BLER and Throughput Analysis of Power Beacon-Based Energy Harvesting UAV-Assisted NOMA Relay Systems for Short-Packet Communications

In this paper, we analyze an unmanned aerial vehicle (UAV) that harvests energy from a power beacon to support the transmissions of finite blocklength packets from a source to two destinations using non-orthogonal multiple access (NOMA) scheme. The impacts of line-of-sight (LoS) and non-LoS communications on the block error rate (BLER) and throughput of the system are presented via the Rician factor. For the evaluation of system performance, we derive the closed-form and approximate expressions of the BLERs of each hop and the end-to-end communication path between source and destination. Based on these derived BLER expressions, we calculate the throughput of the considered UAV-NOMA-short packet communication (SPC) system. Analytical results are verified by Monte-Carlo simulations. Numerical results show that the best system performance can be achieved by selecting a reasonable time switching duration for energy harvesting, blocklength, and the number of training bits. The minimal BLERs can be obtained by controlling the trajectory of the UAV, i.e., its altitude and location on the horizontal plane. The BLER performance of the UAV-NOMA-SPC system is compared with that of the UAV-orthogonal multiple access (OMA)-SPC system, showing that the UAV-NOMA-SPC gives a lower average BLERs than the UAV-OMA-SPC system.

SECTION I. Introduction

Nowadays and soon, electrical devices are growing fast, as reported by Ericsson [1]. This report states that there will be around 36,048 connected devices in wireless communication by the year 2025, which will form the Internet-of-Thing (IoT) systems. Additionally, more interconnected devices operate intelligently, and the electric industry will become a hot trend and continue beyond 2030. Thus, wireless communication systems require much more resources and higher criteria such as latency lower than 0.1 ms, reliability higher than 99.99999%, and sensing accuracy at the millimeter-level. Besides, energy efficiency is a major problem in the fifth-generation (5G) and sixth-generation (6G) networks. To deal with these requirements, several solutions are suggested: 1) using the non-orthogonal multiple access (NOMA) to improve the spectrum efficiency in 5G and 6G systems [2], [3], 2) utilizing short packets to achieve ultra-reliable low latency communications (URLLC), which is helpful in the IoT systems, 3) employing unmanned aerial vehicles (UAVs) to provide dynamic mobility for the communication devices in disaster and emergency response, 4) applying energy harvesting to overcome the limited battery capacity of electric devices and UAV [4]. On the other hand, in the command-control and monitor systems for toxic areas, the usage of UAVs to communicate becomes the trend because it does not require fixed infrastructure [5], [6], [7], [8].

A. Related Work

Recent investigations into UAV-assisted short-packet communications combined with energy harvesting have attracted much attention from researchers. In particular, Ranjha et al. [4] formulated a non-convex joint resource allocation, trajectory design, and energy harvesting problem to minimize the total decoding error rate in URLLC-enable laser-powered UAV relay systems. Raut et al. [9] investigated a nonlinear energy harvesting (EH)-based UAV-assisted FD relay network with infinite and finite blocklength codes, where source UAV harvested energy transmitted from a laser transmitter. The closed-form expression of block error rate (BLER) was derived under the impacts of imperfect channel state information (CSI). Agrawal et al. [10] analyzed the average outage probability, BLER, and goodput of reconfigurable intelligent surface (RIS)-assisted UAV-based multiuser communication system with finite blocklength codes and nonlinear EH of the ground user. Moreover, the investigation harvested energy for the UAV-assisted infinite blocklength communications have been studied in many works such as [11], [12], [13], [14], and [15]. These works stated that the harvested energy was enough for signal transmission.

On the other hand, UAV-assisted short-packet communication (SPC) systems without energy harvesting have also been considered in the literature. Particularly, Hu et al. [16] studied the influence of SPC on the performance of UAV-enable cognitive networks. The authors formulated the optimization problem for maximizing the energy efficiency under the constraints of packet error rate, spectrum sensing duration and threshold, and the transmit power of the UAV. Basnayaka et al. [17] studied the age of information of an URLLC-enable UAV wireless network by deriving the closed-form expression of this performance metric. Yuan et al. [18] derived the closed-form expression of BLER and applied both bisection search and one-dimensional search algorithm to maximize the energy efficiency of the UAV in a UAV-assisted URLLC between a base station and a user. The usage of short packets for remote control of the UAV was considered in [19], [20], and [22]. The authors derived closed-form expressions of the average packet error probability (APEP) and throughput of these systems. Ranjha et al. [21] performed passive beamforming of RIS antenna elements and nonlinear nonconvex optimization to minimize the overall decoding error rate and find the optimal UAV’s position and blocklength. From Table 1, 1 we see that most of the above UAV-assisted SPC systems utilized orthogonal multiple access (OMA) schemes and did not use the energy harvesting technique. Only several works applied simultaneous wireless information and power transfer (SWIPT) to the UAV. However, SWIPT may not be an efficient EH technique for UAV-assisted communication systems.

TABLE 1 Literature Review on Some Related UAV-Assited SPC Systems

Presently, incorporating the NOMA technique into SPC helps to improve the spectral efficiency of finite blocklength transmissions. Specifically, Yao et al. [23] proposed a join decoding for downlink NOMA systems with finite blocklength transmissions. Le et al. [24] studied an uplink NOMA wireless system with SPC under channel estimation errors and residual transceiver hardware impairments. The authors provided the average BLER in the finite blocklength regime, then formulated a maximum throughput optimization problem subject to the packet length. Comparisons between NOMA-SPC and OMA-SPC were also presented. Tran et al. [25] investigated a multi-user downlink MIMO-NOMA system with SPC. For evaluating the system performance, the authors derived the asymptotic and approximate closed-form expressions of the average BLER. Vu et al. [26] studied the performance of IRS-aided short-packet NOMA systems under perfect and imperfect SIC by deriving the closed-form approximated expressions of BLER with random and optimal phase shifts. Yin et al. [27] proposed a packet re-management framework for a cooperative NOMA scheme for SPC and gave a linear searching method to solve the problem of minimizing power consumption. It is worth noticing that the NOMA-SPC systems in [23], [24], [25], [26], and [27] did not consider the usage of UAVs despite of the advantages of using UAV such as high mobility and versatility. Additionally, energy charging for the UAV using a power beacon was also not considered in the above works. The main reason for these matters is the extreme difficulty in obtaining the closed-form mathematical expressions of the BLER and throughput of the EH-enable UAV-assisted NOMA-SPC system over Rician channels. Furthermore, since the UAV’s propulsion energy is much higher than the UAV’s communication energy in practice, the issue of compatibility between them occurs. On the other hand, utilizing the propulsion energy of a UAV for communication might introduce fluctuations in the signals. Moreover, using the energy of a UAV for information transmission would further drain the battery, reducing its operational time. To overcome these limitations, we mathematically study a UAV-assisted NOMA relay system where a UAV harvests energy from a power beacon to support short-packet transmissions over Rician channels. The main contributions of this paper are summarized as follows:

• Unlike previous works which studied UAV-SPC systems without NOMA [4], [9], [10], [16], [17], [18], [19], [20], [21] or NOMA-SPC systems without UAV and EH [23], [24], [25], [26], [27], we fill these research gaps by investigating a UAV-assisted NOMA relay system, where an EH-enable UAV employs NOMA technique to forward finite blocklength packets from a source to two destinations (now called the UAV-NOMA-SPC system).

• We first derive the signal-to-interference-plus-noise ratio (SINR) expressions corresponding to the light-of-sight (LoS) and NLoS communications. Based on these SINR expressions, we provide the closed-form expressions of the performance in terms of the BLER and throughput of the considered UAV-assisted NOMA relay system with finite packet size. For practical UAV-assisted communications, we investigate the system performance over Rician fading channel and different urban environments.

• We extensively study the effects of energy harvesting duration, environments, the number of training bits, the 3D trajectory of the UAV, the Rician factor, and blocklength on the BLERs and throughput of the considered UAV-NOMA-SPC system. We show that the best system performance can be obtained by determining the appropriate energy harvesting duration, UAV coordinate, the number of training bits, and power allocation coefficient. The comparisons between the BLERs of the UAV-NOMA-SPC and UAV-OMA-SPC systems are also provided to show the advantage of the considered system.

The remainder of this paper is structured as follows. Section II describes the system model. Section III presents the mathematical analysis of the BLER and the throughput. The main results and relevant discussions are given in detail in Section IV. Finally, Section V concludes the paper.

Notation: a represents a vector while $\|\textbf {a}\|$ is the Euclidean norm of a. $[\cdots]^{T}$ denotes the transpose of $[\cdots]$ . $f_{X}(x)$ and $F_{X}(x)$ refer to the probability density function (PDF) and the cumulative distribution function (CDF) of a random variable $X$ , respectively; $\|\cdot \|^{2}$ means the Frobenius norm; the expectation operator is $\mathbb {E}[\cdot]$ .

SECTION II. System Model

Fig. 1 illustrates the wireless-powered UAV-NOMA-SPC system considered in this paper. It consists of four nodes: a source node (S) as a controlling station, two destinations ($\rm D_{1}$ and $\rm D_{2}$ ), a UAV as an air relay (AR), and a power beacon (PB). It is assumed that the direct links between S and D_i are unavailable due to mountains or high-rise buildings. Thus, the communication between S and D_i is assisted by the UAV operating as a decode-and-forward (DF) relay. Moreover, the power for S and D_i come from a fixed power source, while the operating energy of the UAV is harvested from the PB [4]. Without loss of generality, we assume that S, PB, and D_i are located on a horizontal plane and the UAV hovers above them. Specifically, the locations of all nodes are $\textbf {w}_{\mathrm{ s}}={\mathrm{ S}}(x_{\mathrm{ s}}, y_{\mathrm{ s}}, 0)$ , $\textbf {w}_{\mathrm{ p}}={\mathrm{ PB}}(x_{\mathrm{ p}}, y_{\mathrm{ p}}, 0)$ , $\textbf {w}_{1}={\rm D_{1}}(x_{w_{1}}, y_{w_{1}}, 0)$ , $\textbf {w}_{2}={\rm D_{2}}(x_{w_{2}}, y_{w_{2}}, 0)$ , and UAV $(x_{u},y_{u}, H)$ . In this paper, the flying time of the UAV from the initial point $\textbf {q}_{I}=(x_{I}, y_{I}, H)$ to the final point $\textbf {q}_{F}=(x_{F}, y_{F}, H)$ is $T$ , and it is divided into $N$ sub-timeslots, i.e., each incremental time of UAV is $\delta _{t}=\frac {T}{N}$ . The location of the UAV at the $n$ th time slot is expressed as UAV $\textbf {q}[n]=(x_{u}[n],y_{u}[n], H)$ , $n\in \{1, \cdots, N\}$ . The distance between PB and UAV at the $n$ th time slot is$d_{p}=\sqrt {\|\textbf {q}[n]-\textbf {w}_{\mathrm{ p}}\|^{2}+v \delta _{t}+H^{2}}$ while the distance between UAV and D_i is $d_{i}=\sqrt {\|\textbf {q}[n]-\textbf {w}_{i}\|^{2}+v \delta _{t}+H^{2}}$ . The sub-timeslot is formulated as $\delta _{t}=\frac {\textbf {q}[n+1]-\textbf {q}[n]}{v}$ , where $v$ is the velocity of UAV. When $n=1$ , the UAV stays right above S; when $n=N$ , the UAV stays right above D_i. The incremental distance during each sub-timeslot $\delta _{t}$ must satisfy $\|\textbf {q}[n+1]-\textbf {q}[n]\|^{2}\le v \delta _{t}$ so that the location of UAV is almost unchanged during each sub-timeslot.

