# Ground-to-UAV Integrated Network: Low Latency Communication over
Interference Channel

Sudhanshu Arya, and Ying Wang (Corresponding author: Ying Wang.)Sudhanshu Arya
and Ying Wang are with the School of Systems and Enterprises, Stevens
Institute of Technology, Hoboken, USA (e-mail<EMAIL_ADDRESS>ywang6@stevens.edu).

###### Abstract

We present a novel and first-of-its-kind information-theoretic framework for
the key design consideration and implementation of a ground-to-UAV (G2U)
communication network with an aim to minimize end-to-end transmission delay in
the presence of interference. The proposed framework is useful as it describes
the minimum transmission latency for an uplink ground-to-UAV communication
must satisfy while achieving a given level of reliability. To characterize the
transmission delay, we utilize Fano’s inequality and derive the tight upper
bound for the capacity for the G2U uplink channel in the presence of
interference, noise, and potential jamming. Subsequently, given the
reliability constraint, the error exponent is obtained for the given channel.
In addition, as a function of the location information of the UAV, a tight
lower bound on the transmit power is obtained subject to the reliability
constraint and the maximum delay threshold. Furthermore, a relay UAV in the
dual-hop relay mode, with amplify-and-forward (AF) protocol, is considered,
for which we jointly obtain the optimal positions of the relay and the
receiver UAVs in the presence of interference, and is compared with the point-
to-point G2U link which ignores the relay. Interestingly, in our study, we
find that, for both the point-to-point and relayed links, increasing the
transmit power may not always be an optimal solution for delay minimization
problems. In particular, we show that increasing the power gives a negligible
gain in terms of delay minimization, though may greatly enhance the outage
performance. Moreover, we prove that there exists an optimal height that
minimizes the end-to-end transmission delay in the presence of interference.
The proposed framework can be used in practice by a network controller as a
system parameters selection criteria, where among a set of parameters, the
parameters leading to the lowest transmission latency can be incorporated into
the transmission. The based analysis further set the baseline assessment when
applying Command and Control (C2) standards to mission-critical G2U and UAV-
to-UAV (U2U) services.

###### Index Terms:

Delay, latency, information-theoretic, interference, UAV.

##  I Introduction

###  I-A Background and Motivation

5G and beyond wireless systems are expected to support an ultra-reliable and
low-latency communication link (URLLC) to enable uninterrupted and ubiquitous
connectivity to mission-critical services where robust information exchange is
important [1]. Indeed, the third generation partnership project (3GPP) aims to
cover three generic connectivity technologies for 5G and beyond systems:
URLLC, enhanced mobile broadband (eMBB), and massive machine-type
communication (mMTC) [2]. eMBB is designed for applications that have high
requirements for the data rate, such as high-resolution video streaming
whereas mMTC focuses on applications that support the massive number of
machine-type devices with ultra-low power consumption. In contrast, URLLC is
designed for delay-sensitive services and applications. For example, the
services including intelligent transportation systems and tactile internet
have reliability requirements of $(1-10^{-3})\sim(1-10^{-9})$ at latency
between $1$ ms to $100$ ms [3]. However, due to the dynamic and fast-fading
channel, shadowing, and path loss over wireless links, it becomes challenging
to meet the quality-of-service (QoS) requirement for URLLC systems.

To meet the challenging demands of 5G wireless networks, 3GPP has recommended
integrating unmanned aerial vehicles (UAVs) into wireless cellular networks
[4]. With a high probability of establishing a LOS communication link, UAVs
are more likely to provide better link quality over short-distance. Moreover,
UAVs are generally cost-effective and offer more flexibility for on-demand
communication systems in low-altitude environments. As a result, UAVs have
received significant research attention in wireless communications [5, 6].

A system model for UAV relay-assisted IoT networks was presented to primarily
explored the impact of requested timeout constraints for uplink and downlink
transmissions [7]. A problem to maximize the total number of served IoT
devices was formulated. The authors jointly optimized the transmission power,
trajectory, system bandwidth, and latency constraints for the full-duplex UAV-
assisted IoT devices. In another work, a single-hop and multi-hop relay-
assisted UAV network with a ground-based transmitter and receiver, the problem
of finding the optimal UAV location to minimize the impact of interference was
investigated [8]. A theoretical framework was presented to determine the
minimum number of UAVs required and their optimal locations to meet the
minimum requirement of the received average signal-to-interference ratio.
Considering a heterogeneous wireless network of a ground base station, relay
aerial vehicles, and a high-altitude platform, the end-to-end delay and
reliability analysis of downlink communication links was investigated [9]. It
was demonstrated that even with a very efficient interference mitigation
technique employed, the current wireless networks designed for terrestrial
users are not suitable to meet the URLLC requirements for downlink control
communication to aerial vehicles.

To improve the communication performance and network connectivity for the
ground nodes or vehicles in a 3D urban scenario, particle swarm optimization
was utilized to find the optimal UAV positions functioning as relay nodes
[10]. To validate the proposed approach, an indoor experiment was also
conducted. However, it is to be noted that, there were no communications among
the relayed UAVs. In another work, a model predictive control approach was
utilized to construct an energy-efficient communication link between the
ground nodes with UAV operating as a relay node [11]. In particular, a problem
was formulated to optimize the source’s transmit energy and the UAV’s
propulsion energy by jointly optimizing the UAV’s mobility and transmission
across the multiple access channel. However, the approach was limited to a
linear trajectory where it was assumed that the UAV is moving along a single
dimension at a fixed height with no lateral displacement. In [12], a
throughput maximization problem was formulated for a UAV-assisted mobile relay
network. The problem was based on jointly optimizing the trajectory of the
relay UAV node and the power allocation of the source and relay node, subject
to the information-causality constraints. In particular, the authors
considered two scenarios. In the first scenario, the relay trajectory was kept
fixed, and it was shown that the optimal source and UAV power allocations obey
a staircase water-filling structure with non-increasing and non-decreasing
power levels at the source and UAV, respectively. Whereas, in another
scenario, an iterative algorithm was presented to jointly optimize the UAV
trajectory and the power allocation in an alternating way. In another similar
work [13], however, considering the circling operation of the fixed-wing UAV,
a rate optimization approach is presented for UAV enabled wireless relay
network. It was assumed that there is no direct link between the ground-based
source and the destination. A variable rate protocol was developed to
optimally adjust the data rate depending on the location of the UAV. It was
shown that the proposed approach significantly outperforms the conventional
fixed-rate relaying network.

