# Self-Sustainable Metasurface-Assisted mmWave Indoor Communication System

Zhenyu Li, Ozan Alp Topal, Özlem Tuğfe Demir, Emil Björnson, Cicek Cavdar Z.
Li, O. A. Topal, E. Björnson, and C. Cavdar are with the School of Electrical
Engineering and Computer Science, KTH Royal Institute of Technology,
Stockholm, Sweden (e-mail: {zhenyuli, oatopal, emilbjo, cavdar}@kth.se).Ö. T.
Demir is with the Department of Electrical and Electronics Engineering, TOBB
University of Economics and Technology, Ankara, Türkiye, (e-mail:
ozlemtugfedemir@etu.edu.tr). This work has been funded by Celtic-Next project
RAI-6Green partly supported by Swedish funding agency Vinnova and by Horizon
2020 Beyond5 Project from the ECSEL Joint Undertaking (JU) under grant
agreement No 876124. The JU receives support from the EU Horizon 2020 research
and innovation programme and Vinnova in Sweden. Ö. T. Demir was supported by
2232-B International Fellowship for Early Stage Researchers Programme funded
by the Scientific and Technological Research Council of Türkiye. E. Björnson
was supported by the FFL18-0277 grant from the Swedish Foundation for
Strategic Research.

###### Abstract

In the design of a metasurface-assisted system for indoor environments, it is
essential to take into account not only the performance gains and coverage
extension provided by the metasurface but also the operating costs brought by
its reconfigurability, such as powering and cabling. These costs can present
challenges, particularly in indoor dense spaces (IDSs). A self-sustainable
metasurface (SSM), which retains reconfigurability unlike a static metasurface
(SMS), achieves a lower operating cost than a reconfigurable intelligent
surface (RIS) by being self-sustainable through power harvesting. In this
paper, in order to find a better trade-off between metasurface gain, coverage,
and operating cost, the design and performance of an SSM-assisted indoor
mmWave communication system are investigated. We first simplify the design of
the SSM-assisted system by considering the use of SSMs in a preset-based
manner and the formation of coverage groups by associating SSMs with the
closest user equipments (UEs). We propose a two-stage iterative algorithm to
maximize the minimum data rate in the system by jointly deciding the
association between the UEs and the SSMs, the phase-shifts of the SSMs, and
allocating time resources for each UE. The non-convexities that exist in the
proposed optimization problem are tackled using the feasible point pursuit
successive convex approximation method and the concave-convex procedure. To
understand the best scenario for using SSM, the resulting performance is
compared with that achieved with RIS and SMS. Our numerical results indicate
that SSMs are best utilized in a small environment where self-sustainability
is easier to achieve when the budget for operating costs is tight.

###### Index Terms:

Self-sustainable metasurface, mmWave communication, ray tracing, indoor
communication system.

##  I Introduction

The increasing demand for wireless connectivity in indoor venues presents a
significant challenge due to the dense user equipment (UE) population, which
is 1.5 times higher than the average traffic per UE in outdoor urban areas
[1]. Constrained by the dense obstructions within indoor environments,
achieving adequate coverage for indoor communication systems can further
exacerbate the challenge. We refer to the indoor environment, which has
blockage-rich properties and high UE density, as the indoor dense space (IDS)
[2]. Due to both blockage and the dense population, UE mobility is restricted,
leading to relatively static channel conditions in an IDS. Some commonly seen
IDSs in our daily lives include aircraft cabins and metro wagons. Benefiting
from the high available bandwidth, millimeter wave (mmWave) is considered to
be one of the promising solutions for providing high-quality wireless
connections to many UEs in indoor environments [3]. However, considering the
high blocking attenuation of the mmWave signals [4], it could still be
challenging to design a mmWave communication system in an IDS with a large
coverage area.

One possible solution to the above problem is to use reflecting metasurfaces
[5, 6, 7, 8, 9]. By phase-shifting the impinging signal before it is
reflected, the metasurface can manipulate the beam direction, enabling
indirect connectivity for UEs situated in non-line-of-sight areas relative to
the transmitter. We refer the gain a UE can achieve from this indirect linkage
provided by the metasurface as the metasurface gain. This gain closely depends
on the capability of the metasurface to dynamically adjust its inserted phase-
shifts according to the target’s channel condition, a capability referred to
as reconfigurability. However, according to the existing studies,
reconfigurability comes with the cost of cabling for power supply and
signaling for phase controlling [10], which we refer to as the operating cost.
Given the potential for disruption due to the cabling to the original function
of the indoor environment and even the safety risks associated with powering,
these operating costs are not always affordable when designing a metasurface-
assisted IDS communications system. Thus, there is a trade-off between system
coverage, metasurface gain, and operating costs.

In our previous work [11], we evaluated the trade-off between the operating
costs and the metasurface gain. The system minimum data rate is maximized by
mixedly deploying reconfigurable intelligent surfaces (RISs) and static
metasurfaces (SMSs) under a given operating cost budget. Here, RIS denotes the
metasurface with reconfigurability requiring a high operating cost, while SMS
operates in a static manner with almost no operating costs. The results have
suggested that the performance gap resulting from having reconfigurability can
be mitigated by densely deploying more SMSs. However, this could still be
unaffordable for indoor environments such as IDS, where available space for
deploying large numbers of such devices is limited. In pursuit of achieving a
more optimal balance between trade-offs, as discussed in [12, 13, 14, 15], a
reconfigurable metasurface with the capability to perform energy harvesting to
compensate for the power consumption required by its reconfigurability is
investigated. With reconfigurability yet a much lower operating cost compared
to RIS, this type of metasurface could potentially promise a better solution
within this trade-off. In this paper, we refer to this type of metasurface as
the self-sustainable metasurface (SSM).

The existing study on SSM predominantly concentrates on enhancing system
performance while maintaining a low total system power consumption. [12]
optimizes the sum rate in an SSM-assisted multi-user multiple-input single-
output (MISO) downlink system by jointly designing the beamformer, SSM phase-
shifts, and harvesting scheme. In [13], to improve the efficiency of wireless
power transfer, an absorb-then-reflect scheme is designed to utilize the SSM
in a multi-user system. [14] minimizes the transmit power by jointly selecting
the beamformer, SSM phase-shifts, and reflection amplitude while satisfying
the signal-to-noise ratio at the receiver side and adhering to the energy
budget of the SSM. In [15], the authors propose a performance metric named
energy-data rate outage probability to describe the performance of an SSM-
assisted system. Different harvesting schemes are evaluated in terms of
minimizing the energy-data rate outage probability under different SSM
placement strategies. Advanced as they are, the necessity of using SSM
compared to RIS and SMS still lacks thorough investigation. Moreover, there
remains a lack of research providing synthesized evaluations of SSM in terms
of its benefits and drawbacks. To address this research gap, this paper
introduces an SSM-assisted communication system design within IDS. We propose
a novel optimization algorithm aimed at maximizing the minimum data rate by
jointly selecting SSM’s phase-shifts, SSM-UE associations, and allocating time
resources for UEs. Additionally, we investigate both the necessity and
feasibility of the SSM-assisted system and perform a comprehensive comparison
with RIS-assisted and SMS-assisted systems.

###  I-A Contributions

Considering the capability of maintaining a low operating cost while
preserving the reconfigurability of the metasurface, SSM can be a better
passive alternative to RIS than SMS. In this paper, in terms of finding a
better solution from the trade-off among metasurface gain, operating cost, and
system coverage, we investigate the performance of an SSM-assisted indoor
mmWave communication system. The contributions of this paper are outlined as
follows:

  * •

We present a large-scale SSM-assisted system designed with low computational
complexity for optimization. In this system, the utilization of the SSM is
simplified by contemplating its use by following in a pre-designed pattern and
by considering the associations between the SSMs and UEs based on the
feasibility of achieving self-sustainability.

  * •

We formulate a mixed-integer programming (MIP) optimization problem to
maximize the minimum achievable data rate in the system by jointly deciding
the SSM-UE association, phase-shifts of the SSMs, and allocating time
resources for each UE. To tackle the non-convexity that comes with the
optimization problem, the feasible point pursuit successive convex
approximation (FPP-SCA) and the concave-convex procedure (CCP) are utilized to
convexify the constraints. Moreover, we propose a two-stage iterative
algorithm based on the convexified optimization problem.

  * •

We investigate the feasibility and necessity of utilizing SSM by checking the
SSM contribution, the SSM-UE association, and the self-sustainability
condition for SSMs in different positions.

  * •

Extending our previous work [11], where we investigate the RIS- and SMS-
assisted system, in this work we focus on SSM, which is reconfigurable yet
requires a much lower operating cost than RIS. In assessing the trade-offs
between metasurface gain, operating costs, and system coverage, we conducted a
thorough comparison of resulting data rates employing SSM, RIS, and SMS across
indoor environments with varying sizes. Through our research, we provide a
fresh perspective on better scenarios and methodologies for deploying SSM over
other types of metasurfaces.