FIGURE 1.

Illustration of a power beacon-based energy harvesting UAV-assisted NOMA relay system for short-packet communications.

A. Channel Model

The wireless channels are modeled by two kinds of fading, i.e., large-scale fading and small-scale fading. The large-scale fading depends on the distance between transmitter and receiver and the path loss of the propagation environment. The amplitude of small-scale fading is fixed in one symbol duration but may be changed randomly over the next symbol. We denote the large-scale fading between S and the UAV as ${\mathcal{ L}}_{s}[n]=\frac {\beta _{0}}{(\|\textbf {q}[n]-\textbf {w}_{\mathrm{ s}}\|^{2}+H^{2})^{\alpha }}$ , and from the UAV to D_i as ${\mathcal{ L}}_{d}[n]=\frac {\beta _{0}}{(\|\textbf {q}[n]-\textbf {w}_{i}\|^{2}+H^{2})^{\alpha }}$ , and the distance between the PB and the UAV as ${\mathcal{ L}}_{p}[n]=\frac {\beta _{0}}{(\|\textbf {q}[n]-\textbf {w}_{\mathrm{ p}}\|^{2}+H^{2})^{\alpha }}$ , where $\beta _{0}$ refers to the channel gain at the reference distance $d_{0}=1$ m [28] and $\alpha (\theta _{\ell})=\left[{\alpha \left({\frac {\pi }{2}}\right)-\alpha (0)}\right]\eta +\alpha (0)$ , $\ell \in \{\rm {S},\rm {PB},\rm {D}_{i} \}$ represents the path-loss coefficient from the UAV to ground users, $0\le \eta \le 1$ is an additional attenuation factor to show the relationship between the LoS probability and the elevation angle of the UAV and ground terminals. The probabilities of LoS and NLoS links are, respectively, characterized as [29]

$$\begin{align*} P_{\mathrm{ LoS}}(\theta _{\ell})&= c_{1}-\frac {c_{1}-c_{2}}{1+\left({\frac {\theta _{\ell} -c_{3}}{c_{4}}}\right)^{c_{5}}}, \tag{1}\\ P_{\mathrm{ NLoS}}(\theta _{\ell})&=1- P_{\mathrm{ LoS}}(\theta _{\ell}), \tag{2}\end{align*}$$ where $\theta _{\ell} =\frac {180^{o}}{\pi }{\mathrm{ arcsin}}\left({\frac {H}{d_{\ell} }}\right)$ ; $c_{k}$ with $k\in \{1, \cdots, 5\}$ are constants, which depend on the types of urban environments. Their empirical values are presented in Table 2.

TABLE 2 Parameters for LoS Probability Calculation [29]

Obviously, the altitude of the UAV influences the quality of signal propagation from ground users to the UAV, i.e., the LoS probability between ground users and the UAV depends on $\theta _{\ell}$ . In particular, larger $\theta _{\ell}$ (which also means higher altitude of the UAV) leads to higher LoS probability; however, the path losses of wireless links also increase.2

Due to the effects of LoS connections and multi-path scattering at ground users, the fluctuation of small-scale fading can be modeled by Rician distribution [31]. Let us denote $h_{s}$ , $h_{p}$ , $h_{d_{i}}$ as the small-scale fading coefficients between S and UAV, PB and UAV, UAV and D_i, respectively. These are modeled as

$$\begin{equation*} h_{\ell} = \sqrt { \frac {K_{\ell} }{K_{\ell} +1}}\sigma _{\ell} e^{j\phi } + \sqrt {\frac {1}{K_{\ell} +1}}{\mathcal{ CN}}(0, \sigma _{\ell} ^{2}), \tag{3}\end{equation*}$$ where the first term represents the specular path arriving with uniform phase $\phi$ and the second term refers to the aggregation of large numbers of reflected and scattered paths, which is independent of $\phi$ ; $K_{\ell} = \frac {P_{\mathrm{ LoS}}(\theta _{\ell})}{1-P_{\mathrm{ LoS}}(\theta _{\ell})}$ is the Rician factor defined as the ratio of the power in the LoS component to the power in the multipath scatters [17], and ${\mathcal{ CN}}(0, \sigma _{\ell} ^{2})$ is the random scattering component, which is a circularly symmetric complex Gaussian (CSCG) random variable with zero mean and unit variance. Note that $K_{\ell}$ only represents the possibility of having LoS communications in small-scale fading but does not include the path loss in large-scale fading. The probability distribution function (PDF) of $h_{\ell}$ is $$\begin{align*} f_{|h_{\ell} |^{2}}(x) &= \frac {(K_{\ell} +1) e^{-K_{\ell} }}{\lambda _{\ell} }\exp \left({-\frac {(K_{\ell} +1)x}{\lambda _{\ell} }}\right) \\ &\quad \times I_{0}\left({2 \sqrt {\frac {K_{\ell} (K_{\ell} +1)x}{\lambda _{\ell} }}}\right), x\ge 0, \tag{4}\end{align*}$$ where $I_{0}(\cdot)$ denotes the zero-order modified Bessel function of the first kind, $\lambda _{\ell} = \mathbb {E}{|h_{\ell} |^{2}}$ is the average channel gain of small-scale fading. Without loss of generality, the average channel gain of the system under the effects of both large-scale and small-scale fading can be presented as $\hat \Omega _{\ell} =\omega \lambda _{\ell} |{\mathcal{ L}}_{\ell} [n]|$ , where $\omega$ is an additional attenuation factor caused by the LoS or NLoS communications in large-scale fading. Notably, $\omega =1$ for LoS communication and $0 < \omega < 1$ for NLoS communication [32]. This means that the average channel gains corresponding to LoS and NLoS communications between ground terminals and the UAV are given by [33] $$\begin{align*} {\Omega _{\ell} }= \begin{cases} \displaystyle (1-\rho)\hat \Omega _{\ell} & \text {LoS link},\\ \displaystyle \omega (1-\rho)\hat \Omega _{\ell} & \text {NLoS link}, \end{cases} \tag{5}\end{align*}$$ where $0 < \rho \le 1$ reflects the impact of moving speed of the UAV [9], [34], [35]. In the case of uniform scattering, $\rho$ can be calculated as $\rho = J_{0}\left({\frac {2\pi f_{c} v}{r_{s} c}}\right)$ , where $f_{c}$ is the carrier frequency, $c$ is the speed of light, $r_{s}$ is the symbol rate, and $J_{0}(\cdot)$ is the for zeroth order Bessel function of the first kind. We can see that $\rho$ affects the average channel gain, i.e., a larger $\rho$ means the velocity $v$ of the UAV is higher, leading to a smaller average channel gain. It is also assumed that the Doppler effect caused by the UAV mobility is perfectly compensated at the receivers [36], and the coherence time of a signal over a channel use is perfect.

Therefore, under the effects of both large-scale and small-scale fading, the channel between PB and UAV is $h_{\mathrm{ pu}}={\mathcal{ L}}_{p}[n]h_{p}$ , between S and UAV is $h_{\mathrm{ su}}={\mathcal{ L}}_{s}[n]h_{s}$ and from UAV to D_i is $h_{\rm ud_{i}}={\mathcal{ L}}_{d}[n]h_{\rm d_{i}}$ .

B. Energy Harvesting and Communication Model

Regarding the energy harvesting process, the interval $\kappa$ of each transmit symbol period is used to harvest energy from PB. During this duration, the UAV harvests energy transmitted from PB by using time switching (TS) protocol.3 Thus, the harvested energy at the UAV in $m$ channel use (cu) is [14], [37], and [38]

$$\begin{equation*} E_{A}=\kappa m P_{\mathrm{ b}}|h_{\mathrm{ pu}}|^{2}\xi, \tag{6}\end{equation*}$$ where $P_{\mathrm{ b}}$ is the transmit power of PB and $\xi$ denotes the energy conversion efficiency at the UAV. Since the UAV operates in half-duplex (HD) mode, the transmit power of the UAV is $$\begin{equation*} P_{\mathrm{ r}}=\frac {2\kappa m\xi P_{\mathrm{ b}}|h_{\mathrm{ pu}}|^{2}}{1-\kappa }=\Delta |h_{\mathrm{ pu}}|^{2},\,\, with \,\, \Delta = \frac {2\kappa m \xi P_{\mathrm{ b}}}{1-\kappa }. \tag{7}\end{equation*}$$

During the remaining symbol period, S transmits control signals to D_i in HD mode, i.e., S sends data blocks with the sizes of $m(1-\kappa)/2$ symbol to the UAV in the first phase; then the UAV forwards the control signals to D₁ and D₂ over remaining $m(1-\kappa)/2$ symbol. For the transmission of control signals, the number of codewords needs to be small to satisfy low latency. Thus, the length of the transmitted packet is finite. This setting is suitable for the communications and controls of coastal ships using finite blocklength codes.4

For the power-domain NOMA technique, in the first phase, S superposes $x_{1}$ and $x_{2}$ according to the power allocation coefficients $a_{1}$ and $a_{2}$ , respectively, i.e, $x_{\mathrm{ S}}=\sqrt {a_{1}P_{\mathrm{ s}}}x_{1}+\sqrt {a_{2}P_{\mathrm{ s}}}x_{2}$ , $a_{1}\le a_{2}$ and $a_{1}+a_{2}=1$ , where $P_{\mathrm{ s}}$ is the transmit power of S. Consequently, the received signal at the UAV is

$$\begin{equation*} y_{\mathrm{ U}} = h_{\mathrm{ su}}(\sqrt {a_{1} P_{\mathrm{ s}}}x_{1} + \sqrt {a_{2} P_{\mathrm{ s}}} x_{2}) + w_{u}, \tag{8}\end{equation*}$$ where $w_{u}={\mathcal{ CN}}(0,\sigma _{\mathrm{ U}}^{2})$ is the additive white Gaussian noise (AWGN) with zero mean and standard variance of $\sigma _{\mathrm{ U}}^{2}$ at the UAV. The signal $x_{2}$ is first decoded at the UAV and treated $x_{1}$ as noise, then the UAV decodes $x_{1}$ by applying SIC technique to subtract $x_{2}$ . Thus, the signal-to-interference plus noise ratios (SINRs) of $x_{1}$ and $x_{2}$ in the case of perfect SIC are, respectively, expressed as $$\begin{align*} \gamma _{\mathrm{ AR}}^{x_{2}} &=\frac {a_{2} P_{\mathrm{ s}}|h_{\mathrm{ su}}|^{2}}{a_{1} P_{\mathrm{ s}}|h_{\mathrm{ su}}|^{2}+\sigma _{\mathrm{ AR}}^{2}}, \tag{9}\\ \gamma _{\mathrm{ AR}}^{x_{1}} &=\frac {a_{1} P_{\mathrm{ s}}|h_{\mathrm{ su}}|^{2}}{\sigma _{\mathrm{ AR}}^{2}}. \tag{10}\end{align*}$$

In the second phase, the UAV re-superimposes $\hat x_{1}$ and $\hat x_{2}$ according to the power coefficients $a_{1}$ and $a_{2}$ , respectively, i.e., $x_{\mathrm{ U}}=\sqrt {a_{1}P_{\mathrm{ r}}}\hat x_{1}+\sqrt {a_{2}P_{\mathrm{ r}}}\hat x_{2}$ . Thus, the received signals at both $\rm D_{1}$ and $\rm D_{1}$ are, respectively, given by

$$\begin{align*} y_{\rm D_{1}} &= h_{\rm ud_{1}}(\sqrt {a_{1} P_{\mathrm{ r}}}\hat x_{1} + \sqrt {a_{2} P_{\mathrm{ r}}} \hat x_{2})+w_{\rm D_{1}}, \tag{11}\\ y_{\rm D_{2}} &= h_{\rm ud_{2}}(\sqrt {a_{1} P_{\mathrm{ r}}}\hat x_{1} + \sqrt {a_{2} P_{\mathrm{ r}}} \hat x_{2})+w_{\rm D_{2}}, \tag{12}\end{align*}$$ where $w_{\rm D_{i}}= {\mathcal{ CN}}(0,\sigma _{\rm D_{i}}^{2})$ is the AWGN at D_i and $P_{\mathrm{ r}}$ is the transmit power of the UAV given in (7).

