# Channel Characterization of UAV-RIS-aided Systems with Adaptive Phase-shift
Configuration

Thanh Luan Nguyen, Georges Kaddoum, Tri Nhu Do, and Zygmunt J. Haas This work
was supported by the Tier 2 Canada Research Chair program and the Natural
Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant
program. The work of Z. J. Haas was supported in part by the U.S. National
Science Foundation under the grant number CNS-1763627.Thanh Luan Nguyen,
Georges Kaddoum and Tri Nhu Do are with the Department of Electrical
Engineering, the École de Technologie Supérieure (ÉTS), Université du Québec,
Montréal, QC H3C 1K3, Canada. Georges Kaddoum is also with Cyber Security
Systems and Applied AI Research Center, Lebanese American University (emails:
<EMAIL_ADDRESS><EMAIL_ADDRESS>tri-
nhu.do@etsmtl.ca).Z. J. Haas is with the Department of Computer Science,
University of Texas at Dallas, Richardson, TX 75080, USA, and also with the
School of Electrical and Computer Engineering, Cornell University, Ithaca, NY
14853, USA (e-mail: zhaas@cornell.edu).

###### Abstract

This letter considers a UAV aiding communication between a ground transmitter
and a ground receiver in the presence of co-channel interference. A discrete-
time Markov process is adopted to model the complex nature of the Air-to-
Ground (A2G) channel, including the occurrence of Line-of-Sight, Non-Line-of-
Sight, and blockage events. Moreover, an adaptive phase-shift-enabled
Reconfigurable Intelligent Surface (RIS) is deployed to combat A2G blockage
events. Novel frameworks based on the shadowed Rician distribution are
proposed to derive closed-form expressions for Ground-to-Air/A2G SINR’
distributions. Numerical results show that RISs with large numbers of
elements, e.g., 256 RIS elements, improve end-to-end Outage Probability (OP)
and reduce blockages.

###### Index Terms:

Reconfigurable Intelligent Surfaces (RIS); UAV; adaptive phase-shift
configuration; channel characterization

##  I Introduction

Reconfigurable intelligent surfaces (RISs) have emerged as a promising
technology for enhancing the communication performance of non-terrestrial
networks [1], especially in the context of Unmanned Aerial Vehicles
(UAVs)-based wireless communication systems [2]. RISs, composed of
programmable RIS elements, can manipulate the direction of incident signals,
improve the received signal strength, mitigate interference, and enhance
coverage rate. These advantages make RISs specifically useful in UAVs-based
systems to enable remote communication even in interference-limited and
disaster-stricken areas. However, the inherently dynamic nature of UAVs
presents distinct challenges when it comes to accurately characterizing both
Air-to-Ground (A2G) and Ground-to-Air (G2A) communication channels.

In [3], the authors provided an accurate Outage Probability (OP) expression
for RIS-aided interference-free wireless networks under Rician fading.
Moreover, [4] recently presented closed-form OP expressions considering
disorientation, misalignment, and hardware imperfections. Similarly, [5]
recently derived closed-form expressions for OP and channel capacity in
interference scenarios. The work in [6] expanded this focus, investigating
performance in UAV-RIS-aided interference-free networks. Alongside these, [7]
introduced a three-dimensional (3D) LoS/NLoS channel model for UAV-to-ground
communications, using a dual RIS setup on building facades. Finally, [8]
contributed a new UAV deployment algorithm for 3D non-homogeneous terrain.
While the studies in [3, 4, 5, 6, 7] consider LoS and NLoS conditions, the
concurrent blockages that disrupt communication is overlooked. Our work
bridges this gap by deriving a comprehensive model accounting for LoS, NLoS,
and blockages, aiming to accurately represent challenges in UAV-RIS-aided
network performance.

Different from existing works, in our work, the elevation angle in A2G
communication links can experience stochastic behaviors, leading to
significant impacts on the channel quality. Specifically, variations in the
elevation angle can lead to unpredictable blockages and communication link
disruptions, which poses challenges in UAV-aided systems. To directly combat
the impact of varying elevation angles, this paper proposes an adaptive phase-
shift configuration-enabled RIS, which mitigates blockage effects in the A2G
communication and ensures a reliable communication link. This letter’s key
contributions are as follows:

  * •

We propose a technical framework that assigns the non-central chi-square
(NCCS) distribution of Rician fading channel power gain to the sum of
independent shadowed Rician (SISR) distribution.

  * •

We propose a stochastic A2G channel model that integrates both blockage and
non-blockage conditions. The blockage condition represents high attenuation
scenarios due to obstructions, which leads to A2G link disruptions.

  * •

With our framework, tractable A2G and G2A SINR results are provided in a
Rician fading environment in the presence of non-IID Co-Channel Interference
(CCI).

  * •

Numerical results indicate that larger RIS elements, specifically 256, not
only improve end-to-end (e2e) OP, but also effectively mitigate blockage
impact.

Notations: Vectors are represented by boldface lowercase letters. Matrices are
denoted by boldface uppercase letters. The transpose, conjugate transpose, and
diagonal matrices are denoted as $(\cdot)^{\sf T}$, $(\cdot)^{\sf H}$,
$\operatorname{diag}(\cdot)$, respectively. ${{\cal CN}(0,\sigma^{2})}$
denotes the circularly symmetric complex Gaussian Random Variable (RV) with
zero mean and variance $\sigma^{2}$, ${\mathbb{E}[\cdot]}$ denotes the
expectation operator, ${\mu_{k,X}=\mathbb{E}[X^{k}]}$ denotes the $k$th moment
of RV $X$, and ${\mathfrak{L}_{X}(s)=\mathbb{E}[e^{-sX}]}$ denotes the Laplace
transform of $X$.

##  II System Model and Proposed Markov Process-based Phase-Shift
Configuration

We consider an interference-limited UAV-RIS-aided system involving an
$M$-antenna ground transmitter (GT), a single-antenna UAV, a single-antenna
ground receiver (GR), and an $N=N_{x}\times N_{y}$ element RIS, as depicted in
Fig. 1. The system contends with $L$ aerial and $K$ ground interferers
impacting G2A and A2G communications. Hereafter, we use S,
$\textnormal{S}_{m}$, U, R, $\textnormal{R}_{n}$, D, $\textnormal{I}_{l}$, and
$\textnormal{J}_{k}$ to represent the GT, its $m$-th antenna, the UAV, the
RIS, RIS’s $n$-th element, the GR, the $l$-th aerial, and the $k$-th ground
interferer, respectively.

Figure 1: Illustrations of a UAV-RIS-aided wireless system in the presence of
non-IID aerial and ground CCIs.