###  I-B Structure

The paper is organized as follows. In Section II we describe the SSM-assisted
system and model the self-sustainable constraint of an SSM. In Section III, we
formulate the SSM-assisted data rate optimization problem and describe the
simplifications we use to lower the computational complexity of the formulated
optimization problem. Later in Section IV, we describe the convexification
process of the optimization problem and propose an iterative SSM-assisted data
rate optimization algorithm that maximizes the minimum achievable data rate. A
ray tracing (RT) environment and the design of the coverage group and preset
based on the obtained channel conditions are provided in Section V. The
numerical results after optimizing the SSM-assisted system are described in
Section VI. Lastly, we conclude the paper in Section VII.

The mathematical notations used in this paper are summarized as follows.
$\mathbb{B}$ denotes the binary domain, and $\mathbb{C}$ denotes the complex
domain. $\|\cdot\|$ represents the $l_{2}$ norm, $\|\cdot\|_{F}$ represents
the Frobenius norm, $|\cdot|$ represents the magnitude of a complex value.
Additionally, $(\cdot)^{*}$ and $(\cdot)^{T}$ represent the conjugate and
transpose operation. Moreover, $\operatorname{diag}(\cdot)$ represents the
diagonal matrix extension of a vector, and $\mathbb{E}\\{\cdot\\}$ represents
the expected value of a random variable. Lastly, $\mathbf{1}_{M}$ and
$\mathbf{0}_{M}$ indicate a $M\times 1$ vector with all its entries equal to 1
and 0 respectively. $[\cdot]_{m}$ takes the value of the $m$-th entry of the
vector.

##  II System model

We consider the downlink of a mmWave communication system that consists of one
BS that serves multiple UEs located in an IDS. The BS is equipped with $N$
antennas in the form of an $\sqrt{N}\times\sqrt{N}$ uniform planar array
(UPA), while each UE is equipped with a single antenna. We denote the UE
indices as $k\in\\{1,2,\ldots,K\\}=\mathcal{K}$ where $K$ is the total number
of UEs. To alleviate interference, the UEs are assumed to be served
orthogonally in time, where $\tau_{k}$ is the time portion allocated to UE
$k$, and $\sum_{k=1}^{K}\tau_{k}=1$. Taking into account the hardware
regulations, the phase-shifts switching speed of a metasurface is restricted.
Thus the allocated time portion is limited in the range of
$\tau_{k}\in[\tau_{\text{min}},1]$ where $0<\tau_{\text{min}}<1$ is the
minimum allowable time portion.

$L$ metasurfaces are deployed in the IDS to improve the downlink rates of the
UEs, where each of them has $M$ elements in the form of an
$\sqrt{M}\times\sqrt{M}$ UPA. The indices of the elements are denoted as
$m\in\\{1,2,\ldots,M\\}=\mathcal{M}$. The index of an arbitrary surface is
denoted as $l\in\\{1,2,\ldots,L\\}=\mathcal{L}$. Those metasurfaces could be
SSMs, but also SMSs or RISs as will be explained in the later analysis. As
illustrated in Fig. 1, the direct channel from the BS to UE $k$ is denoted as
$\mathbf{h}_{k}\in\mathbb{C}^{N}$. Moreover,
$\mathbf{G}_{l}\in\mathbb{C}^{M\times N}$ and
$\mathbf{g}_{l,k}\in\mathbb{C}^{M}$ denote the channel from the BS to the
metasurface $l$ and from the metasurface $l$ to the UE $k$, respectively. Due
to the static UEs and geometry dependence of the mmWave links in the IDS, we
consider the channels to be deterministic and fixed [2]. Later, we obtain
realistic channel coefficients for a specific environment by using RT
simulations. To achieve self-sustainability, as illustrated in Fig. 2,
metasurfaces with the element splitting scheme [15] are considered. Each
element of the metasurface can select its working mode between _reflecting_
and _harvesting_. We define the element working mode indicator vector as
$\boldsymbol{\beta}_{l,k}\in\mathbb{B}^{M}$, the $m$-th entry of the vector,
$\beta_{l,k,m}$, indicates the working mode of the $m$-th element of
metasurface $l$ during the serving time of UE $k$. $\beta_{l,k,m}=1$ indicates
that the $m$-th element of metasurface $l$ is in the reflecting mode when
serving UE $k$, and $\beta_{l,k,m}=0$ indicates it is in the harvesting mode.
When the reflecting mode is selected for a specific element, the element will
be configured to shift the phase of the impinging signal. The power cost of a
metasurface element to work under the reflecting mode is denoted as
$P^{\text{Rf}}$.

When the harvesting mode is selected for an element, the energy of the
impinging signal will be harvested via its harvesting circuit. We assume the
signals that impinge on the harvesting elements can be fully captured and no
signal will be reflected back. The resulting response vector for metasurface
$l$ when serving UE $k$ is given as $\boldsymbol{\phi}_{l,k}\in\mathbb{C}^{M}$

$\boldsymbol{\phi}_{l,k}=\operatorname{diag}(\boldsymbol{\varphi}_{l,k})\boldsymbol{\beta}_{l,k}$
(1)

where $\boldsymbol{\varphi}_{l,k}\in\mathbb{C}^{M}$ is the complex response
vector of the phase-shift. In this work, the metasurface reflection loss is
neglected, thus $|\varphi_{l,k,m}|=1,\forall l,k,m$ holds, where
$\varphi_{l,k,m}$ is the $m$-th entry of $\boldsymbol{\varphi}_{l,k}$.

Figure 1: Metasurface-assisted indoor communication system model. Figure 2:
Illustration of the SSM with element splitting scheme.

We aggregate the BS-metasurface-UE channel for mathematical convenience.
Defining the aggregated channel from BS via metasurface $l$ to UE $k$ as
$\mathbf{H}_{l,k}=\mathbf{G}_{l}^{T}\bar{\mathbf{G}}_{l,k}\in\mathbb{C}^{N\times
M}$, where $\bar{\mathbf{G}}_{l,k}=\operatorname{diag}(\mathbf{g}_{l,k})$, the
received signal at UE $k$ becomes

$y_{k}=\bigg{(}\mathbf{h}_{k}+\sum_{l=1}^{L}\underbrace{\mathbf{G}_{l}^{T}\bar{\mathbf{G}}_{l,k}}_{\triangleq\mathbf{H}_{l,k}}\boldsymbol{\phi}_{l,k}\bigg{)}^{T}\mathbf{w}_{k}x_{k}+n_{k},$
(2)

where $x_{k}\in\mathbb{C}$ is the transmitted data symbol for UE $k$ and the
transmit power is $\mathbb{E}\\{|x_{k}|^{2}\\}=P$, $\forall k$. The unit-norm
precoding vector $\mathbf{w}_{k}\in\mathbb{C}^{N}$ is selected when serving UE
$k$. The independent receiver noise is denoted as
$n_{k}\sim\mathcal{CN}(0,BN_{0})$, where $B$ is the communication bandwidth in
Hz and $N_{0}$ is the noise spectral density in W/Hz.

Since there is no interference due to the orthogonal scheduling of UEs, a good
precoding strategy for the BS is maximum ratio transmission (MRT). The MRT
precoder for the overall channel from the BS to UE $k$ is

$\mathbf{w}_{k}=\frac{\left(\mathbf{h}_{k}+\sum_{l=1}^{L}\mathbf{H}_{l,k}\boldsymbol{\phi}_{l,k}\right)^{*}}{\left\|\mathbf{h}_{k}+\sum_{l=1}^{L}\mathbf{H}_{l,k}\boldsymbol{\phi}_{l,k}\right\|}.$
(3)

The resulting signal-to-noise ratio (SNR) of UE $k$ is

$\Gamma_{k}=\frac{\left\|\mathbf{h}_{k}+\sum_{l=1}^{L}\mathbf{H}_{l,k}\boldsymbol{\phi}_{l,k}\right\|^{2}P}{BN_{0}},$
(4)

and

$R_{k}=\tau_{k}B\log_{2}(1+\Gamma_{k})\quad\text{bit/s}$ (5)

is the data rate achieved by UE $k$.

###  II-A Self-sustainability of the metasurface

In this work, we do not consider any power storage device or centralized power
distribution device for the system for minimizing the operating cost purpose.
Thus, to achieve self-sustainability, a metasurface should harvest enough
power from the impinging radio waves to support its operation. The signals
that are received by the harvesting elements of the metasurface $l$ when the
system is serving UE $k$ can be expressed as

$\mathbf{y}_{l,k}^{\text{Rc}}=\operatorname{diag}(\mathbf{1}_{M}-\boldsymbol{\beta}_{l,k})\mathbf{G}_{l}\mathbf{w}_{k}x_{k}$
(6)

and the power received by the metasurface $l$ when the system is serving UE
$k$ is given as

$P^{\text{Rc}}_{l,k}=\mathbb{E}\left\\{\|\mathbf{y}_{l,k}^{\text{Rc}}\|^{2}\right\\}=\sum_{m=1}^{M}(1-\beta_{l,k,m})\left|\mathbf{\bar{g}}_{l,m}^{T}\mathbf{w}_{k}\right|^{2}P.$
(7)

Here, the power received from the thermal noise is neglected. Moreover,
$\mathbf{\bar{g}}^{T}_{l,m}\in\mathbb{C}^{1\times N}$ denotes the $m$-th row
of $\mathbf{G}_{l}$. Note that due to the dependency between the selection of
$\boldsymbol{\phi}_{l,k}$ and the beamforming vector $\mathbf{w}_{k}$,
according to (7) the received power will also depend on the phase-shifts
selection.