Since there is no regulatory and well-defined pre-allocated spectrum band for
UAV communications, it makes constructing a UAV-assisted communication network
a non-trivial task. Therefore, UAV-assisted network generally coexists with
other wireless networks, e.g., cellular networks [14]. Thus, formulating the
problem of delay minimization given the reliability constraint is critical.
This is the main fact that motivated us in formulating the information-
theoretic-based framework to investigate the end-to-end transmission delay in
the presence of interference. Utilizing our existing outdoor 5G testbed
[Wang2021DevelopmentResearch], Over the Air (OTA) verification and extension
of the proposed work is being performed.

###  I-B Contributions

Unlike traditional terrestrial or satellite communication networks, the air-
to-ground integrated network is affected by the limitations arising
simultaneously from the following segments, i.e., from the aspects of mobility
management, power control, and end-to-end QoS requirements. Therefore, given
the practical resource constraints of an air-to-ground integrated network, it
is critically important for an integrated network to achieve optimal and
reliable performance given the restriction in power consumption. Therefore,
optimal system integration and network design and configuration are of great
significance in an air-to-ground and air-to-air integrated network. To this
end, we present a novel information-theoretic approach to the design of an
optimal air-to-ground integrated network that assesses and minimizes end-to-
end transmission latency in the presence of interference, white noise, and
potential jamming. The main contributions of this paper are listed below.

  * •

We present an information-theoretic framework for the design of an optimal
ground-to-UAV communication network that minimizes end-to-end transmission
delay. The proposed framework is useful as it describes the minimum
transmission latency a UAV network must satisfy while achieving a given level
of reliability in terms of the average error probability. In particular, we
first derive a tight upper bound for the capacity of the air-to-ground
integrated channel in the presence of interference and white noise.
Subsequently, given the reliability constraint, a framework is developed to
analyze the delay introduced in the channel.

  * •

Further, considering the general air-to-ground network with inter-relay
communication, the results are presented to comprehensively characterize the
optimal performance of the system towards end-to-end delay minimization. We
show that despite the simplicity of the point-to-point direct link, the
latency of the amplify-and-forward (AF) relayed-based link can be lower if the
relay is properly located.

  * •

In addition, we considered the power consumption restriction in the air-to-
ground and air-to-air networks. Within the range of allowable system latency,
the optimal transmit power is derived based on the location information of a
UAV. From the analysis, it is shown that increasing the transmission power is
not always the proper solution to the delay minimization problem.

The rest of the paper is organized as follows. In Section II, we present the
system model. The delay analysis is presented in Section III whereas results
and discussions are depicted in Section IV. Finally, the conclusion is drawn
in Section V.

TABLE I: Notations Parameter |  Description
---|---
$d_{c}$ |  Block length of the code word $s_{T}$
$R_{c}$ |  Code rate
$P_{T}$ |  Transmit power
$P_{I}^{(i)}$ |  Transmit power of the $i$th interfering node
$P_{N}$ |  Power of the amplifying node
$\phi_{e}$ |  Reliability constraint
$\Phi_{I}$ |  Number of active interfering nodes at any time $t$
$h_{u,v}$ |  channel gain between the nodes $u$ and $v$
$\alpha_{u,v}$ |  Path loss exponent of the link $u\rightarrow v$
$\alpha_{I,v}^{(i)}$ |  Path loss exponent of the link between the $i$th interferer and the $v$th receiver
$\theta_{u,v}$ |  Elevation angle between the transmitting node $u$ and the receiving node $v$
$\theta_{I,v}^{(i)}$ |  Elevation angle between the receiver $v$ and the $i$th interfering node
$d_{u,v}$ |  3D distance between the nodes $u$ and $v$
$w_{R}(t)$ |  AWGN noise at the input of the receiver
$w_{N}(t)$ |  AWGN noise at the input of the relay node
$N_{0}$ |  Noise spectral density
$B_{uv}$ |  Link bandwidth between the nodes $u$ and $v$
$d_{u,v}$ |  3D distance between the nodes $u$ and $v$
$B$ |  Information bits in a codeword message of length $d_{c}$

##  II System Model

Figure 1: Illustration of the system model.

We consider an interference network where a transmitted signal is corrupted by
the interference signal plus the additive white Gaussian noise (AWGN). In this
noise-plus interference-limited network, there can be two types of
communication links: a direct link from the ground base station (BS) to the
receiver UAV and a relay-assisted link from the BS to the receiver UAV. Let
the ground BS, located at $(x_{T},y_{T},0)$, communicates to the receiver
$(R_{x})$ located at $(x_{R},y_{R},z_{R})$ in the presence of interfering
nodes located at $(x_{I}^{(i)},y_{I}^{(i)},0),i\in\Phi_{I}$. $\Phi_{I}$ is a
subset containing all interfering nodes transmitting at time $t$.

Let the BS transmits with power $P_{T}$ over a channel with path loss exponent
$\alpha_{T}$ and fading coefficient $h_{T}$.
$d_{T}=\sqrt{\left({x_{T}-x_{R}}\right)^{2}+\left({y_{T}-y_{R}}\right)^{2}+\left({z_{T}-z_{R}}\right)^{2}}$
is the distance between the BS and $R_{x}$. $s_{T}$ is a code word of length
$N$ and is a sequence of $N$ numbers such that
$s_{T}=(s_{T_{1}},s_{T_{2}},\cdot\cdot\cdot,s_{T_{N}}).$ The code word $s_{T}$
for an underlying ground-to-UAV channel may be thought of geometrically as a
point in $N-$dimensional Euclidean space. The impact of the channel impairment
due to the Gaussian noise and interference is then to move $s_{T}$ to a nearby
point according to a spherical Gaussian distribution. $s_{T}(t)$ are random
symbols drawn from a constellation size $M$ with unitary mean power.
$P_{I}^{(i)}$ denotes the transmit power of the $i$th interfering node.