$\rm D_{2}$ decodes $\hat x_{2}$ by considering $\hat x_{1}$ as noise, while $\rm D_{1}$ first decodes $\hat x_{2}$ and then $\hat x_{1}$ . Therefore, the SINR of $\hat x_{2}$ at $\rm D_{1}$ is expressed as in (13)

$$\begin{equation*} \gamma _{\rm D_{1}}^{\hat x_{2}} =\frac {a_{2} P_{\mathrm{ r}}|h_{\rm ud_{1}}|^{2}}{a_{1} P_{\mathrm{ r}} |h_{\rm ud_{1}}|^{2}+\sigma _{\rm D_{1}}^{2}}. \tag{13}\end{equation*}$$

After $\rm D_{1}$ performs SIC successfully on $\hat x_{2}$ , the SINRs of $\hat x_{1}$ at $\rm D_{1}$ and $\hat x_{2}$ at $\rm D_{2}$ are, respectively, given by

$$\begin{align*} \gamma _{\rm D_{1}}^{\hat x_{1}} &=\frac {a_{1}P_{\mathrm{ r}}|h_{\rm ud_{1}}|^{2}}{\sigma _{\rm D_{1}}^{2}}, \tag{14}\\ \gamma _{\rm D_{2}}^{\hat x_{2}} &=\frac {a_{2} P_{\mathrm{ r}}|h_{\rm ud_{2}}|^{2}}{a_{1} P_{\mathrm{ r}}|h_{\rm ud_{2}}|^{2}+\sigma _{\rm D_{2}}^{2}}. \tag{15}\end{align*}$$

SECTION III. Performance Analysis

A. Backgrounds on Blocklength Error Rate

In conventional channel coding theorem, the error probability of the communication system is often investigated with the infinite blocklength. However, in practice, the system’s data rate may be limited due to fixed finite blocklength to ensure target error probability. According to [39], the data rate $r_{i}$ of finite blocklength $m$ is approximated as5

$$\begin{align*} r_{i}(m, \gamma _{i}, \epsilon _{i}) \approx C(\gamma _{i})-\sqrt {\frac {V(\gamma _{i})}{m}}Q^{-1}(\epsilon _{i})+O\left({\frac {\log _{2} m}{m} }\right), \tag{16}\end{align*}$$ where $C(\gamma _{i})=\log _{2}(1+\gamma _{i})$ is the Shannon capacity, $V(\gamma _{i})=\left({1-\frac {1}{(1+\gamma _{i})^{2}}}\right)(\log _{2}e)^{2}$ refers to the channel dispersion, measured in squared information units per channel use, which represents the variation of channel compared with deterministic channel for the same capacity, $\epsilon _{i}$ is the expected error probability, $Q^{-1}(\cdot)$ is the inverse Gaussian Q-function $Q(x)=\frac {1}{2\pi }\int _{x}^{\infty }\exp \left({-\frac {t^{2}}{2}}\right)dt$ , and $O\left({\frac {\log _{2} m}{m} }\right)$ is the remainder term of order $\frac {\log _{2} m}{m}$ . Since the blocklength6 is large enough, i.e., $m\ge 100$ as given in [39], from (16), we can rewrite the instantaneous BLER as $$\begin{equation*} \epsilon _{i}\approx Q \left ({(C(\gamma _{i})-r_{i})/\sqrt {V(\gamma _{i})/m}}\right). \tag{17}\end{equation*}$$

Let $\phi _{\mathcal{ A}}^{x_{i}}$ be the event that decoding error of $x_{i}$ occurs at node ${\mathcal{ A}}\in \{\rm UAV, D_{1}, D_{2} \}$ and $\bar \phi _{\mathcal{ A}}^{x_{i}}$ is the complement of $\phi _{\mathcal{ A}}^{x_{i}}$ . From (9) and (10), the instantaneous BLER when decoding $x_{2}$ at the UAV is calculated as

$$\begin{equation*} \Pr (\phi _{\mathrm{ AR}}^{x_{2}})= \epsilon _{\mathrm{ AR}}^{x_{2}}\approx Q\left({\frac {C(\gamma _{\mathrm{ U}}^{x_{2}})-r_{2}}{\sqrt {V(\gamma _{\mathrm{ U}}^{x_{2}})/m(1-\kappa)}}}\right), \tag{18}\end{equation*}$$ where $\gamma _{\mathrm{ U}}^{x_{2}}$ is given in (9) and $r_{2}= 2M/m(1-\kappa)$ with $M$ is the total number of bits of $x_{2}$ . After the UAV decodes and removes $x_{2}$ in (8) successfully, the instantaneous BLER when decoding $x_{1}$ at the UAV is $$\begin{equation*} \Pr (\phi _{\mathrm{ AR}}^{x_{1}}| \bar \phi _{\mathrm{ AR}}^{x_{2}})=\epsilon _{\mathrm{ AR}}^{x_{1}}\approx Q\left({\frac {C(\gamma _{\mathrm{ U}}^{x_{1}})-r_{1}}{\sqrt {V(\gamma _{\mathrm{ U}}^{x_{1}})/m(1-\kappa)}}}\right), \tag{19}\end{equation*}$$ where $\gamma _{\mathrm{ U}}^{x_{1}}$ is given in (10) and $r_{1}= 2M/m(1-\kappa)$ with $M$ is the number of bits of user $\rm D_{1}$ . On the other hand, the UAV can decode $x_{1}$ in the condition of SIC error at $x_{2}$ , i.e., $\Pr (\phi _{\mathrm{ AR}}^{x_{1}}| \phi _{\mathrm{ AR}}^{x_{2}})$ . Thus, the probability of error decoding $x_{1}$ at the UAV is expressed as $$\begin{align*} \Pr (\phi _{\mathrm{ AR}}^{x_{1}})= \Pr (\phi _{\mathrm{ AR}}^{x_{1}}| \phi _{\mathrm{ AR}}^{x_{2}})\Pr (\phi _{\mathrm{ AR}}^{x_{2}})+\Pr (\phi _{\mathrm{ AR}}^{x_{1}}| \bar \phi _{\mathrm{ AR}}^{x_{2}})\Pr (\bar \phi _{\mathrm{ AR}}^{x_{2}}), \tag{20}\end{align*}$$ where $\Pr (\phi _{\mathrm{ AR}}^{x_{1}}| \phi _{\mathrm{ AR}}^{x_{2}})$ is the conditional probability of $\phi _{\mathrm{ R}}^{x_{1}}$ for a given $\phi _{\mathrm{ AR}}^{x_{2}}$ . Due to the high interference level from $x_{2}$ when decoding $x_{1}$ , we have $\Pr (\phi _{\mathrm{ AR}}^{x_{1}}| \phi _{\mathrm{ AR}}^{x_{2}})\approx 1$ . From (20), the total BLER when detecting $x_{1}$ at the UAV is calculated as $$\begin{align*} \tilde {\epsilon }_{\mathrm{ AR}}^{x_{1}}&=1\times {\epsilon }_{\mathrm{ AR}}^{x_{2}}+{\epsilon }_{\mathrm{ AR}}^{x_{1}}(1-{\epsilon }_{\mathrm{ AR}}^{x_{2}}) \\ &={\epsilon }_{\mathrm{ AR}}^{x_{2}}+{\epsilon }_{\mathrm{ AR}}^{x_{1}}- {\epsilon }_{\mathrm{ AR}}^{x_{1}}{\epsilon }_{\mathrm{ AR}}^{x_{2}}\approx {\epsilon }_{\mathrm{ AR}}^{x_{2}}+{\epsilon }_{\mathrm{ AR}}^{x_{1}}. \tag{21}\end{align*}$$ Note that in (21), the error in URLLC usually small, ranging from 10⁻³ to 10⁻⁵. Therefore, the term ${\epsilon }_{\mathrm{ AR}}^{x_{1}} {\epsilon }_{\mathrm{ AR}}^{x_{2}}\to 0$ and can be ignored.

Since the UAV utilizes DF protocol, after receiving the superposed signal, $\rm D_{2}$ detects $x_{2}$ directly and $\rm D_{1}$ applies SIC on $x_{2}$ to obtain $x_{1}$ . Thus, it is assumed that D_i cannot detect $x_{i}$ if the UAV cannot detect it, i.e., $\Pr (\phi _{\rm D_{i}}^{x_{i}}|\phi _{\mathrm{ AR}}^{x_{i}})=1$ . Thus, the BLER when decoding $x_{1}$ at D₁ and decoding $x_{2}$ at D₂ are, respectively, given by

$$\begin{align*} \Pr (\phi _{\rm D_{1}}^{x_{1}}) &= \tilde {\epsilon }_{\mathrm{ AR}}^{x_{1}} +\Pr (\phi _{\rm D_{1}}^{x_{2}})+\Pr (\phi _{\rm D_{1}}^{x_{1}}|\bar \phi _{\rm D_{1}}^{x_{2}})\Pr (\bar \phi _{\rm D_{1}}^{x_{2}}) \\ &=\tilde {\epsilon }_{\mathrm{ AR}}^{x_{1}}+{\epsilon }_{\rm D_{1}}^{x_{2}}+\Pr (\phi _{\rm D_{1}}^{x_{1}}|\bar \phi _{\rm D_{1}}^{x_{2}}). \tag{22}\\ \Pr (\phi _{\rm D_{2}}^{x_{2}})&=\Pr (\phi _{\rm D_{2}}^{x_{2}}|\phi _{\mathrm{ AR}}^{x_{2}})\Pr (\phi _{\mathrm{ AR}}^{x_{2}})+\Pr (\phi _{\rm D_{2}}^{x_{2}}|\bar \phi _{\mathrm{ AR}}^{x_{2}})\Pr (\bar \phi _{\mathrm{ AR}}^{x_{2}}) \\ &=1\times \Pr (\phi _{\mathrm{ AR}}^{x_{2}})+\Pr (\phi _{\rm D_{2}}^{x_{2}}|\bar \phi _{\mathrm{ AR}}^{x_{2}})(1-\Pr (\phi _{\mathrm{ AR}}^{x_{2}})). \tag{23}\end{align*}$$