We consider that the direct links $\textnormal{S}_{m}$-D, ${\forall m}$, are
unavailable due to multiple impenetrable obstacles, such as concrete-walled
structures in the 3D-urban micro (UMi) environment. The relative location of
node X with respect to node Y, for
${\textnormal{X},\textnormal{Y}\\!\in\\!\\{\textnormal{S}_{m},\textnormal{U},\textnormal{D},\textnormal{R}_{n},\textnormal{I}_{l},\textnormal{J}_{k}\\}}$,
in the three-dimensional (3D) Cartesian coordinates system is denoted as
${{\bf
p}_{\textnormal{XY}}\triangleq(x_{\textnormal{XY}},y_{\textnormal{XY}},z_{\textnormal{XY}})}$,
and the corresponding spherical coordinates are
${(d_{\textnormal{XY}},\theta_{\textnormal{XY}},\varphi_{\textnormal{XY}})}$,
where $d_{\textnormal{XY}}$, ${\theta_{\textnormal{XY}}}$, and
${\varphi_{\textnormal{XY}}}$ are the radial distance, the elevation angle,
and the azimuth angle, respectively. The channel coefficient between nodes X
and Y is denoted by
${g_{\textnormal{XY}}=\sqrt{{\ell}_{\textnormal{XY}}}h_{\textnormal{XY}}e^{j2\pi
f_{d,\max}\cos\varpi_{\textnormal{XY}}t}}$, where $h_{\textnormal{XY}}$ is the
complex small-scale fading, ${\ell}_{\textnormal{XY}}$ represents the path
loss, $f_{d,\max}$ [Hz] is the maximum Doppler frequency shift,
$\varpi_{\textnormal{XY}}$ [rad] is the Angle-of-Arrival (AoA) angle, and $t$
is the discrete-time instance. We consider that the UAV is moving with
velocity ${\boldsymbol{\mathrm{v}}\in\mathbb{R}^{3}}$. Herein,
$\|\boldsymbol{\mathrm{v}}\|$ [m/s] denotes the UAV’s movement speed and
$\frac{\boldsymbol{\mathrm{v}}}{\|\boldsymbol{\mathrm{v}}\|}$ specifies the
unit-size movement direction. The maximum Doppler frequency shift and AoA
cosine are given by
${f_{d,\max}\\!=\\!\frac{f_{c}\|\boldsymbol{\mathrm{v}}\|}{c}}$ with $f_{c}$
[Hz] being the carrier frequency, and
${\cos\varpi_{\textnormal{XY}}\\!=\\!\frac{\boldsymbol{\mathrm{v}}^{T}\boldsymbol{\mathrm{p}}_{\textnormal{XY}}}{\|\boldsymbol{\mathrm{v}}\|\|\boldsymbol{\mathrm{p}}_{\textnormal{XY}}\|}}$,
respectively, with $c$ [m/s] being the speed of light. For simplicity, we
assume ${t\\!=\\!1}$ throughout the rest of the paper. In addition, we assume
that all channels experience Rician-$K$ fading, where the small-scale fading
is modeled as
${h_{\textnormal{XY}}\\!=\\!\frac{\sqrt{K_{\textnormal{XY}}}\check{h}_{\textnormal{XY}}+\tilde{h}_{\textnormal{XY}}}{\sqrt{K_{\textnormal{XY}}+1}}}$
with $\check{h}_{\textnormal{XY}}$ being the deterministic LoS component with
unit modulus and $\tilde{h}_{\textnormal{XY}}$ representing the scattering
component. The $K$-factor is defined as
${K_{\textnormal{XY}}\\!=\\!K_{1}e^{K_{2}\theta_{\textnormal{XY}}}}$, where
${\theta_{\textnormal{XY}}\\!=\\!\sin^{-1}\frac{|z_{\textnormal{XY}}|}{d_{\textnormal{XY}}}}$
[rad] is the elevation angle of X relative to Y, and $K_{1}$ and $K_{2}$ are
determined by the environment [9]. The power gain of the X-Y channel, i.e.,
$|g_{\textnormal{XY}}|^{2}$, is an NCCS RV with the following PDF [10]:

$\displaystyle
f_{|g_{\textnormal{XY}}|^{2}}(z)=e^{-\lambda_{\textnormal{XY}}z-K_{\textnormal{XY}}}\lambda_{\textnormal{XY}}I_{0}\big{(}2{{\textstyle{\sqrt{K_{\textnormal{XY}}\lambda_{\textnormal{XY}}z}}}}\big{)},$
(1)

for ${z\\!>\\!0}$, where
${\lambda_{\textnormal{XY}}\triangleq\frac{K_{\textnormal{XY}}+1}{{{\ell}}_{\textnormal{XY}}}}$
and $I_{v}(x)$ is the $v$-th order modified Bessel function of the first kind
[11].

####  II-1 Ground-to-Air (G2A) Communication

Since the $\textnormal{S}_{m}$-D links are unavailable, S steers the
beamforming vector $\boldsymbol{\mathrm{w}}\in\mathbb{C}^{M}$ towards U.
Hence, the received signal at U is given by

$\displaystyle
y_{\textnormal{U}}=\sqrt{P_{\textnormal{S}}}\boldsymbol{\mathrm{g}}_{\textnormal{S}\textnormal{U}}^{\sf
H}\boldsymbol{\mathrm{w}}s+\\!\\!\mathop{\textstyle\sum}_{l=1}^{L}{{\textstyle{\sqrt{P_{\textnormal{I}_{l}}}}}g_{\textnormal{I}_{l}\textnormal{U}}s_{\textnormal{I}_{l}}}+\\!\\!\mathop{\textstyle\sum}_{k=1}^{K}{{\textstyle{\sqrt{P_{\textnormal{J}_{k}}}}}g_{\textnormal{J}_{k}\textnormal{U}}s_{\textnormal{J}_{k}}}+n_{\textnormal{U}},$
(2)

where
${\boldsymbol{\mathrm{g}}_{\textnormal{S}\textnormal{U}}\\!=\\![g_{\textnormal{S}_{1}\textnormal{U}},\ldots,g_{\textnormal{S}_{M}\textnormal{U}}]^{T}\in\mathbb{C}^{M\times
1}}$ is the complex channel vector from S to U with
${g_{\textnormal{S}_{1}\textnormal{U}},\ldots,g_{\textnormal{S}_{M}\textnormal{U}}}$
being statistically independent and identically distributed (IID),
$P_{\textnormal{S}}$, $P_{\textnormal{I}_{l}}$, and $P_{\textnormal{J}_{k}}$
are the transmit powers of S, the $l$-th aerial interferer, and the $k$-th
ground interferer, respectively, $s$, $s_{\textnormal{I}_{l}}$, and
$s_{\textnormal{J}_{k}}$ are unit-energy signals of the respective elements,
and ${n_{\textnormal{U}}}$ is the complex white Gaussian noise (AWGN) with
zero mean and variance $\sigma_{\textnormal{U}}^{2}$.

####  II-2 Air-to-Ground (A2G) Communication

After receiving and decoding the information signal from S, U sends $\hat{s}$
to D, which is reflected by R. Unlike most works, we consider that the U-D
link, i.e., the direct A2G link, is stochastically blocked. Specifically, the
received signal strength over a blocked link is below an acceptable threshold
due to propagation conditions, which is different from the NLoS state where
the signal received via the NLoS paths may still be utilized for reception.

We consider that the blockage state of the direct A2G link follows a discrete-
time Markov process ${{\Phi}=\\{\phi_{0},\phi_{1},\dots,\phi_{\infty}\\}}$,
where $\phi_{t}\\!\in\\!\\{0,1\\}$, ${\forall t\geq 0}$, is the instantaneous
channel state, ${\phi_{t}\\!=\\!0}$ (blocked) for being disrupted due to
blockage and ${\phi_{t}\\!=\\!1}$ (unblocked) for the LoS/NLoS states. The
process’s transition matrix is ${[{\bf P}]_{ij}=p_{ij}}$, where $p_{ij}$
denotes the probability of transitioning from state ${i}$ to state ${j}$. In
addition, let $\pi_{0}$ and $\pi_{1}$ be the steady-state probabilities of the
blockage and non-blockage states, respectively, and
${{\boldsymbol{\pi}}=[\pi_{0},\pi_{1}]^{\sf T}}$, the relationship between
${\boldsymbol{\pi}}$ and the transition matrix can be expressed as
${{\boldsymbol{\pi}}^{\sf T}={\boldsymbol{\pi}}^{\sf T}{\bf P}}$ [10, Eq.
(9.13)]. Hence,

$\displaystyle\pi_{0}=\frac{p_{10}}{p_{01}+p_{10}},\quad\pi_{1}=\frac{p_{01}}{p_{01}+p_{10}},\quad\pi_{0}+\pi_{1}=1.$
(3)

In addition, the considered Markov chain $\Phi$ satisfies the following
memoryless property:

$\displaystyle\Pr(\phi_{t}\mid\phi_{t-1},\phi_{t-2},\dots)$
$\displaystyle\quad=\Pr(\phi_{t}\mid\phi_{t-1})=\Bigg{\\{}\\!\\!\\!\begin{array}[]{cl}\beta,&\phi_{t-1}>\phi_{t},\\\
\beta^{1-\phi_{t}}(1-\beta)^{\phi_{t}},&\phi_{t-1}=\phi_{t},\\\
1-\beta,&\phi_{t-1}<\phi_{t},\end{array}$ (7)

where $\beta$ denotes the blockage probability, which will be detailed in
Section III-E. The received signal at D is

$\displaystyle y_{\textnormal{D}}$
$\displaystyle={\textstyle{\sqrt{P_{\textnormal{U}}}}}\boldsymbol{\mathrm{g}}_{\textnormal{U}\textnormal{R}}^{\sf
T}\boldsymbol{\mathrm{\Theta}}\boldsymbol{\mathrm{g}}_{\textnormal{R}\textnormal{D}}\hat{s}+\mathop{\textstyle\sum}\nolimits_{k=1}^{K}{{\textstyle{\sqrt{P_{\textnormal{J}_{k}}}}}g_{\textnormal{J}_{k}\textnormal{D}}s_{\textnormal{J}_{k}}}$
$\displaystyle\quad+{\textstyle{\sqrt{P_{\textnormal{U}}}}}\check{g}_{\textnormal{U}\textnormal{D}}\hat{s}\\!+\\!\mathop{\textstyle\sum}\nolimits_{l=1}^{L}{\\!{\textstyle{\sqrt{P_{\textnormal{I}_{l}}}}}g_{\textnormal{I}_{l}\textnormal{D}}s_{\textnormal{I}_{l}}}\\!+\\!n_{\textnormal{D}},\\!\\!\\!$
(8)

where
${\check{g}_{\textnormal{U}\textnormal{D}}\\!\triangleq\\!\phi_{0}g_{\textnormal{U}\textnormal{D}}}$
is the direct A2G channel coefficient under the blockage effect,
$P_{\textnormal{U}}$ is the UAV’s transmit power,
$\boldsymbol{\mathrm{g}}_{\textnormal{U}\textnormal{R}},\boldsymbol{\mathrm{g}}_{\textnormal{R}\textnormal{D}}\in\mathbb{C}^{M\times
1}$ are the incident and reflected complex channel vectors from U to R and
from R to D, respectively,
${n_{\textnormal{D}}\sim\mathcal{CN}(0,\sigma_{\textnormal{D}}^{2})}$ is the
AWGN,
${\bf\Theta}\triangleq\mathrm{diag}([\kappa_{1}e^{j\vartheta_{1}},\ldots,\kappa_{N}e^{j\vartheta_{N}}])\in\mathbb{C}^{N\times
N}$ is the phase-shift matrix of R, where ${\kappa_{n}\\!\in\\!(0,1]}$ and
${\vartheta_{n}\\!\in\\![0,2\pi)}$ denotes the amplitude reflection
coefficient and the phase-shift of the $n$-th reflecting element of R. In
addition, we assume that
${\kappa_{1}\\!=\\!\ldots\\!=\\!\kappa_{N}\\!=\\!\kappa}$ [12, 4].

####  II-3 SINR Modeling

From (2) and (8), the G2A and A2G SINRs are respectively formulated as
${\Gamma}_{\mathtt{g2a}}\\!=\\!\frac{\frac{P_{\textnormal{S}}}{\sigma^{2}_{\textnormal{U}}}\|\boldsymbol{\mathrm{g}}_{\textnormal{S}\textnormal{U}}^{\sf
H}\boldsymbol{\mathrm{w}}\|^{2}}{\gamma_{\textnormal{I}\textnormal{U}}+\gamma_{\textnormal{J}\textnormal{U}}+1}$
and
${\Gamma}_{\mathtt{a2g}}\\!=\\!\frac{\frac{P_{\textnormal{U}}}{\sigma^{2}_{\textnormal{D}}}\left|{\bf
g}_{\textnormal{U}\textnormal{R}}^{\sf T}\boldsymbol{\mathrm{\Theta}}{\bf
g}_{\textnormal{R}\textnormal{D}}+\check{g}_{\textnormal{U}\textnormal{D}}\right|^{2}}{\gamma_{\textnormal{I}\textnormal{D}}+\gamma_{\textnormal{J}\textnormal{D}}+1}$,
where
${\gamma_{\textnormal{I}\textnormal{X}}\\!\triangleq\\!\sum_{l=1}^{L}\\!\\!\bar{\gamma}_{\textnormal{I}_{l}\textnormal{X}}|h_{\textnormal{I}_{l}\textnormal{X}}|^{2}}$
and
${\gamma_{\textnormal{J}\textnormal{X}}\\!\triangleq\\!\sum_{k=1}^{K}\\!\\!\bar{\gamma}_{\textnormal{J}_{k}\textnormal{X}}|h_{\textnormal{J}_{k}\textnormal{X}}|^{2}}$
are the aggregated aerial and ground Interference-to-Noise Ratios (INRs) at
${\textnormal{X}\\!\in\\!\\{\textnormal{U},\textnormal{D}\\}}$, respectively,
${\bar{\gamma}_{\textnormal{I}_{l}\textnormal{X}}\\!\triangleq\\!\frac{P_{\textnormal{I}_{l}}{\ell}_{\textnormal{I}_{l}\textnormal{X}}}{\sigma_{\textnormal{X}}^{2}}}$,
${l\\!\in\\![1,L]}$, and
${\bar{\gamma}_{\textnormal{J}_{k}\textnormal{X}}\\!\triangleq\\!\frac{P_{\textnormal{J}_{k}}{\ell}_{\textnormal{J}_{k}\textnormal{X}}}{\sigma_{\textnormal{X}}^{2}}}$,
${k\\!\in\\![1,K]}$. The e2e SINR is determined via the weakest communication
link, specified by the minimum between the G2A and A2G SINRs, and is
formulated as ${\Gamma}_{\rm
e2e}\triangleq\min\\{\Gamma_{\mathtt{g2a}},\Gamma_{\mathtt{a2g}}\\}$.

To obtain ${\\{{\bf
w}^{\star},{\bf\Theta}^{\star}\\}=\operatorname{arg}\max_{{\bf
w},{\bf\Theta}}~{}{\Gamma_{\rm e2e}}}$, we use the fact that
$\max_{{\boldsymbol{\theta}}}\\{\min\\{A_{\boldsymbol{\theta}},B_{\boldsymbol{\theta}}\\}\\}\\!=\\!\min\\{\max_{\boldsymbol{\theta}}\\{A_{\boldsymbol{\theta}}\\},\max_{\boldsymbol{\theta}}\\{B_{\boldsymbol{\theta}}\\}\\}\\}$.
As the SINRs at U and D are independent of ${\bf\Theta}$ and ${\bf w}$,
respectively, the optimal beamforming vector ${\bf w}$ and phase-shift matrix
${\bf\Theta}$ that maximize the SINRs at U and D also maximize the e2e SINR.
As a result, we use the optimal active beamforming, i.e., Maximum-Ratio
Transmission (MRT) beamforming, where the $m$-th weight of the beamforming is
given by
${[\boldsymbol{\mathrm{w}}]_{m}\\!=\\!\frac{\left|[\boldsymbol{\mathrm{g}}_{\textnormal{S}\textnormal{U}}]_{m}\right|}{\|\boldsymbol{\mathrm{g}}_{\textnormal{S}\textnormal{U}}\|}e^{j\angle[\boldsymbol{\mathrm{g}}_{\textnormal{S}\textnormal{U}}]_{m}}}$.
To maximize the A2G SINR, we use the fact that $\big{|}{\bf
g}_{\textnormal{U}\textnormal{R}}^{\sf T}\boldsymbol{\mathrm{\Theta}}{\bf
g}_{\textnormal{R}\textnormal{D}}+\check{g}_{\textnormal{U}\textnormal{D}}\big{|}\leq\big{|}\sum_{n=1}^{N}{|g_{\textnormal{U}\textnormal{R}_{n}}g_{\textnormal{R}_{n}\textnormal{D}}|}+|\check{g}_{\textnormal{U}\textnormal{D}}|\big{|}$
where the equality occurs when $\vartheta_{n}+\angle
g_{\textnormal{U}\textnormal{R}_{n}}+\angle
g_{\textnormal{R}_{n}\textnormal{D}}=\angle\check{g}_{\textnormal{U}\textnormal{D}}$
for all $n\in[1,N]$. It is noted that the phase-shift configuration depends on
the availability of the direct A2G link. Hence, we obtain the following
adaptive phase-shift configuration:

$\displaystyle\vartheta^{\ast}_{n}=\phi_{0}\big{[}\varphi_{\textnormal{U}\textnormal{D}}+2\pi
f_{d,\max}{\boldsymbol{\mathrm{v}}^{T}\boldsymbol{\mathrm{p}}_{\textnormal{U}\textnormal{R}_{n}}}/({\|\boldsymbol{\mathrm{v}}\|\|\boldsymbol{\mathrm{p}}_{\textnormal{U}\textnormal{D}}\|})\big{]}$
$\displaystyle\hskip
20.0pt-\varphi_{\textnormal{U}\textnormal{R}_{n}}\\!-\\!\varphi_{\textnormal{R}_{n}\textnormal{D}}\\!-\\!2\pi
f_{d,\max}{\boldsymbol{\mathrm{v}}^{T}\boldsymbol{\mathrm{p}}_{\textnormal{U}\textnormal{R}_{n}}}/({\|\boldsymbol{\mathrm{v}}\|\|\boldsymbol{\mathrm{p}}_{\textnormal{U}\textnormal{R}_{n}}\|}).$
(9)

By adaptively adjusting $\vartheta^{\ast}_{n}$ based on the instantaneous
state of the direct A2G link, the proposed two-state Markov process-based
adaptive phase shift can maintain reliable communication, even in the presence
of blockages. Subsequently, the optimal e2e SINR is obtained as
${{\Gamma}_{\rm
e2e}^{\star}\\!=\\!\min\\{{\Gamma}_{\mathtt{g2a}}^{\star},{\Gamma}_{\mathtt{a2g}}^{\star}\\}}$
where

$\displaystyle{\Gamma}_{\mathtt{g2a}}^{\star}\\!=\\!\frac{\bar{\gamma}_{\mathtt{g2a}}\gamma_{\mathtt{g2a}}}{\gamma_{\textnormal{I}\textnormal{U}}\\!+\\!\gamma_{\textnormal{J}\textnormal{U}}\\!+\\!1},{\Gamma}_{\mathtt{a2g}}^{\star}\\!=\\!\frac{\big{|}\sqrt{\bar{\gamma}_{\mathtt{cas}}\gamma_{\mathtt{cas}}}+\sqrt{\bar{\gamma}_{\mathtt{dir}}\gamma_{\mathtt{dir}}}\big{|}^{2}}{\gamma_{\textnormal{I}\textnormal{D}}\\!+\\!\gamma_{\textnormal{J}\textnormal{D}}\\!+\\!1},$
(10)

where
${\bar{\gamma}_{\mathtt{g2a}}\\!\triangleq\\!\frac{P_{\textnormal{S}}{\ell}_{\textnormal{S}\textnormal{U}}}{\sigma^{2}_{\textnormal{U}}}}$,
${\gamma_{\mathtt{g2a}}\\!\triangleq\\!\sum_{m=1}^{M}{\\!\\!(h_{\textnormal{S}_{m}\textnormal{U}})^{2}}}$,
${\bar{\gamma}_{\mathtt{cas}}\\!\triangleq\\!\frac{P_{\textnormal{U}}{\ell}_{\textnormal{U}\textnormal{R}}{\ell}_{\textnormal{R}\textnormal{D}}|\kappa|^{2}\\!\\!}{\sigma^{2}_{\textnormal{D}}}}$,
${\bar{\gamma}_{\mathtt{dir}}\\!\triangleq\\!\frac{P_{\textnormal{U}}{\ell}_{\textnormal{U}\textnormal{D}}}{\sigma^{2}_{\textnormal{D}}}}$,
${{\gamma}_{\mathtt{cas}}\\!\triangleq\\!(\sum_{n=1}^{N}{\\!\\!h_{\textnormal{U}\textnormal{R}_{n}}h_{\textnormal{R}_{n}\textnormal{D}}})^{2}}$,
and
${\gamma_{\mathtt{dir}}\\!\triangleq\\!\phi_{0}|h_{\textnormal{U}\textnormal{D}}|^{2}}$.

##  III Aerial Chanel Characterization and End-to-End Performance Analysis

Characterizing the stochastic behavior of Rician fading channels in UAV-RIS-
aided communication presents mathematical challenges due to the use of Bessel
functions and the presence of non-IID interference. Additionally, modeling
blockage events with a two-state Markov chain further increases the complexity
of the performance analysis.

###  III-A Sum of independent shadowed Rician (SISR) distribution

To assist in the performance analysis, we propose alternative expressions of
(1) with the help of the following the Lemma.

######  Lemma .

Let ${Y\\!=\\!\sum_{i=1}^{I}{X_{i}}}$ be the SISR RV, where each $X_{i}$ is a
shadowed-Rican RV whose parameters are $\kappa$, $m$, and $\omega$. The PDF
and CDF of $Y$ are respectively given by:

$\displaystyle f_{Y}(I,\psi;x)$
$\displaystyle=\\!\\!\\!\mathop{\textstyle\sum}_{k=0}^{I\psi-I}\frac{\chi_{I,k}}{(I\psi-k-1)!}\frac{x^{I\psi-k-1}}{\alpha^{I\psi-k}}e^{-\frac{x}{\alpha}},{x>0},\\!$
(11) $\displaystyle F_{Y}(I,\psi;x)$
$\displaystyle=1-\\!\\!\\!\mathop{\textstyle\sum}_{k=0}^{I\psi-I}\chi_{I,k}\frac{\gamma(I\psi-k,x/\alpha)}{(I\psi-k-1)!},{x>0},$
(12)
$\displaystyle=1-\\!\\!\\!\mathop{\textstyle\sum}_{k=0}^{I\psi-I}\mathop{\textstyle\sum}_{n=0}^{I\psi-k-1}\frac{\chi_{I,k}}{n!}\frac{x^{n}}{\alpha^{n}}e^{-\frac{x}{\alpha}},{x>0},$
(13)

where $\gamma(a,x)$ is the lower incomplete Gamma function [11],
${\alpha\\!\triangleq\\!\frac{\xi+1}{\kappa+1}}$,
${\chi_{I,k}\\!\triangleq\\!\binom{I\psi-I}{k}\frac{1}{(\xi+1)^{k}}(\frac{\xi}{\xi+1})^{I\psi-
I-k}}$, ${\xi\\!\triangleq\\!\frac{\kappa\omega}{\psi}}$,
${\kappa\\!\triangleq\\!\frac{1\\!-\\!\sigma^{2}}{\sigma^{2}}}$,
${\omega\\!\triangleq\\!\frac{\Omega(\kappa+1)}{\kappa}}$,
${\Omega\\!\triangleq\\!\frac{\sqrt{\psi}}{\sqrt{\psi-1}}\frac{\sqrt{\mu_{1}^{2}(I+1)-I\mu_{2}}}{I}}$,
and ${\sigma^{2}\\!\triangleq\\!\frac{\mu_{2}-I\Omega}{I}}$.

