9) is upper bounded
by

$\displaystyle\text{E}\left[P_{R}^{\text{Dis-SC}}\right]$
$\displaystyle<C_{0}^{2}\left(\sum_{n=1}^{N}\left(\left[\mathbf{d}_{R}\right]_{n}\left[\mathbf{d}_{T}\right]_{n}\right)^{-\frac{a}{2}}\right)^{2},$
(13)

we obtain that

$\displaystyle\mathcal{G}^{\text{Dis}}$
$\displaystyle>\frac{\sum_{n=1}^{N}\left[\mathbf{d}_{R}\right]_{n}^{-a}\sum_{n=1}^{N}\left[\mathbf{d}_{T}\right]_{n}^{-a}}{\left(\sum_{n=1}^{N}\left(\left[\mathbf{d}_{R}\right]_{n}\left[\mathbf{d}_{T}\right]_{n}\right)^{-\frac{a}{2}}\right)^{2}}$
(14)
$\displaystyle=\left(\frac{M_{-\frac{a}{2}}\left(\mathbf{d}_{R}\odot\mathbf{d}_{T}\right)}{M_{-a}\left(\mathbf{d}_{R}\right)M_{-a}\left(\mathbf{d}_{T}\right)}\right)^{a},$
(15)

where
$M_{p}(\boldsymbol{\xi})=(\sum_{n=1}^{N}[\boldsymbol{\xi}]_{n}^{p}/N)^{1/p}$
denotes the generalized mean of the elements of
$\boldsymbol{\xi}\in\mathbb{R}_{+}^{N\times 1}$ with exponent
$p\in\mathbb{R}_{*}$, and $\odot$ denotes the Hadamard product. Thus, we have
$\mathcal{G}^{\text{Dis}}\geq 1$ from the Cauchy-Schwarz inequality.
Furthermore, since
$\text{min}(\boldsymbol{\xi})<M_{-p}(\boldsymbol{\xi})<\sqrt[p]{N}\text{min}(\boldsymbol{\xi})$,
we have

$\displaystyle\mathcal{G}^{\text{Dis}}$
$\displaystyle>\left(\frac{\text{min}\left(\mathbf{d}_{R}\odot\mathbf{d}_{T}\right)}{\sqrt[a]{N^{2}}\text{min}\left(\mathbf{d}_{R}\right)\text{min}\left(\mathbf{d}_{T}\right)}\right)^{a},$
(16)

indicating that $\mathcal{G}^{\text{Dis}}$ increases exponentially with $a$
and can be significantly high when both
$\text{min}\left(\mathbf{d}_{R}\right)$ and
$\text{min}\left(\mathbf{d}_{T}\right)$ are reduced, i.e., there is at least
an RIS element close to the receiver and another one close to the transmitter.

We numerically quantify $\mathcal{G}^{\text{Dis}}$ in Fig. 2(a), where we
consider the system represented in Fig. 1 with different values of path-loss
exponent $a$ and number of RIS elements $N$. From Fig. 2(a), we observe that
$\mathcal{G}^{\text{Dis}}$ can be as high as several orders of magnitude,
differently from $\mathcal{G}^{\text{Loc}}$, which is limited to $1.62$. Fig.
2(a) confirms that $\mathcal{G}^{\text{Dis}}$ grows exponentially with $a$,
and shows that the gain is only slightly affected by $N$.

Besides, in Fig. 3(a), we report the gain $\mathcal{G}^{\text{Dis}}$ for
different locations of the receiver, located in $(x,y,0)$, where
$x\in[-10,70]$ and $y\in[-40,40]$, with $a=4$ and $N=64$. From Fig. 3(a), we
observe that $\mathcal{G}^{\text{Dis}}$ is particularly high when the receiver
is close to the distributed RIS and at the same time far from the transmitter.
As visible from $\eqref{eq:G-Dis-LB}$, $\mathcal{G}^{\text{Dis}}$ assumes high
values when both the transmitter and the receiver are close to the distributed
RIS. Thus, the high gains in Fig. 3(a) can be observed also for different
locations of the transmitter, as long as it is located in proximity to an RIS
element. Conversely, $\mathcal{G}^{\text{Dis}}$ decreases, approaching $1$,
when both the transmitter and the receiver are located far from all the RIS
elements.

(a) Gain of fully- over single-connected distributed RIS
$\mathcal{G}^{\text{Dis}}$.

(b) Gain of distributed over localized single-connected RIS
$\mathcal{G}^{\text{SC}}$.

(c) Gain of distributed over localized fully-connected RIS
$\mathcal{G}^{\text{FC}}$.

Figure 2: (a) $\mathcal{G}^{\text{Dis}}$, (b) $\mathcal{G}^{\text{SC}}$, and
(c) $\mathcal{G}^{\text{FC}}$ for different values of path-loss exponent and
number of RIS elements.

(a) Gain of fully- over single-connected distributed RIS
$\mathcal{G}^{\text{Dis}}$.

(b) Gain of distributed over localized single-connected RIS
$\mathcal{G}^{\text{SC}}$.

(c) Gain of distributed over localized fully-connected RIS
$\mathcal{G}^{\text{FC}}$.

Figure 3: (a) $\mathcal{G}^{\text{Dis}}$, (b) $\mathcal{G}^{\text{SC}}$, and
(c) $\mathcal{G}^{\text{FC}}$ for different locations of the receiver.

###  III-C Gain of Distributed over Localized Single-Connected RIS

Consider now the gain of distributed over localized single-connected RIS,
defined as $\mathcal{G}^{\text{SC}}=\text{E}[P_{R}^{\text{Dis-
SC}}]/\text{E}[P_{R}^{\text{Loc-SC}}]$. To gain useful insights into
$\mathcal{G}^{\text{SC}}$, we approximate this gain by deriving an upper and a
lower bound. First, by noticing that (9) is lower bounded by

$\displaystyle\text{E}\left[P_{R}^{\text{Dis-SC}}\right]$
$\displaystyle>C_{0}^{2}\frac{\pi^{2}}{16}\left(\sum_{n=1}^{N}\left(\left[\mathbf{d}_{R}\right]_{n}\left[\mathbf{d}_{T}\right]_{n}\right)^{-\frac{a}{2}}\right)^{2},$
(17)

and that (5) is upper bounded by

$\text{E}\left[P_{R}^{\text{Loc-
SC}}\right]<N^{2}C_{0}^{2}d_{R}^{-a}d_{T}^{-a},$ (18)

we obtain that

$\displaystyle\mathcal{G}^{\text{SC}}$
$\displaystyle>\frac{\frac{\pi^{2}}{16}\left(\sum_{n=1}^{N}\left(\left[\mathbf{d}_{R}\right]_{n}\left[\mathbf{d}_{T}\right]_{n}\right)^{-\frac{a}{2}}\right)^{2}}{N^{2}d_{R}^{-a}d_{T}^{-a}}$
(19)
$\displaystyle=\frac{\pi^{2}}{16}\left(\frac{d_{R}d_{T}}{M_{-\frac{a}{2}}\left(\mathbf{d}_{R}\odot\mathbf{d}_{T}\right)}\right)^{a}.$
(20)

