# Intelligent Omni-Surfaces (IOSs) for the MIMO Broadcast Channel ††thanks: A.
Mohamed, N. S. Perović, M. Di Renzo are with Université Paris-Saclay, CNRS,
CentraleSupélec, Laboratoire des Signaux et Systèmes, 3 Rue Joliot-Curie,
91192 Gif-sur-Yvette, France. Email: marco.di-renzo@universite-paris-
saclay.fr. This work was supported in part by the European Commission through
the H2020 PAINLESS project under grant agreement number 812991, the H2020
SURFER project under grant agreement number 101030536, the H2020 ARIADNE
project under grant agreement number 871464, and the H2020 RISE-6G project
under grant agreement number 101017011.

Abdelhamed Mohamed, Nemanja Stefan Perović, , Marco Di Renzo

###### Abstract

In this paper, we consider intelligent omni-surfaces (IOSs), which are capable
of simultaneously reflecting and refracting electromagnetic waves. We focus
our attention on the multiple-input multiple-output (MIMO) broadcast channel,
and we introduce an algorithm for jointly optimizing the covariance matrix at
the base station, the matrix of reflection and transmission coefficients at
the IOS, and the amount of power that is reflected and refracted from the IOS.
The distinguishable feature of this work lies in taking into account that the
reflection and transmission coefficients of an IOS are tightly coupled.
Simulation results are illustrated to show the convergence of the proposed
algorithm and the benefits of using surfaces with simultaneous reflection and
refraction capabilities.

##  I Introduction

Reconfigurable intelligent surfaces (RISs) have emerged as a promising
approach to improve the wireless communication channel quality and to extend
the network coverage [1, 2]. However, the vast majority of works consider
surfaces that can only reflect the incident signals, which limits the coverage
capabilities offered by RISs [3, 4]. To tackle this problem, the recently
proposed concept of intelligent omni-surface (IOS) provides $360^{\circ}$
coverage thanks to surfaces that can simultaneously reflect and refract the
incident signals [5]. The reflection and refraction capabilities of the
incident electromagnetic waves are controlled through the optimization of two
interlinked sets of reflection and transmission coefficients. In general, in
other words, it is not possible to control the reflection and transmission
coefficients independently.

Recently, a few research works have analyzed the performance of IOSs. In [6],
the authors investigate the weighted sum-rate maximization under quality of
service requirements and unit modulus constraints for the IOS elements, by
utilizing the successive convex approximation method. In [7], the energy
efficiency maximization problem is studied, and an optimization algorithm is
proposed for jointly optimizing the transmit power and the passive beamforming
at the IOS. In [8], the weighted sum-rate of an IOS-aided multiple-input
multiple-output (MIMO) system is maximized by using the alternating
optimization method. The precoding matrices are obtained by the Lagrange dual
method, while the reflection and transmission coefficients are obtained by the
penalty concave-convex method. Interested readers can consult [9], [10], [11],
[12] for further information on IOSs.

In contrast to the available works, we focus our attention on optimizing the
IOS in the MIMO broadcast channel. Also, for the first time, we explicitly
take into account the dependence between the reflection and transmission
coefficients, based on a recently implemented IOS prototype [13].
Specifically, the main contributions of this paper are as follows:

  * •

We utilize the duality between the broadcast channel and the multiple access
channel to maximize the achievable sum-rate. We formulate a joint optimization
problem for the users’ covariance matrices, the reflection and transmission
coefficients, and the power ratio between the reflected and transmitted power.
We analyze the case studies with continuous-valued and discrete-valued phase
shifts for the reflection and transmission coefficients, and we assume that
they are not independent of each other.

  * •

Due to the non-convexity of the formulated optimization problem, and the
coupling between the optimization variables in the objective function, we
propose an alternating optimization algorithm to solve the aforementioned
problem. The optimal users’ covariance matrices are obtained by applying the
dual decomposition and the block coordinate maximization (BCM) method, while
the optimal phase shifts of the reflection and transmission coefficients of
the IOS elements are formulated in a simple expression. In addition, the power
ratio between the reflected and refracted power is computed iteratively by
utilizing the gradient ascent method.

  * •

Simulation results show that the proposed algorithm converges relatively fast
(i.e., within a few iterations) to a local optima. Moreover, we quantify the
impact of discretizing the reflection and transmission coefficients for a two-
state IOS testbed platform.

The paper is organized as follows. In Section II, we present the system model
for the IOS-aided MIMO broadcast channel. In Section III, we formulate the
optimization problem to maximize the achievable sum-rate. In Section IV, we
describe the optimization algorithm to solve the optimization problem. In
Section V, we provide simulation results that illustrate the achievable sum-
rate. Conclusions are drawn in Section VI.

Notation: Bold upper and lower case letters denote matrices and vectors,
respectively. $\mathbb{C}^{m\times n}$ denotes the space of $m\times n$
complex matrices. $\mathbf{H}^{\mathsf{T}}$, $\mathbf{H}^{H}$, $|\mathbf{H}|$
and $\operatorname{Tr}(\mathbf{H})$ represent the transpose, Hermitian
transpose, the determinant and the trace of $\mathbf{H}$, respectively. The
rank of $\mathbf{H}$ is denoted by $\operatorname{rank}(\mathbf{H})$ and
$\lambda_{\max}(\mathbf{H})$ is the largest singular value of $\mathbf{H}$,
while $\left(\cdot\right)^{\ast}$ denotes the complex conjugate. The identity
matrix is denoted by $\mathbf{I}$ and the matrix inversion operation is
denoted by ${INV({\bf{X}})}$.