###### Proof:

Provided in the Appendix A. ∎

The above Lemma allows us to represent (1) via the SISR distribution with
${\mu_{1}\\!=\\!\mathbb{E}\\{|g_{\textnormal{XY}}|^{2}\\}}$ and
${\mu_{2}\\!=\\!\mathbb{E}\\{|g_{\textnormal{XY}}|^{4}\\}}$. The parameter $I$
should be proportional to the degrees of freedom of
$|g_{\textnormal{XY}}|^{2}$, e.g., ${I\\!=\\!1}$ for 2 degrees of freedom, and
should satisfy
${I\\!\leq\\!\lfloor\frac{\mu_{1}^{2}}{\mu_{2}-\mu_{1}^{2}}\rfloor}$. As the
LoS component is dominant, the complexity-accuracy trade-off coefficient
$\psi$ should be relatively large to achieve high accuracy. It is worth noting
that the statistical nature of the channel remains Rician, not shadowed-
Rician, and the above Lemma is proposed to be later utilized in Sections III-B
and III-C.

###  III-B Distribution of the G2A SINR

In this subsection, we derive the closed-form expression of the G2A SINR’s
CDF, which is presented in Theorem 1.

######  Theorem 1.

The CDF of the G2A SINR is

$\displaystyle
F_{{\Gamma}_{\mathtt{g2a}}^{\star}}(x)=1-\mathop{\textstyle\sum}_{k=0}^{M\psi_{\textnormal{S}\textnormal{U}}-M}\\!\\!\chi_{M,k}\\!\\!\\!\\!\\!\mathop{\textstyle\sum}_{n=0}^{M\psi_{\textnormal{S}\textnormal{U}}-k-1}\frac{\Delta^{(n)}_{\textnormal{U}}(\frac{x}{\alpha_{\mathtt{g2a}}\bar{\gamma}_{\mathtt{g2a}}})}{n!},$
(14)

for $x>0$, where $\Delta^{(n)}_{\textnormal{U}}(s)$ is given by (17).

###### Proof:

First, the CDF of ${\Gamma}_{\mathtt{g2a}}^{\star}$ can be derived as
${F_{{\Gamma}_{\mathtt{g2a}}^{\star}}(x)\\!=\\!\mathbb{E}\big{\\{}F_{\gamma_{\mathtt{g2a}}}({x(\gamma_{\textnormal{I}\textnormal{U}}\\!+\\!\gamma_{\textnormal{J}\textnormal{U}}\\!+\\!1)}/{\bar{\gamma}_{\mathtt{g2a}}})\big{\\}}}$.
Using the proposed Lemma, we rewrite the CDF of $\gamma_{\mathtt{g2a}}$ with
${I\\!\triangleq\\!M}$ and a pre-defined trade-off parameter
${\psi\triangleq\psi_{\textnormal{S}\textnormal{U}}}$. Here, the $k$th moment
of $|h_{\textnormal{S}_{m}\textnormal{U}}|^{2}$ is required, where
${\mu_{k,|h_{\textnormal{XY}}|^{2}}\\!\triangleq\\!\frac{\Gamma(1+k)L_{k}(-K_{\textnormal{XY}})}{(1+K_{\textnormal{XY}})^{k}}}$
and $L_{k}(x)$ is the Laguerre polynomial of order $k$. Moreover, the $k$th
moment of $\gamma_{\mathtt{g2a}}$ is determined as

$\displaystyle\mu_{k,\gamma_{\mathtt{g2a}}}=\mathop{\textstyle\sum}_{r=1}^{k}\frac{k!M!}{(M\\!-\\!r)!}\mathop{\widetilde{\textstyle\sum}}\limits_{r,k}\bigg{(}\mathop{\textstyle\prod}_{j=1}^{r}{\frac{1}{p_{j}!}}\bigg{)}\mathop{\textstyle\prod}_{i=1}^{\varrho({\bf\Sigma})}{\frac{(\mu_{p_{\langle
i\rangle}})^{\nu_{i}}}{\nu_{i}!}},$ (15)

where $\mu_{i}\triangleq\mu_{i,|h_{\textnormal{XY}}|^{2}}$,
${\mathop{\widetilde{\Sigma}}_{r,k}}$ is the shorthand notation for the
summation over $r$ integers that satisfy
${p_{1}\\!+\\!\cdots\\!+\\!p_{r}\\!=\\!k}$ and ${p_{1},\dots,p_{r}\geq 1}$,
${{\bf\Sigma}=\operatorname{diag}([p_{1},\dots,p_{r}])}$,
$\varrho({\bf\Sigma})$ is the number of distinct diagonal elements in
$\bf\Sigma$, ${p_{\langle
1\rangle}>\cdots>p_{\langle\varrho({\bf\Sigma})\rangle}}$ are the distinct
diagonal elements of $\bf\Sigma$ in decreasing order, and $\nu_{i}$ is the
multiplicity of $p_{\langle i\rangle}$. We can further derive
$F_{{\Gamma}_{\mathtt{g2a}}^{\star}}(x)$ as

$\displaystyle
F_{{\Gamma}_{\mathtt{g2a}}^{\star}}(x)=1-\mathop{\textstyle\sum}_{k=0}^{M\psi_{\textnormal{S}\textnormal{U}}-M}\mathop{\textstyle\sum}_{n=0}^{M\psi_{\textnormal{S}\textnormal{U}}-k-1}\chi_{M,k}(-s_{0})^{n}$
$\displaystyle\qquad\times\frac{1}{n!}\bigg{[}\frac{{\rm d}^{n}}{{\rm
d}s^{n}}e^{-s}\mathop{\textstyle\prod}_{l=1}^{L}{{\mathfrak{L}}_{\gamma_{\textnormal{I}_{l}\textnormal{U}}}(s)}\mathop{\textstyle\prod}_{k=1}^{K}{{\mathfrak{L}}_{\gamma_{\textnormal{J}_{k}\textnormal{U}}}(s)}\bigg{]}\bigg{|}_{s\to
s_{0}},$ (16)

where ${s_{0}\triangleq\frac{x}{\alpha\bar{\gamma}_{\mathtt{g2a}}}}$. Using
[13, Table I] for expanding
${\mathfrak{L}}_{\gamma_{\textnormal{I}_{l}\textnormal{U}}}(s)$ and
${\mathfrak{L}}_{\gamma_{\textnormal{J}_{k}\textnormal{U}}}(s)$, we can
rewrite the above $n$-th order derivatives as follows

$\displaystyle\Delta^{(n)}_{\textnormal{U}}(s)$ $\displaystyle=\frac{{\rm
d}^{n}}{{\rm
d}s^{n}}e^{-s}\mathop{\textstyle\prod}_{l=1}^{L}\frac{e^{-K_{\textnormal{I}_{l}\textnormal{U}}+\frac{K_{\textnormal{I}_{l}\textnormal{U}}}{1+\bar{\gamma}_{\textnormal{I}_{l}\textnormal{U}}s}}}{1+\bar{\gamma}_{\textnormal{I}_{l}\textnormal{U}}s}\mathop{\textstyle\prod}_{k=1}^{K}\frac{e^{-K_{\textnormal{J}_{k}\textnormal{U}}+\frac{K_{\textnormal{J}_{k}\textnormal{U}}}{1+\bar{\gamma}_{\textnormal{J}_{k}\textnormal{U}}s}}}{1+\bar{\gamma}_{\textnormal{J}_{k}\textnormal{U}}s}$
$\displaystyle=\Delta^{(0)}_{\textnormal{U}}(s)\bigg{\\{}1+\\!\mathop{\textstyle\sum}_{r=1}^{n}\mathop{\widetilde{\textstyle\sum}}\limits_{r,n}\frac{n!\mathop{\scriptstyle\prod}_{i=1}^{\varrho({\bf\Sigma})}{S_{p_{\langle
i\rangle}}^{\nu_{i}}(s)}/{\nu_{i}!}}{\mathop{\scriptstyle\prod}_{i=1}^{r}{p_{i}!}}\bigg{\\}},\\!\\!\\!$
(17)