The harvested power is calculated by referring to the non-linear energy
harvesting model proposed in [16] as

$P_{l,k}^{\text{Hr}}=\frac{q_{1}P_{l,k}^{\text{Rc}}}{q_{2}P_{l,k}^{\text{Rc}}+q_{3}},$
(8)

where $q_{1}$, $q_{2}$, and $q_{3}$ are positive constants that are related to
the efficiency of the energy harvesting hardware. This model not only
accurately captures the non-linear characteristic of a real energy harvesting
circuit [17], but also provide convenience for mathematical processing.

##  III Data rate optimization problem with SSM

For fairness, we aim to maximize the minimum achievable data rate in the IDS.
It will be maximized by tuning the phase-shifts of the metasurfaces, selecting
working modes for each metasurface, and allocating time resources with respect
to each UE. The max-min rate optimization problem is formulated as

$\displaystyle\textbf{P1}:\operatornamewithlimits{maximize}_{\\{\boldsymbol{\varphi}_{l,k},\boldsymbol{\beta}_{l,k},\tau_{k}\\}}\,\operatornamewithlimits{min}_{k\in\mathcal{K}}\,R_{k}$
(9a) $\displaystyle\text{s.t.}\ \sum_{k=1}^{K}\tau_{k}\leq 1,$ (9b)
$\displaystyle\hskip 17.07164pt\tau_{k}\geq\tau_{\text{min}},\quad\forall k,$
(9c) $\displaystyle\hskip 17.07164pt|\varphi_{l,k,m}|=1,\quad\forall l,k,m,$
(9d) $\displaystyle\hskip 17.07164pt\beta_{l,k,m}\in\\{0,1\\},\quad\forall
l,k,m,$ (9e) $\displaystyle\hskip
17.07164ptP_{l,k}^{\text{Hr}}\geq\sum_{m=1}^{M}\beta_{l,k,m}P^{\text{Rf}},\quad\forall
l,k.$ (9f)

By maximizing the objective function in (9a) the minimum achievable data rate
among the UEs improved. The constraint in (9b) makes sure the sum of the
allocated time portions does not exceed one and the constraints in (9c) ensure
that the individual allocated time portion is not shorter than the minimum
portion $\tau_{\text{min}}$. The lossless reflection of the metasurface is
ensured by the constraints in (9d). The binary working mode selection is
guaranteed by the constraints in (9e). Finally, the self-sustainability is
achieved by fulfilling the constraints in (9f). Due to the involvement of the
binary variable, P1 is formulated as a mixed-integer programming problem.

###  III-A Simplifying the utilization of SSM

Despite the non-convexity associated with the constraints in P1, the main
issue that makes it challenging to solve is the computational complexity. The
computational complexity of the mixed-integer programming problem P1 increases
with the number of integer variables such as $\beta_{l,k,m}$. Given the dense
population of the IDS, a large number of integer variables is anticipated,
potentially rendering the problem computationally infeasible to solve. To
tackle those issues, in this subsection, we consider the working mode
selection of the SSM is determined according to a pre-designed pattern, which
we will refer to later as the preset. Moreover, we recast the working mode
selection problem into an SSM-UE association problem, which greatly reduces
the computational complexity.

####  III-A1 Preset-based SSM

For the sake of reducing the number of binary variables in the problem, we
consider the mode switching of the SSMs to follow a pre-given
$\boldsymbol{\beta}_{l,k}$ for each SSM $l$ when serving different UE $k$,
which we refer to as the _preset_. The same preset is considered to be
utilized when SSMs are serving different UEs. Thus,
$\boldsymbol{\beta}_{l,k}=\boldsymbol{\beta}_{l},\forall k$. Moreover, for
simplicity and ease of analysis, we consider the preset design shown in Fig.
3, where the SSM is divided into the _reflecting area_ and the _harvesting
area_. The reflecting area of the SSM $l$ consists of $\bar{M}_{l}$ number of
reflecting elements that are of the form of a
$\sqrt{\bar{M}_{l}}\times\sqrt{\bar{M}_{l}}$ UPA. The remaining elements,
which are not in the reflective area, work to harvest power and form the
harvesting area. We assume that the reflecting area is always located at the
lower left corner of the SSM regardless of their serving target. Notice that
only as a simple yet efficient way of using SSM, with this preset-based
simplification, optimality of performance is not guaranteed. However, this not
only eliminates the binary variable $\beta_{l,k,m}$, which greatly reduces the
computational complexity but also makes it easier to compare performance with
other types of metasurfaces for later analysis.

Figure 3: Illustration of the preset design.

Considering the SSM is used based on a given preset, $\beta_{l,k,m}$ is
treated as a constant in the later content, and the equation
$\sum_{m=1}^{M}\beta_{l,k,m}=\bar{M}_{l}$ holds $\forall l$ and $\forall k$.
The sets that consist of the indices of the elements in the reflecting area
and in the harvesting area are denoted as $\bar{\mathcal{M}}_{l}$ and
$\hat{\mathcal{M}}_{l}$, respectively. Those sets satisfy that
$\bar{\mathcal{M}}_{l}\cup\hat{\mathcal{M}}_{l}=\mathcal{M}$ and
$\bar{\mathcal{M}}_{l}\cap\hat{\mathcal{M}}_{l}=\varnothing$.

####  III-A2 SSM-UE association

To avoid violating the self-sustainable constraint for a given preset, we
introduce a binary SSM-UE association indicator $\alpha_{l,k}\in\mathbb{B}$ to
indicate whether SSM $l$ will be assigned to serve UE $k$. Note that the
number of SSM-UE association indicators can still be very high in an IDS
communication network. Considering the obstacle-rich property of the IDS and
the high attenuation of the mmWave propagation, metasurfaces generally will
not be able to offer significant help to a UE that is far away from them. Due
to the transmit beamforming, it is also hard for them to capture enough energy
to achieve self-sustainability when the system is serving a UE that is in the
opposite direction of them. To further reduce the computational complexity, we
first manually assign SSMs to UEs based on the coverage of the SSM to form
multiple groups, which we refer to as the _coverage group_. Then the SSM-UE
association inside the coverage group will be decided jointly while optimizing
the minimum data rate.

We denote the index of the coverage group as $i\in\\{1,\ldots,I\\}$, where
there are $I$ coverage groups in total. Metasurfaces and UEs in group $i$ are
indexed as $l\in\mathcal{L}_{i}\subseteq\mathcal{L}$ and
$k\in\mathcal{K}_{i}\subseteq\mathcal{K}$ respectively. In particular, we
point out that the same SSM can be assigned to serve UEs in different coverage
groups to make the best use of their coverage. In other words, the
intersection of two index sets $\mathcal{L}_{i}~{}\mathcal{L}_{j},~{}\forall
i\neq j$ does not necessarily have to be an empty set. Each group $i$ contains
$L_{i}$ metasurfaces and $K_{i}$ UEs. The SNR of UE $k$, which is in coverage
group $i$, can be expressed as

$\Gamma_{k}=\frac{\left\|\mathbf{h}_{k}+\sum_{l\in\mathcal{L}_{i}}\alpha_{l,k}\mathbf{H}_{l,k}\boldsymbol{\phi}_{l,k}\right\|^{2}P}{BN_{0}}.$
(10)

###  III-B Joint data rate optimization algorithm with SSM

With the above mechanisms considered, P1 is recast as two sub-problems P2-1
and P2-2. The objective of the sub-problem P2-1 is to maximize the minimum
achievable data rate in each coverage group by jointly deciding the phase-
shifts, SSM-UE association, and allocation of time resources inside the
working group. The sub-problem P2-1 is given as

$\displaystyle\textbf{P2-1}:\operatornamewithlimits{maximize}_{\\{\boldsymbol{\varphi}_{l,k},\alpha_{l,k},\tilde{\tau}_{k}\\}}\min_{k\in\mathcal{K}_{i}}\tilde{R}_{k}\
$ (11a) $\displaystyle\text{s.t.
}\sum_{k\in\mathcal{K}_{i}}\tilde{\tau}_{k}\leq 1,$ (11b) $\displaystyle\hskip
17.07164pt\tilde{\tau}_{k}\geq\tau_{\text{min}},\quad\forall
k\in\mathcal{K}_{i},$ (11c) $\displaystyle\hskip
17.07164pt|\varphi_{l,k,m}|=1,\quad\forall
l\in\mathcal{L}_{i},k\in\mathcal{K}_{i},m,$ (11d) $\displaystyle\hskip
17.07164pt\alpha_{l,k}\in\\{0,1\\},\quad\forall
l\in\mathcal{L}_{i},k\in\mathcal{K}_{i},$ (11e) $\displaystyle\hskip
17.07164ptP_{l,k}^{\text{Hr}}\geq\alpha_{l,k}\bar{M}_{l}P^{\text{Rf}},\quad\forall
l\in\mathcal{L}_{i},k\in\mathcal{K}_{i},$ (11f)

where $\tilde{\tau}_{k}$ is the allocated time portion inside the coverage
group. Self-sustainability is not a concern for the unassigned SSM, so when
$\alpha_{l,k}=0$, the constraint (11f) is automatically satisfied.
$\tilde{R}_{k}$ is the resulting data rate after allocating the time resources
in the coverage group, and is given as

$\displaystyle\tilde{R}_{k}(\boldsymbol{\phi}_{l,k},\alpha_{l,k},\tilde{\tau}_{k})=\tilde{\tau}_{k}B\log_{2}\bigg{(}1+$
$\displaystyle\hskip
17.07164pt\frac{\|\mathbf{h}_{k}+\sum_{l\in\mathcal{L}_{i}}\alpha_{l,k}\mathbf{H}_{l,k}\boldsymbol{\phi}_{l,k}\|^{2}P}{BN_{0}}\bigg{)},\quad\forall
k\in\mathcal{K}_{i}.$ (12)

Notice that the binary variable $\alpha_{l,k}$ is coupled with the complex
variables $\boldsymbol{\varphi}_{l,k}$ inside the expression of
$\tilde{R}_{k}$ which brings non-convexity to the problem. In the same
expression, the quadratic form inside the logarithm and the coupling related
to $\tilde{\tau}_{k}$ makes the problem non-convex. Besides, in (11f) due to
the complex form of the MRT precoder and the energy harvesting model, the
binary SSM-UE association indicator $\alpha_{l,k}$ and metasurface responses
$\boldsymbol{\phi}_{l,k}$ are coupled in a non-convex way.