From (22), the instantaneous BLER of $x_{1}$ at $\rm D_{1}$ can be calculated as

$$\begin{equation*} \tilde {\epsilon }_{\rm D_{1}}^{x_{1}}=\tilde {\epsilon }_{\mathrm{ AR}}^{x_{1}}+ {\epsilon }_{\rm D_{1}}^{x_{2}}+{\epsilon }_{\rm D_{1}}^{x_{1}}, \tag{24}\end{equation*}$$ and from (23), the instantaneous BLER of $x_{2}$ at $\rm D_{2}$ can be computed as $$\begin{equation*} \tilde {\epsilon }_{\rm D_{2}}^{x_{2}} ={\epsilon }_{\mathrm{ AR}}^{x_{2}}+{\epsilon }_{\rm D_{2}}^{x_{2}}(1-{\epsilon }_{\mathrm{ AR}}^{x_{2}})\approx {\epsilon }_{\mathrm{ AR}}^{x_{2}}+{\epsilon }_{\rm D_{2}}^{x_{2}}, \tag{25}\end{equation*}$$ where ${\epsilon }_{\rm D_{1}}^{x_{2}}\approx Q\left({\frac {(C(\gamma _{\rm D_{1}}^{\hat x_{2}})-r_{2})}{\sqrt {V(\gamma _{\rm D_{1}}^ {\hat x_{2} \to \hat x_{1}})/m(1-\kappa)} } }\right)$ , ${\epsilon }_{\rm D_{1}}^{x_{1}}\approx Q \left({\frac {(C(\gamma _{\rm D_{1}}^{\hat x_{1}})-r_{1})}{\sqrt {V(\gamma _{\rm D_{1}}^{\hat x_{1}})/m(1-\kappa)} }}\right)$ , and ${\epsilon }_{\rm D_{2}}^{x_{2}}\approx Q\left({\frac {(C(\gamma _{\rm D_{2}}^{\hat x_{2}})-r_{2})}{\sqrt {V(\gamma _{\rm D_{2}}^{\hat x_{2}})/m(1-\kappa)}} }\right)$ , $\tilde {\epsilon }_{\mathrm{ AR}}^{x_{1}}$ and $\epsilon _{\mathrm{ AR}}^{x_{2}}$ are, respectively, given in (21) and (18).

B. Average BLER in Finite Blocklength Regime

From (17), the average BLER at each node can be presented as

$$\begin{equation*} \bar \epsilon _{i}\approx \int _{0}^{\infty } Q\left({\frac {C(\gamma _{i})-r_{i}}{\sqrt {V(\gamma _{i})/m(1-\kappa)}}}\right)f_{\gamma _{i}}(x) dx, \tag{26}\end{equation*}$$ where $f_{\gamma _{i}}(x)$ denotes the PDF of the random variable $\gamma _{i}$ .

Since it is challenging to derive the exact closed-form expression of (26), we apply the approximate Q-function to solve (26) as similar to [25], [40], [41], and [42], i.e.,

$$\begin{align*} {Q\left({\frac {C(\gamma _{i})-r_{i}}{\sqrt {V(\gamma _{i})/m}}}\right)=} \begin{cases} \displaystyle 1, &\gamma _{i}\le \rho _{L}\\ \displaystyle 0.5 - \chi _{i} (\gamma _{i}-\tau _{i}), &\rho _{L} < \gamma _{i} < \rho _{H},\\ \displaystyle 0, &\gamma _{i} \ge \rho _{H} \end{cases} \tag{27}\end{align*}$$ where $\chi _{i} = [2\pi (2^{2r_{i}}-1)/m(1-\kappa)]^{-1/2}$ , $\tau _{i}=2^{r_{i}}-1$ , $\rho _{L}= \tau _{i}-1/(2\chi _{i})$ , and $\rho _{H}=\tau _{i}+1/(2\chi _{i})$ .

From (26) and (27), we can rewrite the average BLER as

$$\begin{equation*} \bar \epsilon _{i}(\Omega _{\ell})\approx \chi _{i}\int _{\rho _{L}}^{\rho _{H}}F_{\gamma _{i}|\Omega _{\ell} }(x|\Omega _{\ell}) dx, \tag{28}\end{equation*}$$ where $F_{\gamma _{i}|\Omega _{\ell} }(x|\Omega _{\ell})$ is the conditional CDF of $\gamma _{i}$ . To obtain the closed-form of BLER, we first derive $F_{\gamma _{i}|\Omega _{\ell} }(x|\Omega _{\ell})$ , which depends on the SINRs. The CDFs of these SINRs are given in detail in Appendix.

Replacing the CDFs from (47) and (48) into (28), the average BLERs can be given by the following Propositions.

Proposition 1:

From (21) and (28), the closed-form expression of the average BLERs of $x_{1}$ at the UAV is

$$\begin{align*} \mathbb {E}\{\tilde {\epsilon }_{\mathrm{ AR}}^{x_{1}}\} &\approx ({\bar \epsilon }_{\mathrm{ AR, LoS}}^{x_{2}}+{\bar \epsilon }_{\mathrm{ AR, LoS}}^{x_{1}})P_{\mathrm{ LoS}} \\ &\quad +({\bar \epsilon }_{\mathrm{ AR, NLoS}}^{x_{2}}+{\bar \epsilon }_{\mathrm{ AR, NLoS}}^{x_{1}})P_{\mathrm{ NLoS}}, \tag{29}\end{align*}$$ and from (18), (27) and (28) the closed-form expression of the average BLERs of $x_{2}$ at the UAV is $$\begin{equation*} \mathbb {E}\{\epsilon _{\mathrm{ AR}}^{x_{2}}\}= \bar \epsilon _{\mathrm{ AR, LoS}}^{x_{2}}P_{\mathrm{ LoS}}+\bar \epsilon _{\mathrm{ AR, NLoS}}^{x_{2}}P_{\mathrm{ NLoS}}, \tag{30}\end{equation*}$$ where ${\bar \epsilon }_{\mathrm{ AR,LoS}}^{x_{1}}$ and ${\bar \epsilon }_{\mathrm{ AR,LoS}}^{x_{2}}$ are, respectively, given in Chebyshev-Gauss approximation7 as $$\begin{align*} \bar \epsilon _{\mathrm{ AR,LoS}}^{x_{2}}&=1-\chi _{2}\sum _{l=0}^{L_{\mathrm{ max}}}\sum _{n=0}^{l}\sum _{\mu =0}^{N}\frac {\pi \beta _{l,n}}{N}\left({\frac {(K+1)u\sigma _{\mathrm{ R}}^{2}}{\Omega _{\mathrm{ su}}P_{\mathrm{ s}}(a_{2}-a_{1}u)} }\right)^{n} \\ &\quad \times \exp \left({-\frac {(K+1)u\sigma _{\mathrm{ R}}^{2}}{\Omega _{\mathrm{ su}}P_{\mathrm{ s}}(a_{2}-a_{1}u)} }\right)\sqrt {1-\psi ^{2}}, \tag{31}\\ {\bar \epsilon }_{\mathrm{ AR, LoS}}^{x_{1}}& =1-\chi _{1}\sum _{l=0}^{L_{\mathrm{ max}}}\sum _{n=0}^{l}\sum _{\mu =0}^{N}\frac {\pi \beta _{l,n}}{N}\left({\frac {(K+1)u\sigma _{\mathrm{ R}}^{2}}{\Omega _{\mathrm{ su}}P_{\mathrm{ s}}a_{1}} }\right)^{n} \\ &\quad \times \exp \left({-\frac {(K+1)u\sigma _{\mathrm{ R}}^{2}}{\Omega _{\mathrm{ su}}P_{\mathrm{ s}}a_{1}} }\right)\sqrt {1-\psi ^{2}}, \tag{32}\end{align*}$$ where $u= \frac {\rho _{H}-\rho _{L}}{2}\psi +\frac {\rho _{H}+\rho _{L}}{2}$ , $\psi =cos\left({\frac {(2~\mu -1)\pi }{2N} }\right)$ , $L_{\mathrm{ max}}\in \{1, \infty \}$ , and $N$ is the number of terms in Chebyshev-Gauss quadrature. To obtain ${\bar \epsilon }_{\mathrm{ AR, NLoS}}^{x_{i}}$ , we only replace $\Omega _{\ell}$ by $\omega \Omega _{\ell}$ from (5) into (31) and (32). It is also used in [32, Eq. (7)].

To calculate $\bar \epsilon _{\mathrm{ AR, LoS}}^{x_{2}}$ and $\bar \epsilon _{\mathrm{ AR, NLoS}}^{x_{2}}$ , we replace (47) into (28) and then use the Chebyshev-Gauss approximation. Moreover, we use (48) to calculate $\bar \epsilon _{\mathrm{ AR, LoS}}^{x_{1}}$ . Note that, when calculating the detecting error of $x_{1}$ at the UAV, the SIC error of $x_{2}$ should be considered, as given in (21).

Remark 1:

Equations (31) and (32) indicate that the BLERs of $x_{1}$ and $x_{2}$ at the UAV depend on the size $m(1-\kappa)$ of data block over S-UAV link, the average channel gain $\Omega _{\mathrm{ su}}$ , and the power allocation coefficients $a_{1}, a_{2}$ . When $u> {a_{2}}/{a_{1}}$ , the upper bound of the BLER of $x_{2}$ at the UAV is $\bar \epsilon _{\mathrm{ R,LoS/NLoS}}^{x_{2}}=1$ .

Proposition 2:

From (21) and (24), the closed-form expression of the average BLER at $\rm D_{1}$ is

