# Unleashing 3D Connectivity in Beyond 5G Networks with Reconfigurable
Intelligent Surfaces

Jiguang He1, Aymen Fakhreddine1,2, Arthur S. de Sena1, Yu Tian1, and Merouane
Debbah1 1Technology Innovation Institute, Abu Dhabi, United Arab Emirates
2Institute of Networked and Embedded Systems, University of Klagenfurt,
Klagenfurt, Austria  E-mails: {jiguang.he, aymen.fakhreddine, arthur.sena,
yu.tian<EMAIL_ADDRESS>
###### Abstract

Reconfigurable intelligent surfaces (RISs) bring various benefits to the
current and upcoming wireless networks, including enhanced spectrum and energy
efficiency, soft handover, transmission reliability, and even localization
accuracy. These remarkable improvements result from the reconfigurability,
programmability, and adaptation capabilities of RISs for fine-tuning radio
propagation environments, which can be realized in a cost- and energy-
efficient manner. In this paper, we focus on the upgrade of the existing
fifth-generation (5G) cellular network with the introduction of an RIS owning
a full-dimensional uniform planar array structure for unleashing advanced
three-dimensional connectivity. The deployed RIS is exploited for serving
unmanned aerial vehicles (UAVs) flying in the sky with ultra-high data rate, a
challenging task to be achieved with conventional base stations (BSs) that are
designed mainly to serve ground users. By taking into account the line-of-
sight probability for the RIS-UAV and BS-UAV links, we formulate the average
achievable rate, analyze the effect of environmental parameters, and make
insightful performance comparisons. Simulation results show that the
deployment of RISs can bring impressive gains and significantly outperform
conventional RIS-free 5G networks.

###### Index Terms:

Beyond 5G, 3D connectivity, achievable rate, UAV, RIS

##  I Introduction

Drones, also known as unmanned aerial vehicles (UAVs), have recently undergone
a tremendous expansion, generating a wide range of emerging applications, such
as goods delivery, urban air taxis, remote surveillance, border control,
agricultural or industrial monitoring, and disaster relief [1]. Even though
the aforementioned applications span distinct domains, the commonality among
them is the demanding need for three-dimensional (3D) wireless connectivity
for the transfer of real-time sensor data and control commands. Such
connectivity must be reliable, secure, and supports high data rates, up to
several hundred megabits per second (Mbps).

The current commercial fifth generation (5G) base stations (BSs) are
customized with the primary purpose to provide a two-dimensional (2D) coverage
to users on the ground. Providing ubiquitous three-dimensional (3D) coverage
and even full earth coverage is a hot topic for the next generations of
wireless networks, i.e., beyond 5G and sixth generation (6G) [2]. To mitigate
the lack of a dedicated infrastructure to serve flying UAVs, approaches that
do not require substantial hardware upgrades or additional deployments to be
borne by mobile network operators should be adopted. In other words, one
should heavily rely on the existing 5G cellular networks and focus on its
upgrade to meet the quality-of-service (QoS) requirements of UAV
communications. To the best of our knowledge, the adoption of reconfigurable
intelligent surfaces (RISs) to extend the 3D coverage of cellular networks for
serving aerial users has not yet been fully explored.

In the recent literature, the works in [3, 4, 5, 6] focus on how the RIS can
be applied to enhance UAV-enabled wireless networks. However, their purpose is
also to serve ground users. To be specific, a terrestrial BS is able to
communicate with distant ground users in the absence of line-of-sight (LoS)
links by mounting an RIS on a UAV, which enables intelligent reflection from
the sky [3]. In [4, 5, 6], UAVs act as aerial BSs with flexible deployment.
The authors in [7] dedicated a whole section reviewing UAV and RIS, and listed
relevant publications that explored UAV-mounted RISs.

In this paper, we leverage the reconfigurability capabilities of RISs to
enhance the existing cellular networks by beamforming toward the sky to
provide 3D connectivity to the UAVs. This offers an “easy to implement”
upgrade to the existing 5G cellular infrastructure, rendering beyond 5G (B5G),
and enables an improved cellular architecture that incorporates large RISs in
key locations at altitudes slightly lower than those of the serving BS
antennas of currently deployed cellular networks. In this sense, unlike the
existing works that consider RIS-mounted UAVs as part of the network either as
relays or aerial BSs, our study considers the UAVs that are in fact cellular
user equipment (UE) served by the network. Moreover, we take into account the
LoS probability and study the average achievable rate under different
propagation environments. Simulation results illustrate how the RIS-assisted
B5G networks outperform their RIS-free counterparts in terms of average
achievable rate and sky coverage.

Notations: A bold lowercase letter ${\mathbf{a}}$ denotes a vector, and a bold
capital letter ${\mathbf{A}}$ denotes a matrix. $(\cdot)^{\mathsf{T}}$ and
$(\cdot)^{\mathsf{H}}$ denote the matrix or vector transpose and Hermitian
transpose, respectively. $\mathrm{Tr}(\cdot)$ denotes the trace operator,
$\mathrm{diag}({\mathbf{a}})$ and $\mathrm{det}(\cdot)$ denote a diagonal
matrix with the entries of ${\mathbf{a}}$ on its diagonal and the determinant
of a matrix, $\phase{a}$ returns the phase of the complex scalar $a$,
${\mathbf{a}}\otimes{\mathbf{b}}$ denotes the Kronecker product of
${\mathbf{a}}$ and ${\mathbf{b}}$, ${\mathbf{I}}_{M}$ denotes the $M\times M$
identity matrix, $\mathbf{0}$ is an all-zero matrix, $j=\sqrt{-1}$, and
$\|\cdot\|_{2}$ denotes the Euclidean norm of a vector. $[{\mathbf{a}}]_{i}$
and $[{\mathbf{A}}]_{ij}$ denote the $i$th element of ${\mathbf{a}}$ and the
$(i,j)$th element of ${\mathbf{A}}$, respectively. Finally, $|\cdot|$ returns
the absolute value of a complex number.