We consider an AF relay channel with one hop, as illustrated in Fig. 1. To
avoid interference between the ground BS and the relay, it is assumed that the
information transmission is conducted via time division where the transmission
from the ground BS to the receiver is divided into two time slots. In the
first slot, the signal is received by the relaying UAV, which is then
amplified, and in the second slot, the amplified signal is received by the
receiver UAV. An arbitrary relay UAV can normalize and re-transmit the
received signal. With the power $P_{N}$, the re-transmitted signal $s_{N}$ is
represented by

$\begin{split}&s_{N}(t)=\frac{{\sqrt{P_{N}}Y_{N}\left(t\right)}}{{\sqrt{E\left[{\left|{Y_{N}\left(t\right)}\right|^{2}}\right]}}}\\\
&=\frac{{\sqrt{P_{N}}\left({\sqrt{P_{T}d_{T,N}^{-\alpha_{T,N}\left({\theta_{T,N}}\right)}}h_{T,N}s_{T}\left(t\right)+w_{N}\left(t\right)}\right)}}{{\sqrt{P_{T}d_{T,N}^{-\alpha_{T,N}\left({\theta_{T,N}}\right)}\left|{h_{T,N}}\right|^{2}+B_{TN}N_{0}}}}\\\
\end{split},$ (1)

where $P_{T}$ is the transmit power of the ground BS and $w_{N}$ denotes the
white noise at the input of the relaying UAV. $E[\cdot]$ represents the
expectation operation. Following (2), the signal received at the receiver is
expressed as

$\begin{split}&Y_{R}\left(t\right)=s_{N}(t)\sqrt{d_{N,R}^{-\alpha_{N,R}\left({\theta_{N,R}}\right)}}h_{N,R}\\\
&+\sum\limits_{i\in\Phi_{I}}{\sqrt{P_{I}^{\left(i\right)}d_{i,R}^{-\alpha_{I}^{\left(i\right)}\left({\theta_{I}^{\left(i\right)}}\right)}}h_{I,R}^{\left(i\right)}s_{I}^{\left(i\right)}\left(t\right)}+w_{R}\left(t\right)\\\
\end{split}$ (2)

We consider that the G2A and the A2A communication links experience LOS
propagation with LOS probability $P_{LOS,h}$,
$h\in\\{h_{T},h_{T,N},h_{N,R},h_{I,R},h_{I,N}\\}$, expressed as [16]

$P_{LOS,h}=\frac{1}{{1+f_{1}\exp\left({-f_{2}\left({\theta_{h}-f_{1}}\right)}\right)}},$
(3)

where $f_{1}$ and $f_{2}$ are environment-dependent parameters determined by
the building density and heights. Due to the LOS path for all the described
links, the small-scale channel fading gain $h$ can be assumed to follow the
Rician model with PDF given by

$f_{h}\left(h\right)=\frac{h}{{\sigma_{h}^{2}}}\exp\left({-\frac{{h^{2}+\rho_{h}^{2}}}{{2\sigma_{h}^{2}}}}\right)I_{0}\left({\frac{{h\rho_{h}}}{{\sigma_{h}^{2}}}}\right),h\geq
0,$ (4)

where $\sigma_{h}$ and $\rho_{h}$ represent the strength of the LOS and NLOS
components. $I_{0}(\cdot)$ is the zeroth order modified Bessel function of the
first kind. Following (4), the Rice factor $K_{h}$(dB) of any link
$h\in\\{h_{T,R},h_{T,N},h_{N,R},h_{I,R},h_{I,N}\\}$, can then be defined as
$K_{h}$(dB) $=10\log_{10}(\frac{\rho_{h}^{2}}{2\sigma_{h}^{2}})$. We like to
point out that the Rice factor $K_{h}$ is a function of the parameters such as
the carrier frequency and the elevation angle $\theta_{h}$ [9].

We define the instantaneous signal-to-noise ratio (SNR) of arbitrary link
$u\rightarrow v$, from the transmitter $u,u\in\\{T,N\\}$ to the receiver
$v,v\in\\{N,R\\}$, over the known interference channel as

$\lambda_{uv}=\frac{{\left|{\sqrt{P_{u}d_{u,v}^{-\alpha_{uv}\left({\theta_{uv}}\right)}}h_{u,v}}\right|^{2}}}{{B_{uv}N_{0}}},$
(5)

and the power ratio between the intended received signal at $v,v\in\\{N,R\\}$
from $u,u\in\\{T,N\\}$, and the interference signal from the $i$th interfering
node is expressed as

$\gamma_{uv}^{(i)}=\frac{{\left|{\sqrt{P_{u}d_{u,v}^{-\alpha_{u,v}\left({\theta_{u,v}}\right)}}h_{u,v}}\right|^{2}}}{{\left|{\sum\limits_{i\in\Phi_{I}}{\sqrt{P_{I}^{\left(i\right)}d_{i,v}^{-\alpha_{I,v}^{\left(i\right)}\left({\theta_{I,v}^{\left(i\right)}}\right)}}h_{I,v}^{\left(i\right)}}}\right|^{2}}}.$
(6)

The expression in (6) is general enough to be treated as a signal-to-
interference ratio (SIR).

Averaging over the fading statistics described in (4), we obtain the average
received SNR $\bar{\lambda}_{uv}$ as illustrated in (7).
$G_{m,n}^{s,t}[\cdot]$ in (7) represents the Meijer G-function [17]. The
derivation steps are provided in Appendix V.