$$\begin{align*} \mathbb {E}\{\tilde {\epsilon }_{\rm D_{1}}^{\mathrm{ e2e}}\}& = ({\bar \epsilon }_{\mathrm{ AR,LoS}}^{x_{2}}+{\bar \epsilon }_{\mathrm{ AR,LoS}}^{x_{1}}+ {\bar \epsilon }_{\rm D_{1},LoS}^{x_{2}}+{\bar \epsilon }_{\rm D_{1},LoS}^{x_{1}})P_{\mathrm{ LoS}} \\ &\quad +({\bar \epsilon }_{\mathrm{ AR,NLoS}}^{x_{2}}+{\bar \epsilon }_{\mathrm{ AR,NLoS}}^{x_{1}}+ {\bar \epsilon }_{\rm D_{1},NLoS}^{x_{2}} \\ &\quad +{\bar \epsilon }_{\rm D_{1},NLoS}^{x_{1}})P_{\mathrm{ NLoS}}, \tag{33}\end{align*}$$ and from (25) the average BLER at $\rm D_{2}$ is $$\begin{align*} \mathbb {E}\{\tilde {\epsilon }_{\rm D_{2}}^{\mathrm{ e2e}}\} &\approx ({\bar \epsilon }_{\mathrm{ AR,LoS}}^{x_{2}}+{\bar \epsilon }_{\rm D_{2},LoS}^{x_{2}})P_{\mathrm{ LoS}} \\ &\quad +({\bar \epsilon }_{\mathrm{ AR,NLoS}}^{x_{2}}+{\bar \epsilon }_{\rm D_{2},NLoS}^{x_{2}})P_{\mathrm{ NLoS}}, \tag{34}\end{align*}$$ where ${\bar \epsilon }_{\mathrm{ R,LoS/NLoS}}^{x_{1}}$ and ${\bar \epsilon }_{\mathrm{ R,LoS/NLoS}}^{x_{2}}$ are given in Proposition 1. The closed-form expressions of ${\bar \epsilon }_{\rm D_{1},LoS}^{x_{2}}$ , ${\bar \epsilon }_{\rm D_{1},LoS}^{x_{1}}$ , and ${\bar \epsilon }_{\rm D_{2},LoS}^{x_{2}}$ are given in (35), (36) and (37), as shown at the bottom of next the page, $$\begin{align*} {\bar \epsilon }_{\rm D_{1},LoS}^{x_{2}}&=1-\chi _{2}\sum _{l=0}^{L_{\mathrm{ max}}} \sum _{n=0}^{l}\sum _{j=0}^{J_{\mathrm{ max}}}\frac { \beta _{l,n}}{(j!)^{2}}\left({\frac {K(K+1)}{\Omega _{\mathrm{ bu}}} }\right)^{j}\frac {K+1}{\Omega _{\mathrm{ bu}}e^{K}}\sum _{\mu =0}^{N}\frac {\pi }{N} \left({\frac {(K+1)\sigma _{\rm D_{1}}^{2} \Psi (u) }{\Delta \Omega _{\rm ud_{1}}} }\right)^{n} \\ &\quad \times 2\left({\frac {\sigma _{\rm D_{1}}^{2}\Omega _{\mathrm{ bu}} }{\Delta \Omega _{\rm ud_{1}}} \Psi (u) }\right)^{\frac {j-n+1}{2}} {\mathcal{ K}}_{j-n+1}\left({2 \sqrt {\frac {(K+1)^{2} \Psi (u) \sigma _{\rm D_{1}}^{2} }{\Delta \Omega _{\rm ud_{1}} \Omega _{\mathrm{ bu}}} }}\right)e^{-\frac {(K+1)\sigma _{\rm D_{1}}^{2} \Psi (u)}{\Delta \Omega _{\rm ud_{1}}} }\sqrt {1-\psi ^{2}},. \tag{35}\end{align*}$$ $$\begin{align*} {\bar \epsilon }_{\rm D_{1},LoS}^{x_{1}}&=1-\chi _{1}\sum _{l=0}^{L_{\mathrm{ max}}} \sum _{n=0}^{l}\sum _{j=0}^{J_{\mathrm{ max}}}\frac {\beta _{l,n}}{(j!)^{2}}\left({\frac {K(K+1)}{\Omega _{\mathrm{ bu}}} }\right)^{j}\frac {K+1}{\Omega _{\mathrm{ bu}}e^{K}}\sum _{\mu =0}^{N}\frac {\pi }{N}\left({\frac {(K+1)u\sigma _{\rm D_{1}}^{2} }{\Delta \Omega _{\rm ud_{1}}a_{1}} }\right)^{n} e^{-\frac {(K+1)u\sigma _{\rm D_{1}}^{2} }{\Delta \Omega _{\rm ud_{1}}a_{1}} } \\ &\quad \times 2\left({\frac {u\sigma _{\rm D_{1}}^{2}\Omega _{\mathrm{ bu}} }{\Delta \Omega _{\rm ud_{1}}a_{1}} }\right)^{\frac {j-n+1}{2}} {\mathcal{ K}}_{j-n+1}\left({2 \sqrt {\frac {(K+1)^{2}u\sigma _{\rm D_{1}}^{2} }{\Delta \Omega _{\rm ud_{1}}\Omega _{\mathrm{ bu}}a_{1}}} }\right) \sqrt {1-\psi ^{2}}, \tag{36}\end{align*}$$ $$\begin{align*} {\bar \epsilon }_{\rm D_{2},LoS}^{x_{2}}&=1-\chi _{2}\sum _{l=0}^{L_{\mathrm{ max}}} \sum _{n=0}^{l}\sum _{j=0}^{J_{\mathrm{ max}}}\frac {\beta _{l,n}}{(j!)^{2}}\left({\frac {K(K+1)}{\Omega _{\mathrm{ bu}}} }\right)^{j}\frac {K+1}{\Omega _{\mathrm{ bu}}e^{K}}\sum _{\mu =0}^{N}\frac {\pi }{N}\left({\frac {(K+1)\sigma _{\rm D_{2}}^{2} }{\Delta \Omega _{\rm ud_{2}}}\Psi (u) }\right)^{n} \\ &\quad \times 2\left({\frac { \sigma _{\rm D_{2}}^{2}\Omega _{\mathrm{ bu}} }{\Delta \Omega _{\rm ud_{2}} } \Psi (u) }\right)^{\frac {j-n+1}{2}} {\mathcal{ K}}_{j-n+1}\left({2 \sqrt {\frac {(K+1)^{2} \sigma _{\rm D_{2}}^{2} }{\Delta \Omega _{\rm ud_{2}} \Omega _{\mathrm{ bu}}}\Psi (u) }}\right)e^{-\frac {(K+1) \sigma _{\rm D_{2}}^{2} }{\Delta \Omega _{\rm ud_{2}} } \Psi (u) } \sqrt {1-\psi ^{2}}, \tag{37}\end{align*}$$ respectively. To obtain ${\bar \epsilon }_{\rm D_{1},LoS}^{x_{2}}$ , ${\bar \epsilon }_{\rm D_{1},NLoS}^{x_{1}}$ , and ${\bar \epsilon }_{\rm D_{2},NLoS}^{x_{2}}$ , we only replace $\Omega _{\ell}$ by $\omega \Omega _{\ell}$ from (5) into (35), (36) and (37), where $\Psi (u) =\frac {u}{a_{2}-a_{1}u}$ , $\Delta = \frac {2\kappa m \xi P_{\mathrm{ b}}}{1-\kappa }$ , $\beta _{l,n}=\frac {K^{l}}{l!n!e^{K}}$ , $u= \frac {\rho _{H}-\rho _{L}}{2}\psi +\frac {\rho _{H}+\rho _{L}}{2}$ , $\psi =cos\left({\frac {(2~\mu -1)\pi }{2N} }\right)$ , $L_{\mathrm{ max}}\in \{1, \infty \}$ , and $J_{\mathrm{ max}}\in \{1, \infty \}$ .

To obtain (35), (36), and (37), we replace (57), (59), and (61), as shown at the top of page 13, into (28), respectively, and then do some manipulations. In contrast, after replacing (58), (60) and (62), as shown at the top of page 13, into (28) we obtain ${\bar \epsilon }_{\rm D_{1},NLoS}^{x_{2}}$ , ${\bar \epsilon }_{\rm D_{1},NLoS}^{x_{1}}$ , and ${\bar \epsilon }_{\rm D_{2},NLoS}^{x_{2}}$ , respectively. Note that when considering the BLER at D_i, we must investigate the error at the UAV transferred to D_i.

Remark 2:

Equations (33) and (34) show that the BLER at D_i is a function of the transmit power of PB, the average channel gain $\Omega _{\rm UD_{i}}$ , the blocklength $m(1-\kappa)$ over ${\mathrm{ UAV}}-{\mathrm{ D}}_{i}$ , link. When $u> {a_{2}}/{a_{1}}$ , the upper bounds of the BLERs of $x_{2}$ at $\rm D_{1}$ and $\rm D_{2}$ are, respectively, ${\bar \epsilon }_{\rm D_{1}, LoS/NLoS}^{x_{2}}=1$ and ${\bar \epsilon }_{\rm D_{2},LoS/NLoS}^{x_{2}}=1$ .

C. Asymptotic Expression of The Average BLER

For insights into the impacts of system parameters on the BLER performance, we present the asymptotic of the average BLER. As mentioned in [25], the integral in (28) can be approximated by using the first-order Riemann integral, i.e., $\int _{a}^{b}f(x)dx=(b-a)f\left({\frac {a+b}{2}}\right)$ . Thus, we can rewrite the average BLER as

$$\begin{equation*} \bar \epsilon _{i}(\Omega _{\ell})\approx \chi _{i} (\rho _{H}-\rho _{L})F_{\gamma _{i}|\Omega _{\ell} }\left({\frac {\rho _{H}+\rho _{L}}{2}|\Omega _{\ell} }\right), \tag{38}\end{equation*}$$ where $\chi _{i}$ , $\rho _{H}$ , and $\rho _{L}$ are given in (27). After replacing $\rho _{H}$ and $\rho _{L}$ into (38), the BLER can be written as $$\begin{equation*} \bar \epsilon _{i}(\Omega _{\ell})\approx F_{\gamma _{i}|\Omega _{\ell} }(\tau _{i}|\Omega _{\ell}), \tag{39}\end{equation*}$$ where $\tau _{i}=2^{r_{i}}-1$ . Since $\tau _{i}$ is small enough and $a_{2}$ is much larger than $a_{1}$ , we can present the series of $e^{x}=\sum _{\mu =0}^{\infty }\frac {(-1)^{\mu} x^{\mu} }{\mu !}$ and the approximate of the BLERs at the UAV as $$\begin{align*} \bar \epsilon _{\mathrm{ AR,LoS}}^{x_{2}}&\approx 1-\sum _{l=0}^{L_{\mathrm{ max}}}\sum _{n=0}^{l}\sum _{\mu =0}^{N_{\mathrm{ max}}}\frac {(-1)^{\mu} \beta _{l,n} \left({\frac {(K+1)\tau _{i}\sigma _{\mathrm{ R}}^{2}}{\Omega _{\mathrm{ su}}P_{\mathrm{ s}}(a_{2}-a_{1}\tau _{i})} }\right)^{n+\mu }}{\mu !}, \tag{40}\\ {\bar \epsilon }_{\mathrm{ AR, LoS}}^{x_{1}}& \approx 1-\sum _{l=0}^{L_{\mathrm{ max}}}\sum _{n=0}^{l}\sum _{\mu =0}^{N_{\mathrm{ max}}}\frac {(-1)^{\mu} \beta _{l,n}}{\mu !}\left({\frac {(K+1)\tau _{i}\sigma _{\mathrm{ R}}^{2}}{\Omega _{\mathrm{ su}}P_{\mathrm{ s}}a_{1}} }\right)^{n+\mu }. \tag{41}\end{align*}$$

For calculating the BLERs of UAV-D_i link, we replace $\tau _{i}$ by $x$ into the CDFs in (57), (59) and (61). Similar to Proposition 1, we replace $\Omega _{\ell}$ by $\omega \Omega _{\ell}$ to obtain the BLER at D_i in NLoS condition.

Remark 3:

From (40) and (41), we can get the diversity order at the UAV, $d=\lim \limits _{P_{\mathrm{ s}}\to \infty }\frac {\log _{2}({\bar \epsilon }_{\mathrm{ AR, LoS}}^{x_{i}})}{\log _{2}(P_{\mathrm{ s}})}=1$ . Moreover, the BLERs at D_i given in (35), (36) and (37) do not depend on $P_{\mathrm{ s}}$ . In contrast, since the transmit power of the UAV is harvested from PB, the BLERs at D_i are constants that depend on $P_{\mathrm{ b}}$ .

SECTION IV. Throughput Analysis

When the blocklength is very large, the ergodic capacity is always used as an evaluating metric. In contrast, if the blocklength is short, the throughput will be used instead of the ergodic capacity. In this section, we derive the throughput of the investigated NOMA-UAV-SPC system. The throughput of a short-packet communication system is defined as the number of successfully decoded packets per second [39]. Throughput measures the efficiency and effectiveness of the data processing in the considered system. As the block error rate increases, the throughput tends to decrease or vice versa. For a given size $M=M_{I}+M_{e}$ bits propagated over $m$ channels with error probability $\epsilon$ , the throughput is calculated as [10]

$$\begin{align*} \tau _{\rm D_{i}}^{\mathrm{ e2e}} =\frac {M-M_{e}}{M}r_{i}(1-\bar \epsilon _{\rm D_{i}}^{\mathrm{ e2e}})=\left({1-\frac {M_{e}}{M}}\right)r_{i}(1-\bar \epsilon _{\rm D_{i}}^{\mathrm{ e2e}}), \tag{42}\end{align*}$$ where $M_{I}$ is the number of information bits, $M_{e}$ is the number of training bits, and $\bar \epsilon _{\rm D_{i}}^{\mathrm{ e2e}}$ is the average BLER given (33) and (34).