In sub-problem P2-2, the time resources are allocated in the whole network in
terms of maximizing the minimum data rate among the whole system. We denote
the output phase-shifts, SSM-UE association, and allocated intra time portion
after finding the solution to P2-1 for each coverage group $i$ as
$\boldsymbol{\phi}_{l,k}^{*}$, $\alpha_{l,k}^{*}$, and $\tilde{\tau}_{k}^{*}$,
respectively. We denote the network-wide allocated time portion as
$\hat{\tau}_{k}\in\mathbb{R}$. The resulting achievable data rate for each UE
in the network is expressed as

$R_{k}=\hat{\tau}_{k}\tilde{R}_{k}(\boldsymbol{\phi}_{l,k}^{*},\alpha_{l,k}^{*},\tilde{\tau}_{k}^{*}),\quad\forall
k.$ (13)

Sub-problem P2-2 is given as

$\displaystyle\textbf{P2-2}:\operatornamewithlimits{maximize}_{\\{\hat{\tau}_{k}\\}}\quad\min_{k\in\mathcal{K}}R_{k}$
(14a) $\displaystyle\text{s.t.
}\hat{\tau}_{k}\tilde{\tau}_{k}^{*}\geq\tau_{\text{min}},\quad\forall k,$
(14b) $\displaystyle\hskip
14.22636pt\sum_{k\in\mathcal{K}}\hat{\tau}_{k}\tilde{\tau}_{k}^{*}\leq 1.$
(14c)

Except for the non-convexities in P2-1, sub-problem P2-2 is formatted as a
linear programming (LP) problem, which can be easily solved by any convex
programming solver.

In the following section, we will tackle the non-convexities in P2-1 and
introduce a solvable iterative optimizing algorithm.

##  IV Iterative joint data rate optimization algorithm

We introduce the variable $\tilde{R}_{i}^{\text{min}}$ that represents the
minimum data rate after allocating time resources within the coverage group
$i$ and recast P2-1 as

$\displaystyle\textbf{P3-1}:\operatornamewithlimits{maximize}_{\\{\boldsymbol{\varphi}_{l,k},\alpha_{l,k},\tilde{\tau}_{k}\\}}\tilde{R}_{i}^{\text{min}}$
(15a) $\displaystyle\text{s.t.
}\tilde{R}_{k}(\boldsymbol{\phi}_{l,k},\alpha_{l,k},\tilde{\tau}_{k})\geq\tilde{R}_{i}^{\text{min}},\quad\forall
k\in\mathcal{K}_{i},$ (15b) $\displaystyle\hskip
14.22636pt\sum_{k\in\mathcal{K}_{i}}\tilde{\tau}_{k}\leq 1,\quad$ (15c)
$\displaystyle\hskip
14.22636pt\tilde{\tau}_{k}\geq\tau_{\text{min}},\quad\forall
k\in\mathcal{K}_{i},$ (15d) $\displaystyle\hskip
14.22636pt|\varphi_{l,k,m}|=1,\quad\forall
l\in\mathcal{L}_{i},k\in\mathcal{K}_{i},m,$ (15e) $\displaystyle\hskip
14.22636pt\alpha_{l,k}\in\\{0,1\\},\quad\forall
l\in\mathcal{L}_{i},k\in\mathcal{K}_{i},$ (15f) $\displaystyle\hskip
14.22636ptP_{l,k}^{\text{Hr}}\geq\alpha_{l,k}\bar{M}_{l}P^{\text{Rf}},\quad\forall
l\in\mathcal{L}_{i},k\in\mathcal{K}_{i}.$ (15g)

To remove the coupling between the binary preset selector $\alpha_{l,k}$ and
the complex continuous phase-shifts $\boldsymbol{\phi}_{l,k}$, we introduce an
auxiliary variable $\mathbf{z}_{l,k}\in\mathbb{C}^{M}$ to replace the term
$\alpha_{l,k}\boldsymbol{\phi}_{l,k}$ with following constraint

$\displaystyle\left|\left[\mathbf{z}_{l,k}\right]_{m}\right|\leq\alpha_{l,k},\quad\forall
l\in\mathcal{L}_{i},k\in\mathcal{K}_{i},m.$ (16)

When $\alpha_{l,k}=1$, each entry of $\mathbf{z}_{l,k}$ will be bounded inside
the unit circle. The optimal solution is observed to be found on the unit
circle, thus fulfilling the constraint (15e). When $\alpha_{l,k}=0$,
$\mathbf{z}_{l,k}$ will be forced to be $\mathbf{0}_{M}$, thus canceling the
contribution from SSM $l$ to UE $k$.

Next, we resolve the non-convexities in (15b). We replace
$\tilde{R}_{i}^{\text{min}}$ with $\tilde{r}_{i}^{\text{min}}$ in (15a) and
(15b), where $\tilde{r}_{i}^{\text{min}}=\sqrt{\tilde{R}_{i}^{\text{min}}}$.
This will enable the construction of a convex form below. Moreover, since the
square root is a monotonically increasing function, this replacement will not
change the solution to the optimization problem. Next, we introduce the
auxiliary variable $e_{k}\in\mathbb{R}$, and (15b) can be written as

$e_{k}\geq
2^{\frac{\left(\tilde{r}_{i}^{\text{min}}\right)^{2}}{B\tilde{\tau}_{k}}},\quad\forall
k\in\mathcal{K}_{i},$ (17)
$1+\frac{\left\|\mathbf{h}_{k}+\sum_{l\in\mathcal{L}_{i}}\mathbf{H}_{l,k}\mathbf{z}_{l,k}\right\|^{2}P}{BN_{0}}\geq
e_{k},\quad\forall k\in\mathcal{K}_{i}.$ (18)

Furthermore, we introduce another auxiliary variable $f_{k}\in\mathbb{R}$ in
place of the quadratic-over-linear term that appears at the right side of (17)
as

$e_{k}\geq 2^{f_{k}},\quad\forall k\in\mathcal{K}_{i},$ (19)
$f_{k}\geq\frac{\left(\tilde{r}^{\text{min}}_{i}\right)^{2}}{B\tilde{\tau}_{k}},\quad\forall
k\in\mathcal{K}_{i},$ (20)

where (19) is in the form of an exponential cone constraint and (20) can be
further formatted as a second-order cone (SOC) constraint as

$f_{k}+\tilde{\tau}_{k}\geq\left\|\begin{bmatrix}\sqrt{2/B}\tilde{r}^{\text{min}}_{i}&f_{k}&\tilde{\tau}_{k}\end{bmatrix}\right\|,\quad\forall
k\in\mathcal{K}_{i}.$ (21)

With the above processes, the coupling related to $\tilde{\tau}_{k}$ has now
been removed. However with the quadratic form appearing at the left side of
the inequality in (18), the constraint is still not convex. To deal with this
issue, we first expand the norm-square term as

$\displaystyle\mathbf{z}_{k}^{H}\underbrace{\mathbf{H}_{k}^{H}\mathbf{H}_{k}}_{\triangleq\mathbf{A}_{k}}\mathbf{z}_{k}+2\Re\left(\mathbf{z}_{k}^{H}\underbrace{\mathbf{H}_{k}^{H}\mathbf{h}_{k}}_{\triangleq\mathbf{b}_{k}}\right)+\underbrace{\mathbf{h}_{k}^{H}\mathbf{h}_{k}}_{\triangleq
c_{k}}$ $\displaystyle\hskip
11.38109pt\geq\frac{BN_{0}(e_{k}-1)}{P},\quad\forall k\in\mathcal{K}_{i},$
(22)

where
$\mathbf{H}_{k}=[\mathbf{H}_{l_{i,1},k},\ldots,\mathbf{H}_{l_{i,L_{i}},k}]\in\mathbb{C}^{N\times
L_{i}M}$ and
$\mathbf{z}_{k}=[\mathbf{z}^{T}_{l_{i,1},k},\ldots,\mathbf{z}^{T}_{l_{i,L_{i}},k}]^{T}\in\mathbb{C}^{L_{i}M}$.
Here, $l_{i,j}$ denotes the index of the $j$th element in $\mathcal{L}_{i}$.
Next, we utilize the FPP-SCA algorithm [18] to relax (22). With
$\mathbf{A}_{k}$ as a postive semi-definite matrix, for any arbitrary vector
$\boldsymbol{\zeta}_{k}$ it holds that