$\bar{\lambda}_{uv}=\frac{{\sqrt{P_{u}d_{u,v}^{-\alpha_{u,v}\left({\theta_{u,v}}\right)}}\left({-2\pi\sigma_{h_{u,v}}^{2}}\right)}}{{B_{uv}N_{0}}}\exp\left({-\frac{{\rho_{h_{u,v}}^{2}}}{{2\sigma_{h_{u,v}}^{2}}}}\right)G_{3,4}^{2,1}\left[{\frac{{2\rho_{h_{u,v}}^{2}}}{{\sigma_{h_{u,v}}^{2}}}\left|{\begin{array}[]{*{20}c}{-2,-1,\frac{1}{2}}\\\
{0,-2,0,\frac{1}{2}}\\\ \end{array}}\right.}\right].$ (7)

##  III Delay Analysis

The critical and important questions we would like to answer in this work are:
What is the transmission latency of the G2U communication link under the given
system parameters$?$ And what are the optimal locations of the relay and the
UAV to have a positive impact in minimizing the transmission delay$?$ More
precisely, our objective is to minimize the delay in transmitting messages
between the transmitter and the receiver in UAV communication while
guaranteeing a required level of reliability $\phi_{e}$, such that,

$P_{e}\left({d_{c},R_{c}}\right)\leq\phi_{e},$ (8)

where $P_{e}\left({d_{c},R_{c}}\right)$ represents the average error
probability of the code with length $d_{c}$ and rate $R_{c}$. The channel code
rate $R$ is a function of the input target signal distribution $f_{T}$, the
input interference signal distribution $f_{I}$, $\lambda$, and $\gamma$. The
channel code rate $R_{c}$ is achievable if there is an encoding function to
map each transmitted message $S$ to $s_{T}$ and there is another decoding
function to map each received signal $Y$ and interference information to a
transmitted message as $\hat{S}$, such that the average error probability
$P_{e}(d_{c},R_{c})\triangleq P_{r}[S\neq\hat{S}]\rightarrow 0$ when $N$ goes
large. Following this, we define the reliability function in terms of the
error exponent of the channel as [18]

$E\left(R_{c}\right)=-\mathop{\lim}\limits_{d_{c}\to\infty}\sup\frac{{\ln
P_{e,\min}\left({d_{c},R_{c}}\right)}}{d_{c}},$ (9)

where ${\rm sup}(\cdot)$ represents the supermum function and
$P_{e,\min}\left({d_{c},R_{c}}\right)$ denotes the infimum or the greatest
lower bound of the error probability over all $\left({d_{c},R_{c}}\right)$
codes for a given $d_{c}$ and $R_{c}$. It can be readily shown that the error
exponent depicted in (3) is upper bounded by the sphere packing exponent

$E\left(R_{c}\right)\geq\mathop{\max}\limits_{\rho>0}\left\\{{E_{0}\left(\rho\right)-\rho
R_{c}}\right\\},$ (10)

where

$E_{0}\left(\rho\right)=-\log\int\limits_{-\infty}^{\infty}{\left({\int{\left({P\left({Y_{R}\left|s_{T}\right.}\right)}\right)^{\frac{1}{{1+\rho}}}dP\left(s_{T}\right)}}\right)^{{{1+\rho}}}}dY_{R}$
(11)

where $P\left(s_{T}\right)$ denotes the cumulative distribution function of
the input $s_{T}$ and $P\left({Y_{R}\left|s_{T}\right.}\right)$ is the
probability density (or mass) function of the output $Y$ given the input
$s_{T}$. $\rho$ is a parameter, $0\leq\rho\leq 1$, and should be selected such
that $d_{c}$ is minimized. Utilizing (7), (8), and (10), and applying the
identity [19, eq. 11], the minimum transmission delay introduced into the
channel in order to satisfy the reliability constraint is given by

$d_{c}\geq\frac{{\rho
B-\log\phi_{e}}}{{G_{2,2}^{1,2}\left[{\bar{\lambda}_{uv}\left|{\begin{array}[]{*{20}c}{1,1}\\\
{1,0}\\\ \end{array}}\right.}\right]}},$ (12)

where $B$ represents the information bits in the codeword of length $d_{c}$.

We define the capacity of the uplink ground-to-UAV channel impaired by the
known interference as

$\begin{split}&C\left({\lambda,\gamma}\right)=\mathop{\sup}\limits_{f_{T}}\left[{\mathop{\inf}\limits_{f_{I}}R_{c}\left({f_{T},f_{I},\lambda,\gamma}\right)}\right]\\\
&\hskip 14.22636pt{\rm subject\hskip 2.84544ptto}\hskip
2.84544pt\mathbb{E}\left[{s_{I}}\right]^{2}=1,\mathbb{E}\left[{s_{T}}\right]^{2}=1\\\
\end{split}$ (13)

Next, we present a tight upper bound for $C\left({\lambda,\gamma}\right)$
defined in (13).

To decode the transmitted message $S$ correctly with a low probability of
error, the conditional entropy $H(S|Y_{R},s_{I})$ has to be close to zero
[20]. Following this, and applying Fano’s inequality [21] into the definition
of channel entropy, it can be readily shown that

$d_{c}R_{c}\left({f_{T},f_{I},\lambda,\gamma}\right)\leq
I\left({S;Y_{R},s_{I}}\right)+d_{c}\zeta_{d_{c}},$ (14)

where $\zeta_{N}$ is the error detection parameter satisfying the condition
$\mathop{\lim}\limits_{d_{c}\to\infty}\zeta_{d_{c}}=0$. Applying the
definition of mutual information and utilizing the fact that $S$ and $s_{I}$
are independent, (14) can be written as

$d_{c}R_{c}\left({f_{T},f_{I},\lambda,\gamma}\right)\leq
H\left({Y_{R}\left|{s_{I}}\right.}\right)-H\left({Y_{R}\left|{S,s_{I}}\right.}\right)+\zeta_{d_{c}},$
(15)

where $H\left({Y_{R}\left|{s_{I}}\right.}\right)$ can be interpreted as the
uncertainty in $Y$ conditional on $s_{I}$.
$H\left({Y_{R}\left|{S,s_{I}}\right.}\right)$ accounts for the reduction in
uncertainty of $Y$ conditional on $s_{I}$ from the observation of $s_{I}$.
Substituting the expressions for $H\left({Y_{R}\left|{s_{I}}\right.}\right)$
and $H\left({Y\left|{S,s_{I}}\right.}\right)$ into (15), the tight upper bound
for the capacity of the air-to-ground channel is derived as illustrated in
(16). $N_{p}$ in the summation term in (16) denotes the packet length. $T_{c}$
represents the block of symbols over which the channel is assumed to be
constant. The detailed derivation steps of (16) are provided in Appendix V.