For the investigated UAV-NOMA-SPC system, the total throughput is computed as

$$\begin{align*} \tau (M_{e}) &= \tau _{\rm D_{1}}^{\mathrm{ e2e}}+\tau _{\rm D_{2}}^{\mathrm{ e2e}} \\ & =\left({1-\frac {M_{e}}{M}}\right)\big [r_{1}(1-\bar \epsilon _{\rm D_{1}}^{\mathrm{ e2e}})+r_{2}(1-\bar \epsilon _{\rm D_{2}}^{\mathrm{ e2e}})\big]. \tag{43}\end{align*}$$ It is noted that, for a fixed size of information data, increasing $M_{e}$ will reduce the rate $(M-M_{e})/M$ , which then improves the system throughput. However, a larger $M_{e}$ leads to less spectrum efficiency, i.e., the number of information bits is decreased over the same channel uses. On the other hand, reducing the number of information bits, in turn, reduces the throughput. Hence, there is a trade-off between the number of information bits and the number of training bits that maximizes the system throughput.

SECTION V. Numerical Results

In this section, we provide the analytical and simulation results of the average BLER and throughput of the considered NOMA-UAV relay system to evaluate the performance of the investigated system and validate the analytical expressions of BLER and throughput in the previous sections. We use $10\times 2^{14}$ independent trials for Monte-Carlo simulations. Since the LoS links dominate the reflection and scattering links, the Rician factor $K$ is integer [43]. Unless otherwise stated, we let the number of transmitted bits $M=256$ and $M=128$ , the packet length $m=200$ . The locations of all nodes are set as follows: $\rm S(-300, 0, 0)$ , $\rm PB(-100, -100, 0)$ , ${\rm D_{1}}$ (200, 0, 0), ${\rm D_{2}}$ (250, 100, 0), the starting location of the UAV is $\textbf {q}_{I}=(-200, 0, 150)$ and the ending location of the UAV is $\textbf {q}_{F}=(200, 0, 150)$ . The coefficients for different urban environments are given in Table 2. Unless otherwise indicated in the figures, parameter settings are presented in Table 3. For achieving the BLERs threshold of the system, we set SNR = 30 dB [14], [22] and the UAV velocity $v$ = 10 m/s.

TABLE 3 System Parameters Used for Simulations

Fig. 2 plots the average BLERs of $x_{1}$ and $x_{2}$ at the UAV using Proposition 1 and the average BLERs of $x_{1}$ and $x_{2}$ at $D_{1}$ and $D_{2}$ using Proposition 2. Moreover, the BLERs of UAV–$\rm D_{1}$ and UAV–$\rm D_{2}$ channels are also given. As observed from Fig. 2, the BLERs of UAV–$\rm D_{1}$ and UAV–$\rm D_{2}$ channels do not change with the SNR because the transmit power at the UAV is fixed. In contrast, the BLERs of $x_{1}$ and $x_{2}$ at the UAV decrease with the increase of the SNR. In contrast, the BLERs of the average BLERs of $x_{1}$ and $x_{2}$ at $D_{1}$ and $D_{2}$ (Proposition 2) are first decreased as the SNR increases and then are saturated as the SNR becomes higher. This feature is because the errors of the first hop are transferred to the second hop, i.e., the end-to-end BLERs are the cumulative errors at UAV and D_i. In addition, the error of $x_{1}$ at $\rm D_{1}$ is lower than that of $x_{2}$ at $\rm D_{2}$ because $\rm D_{1}$ detects $x_{1}$ after performing SIC on $x_{2}$ , i.e., without interference, while $\rm D_{2}$ detects $x_{2}$ by considering $x_{1}$ as the interference. Finally, the approximate and simulation results closely match the exact ones.

FIGURE 2.

Average BLERs of $x_{1}$ and $x_{2}$ at the UAV and $\rm D_{1}$ and $\rm D_{2}$ versus the SNR; $K=1, \omega = 0.7$ , and $m=200$ .

Fig. 3 demonstrates the average BLER of $x_{1}$ and $x_{2}$ at $\rm D_{1}$ and $\rm D_{2}$ versus the SNR for different values of Rice factor $K$ . Firstly, we can see that the average BLERs decrease when $K$ increases because increasing $K$ makes the LoS probabilities higher. Secondly, these average BLERs linearly reduce when SNR < 25 dB, and then the floor in the high SNR regime. The reason behind this feature is that there always exist interferences among signals in power-domain NOMA systems that limit the system performance. Similar behaviors were also mentioned in [3] and [8]. Moreover, the BLER of $x_{1}$ at $\rm D_{1}$ is lower than that of $x_{2}$ at $\rm D_{2}$ because the duration for detecting $x_{1}$ only takes one time slot.

FIGURE 3.

Average BLERs of $x_{1}$ and $x_{2}$ at $\rm D_{1}$ and $\rm D_{2}$ versus the SNR for different $K$ ; $\omega =0.7, m=200$ , urban environment.

Fig. 4 shows the average BLERs of $x_{1}$ and $x_{2}$ at $\rm D_{1}$ and $\rm D_{2}$ versus the altitude $H$ of the UAV for different urban environments and Rice factor $K$ with SNR $= 30$ dB, $P_{\mathrm{ b}}= 100$ W, packet size $M = 256$ bits, and blocklength $m=200$ . When the altitude of the UAV is low (i.e., $H=5$ m), the path-loss is high because the signal propagation on the ground is affected by severe blockages, $\alpha (0)=3.5$ , and $P_{\mathrm{ Los}}(0)\to 0$ . When $H$ increases, the BLERs decrease to the minimum value and then increase. It implies that there are the optimal altitudes of the UAV that provide the lowest BLERs. We observe that these minimum values are different and depend on the urban environments. The reason behind this feature is that when the altitude of the UAV becomes higher, the LoS probability is also higher, which in turn improves the channel gain. In contrast, a higher UAV’s altitude makes the communication link longer, resulting in higher path loss. On the other hand, the error at $\rm D_{1}$ is higher than that at $\rm D_{2}$ because the SIC technique is applied at $\rm D_{1}$ , providing interference-free at $\rm D_{1}$ . Finally, the urban environment gives similar BLERs as dense urban.

FIGURE 4.

Average BLERs of $x_{1}$ and $x_{2}$ at $\rm D_{1}$ and $\rm D_{2}$ versus the UAV’s altitude $H$ for different urban environments; $K=1, \omega =0.7$ , and $m=200$ .

Fig. 5 depicts the average BLERs of $x_{1}$ and $x_{2}$ at $\rm D_{1}$ and $\rm D_{2}$ versus the location of the UAV for different environments and locations of PB. As shown in Fig. 5, since the path-loss of the urban environment is higher than that of the suburban environment, the urban environment has worse BLER performance. On the other hand, the BLER of $x_{2}$ at $\rm D_{2}$ is lower than that of $x_{1}$ at $\rm D_{1}$ . It is because the detecting error of $x_{2}$ impacts the BLER of $x_{1}$ at $\rm D_{1}$ , and the error at the UAV is transferred to $\rm D_{1}$ . Moreover, the location of PB greatly influences the minimal BLERs. Particularly, for $[x_{\mathrm{ p}}, y_{\mathrm{ p}}]=[{50, 0}]$ , the minimal BLER occurs at $x_{\mathrm{ U}}=50$ (m), while $[x_{\mathrm{ p}}, y_{\mathrm{ p}}]=[{100, 0}]$ , the minimal BLER happens at $x_{\mathrm{ U}}=100$ (m). It means the best BLER performance is achieved when the UAV is right above the PB, which is reasonable because the distance between the PB and the UAV is shortest, leading to the highest amount of harvested energy. Moreover, there exists a location of the UAV that gives the best BLER performance.

FIGURE 5.

Average BLERs of $x_{1}$ and $x_{2}$ at $\rm D_{1}$ and $\rm D_{2}$ versus $x_{\mathrm{ U}}$ for different urban environments and locations of PB; $K=1, M=256, m=200$ , and $\omega =0.7$ .

Fig. 6 presents the average BLERs of $x_{1}$ and $x_{2}$ at $\rm D_{1}$ and $\rm D_{2}$ versus the blocklength (the channel utilization) $m$ with the fixed number of training bits and information bits. For comparison, we also give the BLER of the UAV-OMA-SPC system. From Fig. 6, we see that the BLERs are reduced as the blocklength gets larger. It means that, for a fixed number of transmitted information bits, increasing the number of channel utilization, i.e., reducing the number of bits propagating over the channel, results in improved BLER performance. However, it may reduce the spectrum efficiency. Furthermore, increasing $K$ means the LoS channel gain is higher, thus improving BLER performance. Note that when $K=0$ , the channel fading follows the Rayleigh distribution. On the other hand, the BLER of the UAV-NOMA-SPC system is better than that of the UAV-OMA-SPC system because the system performance is linearly proportional to the bandwidth usage.

FIGURE 6.

Average BLERs of $x_{1}$ and $x_{2}$ at $\rm D_{1}$ and $\rm D_{2}$ versus the blocklength $m$ for different $K$ ; $M=256$ and $\omega =0.7$ .

Fig. 7 shows the average BLERs of $x_{1}$ and $x_{2}$ at $\rm D_{1}$ and $\rm D_{2}$ versus the switching ratio $\kappa$ for different transmit power of PB. From Fig. 7, we see that there exists the optimal values $\kappa _{\mathrm{ opt}}$ of $\kappa$ , which minimizes the average BLERs. Moreover, different $\kappa _{\mathrm{ opt}}$ minimal BLERs can be achieved for different transmit power of PB. On the other hand, the BLER performance of $x_{1}$ at $\rm D_{1}$ is lower than that of $x_{2}$ at $\rm D_{2}$ . In addition, the energy harvesting time must be appropriately determined to ensure enough time for signal processing. Moreover, larger $\kappa$ means the transmit power of PB is higher, i.e., the signal processing time is shorter. In other words, the signal processing time can be reduced for a fixed error threshold when the transmit power is higher.

FIGURE 7.

Average BLERs of $x_{1}$ and $x_{2}$ at $\rm D_{1}$ and $\rm D_{2}$ versus the time switching ratio $\kappa$ for different transmit power of PB, $M=256, K=1, m=200$ , and SNR $= 20$ dB.

Fig. 8 illustrates the average BLERs of $x_{1}$ and $x_{2}$ at the UAV, $\rm D_{1}$ , and $\rm D_{2}$ versus the LoS probability for different packet sizes, SNR $= 30$ dB and $m=200$ . Note that only the channels used for actual data transmission are considered; thus, the number of used channels for training is out of the scope. As shown in Fig. 8, the BLERs decrease as the LoS probability increases. Moreover, when $\omega =1$ , the system achieves ideal transmission as free-space communication, and when $\omega =0$ , BLERs are almost equal to one. On the other hand, the BLERs of $x_{1}$ are lower than those of $x_{2}$ because the interference from $x_{1}$ impacts the signal detection of $x_{2}$ . Additionally, for the same number of used channels, the BLERs in the case of a small number of transmitted bits (i.e., $M=128$ ) is better than in the case of a large number of transmitted bits (i.e., $M=256$ ).

FIGURE 8.

Average BLERs of $x_{1}$ and $x_{2} \rm D_{1}$ and $\rm D_{2}$ versus LoS probability in urban environment for different packet sizes; SNR $= 30$ dB, $K=1$ .