$\mathbf{z}_{k}^{H}(-\mathbf{A}_{k})\mathbf{z}_{k}\leq
2\Re\left(\boldsymbol{\zeta}_{k}^{H}(-\mathbf{A}_{k})\mathbf{z}_{k}\right)-\boldsymbol{\zeta}_{k}^{H}(-\mathbf{A}_{k})\boldsymbol{\zeta}_{k},\forall
k\in\mathcal{K}_{i}.$ (23)

At each iteration $\epsilon$, by inserting the non-negative slack variable
$s_{k}$ and replacing $\boldsymbol{\zeta}_{k}$ with the previously obtained
solution $\mathbf{z}_{k}^{(\epsilon-1)}$, we can replace the quadratic term in
(18) with its affine approximation as

$\displaystyle-2\Re\left(\left(\mathbf{z}_{k}^{(\epsilon-1)}\right)^{H}\mathbf{A}_{k}\mathbf{z}_{k}\right)+\left(\mathbf{z}_{k}^{(\epsilon-1)}\right)^{H}\mathbf{A}_{k}\mathbf{z}_{k}^{(\epsilon-1)}$
$\displaystyle\hskip
2.84526pt-2\Re\left(\mathbf{z}_{k}^{H}\mathbf{b}_{k}\right)\leq
s_{k}+c_{k}+\frac{BN_{0}}{P}(1-e_{k}),\quad\forall k\in\mathcal{K}_{i},$ (24)
$s_{k}\geq 0,\quad\forall k\in\mathcal{K}_{i}.$ (25)

Next, we tackle the non-convexity in the self-sustainable constraint.
Introducing an auxiliary variable $u_{l,k}$ and making it larger than or equal
to the consumed power but less than or equal to the harvested power, the
inequality (15g) is equivalent to the following constraints:

$\alpha_{l,k}\bar{M}_{l}P^{\text{Rf}}\leq u_{l,k},\quad\forall
l\in\mathcal{L}_{i},k\in\mathcal{K}_{i},$ (26)
$\displaystyle\frac{q_{1}\sum_{m=1}^{M}(1-\beta_{l,k,m})|\bar{\mathbf{g}}_{l,m}^{T}\mathbf{w}_{k}|^{2}P}{q_{2}\sum_{m=1}^{M}(1-\beta_{l,k,m})|\bar{\mathbf{g}}_{l,m}^{T}\mathbf{w}_{k}|^{2}P+q_{3}}$
$\displaystyle\hskip 8.53581pt\geq u_{l,k},\quad\forall
l\in\mathcal{L}_{i},k\in\mathcal{K}_{i}.$ (27)

Notice that $\mathbf{w}_{k}$ is in a non-convex functional relationship with
$\mathbf{z}_{l,k}$, thus (27) is not convex. To tackle this, we multiply both
the denominator and numerator of the left side of (27) by the term
$\|\mathbf{h}_{k}+\mathbf{H}_{k}\mathbf{z}_{k}\|^{2}$. We further introduce
auxiliary variables $t_{l,k,m}\in\mathbb{R}$ and $d_{k}\in\mathbb{R}$ with
following constraints:

$\displaystyle|\bar{\mathbf{g}}_{l,m}^{H}(\mathbf{h}_{k}+\mathbf{H}_{k}\mathbf{z}_{k})|^{2}P\geq
t_{l,k,m},\quad\forall l\in\mathcal{L}_{i},k\in\mathcal{K}_{i},m,$ (28)
$\displaystyle\|\mathbf{h}_{k}+\mathbf{H}_{k}\mathbf{z}_{k}\|^{2}\leq
d_{k},\quad\forall k\in\mathcal{K}_{i},$ (29) $\displaystyle
t_{l,k,m}\leq\mathcal{C}(1-\beta_{l,k,m}),\quad\forall
l\in\mathcal{L}_{i},k\in\mathcal{K}_{i},m,$ (30)

where $\mathcal{C}$ is a sufficiently large positive number. When
$\beta_{l,k,m}=1$, the constraint (30) will force $t_{l,k,m}=0$, thus avoiding
that any reflecting element contributes to the energy harvesting. When
$\beta_{l,k,m}=0$, the large constant $\mathcal{C}$ makes sure that
$t_{l,k,m}$ will not be bounded by (30). With all the newly introduced
auxiliary variables injected, (27) can be expressed as

$\frac{q_{1}\sum_{m=1}^{M}t_{l,k,m}}{q_{2}\sum_{m=1}^{M}t_{l,k,m}+q_{3}d_{k}}\geq
u_{l,k},\quad\forall l\in\mathcal{L}_{i},k\in\mathcal{K}_{i}.$ (31)

However, after replacement, there are still constraints expressed in a non-
convex way. The quadratic term appears at the left side of the inequality
constraint (28) still brings non-convexity, and the fractional term at the
left side of the inequality constraint (31) also introduces non-convexity.

We first handle the non-convexity in (28). Similar to (18), where the
inequality is formulated as a quadratic term greater than or equal to an
affine term, (28) can also be convexified using FPP-SCA. We first expand the
quadratic term as

$\displaystyle\underbrace{\mathbf{h}_{k}^{H}\mathbf{\bar{g}}_{l,m}\mathbf{\bar{g}}_{l,m}^{H}\mathbf{h}_{k}}_{\triangleq\hat{c}_{l,k,m}}+2\Re\left(\mathbf{z}_{k}^{H}\underbrace{\mathbf{H}_{k}^{H}\mathbf{\bar{g}}_{l,m}\mathbf{\bar{g}}_{l,m}^{H}\mathbf{h}_{k}}_{\triangleq\hat{\mathbf{b}}_{l,k,m}}\right)$
$\displaystyle+\mathbf{z}_{k}^{H}\underbrace{\mathbf{H}_{k}^{H}\mathbf{\bar{g}}_{l,m}\mathbf{\bar{g}}_{l,m}^{H}\mathbf{H}_{k}}_{\hat{\mathbf{A}}_{l,k,m}}\mathbf{z}_{k}\geq\frac{t_{l,k,m}}{P},\quad\forall
l\in\mathcal{L}_{i},k\in\mathcal{K}_{i},m.$ (32)

Then (28) can be approximated as an affine inequality with the non-negative
slack variable $s_{l_{i},k_{i},m}^{\prime}$ as

$\displaystyle-2\Re\left(\left(\mathbf{z}_{k}^{(\epsilon-1)}\right)^{H}\hat{\mathbf{A}}_{l,k,m}\mathbf{z}_{k}\right)$
$\displaystyle\hskip
5.69054pt+\left(\mathbf{z}_{k}^{(\epsilon-1)}\right)^{H}\hat{\mathbf{A}}_{l,k,m}\mathbf{z}_{k}^{(\epsilon-1)}-2\Re\left(\mathbf{z}_{k}^{H}\hat{\mathbf{b}}_{l,k,m}\right)\leq
s_{l,k,m}^{\prime}$ $\displaystyle\hskip
5.69054pt+\hat{c}_{l,k,m}-\frac{t_{l,k,m}}{P},\quad\forall
l\in\mathcal{L}_{i},k\in\mathcal{K}_{i},m$ (33) $s_{l,k,m}^{\prime}\geq
0,\quad\forall l\in\mathcal{L}_{i},k\in\mathcal{K}_{i},m.$ (34)

Then we tackle the non-convexity in (31). We reformulate (31) as

$\displaystyle\left(u_{l,k}+q_{2}\sum_{m=1}^{M}t_{l,k,m}+q_{3}d_{k}\right)^{2}\leq
2q_{1}\sum_{m=1}^{M}t_{l,k,m}$
$\displaystyle+u_{l,k}^{2}+\left(q_{2}\sum_{m=1}^{M}t_{l,k,m}+q_{3}d_{k}\right)^{2},\quad\forall
l\in\mathcal{L}_{i},k\in\mathcal{K}_{i},m.$ (35)

We introduce the auxiliary variable $v_{l,k}\in\mathbb{R}$ and let

$v_{l,k}=q_{2}\sum_{m=1}^{M}t_{l,k,m}+q_{3}d_{k}\quad\forall
l\in\mathcal{L}_{i},k\in\mathcal{K}_{i}.$ (36)

Then (35) can be expressed with the help of $v_{l,k}$ as

$\displaystyle(u_{l,k}+v_{l,k})^{2}\leq
2q_{1}\sum_{m=1}^{M}t_{l,k,m}+u_{l,k}^{2}+v_{l,k}^{2}\quad
l\in\mathcal{L}_{i},k\in\mathcal{K}_{i}.$ (37)

The quadratic terms that appear at the right side of the inequality can be
convexified by utilizing the concave-convex procedure (CCP) [19]. Replacing
the $u_{l,k}^{2}$ and $v_{l,k}^{2}$ with their first-order Taylor expansion at
their optimal solution obtained in the previous iteration as

$\displaystyle(u_{l,k}+v_{l,k})^{2}\leq 2q_{1}\sum_{m=1}^{M}t_{l,k,m}$ (38)
$\displaystyle+\left(u_{l,k}^{(\epsilon-1)}\right)^{2}+2\left(u_{l,k}^{(\epsilon-1)}\right)\left(u_{l,k}-u_{l,k}^{(\epsilon-1)}\right)$
$\displaystyle+\left(v_{l,k}^{(\epsilon-1)}\right)^{2}+2\left(v_{l,k}^{(\epsilon-1)}\right)\left(v_{l,k}-v_{l,k}^{(\epsilon-1)}\right)+s_{l,k}^{\prime\prime},$
$\displaystyle\forall l\in\mathcal{L}_{i},k\in\mathcal{K}_{i}$
$s_{l,k}^{\prime\prime}\geq 0,\quad\forall
l\in\mathcal{L}_{i},k\in\mathcal{K}_{i},$ (39)

where $s_{l_{i},k_{i}}^{\prime\prime}$ is the introduced non-negative slack
variable for taking the affine approximation based on CCP.