##  II System Model

We consider an IOS-assisted multi-user MIMO broadcast channel, where a multi-
antenna base station (BS) with $N_{t}$ antennas communicate with $K$ users,
and each user is equipped with $N_{r}$ antennas. The total number of users in
the reflection and transmission sides of the IOS are denoted by $KR$ and $KT$,
respectively, with $K=KR+KT$. We assume that the BS and the users’ antennas
form a uniform linear array (ULA), and the inter-antenna separations are
$s_{t}$ and $s_{r}$, respectively. In addition, we assume that the IOS
comprises $N$ elements that can simultaneously reflect and refract the
incident signals. Since the users can be located either in the reflection or
the transmission sides of the IOS, a single user receives either the signal
reflected or the signal refracted from the IOS.

Figure 1:  Aerial view of the considered communication system.

To serve the users located in the reflection and transmission sides
simultaneously, each element of the IOS applies a complex-valued reflection
and transmission coefficient. The matrix of $N$ reflection or transmission
coefficients is denoted by
$\mathbf{F}(\boldsymbol{\theta})=\mathrm{diag}(\boldsymbol{\theta})\in\mathbb{C}^{N\times
N}$, where
$\boldsymbol{\theta}=\\{\boldsymbol{\theta}^{r},\boldsymbol{\theta}^{t}\\}$,
and
$\boldsymbol{\theta^{r}}=\sqrt{\rho}\left[{\tilde{\beta}^{r}_{1}},{\tilde{\beta}^{r}_{2}},\dots\newline
,{\tilde{\beta}^{r}_{N}}\right]^{T}\in\mathbb{C}^{N\times 1}$ and
$\boldsymbol{\theta^{t}}=\sqrt{1-\rho}\left[{\tilde{\beta}^{t}_{1}},{\tilde{\beta}^{t}_{2}},\dots\newline
,{\tilde{\beta}^{t}_{N}}\right]^{T}\in\mathbb{C}^{N\times 1}$. Specifically,
the parameter $\rho\in\left[0,1\right]$ is an optimization variable that
accounts for the amount of power that the IOS directs towards the users in the
reflection side ($1-\rho$ is the amount of power directed towards the
transmission side). Also, $\tilde{\beta}^{r}_{n}$ and $\tilde{\beta}^{t}_{n}$
are the reflection and transmission coefficients of the IOS. These reflection
and transmission coefficients are interlinked and are optimized in pairs for
each IOS element, i.e.,
$\left\\{\tilde{\beta}^{r}_{n},\tilde{\beta}^{t}_{n}\right\\}\in\left\\{({\tilde{\beta}}^{r}_{s1},{\tilde{\beta}}^{t}_{s1}),({\tilde{\beta}}^{r}_{s2},{\tilde{\beta}}^{t}_{s2}),\dots,({\tilde{\beta}}^{r}_{sQ},{\tilde{\beta}}^{t}_{sQ})\right\\}=\Psi$,
where $Q$ is the number of (reflection, transmission) coefficient pairs.

Due to the presence of the IOS, the end-to-end channel for the
$k_{r}\in[1,\dots KR]$-user located in the reflection side is:

$\displaystyle{\mathbf{H}_{{k_{r}}}}={\mathbf{D}_{{k_{r}}}}+{\mathbf{G}_{{k_{r}}}}\mathbf{F}(\boldsymbol{\theta}^{r})\mathbf{U}={\mathbf{D}_{{k_{r}}}}+{\mathbf{G}_{{k_{r}}}}\sqrt{\rho}$
(1)
$\displaystyle\times\operatorname{diag}\left({\tilde{\beta}_{1}^{r},\tilde{\beta}_{2}^{r},...,\tilde{\beta}_{N}^{r}}\right)\mathbf{U}={\mathbf{D}_{{k_{r}}}}+\sqrt{\rho}\sum\limits_{n=1}^{N}{\tilde{\beta}_{n}^{r}}{\mathbf{g}_{n,{k_{r}}}}\mathbf{u}_{n}$

and the end-to-end channel for the $k_{t}\in[1,\dots KT]$-user located in the
transmission side is:

$\displaystyle{\mathbf{H}_{{k_{t}}}}={\mathbf{D}_{{k_{t}}}}+{\mathbf{G}_{{k_{t}}}}\mathbf{F}(\boldsymbol{\theta}^{t})\mathbf{U}={\mathbf{D}_{{k_{t}}}}+{\mathbf{G}_{{k_{t}}}}\sqrt{1-\rho}$
(2)
$\displaystyle\times\operatorname{diag}\left({\tilde{\beta}_{1}^{t},\tilde{\beta}_{2}^{t},...,\tilde{\beta}_{N}^{t}}\right)\mathbf{U}={\mathbf{D}_{{k_{t}}}}+\sqrt{1-\rho}\sum\limits_{n=1}^{N}{\tilde{\beta}_{n}^{t}}{\mathbf{g}_{n,{k_{t}}}}\mathbf{u}_{n}$

In a compact form, we have:

$\mathbf{H}_{k}(\boldsymbol{\theta})=\mathbf{D}_{k}+\mathbf{G}_{k}\mathbf{F}(\boldsymbol{\theta})\mathbf{U}$
(3)

where $\mathbf{D}_{k}\in\mathbb{C}^{N_{r}\times N_{t}}$ denotes the channel
matrix between the BS and the _k_ -th user, $\mathbf{U}\in\mathbb{C}^{N\times
N_{t}}$ denotes the channel matrix between the BS and the IOS, and
$\mathbf{G}_{k}\in\mathbb{C}^{N_{r}\times N}$ denotes the channel matrix
between the IOS and the _k_ -th user.