Fig. 9 plots the throughput of D_i and the whole system versus the SNR in dB for different velocities of the UAV. From Fig. 9, we see that the throughput of $x_{1}$ is higher than that of $x_{2}$ , which means the possibility of successfully detecting $x_{1}$ is higher. It is because $x_{2}$ is detected by considering $x_{1}$ as interference, while $x_{1}$ is detected after successfully performing SIC on $x_{2}$ . On the other hand, the velocity of the UAV significantly affects the achieved throughput. Specifically the throughput when the UAV is stationary ($v=0$ ) is better than the throughput when $v=20$ , i.e., $\tau =12$ bit/s/Hz versus $\tau = 8$ bit/s/Hz. Moreover, the throughput is saturated in the high SNR region (i.e., SNR > 20 dB), confirming Remark 3, i.e., the system performance is constrained by the transmit power of PB.

FIGURE 9.

Throughput of D_i and the whole system versus the SNR different velocities $v$ of the UAV; $M = 256, m=200$ .

Fig. 10 depicts the throughput of D_i and the whole system versus the number of training bits for different data rates. As observed in Fig.10, higher data rate $r$ leads to better throughput for the same number of training bits $M_{e}$ . In addition, a higher data rate needs more training bits, i.e., for the case $r=2$ , the maximal throughput is obtained at 80 training bits, while for the case $r=4$ , nearly 130 training bits are needed to achieve the maximal throughput. Moreover, the throughputs increase with $M_{e}$ to the maximal values and then decrease as $M_{e}$ increases further. It is because for a given bit stream, increasing the number of training bits reduces the number of information bits, i.e., lower throughput. On the other hand, lowering the number of training bits increases the decoding error, i.e., lower system throughput. Furthermore, the analysis results are consistent with the simulation results, confirming the correctness of the mathematical analysis.

FIGURE 10.

Throughput of D_i and whole system versus the number of training bits for different data rates; SNR $= 30$ dB, $M=256$ .

Fig. 11 depicts the throughputs versus the power allocation for $\rm D_{1}$ , i.e., $a_{1}$ for fixed transmit power SNR = 30 dB and data rate $r=4$ . It is noted that the power allocation for $\rm D_{2}$ is $a_{2}=1-a_{1}$ . When the power allocation coefficient $a_{1}$ increases from 0.4 to 1, the sum throughput and throughput of $\rm D_{2}$ increase, reach the maximal value, and then reduce. Meanwhile, the throughput of $\rm D_{1}$ continuously increases. It is because higher $a_{1}$ (or lower $a_{2}$ ) means more power is allocated to $\rm D_{1}$ , leading to higher throughput. On the other hand, we also see that the optimal system throughput is obtained at $a_{1}=0.7$ and the balanced throughputs of $\rm D_{1}$ and $\rm D_{2}$ are achieved at $a_{1}\approx 0.63$ .

FIGURE 11.

Throughput of D_i and whole system versus the power allocation coefficient ($a_{1}$ ); SNR $= 30$ dB, $M=256, m$ = 200, $r = 4$ .

SECTION VI. Conclusion

In this paper, we have analyzed the performance of the UAV-assisted NOMA relay system for SPC, where the UAV harvested energy from a power beacon to support the transmissions of finite blocklength packets from a source to two destinations. The system performance in terms of BLER and throughput over Rician fading channels was investigated. Based on the approximate Chebyshev-Gauss quadrature and the first-order Riemann integral, we obtained the closed-form expressions of the BLERs of each hop and the whole path. Simulation results verified all analytical results. The results indicated that the optimal altitude of the UAV is $H\approx 100$ m provided the best system BLERs. Moreover, the time switching ratio energy harvesting $\kappa = 0.35$ gave the smallest BLER. The number of training bits was chosen in the range $80 < M_{e} < 130$ to maximize the throughput. The BLER of the UAV-NOMA-SPC was compared with that of the UAV-OMA-SPC systems, indicating that the UAV-NOMA-SPC had lower BLERs. The considered system is suitable for narrow-band systems in the cases the infrastructure systems cannot be deployed, is overloaded, or broken. The considered system can be applied to various IoT applications. For example, in a smart traffic system, $\rm D_{1}$ can be a remote-control car receiving control messages that usually contain a few bytes but are time-critical. At the same time, $\rm D_{2}$ can be a self-driving car receiving the confidential driving routes from S. Moreover, the considered system model can be used for an artillery targeting center controlling many artillery units through a UAV.

Appendix

This appendix provides step-by-step derivations to obtain the CDF of the SINR at each node with the Rician fading channel. Since $X$ has Rician distribution, after applying the first-order Marcum Q-function, we have

$$\begin{equation*} F_{X}(x) = 1-Q_{1}(\sqrt {2K}, \sqrt {2\beta _{i} x}), \tag{44}\end{equation*}$$ where $\beta _{i}=\frac {K+1}{\Omega _{i}}$ with $\Omega _{i}=\mathbb {E}\{X\}$ is the expected value of $X$ . Using [43, Eq. (4.18)] and changing the zero-order modified Bessel function into the series formulas, we obtain $F_{X}(x)$ and $f_{X}(x)$ as $$\begin{align*} F_{X}(x) & = 1-\sum _{l=0}^{\infty }\sum _{n=0}^{l}\beta _{l,n}(\beta _{i} x_{i})^{n}\exp (-\beta _{i} x_{i}), \tag{45}\\[3pt] f_{X}(x) & = \frac {\beta _{i}}{e^{K}}\exp (-\beta _{i}x)\sum _{j=0}^{\infty }\frac {1}{(j!)^{2}}\left({\frac {K(K+1)x}{\Omega _{i}} }\right)^{j}, \tag{46}\end{align*}$$ where $\beta _{l,n}=\frac {K^{l}}{l!n!e^{K}}$ .

From (5), (9), and (10), we have $F_{\gamma _{\mathrm{ U}}^{x_{1}}}(x)$ and $F_{\gamma _{\mathrm{ U}}^{x_{2}}}(x)$ as

$$\begin{align*} F_{\gamma _{\mathrm{ U}}^{x_{2}}}(x) &= P_{\mathrm{ LoS}} \underbrace {\Pr \left({|h_{\mathrm{ su}}|^{2} < \frac {x\sigma _{\mathrm{ R}}^{2}}{P_{\mathrm{ s}}(a_{2}-a_{1}x)}|_{\omega =1}}\right)}_{\cal I_{1}} \\[3pt] &\quad + P_{\mathrm{ NLoS}} \underbrace {\Pr \left({|h_{\mathrm{ su}}|^{2} < \frac {x\sigma _{\mathrm{ R}}^{2}}{P_{\mathrm{ s}}(a_{2}-a_{1}x)}|_{\omega < 1}}\right)}_{\cal I_{2}}, \tag{47}\\[3pt] F_{\gamma _{\mathrm{ U}}^{x_{1}}}(x) &= P_{\mathrm{ LoS}}\underbrace {\Pr \left({|h_{\mathrm{ su}}|^{2} < \frac {x\sigma _{\mathrm{ R}}^{2}}{a_{1}P_{\mathrm{ s}}}|_{\omega =1}}\right)}_{\cal I_{3}} \\[3pt] &\quad +P_{\mathrm{ NLoS}} \underbrace {\Pr \left({|h_{\mathrm{ su}}|^{2} < \frac {x\sigma _{\mathrm{ R}}^{2}}{a_{1}P_{\mathrm{ s}}}|_{\omega < 1}}\right)}_{\cal I_{4}}. \tag{48}\end{align*}$$

Applying (45) for (47) and (48), we have the CDFs of the SNRs of $x_{1}$ and $x_{2}$ at the UAV in the cases of LoS and NLoS communications and with the average power gain $\Omega _{\ell}$ given in (5) as

$$\begin{align*} {\mathcal{ I}}_{1}&= 1-\sum _{l=0}^{\infty }\sum _{n=0}^{l}\beta _{l,n}\left({\frac {(K+1)x\sigma _{\mathrm{ R}}^{2}}{\Omega _{\mathrm{ su}}P_{\mathrm{ s}}(a_{2}-a_{1}x)} }\right)^{n} e^{-\frac {(K+1)x\sigma _{\mathrm{ R}}^{2}}{\Omega _{\mathrm{ su}}P_{\mathrm{ s}}(a_{2}-a_{1}x)}}, \tag{49}\\ {\mathcal{ I}}_{2}&= 1-\!\!\sum _{l=0}^{\infty }\sum _{n=0}^{l}\beta _{l,n}\left({\frac {(K+1)x\sigma _{\mathrm{ R}}^{2}}{\omega \Omega _{\mathrm{ su}}P_{\mathrm{ s}}(a_{2}-a_{1}x)} }\right)^{n}\!\! e^{-\frac {(K+1)x\sigma _{\mathrm{ R}}^{2}}{\omega \Omega _{\mathrm{ su}}P_{\mathrm{ s}}(a_{2}-a_{1}x)}}, \tag{50}\\ {\mathcal{ I}}_{3}&= 1-\sum _{l=0}^{\infty }\sum _{n=0}^{l}\beta _{l,n}\left({\frac {(K+1)x\sigma _{\mathrm{ R}}^{2}}{\Omega _{\mathrm{ su}}P_{\mathrm{ s}}a_{1}} }\right)^{n}e^{-\frac {(K+1)x\sigma _{\mathrm{ R}}^{2}}{\Omega _{\mathrm{ su}}P_{\mathrm{ s}}a_{1}}}, \tag{51}\\ {\mathcal{ I}}_{4}&= 1-\sum _{l=0}^{\infty }\sum _{n=0}^{l}\beta _{l,n}\left({\frac {(K+1)x\sigma _{\mathrm{ R}}^{2}}{\omega \Omega _{\mathrm{ su}}P_{\mathrm{ s}}a_{1}} }\right)^{n}e^{-\frac {(K+1)x\sigma _{\mathrm{ R}}^{2}}{\omega \Omega _{\mathrm{ su}}P_{\mathrm{ s}}a_{1}}}. \tag{52}\end{align*}$$

Next, we calculate the CDFs for $\gamma _{\rm D_{1}}^{\hat x_{2}}, \gamma _{\rm D_{1}}^{\hat x_{1}}$ and $\gamma _{\rm D_{2}}^{\hat x_{2}}$ given in (13), (14), and (15), respectively. From (13), we can rewrite the CDF of $\gamma _{\rm D_{1}}^{\hat x_{2}}$ as

$$\begin{align*} F_{\gamma _{\rm D_{1}}^{\hat x_{2}}}(x) &= \underbrace {P_{\mathrm{ LoS}}\Pr \left({|h_{\mathrm{ pu}}|^{2}|h_{\rm ud_{1}}|^{2} < \frac {x\sigma _{\rm D_{1}}^{2}}{\Delta (a_{2}-a_{1}x)}|_{\omega =1}}\right)}_{\cal I_{5}} \\ &\quad +P_{\mathrm{ NLoS}}\underbrace {\Pr \left({|h_{\mathrm{ pu}}|^{2}|h_{\rm ud_{1}}|^{2} < \frac {x\sigma _{\rm D_{1}}^{2}}{\Delta (a_{2}-a_{1}x)}|_{\omega < 1}}\right)}_{\cal I_{6}}, \tag{53}\end{align*}$$ where $\Delta$ is presented in (7).