where
${S_{i}(s)\\!=\\!\sum_{l=1}^{L}{\delta^{(i)}_{\textnormal{I}_{l}\textnormal{U}}(s)}\\!+\\!\sum_{k=1}^{K}{\delta^{(i)}_{\textnormal{J}_{k}\textnormal{U}}(s)}-C_{i}}$
with ${C_{1}\\!=\\!1}$ and ${C_{j\geq 2}\\!=\\!0}$, and
${\delta^{(n)}(s)\\!=\\!(n\\!-\\!1)!\frac{(-\bar{\gamma})^{n}}{(1+\bar{\gamma}s)^{n}}(1\\!+\\!\frac{nK}{1+\bar{\gamma}s})}$.
The detailed derivation of (17) is omitted here due to space constraint111The
full proof of Theorem 1 is available at
https://github.com/thanhluannguyen/UAV-RIS-blockage.. This completes the proof
of Theorem 1. ∎

###  III-C Distribution of A2G SINR

In this subsection, we investigate the distribution of the A2G SINR and derive
its CDF in the following Theorem.

######  Theorem 2.

The CDF of the A2G SINR is given by

$\displaystyle
F_{{\Gamma}_{\mathtt{a2g}}^{\star}}(x)=1-\\!\\!\\!\mathop{\textstyle\sum}_{k=0}^{I\psi-I}\\!\\!\chi_{I,k}\\!\\!\\!\\!\\!\mathop{\textstyle\sum}_{n=0}^{I\psi-k-1}\frac{\Delta^{(n)}_{\textnormal{D}}(\frac{x}{\alpha_{\mathtt{cas}}\bar{\gamma}_{\mathtt{cas}}})}{n!},x>0.$
(18)

###### Proof:

Let
${\gamma_{\mathtt{a2g}}\\!\triangleq\\!\big{|}\sqrt{\gamma_{\mathtt{cas}}}+\sqrt{\gamma_{\mathtt{dir}}\bar{\gamma}_{\mathtt{dir}}/\bar{\gamma}_{\mathtt{cas}}}\big{|}^{2}}$
be the normalized A2G SNR. Next, we adopt the proposed Lemma to approximate
the distribution of $\gamma_{\mathtt{a2g}}$, which requires the $k$th moments
of
${\gamma_{\mathtt{dir}}\bar{\gamma}_{\mathtt{dir}}/\bar{\gamma}_{\mathtt{cas}}}$
and of $\gamma_{\mathtt{cas}}$. Under blockage, the PDF of
$|\check{g}_{\textnormal{U}\textnormal{D}}|^{2}$, defined as
${f_{|\check{g}_{\textnormal{U}\textnormal{D}}|^{2}}(x)\\!=\\!\mathbb{E}[f_{\phi_{0}|g_{\textnormal{U}\textnormal{D}}|^{2}}(x)]}$,
is

$\displaystyle f_{|\check{g}_{\textnormal{U}\textnormal{D}}|^{2}}(x)$
$\displaystyle=\frac{\pi_{1}}{1-\beta}e^{-\lambda_{\textnormal{U}\textnormal{D}}x-K_{\textnormal{U}\textnormal{D}}}\lambda_{\textnormal{U}\textnormal{D}}I_{0}\big{(}2{{\textstyle{\sqrt{K_{\textnormal{U}\textnormal{D}}\lambda_{\textnormal{U}\textnormal{D}}x}}}}\big{)},$
(19)

for ${x\\!>\\!\tau}$. We first derive the $k$th moment of
${\gamma\triangleq\frac{\gamma_{\mathtt{dir}}\bar{\gamma}_{\mathtt{dir}}}{\bar{\gamma}_{\mathtt{cas}}}}$
as

$\displaystyle{\mu_{k,\gamma}\\!=\\!\mathbb{E}[\gamma^{k}]\\!=\\!\scalebox{1.5}{$\int$}_{\tau}^{\infty}{f_{|\check{g}_{\textnormal{U}\textnormal{D}}|^{2}}(x)x^{k}/{(\bar{\gamma}_{\mathtt{cas}}\sigma_{\textnormal{D}}^{2}})^{k}{\rm
d}x}}.$ (20)

Then, utilizing (19) and invoking the Nuttall-$Q$ function $Q_{m,n}(a,b)$, we
obtain

$\displaystyle{\mu_{k,\gamma}\\!=\\!\frac{\pi_{1}}{1-\beta}\frac{Q_{2k+1,0}(\sqrt{2K_{\textnormal{U}\textnormal{D}}},\sqrt{2\lambda_{\textnormal{U}\textnormal{D}}\tau})}{(2\lambda_{\textnormal{U}\textnormal{D}}\sigma_{\textnormal{D}}^{2}\bar{\gamma}_{\mathtt{cas}})^{k}}}.$
(21)

Next, we use (15) to calculate the $k$th moment of $\gamma_{\mathtt{cas}}$ as

$\displaystyle{\mu_{k,\gamma_{\mathtt{cas}}}\\!=\\!\mathop{\textstyle\sum}_{r=1}^{2k}\frac{(2k)!N!}{(N-r)!}\mathop{\widetilde{\mathop{\textstyle\sum}}}_{r,2k}\Big{(}\mathop{\textstyle\prod}_{j=1}^{r}{\frac{1}{p_{j}!}}\Big{)}\mathop{\textstyle\prod}_{i=1}^{\varrho({\bf\Sigma})}{\frac{(\mu_{p_{\langle
i\rangle}})^{\nu_{i}}}{\nu_{i}!}}},$ (22)

where, in this case,
$\mu_{i}\triangleq\mu_{\frac{i}{2},|h_{\textnormal{U}\textnormal{R}_{n}}|^{2}}\mu_{\frac{i}{2},|h_{\textnormal{R}_{n}\textnormal{D}}|^{2}}$.

Afterwards, we rewrite the CDF of $\gamma_{\mathtt{a2g}}$ using the Lemma
Lemma, which requires its $k$th moment to be

$\displaystyle{\mu_{k,\gamma_{\mathtt{a2g}}}\\!=\\!\mathop{\textstyle\sum}_{i=0}^{2k}{\textstyle\binom{2k}{i}}(\mu_{k-\frac{i}{2},\gamma_{\mathtt{cas}}}){(\mu_{\frac{i}{2},\gamma})}}.$
(23)

After a series of mathematical manipulations, we obtain (18). This completes
the Proof of Theorem 2. ∎

###  III-D Asymptotic analysis

We have
${F_{Y}(I,\psi;x)\\!\to\\!\sum_{k=0}^{I\psi-I}\frac{\chi_{I,k}}{(I\psi-k)!}(\frac{x}{\alpha})^{I\psi-k}}$
as ${x\\!\to\\!0}$ by using [11, Eq. (8.354.1)]. In high transmit power
regimes, where
${P_{\textnormal{S}}\\!=\\!P_{\textnormal{U}}\\!=\\!P\\!\to\\!\infty}$,
$F_{\Gamma_{\mathtt{g2a}}^{\star}}(x)$ can be further simplified as

$\displaystyle
F_{\Gamma_{\mathtt{g2a}}^{\star}}(x)\to\mathop{\textstyle\sum}_{k=0}^{M\psi_{\textnormal{S}\textnormal{U}}-M}\frac{\mu_{M\psi_{\textnormal{S}\textnormal{U}}-k}\chi_{M,k}}{(M\psi_{\textnormal{S}\textnormal{U}}-k)!}\Big{(}\frac{x}{\alpha_{\mathtt{g2a}}\bar{\gamma}_{\mathtt{g2a}}}\Big{)}^{M\psi_{\textnormal{S}\textnormal{U}}-k}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!,$
(24)

where, for this equation,
${\mu_{k}\\!=\\!\mathbb{E}[(\gamma_{\textnormal{I}\textnormal{U}}\\!+\\!\gamma_{\textnormal{J}\textnormal{U}}\\!+\\!1)^{k}]}$
and is derived as ${\mu_{k}\\!=\\!(-1)^{k}[\frac{{\rm d}^{k}}{{\rm
d}s^{k}}\mathfrak{L}_{\gamma_{\textnormal{I}\textnormal{U}}\\!+\\!\gamma_{\textnormal{J}\textnormal{U}}\\!+\\!1}(s)]|_{s\to
0}\\!=\\!(-1)^{k}\Delta_{\textnormal{U}}^{(k)}(0)}$.