To simplify the notation, we write $\mathbf{H}_{k}$ instead of
$\mathbf{H}_{k}(\boldsymbol{\theta})$, where the dependence on
$\boldsymbol{\theta}$ is implicit. Thus, the received signal at the k-th user
is written as:

$\mathbf{y}_{k}=\mathbf{H}_{k}\mathbf{x}_{k}+\sum\nolimits_{j=1,j\neq
k}^{K}\mathbf{H}_{k}\mathbf{x}_{j}+\mathbf{n}_{k}$ (4)

where $\mathbf{H}_{k}\in\mathbb{C}^{N_{r}\times N_{t}}$ is the channel matrix
for the _k_ -th user, $\mathbf{x}_{k}\in\mathbb{C}^{N_{t}\times 1}$ is the
transmitted signal intended for the _k_ -th user, and
$\mathbf{x}_{j}\in\mathbb{C}^{N_{t}\times 1}$ for $j\neq k$ are the
transmitted signals intended for other users, which act as interference for
the detection of $\mathbf{x}_{k}$. The noise vector
$\mathbf{n}_{k}\in\mathbb{C}^{N_{r}\times 1}$ consists of independent and
identically distributed (i.i.d.) elements whose distribution is
$\mathcal{CN}(0,N_{0})$, where $N_{0}$ is the noise variance.

##  III Problem Formulation

We are interested in maximizing the achievable sum-rate of the considered IOS-
aided wireless communication system. To accomplish this, we exploit the fact
that the achievable rate region of a Gaussian MIMO broadcast channel can be
achieved by dirty paper coding (DPC) [14]. DPC enables us to reduce the
interference in a communication system, i.e., to perfectly eliminate the
interference term $\sum_{j<k}\mathbf{H}_{k}\mathbf{x}_{j}$ for the _k_ -th
user. In this regard, the ordering of the users clearly matters. Let $\pi$ be
an ordering of users, i.e., a permutation of the set $\\{1,2,\ldots,K\\}$.
Then, for this ordering, the achievable rate for the _k_ -th user can be
computed as [15, Eq. (3)]:

$R_{\pi(k)}=\log_{2}\frac{\Bigl{|}\mathbf{I}+\mathbf{H}_{\pi(k)}\bigl{(}\sum_{j\geq
k}\mathbf{S}_{\pi(j)}\bigr{)}\mathbf{H}_{\pi(k)}^{H}\Bigr{|}}{\Bigl{|}\mathbf{I}+\mathbf{H}_{\pi(k)}\bigl{(}\sum_{j>k}\mathbf{S}_{\pi(j)}\bigr{)}\mathbf{H}_{\pi(k)}^{H}\Bigr{|}}$
(5)

where
$\mathbf{S}_{k}=\mathbb{E}\bigl{\\{}\mathbf{x}_{k}\mathbf{x}_{k}^{H}\bigr{\\}}\succeq\mathbf{0}$
is the input covariance matrix of user $k$. We assume a sum-power constraint
at the BS, i.e.:

$\sum\nolimits_{k=1}^{K}\operatorname{Tr}\bigl{(}\mathbf{S}_{k}\bigr{)}\leq P$
(6)

where $P$ is the maximum total power at the BS. Therefore, the achievable sum-
rate optimization problem for the RIS-assisted MIMO broadcast channel can be
expressed as:

$\displaystyle\underset{\mathbf{S},\boldsymbol{{\tilde{\beta}}}^{l},{\rho}}{\text{max}}$
$\displaystyle~{}\sum_{k=1}^{K}\log_{2}\frac{\Bigl{|}\mathbf{I}+\mathbf{H}_{\pi(k)}\bigl{(}\sum_{j\geq
k}\mathbf{S}_{\pi(j)}\bigr{)}\mathbf{H}_{\pi(k)}^{H}\Bigr{|}}{\Bigl{|}\mathbf{I}+\mathbf{H}_{\pi(k)}\bigl{(}\sum_{j>k}\mathbf{S}_{\pi(j)}\bigr{)}\mathbf{H}_{\pi(k)}^{H}\Bigr{|}}$
(7a) $\displaystyle\mathrm{s.t.}$
$\displaystyle~{}\sum\nolimits_{k=1}^{K}\operatorname{Tr}\bigl{(}\mathbf{S}_{k}\bigr{)}\leq
P;\mathbf{S}_{k}\succeq\mathbf{0},\forall k,$ (7b)
$\displaystyle\quad\left\\{\tilde{\beta}^{r}_{n},\tilde{\beta}^{t}_{n}\right\\}\in\Psi;\quad\forall
n\in[1,2,\dots,N],$ (7c) $\displaystyle\quad 0\leq\rho\leq 1$ (7d)

where
$\boldsymbol{\tilde{\beta}}^{l}=\\{\boldsymbol{\tilde{\beta}}^{r},\boldsymbol{\tilde{\beta}}^{t}\\}$
and $l\in\left\\{r,t\right\\}$,.

To solve this problem, we exploit the duality between the MIMO broadcast
channel and the multiple access channel, as recently done in [16].
Accordingly, we reformulate the optimization problem in (7) as follows:

$\displaystyle\underset{\mathbf{\bar{S}},\boldsymbol{{\tilde{\beta}}}^{l},{\rho}}{\max}$
$\displaystyle\quad
f(\mathbf{\bar{S}},\boldsymbol{{\tilde{\beta}}},{\rho})=\log_{2}{\Bigl{|}\mathbf{I}+\sum_{k=1}^{K}{\mathbf{H}_{k}^{H}}\mathbf{\bar{S}}\mathbf{H}_{k}\Bigr{|}}$
(8) $\displaystyle\mathrm{s.t.}$
$\displaystyle\quad\sum\nolimits_{k=1}^{K}\operatorname{Tr}\bigl{(}\mathbf{\bar{S}}_{k}\bigr{)}\leq
P;\mathbf{\bar{S}}_{k}\succeq\mathbf{0},\forall k,$
$\displaystyle\quad\left\\{\tilde{\beta}^{r}_{n},\tilde{\beta}^{t}_{n}\right\\}\in\Psi;\quad\forall
n\in[1,2,\dots,N],$ $\displaystyle\quad 0\leq\rho\leq 1,$

where $\mathbf{H}_{k}^{H}$ represents the dual multiple access channel
corresponding to $\mathbf{H}_{k}$ and $\mathbf{\bar{S}_{k}}$ is the dual
multiple access input covariance matrix of the _k_ -th user. Once the input
covariance matrices $(\bar{\mathbf{S}}_{k})_{k=1}^{K}$ in the dual multiple
access channel are found, the corresponding covariance matrices
$(\mathbf{S}_{k})_{k=1}^{K}$ in the broadcast channel are obtained from [15,
Eq. (11)], as:

$\mathbf{S}_{k}=\mathbf{B}_{k}^{-1/2}\mathbf{F}_{k}\mathbf{G}_{k}^{H}\mathbf{A}_{k}^{1/2}\bar{\mathbf{S}}_{k}\mathbf{A}_{k}^{1/2}\mathbf{G}_{k}\mathbf{F}_{k}^{H}\mathbf{B}_{k}^{-1/2}$
(9)

where
$\mathbf{A}_{k}=\mathbf{I}+\mathbf{H}_{k}(\sum_{i=1}^{k-1}\mathbf{S}_{i})\mathbf{H}_{k}^{H}$,
$\mathbf{B}_{k}=\mathbf{I}+\sum_{i=k+1}^{K}\mathbf{H}_{i}^{H}\bar{\mathbf{S}}_{i}\mathbf{H}_{i}$,
and $\mathbf{F}_{k}\boldsymbol{\Lambda}_{k}\mathbf{G}_{k}^{H}$ is the singular
value decomposition of
$\mathbf{B}_{k}^{-1/2}\mathbf{H}_{k}^{H}\mathbf{A}_{k}^{-1/2}$.

##  IV Proposed Optimization Method

To solve the formulated problem, we propose an alternating optimization
method. The users’ covariance matrices are optimized by exploiting the dual
decomposition method in [16]. The reflection and transmission coefficients of
the IOS are obtained by generalizing the method in [17], which is applicable
to reflecting surfaces, under the assumption of continuous-valued
coefficients. The corresponding discrete-valued reflection and transmission
coefficients are obtained by projecting the obtained solutions onto the
feasible set of possible discrete values. Moreover, we present a gradient-
based method for optimizing the power ratio.

###  IV-A Covariance Matrix Optimization

For given values of the power ratio and the reflection and transmission
coefficients, the achievable sum-rate optimization problem in (8) is
simplified as follows:

$\displaystyle\underset{\bar{\mathbf{S}}}{\max}$
$\displaystyle\quad\log_{2}\Bigl{|}\mathbf{I}+\sum\limits_{k=1}^{K}\mathbf{H}_{k}^{H}\bar{\mathbf{S}}_{k}\mathbf{H}_{k}\Bigr{|}$
(10a) $\displaystyle\mathrm{s.t.}$
$\displaystyle\quad\bar{\mathbf{S}}\in\mathcal{S}$ (10b)

where $\bar{\mathbf{S}}\triangleq(\bar{\mathbf{S}}_{k})_{k=1}^{K}$ for
$\mathcal{S}=\\{\bar{\mathbf{S}}\ |\
\sum\nolimits_{k=1}^{K}\operatorname{Tr}\bigl{(}\bar{\mathbf{S}}_{k}\bigr{)}\leq
P;\bar{\mathbf{S}}_{k}\succeq\mathbf{0}\thinspace\thinspace\forall k\\}$. As
described in [16, 18], the optimization problem in (10) is solved by using the
dual decomposition Lagrangian and the accelerated block coordinate
maximization methods.

###  IV-B IOS Optimization

For given $\left\\{{{{\bar{S}}_{k}}}\right\\}_{k=1}^{K}$ and $\rho$, the
optimization problem in (8) with respect to $\tilde{\beta}_{n}^{l}$,
$l\in\left\\{r,t\right\\}$, can be written as follows:

$\displaystyle\underset{{{\tilde{\beta}_{n}^{l}}}}{\max}$ $\displaystyle\quad
f({{\tilde{\beta}_{n}^{l}}})=\log_{2}{\Bigl{|}\mathbf{I}+\sum_{k=1}^{K}{\mathbf{H}_{k}^{H}}\mathbf{\bar{S}}\mathbf{H}_{k}\Bigr{|}}$
(11) $\displaystyle\mathrm{s.t.}$
$\displaystyle\quad\left\\{\tilde{\beta}^{r}_{n},\tilde{\beta}^{t}_{n}\right\\}\in\Psi;\quad\forall
n\in[1,2,\dots,N]$

To tackle this problem, we first optimize both $\tilde{\beta}^{r}_{n}$ and
$\tilde{\beta}^{t}$ by assuming that they are continuous-valued and
independent coefficients, and we then project the obtained solutions onto the
set of feasible discrete phase shifts, by taking into account that the
reflection and transmission coefficients are interlinked.

For ease of writing, we define the variables
${{\bf{G}}_{{k}}}=[{\bf{g}}_{1},\ldots,{\bf{g}}_{n}]$,
${\bf{U}}={[{\bf{u}}_{1}^{T},...,{\bf{u}}_{n}^{T}]^{T}}$, and

${{{\bf{\bar{D}}}}_{k}}=\left\\{\begin{array}[]{l}{\bf{D}}_{i}^{r}~{}~{}\forall
i\in\left\\{{1,...,{KR}}\right\\}\\\ {\bf{D}}_{j}^{t}~{}~{}\forall
j\in\left\\{{1,...,{KT}}\right\\}\end{array}\right.$ (12)
${{{\bf{\bar{g}}}}}_{n,k}=\left\\{\begin{array}[]{l}{\bf{g}}_{n,i}^{r}~{}~{}\forall
i\in\left\\{{1,...,{KR}}\right\\}\\\ {\bf{g}}_{n,j}^{t}~{}~{}\forall
j\in\left\\{{1,...,{KT}}\right\\}\end{array}\right.$ (13)