From (14) and (15), we have the CDF of $\gamma _{\rm D_{1}}^{\hat x_{1}}$ and the CDF of $\gamma _{\rm D_{2}}^{\hat x_{2}}$ in the case of perfect SIC as

$$\begin{align*} F_{\gamma _{\rm D_{1}}}^{\hat x_{1}}(x) &= P_{\mathrm{ LoS}}\underbrace {\Pr \left({|h_{\mathrm{ pu}}|^{2}|h_{\rm ud_{1}}|^{2} < \frac {x\sigma _{\rm D_{1}}^{2}}{\Delta a_{1}}|{\omega =1}}\right)}_{\cal I_{7}} \\ &\quad +P_{\mathrm{ NLoS}}\underbrace {\Pr \left({|h_{\mathrm{ pu}}|^{2}|h_{\rm ud_{1}}|^{2} < \frac {x\sigma _{\rm D_{1}}^{2}}{\Delta a_{1}}|{\omega < 1}}\right)}_{\cal I_{8}}, \tag{54}\\ F_{\gamma _{\rm D_{2}}^{\hat x_{2}}}(x) &= P_{\mathrm{ LoS}}\underbrace {\Pr \left({|h_{\mathrm{ pu}}|^{2}|h_{\rm ud_{2}}|^{2} < \frac {x\sigma _{\rm D_{2}}^{2}}{\Delta (a_{2}-a_{1}x)}|_{\omega =1}}\right)}_{\cal I_{9}} \\ &\quad +P_{\mathrm{ NLoS}}\underbrace {\Pr \left({|h_{\mathrm{ pu}}|^{2}|h_{\rm ud_{2}}|^{2} < \frac {x\sigma _{\rm D_{2}}^{2}}{\Delta (a_{2}-a_{1}x)}|_{\omega < 1}}\right)}_{\cal I_{10}}. \tag{55}\end{align*}$$

Thanks to the fundamental of the conditional probability given in [44, Chapter 4], we can rewrite ${\mathcal{ I}}_{5}$ as

$$\begin{equation*} {\mathcal{ I}}_{5} = \int \nolimits _{0}^{\infty } F_{|h_{\rm ud_{1}}|^{2}}\left({\frac {x\sigma _{\rm D_{1}}^{2}}{y\Delta (a_{2}-a_{1}x)}}\right)f_{|h_{\mathrm{ pu}}|^{2}}(y)dy. \tag{56}\end{equation*}$$

Substituting (45) and (46) into (56), and then using [45, Eq. (3. 4719)], we obtain the closed-form of ${\mathcal{ I}}_{5}$ in the case of LoS communication as in (57). Plugging $\Omega _{\mathrm{ bu}}$ and $\Omega _{\rm ud_{1}}$ by $\omega \Omega _{\mathrm{ bu}}$ and $\omega \Omega _{\rm ud_{1}}$ into (57) we get ${\mathcal{ I}}_{6}$ as in (58). These expressions are shown at the top of the next page.

$$\begin{align*} {\mathcal{ I}}_{5}&=1-\sum _{l=0}^{\infty } \sum _{n=0}^{l}\sum _{j=0}^{\infty }\frac {\beta _{l,n}}{(j!)^{2}}\left({\frac {K(K+1)}{\Omega _{\mathrm{ bu}}} }\right)^{j}\frac {K+1}{\Omega _{\mathrm{ bu}}e^{K}}\left({\frac {(K+1)x\sigma _{\rm D_{1}}^{2} }{\Delta \Omega _{\rm ud_{1}}(a_{2}-a_{1}x)} }\right)^{n} e^{-\frac {(K+1)x\sigma _{\rm D_{1}}^{2} }{\Delta \Omega _{\rm ud_{1}}(a_{2}-a_{1}x)} } \\ &\quad \times 2\left({\frac {x\sigma _{\rm D_{1}}^{2}\Omega _{\mathrm{ bu}} }{\Delta \Omega _{\rm ud_{1}}(a_{2}-a_{1}x)} }\right)^{\frac {j-n+1}{2}} {\mathcal{ K}}_{j-n+1}\left({2 \sqrt {\frac {(K+1)x\sigma _{\rm D_{1}}^{2} }{\Delta \Omega _{\rm ud_{1}}(a_{2}-a_{1}x)} \frac {K+1}{\Omega _{\mathrm{ bu}}}} }\right), \,\,\,\beta _{l,n}=\frac {K^{l}}{l!n!e^{K}}, \tag{57}\\ {\mathcal{ I}}_{6}&=1-\sum _{l=0}^{\infty } \sum _{n=0}^{l}\sum _{j=0}^{\infty }\frac {\beta _{l,n}}{(j!)^{2}}\left({\frac {K(K+1)}{\omega \Omega _{\mathrm{ bu}}} }\right)^{j}\frac {K+1}{\omega \Omega _{\mathrm{ bu}}e^{K}}\left({\frac {(K+1)x\sigma _{\rm D_{1}}^{2} }{\Delta \omega \Omega _{\rm ud_{1}}(a_{2}-a_{1}x)} }\right)^{n} e^{-\frac {(K+1)x\sigma _{\rm D_{1}}^{2} }{\Delta \omega \Omega _{\rm ud_{1}}(a_{2}-a_{1}x)}} \\ &\quad \times 2\left({\frac {x\sigma _{\rm D_{1}}^{2}\omega \Omega _{\mathrm{ bu}} }{\Delta \omega \Omega _{\rm ud_{1}}(a_{2}-a_{1}x)} }\right)^{\frac {j-n+1}{2}} {\mathcal{ K}}_{j-n+1}\left({2 \sqrt {\frac {(K+1)x\sigma _{\rm D_{1}}^{2} }{\Delta \omega \Omega _{\rm ud_{1}}(a_{2}-a_{1}x)} \frac {K+1}{\omega \Omega _{\mathrm{ bu}}}} }\right). \tag{58}\end{align*}$$ $$\begin{align*} {\mathcal{ I}}_{7}&=1-\sum _{l=0}^{\infty } \sum _{n=0}^{l}\sum _{j=0}^{\infty }\frac {\beta _{l,n}}{(j!)^{2}}\left({\frac {K(K+1)}{\Omega _{\mathrm{ bu}}} }\right)^{j}\frac {K+1}{\Omega _{\mathrm{ bu}}e^{K}}\left({\frac {(K+1)x\sigma _{\rm D_{1}}^{2} }{\Delta \Omega _{\rm ud_{1}}a_{1}} }\right)^{n}\exp \left({-\frac {(K+1)x\sigma _{\rm D_{1}}^{2} }{\Delta \Omega _{\rm ud_{1}}a_{1}} }\right) \\ &\quad \times 2\left({\frac {x\sigma _{\rm D_{1}}^{2}\Omega _{\mathrm{ bu}} }{\Delta \Omega _{\rm ud_{1}}a_{1}} }\right)^{\frac {j-n+1}{2}} {\mathcal{ K}}_{j-n+1}\left({2 \sqrt {\frac {(K+1)x\sigma _{\rm D_{1}}^{2} }{\Delta \Omega _{\rm ud_{1}}a_{1}} \frac {K+1}{\Omega _{\mathrm{ bu}}}} }\right), \tag{59}\\ {\mathcal{ I}}_{8}&=1-\sum _{l=0}^{\infty } \sum _{n=0}^{l}\sum _{j=0}^{\infty }\frac {\beta _{l,n}}{(j!)^{2}}\left({\frac {K(K+1)}{\omega \Omega _{\mathrm{ bu}}} }\right)^{j}\frac {K+1}{\omega \Omega _{\mathrm{ bu}}e^{K}}\left({\frac {(K+1)x\sigma _{\rm D_{1}}^{2} }{\Delta \omega \Omega _{\rm ud_{1}}a_{1}} }\right)^{n}\exp \left({-\frac {(K+1)x\sigma _{\rm D_{1}}^{2} }{\Delta \omega \Omega _{\rm ud_{1}}a_{1}} }\right) \\ &\quad \times 2\left({\frac {x\sigma _{\rm D_{1}}^{2} \omega \Omega _{\mathrm{ bu}} }{\Delta \omega \Omega _{\rm ud_{1}}a_{1}} }\right)^{\frac {j-n+1}{2}} {\mathcal{ K}}_{j-n+1}\left({2 \sqrt {\frac {(K+1)x\sigma _{\rm D_{1}}^{2} }{\Delta \omega \Omega _{\rm ud_{1}}a_{1}} \frac {K+1}{\omega \Omega _{\mathrm{ bu}}}} }\right). \tag{60}\end{align*}$$ $$\begin{align*} {\mathcal{ I}}_{9}&=1-\sum _{l=0}^{\infty } \sum _{n=0}^{l}\sum _{j=0}^{\infty }\frac {\beta _{l,n}}{(j!)^{2}}\left({\frac {K(K+1)}{\Omega _{\mathrm{ bu}}} }\right)^{j}\frac {K+1}{\Omega _{\mathrm{ bu}}e^{K}}\left({\frac {(K+1)x\sigma _{\rm D_{2}}^{2} }{\Delta \Omega _{\rm ud_{2}}(a_{2}-a_{1}x)} }\right)^{n} e^{-\frac {(K+1)x\sigma _{\rm D_{2}}^{2} }{\Delta \Omega _{\rm ud_{2}}(a_{2}-a_{1}x)}} \\ &\quad \times 2\left({\frac {x\sigma _{\rm D_{2}}^{2}\Omega _{\mathrm{ bu}} }{\Delta \Omega _{\rm ud_{2}}(a_{2}-a_{1}x)} }\right)^{\frac {j-n+1}{2}} {\mathcal{ K}}_{j-n+1}\left({2 \sqrt {\frac {(K+1)x\sigma _{\rm D_{2}}^{2} }{\Delta \Omega _{\rm ud_{2}}(a_{2}-a_{1}x)} \frac {K+1}{\Omega _{\mathrm{ bu}}}} }\right), \tag{61}\\ {\mathcal{ I}}_{10}&=1-\sum _{l=0}^{\infty } \sum _{n=0}^{l}\sum _{j=0}^{\infty }\frac {\beta _{l,n}}{(j!)^{2}}\left({\frac {K(K+1)}{\omega \Omega _{\mathrm{ bu}}} }\right)^{j}\frac {K+1}{\omega \Omega _{\mathrm{ bu}}e^{K}}\left({\frac {(K+1)x\sigma _{\rm D_{2}}^{2} }{\Delta \omega \Omega _{\rm ud_{2}}(a_{2}-a_{1}x)} }\right)^{n} e^{-\frac {(K+1)x\sigma _{\rm D_{2}}^{2} }{\Delta \omega \Omega _{\rm ud_{2}}(a_{2}-a_{1}x)}} \\ &\quad \times 2\left({\frac {x\sigma _{\rm D_{2}}^{2} \omega \Omega _{\mathrm{ bu}} }{\Delta \omega \Omega _{\rm ud_{2}}(a_{2}-a_{1}x)} }\right)^{\frac {j-n+1}{2}} {\mathcal{ K}}_{j-n+1}\left({2 \sqrt {\frac {(K+1)x\sigma _{\rm D_{2}}^{2} }{\Delta \omega \Omega _{\rm ud_{2}}(a_{2}-a_{1}x)} \frac {K+1}{\omega \Omega _{\mathrm{ bu}}}} }\right) \tag{62}\end{align*}$$ Similar to calculating for (53), we can derive the CDFs of ${\gamma _{\rm D_{1}}}^{\hat x_{1}}$ and ${\gamma _{\rm D_{2}}^{\hat x_{2}}}$ for (54) and (55).