Hence, applying the above steps, $F_{\Gamma_{\mathtt{a2g}}^{\star}}(x)$ when
${P\\!\to\\!\infty}$ can be simplified as

$\displaystyle
F_{\Gamma_{\mathtt{a2g}}^{\star}}(x)\to\mathop{\textstyle\sum}_{k=0}^{I\psi-I}\frac{\mu_{I\psi-I}\chi_{I,k}}{(I\psi-I)!}\Big{(}\frac{x}{\alpha_{\mathtt{a2g}}\bar{\gamma}_{\mathtt{a2g}}}\Big{)}^{I\psi-k},$
(25)

where, for (25),
${\mu_{k}=\mathbb{E}[(\gamma_{\textnormal{I}\textnormal{D}}+\gamma_{\textnormal{J}\textnormal{D}}+1)^{k}]=(-1)^{k}\Delta_{\textnormal{D}}^{(k)}(0)}$.

###  III-E Blockage probability of direct A2G channel

The Markov stochastic process serves as a dynamic representation of the direct
A2G link’s availability. In scenarios where the direct A2G link encounters a
blockage (${\phi_{0}=0}$), the value of the direct A2G channel power gain
$|g_{\textnormal{U}\textnormal{D}}|^{2}$ falls below a predefined threshold
$\tau$, rendering it unavailable for communication purposes. In this case, the
e2e communication is redirected through a cascaded link. From (7), the
blockage probability can be obtained as

$\displaystyle\beta$
$\displaystyle=\Pr(|g_{\textnormal{U}\textnormal{D}}|^{2}<\tau)$ (26)
$\displaystyle=1-Q\big{(}\sqrt{2K_{\textnormal{U}\textnormal{D}}},\sqrt{2\lambda_{\textnormal{U}\textnormal{D}}\tau}\big{)},$
(27)

where $Q(a,b)$ is the first order Marcum Q-function.

The impact of the blockage probability on the system performance is presented
in the following section.

##  IV Numerical Results

In this section, we validate the accuracy of our derived analytical
expressions based on numerous numerical simulations. Here, we consider a noise
spectral density of $-174$ dBm/Hz and system bandwidth of $10$ MHz [12]. We
consider that S is located at the origin of a normalized 3D Cartesian
coordinate system with the coordinates of U, D, and R being $(0.5,0.5,1.0)$,
$(0.5,0.5,0)$, and $(1.0,1.0,0)$, respectively. We consider the G2A/A2G path
loss model as
${\ell_{\textnormal{XY}}\\!=\\!-G_{t}\\!-\\!G_{r}\\!+\\!22.7\\!+\\!26\log(f_{c})\\!-\\!36.7\log(d_{\textnormal{XY}})}$
dB, where ${G_{t}\\!=\\!G_{r}\\!=\\!0}$ dBi as transmit and receive antenna
gains, and ${f_{c}\\!=\\!3}$ GHz as the carrier frequency [14, Table
B.1.2.1-1], and
${\ell_{\textnormal{R}\textnormal{D}}\\!=\\!\\!-\\!G_{t}\\!-\\!G_{r}\\!-\\!37.3+26\log(f_{c})-36.7\log(d_{\textnormal{R}\textnormal{D}})~{}\text{dB}}$.
With Rician fading, we have ${K_{1}\\!=\\!0}$ dB and
${K_{2}\\!=\\!\frac{2}{\pi}\log(\frac{K_{\pi}}{K_{0}})}$, where
${K_{\pi}\\!=\\!5}$ dB [9]. Additionally, we set ${{\bf
p}_{\textnormal{J}_{1}\textnormal{S}}\\!=\\!(0.4,0.4,0)}$, ${{\bf
p}_{\textnormal{J}_{2}\textnormal{S}}\\!=\\!(0.4,0.8,0)}$, ${{\bf
p}_{\textnormal{I}_{1}\textnormal{S}}\\!=\\!(0.2,0.2,0.6)}$, ${{\bf
p}_{\textnormal{I}_{2}\textnormal{S}}\\!=\\!(0.4,0.8,0.4)}$, and
${P_{\textnormal{I}_{l}}\\!=\\!P_{\textnormal{J}_{k}}\\!=\\!0}$ dBm for all
${l\\!\in\\![1,L]}$ and ${k\\!\in\\![1,K]}$.

Fig. 2a and Fig. 2b demonstrate the close match between analytical and
simulated results across different numbers of transmit antennas ($M$) and RIS
elements ($N$). It is noted that we set
${P_{\textnormal{S}}\\!=\\!P_{\textnormal{U}}\\!=\\!0}$ dBm for normalization
purposes. Moreover, the CDF of the e2e SINR is depicted in Fig. 2b, as a
function of $M$ and $N$ at a spectral efficiency ($R_{\tt SE}$) of 0.5 bps/Hz,
${P_{\textnormal{S}}\\!=\\!P_{\textnormal{U}}\\!=\\!23}$ dBm, and
$\beta\\!=\\!0$. Notably, increasing the number of transmit antennas or RIS
elements decreases the SINR’s CDF, suggesting the system is better suited for
harsher environment conditions and stricter QoS requirements. In both figures,
the analytical and simulation results match well, which validates the accuracy
of our analysis.

Fig. 5 presents the relationship between e2e OP and the transmission power. It
is noted that the e2e OP is formulated as ${{\rm OP}_{\rm
e2e}\\!=\\!1-(1-F_{\Gamma^{\star}_{\mathtt{g2a}}}(\tau_{\rm
e2e}))(1-F_{\Gamma^{\star}_{\mathtt{a2g}}}(\tau_{\rm e2e}))}$. With an
increase in transmit power, the e2e OP generally decreases. Comparing the
decay rate among the three configurations, we observe that $N=64$ elements
shows the fastest decay, whereas $N=24$ elements shows the lowest.

Fig. 5 demonstrates the RIS’s notable effect on the e2e OP across various
transmission powers, where ${R_{\tt SE}\\!=\\!0.5}$ bps/Hz, $N=6$, and
$M=144$. The use of RIS evidently reduces OP in both unblocked and highly
blocked (blockage probability of 0.95) scenarios. Without RIS (w/o RIS), a
minor blockage probability of 0.01 considerably escalates OP, underscoring
RIS’s critical role in system resilience and performance amidst non-negligible
blockages.

In Fig. 5, where ${M\\!=\\!4}$ and ${R_{\tt SE}\\!=\\!1}$ bps/Hz, we observe
that as $\beta$ increases from 0 to 1, the OP generally increases, resulting
in lower e2e communication reliability. However, such a reduction in OP is
alleviated by deploying more RIS elements. Noticeably, increasing $N$ to 196
yields OP of only 0.01%. Furthermore, the impact of blockage is mitigated at
high values of $N$, especially in the range of 144 to 196 elements. When the
number of RIS’s elements is relatively large, the OP stabilizes. In this
regard, the RIS demonstrates its robustness against blockages as it guarantees
stable and reliable performance with large numbers of elements.