Under the assumption that $\tilde{\beta}^{r}_{n}$ and $\tilde{\beta}^{t}_{n}$
can be optimized independently (this is removed next by projecting on the
feasible set), the objective function can be rewritten as follows:

${\small
f({{\tilde{\beta}_{n}^{l}}})={\log_{2}}\left|\begin{array}[]{l}{\bf{I}}+\left({\begin{array}[]{*{20}{l}}{\sum\limits_{k=1}^{KR}{\left({{{{\bf{\bar{D}}}}_{k}}+\sum\limits_{n=1}^{N}{\sqrt{\rho}\tilde{\beta}_{n}^{r}}{{{\bf{\bar{g}}}}_{n,k}}{{\bf{u}}_{n}}}\right)^{H}}}\\\
{\times{{\overline{\bf{S}}}_{k}}{{\left({{{{\bf{\bar{D}}}}_{k}}+\sum\limits_{n=1}^{N}{\sqrt{\rho}\tilde{\beta}_{n}^{r}}{{{\bf{\bar{g}}}}_{n,k}}{{\bf{u}}_{n}}}\right)}}}\end{array}}\right)\\\
+\left({\begin{array}[]{*{20}{l}}{\sum\limits_{k=1}^{KT}{\left({{{{\bf{\bar{D}}}}_{k}}+\sum\limits_{n=1}^{N}{\sqrt{1-\rho}\tilde{\beta}_{n}^{t}}{{{\bf{\bar{g}}}}_{n,k}}{{\bf{u}}_{n}}}\right)^{H}}}\\\
{\times{{\overline{\bf{S}}}_{k}}{{\left({{{{\bf{\bar{D}}}}_{k}}+\sum\limits_{n=1}^{N}{\sqrt{1-\rho}\tilde{\beta}_{n}^{t}}{{{\bf{\bar{g}}}}_{n,k}}{{\bf{u}}_{n}}}\right)}}}\end{array}}\right)\end{array}\right|}$
(14)

Thus, for optimizing $\tilde{\beta}_{n}^{r}$, we have:

$f(\tilde{\beta}_{n}^{r})={\log_{2}}\left|{{\bf{A}}_{n}^{r}+\tilde{\beta}_{n}^{r}{\bf{B}}_{n}^{r}+\tilde{\beta}_{n}^{r*}{\bf{B}}_{n}^{rH}}\right|$
(15)

and, for optimizing $\tilde{\beta}_{n}^{t}$, we have:

$f(\tilde{\beta}_{n}^{t})={\log_{2}}\left|{{\bf{A}}_{n}^{t}+\tilde{\beta}_{n}^{t}{\bf{B}}_{n}^{t}+\tilde{\beta}_{n}^{t*}{\bf{B}}_{n}^{tH}}\right|$
(16)

where:

${\small{\bf{B}}_{n}^{r}=\sqrt{\rho}\sum\limits_{k=1}^{KR}{({{{{\bf{\bar{D}}}}^{H}}_{k}+\sum\limits_{l=1,n\neq
l}^{N}{\sqrt{\rho}\tilde{\beta}_{l}^{r*}}{\bf{u}}_{l}^{H}{{{\bf{\bar{g}}}}^{H}}_{l,k}}){{\overline{\bf{S}}}_{k}}{{{\bf{\bar{g}}}}_{n,k}}{{\bf{u}}_{n}}}}$
(17)
${\small{\bf{B}}_{n}^{t}=\sum\limits_{k=1}^{KT}{(\sqrt{1-\rho}{{{{\bf{\bar{D}}}}^{H}}_{k}+\sum\limits_{n\neq
l}^{N}{({1-\rho)}\tilde{\beta}_{l}^{t*}}{\bf{u}}_{l}^{H}{{{\bf{\bar{g}}}}^{H}}_{l,k}}){{\overline{\bf{S}}}_{k}}{{{\bf{\bar{g}}}}_{n,k}}{{\bf{u}}_{n}}}}$
(18)
${\small\begin{array}[]{l}{\bf{A}}_{n}^{r}={\bf{I}}+\sum\limits_{k=1}^{KT}{({{{{\bf{\bar{D}}}}_{k}}+\sum\limits_{n=1}^{N}{\sqrt{1-\rho}\tilde{\beta}_{n}^{t}}{{{\bf{\bar{g}}}}_{n,k}}{{\bf{u}}_{n}}})^{H}}\\\
\times{\overline{\bf{S}}_{k}}{({{{{\bf{\bar{D}}}}_{k}}+\sum\limits_{n=1}^{N}{\sqrt{1-\rho}\tilde{\beta}_{n}^{t}}{{{\bf{\bar{g}}}}_{n,k}}{{\bf{u}}_{n}}})}+\sum\limits_{k=1}^{KR}\rho{\bf{u}}_{n}^{H}{{{\bf{\bar{g}}}}^{H}}_{n,k}{\overline{\bf{S}}_{k}}{{{\bf{\bar{g}}}}_{n,k}}{{\bf{u}}_{n}}\\\
+\sum\limits_{k=1}^{KR}{({{{{\bf{\bar{D}}}}^{H}}_{k}+\sum\limits_{l=1,n\neq
l}^{N}{\sqrt{\rho}\tilde{\beta}_{l}^{r*}{\bf{u}}_{l}^{H}{{{\bf{\bar{g}}}}^{H}}_{l,k}}})}\\\
\times{\overline{\bf{S}}_{k}}({{{{\bf{\bar{D}}}}_{k}}+\sum\limits_{l=1,n\neq
l}^{N}{\sqrt{\rho}\tilde{\beta}_{l}^{r}{{{\bf{\bar{g}}}}_{l,k}}{{\bf{u}}_{l}}}})\end{array}}$
(19)
${\small\begin{array}[]{*{20}{l}}\begin{array}[]{l}{\bf{A}}_{n}^{t}={\bf{I}}+\sum\limits_{k=1}^{KR}{({{{{\bf{\bar{D}}}}_{k}}+\sum\limits_{n=1}^{N}{\sqrt{\rho}\tilde{\beta}_{n}^{r}}{{{\bf{\bar{g}}}}_{n,k}}{{\bf{u}}_{n}}})^{H}}\\\
\times{\overline{\bf{S}}_{k}}{({{{{\bf{\bar{D}}}}_{k}}+\sum\limits_{n=1}^{N}{\sqrt{\rho}\tilde{\beta}_{n}^{r}}{{{\bf{\bar{g}}}}_{n,k}}{{\bf{u}}_{n}}})}+\sum\limits_{k=1}^{KT}{({1-\rho})}{\bf{u}}_{n}^{H}{{{\bf{\bar{g}}}}^{H}}_{n,k}{\overline{\bf{S}}_{k}}{{{\bf{\bar{g}}}}_{n,k}}{{\bf{u}}_{n}}\\\
+\sum\limits_{k=1}^{KT}{({{{{\bf{\bar{D}}}}^{H}}_{k}+\sum\limits_{l=1,n\neq
l}^{N}{\sqrt{1-\rho}\tilde{\beta}_{l}^{t*}{\bf{u}}_{l}^{H}{{{\bf{\bar{g}}}}^{H}}_{l,k}}})}\end{array}\\\
{\times{{\overline{\bf{S}}}_{k}}({{{{\bf{\bar{D}}}}_{k}}+\sum\limits_{l=1,n\neq
l}^{N}{\sqrt{1-\rho}\tilde{\beta}_{l}^{t}{{{\bf{\bar{g}}}}_{l,k}}{{\bf{u}}_{l}}}})}\end{array}}$
(20)