$\begin{split}&R_{c,uv}\leq\frac{{\left({T_{c}-1}\right)}}{T_{c}}\log\left({1+\frac{{P_{u}d_{u,v}^{-\alpha_{u,v}\left({\theta_{u,v}}\right)}\left\|{h_{u,v}}\right\|^{2}}}{{B_{uv}N_{0}}}}\right)\\\
&\hskip
11.38092pt+\frac{1}{N_{p}}\sum\limits_{j=1}^{N_{p}/T_{c}}{\mathop{E}\limits_{s_{I,j}}\left[{\log\left({1+\frac{{P_{u}d_{u,v}^{-\alpha_{u,v}\left({\theta_{u,v}}\right)}\left\|{h_{u,v}}\right\|^{2}}}{{\sum\limits_{i\in\Phi_{I}}{P_{I}^{\left(i\right)}d_{i}^{-\alpha_{I,v}^{\left(i\right)}\left({\theta_{I,v}^{\left(i\right)}}\right)}\left\|{h_{I,v}^{\left(i\right)}}\right\|^{2}\left\|{s_{I,j}^{\left(i\right)}}\right\|^{2}+{B_{uv}N_{0}}}}}}\right)}\right]}+\zeta_{d_{c}}\end{split}.$
(16)

$P_{u,{\rm min}}\geq\left\\{{\left({1+\rho}\right)\exp\left({\frac{{\rho
B-\log\phi_{e}}}{{d_{c,\max}}}}\right)\left({\left|{\sum\limits_{i\in\Phi_{I}}{\sqrt{P_{I}^{\left(i\right)}d_{I,v}^{-\alpha_{I,v}^{\left(i\right)}\left({\theta_{I,v}^{\left(i\right)}}\right)}}h_{I,v}^{\left(i\right)}}}\right|^{2}+B_{uv}N_{0}}\right)-1}\right\\}\frac{1}{{d_{u,v}^{-\alpha_{u,v}\left({\theta_{u,v}}\right)}\left\|{h_{u,v}}\right\|^{2}}}.$
(17)

$\begin{split}\bar{\lambda}_{u,v}=\frac{{\sqrt{P_{x}d_{u,v}^{-\alpha_{u,v}\left({\theta_{u,v}}\right)}}}}{{B_{uv}N_{0}}}\left({\frac{\pi}{{\sigma_{h_{u,v}}^{2}}}}\right)\exp\left({-\frac{{\rho_{h_{u,v}}^{2}}}{{2\sigma_{h_{u,v}}^{2}}}}\right)\left\\{{\int\limits_{0}^{\infty}{\left\\{{\underbrace{h_{u,v}^{3}G_{1,3}^{1,0}\left[{h_{u,v}^{2}\frac{{\rho_{h_{u,v}}^{2}}}{{\sigma_{h_{u,v}}^{2}}}\left|{\begin{array}[]{*{20}c}{\frac{1}{2}}\\\
{0,0,\frac{1}{2}}\\\ \end{array}}\right.}\right]}_{{\rm Term\hskip
4.26773ptI}}}\right.}}\right.\\\
\left.{-\left.{\underbrace{h_{u,v}^{3}G_{1,2}^{1,1}\left[{\frac{{h_{u,v}^{2}}}{{2\sigma_{h_{u,v}}^{2}}}\left|{\begin{array}[]{*{20}c}1\\\
{1,0}\\\
\end{array}}\right.}\right]G_{1,3}^{1,0}\left[{h_{xy}^{2}\frac{{\rho_{h_{u,v}}^{2}}}{{\sigma_{h_{u,v}}^{2}}}\left|{\begin{array}[]{*{20}c}{\frac{1}{2}}\\\
{0,0,\frac{1}{2}}\\\ \end{array}}\right.}\right]}_{{\rm Term\hskip
4.26773ptII}}}\right\\}dh_{u,v}}\right\\}\\\ \end{split}.$ (18)

Figure 2: Signal-to-interference ratio relative to the receiver height for
different $y_{R}$ along the line joining the transmitter and the interferer.

We like to point out that, since the channel is random over a single
transmission interval with the assumption that the channel state information
is not available to the BS, and considering the capacity as defined in (13),
it is now possible that $C\left({\lambda,\gamma}\right)=0$ with nonzero
probability irrespective of the link range and transmission power. Therefore,
the power allocation approach may not be an optimal solution for the uplink
G2U networks. Moreover, it is now impossible to guarantee successful
information transmission with $\phi_{e}=0$ and
$C\left({\lambda,\gamma}\right)>0$.

Next, we obtain the minimum transmit power $P_{u,\rm min}$ required at node
$u$, given the maximum allowable delay $d_{c,\rm max}$ and the reliability
constraint $\phi_{e}$ at the receiver $v$. Following (2), (10), and (11), the
tight upper bound on the transmit power requirement, subject to the
reliability constraint and the maximum delay threshold, can readily be derived
as shown in (17).

##  IV Results and Analysis

The simulation parameters are summarized in Table II. We consider a slow-
fading channel, where the actual channel gains are assumed to be constant,
however, random over a single transmission interval. Unless otherwise stated,
the interferer coordinates are set to $(0,500,0)$. To get a satisfactory
statistical average, for each link configuration parameter set, 1000
realizations of normal distributed independent random variables are generated.
The delay is then obtained by averaging the results over 1000 realizations.
For brevity, we assume one interfering node. For simplicity, we consider the
following three cases to describe the link geometries.

TABLE II: System Parameters Parameter |  Value
---|---
Packet size |  $32$ bytes
Rice factor of G2A link |  $5\sim 12$ dB
Rice factor of A2A link |  $10\sim 12$ dB
Noise spectral density $N_{0}$ [9] |  $-174$ dBm/Hz
LOS (NLOS) shadow fading standard deviation |  $4$ $(6)$ dB
Model parameter $f_{1}$ [22] |  12.08
Model parameter $f_{2}$ [22] |  0.11
Carrier frequency $f_{c}$ |  $2$ GHz

  * •

Case I: It considers a scenario where the receiver is located above in a
region near the vicinity of the transmitter

  * •

Case II: It considers a scenario where the receiver is located above in a
region near the midway of the line joining the transmitter and the receiver

  * •

Case III: It considers a scenario where the receiver is located above in a
region near the vicinity of the interfering node

Fig. 2 shows the SIR variation with receiver height $(z_{R})$ at different
positions along the line joining the BS and the interferer. For Case I, as
$z_{R}$ increases, the signal of interest becomes weaker, and hence, the
interference becomes relatively stronger. On the contrary, for Case III,
increasing $z_{R}$ results in a weaker interference thereby making the signal
of interest relatively stronger and hence higher SIR. However, the nearly
constant SIR for case II can be attributed to the fact that both, the signal
of interest and the interference signal, become weak proportionally with
increasing $z_{R}$.