(a)

(b)

Figure 2: CDF of the e2e SINR as a function of a) the target SE [bps/Hz] and
the number of b) transmit antenna ($M$) and RIS element ($N$).

Figure 3: e2e OP versus the transmit powers versus the number of RIS elements.

Figure 4: e2e OP with (w/) and without (w/o) the help of RIS under varying
transmission powers.

Figure 5: e2e OP versus the blockage probability ($\beta$).

##  V Conclusion

In this letter, we presented stochastic A2G channel models for UAV-RIS-aided
systems. Specifically, using a discrete-time Markov process, the A2G
communication with ground UEs under varying conditions is modeled, including
blocked and non-blockage scenarios. We provided tractable closed-form
expressions for the CDF of the G2A and A2G SINRs in Rician fading environments
in the presence of non-IID CCI by adopting the proposed technical framework
that matched the distribution of the channel power gain to the sum of
independent shadowed Rician distribution. The results demonstrated that
increasing the number of RIS elements significantly improves the e2e OP and
mitigates the impact of blockage in the direct A2G link.

##  Appendix A Proof of the Lemma

Using the physical model of the shadowed Rician distribution, a signal power
$X_{i}$ can be expressed in terms of the in-phase and quadrature-phase
components of the fading signals as
${X_{i}=(P_{i}+\xi_{i}p_{i})^{2}+(Q_{i}+\xi_{i}q_{i})^{2}}$, where $P_{i}$ and
$Q_{i}$ are IID Gaussian RVs with zero mean and variance
$\frac{\sigma^{2}}{2}$; $p_{i}$ and $q_{i}$ are real numbers satisfying
${p_{i}^{2}\\!+\\!q_{i}^{2}\\!=\\!1}$; and $\xi_{i}$ is a
${\text{Nakagami-}m}$ RV with shaping parameter $\psi$ and scale $\Omega$. The
first and the second moments of $X_{i}$ are given by
${\mathbb{E}[X_{i}]=\Omega+\sigma^{2}}$ and
${\mathbb{E}[X_{i}^{2}]=2\sigma^{4}+4\sigma^{2}\Omega+\frac{\mu+1}{\mu}\Omega^{2}}$,
respectively. For chosen $I$ and the trade-off parameter $\psi$, by solving
the following system of equations i)
${(\Omega\\!+\\!\sigma^{2})Q\\!=\\!\mu_{1}}$ and ii)
${(2\sigma^{4}\\!+\\!4\sigma^{2}\Omega\\!+\\!\frac{\psi\\!+\\!1}{\psi}\Omega^{2})I+I(I\\!-\\!1)(\Omega\\!+\\!\sigma^{2})^{2}\\!=\\!\mu_{2}}$,
where $\mu_{k}\triangleq\mathbb{E}[Y^{k}]$, we obtain $\Omega$ and
$\sigma^{2}$ as follows

$\displaystyle\Omega$
$\displaystyle={\textstyle{\sqrt{\mu_{1}^{2}(I+1)-I\mu_{2}}}}\sqrt{\psi/(\psi-1)}/I,$
(28) $\displaystyle\sigma^{2}$
$\displaystyle=(\mu_{1}-{\textstyle{\sqrt{\mu_{1}^{2}(I+1)-I\mu_{2}}}}\sqrt{\psi/(\psi-1)})/I.$
(29)

The parameters $\kappa$ and $\omega$ of shadowed Rician are obtained as
$\kappa=1/\sigma^{2}-1$ and $\omega=\Omega(\kappa+1)/\kappa$, respectively.
The rest of the analysis, including the CDF and PDF of $Y$, is presented in
[15]. This completes the proof of the Lemma.

## References

  * [1] Z. Lin, H. Niu, K. An, Y. Wang, G. Zheng, S. Chatzinotas, and Y. Hu, “Refracting RIS-aided hybrid satellite-terrestrial relay networks: Joint beamforming design and optimization,” _IEEE Trans. Aerosp. Electron. Syst._ , vol. 58, no. 4, pp. 3717–3724, Aug 2022.
  * [2] Z. Chen et al, “Reconfigurable intelligent surface assisted B5G/6G wireless communications: Challenges, solution, and future opportunities,” _IEEE Commun. Mag._ , vol. 61, no. 1, pp. 16–22, Jan. 2023.
  * [3] A. M. Salhab and M. H. Samuh, “Accurate performance analysis of reconfigurable intelligent surfaces over Rician fading channels,” _IEEE Wirel. Commun. Lett._ , vol. 10, no. 5, pp. 1051–1055, May 2021.
  * [4] A.-A. A. Boulogeorgos, A. Alexiou, and M. D. Renzo, “Outage performance analysis of RIS-assisted UAV wireless systems under disorientation and misalignment,” _IEEE Trans. Veh. Technol._ , vol. 71, no. 10, pp. 10 712–10 728, Oct. 2022.
  * [5] Y. Bian, D. Dong, J. Jiang, and K. Song, “Performance analysis of reconfigurable intelligent surface-assisted wireless communication systems under co-channel interference,” _IEEE Open J. Commun. Soc._ , vol. 4, pp. 596–605, Feb. 2023.
  * [6] L. Yang, F. Meng, J. Zhang, M. O. Hasna, and M. D. Renzo, “On the performance of RIS-assisted dual-hop UAV communication systems,” _IEEE Trans. Veh. Technol._ , vol. 69, no. 9, pp. 10 385–10 390, Sep. 2020.
  * [7] H. Jiang, B. Xiong, H. Zhang, and E. Basar, “Physics-based 3D end-to-end modeling for double-RIS assisted non-stationary UAV-to-ground communication channels,” _IEEE Trans. Commun._ , pp. 1–1, Apr. 2023.
  * [8] A. V. Savkin, C. Huang, and W. Ni, “On-demand deployment of aerial base stations for coverage enhancement in reconfigurable intelligent surface-assisted cellular networks on uneven terrains,” _IEEE Commun. Lett._ , vol. 27, no. 2, pp. 666–670, Feb. 2023.
  * [9] C. You and R. Zhang, “3D trajectory optimization in Rician fading for UAV-enabled data harvesting,” _IEEE Trans. Wirel. Commun._ , vol. 18, no. 6, June 2019.
  * [10] P. Z. Peebles, _Probability, Random Variables and Random Signal Principles_ , 4th ed. New York, NY, USA: McGraw-Hill Science, 2000.
  * [11] I. S. Gradshteyn and I. M. Ryzhik, _Tables of Integrals, Series, and Products_ , 7th ed. New York, NY, USA: Academic Press, 2007.
  * [12] T. N. Do, G. Kaddoum, T. L. Nguyen, D. B. da Costa, and Z. J. Haas, “Multi-RIS-aided wireless systems: Statistical characterization and performance analysis,” _IEEE Trans. Commun._ , vol. 69, no. 12, pp. 8641–8658, Dec. 2021.
  * [13] F. J. Lopez-Martinez, J. M. Romero-Jerez, and J. F. Paris, “On the calculation of the incomplete MGF with applications to wireless communications,” _IEEE Trans. Commun._ , vol. 65, no. 1, pp. 458–469, Jan. 2017.
  * [14] G. T. RP-150496, “Study on Downlink Multiuser Superposition Transmission,” Mar. 2015.
  * [15] G. Alfano and A. De Maio, “Sum of squared shadowed-Rice random variables and its application to communication systems performance prediction,” _IEEE Trans. Wirel. Commun._ , vol. 6, no. 10, Oct. 2007.