Input: $\bar{\mathbf{S}}^{(0)}$,
$\left\\{\tilde{\beta}^{r(0)},\tilde{\beta}^{t(0)}\right\\}$ and $\rho^{(0)}$
with feasible values

1

2Set $i\leftarrow 0$

3repeat

4

5 Compute $(\bar{\mathbf{S}}_{k}^{(i+1)})_{k=1}^{K}$ according to Algorithm 2
in [16]

6 for  _$n=1,2,\ldots,N_{\mathrm{ris}}$_ do

7

8 $\tilde{\beta}_{n}^{*(i+1)}=\exp(-j\arg(\sigma_{n}))$ using (21)

9 Apply the phase shift projection procedure, if required, based on (22)

10 end for

11

12
$\rho^{{(i+1)}}=\mathbb{P}_{D}\left({\rho}^{(i)}+{\upsilon^{(i)}}\nabla_{\rho}f(\rho)|_{\rho={\rho^{(i)}}}\right)$

13 $i\leftarrow i+1$;

14 until  _a stopping criterion is met_

Output: ${\mathbf{S}}^{\bullet}={\mathbf{S}}^{(i)}$,
$\rho^{\bullet}=\rho^{(i)}$ and
$\left\\{{{{{\bf{\tilde{\beta}}}}^{r\bullet(i)}},{{{\bf{\tilde{\beta}}}}^{t\bullet(i)}}}\right\\}$

Algorithm 1 AO algorithm to solve (8)

The optimal solution of the optimization problem in (14) is then given by
[16]:

$\displaystyle\tilde{\beta}_{n}^{*r}=\exp(-j\arg(\sigma_{n}^{r}))$ (21a)
$\displaystyle\tilde{\beta}_{n}^{*t}=\exp(-j\arg(\sigma_{n}^{t}))$ (21b)

where $\sigma_{n}^{r}$ and $\sigma_{n}^{t}$ are the only non-zero eigenvalues
of ${\left({{\bf{A}}_{n}^{r}}\right)^{-1}}{\bf{B}}_{n}^{r}$ and
${\left({{\bf{A}}_{n}^{t}}\right)^{-1}}{\bf{B}}_{n}^{t}$, respectively.

The obtained continuous-valued solution of the reflection and transmission
coefficients is denoted by
$\left\\{\tilde{\beta}_{n}^{*r},\tilde{\beta}_{n}^{*t}\right\\}=SC$. The
corresponding discrete-valued solution in the considered feasible set is
obtained by projecting $SC$ on the discrete set $\Psi$. As a case study, we
consider the IOS prototype in [13], where each IOS element can be configured
in two states, where each state is identified by a pair of reflection and
transmission coefficients, i.e.,
$\left\\{({{{\bf{\tilde{\beta}}}}^{r}_{S1}},{{{\bf{\tilde{\beta}}}}^{t}_{S1}}),({{{\bf{\tilde{\beta}}}}^{r}_{S2}},{{{\bf{\tilde{\beta}}}}^{t}_{S2}})\right\\}=\left\\{S1,S2\right\\}=\Psi$.

In this case, the projection can be formulated as follows:

${\left\\{{{{{\bf{\tilde{\beta}}}}^{r\bullet}},{{{\bf{\tilde{\beta}}}}^{t\bullet}}}\right\\}=\left\\{\begin{array}[]{l}\left({\beta_{S1}^{r},\beta_{S1}^{t}}\right),{\left\|{SC-S1}\right\|_{2}}<{\left\|{SC-S2}\right\|_{2}}\\\
\left({\beta_{S2}^{r},\beta_{S2}^{t}}\right),\quad\text{otherwise}\end{array}\right.}$
(22)

where
$\left\\{{{{{\bf{\tilde{\beta}}}}^{r\bullet}},{{{\bf{\tilde{\beta}}}}^{t\bullet}}}\right\\}$
denotes the reflection and transmission coefficients that minimize the
distance between the solution of the continuous-valued case and the feasible
discrete set.