Second, by recalling (13) and noticing that (5) is lower bounded by

$\text{E}\left[P_{R}^{\text{Loc-
SC}}\right]>\frac{\pi^{2}}{16}N^{2}C_{0}^{2}d_{R}^{-a}d_{T}^{-a},$ (21)

we have

$\displaystyle\mathcal{G}^{\text{SC}}$
$\displaystyle<\frac{\left(\sum_{n=1}^{N}\left(\left[\mathbf{d}_{R}\right]_{n}\left[\mathbf{d}_{T}\right]_{n}\right)^{-\frac{a}{2}}\right)^{2}}{\frac{\pi^{2}}{16}N^{2}d_{R}^{-a}d_{T}^{-a}}$
(22)
$\displaystyle=\frac{16}{\pi^{2}}\left(\frac{d_{R}d_{T}}{M_{-\frac{a}{2}}\left(\mathbf{d}_{R}\odot\mathbf{d}_{T}\right)}\right)^{a}.$
(23)

Thus, considering (20) and (23), we can approximate $\mathcal{G}^{\text{SC}}$
as

$\mathcal{G}^{\text{SC}}\approx\left(\frac{d_{R}d_{T}}{M_{-\frac{a}{2}}\left(\mathbf{d}_{R}\odot\mathbf{d}_{T}\right)}\right)^{a}\triangleq\tilde{\mathcal{G}}^{\text{SC}},$
(24)

which can be greater or less than one, depending on the positions of the
localized and distributed RISs.

Since distributed single-connected RIS is not always beneficial over localized
single-connected RIS, we provide intuitive sufficient and necessary conditions
for $\tilde{\mathcal{G}}^{\text{SC}}>1$. First, since
$M_{-\frac{a}{2}}(\mathbf{d}_{R}\odot\mathbf{d}_{T})<\sqrt[a]{N^{2}}\text{min}\left(\mathbf{d}_{R}\odot\mathbf{d}_{T}\right)$,
we have

$\tilde{\mathcal{G}}^{\text{SC}}>\left(\frac{d_{R}d_{T}}{\sqrt[a]{N^{2}}\text{min}\left(\mathbf{d}_{R}\odot\mathbf{d}_{T}\right)}\right)^{a},$
(25)

and a sufficient condition for $\tilde{\mathcal{G}}^{\text{SC}}>1$ is given by
$d_{R}d_{T}>\sqrt[a]{N^{2}}\text{min}\left(\mathbf{d}_{R}\odot\mathbf{d}_{T}\right)$.
Second, since
$M_{-\frac{a}{2}}(\mathbf{d}_{R}\odot\mathbf{d}_{T})>\text{min}(\mathbf{d}_{R}\odot\mathbf{d}_{T})$,
we have

$\tilde{\mathcal{G}}^{\text{SC}}<\left(\frac{d_{R}d_{T}}{\text{min}\left(\mathbf{d}_{R}\odot\mathbf{d}_{T}\right)}\right)^{a},$
(26)

and a necessary condition for $\tilde{\mathcal{G}}^{\text{SC}}>1$ is given by
$d_{R}d_{T}>\text{min}(\mathbf{d}_{R}\odot\mathbf{d}_{T})$.

In Fig. 2(b), we numerically evaluate $\mathcal{G}^{\text{SC}}$ considering
the systems represented in Fig. 1. Interestingly, we always have
$\mathcal{G}^{\text{SC}}<1$, meaning that the distributed single-connected RIS
is not beneficial over the localized single-connected RIS in the considered
scenario. This is because the receiver is located close to the localized RIS,
yielding a particularly small $d_{R}d_{T}$, which deteriorates the gain, as
given in (24).

In addition, in Fig. 3(b), we report $\mathcal{G}^{\text{SC}}$ for different
locations of the receiver. Remarkably, we observe that
$\mathcal{G}^{\text{SC}}<1$ only when the receiver is in close proximity to
the localized RIS, while it can reach high values when the receiver is far
from the localized RIS. Thus, distributed single-connected RIS offers better
coverage than localized single-connected RIS, improving the received signal
power over a much wider area.

###  III-D Gain of Distributed over Localized Fully-Connected RIS

Finally, we consider the gain of distributed over localized fully-connected
RIS, defined as $\mathcal{G}^{\text{FC}}=\text{E}[P_{R}^{\text{Dis-
FC}}]/\text{E}[P_{R}^{\text{Loc-FC}}]$. From (7) and (11), we directly obtain

$\displaystyle\mathcal{G}^{\text{FC}}$
$\displaystyle=\frac{\sum_{n=1}^{N}\left[\mathbf{d}_{R}\right]_{n}^{-a}\sum_{n=1}^{N}\left[\mathbf{d}_{T}\right]_{n}^{-a}}{N^{2}d_{R}^{-a}d_{T}^{-a}}$
(27)
$\displaystyle=\left(\frac{d_{R}d_{T}}{M_{-a}\left(\mathbf{d}_{R}\right)M_{-a}\left(\mathbf{d}_{T}\right)}\right)^{a},$
(28)

indicating that $\mathcal{G}^{\text{FC}}\geq\mathcal{G}^{\text{SC}}$ by the
Cauchy-Schwarz inequality. Since

$\mathcal{G}^{\text{FC}}>\left(\frac{d_{R}d_{T}}{\sqrt[a]{N^{2}}\text{min}\left(\mathbf{d}_{R}\right)\text{min}\left(\mathbf{d}_{T}\right)}\right)^{a},$
(29)

a sufficient condition for $\mathcal{G}^{\text{FC}}>1$ is
$d_{R}d_{T}>\sqrt[a]{N^{2}}\text{min}\left(\mathbf{d}_{R}\right)\text{min}\left(\mathbf{d}_{T}\right)$,
which is more relaxed than the sufficient condition for
$\tilde{\mathcal{G}}^{\text{SC}}>1$. In addition, since

$\mathcal{G}^{\text{FC}}<\left(\frac{d_{R}d_{T}}{\text{min}\left(\mathbf{d}_{R}\right)\text{min}\left(\mathbf{d}_{T}\right)}\right)^{a},$
(30)

a necessary condition for $\mathcal{G}^{\text{FC}}>1$ is
$d_{R}d_{T}>\text{min}\left(\mathbf{d}_{R}\right)\text{min}\left(\mathbf{d}_{T}\right)$,
which is more relaxed than the necessary condition for
$\tilde{\mathcal{G}}^{\text{SC}}>1$.

In Fig. 2(c), we numerically evaluate $\mathcal{G}^{\text{FC}}$ considering
the systems represented in Fig. 1. Remarkably, we always observe
$\mathcal{G}^{\text{FC}}>1$, meaning that the distributed fully-connected RIS
always outperforms localized fully-connected RIS in the considered scenario.
Furthermore, Fig. 2(c) shows that $\mathcal{G}^{\text{FC}}$ grows
exponentially with $a$, reaching values as high as several orders of
magnitude, and is only minimally impacted by $N$.

Besides, we report $\mathcal{G}^{\text{FC}}$ for different locations of the
receiver in Fig. 3(c). Remarkably, we observe that
$\mathcal{G}^{\text{FC}}>15$ dB for all receiver locations, indicating that
distributed fully-connected RIS offers better performance than localized
fully-connected RIS over the whole considered area. In particular,
$\mathcal{G}^{\text{FC}}$ is significantly high when the receiver is far from
the localized RIS while being close to the distributed RIS.

##  IV General Lossy BD-RIS Model

We have introduced the concept of distributed RIS, showing that distributed
BD-RIS can enable significant gains over distributed single-connected RIS and
localized BD-RIS. In this section, we study the impact of the losses within
the BD-RIS architecture by developing a model for BD-RIS that accounts for
lossy interconnections.