All the non-convexities except for the binary constraint (11e) in P3-1 have
now been tackled. The (11e) can be handled by MIP solver MOSEK [20] with the
branch-and-bound method. Together, the solvable optimization problem can be
formulated as

$\displaystyle\textbf{P4-1}:\operatornamewithlimits{maximize}_{\left\\{\begin{tabular}[]{c
c c}$\mathbf{z}_{l,k}$&$\alpha_{l,k}$&$\tilde{\tau}_{k}$\\\
$e_{k}$&$f_{k}$&$u_{l,k}$\\\ $t_{l,k,m}$&$d_{k}$&$v_{l,k}$\\\
$s_{k}$&$s_{l,k,m}^{\prime}$&$s_{l,k}^{\prime\prime}$\end{tabular}\right\\}}\tilde{R}_{i}^{\text{min}}-\bar{s}_{i}$
(44) $\displaystyle\text{s.t.
}\eqref{eq:11e},\eqref{eq:15c},\eqref{eq:15d},\eqref{eq:16},\eqref{eq:21},\eqref{eq:21a},\eqref{eq:fppsca2},\eqref{eq:slack1},$
$\displaystyle\hskip
15.649pt\eqref{eq:26},\eqref{eq:29a},\eqref{eq:32},\eqref{eq:33a},\eqref{eq:34a},\eqref{eq:38},\eqref{eq:39},$

where $\bar{s}_{i}$ is the penalty term for introducing the slack variables
after utilizing approximation of some terms when casting P4-1 over coverage
group $i$. The value of the non-negative slack variables is positively related
to the gap between the approximation and the original term. By taking
penalties for the slack variables, the algorithm will push to find an optimal
solution that is close to the optimal solution of the original problem.
Defining the $\omega$, $\omega^{\prime}$, and $\omega^{\prime\prime}$ as the
positive penalty coefficients for $s_{k}$, $s_{l,k,m}^{\prime}$, and
$s_{l,k}^{\prime\prime}$ respectively, $\bar{s}_{i}$ is defined as

$\displaystyle\bar{s}_{i}=$
$\displaystyle\omega\sum_{k\in\mathcal{K}_{i}}s_{k}+\omega^{\prime}\sum_{l\in\mathcal{L}_{i}}\sum_{k\in\mathcal{K}_{i}}\sum_{m=1}^{M}s_{l,k,m}^{\prime}$
$\displaystyle+\omega^{\prime\prime}\sum_{l\in\mathcal{L}_{i}}\sum_{k\in\mathcal{K}_{i}}s_{l,k}^{\prime\prime}.$
(45)

Figure 4: Flowchart of the two-stage iterative data rate optimization
algorithm.

The iterative joint data rate optimization algorithm is proposed as a two-
stage process. The flowchart of the two-stage optimization algorithm is
illustrated in Fig. 4. In the first stage, the whole communication system is
divided into coverage groups manually. The index sets of the consisted SSMs
and UEs $\mathcal{L}_{i}$ and $\mathcal{K}_{i}$ are thereby obtained. Then by
inputting the obtained index sets along with the channel information
associated with the coverage group, Algorithm 1 is utilized for all the formed
coverage groups of the system. Using the outputs from the Algorithm 1 with
respect to all the coverage groups, optimized $\tilde{R}_{k},\forall
k\in\mathcal{K}$ can be obtained. In the second stage, by inserting
$\tilde{R}_{k},\tilde{\tau}_{k}^{(E)},\forall k\in\mathcal{K}$ into P2-2, and
solving the optimization problem, the time resources will be allocated
network-wide to maximize the minimum data rate in the whole system.

Algorithm 1 Joint coverage group minimum data rate maximization algorithm

1:Given: The manual SSM-UE association made to coverage group
$\mathcal{K}_{i}$, $\mathcal{L}_{i}$. The channel information of the coverage
group $\mathbf{h}_{k}$, $\mathbf{H}_{k}$

2:Initialization: Initialize $\mathbf{z}_{k}^{(0)}$ randomly while keeping
$|\mathbf{z}_{l,k,m}^{(0)}|=1,~{}\forall l,k,m$. Set the iteration counter to
$\epsilon=0$. Set the maximum iteration number to $E$.

3:while $\epsilon<E$ do

4: $\epsilon\leftarrow\epsilon+1$

5: Solve P4-1 and set $\mathbf{z}_{k}^{(\epsilon)},~{}\forall
k\in\mathcal{K}_{i}$ to its solution

6:end while

7:Output: The SSM-UE association $\alpha_{l,k}^{(E)}$, phase-shift
configurations $\boldsymbol{\phi}_{l,k}^{(E)}$ of SSMs, and allocated time
portions $\tilde{\tau}_{k}^{(E)}$

##  V Simulation design

Considering the static property of the IDS channel and the complexity of the
IDS environment, RT simulations are particularly well suited to capture
realistic channel conditions. In this paper, the RT simulation is performed
using the commercial RT platform Wireless Insite [21].

As an example of an IDS, we consider an aircraft cabin with 31 rows and 186
seats full of passengers. Fig. 5 shows the considered aircraft cabin
environment. The BS is equipped with an $8\times 8$ UPA of isotropic antennas
and is placed close to the ceiling in the middle of the cabin. The antenna
array surface of the BS is parallel to the cabin floor. We use a single
isotropic antenna to represent the UE held by the passenger. All UEs are set
to be higher than the seats to simulate the case when passengers are sitting
in their seats and using their cell phones to require service from the BS. The
detailed geometry condition of the considered environment can be found in our
previous work [22].

Metasurfaces are not provided as a module by the RT simulator. Hence, we
modeled them as antenna sets in the simulator and synthesized the cascaded
channel. The detailed metasurface structure we modeled is given in Fig. 5b. As
indicated in [23], the radiation pattern of the metasurface element is similar
to that of a cosine antenna. Therefore, we use an $32\times 32$ transceiver
cosine antenna array to model the metasurface where each cosine antenna
represents one metasurface element. The metasurface elements are placed on an
impenetrable substrate that is modeled as a perfect wave absorber.
Correspondingly, we align the direction in which the antenna gain of each
cosine antenna reaches its maximum perpendicular to the array surface. As
shown in Fig. 5b, the cosine antenna only serves half of the sphere, which we
will refer to as the reflecting side in this paper. The other half sphere
where the antenna gain is zero mimics the wave-absorbing effect of the
substrate. After the RT simulation, the channel impulse responses (CIRs)
obtained from the BS-metasurface link and the metasurface-UE link are regarded
as $\mathbf{G}_{l}$ and $\mathbf{g}_{l,k}$ respectively.

(a)

(b)

(c)

Figure 5: Illustration of the metasurface, BS, and UE placement in the IDS.
(a) The front view of the cabin; (b) the detailed metasurface structure, (c)
the device layout in the cabin.

In our previous work, we observed that the highest SNR to the UEs are obtained
when the metasurfaces are placed above the middle seats on both sides of the
corridor in each row, with the reflective side perpendicular to the ground and
facing the BS [22]. With this deployment, all metasurfaces will be line-of-
sight to the BS. With two SSMs deployed in each row, there are $L=62$ SSMs
deployed in the cabin. Fig. 5 illustrates the geometric placement of the UEs,
metasurfaces, and BS in the cabin, where the material details of the airplane
cabin can be found in our previous work [22]. The detailed simulation settings
are listed in Table I.

TABLE I: Simulation parameters for the RT. Carrier frequency |  $28$ GHz
---|---
Bandwidth |  $1$ GHz
BS antenna type | Isotropic
Polarization | V - V
Transmit power |  $30$ dBm
Number of BS antennas | $64$
BS antenna spacing | $\lambda/2$
Metasurface element spacing | $\lambda/4$

###  V-A Coverage group design

Figure 6: The cascaded BS-SSM-UE channel gain with respect to a SSM that is
(a) far-away (b) near to the BS. The red rectangular represents the checked
SSM, and the pink circle represents the BS.