##  II System model

We focus on the 3D connectivity for UAV communications with the aid of a
static RIS along with a legacy 5G BS, depicted in Fig. 1. The BS is equipped
with a uniform linear array (ULA) structure, consisting of $N_{\text{B}}$
vertical antenna elements, down-tilted by a clockwise rotation of $\beta$
radians. To offer 3D connectivity for the flying UAV without additional
deployment of costly next-generation BSs, we rely on the cost-efficient
deployment of an RIS, which is capable of performing 3D beamforming thanks to
its massive number of sub-wavelength meta-atoms. The RIS can generate a group
of candidate beams to cover the whole sky, where the UAV is supposed to be
located. Specifically, one RIS composed of
$N_{\text{R}}=N_{\text{R},x}N_{\text{R},y}$ meta-atoms is deployed in the
close proximity of the BS with LoS availability [8], owning a uniform planar
array (UPA) structure, which is parallel to the $x\text{-}y$ plane;
$N_{\text{R},x}$ and $N_{\text{R},y}$ denote the number of RIS meta-atoms
across $x$ and $y$ axes, respectively. The flying UAV is also supposed to
employ a UPA with $N_{\text{U}}=N_{\text{U},x}N_{\text{U},y}$ antennas, where
$N_{\text{U},x}$ and $N_{\text{U},y}$ are the number of antennas across $x$
and $y$ axes, respectively. We assume that the antenna planes of RIS and UAV
are parallel to each other. The coordinates of the BS, the RIS, and the UAV
are
${\mathbf{p}}_{\text{B}}=(x_{\text{B}},y_{\text{B}},z_{\text{B}})^{\mathsf{T}}\in\mathbb{R}^{3}$,
${\mathbf{p}}_{\text{R}}=(x_{\text{R}},y_{\text{R}},z_{\text{R}})^{\mathsf{T}}\in\mathbb{R}^{3}$,
and
${\mathbf{p}}_{\text{U}}=(x_{\text{U}},y_{\text{U}},z_{\text{U}})^{\mathsf{T}}\in\mathbb{R}^{3}$,
respectively.

Figure 1: 3D connectivity enabled by an RIS along with a legacy 5G BS.

###  II-A Channel Model

We consider the far-field propagation for all the channels, i.e., BS-RIS, RIS-
UAV, and BS-UAV channels, which are modeled by taking into consideration the
channel parameters, calculated by following the geometrical relationship
between any pair of network nodes. For the purpose of simplicity, we only
consider the LoS path component for each channel, which is well justified for
UAV communications [9]. Even though the multi-path components may exist, their
sum power is not comparable to the power in the LoS path, especially in
millimeter wave (mmWave) frequency bands [10]. Characterized by the extended
Saleh-Valenzuela channel model [11], the BS-RIS channel
${\mathbf{H}}_{\text{B,R}}\in\mathbb{C}^{N_{\text{R}}\times N_{\text{B}}}$ is
modeled as

${\mathbf{H}}_{\text{B,R}}=\sqrt{\frac{N_{\text{B}}N_{\text{R}}}{\rho_{\text{B,R}}}}\exp(-j2\pi\tau_{\text{B,R}})\boldsymbol{\alpha}_{\text{R}}(\theta^{r}_{\text{B,R}},\phi^{r}_{\text{B,R}})\boldsymbol{\alpha}_{\text{B}}^{\mathsf{H}}(\phi^{t}_{\text{B,R}}-\beta),$
(1)

where $\rho_{\text{B,R}}\in\mathbb{R}$ is the path loss, dependent on the BS-
RIS distance, carrier frequency, and shadowing effect, $\tau_{\text{B,R}}$ is
the propagation delay, $\theta^{r}_{\text{B,R}}\in\mathbb{R}$,
$\phi^{r}_{\text{B,R}}\in\mathbb{R}$, and $\phi^{t}_{\text{B,R}}\in\mathbb{R}$
are the azimuth, the elevation angle of arrival (AoA), and the elevation angle
of departure (AoD), respectively. The channel parameters are calculated based
on the centroids of the BS antenna array and the RIS plane, as [12]

$\displaystyle x_{\text{R}}$
$\displaystyle=x_{\text{B}}+d_{\text{B,R}}\cos(\theta^{r}_{\text{B,R}})\cos(\phi^{r}_{\text{B,R}}),$
(2) $\displaystyle y_{\text{R}}$
$\displaystyle=y_{\text{B}}+d_{\text{B,R}}\sin(\theta^{r}_{\text{B,R}})\cos(\phi^{r}_{\text{B,R}}),$
(3) $\displaystyle z_{\text{R}}$
$\displaystyle=z_{\text{B}}+d_{\text{B,R}}\sin(\phi^{r}_{\text{B,R}}),$ (4)
$\displaystyle\phi^{t}_{\text{B,R}}$ $\displaystyle=\phi^{r}_{\text{B,R}},$
(5)

where
$d_{\text{B,R}}=\|{\mathbf{p}}_{\text{B}}-{\mathbf{p}}_{\text{R}}\|_{2}\in\mathbb{R}$
is the distance between the BS and the RIS. The normalized array response
vectors
$\boldsymbol{\alpha}_{\text{R}}(\theta^{r}_{\text{B,R}},\phi^{r}_{\text{B,R}})\in\mathbb{C}^{N_{\text{R}}}$,
at the RIS, and
$\boldsymbol{\alpha}_{\text{B}}(\phi^{t}_{\text{B,R}}-\beta)\in\mathbb{C}^{N_{\text{B}}}$,
at the BS, i.e.,
$\|\boldsymbol{\alpha}_{\text{R}}(\theta^{r}_{\text{B,R}},\phi^{r}_{\text{B,R}})\|_{2}=\|\boldsymbol{\alpha}_{\text{B}}(\phi^{t}_{\text{B,R}}-\beta)\|_{2}=1$,
are