(a) Interference-limited channel.

(b) Interference channel with end-to-end received signal-to-noise power ratio
$\lambda=30$ dB.

(c) Interference channel with end-to-end received signal-to-noise power ratio
$\lambda=0$ dB.

Figure 3: Delay profile as a function of the receiver height over different
channel conditions (Case I).

In Fig. 3(a), 3(b), and 3(c), we show the results for the delay, while varying
the receiver height $z_{R}$ for several channel conditions, i.e., over the
interference-limited channel, and when $\bar{\lambda}=30$ dB and
$\bar{\lambda}=0$ dB, respectively. In Fig. 3(a), as expected, we can observe
that for an interference-limited channel, when the receiver is located above
the transmitter, the delay is an increasing function of the receiver height.
In contrast, when the receiver moves in either direction in a line joining the
transmitter and the interferer, the delay shows non-monotonic behaviors with
the receiver height. Indeed, for the given link configuration and the
placements of the ground BS and the interferer, the delay first decreases with
increasing the height till the optimal height is obtained and then increases
with the height. This quasi-monotonic behavior can be attributed to the fact
that increasing the receiver height beyond the optimal height causes higher
interference and path loss. In contrast to the interference-limited channel,
slightly different trends are observed in the interference-plus-noisy channel.
For example, in Fig. 3(b), when $\bar{\lambda}=30$ dB, the delay increases
with $z_{R}$ irrespective of $x_{R}$ and $y_{R}$, however, in Fig. 3(c), when
$\bar{\lambda}$ is very low, the three curves, however, cannot be
distinguished. It shows that noise is a dominating factor in this case.

(a) Interference-limited channel.

(b) Interference channel with end-to-end received signal-to-noise power ratio
$\lambda=0$ dB.

Figure 4: Delay profile as a function of the receiver height over different
channel conditions (Case II).

Figure 4 presents the end-to-end delay profile relative to the receiver height
for the direct link over different channel conditions for Case II. The
different curves in Fig. 4(a) and 4(b) demonstrate the impact of varying the
receiver along the mid-point on the line joining the transmitter and the
interferer. Based on the results, a few important observations can be made.

  * •

If the UAV is located near the midpoint such that, $d_{T,N}>d_{i,R}$, the
delay first increases rapidly with the height and then decreases gradually
with further increasing the UAV height. This monotonic behavior can be
attributed to the fact that increasing the height initially causes received
signal power to decrease relatively at a higher rate than interference and
thereby reduces the number of packets successfully received. However, at
larger heights, the impact of the interference signal also reduces, thus
causing the delay to decrease gradually.

  * •

Interestingly, and contrary to the above point, if the UAV is located near the
midpoint such that $d_{T,N}\leq d_{i,R}$, increasing the height does not
reflect significant changes in the delay.

(a) Interference-limited channel.

(b) Interference channel with end-to-end received signal-to-noise power ratio
$\lambda=0$ dB.

Figure 5: Delay profile as a function of the receiver height over different
channel conditions (Case III).

Figure 5 depicts the delay performance relative to the receiver heights over
different channel types for Case III. The results reveal important
observations on the delay characteristics, that when the receiver is located
in a region near the vicinity of the interfering node, the performance is
primarily dominated by interference. Interestingly, as can be seen, the delay
decreases with the increasing receiver height for all the curves under all
channel conditions. Where the delay values are finite but very high. The
higher delay is due to the increasing interference that reduces the success
probability of transmission. Moreover, it is to be noted that, irrespective of
the channel conditions, the performance is primarily dominated by interference
and not by the Gaussian noise.

Figure 6: Minimum delay relative to received SNR for different receiver
locations.

Given the location of the receiver and the interferer, the minimum delay a G2U
communication system must encounter for different values of $\bar{\lambda}$ is
illustrated in Fig. 6. In obtaining these curves, the interference power is
kept constant at $30$ dBm whereas, the reliability constraint is set to
$10^{-4}$. Following important observations can be made.

  * •

As can be seen, the impact of increasing the signal power in minimizing the
transmission delay dominates only if the UAV is located near a region above
the transmitter. However, it is to be noted that the negative rate of change
of delay with respect to the change in $\bar{\lambda}$ approaches zero for
larger values of $\bar{\lambda}$. Moreover, it is also important to note that,
as the UAV approaches a region away from the transmitter and near the
interferer, the rate $\frac{dd_{c,\rm min}}{d\bar{\lambda}}\rightarrow 0$ even
for smaller values of $\bar{\lambda}$. It shows that increasing
$\bar{\lambda}$ gives a negligible gain in terms of minimizing the
transmission delay, though may greatly enhance the outage performance.

  * •

As low $\bar{\lambda}$ values, increasing the height of the UAV does not
necessarily impact the delay performance. However, as $\bar{\lambda}$
increases, the delay performance degrades with increasing the height.

Figure 7: Delay profile relative to the relay node location.

Given the range of allowable system latency that guarantees that the
reliability constraint is met, the minimum transmission power required at the
BS for different noise-to-interference power (NIP) levels is illustrated in
Fig. 7. In obtaining these results, we fix the location of the receiver and
the interferer at $(0,250,250)$ and $(0,500,0)$, respectively. The reliability
constraint is set to $P_{e}\leq\phi_{e}=10^{-4}$. The key observations from
the results obtained are as follows:

  * •

Given the reliability constraint and fixed NIP value, there exists a non-
linear relation between the allowable transmission delay and the corresponding
transmit power requirements. As can be seen, the required transmit power
reduces significantly when the allowable transmission delay is relaxed from
its initial value. However, on further relaxing the delay requirement, the
required transmit power reduces gradually. This phenomenon validates our claim
that increasing the signal power beyond a certain limit may not be an optimal
solution to the delay minimization problem.