We model a BD-RIS with lossy interconnections as an $N$-port network with a
circuit topology described as follows. First, port $m$ is connected to ground
through a tunable impedance $Z_{m}$, for $m=1,\ldots,N$, as represented in
Fig. 4(a). Second, port $m$ and port $n$, if interconnected in the BD-RIS
architecture, are interconnected through a tunable impedance $Z_{n,m}$ in
series with a transmission line of length $\ell_{n,m}$, for $n,m=1,\ldots,N$,
as represented in Fig. 4(b).

Figure 4: (a) Port $m$ connected to ground through $Z_{m}$ and (b) ports $m$
and $n$ interconnected through $Z_{n,m}$ and a transmission line of length
$\ell_{n,m}$.

Given an $N$-port network implemented with this circuit topology, our goal is
to characterize its scattering matrix $\boldsymbol{\Theta}$. Since it is hard
to directly derive $\boldsymbol{\Theta}$ from the BD-RIS circuit topology, we
characterize the BD-RIS admittance matrix $\mathbf{Y}\in\mathbb{C}^{N\times
N}$, which is related to $\boldsymbol{\Theta}$ by

$\boldsymbol{\Theta}=\left(\mathbf{I}+Z_{0}\mathbf{Y}\right)^{-1}\left(\mathbf{I}-Z_{0}\mathbf{Y}\right),$
(31)

where $Z_{0}$ is the reference impedance, typically set to $Z_{0}=50$ $\Omega$
[38, Chapter 4]. According to the definition of $\mathbf{Y}$, each entry
$[\mathbf{Y}]_{n,m}$ is obtained by driving port $m$ with a voltage $V_{m}$,
short-circuiting all other ports, measuring the short-circuit current $I_{n}$
at port $n$, and computing the ratio

$[\mathbf{Y}]_{n,m}=\left.\frac{I_{n}}{V_{m}}\right|_{V_{k}=0,\forall k\neq
m},$ (32)

for $n,m=1,\ldots,N$ [38, Chapter 4]. In the following, we separately model
the off-diagonal entries $[\mathbf{Y}]_{n,m}$, with $n\neq m$, and the
diagonal entries $[\mathbf{Y}]_{m,m}$.

###  IV-A Modeling the Admittance Matrix: Off-Diagonal Entries

To derive the off-diagonal entries $[\mathbf{Y}]_{n,m}$, with $n\neq m$, given
by (32), we make two observations that allow us to study a simplified circuit.
First, the current flowing on the transmission lines interconnecting port $j$
and port $k$, with $j,k\neq m$, is zero since these transmission lines are
short-circuited at both ends according to (32). Consequently, these
transmission lines can be removed from the circuit when computing (32).
Second, the current flowing on the transmission lines interconnecting port $m$
and port $k$, with $k\neq n$, do not influence the current $I_{n}$. Thus, also
these transmission lines can be removed from the circuit as they do not impact
on (32). Following these two observations, we have that each off-diagonal
entry $[\mathbf{Y}]_{n,m}$, with $n\neq m$, can be derived by studying an
equivalent circuit made of a single transmission line, as represented in Fig.
5.

Figure 5: Equivalent circuit to be studied to compute (a) the off-diagonal
entry $[\mathbf{Y}]_{n,m}$ and (b) the diagonal entry $[\mathbf{Y}]_{m,m}$.

To study the circuit in Fig. 5 and determine $[\mathbf{Y}]_{n,m}$, we begin by
solving the telegrapher equations in the sinusoidal steady-state condition
[38, Chapter 2], giving the voltage $V(z)$ and current $I(z)$ on the
transmission line interconnecting port $m$ and port $n$ as

$V(z)=V_{0}^{+}e^{-\gamma z}+V_{0}^{-}e^{\gamma z},$ (33)
$I(z)=\frac{V_{0}^{+}e^{-\gamma z}-V_{0}^{-}e^{\gamma z}}{Z_{0}},$ (34)

respectively, where $V_{0}^{+}\in\mathbb{C}$ and $V_{0}^{-}\in\mathbb{C}$ are
constants, $\gamma\in\mathbb{C}$ is the propagation constant of the
transmission line, and $Z_{0}\in\mathbb{R}$ is the characteristic impedance of
the transmission line222Note that $Z_{0}$ is real for both lossless and the
so-called low-loss transmission lines [38, Chapter 2], which are considered in
this study.. In the following, we characterize the voltage and current on the
transmission line by determining the constants $V_{0}^{+}$ and $V_{0}^{-}$.

Since the transmission line in Fig. 5 is terminated in a short circuit, the
voltage at $z=0$ is zero. Thus, by using (33), we have
$V(0)=V_{0}^{+}+V_{0}^{-}=0$, giving $V_{0}^{-}=-V_{0}^{+}$. By substituting
$V_{0}^{-}=-V_{0}^{+}$ into (33) and (34), $V(z)$ and $I(z)$ simplify to

$V(z)=V_{0}^{+}\left(e^{-\gamma z}-e^{\gamma z}\right),$ (35)
$I(z)=\frac{V_{0}^{+}}{Z_{0}}\left(e^{-\gamma z}+e^{\gamma z}\right),$ (36)

respectively. Furthermore, we can determine $V_{0}^{+}$ by using the input
impedance of the transmission line, defined as

$Z_{\text{in}}=\frac{V\left(-\ell_{n,m}\right)}{I\left(-\ell_{n,m}\right)},$
(37)

and that can be expressed as

$Z_{\text{in}}=Z_{0}\frac{e^{\gamma\ell_{n,m}}-e^{-\gamma\ell_{n,m}}}{e^{\gamma\ell_{n,m}}+e^{-\gamma\ell_{n,m}}},$
(38)

by using (35) and (36). Specifically, we determine $V_{0}^{+}$ from the
voltage at $z=-\ell_{n,m}$, given by

$\displaystyle V(-\ell_{n,m})$
$\displaystyle=V_{m}\frac{Z_{\text{in}}}{Z_{\text{in}}+Z_{n,m}}$ (39)
$\displaystyle=V_{0}^{+}\left(e^{\gamma\ell_{n,m}}-e^{-\gamma\ell_{n,m}}\right),$
(40)

where (39) is a direct consequence of (37), and (40) is obtained by expressing
$V(-\ell_{n,m})$ through (35). By equating (39) and (40), we obtain the
expression of $V_{0}^{+}$ as

$\displaystyle V_{0}^{+}$
$\displaystyle=V_{m}\frac{Z_{\text{in}}}{\left(Z_{\text{in}}+Z_{n,m}\right)\left(e^{\gamma\ell_{n,m}}-e^{-\gamma\ell_{n,m}}\right)}$
(41)
$\displaystyle=V_{m}\frac{Z_{0}}{Z_{n,m}\left(e^{\gamma\ell_{n,m}}+e^{-\gamma\ell_{n,m}}\right)+Z_{0}\left(e^{\gamma\ell_{n,m}}-e^{-\gamma\ell_{n,m}}\right)},$
(42)

where (42) is derived by using (38).