$\displaystyle\boldsymbol{\alpha}_{\text{R}}(\theta^{r}_{\text{B,R}},\phi^{r}_{\text{B,R}})\triangleq\frac{1}{\sqrt{N_{\text{R}}}}\Big{[}1,\exp\big{(}j\frac{2\pi
d_{x}}{\lambda}\cos(\theta^{r}_{\text{B,R}})\sin(\phi^{r}_{\text{B,R}})\big{)},$
$\displaystyle\;\;\;\cdots,\exp\big{(}j\frac{2\pi
d_{x}}{\lambda}(N_{\text{R},x}-1)\cos(\theta^{r}_{\text{B,R}})\sin(\phi^{r}_{\text{B,R}})\big{)}\Big{]}^{\mathsf{T}}$
$\displaystyle\otimes\Big{[}1,\exp\big{(}j\frac{2\pi
d_{y}}{\lambda}\sin(\theta^{r}_{\text{B,R}})\sin(\phi^{r}_{\text{B,R}})\big{)},$
$\displaystyle\;\;\;\cdots,\exp\big{(}j\frac{2\pi
d_{y}}{\lambda}(N_{\text{R},y}-1)\sin(\theta^{r}_{\text{B,R}})\sin(\phi^{r}_{\text{B,R}})\big{)}\Big{]}^{\mathsf{T}},$
(6)
$\displaystyle\boldsymbol{\alpha}_{\text{B}}(\phi^{t}_{\text{B,R}}-\beta)\triangleq\frac{1}{\sqrt{N_{\text{B}}}}\Big{[}1,\exp\big{(}j\frac{2\pi
d_{z}}{\lambda}\cos(\phi^{t}_{\text{B,R}}-\beta)\big{)},$
$\displaystyle\;\;\;\cdots,\exp\big{(}j\frac{2\pi
d_{z}}{\lambda}(N_{\text{B}}-1)\cos(\phi^{t}_{\text{B,R}}-\beta)\big{)}\Big{]}^{\mathsf{T}},$
(7)

where $d_{x}\in\mathbb{R}$, $d_{y}\in\mathbb{R}$, and $d_{z}\in\mathbb{R}$
denote the inter-element spacing across $x$, $y$, and $z$ axes, respectively,
and $\lambda$ is the wavelength. The effect of orientation angle
$\beta\in\mathbb{R}$ is integrated into the array response vector
$\boldsymbol{\alpha}_{\text{B}}(\cdot)$. The RIS-UAV channel
${\mathbf{H}}_{\text{R,U}}\in\mathbb{C}^{N_{\text{U}}\times N_{\text{R}}}$ and
BS-UAV channel ${\mathbf{H}}_{\text{B,U}}\in\mathbb{C}^{N_{\text{U}}\times
N_{\text{B}}}$ can be modeled in the same manner, as

$\displaystyle{\mathbf{H}}_{\text{B,U}}$
$\displaystyle=\sqrt{\frac{N_{\text{B}}N_{\text{U}}}{\rho_{\text{B,U}}}}\exp(-j2\pi\tau_{\text{B,U}})\boldsymbol{\alpha}_{\text{U}}(\theta^{r}_{\text{B,U}},\phi^{r}_{\text{B,U}})$
$\displaystyle\times\boldsymbol{\alpha}_{\text{B}}^{\mathsf{H}}(\phi^{t}_{\text{B,U}}-\beta),$
(8) $\displaystyle{\mathbf{H}}_{\text{R,U}}$
$\displaystyle=\sqrt{\frac{N_{\text{R}}N_{\text{U}}}{\rho_{\text{R,U}}}}\exp(-j2\pi\tau_{\text{R,U}})\boldsymbol{\alpha}_{\text{U}}(\theta^{r}_{\text{R,U}},\phi^{r}_{\text{R,U}})$
$\displaystyle\times\boldsymbol{\alpha}_{\text{R}}^{\mathsf{H}}(\theta^{t}_{\text{R,U}},\phi^{t}_{\text{R,U}}),$
(9)

where $\boldsymbol{\alpha}_{\text{U}}(\cdot)\in\mathbb{C}^{N_{\text{U}}}$ is
the array response vector at the UAV, $\rho_{\text{B,U}}$,
$\rho_{\text{R,U}}$, $\tau_{\text{B,U}}$, $\tau_{\text{R,U}}$,
$\theta^{r}_{\text{B,U}}$, $\phi^{r}_{\text{B,U}}$, $\phi^{t}_{\text{B,U}}$,
$\theta^{r}_{\text{R,U}}$, $\phi^{r}_{\text{R,U}}$, $\theta^{t}_{\text{R,U}}$,
and $\phi^{t}_{\text{R,U}}$ are defined in the same way as those in (1). Since
we assume that the antenna plane of the UAV is parallel to that of the RIS, we
have $\theta^{r}_{\text{R,U}}=\theta^{t}_{\text{R,U}}$ and
$\phi^{r}_{\text{R,U}}=\phi^{t}_{\text{R,U}}$.