  * •

Given the delay requirement and the reliability constraint, the required
signal power increases with increasing NIP. Importantly, it is to be noted
that, this behavior follows a constant rate of change of required transmit
power with respect to the change in NIP, irrespective of the delay
requirement.

(a) Received signal-to-noise power ratio $\lambda=-2$ dB.

(b) Received signal-to-noise power ratio $\lambda=0$ dB.

(c) Received signal-to-noise power ratio $\lambda=5$ dB.

Figure 8: Delay profile relative to the relay node location for different
receiver heights (without noise amplification at the relay node).

Figure 8 shows the delay profile against the receiver height for different
relay node locations. The $x_{R}$ and $y_{R}$ coordinates of the receiver are
fixed to $(0,250)$ and are located midway of the line joining the transmitter
and the interferer. The amplification gain of the relay UAV is set to $-3$ dB.
However, it is assumed that there is no noise at the input of the relay. For
the relay-assisted network, it can be seen that the delay increases with the
height initially and then decreases. This phenomenon is more dominating when
the relay node is located on the opposite side of the line joining the
transmitter and the receiver. Moreover, it can be seen that, as the relay node
moves closer to the region between the transmitter and the receiver, the delay
performance improves. However, it is to be noted that, as the relay node moves
closer to the receiver, the delay though may be lower, but increases with
increasing the receiver height. Moreover, for the direct link, as the receiver
height increases, there is no impact on the delay performance. Intuitively,
this behavior can be attributed to the fact that, if the UAV is located near
the region in the middle of the BS and the interferer, changing the UAV height
vary the signals strengths from the transmitter and the interferer
identically. In obtaining the results in Figs. 8(a), 8(b), 8(c), a few points
are important to note.

  * •

For the relayed-assisted network, we assume that there is no direct path
between the BS and the receiver UAV. Thus, the relayed UAV in our case serves
primarily as compensation for channel degradation due to the path loss and
fading between the ground BS and the receiver UAV. Therefore, no additional
diversity is achieved in this case.

  * •

The relayed UAV utilizes an amplify-and-forward relaying protocol. Therefore,
these results will serve as a lower bound when compared with the conventional
decode-and-forward protocol.

(a) Received signal-to-noise power ratio $\lambda=5$ dB.

(b) Received signal-to-noise power ratio $\lambda=5$ dB.

(c) Received signal-to-noise power ratio $\lambda=5$ dB.

Figure 9: Delay profile relative to the relay node location for different
receiver heights (with noise amplification at the relay node).

The results presented in Fig. 9 are obtained with identical conditions as set
in obtaining the results in Fig. 8, however, the relay node is assumed to have
Gaussian noise at its input with noise amplification considered.

Figure 10: Optimal link configuration for minimum delay. Figure 11: Delay
profile relative to the relay node location.

In Fig. 10, the joint optimization of the relay node and location and the
receiver height is presented with the aim to minimize the delay. In obtaining
the results in Fig. 10, the end-to-end received average signal-to-noise power
ratio is set to $-2$ dB whereas the amplification gain of the relay node is
set to $-3$ dB. The relay node location index $\\{1,2,3,4,5\\}$ corresponds to
the location coordinates as $(0,-50,50)$, $(0,00,50)$, $(0,50,50)$,
$(0,100,50)$, $(0,150,50)$, respectively. $x_{R}$ and $y_{R}$ are set to $0$
and $250$.

Figure 11 shows the impact of the relay node location in minimizing the delay.
Interestingly, it is to be noted that due to the hammock shape of the delay
profile, as illustrated in Fig. 11, there is only one global minimum, and the
optimal relay node location corresponding to this global minima can readily be
obtained using convex optimization. Recall that in obtaining the results in
Fig. 11, our dual-hop AF scheme does not exploit the existence of the direct
link from the BS to the UAV for the coherent combining of signals at the
receiver. Ignoring the direct link in this analysis provides an upper bound on
the delay that can be achieved over an interference plus noise G2U channel.

##  V Conclusion

In this work, we have presented an information-theoretic framework to study
the delayed performance of the G2U integrated network. Utilizing Fano’s
inequality, we have derived the tight upper bound for the channel capacity of
the G2U network Based on the location information of the UAV for a point-to-
point G2U, we have also obtained the tight lower bound on the transmit power
requirement subject to the reliability constraint and the maximum delay
threshold. Interestingly, it has been shown that, despite the simplicity of
the point-to-point direct link, the latency of the AF relayed-based link can
be lower, if the relay is properly placed. Our results show that increasing
the transmit power may not always be an optimal solution for latency
minimization problems, though it may significantly improve the outage
performance. A possible extension of this work relates to investigating the
ground-to-multi-relayed UAV networks, which can offer higher reliability at
the cost of even more system complexity.

[Proof of (7)] Utilizing (4) and (5) and applying the identities [23, eq.
03.02.26.0006.01] and [23, eq. 01.03.26.0007.01], the average received SNR for
the link $u\rightarrow v$ can be expressed as depicted in (18).

Applying the identity [23, eq. 07.34.21.0009.01], and by utilizing the fact
that the gamma function has simple poles at negative integers, it can be
readily shown that the integral in the Term I in (18) is guaranteed to
approach zero irrespective of the values of $\sigma_{h_{u,v}}^{2}$ and
$\rho_{h_{u,v}}^{2}$. We like to point out that the gamma function is a
meromorphic function that has simple poles at negative integers [24]. Finally,
applying the identity [23, eq. 07.34.21.0011.01], the integral depicted in
Term II can be computed to a closed form as
$2\left({-\sigma_{h_{u,v}}^{4}}\right)G_{3,4}^{2,1}\left[{\frac{{2\rho_{h_{u,v}}^{2}}}{{\sigma_{h_{u,v}}^{2}}}\left|{\begin{array}[]{*{20}c}{-2,-1,\frac{1}{2}}\\\
{0,-2,0,\frac{1}{2}}\\\ \end{array}}\right.}\right]$. Substituting the
equivalent closed-form expressions corresponding to Term I and Term II in
(18), the expression for the average received SNR is obtained as shown in (7).