We can now derive the off-diagonal entry $[\mathbf{Y}]_{n,m}$, with $n\neq m$,
given by (32). Specifically, $I_{n}$ is obtained through (36) as

$I_{n}=-I(0)=-2\frac{V_{0}^{+}}{Z_{0}},$ (43)

with $V_{0}^{+}$ given by (42). Thus, by substituting (43) into (32), we
finally obtain

$[\mathbf{Y}]_{n,m}=\frac{-2}{Z_{n,m}\left(e^{\gamma\ell_{n,m}}+e^{-\gamma\ell_{n,m}}\right)+Z_{0}\left(e^{\gamma\ell_{n,m}}-e^{-\gamma\ell_{n,m}}\right)}.$
(44)

Remarkably, (44) provides the general expression of the off-diagonal entry
$[\mathbf{Y}]_{n,m}$ as a function of the tunable impedance $Z_{n,m}$ and the
transmission line parameters $\ell_{n,m}$, $\gamma$, and $Z_{0}$. Because of
the reciprocity of the circuit, $\mathbf{Y}$ is symmetric and we have
$[\mathbf{Y}]_{n,m}=[\mathbf{Y}]_{m,n}$. Furthermore, if port $m$ and $n$ are
not interconnected in the BD-RIS architecture, $Z_{n,m}$ is an open circuit,
i.e., $Z_{n,m}=\infty$, yielding $[\mathbf{Y}]_{n,m}=[\mathbf{Y}]_{m,n}=0$.

###  IV-B Modeling the Admittance Matrix: Diagonal Entries

To derive the diagonal entries $[\mathbf{Y}]_{m,m}$, we observe that the
current flowing on the transmission lines interconnecting port $j$ and port
$k$, with $j,k\neq m$, is zero since these transmission lines are short-
circuited at both ends according to (32). Thus, these transmission lines do
not impact on (32) and can be removed from the circuit. Following this
observation, we have that each diagonal entry $[\mathbf{Y}]_{m,m}$ can be
derived by studying the equivalent circuit represented in Fig. 5 to find the
ratio $[\mathbf{Y}]_{m,m}=I_{m}/V_{m}$.

From the circuit in Fig. 5, we directly notice by Kirchhoff’s first law that

$I_{m}=\frac{V_{m}}{Z_{m}}+\sum_{n\neq m}I_{n}^{\prime},$ (45)

where $I_{n}^{\prime}$ can be computed through (36) as

$I_{n}^{\prime}=I(-\ell_{n,m})=\frac{V_{0}^{+}}{Z_{0}}\left(e^{\gamma\ell_{n,m}}+e^{-\gamma\ell_{n,m}}\right),$
(46)

$\forall n\neq m$, with $V_{0}^{+}$ given by (42). Thus, by substituting (45)
into (32), we obtain

$[\mathbf{Y}]_{m,m}=\frac{1}{Z_{m}}+\sum_{n\neq
m}\frac{I_{n}^{\prime}}{V_{m}},$ (47)

readily giving

$[\mathbf{Y}]_{m,m}=\frac{1}{Z_{m}}-\sum_{n\neq
m}\frac{e^{\gamma\ell_{n,m}}+e^{-\gamma\ell_{n,m}}}{2}[\mathbf{Y}]_{n,m},$
(48)

by using (46), where $[\mathbf{Y}]_{n,m}$ is given by (44). Remarkably, the
diagonal entry $[\mathbf{Y}]_{m,m}$ is a function of $Z_{m}$, $Z_{n,m}$,
$\forall n\neq m$, and the transmission line parameters.

##  V Simplified Lossy BD-RIS Models

We have modeled the RIS admittance matrix $\mathbf{Y}$ by determining its
entries in (44) and (48). In (44) and (48), the propagation constant of the
transmission line $\gamma=\alpha+j\beta$ is in general a complex number, with
real part $\alpha$ denoted as the attenuation constant and imaginary part
$\beta$ denoted as the phase constant [38, Chapter 2]. Given the complex
nature of $\gamma$, it is hard to gain engineering insights into the entries
of $\mathbf{Y}$ from (44) and (48). For this reason, we simplify this general
model into three simplified models by making different assumptions on the
interconnection lengths $\ell_{n,m}$ and the attenuation constant $\alpha$.

###  V-A Model with Interconnection Lengths Multiple of $\lambda/2$

Consider the case when $\ell_{n,m}$ is a multiple of $\lambda/2$, with the
wavelength $\lambda$ defined as $\lambda=\frac{2\pi}{\beta}$, i.e., when
$\ell_{n,m}=\frac{\pi}{\beta}K_{n,m}$, for some $K_{n,m}\in\mathbb{Z}$. In
this case, we have
$e^{j\beta\ell_{n,m}}=e^{-j\beta\ell_{n,m}}=(-1)^{K_{n,m}}$. In addition, we
assume the tunable impedance components of the RIS to be purely reactive,
i.e., $Z_{m}=jX_{m}$ with $X_{m}\in\mathbb{R}$ and $Z_{n,m}=jX_{n,m}$ with
$X_{n,m}\in\mathbb{R}$. Thus, (44) can be simplified into

$\displaystyle[\mathbf{Y}]_{n,m}$
$\displaystyle=\frac{-2(-1)^{K_{n,m}}}{jX_{n,m}\left(e^{\alpha\ell_{n,m}}+e^{-\alpha\ell_{n,m}}\right)+Z_{0}\left(e^{\alpha\ell_{n,m}}-e^{-\alpha\ell_{n,m}}\right)}$
(49)
$\displaystyle=\frac{-(-1)^{K_{n,m}}}{jX_{n,m}\cosh\left(\alpha\ell_{n,m}\right)+Z_{0}\sinh\left(\alpha\ell_{n,m}\right)}.$
(50)

Remarkably, in (50) the term $jX_{n,m}\cosh\left(\alpha\ell_{n,m}\right)$ is
purely imaginary and the term $Z_{0}\sinh\left(\alpha\ell_{n,m}\right)$ is
purely real. Thus, if $[\mathbf{Y}]_{n,m}\neq 0$, $[\mathbf{Y}]_{n,m}$ must
have a non-zero real part, which dissipates real power. More interestingly, it
can be easily verified that the set of possible values of $[\mathbf{Y}]_{n,m}$
given by (50) is a circle in the complex plane with radius

$r=\frac{1}{2Z_{0}\sinh\left(\alpha\ell_{n,m}\right)},$ (51)

and centered in $c=(-(-1)^{K_{n,m}}r,0)$, where any point on the circle can be
achieved by properly tuning $X_{n,m}$. Thus, $[\mathbf{Y}]_{n,m}$ has always a
negative real part when $K_{n,m}$ is even, i.e., the interconnection length
$\ell_{n,m}$ is multiple of the wavelength $\lambda$, and has always a
positive real part when $K_{n,m}$ is odd.

For better clarity, we report in Fig. 6 the possible values of
$[\mathbf{Y}]_{n,m}$ considering different parameters, by fixing $Z_{0}=50$
$\Omega$ and $\ell_{n,m}=0.1$ m. From Fig. 6, we can make the following two
observations. First, when $\alpha\ell_{n,m}=0$, we have $r=+\infty$ according
to (51), meaning that $[\mathbf{Y}]_{n,m}$ can assume an arbitrary imaginary
part with always zero real part. In this ideal case, the BD-RIS has full
flexibility and does not dissipate real power. Second, $r$ decreases as the
value of $\alpha\ell_{n,m}$ increases, reducing the flexibility of the BD-RIS
since $[\mathbf{Y}]_{n,m}$ has imaginary part constrained within the interval
$[-r,+r]$.

Figure 6: Possible values of $[\mathbf{Y}]_{n,m}$ modeled as in (50), with
$Z_{0}=50$ $\Omega$ and $\ell_{n,m}=0.1$ m.