To understand the coverage of an SSM in our considered scenario, the cascaded
BS-SSM-UE channel $\mathbf{H}_{l,k}$ is analyzed for all the SSMs and with
respect to every UE $k$ in the network. Due to the high symmetry in the
considered IDS environment, SSMs show similar coverage patterns regardless of
their location. Fig. 6 shows the coverage condition of a far-away SSM and a
nearby SSM. Approximately 3 rows of UE in front of the reflective side of the
SSM are covered for either the far-end SSM case or the near-end SSM case. Thus
we consider grouping 1 row of UEs with their nearest 1, 2, or 3 rows of SSMs
to form a coverage group. And we index the coverage group with the location
row of the UE. Also, in terms of the nearest rows of SSMs, the considered row
of UEs should be at the reflective side of the SSMs. As illustrated in Fig. 7,
when we consider grouping 2 rows of SSMs with 1 row of UEs, UEs at row $i$
will be grouped with 4 SSMs that are at row $i$ and row $i-1$ to form coverage
group $i$.

To determine the size of the coverage group, we consider an over-optimistic
case where the self-sustainable constraints of the SSMs are neglected, and all
the elements are used to enhance communication. In this case, the SSMs can be
assigned to serve all the UEs in their coverage group, in other words,
$\alpha_{l,k}=1,\forall l,k$ holds. One upper bound on the SNR under this
hypothesis is

$\frac{\left\|\mathbf{h}_{k}+\mathbf{H}_{k}\boldsymbol{\phi}_{k}\right\|^{2}P}{BN_{0}}\leq\frac{(\left\|\mathbf{h}_{k}\|+\|\mathbf{H}_{k}\right\|_{F})^{2}P}{BN_{0}}=\hat{\Gamma}_{k}$
(46)

Fig. 8 illustrates $\hat{\Gamma}_{k}$ when considering grouping 1 row of UEs
with 2, 4, and 6 nearest of SSMs. The results indicate that most of the
indirect components received by the UE originate from the SSMs located within
the same row. While including additional SSMs in the coverage group can yield
some benefits, these advantages diminish when more than 4 SSMs are
incorporated. Therefore, in this paper, we design the coverage group as
comprising 6 UEs with 4 nearest SSMs (equivalent to group 1 row of UEs and 2
rows of SSMs).

Figure 7: Illustration of different coverage group designs. Coverage group
consists of (a) 6 SSMs, 6 UEs; (b) 4 SSMs, 6 UEs; (c) 2 SSMs, 6 UEs. Figure 8:
The upper bound of the achievable SNR under different scales of the coverage
group.

###  V-B Preset design

Figure 9: (a) Averaged received power, harvested power, and consumed power
under the searched-out reflecting area size with randomized phase-shifts; (b)
averaged and rounded down of searched-out reflecting area size and the
distance between the SSM and the BS.

According to (7), due to the MRT beamforming, the self-sustainability of an
SSM depends not only on the reflecting area size of a given preset and the
distance to the BS, but also on the phase-shifts of all SSMs assigned to serve
the target UE. However, these phase-shifts can be only optimized and
determined after the reflecting area size of the SSMs is decided. To tackle
this issue, we determine the size of the reflecting area by considering an MRT
beamformer by randomizing the SSM phase-shifts. The MRT beamformer when
serving UE $k$ under the randomized phase-shifts is denoted as
$\hat{\mathbf{w}}_{k}\in\mathbb{C}^{N}$, and is obtained by replacing
$\boldsymbol{\phi}_{l,k}$ in (3) with $\hat{\boldsymbol{\phi}}_{l,k}$ where
$\hat{\boldsymbol{\phi}}_{l,k}=e^{-j\hat{\boldsymbol{\varphi}}_{l,k}}$, and
the entries of $\hat{\boldsymbol{\varphi}}_{l,k}$ are all uniformly
distributed in the range of $(0,2\pi]$. The received power, harvested power,
and consumed power of SSM $l$ when serving UE $k$ given $\hat{\mathbf{w}}_{k}$
are denoted as $\hat{P}_{l,k}^{\text{Rc}}$, $\hat{P}_{l,k}^{\text{Hr}}$, and
$\hat{P}_{l}^{\text{Rf}}$ respectively. We search for a largest possible
$\bar{M}_{l}$ for each SSM $l$ from the given set of reflecting area size
$\bar{M}_{l}\in\\{16,64,81,121,144,169,225,256\\}$ from 10 rounds of Monte-
Carlo simulations referring to the utilization criteria of the SSM in the
searched situation. We further claim that the utilization criteria of SSM
under a given $\bar{M}_{l}$ refers to the ability to cover all the UEs in the
same row as it while maintaining a self-sustainable condition for the MRT
beamformer with randomized phase-shifts. We highlight that even though the
size of the reflecting area we found is not optimal, it is still a meaningful
size that allows the SSM to have a higher chance of being self-sustainable
even without optimizing its phase-shifts. For the convenience of evaluating
the self-sustainability condition, we denote $\hat{P}^{\text{Rc}}_{l}$ and
$\hat{P}^{\text{Hr}}_{l}$ as the averaged minimum received and harvested power
over all Monte-Carlo simulations, respectively, across various serving UE
within all the coverage groups associated with SSM $l$. Fig. 9 illustrates
powers and the reflecting area sizes of the SSMs. Fig. 9(a) shows all the
metasurfaces with searched-out reflecting area sizes that can achieve self-
sustainability. The relationship between the distance from SSM $l$ to the BS,
which is denoted as $d_{l}$, and the reflecting area size $\bar{M}_{l}$ is
shown in Fig. 9(b). The results show that the reflecting area size is smaller
at the SSMs that are close to the BS, whereas it is larger for SSMs positioned
at the cabin’s edge and farther away from the BS. The SSM with the largest
reflecting area size is situated between the far-end and middle of the cabin.
This positioning is attributed to the utilization of the MRT beamforming
approach we have taken. The cascaded channel is dominated by the direct
channel for UEs that are close to the BS, power will mainly be distributed to
the direct channel, thus the insufficient received power leads to a smaller
reflecting area of SSM. When the direct channel is not dominant, SSMs receive
more power yet the received power decays with the distance to the BS.
Therefore, the largest reflecting area appears in between the middle and the
edge of the cabin. A detailed analysis of this observation will be presented
in Section VI-A. The reflecting area size determined in this section will be
used for the later analysis of this work.

##  VI Numerical Results

Figure 10: Convergence condition of applying Algorithm 1 concerning the inputs
from coverage group 1, 10, and 30.

Once the channel coefficients have been obtained by RT simulation, the
proposed two-stage iterative data rate optimization algorithm can be applied
in conjunction with the preset and coverage group design. The power
consumption of one reflecting element $P^{\text{Rf}}=2\mu W$ [24]. The
coefficients featuring power harvesting $q_{1}$, $q_{2}$, and $q_{3}$ are set
to be 0.3904, 0.8260, and 0.6823 [25].

We set the maximum number of iteration rounds at $E=5$. Based on our tests, we
set the penalty coefficients as $\omega=100$, $\omega^{\prime}=10^{7}$, and
$\omega^{\prime\prime}=10^{7}$ to guarantee that constraints are not violated.
Fig. 10 illustrates a snapshot case of the convergence condition of applying
Algorithm 1 with respect to the inputs from coverage groups 1, 10, and 30.
Comparable convergence conditions are observed regardless of the coverage
group, with convergence typically achieved within approximately 2 to 3 rounds
of iteration for the algorithm.

###  VI-A Feasibility and necessity of utilizing SSM

Figure 11: (a) BS contribution for each UE; (b) SSM-UE association condition.

Due to propagation attenuation, in some locations, the harvested power may be
insufficient to support the operation of the SSM. We define the capability of
SSMs to maintain self-sustainability as the _feasibility_. Additionally, SSMs’
contributions vary based on their deployment locations when serving UEs
directly connected to the BS. We define the capability of SSMs to outperform
in terms of contributing to enhancing the UE’s SNR over direct connectivity as
the _necessity_. To better understand the feasibility and the necessity of
utilizing the SSMs at different locations of the cabin, we first evaluate how
much gain we obtain by introducing the SSM to the system. To this end, we
define a contribution indicator metric $\eta_{k}\in\mathbb{R}$. We denote the
SNR of the UE $k$ considering only the BS with MRT beamforming as
$\Gamma_{k}^{\text{BS}}$, and give the closed-form expression as
$\Gamma_{k}^{\text{BS}}=\frac{\|\mathbf{h}_{k}\|^{2}P}{BN_{0}}$. We can define
the $\eta_{k}$ as

$\eta_{k}=\frac{\Gamma_{k}^{\text{BS}}}{\Gamma_{k}},$ (47)

which is directly proportional to the magnitude of the direct component
$\|\mathbf{h}_{k}\|^{2}$. As the strength of the direct component received by
UE $k$ increases, $\eta_{k}$ approaches 1. For a UE that does not have a
direct connection to the BS, $\eta_{k}$ will be 0.