###  II-B LoS Probability

The LoS probability of the channels relies on the complex environmental
propagation conditions, e.g., the height of the buildings, density of the
buildings, and spatial distribution of the scatterers. Therefore, different
LoS probabilities should be applied for different scenarios, such as urban,
suburban, and rural. We, in this work, resort to the International
Telecommunication Union Radiocommunication (ITU-R) for the modeling of LoS
availability, introduced in [13, 14] as

$p_{\text{LoS}}^{\text{ITU}}(h_{\text{T}},h_{\text{R}},\gamma)=\prod_{m=0}^{M}\bigg{(}1-\exp\bigg{(}-\frac{\big{(}h_{\text{T}}-\frac{(m+0.5)(h_{\text{T}}-h_{\text{R}})}{M+1}\big{)}^{2}}{2\gamma^{2}}\bigg{)}\bigg{)},$
(10)

where $M=\lfloor r\sqrt{\alpha\kappa}\rfloor-1$ denotes the number of
buildings between the pair of network nodes. The variables $r$, $\alpha$, and
$\kappa$ denote the ground distance in kilometers between the pair of nodes,
the fraction of the area covered by buildings to the total area, and the
average number of buildings per unit area, respectively, $\gamma$ is a height
distribution parameter, and $h_{\text{T}}$ and $h_{\text{R}}$ are the heights
for the transmitter and receiver, respectively. By tuning the parameters
$\alpha$, $\kappa$, and $\gamma$, we can model the LoS probabilities for all
the aforementioned scenarios. Thus, we apply the expression (10) to both of
the channels ${\mathbf{H}}_{\text{B,U}}$ and ${\mathbf{H}}_{\text{R,U}}$. In
addition, we assume that the RIS is placed in the proximity of the BS, so the
LoS availability is always guaranteed. Note that for characterizing
${\mathbf{H}}_{\text{R,U}}$, the RIS is deemed as a virtual transmitter.

##  III Achievable Rate and Its Average

###  III-A Achievable Rate

The benefits of introducing RIS for 3D connectivity are multi-fold: i) When
the direct BS-UAV channel exists, tremendous multiplexing gains can be
obtained aiming for extremely high data-rate transmission. ii) When the BS-UAV
channel is temporally unavailable, the BS can maintain the connectivity via
the RIS. We treat the two cases separately in this subsection. The entire end-
to-end channel between the BS and the UAV is summarized as

${\mathbf{H}}=\mathbb{I}({\mathbf{H}}_{\text{B,U}}){\mathbf{H}}_{\text{B,U}}+\mathbb{I}({\mathbf{H}}_{\text{R,U}}){\mathbf{H}}_{\text{R,U}}\boldsymbol{\Omega}{\mathbf{H}}_{\text{B,R}},$
(11)

where $\mathbb{I}(\cdot)\in\\{0,1\\}$ is the indicator function; if
${\mathbf{H}}_{\text{B,U}}\neq\mathbf{0}$,
$\mathbb{I}({\mathbf{H}}_{\text{B,U}})=1$; otherwise,
$\mathbb{I}({\mathbf{H}}_{\text{B,U}})=0$. The same principle is applied to
$\mathbb{I}({\mathbf{H}}_{\text{R,U}})$. The diagonal matrix
$\boldsymbol{\Omega}\in\mathbb{C}^{N_{\text{R}}\times N_{\text{R}}}$ is the
RIS phase control matrix. Unlike the conventional amplify-and-forward (AF)
relays, the RIS is supposed to change only the phase shifts of the impinging
signals, realizing analog/passive beamforming [15]. Namely, strict constraints
are imposed for the diagonal entries of $\boldsymbol{\Omega}$, i.e.,
$|[\boldsymbol{\Omega}]_{kk}|=1$, $\forall k\in\\{1,2,\cdots,N_{\text{R}}\\}$.
By referring to (10), the LoS probabilities of ${\mathbf{H}}_{\text{B,U}}$ and
${\mathbf{H}}_{\text{R,U}}$ are

$\displaystyle\text{Pr}(\mathbb{I}($
$\displaystyle{\mathbf{H}}_{\text{B,U}})=1)=$
$\displaystyle\prod_{m=0}^{M}\bigg{(}1-\exp\bigg{(}-\frac{\big{(}z_{\text{B}}-\frac{(m+0.5)(z_{\text{B}}-z_{\text{U}})}{M+1}\big{)}^{2}}{2\gamma^{2}}\bigg{)}\bigg{)},$
(12) $\displaystyle\text{Pr}($
$\displaystyle\mathbb{I}({\mathbf{H}}_{\text{R,U}})=1)=$
$\displaystyle\prod_{m=0}^{M}\bigg{(}1-\exp\bigg{(}-\frac{\big{(}z_{\text{R}}-\frac{(m+0.5)(z_{\text{R}}-z_{\text{U}})}{M+1}\big{)}^{2}}{2\gamma^{2}}\bigg{)}\bigg{)}.$
(13)