In the case $\ell_{n,m}$ is a multiple of $\lambda/2$ for all the
interconnections departing from port $m$, i.e.,
$\ell_{n,m}=\frac{\pi}{\beta}K_{n,m}$, for some $K_{n,m}\in\mathbb{Z}$,
$\forall n\neq m$, (48) simplifies as

$\displaystyle[\mathbf{Y}]_{m,m}$ $\displaystyle=\frac{1}{jX_{m}}-\sum_{n\neq
m}(-1)^{K_{n,m}}\cosh\left(\alpha\ell_{n,m}\right)[\mathbf{Y}]_{n,m}$ (52)
$\displaystyle=\frac{1}{jX_{m}}+\sum_{n\neq
m}\frac{1}{jX_{n,m}+Z_{0}\tanh\left(\alpha\ell_{n,m}\right)}.$ (53)

Remarkably, $[\mathbf{Y}]_{m,m}$ has always a positive real part, $\forall
K_{n,m}\in\mathbb{Z}$, given by

$\Re\left\\{[\mathbf{Y}]_{m,m}\right\\}=\sum_{n\neq
m}\frac{Z_{0}\tanh\left(\alpha\ell_{n,m}\right)}{X_{n,m}^{2}+Z_{0}^{2}\tanh^{2}\left(\alpha\ell_{n,m}\right)},$
(54)

while it can have an arbitrary imaginary part depending on the tunable value
$X_{m}$.

###  V-B Model with Lossless Interconnections

Consider the case of lossless interconnections, i.e., $\alpha=0$, and do not
assume any constraints on the lengths $\ell_{n,m}$. In addition, we assume
purely reactive tunable impedance components, i.e., $Z_{m}=jX_{m}$ and
$Z_{n,m}=jX_{n,m}$. In this case, (44) can be simplified into

$[\mathbf{Y}]_{n,m}=\frac{-1}{jX_{n,m}\cos\left(\beta\ell_{n,m}\right)+jZ_{0}\sin\left(\beta\ell_{n,m}\right)},$
(55)

which is purely imaginary. Interestingly, $[\mathbf{Y}]_{n,m}$ can be any
imaginary number according to the tunable impedance component $X_{n,m}$.

Besides, with $\alpha=0$, (48) boils down to

$\displaystyle[\mathbf{Y}]_{m,m}$ $\displaystyle=\frac{1}{jX_{m}}-\sum_{n\neq
m}\cos\left(\beta\ell_{n,m}\right)[\mathbf{Y}]_{n,m}$ (56)
$\displaystyle=\frac{1}{jX_{m}}+\sum_{n\neq
m}\frac{1}{jX_{n,m}+jZ_{0}\tan\left(\beta\ell_{n,m}\right)},$ (57)

which is also purely imaginary. Note that $[\mathbf{Y}]_{m,m}$ can be set
independently of the values of $[\mathbf{Y}]_{n,m}$ by properly choosing
$X_{m}$. Thus, the whole admittance matrix $\mathbf{Y}$ is purely imaginary,
indicating that the RIS is lossless and does not dissipate any real power.
This is consistent with the fact that the RIS is assumed to be made of
lossless tunable components and lossless transmission lines. Remarkably, when
the transmission lines of the RIS interconnections are lossless, their only
effect is the variation in phase of voltages and currents along them.

###  V-C Model with Interconnection Lengths Multiple of $\lambda/2$ and
Lossless Interconnections

Consider the case when $\ell_{n,m}$ is a multiple of $\lambda/2$, i.e., when
$\ell_{n,m}=\frac{\pi}{\beta}K_{n,m}$, for some $K_{n,m}\in\mathbb{Z}$, and
the interconnections are lossless, i.e., $\alpha=0$. In this case, we have
$e^{\gamma\ell_{n,m}}=e^{-\gamma\ell_{n,m}}=(-1)^{K_{n,m}}$. In addition, we
assume purely reactive tunable impedance components, i.e., $Z_{m}=jX_{m}$ and
$Z_{n,m}=jX_{n,m}$. Thus, (44) can be simplified into

$[\mathbf{Y}]_{n,m}=\frac{-(-1)^{K_{n,m}}}{jX_{n,m}},$ (58)

which can be an arbitrary imaginary number depending on the tunable impedance
component $X_{n,m}$.

In the case $\ell_{n,m}$ is a multiple of $\lambda/2$ for all the
interconnections departing from port $m$, i.e.,
$\ell_{n,m}=\frac{\pi}{\beta}K_{n,m}$, for some $K_{n,m}\in\mathbb{Z}$,
$\forall n\neq m$, (48) simplifies as

$\displaystyle[\mathbf{Y}]_{m,m}$ $\displaystyle=\frac{1}{jX_{m}}-\sum_{n\neq
m}(-1)^{K_{n,m}}[\mathbf{Y}]_{n,m}$ (59)
$\displaystyle=\frac{1}{jX_{m}}+\sum_{n\neq m}\frac{1}{jX_{n,m}},$ (60)

which can be an arbitrary imaginary number. Remarkably, when all the
interconnection lengths $\ell_{n,m}$ are a multiple of $\lambda$, i.e.,
$K_{n,m}$ is even, $\forall n,m$, the transmission lines have no visible
effect as they apply a phase shift of a multiple of $2\pi$ to the voltages and
currents propagating along them. In this case, the simplified model in (58)
and (60) is equivalent to the model with no transmission lines, i.e., with
$\ell_{n,m}=0$, widely considered in previous literature on BD-RIS [19, 21].

##  VI Lossy BD-RIS Optimization

We have modeled the admittance matrix $\mathbf{Y}$ of BD-RIS with lossy
interconnections and derived three simplified models. In this section, we
optimize BD-RIS characterized by these simplified models to assess the impact
of losses on the performance of BD-RIS.

###  VI-A Optimizing Lossless BD-RIS

Consider the lossless BD-RIS model of Section V-B. The received signal power
maximization problem in a BD-RIS-aided MIMO system writes as

$\displaystyle\underset{X_{n,m},X_{m},\mathbf{g},\mathbf{w}}{\mathsf{\mathrm{max}}}\;\;$
$\displaystyle
P_{T}\left|\mathbf{g}\left(\mathbf{H}_{RT}+\mathbf{H}_{R}\boldsymbol{\Theta}\mathbf{H}_{T}\right)\mathbf{w}\right|^{2}$
(61) $\displaystyle\mathsf{\mathrm{s.t.}}\;\;\;$
$\displaystyle\eqref{eq:T(Y)},\;\eqref{eq:Ynm-lossless},\;\eqref{eq:Ym-
lossless2},$ (62)
$\displaystyle\left\|\mathbf{g}\right\|=1,\;\left\|\mathbf{g}\right\|=1,$ (63)

which is solved by jointly optimizing $X_{n,m}$, $X_{m}$, $\mathbf{g}$,
$\mathbf{w}$. To this end, we initialize $\mathbf{g}$ and $\mathbf{w}$ to
feasible values and alternate between the following two steps.