Fig. 11(a) shows the $\eta_{k}$ distribution of each UE in the cabin. It can
be seen that almost all UEs that have direct links to the BS have a
$\eta_{k}\geq 0.8$, which are distributed in the closest 6 rows to the BS.
This indicates in that area, the transmit power is more aligned to the direct
channel which leads to a much higher contribution from the direct component
rather than via the SSMs. This observation well aligns with the conclusion
given in [26]. Fig. 11(b) shows the number of associated UEs to a specific
SSM. It can be observed that SSMs at the edge of the cabin are associated with
more UEs, whereas SSMs located in the middle of the cabin, closer to the BS,
are associated with fewer UEs. Combining with the results in Fig. 11(a), it
can be concluded that SSMs positioned in the middle of the cabin neither
provide comparable gains nor are associated with serving more UEs.
Consequently, placing SSMs in that area can be a low necessity when designing
the SSM-assisted system.

The self-sustainability conditions to the solutions to Algorithm 1 are
analyzed for the sake of understanding the feasibility of using SSMs. All of
the SSMs are observed to satisfy the self-sustainability conditions after the
simulations. Fig. 12 illustrates a snapshot case, where we show the power
harvested and the power consumed by SSM $12$ and $29$ when they are serving
different UEs in their coverage groups. The former is in the middle of the
cabin and closer to the BS, and the latter is at the edge of the cabin and far
away from the BS.

Figure 12: Self-sustainability condition of (a) SSM 12, close to the BS,
$\bar{M}_{12}=64$; (b) SSM 29, far from the BS, $\bar{M}_{29}=169$.

From the figure it can be seen that both SSM 12 and SSM 29 harvest more power
than they consume for all the assigned UEs, showing that they are self-
sustainable. Interestingly, as shown in Fig. 12(a), despite the shorter
distance to the BS, SSM 12 could harvest smaller power than SSM 29. This could
also be explained by checking $\eta_{k}$ given in Fig. 11(a). When serving UEs
that have direct connections to the BS, most of the transmit power is directed
toward the direct channel. As a result, only a small portion of power is
captured by the SSMs. Moreover, the SSMs close to the BS are limited by the
lowest harvested power and, as a result, cannot support a larger reflecting
area, resulting in the harvested power not being sufficiently used in some
cases. In the case of SSM 29, all its serving UEs do not have direct
connection to the BS, thus power is only allocated to the indirect channel. As
shown in Fig. 12(b), most of the harvested power is used to support its
operation regardless of its serving target, suggesting wellness both in the
design of the preset and in the use of power.

Figure 13: (a) Time allocation and the resulting SNR of the whole cabin; (b)
intra-group time allocation and the optimized SNR of the coverage group that
consists of UEs 13-18.

Finally, the time resource allocation condition is evaluated, and is
illustrated in Fig. 13. Fig. 13(a) shows that all UEs achieve an SNR larger
than 0 dB, which proves the feasibility of employing SSM to guarantee
connectivity for the whole cabin. The same figure shows that the resulting SNR
decreases roughly as the distance between the UE and BS increases. Moreover,
more time resources are allocated to UEs with lower SNR values, while those
with better SNRs are allocated with less time resources. UEs near the BS
receive the least amount of time resources, yet still more than the minimum
allocable portion. This means that the resulting data rate of each UE in the
network is expected to be similar, which later gets validated as shown in Fig.
15. Fig. 13(b) shows the time allocation condition outputted by applying
Algorithm 1 concerning one coverage group. Similar to what is shown in Fig.
13(a), the allocated time portion inside the coverage group is also negatively
related to the SNR of the UEs.

###  VI-B Achievable data rate

(a)

(b)

Figure 14: Illustrations of metasurface performance test. (a) Setups for
different types of metasurface; (b) considered cabins of different sizes and
their relative position.

To determine the placement of SSM, SMS, and RIS within the trade-offs
involving metasurface gain, system coverage, and operating cost, we conduct a
comparison of the minimum data rate achieved with SSM-assisted, SMS-assisted,
and RIS-assisted systems. To fairly compare the performance of SSM with the
performance of other types of metasurfaces, we first consider a scenario
illustrated in Fig. 14a. We consider the size of the RIS and SMS to be equal
to the reflective area of the SSM. We also place them geographically in the
same position. Considering this, the channel coefficient received by every
element within the RIS and the SMS would align with what is received by each
reflecting element within the reflecting area of the SSM. Since RIS and SMS
are not restricted by self-sustainability, they are utilized referring to the
system model mentioned in Fig. 1, where all the RISs/SMSs will jointly serve
every UE. The remaining simulation settings are the same for both the RIS and
SMS cases.

Considering the huge attenuation of the mmWave signals, the feasibility of
achieving self-sustainability in different sizes of indoor environments for
SSM is also expected to be different. It can be expected that when the SSM is
barely reaching self-sustainable, the received signal will be prioritized to
align with the harvesting channel to ensure its operation, thus less power
will be reflected to the UE by the reflective part of the SSM. To explore the
influence of self-sustainable feasibility, a performance comparison is
conducted across various scenarios employing different types of metasurfaces,
considering varying cabin sizes. As illustrated in Fig. 14b, three cases,
where the cabin consists of UEs and SSMs in 31 rows, 21 rows, and 11 rows,
respectively, are considered. In all three cases, the BS is always placed in
the center of the cabin.

Figure 15: Optimized data rate under the assistance of different types of
metasurface and under different cell sizes.

After obtaining the channel coefficients, the system performance with the
assistance of RIS and SMS is evaluated by utilizing the algorithm we proposed
in our previous work [11], where, the minimum data rate is optimized by
jointly allocating time resources and selecting the phase-shifts. The results
are shown in Fig. 15 along with the system performance we obtained with SSM.
This observation underscores the trade-off between reconfigurability and
operating cost. Under the same channel conditions, RIS outperforms SMS
benefiting notably from its reconfigurable nature. The performance gap of that
RIS over SMS is $15.2$, $19.8$, and $22.4$ Mbps on average for each UE when
the cabin gets smaller. As a self-sustainable alternative of RIS, SSM, has
shown a performance better than SMS due to its reconfigurability, yet a
performance worse than RIS due to the extra limitation of self-sustainability.
The performance gap of SSM over SMS is $3.6$, $9.9$, and $19.8$ Mbps for a
cabin consisting of 31, 21, and 11 rows, respectively. It is worth noting that
in the 11-row cabin case, due to the short distance from the SSMs to the BS,
the self-sustainability for the SSMs is much easier to achieve. Since the
reflecting areas of the SSMs share the same channel condition as the RISs, the
RIS-assisted system performance can be considered as a good upper bound for
the SSM. The small gap shows that the simplifications we make do not degrade
the performance of the SSM much when the feasibility of achieving self-
sustainability is not tight.

It is also worth noticing that the data rate gap obtained with SSM over SMS is
getting less significant as the cabin size gets larger. With increasing
distance to the BS, it becomes more challenging for SSMs to achieve self-
sustainability. Thus, the resulting phase-shifts that are applied to SSM will
differ from those in the RIS case. In this instance, phase-shifts are selected
to affect the transmitting beamforming, which further helps to improve the
amount of harvesting power yet losing the metasurface gain SSMs can provide.
Thus, as the coverage requirement increases, the performance gap between RIS
and SSM widens, while the gap between SSM and SMS narrows. It can be expected
that the coverage of SSM stops when it is infeasible to be self-sustainable.
In summary, even though SSM shows a better performance than SMS, limited by
the self-sustainable constraint, its coverage will be worse than using SMS,
which is passive by its nature. Thus, when the system coverage requirement is
high, SMS is still a possible solution to enhance the communication quality.

##  VII Conclusion

Figure 16: General comparison in terms of operating cost, metasurface gain,
and system coverage among SSM, SMS, and RIS.

In this paper, in terms of finding a better balance between the metasurface
gain, system coverage, and operating cost such as cabling and powering
required by the reconfigurability, we investigate the performance of an SSM-
assisted system. The metasurface is considered to take the element splitting
scheme to achieve self-sustainability, where part of the elements in the
metasurface are used only to harvest power from the electromagnetic wave, and
the rest part of the elements are used to reflect the signal. To lower the
computational complexity of optimizing the SSM-assisted system, certain
simplifications are done. Instead of deciding the working mode of each element
we consider using the SSM in a preset-based manner, where the whole SSM is
divided into the reflecting area and the harvesting area. The preset of each
SSM is designed and given beforehand by considering an MRT beamformer in which
the phase-shifts of each SSM are randomly selected. Moreover, the SSMs and UEs
are associated to form a coverage group based on the coverage of the SSM to
further lower the computational complexity. The feasibility and the necessity
of using SSMs are evaluated by checking the self-sustainable and the
association conditions for SSMs at various locations. Our findings suggest
that in areas where UEs benefit from a strong direct channel from the BS, the
transmit power tends to align more with this direct channel. Consequently,
less power will be harvested by SSMs, and limited by the minimum harvested
power employing SSMs in such areas becomes less feasible. Furthermore, for the
same reason the contribution of employing SSM is less significant. This
indicates a lower necessity for enhancing UEs in those areas with SSM
technology. The performance of using RIS, SMS, and SSM is compared and
concluded in Fig. 16. The results show that SSM brings a performance better
than SMS and worse than RIS. The results also show the coverage shortfall of
SSM, while trying to use SSM to improve the UEs that are far from the BS, the
difficulty for SSM to maintain its self-sustainable condition increases, and
this in the end will result in a smaller coverage compared to other types of
metasurfaces.

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