$R=\begin{cases}\log_{2}\Big{(}\det\big{(}{\mathbf{I}}+\frac{P}{\sigma^{2}}{\mathbf{W}}^{\mathsf{H}}{\mathbf{H}}{\mathbf{F}}{\mathbf{F}}^{\mathsf{H}}{\mathbf{H}}^{\mathsf{H}}{\mathbf{W}}\big{)}\Big{)},&\text{if
}\mathbb{I}({\mathbf{H}}_{\text{B,U}})=1,\mathbb{I}({\mathbf{H}}_{\text{R,U}})=1,\\\
\log_{2}\Big{(}\det\big{(}1+\frac{P}{\sigma^{2}}{\mathbf{w}}^{\mathsf{H}}{\mathbf{H}}_{\text{B,U}}{\mathbf{f}}{\mathbf{f}}^{\mathsf{H}}{\mathbf{H}}_{\text{B,U}}^{\mathsf{H}}{\mathbf{w}}\big{)}\Big{)},&\text{if
}\mathbb{I}({\mathbf{H}}_{\text{B,U}})=1,\mathbb{I}({\mathbf{H}}_{\text{R,U}})=0,\\\
\log_{2}\Big{(}\det\big{(}1+\frac{P}{\sigma^{2}}{\mathbf{w}}^{\mathsf{H}}{\mathbf{H}}_{\text{R,U}}\boldsymbol{\Omega}{\mathbf{H}}_{\text{B,R}}{\mathbf{f}}{\mathbf{f}}^{\mathsf{H}}{\mathbf{H}}_{\text{B,R}}^{\mathsf{H}}{\mathbf{H}}_{\text{R,U}}^{\mathsf{H}}\boldsymbol{\Omega}^{\mathsf{H}}{\mathbf{w}}\big{)}\Big{)},&\text{if
}\mathbb{I}({\mathbf{H}}_{\text{B,U}})=0,\mathbb{I}({\mathbf{H}}_{\text{R,U}})=1,\\\
0,&\text{if
}\mathbb{I}({\mathbf{H}}_{\text{B,U}})=0,\mathbb{I}({\mathbf{H}}_{\text{R,U}})=0.\end{cases}$
(14)

The achievable rate between the BS and the UAV depends on the LoS availability
of ${\mathbf{H}}_{\text{B,U}}$ and ${\mathbf{H}}_{\text{R,U}}$. Thus, a
segmented function is considered for the achievable rate in bps/Hz, as shown
in (14) on the top of the next page. For case i)
$\mathbb{I}({\mathbf{H}}_{\text{B,U}})=\mathbb{I}({\mathbf{H}}_{\text{R,U}})=1$,
it also depends on the design of the combining matrix
${\mathbf{W}}\in\mathbb{C}^{N_{\text{U}}\times 2}$, the precoder
${\mathbf{F}}\in\mathbb{C}^{N_{\text{B}}\times 2}$, and $\boldsymbol{\Omega}$.
For case ii)
$\mathbb{I}({\mathbf{H}}_{\text{B,U}})=1,\mathbb{I}({\mathbf{H}}_{\text{R,U}})=0$,
it also depends on the design of the combining vector
${\mathbf{w}}\in\mathbb{C}^{N_{\text{U}}}$ and the beamforming vector
${\mathbf{f}}\in\mathbb{C}^{N_{\text{B}}}$. For the case iii)
$\mathbb{I}({\mathbf{H}}_{\text{B,U}})=0,\mathbb{I}({\mathbf{H}}_{\text{R,U}})=1$,
it also depends on the design of ${\mathbf{w}}$, ${\mathbf{f}}$, and
$\boldsymbol{\Omega}$. The notations $P$ and $\sigma^{2}$ denote the transmit
power at the BS and the noise variance at the UAV.

####  III-A1 Case i)
$\mathbb{I}({\mathbf{H}}_{\textsc{B,U}})=\mathbb{I}({\mathbf{H}}_{\textsc{R,U}})=1$

In this case, spatial multiplexing gains can be achieved, since two different
routes from the BS to the UAV can be concurrently available with the end-to-
end channel
${\mathbf{H}}={\mathbf{H}}_{\text{B,U}}+{\mathbf{H}}_{\text{R,U}}\boldsymbol{\Omega}{\mathbf{H}}_{\text{B,R}}$,
where ${\mathbf{H}}_{\text{B,U}}\neq\mathbf{0}$ and
${\mathbf{H}}_{\text{R,U}}\neq\mathbf{0}$. We focus on the design of
${\mathbf{W}}$, ${\mathbf{F}}$, and $\boldsymbol{\Omega}$ as to maximize the
achievable rate. The optimization problem is formulated as [16]

$\displaystyle\mathcal{P}_{1}:\max\limits_{{\mathbf{W}},{\mathbf{F}},\boldsymbol{\Omega}}\;\;\;$
$\displaystyle\log_{2}\Big{(}\det\big{(}{\mathbf{I}}_{2}+\frac{P}{\sigma^{2}}{\mathbf{W}}^{\mathsf{H}}{\mathbf{H}}{\mathbf{F}}{\mathbf{F}}^{\mathsf{H}}{\mathbf{H}}^{\mathsf{H}}{\mathbf{W}}\big{)}\Big{)},$
(15a) s.t.
$\displaystyle\mathrm{Tr}({\mathbf{W}}^{\mathsf{H}}{\mathbf{W}})=1,$ (15b)
$\displaystyle\mathrm{Tr}({\mathbf{F}}^{\mathsf{H}}{\mathbf{F}})=1,$ (15c)
$\displaystyle|[\boldsymbol{\Omega}]_{kk}|=1,\forall k,$ (15d)

which is non-convex due to the non-convex constraint in (15d). Therefore, it
is challenging to find the optimal solution. In order to ease the analysis
that follows, we use a simplified yet intuitive two-stage approach to
sequentially optimize $\boldsymbol{\Omega}$ and
$\\{{\mathbf{W}},{\mathbf{F}}\\}$. We first focus on the term
${\mathbf{H}}_{\text{R,U}}\boldsymbol{\Omega}{\mathbf{H}}_{\text{B,R}}$, and
define
$\eta=\sqrt{\frac{N_{\text{R}}N_{\text{U}}}{\rho_{\text{R,U}}}}\sqrt{\frac{N_{\text{B}}N_{\text{R}}}{\rho_{\text{B,R}}}}\exp(-j2\pi\tau_{\text{R,U}})\exp(-j2\pi\tau_{\text{B,R}})\boldsymbol{\alpha}_{\text{R}}^{\mathsf{H}}(\theta^{t}_{\text{R,U}},\phi^{t}_{\text{R,U}})\\\
\boldsymbol{\Omega}\boldsymbol{\alpha}_{\text{R}}(\theta^{r}_{\text{B,R}},\phi^{r}_{\text{B,R}})$.
In this regard, ${\mathbf{H}}$ in (11) can be reformulated as