####  VI-A1 Optimizing $X_{n,m}$ and $X_{m}$ with Fixed $\mathbf{g}$ and
$\mathbf{w}$

When $\mathbf{g}$ and $\mathbf{w}$ are fixed, we update $X_{n,m}$ and $X_{m}$
by solving

$\underset{X_{n,m},X_{m}}{\mathsf{\mathrm{max}}}\;\;P_{T}\left|\bar{h}_{RT}+\bar{\mathbf{h}}_{R}\boldsymbol{\Theta}\bar{\mathbf{h}}_{T}\right|^{2}\;\;\mathsf{\mathrm{s.t.}}\;\;\eqref{eq:T(Y)},\;\eqref{eq:Ynm-
lossless},\;\eqref{eq:Ym-lossless2},$ (64)

where $\bar{h}_{RT}=\mathbf{g}\mathbf{H}_{RT}\mathbf{w}$,
$\bar{\mathbf{h}}_{R}=\mathbf{g}\mathbf{H}_{R}$, and
$\bar{\mathbf{h}}_{T}=\mathbf{H}_{T}\mathbf{w}$, which is apparently hard
given the non-convex constraints. Nevertheless, (64) can be simplified by
noticing that $\mathbf{Y}$ given by (55) and (57) can be any purely imaginary
symmetric matrix with $[\mathbf{Y}]_{n,m}=0$ if ports $m$ and $n$ are not
interconnected, i.e., $\mathbf{Y}=j\mathbf{B}$, $\mathbf{B}=\mathbf{B}^{T}$,
with $[\mathbf{B}]_{n,m}=0$ if $Z_{n,m}=\infty$, where
$\mathbf{B}\in\mathbb{R}^{N\times N}$ denotes the RIS susceptance matrix.
Thus, (64) can be rewritten as

$\displaystyle\underset{\mathbf{B}}{\mathsf{\mathrm{max}}}\;\;$ $\displaystyle
P_{T}\left|\bar{h}_{RT}+\bar{\mathbf{h}}_{R}\left(\mathbf{I}+jZ_{0}\mathbf{B}\right)^{-1}\left(\mathbf{I}-jZ_{0}\mathbf{B}\right)\bar{\mathbf{h}}_{T}\right|^{2}$
(65) $\displaystyle\mathsf{\mathrm{s.t.}}\;\;\;$
$\displaystyle\mathbf{B}=\mathbf{B}^{T},\;[\mathbf{B}]_{n,m}=0\text{ if
}Z_{n,m}=\infty,$ (66)

which is an unconstrained problem in the upper triangular entries of
$\mathbf{B}$ that are not forced to zero. This problem has been solved through
the quasi-Newton method in [19] for group-/fully-connected RISs, and through
global optimal closed-form solutions in [26] and [21], for group-/fully- and
forest-/tree-connected RISs, respectively. Given the optimal susceptance
matrix $\mathbf{B}^{\star}$ solution to (65)-(66), the values of the tunable
reactance components $X_{n,m}$ and $X_{m}$ can be obtained by inverting (55)
and (56) with $\mathbf{Y}=j\mathbf{B}^{\star}$, respectively, i.e.,

$X_{n,m}=\frac{1}{\cos\left(\beta\ell_{n,m}\right)[\mathbf{B}^{\star}]_{n,m}}-Z_{0}\tan\left(\beta\ell_{n,m}\right),$
(67) $X_{m}=\frac{-1}{[\mathbf{B}^{\star}]_{m,m}+\sum_{n\neq
m}\cos\left(\beta\ell_{n,m}\right)[\mathbf{B}^{\star}]_{n,m}}.$ (68)

####  VI-A2 Optimizing $\mathbf{g}$ and $\mathbf{w}$ with Fixed $X_{n,m}$ and
$X_{m}$

When $X_{n,m}$ and $X_{m}$ are fixed, also $\boldsymbol{\Theta}$ is fixed, as
given by (31), where $\mathbf{Y}$ is determined by (55) and (57). Thus,
$\mathbf{g}$ and $\mathbf{w}$ are updated as the dominant left and right
singular vectors of
$\mathbf{H}_{RT}+\mathbf{H}_{R}\boldsymbol{\Theta}\mathbf{H}_{T}$,
respectively, which is globally optimal.

These two steps are alternatively repeated until convergence of the objective
(61). Notably, this optimization framework is also valid for the model of
Section V-C as it is a special case of the model of Section V-B.

###  VI-B Optimizing Lossy BD-RIS

Considering the lossy BD-RIS model of Section V-A, the received signal power
maximization problem writes as

$\displaystyle\underset{X_{n,m},X_{m},\mathbf{g},\mathbf{w}}{\mathsf{\mathrm{max}}}\;\;$
$\displaystyle
P_{T}\left|\mathbf{g}\left(\mathbf{H}_{RT}+\mathbf{H}_{R}\boldsymbol{\Theta}\mathbf{H}_{T}\right)\mathbf{w}\right|^{2}$
(69) $\displaystyle\mathsf{\mathrm{s.t.}}\;\;\;$
$\displaystyle\eqref{eq:T(Y)},\;\eqref{eq:Ynm-mult2},\;\eqref{eq:Ym-mult2},$
(70) $\displaystyle\left\|\mathbf{g}\right\|=1,\;\left\|\mathbf{g}\right\|=1,$
(71)

which is solved by initializing $\mathbf{g}$ and $\mathbf{w}$ to feasible
values and by alternating between the following two steps.

####  VI-B1 Optimizing $X_{n,m}$ and $X_{m}$ with Fixed $\mathbf{g}$ and
$\mathbf{w}$

When $\mathbf{g}$ and $\mathbf{w}$ are fixed, $X_{n,m}$ and $X_{m}$ are
updated by solving

$\underset{X_{n,m},X_{m}}{\mathsf{\mathrm{max}}}\;\;P_{T}\left|\bar{h}_{RT}+\bar{\mathbf{h}}_{R}\boldsymbol{\Theta}\bar{\mathbf{h}}_{T}\right|^{2}\;\;\mathsf{\mathrm{s.t.}}\;\;\eqref{eq:T(Y)},\;\eqref{eq:Ynm-
mult2},\;\eqref{eq:Ym-mult2}.$ (72)

Interestingly, the constraints of problem (72) can be tightened as a
consequence of the following proposition.

######  Proposition 1.

The average real power dissipated by an RIS with admittance matrix
$\mathbf{Y}$ modeled as in Section V-A is reduced if any
$|\Re\\{[\mathbf{Y}]_{i,j}\\}|$ is reduced, with $i\neq j$.

###### Proof.

Consider the following two RISs: the first has admittance matrix $\mathbf{Y}$
and dissipates a power $P_{\text{av}}$; the second has admittance matrix
$\mathbf{Y}^{\prime}$ obtained from $\mathbf{Y}$ by reducing
$|\Re\\{[\mathbf{Y}]_{i,j}\\}|$ by $\Delta>0$ for some $i,j$, i.e.,
$|\Re\\{[\mathbf{Y}^{\prime}]_{i,j}\\}|=|\Re\\{[\mathbf{Y}]_{i,j}\\}|-\Delta$,
and dissipates a power $P_{\text{av}}^{\prime}$. According to (50) and (52),
introducing $\mathbf{G}=\Re\\{\mathbf{Y}\\}$ and
$\mathbf{G}^{\prime}=\Re\\{\mathbf{Y}^{\prime}\\}$, we have
$[\mathbf{G}^{\prime}]_{n,m}=[\mathbf{G}]_{n,m}$ for $m,n\notin\\{i,j\\}$, and