###  IV-C Power Ratio Optimization

For given values of $\left\\{{{{\bar{S}}_{k}^{*}}}\right\\}_{k=1}^{K}$ and
$\left\\{{{{{\bf{\tilde{\beta}}}}^{r\bullet}},{{{\bf{\tilde{\beta}}}}^{t\bullet}}}\right\\}$,
the optimization problem in (8) can be explicitly rewritten as:

$\displaystyle\underset{\rho}{\max}$ $\displaystyle\quad
f(\rho)=\log_{2}\Bigl{|}\mathbf{I}+\sum\limits_{k=1}^{K}\mathbf{H}_{k}^{H}\bar{\mathbf{S}}_{k}\mathbf{H}_{k}\Bigl{|}$
(23a) $\displaystyle\mathrm{s.t.}$ $\displaystyle\quad 0\leq\rho\leq 1$ (23b)

The objective function $f(\rho)$ can be rewritten as:

$\displaystyle
f(\rho)={\log_{2}}\left|{{\bf{I}}+\sum\limits_{k=1}^{K}{{{\bf{H}}_{k}^{H}}{{\overline{\bf{S}}}_{k}}{\bf{H}}_{k}}}\right|$
(24)
${\small=\log_{2}\left|\begin{array}[]{c}\begin{array}[]{*{20}{l}}\begin{array}[]{l}{\bf{I}}+2\sqrt{\rho}\sum\limits_{k=1}^{KR}{\sum\limits_{n=1}^{N}{{{\tilde{\beta}}_{n}}^{r*}{{\bf{u}}_{n}^{H}}{{{\bf{\bar{g}}}}_{n,k}^{H}}{{\overline{\bf{S}}}_{k}}{{{\bf{\bar{D}}}}_{k}}}}\\\
+2\sqrt{1-\rho}\sum\limits_{k=1}^{KT}{\sum\limits_{n=1}^{N}{{{\tilde{\beta}}_{n}}^{t*}}{{\bf{u}}_{n}^{H}}{{{\bf{\bar{g}}}}_{n,k}^{H}}{{\overline{\bf{S}}}_{k}}{{{\bf{\bar{D}}}}_{k}}}\end{array}\\\
+{\sum\limits_{k=1}^{KR}{\left({\begin{array}[]{*{20}{l}}{\sum\limits_{n=1}^{N}{\sum\limits_{n=1}^{N}{\rho{{\tilde{\beta}}_{n}^{r*}}\tilde{\beta}_{n}^{r}{{\bf{u}}_{n}^{H}}}}{{{\bf{\bar{g}}}}_{n,k}^{H}}{{\overline{\bf{S}}}_{k}}{\bf{\bar{g}}}_{n,k}{\bf{u}}_{n}}\\\
{+{{{\bf{\bar{D}}}}_{k}^{H}}{{\overline{\bf{S}}}_{k}}{{{\bf{\bar{D}}}}_{k}}}\end{array}}\right)}}\end{array}\\\
+\sum\limits_{k=1}^{KT}{\left({\begin{array}[]{*{20}{l}}{\sum\limits_{n=1}^{N}{\sum\limits_{n=1}^{N}{(1-\rho){{\tilde{\beta}}_{n}^{t*}}\tilde{\beta}_{n}^{t}}{\bf{u}}_{n}^{H}{\bf{\bar{g}}}_{n,k}^{H}{{\overline{\bf{S}}}_{k}}}{{{\bf{\bar{g}}}}_{n,k}}}{{\bf{u}}_{n}}\\\
{+{{{\bf{\bar{D}}}}_{k}^{H}}{{\overline{\bf{S}}}_{k}}{{{\bf{\bar{D}}}}_{k}}}\end{array}}\right)}\end{array}\right|}$
(25)

In order to find the optimal value of the power ratio $\rho$, we solve the
equation $f^{{}^{\prime}}(\rho)=0$, where $f^{{}^{\prime}}(\rho)$ is the
first-order derivative of $f(\rho)$. For ease of notation, let us define:

${\small{{\bf{X}}_{\bf{1}}}=\sum\nolimits_{k=1}^{KR}{\sum\nolimits_{n=1}^{N}{{{\tilde{\beta}}_{n}}^{r*}{{\bf{u}}_{n}^{H}}{{{\bf{\bar{g}}}}_{n,k}^{H}}{{\overline{\bf{S}}}_{k}}{{{\bf{\bar{D}}}}_{k}}}}}$
(26)
${\small{{\bf{X}}_{\bf{2}}}=\sum\nolimits_{k=1}^{KT}{\sum\nolimits_{n=1}^{N}{{{\tilde{\beta}}_{n}}^{t*}}{{\bf{u}}_{n}^{H}}{{{\bf{\bar{g}}}}_{n,k}^{H}}{{\overline{\bf{S}}}_{k}}{{{\bf{\bar{D}}}}_{k}}}}$
(27)
${\small{{\bf{Y}}_{\bf{1}}}=\sum\nolimits_{k=1}^{KR}{\sum\nolimits_{n=1}^{N}{\sum\nolimits_{n=1}^{N}{{{\tilde{\beta}}_{n}^{r*}}\tilde{\beta}_{n}^{r}{{\bf{u}}_{n}^{H}}}}{{{\bf{\bar{g}}}}_{n,k}^{H}}{{\overline{\bf{S}}}_{k}}{\bf{\bar{g}}}_{n,k}{\bf{u}}_{n}}}$
(28)
${\small{{\bf{Y}}_{2}}=\sum\nolimits_{k=1}^{KT}{\sum\nolimits_{n=1}^{N}{\sum\nolimits_{n=1}^{N}{{{\tilde{\beta}}_{n}^{t*}}\tilde{\beta}_{n}^{t}}{\bf{u}}_{n}^{H}{\bf{\bar{g}}}_{n,k}^{H}{{\overline{\bf{S}}}_{k}}}{{{\bf{\bar{g}}}}_{n,k}}}{{\bf{u}}_{n}}}$
(29) $\displaystyle{\bf{\bar{X}}}$
$\displaystyle={{\bf{I}}+\sum\nolimits_{i=1}^{{KR}}{{{\bf{H}}_{i}}{{\overline{\bf{S}}}_{i}}{\bf{H}}_{i}^{H}}+\sum\nolimits_{j=1}^{{KT}}{{{\bf{H}}_{j}}{{\overline{\bf{S}}}_{j}}{\bf{H}}_{j}^{H}}}$
(30)

Figure 2: Continuous-valued coefficients.