${\mathbf{H}}(\eta)=\underbrace{{\mathbf{H}}_{\text{B,U}}}_{\text{Rank
One}}+\underbrace{\eta\boldsymbol{\alpha}_{\text{U}}(\theta^{r}_{\text{R,U}},\phi^{r}_{\text{R,U}})\boldsymbol{\alpha}_{\text{B}}^{\mathsf{H}}(\phi^{t}_{\text{B,R}}-\beta)}_{\text{Rank
One}},$ (16)

which is a summation of two rank-one matrices. Here, we replace ${\mathbf{H}}$
as ${\mathbf{H}}(\eta)$ to show its dependence on $\eta$. Based on the
following facts:

$\lim_{N_{\text{B}}\rightarrow\infty}\boldsymbol{\alpha}_{\text{B}}^{\mathsf{H}}(\phi^{t}_{\text{B,U}}-\beta)\boldsymbol{\alpha}_{\text{B}}(\phi^{t}_{\text{B,R}}-\beta)\approx
0,\;\text{if }\;\phi^{t}_{\text{B,U}}\neq\phi^{t}_{\text{B,R}},$ (17)
$\displaystyle\lim_{N_{\text{U}}\rightarrow\infty}\boldsymbol{\alpha}^{\mathsf{H}}_{\text{U}}(\theta^{r}_{\text{B,U}},\phi^{r}_{\text{B,U}})\boldsymbol{\alpha}_{\text{U}}(\theta^{r}_{\text{R,U}},\phi^{r}_{\text{R,U}})\approx
0,\;\text{if}\;$
$\displaystyle\theta^{r}_{\text{R,U}}\neq\theta^{r}_{\text{B,U}}$
$\displaystyle\&\phi^{r}_{\text{R,U}}\neq\phi^{r}_{\text{B,U}},$ (18)

which can be readily proven. Thus, the singular value decomposition (SVD) of
${\mathbf{H}}(\eta)$ is approximated by

${\mathbf{H}}(\eta)\approx{\mathbf{U}}\mathrm{diag}([\sqrt{N_{\text{B}}N_{\text{U}}/\rho_{\text{B,U}}}\;\;|\eta|]){\mathbf{V}}^{\mathsf{H}},$
(19)

where
${\mathbf{U}}=[\exp(-j2\pi\tau_{\text{B,U}})\boldsymbol{\alpha}_{\text{U}}(\theta^{r}_{\text{B,U}},\phi^{r}_{\text{B,U}});\boldsymbol{\alpha}_{\text{U}}(\theta^{r}_{\text{R,U}},\phi^{r}_{\text{R,U}})]$
and
${\mathbf{V}}=[\boldsymbol{\alpha}_{\text{B}}(\phi^{t}_{\text{B,U}}-\beta);\boldsymbol{\alpha}_{\text{B}}(\phi^{t}_{\text{B,R}}-\beta)]$.
According to (17) and (18), ${\mathbf{U}}$ and ${\mathbf{V}}$ are nearly
partial of unitary matrices, i.e.,
${\mathbf{U}}^{\mathsf{H}}{\mathbf{U}}\approx{\mathbf{I}}_{2}$ and
${\mathbf{V}}^{\mathsf{H}}{\mathbf{V}}\approx{\mathbf{I}}_{2}$.

To this end, we first maximize $|\eta|$ in the first stage, which is equal to
$\sqrt{\frac{N_{\text{R}}N_{\text{U}}}{\rho_{\text{R,U}}}}\sqrt{\frac{N_{\text{B}}N_{\text{R}}}{\rho_{\text{B,R}}}}$.
Afterwards, we perform SVD of
${\mathbf{H}}(\eta=\sqrt{\frac{N_{\text{R}}N_{\text{U}}}{\rho_{\text{R,U}}}}\sqrt{\frac{N_{\text{B}}N_{\text{R}}}{\rho_{\text{B,R}}}})$
to design the optimal ${\mathbf{W}}$ and ${\mathbf{F}}$ in the second stage.
The optimal closed-form solutions for $\boldsymbol{\Omega}$, ${\mathbf{W}}$,
and ${\mathbf{F}}$ are summarized as

$\displaystyle[\boldsymbol{\Omega}]_{kk}$
$\displaystyle=\exp\bigg{\\{}j\Big{(}\phase{[\boldsymbol{\alpha}_{\text{R}}(\theta^{t}_{\text{R,U}},\phi^{t}_{\text{R,U}})]_{k}}-\phase{[\boldsymbol{\alpha}_{\text{R}}(\theta^{r}_{\text{B,R}},\phi^{r}_{\text{B,R}})]_{k}}\Big{)}\bigg{\\}}$
$\displaystyle\times\exp\big{(}j2\pi(\tau_{\text{R,U}}+\tau_{\text{B,R}})\big{)},$
(20) $\displaystyle{\mathbf{W}}$
$\displaystyle=\frac{\sqrt{2}}{2}{\mathbf{U}},{\mathbf{F}}=\frac{\sqrt{2}}{2}{\mathbf{V}},$
(21)

where ${\mathbf{W}}$ and ${\mathbf{F}}$ meet the constraints specified in
(15b) and (15c).