$\left[\mathbf{G}^{\prime}\right]_{i,j}=\left[\mathbf{G}\right]_{i,j}+(-1)^{K_{i,j}}\Delta,$
(73)
$\left[\mathbf{G}^{\prime}\right]_{j,i}=\left[\mathbf{G}\right]_{j,i}+(-1)^{K_{i,j}}\Delta,$
(74)
$\left[\mathbf{G}^{\prime}\right]_{k,k}=\left[\mathbf{G}\right]_{k,k}-\cosh\left(\alpha\ell_{i,j}\right)\Delta,$
(75)

for $k\in\\{i,j\\}$. Given these two RISs, our goal is to prove that
$P_{\text{av}}^{\prime}\leq P_{\text{av}}$. To this end, we first recall that
average real power dissipated by an $N$-port network is defined as
$P_{\text{av}}=\Re\\{\mathbf{v}^{T}\mathbf{i}^{*}\\}/2$, where
$\mathbf{v},\mathbf{i}\in\mathbb{C}^{N\times 1}$ are the voltage and current
vectors at the $N$ ports, respectively [38, Chapter 4]. Thus, if the
admittance matrix $\mathbf{Y}$ of this $N$-port network is symmetric, it is
possible to show that

$P_{\text{av}}=\frac{1}{2}\mathbf{v}^{T}\mathbf{G}\mathbf{v}^{*}=\frac{1}{2}\sum_{m,n}v_{n}v_{m}^{*}\left[\mathbf{G}\right]_{n,m},$
(76)

where $\mathbf{v}=[v_{1},\ldots,v_{N}]^{T}$, giving the power dissipated by
the first RIS. Similarly, the power dissipated by the second RIS is

$\displaystyle P_{\text{av}}^{\prime}$
$\displaystyle=\frac{1}{2}\sum_{m,n}v_{n}v_{m}^{*}\left[\mathbf{G}^{\prime}\right]_{n,m}$
(77)
$\displaystyle=\frac{1}{2}\sum_{m,n\notin\\{i,j\\}}v_{n}v_{m}^{*}\left[\mathbf{G}\right]_{n,m}+\frac{1}{2}\sum_{m,n\in\\{i,j\\}}v_{n}v_{m}^{*}\left[\mathbf{G}^{\prime}\right]_{n,m},$
(78)

since $\left[\mathbf{G}^{\prime}\right]_{n,m}=\left[\mathbf{G}\right]_{n,m}$
for $m,n\notin\\{i,j\\}$. By substituting (73), (74), and (75) into (78), we
obtain

$P_{\text{av}}^{\prime}=P_{\text{av}}-\frac{1}{2}\left(|v_{i}|^{2}\cosh\left(\alpha\ell_{i,j}\right)\Delta+\right.\\\
\left.|v_{j}|^{2}\cosh\left(\alpha\ell_{i,j}\right)\Delta-(-1)^{K_{i,j}}\left(v_{i}v_{j}^{*}+v_{i}^{*}v_{j}\right)\Delta\right).$
(79)

Finally, since $\cosh\left(\alpha\ell_{i,j}\right)\geq 1$, we have

$P_{\text{av}}^{\prime}\leq
P_{\text{av}}-\frac{1}{2}|v_{i}-(-1)^{K_{i,j}}v_{j}|^{2}\Delta,$ (80)

indicating that $P_{\text{av}}^{\prime}\leq P_{\text{av}}$. ∎

From Proposition 1, we deduce that it is convenient to select the off-diagonal
entries $[\mathbf{Y}]_{n,m}$ with a small real part in magnitude to reduce the
power dissipated by the RIS. Interestingly, Fig. 6 shows that we can select
$[\mathbf{Y}]_{n,m}$ with any imaginary part in the interval $[-r,+r]$, with
the real part constrained by $|\Re\\{[\mathbf{Y}]_{n,m}\\}|\leq r$. This
constraint means that we can select $[\mathbf{Y}]_{n,m}$ in the right
semicircle of its possible values if $K_{n,m}$ is even, and in the left
semicircle if $K_{n,m}$ is odd, to reduce the dissipated power.

By adding the constraint $|\Re\\{[\mathbf{Y}]_{n,m}\\}|\leq r$ for $n\neq m$
to (72), the resulting problem can be reformulated by expressing $\mathbf{Y}$
as a function of an auxiliary variable $\mathbf{A}\in\mathbb{R}^{N\times N}$
as

$[\mathbf{Y}]_{n,m}=(-1)^{K_{n,m}}\left(\frac{r}{\sqrt{\frac{[\mathbf{A}]_{n,m}^{2}}{r^{2}}+1}}-r\right)+j\frac{[\mathbf{A}]_{n,m}}{\sqrt{\frac{[\mathbf{A}]_{n,m}^{2}}{r^{2}}+1}},$
(81) $[\mathbf{Y}]_{m,m}=\sum_{n\neq
m}\cosh\left(\alpha\ell_{n,m}\right)\left|\Re\\{[\mathbf{Y}]_{n,m}\\}\right|+j[\mathbf{A}]_{m,m},$
(82)

where $\mathbf{A}$ has been introduced such that $\Im\\{[\mathbf{Y}]_{n,m}\\}$
varies in the interval $[-r,+r]$ as $[\mathbf{A}]_{n,m}$ varies in
$[-\infty,+\infty]$, with $n\neq m$, and
$\Im\\{[\mathbf{Y}]_{m,m}\\}=[\mathbf{A}]_{m,m}$. Furthermore, in the case of
a lossless RIS, i.e., $r=+\infty$, (81) and (82) boil down to
$[\mathbf{Y}]_{n,m}=j[\mathbf{A}]_{n,m}$ and
$[\mathbf{Y}]_{m,m}=j[\mathbf{A}]_{m,m}$, respectively. Thus, the resulting
problem writes as

$\displaystyle\underset{\mathbf{A}}{\mathsf{\mathrm{max}}}\;$ $\displaystyle
P_{T}\left|\bar{h}_{RT}+\bar{\mathbf{h}}_{R}\left(\mathbf{I}+Z_{0}\mathbf{Y}\right)^{-1}\left(\mathbf{I}-Z_{0}\mathbf{Y}\right)\bar{\mathbf{h}}_{T}\right|^{2}$
(83) $\displaystyle\mathsf{\mathrm{s.t.}}\;\;$
$\displaystyle\eqref{eq:Anm},\;\eqref{eq:Am},\;\mathbf{A}=\mathbf{A}^{T},\;[\mathbf{A}]_{n,m}=0\text{
if }Z_{n,m}=\infty,$ (84)

which is an unconstrained problem in the upper triangular entries of
$\mathbf{A}$ that are not forced to zero. Remarkably, (83)-(84) is a
generalization of (65)-(66) since in the case of a lossless RIS, constraints
(81) and (82) become $[\mathbf{Y}]_{n,m}=j[\mathbf{A}]_{n,m}$ and
$[\mathbf{Y}]_{m,m}=j[\mathbf{A}]_{m,m}$, respectively, reducing (83)-(84) to
(65)-(66). Since (83)-(84) is unconstrained, it can be solved with the quasi-
Newton method to find a stationary point. Given the optimal
$\mathbf{A}^{\star}$ solution to (83)-(84), the admittance matrix $\mathbf{Y}$
is first obtained throguh (81) and (82). Then, the values of the tunable
reactance components $X_{n,m}$ and $X_{m}$ are obtained from $\mathbf{Y}$ by
inverting (50) and (52), respectively.