[Proof of (16)] Applying the chain rule and utilizing the fact that
conditioning reduces entropy, $H\left({Y_{R}\left|{s_{I}}\right.}\right)$ can
readily be expressed as

$\begin{split}&H\left({Y_{R}\left|{s_{I}}\right.}\right)\leq\sum\limits_{j=1}^{N_{p}/T_{c}}{\left\\{{\mathop{\max}\limits_{{\rm
trace}\left[{\mathbb{E}\left[{s_{T,j},s_{T,j}^{*}}\right]}\right]\leq
T}\mathop{\mathbb{missing}}{E}\limits_{s_{I,j}}\left[{\log\left({\pi
e}\right)^{T}}\right.}\right.}\\\ &\hskip
11.38092pt\times\det\left[{P_{T}d_{T,R}^{-\alpha_{T,R}\left({\theta_{T,R}}\right)}h_{T,R}\mathbb{E}\left[{s_{T,j},s_{T,j}^{*}}\right]}\right.\\\
&\hskip
9.95863pt\left.{\left.{\left.{+\sum\limits_{i\in\Phi_{I}}{P_{I}^{\left(i\right)}d_{i,R}^{-\alpha_{I,R}^{\left(i\right)}\left({\theta_{I,R}^{\left(i\right)}}\right)}\left[{s_{I,j}^{\left(i\right)}\left({s_{I,j}^{\left(i\right)}}\right)^{*}}\right]+B_{TR}N_{0}I}}\right]}\right]}\right\\}\\\
\end{split}.$ (19)

where $I$ is the identity matrix. Subscript $j$ represents the block index. It
follows that $N_{p}/T_{c}$ should be a positive integer. Following a similar
approach as illustrated in [25], with some simple mathematical manipulations,
(19) can be written as illustrated in (20).

Utilizing the chain rule for entropy,
$H\left({Y\left|{S,s_{I}}\right.}\right)$ can readily be expressed as shown in
(21). Applying the Markov chain, the expression in (21) is reduced to the
expression as illustrated in (22).

$\begin{split}&H\left({Y_{R}\left|{s_{I}}\right.}\right)\leq
N_{p}\log\left({\pi
e}\right)+\frac{{N_{p}\left({T_{c}-1}\right)}}{T_{c}}\log\left({P_{T}d_{T,R}^{-\alpha_{T,R}\left({\theta_{T,R}}\right)}+B_{TR}N_{0}}\right)\\\
&+\sum\limits_{j=1}^{N_{p}/T_{c}}{\mathop{E}\limits_{s_{I,j}}\left[{\log\left({P_{T}d_{T,R}^{-\alpha_{T,R}\left({\theta_{T,R}}\right)}+\sum\limits_{i\in\Phi_{I}}{P_{I}^{\left(i\right)}d_{i,R}^{-\alpha_{I,R}^{\left(i\right)}\left({\theta_{I,R}^{\left(i\right)}}\right)}\left\|{s_{I,j}^{\left(i\right)}}\right\|^{2}+B_{TR}N_{0}}}\right)}\right]}\\\
\end{split}.$ (20)

$\begin{split}&H\left({Y_{R}\left|{S,s_{I}}\right.}\right)=\sum\limits_{j=1}^{N_{p}/T_{c}}{H\left({y_{j}\left|{y_{1},\cdot\cdot\cdot,y_{j-1},S,s_{I}}\right.}\right)}\\\
&\hskip
51.21504pt=\sum\limits_{j=1}^{N_{p}/T_{c}}{H\left({\left.{\sum\limits_{i\in\Phi_{I}}{\sqrt{P_{I}^{\left(i\right)}d_{I,R}^{-\alpha_{I,R}^{\left(i\right)}\left({\theta_{I,R}^{\left(i\right)}}\right)}}h_{I,j}^{\left(i\right)}s_{I,j}^{\left(i\right)}}+w_{R}}\right|y_{1},\cdot\cdot\cdot,y_{j-1},S,s_{T},s_{I}}\right)}\\\
\end{split}.$ (21)

$\begin{split}&H\left({Y\left|{S,s_{I}}\right.}\right)=\\\ &\hskip
28.45274ptN_{p}\log\left({\pi
e}\right)+\frac{{N_{p}\left({T_{c}-1}\right)}}{T_{c}}\log\left({B_{TR}N_{0}}\right)\\\
&\hskip
28.45274pt+\sum\limits_{j=1}^{N_{p}/T_{c}}{\mathop{\mathbb{missing}}{E}\limits_{s_{I},j}\log\left({\sum\limits_{i\in\Phi_{I}}{P_{I}^{\left(i\right)}d_{I,R}^{-\alpha_{I,R}^{\left(i\right)}\left({\theta_{I,R}^{\left(i\right)}}\right)}\left\|{s_{I,j}^{\left(i\right)}}\right\|}^{2}+w_{R}}\right)}\\\
\end{split}.$ (22)

Substituting (20) and (22) into (10), and applying simple mathematical
manipulations, yields (16).

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|  Sudhanshu Arya (Member, IEEE) is a researcher in the School of System and
Enterprises at Stevens Institute of Technology, NJ, USA. He received his
M.Tech. degree in communications and networks from the National Institute of
Technology, Rourkela, India, in 2017, and the Ph.D. degree from Pukyong
National University, Busan, South Korea, in 2022. He worked as a Research
Associate with the Department of Artificial Intelligence Convergence, at
Pukyong National University. His research interests include wireless
communications and digital signal processing, with a focus on free-space
optical communications, optical scattering communications, optical spectrum
sensing, computational game theory, and artificial intelligence. He received
the Best Paper Award in ICGHIT 2018 and the Early Career Researcher Award from
the Pukyong National University in 2020.
---|---
|  Ying Wang (Member, IEEE) received the B.E. degree in information
engineering at Beijing University of Posts and Telecommunications, M.S. degree
in electrical engineering from University of Cincinnati and the Ph.D. degree
in electrical engineering from Virginia Polytechnic Institute and State
University. She is an associate professor in the School of System and
Enterprises at Stevens Institute of Technology. Her research areas include
cybersecurity, wireless AI, edge computing, health informatics, and software
engineering.
---|---