The achievable rate, rewritten as $R_{1}$, for this case is

$R_{1}\approx\log_{2}\Big{(}1+\frac{P}{4\sigma^{2}}\frac{N_{\text{B}}N_{\text{U}}}{\rho_{\text{B,U}}}\Big{)}+\log_{2}\Big{(}1+\frac{P}{4\sigma^{2}}\frac{N_{\text{R}}N_{\text{U}}}{\rho_{\text{R,U}}}\frac{N_{\text{B}}N_{\text{R}}}{\rho_{\text{B,R}}}\Big{)}.$
(22)

####  III-A2 Case ii)
$\mathbb{I}({\mathbf{H}}_{\textsc{B,U}})=1,\mathbb{I}({\mathbf{H}}_{\textsc{R,U}})=0$

For this case, we only need to perform SVD of ${\mathbf{H}}_{\text{B,U}}$ to
design the optimal beamforming vectors ${\mathbf{w}}$ and ${\mathbf{f}}$. The
calculation of the achievable rate is straightforward. The achievable rate,
redefined as $R_{2}$, for this case is
$R_{2}=\log_{2}\Big{(}1+\frac{P}{\sigma^{2}}\frac{N_{\text{B}}N_{\text{U}}}{\rho_{\text{B,U}}}\Big{)}$.

####  III-A3 Case iii)
$\mathbb{I}({\mathbf{H}}_{\textsc{B,U}})=0,\mathbb{I}({\mathbf{H}}_{\textsc{R,U}})=1$

For this case, we can follow the two-stage approach, already applied for case
i). The achievable rate, redefined as $R_{3}$, for this case is
$R_{3}=\log_{2}\Big{(}1+\frac{P}{\sigma^{2}}\frac{N_{\text{R}}N_{\text{U}}}{\rho_{\text{R,U}}}\frac{N_{\text{B}}N_{\text{R}}}{\rho_{\text{B,R}}}\Big{)}$.

###  III-B Average Achievable Rate

We take the LoS probabilities and their associated achievable rates into
consideration and calculate the average achievable rate as

$\displaystyle\bar{R}$
$\displaystyle=R_{1}\text{Pr}(\mathbb{I}({\mathbf{H}}_{\text{B,U}})=1)\text{Pr}(\mathbb{I}({\mathbf{H}}_{\text{R,U}})=1)$
$\displaystyle+R_{2}\text{Pr}(\mathbb{I}({\mathbf{H}}_{\text{B,U}})=1)\text{Pr}(\mathbb{I}({\mathbf{H}}_{\text{R,U}})=0)$
$\displaystyle+R_{3}\text{Pr}(\mathbb{I}({\mathbf{H}}_{\text{B,U}})=0)\text{Pr}(\mathbb{I}({\mathbf{H}}_{\text{R,U}})=1).$
(23)

##  IV Simulation Results

In this section, we study the average achievable rate for an RIS-assisted B5G
network and compare it with its RIS-free counterpart. For simplicity, we only
consider the free-space path loss, which is modeled as:
$\rho=d^{2}f_{c}^{2}/10^{8.755}$, where $f_{c}$ (in KHz) is the carrier
frequency, defined as $f_{c}=\frac{c}{\lambda}$ with $c$ being the speed of
light ($3\times 10^{8}$ m/s). We omit the subscripts of $\rho$ and $d$ to make
them general and applicable to all the path losses and distances. The system
parameters are set as: $f_{c}\in\\{28,3.5\\}$ GHz, $\beta=\pi/3$,
$N_{\text{B}}=8\times 8$, $N_{\text{R}}=20\times 20$, $N_{\text{U}}=8\times
8$, ${\mathbf{p}}_{\text{B}}=(0,0,10)$,
${\mathbf{p}}_{\text{R}}=(0.5,0.5,9.5)$, and bandwidth $B=20$ MHz. The UAV is
supposed to be located at any possible position with
$x_{\text{U}}>0,y_{\text{U}}>0,z_{\text{U}}>0$. To model the LoS probability,
we follow [13] for the setup for $(\alpha,\kappa,\gamma)$: suburban (0.1, 750,
8), urban (0.3, 500, 15), dense urban (0.5, 300, 20), and highrise urban (0.5,
300, 50). We fix the height for the UAV as $100$ meters, i.e.,
$z_{\text{U}}=100$, while its $x$-axis and $y$-axis are uniformly distributed
within $[100,1000]$ meters when $f_{c}=28$ GHz and $[100,2000]$ meters when
$f_{c}=3.5$ GHz. The cumulative distribution functions (CDFs) of the average
achievable rate are shown in Fig. 2 for different propagation environments
with $f_{c}=28$ GHz. It can be seen from the simulation curves that the higher
LoS availability (e.g., suburban) the better the average achievable rate. By
following $B=20$ MHz, the BS can offer up to several hundred Mbps data rate
for the flying UAVs in the suburban scenario.

Figure 2: CDFs of average achievable rate for different environments, i.e.,
suburban, urban, dense urban, and highrise urban, with $f_{c}=28$ GHz.