Figure 3: Discrete-valued coefficients.

The first-order and second-order derivatives, $f^{{}^{\prime}}(\rho)$ and
$f^{{}^{\prime\prime}}(\rho)$, respectively, can be formulated as:

${f^{\prime}(\rho)=\nabla_{\rho}f(\rho)=\operatorname{Tr}\left({{{{\bf{\bar{X}}}}^{-1}}\left(\begin{array}[]{l}\left({\frac{{{{\bf{X}}_{1}}}}{{\sqrt{\rho}}}-\frac{{{{\bf{X}}_{2}}}}{{\sqrt{1-\rho}}}}\right)\\\
+\left({{{\bf{Y}}_{1}}-{{\bf{Y}}_{2}}}\right)\end{array}\right)}\right)}$ (31)
${\begin{array}[]{l}f^{\prime\prime}(\rho)=-\frac{1}{{\ln
2}}{\left({INV({\bf{\bar{X}}})}\right)^{2}}\\\
\times\left(\begin{array}[]{l}0.5{\left({\sqrt[-3/2]{\rho}{{\bf{X}}_{\bf{1}}}+\sqrt[-3/2]{{1-\rho}}{{\bf{X}}_{\bf{2}}}}\right){{\bf{\bar{X}}}}}\\\
+{\left({\left({\frac{{{{\bf{X}}_{1}}}}{{\sqrt{\rho}}}-\frac{{{{\bf{X}}_{2}}}}{{\sqrt{1-\rho}}}}\right)+\left({{{\bf{Y}}_{1}}-{{\bf{Y}}_{2}}}\right)}\right)^{2}}\end{array}\right)\end{array}}$
(32)

where $\nabla_{\rho}f(\rho)$ is the gradient of $f(\rho)$ with respect to
$\rho$.

The objective function in (23a) is a concave function with respect to $\rho$,
since the second derivative in (32) is no greater than zero. Therefore, the
optimization problem in (23a) has a single optimum value. However, it is hard
to find a closed-form solution by solving $f^{{}^{\prime}}(\rho)=0$. Thus, we
utilize the projected gradient (PG) method. Specifically, $\rho$ in the
$(i+1)$-th iteration is updated as follows:

$\rho^{i+1}=\mathbb{P}_{D}\left({\rho}^{i}+{\upsilon^{i}}\nabla_{\rho}f(\rho)|_{\rho={\rho^{i}}}\right)$
(33)

where $\upsilon^{i}$ in (33) is the step size that is updated by using the
backtracking line search method in [16], and the projection operator
$\mathbb{P}_{D}$ is defined as:

${\small\hat{\rho}=\mathbb{P}_{D}(\rho)=\left\\{{\begin{array}[]{*{20}{l}}{\rho^{min}}&{\rho^{\bullet}<\rho^{min}}\\\
{\rho^{\bullet}}&{\rho^{min}\leqslant\rho^{\bullet}\leqslant\rho^{max}}\\\
{\rho^{max}}&{{\rho}>\rho^{max}}\end{array}}\right.}$ (34)

where $\rho^{\bullet}$ and $\hat{\rho}$ are the solutions obtained before and
after implementing the projection, and $\rho^{min}$ and $\rho^{max}$ are the
minimum and maximum values for $\rho$, respectively.

##  V Simulation Results

In this section, we evaluate the achievable sum-rate of the proposed
algorithms with the aid of Monte Carlo simulations. Specifically, we compare
the sum-rates under the assumption that the reflection and transmission
coefficients are continuous-valued and independent values, and under the
assumption that they are interlinked and belong to a discrete set.

The simulation setup is the following: the carrier frequency is
$f=2\,\mathrm{GHz}$ (the wavelength is $\lambda=15\,\mathrm{cm}$),
$s_{t}=s_{r}=s_{\mathrm{ris}}=\lambda/2=7.5\,\mathrm{cm}$, the network
topology is given in Fig. 1, $N_{t}=8$, $P=1\,\mathrm{W}$, and
$N_{0}=-110\thinspace\mathrm{dB}$. The IOS consists of
$N_{\mathrm{ris}}=15\times 15=225$ elements placed in a $15\times 15$ square
formation. The users are equipped with $N_{r}=2$ antennas and are randomly
distributed within the disks shown in Fig. 1. The results are averaged over
100 independent channel realizations.

The achievable sum-rate is reported in Fig. 3 and Fig. 3. The proposed
algorithm converges in a few iterations. By closing analyzing the sum-rate of
the users located in the reflection and transmission sides of the IOS, we
observe that the sum-rate loss in Fig. 3 as compared to Fig. 3 is due to the
non-unitary value of the reflection and transmission coefficients for the
considered feasible set in [13, Table 1].

##  VI Conclusion

In this paper, we have proposed optimization algorithms for maximizing the
sum-rate in IOS-aided MIMO broadcast channels. We have optimized the
covariance matrices at the transmitter, the reflection and transmission
coefficients of the IOS, and the ratio between the reflected and refracted
power. The simulations results demonstrate that the presented algorithm
provides a rapid convergence rate in a few iterations.

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  *[BCM]: block coordinate maximization
  *[BS]: base station
  *[ULA]: uniform linear array
  *[i.i.d.]: independent and identically distributed
  *[DPC]: dirty paper coding