####  VI-B2 Optimizing $\mathbf{g}$ and $\mathbf{w}$ with Fixed $X_{n,m}$ and
$X_{m}$

When $X_{n,m}$ and $X_{m}$ are fixed, also $\boldsymbol{\Theta}$ is fixed, as
given by (31), where $\mathbf{Y}$ is determined by (50) and (53). Thus,
$\mathbf{g}$ and $\mathbf{w}$ are updated as the dominant left and right
singular vectors of
$\mathbf{H}_{RT}+\mathbf{H}_{R}\boldsymbol{\Theta}\mathbf{H}_{T}$,
respectively, which is globally optimal.

These two steps are repeated until convergence of (69).

##  VII Numerical Results

Figure 7: (a) Received signal power and (b) gain of BD-RIS over conventional
RIS versus the number of RIS elements, for localized RIS.

Figure 8: (a) Received signal power and (b) gain of BD-RIS over conventional
RIS versus the number of RIS elements, for distributed RIS.

Figure 9: (a) Received signal power and (b) gain of BD-RIS over conventional
RIS versus the number of transmit antennas, for distributed RIS.

Figure 10: (a) Received signal power and (b) gain of BD-RIS over conventional
RIS versus the $x$ coordinate of the receiver, for distributed RIS.

To evaluate the performance of localized and distributed BD-RIS with lossy
interconnections, we consider the system represented in Fig. 1, where the
transmitter and receiver are located at $(0,0,0)$ and $(20,0,0)$,
respectively. In the case of localized RIS, the RIS is a ULA centered in
$(20,0,2)$ and with an inter-element distance $0.05$ m, as shown in Fig. 1. We
model the path-gain of the channels $\mathbf{H}_{R}^{\text{Loc}}$ and
$\mathbf{H}_{T}^{\text{Loc}}$ through the distance-dependent model and their
small-scale fading as i.i.d. Rayleigh distributed, i.e.,
$\mathbf{H}_{i}^{\text{Loc}}=\sqrt{\rho_{i}}\widetilde{\mathbf{H}}_{i}$, where
$\rho_{i}=C_{0}d_{i}^{-a}$ and
$\widetilde{\mathbf{H}}_{i}\sim\mathcal{CN}(\mathbf{0},\mathbf{I})$, for
$i\in\\{R,T\\}$. In the case of distributed RIS, the RIS is a ULA with
elements uniformly placed between $(0,0,2)$ and $(40,0,2)$, as shown in Fig.
1. Accordingly, the channels are given by
$\mathbf{H}_{R}^{\text{Dis}}=\widetilde{\mathbf{H}}_{R}\mathbf{R}_{R}^{1/2}$
and
$\mathbf{H}_{T}^{\text{Dis}}=\mathbf{R}_{T}^{1/2}\widetilde{\mathbf{H}}_{T}$,
where $\mathbf{R}_{i}=\text{diag}(\boldsymbol{\rho}_{i})$,
$[\boldsymbol{\rho}_{i}]_{n}=C_{0}[\mathbf{d}_{i}]_{n}^{-a}$, and
$\widetilde{\mathbf{H}}_{i}\sim\mathcal{CN}(\mathbf{0},\mathbf{I})$, for
$i\in\\{R,T\\}$. The direct channel is assumed to be fully obstructed, i.e.,
$\mathbf{H}_{RT}=\mathbf{0}$, and we set $C_{0}=-30$ dB, $a=4$, and $P_{T}=10$
W. In the following, we report the performance of tridiagonal BD-RIS, a tree-
connected BD-RIS that is proven to be the least complex BD-RIS architecture
able to achieve the performance upper bound in single-user systems [21], and
compare it with conventional RIS. Recalling that tridiagonal BD-RIS presents
interconnections only between adjacent RIS elements [21], its interconnection
lengths are all equal to the inter-element distance.

In Fig. 7, we report the received signal power obtained with localized RIS
versus the number of RIS elements, with $N_{R}=1$ and $N_{T}=1$ for
simplicity. We can make the following observations. First, BD-RIS achieves a
higher received signal power than conventional RIS thanks to the additional
flexibility, both with lossless and lossy interconnections. Second, lossless
BD-RIS achieves the highest received signal power since no power is dissipated
by the BD-RIS circuit. Specifically, Fig. 7 shows that lossless BD-RIS offers
a gain over conventional RIS of up to 1.62, as discussed in Section III-A.
Third, the received signal power achieved by BD-RIS decreases as the
attenuation constant $\alpha$ increases. However, the performance of localized
BD-RIS is only slightly impacted by the losses given the short inter-element
distance.

In Fig. 8, we report the received signal power obtained with distributed RIS
versus the number of RIS elements, with $N_{R}=1$ and $N_{T}=1$. We can make
the following observations. First, BD-RIS always achieves a higher received
signal power than conventional RIS. Second, lossless BD-RIS achieves the
highest received signal power, offering gains of up to 30 dB over conventional
RIS. These massive gains are because the EM signal can propagate without
losses between the BD-RIS elements, which are spread over a long line. Third,
the received signal power of distributed BD-RIS visibly decreases as the
attenuation constant $\alpha$ increases, given the significantly long inter-
element distance. Nevertheless, lossy BD-RIS can still achieve more than an
order of magnitude of gain over conventional RIS.

To investigate the impact of $N_{R}$ and $N_{T}$ on the system performance, we
report in Fig. 9 the received signal power of distributed RIS versus the
number of transmit antennas, with $N=32$ and $N_{R}=2$. We observe that the
received signal power of BD-RIS and conventional RIS increases with $N_{T}$,
given the diversity gain offered by the multiple transmit antennas. Besides,
the gain of BD-RIS over conventional RIS is approximately stable for any value
of $N_{T}$.

In Fig. 10, we report the received signal power obtained with distributed RIS
for different locations of the receiver. Specifically, the receiver is located
in $(x,0,0)$, where $x\in[10,50]$, while we fix $N=32$, $N_{R}=2$, and
$N_{T}=2$. We observe that the received signal power decreases as $x$
increases since $x$ also represents the distance between the transmitter and
receiver. The received signal power decreases significantly after $x=40$ m, as
this is the location of the last distributed RIS element. Besides, Fig. 10
shows that the gain of lossless BD-RIS over conventional RIS increases with
$x$ reaching values as high as several orders of magnitude until $x=40$ m.

##  VIII Conclusion

We propose the concept of distributed RIS, whose elements are distributed over
a wide region, in opposition to localized RIS, as commonly considered in
previous literature. We derive its scaling laws and analyze the gain of
distributed RIS over localized RIS, and of distributed BD-RIS over distributed
conventional RIS. The derived gains show that lossless distributed BD-RIS can
offer gains as high as several orders of magnitude over distributed
conventional RIS and localized BD-RIS. In particular, these substantial gains
are enabled by the tunable impedance components interconnecting the BD-RIS
elements, which allow the EM signal to effectively reach the receiver by
propagating within the BD-RIS circuit.

To assess the practical performance of distributed BD-RIS, we model BD-RIS
with lossy interconnections by using transmission line theory. More precisely,
we model the BD-RIS interconnections as tunable impedance components in series
with lossy transmission lines and derive the expression of the BD-RIS
admittance matrix. Since it is hard to gain engineering insights from the
obtained model, we also derive three simplified models through different
assumptions. Finally, we optimize lossy BD-RIS based on the proposed models
and evaluate its performance. Numerical results show that the performance of
localized BD-RIS is only slightly impacted by losses because of the short
interconnection lengths. Furthermore, distributed BD-RIS can achieve orders of
magnitude of gains over conventional RIS, even with low losses.

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