Comparisons of the average achievable rates of the RIS-assisted and RIS-free
UAV communication networks are shown in Figs. 3 and 4 for both carrier
frequencies, i.e., mmWave and C band. As we can see, significant rate gains
can be obtained by introducing the RIS to the existing 5G cellular networks,
confirming the 3D connectivity enhancements that can be unleashed by RISs to
UAVs. It also becomes clear that higher frequencies offer a shorter
communication range due to more server path loss.

Figure 3: Comparison between RIS-assisted (left) and RIS-free (right) UAV
communication networks in terms of average achievable rate at 28 GHz (mmWave).
The urban scenario is evaluated. Figure 4: Comparison between RIS-assisted
(left) and RIS-free (right) UAV communication networks in terms of average
achievable rate at 3.5 GHz (C band). The urban scenario is evaluated.

##  V Conclusions

In this paper, we have studied the 3D connectivity enabled by exploiting the
existing 5G BS together with an RIS. We have investigated the average
achievable rate while taking into consideration of the LoS probabilities under
different propagation environments. Simulation results have shown that with
the introduction of RIS, substantially higher average achievable rates can be
obtained compared to its RIS-free counterpart.

## Acknowledgement

This contribution of Aymen Fakhreddine has been partly funded by FWF – Der
Wissenschaftsfonds (Austrian Science Fund) ESPRIT program under grant number
ESP-54.

## References

  * [1] C. Zhang, W. Zhang, W. Wang, L. Yang, and W. Zhang, “Research challenges and opportunities of UAV millimeter-wave communications,” _IEEE Wireless Commun._ , vol. 26, no. 1, pp. 58–62, 2019.
  * [2] Z. Zhang, Y. Xiao, Z. Ma, M. Xiao, Z. Ding, X. Lei, G. K. Karagiannidis, and P. Fan, “6G wireless networks: Vision, requirements, architecture, and key technologies,” _IEEE Veh. Technol. Mag._ , vol. 14, no. 3, pp. 28–41, 2019\.
  * [3] H. Lu, Y. Zeng, S. Jin, and R. Zhang, “Aerial intelligent reflecting surface: Joint placement and passive beamforming design with 3D beam flattening,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 7, pp. 4128–4143, 2021.
  * [4] X. Mu, Y. Liu, L. Guo, J. Lin, and H. V. Poor, “Intelligent reflecting surface enhanced multi-UAV NOMA networks,” _IEEE J. Sel. Areas Commun._ , vol. 39, no. 10, pp. 3051–3066, 2021.
  * [5] X. Liu, Y. Liu, and Y. Chen, “Machine learning empowered trajectory and passive beamforming design in UAV-RIS wireless networks,” _IEEE J. Sel. Areas Commun._ , vol. 39, no. 7, pp. 2042–2055, 2020.
  * [6] S. Li, B. Duo, X. Yuan, Y.-C. Liang, and M. Di Renzo, “Reconfigurable intelligent surface assisted UAV communication: Joint trajectory design and passive beamforming,” _IEEE Wireless Commun. Lett._ , vol. 9, no. 5, pp. 716–720, 2020.
  * [7] Y. Liu, X. Liu, X. Mu, T. Hou, J. Xu, M. Di Renzo, and N. Al-Dhahir, “Reconfigurable intelligent surfaces: Principles and opportunities,” _IEEE Commun. Surveys Tuts._ , vol. 23, no. 3, pp. 1546–1577, 2021.
  * [8] E. Björnson, H. Wymeersch, B. Matthiesen, P. Popovski, L. Sanguinetti, and E. de Carvalho, “Reconfigurable intelligent surfaces: A signal processing perspective with wireless applications,” _IEEE Signal Process. Mag._ , vol. 39, no. 2, pp. 135–158, 2022.
  * [9] L. Yang and W. Zhang, “Beam tracking and optimization for UAV communications,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 11, pp. 5367–5379, 2019.
  * [10] M. R. Akdeniz, Y. Liu, M. K. Samimi, S. Sun, S. Rangan, T. S. Rappaport, and E. Erkip, “Millimeter wave channel modeling and cellular capacity evaluation,” _IEEE J. Sel. Areas Commun._ , vol. 32, no. 6, pp. 1164–1179, 2014.
  * [11] O. E. Ayach, S. Rajagopal, S. Abu-Surra, Z. Pi, and R. W. Heath, “Spatially sparse precoding in millimeter wave MIMO systems,” _IEEE Trans. Wireless Commun._ , vol. 13, no. 3, pp. 1499–1513, 2014.
  * [12] J. He, H. Wymeersch, L. Kong, O. Silvén, and M. Juntti, “Large intelligent surface for positioning in millimeter wave MIMO systems,” in _proc. IEEE 91st Vehicular Technology Conference (VTC2020-Spring)_ , 2020, pp. 1–5.
  * [13] A. Al-Hourani, S. Kandeepan, and S. Lardner, “Optimal LAP altitude for maximum coverage,” _IEEE Wireless Commun. Lett._ , vol. 3, no. 6, pp. 569–572, 2014.
  * [14] M. Gapeyenko, D. Moltchanov, S. Andreev, and R. W. Heath, “Line-of-sight probability for mmWave-based UAV communications in 3D urban grid deployments,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 10, pp. 6566–6579, 2021.
  * [15] C. Huang, A. Zappone, G. C. Alexandropoulos, M. Debbah, and C. Yuen, “Reconfigurable intelligent surfaces for energy efficiency in wireless communication,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 8, pp. 4157–4170, 2019.
  * [16] R. Li, S. Sun, Y. Chen, C. Han, and M. Tao, “Ergodic achievable rate analysis and optimization of RIS-Assisted millimeter-wave MIMO communication systems,” _IEEE Trans. Wireless Commun._ , vol. 22, no. 2, pp. 972–985, 2023.